Long-time self-diffusion of charged spherical colloidal particles in parallel planar layers.

Long-time self-diffusion of charged spherical colloidal particles in parallel planar layers Claudio Contreras-Aburto, César A. Báez, José M. Méndez-Alcaraz, and Ramón Castañeda-Priego Citation: The Journal of Chemical Physics 140, 244116 (2014); doi: 10.1063/1.4884822 View online: http://dx.doi.org/10.1063/1.4884822 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/24?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Short-time self-diffusion coefficient of a particle in a colloidal suspension bounded by a microchannel: Virial expansions and simulation J. Chem. Phys. 135, 164104 (2011); 10.1063/1.3653941 Long-time self-diffusion of charged colloidal particles: Electrokinetic and hydrodynamic interaction effects J. Chem. Phys. 127, 034906 (2007); 10.1063/1.2753839 Self-diffusion of rodlike and spherical particles in a matrix of charged colloidal spheres: A comparison between fluorescence recovery after photobleaching and fluorescence correlation spectroscopy J. Chem. Phys. 121, 7022 (2004); 10.1063/1.1791631 Long time self-diffusion in suspensions of highly charged colloids: A comparison between pulsed field gradient NMR and Brownian dynamics J. Chem. Phys. 114, 975 (2001); 10.1063/1.1326909 Lubrication corrections for three-particle contribution to short-time self-diffusion coefficients in colloidal dispersions J. Chem. Phys. 111, 3265 (1999); 10.1063/1.479605

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THE JOURNAL OF CHEMICAL PHYSICS 140, 244116 (2014)

Long-time self-diffusion of charged spherical colloidal particles in parallel planar layers Claudio Contreras-Aburto,1 César A. Báez,2 José M. Méndez-Alcaraz,2 and Ramón Castañeda-Priego1 1

División de Ciencias e Ingenierías, Campus León, Universidad de Guanajuato, Loma del Bosque 103, 37150 León, Guanajuato, Mexico 2 Departamento de Física, Cinvestav, Av. IPN 2508, Col. San Pedro Zacatenco, 07360 México, D. F., Mexico

(Received 7 February 2014; accepted 11 June 2014; published online 26 June 2014) The long-time self-diffusion coefficient, DL , of charged spherical colloidal particles in parallel planar layers is studied by means of Brownian dynamics computer simulations and mode-coupling theory. All particles (regardless which layer they are located on) interact with each other via the screened Coulomb potential and there is no particle transfer between layers. As a result of the geometrical constraint on particle positions, the simulation results show that DL is strongly controlled by the separation between layers. On the basis of the so-called contraction of the description formalism [C. Contreras-Aburto, J. M. Méndez-Alcaraz, and R. Castañeda-Priego, J. Chem. Phys. 132, 174111 (2010)], the effective potential between particles in a layer (the so-called observed layer) is obtained from integrating out the degrees of freedom of particles in the remaining layers. We have shown in a previous work that the effective potential performs well in describing the static structure of the observed layer (loc. cit.). In this work, we find that the DL values determined from the simulations of the observed layer, where the particles interact via the effective potential, do not agree with the exact values of DL . Our findings confirm that even when an effective potential can perform well in describing the static properties, there is no guarantee that it will correctly describe the dynamic properties of colloidal systems. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4884822] (iv)

I. INTRODUCTION

Colloidal suspensions are very important systems in soft condensed matter, since they are found in many challenging situations in nature and used in an enormous amount of applications.1 They also involve open fundamental questions, such as the physical mechanisms that lead to the arrested states of matter,2 and are excellent models for some molecular systems, with the advantage of having length and time scales orders of magnitude larger than the molecular ones.3 It is in this context that we report here on a special case of model systems formed by charged spherical colloidal particles in parallel planar layers.4 There are several good reasons to study these model systems, for example: (i)

They are open and continuous in two dimensions, but closed and discrete in the third dimension. This promotes the rise of geometrical effects never seen before in either open or closed systems.4 (ii) The equations describing their static microstructure are mathematically equivalent to the equations that describe the microstructure of colloidal mixtures in open spaces. Therefore, theoretical schemes originally developed for homogeneous mixtures can be used to describe the layered systems, as it is the case of the schemes formulated to evaluate the effective interaction potentials.4 (iii) The equations describing particle dynamics in the layered systems are also mathematically equivalent to the corresponding equations in homogeneous colloidal mixtures. 0021-9606/2014/140(24)/244116/11/$30.00

They allow for a convenient control of thermodynamics, structure, and dynamics by means of a single geometrical parameter: The separation between layers.4 (v) They may serve as model systems for several interesting systems occurring in nature, such as the lipid bilayers.5–8 (vi) Their experimental realization is possible. For instance, colloidal particles trapped by parallel linear optical tweezers constitute a one-dimensional version of the model.9 In a previous article, we reported on the (i), (ii), and (iv) points of this list.4 It is shown therein that the static structure can be conveniently controlled by changing the separation between layers. The mechanism behind this control is of pure geometrical nature: When the layers are very close to, or very far from each other, they behave like two-dimensional systems. At intermediate separations, however, they behave like three-dimensional systems.10 The geometrical effect is strong enough to induce an effective attraction between like-charged particles, and even to provoke their aggregation.4 This phenomenon was explained in terms of the effective interaction between the particles in one of the layers, induced by the particles in the other layers, after implementing the contraction of the description formalism in the framework of the integral equations theory of simple liquids.11, 12 The latter implementation was made possible thanks to the feature listed in point (ii) of the foregoing list. In the present work, we extend the analysis to the study of the long-time self-diffusion coefficient. As in our previous

