Correlation of Two Coupled Particles in Viscoelastic Medium

Transcription

1 Commun. Theor. Phys. (Beijing, China 46 (006 pp c International Academic Publishers Vol. 46, No., August 5, 006 Correlation of Two Coupled Particles in Viscoelastic Medium XIE Bai-Song,,,4 WU Hai-Cheng,, YAN Shi-Wei,, ZHANG Feng-Shou,, and Sayipjamal Dulat 3,4 Key Laboratory of Beam Technology and Materials Modification of Ministry of Education, Beijing Normal University, Beijing 00875, China Institute of Low Energy Nuclear Physics, Beijing Normal University, Beijing 00875, China 3 Department of Physics, Xinjiang University, Urumqi , China 4 The Abdus Salam International Centre for Theoretical Physics, Strada Costiera, 3404 Trieste, Italy (Received November 5, 005 Abstract Considering the viscoelastic memory effect, we study the correlated motion of two hydrodynamically coupled colloidal particles, each of which confined in a harmonic potential well, in a Kelvin-type and Maxwell-type viscoelastic medium. We find that viscoelastic relaxation plays a significant role in modifying the correlation, particularly the cross correlation. We also find that both the real and imaginary parts of the response function are significantly different from the viscous medium case. In particular there is a phase shift between the vanishing imaginary part and the maximal real part of the response function in a viscoelastic medium. In addition imaginary part of the cross correlation response function exhibits a net energy loss (gain behavior when the elasticity parameter of the medium is larger (smaller than the critical value for Kelvin (Maxwell viscoelastic fluid. Some implication of our results and their connection with previous works are discussed. PACS numbers: Bc, 83.0.Pp, 8.70.Dd Key words: linear viscoelasticity, particle dynamics, colloids Introduction Complex fluid exists ubiquitously in physical, chemical, biological, and artificial systems, for example, colloidal solution, material processing and polymer productions. [,] Due to interactions among its components and medium inhomogeneity, it exhibits intricate viscoelastic rheological behavior. Viscoelasticity and rheology are the most important properties of the soft materials and are studied extensively in recent years. [3 5] The experimental and theoretical study of viscoelasticity is both fundamental and immense practical importance since it intrinsically reveals the rheology of soft materials. For example, shear flow has been reported to induce the formation of particle strings in viscoelastic liquids. However, if the viscosity of the viscoelastic medium is large enough, shear flow will induce non-equilibrium microstructure of the particles. Recently, Microrheology technique has been developed to determine the viscoelastic properties of the soft materials of systems ranging from polymer solutions to the interior of living cells on microscopic scales. [5,6,0] This technique increases our microscopic understanding of the complicated soft materials. It has been proved to be very effective tool in examining and studying the structure and dynamics of the system and can be used to measure viscoelastic parameters. By using laser tweezers one can capture and manipulate a small dielectric particles that is confined into a potential well in a viscoelastic medium. The change of particle position can be tracked, monitored and examined experimentally. One can also associate it to compliance-tensor (or response function. The Brownian dynamics of a single particle and two particles pair dispersed in viscous medium have been studied by a number of researchers. [7 9,,3] However, the single particle microrheology technique has some flaws because of inhomogeneity of complex materials in small scales. On the other hand the problem can be solved by examining the correlated motion of two particles located at different places. Indeed, the two particles microrheology technique has been developed and proven as an effective tool for understanding the microrheology properties of the soft materials. The displacement r(ω of a spherical particle of radius a in response to a force, F (ω, at oscillation frequency ω in a viscous fluid is given by the generalized Stokes Einstein relation, r(ω F (ω 6πaG(ω, ( where /6πaG(ω is the frequency-dependent response function (G(ω is complex shear modulus. For a pure viscous fluid it is known as G(ω iωη, where η is the viscosity of the fluid. The two-point response function for two coupled particles, each of which is confined in harmonic potential in a viscous medium has recently been studied by several groups. [7 9,,3] For the interparticle correlation of two spherical particles, the general- The project supported by National Natural Science Foundation of China under Grant Nos and , the Abdus Salam International Centre for Theoretical Physics (ICTP, Trieste, Italy under the Associate Programm, and the Foundation for New Century Excellent Talents in University of China

