Friction, adhesion, and deformation: dynamic measurements with the atomic force microscope

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Friction adhesion anddeformation dynamicmeasurements with theatomic force microscopePhil AttardVersion of record first published 02 Apr 2012

To cite this article Phil Attard (2002) Friction adhesion and deformationdynamic measurements with the atomic force microscope Journal of AdhesionScience and Technology 167 753-791

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J Adhesion Sci Technol Vol 16 No 7 pp 753ndash791 (2002)Oacute VSP 2002

Friction adhesion and deformation dynamicmeasurements with the atomic force microscope

PHIL ATTARD curren

Ian Wark Research Institute University of South Australia Mawson Lakes SA 5095 Australia

Received in nal form 6 December 2001

AbstractmdashA summary is given of the authorrsquos recent experimental and theoreticalwork involving theatomic force microscope with the focus being upon dynamic measurements Four topics are coveredcalibration techniques for the friction force microscope quantitative measurements of friction and theeffect of adhesion measurement and theory for the deformation and adhesion of viscoelasticparticlesand the interaction and adhesion of hydrophobic surfaces due to bridging nanobubbles

Keywords Atomic force microscope friction adhesion deformation nanobubble

1 INTRODUCTION

The atomic force microscope (AFM) [1] is commonly used to image surfacesand to study the interaction and adhesion of particles The widespread adoptionof the AFM is due to its ease of use the molecular-level information that itprovides and the variety of surfaces that can be studied in a broad range ofenvironments In addition the computer interface allows exible control of thedevice and the automated acquisition of large volumes of data it facilitates multiplerepeat experiments to check reproducibility and to minimize statistical error andit enables detailed data analysis This computer control opens up the possibility ofreal-time monitoring of experiments and the exploration of time-dependent effectsThe AFM is well suited to studying the latter whereas the original surface forceapparatus [2] and its variants [3 4] either lack automated data acquisition or sufferfrom inertial and other artefacts that must be taken into account in the quantitativeinterpretation of dynamic force measurements [5 6]

The distinction between equilibrium and non-equilibrium forces is quite impor-tant To some extent the primary concern with the AFM has been (and shouldbe) to ensure that the experiments are carried out slowly enough that equilibrium

currenPhone +062 8 8302 3564 Fax +062 8 8302 3683 E-mail philattardunisaeduau

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is established at each instant so that the measured forces are comparable to thosemeasured statically Beyond that an exciting eld of research exploits the dynamiccapabilities of the AFM to measure non-equilibrium phenomena in a controlledfashion We review two examples from our laboratory that show the utility of dy-namic AFM measurements for non-equilibrium systems Results and quantitativeanalyses are presented for the deformation interaction and adhesion of viscoelasticdroplets (Section 4) and for the interaction and adhesion of surfaces via bridgingnanobubbles during their spreading (Section 5)

The most obvious technique that utilizes the dynamic capability of the AFMis the measurement of friction which is also called friction force microscopy orlateral force microscopy Since the original work of Mate et al [7] the elds offriction force mapping (sometimes called chemical imaging) and of nanotribologyhave grown greatly (see for example papers in Refs [8] and [9]) Althoughsome progress has been made this research has been limited by approximations orrestrictions on the lateral calibration method for the AFM This de ciency has beenrecti ed quite recently by two techniques that yield the torsional spring constant ofthe cantilever and the voltage response of the lateral photodiode to cantilever twist[10ndash12] This review begins by summarizing the limitations of previous calibrationtechniques and by detailing the procedures involved in the newer quantitativemethods (Section 2) The results that we have obtained in our laboratory [13] for thequantitative dependence of friction on adhesion in a system with electrical double-layer interactions are then reviewed (Section 3)

2 CALIBRATION OF THE FRICTION FORCE MICROSCOPE

21 Critical review

In order to use the AFM various calibrations have to be performed The lateralmovement of the piezo-crystal is often calibrated using model substrates Theexpansion factor that relates the applied voltage to the distance that the piezo-crystalexpands in the vertical direction normal to the substrate 1z can be measuredfrom the interference fringes due to the re ection of the laser from both thecantilever and the substrate The normal spring constant of the cantilever kx canbe obtained gravitationally thermally or by resonance techniques [14ndash16] Thenormal photodiode sensitivity factor reg0 relates the measured vertical differentialphotodiode voltage 1Vvert to the vertical de ection of the cantilever 1x which inthe constant compliance regime is equal to the piezo-drive movement 1x D 1zFor the quantitative measurement of friction in addition to these one has toobtain the torsional spring constant of the cantilever kmicro and the lateral photodiodesensitivity factor macr which relates the measured lateral differential photodiodevoltage 1Vlat to the twist angle of the cantilever 1micro

Unfortunately almost all lateral calibration techniques that have been used todate are approximate in one way or another and the measurements of friction that

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Friction adhesion and deformation 755

utilize them must be regarded as semi-quantitative Brie y a critical review ofthe literature reveals that in most cases [17ndash21] the torsional spring constant iscalculated not measured using an analytic approximation [22] that idealizes theactual geometry of the cantilever In addition it ignores the effects of coatings andthickness variations which in the case of the normal spring constant can alter thevalue by an order of magnitude The lateral sensitivity factor which relates thephotodiode voltage to the twist angle has also been obtained by assuming it to beproportional to the vertical sensitivity [18] by modelling the beam path and pro le[19] and by assuming that the tip is pinned during the initial part of the frictionloop [17 23] The assumed pinning does not take into account any slippage ordeformation of the tip which can make the method inaccurate and others haveattempted to improve the method by invoking certain simple models of frictionand deformation [20 21] Measurements of friction parallel to the long axis ofthe cantilever using the normal spring constant and sensitivity [24 25] erroneouslyneglect the bending moment of the cantilever [6 21] Toikka et al [23] attempted touse gravity acting on an attached lever but the torque that they applied was shownto give negligible cantilever twist [12] and it appears that what they measured wasin fact photodiode saturation And nally the commonly used calibration method ofOgletree et al [26] is restricted by the need for a specialized terraced substrate andan ultra-sharp tip For the calibration this method makes two assumptions aboutthe friction law namely that friction is a linear function of the applied load and thatit vanishes when the applied load is the negative of the adhesion (ie the pull-offpoint) Counter-examples showing non-linear behaviour are known [13 27] andone should check whether or not such a linear regime exists before using it for thecalibration That none of the previous calibration methods is entirely satisfactoryis con rmed by the fact that many friction force microscopy (FFM) papers givefriction in terms of volts rather than Newtons [27ndash29] Almost all friction forcemaps are similarly uncalibrated and the images are given in terms of volts ratherthan the physical friction coef cient

Feiler et al [12] have developed a direct technique that simultaneously measuresthe cantilever spring constant and the lateral sensitivity of the photodiode Thatparticular method is discussed in detail below

Meurk et al [10] have given a method for directly calibrating the lateral sensitivityof the photodiode Basically the angle of a re ective substrate is varied with respectto the laser beam In some AFM scanners there is a stepper motor that facilitatesthe tilt of the head From the geometry and the amount of movement the degreeof tilt 1micro can be calculated The change in the lateral photodiode voltage 1Vlatis linear in the tilt angle and the ratio of the two gives the lateral sensitivity of theAFM

The torsional spring constant of the cantilever can be obtained directly by thetechnique developed by Bogdanovic et al [11] Here a protuberance (eg anupturned tipped cantilever) is glued to the substrate and force measurements areperformed against it with the protuberance making contact off the central axis of

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the tipless force measuring cantilever which causes the latter to bend and to twistRecording the normal and lateral photodiode voltages in the constant complianceregime at several different lateral off-sets allows the spring constant divided by thelateral sensitivity to be obtained Combined with the method of Meurk et al [10]this allows a full calibration of the AFM (In principle one can also obtain the lateralsensitivity with this method However the small leverage and high torsional springconstant make it impractical to do so)

22 Quantitative calibration technique

We now describe in detail a one-step method that simultaneously measures both thelateral photodiode sensitivity and the torsional spring constant of the cantilever thathas been developed in our laboratory [12] A glass bre 50ndash200 sup1m in length isglued perpendicular to the long axis of the cantilever and parallel to the substrate Toensure that the substrate pushes on the end of the bre a colloid sphere is attachedat its tip (see Fig 1) Using the well-known colloid probe attachment procedure ofDucker et al [30] an epoxy resin is used to attach the sphere and a heat-settingadhesive is used to attach the bre This allows the bre to be removed later andthe cantilever used for friction measurements (ie the method is non-destructive)Attaching the sphere is convenient but not essential other ways to ensure that it isthe end of the bre that touches the substrate include gluing the bre to the cantileverat a slight angle having a ledge or colloid probe on the substrate or performing themeasurement with the head or substrate tilted a small amount (eg by using thestepper motor)

The essence of the method is that pushing on the tip of the bre with a forceF produces a torque iquest D F L where L is the length of the bre The cantileversimultaneously de ects 1x D F=kx and twists 1micro D iquest=kmicro The de ection andhence the force and torque is obtained from the differential vertical photodiodevoltage 1x D reg01Vvert where the bare sensitivity factor reg0 is measured inthe constant compliance regime without the attached bre The actual sensitivityfactor with the attached bre regL is greater than this because only part of thepiezo-drive movement goes into de ecting the cantilever 1x lt 1z (the rest issoaked up by the twist) The bare vertical sensitivity factor has to be measured ina separate experiment and depends on the positions of the laser the photodiodeand the cantilever mount With practice it is possible to obtain better than 10

Figure 1 Rectangular cantilever with attached bre and sphere When the substrate is moved adistance 1z the cantilever de ects a distance 1x and twists an amount 1micro The correspondingchanges in the differential photodiode voltages 1Vvert and 1Vlat are measured

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reproducibility in this quantity between different experiments and after remountingthe cantilever The best way to ensure this is to maximize the total vertical signaland to minimize the differential lateral signal each time

The calibration factor of primary interest is the one that relates the differentiallateral photodiode voltage to an applied torque iquest D deg 1Vlat This is given by

deg Diquest

1VlatD

kx1xL

1VlatD kxreg0L

1Vvert

1Vlat (1)

This equation predicts a linear relationship between the two photodiode signalswhich as can be seen in Fig 2 is indeed the case The slope of this line combinedwith the measured values for the vertical spring constant the bare vertical sensitivityfactor and the length of the bre gives the factor that converts the differential lateralphotodiode voltage to the applied torque in general (ie independent of the attached bre) Figure 3 shows the lateral sensitivity factor obtained using a number ofdifferent bres That the same value is obtained each time shows that it is an intrinsicproperty of the cantilever and AFM set-up It also con rms that remounting thecantilever does not preclude reproducible results from being obtained

We found that the calibration procedure was straightforward and relatively robustThe method was less successful whenever there was signi cant adhesion betweenthe substrate and the tip of the bre or the attached sphere We minimizedsuch adhesion by using silica surfaces and conducting the calibration in water atnatural pH

Figure 2 Lateral differential photodiode voltage as a function of the vertical voltage Both weremeasured for a cantilever with an attached bre over the whole approach regime of a single forcemeasurement The data are from Ref [12]