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work,4 the systems are simulated using Brownian dynamics (BD) with a Yukawa-like potential and, as a first approximation, the solvent-mediated hydrodynamic interactions are ignored. The static structure functions and the long-time selfdiffusion coefficients are calculated from the simulation data. We explore the cases of two and three layers, with one or two species of particles. The choice of parameters is explained in Sec. II. Since we frequently quote the aforementioned article, in the following we refer to it as P1. The simulation results show that DL is strongly controlled by the layer spacing. However, we will show that this phenomenon cannot be described by means of an effective interaction alone, as it was the case in P1 for the static structure. We came to this conclusion after running Brownian dynamics computer simulations for only one layer with the particles interacting through the effective interaction potential obtained from the contraction of the description formalism,11, 12 which in P1 was shown to accurately reproduce the static structure.4 In addition to our simulation studies of the full and of the contracted systems, we implement a simplified modecoupling theory (SMCT) for long-time self-diffusion in parallel planar layers. This implementation is feasible thanks to the characteristic denoted in point (iii) of the previous list, as it is demonstrated in Sec. III of this work. The SMCT produces results that are in qualitative agreement with the simulations, except in the cases of layers of particles with opposite charge sign, where more sophisticated models and solution schemes are required to properly account for the strong correlations between oppositely charged particles. In the cases where we find satisfactory agreement between the SMCT and the simulation results, a simple rescaling of the predicted excess friction can turn the SMCT results into good quantitative agreement with the simulations. Our aim in implementing the SMCT is twofold: First, it serves as an independent dynamical theory to calculate the long-time self-diffusion coefficient (note, however, that the SMCT needs as input the static structure functions obtained from the simulations). Second, the SMCT has been used with success in the study of both self- and collective transport properties in homogeneous mixtures;13 here, we show that the layered system constitutes another example in which the SMCT can be straightforwardly applied. This paper is organized as follows: In Sec. II, the model system, notation details, and the essentials of the calculation of the self-diffusion coefficients and the correlation functions from the Brownian dynamics computer simulations are briefly introduced. Section III explains the simplified mode-coupling theory. In Sec. IV, results are presented and discussed for the long-time self-diffusion coefficient corresponding to several cases of two and three layers, with one or two particle species. Our concluding remarks are summarized in Sec. V. II. MODEL SYSTEMS, NOTATION, AND SIMULATION DETAILS

The model systems consist of q equally spaced parallel planar layers with p species of particles. The distance between adjacent layers is denoted by the symbol . A schematic representation of a three layer system is drawn in Figure 1 of P1. The layers are assumed to be parallel to the xy-plane. In order

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FIG. 1. The scaled mean square displacement, Wα (t)/d 2 of Case I, is shown as a function of t/τ for the three separations κ: 0, 0.54, and 1.80. The orange line corresponds to the ideal diffusion case. The lines of reduced opacity represent the asymptotes that describe long-time self-diffusion.

to explain our notation let us first use the capital letters A, B, . . . = 1, . . . , q as superscripts to denote the layers, and the letters a, b, . . . = 1, . . . , p as subscripts to label the species. Thus, for instance, nA a (defined as the thermodynamic limit of the quotient NaA /S), represents the surface number density of NaA particles of species a in layer A which has the surface area S. Similarly, uAB ab (R) is the pair potential between a particle of species a in layer A and a particle of species b in layer B, which are separated by the distance R. The pairs (A, a), (B, b), . . . can now be replaced by the Greek letters α, β, . . . = 1, . . . , qp in the form of subscripts, as in nα and uαβ (R), keeping the meaning of these quantities. For instance, with 3 layers and 2 species the Greek subscripts have 6 possible values. As it is shown in P1 and in Sec. III, with this notation the equations describing the static structure and the dynamics of the layered model systems become mathematically identical to the corresponding equations describing homogeneous colloidal mixtures. The screened Coulomb potential,14, 15 uαβ (R) = Kα Kβ

exp(−κR) , R

(1)

is used for the interaction between particles. Here, R is the distance between the centers of the particles, Kα are the electrostatic coupling parameters, and κ is the Debye-Hückel screening parameter. The distance Rij between the centers of particle i of species a in layer A and particle j of species b in layer B is given by Rij = [(xi − xj )2 + (yi − yj )2 + (zi − zj )2 ]1/2 . It can be written as Rij = [rij2 + (zA − zB )2 ]1/2 , where rij is the magnitude of the projection of the vector Rij on the xy-plane, and zA and zB are the positions of layers A and B, respectively. Since zA and zB are fixed parameters in the simulations, which are unambiguously indicated by the Greek subscripts, the notation uαβ (r) ≡ uαβ (R) is used through this work.


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Besides and the screening length κ −1 , the lengths dα qp = and d = ( α=1 nα )−1/2 are used to interpret the results. Here, dα is the average distance between particles of species a in layer A, and d estimates the same quantity for all particles in the system. Both lengths are related by dα = d/(xα )1/2 , where xα = nα /n is the molar fraction of“species” qp α, i.e., of particles of species a in layer A, and n = α=1 nα is the total number density. Throughout this work the screening parameter is fixed to the value κd = 1.0. This is a typical value of suspensions from moderately to strongly correlated chargestabilized colloidal particles.12, 16, 17 The time scale used in the simulations is τ = d2 /D0 , where D0 is the single-particle diffusion coefficient. As usual, the energy scale is the thermal energy kB T, with kB and T being the Boltzmann’s constant and the absolute temperature, respectively. Following the procedure explained in detail in P1, BD computer simulations of the model systems are performed in order to calculate the partial radial distribution functions gαβ (r), the partial structure factors Sαβ (k), and the mean square displacement Wα (t). From Wα (t), the long-time selfdiffusion coefficient −1/2 nα

DαL = lim

t→∞

Wα (t) 4t

FIG. 3. Schematic representation of the nearest neighbor ring around a particle for different layer separations. When κ = 0 the nearest neighbor ring is very compact. When κ → ∞ it expands and particles are able to diffuse faster as compared to κ = 0. In R2, there is a nearest neighbor ring contribution coming from the same layer and a second contribution coming from the adjacent layer.

(2)

is obtained. Here, k is the two-dimensional wave number reciprocal to r, and t stands for time. The analyzed cases are the following:

r Case I: Two layers, numbered 1 and 2, and one species; the Greek subscripts get the values 1 and 2 for the particles in layer 1 and in layer 2, √ respectively. The system √ parameters are: K1 = K11 = 50, K2 = K12 = 50, n1 = n11 = 1/2, and n2 = n21 = 1/2. The partial densities are the same in both layers. The number of particles used in the simulations to calculate the structure functions (diffusion coefficients) are N1 = 7200(1250) and N2 = 7200(1250). The results are shown in Figs. 1–5.

