Langmuir 1992,8, 2889-2897
2889
Time-Dependent Self-Diffusion of Brownian Particles with Square Well Interaction B. Cichockit Institute of Theoretical Physics, Warsaw University, Hoza 69, 00-618 Warsaw, Poland
B. U. Felderhof Institut fiir Theoretische Physik A, R. W. T.H. Aachen, Templergraben 55,5100 Aachen, Germany Received April 1, 1992 We study the time-dependentself-diffusion coefficientfor a semidilutesuspension of sphericalBrownian particles interacting via a square well potential. Under neglect of hydrodynamic interactions, we derive an exact expression for the Fourier transform of the memory function, valid to f i s t order in the volume fraction. We analyze the time dependence of the memory function and the corresponding distribution of relaxation times. We compare with simple approximations involving a small number of parameters which can be determined by calculation. 1. Introduction
The mean square displacement of a selected particle in a suspension of spherical Brownian particles does not simply grow linearly with time in proportion to a constant diffusion coefficient. Rather it shows a more complicated behavior due to direct and hydrodynamic interactions. One may define a time-dependentself-diffusioncoefficient from the time derivative of the mean square displacement.14 In this article we study this time-dependent coefficient for a suspension of Brownian particles interacting via a square well potential. Under neglect of hydrodynamicinteractions we derive an exact expression for the memory function, valid to first order in the volume fraction. The same model was studied earlier by Van Den B r o e ~ k . ~ The diffusion coefficient displays an interesting dependence on the parameters specifying the square well. We find that the mean relaxation time, characterizing the rate of change of the memory function, drops dramatically as the depth of the well is increased. It reaches a minimum near the point where the steady state perturbed distribution function has no long range tail. At the same time the distribution of relaxation times develops a rich structure. We show that a four-pole approximations provides a reasonably accurate descriptionof the spectrum and the corresponding memory function. A number of experimentalsystems may be modeled with a square well interaction potential, for example silica dispersions with added polymer,4*'v8 or proteinlwater solutions."" The interaction between silica particles with a hard core and stabilizingpolymer coatingmay be modeled + Also at the Institute of Fundamental Technological Fiesearch, Polish Academy of Sciences, Swigtokrzyska 21, 00-049 Warsaw, Poland. (1) Ackerson, B. J. J. Chem. Phys. 1978,69,684. (2) Dieterich, W.; Peschel, I. Physica A 1979, 95,208. (3) Pueey, P. N.; Tough, R. J. A. In Dynamic Light Scattering and Velocimetry: Applications of Photon Correlation Spectroscopy;Pecora, R., Ed.;Plenum: New York, 1982. (4) Pueey, P. N. In Liquids, Freezing and Class Transition;Hansen, J. P., Leveaque, D., Zinn-Juetin, J., Eds.;North-Holland: Amsterdam, 1991; p 763. (5) Van Den Broeck, C. J. Chem. Phys. 19811,84,4248. (6) Cichocki, B.; Felderhof, B. U. J. Chem. Phys. 1992,96,9055. (7) Aaakura, 5.;Ooeawa, F. J. Polym. Sci. 19118, 33, 183. (8) de Hek, H.; Vrij, A. J. Colloid Interface Sci. 1981,84, 409. (9) Thomson, J. A.; Schurtenberger,P.;Thurston,G.M.; Benedek,G. B. Roc. Natl. Acad. Sci. U S A . 1987,84,7079.
with a square well potential with a deep and narrow well.12 In the limit where the depth becomes infinite and the width goes to zero such that the product of the Boltzmann factor and width remains constant, one speaks of sticky spheres.I3 In the sticky sphere limit the rather complicated expressions derived for the general case simplify considerably. 2. Self-Diffusion
We consider a suspension of identical spherical Brownian particles of radius a. At average number density n the volume fraction occupied by particles is #J = (4r/3)na3. In the dilute limit the particles diffuse independently with bare diffusion coefficient given by the Stokes-Einstein expression Do = k~TIGrqa,where q is the shear viscosity of the solvent. At higher density the diffusionis influenced by direct and hydrodynamic interactions. These are incorporated in the generalized Smoluchowski equation which describes the time evolution of the probability distribution of c ~ n f i g u r a t i o n s . ~ ~ ~ J ~ To study self-diffusionwe consider the Brownian motion of a selected particle labeled 1. Ita mean square displacement is defined as W(t) = i([Rl(t) - R,(0)12)
(2.1)
where the angle brackets indicate an average over the equilibrium ensemble, and the time evolution is governed by the adjoint Smoluchowskioperator. The rate of change of the mean square displacement defines a time-dependent self-diffusion coefficient D,(t) = d Wldt
(2.2)
The short-time diffusion coefficient DsS and the longtime diffusion coefficientDsLare defined from the limiting values (10) Schurtenberger,P.;Chamberlin,R. A.; Thurston,G.M.; Thomson, J. A.; Benedek, G. B. Phys. Rev. Lett. 1989, 63, 2064. (11) Broide, M. L.; Berlaud, C. R.; Pande, J.; Ogun, 0.0.; Benedek, G. B. R o c . Natl. Acad. Sci. U S A . 1991,88,5660. (12) Jansen, J. W.; de Kruif, C. G.; Vrij, A. J. Colloid Interface Sci. 1986,114,492. (13) Baxter, R. J. J. Chem. Phys. 1968,49,2770. (14) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge Univeriity Press: Cambridge, 1989.
