Adsorption trajectories and free-energy separatrices for ......Adsorption trajectories and...

Adsorption trajectories and free-energy separatrices for colloidal particlesin contact with a liquid-liquid interface

Joost de Graaf,1,a� Marjolein Dijkstra,1 and René van Roij2,b�1Soft Condensed Matter, Debye Institute for Nanomaterials Science, Utrecht University,Princetonplein 5, 3584 CC Utrecht, The Netherlands2Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands

�Received 18 February 2010; accepted 23 March 2010; published online 26 April 2010�

We apply the recently developed triangular tessellation technique as presented by J. de Graaf et al.�Phys. Rev. E 80, 051405 �2009�� to calculate the free energy associated with the adsorption ofanisotropic colloidal particles at a flat interface. From the free-energy landscape, we analyze theadsorption process, using a simplified version of Langevin dynamics. The present result is a firststep towards understanding the time-dependent behavior of colloids near interfaces. This studyshows a wide range of adsorption trajectories, where the emphasis lies on a strong dependence ofthe dynamics on the orientation of the colloid at initial contact with the interface. We believe thatthe observed orientational dependence in our simple model can be recovered in suitableexperimental systems. © 2010 American Institute of Physics. �doi:10.1063/1.3389481�

I. INTRODUCTION

The adsorption of colloidal �nano�particles at liquid-liquid interfaces is not only of scientific interest, but alsorelevant for industry. The formation of two-dimensional �2D�structures,1,2 which may be utilized in photonic bandgap ma-terials, and the stabilization of Pickering emulsions3 are twoexamples of possible applications. There is also an impetusto theoretically describe the colloidal adsorption processmore accurately, in order to gain a deeper insight into themechanisms at work in experimental systems. Especially 2Dfluid phenomena and phase transitions in 2D fluids of aniso-tropic particles4,5 are of interest. The tunability and variety ofcolloidal particles currently available, coupled with our stilllimited knowledge on colloidal adsorption, leave the study oftheir interfacial phenomena an open field.

To better understand the complex systems4–9 which arisewhen colloidal particles are brought in contact with an inter-face, we developed a method to determine the free energyassociated with the adsorption of a single particle.10 Thisfree-energy calculation is based on similar surface tensionarguments as proposed by Pieranski,1 in his ground breakingstudy of colloidal adsorption phenomena. The model in ourinvestigations encompasses surface and line tension, but dis-regards interfacial deformation and electrostatic effects. Ourcontribution to this theoretical description is the triangulartessellation scheme,10 which allows us to efficiently deter-mine the adsorption free energy of an anisotropic colloid. Inthis paper we apply our technique to perform an initial in-vestigation into the dynamics that occur when a colloid at-taches at the interface and relaxes to its preferred configura-tion.

The dynamics we study in our model is a simplifiedversion of Langevin dynamics.11,12 From the adsorption free

energy, which acts as a potential energy on the colloid, avector field of the adsorption force is determined. This vectorfield is then studied using these dynamics to derive adsorp-tion trajectories. For convenience, the anisotropy of the par-ticle is only considered in the free energy and not in thefriction tensor, which for interfacial systems is also positiondependent. Our choice allows us to examine general trendsand gross features and to showcase our numerical method,while at the same time give results which resemble those onewould expect for actual physical systems. We consider thetime scale associated with the behavior predicted by ourmodel and conclude that there are parameters for which thedynamics are sufficiently slow to allow adsorption aniso-tropy effects to be observed in experimental systems.

In this paper we focus on three systems to point out thecomplexities that anisotropy can induce in the adsorptionprocess. As a reference system we discuss the adsorption ofan ellipsoidal particle for several conveniently chosen systemparameters. Subsequently, we proceed to describe the samesystem for a cylindrical particle to show the effects of shape.The free-energy landscape for cylinders is more complicated,leading to metastable adsorption configurations with a largestability domain in the adsorption region. Finally, we study acylindrical colloid with aspect ratio of 1. The occurrence ofinterfacial adsorption here is strongly dependent on the wayin which the particle makes initial contact with the interface.For some initial configurations there will be no adsorption atthe interface, depending on the value of the three-phase con-tact angle of the particle and the two liquids.

In conclusion, we apply our triangular tessellationscheme to determine the adsorption behavior of nonconvexparticles via Langevin-like dynamics. We show that for thesystems considered the adsorption mechanism is far morecomplex than was previously believed. The effects of includ-

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THE JOURNAL OF CHEMICAL PHYSICS 132, 164902 �2010�

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ing interfacial deformation, and anisotropic friction, certainlymerit further investigation, but go beyond the scope of ourinitial investigation.

II. METHOD

A. Theoretical considerations

We consider the adsorption of a solid uniaxial convexcolloid at a planar oil-water interface separating two homo-geneous half spaces of oil and water. The interfacial normalcoincides with the z axis of our coordinate frame, see Fig. 1.The depth z of the interface is measured with respect to thecenter of the particle. We locate the origin of our system atthe particle’s center. Medium 1 �M1� is the half space aboveand medium 2 �M2� is the half space below the interface. Thepolar angle �� 0,� /2� gives the angle between the col-loid’s rotational symmetry axis and interfacial normal. Forsimplicity, capillary effects due to the presence of the colloidat the interface are neglected. A simple calculation10 showsthat gravity does not play a significant role in interfacialadsorption for most colloidal systems, and it is hence ig-nored.

The adsorption free energy of the colloid contains termsdepending on three surface areas with corresponding surfacetensions: �i� the surface area of the colloid above the inter-face S1, �ii� the colloid’s surface area below the interface S2,

and �iii� the surface area excluded from the interface by thepresence of the colloid S12. The contact line, of length L,where the three phases meet, also contributes to the adsorp-tion free energy. The dependence of these quantities on z and� is implicit for notational convenience. We denote the totalsurface area of the colloid by S, therefore S1 ,S2� �0,S�, un-der the constraint S=S1+S2.

Following Pieranksi,1 the adsorption free energy can bewritten as

F�z,�� = �12��S1 − S�cos� − S12� + �L , �1�

where �12 is the M1-M2 surface tension and � is the linetension. This free energy, Eq. �1�, is defined with respect to areference point, namely, F=0 when the colloid is completelyimmersed in M1. The contact angle � is introduced viaYoung’s equation �12 cos�=�1c−�2c,

13 with �1c and �2c theM1-colloid and the M2-colloid surface tension, respectively.F�z� ,�� is made dimensionless and scale invariant by divid-ing the free energy by �12S ��12�0� and writingz=z��a2+2b2. Here a is the rotational symmetry semiaxis,b the perpendicular semiaxis, and m�a /b the aspect ratio.We thus obtain

f�z�,�� =F�z,��12S

= cos��r1 − 1� − r12 + ��l , �2�

where r1�S1 /S and r12�S12 /S are surface area ratios,

��

�12�S�3�

is the dimensionless line tension, and l�L /�S is a dimen-sionless contact line length. An extended adsorption free-energy expression, which can handle nonconvex patternedcolloids, is given in Ref. 10.

