To describe the full spectrum of surface fluctuations of the interface
between phase-separated colloid-polymer mixtures from low scattering vector q
(classical capillary wave theory) to high q (bulk-like fluctuations), one must
take account of the interface's bending rigidity. We find that the bending
rigidity is negative and that on approach to the critical point it vanishes
proportionally to the interfacial tension. Both features are in agreement with
Monte Carlo simulations.Comment: 5 pages, 3 figures, 1 table. Accepted for publication in Phys. Rev.
  Let

Edgar M. Blokhuis

J. D. Weeks

J. S. Rowlinson

J. W. Gibbs

Joris Kuipers

Richard L. C. Vink

S. Asakura

W. Helfrich

English

arXiv

Crossref

Description of the Fluctuating Colloid-Polymer Interface

Blokhuis, E.M.

Kuipers, J.

Vink, R.L.C.

Leiden University Scholary Publications

Description of the Fluctuating Colloid-Polymer InterfaceEdgar M. Blokhuis,1 Joris Kuipers,1 and Richard L. C. Vink21Colloid and Interface Science, Leiden Institute of Chemistry, Gorlaeus Laboratories,P.O. Box 9502, 2300 RA Leiden, The Netherlands2Institute of Theoretical Physics, Georg-August-Universität, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany(Received 26 May 2008; published 18 August 2008)To describe the full spectrum of surface fluctuations of the interface between phase-separated colloid-polymer mixtures from low scattering vector q (classical capillary wave theory) to high q (bulklikefluctuations), one must take account of the interface’s bending rigidity. We find that the bending rigidity isnegative and that on approach to the critical point it vanishes proportionally to the interfacial tension. Bothfeatures are in agreement with Monte Carlo simulations.DOI: 10.1103/PhysRevLett.101.086101 PACS numbers: 68.03.Cd, 68.05.Cf, 68.35.CtOne of the outstanding theoretical problems in theunderstanding of the structure of a simple liquid surfaceis the description of the full spectrum of surface fluctua-tions obtained in light scattering experiments [1,2] andcomputer simulations [3–5]. Insight into the structure ofa simple liquid surface is provided by molecular theories[6,7], such as the van derWaals squared-gradient model, onthe one hand, and the capillary wave model [8,9] on theother hand. The theoretical challenge is to incorporate boththeories and to describe the spectrum of fluctuations of aliquid surface from the molecular scale to the scale ofcapillary waves.Here, we report on a theoretical description ofMonte Carlo (MC) simulations [4] of a system consistingof a mixture of colloidal particles with diameter d andpolymers with a radius of gyration Rg. The presence ofpolymer induces a depletion attraction [10] between thecolloidal particles which may ultimately induce phaseseparation [11,12]. The resulting interface of the demixedcolloid-polymer system is studied for a number of polymerconcentrations and for a polymer-colloid size ratio " 1þ 2Rg=d ¼ 1:8.The quantity studied in the simulations is the (surface)density-density correlation function:SðrkÞ  1ð‘  vÞ2Z LLdz1Z LLdz2h½ð~r1Þ  stepðz1Þ ½ð ~r2Þ  stepðz2Þi; (1)where ð~rÞ is the colloidal density, ~rk ¼ ðx; yÞ is the direc-tion parallel to the surface, and where we have definedstepðzÞ  ‘ðzÞ þ vðzÞ with ðzÞ the Heavisidefunction and ‘;v the bulk density in the liquid and vaporregion, respectively, where by ‘‘liquid’’ we mean the phaserelatively rich in colloids and by ‘‘vapor’’ the phase rela-tively poor in colloids. Its Fourier transform is termed thesurface structure factorSðqÞ ¼Zd~rkei ~q ~rkSðrkÞ: (2)In Fig. 1, MC simulation results [4] for SðqÞ are shown forvarious values of the integration limit L. The figure showsthat the contribution to SðqÞ from short wavelength fluctu-ations (high q) increases with L.To analyze SðqÞ, one needs to model the density fluctua-tions in the interfacial region. In the capillary wave model(CW) [8], the fluctuating interface is described in terms ofa two-dimensional surface height function hð ~rkÞð ~rÞ ¼ 0ðzÞ  00ðzÞhð ~rkÞ þ    ; (3)where 0ðzÞ ¼ hð ~rÞi. In the extended capillary wavemodel (ECW), the expansion in gradients of hð~rkÞ is con-tinued [13–15]:ð ~rÞ ¼ 0ðzÞ  00ðzÞhð ~rkÞ 1ðzÞ2hð ~rkÞ þ    : (4)011q110S(q)L/W = 4L/W = 3L/W = 2L/W = 1CWFIG. 