FIG. 2. Number of nearest neighbors in cases I (black line), II (red lines), 1 = N 2 , because the layers are identical. In and IV (blue lines). In case I, Nnb nb 1 2 1 > N 2 , through case II Nnb = Nnb , because n1 < n2 . Note that in case II Nnb nb 1 = N 3 , because the outer layers are identical. In most of R2. In case IV Nnb nb 2 is much larger than N 1 = N 3 in part of R2. case IV n2 < n1 = n3 , and Nnb nb nb

r Case II: Two layers, numbered 1 and 2, and one species; the Greek subscripts get the values 1 and 2 for the particles in layer 1 and in layer 2, respectively. √ The system parameters are: K1 = K11 = 50, K2 √ = K12 = 50, n1 = n11 = 2/5, and n2 = n21 = 3/5. The partial densities are different; layer 1 is more diluted. The number of particles used in the simulations to calculate the structure functions (diffusion coefficients) are N1 = 5760(1000) and N2 = 8640(1500). The results are shown in Fig. 6. r Case III: Two layers, numbered 1 and 2, and two species, numbered 1 and 2, but in layer 1 there are only particles of species 1, and in layer 2 there are only particles of species 2; the Greek subscripts get the values 1 and 2 for the particles in layer 1 and in layer 2, respectively. The system parameters are: K1

FIG. 4. The partial potentials of mean force (case I), evaluated at the position of the first peak of the partial radial distribution functions, are plotted as a function of the layer separation.



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FIG. 5. Reduced long-time self-diffusion coefficient DαL /D 0 as function of κ for case I. Since both layers are identical (they have the same species and densities) their DαL /D 0 overlap. The BD results are shown with blue lines, the MCT results with cyan lines, and the BD results for the contracted systems with red lines. A schematic representation of the system is drawn in the top part of the figure.

FIG. 7. Reduced long-time self-diffusion coefficients D1L /D 0 (solid lines) and D2L /D 0 (dashed lines) for case III. The BD results are shown with blue lines, the MCT results with cyan lines, and the BD results for the contracted systems with red lines. A schematic representation of the system is drawn in the top part of the figure.

√ √ = = − 15, K2 = K22 = 50, n1 = n11 = 3/10, and n2 = n22 = 7/10. The number of particles used in the simulations to calculate the structure functions (diffusion coefficients) are N1 = 4320(750) and N2 = 10 080(1750). The results are shown in Fig. 7. r Case IV: Three layers, numbered 1, 2, and 3, and one species; the Greek subscripts get the values 1, 2, and 3 for the particles in layer 1, in layer 2, and in layer 3, √ respectively. The system K1 = K11 √ parameters 3are:√ 2 = 50, K2 = K1 = 50, K3 = K1 = 50, n1 = n11 = 3/8, n2 = n21 = 2/8, and n3 = n31 = 3/8. The outermost layers (1 and 3) have the same density and the layer in the middle (layer number 2) is more di-

luted. The number of particles used in the simulations to calculate the structure functions (diffusion coefficients) are N1 = 5400(937), N2 = 3600(625), and N3 = 5400(938). The results are shown in Fig. 8. r Case V: Three layers, numbered 1, 2, and 3, and two species, numbered 1 and 2, but in layers 1 and 3 there are only particles of species 1, and in layer 2 there are only particles of species 2; the Greek subscripts get the values 1, 2, and 3 for the particles in layer 1, 2, and 3, respectively. The √ 1 = K = 50, K = K22 system parameters are: K 1 2 1 √ √ = − 30, K3 = K13 = 50, n1 = n11 = 0.2812, n2 = n22 = 0.4688, and n3 = n31 = 0.2500. All partial

FIG. 6. Reduced long-time self-diffusion coefficients D1L /D 0 (solid lines) and D2L /D 0 (dashed lines) for case II. The BD results are shown with blue lines, the MCT results with cyan lines, and the BD results for the contracted systems with red lines. A schematic representation of the system is drawn in the top part of the figure.

FIG. 8. Reduced long-time self-diffusion coefficients D1L /D 0 = D3L /D 0 (solid lines) and D2L /D 0 (dashed lines) for case IV. The BD results are shown with blue lines, the MCT results with cyan lines, and the BD results for the contracted systems with red lines. A schematic representation of the system is drawn in the top part of the figure.

K11


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FIG. 9. Reduced long-time self-diffusion coefficients D1L /D 0 (solid lines), D2L /D 0 (dashed lines), and D3L /D 0 (dotted lines) for case V. The BD results are shown with blue lines, the MCT results with cyan lines, and the BD results for the contracted systems with red lines. A schematic representation of the system is drawn in the top part of the figure.

densities are different. The number of particles used in the simulations to calculate the structure functions (diffusion coefficients) are N1 = 4049(703), N2 = 6751(1172), and N3 = 3600(625). The results are shown in Fig. 9. The layers are always numbered from the bottom to the top (this is just a convention, without any physical meaning). All quantities are scaled with d, kB T, and τ = d2 /D0 . The values of the system parameters mentioned in the previous list are therefore given in reduced units (for a simpler notation, we have omitted any decoration in the corresponding symbols, such as n∗α = nα d 2 ). According to this, the values of the electrostatic coupling constants are given already in units of (dkB T)−1/2 , so that the values of the pair potential are scaled with kB T. As mentioned before, throughout this work the scaled screening parameter is fixed to the value κd = 1.0. In order to estimate the long-time limit of Wα (t), the simulations of the full system are allowed to run until 25τ with steps of 10−4 τ . Error bars for the figures were obtained from the standard deviation of the results obtained from 96 different simulation runs, using 96 different seeds in the random number generator. However, they are not shown in the figures because they are all of the same size or even smaller than the depicted symbols. The structure functions gαβ (r) and Sαβ (k) are not shown in this paper; they are calculated just to be used as inputs in the mode-coupling scheme and for the evaluation of the effective interaction potentials according to the procedure explained in P1. We will discuss more about this point in Sec. IV. After the effective interaction potential between the particles in one of the layers is obtained, it is used in BD simulations of the effective one-layer system. These latter simulations were performed with a total of 4225 particles in the simulation box of the effective one-layer system. They were allowed to run until 20τ with steps of 10−4 τ . Error bars were