The adsorption configuration is the location of the free-energy minimum in Eq. �2� and is denoted by �zad

� ,�ad�. Theadsorption free energy for this configuration is given by fad� f�zad

� ,�ad�. For multiple �metastable� minima, the various�zad

� ,�ad� are labeled with a subscript i=1,2 , . . ., where thedeepest minimum is given the lowest index. We denote thepositive value of z�, for which the interface just touches thetop of the particle by zdet

� �� for given �; as it is also thedetachment position. Note that this value is always positive,whenever we consider −zdet

� �� the interface just touches thebottom of the particle, because of the symmetry of the prob-lem. When z��−zdet

� �� or z��zdet� �� the colloid is com-

pletely immersed in M1 and M2, respectively. For a given �,f�z� ,�� has a minimum as a function of z�, the location ofwhich is denoted by zmin

� �� and the corresponding free en-ergy by fmin��. It is possible that a single equi-�-curvef�z� ,�� has multiple minima. This in turn leads to multiplezmin

� ��, which are labeled with indices as above. Analo-gously, �min�z�� gives the location of the minimum inf�z� ,�� as a function of � for a fixed z�, the value of whichwe denote by fmin�z��. Here multiple minimum curves canexist as well. Often we abbreviate zdet

� ��, �min�z��, andzmin

� �� to zdet� , �min, and zmin

� , respectively.

S1 1cγ,

Μ2

2cγ,S2

S12 12,γ

τL,

Μ1

xz−view

φ

(0,0)

z

xy−view(b)

(a)

FIG. 1. Two representations of an ellipsoidal colloid adsorbed at a flatinterface, located at depth z measured from the center of the colloid �0,0�.The xz view �a� shows the two media, their dividing interface, and the polarangle � which the colloid’s rotational symmetry axis makes with the inter-facial normal. The surface area of the colloid above the interface is denotedby S1, with �1c the M1-colloid surface tension, and the surface area of thecolloid below the interface is denoted by S2, with �2c the M2-colloid surfacetension. The colloid excludes an area S12 from the interface, the regionenclosed by dashed curve in the xy view �b�, of which the surface tension isgiven by �12. The length L of this dashed curve is the contact line length andthe corresponding line tension is denoted by �. The solid curve in the xyview indicates the colloid’s outline.

164902-2 de Graaf, Dijkstra, and van Roij J. Chem. Phys. 132, 164902 �2010�


B. Numerical approximation scheme

Deriving analytic expressions for the dependence of S1,S2, S12, and L on z

� and � is highly nontrivial, in general, ifnot impossible.10 To analyze colloids adsorbed at an interfacewe have developed the numerical technique called triangulartessellation, which we briefly outline here, see Fig. 2. Fulldetails are given in Ref. 10, including the application to morecomplex colloid shapes than discussed in this paper.

Using a parametrization of the colloid at the interface,the surface and shape of the particle are approximated by apolyhedron, the faces of which are triangles, see Fig. 2�b�.This approximation can be improved upon by reducing theoverall triangle size, thus achieving better correspondencebetween the actual particle shape and the approximatingpolyhedron. The fineness of the tessellation, which is oftennonequidistant, is indicated by an n�m vertex notation.10

The value nm is the number of vertices in the mesh, each ofwhich is a member of six adjacent triangles. Modeling a 2Dor three-dimensional �3D� object in this way is a well-knowncomputer science method and it has been successfully ap-plied to various surface tension problems in physics.14–16 The

area of each of the triangles composing the polyhedron canbe determined by a simple cross product. Summing overthese areas results in an approximated value for the total

surface area, we denote this value by S̃.When the tessellated object is intersected by a plane,

some triangles of the approximating polyhedron are inter-sected by it, see Fig. 2�b�. These triangles are subdivided,Fig. 2�c�, into at most three subtriangles, which are notintersected.10 After this subdivision, we determine which tri-

angles lie above, given by the set ̃↑, and below the plane,

given by the set ̃↓, see Fig. 2�d�. Using these sets it ispossible to determine the approximated surface areas S̃1=�ĩ↑,i and S̃2=�ĩ↓,i, where ̃�,i is the surface area of theith triangle in the set ̃�. It follows that S̃1+ S̃2= S̃.

From the polyhedral approximation the points are ex-tracted where the plane intersects ribs of the triangles thattessellate the colloid, see Figs. 2�c� and 2�f�. These are usedto obtain the approximate surface area excluded from the

interface by the colloid, say S̃12, and the approximate contact

line length, say L̃. The points of intersection form a polygonin the plane, Fig. 2�f�, the area of which is determined usinga trapezium rule-type integration or a modified version ofGreen’s integral theorem.10 With either method, it is impor-tant to take steps to minimize numerical uncertainty.10

C. Adsorption trajectories and separatrices

Applying the triangular tessellation technique to solveEq. �2� as a function of z� and � yields a free-energy land-scape �on a grid�. This landscape is studied in various ways,using z�- or �-sections,10 and by zmin

� �� and �min�z�� curveswith respective corresponding fmin. It is also possible to ex-amine the negative gradient at each point of the adsorptionfree-energy landscape,

F�z�,�� = − �f��z�,�� − �ẑ� ��z�

+ �̂��

�� f��z�,�� , �4�

with ẑ� and �̂� unit vectors, ��, and f��z� ,�� f�z� ,��; thus obtaining a vector field of adsorption forceF. Note that with this choice for �� and f� the domain isbounded by �z� ,�� −1,1�� 0,0.5�. It should be empha-sized that the adsorption free energy acts as a potential en-ergy for the colloid, which can be differentiated with respectto its macroscopic coordinates, obtaining a force; the micro-scopic coordinates of the fluid have been integrated out toyield the tension terms.

The vector field is studied by examining four features:�i� flow lines, �ii� minima and maxima, �iii� saddle points,and �iv� separatrices. We will show that for certain particlespecies there are multiple minima in the adsorption free-energy landscape, also see Ref. 10. Each of these minima issurrounded by a region to which that particular minimum isattractive. That is to say, all flow lines which originate frompoints in this region reach that minimum when t→, with ttime. A flow line, ��t�= �z��t� ,��t��, is defined to follow thepath of steepest descent from its starting point at ��t=0�. It isa solution to the differential equation,

(a) (b)

(c) (d)

(e) (f)

I

II

III

FIG. 2. A sketch of the triangular tessellation scheme in a xz plane projec-tion ��a�–�d�� and in a xy plane projection ��e� and �f��. A colloid at a planarinterface �a� is approximated by a polyhedron of which the faces are tri-angles �b�. The surface of the colloid is thus effectively tessellated by tri-angles, which can be used to determine the approximate total surface area S̃.Triangles which are intersected by the interface are divided into at mostthree subtriangles �c�. Upon this subdivision, both the area above and belowthe interface is approximated more accurately �d�. Using the sets of trianglesin graph �d�, it is possible to determine S̃1 and S̃2, which approximate thevalue of S1 and S2, respectively. The boundary of the area cut out from theinterface, the dashed line in �e�, is approximated by a polygon �f�. Thispolygon is formed by the points where the interface intersects triangles of

the tessellation ��b� and �c��. The surface area S̃12 and contact line length L̃can be determined from �f�.

164902-3 Colloidal adsorption trajectories J. Chem. Phys. 132, 164902 �2010�


�̇�t� ��t�

�t= F��t�� , �5�

where the dot denotes the time derivative and the solution isfixed by imposing the initial point ��t=0�. We refer to theseflow lines as adsorption trajectories to stress their relationwith the physical path followed by a particle when adsorbingto the interface. The dividing line between two attractiveregions is referred to as the separatrix. This separatrix cancontain �local� maxima and saddle points.

D. Timescale of colloid motion

It is desirable to review the concept of time as intro-duced by solving the differential equation �̇�t�=F��t��. Thevariable t in this equation is mathematically a parametriza-tion parameter. This parameter is, however, related to physi-cal time, since it is relative to the size of the force exerted ata point ��t�, but not scaled according to a friction tensor. Inthe Langevin equation11,12 for a sphere in a homogeneousmedium, the time to travel over a short distance l is equalto M�l / A, where M is the mass, �−1 the Brownian time,and A is the force. We have a similar situation for our pa-rametrization, save the mass and friction coefficient. The1 / F is the reduced time that it takes to move along a flowline through a specific point. Note that it diverges for ex-trema in the free-energy landscape, which is in accordancewith the particle being stationary there. We analyze the rela-tion between the reduced time t and the actual time tr in moredetail in Sec. IV.