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758 P Attard

Figure 3 Lateral sensitivity factor for different bres The cantilevers were taken from the samebatch Circles indicate bres with an end-attached sphere diamonds indicate bare bres lledsymbols are for approach and open symbols are for retraction The data are from Ref [12]

It is possible to verify independently the procedure by obtaining the sensitivityfactor that relates the change in angle to the change in the lateral photodiode signaland comparing this with the value obtained by the method of Meurk et al [10]From the slope of the constant compliance region of the force curve with theattached bre one can obtain the constants

regL D1z

1Vvertand macrL D

1z

1Vlat (2)

for the vertical and lateral de ections respectively With these the lateral sensitivitycan be shown to be given by [12]

1micro

1VlatD

macrL1 iexcl reg0=regL

L (3)

A value of 3 pound 10iexcl4 radV was obtained using our method [12] compared with17 pound 10iexcl4 radV using the method of Meurk et al [10]

The torsional spring constant itself is given by [12]

kmicro DiexclkxL2

1 iexcl regL=reg0 (4)

A value of 2 pound 10iexcl9 N m was obtained using our method [12] compared with12 pound 10iexcl9 N m calculated from the method of Neumeister and Ducker [22]

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3 ADHESION AND FRICTION

31 Intrinsic force

One of the oldest ideas concerning the nature of friction is embodied in Amontonsrsquolaw which states that the friction force f is proportional to the applied load Lf D sup1L where sup1 is the coef cient of friction For the case of adhering surfaceswhere a negative load needs to be applied to separate them it is known that therecan be substantial friction even when the load is zero Hence Amontonsrsquo law maybe slightly modi ed

f Draquo

sup1L C A L gt iexclA

0 L lt iexclA(5)

where A gt 0 is the adhesion which is the greatest tension that the surfaces cansustain (ie it is equal to the absolute value of the force required to pull the surfacesapart) This modi ed version re ects the plausible idea that friction only occurswhen the surfaces are in contact Amontonsrsquo law raises several questions Is frictiona linear function of load Is the role of adhesion only to shift the effective loadWhat is the law for non-adhering surfaces Is friction zero for surfaces not incontact And what does contact mean on a molecular scale

The AFM is an ideal tool to test the fundamental nature of friction and we setout to answer quantitatively these and other questions [13] We chose a system thatwould allow us to alter the adhesion in a controlled manner so that as far as possibleall other variables were kept constant We used a titanium dioxide substrate (rootmean square roughness of 03 nm) and a silicon dioxide colloid probe (root meansquare roughness of 08 nm 7 sup1m diameter) The measurements were carried out inan aqueous electrolyte (10iexcl3 M KNO3) as a function of pH The SiO2 is negativelycharged at practically all pHs (its point of zero charge is frac14 pH 2) whereas TiO2

is positively charged at low pH and negatively charged at high pH (its point ofzero charge is frac14 pH 45) Hence at low pH the attractive double-layer interactionbetween the surfaces causes them to adhere and at high pH they repel each otherand do not adhere

There have been several other AFM studies of friction between surfaces withelectrical double-layer interactions [27 31 32] In some cases an applied voltagehas been used to modify the adhesion but the friction coef cients and force lawshave all been qualitative in the sense of the preceding section A critical discussionof these results is given in Ref [13]

The load which is the applied normal force is shown in Figs 4 and 5 as a functionof the separation for various pHs It can be seen that the surfaces do indeed interactwith an electrical double-layer interaction and that the pH controls the sign and themagnitude of the force For pH 4 and 5 the attractive double-layer interaction givesan adhesion of A D 105 and 44 nN respectively However at higher pH thesurfaces do not adhere

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Figure 4 Force on approach as a function of the separation The substrate is TiO2 the 7 sup1m diametercolloid probe is SiO2 and the background electrolyte is 1 mM KNO3 From top to bottom the curvescorrespond to pH 8 7 6 5 and 4 The inset shows constant potential (AtildeSiO2

D iexcl50 mV andAtildeTiO2 D iexcl43 mV) and constant charge ts to the pH 8 case on a log scale Reprinted with permissionfrom Ref [13] Copyright (2000) American Chemical Society

Figure 5 Same as Fig 4 on retraction The inset magni es the three highest pHs at small separationsReprinted with permission from Ref [13] Copyright (2000) American Chemical Society

In view of equation (5) we are motivated to de ne the detachment force Fdetach as the minimum applied force necessary to keep the surfaces in contact [13] Fornon-adhering surfaces this is a positive quantity and for adhering surfaces it isnegative (in fact it is the negative of the adhesion) The detachment force at pH6 7 and 8 was Fdetach D 14 26 and 35 nN respectively (Fig 5) In view of theclose relationship between adhesion and the detachment force one may de ne anintrinsic force

Fintrinsic D L iexcl Fdetach (6)

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Figure 6 Friction force as a function of the applied load Reprinted with permission from Ref [13]Copyright (2000) American Chemical Society

which may be thought of as the force in excess of that when the surfaces are justin contact In this language Amontonsrsquo law generalized to non-adhering surfaceswould read f D sup1Fintrinsic

We measured friction as a function of the applied load at various pHs This wasdone in the usual fashion [7] by moving the substrate back and forth in the directionperpendicular to the long axis of the cantilever and recording friction loops Thelength of the scan in each direction was 05 sup1m and the velocity was 1 sup1msThe lateral calibration factor obtained as detailed above [12] was used to convert(half) the voltage difference between the two arms of the friction loop to the appliedtorque iquest The friction force was obtained as f D iquest=2r where r D 7 sup1m is theradius of the colloid probe The applied load was xed by using the set-point featureof the AFM (ie the vertical de ection signal was held constant during the frictionloop)

Friction is plotted as a function of the applied load in Fig 6 In general frictionincreases with increasing load At a given applied load friction is also larger thelower the pH Since the adhesion increases with decreasing pH one may restatethis fact as the higher the adhesion the higher the friction at a given applied loadMoreover friction is non-zero at zero loads for adhering surfaces For non-adheringsurfaces friction is zero for small but non-zero applied loads

The quantitative behaviour of friction with pH is not obvious when plotted as afunction of the applied load But when plotted against intrinsic load Fig 7 theutility of the detachment force becomes evident The functional form of the frictionforce law is fundamentally independent of pH and all the measurements lie on asingle universal curve In other words the major role of pH is to determine theadhesion (or more precisely the detachment force) Once this parameter has beenexperimentally determined from a normal force measurement at a given pH thefriction at that pH may be predicted from the friction measured at any other pHmerely by shifting the load by the detachment force

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Figure 7 Friction force as a function of the intrinsic load Reprinted with permission from Ref [13]Copyright (2000) American Chemical Society

These experiments show that for this system friction is not a linear function of theload (ie the friction coef cient sup1 D df=dL is not independent of load) There isa noticeable curvature in the plot with friction increasing more rapidly at higherloads The loads that have been applied here are relatively weak [the averagepressure in the contact region (see below) is less than about 10 MPa and the peakpressure is less than about 100 MPa [13]] and it is not clear what will happen athigher loads than these

Whilst it is not implausible that the friction should be zero for negative intrinsicforces in all cases (this corresponds to the surfaces being out of contact) it is a littlesurprising that for positive intrinsic forces the increase in friction is the same in allcases After all not only are the adhesion and the normal force laws different atdifferent pHs but also the surface chemistry varies due to the different amount ofion binding that occurs The fact that the latter has almost no effect on friction isperhaps not unexpected since over the range of pH studied for TiO2 only about1 of the surface sites are converted from HC at low pH to OHiexcl at high pH andfor SiO2 the change is about 10 [33] Nevertheless it is not immediately obviouswhy surfaces with different adhesion display quantitatively the same friction for thesame intrinsic force

32 Elastic deformation

In order to investigate the above question further we carried out elastic deformationcalculations on the sphere and substrate under the experimental conditions [13]Elastic deformation has long been thought to play a dominant role in the friction ofmacroscopic bodies mainly in the context of using contact mechanics to account forasperity attening [34 35] We however were in a position to go beyond contacttheories such as that due to Johnson Kendall and Roberts (JKR) [36] or that dueto Derjaguin Muller and Toporov (DMT) [37] We used the soft-contact algorithmof Attard and Parker [38 39] and invoked the actual experimentally measured force

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Figure 8 Calculated surface pro les for an applied load of 5 nN From top to bottom the pH is 87 6 5 and 4 and in each case the measured force law has been used in the calculations Youngrsquosmodulus and Poissonrsquos ratio for SiO2 E=1 iexcl ordm2 D 77 pound 1010 Nm2 have also been used Thebottom dashed curve is for an applied load of 720 nN for the pH 4 case The abscissa is the distancefrom the central axis in micrometres and the ordinate is the local separation in nanometres The dataare from Ref [13]

law which has of course an extended range (ie it is not a contact force) Thealgorithm self-consistently calculates the surface shape of the elastically deformedbodies due to the local pressure which in turn depends on the local separationof the deformed bodies In this way we obtain the actual surface shape and theactual pressure pro le whereas contact mechanics assumes simpli ed and non-physical forms for both We tted a smooth curve to the measured force law atdifferent pHs and using the Derjaguin approximation differentiated this to obtainthe pressure as a function of the surface separation The latter is required by thealgorithm [38 39] as discussed in the following The calculations presented inRef [13] are the rst elastic deformation calculations using an actual experimentallymeasured force law For the present calculations there was no hysteresis betweenthe loading and unloading cycles (The hysteresis observed in the original papers[38 39] for soft adhering bodies has since been attributed to a non-equilibriumviscoelastic effect [40 41] see Section 4)

Figure 8 shows the resultant surface shape at an applied load of 5 nN This load isgreater than all the detachment forces and in all cases the surfaces showed non-zerofriction It can be seen that very little surface attening has occurred and that thesurfaces at different pHs are effectively displaced parallel to each other

Also included in Fig 8 is a high load (720 nN) case which shows substantial attening However there is no well-de ned contact region and there is certainly

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Figure 9 Calculated surface pro les for an intrinsic force of 15 nN From top to bottom the virtuallyindistinguishable curves correspond to pHs of 8 7 6 5 and 4 respectively The data are fromRef [13]

no sharp change in the surface pro le to demark contact despite the fact that thesecalculations are for the adhering pH 4 surfaces

The tted force law includes a LennardndashJones soft repulsion with length scale05 nm [13] and one could de ne contact as local separations smaller than this Suchan arbitrary de nition is somewhat problematic particularly since the curves at 5 nNload which are not in contact by the de nition display non-zero friction In viewof this discussion of the meaning of contact for systems with realistic surface forcesof non-zero range the inapplicability of simple contact theories such as Hertz JKRor DMT is clear One might also conclude that the experimental veri cation orrefutation of Amontonsrsquo second law (for a given load friction is independent of thecontact area) at the molecular level will be dif cult

