Simple liquids confined to molecularly thin layers. I. Confinement-induced liquid-to-solid phase...

Simple liquids confined to molecularly thin layers. I. Confinement-inducedliquid-to-solid phase transitionsJacob Klein and Eugenia Kumacheva Citation: J. Chem. Phys. 108, 6996 (1998); doi: 10.1063/1.476114 View online: http://dx.doi.org/10.1063/1.476114 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v108/i16 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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JOURNAL OF CHEMICAL PHYSICS VOLUME 108, NUMBER 16 22 APRIL 1998

Simple liquids confined to molecularly thin layers. I. Confinement-inducedliquid-to-solid phase transitions

Jacob Kleina) and Eugenia Kumachevab)

Weizmann Institute of Science, Rehovot 76100, Israel

~Received 2 September 1997; accepted 14 January 1998!

A surface force balance with extremely high resolution in measuring shear forces has been used tostudy the properties of films of the simple organic solvents cyclohexane,octamethylcyclotetrasiloxane, and toluene, confined in a gap between smooth solid surfaces. Wewere able to probe in detail the transition between liquidlike and solidlike behavior of the films asthe gap thickness decreased. Our results reveal that in such confined layers the liquids are fluid downto a film thickness of few molecular layers~typically seven, depending on the particular liquidexamined!. On further decreasing the gap thickness by a single molecular layer, the films undergoan abrupt transition to become solidlike in the sense that they are able to sustain a finite shear stressfor macroscopic times. At the transition, the effective rigidity of the films, quantified in terms of aneffective creep viscosity, increases by at least seven orders of magnitude. This sharp transition isreversible and occurs as a function of the confinement alone: it does not require external appliedpressure. Following the transition the confined films behave under shear in a manner resemblingductile solids. ©1998 American Institute of Physics.@S0021-9606~98!51415-8#

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I. INTRODUCTION

In recent years the properties of simple liquids in tvicinity of solid surfaces, both for bulk liquids and particularly when they are confined to thin films, have been studextensively. This is partly because of obvious practical cnections~friction and lubrication,1 wetting properties,2 adhe-sion and wear3,4!, but at the same time because the behavof liquids in ultrathin films is not yet well understood. Aoutstanding issue, on which we shall focus in this papconcerns the manner in which liquid films change from befluid in macroscopically thick films, to behaving like solidwhen confined to films only a monolayer or twothickness.1,5–7 For bulk liquids in contact with a singlesmooth solid surface, layering of the molecules is inducdecaying over a few molecular diameters from tsurface.8–10 The question of the fluidity of liquids nearsurface has also been examined.8,9 Careful determinations othe plane of zero shear when liquids flow past a single ssurface indicate that bulk simple liquids near a surfacemain fluid right up to the solid–liquid interface.11–13Surfaceforce balances~SFB! have been used to probe both the dnamic and the structural properties of ultrathin films and sgest that, even when confined, liquids retain their bulk vcosity as long as the films are thicker than aboutmolecular diameters.11–14 At the other extreme, highly compressed films of simple liquids confined to 1–3 moleculayers between smooth solid surfaces have been showdisplay solidlike features, manifested by a finite yiestress5,6,15,16and characteristic stick-slip behavior,1 as well asa marked layering of the confined molecules.17,18

a!Author to whom correspondence should be addressed.b!Present address: Department of Chemistry, University of Toronto, T

onto, ON M5S 3H6, Canada.

6990021-9606/98/108(16)/6996/14/$15.00

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Few experimental studies have addressed the proceswhich a liquid changes to a solid as it becomes confinedincreasingly smaller gaps. Some studies based on the sof highly compressed layers between parallel plates haveported that these undergo a continuous increase in theircosity as the plate separation decreases below ananometers.19–21 This regime has been described as ‘‘intemediate’’ between a bulk liquid and a solid, and it has beremarked19,20 that the effective viscosityheff of some liquidsin this regime varies with the shear rateg asheff}g22/3. Incontrast, a different picture, where the liquid-to-solid trantion in simple liquids is a very sharp one and occursincreasing confinement at a well-defined film thickness, wno intermediate regime, has recently been reported.7

Computer simulations and theoretical investigatiohave shed much light on the molecular details underlyboth structural and dynamic behavior of liquids in the highconfined regime, as will be discussed in more detail laSeveral studies suggest that liquids composed of sphemolecules may undergo a sharp liquid-to-solid transitionconfinement to a few monolayers.22–27 In others, the varia-tion of liquid viscosity with shear rate in ultrathin confinefilms is explored, as is the question of stick-slip sliding btween the confining surfaces.25,26,28–30Very recently, the na-ture of the confinement-induced liquid-to-solid transitionself has been addressed directly both analytically31,32and viacomputer simulation studies.27,33

The present study extends our earlier brief reportconfinement-induced phase transitions in a simple liqu7

Using a surface force balance with uniquely high resolutin directly measuring shear forces across thin films, wevestigate a number of simple nonpolar liquids with quaspherical molecules, and examine comprehensively thenamical and mechanical behavior of the confined films. O

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6997J. Chem. Phys., Vol. 108, No. 16, 22 April 1998 J. Klein and E. Kumacheva

study reveals several qualitatively new features of thin-fiproperties. In this first paper~I! we focus on the issue of thtransition in the behavior of the confined films as the fithickness is reduced from macroscopic dimensions topoint where the liquid behaves in a solidlike fashion. In SII we describe in detail the surface force balance andoperation. The effect of increasing confinement on the shresponse of the liquids is described in Sec. III. In the fisection we consider this behavior in the light of theoretiand computer simulation studies, as well as of earlier invtigations on confined simple liquids. In the following pap~II ! we report on the properties of the solidlike confinfilms, that is, after they have undergone the liquid-to-sotransition.

II. EXPERIMENT

A. Apparatus and measurements

The use of mica as a model smooth substrate to meafriction was first reported by Bailey and Courtney-Prat34

who deposited boundary lubricants on the mica surfacSurface force balances~SFB!, where normal forces betweeatomically smooth, curved mica surfaces are directly msured as a function of their separation, have been utiliextensively since their first descriptions in the late 1960sTabor and co-workers,35 and especially following their extension by Israelachvili and Adams to the case of foracross liquid media.36 Force balances have also been usedinvestigate friction and shear forces across mica sheetstween which different materials~such as polymers, surfactants, or simple liquids! could be compressed.5,6,37–40In ourprevious investigations we examined the shear forcestween polymer-bearing mica surfaces~both adsorbed andend-tethered chains! immersed in a solvent.40–42 Due to theremarkable lubricating properties of polymer brushes in gosolvents, the shear forces during steady sliding, even usubstantial normal loads, can be several orders of magniweaker than the smallest forces measureable with eaforce balances. The apparatus used in the present studyoriginally designed to measure these very weak forces.

Figure 1 shows the surface force balance schematic~a brief description has appeared earlier40!. The normalforcesF(D) and lateral~or shear! forcesFs(D) between thetwo curved mica sheets a closest distanceD apart are mea-sured by monitoring the bending of two orthogonal setsleaf springs, vertical springsS1 ~spring constant K1

5300 N/m), and a horizontal springS2 ~spring constantK2

5150 N/m!. The bending ofS2 ~on which the lower micasurface is mounted! is determined, as in earlieversions,35,36,43 using multiple beam interferometry. Thisdone by monitoring the change in wavelength of fringesequal chromatic order~FECO! in response to applied motioin the z direction.

