Comparison of Approximations between Additivity of Velocities and Additivity of Forces for Stokesian Dynamics Methods

Comparison of Approximations between Additivity of Velocities and Additivity of Forces for Stokesian Dynamics Methods

Journal of Colloid and Interface Science 243, 342–350 (2001) doi:10.1006/jcis.2001.7852, available online at http://www.idealibrary.com on Comparison...

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Journal of Colloid and Interface Science 243, 342–350 (2001) doi:10.1006/jcis.2001.7852, available online at http://www.idealibrary.com on

Comparison of Approximations between Additivity of Velocities and Additivity of Forces for Stokesian Dynamics Methods Akira Satoh1 Department of Machine Intelligence and System Engineering, Faculty of System Science and Technology, Akita Prefectural University, 84-4, Ebinokuchi, Tsuchiya-aza, Honjo 015-0055, Japan Received March 20, 2001; accepted July 16, 2001; published online October 5, 2001

The characteristics and differences in the two approximations, i.e., the additivity of forces and the additivity of velocities, on which Stokesian dynamics methods are based, have been investigated. Stokesian dynamics simulations of a ferromagnetic colloidal dispersion have been carried out for a simple shear flow, and the aggregate structures and averaged viscosities have been evaluated. From the results of aggregate structures in equilibrium, the correlation functions obtained by the additivity of forces are quantitatively different from those by the approximation of ignoring hydrodynamic interactions, although a qualitative agreement is recognized. The results obtained by the additivity of velocities give medium characteristics between the two above approximations. The shape of the pair correlation function for the additivity of velocities approaches that for the additivity of forces as the influence of magnetic interactions decreases. From the results of transient characteristics from an initial state, the results by the additivity of velocities agree well with those by the additivity of forces. In contrast, the results without hydrodynamic interactions deviate from those of the two additivity approximations at a time step when a nearly particle– particle touching situation starts to appear. °C 2001 Academic Press Key Words: ferromagnetic colloidal dispersion; aggregation phenomena; Stokesian dynamics method; additivity of forces; additivity of velocities; viscosity; correlation function.

1. INTRODUCTION

Functional fluids, which behave as if the fluids themselves had functional properties under certain conditions, are highly attractive from an application point of view in the colloid physics engineering field. Modern colloidal dispersion techniques enable us to generate various new types of functional fluids. Typical functional fluids which have already been generated to date are magnetic fluids (or ferrofluids) (1), electrorheological (ER) fluids (2), and magnetorheological (MR) suspensions (2). Ferrofluids and magnetorheological suspensions exhibit their functional properties in an applied magnetic field, and electrorheological fluids exhibit their functional properties in an electrical field. If such dispersion techniques are applied to the surface-reforming

1

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engineering field, we may control the surface properties of materials during fabrication by means of external magnetic or electrical fields. For example, recording materials (tapes) with much higher quality may be developed using such techniques (3–5). In these functional fluids, particles dispersed in a solvent aggregate to form chain-like or column-like clusters in an applied field (6–10). These particle clusters offer a large resistance to a flow field and so cause a significant increase in the apparent viscosity (11–15). Stokesian dynamics and Brownian dynamics methods are powerful microsimulation techniques to investigate the relationships between the aggregate structures of particles and the rheological properties such as non-Newtonian viscosities and viscoelastic properties (16–25). For dilute colloidal dispersions, even consideration of friction forces of particles alone does not cause serious errors in determining the particle motion in the process of simulations. In contrast, hydrodynamic interactions between particles have to be taken into account for nondilute dispersions. Generally speaking, Stokesian dynamics simulations with hydrodynamic interactions are much more difficult than molecular-dynamicslike simulations for colloidal dispersions. There are mainly two approximations to take into account multibody hydrodynamic interactions between particles, i.e., the additivity of forces (16–21) and the additivity of velocities (22– 25). The former approximation can reproduce the lubrication effect more accurately, which arises when particles nearly touch. However, since the inverse of the resistance matrix has to be calculated in the method based on this approximation, the simulations are usually restricted to a small system. In contrast, although the additivity of velocities is inferior concerning the accuracy of the lubrication effect, the calculation of an inverse matrix is unnecessary in simulations. One can, therefore, expand the simulations to a larger system (for example, N = 1000 or 10000). Due to the above-mentioned drawback of the method based on the additivity of velocities, there are only a few examples to date: the application to ferromagnetic colloidal dispersions being one of its successes (23–25). It is noted, however, that this simulation method is under development at this time and is not a fully established method. It is for this reason that the objective of the present study is to investigate the characteristics and differences in the