Figure 9 shows the surface shapes at different pHs at an intrinsic load of 15 nNwhich corresponds to an applied load of 5 nN for the pH 4 case The changefrom Fig 8 is quite dramatic and one can see that the pro les have coalescedIn other words surfaces at a given intrinsic load have the same shape and localsurface separation Given that friction is also a universal function of intrinsic load(Fig 7) one may conclude that friction is a function of the local separation andindependent of the force law In so far as the short-range interactions between theatoms on the two surfaces can be expected to be independent of the pH one cansay that these are the interactions that determine friction Friction occurs betweentwo bodies when energy can be transferred from one to another which means thatthey have to be close enough for the interaction between atoms on the two surfacesto be comparable to the thermal energy [13] One concludes that the only role of

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adhesion in friction is to decrease the amount of applied load that is necessary tobring the surfaces to a given separation

4 VISCOELASTIC DEFORMATION AND ADHESION

41 Viscoelastic theory

The shapes of the deformed surfaces given above were obtained by solving theequations of continuum elasticity theory in the semi-in nite half-space approxima-tion [38 42]

ur Diexcl2frac14E

Zds

phs

jr iexcl sj (7)

Here the elasticity parameter E is given in terms of Youngrsquos moduli and Poissonrsquosratios of the two bodies 2=E D 1 iexcl ordm2

1=E1 C 1 iexcl ordm22=E2 r D jrj and s D jsj

are the lateral distances from the central axis connecting the centres of the bodies(the integration is over the two-dimensional plane bisecting the two bodies) andph is the pressure between two in nite planar walls at a separation of h Thetotal deformation normal to the surfaces at each position is ur and hence the localseparation between the two bodies is hr D h0riexclur Here the local separationof the undeformed surfaces is h0r D h0 Cr2=2R where h0 is the separation on theaxis and where Riexcl1 D Riexcl1

1 C Riexcl12 is the effective radius of the interacting bodies

in general the Ri is related to the principal radius of curvature of each body [43]For contact theories such as Hertz JKR or DMT the local pressure phr that

appears in the integrand of equation (7) is replaced by a contact pressure pcr thatis a speci ed function of radius which when integrated gives ur D r2=2R whichcorresponds to a at contact region hr D 0 In contrast for realistic force lawsthat have an extended range such as van der Waals electrical double layer or theactual measured ph discussed above the integral must be evaluated numericallyBecause in this case the local separation depends on the deformation equation (7)represents a non-linear integral equation that must be solved by iteration for eachnominal separation h0

An ef cient algorithm for the solution of the non-contact elastic equation hasbeen given by the Attard [38 40] and it has been used to analyse a variety of forcelaws [13 38ndash41] Other workers have also calculated the elastic deformation ofthe solids using realistic surface forces of nite range [44ndash51] There have ofcourse been a large number of experimental studies to measure the interaction ofdeformable solids These include AFM measurements [52ndash62] as well as resultsobtained with the surface force apparatus and the JKR device [63ndash72] Thesestudies in general show that the adhesion and interaction are hysteretic and time-dependent particularly for highly deformable solids with high surface energiesSuch behaviour is characteristic of viscoelastic materials Maugis and Barquinshave given a review of adhesion experiments on viscoelastic materials which they

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766 P Attard

attempt to interpret in quasi-JKR terms introducing a somewhat ill-de ned time-dependent surface energy [73]

A proper theoretical treatment of the deformation and adhesion of viscoelasticmaterials involves replacing the elasticity parameter which gives the instantaneousresponse to the pressure by the creep compliance function which gives the responseto past pressure changes In this way the prior history of the sample is accountedfor Hence the generalization of the elastic half-space equation involves a timeconvolution integral [74 75]

ur t iexcl ur t0 DZ t

t0

dt 0 iexcl2frac14Et iexcl t 0

Zds

Pphs t 0

jr iexcl sj (8)

Here Pphr t is the time rate of change of the local pressure at a distance r D jrjfrom the axis and at time t The bodies are assumed to be stationary up to time t0and if interacting or in contact have at that time xed deformation corresponding tostatic elastic equilibrium ur t0 D u1r This expression is essentially equivalentto that used by a number of authors [76ndash79] with the difference being that thelatter have treated contact problems with Pphs t replaced by a speci ed analyticPpcs t whereas here Pphs t is determined by the physical force law and therate of change of separation over earlier times

An algorithm has been developed for solving the full non-contact problem for thecase where the creep compliance function has an exponential form [74]

1

EtD

1

E1C

E1 iexcl E0

E1E0eiexclt=iquest (9)

Here E0 and E1 are the short- and long-time elasticity parameters respectivelyand iquest is the relaxation time The algorithm can be generalized to more complexmaterials with multiple relaxation times [74] The present three-parameter model isperhaps the simplest model of viscoelastic materials although an alternative three-parameter expression Etiexcl1 D C0 C C1tm 0 lt m lt 1 has also been used tomodel liquid-like materials [78ndash80]

With the exponential creep compliance function differentiation of the deforma-tion yields [74]

Pur t Diexcl1iquest

[ur t iexcl u1r t]

iexcl2

frac14E0

Zds

Pphs t

jr iexcl sj (10)

where u1 is the static deformation that would occur in the limit t 1 if thepressure pro le were xed at its current value

u1r t Diexcl2

frac14E1

Zds

phs t

jr iexcl sj (11)

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Figure 10 Surface pro les for adhering viscoelastic spheres The pro les are plotted everymillisecond or every 2 nm from h0 D 10 nm (top) to iexcl10 nm (bottom) The drive speed isj Ph0j D 2 sup1ms and the Hamaker constant is A D 10iexcl19 J with z0 D 05 nm and R D 10 sup1mThe viscoelastic parameters are E0 D 1010 Nm2 E1 D 109 Nm2 and iquest D 1 ms The right-handpanel is for loading and the left-hand panel is for unloading The data are from Ref [75]

The rate of change of the pressure is

Pphr t D p0hr tpound Ph0t iexcl Pur t

curren (12)

where Ph0t is the speci ed drive trajectory Accordingly equation (10) representsa linear integral equation for the rate of change of deformation It can be solvedusing the same algorithm that has been developed for the elastic problem [38 40]It is then a simple matter to solve the differential equation for the deformation bysimple time stepping along the trajectory ur t C 1t D ur t C 1t Pur t

The algorithm has been used to obtain results for an electrical double-layerrepulsion [74] and for a van der Waals attraction [75] The latter is

ph DA

6frac14h3

microz6

0

h6iexcl 1

para (13)

where A is the Hamaker constant and z0 characterizes the length scale of the soft-wall repulsion Figure 10 shows the shape of viscoelastic spheres during theirinteraction The total time spent on the loading branch is ten times the relaxationtime so that one expects to see viscoelastic effects At the largest separationprior to approach the surfaces are undeformed Prior to contact on approachthey bulge towards each other under the in uence of the van der Waals attractionThere is a relatively rapid jump into contact and initially a fast spreading of the attened contact region which continues to grow as the bodies are driven furthertogether At the edges of the contact region there is a noticeable rounding of

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768 P Attard

the surface pro les on the approach branch Following the reversal of the motion(unloading) the surfaces become extended as they are pulled apart and there is asharper transition between contact and non-contact than on the loading branch Itshould be noted however that even in this case the slopes at the edge of the contactregion are not discontinuous as predicted by the JKR theory Following the turningpoint the surfaces are effectively pinned in contact for a time and then the contactregion begins to recede After the surfaces jump apart there remains a memoryof the stretching that occurred during unloading and for a time comparable to therelaxation time of the material the deformed separation is smaller on the unloadingbranch out of contact than at the corresponding position upon loading

This hysteresis in surface shape is re ected in the difference in force versusnominal separation curves on the loading and unloading branches (Fig 11) Onapproach prior to contact a given attraction occurs at larger nominal separationfor slower driving speeds In these cases there is an increased bulge leading tosmaller actual separations a consequence of the fact that viscoelastic materialssoften over longer time-scales The jump of the surfaces into contact is re ected ina sharp decrease in the force Once in contact the force increases and the nominalseparation becomes negative which is a re ection of the deformation and growth ofthe attened contact region under increasing load The faster the bodies are driventogether the steeper is the slope of the force curve as one might expect since thiscorresponds to materials that are in effect stiffer since they do not have time to relax

Figure 11 Interaction forces for adhering viscoelastic spheres From small to large the hysteresisloops correspond to driving velocities of j Ph0j D 1 2 and 5 sup1ms using the viscoelastic parameters ofFig 10 The crosses represent the static equilibriumelastic result for E1 D 109 Nm2 Inset loadingcurves in the region near initial contact The diamonds represent the static equilibrium elastic resultfor E0 D 1010 Nm2 and the bold curve is the force for rigid particles The data are from Ref [75]

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Following the reversal of the direction of motion in Fig 11 a small increase inthe nominal separation gives a large decrease in the applied load which causes theunloading branch to lie beneath the loading branch This behaviour is re ected in thesurface pro les (Fig 10) where on the loading branch increasing the load causesthe contact area to grow In contrast immediately following the turning pointdecreasing the load stretches the surfaces at a xed contact area The hysteresis inthe force curves manifests the fact that a certain energy has to be put into the systemto move the surfaces a nominal distance on loading and less energy is recoveredfrom the system in moving the same distance on unloading This is preciselywhat one would expect from a viscoelastic system The size of the hysteresis loopincreases with the speed with which the bodies are driven together or pulled apartAs the speed is decreased both loops appear to coalesce on the long-time elasticresult which corresponds to static equilibrium equation (7)

Figure 11 also shows that the adhesion which is the maximum tension on the forceloop increases with the drive velocity Ph0 Because the position here is controlledwe are able to calculate the trajectory past the force minimum and beyond the out ofcontact jump In an experiment that controlled the load the force minimum wouldbe the last point measured in contact The position of the minimum force moves tosmaller (more negative) nominal separations as the velocity is increased It can beseen that the adhesion of the viscoelastic bodies is signi cantly greater than that ofelastic bodies

The velocity dependence of the adhesion is explored in more detail in Fig 12As the velocity is decreased the curves asymptote to the static equilibrium elastic

Figure 12 Adhesion force The maximum tension normalized by the JKR elastic adhesion is plottedas a function of the drive velocity (logarithmic scale) The parameters are as in Fig 10 except that theHamaker constant is A D 1 5 and 10 pound10iexcl20 J (the surface energy is deg acute A=16frac14z2

0 D 080 398and 796 mJm2) for the dotted dashed and solid curves respectively The data are from Ref [75]

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770 P Attard

result calculated from equation (7) It should be noted that the elastic result isnot given by the JKR prediction which as a contact approximation that neglectsthe range of the van der Waals interaction is not exact It can be seen that forelastic materials the JKR approximation is more accurate for bodies with largersurface energies As the velocity increases and the system is given less time toequilibrate viscoelastic effects become more evident and the adhesion increasesFor the present parameters at speeds greater than about 10 sup1ms there occursa noticeable dependence of the normalized adhesion on the surface energy withhigher energy bodies showing less (normalized) adhesion The actual adhesionincreases with the surface energy at all driving velocities This suggests that atvery high speeds the adhesion will be independent of the surface energy