Application of shear motion and measurements ofshear force are carried out as follows. The top mica surfacmounted on a piezoelectric tube~PZT, diameter 12.7 mmlength 12.7 mm, and wall thickness ca. 1 mm, ceramic mterial 5-H, supplied by Morgan Matroc Ltd., UK, cat. n8531-5H! whose outside silvered surface is divided, by


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moval of four narrow conducting strips at 90° intervals, infour equal sectorsA–D. By applying equal and oppositvoltages~with respect to the inner conducting surfaceE ofthe PZT! to two of the opposing outside sectors~sayA andCwhich oppose each other alongx in Fig. 1!, one expandswhile the other contracts. This provides a lateral moti~in thex direction! to the top mica surface, as indicated in thright of Fig. 1. The range of lateral motion is some 15 000in either the1x or 2x directions, for a total lateral motionamplitude xm'30 000 Å. The resonance frequency of thPZT in its shear mode is over 100 kHz, but in the configration shown in Fig. 1 the maximum usable shear frequenm is limited by the mountingBO to nm'550 Hz. The fullrange of accessible mean shear velocitiesvs is thus from rest(vs50) to a maximal valuevs52xmnm'3.53107 Å/s ~asteplike voltage pulse can result in even higher instantaneshear velocities!. Any shear forces acting between the toand bottom mica surfaces, as the former are made to movthe x direction, are transmitted to the vertical springsS1 viathe rigid stainless steel mountBO on which the top surfaceis mounted~the total mass of the PZT and upper lens andboatBO on which they are mounted is ca. 20 g!.44

The sectored PZT is especially versatile in providiboth shear and normal motion of the upper mica surfapure normal motion~in the z direction! occurs when thepotentials of all four outer sectors are equal. A change inpotential then produces pure axial extension or contractthis is used particularly for the measurement of the profiof normal interactions between the surfaces. The PZT

FIG. 1. Schematic~approximately to scale! of the shear force balance usein the present experiments. The two mica sheets are mounted on cylindquartz lenses in a crossed-cylinder configuration~inset!. The top lens ismounted on a sectored piezoelectric tube~PZT, shown blown up on the righand illustrating the sideways motion induced when opposing sectors inx-direction experience equal and opposite potentials!. The PZT~see text forspecifications! is mounted via a rigid stainless steel boatBO onto two ver-tical copper-berrylium leaf springsS1 ~spring constantK15300 N/m! whichare rigidly mounted into the main body of the balance. Lateral forcestween the mica surfaces are transmitted to the springsS1 , whose bendingchanges the thicknessx of the air-gapG between the boatBO and a ca-pacitance probe CP. This is monitored by the change in capacitanceG~measured with a capacitor bridge, see text for details!. Normal forces aremeasured via the bending of the horizontal stainless steel leaf sprinS2

~spring constantK25150 N/m!, whose motion is a perfect up–down onwithout tilt ~the single cantilever representation ofS2 is for simplicity!. ThedistanceD between the mica surfaces is measured as usual via the fringequal chromatic order origination from the white light after projection onthe slit of the spectroscope.


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6998 J. Chem. Phys., Vol. 108, No. 16, 22 April 1998 J. Klein and E. Kumacheva

also provide simultaneous normal and shear motion, bydependent adjustment of the voltages applied to the innerouter sectors~E and A–D, respectively!, which enablesmeasurement of shear forces while very delicately varythe normal forces. In addition, it can provide motion of adesired form in thex-y plane~not only alongx! by suitablesynchronized inputs to the four sectors~for example, motionat 45° to x, at right angles tox, i.e. alongy, or circularmotion in thex-y plane!. This can be especially revealinwhere the shear forces are anisotropic with respect torelative shear direction, as in the case of confined liqcrystals.45 In the present experiments only motion alongxwas utilized.

The bending ofS1 in response to the shear forcesdetected by the changes in the capacitance of a small gaGbetween a fixed capacitance probeCP and the moving boaBO as shown. The capacitance bridge~Accumeasure 1000probe ASP-1-ILA, MTI Ltd., Albany, NY! can detectchangesdx in the gapG as low asdx562 to 3 Å, provid-ing an optimal sensitivityK1dx of better than60.1 mN inmeasuringFs(D). These values of the sensitivity and reslution in measuring the shear forces in monotonic slidmotion are comparable to those achievable in measuringnormal forcesF(D). They are about 100–1000 times bettthan values reported in earlier studies5,6,15 where frictionalforces were measured between mica surfaces sliding stepast each other across simple liquids. Such sensitivity issential for the measurement of the weak shear forcesported in the present study~and, particularly, for measuringthe ultraweak shear forces between sliding polymer brusreported earlier41!. The actual output of the capacitancbridge is in volts, linear in the dimensions of the capacitangap G; it is calibrated in the factory~1 mV525.0 Å! andchecked by us prior to each experiment~see below!. In anexperiment, following mounting of the surfaces, the capator gapG is adjusted to be parallel, and set, using a diffential micrometer~not shown in Fig. 1!, to the appropriatedimensions, within the thickness range 20610 mm.

Random vibrations due to ambient noise are minimizby a custom-built electronic vibration isolation unit speciatuned to eliminate the low-frequency tilting motion of thbuilding ~MOD 2A, JRS Ltd, Zurich!. All experiments werecarried out at 23.5 °C. The rooms in which experiments wperformed were thermally insulated and actively temperacontrolled to60.25 °C in order to minimize thermal drift.

A central requirement in measuring the shear forcethat the surfaces move parallel to each other, at any gseparationD, when lateral motion is applied to the top suface. This is because changes inD can lead to rapid changein the normal force, which in turn result in changesFs(D). In general it is not possible to ensure that mechanmounting of the PZT, boatBO, and springsS1 will result incompletely parallel motion of the surfaces when they arein contact with each other under strong compression. Torect for this, a small proportion of the voltage appliedeither sectorA or sectorC of the PZT~whenA andC are theactive sectors! is fed back to the inner sectorE. Since thevoltage on sectorE provides for pure axial extension or contraction ~z direction! of the PZT, independent of the she


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motion, the adjustable feedback enables elimination ofchanges in the surface separationD, even when the surfaceare not in contact, when the top surface is moved laterallypractice, it is possible to ensure thatD changes by no morethan ca. 10 Å as the top surface traverses the entire ranglateral motion of 30 000 Å. Over the more usual rangelateral motion in the experiments of order 103 Å, thechanges inD during the lateral motion can be adjusted toless than 1 Å. At the same time, the extent of tilt due tocantilever motion of the PZT is extremely small, corresponing to a change inD of less than 1 Å for a lateral motion of104 Å of the top surface.

In a shear experiment, in addition to measuring the dtanceD and the normal forceF(D) between the surfacestwo additional outputs are recorded simultaneously. Thare the voltage applied to the PZT, which gives a dirmeasure of the applied lateral motion, and the signal frthe capacitor bridge, which monitors the bending of the vtical springsS1 , and thereby provides a direct measure of tshear forceFs . These are recorded simultaneously eitheran XYt double pen chart recorder or via a storage osciscope. In the latter case signal averaging is possible~if de-sired! to enhance the signal-to-noise level. In Fig. 2 we illutrate typical outputs from the surface force balance.

In Fig. 2~a! the surfaces are in contact with no latermotion applied to the top surface. The level of this null snal, trace 2~a! ~from the capacitor bridge!, is a measure of thesensitivity and resolution to which forces may be measuras noted, the noise leveldx sets this resolution atK1dx'60.1mN. When the surfaces are far from contact the nolevel can be larger~examples are given later!. Trace 2~b!shows the voltage applied to the PZT when a back-and-foshear motionDx0 is applied to the top surface: here thsurfaces are in adhesive contact~in air!, and trace 2~c! showsthe bendingDx of the vertical springs in response to thshear force between them. Whenever this shear forceK1Dxis less than the frictional force between the surfaces, tmove together, as shown in trace 2~c!. That is, Dx5Dx0 .~This type of measurement is carried out in each experimto calibrate the shear motion.!

Finally, trace 2~d! shows the shear response when tshear forceK1Dx exceeds the frictional force. In this example, four monolayers of cyclohexane are confinedtween the surfaces: the surfaces adhere and move togethlong as the shear force is less than the static friction fobetween them. When the shear forceK1Dx exceeds the staticfriction they slide via a characteristic stick-slip motion. Thtype of motion occurs when the confined material, whichable to sustain a finite shear stress, breaks down at scritical value of the increasing stress, at the top of the spiindicated by C in Fig. 2~d!. The surfaces then slide or slirapidly relative to each other to relieve some of the stressin the vertical portions following the spike tips, but at somfinite shear stress, as shown by point D in Fig. 2~d!, theystick together again. The confined material is then once mcapable of sustaining a shear stress, and the cycle repitself. We note especially in Fig. 2~d! the very high repro-ducibility of the stick-slip pattern between consecutive baand-forth sliding cycles.