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obtained from 12 different simulations runs and are always smaller than the depicted symbols in the figures. Two final remarks about our model system and calculations are now given. The screened Coulomb potential, as written in Eq. (1), does not include the size of the colloidal particles. Particle size can be introduced by employing a piecewise pair potential with hard-core and screened Coulomb parts. Many specialized simulation techniques have been developed in order to deal with particle size effects.18 In this study, we are more interested in analyzing dimensionality effects (or for that matter, the effects of changing ), and do not consider particle size effects in our simulations. We are well aware of the fact that particle size effects play an important role in determining transport properties.13 However, it is fair to say that the probability of contact is very small in systems of particles with equal charge sign that have Kα values not too small and κ value not too large. In P1, we verified that the structure functions of Yukawa-like particles for various sets of parameters (nα , Kα , κ), with hard-core corresponding to a particle diameter σ /d = 0.357 (calculated by means of Monte Carlo simulations and the hypernetted chain integral equation), and without hard-core (calculated by means of BD simulations), are in good agreement. The same is true for the structure functions of the cases presented in this work, simply because in all cases gαα (r σ ) = 0. Thus, since any other diameter value such that σ /d 0.36 would also be a valid choice, any concrete reference to a particular diameter value is unnecessary in this work. Notwithstanding this, the horizontal axis of Figs. 2 and 5–9 has a grid divided in intervals of width 0.36, as in the figures presented in P1 for the case σ /d = 0.357. The fact that the diffusion coefficient at infinite dilution, Dα0 , varies as the inverse of the particle diameter is not a cause of conflict, because we present and analyze results for the dimensionless quantity DαL /Dα0 . We should remark that in those systems where all layers have the same charge sign the layer spacing can be varied starting from κ = 0 (because the probability of particle contacts is very small). However, in the cases III and V it should be varied starting from some κ > 0, let us say κ = 0.36. We do this in order to avoid the possibility of an overlapping configuration in cases III and V, because the overlapping configuration in such cases cannot be studied without including particle size effects. III. THEORY

The central quantity that describes the dynamics of a particle of species α in an homogeneous colloidal mixture is the self-intermediate scattering function19, 20 Gα (k, t) =

Nα ık·[Rα (t)−Rα (0)] 1 i i e , Nα i=1

(3)

where k is the wave-vector of magnitude k, and Rαi the position vector of the ith particle of species α, with Cartesian components xiα , yiα , and ziα . The brackets represent the equilibrium ensemble average. In the layered systems, there is no


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movement of the particles in the stacking direction; the only components of the vectors k and Rαi entering in Gα (k, t) are their projections on the plane of the layers. Therefore, Gα (k, t) becomes a function defined over the two-dimensional vector k = (kx , ky ) alone. This means that the equations describing the dynamics of a particle in the layered systems have the same mathematical structure of the ones for homogeneous colloidal mixtures, as far as the notation described in Sec. II is used. This allows for the use of well-developed theories that describe the Gα (k, t) in multi-component systems.19 Following Nägele and Dhont,19 DαL is given by DαL 1 = lim lim , z→0 k→0 1 + Mαirr (k, z) Dα0

(4)

where Dα0 is the single-particle diffusion coefficient, and Mαirr (k, z) is the Laplace transform of the irreducible memory function associated to Gα (k, t). The excess friction ζαL due to the inter-particle interactions, appearing in the generalized Einstein relation DαL =

kB T , ζα0 + ζαL

(5)

is immediately identified as ζαL = lim lim Mαirr (k, z) . z→0 q→0 ζα0

(8)

and F (k, t) ≈ exp[−k 2 D 0 t S−1 (k)] · S (k) ,

(9)

where S is the matrix with the partial static structure factors Sαβ (k) = Fαβ (k, t = 0) as elements, which are related to C through the Ornstein-Zernike equation C = 1 − S−1 . Substituting Eqs. (8) and (9) in Eq. (7), the transcendental equation ∞ 1 DαL = 1 + dk k D0 4π nα 0 −1

−1 L Dα S (k) + 1 · (S (k) − 1) × (S (k) − 1) · D0 αα

(10) is obtained, where the term containing the integral can be identified with the reduced excess friction ζαL /ζ 0 . The partial static structure factors needed as input are directly obtained from the BD computer simulations. Equation (10) is then iteratively solved up to a precision of 6 digits, which is achieved after about 10 iteration steps.

(6) IV. RESULTS

= kB T /Dα0 is the free-particle friction coefficient associated to Dα0 . We assume that in the model systems ζα0 = ζ 0 , and thus Dα0 = D 0 . Solvent-mediated hydrodynamic interactions are being neglected, and that is the reason why Dα0 ap-

Here, ζα0

pears in Eq. (4), instead of the short-time self-diffusion coefficient DαS . Equations (4)–(6) cast the problem of calculating DαL /D 0 (or equivalently ζαL /ζ 0 ) as the problem of evaluating the limit indicated in Eq. (6). The MCT provides us with a method to obtain Mαirr (k, t), and the limit leads to (written here in two-dimensions)19 ∞ ζαL D0 = dk k 3 ζ0 4π nα 0

∞ × C (k) · dt Gα (k, t) F (k, t) · C (k) , 0

tems, one can make the approximations19 Gα (k, t) ≈ exp −k 2 DαL t

αα

(7) where C and F are matrices of elements Cαβ (k) = (nα nβ )1/2 cαβ (k) and Fαβ (k, t), respectively. Here, cαβ (k) is the Fourier transform of the partial direct correlation function cαβ (r), and Fαβ (k, t) is the partial dynamic structure factor.20 This equation needs the structure functions C (k), Gα (k, t), and F (k, t) as input. The exact calculation of these input functions is a very cumbersome task which must be faced when dealing with highly correlated systems, such as those near the glass transition.21, 22 In this work, we deal exclusively with liquid systems and implement a simplified solution scheme that has been successfully applied to describe both the self-23 and collective dynamics of colloids and ions.13, 24 For moderately dense sys-