42 Central deformation approximation

For the case of elastic particles or bodies a relatively accurate analytic approxima-tion for the elastic integral has been developed to treat the pre-contact situation [38]The elastic central deformation approximation (CDA) consists of replacing the de-formation ur everywhere by its value on the central axis u0 An analogousapproximation can be made for the viscoelastic case and results in the form of ananalytic differential equation have been presented for the van der Waals attractionused above [75] and for an electrical double-layer repulsion [74] The latter has theform

ph D P eiexclmiddoth (14)

In this case the analytic approximation for the central deformation ut acute u0 t

is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq

8frac14middotRP 2=E20 exp iexclmiddot[h0t iexcl ut] and u1t D iexclE0f t=E1middot

For a given trajectory h0t the deformation ut is readily obtained from the pre-ceding equation for Put by simple time-stepping The force in this approximationis essentially as given by Derjaguin except of course that the actual deformed sepa-ration is used rather than the nominal separation that would be appropriate for rigidparticles or bodies That is F t D 2frac14Rmiddotiexcl1P exp iexclmiddot[h0t iexcl ut]

This central deformation approximation is tested against the exact results for thepre-contact deformation of a viscoelastic sphere being driven towards a substrate inFig 13 The deformation is negative which corresponds to attening of the particlesor bodies under their mutual repulsion It may be seen that the differential equationis quantitatively accurate for the deformation It correctly shows that at a givenposition h0 the deformation is greater at the slower driving speed because the long-time component of the elasticity which is softer than the short-time component hasmore time to take effect Conversely the force is greater at the faster driving speed

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Figure 13 Pre-contact attening for repulsive forces The symbols represent the exact calculationand the solid curves are the central deformation approximation equation (15) The parameters areas in Fig 10 with P D 107 Nm2 and middotiexcl1 D 1 nm being used in the pressure law equation (14)A constant driving velocity of Ph0 D 5 (upper) and of 1 sup1ms (lower) is used The inset shows thecorresponding forces normalized by the radius for Ph0 D 1 sup1ms with the bold curve representing thein nitely rigid case (no deformation) The data are from Ref [74]

because the surface separation of the effectively stiffer material is smaller at a givenposition (not shown)

The inset of Fig 13 compares the load on a viscoelastic sphere to that on anundeformable one at a given position It can be seen that the load required to movethe deformable particle or body a nominal amount (the drive distance) is less thanthat required for a rigid particle because the surface separation between deformedparticles is greater than that between undeformed particles The agreement betweenthe central deformation approximation equation (15) and the exact calculations inthe inset con rms the validity of the elastic Derjaguin approximation As the latterapproximation shows the major effect of deformation on the force arises from thechange in surface separation rather than from any increase in contact area due to attening

It should be clearly understood that the CDA is only valid for small deformationswhich is the case prior to contact for weak loads and for stiff bodies In thesecases it does not matter that it is a poor approximation to take the deformationto be constant everywhere because the region away from the central axis wherethe approximation fails contributes negligibly to the integration for the force or forthe deformation because the local separation is large In this regime the Derjaguinapproximation is indeed accurate and the major effect of deformation on the force isthat it alters the separation on the central axis Conversely the CDA is not accuratein the post-contact situation where there is a attened contact region Because the

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local separation is small throughout the contact region the local pressure is rapidlyvarying and this must be accurately calculated It is not permissable to take thedeformation to be constant here and the CDA is inapplicable In this regime theDerjaguin approximation also fails

43 Deformation and adhesion measurements

The AFM is an ideal tool for the study of viscoelastic effects because of its real-time acquisition of data during controlled dynamic measurements The data that aredirectly obtainable are the force as a function of the drive distance for both loadingand unloading and the adhesion Detailed analysis of these data using the elasticand viscoelastic theories described above should allow the extraction of the amountof deformation and the values of the elastic parameters and relaxation times

In our laboratory we have recently commenced a research programme of quantita-tive AFM measurements and analyses of the interaction deformation and adhesionof viscoelastic materials [81] We use an emulsion polymerization process to makepoly(dimethylsiloxane) (PDMS) droplets or particles [82 83] The deformabilityranges from liquid- to solid-like and is controlled by the ratio of trimer to monomercross-linker used in the synthesis Depending on the conditions micrometre-sizeddroplets form and subsequently adhere to a hydrophobic glass slide which is latertransferred to the AFM A 7 sup1m silica colloid probe is attached to the cantileverthe well-de ned and known geometry and surface chemistry of the probe enable aquantitative analysis of the measurements The zeta potential of the droplets is mea-sured by electrophoresis [84] The surface chemistry of the droplets is very similarto that of the silica probe at pH 96 the zeta potential of a droplet is iexcl46 mV andthat of the silica colloid is iexcl62 mV

There have been a number of previous AFM studies of deformable solid surfaces[52ndash62] In addition the AFM has been applied to air bubbles [85ndash88] and tooil droplets [89ndash92] Measurements of such systems raise two immediate issuesthe determination of the normal sensitivity factor which relates the measuredvertical photodiode voltage to the de ection of the cantilever and the determinationof the zero of separation Two further issues of analysis arise the conversionof the nominal separation to the actual separation (ie the determination of thedeformation) and the relationship of the material and surface properties of thesubstrate to the measured interaction

One can perform the vertical calibration by a prior measurement on a hardsubstrate in the constant compliance regime We performed this calibration insitu by simply moving away from the droplet and pressing the cantilever againstthe substrate [81] If this is not possible (because either the drop is macroscopicor a deformable probe is attached to the cantilever) then one can perform thecalibration on another cantilever provided that one takes care with the remountingand alignment of the laser beam as described in Section 2 above and in Ref [12]

The matter of determination of the zero of separation can only be done if the forcelaw is known At large separations the deformation is always negligible because

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Figure 14 AFM measurement of the force between a PDMS droplet (iexcl46 mV) and a silica sphere(iexcl62 mV) in 1 mM KNO3 at pH 98 The drive speed is 12 sup1ms and the drive distance is withrespect to an arbitrary zero The at force extrema arise from photodiode saturation Inset force ona logarithmic plot The zero of the nominal separation is determined by shifting the data to coincidewith the electricaldouble-layer force at large separation calculatedusing the measured zeta potentialsThe straight line is the linear PoissonndashBoltzmann law for rigid particles and the partly obscured curveis the elastic central deformation approximation equation (18) with a tted elasticity parameterE1 D 7 pound 105 Jm3 The CDA is shown dashed for h0 lt iexcl19 nm which for a pure double-layerinteraction is the point of actual contact h D 0 The data are from Ref [81]

here the force is weak In practical terms of course it is a matter of whether ornot one has the instrumental resolution to measure weak enough forces and thisis determined by the ratio of the cantilever spring constant to the deformabilityof the substrate or particle Assuming that this regime is accessible then at largeseparations the measured force must equal that between rigid particles If the latteris known then this fact can be used to shift the experimental data so that theycoincide with the known force law at large separations When this is done the drivedistance which has arbitrary zero is converted to a nominal separation which isthe separation between rigid particles This procedure is now illustrated as is themethod of calculating the deformation of the particles which allows the conversionof the nominal separation to the actual separation

Figure 14 shows the force between a silica sphere (diameter 7 sup1m) and a solid-like PDMS droplet (diameter 12 sup1m 50 trimer) measured as a function of thedrive distance [81] After the initial zero force regime one can see the electricaldouble-layer repulsion due to the interaction of the two negatively charged surfacesAt a force of around 20 nN there is a jump into contact due to the van der Waalsattraction followed by a soft compliance regime The latter is characterized by

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a nite slope and a non-zero curvature Upon reversing the direction (ignoringthe instrumental saturation at about 35 nN force) the soft compliance is againevident with the change in slope indicating hysteresis The adhesion of the surfacescontributes to this hysteresis and they do not jump apart until being driven a distanceof several hundred nanometres from the point of maximum load (Again theinstrumental saturation at about iexcl35 nN is ignored)

The analysis of the data is illustrated in the inset of Fig 14 The zero of separationis established by shifting the measured data horizontally to coincide with the linearPoissonndash Boltzmann law at large separations It can be seen that over a limitedregime the data are indeed linear on the log plot with a slope corresponding tothe expected Debye length The relatively short range of this regime is due to acombination of the large deformability of the PDMS droplet and the stiffness of thecantilever k D 058 Nm chosen in order to measure large applied loads and asmuch of the adhesion as possible The magnitude of the data at the largest separationis slightly greater than the resolution of the AFM as can be judged from the noiseThe apparent accelerated decay of the data compared with the Debye length at theextremity of the range exhibited is due to the laser beam re ected from the cantileverbeing interfered with by re ections from the substrate

The linear PoissonndashBoltzmann law used here is given by F h0 D 2frac14Rmiddotiexcl1D pound

P0eiexclmiddotDh0 where middotiexcl1D D 96 nm is the Debye screening length h0 is the nominal

separation (between rigid particles) and R D 06 sup1m is the radius of the PDMSdroplet In linear Poissonndash Boltzmann theory the pre-factor in the pressure lawequation (14) is given by

P D 20rmiddot2DAtilde1Atilde2 (16)

where 0 D 8854 pound 10iexcl12 is the permittivity of free space r D 78 is thedielectric constant of water and Atilde1 D iexcl46 mV and Atilde2 D iexcl62 mV are the surfacepotentials of the PDMS and the silica sphere respectively which are measuredindependently by electrophoresis [84] In practice an effective surface potential isused which essentially converts this into the non-linear PoissonndashBoltzmann law inthe asymptotic regime [93 94] One replaces Atilde by 4deg kBT =q where q D 16 pound10iexcl19 C is the charge on the monovalent electrolyte ions kB D 138 pound 10iexcl23 JK isBoltzmannrsquos constant T D 300 K is the temperature and

deg DeqAtilde=2kBT iexcl 1

eqAtilde=2kBT C 1 (17)

As discussed in Section 42 the central deformation approximation (CDA) forelastic particles or bodies gives for the pre-contact deformation [38]

u D iexclp

8frac14R=middotDE2P0eiexclmiddotD [h0iexclu]

acute iexcleiexclmiddotD[h0iexclu] (18)

Although this can be solved by iteration to obtain the deformation u for anynominal separation h0 for the purposes of plotting it is easier to specify h and

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to calculate directly the corresponding u and h0 The resultant force is F h0 D2frac14Rmiddotiexcl1

D P0eiexclmiddotDh where the actual separation is h D h0 iexcl uThe inset of Fig 14 compares this elastic CDA with the measured data using

a tted elasticity of E1 D 7 pound 105 Nm2 At large separations in the weakforce regime it coincides with the rigid particle result but due to the extremesoftness of the particles the force increases much less rapidly than the linearPoissonndash Boltzmann law predicts The CDA predicts that the surfaces come intoactual contact (h D 0) at a nominal separation of h0 D iexcl19 nm and the theoryis continued past this point as a dashed line There is a noticeable increase in thesteepness of the data beyond this point which suggests that the force is no longera pure double-layer interaction The agreement between the approximation and themeasurements is quite good which con rms the utility of the former and the role ofdeformation in the latter