342

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ADDITIVITY OF VELOCITIES

simulations methods based on the two approximations, i.e., the additivity of forces and the additivity of velocities. To do so, a ferromagnetic colloidal dispersion is considered as a model system, and Stokesian dynamics simulations have been conducted for a simple shear flow to evaluate the aggregate structures of particles, the pair correlation functions, and the averaged viscosities.

is γ˙ a, the force is 6π ηa γ˙ a, the angular velocity is γ˙ , and the torque is 8π ηa 3 γ˙ . 2.2. Approximation of Additivity of Forces For the above-mentioned simple shear flow, the velocity vi and angular velocity ω i of particle i on the level of the approximation of the additivity of forces are written as (16–22) (

2. STOKESIAN DYNAMICS METHODS

I+

2.1. Approximation of Additivity of Velocities In nondilute colloidal dispersions, multibody hydrodynamic interactions are a main factor for governing the particle motion. If a simple shear flow is considered, the velocity vi and angular velocity ω i of particle i are written as (22, 23) N X

vi∗ = U∗ + Fi∗ +

j=1(6=i)

Ã

N X

+2

˜b∗ ii

Ti∗

·

j=1(6=i)

+2

N X

(a∗ii − I) · Fi∗ +

N X

g˜ i∗

ai∗j · F∗j

j=1(6=i)

+

!

N X

˜b∗ ij

·

T∗j

j=1(6=i)

ω i∗ = Ω∗ +

3 2

:E ,

N X

b∗ii · Fi∗ +

j=1(6=i) N X

+ Ti∗ +

[1]

j=1(6=i)

+

N X

N X

˜h∗ : E∗ , i

2 + 3

(

N X

− I) · (vi∗ − U∗ ) +

N X

Ai∗j · (v∗j − U∗ )

j=1(6=i)

˜ ∗ · (ω ∗ − Ω∗ ) + B i ii

j=1(6=i)

N X

)

˜ ∗ · (ω ∗ − Ω∗ ) B j ij

j=1(6=i)

N 2 X ˜ ∗ : E∗ = F∗ , G − i 3 j=1(6=i) i ( ) N N X X 1 ∗ ∗ ∗ ∗ ∗ ∗ B · (vi − U ) + Bi j · (v j − U ) 2 j=1(6=i) ii j=1(6=i) ) ( N X ∗ (Cii − I) · (ω i∗ − Ω∗ ) + I+

! bi∗j · F∗j

N X

+

N X j=1(6=i)

j=1(6=i)

(c∗ii − I) · Ti∗ +

(A∗ii j=1(6=i)

[3]

j=1(6=i)



j=1(6=i)

Ã

)

N X

ci∗j · T∗j

j=1(6=i)