In this section, the results for the reduced long-time selfdiffusion coefficients, DαL /D 0 shown in Figs. 5–9, are reported as functions of κ for the five cases mentioned in Sec. II. They are displayed according to the following general context:

r The dimensionless product κ was selected for the hor-

izontal axes, because the screening length κ −1 is a measure of the range of the potential and the layers become uncorrelated when κ 1. r The values of κ are distributed in three regimes: Regime R1 where 0 ≤ κ ≤ 0.18, regime R2 where 0.18 ≤ κ ≤ 1.44, and regime R3 where 1.44 ≤ κ ≤ ∞. In R1, the layers behave pretty much like a twodimensional system with number density n. In R3, the layers become uncorrelated and they are expected to behave, again, like independent two-dimensional systems, each with the corresponding number density nα . In R2, there are large cross correlations between distinct layers, and if we count the number of nearest neighbors to a particle, we will see that there are some of them on the adjacent layers besides the ones belonging to the same layer. This plays a very important role in the dynamics of each particle, as explained further below. r The results for D L /D 0 obtained from BD simulations α of the full system are displayed in the figures with blue lines using solid, dashed, and dotted strokes. They are not displayed for 0 ≤ κ ≤ 0.36 in the cases where colloids with opposite charge sign are present (see last remarks in Sec. II). In that regime, the correlation between colloids of opposite charge sign is expected to


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r

r

r

r


overcome the dimensional effect we are trying to describe. The SMCT predictions for the DαL /D 0 are shown in the figures with cyan lines using solid, dashed, and dotted strokes, and in the same regimes where the BD results are shown. The effective interaction potential between the particles in one of the layers is obtained by contracting the other layers of the description, according to the procedure described in P1. This is done for each one of the layers. Those effective potentials are then used in a BD simulation of the corresponding effective one-layer system (the observed layer). The results for DαL /D 0 obtained from these simulations are shown in the figures with red lines using solid, dashed, and dotted strokes, and in the same regimes where the BD results for the full systems are shown. After a detailed comparison between the static structure functions obtained from both simulation routes (the one with all layers and the bare potentials, and the contracted one with only the observed layer and the effective interaction), we observe that they are the same in R3. This is not surprising, since in R3 the layers are uncorrelated and the contraction of the description is superfluous. This is also the reason for the excellent agreement between the blue and red lines in Figs. 5–9 throughout R3. In R1, however, the structure functions are very different. The approximations used in P1 do not work in this regime, where the contraction of some layers roughly corresponds to the contraction of a part of the particles in homogeneous twodimensional systems. This is a new issue in the contraction of the description formalism implemented in P1, which demands our attention, but is not analyzed in this work. Finally, the structure functions show a satisfactory agreement in R2, in the same sense as in P1. The differences between blue and red lines in Figs. 5–9 throughout R2 are, therefore, imputable to effects beyond those contained in the effective interactions. R2 is most important to show the dimensionality effects behind the behavior of DαL /D 0 as function of κ. The results for R1 and R3 are shown only for completeness.

Some figures for the mean-square displacement, the number of nearest neighbors, and the potential of mean-force evaluated in the first peak of the radial distribution function are shown as an aid to explain the results for DαL /D 0 . Note, however, that in these additional figures the color code differs from the one used to display DαL /D 0 .

A. Two layers and one species (case I)

The static structure of this case is described in Figures 2–5 of P1. Since both layers are identical (they have the same species and densities), their results for the mean square displacement and the long-time self-diffusion coefficient overlap in Figs. 1–5. Figure 1 shows the scaled mean square displacement, Wα (t)/d 2 , as function of t/τ for the

J. Chem. Phys. 140, 244116 (2014)

three layer separations κ = 0, 0.54, and 1.80, which are located in R1, R2, and R3, respectively. The orange line corresponds to the ideal diffusion case (or diffusion at infinite dilution). A log-log scale was selected in order to resolve the short (blueberry background), transient (aqua background), and long-time (teal background) regimes in one plot. The following analysis applies only to the long-time regime. The lines with reduced opacity in Fig. 1 represent the long-time asymptotes of the mean square displacement. They all have unit slope (as the orange line) and are shifted downwards with respect to the ideal diffusion curve. The more shifted downwards the asymptote is, the smaller the corresponding value of Log(4DαL /D 0 ). This shows that DαL /D 0 must be a nonmonotonic function of κ. As expected, the particles diffuse slower when κ = 0 than in the case κ = 1.80, because the layers overlap when κ = 0 and are practically uncorrelated in the case κ = 1.80. However, it is surprising that the particles diffuse even slower in the case κ = 0.54. This rather unexpected behavior can be explained after counting the number of nearest neighbors, and analyzing the interactions of a particle with the particles in the annuli of nearest neighbors, as it has been done by other authors.16, 25 α Let Nnb be the number of nearest neighbors. It has two contributions which consist of the surface integrals of the quantities n1 g11 (r) (same layer contribution) and n2 g12 (r) (adjacent layer contribution), over the region of their first peak. The same layer contribution (n1 g11 ) is considered for all values of κ. The adjacent layer contribution (n2 g12 ) is considered only in the R1 and R2 regimes and is ignored in R3. The α as a function of κ for case I is shown with the number Nnb α is about 6 in R1 black line in Fig. 2. As one might expect, Nnb and in R3. In R2, however, it is much larger, and even reaches the value of 12. This is what we call dimensionality effect, in R2 there are additional particles belonging to the adjacent layer which are located in the region of nearest neighbors. This leads to a slower diffusion in R2 than in R1 and R3. The involved processes are schematically illustrated in Fig. 3. A more physically insightful route to explain the dimensionality effect is provided by the partial potentials of min /kB T be the value mean force (PPMF), wαβ /kB T . Let wαβ that the PPMF takes at the position of the first peak, rfp , of the corresponding partial radial distribution function. Each PPMF shows a potential well around rfp , of depth quantimin fied by the absolute value of wαβ /kB T . The larger the value min of |wαβ |/kB T , the more attractive are the average interactions between a particle and the particles located in the nearest neighbor ring. Fig. 4 shows the PPMF corresponding to case I. The blue (red) curve tells us that as κ increases from zero, the average interactions between a particle and its nearest neighbor particles in the same (adjacent) layer become less (more) attractive and, as a result, the ring of nearest neighbors in the same (adjacent) layer slightly expands (contracts) as κ continues to increase. This process is schematically illustrated with the arrows pointing out wards (inwards) in Fig. 3. This can be interpreted as an induced repulsive min min (κ[ + ]) − w11 (κ))/(κ), which slightly force (w11 expands the ring in the same layer in order to make room for the particles in the adjacent nearest neighbor ring. In fact,