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B. Materials

Three different liquids were examined: cyclohexantoluene, and octamethylcyclotetrasiloxane~OMCTS!. Bothcyclohexane and OMCTS, especially the latter, due tononpolar nature and the large size of its quasispherical mecules~diameter ca. 9 Å!, have been used as model liquidsseveral earlier investigations.15–17,20,21Their structures, mo-lecular dimensions, and melting temperatures are givenTable I.

Analytical grade toluene~Fluka! was used as receivefrom freshly opened bottles. Cyclohexane~Fluka, analyticalgrade! and OMCTS~Fluka, purum grade, 99% pure! werestored for 2 days above 4A molecular sieves and thentilled over pure, dry, filtered nitrogen. In several of the eperiments the liquids were twice distilled over nitrogen. FOMCTS the middle fraction boiling at 175 °C~the literatureboiling temperature! was collected, and injected into the ba

FIG. 2. Traces corresponding to typical output of the shear experiment~a!Variation with time of the capacitor bridge signal~corresponding to thebending of the vertical leaf springsS1!, giving the equivalent shear forcbetween the upper and lower surfaces, when the two are in adhesive coin air. No lateral motion is applied to the top surface.~b! Variation with timeof the voltage applied to the sectored PZT, giving the back-and-forth latmotion of the upper surface.~c! Variation of the shear force between thsurfaces~derived from the changes in capacitor bridge signal! correspondingto the applied motion in~b! when the surfaces are in adhesive air contaThe frictional force exceeds the shear force at all points and the surfmove together without sliding.~d! Upper trace: applied back-and-forth motion of the upper surface. Lower trace: corresponding variation of the sforce between the surfaces, separated byn54 layers of cyclohexane undea normal loadF/R53.5 mN/m. When the shear force exceeds the frictioforce slip occurs between the surfaces~e.g., at pointC to point D! whichthen stick together again~at pointD! when the stress has partially relaxe


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of the force balance~Fig. 1! under dry filtered nitrogen. Theinside of the force balance was kept dry with P2O5. Wefound that for cyclohexane this distillation procedure did nresult in noticeable changes relative to using cyclohexdirectly from a freshly opened bottle, and in a number of texperiments this latter procedure was used. For the casOMCTS, however, we found that distillation and transferthe liquid under dry nitrogen to a balance in which a datmosphere was maintained influenced the value of structforces, and were essential for observing the behavior unconfinement described in this report.

All organic liquids used to clean the apparatus were alytical grade. Water used was purified~ELGA-Q water puri-fication system!. Sym~diphenyl carbazide! was used to gluethe mica sheets to the cylindrical glass lenses for the cyhexane and OMCTS experiments, and sucrose was usethe glue for the toluene experiments~both glues were BDHanalytical grade!.

Results shown are from several different experimenwith data often taken from a number of different contapoints between the mica surfaces in each experiment.

III. RESULTS

A. Normal force profiles

Prior to shear measurements the normal force profiF(D) were determined in each experiment. Figures 3 anshow typical normalized force-distance profiles for cychexane and for OMCTS. The profiles show characterisoscillations, with each maximum corresponding to a discrnumbern of monolayers, as marked. They are closely simito those reported earlier for these liquids~see captions toFigs. 3 and 4, and also Fig. 5!. The issue of sample purity iscrucial, as impurities can result in partial or total suppressof the structural oscillations~especially if the impurities aresurface active! and, as we see later, in strong modificationthe shear behavior of the confined liquids. In Fig. 5 our dfor the structural oscillations in OMCTS are compared wseveral independent earlier determinations17,46–48 of struc-tural forces in this liquid which had been purified and dtilled or double distilled in a similar manner to that in thpresent study. While there is appreciable scatter, for reaswhich have been discussed,48 nonetheless data from all sucstudies fall within the shaded bands indicated.49 In contrast,where the OMCTS is not distilled, the magnitudes of trepulsive structural forces are very much weaker~opencircles in Fig. 5!, presumably because of the presenceforeign ~non-OMCTS! molecules.50 Studies on OMCTSwhere the effect of added foreign molecules or of sligroughening of the confining surfaces has been explicinvestigated46,47,51also show a much reduced magnitudethe structural forces. Taken together, these observationsambiguously show that the effect of foreign molecules isdisrupt the ordering/layering of the molecules of the confinOMCTS, and that in order to eliminate such a disruption itimportant to distil carefully the as-received liquid as wellto dry it thoroughly. Determination of the oscillatory forcprofiles at the beginning of each experiment was thus a

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TABLE I. Structure and properties of liquids used.

Liquid Structure and formulaApproximatediameter~Å!

Meltingtemp.~°C!

Octamethylcyclo-tetrasiloxane~OMCTS!

(SiO!4~CH3)8 8.5–9 17

Cyclohexane C6H12 5.5 6

Toluene C6H5CH3 ~6! 295

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condition for its continuation, as absence or reduced matude of the oscillations generally indicated the presenceimpurities.52

Similar oscillations could be observed for toluenthough these could be observed to some four or five layonly, and were not accurately measured in this study. Oslations were monitored frequently during the course ofexperiment for control of its continuing integrity. At thhighest normal compressions applied in this study the liquwere squeezed down ton53 layers, though on occasionwhen sheared at high compressions, a further layer coul‘‘squeezed out’’, ton52. At the highest compressions wwere able to observe some flattening of the surfaces neacontact region~see below!.

B. Confinement-induced liquid-to-solid transitions

We discovered that on progressively bringing the sfaces closer together, we could induce, by increased conment alone, sharp transitions in the dynamic properties ofliquids in the gap. Figure 6 shows a qualitative manifestatof the sharpness of this transition~for the case of OMCTS!as the two mica surfaces approach. The traces shown redirectly the change in capacitance of the air gapG ~Fig. 1!,and thus monitor the variation with time of the relative laeral ~shear! motion Dx of the surfaces across the confinliquid. At large surface separationsD, and with no motionapplied to the top surface,Dx responds to external ambien

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ordFIG. 3. Force (F)-distance (D) profile between curved mica surfaces~plot-ted asF(D)/R whereR is the mean radius of curvature! in a crossed cyl-inder configuration in cyclohexane. Solid triangles: data from Ref. 17. Aearlier investigations of structural forces Refs. 17, 21, and 46–48, the vaof D are with respect to zero determined for dry air contact betweensurfaces.


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noise at the characteristic frequency~ca. 17 Hz! of the springS1 , as shown in trace 6~a! (D51160 Å). When the surfaceare brought closer together, these random vibrations pedown toD56262 Å, trace 6~b!; this corresponds to pointbon the normalized force profile, as shown on the right oftraces, atn57 molecular layers of OMCTS. Upon slighfurther compression, the surface separation decreasedcontinuously toD55462 Å ~point c on the normal forceprofile, corresponding ton56!. At this point, as seen in trac6~c!, the 17 Hz vibrations cease abruptly. This transitioncurs over time scales shorter than we were able to mea~, 0.5 s!. The change in confinement from seven to smolecular layers has all at once rendered the film soenough to resist the random lateral shearing motion induby the external noise. The transition is reversible, that is,increasing the separation toD.6 monolayers, random vibrations ~at ca. 17 Hz! are observed again. Moreover, th

FIG. 4. F(D)/R profiles in OMCTS. Solid triangles: data from Ref. 17

FIG. 5. Comparison of force-distance profiles in OMCTS from severaldependent studies~., Ref. 17;m, Ref. 46;j, Ref. 47;1, Ref. 48! with thepresent data~d!. The solid points are the maxima and minima of tF(D)/R variation for samples that have been distilled or double-distillThe shaded bands include scatter in all data. The open circles~s! are froma study~Ref. 21! where the OMCTS was used as received, under dry cditions but without distillation~no data for minima were reported in thistudy!.


