Faraday Discuss. Chem. SOC., 1987, 83, 69-73 Brownian Motion of Colloidal Crystals B. Ubbo Felderhof Institut f u r Theoretische Physik A, R. W. T. H. Aachen, Templergraben 55, 5100 Aachen, West Germany Robert B. Jones* Department of Physics, Queen Mary College, Mile End Road, London El 4NS We study the dynamics of colloidal crystals taking account of the retardation of the interactions due to the finite diffusivity of the counterions. It is shown that the retardation gives rise to electric dipole interactions proportional to the velocities of the macroions. As a consequence the damping rate of the overdamped transverse phonons tends to zero at small wavenumber. This agrees with observation in dynamic light scattering. 1. Introduction Charged spheres immersed in electrolyte solution can form colloidal crystals with a lattice distance much larger than the diameter of the spheres if the salt content of the solution is sufficiently low.' The dynamics of colloidal crystals differs from that of solid-state crystals due to the presence of the solvent fluid.The fluid causes friction which strongly dampens the motion. The long-range Coulomb forces between equally charged spheres are shielded by small counterions which diffuse fast in the ambient fluid. In this respect there is some similarity with the situation in metals where the conduction electrons move much faster than the ions, and the interactions between the ions may be treated in adiabatic approximation. We shall show that in colloidal crystals the finite diffusivity of the counterions gives rise to important effects.The dynamics of colloidal crystals has been studied experimentally in dynamic light-scattering experiments.* One observes dispersion curves of the damping rates of the overdamped longitudinal and transverse phonons very similar to the phonon disper- sion curves of solid-state crystals. Theoretically the lattice dynamics has been treated by analogy with solid-state theory, the only modification being the flow of the ambient fluid and the associated friction on the spheres. This theory was first developed by Hurd et al.,293 who assumed central-nearest-neighbour and next-nearest-neighbour interactions. In this theory the damping rate of the transverse modes TT(q) does not tend to zero as the wavenumber q tends to zero, in disagreement with the experimental data.In an earlier article4 we extended the theory by allowing more general potential interactions and showed that this approach cannot produce a damping rate T r ( q ) which tends to zero at small q. We suggested earlier4 that a qualitative feature is missing from the theory, namely the retardation of the interactions due to the finite diffusivity of the counterions. The idea was that in colloidal crystals the adiabatic approximation breaks down and that the instantaneous potential interactions must be replaced by retarded interactions. Here we show in a simplified model that the retardation has drastic effects and gives rise to long-range dipolar forces proportional to the velocities of the spheres. This makes a qualitative difference in the dispersion equation and leads to damping rates r,( q ) which tend to zero for small q, in agreement with experiment.6970 Brownian Motion of Colloidal Crystals 2. Retarded Interactions In order to derive the interactions we must consider the distortion of the Debye clouds about moving charged spheres. We begin by considering a static configuration of N spheres of radius a, with centres at (R, , . . . , R N ) and with charges { Z , e , . . . , Z , e } . Later we shall assume the positions {Rj} to be the sites of a regular lattice and the charges ( 2 , e ) to be identical, but at present we allow greater generality. We assume that on a macroscopic scale the centres {R,> are distributed uniformly in the container. Further we assume identical counterions of charge ze and diffusion coefficient D.In our model we replace the spatially varying density of the total Debye cloud in the static configuration by a constant no and describe the static potential by the linear Poisson- Boltzmann expression where K ? is the ionic strength K* = ( 4 m , z 2 e 2 / s k T ) and E is the dielectric constant of the solvent. Next we consider small oscillations of the spheres so that their centres have positions r,(t)=R,+s,,exp(-iot) j = 1 , ..., N. (2.2) Correspondingly, the perturbed counterion density varies harmonically with time. It satisfies the continuity equation -ion, + V * j , = 0 (2.3) where the current density is proportional to the gradient of the electrochemical potential n, pcL, = kT-+ ze+,. no This leads to the generalized diffusion equation n ze kT -ion, = DV'n, fL DO2+,.The electrostatic potential 4, satisfies the Poisson equation GV*C#I, = - 4 ~ z e n , + 417 1 pjw V 6 ( r - Rj) j (2.4) where p,, = Zjesj, is the dipole moment due to the displacement of the j t h sphere. The solution of the coupled equations (2.5) and (2.6) is given by with A2=K2-io/D. (2.8) The last term in eqn (2.7) shows that a moving sphere gives rise to a long-range electrostatic potential proportional to its velocity. The force of the ith sphere is given byB. U. Felderhof and R. B. Jones 71 where Eiu is the self-field acting on sphere i due to its own distorted Debye cloud and Ed, involves the positions of the other spheres. The self-field is given by K 2 I - 3& Es =-( K - >Pi- (2.10) and gives rise to ionic friction. The total friction coefficient of the sphere becomes K~ K - A 3~ iw &(w)=6mp0+- ----Zfe2= Jh+Ye,;(w) (2.11) where 7 is the viscosity of the solvent.The remaining electric field may be written where Uj., = -iosj, is the velocity of the j t h sphere. The tensor G, is found from the shielded Coulomb potential exp ( - A r ) r G,(A) = G(R; - R,, A ) , G ( r , A ) = VV (2.13) and F, is the dipole tensor -1 + 3 i i F , = F(Ri - R j ) F(r) =- (2.14) r3 * Owing to the relaxation effect the Debye cloud about a sphere lags behind when the sphere is moving, and this creates an electric dip01e.