2 354 XIE Bai-Song, WU Hai-Cheng, YAN Shi-Wei, ZHANG Feng-Shou, and Sayipjamal Dulat Vol. 46 ized Stokes Einstein relation is x n (ω χ nm (ωf m (ω, (n, m,, ( where x n (ω is the displacement of the n-th probe particle, F m (ω is the external force on the m-th particle, χ nm (ω is the complex (χ nm (ω χ nm(ω + i nm(ω, symmetric (χ nm (ω χ mn (ω response function of the system. Mason and Weitz [3] have studied the linear viscoelastic moduli of complex fluid over a greatly extended frequency range and found that the shear moduli behaves as G(s g g s g 3 s g 4 s g 5 s, where g i are positive fitting parameters. For simplicity we study theoretically the response function for Kelvin viscoelastic medium with the complex shear modulus G(s g +g 5 s. In addition, Mason et al. [4] have also obtained the other moduli form by fitting data as G(s j G js/(s + /τ j. Obviously this moduli form is the summation of several different moduli in Maxwell viscoelastic medium. Thus we also study the corresponding response function with the complex shear modulus G(s ηs/(s + /τ in Maxwell viscoelastic medium. This paper is organized in the following way. In Sec. we examine the correlated motions of two hydrodynamically coupled colloidal particles confined in separate harmonic potential well, in a Kelvin viscoelastic medium. First by solving the coupled equations of motion for the two particles, we calculate the autocorrelation and cross-correlation function, and then by using the fluctuation-dissipation theorem [6] and the Kramers Kronig relation, [6] we determine two particles response function as a function of the frequency. Moreover we analyze the effect of the elasticity of the viscoelastic medium on the response function. In Sec. 3, in a similar way as in Kelvin viscoelastic medium, we investigate the correlated motions of two hydrodynamically coupled colloidal particles confined in separate harmonic potential well, in a Maxwell viscoelastic medium. In Sec. 4, we give conclusions, and some implication of our results, as well as their connection with previous works. Response Function in Kelvin Viscoelastic Medium In this section we study a system, the two hydrodynamically coupled particles confined into independent harmonic potential well in a viscoelastic medium. Here, at first we consider the general case where the viscoelastic effect is described by the inherent integral kernel associated with the velocity that attributes the force. The basic coupled equations are [3] m i du i (t dt t t ξ ii (t t u i (t dt ξ ij (t t u j (t dt k i x i (t + f i (t, (i,, (3 where m i is the i-th particle mass, u i (t( ẋ i (t is the i-th particle velocity, x i (t is the i-th particle displacement, k i is the elastic constant of harmonic potential well, f i (t is the random Brownian forces acting on i-th particle, the time-dependent frictional coefficient ξ ij (t is the generalization of the viscoelastic memory to the case of two interacting particles. The first term associated with ξ ii (t represents the force acting on the first moving particle while the second one is stationary, the second term associated with ξ ij (t describes the force acting on i-th particle induced by the motion of the second j-th one. By ignoring the inertial effect of particles, and by making Laplace transformation x(s x(t e st dt, here 0 s is the Laplace transform variable, we have two coupled equations (sξ (s + k x (s + sξ (sx (s ξ (sx (0 + ξ (sx (0 + f (s, sξ (sx (s + (sξ (s + k x (s ξ (sx (0 + ξ (sx (0 + f (s. (4 Reference [9] has studied the one- and two-point microrheology of viscoelastic medium and shown that, under typical experimental conditions, the one-particle friction coefficient, ξ (s, and the two-particle friction coefficient, ξ (s, satisfy the generalized Stokes Einstein relation (see also Appendix: sξ (s 6πaG(s ɛ, ξ (s ɛξ (s (5 with ɛ 3a/R, G(s is the Laplace transform of the shear modulus G(t. For Kelvin solid-like viscoelastic medium we have G(s µ + ηs η(s + Ω, (6 where µ is the Lame elastic constant, [7] Ω µ/η is the inverse of relaxation time of medium. By Eqs. (5 and (6 equations (4 can be rewritten as ( τ(s + Ω ɛτ(s + Ω τ(s + Ω ɛ + x (s ɛ x (s s( ɛ x (0 ɛτ(s + Ω ɛ x (s + ( τ(s + Ω ɛ + x (s ɛτ(s + Ω s( ɛ x (0 + f (s, k ɛτ(s + Ω s( ɛ x τ(s + Ω (0 + s( ɛ x (0 + f (s, (7 k where k /k, and τ 6πηa/k. By solving Eqs. (7 one first gets x i (s (i, and then by using x i (0x j (0 k BT k i δ ij, where k B is the Boltzmann s constant, and T is the absolute temperature, δ ij is the Kronecker δ function, one obtains