0743-7463/92/2408-2889$03.00/00 1992 American Chemical Society
Cichocki and Felderhof
2890 Langmuir, Vol. 8, No. 12, 1992
D Z = D,(O),
= Ds(-)
D :
(2.3)
If hydrodynamic interactions are neglected,then the shorttime diffusion coefficient DsS is identical with the bare diffusion coefficientDo. We definethe relaxationfunction I L S W by DS(t) = D,L + pS(t)
(2.4)
This function decays to zero from its initial value ps(0) = Dss - DsL. The diffusion coefficient Ds(t) may be related to the time-dependent self-scattering function observed in dynamic light scattering from tracer particles. The Fourier transform of the scattering function Fs(q,t) leads to the definition of a wavenumber-and frequency-dependentselfdiffusion coefficient Ds(q,w). It may be expressed a s 3 s 4 Ds(q,m) + fiS(q,w)
DS(q,w)
(2.5)
with a memory function &(q,w) which tends to zero at high frequency. It may be shown that the Fourier transform of the relaxation function &(w) = JOmeiYtps(t) dt
(2.6)
is related to the memory function at zero wavenumber by 1
cLs(w) = j - p S ( O , O )
- fis(0,O)l
(2.7)
;L&)
+ L1-l U'I)
(2.8)
where L is the adjoint of the Smoluchowskioperator, and U'1 = LR1 is the velocity of the selected particle on the Smoluchowski time scale." The time scale characterizing the rate of change of the memory function is defined by18
TM
rs(z)
(2.15)
where r s ( z )is the Laplace transform of the function ~ s ( T ) (2.16)
with the variable z = -iWTM. Substituting eq 2.13, we fiid that I's(z) is given by the Stieltjes integral (2.17)
Thus I's(z) is analytic in the complexz plane with a branch cut along the negative real axis. From the normalization properties (2.14) it follows that rye) = 1,
rs(z) = i / z
as z
-
m
(2.18)
The fact that ps(u) is positive imposes an important constraint on the behavior of the function rs(z). 3. Semidilute Suspension with Square Well Interaction In the followingwe evaluatethe function I's(z) explicitly for a semidilute suspension of particles with square well interaction potential. The hydrodynamicinteraction will be neglected. To first order in the volume fraction the memory function, given by eq 2.8, may be expressed as
The memory function is given by the expre~sion'~J~ fis(0,w) = 1/3(u'1.(iw
(DZ - )D :
2Cl,(O,w) = DoaS(wM
(3.1)
with a dimensionless coefficienta s ( w ) which may be found from the solution of the pair Smoluchowskiequation. We have shown elsewherelg that the coefficient may be calculated from a one-dimensional integral as(0)= Jomx2gfUdx
(3.2)
where x = r/2a is the dimensionlesspair distance andg(x) is the low density radial distribution function From the identity
g(x) = exp[-Bu(2ax)I (2.10)
we find that it may also be expressed as
We write the relaxation function as
for interaction potential u(r). Furthermore, f(x) is the radial part of the perturbed pair distribution function, and U(x) is a given function expressed in terms of the interactions. In the absence of hydrodynamic interactions f ( x ) satisfies the radial differential equation d -
It follows from the general properties of the Smoluchowski equation that the dimensionless function T S ( T ) may be expressed ad8 (2.13)
g - 2 g f -a"x"gf = 2x2 & d x x gdx dx with the variable a2 = -2iwa2/Do
J;F
du = 1
(2.14)
(3.4)
(3.5)
For vanishing hydrodynamic interaction the function U( x ) is given by
with a spectral density ps(u) which has been normalized to l m p s ( u )du = 1,
(3.3)
(3.6)
At zero frequency the coefficient as(0) may be calculated conveniently from the electricpolarizabilitya of a related dielectric problem according to the relationP6
The relaxation function decays monotonically with time.
Ita Fourier transform may be expressed as (15) Hanna, S.; Hess, W.; Klein, R. Physica A 1982, 111, 181. (16) Cichocki, B.; Felderhof, B. U. Phys. Rev. A 1990,42, 6024. (17) Felderhof, B. U. Physica A 1987,147, 203. (18) Cichocki, B.; Felderhof, B. U. Phys. Rev. A 1991,44,6551.
(3.7)
where the coefficient Js is given by the integral (19) Cichocki, B.; Felderhof, B.
U.J. Chem. Phys. 1992,96,6978.
Self-Diffusion of Brownian Particles
Langmuir, Vol. 8, No. 12,1992 2891 (3.8)
In the absence of hydrodynamicinteractions JS is related to the second virial coefficient B2 by the identity B2 = (4d3)a3Js. In the followingwe consider specifically the square well potential
u(r) =
{
for 0 < r < 2a for 2a < r < 2b for2b