~ The effect is reduced by the electrophoretic effect, which has been left out of the above calculation and should be included as a convective flow term in eqn (2.5).However, despite the electrophoretic effect, long-range electric dipole interactions remain present, and these drastically affect the dynamics of the system. In the following we study the dynamics on the basis of the above equations. 3. Colloidal Crystal Dynamics We specialize to the case where the charges { z j e } are all equal and the positions {Rj} are located at the sites of a regular lattice. We have shown previously4 that the characteristic damping rates follow from the dispersion equation liwl - M(q, w ) H ( q , w ) l = 0. Miq, 0) = Mob, o)+Mc(q, 0) (3.1) (3.2) The 3 x 3 mobility matrix M( q, w ) is given by where M,(q, w ) is independent of the lattice structure and M,(q, w ) involves a lattice sum of hydrodynamic interactions.The first matrix Mo( q, o) incorporates a mean-field treatment of the fluid flow. It is given by where uc is the volume of a unit cell and a' = -iup/V, with p the mass density of the solvent. The second matrix in eqn (3.2) is well approximated by M c ( q , 0) = - 5 h ' V 3 K ( 4 ) (3.4)72 Brownian Motion of Colloidal Crystals where 4 is the volume fraction occupied by spheres and K ( q ) is a dimensionless lattice sum evaluated by Hurd et al.3 The matrix H(q, w > in eqn (3.1) involves the Coulomb interactions between spheres, as affected by the counterions. The matrix is given by H(q, w ) = -w2ml + D(q, o) (3.5) where m is the mass of a sphere and the dynamical matrix D(q, 0) may be written D(q, o) =E(q, w)+iwF(q, o ) - i ~ ~ ~ ( o ) l (3.6) where E(q, o) refers to the first term in eqn (2.12) and F(q, w ) to the second term.The matrix E(q, o) may be expressed as EV, (3.7) where S(q, A ) is given by the lattice sum where the prime indicates that the origin is excluded. The matrix F(q, w ) may be expressed as The dispersion equation (3.1) may now be cast in the form IIo’+iA(q, w ) w ‘ - B ( q , w)w-iC(q,w)(=O where the matrices A, B and C are given by with mf = pv, the mass of a unit cell of fluid and with R ( 4 ) = [I - 4’”K(q)]-’. (3.9) (3.10) (3.1 1) (3.12) For the b.c.c. lattice and special directions in q-space the dispersion equation decomposes into separate equations for purely longitudinal and purely transverse modes. The damping rates of the slowest modes are well approximated by (3.13) A striking difference with the corresponding result in the theory with instantaneous potential interactions is that BT( q, 0 ) tends to a non-vanishing constant at small y.This may be seen from the behaviour of the lattice sum (3.9) at small q, which is given by‘ (3.14)B. U. Felderhof and R. B. Jones 73 Q 3 Fig. 1. Plot of the damping rates rL(4) and IYT(4) of overdamped longitudinal and transverse phonons in a b.c.c. colloidal crystal for wavevectors q in the [ l , O,O] direction. The damping rates are given in Hz and the dimensionless wavenumber Q is defined by Q = q a / h , where a is the lattice distance. The matrix E ( q , 0) vanishes as q’ at small q. As a consequence the transverse damping rate rT( q ) vanishes at small q. This theoretical result agrees with the experimental data.’ In fig.1 we plot the dispersion curves for T , - ( q ) and Tr(q) as calculated in the approximation (3.13) for wavevectors q in the [ 1, 0,0] direction of a b.c.c. lattice. We have used the parameters 2 = 1350, sphere radius a, = 0.1 17 pm, diffusion coefficient D = 2.2 x lo-‘ cm’ s-’, lattice distance a = 1.45 pm, Debye parameter K a = 4.04 and fluid viscosity q = 0.01 poise. Where possible we have chosen the parameters in accordance with the experiment of Hurd et aL2 The behaviour of the damping rates rL,T(q) near the boundary of the Brillouin zone may be changed slightly in a more accurate determina- tion of the roots of eqn (3.10). 4. Discussion We have shown above that the dynamics of colloidal crystals is drastically affected by electric dipole interactions caused by the lag of the Debye cloud about the moving macroions. The damping rate of transverse phonons is predicted to tend to zero at long wavelengths. This is in agreement with light scattering experiments in which the Brownian motion of coiloidal crystals is observed.‘ Our theory has been simplified through our use of the linear Poisson-Boltzmann equation and the omission of the electrophoretic effect. We expect that an improved theory would lead to qualitatively the same results. References 1 P. Pieranski, Contemp. Phyy., 1983, 24, 25. 2 A. J. Hurd, N. A. Clark, R. C. Mockler and W. J . O’Sullivan, Phys. Rev. A, 1982, 26, 2869. 3 A. J. Hurd, N. A. Clark, R. C. Mockler and W. J. O’Sullivan, J. Fluid Mech., 1985, 153, 401 4 B. U. Felderhof and R. B. Jones, 2. Phys., Teil B, 1986, 64, 393. 5 F. Booth, J. Chem. Php., 1954, 22, 1956. 6 B. R. A. Nijboer and F. W. de Wette, Physica, 1958, 24, 422. Received 9th December, 1986