3 No. Correlation of Two Coupled Particles in Viscoelastic Medium 355 the following autocorrelation and cross-correlation functions: x (sx (0 k BT (s + Ω[τ(s + Ω + ] k τ s(s s (s s where are two poles with x (sx (0 x (sx (0 ɛk BT k τ, x (sx (0 k BT k τ (s + Ω[τ(s + Ω + ] s(s s (s s s + Ω s(s s (s s, (8 s, Ω + s +, (9 s +, ( + ± ( + 4ɛ. (0 τ The time-domain autocorrelation and cross-correlation functions of the two particles positions are obtained by the inverse Laplace transformation of x i (sx j (0, for example, the cross-correlation function is ɛk B T ( s+ e st x (tx (0 x (tx (0 s e st. ( k τ(s s s s We can calculate the imaginary part of the response function by using the fluctuation-dissipation theorem, [6] ij(ω ω k B T x i(ωx j (0, ( where x i (ωx j (0 is the Fourier transform of x i (tx j (0. We can determine the real parts of the response function from the imaginary parts of the response function by using the Kramers Kronig relation [6] + χ dξ ij ij(ω P (ξ π ξ ω. (3 By Eqs. ( and (3 we obtain the following real and imaginary components of the elements of the response function: χ ( s s + [ + τs + ] (ω (s s k τ ω + s s s [ + τs ] ω + s, ( ωs [ + τs ] (ω (s s k τ ω + s ωs +[ + τs + ] ω + s, χ ( s s + [ + τs + ] (ω (s s k τ ω + s s s [ + τs ] ω + s, ( ωs [ + τs ] (ω (s s k τ ω + s ωs +[ + τs + ] ω + s, χ (ω χ ɛ ( s s (ω (s s k τ ω + s s s + ω + s, (ω ɛ ( ωs+ (ω (s s k τ ω + s ωs ω + s. (4 We should emphasize that for Ω 0, equation (4 can be reduced to the results of Ref. []. It is important to note that the imaginary part (ω c vanishes, at nontrivial frequency ω c ( ɛ s s + Ω(s + s τ Ω, (5 and at ω 0 and ω. The nonzero real part is χ ɛ (ω c k τ(s + s ɛ k [( + + Ωτ]. On the other hand the maximum of χ (ω occurs at frequency s ω c s (s + Ω s s (s + Ω s (s + Ω. (6 s (s + Ω Thus the maximum value of χ (ω is where χ ɛ (ω c k τ(s + s Q, (7 Q + Ω(s + s + (s s s s (s + Ω(s + Ω (s s. (8 It is easy to prove that if Ω > 0, we always have ω c > ω c, which means the frequency has a right shift between the vanishing imaginary part and the maximum real part. In general, Q > and χ (ω c > χ (ω c always hold. In,

4 356 XIE Bai-Song, WU Hai-Cheng, YAN Shi-Wei, ZHANG Feng-Shou, and Sayipjamal Dulat Vol. 46 addition, it is easy to show that the maximum of χ (ω c always decrease when Ω increases. From Eq. (5 we see that the nontrivial frequency which leads the zero imaginary part would not exist when ( ɛ Ω > Ω c. (9 τ The above equation implies that when the elasticity of the medium is strong enough, the parameter Ω passes through the critical parameter Ω c, such that the imaginary part is always positive, which means the cross correlation effect yields the energy loss of the particle. Now in order to see clearly the effect of elasticity of the medium on the response function we illustrate some examples numerically for various Ω in Figs. 4. The real and imaginary components of the elements of the response function, χ (ω, (ω, χ (ω, (ω, χ (ω, and (ω for two particles in a harmonic potential well in a Kelvin viscoelastic medium are given in Figs. 4, for separation distance of a/r 0.60, k 0.0 dyn/cm and From Fig. (a (Ω 0 we see that the imaginary part (ω has extremum, the vanishing imaginary part (ω of the cross correlation function corresponds to the maximum real part χ (ω of the cross correlation function. This is consistent with the analysis of Ref. []. From Fig. (b we see that χ (ω and χ (ω have no extremum, and (ω and (ω are always positive and have maximum. Figure (a (Ω 300 shows that the frequency, at which the real part χ (ω acquires the maximum, is a slightly larger than the frequency, at which the imaginary part (ω vanishes. This means there are different phase shifts between real and imaginary parts of response function for viscoelastic and viscous medium. This point is consistent with the theoretical analysis that we discussed above. One also notes that