244116-8


if one looks at the cross radial distribution function, g12 (r), one can see that in R2 a particle is effectively surrounded by particles in the adjacent layer as much as it is surrounded by particles in the same layer. Thus, the particle can escape from the ring of first neighbors when it is able to overcome the additional energy difference due to the presence of the nearest neighbor particles in the adjacent layer. The red line in Fig. 4 indicates that in R2 a particle is strongly bounded to the cage created by the particles in the ring of first neighbors of the adjacent layers and, in consequence, in this regime the long-time self-diffusion coefficient should decrease (because the long-time self-diffusion coefficient is related to the process in which a particle and its nearest neighbors reconfigure). For layer separations near κ 0.5, the additional energetic obstacle starts to fade, explaining thus the recovery of min /kB T reaches its DαL /D 0 . As κ continues increasing, w11 constant value corresponding to κ = ∞, which of course is above the value corresponding to κ = 0, and simultaneously min /kB T goes to zero. w12 An additional way to explain the dimensionality effect could be the following. Consider the straight line describing the short-time behavior of Wα (t), and the asymptote corresponding to the long-time behavior. The time at which both lines intercept each other, λ2 /D0 , defines thus the length scale λ (they would not cross of course in Fig. 1, but in a linear scale). A plot of λ versus κ (not shown) is expected to behave as the position of the first peak of the radial distribution function shown in Fig. 3 of P1; behaving as n−1/2 in R1 and R3, and as n−1/3 in R2 (the latter corresponds to the natural scaling of three-dimensional repulsive colloidal systems). It is expected because the particles pass from the short to the long-time regime when they leave the cage of first neighbors, whose size is roughly given by the position of the first peak of the radial distribution function shown in Fig. 3 of P1. All results of case I are summarized in Fig. 5, where the reduced DαL is shown as function of κ. A non-monotonic behavior of DαL /D 0 is observed in R2, with a minimum located around κ ≈ 0.5. The separation between layers can therefore be used to control the diffusion of the particles, in the same way that it can be used to control the structure of the layers (see P1). The SMCT predictions displayed in Fig. 5 show more or less the correct qualitative behavior, but they considerably overestimate the values of DαL /D 0 . We note that a factor of 2 in front of the integral expression for the excess friction in Eq. (10) leads to a very good quantitative agreement between the blue and cyan lines. This is also true for Figs. 6 and 8, but only partially for Figs. 7 and 9 which correspond to the cases of layers of particles with opposite charge sign. We do not have an explanation for this numerical factor, other than it has to be related to the non-exponential relaxation terms ignored in the approximations implicit in Eqs. (8) and (9). The BD results for the effective one-layer system shown with red lines in Fig. 5 are not even in qualitative agreement with the results from simulations of the full system neither in R1 or R2. Although the static structure is satisfactorily reproduced by the effective interaction potentials in R2, as shown in P1, the self-diffusion coefficients are not reproduced. Note that the red line in Fig. 5 does not constitute an

J. Chem. Phys. 140, 244116 (2014)

attempt to explain the dimensionality effects leading to the non-monotonous behavior of DαL /D 0 ; what these results tell us is that the effective potential does not contain enough information to reproduce the dynamics of the effective systems. We close this subsection with some comments related to this last issue. After the contraction of the description is performed, some particles of the full system are not longer explicitly included in the equations describing the physics of the remaining particles. For the latter ones, the static structure is reproduced by the effective interaction potentials, but their dynamics also depends on the interactions mediated by the effective medium. The effective medium, however, changes after the contraction from being a pure solvent to incorporate the contracted particles too. Those indirect interactions between the remaining particles show up as memory effects, which are not contained in the effective direct interactions. The short time dynamics of an homogeneous binary colloidal mixture, for instance, is given by26 2 ∂Fαβ (q, t) =− Kαγ (q)Oγ δ (q)Fδβ (q, t), ∂t γ ,δ=1

(11)

where the matrix O of elements Oγ δ (q) is the inverse of the matrix of static partial structure factors S, and Kαγ (q) is the first cumulant of Fαγ (q, t). Both quantities are given in terms of the radial distribution functions gαβ (r).27 After contracting species 2 from Eq. (11), the evolution equation for F11 (q, t) in the effective system of species 1 is obtained ∂F11 (q, t) = −D11 (q)F11 (q, t) ∂t t M11 (q, t − t )F11 (q, t )dt , + where Dαδ (q) =

2

γ =1

(12)

0

Kαγ (q)Oγ δ (q) and

M11 (q, t) = D12 (q)D21 (q)e−D22 (q)t .

(13)

Equation (12) is neither optimized nor suitable for further calculations, but it undoubtedly shows that a contraction of the description of the short-time dynamics may give rise to a new short- and intermediate-time dynamics with exponential memory for the remaining particles. The parameters of the memory kernel, however, depend on the diffusivity tensors of the contracted species and, therefore, are not contained in the effective interaction potential. The difference between the blue and red lines in Fig. 5 throughout R2 may be imputable to this kind of issues and, therefore, represents a challenge for future attempts to develop a contraction of the description formalism with dynamic equations. In the following cases, we do not show more figures for either the mean-square displacement or the potential of meanforce, since they all manifest the same qualitative behavior observed in this case. B. Two layers and one species (case II)

Case II is very similar to case I, but in this case n1 = 2/5 and n2 = 3/5. Fig. 6 shows D1L /D 0 (solid lines) and D2L /D 0


244116-9


(dashed lines) as functions of κ. Again, a non-monotonous behavior is observed in the blue lines inside R2, but the particles in the more diluted layer diffuse slower than the particles in the denser layer (this ceases to happen when the layers become uncorrelated at κ ≈ 1). This apparently counterα as intuitive result can be explained after evaluating Nnb function of κ. We obtained that the particles in the more dilute layer have more nearest neighbors in R2 than the particles in the denser layer, as it is shown with the red lines in Fig. 2. Once again, the predictions of the SMCT severely overestimate the values of DαL /D 0 . However, a good quantitative agreement is obtained between the SMCT predictions and the simulations results if the excess friction is increased by a factor of two, as it happened in case I. It is remarkable that the SMCT (cyan lines) and the contracted description (red lines) also predict a faster diffusion in the denser layer in part of R2. The huge difference between the red and blue lines in part of R2 (for κ 1) also indicates that the effective interaction potential alone is not able to reproduce the dynamics of the contracted system.

