Dynamics of colloidal particles

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1 Dynamics of colloidal particles J. M. Méndez-Alcaraz. Depto. de Física, CINESTA-IPN, Av. IPN 258, Col. San Pedro Zacatenco, 736 México D. F., Mexico. O. Alarcón-Waess. Depto. de Física y Matemáticas, UDLA, Puebla, Sta. Catarina Mártir, Cholula, 7282 Puebla, Mexico. We present a general method for constructing effective one particle dynamics in colloidal systems, which results from a contraction of the description of colloidal mixtures based on the many particles Langevin and Smoluchowski equations. In order to illustrate its applicability, we calculate the long-time self-diffusion coefficient of charged particles, as well as its electrophoretic mobility. The accuracy of our approaches is pointed out by comparison with available experimental data. I. INTRODUCTION The experimental access to complex fluids is always limited. For example, a clean salty aqueous suspension of polystyrene spheres contains obviously polystyrene spheres and water molecules, but also ions dissociated from the polystyrene molecules on the surface of the spheres, called counterions, and at least two different species of salt ions. Thesizeofthefirstonesrangesfromaboutsometens of nanometers, up to several microns. The other particles are of subnanometer dimensions. Observing the suspension by means of light scattering, only the polystyrene spheres are detected by the experiment, because they are the only particles able to affect the photons of micrometers wave lengths. The explanation of the experimental results requires therefore the theoretical formulation of effective models able to capture the effects induced by the unobserved particles on the physics of the observed ones. In our example, it leads to effective screened coulombic pair interaction potentials between polystyrene particles, containing parameters related to the solvent molecules and small ions. As explained in another paper in this book, it results from a contraction of the description of colloidal mixtures based on the integral equations theory of simple liquids[1]. The same integral equations theory accounts for the structural properties of the observed suspension by taking the effective interaction potentials as input[2]. The explanation of the dynamic properties of the observed particles is more complicated. They are immersed in an invisible medium composed by, for example, the unobserved solvent molecules and small ions, with which they exchange energy and momentum. Their dynamics is therefore non-conservative, which is a physical feature noncontained in the effective pair interaction potentials. In this paper we present a general method for constructing effective one particle dynamics in colloidal systems, which results from a further contraction of the description of colloidal mixtures based on the many particles Langevin and Smoluchowski equations. In order to illustrate its applicability, we calculate the long-time self-diffusion coefficient of charged particles, as well as its electrophoretic mobility. The accuracy of our approaches is tested by their comparison with available experimental data. II. CONTRACTION OF THE DESCRIPTION AND EFFECTIE DYNAMICS Let us think further about a salty aqueous suspension of polystyrene spheres. It means, a liquid mixture of five components or species; the polystyrene spheres or macroions (species m), the ions dissociated from the polystyrene molecules in the spheres or counterions (species c), the positive salt ions (species +), the negative salt ions(species ) and the water molecules (species w). Globally, the suspension is electroneutral. Locally, however, the system is charged and the particles interact between them with coulombic potentials. The species w looks like electric dipoles (molecular model). In a first contraction of the description, the water molecules can be included only through the dielectric constant of the solvent, leading to the primitive model (PM) for the pair interaction potential u (r) between two particles of species α and β separated by the distance r: u (r) = + for r<a α + a β = Q αq β 1 for r a α + a β, (1) r with α, β = m, c, +,. Here, the hard core prevents the interpenetration and Q α (a α ) is the charge (radius) of a particle of species α. A second contraction of the description can be carried out from the PM, so that the small ions