The force given by both the CDA and the measurements becomes relatively linearon the log plot at negative nominal separations (Recall that a negative nominalseparation would correspond to interpenetration of the bodies or particles if they didnot deform the actual separation is always positive) Effectively the Debye lengthhas been renormalized due to the elasticity of the substrate It is straightforward toobtain from equation (18) an expression for the CDA decay length in this regimeThe limiting force is given by

F h0 D 2frac14Rmiddotiexcl1D P 0

0eiexclmiddoth0 (19)

where the decay length is

middot DmiddotD

1 C middotD (20)

and the renormalized pressure coef cient is

P 00 D P0eiexclmiddot (21)

The length was de ned above and the regime of validity of this result is iexcl lt

h0 iquest middotiexcl1D

The amount of deformation is substantial being of the order of 100 nm at thelargest applied loads compared with a particle diameter of 1200 nm It is possiblethat the turn up in the force just prior to the van der Waals jump could be dueto the contribution from the underlying rigid substrate at these large deformationsAlternatively there is some evidence that this is instead due to a steric repulsion dueto extended polymer chains (see above and below)

The viscoelastic nature of the PDMS droplet is clearly exhibited in Fig 15 whichshows the velocity dependence of the interaction (The hydrodynamic drainageforce is negligible here) In general the repulsive force at a given drive positionincreases with increasing drive velocity This is consistent with the notions thatunderlie the creep compliance function namely that viscoelastic materials areinitially stiff and soften over time One may conclude from the data that relaxation

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776 P Attard

Figure 15 Velocity dependence of the PDMS loading curve From top to bottom the velocities are3 1 and 05 sup1ms The curves are the viscoelastic central deformation approximation using ttedparameters E0 D 5 pound 106 Jm3 E1 D 5 pound 105 Jm3 and iquest D 003 s The bold curve is thedouble-layer force between rigid particles Inset force on a logarithmic scale The data are fromRef [81]

processes decrease the force at a given nominal separation for particles that arebeing more slowly loaded The physical mechanism by which this occurs is the attening of the particle which increases the actual separation and consequentlydecreases the force Driving more slowly allows time for this deformation to occur

The viscoelastic CDA has been tted to the data in Fig 15 The long-timeelasticity E1 D 5 pound 105 Nm2 is slightly less than that used in the elastic CDA tted in Fig 14 evidently the latter incorporates some of the initial stiffness The tted short-time elasticity E0 D 5pound106 Nm2 is substantially greater than the shorttime one which indicates a large viscoelastic effect At the fastest driving velocityshown the loading curve approaches that between rigid surfaces The relaxationtime used in the approximation is iquest D 003 s and it is suf cient to describe thetransition from short- to long-time behaviour observed in the experiments

The viscoelastic CDA may be described as semi-quantitative There are a numberof reasons for the evident discrepancies between the theory and the experimentsFirst is the obvious fact that the CDA is an approximation to the full viscoelastictheory In particular it is not accurate when there is substantial surface attening asoccurs for example in the post-contact regime Second of course is the simplicityof the three-parameter viscoelastic model Doubtless there are multiple relaxationmodes in the PDMS droplet and the model is only useful in so far as one ofthese modes dominates the experiment Third is the use of the purely exponentialdouble-layer force law Close to actual contact this is not correct (due for example

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Figure 16 Hysteresis and adhesion of the PDMS droplet The velocitiesare j Ph0j = 4 2 and 05 sup1msfrom top to bottom at the point of reversal The data are from Ref [81]

to the non-linear nature of the PoissonndashBoltzmann equation and also to chargeregulation effects such as constant potential boundary conditions) Despite thesesimpli cations the CDA represents a viable approximate theory that can be used toextract the material parameters of viscoelastic materials

An additional consideration is that close to contact other forces will start tocontribute as discussed in connection with the CDA prediction of contact in Fig 14In particular the kink in the data in Fig 15 at a load of 15ndash2 nN is evidence of sucha non-electrical double-layer force This and the subsequent steeper gradient in themeasured data likely indicate actual steric contact of polymer tails that extend outfrom the surface of the particle into the solution (Miklavcic and Mar Iumlcelja haveused a mean- eld theory to model the interaction of polyelectrolytes and obtained asimilar initial softening of the double-layer repulsion followed by a steeper steric in-teraction [95]) That this kink occurs at a substantially lower load than the putativevan der Waals jump identi ed in Fig 14 and is of different character supportsa model of the PDMS droplet as a dense core surrounded by a diffuse corona ofpolymer tails

Figure 16 shows the velocity dependence of the hysteresis and the adhesion ofthe PDMS droplet The area of the hysteresis loop which gives the amount ofenergy dissipation increases with the drive speed as one would expect for a viscoussystem The maximum load drops with decreasing speed as predicted by theviscoelastic theory Fig 11 The difference between Fig 11 and Fig 16 is that in theformer the turning point is at a xed nominal separation whereas in the latter it isat a xed drive distance the nominal separation at a xed drive distance decreaseswith speed due to the decreased cantilever de ection

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778 P Attard

The adhesion which is the minimum load or equivalently the maximum tensionalso increases with the drive speed What is also noticeable on the retractioncurves are the long-range attractions that increase with separation and that appear asdiscrete steps These may be attributed to individual bridging polymers with the atregions corresponding to the peeling of the polymer from the silica sphere segmentby segment and the regions of increasing force corresponding to the stretching ofthe individual polymer chains Such forces between individual bridging polymershave been explored in other AFM measurements [62 96ndash100] Between one andthree bridging chains can be seen in the individual force curves in Fig 16 The forcedue to the longest bridging polymer is remarkably independent of velocity

5 BRIDGING NANOBUBBLE DYNAMICS

51 Experimental evidence

In 1972 Blake and Kitchener [101] found that bubbles ruptured at inexplicably largeseparations from hydrophobic surfaces but it took a decade before the existenceof a long-range attraction between such surfaces was con rmed by direct forcemeasurements [102ndash104] The force appeared to be universally present betweenhydrophobic surfaces (ie those on which water droplets had a high contact angle)and was much stronger than the van der Waals attraction which was the only otherknown attractive force between identical surfaces It produced an extremely largeadhesion and it had a measurable range of hundreds of nanometres [105 106]which is orders of magnitude larger than that of most surface forces

The broad features of this unusual force were reproduced in a number oflaboratories and many efforts were made to explain its origin The earliest attempt ata quantitative theory suggested that the surfaces coupled by correlated electrostatic uctuations with the consequence that the decay length of the attraction should behalf the Debye length [107] This idea was subsequently taken up and developedby a number of authors [108ndash111] Although several experiments appear to showthe predicted dependence on the electrolyte concentration [103 104 112] thevast majority are insensitive to the concentration or valence of the electrolyte[106 113ndash116] One must conclude that the proposed electrostatic mechanismis not in general the origin for the measured hydrophobic attraction It had alsobeen proposed that the surface-induced structure in the water was responsible forthe long-range interaction [117] This poly-structural theory is contradicted bythe evidence from computer simulations which show that the structure inducedby surfaces propagates less than about 1 nm into the water [118 119] Furthermorethe fact that the solvophobic force measured in non-hydrogen-bonding organicliquids is almost identical to that measured in water has also been taken as evidenceagainst the theory [120] Finally vapour cavities had been observed between thehydrophobic surfaces when they were in contact [121] and a theory for the force interms of separation-induced spinodal cavitation has been developed [122ndash124] Itis dif cult to design an experimental test of this theory

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Figure 17 Force measured between hydrophobic glass surfaces in water (R D 21 mm) Threeseparate approach curves are shown Inset magni cation at large separations showing steps in thedata The data are from Ref [106]

In 1994 Parker et al [106] explored the phenomenon with the Measurement andAnalysis of Surface and Interaction Forces (MASIF) device [3 4] This is a type ofAFM that uses macroscopic surfaces (radii 2 mm) and like the AFM electronicallycollects large volumes of data at high resolution Some of these data are reproducedin Fig 17 where the extreme range and strength of the attraction are evident Thesteps in the force at large separations had not previously been seen with the surfaceforces apparatus because of its low resolution and few data points (They are alsodif cult to see with the AFM because the low inertia and weak spring constantof the cantilever lead to a rapid jump into contact and there are normally no dataavailable between the onset of the attraction and the jump) These steps in the dataprovided the key to understanding the physical origin of the force It was proposedthat there were sub-microscopic bubbles present on the hydrophobic surfaces andthat each step represented the instant of attachment of a bubble on one surface to theother surface [106 125] These bridging bubbles spread along the surfaces and giverise to the measured force An attractive feature of the lsquonanobubblersquo theory is thatthe range of the interaction between hydrophobic surfaces is set by the height of thebubbles on the isolated surface and there is no need to invoke any new long-rangeforce to account for the data The fact that calculations of the force due to multiplebridging bubbles were in quantitative agreement with the measured data providedstrong support for the proposed physical origin [106]

Further support for the notion that nanobubbles pre-existed on the hydrophobicsurfaces and that their bridging was responsible for the measured attractionssubsequently came from de-aeration experiments which showed that the forcetended to be more short-ranged when measured in de-aerated water [115 126]Wood and Sharma [126] showed that the force was also of shorter range whenmeasured between surfaces that had never been exposed to the atmosphere which

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780 P Attard

Figure 18 Force between a silica colloid (R D 103 sup1m) and glass surface Both surfaces werehydrophobized by exposure to silane vapour and the AFM measurements were performed in 95 mM(crosses) and 019 mM (triangles) NaCl at a drive velocity of 45 sup1ms Inset large separationrepulsion on a logarithmic scale The curve is the calculated hydrodynamic drainage force The dataare from Ref [116]

suggests that the bubbles attached to defects on the surfaces when they were takenthrough the airndashwater interface

In 1998 Carambassis et al [116] obtained AFM results that by virtue of thedetail of the force curves provided signi cant support for nanobubbles as the originof the long-range attraction By using a colloid sphere attached to the cantileverthey were able to obtain the force due to a single nanobubble in the contact regionand their results were more readily interpretable than the multiple bubble results ofParker et al [106] Perhaps the most striking new feature that appears in Fig 18is the short-range repulsion that appears prior to the jump into contact The datasuggest that prior to interaction there is a nanobubble on one of the surfaces in onecase it has a height of about 60 nm and in the other case it has a height of about150 nm The evident repulsion prior to the jump into contact is in part a double-layerinteraction between the liquidndashvapour interface and the approaching solid surfaceA quantitative theory for the data following the jump has been made by Attard [127]and is discussed in more detail below According to the theory the jump into contactfollowing the initial repulsion is due to the bridging of the bubble between the twosurfaces and the extended soft-contact varying-compliance region is a dynamiceffect due to its lateral spreading The results of Carambassis et al [116] havebeen con rmed by a number of similar AFM measurements [128ndash131] Theselater papers include measurements of forces in de-aerated water and concur withthe earlier conclusion of Wood and Sharma that the force was on average shorter-