The flow line for a specific initial point is determined bylinear differential solver which employs steepest descent. Inthis scheme the time parameter is determined as follows. Thetime in which step i is taken, ti, is defined as the ratio of thestep size li and the force at the initial point of the step, i.e.,ti=li / �f i. We thus obtain a series of ti. The time re-quired to arrive at point n along the flow line, say t�n�, fromthe initial point is then given by

t�n� = �i=0

n−1

ti. �6�

In the limit of infinitesimally small step size, the solution tothe vector field with the initial condition ��t=0� and theproper time dependence are obtained.

III. RESULTS

In this paper we focus on anisotropic particles. We donot give explicit expressions for the case of a sphere, whichis a natural system for comparison, in the interest of concise-ness. The adsorption free energy for a sphere is easily shownto be a simple parabola ��=0�, with a free-energy minimumat depth z=a cos�, where a is the radius of the sphere. Forthe purpose of comparison, however, we demonstrate belowthat the adsorption process for spheres yields similar adsorp-tion times to those predicted for our anisotropic particles.

All data have been produced using the triangular tessel-lation technique10 on 5002-vertex grids heterogeneouslymapped to the object under consideration. To determine the

landscape, the value of f��z� ,�� is calculated on 250 by 250nonequidistant grid points in z�� −zdet

� ,zdet� � and ��

� �0,0.5�. The data on this mesh are then interpolated with athird order interpolation scheme to yield the full landscape.The accuracy of the triangular tessellation has been verifiedby a semianalytic method.10 In this case, the technique re-sults in a relative uncertainty of less than 10−4 per data point.

The flow lines are determined by means of a linear dif-ferential solver and contain over 1000 steps per line. Thestep size is reduced until subsequent reductions result in arelative difference between the curves of less than 10−3. Thisuncertainty is determined by summing over all points, thedifference in position between two successive reductions,and dividing by the total path length. The differential solveris considered to have converged on the solution of the vectorfield when the difference is less than 10−3. For some extremecases we are forced to work with less precision, as we willindicate in the text.

A. Adsorption of ellipsoidal particles

Figure 3�a� shows adsorption trajectories ��t� for an el-lipsoidal particle with aspect ratio m=6 for cos�=−0.5 and��=−0.1. In Fig. 3�b� the zdet

� ��, the zmin� ��, the �min�z��,

and the attractor A�z� ,�� curves for this landscape are given.The location of the minimum is indicated with a dot. Theterm attractor is introduced here to describe a feature in thevector field of adsorption force to which the flow lines areattracted, as can be seen in Fig. 3�a�. Mathematical analysisshows that for points on the attractor one of the eigenvectorsof the Hessian matrix ��T�f� is in the direction of thegradient, with a positive sign, and the other is perpendicularto it, with a negative sign. However, the gradient is nonzerofor points on the attractor, which are therefore not saddlepoints. Note that the attractor lies between the zmin

� �� and�min�z�� curves, for which one of the components of thegradient vector is zero.

By analyzing all points on the landscape, we concludethat the entire region between the zdet

� �� curves with �� 0,� /0.5� is attracted to the single minimum with�zad

� ,�ad��−0.0368,0.5��. This was to be expected on thebasis of our observations in Ref. 10. The flow lines in Fig.3�a� give a rather abstract picture of the colloidal motionthrough the interface. The main point of introducing this el-lipsoidal system is to have a basis for comparison, when westudy cylinders in Sec. III B.

To illustrate the behavior of the particle along one of theadsorption trajectories in Fig. 3�a�, we have included Fig. 4,which shows the time-dependent movement. Figure 4�a�shows several snapshots of the motion along the flow linewith ��t=0��0.930,0.0965�� in Fig. 3. Figure 4�b� showsthe location of the snapshots on the reproduced flow line.Here it should be stressed that the minimum is reached onlyat t→. The final snapshot shows a configuration, which isreached at a finite time, very close to the minimum.

From Fig. 4�a� it becomes clear that the movement of thecolloid through the interface, when it is close to vertical, isquite slow when compared to the rotational part of the move-ment before it reaches its equilibrium configuration. The fi-



nal part of the movement, reaching the equilibrium position,takes infinitely long, however. Slowing down at the end oc-curs only when the colloid is very close to its adsorptionconfiguration. The difference in speed between the rotationaland vertical motion parts of the adsorption is caused by thedifference in free-energy decay between points 0 to 7 and 8to 15 in Fig. 4�b�, respectively. In part it is caused by ourimplicit choice of the ratio between friction coefficients forthe translational and rotational movements. We return to thisin Sec. IV.

It should be noted that the choice of cos� and �� usedhere is not coupled to a particular physical system. The valuecos�=−0.5 is inspired by experimentally reasonable param-eters, see, for instance, Ref. 17, in which contact angles foroil-water interfaces with gold or silica colloids are deter-mined. The numerical value of the line tension was chosen

solely to make the flow lines emanating from M1 moreclearly visible. For ��=0 the attractor is so close to the −zdet

�

curve that they are obscured. For typical values of the inter-facial tension �12 and for the surface area of 1 �m sizedcolloids our value of � is in the nN range. This is within therange given in literature Refs. 18–20, which spans severaldecades from 10−12 to 10−6 N. Finally, it should be stressedthat the line tension and contact angle are related to eachother via the material properties of media present at the three

0

0.1

0.2

0.3

0.4

0.5

−1 −0.5 0 0.5 1

φ/π

z*

M1

(z*ad,φad)

z*min(φ)

φmin(z*)

z*det(φ)

A(z*,φ)

0

0.1

0.2

0.3

0.4

0.5φ/π

M1

M2

(z*ad,φad)

z*det(φ)

η(t)

(a)

(b)

FIG. 3. Graph �a� shows adsorption trajectories ��t� for an ellipsoidal par-ticle with aspect ratio m=6 for cos�=−0.5 and ��=−0.1. The arrow headsindicate the direction of colloid motion through the free-energy landscape,the dot gives the location of the minimum, i.e., the adsorption configuration,and the thick gray lines the zdet

� �� curves. The symbols M1 and M2 indi-cate that the colloid is completely immersed in the respective media, when-ever its z� value is to the right or left of the zdet�� curve. Graph �b� showsthe zdet

� �� curves in gray �thick, solid�, the zmin� �� curve in red �thick,

dots�, the �min�z�� curve in green �thick, dash-dot�, and the attractor A�z� ,��in blue �thin, solid�. The location of the minimum is again given by a dot.Note that the zmin

� �� and the �min�z�� curves in �b� indeed intersect withpoints on the adsorption trajectories where the tangent is vertical and hori-zontal, respectively, when �a� and �b� are superimposed.

t � 0.000 t � 0.203 t � 0.552 t � 1.181

t � 1.809 t � 2.577 t � 3.136 t � 3.834

t � 4.603 t � 5.029 t � 5.517 t � 5.936

t � 6.286 t � 6.565 t � 6.774 t � 6.984

0

0.1

0.2

0.3

0.4

0.5

−1 −0.5 0 0.5 1

φ/π

z*

M1

M2

(z*ad,φad)

z*det(φ)

η(t)

0 1 2 3

7654

111098

12 14 15

15

14

13

12

11

10

9

8

7 6

5

4

3

21

0

(b)

(a)

13

2M

1M

FIG. 4. Graph �a� shows several snapshots of the motion of an ellipsoidalcolloid with m=6, cos�=−0.5, and ��=−0.1 through the interface along aflow line from Fig. 3. An xz view of the colloid is represented by a blacksilhouette and the interface is indicated by a thin black line. The time, atwhich a snapshot is taken for each frame, is given in the top left corner. Thenumbers at the bottom right of the interfacial line correspond to the numbersin graph �b�. The first frame also shows the location of the two media M1and M2. Graph �b� shows the zdet

� �� curves, thick gray lines; the adsorp-tion free-energy minimum, black dot �overlapped by point 15�; the flow linefor which the snapshots are taken, thin black curve; and the �z� ,�� for whichthe snapshots in graph �a� are taken, numbered black squares. Again thelocation of the media is indicated by the symbols M1 and M2.



phase contact line. This is not presently included in ourmodel, since we are not aware of any study which quantifiesthe relation between cos� and ��.