as Ω increases, the energy gain frequency regime becomes narrow, and the energy gain decreases. Figure 3(a implies that when the viscoelastic parameter Ω reaches to a certain critical value Ω c (see Eq. (9 then the negative part (ω vanishes. It means that the energy gain of particles from the oscillating potential is suppressed when the elasticity of viscoelastic medium is strong enough. There is always a negative phase shift between the displacement of the first particle, r, and F, the force acting on the first particle. Therefore the work is done by F is always negative, this results in a net energy loss of the particle in the stationary trap. From Fig. 4(a we note that the extremum property of the real part of the response function, χ (ω, would vanish when the viscoelastic parameter Ω approaches to a certain critical value Ω c 789., and χ (ω is almost constant at a wide range of frequency. We also see that the self-response functions of particles and, χ and χ are gradually approaching with each other. For Ω > Ω c, our numerical studies show that the maximum energy dissipation occurs in the resonance-like frequency ω Ω, and the effective energy dissipation frequency is from Ω/0 to 0Ω. For large Ω, from Eq. (9 we have s, Ω. In this case the maximum imaginary part ij of the response function approximately coincides with the turning point of the real part χ ij of the corresponding response function. Fig. The real and imaginary parts of the components of the response function for two particles confined in the harmonic potential well in a Kelvin viscoelastic medium with Ω 0, a/r 0.60, k 0.0 dyn/cm and 0.45 from Eqs. (4. Fig. The real and imaginary parts of the components of the response tensor for two particles confined in the harmonic potential well in a Kelvin viscoelastic medium. The system parameters are the same as in Fig. except with Ω 300.

5 No. Correlation of Two Coupled Particles in Viscoelastic Medium 357 Fig. 3 The real and imaginary parts of the components of the response for two particles confined in the harmonic potential well in a Kelvin viscoelastic medium. The system parameters are the same as in Fig. except with Ω from Eqs. (4. Fig. 4 The real and imaginary parts of the components of the response function for two particles confined in the harmonic potential well in a Kelvin viscoelastic medium. The system parameters are the same as in Fig. except with Ω 789. from Eqs. (4. 3 Response Function in Maxwell Viscoelastic Medium In this section we study a system, the two hydrodynamically coupled particles, each of which is confined in a harmonic potential well in a viscoelastic medium where the elastic unit is associated with viscous unit in the Maxwell fluid-like behavior. In this case the shear modulus is G(s ηs τ m s +, (0 where τ m η/µ is the relaxation time. Thus by using Eqs. (5 and (0, equation (4 can be rewritten as ( τg0 ɛ + x (s ɛτg 0 ɛ x (s τg 0 s( ɛ x (0 ɛτg 0 s( ɛ x (0 + f (s, k ( τg0 ɛτg 0 ɛ x (s + ɛ + x (s ɛτg 0 s( ɛ x (0 + τg 0 s( ɛ x (0 + f (s, ( k where G 0 (s s/(τ m s +. In an analogous way as in Sec., we obtain the following real and imaginary components of the elements of the response function π ij π ij (ω + iπ ij (ω: π (ω ( s [ + (τ + τ m s ] (s + s k τ ω + s s [ + (τ + τ ms ] ω + s, π (ω ( ωs [ + (τ + τ m s ] (s + s k τ ω + s ωs [ + (τ + τ m s ] ω + s, π (ω ( s [ + (τ + τ m s ] (s + s k τ ω + s s [ + (τ + τ ms ] ω + s, π (ω ( ωs [ + (τ + τ m s ] (s + s k τ ω + s ωs [ + (τ + τ m s ] ω + s, π (ω π (ω ɛ ( s ( + τ m s (s + s k τ ω + s s ( + τ m s ω + s, π (ω π (ω ɛ ( ωs ( + τ m s (s + s k τ ω + s ωs ( + τ m s ω + s, ( where s, s +, τ m s +,, (3 are two poles of the response function in the complex plane with s +, given in Eq. (0. When τ m 0 our results ( can be reduced to that of Ref. []. One notes that for any τ m the imaginary part π (ω c of the cross correlation