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ranged in this case [126] Finally infra-red spectroscopy has been used to show thepresence of gaseous CO2 between aggregated hydrophobic colloids [132]

Taken in total the evidence in support of the existence of nanobubbles isoverwhelming There is now general consensus that they are responsible forthe long-range attractions measured between hydrophobic surfaces as originallyproposed by Attard and co-workers [106 125]

52 Theory for bridging bubbles

In order to calculate the force due to a bridging bubble one must rst calculatethe bubble shape This is done by optimizing the appropriate constrained thermo-dynamic potential [133 134] In this case the external atmospheric pressure p0the temperature T the liquidndashvapour surface energy deg and the difference in solidsurface energies 1deg gt 0 (the contact angle at equilibrium is micro D cosiexcl1[iexcl1deg=deg ])are xed as is the number of gas molecules N The last condition is importantas assuming diffusive equilibrium of the gas with the atmosphere leads to the pre-diction that all bubbles are unstable [106 125 127] The constrained Gibbs freeenergy for an arbitrary bubble pro le zr is

G[z]jX h0 D p0V iexcl NkBT ln V C deg Alv iexcl 1deg Asv (22)

where kB is Boltzmannrsquos constant V [z] is the volume of the bubble Alv[z] is theliquidndashvapour surface area Asv[z] is the solidndashvapour surface area X representsthe xed variables listed above and h0 is the separation between the solid surfaces

The equilibrium bubble pro le Nzr may be obtained by functional differenti-ation which results in the EulerndashLagrange equations and which was the originalprocedure used to obtain the force due to a bridging bubble [106] Alternativelythe pro le may be parameterized by a suitable polynomial expansion and the opti-mization may be carried out with respect to the coef cients such a procedure hascertain numerical advantages [127] If the coef cients are denoted by ai then thedependence of the pro le on them and on the separation may by symbolized aszrI a h0 The equilibrium pro le Nzr D zrI Na h0 is the one that minimizesthe constrained potential and hence the equilibrium coef cients satisfy

G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)

The thermodynamic potential is the minimum value of the constrained potentialGX h0 acute G[Nz]jX h0 The force between the solids is [127]

F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

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782 P Attard

Figure 19 Gibbs potential for a bridging bubble as a function of the constrained contact radiusThe surface separations are from bottom to top h0 D 30 40 50 60 70 80 and 90 nm Theequilibriumradius which is given by the minimum in the potential is microscopicat small separationsand sub-microscopic at large separations The liquidndashvapour surface tension is deg D 72 mNm theexternal pressure is p0 D 105 Nm2 both hydrophobic surfaces are of radius R D 20 sup1m andhave an equilibrium water contact angle of micro D 100plusmn and the number of gas molecules is xed atN D 14 pound 105 Inset magni cation of the minimum at sub-microscopic radii The data are fromRef [40]

Even though Nai depends on h0 the second equality follows from the variationalnature of the constrained thermodynamic potential as manifest in equation (23)[133 134]

One advantage of the constrained thermodynamic potential approach is that theapproach to equilibrium can be explored by holding particular variables constantThis is illustrated in Fig 19 where the potential is plotted as a function of thecontact radius Minima in the potential correspond to equilibrium values Whetherthese minima are local or global determines whether that particular size is stable ormetastable It can be seen that there are deep minima at microscopic radii and moreshallow minima at sub-microscopic radii Microscopic bubbles are absolutely stableat small separations and sub-microscopic bubbles are absolutely stable at largeseparations and there is an overlapping regime at intermediate separations whereone branch is metastable with respect to the other (All the bridging bubbles arestable with respect to the hemispherical bubble on the isolated surface which has aGibbs free energy of 5035 pJ) Hence the bridging bubble is hysteretic approachingfrom large separations the bubble is initially sub-microscopic before jumping tomicroscopic dimensions and conversely upon retraction with the reverse jumpoccurring at larger separations

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Figure 20 Equilibrium shape of a bridging bubble The bubble shrinks as the separation increasesfrom right to left the microscopic bubbles occur at separations of h0 D 0 10 20 30 40 50 60and 70 nm The other parameters are as in Fig 19 Inset magni cation of the large separation sub-microscopic bubbles with from right to left h0 D 60 70 80 90 and 100 nm The rst two pro lesare metastable with respect to their microscopic counterparts at the same separation The data arefrom Ref [40]

Figure 20 shows the equilibrium shape of the bridging bubble In accord withthe constrained thermodynamic potential calculations Fig 19 one can see that atsmall separations the equilibrium bridging bubble has a microscopic lateral radiuswhereas at larger separations it is sub-microscopic There is a marked distinctionbetween the two sizes On the isolated surface this bubble sits as a hemisphere ofradius 50 nm height 413 nm and contact radius 492 nm Hence it can be seenthat at small separations the bubble has expanded laterally by more than a factor of20 In general the bubbles are concave or saddle-shaped which indicates that theinternal gas pressure is less than the external atmospheric pressure However thedeparture from cylindrical shape is relatively small and it will be shown below thatapproximating the bubble as a cylinder provides simple but accurate results for theforce due to the bridging bubble

The hysteresis due to the local minima in the constrained thermodynamic potentialappears clearly in the force plot Fig 21 The force due to the bridging bubbleis attractive and increases monotonically with separation It is weak on the sub-microscopic branch and much stronger on the microscopic branch The jump onapproach occurs at smaller separations than that on retraction

Also shown in Fig 21 is the force due to a cylindrical bridging bubble In thisapproximation the optimum radius of the cylinder Nrh0 is obtained by minimizingthe constrained thermodynamic potential given above For microscopic cylindersthe pressure inside the bubble may be neglected The inverse formula for the

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784 P Attard

Figure 21 The interaction force due to an unconstrained bridging bubble (parameters as in Fig 19)The attraction is large at small separations where the bubble is microscopic and is weak at largeseparations where the bubble is sub-microscopic Note that the jump between the two branchesoccurs at smaller separations on approach h0 D 52 nm than on retraction h0 D 80 nm whichgives rise to hysteresis in the force The dotted curve that terminates at h0 D 76 nm is the bridgingcylinder approximation equation (25) The horizontal arrow is the classical capillary adhesionequation (26) Inset expansion of the force on the sub-microscopic branch No bridging bubblewith these parameters is stable beyond h0 D 112 nm The data are from Ref [40]

separation as a function of the radius explicitly is [127]

Nh0 D 2p

R2 iexcl r2 iexcl 2R C2Rr1deg iexcl 2r2deg

rp0 C deg p

R2 iexcl r2 (25)

The force is F D iexclfrac14r2p0 iexcl 2frac14rdeg It can be seen in Fig 21 that the bridgingcylinder approximation is quite accurate for the force on the microscopic branch

The adhesion or capillary force due to the bridging bubble is also of interest Thelargest radius occurs at contact h0 D 0 and in the bridging cylinder approximationit is rcurren D iexcl3deg =2p0[1 iexcl

p1 C 8Rp01deg =9deg 2] [127] The capillary adhesion is

given by F curren D iexclfrac14rcurren2p0 iexcl 2frac14rcurrendeg As can be seen in Fig 21 this result is moreaccurate for small colloidal particles than the classical result

F curren D 2frac14Rdeg cos micro (26)

(Both results agree in the limit of large R)

53 Spreading bubble

The calculated force in Fig 21 appears qualitatively different from the measuredforces shown in Fig 18 Although the experiments show a de nite jump into

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contact the attraction is about two orders of magnitude weaker than the calculatedadhesion In addition the pre-jump repulsion and the soft-contact varying-compliance region are not predicted by the calculations

Obviously the calculated force due to the bridging bubble is only relevant afterattachment of the bubble to the approaching surface and no attempt has been madeto describe the force curve prior to this point The repulsion at large separationevident in the inset of Fig 18 is in part due to the hydrodynamic drainageforce between the colloid particle and the substrate F D iexcl6frac14acuteR2 Ph0=h whereacute D 10iexcl3 kg miexcl1 siexcl1 is the viscosity of water The sharp increase in the repulsionimmediately prior to the jump is probably a combination of deformation plus anelectrical double-layer repulsion The decay length of the measured force wasobserved to decrease with increasing electrolyte concentration but was about one- fth the Debye length in pure water and about twice the Debye length in 10 mMmonovalent electrolyte [116]

The soft-contact varying-compliance region prior to the colloid probe cominginto hard contact with the substrate appears to be a dynamic effect due to thespreading of the bubble (ie surface drying) For the case of a liquid drop on asurface it is well known that a growing drop makes a greater contact angle withthe substrate than a shrinking one and that the gap between the advancing andreceding angles increases with increasing velocity [135ndash137] The existence ofhysteresis and dynamic effects indicates that the equilibration of three-phase contactoccurs over macroscopic time-scales and that the thermodynamic driving forcetowards equilibrium is small compared with dissipative forces (see the discussionof viscoelasticity in Section 3) Similar contact angle hysteresis occurs for ahemispherical bubble in contact with a substrate Hence for the present problemof a bridging bubble one expects hysteresis and velocity-dependent effects as thebubble spreads or recedes

Of course in order to have hysteresis one must have dissipation and the simplestmodel is to invoke a drag force that is proportional to both the velocity and thelength of the contact line

Fd D iexcl2frac14arc Prc (27)

Here rc is the contact radius Prc is its velocity and a is the drag coef cientThe physical origin of the contact line friction is not clear although two likelycontributing mechanisms are viscous dissipation due to hydrodynamic ow in thecontact region [136] and jumping of the contact line between asperities [135 137]In the state of steady motion of the contact line the thermodynamic driving forcemust exactly balance the drag force

iexclGrcjX h0

rciexcl 2frac14arc Prc D 0 (28)

The rst term is the derivative of the constrained thermodynamic potential of abridging bubble of xed contact radius rc but otherwise of optimum shape (cf

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786 P Attard

Figure 22 Dynamic force due to a spreading bridging bubble The AFM data are those of Fig 18[116] and the curves are equation (28) using a tted drag parameter of a D 32 kN s miexcl2 [40] Thecurve passing through the crosses is for N such that on the isolated substrate the hemispherical bubblehas radius Rb D 75 nm and height zb D 62 nm the curve passing through the triangles is for N suchthat Rb D 200 nm and zb D 165 nm The other parameters are as in Fig 19

Fig 19) This differential equation for the contact radius may be solved for a giventrajectory h0t by simple time-stepping [127] The force between the probe andthe substrate was taken to be given by equation (24)

Figure 22 shows that this model of contact line motion is able to describequantitatively the measured data in the soft contact regime The rapid jump intocontact upon bubble attachment the minimum in the force and the ever-steepeningrepulsion are all present in the theoretical calculations The origin of the repulsionis that the drag on the contact line prevents the bubble from growing to its optimumsize at a given separation As the colloid particle is driven towards the substrate theconsequent compression of the bubble leads to the repulsive force