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transition to the solid-like~shear-stress sustaining! phase isinduced by the additional confinement alone: little normpressure on the OMCTS is necessary~see caption to Fig. 6!.Once then57→n56 transition has taken place, the cofined OMCTS retains its solidlike characteristics even if tapplied small normal force@corresponding to pointc on theright-hand side~RHS! of Fig. 6# is reduced to zero.

The sharp transition observed for OMCTS at the cofinement changen57→n56 is characteristic also of theother liquids studied, though it could take place at slighdifferent confinements. For cyclohexane the abrupt chamanifested in Fig. 6 was generally observed asn57→n56. However, in one or two instances where ambient nowas especially low~in late night experiments! the transitionwas seen for a change in layer spacingn58→n57, and

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FIG. 6. Left-hand side: Direct recordings~from a recording oscilloscope! ofthe variation in time of the relative lateral displacementDx of the micasurfaces immersed in OMCTS~with no applied lateral motion! at separa-tions as follows: trace~a! D51160 Å; trace~b! D56262 Å; trace~c! D55462 Å. Traces~b! and ~c! correspond to pointsb and c on the forcedistance profile for this liquid reproduced~from Fig. 4! on the right-handside.

FIG. 7. Top trace: Lateral back-and-forth motion applied to the top msurface when sliding past the lower surface across OMCTS. Traces Aand C correspond to the surface separations~or n values! shown; the normalforces for traces B and C correspond to the pointsB andC shown on theright-hand side force profiles for OMCTS reproduced from Fig. 4.


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solidlike behavior in cyclohexane was manifested in thconditions already forn57. For toluene, the abrupt liquidto-solid change was observed at then55→n54 transition.We discuss the nature of these transitions in more delater.

A more quantitative manifestation of this liquid-to-soltransition was obtained by applying a controlled lateral mtion to the top mica surface via the piezoelectric tube ameasuring directly the resulting shear forceFs(D) betweenthe surfaces. This is shown for the case of OMCTS in Figas the top surface is moved laterally at a uniform velocfirst in one direction and then in the other, top zig-zag train Fig. 7, the shear force transmitted to the lower surfacethe confined OMCTS is recorded directly in the tracshown. At surface separations fromD51160 Å down toD56262 Å (n57) no shear response is detected withinnoise level of the signal, as seen from traces A (D51160 Å)and B (D562 Å) of Fig. 7@trace B is taken at pointB on theF(D)/R plot to the right of the trace#. When the surfaces armoved together toD55462 Å (n56), the form of theshear stress transmitted by the OMCTS changes sha@trace C of Fig. 7, taken at pointC on the adjacentF(D)/Rplot#. The response is now characteristic of a solid confinbetween two surfaces: an initial elastic regime, as the shstress rises to a yield point~regiona in curveC!, is followedby a ‘‘plastic’’ deformation of the OMCTS in the gap, as foa ductile solid under shear. The two confining surfaces tslide past each other at a uniform mean velocity~regionb incurve C!. This behavior is similar to that shown in Fig. 2trace D, though the data averaging in trace C of Fig. 7obscured any stick-slip behavior.

1. Effective viscosity prior to the liquid-to-solidtransition

The shear force data of Fig. 7, trace B show thatresponseFs(D) of OMCTS films only seven molecular layers thick is liquidlike and similar to that of much thickefilms, as in trace A, and that both are within the signal relution dFs . Qualitatively similar behavior was observed fconfined cyclohexane. We may use this observation to emate from our data an upper limit on the viscosity corsponding to this response of the liquids confined betweenmica surfaces. We focus on OMCTS, and consider thehavior in the liquidlike regime at film thickness corresponing to n57 monolayers just prior to the transition.

The configuration of the crossed cylindrical mica sufaces near the region of closest approach is equivalent toof a sphere, radiusR ~'1 cm! a distanceD from a plane.The forceFs(D) required to move a sphere in a liquid mdium of Newtonian viscosityh at constant velocityvs par-allel to a plane wall a distanceD away, forD!R, is givenby a series expansion whose leading term is53

Fs~D !5S 16p

5 D vshR ln~R/D !. ~1!

Thus forD,100 nm, the range of the present study, tforce on the moving sphere is dominated by its viscousteractions with the wall. Equation~1! applies for the casewhere the viscosityh is constant. For the case where t


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viscosity may change with the confinement we need a difent approach, since the extent of confinement~and thus theviscosity! changes across the gap due to the curvature ofsurfaces. We need to extract from the shear forceFs(D)between curved surfaces, a closest distanceD apart across aliquid, the effective viscosity of a film of the same liquiconfined between flat parallel plates a distanceD apart. Werequire, in effect, the equivalent for the shear forces of‘‘Derjaguin approximation’’ for normal forces. This is donin Appendix A where it is shown that the effective meaviscosity of the liquid confined to a layer of thicknessD isgiven by

heff'„]Fs~D !/]D…

S 16p

5 DRg

, ~2!

where g5(vs /D) is the shear rate at the point of closeapproach. In our experimentsFs(D) is within the noiselevel udFs(D)u'0.5mN throughout the range DD5(116– 6.2 nm), as seen from curvesA and B of Fig. 7.Approximating ]Fs(D)/]D by „dFs(D)/DD… in thisrange,54 we find that forD56.2 nm~i.e., seven monolayersof OMCTS! andvs51600 nm/s~Fig. 6!, this gives an upperestimate onheff of ca 3 P~poise!.

2. Contact area between surfaces following thetransition

The analysis above treated the material between thefaces prior to the liquid–solid transition as fluid, and tsolid confining surfaces themselves as remaining curvFollowing the transition the confined material is solidlike:can then sustain a shear stress, and a flattening of theconfining mica surfaces under normal load or due to attrtion may occur in the presence of the confined film~see Fig.13 in Appendix B!. We require the areaA of this flattenedregion between the surfaces as they are sheared at diffeloads and film thicknesses. In general, it is only at normloads Fn /R>4 mN/m that measurable flattening of thcurved mica surfaces~and thus a measurableA!, is pro-duced by distortion of the glue/mica layers. Most of trange of interest for the study of films of thickness corsponding ton53 – 7, for both cyclohexane and OMCTS~seeFigs. 3 and 4!, is at lower loads. In this caseA is evaluatedusing the Johnson–Kendall–Roberts~JKR! model55 for con-tact between curved, adhering surfaces of finite compressity. According to this, for a sphere in contact with a rigid fl~equivalent to the geometry of our crossed cylindrical sfaces!, a circular contact area of radiusa is formed at thecontact position when the two are compressed by a forceFn ,such that

a35R

K@Fn12Fp12~FnFp1Fp

2!1/2#. ~3!

Here R is the radius of the undeformed sphere@R5(R1R2)/(R11R2) for contact between crossed cylindeof radii R1 , R2#; K is related to the Youngs modulusE of thesphere, asK5(2/3)„E/(12n2)… wheren is the Poisson ratioof the material~the two contacting solids assumed to be


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the same material!; and Fp is the pull-off force required toseparate the two surfaces against the adhesion force betthem. The contact area is then given byA5pa2. In ourexperiments, the radiiR1 and R2 are measured in the standard way from the shape of the interference fringes nearpoint of closest approach of the~separated! surfaces. Theforce Fp corresponds directly to the depth of the attractwell in theF(D) profile that is associated with the numbermolecular monolayers in the film being studied. The modlus K in our experiments refers to the three-layer compoglass-glue-mica. It was estimated by measuring the acarea of contact between the surfaces under normal folarge enough to produce a measureable flattening of thefaces: a valueK5(160.3) 109 N/m2 was evaluated. Thisvalue is comparable with that determined for the glass-gmica combination employed in earlier experiments,56 wherea different glue~EPON 1004! was used in a similar configuration of the surfaces.