Further analysis and our observations in Ref. 10 showthat qualitatively similar results are obtained for contactangles with −1�cos��0 and ��0.0. For ��0 the forma-tion of adsorption barriers, as discussed in Ref. 10, hindersadsorption to the interface from either medium and conse-quently change the adsorption free-energy landscape suffi-ciently to alter the adsorption trajectories qualitatively.

Additional representations of the movement along theadsorption trajectories shown in Fig. 3�a� are given in Ap-pendix. The motion studied in Fig. 4 will also be revisited.

B. Trajectories for cylindrical colloids

Figure 5 shows the adsorption trajectories and associatedcurves for a cylindrical colloid with m=6 when cos�=−0.5and ��=−0.1. Despite the close similarity of these parametersto those in Sec. III A for ellipsoids, the corresponding free-energy landscape for cylinders has two minima instead ofone: at �zad

� ,�ad��−0.079,0.5�� with fad�−0.397 and at�zad

� ,�ad��−0.962,0.0�� with fad�−0.090. For these pa-rameters the minimum with �ad=0.5� is the absolute mini-mum, not unlike the ellipsoidal case of Sec. III A, the mini-mum with �ad=0.0� is a metastable minimum. It can beshown theoretically that the latter constitutes a configurationwhere one of the caps of the cylinder is flush with the inter-face. Thereby a surface is excluded from the interface, low-ering the free energy, while the rest of the colloid is im-mersed in the preferred medium �M1 in this case�. There isonly contact with M2 on the excluded surface. This configu-ration can be metastable because excluding surface area fromthe interface strongly lowers the free energy.

The kinks in the adsorption trajectories of Fig. 5�a� aredirectly related to ridges in the free-energy landscape causedby the sharp corners of the cylinder. These ridges are indi-cated by the zdsc

� �� curves in Fig. 5�b�, where “dsc” refers tothe discontinuity that occurs in the gradient vector field. Notethat similar to the zdet

� curves these are also symmetric in z�

=0, hence we will use the notation to distinguish betweenthe different branches. In the bottom left corner of Fig. 5�b�the three curves nearly coincide, but it can be shown thatfrom left to right they are �min�z��, zmin

� ��, and −zdsc� ��.

From the figure it is not clearly apparent that the curve�min�z�� is discontinuous at �z� ,��0.315,0.0��. There isa small gap between the “vertical” and “horizontal”branches. The kink in the vertical branch and the startingpoint of the horizontal segment lie above each other. Forzmin

� �� the ridges in the potential landscape also induce akink. The presence of these ridges can strongly influence thebehavior of the flow lines for the cylindrical colloid’s adsorp-tion, as we will see.

In Fig. 5�c�, we show the separatrix S�z� ,�� between thetwo minima and the three attractors Ai�z� ,��, i=1,2 ,3,which are present in this free-energy landscape. There is asaddle point on the separatrix, where it meets with two of theattractors, A1 and A2. The separatrix forms the division be-tween the regions to which the respective minima are attrac-

tive. Remarkably the size of the �z� ,�� domain to which themetastable minimum is attractive is still substantial for anaspect ratio as high as m=6. For a particle adsorbing to theinterface from M1, on the left-hand side of Fig. 5�c�, anycolloid which touches the interface with ��0.224� will ad-sorb to the metastable minimum. For attachment from M2 we

0

0.1

0.2

0.3

0.4

0.5

−1 −0.5 0 0.5 1

φ/π

z*

M1

M2

A1(z*,φ)

A2(z*,φ)

A3(z*,φ)

S(z*,φ)

0

0.1

0.2

0.3

0.4

0.5

φ/π

M1

M2

z*min(φ)

φmin(z*)

z*dsc(φ)

0

0.1

0.2

0.3

0.4

0.5

φ/π

M1

M2

(z*ad,φad)

(z*sa,φsa)

z*det(φ)

η(t)

(a)

(b)

(c)

FIG. 5. Properties of the free-energy landscape for a cylinder with aspectratio m=6 for cos�=−0.5 and ��=−0.1. Graph �a� shows several adsorptiontrajectories ��t� �thin black curves�; the location of free-energy minima�dots�, of the saddle point �diamond�, and of media M1 and M2. Thick graycurves indicate zdet

� ��. Graph �b� displays the vector field discontinuities,zdsc

� ��, in purple �thick, dots�. The zmin� �� curve is represented in red �thick,

dashed� and �min�z�� in green �thick, dash-dot�. Graph �c� shows the attrac-tors, Ai�z� ,��, and the separatrix, S�z� ,��.



find that the metastable minimum is attractive for ��0.014�. Therefore, the colloid has only a very small win-dow to adsorb to the metastable configuration from the en-ergetically unfavorable medium. However, adsorption to themetastable configuration from the preferred medium is quitelikely because almost half of the orientations will lead to thisconfiguration.

The appearance of a tertiary attractor, see Fig. 5�c�, issomewhat surprising. The first and second attractors aremerely the split form of a feature similar to the attractor ofthe ellipsoid from Sec. III A. This “main attractor” is splitbecause of the two minima, causing the separatrix to inter-sect it. The tertiary attractor, leading to the absolute mini-mum of the free-energy landscape, is caused by a subtleinterplay between the rightmost ridge, zdsc

� ��, and the�min�z�� curve. Note that the attractor terminates at exactlythe same point as where the �min�z�� curve has a kink. Alsonote that the presence of the tertiary attractor strongly influ-ences the behavior of the adsorption trajectories around it,see Fig. 5�a�.

For this cylindrical colloid with m=6, we again expectqualitatively similar results for ��0; for ��0 adsorptionbarriers are found. However, since the depth of the minima isstrongly dependent on the value of the contact angle cos�,also see Ref. 10 for this depth when m=4, the position of theseparatrix will vary significantly with cos�. We come back tothis in Sec. IV, when we discuss cylindrical particles withaspect ratio m=1.

The adsorption trajectories in Fig. 5�a� that end up in thestable minimum are similar to those of an ellipsoid, exceptaround the secondary attractor, see Appendix, for examples.In Fig. 6 we consider colloid movement along one flow linewhich leads to the metastable minimum. Figure 6�a� repro-duces several snapshots of the motion along the flow linewith ��t=0��0.979,0.014��, also see Fig. 5. Figure 6�b�shows the location of the snapshots on the flow line. Thefinal snapshot gives a configuration very close to the mini-mum, which is reached only at infinite time. It should benoted that the motion between frames 12 and 15 �see Fig. 6�could only be determined with a relative uncertainty of�10−2. This is due to the close proximity of the �min�z��,zmin

� ��, and −zdsc� �� curves, as can be seen in Fig. 5. Con-

sequently, the time dependence of this part of the motion hassignificant uncertainty, which we estimate to be at most 15%.