2 no longer appear as separated species. This leads to the one component model (OCM) or effective screened coulombic interaction between the macroions[1, 3, 4]: 2 u mm (r) = + for r<2a m µ = Q2 m e κa m 2 e κr 1+κa m r for r 2a m, (2) with κ 2 = 4π k B T α=c,+, n α Q 2 α (3) being the square inverse Debye-Hückel screening length and n α = N α / the number density of particles of species α in the volume. The structure of the suspension can be calculated either in the PM by solving the Ornstein-Zernike (OZ) equation[5] h (r) =c (r)+ Z n γ c αγ (r )h γβ ( r r )dr (4) γ=m,c,+, or in the OCM by solving the more contracted OZ equation Z h mm (r) =c mm (r)+n m c mm (r )h mm ( r r )dr, (5) together with an appropriate closure relation[5], by using (1) or (2) as input for the potential, respectively. Here, c (r) is the direct correlation function, h (r) =g (r) 1 the total correlation function and g (r) the partial radial distribution function. Once g (r) is known, all structural functions and thermodynamical properties of the system can be evaluated, each of them in their respective level of description[2, 5]. The macoion-macroion structure functions obtained by following both routes are the same, equations (1) and (4) or equations (2) and (5), since that is a necessary condition in the construction of the effective interaction potentials[1, 6]. In the PM, the motion of particle i of species α is described by the many particles Langevin equation m α dv α i dt = β j ζ ij vβ j iu + f α i, (6) which corresponds to the second Newton s law for a particle of mass m α moving with velocity vi α at time t. The force is the superposition of the friction P β Pj ζ ij v β j due to the solvent, the interaction iu with the other particles and the Gaussian random force fi α (t) related to ζ ij through a fluctuation-dissipation theorem (FDT). The greek indices refer to the species (m, c, + or ) and the latin ones to the particles (running from 1 to the total number of particles of the corresponding species N m, N c, N + or N ); particle i is of species α and particle j of species β. The quantity U stands for the total interaction potential energy, assumed to be pair-wise additive. An alternative description of the collective motion of the N = P α N α particles in the system is the many particles Smoluchowski equation P N = α,β i,j i D ij j P N + β ( j U) P N, (7) which corresponds to the Liouville equation (after integrating over all variables of the solvent molecules and the momentum of the polystyrene spheres and small ions[7]) for the normalized distribution function P N = P γ N (r 1, r 2,, r N ; t) of finding particles 1 of species α in r 1,, and particle N of species γ in r N at time t. The diffusion tensor D and the friction tensor ζ are connected by the Einstein relation D = k B T ζ 1. Both tensors contain the effects induced by the invisible solvent onto the dynamics of the observed particles, known as hydrodynamic interactions (HI). We take here the most simple approximation for them; ζ ij = ζ α δ ij 1 and D ij = D α δ ij 1, where ζ α is given by the Stokes expression 6πηa α, η being the shear viscosity of the solvent. In order to derive a closed equation for vi α from equations (6) and (7), we make use of reduced distribution functions ρ (n) γ (r 1, r 2,, r n ; t) =N α (N β δ βα ) (N γ δ γα δ γβ ) R P N dr n+1 dr N of finding any particle of species

3 α in r 1,, and any particle of species γ in r n at time t, regardless of the rest of the particles. Carrying out this integration, equation (6) takes the continuous form dv1 α m α = ζ dt α v1 α 1 Z Z [ α 1 u (r 12 )] ρ (2) N (r 1, r 2 ; t)dr 1 dr 2 + f1 α. (8) α β In addition, equation (7) gives rise to a hierarchy of evolution equations involving all reduced distribution function. The member of the hierarchy corresponding to ρ (2) (r 1, r 2 ; t) becomes ρ (2) = D α 2 1ρ (2) + D β 2 2ρ (2) h i +βd α 1 ( 1 u (r 12 )) ρ (2) +βd α 1 Z γ +βd β 2 Z γ h + βd β 2 ( 1 u αγ (r 13 )) ρ (3) γ dr 3 ( 2 u (r 21 )) ρ (2) ( 2 u βγ (r 23 )) ρ (3) γ dr 3. (9) We now pass from the partial to the total time derivative by adding the streaming term v α 1 1 ρ (2) ; dρ (2) dt = v α 1 1 ρ (2) + ρ(2) i. (1) This completes the scheme, which can be recapitulated as following: First, an approximation for ρ (3) γ ρ (2) and vα 1 is necessary (we will come back to this point). Second, equation (1) is solved for ρ (2) 3 in terms of in terms of v1 α. Third, this solution is put in to (8), leading to an equation for the motion of any particle of species α alone. Due to the stochastic nature of the previous equations, the complete procedure can be carried out by applying the Medina-Noyolas s contraction procedure of non-markov processes[8], which leads to the generalized Langevin equation: Z t dv α m α = ζ dt α v α ζ α (t t )v α (t )dt + F α + f α. (11) Here, F α (t) is a colored random force related to ζ α (t) through a FDT. The contraction also results in explicit expressions for ζ α (t), as shown down. Equation (11) refers only to one particle of species α explicitly. The rest of the particles of the same species were contracted, as well as all particles of the other species. Therefore, equation (11) represents an effective one particle dynamics, where the rest of the systems is included only in the friction coefficient ζ α (thewatermolecules)andinits increment ζ α (t) (the macroions, counterions and salt ions). In the following section we apply this scheme in order to calculate the long-time self-diffusion coefficient of polystyrene spheres. III. LONG-TIME SELF-DIFFUSION COEFFICIENT We can take the bulk value n α = N α / for ρ (1) α (r 1 ; t) and rewrite the quantity ρ (2) (r 1, r 2 ; t) ρ (1) α (r 1 ; t)ρ (r 2 /r 1 ; t) as n α n β (r 12 ; t), whereρ(r 2 /r 1 ; t) is the conditional reduced distribution function of finding particle 2 of species β in r 2 at time t, given the particle 1 of species α is in r 1 atthesametime. Also,n β (r 12 ; t) =n eq β (r 12)+δn β (r 12 ; t) is the instantaneous local density of particles of species β around a central particle of species α with equilibrium value n eq β (r 12) and fluctuations δn β (r 12 ; t) around the equilibrium state. Taking into account these definitions, equation (8) reads dv α m α = ζ dt α v α + Z ( u (r)) δn β (r; t)dr + f α. (12) β We take for ρ (2) / one of the simplest approaches available in the literature; the modified Fick s approximation[8]: δn β =(D α + D β ) n eq β (r) γ Z δn γ (r ; t) hδn β (r)δn γ (r )i eq dr, (13)