Several simpli cations have been made in the model calculations The calcu-lations are for two identical spheres of radius 20 sup1m whereas the experimentaldata are for a sphere of radius 103 sup1m interacting with a at substrate Similarlythe calculations are for a symmetric bridging bubble which immediately followingattachment to the approaching surface is likely a poor approximation to reality Ad-ditionally in the latter attachment regime the velocity of the contact line is almostcertainly changing rapidly and assuming steady-state conditions likely introduceserrors here Finally no attempt has been made to include the pre-attachment forcesin the calculations The bubble was taken to attach when the separation equalled itsheight on the isolated surface which was tted to the data and the initial contactradius was chosen to give zero normal force at this point

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Because of the variability in the measured data and because of the limitednumber of force curves analysed one can not yet claim to have con rmed thedrag law (27) Nevertheless it is of interest to compare the tted drag coef cienta D 32 pound 103 N miexcl1 siexcl1 with the value of 6 pound 10iexcl2 N miexcl1 siexcl1 estimated by deRuijter et al [137] from molecular dynamics simulations of a spreading hexadecanedroplet The large discrepancy between the two may be due in part to the lowviscosity of the simulated liquid (two orders of magnitude less than that of water)to the low surface tension (about one- fth that of water) and to a low level ofcoupling between the substrate and the liquid in the simulations The averagespeed of the contact line in the simulations is about 1 ms [137] whereas in theexperiments [116] and in the theory [127] the bubble spreads at about 10 sup1msIn both simulations and theory the product of drag coef cient and velocity is3ndash6 pound 10iexcl2 Nm which is of the same order of magnitude as the surface tension

Despite the caveats outlined above the agreement between theory and experimentsupports the notions that bridging bubbles are responsible for the measured forcesand that it is the motion of the contact line that gives rise to the details ofthe force curve Accordingly the theory combined with the dynamic forcemeasurements allows the phenomenon of dynamic wetting to be followed withmolecular resolution

6 CONCLUSION

The atomic force microscope (AFM) is ideally suited to carrying out dynamicmeasurements that can elucidate a variety of time-dependent and non-equilibriumphenomena Here three examples have been reviewed friction viscoelasticity andwetting

In the case of friction a quantitative method of calibrating the torsional springconstant and the lateral photodiode response was described [12] The method isdirect non-destructive and single-step The friction between metal oxide surfacesin an aqueous electrolyte was measured as a function of the applied load usingthe pH to control the adhesion [13] It was found that with the detachment forceused to shift the applied load friction became a universal function of the intrinsicload independent of the pH Elastic deformation calculations further revealed thatsurfaces with the same intrinsic load were at the same local separation whichsuggests that friction is mediated by the short-range interactions between the atoms

A theory for the deformation and adhesion of viscoelastic materials or particlesinteracting with realistic surface forces of non-zero range was summarized [74 75]A triangular drive trajectory led to hysteretic force loops with the hysteresis andthe adhesion increasing with velocity A central deformation approximation (CDA)was introduced that gave accurate analytic results in the pre-contact regime and thatallowed the zero of separation in AFM force measurements to be established AFMmeasurements on PDMS droplets were shown to be qualitatively in accord with

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788 P Attard

the theory and the viscoelastic material parameters were extracted from the data by tting the theory to it [81]

The force between hydrophobic surfaces has been ascribed to bridging nanobub-bles [106] and the soft-contact varying-compliance region observed in AFM mea-surements has been attributed to the drying of the surface as the bubble spreadslaterally [116] This is a dynamic effect that depends on the drive velocity Thethermodynamic force due to a bridging bubble has been calculated and assumingsteady-state conditions and a simple model of contact line friction a quantitativeaccount of the measured data has been obtained [127]

Acknowledgements

It has been a privilege to work with Archie Carrambassis Adam Feiler GraemeGillies Ian Larson John Parker Mark Rutland and James Tyrrell and I thankthem for their very signi cant contributions to the experimental work reviewed hereDiscussions with Sonja Engels and Clive Prestidge have also been helpful

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614 (1996)90 B A Snyder D E Aston and J C Berg Langmuir 13 590 (1997)91 P G Hartley F Grieser P Mulvaney and G W Stevens Langmuir 15 7282 (1999)92 D E Aston and J C Berg J Colloid Interface Sci 235 162 (2001)93 P Attard J Phys Chem 99 14174 (1995)94 P Attard Adv Chem Phys 92 1 (1996)95 S J Miklavcic and S Mar Iumlcelja J Phys Chem 92 6718 (1988)96 T J Senden J-M di Meglio and P Auroy Eur Phys J B 3 211 (1998)97 X Chacirctellier T J Senden J-F Joanny and J-M di Meglio Europhys Lett 41 303 (1998)98 T J Senden J M Di Meglio and I Silberzan C R Acad Sci Paris Ser IV 1 1143 (2000)99 E-L Florin V T Moy and H E Gaub Science 264 415 (1994)

100 U Dammer O Popescu P Wagner D Anslmetti H-J Guumlntherodt and G N Misevic Science267 1173 (1995)

101 T D Blake and J A Kitchener J Chem Soc Faraday Trans 1 68 1435 (1972)102 J N Israelachvili and R M Pashley J Colloid Interface Sci 98 500 (1984)103 Ya I Rabinovich and B V Derjaguin Colloids Surfaces 30 243 (1988)104 H K Christenson P M Claesson J Berg and P C Herder J Phys Chem 93 1472 (1989)105 K Kurihara S Kato and T Kunitake Chem Phys Lett 9 1555 (1990)106 J L Parker P M Claesson and P Attard J Phys Chem 98 8468 (1994)107 P Attard J Phys Chem 93 6441 (1989)108 R Podgornik J Chem Phys 91 5840 (1989)109 Y H Tsao D F Evans and H Wennerstroumlm Langmuir 9 779 (1993)110 S J Miklavic D Y C Chan L R White and T W Healy J Phys Chem 98 9022 (1994)111 O Spalla and L Belloni Phys Rev Lett 74 2515 (1995)112 P Kekicheff and O Spalla Phys Rev Lett 75 1851 (1995)113 H K Christenson J Fang B W Ninham and J L Parker J Phys Chem 94 8004 (1990)114 H K Christenson P M Claesson and J L Parker J Phys Chem 96 6725 (1992)115 L Meagher and V S J Craig Langmuir 10 2736 (1994)116 A Carambassis L C Jonker P Attard and M W Rutland Phys Rev Lett 80 5357 (1998)117 J C Eriksson S Ljunggrenand P M Claesson J Chem Soc Faraday Trans 2 85 163 (1989)118 C Y Lee J A McCammon and P J Rossky J Chem Phys 80 4448 (1984)119 J P Valleau and A A Gardner J Chem Phys 86 4162 (1987)120 R F Considine and C J Drummond Langmuir 16 631 (2000)121 H K Christenson and P M Claesson Science 239 390 (1988)

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122 P Attard C P Ursenbach and G N Patey Phys Rev A 45 7621 (1992)123 D R Beacuterard P Attard and G N Patey J Chem Phys 98 7236 (1993)124 K Lum D Chandler and J D Weeks J Phys Chem B 103 4570 (1999)125 P Attard Langmuir 12 1693 (1996)126 J Wood and R Sharma Langmuir 11 4797 (1995)127 P Attard Langmuir 16 4455 (2000)128 R F Considine R A Hayes and R G Horn Langmuir 15 1657 (1999)129 V S J Craig B W Ninham and R M Pashley Langmuir 15 1562 (1999)130 J Mahnke J Stearnes R A Hayes D Fornasiero and J Ralston Phys Chem Chem Phys 1

2793 (1999)131 N Ishida M Sakamoto M Miyara and K Higashitani Langmuir 16 5681 (2000)132 W Gong J Stearnes R A Hayes D Fornasiero and J Ralston Phys Chem Chem Phys 1

2799 (1999)133 P Attard J Stat Phys 100 445 (2000)134 P Attard Thermodynamics and Statistical Mechanics Equilibrium by Entropy Maximisation

Academic Press London (2002)135 P G de Gennes Rev Mod Phys 57 827 (1985)136 R G Cox J Fluid Mech 357 249 (1998)137 M J de Ruijter T D Blake and J de Coninck Langmuir 15 7836 (1999)

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damages whatsoever or howsoever caused arising directly or indirectly inconnection with or arising out of the use of this material

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PHIL ATTARD curren





1 INTRODUCTION




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rary

] at

06

23 1

1 M

arch

201

3

754 P Attard




21 Critical review



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

756 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




deg Diquest

1VlatD

kx1xL

1VlatD kxreg0L

1Vvert

1Vlat (1)




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

758 P Attard



regL D1z

1Vvertand macrL D

1z

1Vlat (2)


1micro

1VlatD


L (3)



kmicro DiexclkxL2



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3



31 Intrinsic force


f Draquo


0 L lt iexclA(5)






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

760 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

762 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

764 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






ur Diexcl2frac14E

Zds

phs

jr iexcl sj (7)








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

766 P Attard




t0


Zds

Pphs t 0

jr iexcl sj (8)



1

EtD

1

E1C

E1 iexcl E0




Pur t Diexcl1iquest

[ur t iexcl u1r t]

iexcl2

frac14E0

Zds

Pphs t

jr iexcl sj (10)


u1r t Diexcl2

frac14E1

Zds

phs t

jr iexcl sj (11)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





curren (12)



ph DA

6frac14h3

microz6

0

h6iexcl 1

para (13)


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

768 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

770 P Attard






is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3



PHIL ATTARD curren





1 INTRODUCTION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

754 P Attard




21 Critical review



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

756 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




deg Diquest

1VlatD

kx1xL

1VlatD kxreg0L

1Vvert

1Vlat (1)




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

758 P Attard



regL D1z

1Vvertand macrL D

1z

1Vlat (2)


1micro

1VlatD


L (3)



kmicro DiexclkxL2



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3



31 Intrinsic force


f Draquo


0 L lt iexclA(5)






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

760 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

762 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

764 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






ur Diexcl2frac14E

Zds

phs

jr iexcl sj (7)








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

766 P Attard




t0


Zds

Pphs t 0

jr iexcl sj (8)



1

EtD

1

E1C

E1 iexcl E0




Pur t Diexcl1iquest

[ur t iexcl u1r t]

iexcl2

frac14E0

Zds

Pphs t

jr iexcl sj (10)


u1r t Diexcl2

frac14E1

Zds

phs t

jr iexcl sj (11)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





curren (12)



ph DA

6frac14h3

microz6

0

h6iexcl 1

para (13)


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

768 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

770 P Attard






is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

754 P Attard




21 Critical review



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

756 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




deg Diquest

1VlatD

kx1xL

1VlatD kxreg0L

1Vvert

1Vlat (1)




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

758 P Attard



regL D1z

1Vvertand macrL D

1z

1Vlat (2)


1micro

1VlatD


L (3)



kmicro DiexclkxL2



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3



31 Intrinsic force


f Draquo


0 L lt iexclA(5)






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

760 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

762 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

764 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






ur Diexcl2frac14E

Zds

phs

jr iexcl sj (7)








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

766 P Attard




t0


Zds

Pphs t 0

jr iexcl sj (8)