Similar to the ellipsoid in Fig. 4�a� the cylinder firstmoves almost vertically through the interface �frames 1–8�,before it tilts slightly �frames 9–11�. However, when ittouches the interface with one point of the edge of one of thecaps �frame 12�, it does not continue to tilt in the same di-rection. At this point it is energetically favorable to moveinto the metastable minimum by tilting back to the verticalorientation �frames 13–15�. Finally, the colloid comes to restwith one cap flush with the interface at t→. Tilting backwhen the edge of the cap makes contact with the interface�frame 12� is an indication that the discontinuity ridges in thelandscape act as a dynamical “barrier.”

We have thus shown that if a particle can exclude arelatively large area from the interface by one of its ends, thedynamics and possible adsorption configurations are strongly

influenced. Both an ellipsoid and a spherocylinder do nothave a secondary minimum and therefore cannot exhibit thevertical adsorption of single particles.

C. Separatrices and special configurations

In this section we study the free-energy landscape of ashort cylindrical colloid with aspect ratio m=1, ��=0, andseveral cos�. Figure 7�a� shows the minima, saddle points,and separatrices for five values of the contact angle, cos�=0.0, �0.1, �0.2, �0.4, and �0.6. In Fig. 7�b� these three

0

0.1

0.2

0.3

0.4

0.5

−1 −0.5 0 0.5 1

φ/π

z*

M1

M2

(z*ad,φad)

(z*sa,φsa)

z*det(φ)

η(t)

t � 0.000 t � 0.079 t � 0.554 t � 1.346

t � 2.138 t � 3.043 t � 4.061 t � 5.079

t � 6.211 t � 7.002 t � 7.455 t � 7.681

t � 7.794 t � 10.170 t � 10.962 t � 11.313

0 1 2 3

7654

111098

12 13 14 15

(a)

(b)

M1

2M

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

0

FIG. 6. Graph �a� shows several snapshots of a cylindrical colloid withm=6, cos�=−0.5, and ��=−0.1 adsorbing to the metastable minimum. An xzview of the colloid is represented by a black silhouette and the interface isindicated by a thin black line. The time is given in the top left corner �firsttwo rows� and the bottom left corner �last two rows�. The numbers at thebottom right of the interfacial line correspond to the numbers in graph �b�.The first frame indicates the location of the two media. Graph �b� shows the

zdet

� �� curves, thick gray line; the adsorption free-energy minimum, blackdots �one is overlapped by point 15�; the saddle point, black diamond; theflow line for which the snapshots are taken, thin black curve; and the �z� ,��for which the snapshots in graph �a� are taken, numbered black squares.



properties of the free-energy landscape are given for cos�=−0.7, �0.72, �0.74, and �0.78. The inset shows the sepa-ratrices for a small region of Fig. 7�b�. Finally, Fig. 7�c�indicates the location of the minima, saddle points, and sepa-ratrices for cos�=−0.8, �0.9, and �0.95. The inset is in-

cluded to prove that the various minima in the top left cornerof the graph are all separate points, as opposed to the minimain the bottom left corner of the graph, which all have thesame z� and � value.

For cos�=0 there is a single minimum at �zad� ,�ad�

��0.000,0.271�� and for cos�=−1 there is no minimum.For all other values of cos� one of the minima is located at�zad

� ,�ad��−0.571,0.0��, which is exactly when one of thecylinder caps is flush with the interface. The location of theother minimum changes with the value of cos�. One of theseminima is stable and the other is metastable. The minimumwith �zad

� ,�ad��−0.571,00�� is stable when cos��−0.23.The labeling order of the minima with �ad=0.5� in the topsof Figs. 7�b� and 7�c� indicates the way in which these ap-pear in the graph. As already mentioned, the inset of Fig.7�c� shows that these minima are indeed distinct.

With decreasing cos� �cos� tends toward �1� the sepa-ratrices, denoted by the black lines, move from the left lowercorner toward the center of the region enclosed by the

zdet

� �� curves, the adsorption region. The locationof the secondary minimum shifts closer to �zad

� ,�ad��−0.571,0.5��. For a certain value of cos� the separatrixsplits into two pieces. This is shown in Fig. 7�b�, where theseparatrices for cos�=−0.7, �0.72, �0.74, and �0.78 aregiven �labels 5, 6, 7, and 8, respectively�. Further study in-dicates that for cos��−0.705 there is a transition betweenthe single and double separatrix regimes. From Figs. 7�a� and7�b� it is apparent that for decreasing cos� the position of thesaddle point, denoted by a diamond, moves closer to the−zdet

� �� curve. Around cos�=−0.705 the saddle point lies onthe −zdet

� �� curve and is degenerate. For a normal saddlepoint there are two attractive and two repulsive directions,whereas here there are two repulsive and only one attractivedirection.

For lower values of cos� the saddle point transforms intotwo degenerate saddle points on the boundary of the adsorp-tion region. Both of these have only one attractive and onerepulsive direction. The inset in Fig. 7�b� clearly shows thatthe separatrices are disjoined over the entire adsorption re-gion. This was to be expected since the separatrices are flowlines of a vector field. Figure 7�c� gives a clearer picture ofthe distance between the two separatrices for cos��−0.705.To determine these curves a tolerance of 10−2 was used forthe flow line convergence algorithm. When cos�↓−1 the areabetween the two separatrices tends to encompass the entireadsorption region. The most interesting feature of this in-terseparatrix domain is that a flow line starting in it will flowtoward a point on the −zdet

� �� curve, between the two saddlepoints. Along such a flow line the value of f�z� ,�� is alwayspositive, yet monotonically decreasing.

The effect of an interseparatrix domain for certain valuesof the contact angle is that the adsorption of colloids can bestrongly dependent on the initial configuration. The existenceof adsorption minima does not necessitate adsorption even ifthermal fluctuations are ignored. However, the situation stud-ied here, a cylindrical colloid with aspect ratio m=1 andextreme values of the contact angle, is not representative formost colloidal systems. To illustrate the differences betweenthe various “adsorption” possibilities for a cylinder with m

0

0.1

0.2

0.3

0.4

0.5

−1 −0.5 0 0.5 1

φ/π

z*

M1

M2

0.45

0.5

0.55

−0.6 −0.55 −0.5

0.055

0.06

−0.01 0 0.01

0

0.1

0.2

0.3

0.4

0.5

φ/π

M1

M2

0

0.1

0.2

0.3

0.4

0.5φ/π

M1

M2

(z*ad,φad)

(z*sa,φsa)

z*det(φ)

5−8

11−9

9−11

(a)

(b)

(c)

11 9 7 5

6810

8−5

4

31−4 1 2

41

0

2

4 3

3

2

1

8

7

6

5

6

7

8

1110

9

9

1011

FIG. 7. Properties of the free-energy landscape of a cylinder with aspectratio m=1 for ��=0 and several contact angles, labeled as follows in �a�cos�=0.0 �0�, �0.1 �1�, �0.2 �2�, �0.4 �3�, and �0.6 �4�; in �b� cos�=−0.7 �5�, �0.72 �6�, �0.74 �7�, and �0.78 �8�; in �c� cos�=−0.8 �9�, �0.9�10�, and �0.95 �11�. Minima are given by thick dots, saddle points bydiamonds, separatrices by black lines, and the zdet

� �� curves by thick graycurves. The inset in �b� shows an enlargement of a piece of graph �b�. Thelocations of some of the minima in graphs �b� and �c� are shown in the insetin �c�.