4 where hδn β (r)δn γ (r )i eq means the equilibrium density fluctuations correlation function. Equation (13) is, ad hoc, linear in δn γ and represents a particular case of the general equation (9). We also linearize equation (1) in the small quantities v β and δn β ; dδn β dt = v β n eq β + δn β. (14) After contracting our equations following references [8] and [9] in order to generate an effective dynamics for a polystyrene sphere, we get ζ m = = ζ m 6π 2 ζ m (t)dt α=m,c,+, n α (1 + D α /D m ) h 2 mα(q)q 2 dq. (15) Here, h mα (q) is the Fourier transform of h mα (r). Once ζ m is determined, de long-time self-diffusion coefficient DS L of polystyrene particles follows immediately from the velocity self-correlation function hv m (t) v m ()i /3, because its time integral is related to the mean-square displacement. Therefore, one gets or D L S D m = " 1+ n m 12π 2 D L S = h 2 mm(q)q 2 dq + 1 6π 2 k B T ζ m + ζ m (16) α=c,+, n α (1 + D α /D m ) 4 h 2 mα(q)q 2 dq# 1, (17) with substitution of (15) into (16). The second term in the parenthesis accounts for self-friction (SF, the friction due to the interaction between the tracer particle and the other macroions). In order to evaluate it, we numerically calculate h mm (r) within the OCM and the thermodynamically self-consistent Rogers-Young (RY) approximation[1] g mm (r) =exp[ βu mm (r)] 1+ exp [(h mm(r) c mm (r)) f(r)] 1 f(r). (18) Here, the value of the mixture parameter α in f(r) =1 e αr is fixed by demanding the equality of the isothermal compressibility calculated from the compressibility equation of state and from the virial equation of state[1, 11]. The third term in the parenthesis of equation (17) accounts for electrolyte friction (EF, the friction due to the interaction between the tracer particle and the counterions and salt ions). We evaluate it within the PM and the Debye-Hückel limit; h mα (r) = 1+β Q mq α e κa m e κr Θ(r a m ) 1, (19) 1+κa m r which considers point-like counterions and salt ions. Here, Θ(r a m ) is the Heaviside s step function. Finally, DS L/D m remains " DS L = 1+ n Z # 1 m D m 12π 2 h 2 mm(q)q 2 dq + KQ 2 n α Q 2 α m, (2) 1+a m /a α α=c,+, with K = h 3(k B T ) 2 κ (1 + κa m ) 2 /2πi 1. (21) We now use equation (2) in order to evaluate the long-time self-diffusion coefficient in systems with different volume fractions ϕ m =4πn m a 3 m/3 of polystyrene spheres and salt concentrations n s (in moles/dm 3 ), keeping Q m =5e and a m =5nm constant. For the small ions we take Q c = Q + = Q =e + and the radii a c = a + =.18nm and a =.12nm, obtained from the value of the ionic mobility of Na + and Cl in dilute aqueous solution by means of the Stokes friction. We also take T =3Kand =78.2. The results are shown in Figure 1. Let us first