3. Relaxation under shear stress following the liquid-to-solid transition

Figure 6 and 7 demonstrate that the changes in mechcal properties of the confined phases following the liquisolid transition atn57→n56 are striking. A measure of thrigidity of the solidlike phases following the transition mabe obtained from their resistance to creep under stresprocess which may be quantified in terms of an effect‘‘creep viscosity’’ heff . While it is not in general appropriatto characterize solidlike materials by a viscosity, a measof such a viscosity is nonetheless instructive for highlightthe changes that take place at this transition. An estimatheff was obtained by monitoring the relaxation of the film

FIG. 8. ~A! The top trace shows the shear force between two mica surfacross an OMCTS film of thicknessn56 molecular layers, in response tthe motion applied to the upper surface given by the second trace. Pa– f refer to different stages in the shearing motion as noted in the text.normal force between the surfaces isF/R50.14 mN/m, corresponding tothe pointC on the OMCTS force-distance profile shown; the area of conA for this load for OMCTS forn56, evaluated from Eq.~3!, is A5(1.260.3)•10210 m2. ~B! As for ~A! but with a film ofn56 molecular layers ofcyclohexane. The normal force between the surfaces isF/R51.67 mN/m,corresponding to the pointC on the cyclohexane force-distance profishown; the area of contactA for this load for cyclohexane atn56, evalu-ated from Eq.~3!, is A5(1.960.3)•10210 m2.


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its solidlike phase following an applied shear stress, as illtrated in Fig. 8~A! and 8~B! for OMCTS and cyclohexanerespectively.

In both cases~OMCTS and cyclohexane! shown in Fig.8 the films are six monolayers thick, having just gothrough the liquid-to-solid transition. They are at points idicated byC on the respective normal force profiles showin the lower parts of Figs. 8~A! and 8~B! for the two mate-rials. The mica surfaces interact via a contact areaA ~dif-ferent for the two cases!, evaluated from the JKR model adescribed above. The configuration in each case@Figs. 8~A!and 8~B!# is then one of a Couette-like geometry wherefilm of areaA and thicknessD is subjected to a shear forcFs applied via a spring of constantK1 @see Fig. 13 in Ap-pendix B#. Initially, the film confined between the two sufaces is under zero shear force,Fs50 @regiona in Figs. 8~A!and 8~B!#; at pointsb a steady lateral motion at velocityvs

5(dx/dt) is applied to the top surface. At first the sheforce across the film rises~regionc! to a yield valueFs(y) .Subsequently, the surfaces slide past each other, via a sslip motion, with a mean velocityvs . In this regime@regiond in Figs. 8~A! and 8~B!# the shear force across the filmoscillates due to the stick-slip motion, but its mean varemains essentially constant atFs<Fs(y) . The applied mo-tion is then stopped@pointse#, and the relaxation of the sheaforce due to creep of the confined phase@regionsf in Fig. 8#is monitored by observing the change inFs . For both mate-rials, OMCTS, Fig. 8~A!, and cyclohexane, Fig. 8~B!, theextent of relaxation inFs over the times shown is very lowit is within the noise level of the signals in the two cases.Appendix B the relaxation of the shear forceFs due to creepof the confined film in this Couette geometry is considerfor a decayDFs (!Fs(y)) in Fs over a timeDt, the effectivecreep viscosity is given by

heff5Fs~y!K1DDt

DFsA. ~4!

Focusing again on OMCTS we have, for the trace of F8~A!, Fs(y)59.5mN andA5(1.260.3)310210 m2; over aperiod Dt510 s the decay is within the noise level,DFs

,dFs50.3mN. This gives from Eq.~4! a lower limit on theeffective creep viscosity,heff>6.107 P.

C. Variation of shear stress with shear velocity

The variation with shear rate of the shear stress in cfined films just subsequent to the liquid–solid transition, aunder essentially zero applied pressure, is of particular inest. This is because the variation of shear stress with slidvelocity is qualitatively very different in liquids and solidsThe shear response of such films as a function of the mrelative shear velocity~or, equivalently, shear rate! of thesurfaces is shown in Fig. 9. The data shown is for OMCfilms with n56 monolayers (D56362 Å). Figures 9~a!–9~c!, top traces, show the back-and-forth applied motionthe top mica surface at a number of different velocitiesvs ,together with the corresponding shear forces, lower tracesvs

could be conveniently varied by changing the rate at whthe potential was applied to the PZT. We observe at o

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that althoughvs varies over a factor of 40, there is littlchange in the shear response between the traces. Wethat for all three cases shown the applied normal loadextremely small~, 1 mN!, so that the solidlike behavioobserved is induced effectively by confinement alone. Tclear conclusion from these traces~which are typical of oth-ers! is that the shear stress is independent~or only veryweakly dependent! on the shear velocity. This is true for botcyclohexane and OMCTS~the variation ofFs with vs fortoluene was not investigated! just following the liquid-to-solid transition. This behavior is characteristic of a solsince for a liquid one expects the shear stress to be protional to the shear rate. In the following paper~II !, we de-scribe how thinner films well in the solidlike regime (n55 – 3), respond to shear at much higher pressures anda wider range ofvs . We find, in agreement with Fig. 9, thathe mean stress to slide the surfaces past each other remlargely independent of the sliding velocity for all these laythicknesses, particularly at the highest shear rates used.

IV. DISCUSSION

Two particular features of the surface force balance uin the present study, true parallel motion~to 61 Å over arange of lateral motion of order 1000 Å! and extreme sensitivity in measuring shear forces directly, allow us to probedetail the dynamic properties of thin films between solid sfaces. In particular, these features enable the determinaof shear forces between the surfaces across confined siliquids even in the absence of any applied compressive~nor-mal! forces. This permits us to examine in detail the chanin properties of the progressively confined layers fromliquid to the solidlike states. This was not possible in tearlier pioneering shear studies of simple liquids confinbetween curved surfaces.5,6,15 ~We recall that the present ap

FIG. 9. The top trace in each of~a!–~c! shows the back-and-forth motionapplied to the upper mica surface sliding relative to the lower mica suracross a film ofn56 monolayers of OMCTS (D55263 Å); the lowertrace in each of~a!–~c! shows the corresponding shear forces between thvs are the applied velocities. The scale bar in the force applies to all tcases. The normal loads are~a! F/R50.0160.1 mN/m; ~b! F/R520.0660.1 mN/m; ~c! F/R520.0960.1 mN/m.


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paratus is capable of a sensitivity and resolution in meaing shear forces some three orders of magnitude higherin the earlier studies.! In these, large compressions had toapplied to flatten the surfaces at the region of contact, sothe overall shear forces became large enough to measThese large compressions squeezed the liquids being prto films some three or fewer monolayers thick, well withthe solidlike regime. In our study we were able to investigin detail much thicker films, and so to access the confiment-induced liquid-to-solid transition.