=1 and cos�=−0.95, we have included Fig. 8, which showssnapshots of the movement of the colloid for three differentadsorption trajectories though the free-energy landscape.Figure 8�d� gives the location of the various snapshots inFigs. 8�a�–8�c�. For clarity the rotational symmetry axis isgiven by a white line on the black silhouette of the colloid.

The first trajectory, Fig. 8�a�, shows the colloidadsorbing to the metastable minimum at �zad

� ,�ad��−0.567,0.5��. In its final configuration, the colloid onlybarely penetrates the interface and its rotationally symmetryaxis is perpendicular to the interfacial normal. Figure 8�c�shows the adsorption to the primary minimum at �zad

� ,�ad�= �−0.571,0.0��. Here the final configuration is when one ofthe cylinder caps is flush with the interface, as discussedearlier. Note that the rotational symmetry axis is indeed par-allel to the interfacial normal. The intermediate series ofsnapshots, Fig. 8�b�, shows the colloid along a flow line inthe interseparatrix domain. The colloid moves through theinterface and slows down when it approaches the −zdet

� ��curve, since the gradient tends to zero here. The final con-

figuration included shows the cylinder with only a singleattachment point on the interface, it is effectively detached.The three trajectories shown in Fig. 8 have also been in-cluded in the movie files referred to in Appendix.

IV. DISCUSSION

In the previous we have seen that the dynamics of col-loid motion through the interface can be very rich. In thissection we analyze the relation between the proposed flow-line dynamics and the dynamics associated with experimen-tal systems. We also discuss the elements present in realsystems, which we have not incorporated in our simplemodel and consider how they might affect the obtained re-sults.

A. Estimate of the adsorption time

For a dispersed particle undergoing a force, the dynam-ics are governed by a solution to the complete LangevinEquation.11,12 When we neglect inertia, the random force

t � 0.000 t � 0.005 t � 0.014 t � 0.814

t � 1.513 t � 1.816 t � 2.096 t � 2.331

0

0.1

0.2

0.3

0.4

0.5

−1 −0.5 0 0.5 1

φ/π

z*

M1

M2

(z*ad,φad)

z*det(φ)

S(z*,φ)

η(t)

t � 0.001 t � 0.063 t � 0.109 t � 0.679

t � 1.191 t � 1.476 t � 2.844 t � 5.698

t � 0.002 t � 0.192 t � 0.385 t � 0.530

t � 0.739 t � 1.061 t � 1.608 t � 3.220

(b)(a)

(c) (d)

0

12

3

45

6

7

7 6

5

4

3 2

1

0

0

1

234

5

6

7

0 1 2 3

7654 4

0

5

1 2 3

76

0 1 2 3

7654

(a)

(b)

(c)

M1

M2

M1

M2

2

M1

M

FIG. 8. Graphs ��a�–�c�� show snapshots of the motion of a cylindrical colloid with m=1, cos�=−0.95, and ��=0.0 through the interface along a flow line ofthe corresponding free-energy landscape. An xz view of the colloid is represented by a black silhouette and the interface is indicated by a thin black line. Thewhite line on the silhouette is used to indicate the rotational symmetry axis of the colloid. The time is given in the top left corner �first row� and the bottomleft corner �second row�. The numbers at the bottom right of the interfacial line correspond to the numbers in graph �d�, the graph letter is given near the “7”square on each flow line. For clarity graph �d� shows the three respective flow lines ��t� corresponding to the snapshots in graphs ��a�–�c��. The flow lines areindicated by thin black curves, the separatrices S�z� ,�� also see Fig. 7�c�� by thin yellow curves, the zdet

� �� by thick gray curves, and the minima by thickdots. In all graphs the location of the two media is given by the symbols M1 and M2.



term, and limit ourselves to studying the z and � compo-nents, the equations of motion �EOMs� become

��a��z�tr��tr

= −�

�zF�z�tr�,��tr�� , �7�

��a��tr�

�tr= −

�

��F�z�tr�,��tr�� , �8�

where tr is the “real” unreduced time, F is our unscaledadsorption free energy, and a is the long semiaxis of theparticle. The prefactors � and � represent the translationaland rotational friction coefficients, respectively. These coef-ficients should account for the size a and the shape of theparticle, as well as for the positional z and orientational �dependencies of the system. This includes the difference inviscosity between the two media, which influences the fric-tion force, the way in which friction changes in the z direc-tion according to the colloid’s �, and vice versa. We have notincorporated these dependencies in the notation used herebecause of the simplifications we apply to Eqs. �7� and �8� inthe following.

Let us assume that � and � are independent of z and �;therefore, only changing with scaling and particle types; wereturn to the validity of this later. It can then be easily shownthat the EOMs of Eqs. �7� and �8� allow us to regain theflow-line dynamics of Eq. �5�, �̇�t�=F��t��. We requirethat F�z ,��=�12Sf��z� ,��, z=�a2+2b2z�, �=��, tr=�t,where � is a time scale based on the system parameters, and

��a� =a2 + 2b2

�2��a� , �9�

� =a2 + 2b2

�12S��a� , �10�

where the friction coefficients are now coupled by imposingthe flow-line criterion.

For a sphere we set ��a�=6��0a, with �0 the viscosityof the densest medium. In this way we regain the expectedtranslational friction21 when the particle is free to move inthat medium. Note that for this choice, we do not recover theexpected rotational friction coefficient 8��0a

3 �Refs. 22 and23� in front of the time derivative �� /�tr. Instead we find18�0a

3 /�, which differs by a factor of 9 /4�2�0.23. In ret-rospect, it would have been more appropriate to rescale thesystem such that the flow-line dynamics give the expectedresults for a sphere. However, we investigated the adsorptiontrajectories with this constraint and found qualitatively simi-lar results.

As an example of this qualitative similarity we have in-cluded Fig. 9, in which cylindrical colloid’s separatrices for asystem with m=1, cos�=−0.95, and ��=0 are given for sev-eral choices of the rotational friction coefficient ��a�. Theseparatrix set S0 is generated using the translational ��a��and rotational friction coefficients ��0��a�� used through-out this paper. This leads to the same result as in Fig. 8�d�.For S1 the rotational friction coefficient �1=9�0 / �4�2� isused. The value of �1 is hence approximately 0.051 that ofthe spherical rotation coefficient. This is expressed in faster

� movement, which causes a narrowing of the interseparatrixdomain. For S2 the rotational coefficient is given by �2=4�2�0 /9, and is therefore equal to that of a sphere. Onecan see a corresponding widening of the interseparatrix do-main, since movement in the � direction is retarded.

Note that for each choice of �i the separatrices terminatein the same points on the −zdet

� curve. This is caused by thefact that the location of �degenerate� saddle points is unaf-fected by scaling transformations. The gradient or equiva-lently the shape of the landscape is, however, affected lead-ing to different separatrices. For the set S3 the separatricesare disjoined over the entire adsorption region. This is notclearly visible from the graph due to their close proximity toone another. This disjoined nature is commented on in Sec.III C, when the inset of Fig. 7�b� is discussed. Because oftheir close proximity as well as numerical uncertainty in thefree energy landscape, the set S1 could be determined withfractional uncertainty of at worst 0.05, close to M2.

The qualitative similarity of the result is expressed in thethree part partitioning of the adsorption region and the pres-ence of an interseparatrix domain. For values of ��a� similarto that of a sphere, our model predicts the presence of ainterseparatrix domain, which is always present at the side ofM1. The presence of this domain persists �noticeably� on theM2 side of the interface for values as low as 1/5 of the valueof 8��0a

3 and only increases in size for higher values of therotational friction coefficient. For this particular system, therotational friction coefficients is not likely to be much higheror lower than that of a sphere because of the aspect ratio ofthe particle.