1

EtD

1

E1C

E1 iexcl E0




Pur t Diexcl1iquest

[ur t iexcl u1r t]

iexcl2

frac14E0

Zds

Pphs t

jr iexcl sj (10)


u1r t Diexcl2

frac14E1

Zds

phs t

jr iexcl sj (11)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





curren (12)



ph DA

6frac14h3

microz6

0

h6iexcl 1

para (13)


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

768 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

770 P Attard






is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

756 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




deg Diquest

1VlatD

kx1xL

1VlatD kxreg0L

1Vvert

1Vlat (1)




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

758 P Attard



regL D1z

1Vvertand macrL D

1z

1Vlat (2)


1micro

1VlatD


L (3)



kmicro DiexclkxL2



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3



31 Intrinsic force


f Draquo


0 L lt iexclA(5)






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

760 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

762 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

764 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






ur Diexcl2frac14E

Zds

phs

jr iexcl sj (7)








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

766 P Attard




t0


Zds

Pphs t 0

jr iexcl sj (8)



1

EtD

1

E1C

E1 iexcl E0




Pur t Diexcl1iquest

[ur t iexcl u1r t]

iexcl2

frac14E0

Zds

Pphs t

jr iexcl sj (10)


u1r t Diexcl2

frac14E1

Zds

phs t

jr iexcl sj (11)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





curren (12)



ph DA

6frac14h3

microz6

0

h6iexcl 1

para (13)


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

768 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

770 P Attard






is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

756 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




deg Diquest

1VlatD

kx1xL

1VlatD kxreg0L

1Vvert

1Vlat (1)




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

758 P Attard



regL D1z

1Vvertand macrL D

1z

1Vlat (2)


1micro

1VlatD


L (3)



kmicro DiexclkxL2



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3



31 Intrinsic force


f Draquo


0 L lt iexclA(5)






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

760 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

762 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

764 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






ur Diexcl2frac14E

Zds

phs

jr iexcl sj (7)








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

766 P Attard




t0


Zds

Pphs t 0

jr iexcl sj (8)



1

EtD

1

E1C

E1 iexcl E0




Pur t Diexcl1iquest

[ur t iexcl u1r t]

iexcl2

frac14E0

Zds

Pphs t

jr iexcl sj (10)


u1r t Diexcl2

frac14E1

Zds

phs t

jr iexcl sj (11)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





curren (12)



ph DA

6frac14h3

microz6

0

h6iexcl 1

para (13)


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

768 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

770 P Attard






is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




deg Diquest

1VlatD

kx1xL

1VlatD kxreg0L

1Vvert

1Vlat (1)




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

758 P Attard



regL D1z

1Vvertand macrL D

1z

1Vlat (2)


1micro

1VlatD


L (3)



kmicro DiexclkxL2



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3



31 Intrinsic force


f Draquo


0 L lt iexclA(5)






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

760 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

762 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

764 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






ur Diexcl2frac14E

Zds

phs

jr iexcl sj (7)








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

766 P Attard




t0


Zds

Pphs t 0

jr iexcl sj (8)



1

EtD

1

E1C

E1 iexcl E0




Pur t Diexcl1iquest

[ur t iexcl u1r t]

iexcl2

frac14E0

Zds

Pphs t

jr iexcl sj (10)


u1r t Diexcl2

frac14E1

Zds

phs t

jr iexcl sj (11)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





curren (12)



ph DA

6frac14h3

microz6

0

h6iexcl 1

para (13)


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

768 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

770 P Attard






is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

758 P Attard



regL D1z

1Vvertand macrL D

1z

1Vlat (2)


1micro

1VlatD


L (3)



kmicro DiexclkxL2



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3



31 Intrinsic force


f Draquo


0 L lt iexclA(5)






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

760 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

762 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

764 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






ur Diexcl2frac14E

Zds

phs

jr iexcl sj (7)








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

766 P Attard




t0


Zds

Pphs t 0

jr iexcl sj (8)



1

EtD

1

E1C

E1 iexcl E0




Pur t Diexcl1iquest

[ur t iexcl u1r t]

iexcl2

frac14E0

Zds

Pphs t

jr iexcl sj (10)


u1r t Diexcl2

frac14E1

Zds

phs t

jr iexcl sj (11)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





curren (12)



ph DA

6frac14h3

microz6

0

h6iexcl 1

para (13)


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

768 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

770 P Attard






is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3



31 Intrinsic force


f Draquo


0 L lt iexclA(5)






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

760 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

762 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

764 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






ur Diexcl2frac14E

Zds

phs

jr iexcl sj (7)








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

766 P Attard




t0


Zds

Pphs t 0

jr iexcl sj (8)



1

EtD

1

E1C

E1 iexcl E0




Pur t Diexcl1iquest

[ur t iexcl u1r t]

iexcl2

frac14E0

Zds

Pphs t

jr iexcl sj (10)


u1r t Diexcl2

frac14E1

Zds

phs t

jr iexcl sj (11)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





curren (12)



ph DA

6frac14h3

microz6

0

h6iexcl 1

para (13)


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

768 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

770 P Attard






is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

760 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

762 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

764 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






ur Diexcl2frac14E

Zds

phs

jr iexcl sj (7)








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

766 P Attard




t0


Zds

Pphs t 0

jr iexcl sj (8)



1

EtD

1

E1C

E1 iexcl E0




Pur t Diexcl1iquest

[ur t iexcl u1r t]

iexcl2

frac14E0

Zds

Pphs t

jr iexcl sj (10)


u1r t Diexcl2

frac14E1

Zds

phs t

jr iexcl sj (11)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





curren (12)



ph DA

6frac14h3

microz6

0

h6iexcl 1

para (13)


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

768 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

770 P Attard






is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

762 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

764 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






ur Diexcl2frac14E

Zds

phs

jr iexcl sj (7)








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

766 P Attard




t0


Zds

Pphs t 0

jr iexcl sj (8)



1

EtD

1

E1C

E1 iexcl E0




Pur t Diexcl1iquest

[ur t iexcl u1r t]

iexcl2

frac14E0

Zds

Pphs t

jr iexcl sj (10)


u1r t Diexcl2

frac14E1

Zds

phs t

jr iexcl sj (11)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





curren (12)



ph DA

6frac14h3

microz6

0

h6iexcl 1

para (13)


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

768 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

770 P Attard






is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

762 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

764 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






ur Diexcl2frac14E

Zds

phs

jr iexcl sj (7)








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

766 P Attard




t0


Zds

Pphs t 0

jr iexcl sj (8)



1

EtD

1

E1C

E1 iexcl E0




Pur t Diexcl1iquest

[ur t iexcl u1r t]

iexcl2

frac14E0

Zds

Pphs t

jr iexcl sj (10)


u1r t Diexcl2

frac14E1

Zds

phs t

jr iexcl sj (11)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





curren (12)



ph DA

6frac14h3

microz6

0

h6iexcl 1

para (13)


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

768 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

770 P Attard






is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

764 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






ur Diexcl2frac14E

Zds

phs

jr iexcl sj (7)








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

766 P Attard




t0


Zds

Pphs t 0

jr iexcl sj (8)



1

EtD

1

E1C

E1 iexcl E0




Pur t Diexcl1iquest

[ur t iexcl u1r t]

iexcl2

frac14E0

Zds

Pphs t

jr iexcl sj (10)


u1r t Diexcl2

frac14E1

Zds

phs t

jr iexcl sj (11)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





curren (12)



ph DA

6frac14h3

microz6

0

h6iexcl 1

para (13)


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

768 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

770 P Attard






is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

764 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






ur Diexcl2frac14E

Zds

phs

jr iexcl sj (7)








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

766 P Attard




t0


Zds

Pphs t 0

jr iexcl sj (8)



1

EtD

1

E1C

E1 iexcl E0




Pur t Diexcl1iquest

[ur t iexcl u1r t]

iexcl2

frac14E0

Zds

Pphs t

jr iexcl sj (10)


u1r t Diexcl2

frac14E1

Zds

phs t

jr iexcl sj (11)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





curren (12)



ph DA

6frac14h3

microz6

0

h6iexcl 1

para (13)


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

768 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

770 P Attard






is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






ur Diexcl2frac14E

Zds

phs

jr iexcl sj (7)








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

766 P Attard




t0


Zds

Pphs t 0

jr iexcl sj (8)



1

EtD

1

E1C

E1 iexcl E0




Pur t Diexcl1iquest

[ur t iexcl u1r t]

iexcl2

frac14E0

Zds

Pphs t

jr iexcl sj (10)


u1r t Diexcl2

frac14E1

Zds

phs t

jr iexcl sj (11)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





curren (12)



ph DA

6frac14h3

microz6

0

h6iexcl 1

para (13)


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

768 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

770 P Attard






is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

766 P Attard




t0


Zds

Pphs t 0

jr iexcl sj (8)



1

EtD

1

E1C

E1 iexcl E0




Pur t Diexcl1iquest

[ur t iexcl u1r t]

iexcl2

frac14E0

Zds

Pphs t

jr iexcl sj (10)


u1r t Diexcl2

frac14E1

Zds

phs t

jr iexcl sj (11)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





curren (12)



ph DA

6frac14h3

microz6

0

h6iexcl 1

para (13)


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

768 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

770 P Attard






is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





curren (12)



ph DA

6frac14h3

microz6

0

h6iexcl 1

para (13)


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

768 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

770 P Attard






is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

768 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

770 P Attard






is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

770 P Attard






is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

770 P Attard






is [74]

Put Df t Ph0t iexcl

poundut iexcl u1t

curren=iquest

1 C f t (15)

where f t acuteq




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

772 P Attard








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

774 P Attard











u D iexclp




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









middot DmiddotD

1 C middotD (20)







Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

776 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

778 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

780 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3









G[z]jX h0

ai

shyshyshyshyNa

D 0 (23)


F h0 D iexclsup3

GX h0

h0

acute

X

D iexclsup3

G[Nz]jX h0

h0

acute

NaX

D 1p

sup3V

h0

acute

Naiexcl deg

sup3Alv

h0

acute

Na (24)

Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

782 P Attard




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

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ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

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ded

by [

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e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

784 P Attard



Nh0 D 2p


rp0 C deg p

R2 iexcl r2 (25)







53 Spreading bubble


Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

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ded

by [

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e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








iexclGrcjX h0



Dow

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ded

by [

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e U

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rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

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ded

by [

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e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

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ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

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ded

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rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

786 P Attard





Dow

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ded

by [

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e U

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rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

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e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

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ded

by [

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e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

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ded

by [

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e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3




6 CONCLUSION




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

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ded

by [

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e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

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e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

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ded

by [

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e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

788 P Attard



Acknowledgements


REFERENCES




Dow

nloa

ded

by [

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e U

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rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

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ded

by [

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e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

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ded

by [

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e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

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ded

by [

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e U

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rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3








Dow

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ded

by [

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e U

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rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

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e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

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ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

790 P Attard






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

nloa

ded

by [

Yal

e U

nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3






Dow

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ded

by [

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nive

rsity

Lib

rary

] at

06

23 1

1 M

arch

201

3

Date post:	03-Dec-2016
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Friction, adhesion, and deformation: dynamic measurements with the atomic force microscope

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