Our main new finding is the sharp transition on increaing confinement between films of thickness correspondingn5(nc11) monolayers, where the confined material wliquidlike, and films just one monolayer thinner,n5nc ,where the confined films were able to sustain a shear stover macroscopic times. This ability to sustain a shear stis a fundamental signature of a solid: a liquid, by definitiocannot sustain such a stress. The critical thicknessnc mea-sured in our experiments varied somewhat between theferent liquids:nc56 for OMCTS,nc56 or 7 for cyclohex-ane, andnc54 for toluene. A convenient way of quantifyinthis transition is via the change inheff across the transition~Secs. III B 1 and III B 2!. This is shown in Fig. 10 for OM-CTS, where an increase of at least 761 orders of magnitudein the effective viscosity is observed as the confinement gfrom n5nc11 ~57! layers of the liquid, atD56262 Å, ton5nc ~56! layers. The limits are set by the signal-to-noiratio of our measurements, as described in Sec. III. Simeffects were observed also for cyclohexane and~qualita-tively! for toluene, at the respectivenc values. Within therange of our parameters, there is clearly no evidence wever of an ‘‘intermediate’’ regime of progressively increaing viscosity on increasing confinement for these liquidas reported earlier for a number of liquids,19–21 includingOMCTS. The origins of this discrepancy in the caseOMCTS may be related to different procedures in samprocessing.57

The abruptness and magnitude of the change atconfinement-induced transition seen in Fig. 10 are remin

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FIG. 10. Variation with film thicknessD of the effective mean viscosityheff

of confined OMCTS films~on a double-logarithmic plot! determined asdescribed in text. The broken lines atD,54 Å and atD.62 Å are lowerand upper limits onheff , respectively. The bulk viscosity of OMCTS a23 °C is indicated ashbulk . The reversibility indicated atD562↔54 Årefers to the transition in the dynamical properties of the confined OMC


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cent of a first-order transition, and suggest that at the tration the entire film locks into a solid structure. We note athat this transition is brought about by increasing confiment alone, with little applied external pressure~see, e.g.,data of Fig. 9!. Finally, we remark that this effect is seen fliquids ~cyclohexane, OMCTS, and toluene! whose molecu-lar size~Table I! differs appreciably from each other as weas from the crystal periodicity of the confining mica surfacthis indicates that epitaxiality with the surface lattice is nnecessary for inducing the transition in such simple liquiIt would be extremely interesting, and challenging, to be ato probe the in-plane and normal-to-plane structural featuof the confined solid on both sides of the transition. Hoever, the geometry of the confinement and the extremsmall volume of interest within the gap render this well byond the present capabilities of x-ray techniques.58

Insight into the origin of this transition may be obtainefrom the following considerations. Both experiments acomputer simulations8–10,59suggest that simple bulk liquidin contact with a single planar surface undergo some laing, extending to a few molecular diameters away fromsurface. This layering is due to geometric packing, andsimilar to, but more marked and longer ranged than the dsity oscillations about a single liquid molecule revealedits radial distribution function~it may be viewed as corresponding to the radial distribution function of the liquid moecules about a ‘‘molecule’’ of radiusR, whereR→`!. De-spite this layering, as noted in the Introduction, experimehave revealed11–13 that unconfined liquids@or liquids con-fined to large gaps (n@10)# retain their fluidity, with vis-cosities comparable to bulk viscosity, right up to the liquconfining-solid interface~to within a single molecular layer!.Layering, in other words, does not necessarily result inlidity. This at once implies that we are not dealing with seeral immobilized molecular layers on each mica surfawhen they are far apart, that merely shift the effective solliquid interface away from the surface. Rather, the pictthat emerges is that, as the surfaces approach each otherlarge separations, the confined liquids retain their bulk fluity across the entire gap, until at a critical spacing the enfilm undergoes a liquid-to-solid transition.

This transition appears to be due to a synergistic effproduced by the confinement, and reminiscent, we suggof a Lindemann-like criterion:60 At gap spacings greater thathe critical valuenc , the layered molecules are less orderand have a freedom of displacement greater than someessary fraction of their mean spacing. At gapsnc or thinner,the confinement suppresses this freedom to move to bethe necessary fraction, and the entire confined liq‘‘freezes.’’ Alternatively we may think of the layering aboueach isolated surface as leading to a slightly higher mdensity in the near-surface region. When the surfacesproach, these near-surface regions overlap each other.results in an overall mean density within the intersurfacethat leads to the Lindemann crossover. Thermodynamicathis effect may be viewed in terms of the associated entrochanges. Forn.nc the entropy associated with local liquidlike translational motion of the confined molecules resultsa greater free-energy reduction than the enthalpic gain re


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ing from fixing them in a solidlike lattice. Forn<nc , thesmaller space available to the confined molecules revethis, and a solidlike behavior results. This picture is illutrated schematically in Fig. 11. This effect should dependa sensitive fashion on the excess temperature of the meaments above the bulk melting temperature of the liquid,well as on the detailed surface–liquid interactions.

The above discussion is clearly rather qualitative. Thhave been several detailed theoretical and computer simtion studies of the effects, both on structural and mechanidynamic properties, induced by confinement on liquof both spherical and linear molecules.22,23,25–29,33,59,61–63

These have shed much light on the processes that mataking place as liquids are confined to progressively thinfilms, and as the films are sheared. The details of the modo not exactly tally with the conditions of our experimen~for example, the liquid molecules are often assumed toidentical to the surface molecules, and are taken as pehard spheres!, precluding detailed comparisons, but thehave been several very suggestive findings. Here we fobriefly on those studies that provide clues to the issue ofliquid-to-solid transition under increasing confinement.

As noted, the geometric constraints of smooth walls leto layering of spherical molecules,8,9 even in the vicinity of asingle unconfined surface, though this is not necessarilyrelated with freezing.62 In-plane ordering22 ~in addition to thelayering! and correlated freezing23 were observed by Cushman, Schoen, Diestler and Rhykerd in simulations of ragas fluids confined between two solid surfaces. In relatheoretical work24 these authors have also shown that, unsufficient confinement, liquids can form epitaxially growinsolids with a thermodynamically stable configuration. A covenient measure of in-plane order is the Debye–Waller ftor ~DWF!, defined in terms of the two-dimensional structufactor within a layer, whose value is unity in a perfect crysat zero temperature, and drops to 0.6 at the melting poin64

Molecular dynamics simulations by Robbins, Thompsoand Grest26,65,66 indicate that for confined spherical moecules the DWF rises sharply on increasing confinemfrom a value below 0.6~i.e., liquidlike! to one above it~sol-idlike! when the confined liquid is compressed to belowgiven thickness. In contrast, for chainlike~nonspherical!molecules the DWF remained lower than 0.6~indicating amore liquidlike in-plane order! at all accessible compressionin their study.

Another useful measure of solidity of confined liquids

FIG. 11. Illustrating schematically the transition between a layered but flfilm of thicknessnc11 molecular layers, and one of thicknessnc with alayered but frozen structure.


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the self-diffusion coefficientDs within the layers, which isrelated to the effective viscosity. In simulations10,26,33,65thishas been shown to drop abruptly when films~of sphericalmolecules! are confined below a given film thickness~corre-lated to the point where the DWF exceeds 0.6!, again indi-cating a rather sharp solidification process.

Some of the most suggestive indications concernconfinement-induced liquid-to-solid transition in thin filmhave come from very recent simulation work by Landmaand co-workers,27,33 who use a grand canonical ensembmolecular dynamics method to simulate confined liquiTheir results show clearly that molecules that are ablepack easily~unbranched linear alkanes and also sphermolecules resembling OMCTS! undergo a rather suddechange in their properties~a very sharp drop in the diffusioncoefficient, for example!, from liquidlike to solidlike behav-ior, at film thicknesses corresponding to around five tolayers. Moreover, at this point the confined films in thstudies become capable of sustaining a shear stress. Tobservations suggest an abrupt liquid-to-solid transitionthese confinements, very similar to the experimental obvations of the present study.7 This contrasts with the simulated behavior of branched or irregularly shaped molecu~such as squalane! which exhibit a monotonic and continuous ~liquidlike! decrease in the number of confined moecules as the film thickness decreases.27,33 The models ofthese studies are necessarily cruder than reality evenliquids such as OMCTS and cyclohexane which have quspherical molecules. Their results, however, are fully content with our observation of an abrupt liquid–solid transitiat a given film thickness for these simple liquids.

Recently the issue of confinement induced freezingbeen examined explicitly using different analytical aproaches. Tkachenko and Rabin31 have presented a modebased on a Lindemann criterion to show that at a given laspacing, a confined liquid would solidify abruptly eveabove its bulk melting temperature. Weinstein and Safra32

have used a density functional approach, with the additioconstraint of the two confining surfaces, to determine a phdiagram for the confined liquid, and again a confinemeinduced freezing is predicted to occur abruptly at tempetures where the bulk material would be liquid. Both of theapproaches have the advantage that in principle they candict the effect of temperature on the necessary confinemat which the freezing occurs, and for both models the freing is predicted to occur as a result of confinement alone:in the absence of external pressure. This is in line withobservations of the present study.