Before discussing the accuracy of the simplificationsused in our research, let us gauge the time tr required for acolloid to attach to the interface and relax to its final posi-

0

0.1

0.2

0.3

0.4

0.5

−1 −0.5 0 0.5 1

φ/π

z*

M1

M2

(z*ad,φad)S0(z*,φ)S1(z*,φ)S2(z*,φ)z*det(φ)

FIG. 9. The separatrices Si�z� ,�� for a cylinder with m=1, cos�=−0.95, and��=0 are given for several values of the rotational friction coefficient. Thedots indicate the location of the adsorption minima and the thick gray linesthe zdet

� curves. For each case ��a�=6��0a. For S0 the rotational frictioncoefficient is ��a�= �a2+2b2��a� /�2��0, which is the same value as usedin Fig. 8�d� and the rest of the text. For S1 the value �1=9�0 / �4�2� and forS2 the value �2=4�

2�0 /9 are used, with �i the respective coefficients.Although the details change upon changing the rotational friction, the phe-nomenon of disjoined separatrices is a seen to be robust.



tion. We assume ��a�=6��0a, with �0 the largest of the twoviscosities. For the movements studied in Figs. 4, 6, and 8the duration of the adsorption process �the part in whichsignificant changes occur� is of the order t�5. According toEq. �10�, the physical time is estimated at

tr � 6��0aa2 + 2b2

�12St . �11�

For a sphere, Eq. �7� can be solved analytically ��=0� be-cause of the simple shape of the free-energy landscape. It iseasily shown that moving from z=a to z=a cos� �the equi-librium position� is exponential asymptotic with time, wherethe decay time is given by 3�0a / �2�12�. This is almost thevalue for � in the case of a sphere, save a factor of 3 whichis related to the normalization of z� in our model. Thereforethe time scales will be similar to those found for the aniso-tropic colloids discussed here.

Upon using the approximation of Eq. �11�, we arrive atthe following result, see Table I. Here we have consideredellipsoids with aspect ratios m=1 and m=6, sizes a=125 nm, a=500 nm, and a=2.5 �m, which are typicalcolloidal length scales. For the viscosity we use �0=1.5 Pa s �glycerol�, �0=1.56�10−2 Pa s �cyclohexylchlo-ride�, and 1.0�10−3 Pa s �H2O�,

24 which are commonlyused solvents. Surface tension values between two typicalliquids areof the order �12=10

−2 N m−1 �Ref. 25� or lower than �12=10−5 N m−1 with the addition of surfactants.26,27

Note that it follows from Table I that length of the ad-sorption process can vary significantly with the choice ofsystem parameters. For values of the interfacial tensionwhich are in the order of 10−2 N m−1 the adsorption processtypically takes tens to hundreds of microseconds. However,upon lowering the interfacial tension, e.g., by adding surfac-tants, the process slows down significantly, in some casestaking upward of a second. This is an exciting prospect, sincefor many systems the time scale is thus accessible to experi-mental techniques. It can also be shown that the particularshape of the particle, e.g, ellipsoid, cylinder, or spherocylin-der, does not influence these results significantly, since�a2+2b2� /S is virtually the same for these particle types.

The limiting factor in an experiment to determine theadsorption behavior is the maximum operating frequency ofthe camera and not the optical elements used for imaging.For modern cameras operating frequencies of 10.000 Hz ormore are obtainable.28 However, we should bare in mind that

our time tr is based on a rather crude estimate and that therecan be a substantial deviation from the prediction from thisvalue for an actual experimental system. Nevertheless, a dis-crepancy of a decade or two will still place many of theabove mentioned systems within the experimentally observ-able range.

B. Limitations of the model and experimentalconsiderations

There are several key points of criticism which can beidentified when trying to compare the dynamics predicted byour flow lines in the reduced system to real-world dynamics.These points should be taken into account in followup stud-ies, but go beyond the scope of an initial investigation.

Neglecting the random thermal force component is anacceptable simplification, if we only want to study the aver-age movement, i.e., the trajectory averaged over many ad-sorption events. Some care should be taken though, since itis clear from, for instance, Fig. 7�b� that adsorption trajecto-ries which have almost identical starting points can divergefrom each other quite rapidly. This may prove problematic inestablishing an average for an experimental system, espe-cially when the magnitude of the free-energy landscape fea-tures becomes in the order of a kBT. Neglecting the inertia isalso acceptable, since most colloidal systems are stronglyoverdamped. Even under high potential differences—a1 �m colloidal particle can typically experience an interfa-cial adsorption potential in the order of 104kBT �Ref. 1�—colloids cannot be easily forced into the ballistic regime ofmotion. We come back to this when discussing buoyancyeffects.

Another point of concern is neglecting anisotropic ef-fects in friction force. As we have seen in the previous, thefriction coefficient in part imposes the time scale on the sys-tem. For anisotropic particles, which have a friction coeffi-cient tensor, this time scale changes according to the direc-tion of motion and the position of the particle. However, thedifference in friction coefficient between motion along thelong semiaxis of an ellipsoid and perpendicular to this semi-axis has been shown to be no greater than a factor of2.12,29–31 For the rotational friction, the change is likely to bemore substantial than a factor of 2 for strongly oblong par-ticles. Suppose we assume proportionality with the rotationalradius to the third power, as is the case for a sphere.23 Then,depending on the axis of rotation, the friction coefficient var-

TABLE I. The physical time tr required to complete the motion of the colloid through the interface on the basisof our model. We assume t�5 and study two aspect ratios m=1 and m=6 for ellipsoids. Several values of theviscosity �0 are considered.

�12�N m−1�

�0�Pa s�

m=1, a=125 nmtr �ms�

m=1, a=0.5 �mtr �ms�

m=6, a=0.5 �mtr �ms�

m=6, a=2.5 �mtr �ms�

1.5 4�10−1 2�100 4�100 2�101

10−2 1.56�10−2 4�10−3 2�10−2 5�10−2 2�10−1

1.0�10−3 3�10−4 1�10−3 3�10−3 1�10−2

1.5 4�102 2�103 4�103 2�104

10−5 1.56�10−2 4�100 2�101 5�102 2�102

1.0�10−3 3�10−1 1�100 3�100 1�101



ies between ��a3 and ��b3. Even if the proportionality isless extreme than in our simple estimate, the differences caneasily be very significant.

The difference in friction between the two media border-ing the interface can be several orders of magnitude, see, forinstance, Ref. 24 for the viscosity values of commonly usedsolvents. Such a difference in friction coefficient will play arole in describing a particular system accurately. We assumedthat the timescale of the dynamics is dominated by the me-dium with the highest friction coefficient or, equivalently,viscosity. This assumption holds whenever there is substan-tial penetration of the colloid in that medium. A more accu-rate approach requires us to weight the friction coefficient bythe surface areas of the particle which are in the respectivemedia. Strong differences in viscosity of the respective me-dia most likely induce a resistance for the colloid to moveinto the more viscous medium. This can substantially alterthe qualitative behavior of the flow lines.

The above mentioned points of criticism are related tothe way in which we approximate the Langevin equation toachieve the flow-line dynamics discussed in this paper. Thesecan be considered to be independent of the changes in thedynamics induced by using a more elaborate model to deter-mine the free-energy landscape of the particle’s adsorption.The former concerns the way in which the dynamics areextracted from the landscape, whereas the latter concerns thedetermination of the landscape itself. Of course, it is alsopossible that these two are coupled, whenever the motion isfast enough to allow the particle’s velocity to influence thefree-energy landscape.