1. MC simulation results for the surface structure factor(in units of d4) versus q (in units of 1=d) for various values of theintegration limit L=W ¼ 1; 2; 3; 4 [4]. The dashed line is thecapillary wave model. In this example " ¼ 1:8, p ¼ 1:0.PRL 101, 086101 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending22 AUGUST 20080031-9007=08=101(8)=086101(4) 086101-1  2008 The American Physical SocietyThe function 1ðzÞ is identified as the correction to thedensity profile due to the curvature of the interface,hð~rkÞ  1=R1  1=R2, with R1 and R2 the (principal)radii of curvature.With Eq. (4) inserted into Eq. (1), we find that SðqÞequals the height-height correlation function, SðqÞ ¼ShhðqÞ, whereShhðqÞ Zd~rkei ~q ~rk hhð ~r1;kÞhð~r2;kÞi: (5)Here we have assumed that the location of the interface, asdescribed by the height function hð ~rkÞ, is given by theGibbs equimolar surface [16], which gives for 0ðzÞ and1ðzÞ:Zdz½0ðzÞ  stepðzÞ ¼ 0;Zdz1ðzÞ ¼ 0: (6)Naturally, other choices are possible [5] and equally legiti-mate as long as they lead to a location of the dividingsurface that is ‘‘sensibly coincident’’ [16] with the inter-facial region.The height-height correlation function ShhðqÞ is deter-mined by considering the free energy associated with asurface fluctuation [8,9]. The inclusion of a curvaturecorrection to the free energy is described by the Helfrichfree energy [17]. It gives for  ¼ 12Z d ~qð2Þ2 ðqÞq2hð ~qÞhð ~qÞ; (7)withðqÞ ¼ þ kq2 þ    : (8)The coefficient k is identified as Helfrich‘s bending rigidity[17,18]. It is important to realize that the bending rigidity,defined by Eqs. (7) and (8), depends on the choice made forthe location of the dividing surface (here, the Gibbs equi-molar surface for the colloid component).Using Eq. (7), the height-height correlation function canbe calculated [18]ShhðqÞ ¼ kBTðqÞq2 ¼kBTq2 þ kq4 þ    : (9)Without bending rigidity (k ¼ 0) this is the classical cap-illary wave result in the absence of gravity (dashed line inFig. 1). When L is sufficiently large, the capillary wavemodel accurately describes the behavior of SðqÞ at low q.To model SðqÞ in the whole q range, we also includebulklike fluctuations to the density:ð~rÞ ¼ 0ðzÞ  00ðzÞhð~rkÞ 1ðzÞ2hð~rkÞ þ bð~rÞ:(10)Inserting Eq. (10) into Eq. (1), one now finds thatSðqÞ ¼ ShhðqÞ þN LSbðqÞ: (11)The second term is derived from an integration into thebulk regions (to a distance L) of the bulk structure factorSbðqÞSbðqÞ ¼ 1þ bZd~r12ei ~q~r12½gðrÞ  1: (12)The density correlation function gðrÞ differs in eitherphase, but here we take for it g‘ðrÞ of the bulk liquid.This approximation may be justified by arguing that closeto the critical point there is no distinction between the twobulk correlation functions, whereas far from the criticalpoint the contribution from the bulk vapor can be neglectedsince v  0. The error is further reduced by fitting theL-dependent prefactorN L to the limiting behavior of SðqÞat qd ! 1.In Fig. 2, we show the result from Fig. 1 for L=W ¼ 3.For qd  1 the results asymptotically approach the capil-lary wave model (dotted line). The dashed line is the resultof adding the bulklike fluctuations to the capillary waves:SðqÞ ¼ kBTq2þN LSbðqÞ: (13)Figure 2 shows that Eq. (13) already matches the simula-tion results quite accurately except at intermediate valuesof q, qd  1.Finally, we include a bending rigidity in SðqÞ:SðqÞ ¼ kBTq2 þ kq4 þ    þN LSbðqÞ: (14)The value of the bending rigidity is extracted from thebehavior of SðqÞ at low q. The fact that the simulationresults in Fig. 2 are systematically above the capillarywave model in this region indicates that the bending rigid-ity thus obtained is negative, k < 0. Unfortunately, a nega-tive bending rigidity prohibits the use of SðqÞ in Eq. (14) to011q110S(q)CWCW + bulkECW + bulkFIG. 