6 358 XIE Bai-Song, WU Hai-Cheng, YAN Shi-Wei, ZHANG Feng-Shou, and Sayipjamal Dulat Vol. 46 response function vanishes at frequency s s ω c + τ m (s + s s+ s ( ɛ τms + s τ τm( ɛ (4 except for ω 0 and ω. Thus the nonzero real part is π (ω c ɛ[ + τ m(s + s ] s s ɛ [ τ ] m + k τ(s + s s + s k τ s + + s τ m s + s τ m (s + + s + τms + s ɛ [ k ( + + δ( ɛ δ ] + δ( + + δ ( ɛ, (5 where δ τ m /τ. On the other hand the maximum of π (ω located at frequency, ω c s + τm s s + τm s s s s + s + τ m s s s s s + τm s s + τm s (s + s + τ m s + τm s + τ m s s s + + s τ m s + s s + s ( τm s + ( τ m s (s + + s τ m s + s + τ m s + s ( τ m s + ( τ m s where s+ s H, H ( τ m s + ( τ m s τ ms + s τ ms + s [ ( τ m s + ( τ m s ] s + + s τ m s + s + δ( + + δ ( ɛ δ ( ɛ + δ( ɛ [ + δ( + + δ ( ɛ ] ( + + δ( ɛ. (7 Obviously when δ 0 we have H. One can prove that for any τ m > 0 we always have ω c < ω c, which means the frequency has a left shift between the vanishing imaginary part and the maximum real part. Certainly π (ω c < π (ω c always holds. In addition it is easy to show that the corresponding maximum value of π (ω c always decreases as τ m increases. From Eq. (4 we see that the nontrivial frequency which leads the zero imaginary part of π would not exist when τ τ m > τ mc ( ɛ. (8 (6 The above equation indicates when the elasticity of the Maxwell viscoelastic medium is weak enough, parameter τ m passes through the critical parameter τ mc, as a result the imaginary part π is always negative, which means the cross correlation effect yields the energy gain of the particle. Fig. 5 The real and imaginary parts of the components of the response tensor for two particles confined in the harmonic potential well in a Maxwell viscoelastic medium with τ m , a/r 0.60, k 0.0 dyn/cm and 0.45 from Eqs. (. Fig. 6 The real and imaginary parts of the components of the response tensor for two particles confined in the harmonic potential well in a Maxwell viscoelastic medium. The system parameters are the same as in Fig. except with τ m from Eqs. (.

7 No. Correlation of Two Coupled Particles in Viscoelastic Medium 359 In order to see clearly the effect of the elasticity of the Maxwell viscoelastic medium on the response function we illustrate some examples numerically for various τ m in Figs The behaviors of π (ω and π (ω exhibit also abundant phenomena. From Fig. 5 one notes that when the viscoelastic parameter τ m increases to a certain critical value of τ mc ( see Eq. (8, the positive part π (ω vanishes. This means that the energy loss of particles from the oscillating trap potential is suppressed when the elasticity of the Maxwell viscoelastic medium is weak enough. If the energy-dissipative imaginary part π (ω of the self-response function is smaller than the energy-gain imaginary part π (ω of the cross response function, the particle would get a net energy gain. However, this energy gain of the particle occurs at the cost of the extra dissipation of the other particle in the oscillating trap. From Fig. 6 one notes that when τ m , the positive part π (ω also vanishes. This indicates that the response functions of the two coupled particles always have opposite phase. From Fig. 7 we conclude that when τ m, the resonance-like phenomena start to occur, the energy dissipation due to the imaginary part of the self response function or the energy gain due to the imaginary part of the cross correlation response function reaches maximum at ω /τ m. From Fig. 8, we see surprisingly that the similar resonance-like phenomena occur still at resonance frequency ω /τ m, and the effective energy dissipation frequency is from τ m /0 to 0τ m. In general, the self-response functions of particles and do not approach as τ m increases. Obviously the response functions exist in the regime of frequency lower than /τ m, and they would vanish for large τ m. For very large τ m, from Eq. (3 we have approximately s, /τ m. In this case the extremum of the imaginary part π (ω of the response function coincides approximately with the turning point of the real part π (ω of the corresponding response function. Fig. 7 The real and imaginary parts of the components of the response tensor for two particles confined in the harmonic potential well in viscoelastic medium. The system parameters are the same as in Fig. except with τ m. Fig. 8 The real and imaginary parts of the components of the response tensor for two particles confined in the harmonic potential well in viscoelastic medium. The system parameters are the same as in Fig. except with τ m 0. 4 Discussions and Summary For the Kelvin viscoelastic medium, the imaginary part (ω of the cross response function has an approximate maximum,max(ω ɛ ω k τ, (9 when the viscoelastic relaxation frequency Ω is large enough. Obviously,max(ω is proportional to the ɛ 3a/R, and inversely proportional to the oscillating frequency ω. For the Maxwell viscoelastic medium, the imaginary part π (ω of the cross response function has an approximate maximum π,max(ω ω ɛ k τ( ɛ, (30 when the viscoelastic relaxation time τ m is large enough. π,max(ω depends nonlinearly on the ɛ, and proportional to the oscillating frequency ω. The difference between,max(ω and π,max(ω is important for understanding some features of biopolymer system that is associated with the minimum depth of cross time correlation function, which depends on ɛ. For example reference [4] studied the simple viscous model, and obtained the following theoretical linear relation between the minimum depth of cross correlation function and ɛ x (tx (0 min k BT ek ɛ. (3 One notes the theoretical curve from Eq. (3 is not in good agreement with the experimental measurement curve