5 5 FIG. 1: The figure shows macroions and salt concentration effects on the reduced long-time self-diffusion coefficient of macroions with charge 5e and radius 5nm. The used salt was NaCl. consider salt concentration effects, i.e., the case of only one macroion immersed in an electrolyte solution (ϕ m =). Electroneutrality leads to n c =and n + = n = n s. For very low salt contents the diffusion coefficient is the corresponding to a free particle. By increasing n s the diffusion coefficient is reduced until a minimum is reached at κa m =1. Then, DS L increases again, by increasing n s,towardsd m. The curve looks like an inverted bell. For macroions concentration effects we have that the electroneutrality leads to n c =5n m and n + = n = n s. We find in the region with κa m À 1 that the screening of the interaction between macroions is so large that they behave like isolated particles in an electrolyte solution. In the region with κa m 1 however the macroion-macroion interaction is decisive; the diffusion coefficient decreases with ϕ m. From the scale of Figure 1, it becomes clear that DS L decreases as ln ϕ m, which agrees qualitatively with the available experimental measurements[12]. Moreover, in the transition region (κa m 1) a non-monotonic behavior of DS L as function of n s is found for ϕ m This is a consequence of the competition between self-friction and electrolyte friction, which has been recently observed[12]. For ϕ m & 1 3 the diffusion coefficient is always an increasing function of n s. I. ELECTROPHORETIC MOBILITY Letusturntothedescriptionofthemotionofamacroionundertheinfluence of an external constant electric field E. Equation (6) and (7) must be modified by adding the terms Q α E and β (Q β E) P N to their right hands, respectively[7]. It results in the terms Q α E and βd α Q α E 1 ρ (2) βd βq β E 2 ρ (2),whichmustbeinturn added to the right hands of equations (8) and (9), respectively. The contraction of the modified equations leads to

6 6 the generalized Langevin equation in the external electric field E: m α dv α dt = ζ α v α + Z t Z t ζ α (t t )v α (t )dt + Q α E Q α (t t )Edt + F α + f α. (22) The electrophoretic mobility of polystyrene spheres µ m follows from the steady state average of this equation; µ m µ m = 1+ Q m/q m 1+ ζ m /ζ m, (23) with ζ m ( Q m ) given by the time integral of ζ m (t) ( Q m (t)). The quantity µ m stands for the free particle mobility Q m /ζ m. Making use of the bulk value n α = N α / for ρ (1) α (r 1 ; t) and rewriting the quantity ρ (2) (r 1, r 2 ; t) ρ (1) α (r 1 ; t)ρ (r 2 /r 1 ; t) as n α n β (r 12 ; t), but this time with n β (r 12 ; t) =n ss β (r 12)+δn β (r 12 ; t) the instantaneous local density of particles of species β around a central particle of species α with steady state value n ss β (r 12) and fluctuations δn β (r 12 ; t) around the steady state, equation (12) is modified by adding the term Q α E to its right hand. Moreover, the appropriate extension of the modified Fick s approximation succeeds by changing the superscript equilibrium (eq) by steady state (ss), and by adding the term β (D α + D β ) Q β E n ss β (r) to the right hand of equation (13). After contracting our new equations following references [8] and [13] in order to generate an effective dynamics for a polystyrene sphere, we get µ m µ m = 1+I mm + P α=c,+, Q αi mα /Q m 1+I mm /2+ P α=c,+, I mα/ (1 + D α /D m ), (24) with I = n β 6π 2 h 2 (q)q 2 dq. (25) The friction increment is the same as in the previous section, but this time we also have an expression for the electrokinetic charge. In the derivation of (24) we assumed that the statistical averages in the stationary state can be taken as in equilibrium, which is expected to be valid for weak electric fields. By the way, we also assumed an homogeneous fluid. Making the same approximations for the structure functions as in the previous section, we get µ m µ m 1+I mm + KQ m Pα=c,+, = n αq 3 α 1+I mm /2+KQ 2 m Pα=c,+, n αq 2 α/ (1 + a m /a α ), (26) with K given by (21). Let us now evaluate the electrophoretic mobility of polystyrene spheres for the same parameters as in Figure 1. The results are shown in Figure 2. Let us first consider salt concentration effects, i.e., the case of only one macroion immersed in an electrolyte solution (ϕ m =). Electroneutrality leads to n c =and n + = n = n s. For very low salt contents the mobility is the corresponding to a free particle. By increasing n s, the mobility is reduced until a minimum is reached at κa m =1 and then µ m increases again towards µ m. The curve looks like an inverted bell. Actually, it coincides with that for D L S, because the electrokinetic charge is zero for symmetric salt and ϕ m =. Regarding the response of the electrophoretic mobility to the macroion concentration we first have that the electroneutrality leads to n c =5n m and n + = n = n s. We find in the region with κa m À 1 that the screening of the interaction between macroions is so large that they behave like isolated particles in an electrolyte solution. In the region with κa m 1 however the macroion-macroion interaction is decisive; the mobility increases with ϕ m. From the scale of Figure 2, it becomes clear that µ m increases as ln ϕ m, which agrees with the available experimental measurements[14]. Moreover, the inverted bell becomes more asymmetric due to this enhancement, and its minimum simultaneously moves to larger values of κa m. Its new position is roughly given by the value of n s where the curves for ϕ m 6=meet the one for ϕ m =. From a visual inspection of Figure 2, it seems to occur when n s [in moles/dm 3 ] ϕ m (note that this criterion is valid for the magnitudes, but not for the unities). Surprisingly, it is exactly what Weber observed in his experiments[15]. We also got similar results for other values of Q m. In particular, the position of the minimum does not appreciably depend on the value of Q m. This explains why we can compare with the results of Weber without to know anything about the charge on his particles.