C. Two layers and two species (case III)

In case III, the layer concentrations are also different (n1 = 3/10 and n2 = 7/10), and there are two species with differing electrostatic both in magnitude and √ coupling constants √ sign (K1 = − 15 and K2 = 50). Fig. 7 shows the results for D1L /D 0 (solid lines) and D2L /D 0 (dashed lines) as functions of κ. There is no surprise here; the denser the layer the slower the diffusion of its particles. But not all is that simple; although particles of species 2 are only slightly affected by the presence of particles of species 1, since species 2 is more concentrated and stronger interacting than species 1, the latter are strongly coupled to the former in 0.36 κ 0.54, so that they diffuse almost together. This is the reason for the small values of D1L /D 0 in that part of R2. Once again, the predictions of the SMCT severely overestimate the values of DαL /D 0 , and a good quantitative agreement is obtained if the excess friction is increased by a factor of two, as in cases I and II, but only when the layers become uncorrelated (i.e., only in R3). The difference between the red and blue lines in R2 for D1L /D 0 and κ 1 may also indicate that the effective interaction potential alone is not able to reproduce the dynamics of species 1. The dynamics of species 2 however is well captured, as shown by the dashed red line, because species 2 is only slightly affected by the presence of particles of species 1.

D. Three layers and one species (case IV)

In case IV, there are three layers with particles of only one species. The results for the outer layers are the same, because they are identical (n1 = n3 = 3/8). The inner layer is more diluted (n2 = 2/8). Figure 8 displays the reduced D1L = D3L (solid lines) and D2L (dashed lines) as functions of κ. The blue lines show a non-monotonous behavior of DαL in R2. As in cases I and II, the explanation for this behavior relies on a larger number of nearest neighbors through R2 than in R1

J. Chem. Phys. 140, 244116 (2014)

and R3, as it is shown with blue lines in Fig. 2. Additionally, in part of R2 the particles of the more diluted layer diffuse slower than the particles of the denser layers. Again, this happens because in R2 a particle in the more dilute layer has more nearest neighbors than a particle in the denser layers, as it is shown with the blue lines in Fig. 2. As found in the previous cases, the SMCT predictions overestimate the values of DαL /D 0 , and a good quantitative agreement is obtained if the excess friction is increased by a factor of two, as it happened in the cases I–III. Huge differences between red and blue lines are still found in R2. As in case II, the SMCT (cyan lines) and the contracted description (red lines) also predict a faster diffusion in the denser layer for part of R2. E. Three layers and two species (case V)

In case V, there are three layers, with two species of particles with √ differing charge both √ in magnitude and sign (K1 = K3 = 50 and K2 = − 30). All densities are different: n1 = 0.2812, n2 = 0.4688, and n3 = 0.2500. Figure 9 shows D1L /D 0 (solid lines), D2L /D 0 (dashed lines), and D3L /D 0 (dotted lines) as functions of κ. Although the layer in the middle is more concentrated than the other two layers, the particles in this layer display a diffusion coefficient curve which lies in between the other two curves, because the particles in this layer are less interacting. As in case III, there is a strong coupling between two colloids of opposite charge sign, which force them to diffuse almost together. The structural coupling in this kind of systems has been discussed in P1. The SMCT predictions, again, overestimate the selfdiffusion coefficients, and as in case III a scaling by a factor of two for the excess friction only leads to quantitative agreement when the layers are uncorrelated (i.e., in R3). For κ 1, the MCT predictions are even in qualitatively disagreement with the simulation results of the full system. Huge differences between the red and blue lines are still found in R2. V. CONCLUDING REMARKS

The long-time self-diffusion coefficient, DαL , of charged spherical colloidal particles on parallel planar layers was studied as a function of the dimensionless distance between adjacent layers, κ. To better understand the results, the values of κ were split in three regimes: 0 ≤ κ ≤ 0.18 or R1, 0.18 ≤ κ ≤ 1.44 or R2, and 1.44 ≤ κ ≤ ∞ or R3. Since κ is the inverse screening length, the layers become uncorrelated in R3. In R2, the layers are very correlated and on counting the number of nearest neighbors, Nnb , we found that an important part of Nnb comes from the particles in adjacent layers, besides the ones belonging to the same layer. From Brownian dynamics computer simulation results, the systems were found to behave like two-dimensional systems in R1 and in R3. The combined two-dimensional density in R1 is larger than the density of the separated layers. Therefore, DαL is larger in R3 than in R1. We expected to see a monotonous increase of DαL through R2, connecting the value


244116-10


in R1 with the value in R3. However, in most cases DαL is found to decrease until κ ≈ 0.5 and then to increase until reaching R3. This happened because the systems in R2 have an excess of nearest neighbors, with a larger number of particles in the ring of first neighbors. In all cases studied, the layers are in the fluid state, as it can be straightforwardly corroborated by applying the dynamic freezing criterion of colloidal liquids.28, 29 In P1, we demonstrated that the layer spacing is a convenient control parameter for the structure and effective interactions. In this work, the same conclusion has been extended to the long-time self-diffusion. The simplified mode-coupling theory qualitatively reproduced most of the simulation results, except for the systems made up of particles of opposite charge sign, where it only works in R3. The theoretical predictions for the other cases always overestimated the values of DαL , as compared to the simulation results. An heuristic improvement can be achieved by including a factor C > 1 in front of the integral in Eq. (10). For C ≈ 2, a good quantitative agreement is obtained in most cases. At this moment, we do not have an explanation for this numerical factor, other than it has to be related to the nonexponential relaxation terms ignored in the approximations implicit in Eqs. (8) and (9). The structural changes occurring in the model systems when the separation between layers is varied can be explained in terms of the effective interaction potential among the particles in one layer (the so-called observed layer), induced by the particles in the other layers.4 In order to investigate to what extent in R2 the non-monotonous behavior of the DαL of an observed layer can also be explained in the same way, we performed Brownian dynamics computer simulations of the observed layer with the particles interacting via the effective interaction potential. The latter was obtained from the contraction of the description formalism in the framework of the integral equations theory of simple liquids11, 12 after neglecting the difference between the bare and effective bridge functions.4 Although we corroborated that this approximation describes in a correct way the static structure of the observed layer in R2 and R3, the corresponding DαL is correctly described only in R3. This apparent inconsistency is explained in quite general terms if we realize that integrating out degrees of freedom from the static configurational partition function (this is essentially what we do when applying the contraction of the description formalism4 in the framework of the integral equations theory of simple liquids), is not the same as integrating out the same degrees of freedom starting from the solution of the Smoluchowski equation (this is equivalent to reduce the number of species entering explicitly in the memory equation).27 The latter process of contraction includes the hydrodynamic interactions via the hydrodynamic diffusivity tensors that do not affect the equilibrium static structure.27 Moreover, in the colloidal regime the hydrodynamic interactions can be considered to be instantaneous, because the suspending medium is just a molecular solvent in which the propagation of shear waves is much faster than the structural relaxation of the colloidal particles.27 However, when we contract some colloidal particles of the description, the suspending medium ceases to be just a molecular solvent because the