We make a final brief remark concerning the variationshear stress with shear rate indicated in Fig. 9. This shhow the shear stress between sliding mica surfaces acron56 film of OMCTS ~film thicknessD55362 Å!, that is,just after the liquid–solid transition, varies with sliding vlocity. The velocitiesvs vary by a factor of 40, corresponding to shear ratesg5(vs /D) in the range 8 – 300 s21, whilethe applied normal pressuresP are close to zero~see cap-tion!. Within the scatter, the shear stressSc is essentiallyidentical for all three shear velocities~the small differencesare well within the uncertainty in shear stress expected fr


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the uncertainty in the normal load!. This independence of theshear stress from the shear rate67 ~i.e., Sc}g0! is clearlycharacteristic of shearing of a ductile solid,68 rather than of aliquid or a liquidlike film. We return to this point in thefollowing paper~II ! when we consider a much broader ranof data on liquids confined to even thinner films (n53 – 6).

V. SUMMARY

We have described a surface force balance withtremely high sensitivity and resolution in measuring shforces across thin films. Using this we have extended eainvestigations on the properties of confined simple liquidsthicker films and lower applied pressures. Here we havecus on the process by which simple liquids become sounder increasing confinement.

We discovered that the transition between liquidlike bhavior and a solidlike phase of the liquids under progressconfinement takes place abruptly at a well-defined film thiness~equivalent tonc monolayers of the liquid, wherenc

varied with the liquid but was around six molecular layer!.For thinner films (n<nc) the films behaved in a solidlikefashion, in the sense of requiring a critical stress in ordeshear them. We found no evidence of an intermediate regof increasingly viscous behavior for the three simple liqustudied. Our results are reminiscent of a Lindemann-lmechanism for freezing of a liquid, save that the changesinduced by confinement, rather than by externally applpressure or by changes in temperature. In the followpaper69 ~II ! we describe in detail the mechanical, frictionaand dynamic behavior of such confined films once they hsolidified.

ACKNOWLEDGMENTS

We thank R. Ball, H. Christensen, S. Granick, G. GreJ. Israelachvili, M. Robbins, U. Landmann, Y. Rabin, S. Sfran, and A. Weinstein for useful comments and correspdence, and U. Landman for providing us with a copy of R33 prior to publication. This work was supported by therael Science Foundation, the Ministry of Science and A~Israel-Tashtit programme! and the U.S.-Israel BinationaScience Foundation.

APPENDIX A: A ‘‘DERJAGUIN APPROXIMATION’’FOR SHEAR FORCES BETWEEN CURVED SURFACES

The Derjaguin approximation70 for normal forces be-tween two curved surfaces a closest distanceD apart relatesthe normal forceF(D) between them to the interaction energy per unit area between flat parallel plates a distancDapart obeying the same force-distance law, as

F~D !/2pR5E~D ! ~R@D !, ~A1!

where R is the mean radius of curvature of the surfacWhen two curved surfaces a closest distanceD apart aremoving with velocityvs parallel to each other across a mdium with some effective viscosityheff ~which may be afunction of D!, a shear forceFs(D,vs) will result. In thisAppendix we derive71 an approximate expression for th


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equivalent shear stressS(D,vs) between flat plates slidingparallel to each other at velocityvs a distanceD apart acrossa medium obeying the same viscosity versus thickness~i.e., the same variation ofheff with D!. This is particularlyuseful since theoretical models of shear forces often treatparallel plates rather than curved surfaces. To be explicit,treat the case of a sphere, radiusR, sliding parallel to a planea closest distanceD away ~in the lubrication limitR@D!.

It is instructive to derive first, via a crude model, thshear force between a sphere moving at velocityvs in a fluidof uniform ~Newtonian! viscosityh parallel to a plane a distanceD away, as illustrated in Fig. 12. The main simplifiction of this model is the assumption of laminar flow, i.e.,backflow effects. We consider the forces on an annulusradiusr as shown. From the geometry,r 252R(h2D), i.e.,

h~r !5D1~r 2/2R! . ~A2!

The shear rate atr is „vs /h(r )…, so that the local Newtonianshear stresss(r )5h„vs /h(r )…. Integrating stresses over thannuli for the total shear force,

Fs5E0

R

2prs~r !dr

5E0

R

2prhvs

D1~r 2/2R!dr

52phvsR log„11~R/2D !…

'2phvsR log~R/D ! ~A3!

since in the geometry of our experimentsR/D5105. Now adetailed solution of the force acting on a sphere sliding clto a plane has been carried out by Goldmanet al.53 Theseauthors show, using the Navier–Stoke equations andtaking the correct flow field into account, that the forcegiven by a series expansion whose dominant term~for R@D! is

Fs~D !5~16p/5!vshR ln~R/D !. ~A4!

Comparison of~A3! and ~A4! shows that the value ofFs

derived in a calculation which ignores the backflow effeon the flow field has precisely the right form~in particularthe correct logarithmic dependence!, and differs in magni-tude from a detailed calculation only by some 35%. Tobservation will be useful in what follows.

To extractS(D,vs) in the more general case where teffective viscosity in the gap may vary withD, we proceedas follows. From the geometry of Fig. 12, we haveh5D1(r 2/R). Thusdh5(r /R)dr, and the areadA of an annulusmay be written asdA(h)52prdr 52pRdh. The overall

FIG. 12. Illustrating the geometry of a sphere sliding at velocityvs a closestdistanceD from a plane as discussed in Appendix A.


w

ate

of

e

us

s

s

force between the surfaces is then as before~ignoring anybackflow effects! the sum of the forces over all annuli, giveby

Fs~D,vs!5ED

`

S~h,vs!dA~h!5ED

`

2pRS~h,vs!dh,

~A5!

where we have put the upper integration limit as` sinceR@D.

From ~A5! we have

S~D,vs!521

2pR

]Fs~D,vs!

]D. ~A6!

This is the relation we have been seeking, giving the shstressS(D,vs) between flat parallel plates in terms of thshear forceFs(D,vs) between curved surfaces. Finally wrecall that neglect of backflow effects leads to some erwhich for the simple Newtonian case was equivalent toplacing the prefactor 2p by ~16p/5!, a change only in thenumerical prefactor by some 35%, as discussed followEq. ~A4! above @compare Eqs.~A3! and ~A4!#. If, by ex-trapolation, we make the rough assumption that the inclusof backflow has the same effect also in the case whereviscosity is confinement dependent, we may make the scorrection to Eq.~A6!, to give finally

S~D,vs!'21

~16p/5!R

]Fs~D,vs!

]D. ~A7!

Replacing the shear stress in~A7! by the Newtonian formS(D,vs)5heffg, whereg5(vs /D), leads at once to Eq.~2!in the text.

We note that care should be taken when extractS(D,vs) from shear measurements between curved surfaIn particular, S(D,vs) is obtained from data onFs(D,vs)taken over a range ofD at fixed shear velocityvs .

APPENDIX B: STRESS RELAXATION IN A COUETTECONFIGURATION

The geometry of the contacting surfaces when theone is pulled by a spring~corresponding to the shear sprinS1 in Fig. 1, of constantK1! is shown in Fig. 13. The area ocontact isA, enclosing a film of thicknessD with an effec-tive viscosityheff , in steady motion the spring is stretchedlengthx0 , and the steady-state velocity isvs . During steadymotion the tensionK1x0 in the spring is balanced by thshear forceFs(y) across the film:

FIG. 13. Illustrating the stress relaxation in a couette geometry whenupper surface is under tension due to springS1 and is in contact over a filmof areaA and thicknessD with the lower surface, as for the data of Fig.and as discussed in Appendix B.


cenc

e

hy

n

J.

tur

ture

-ly.

vilipli-ns.

ord-t re-olid

ro-aceatinghich

n onperi-

se toisown inith

to

rec-

er.

ualtrast

that

illa-

er

innort

the


K1x05Fs~y!5A.~shear stress!5Aheffvs /D. ~B1!