One of the effects to be included in a more elaboratefree-energy model is the interfacial deformation, which al-lows for undulation of the contact line. Our model assumes aflat interface and therefore does not accommodate for thecorrect three phase contact angle along the contact line for agiven configuration. Such effects have been studied in litera-ture for a limited number of configurations,32,33 but the avail-able techniques either cannot be used to determine the fullfree-energy landscape or are computationally prohibitivelyexpensive.

For experimental systems the solvent is sometimes den-sity matched to the particle to eliminate the effects of gravity.Matching is, however, more complicated to achieve for athree component system, such as the particle near a liquid-liquid interface. To account for gravity in our model, werequire the buoyancy mass of the particle. This mass can bedetermined by volume integration over the parts which pen-etrate the respective media, assuming a homogeneous den-sity distribution. Such volumetric integrations can be per-formed, using a 3D analogy of our 2D surface integrationscheme based on triangular tessellation. That is to say, somekind of spatial tessellation using polyhedra. Nevertheless, wehave shown in Ref. 10 that for colloidal systems the effectsof gravity will be negligible because of the low Bond num-ber for such particle sizes.

When considering larger sized particles, the effect ofbuoyancy becomes more pronounced and can induce stronginterfacial deformation. In Ref. 34 spherical particles, sub-stantially larger than colloids, are released on an air-water

interface and thereby generate an in-plane interfacial repul-sive force. This force is shown to be caused by capillary riseand vertical underdamped oscillatory motion. Surprisingly,for these systems the inertia term does seem to contributesubstantially. This is in part due to the lack in density match-ing between the particles. It can be shown that the conditionfor overdamping is �0

2 / ��12�c��a, with �c the colloid’s den-sity, whenever the colloid is reasonably density matched withboth media. One can obtain this result by a calculation analo-gous to that presented in the appendix of Ref. 34. It shouldbe noted that for typical values of the particle density, sur-face tension, and viscosity, this condition is met for colloidalparticles, i.e., they are indeed overdamped as assumedthroughout in this paper.

Finally, in many physical system, the particle and theinterface are often charged by self-dissociation of the surfacemolecules on the colloid and of those which compose themedia.35 For a like-charged interface there can be a substan-tial charge repulsion barrier which needs to be overcomebefore adsorption can take place, if this barrier can becrossed at all. Including electrostatic effects into the modelwill substantially change the appearance of the free-energylandscape. We cannot at this time predict the effects on theadsorption dynamics of free-energy landscape altering ef-fects, such as electric interactions and interfacial deforma-tion.

In conclusion, we have identified the time which we ex-pect the adsorption process to take for several experimentallyfeasible systems, based on our model. The results are encour-aging as this time regime seems to be accessible to currentexperimental techniques, even though there are still quite afew caveats to be considered.

V. CONCLUDING REMARKS

We have employed the triangular tessellation techniqueintroduced in Ref. 10 to determine the adsorption free-energylandscape of anisotropic colloids at a flat interface. This freeenergy is composed of surface- and line-tension contribu-tions, but does not take into account interfacial deformationand electrostatics. We analyzed the obtained free-energylandscapes by means of a vector field of adsorption force andits associated flow lines. These flow lines are calculated us-ing a linear steepest descent method and they are param-etrized in units of reduced time. The link between theseflow-line dynamics and Langevin dynamics and the validityof our approximation are examined. For typical viscositiesand surface tensions we predict that time scales to completethe adsorption process are in the microsecond to second re-gime. Such time scales are accessible for observation inmodern experimental setups.

Within our model, we re-establish that there is a strongdependence of the adsorption free-energy landscape on theshape of the colloidal particle. This in turn leads to a widerange of adsorption phenomena. For ellipsoidal particles wefind a single equilibrium adsorption configuration with rela-tively simple dynamics. However, for cylindrical particlesthere can be two adsorption configurations, depending on thevalue of the contact angle. The presence of two minima in



the free-energy landscape leads to a separatrix, which formsthe divide between the regions to which the respectiveminima are attractive. The metastable minimum can have alarge domain in the adsorption region to which it is attrac-tive. This is significant, since it suggests that adsorption inunexpected configurations should be easier than previouslybelieved. The implications of this large domain size on theexperimentally observed buckling and flipping transitions4,5

merit further investigation.We have also shown that for short cylinders the free-

energy landscape allows for another type of colloid interac-tion with the interface, other than adsorption. Depending onthe value of the contact angle, there can be a domain in theadsorption region, for which colloids do not attach to theinterface, but simply move through it unhindered. This is ofparticular interest, since it shows, within the confines of ourmodel, that the presence of stable and metastable adsorptionconfigurations is not a sufficient criterion to guarantee par-ticle adsorption. To the best of our knowledge, this is the firsttime that this phenomenon has been observed theoretically.

Our simple model to calculate the free energy does notinclude interfacial deformation due to capillary, electrostatic,or gravitational forces; nor does it take into account inertia,anisotropy in the friction tensor, and the Brownian randomforce to determine the colloid dynamics. We are, however,optimistic that the phenomenology described here can be re-covered in a more elaborate model. The method presentedhere thus forms the first stepping stone toward obtaining bet-ter correspondence between experiments and their theoreticaldescription, and thus brings us to a closer understanding ofcolloidal adsorption phenomena.

ACKNOWLEDGMENTS

M.D. acknowledges financial support by a “NederlandseOrganisatie voor Wetenschappelijk Onderzoek” �NWO� ViciGrant, and R.v.R. by the Utrecht University High PotentialProgramme.

APPENDIX: COLLOID MOTION THROUGH THEINTERFACE

In this appendix we describe the content of the supple-mentary material36 for this paper. The supplementary mate-rial gives movies for the time-dependent movement along afew adsorption trajectories representative of the colloid’smotion through the free energy landscapes of Figs. 3, 5, and8.

The data repository contains three folders labeled as fol-lows: the particle type appears first, followed by the aspectratio m, followed by the value of the contact angle cos�, andfinally the line tension value �� is given. The lower case mappearing in front of the numbers, is to indicate that it is anegative value, where the m stands for minus. The individualmovie files are labeled similarly: an abbreviation of the type,followed by the value of the aspect ratio, and finally thecoordinates of ��t=0� in the order z��t=0�, ��t=0�. The filewhere the value of the aspect ratio is preceded by the letter Agives the motion of the colloid along the attractor.

Each movie file consists of a panel which shows twoframes side by side. The left frame gives the zdet

� lines ingray as well as the flow line in black. The final configurationis indicated by a black dot and the configuration at time t bya red dot. The time appears in the top left corner of the rightframe. A thin line divides this frame in two and serves toindicate the location of the interface. A silhouette similar tothose used in Figs. 4�a�, 6�a�, and 8�a�–8�c� gives the xzprojection of the colloid. Again a white line is added to thesilhouette of the cylindrical colloid with m=1 to show thecolloids orientation.

A few of the movie files show slight jittering of colloidalong its adsorption trajectory. Especially flow lines to themetastable minimum of a cylinder with m=6, also see themain text. These adsorption trajectories suffered fromslightly higher levels of numerical uncertainty than the oth-ers, since many features in the landscape are close togetherin this case.

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http://dx.doi.org/10.1140/epje/i2007-10314-1http://dx.doi.org/10.1016/0021-9797(69)90399-3http://dx.doi.org/10.1073/pnas.0610589104http://dx.doi.org/10.1063/1.3389481

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