2. MC simulation results [4] (circles) for the surfacestructure factor (in units of d4) versus q (in units of 1=d). Thedotted line is the capillary wave model; the dashed line is thecombination of the capillary wave model and the bulk correla-tion function; the solid line is the combination of the extendedcapillary wave model and the bulk correlation function. In thisexample " ¼ 1:8, p ¼ 1:0, L=W ¼ 3.PRL 101, 086101 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending22 AUGUST 2008086101-2fit the simulation results in the entire q range, since thedenominator becomes zero at a certain value of q. It istherefore convenient to rewrite the expansion in q inEq. (14) asSðqÞ ¼ kBTq21 kq2 þ   þN LSbðqÞ; (15)which is equivalent to Eq. (14) to the order in q2 consid-ered, but which has the advantage of being well behaved inthe entire q range. The above form for SðqÞ, with thebending rigidity used as an adjustable parameter, is plottedin Fig. 2 as the solid line. Exceptionally good agreementwith the MC simulations is now obtained for all q. InTable I, we list the fitted values for the bending rigidityfor a number of different polymer concentrations.Next, we investigate whether the value and behavior of kcan be understood from a molecular theory. One shouldthen consider a microscopic model for the free energy todetermine the density profiles 0ðzÞ and 1ðzÞ. Here, weconsider the free energy density functional based on asquared-gradient expansion [7,13,14,19]:½ ¼Zd~rmj ~rð ~rÞj2  B4½ð ~rÞ2 þ gðÞ; (16)where the coefficients m and B are defined asm   112Zd~r12r2UðrÞ; B   160Zd~r12r4UðrÞ:(17)The integration over ~r12 is restricted to the attractive part(r > d) of the interaction potential UðrÞ, for which weconsider the Asakura-Oosawa-Vrij depletion interactionpotential [10]:UðrÞ ¼ kBTp2ð" 1Þ32"3  3"2rdþrd3; (18)where the intermolecular distance is in the range 1<r=d < ". For explicit calculations, gðÞ is taken to be ofthe Carnahan-Starling form:gðÞ ¼ kBT lnðÞ þ kBT ð4 32Þð1 Þ2  a2;(19)where   ð=6Þd3,  ¼ coex, and the van der Waalsparameter a is given bya   12Zd~r12UðrÞ: (20)The surface tension, to leading order in the squared-gradient expansion, can be determined from the usualexpression [7] ¼ 2 ffiffiffiffimp Z ‘vdffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigðÞ þ pq: (21)In the inset of Fig. 3, the surface tension is shown as afunction of the colloidal volume fraction difference,  ‘  v. The squared-gradient expression (solid line) is insatisfactory agreement [20] with the MC simulations.The (planar) density profile 0ðzÞ is determined fromminimizing the free energy functional ½ in Eq. (16) inplanar symmetry. To also determine the density profile1ðzÞ from a minimization procedure, one should considerthe energetically most favorable density profile for a givencurvature of the surface. To set the curvature to a specificvalue, one adds to the free energy in Eq. (16) an externalfield Vextð~rÞ that acts a Lagrange multiplier. Differentchoices for Vextð ~rÞ can then be made, but we choose itsuch that it acts only in the interfacial region:Vextð ~rÞ ¼ 00ðzÞhð ~rkÞ; (22)with the Lagrange multiplier  set by the imposed curva-ture. This choice for Vextð~rÞ constitutes our fundamental‘‘ansatz’’ for the determination of 1ðzÞ. It improves onearlier choices made [13,14,21] in the sense that the bulkdensities are equal to those at coexistence and the densityprofile remains a continuous function.The minimization of the free energy, with the aboveexternal field added, using the fluctuating density inEq. (4) yields the following Euler-Lagrange (EL) equationsTABLE I. MC simulation results [4] for the polymer volumefraction p, liquid and vapor colloidal volume fractions, ‘ andv, surface tension  (in units of kBT=d2), bending rigidity k (inunits of kBT; in parenthesis the estimated error in the last digit),andffiffiffiffiffiffiffiffiffiffiffiffiffik=p (in units of d).p ‘ v  kffiffiffiffiffiffiffiffiffiffiffiffiffik=p0.9 0.2970 0.0141 0.1532 0:045 (15) 0.541.0 0.3271 0.0062 0.2848 0:07 (2) 0.501.1 0.3485 0.0030 0.4194 0:10 (3) 0.491.2 0.3647 0.0018 0.5555 0:14 (3) 0.500 0.1 0.2 0.3 0.4∆η-0.16-0.12-0.08-0.040k0 0.2 0.4∆η00.20.40.6σFIG. 3. Bending rigidity in units of kBT versus the volumefraction difference . The inset shows the surface tension inunits of kBT=d2. The solid lines are the gradient expansionapproximation; filled circles are the results from the MC simu-lations; the dashed line is the fitffiffiffiffiffiffiffiffiffiffiffiffiffik=p  0:47d.PRL 101, 086101 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending22 AUGUST 2008086101-3for 0ðzÞ and 1ðzÞ:g0ð0Þ ¼ 2m000 ðzÞ;g00ð0Þ1ðzÞ ¼ 2m001 ðzÞ þ 4m00ðzÞþ 2B0000 ðzÞ þ 200ðzÞ:(23)The change in free energy  due to a certain densityfluctuation is determined by inserting ð ~rÞ in Eq. (4) intothe expression for in Eq. (16). One finds that  is thengiven by the expression in Eq. (7), with the bending rigidity[14]k ¼ 2mZdz1ðzÞ00ðzÞ B2Zdz00ðzÞ2; (24)where we have used the EL equations in Eq. (23).To determine 0ðzÞ we assume proximity to the criticalpoint where gðÞ takes on the usual double-well form. Thesolution of the EL equation in Eq. (23) then gives [7]:0ðzÞ ¼ 12 ð‘ þ vÞ 2tanhðz=2Þ; (25)where  is a measure of the interfacial thickness which weshall define as   mðÞ2=ð3Þ, with the value of given by Eq. (21). To determine 1ðzÞ the differentialequation in Eq. (23) is solved using the tanh profile for0ðzÞ, yielding:1ðzÞ ¼ 3B10mf1 ln½2 coshðz=2Þgcosh2ðz=2Þ ; (26)where we have used that  ¼ 2mþ B=ð52Þ.Inserting Eq. (26) into Eq. (24), one finds for kk ¼ BðÞ260¼  B20m: (27)This expression indicates that the bending rigidity vanishesnear the critical point with the same exponent as thesurface tension, i.e.,k / Bm/ d2: (28)This scaling behavior should be contrasted to the usualassumption that k / 2, i.e., that k approaches a finite,nonzero limit at the critical point [18,21].In Fig. 3, the gradient expansion result in Eq. (27) for thebending rigidity is shown as the solid line. The bendingrigidity is negative, in line with the simulation results,although the magnitude is significantly lower.To summarize, we have shown that to account for thesimulated scattering function over the whole range ofscattering vector q, including the intermediate range be-tween low q (classical capillary wave theory) and high q(bulklike fluctuations), one must take account of the inter-face‘s bending rigidity. Two of the important results arethat the bending rigidity k for the interface between phase-separated colloid-polymer mixtures is negative, and that onapproach to the critical point it vanishes proportionally tothe interfacial tension rather than, as had often been sup-posed, varying proportionally to the product of the tensionand the square of the correlation length, thereby approach-ing a finite, nonzero limit. Both features of k are in accordwith what is found in the simulations. The magnitude of kobtained from the molecular theory is lower (ffiffiffiffiffiffiffiffiffiffiffiffiffik=p 0:13d) than in the simulations (ffiffiffiffiffiffiffiffiffiffiffiffiffik=p  0:47d; dashedline in Fig. 3).One of us (R. L. C. V.) acknowledges the DeutscheForschungsgemeinschaft for support (Emmy NoetherGrant No. VI 483/1-1).[1] C. Fradin, A. Braslau, D. Luzet, D. Smilgies, M. Alba, N.Boudet, K. Mecke, and J. Daillant, Nature (London) 403,871 (2000).[2] S. Mora, J. Daillant, K. Mecke, D. Luzet, A. Braslau, M.Alba, and B. Struth, Phys. Rev. Lett. 90, 216101 (2003).[3] J. Stecki and S. Toxvaerd, J. Chem. Phys. 103, 9763(1995).[4] R. L. C. Vink, J. Horbach, and K. Binder, J. Chem. Phys.122, 134905 (2005).[5] P. Tarazona, R. Checa, and E. Chacon, Phys. Rev. Lett. 99,196101 (2007).[6] R. Evans, Adv. Phys. 28, 143 (1979).[7] J. S. Rowlinson and B. Widom, Molecular Theory ofCapillarity (Clarendon, Oxford 1982).[8] F. P. Buff, R.A. Lovett, and F.H. Stillinger, Phys. 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Condens.Matter 11, 10 079 (1999); J. Kuipers and E.M. Blokhuis,J. Colloid Interface Sci. 315, 270 (2007).[21] E.M. Blokhuis and D. Bedeaux, Mol. Phys. 80, 705(1993).PRL 101, 086101 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending22 AUGUST 2008086101-4

Description of the fluctuating colloid-polymer interface

Description of the fluctuating colloid-polymer interface

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