7 7 FIG. 2: The figure shows macroions and salt concentration effects on the reduced electrophoretic mobility of macroions with charge 5e and radius 5nm. The used salt was NaCl.. DISCUSSION AND CONCLUSIONS We presented here a general method for constructing effective one particle dynamics in colloidal systems, which results from a contraction of the description of multicomponent systems based on the many particles Langevin and Smoluchowski equations. Its applicability is illustrated by calculating the long-time self-diffusion coefficient of charged particles, as well as its electrophoretic mobility. The comparison of our results with available experimental data confirms that our theoretical scheme captures macroion and salt concentration effects with qualitative accuracy. This is however a great goal, because some of the explained phenomena were captured here for the first time in a theory. In particular, the shift of the minimum of µ m, as function of salt content, with increasing macroion concentration and the simultaneous enhancement of µ m. Although not shown in this paper, our theoretical scheme can also accounts for other new phenomena, like the saturation of µ m for very structured systems[14] and the macroion charge inversion in presence of asymmetric salt[16]. These results will be published elsewhere[17]. Acknowledgments ery helpful discussions with M. Medina-Noyola and R. Weber are gratefully acknowledged. This work was supported by CONACyT-Mexico (grants E and 1748P-E). [1] R. Castañeda-Priego, A. Rodríguez-López, M. D. Carbajal-Tinoco, P. González-Mozuelos and J. M. Méndez-Alcaraz, Proceedings of the 1st. Mexican Meeting on Mathematical and Experimental Physics, edited by E. Díaz-Herrera, Kluwer

8 Academic/Plenum Press (22). [2] J. M. Méndez-Alcaraz, M. Chávez-Páez, B. D Aguanno and R. Klein, Physica A 22, 173 (1995). [3] P. González-Mozuelos and M. D. Carbajal-Tinoco, J. Chem. Phys. 19, 1174 (1998). [4]M.Medina-NoyolaandD.A.McQuarrie,J.Chem.Phys.73, 6279 (198). [5]J.P.HansenandI.R.McDonald,Theory of Simple Liquids (Academic Press, London, 1986). [6] J. M. Méndez-Alcaraz and R. Klein, Phys. Rev. E 61, 495 (2). [7]T.J.MurphyandJ.L.Aguirre,J.Chem.Phys.57, 298 (1972). [8]M.Medina-Noyola,FaradayDiscuss.Chem.Soc.83, 21 (1987). [9] J. M. Méndez-Alcaraz and O. Alarcón-Waess, Physica A 268, 75 (1999). [1] F. J. Rogers and D. A. Young, Phys. Rev. A 3, 999 (1984). [11] J. M. Méndez-Alcaraz, B. D Aguanno and R. Klein, Langmuir 8, 2913 (1992). [12] D. Rudhardt and P. Leiderer, private communications. [13] O. Alarcón-Waess and R. M. elasco, J. Chem. Phys. 11, 511 (1994). [14]M.Evers,N.Garbow,D.HessingerandT.Palberg,Phys.Rev.E57, 6774 (1998). [15] M. Deggelmann, T. Palberg, M. Hagenbüchle, E. E. Maier, R. Krause, C. Graf and R. Weber, J. Colloid Interface Sci. 143, 318 (1991). [16] M. Lozada-Cassou and E. González-Tovar, J. Colloid Interface Sci. 239, 285 (21). [17] J. M. Méndez-Alcaraz and O. Alarcón-Waess, manuscript in preparation. 8

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