J. Chem. Phys. 140, 244116 (2014)

particles whose degrees of freedom are integrated out become part of the suspending medium. This means that after the contraction of the description the interactions mediated by the suspending medium become non-instantaneous. This was illustrated in Subsection IV A, with the emergence of memory effects after the contraction of the description. This effective non-instantaneous hydrodynamics is not contained in the effective potential, as it is illustrated with the results presented in this work. Thus, we confirm that even when an effective potential can perform well in describing the static properties, it is not guaranteed that it will correctly describe the dynamic properties. This is an important issue, since effective potentials are typically employed to study some dynamic properties, such as the dynamical arrest transition in, for example, colloidpolymer mixtures.30 To make a correct account of the dynamics in colloidal systems it could be necessary to consider the dynamic influence (e.g., through memory effects) of all the species that constitute the suspension. We should stress that the model of colloids in parallel planar layers can be used to study the effects of a structured substrate on the distribution of adsorbed particles, as well as the structural effects of a force field acting on a twodimensional colloidal suspension.31 Particularly, the colloidal particles trapped in parallel lines by linear optical tweezers9 are a one-dimensional version of our model, in which most of the quantities we have calculated in this paper, and in P1, could be experimentally measured.

ACKNOWLEDGMENTS

Financial support by PROMEP and CONACyT (through Grant Nos. 61418/2007, 102339/2008, 60595 and Red Temática de la Materia Condensada Blanda, as well as through scholarships 172596 and 209891) is gratefully acknowledged. The authors also thank the General Coordination of Information and Communications Technologies (CGSTIC) at Cinvestav for providing HPC resources on the Hybrid Cluster Supercomputer “Xiuhcoatl”, that have contributed to the research results reported within this paper. 1 D.

F. Evans and H. Wennerström, The Colloidal Domain, where Physics, Chemistry, Biology and Technology meet (Wiley-VCH, 1999). 2 N. E. Valadez-Pérez, Y. Liu, A. P. R. Eberle, N. J. Wagner, and R. Castañeda-Priego, Phys. Rev. E 88, 060302(R) (2013). 3 G. Nägele, The Physics of Colloidal Soft Matter (Institute of Fundamental Technological Research, Warsaw, 2004). 4 C. Contreras-Aburto, J. M. Méndez-Alcaraz, and R. Castañeda-Priego, J. Chem. Phys. 132, 174111 (2010). 5 W. Pfeiffer et al., Europhys. Lett. 8, 201 (1989). 6 D. Qu et al., Colloids Surf. A 303, 97 (2007). 7 G. Espinosa et al., Proc. Natl. Acad. Sci. U.S.A. 108, 6008 (2011). 8 Y. Gambin et al., Proc. Natl. Acad. Sci. U.S.A. 103, 2098 (2006). 9 M. C. Jenkins and S. U. Egelhaaf, J. Phys.: Condens. Matter 20, 404220 (2008). 10 Strictly speaking, the model of parallel planar layers consists of a set of two-dimensional systems, and there is no three-dimensional movement of the particles. However, the interaction between particles is threedimensional. It is in this sense that we say that the set of layers behaves as a three-dimensional system, in allusion to the remnant on the structural and dynamical properties from the three-dimensional nature of the pairpotential. 11 J. M. Méndez-Alcaraz and R. Klein, Phys. Rev. E 61, 4095 (2000).


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Castañeda-Priego, A. Rodríguez-López, and J. M. Méndez-Alcaraz, Phys. Rev. E 73, 051404 (2006). 13 C. Contreras Aburto and G. Nägele, J. Chem. Phys. 139, 134110 (2013). 14 E. J. W. Verwey and J. T. G. Overbeek, Theory of the Stability of Lyophobic Colloids (Elsevier, Amsterdam, 1948). 15 M. Medina-Noyola and D. A. McQuarrie, J. Chem. Phys. 73, 6279 (1980). 16 L. F. Rojas-Ochoa, R. Castañeda-Priego, V. Lobaskin, A. Stradner, F. Scheffold, and P. Schurtenberger, Phys. Rev. Lett. 100, 178304 (2008). 17 R. Castañeda-Priego, L. F. Rojas-Ochoa, V. Lobaskin, and J. C. MixtecoSánchez, Phys. Rev. E 74, 051408 (2006). 18 F. de J. Guevara-Rodríguez and M. Medina-Noyola, Phys. Rev. E 68, 011405 (2003). 19 G. Nägele and J. K. G. Dhont, J. Chem. Phys. 108, 9566 (1998). 20 J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, 1990).

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Nägele, J. Bergenholtz, and J. K. G. Dhont, J. Chem. Phys. 110, 7037 (1999). 22 A. J. Banchio, J. Bergenholtz, and G. Nägele, Phys. Rev. Lett. 82, 1792 (1999). 23 M. G. McPhie and G. Nägele, J. Chem. Phys. 127, 034906 (2007). 24 G. Nägele, M. Heinen, A. J. Banchio, and C. Contreras Aburto, Eur. Phys. J. Spec. Top. 222, 2855 (2013). 25 W. Schaertl and H. Sillescu, J. Stat. Phys. 77, 1007 (1994). 26 R. B. Jones and G. S. Burfield, Physica A 111, 562 (1982). 27 G. Nägele, Phys. Rep. 272, 215 (1996). 28 H. Löwen, T. Palberg, and R. Simon, Phys. Rev. Lett. 70, 1557 (1993). 29 H. Löwen, Phys. Rev. E 53, R29 (1996). 30 P. J. Lu et al., Nature (London) 453, 499 (2008). 31 S. Herrera-Velarde et al., J. Nanofluids 1, 44 (2012).


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