At time t50 the motion of the spring tip is stopped@corre-sponding to the pointe in Fig. 8~A!, for example#. At sub-sequent timet the top surface will have moved by a distanx, relaxing some of the force in the spring. The new balaof forces is

K1~x02x!5Fs~ t !5~Aheff /D !~dx/dt!, ~B2!

which is readily solved@subject to the condition~B1! at t50# to give

Fs~ t !5~Aheffvs /D !e2t/t5Fs~y!e2t/t, ~B3!

wheret5(Aheff /K1D). The extent of relaxation after a timDt is DFs5Fs(y)2Fs(t), i.e.,

DFs5Fs~y!~12e2Dt/t!5~Dt/t!Fs~y! ~B4!

for DFs!Fs(y) . Rearranging~B4! and substituting fortgives Eq.~4! in the text.

1D. Tabor,Friction ~Doubleday, New York, 1973!.2J. S. Rowlinson and B. Widom,Molecular Theory of Capillarity~Claren-don Press, Oxford, 1982!.

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~1989!.14J. Klein, Y. Kamiyama, H. Yoshizawa, J. N. Israelachvili, G. Fredrickso

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~1987!.23C. L. Rhykerd, M. Schoen, D. J. Diestler, and J. H. Cushman, Na

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~1996!.31A. Tkatchenko and Y. Rabin, Solid State Commun.103, 361 ~1997!;

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~London! 370, 634 ~1994!.42J. Klein, Annu. Rev. Mater. Sci.26, 581 ~1996!.43J. Klein, Nature~London! 288, 248 ~1980!.44While shear motion can be applied to frequencies up tonm , shear forces

can only be measured at frequenciesv,vc'17 Hz, the resonance frequency of the springsK1 when carrying the top lens mounting assemb

45J. Janik, R. Tadmor, and J. Klein, Langmuir13, 4466~1997!.46H. K. Christenson, Chem. Phys. Lett.118, 455 ~1985!.47H. K. Christenson and C. E. Blom, J. Chem. Phys.86, 419 ~1987!.48T. K. Vanderlick, L. E. Scriven, and H. T. Davis, Colloids Surf.52, 9

~1991!.49The data exclude the early pioneering report by Horn and Israelach

~Ref. 18!, where the presence of trace amounts of water was later imcated~Ref. 17! as being responsible for the weakness of the oscillatio

50Such impurities can come from the reactants diethoxydimethylsilane~more probably! hexamethyldisiloxane. An additional possible side prouct is octamethyltrisiloxane. These have a linear structure, and if nomoved they may suppress layering and mask the sharp liquid-to-stransitions.

51H. K. Christenson, J. Phys. Chem.90, 4 ~1986!.52Indeed, in a number of the early experiments where our distillation p

cedure was not adequate, with OMCTS in particular, oscillating surfforces were absent or much reduced. Similar suppression of the oscillforces was observed also after several days of an experiment, by wtime impurities, i.e., foreign molecules, had crept in@similar effects wereseen by A. Berman and J. N. Israelachvili~private communication!#. Inthese cases we also did not observe the sharp liquid-to-solid transitioincreasing confinement described in this paper. As noted, such exments were aborted.

53A. J. Goldman, R. G. Cox, and H. Brenner, Chem. Eng. Sci.22, 637~1967!. We may use Eq.~2! for data as in Fig. 7 down ton57 monolay-ers. This is because the normal forces between the surfaces are clozero ~point B in Fig. 7!, while the adhesion due to oscillating forcessuppressed as long as the surfaces are sliding past each other, as shpaper II ~following!. Thus the sliding surfaces retain their curvature wessentially no distortion@both Fp and Fn in Eq. ~3! are zero or close tozero# and Eq.~2! remains valid.

54This is an approximation, since the change in the shear forcedFs maytake place over a smaller range ofDD. The effect of this approximationmay be to increase the upper limit ofheff just prior to the transition by asmuch as a factor of 10. We note also that the value we have taken fordFs

is itself also an upper limit and is probably much smaller: this actsreduce the upper limit ofheff . Overall, therefore, the upper limit ofheff

evaluated may have an uncertainty of about a factor of 10 in either dition.

55K. L. Johnson, K. Kendall, and A. D. Roberts, Proc. R. Soc. London, SA 324, 301 ~1971!.

56R. G. Horn, J. N. Israelachvili, and F. Pribac, J. Colloid Interface Sci.115,480 ~1987!.

57In the study of OMCTS published in Ref. 21, which indicated a gradincrease in the fluid viscosity on progressive confinement, and, in conto several other studies~Refs. 7, 15, 16, and 69!, an absence of anyzero-shear-rate yield stress upon shear, the liquid used was similar toused in the present investigation~containing up to 1% impurities!, savethat it was not distilled prior to its introduction to the cell~Ref. 72!. Thiscontrasts with the procedure described in Sec. II B. The structural osctions reported for OMCTS in that study~Ref. 21! were also much reducedin magnitude~see open circles in Fig. 5!, strongly suggesting the presencof foreign molecules~Ref. 50!. These may have interfered with moleculaordering~such as layering! in the confined OMCTS films, as noted alsoSec. III A ~Ref. 52!, thereby preventing the sharp liquid–solid transitiofrom occuring in the sample studied in Ref. 21. This interesting rep~Ref. 21! is thus of particular significance in that it may suggest that


n

E. s

oled.

d.sug-


presence of trace amounts of foreign molecules has large effects oproperties of thin OMCTS films.

58S. H. J. Idziak, C. R. Safinya, R. S. Hill, K. E. Kraiser, M. Ruths, H.Warriner, S. Steinberg, K. S. Liang, and J. N. Israelachvili, Science264,1915 ~1994!.

59See review by N. Dan, Curr. Opin. Colloid Interface Sci.1, 48 ~1996!, andreferences therein.

60D. Tabor,Gases, Liquids and Solids, and Other States of Matter~Cam-bridge University Press, Cambridge, 1991!.

61Nanotribology, edited by J. F. Belak, in Mater. Res. Soc. Bull.18, 15~1993!.

62I. Bitsanis, J. Magda, M. Tirrell, and H. T. Davis, J. Chem. Phys.87, 1733~1987!.

63K. Binder, J. Phys. I6, 1271~1996!.64M. J. Stevens and M. O. Robbins, J. Chem. Phys.98, 2319~1993!.65M. O. Robbins, P. A. Thompson, and G. S. Grest, Mater. Res. Bull.18, 45

~1993!.66P. A. Thompson, M. O. Robbins, and G. S. Grest, Isr. J. Chem.35, 93

~1995!.67If we attempt to extract an ‘‘effective viscosity’’heff for the sheared films


thefrom data as in Fig. 9, using the relationSc5gheff ~applicable for New-tonian fluids!, then our observation thatSc}g0 tells us at once thatheff

}g21. This contrasts with the relationheff}g0 expected at these shear ratefor simple liquids. It differs also from theheff}g22/3 relation suggestedearlier ~Ref. 20! for some liquids, including OMCTS (n53) at compa-rable shear rates~see Ref. 57 for possible origins of this difference!.

68R. W. K. Honeycombe,The Plastic Deformation of Metals~Arnold, Lon-don, 1984!.

69E. Kumacheva and J. Klein, J. Chem. Phys.108, 7010~1998!, followingpaper.

70B. V. Derjaguin, Kolloid Zeits.69, 155 ~1934!.71We are indebted to Dr. Robin Ball for suggesting this approach.72S. Granick~private communication! has informed us that similar results t

Ref. 21 were obtained also when their OMCTS samples were distilHowever, in contrast to several other studies~Refs. 7, 15, 16, and 69!, azero-shear-rate yield stress~indicating solidlike behavior! was still notobserved when thin films (n<3) of their OMCTS samples were sheareWe do not have an explanation for this discrepancy, other than ourgestion above~Ref. 57!.


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Simple liquids confined to molecularly thin layers. I. Confinement-induced liquid-to-solid phase...

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