[2]

j=1(6=i)

where Fi and Ti are the force and torque acting on the ambient liquid by the particle i, (δ x , δ y , δ z ) are unit vectors, I is the unit tensor, ai∗j , bi∗j , etc. are the mobility tensors and should be referred to in Refs. (22, 26). The fluid is assumed to flow in the x-direction with the shear rate γ˙ and the velocity vector without particles, U(r), is expressed as U(r) = Ω × r + E · r = yδ x , in which r is the position vector, E is the rate-of-strain tensor, and Ω is the angular velocity vectors. The rate-of-strain tensor E has only E x y and E yx as non-zero components; E x y = E yx = ∗ = 12 ). The angular velocity vector is written as γ˙ /2(E x∗y = E yx Ω = −δ z /2 for the present case. The superscript ∗ denotes dimensionless quantities. In this study Brownian motion of particles is assumed to be negligible. Since the calculation of the inverse of a matrix is unnecessary in the additivity of velocities, this method is applicable to a large system, which has already been pointed out. Each quantity is nondimensionalized in the usual way for a simple shear flow with the following representative values: the length is a (the particle radius), the time is 1/γ˙ , the velocity

Ci∗j · (ω ∗j − Ω∗ ) −

N X

˜ ∗ : E∗ = T∗ , H i i

[4]

j=1(6=i)

in which Ai∗j , Bi∗j , etc. are the resistance tensors and should be referred to in Refs. (22, 26). If Fˆ ∗ describes the column vector containing the forces and torques, vˆ ∗ describes the column vector containing the translational and angular velocities of all N particles, R∗ is the resistance matrix, and Φ∗ is the ˜ ∗ including coefficients, ˜ ∗, . . . , H column vector containing G N 1 then the equations for all particles can be written in one matrix form: Fˆ ∗ = R∗ · vˆ ∗ − Φ∗ : E∗ .

[5]

Hence, vˆ ∗ can be solved from this equation as vˆ ∗ = R∗−1 · (Fˆ ∗ + Φ∗ : E∗ ).

[6]

It is clear from Eq. [6] that the inverse of the resistance matrix has to be calculated to obtain the translational and angular velocities. If we consider a three-dimensional system of N particles, this means that the 6N × 6N resistance matrix is treated in simulations. Hence, the simulations based on this approximation are generally limited to a small system, which has already been pointed out. 2.3. Model of Colloidal Dispersion A ferromagnetic colloidal dispersion is considered as a model dispersion to investigate the characteristics and differences of Stokesian dynamics methods based on the two approximations,

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AKIRA SATOH

i.e., the additivity of forces and the additivity of velocities. A particle is idealized as a spherical particle with central point dipole coated with a uniform surfactant (or steric) layer. A colloidal dispersion is assumed to be composed of many such particles with the same radius. The magnetic interaction energy between (H ) particles i and j, u i(m) j , the particle–field interaction energy, u i , and the interaction energy arising due to the overlapping of the steric layers, u i(Vj ) , are expressed, respectively, as (23) µ ¶3 ds (m) ± u i(m)∗ = u {ni · n j − 3(ni · ti j )(n j · ti j )}, kT = λ j ij ri j u i(H )∗

u i(Vj )∗

=

[7]

± u i(H ) kT

[8] = −ξ ni · h, µ ¶ ½ ¾ u i(Vj ) 2ri j /ds ri j /ds − 1 d = ln = λV 2 − −2 , kT tδ ri j tδ [9]

in which ni is the unit vector denoting the direction of the magnetic moment mi (m = |mi |), ti j is the unit vector given by ri j /ri j , ri j = ri − r j , ri j = |ri j |, h = H/H, H is the applied magnetic field (H = |H|), ds is the diameter of the particle excluding the steric layer (the respective radius is expressed by as ), k is Boltzmann’s constant, T is the absolute temperature of the fluid, d is the diameter of the particle including the steric layer (a is the radius), and tδ is the ratio of the thickness of steric layer δ to the radius of the solid part of the particle, equal to 2δ/ds . The nondimensional parameters, λ, ξ , and λV , appearing in the above equations, are written as follows, λ=

µ0 m 2 π ds2 n s µ0 m H , ξ= , λV = , 3 4π ds kT kT 2

[10]

in which µ0 is the permeability of free space and n s is the number of surfactant molecules per unit area on the particle surface. λ, ξ, and λV are the dimensionless parameters representing the strengths of magnetic particle–particle, magnetic particle–field, and steric particle–particle interactions relative to the thermal energy, respectively. In the above equations, the force and torque of the particle acting on the ambient fluid, Fi∗ and Ti∗ (i = 1, 2, . . . , N ), are written as (23) Fi∗ =