8 360 XIE Bai-Song, WU Hai-Cheng, YAN Shi-Wei, ZHANG Feng-Shou, and Sayipjamal Dulat Vol. 46 because that the measured data exhibit nonlinear relation between the minimum depth of cross correlation function and ɛ. In our model for Kelvin medium Ωτ, however, from Eq. (, we find the following relation between them x (tx (0 min k BT ek ɛ( 3Ωτ( ɛ. (3 And for Maxwell medium τ m /τ we have x (tx (0 min k ( BT ek ɛ τ m + τ m τ τ ɛ. (33 Fig. 9 The dependence of the minimum depth of cross time correlation on the ratio between the radius of the spheres a and their average distance R. The solid-line is for Kelvin medium with Ωτ 0.03, and the dashed-line is for Maxwell medium with τ m/τ 0.. The heavy solid-line and filled solid circles are results from Ref. [4]. In Fig. 9 we have shown modified nonlinear relation between the minimum depth of cross-correlation function and ɛ by choosing appropriate parameters. Because of small ɛ, the viscoelastic memory effect, Ωτ or τ m /τ, plays more important role on modification than the nonlinear term of ɛ or ɛ. By comparing with the results of Ref. [4], obviously our results from Eqs. (3 and (33 are in good agreement with the experimental results, if the viscoelastic relaxation time is introduced appropriately as in Fig. 9. In summary, in this paper, we made an initial effort to understand the interactions between particles in this complex fluid. The correlated motions of two colloidal particles confined in separate harmonic potential well in Kelvin solid-like or Maxwell fluid-like viscoelastic medium were examined theoretically. We considered the viscoelastic memory effect for two coupled particles confined in harmonic potential well in viscoelastic medium and investigated the problem involving of two-particle correlation and the response function. By comparing our results to that of the viscous medium, [4] we found that the viscoelastic relaxation plays a significant role in modifying the correlation especially the cross correlation. Both of the real and imaginary parts of response function are significantly changed. In particular, there is a phase shift between the vanishing imaginary part and the maximum real part of cross response function. Moreover the imaginary part of cross response function exhibits a pure energy dissipation (gaining behavior when the elasticity parameter is larger (smaller than the critical value for a Kelvin solid-like (Maxwell fluid-like complex medium. In some special situation, for example, for biological systems, the shear modulus may be as complex as G(s s, where is a fraction exponent, e.g., 3/4. [5,0] The analytical studies here could not directly be applicable to the biological complex fluid. However, from physical point of view, our study might be helpful to give some indications and insights into the related problems in biological system. The theoretical analysis and the numerical stimulations are under study. Appendix: One- and Two-Particle Friction Coefficients For the pure viscous medium, ξ ij (t t ξ 0 ij δ(t t, by ignoring the inertial effect of particles, we rewrite Eq. (3 as ξ 0 u (t + ξ 0 u (t k x (t + f (t, ξ 0 u (t + ξ 0 u (t k x (t + f (t. (A By solving u (t, u (t, and comparing with Eq. ( in Ref. [] we get ξ 0 ξ 0 [( ɛ H ], H ξ 0 ξ 0 [( ɛ H ] ɛξ0, where ɛ 3a/R, and H H 6πηa, H H 4πηR are the lowest order components, in /R, of the Oseen tensor [] for motions in the longitudinal directions along the line between the centers of the two particles. Here R is the separation distance between two particles. By making Laplace transformation of ξ ij (t ξij 0 δ(t, we get the following one- and two-particle friction coefficients sξ (s 6πaG(s ɛ, ξ (s ɛξ (s, (A where G(s ηs is the Laplace transform of the viscous modulus. Acknowledgments Authors thank Profs. K.F. He and Z.Q. Huang for useful discussions.

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