N X ¡

N X (V )∗ ¢ ∗ + F = Ti(m)∗ + Ti(H )∗ , [11] Fi(m)∗ , T i j ij j

j =1 ( j6=i)

j =1 ( j6=i)

8 [−(ni · n j )ti j + 5(ni · ti j )(n j · ti j )ti j ri∗4j

− {(n j · ti j )ni + (ni · ti j )n j }],

= −Rm Ti(m)∗ j

2 ri∗j



(2/(1 + tδ ) ≤ ri∗j ≤ 2),

[13]

2 {ni × n j − 3(n j · ti j )ni × ti j }, ri∗3j

[14]

Ti(H )∗ = R H ni × h.

[15]

(m) Fi(m) j and Ti j are the magnetic force and torque acting on the particle i by neighboring particles, Ti(H ) is the torque acting on the particle i due to the deviation of the magnetic moment from an applied magnetic field in direction, and Fi(Vj ) is the force due to the overlapping of the steric layers. The nondimensional numbers in Eqs. [12]–[15], Rm , R H , and RV , are due to the forces and torques being normalized by the viscous force of the shear flow and are written as

Rm =

µ0 m 2 µ0 m H kT λV , RH = , RV = . 2 6 3 64π ηs a γ˙ 8π ηs a γ˙ 6π ηs a 2 γ˙ δ [16]

Rm is the ratio of the representative magnetic particle–particle force to the representative hydrodynamic shear force, R H is the ratio of the representative torque due to an applied field to the hydrodynamic torque, and RV is the ratio of the representative repulsion due to steric layers to the hydrodynamic shear force. Although the nondimensional shear rate γ˙ ∗ is always unity, the strength of the shear rate can be changed using the nondimensional number Rm . That is, the change of Rm = 0 → ∞ means that the shear rate changes from ∞ to zero. Viscosity data are used to discuss the characteristics and differences of the simulation methods. For the cases of ferrofluids and MR suspensions, magnetic properties of dispersions are used to control fluid properties in engineering applications. We here, therefore, concentrate our attention on the contribution of the magnetic interactions to the viscosity and do not take into account the contribution of the stresslet of each particle. Hence, the following expression of the viscosity does not have all terms, but only two terms due to magnetic (and steric) forces and torques. The instantaneous viscosity increase due to the magnetic forces, ηmyx , is written as (23)

ηm∗ yx =

ηmyx ηs

=−

N N X N 6π X 4π X ∗ ∗ y F + T ∗, i j i j x V ∗ i=1 j=1 V ∗ i=1 i z

[17]

( j>i)

where = −Rm Fi(m)∗ j

µ Fi(Vj )∗ = RV ti j ln

[12]

in which the first term on the right-hand side is due to the magnetostatic forces and repulsive forces between the steric layers of each particle and the second term is due to the torques acting on the particles including the contribution from an external field.

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ADDITIVITY OF VELOCITIES

3. COMPUTATION DETAILS AND SIMULATION PARAMETERS

The following conditions and values were used in conducting Stokesian dynamics simulations. In the simulations based on the additivity of forces, huge computation time is required because the inverse of the resistance matrix has to be calculated with every time step. As shown later, even a small system of N = 64 is too heavy from a computation time point of view. We consider, therefore, a special case of a highly strong magnetic field (ξ = ∞), which leads to ωi∗ = 0 and ni = h (i = 1, . . . , N ). With this assumption, only translational velocities are evaluated and the resistance matrix becomes 3N × 3N . The number of particles, N , is 64, the volumetric fractions is taken as ϕV = 0.15, and the number density, n ∗ (= na 3 ) is 0.0358. For this case, the length of the simulation box (cube), L ∗ (= L/a), is 12.14. The value of RV /Rm is chosen as 52.89, which corresponds to the case of λ = 9 where thick chain-like clusters are formed in a strong magnetic field (10, 23, 24). The values of Rm are taken as Rm = 1, 5, 20, 50, and 100 to clarify the influence of the shear rate. The cutoff radius for magnetic particle–particle in∗ (=rcoff /a), is L ∗ /2. The time interval, 1t ∗ , used in teractions, rcoff the present simulations is as follows: 1t ∗ = (0.0001, 0.00002) for Rm = (20,100), for example. The simulations were carried out up to 2,500,000 time steps to obtain averaged values with sufficient accuracy. The resistance tensor, A∗11 , for example, is expressed using A∗ and Y11A∗ as (22, 26) the resistance functions of X 11 A∗ ee + Y11A∗ (I − ee), A∗11 = X 11

[18]

in which e is the unit vector between two particles of interest. Figure 1 shows the characteristics of the resistance functions which are dependent only on the particle–particle distance r ∗ . It A∗ increases rapidly (exactly, in is clearly seen that the value of X 11 inverse proportion to the particle–particle distance) as particles approach each other to attain a nearly touching situation. This

type of behavior is undesirable from a simulation point of view as it may lead to the computational instability of the system, arising from the discretization of the translational motion during the simulation. However, the exact form of the resistance function has already been approximated because the ferromagnetic particles are modeled as a solid sphere coated with a soft material of surfactant, whereas, in practice, the steric layer may change in shape due to the large force arising from the lubrication effect. It is, therefore, reasonable to make an additional approximation of restricting the magnitude of the resistance function when particles are nearly in contact to maintain the computational stability. In the present study, the linear approximation between r ∗ = 2.01 and r ∗ = 2.00 is used. That is, each resistance function converges to a value that is for the case of nonhydrodynamic interactions, which is shown in terms of broken lines in Fig. 1. The value of 2.01 is chosen from the values of 2.001, 2.005, 2.01, 2.02, 2.05, and 2.1, for which the simulations have been carried out to check the particle overlap and the equilibrium quantities. In the present model system, the pair correlation function and the viscosity are not so sensitive to this value, but the particle overlap occurred for the small value of 2.001. We therefore adopted the value of 2.01, for which the particle overlap did not A∗ is much larger than unity. occur and the maximum value of X 11 This approximation has been used for both the resistance and mobility functions. We mention the computation time for some typical cases before proceeding to the results. All simulations were conducted on an Alpha workstation with dual CPUs produced by Compaq. A total of about 2,500,000 time steps were needed to obtain the results with sufficient accuracy. For the case of N = 64, the computation time is 12.5 days for the additivity of forces, 1.5 h for the additivity of velocities, and 0.6 h for the case of ignoring hydrodynamic interactions. Also, for N = 512, it is 2.6 days for the additivity of velocities. These data for the computation time were obtained for the case of only the translational velocities being considered. It is noted that the additivity of velocities has a significant advantage in the computation time compared with the additivity of forces. 4. RESULTS

4.1. Influence of Cutoff Distances for Hydrodynamic Interactions

A∗ and Y A∗ . FIG. 1. Properties of resistance functions, X 11 11

Cutoff distances for hydrodynamic interactions between particles may be used to reduce the computation time, but they must be used with considerable caution. Figure 2 shows the influence (hydro) on the averaged viscosities which of the cutoff distance rcoff was obtained by the simulation method based on the additivity of forces. The maximum and minimum values of the subaveraged viscosity are used as an error bar in Fig. 2; such subaveraged viscosities were calculated using the instantaneous viscosity data for each 50,000 time step. It is clearly seen that the results are strongly dependent on the cutoff distance for hydrodynamic interactions. If ferromagnetic particles aggregate to form stable

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AKIRA SATOH

particles is denoted by ri(min)∗ , we define the coefficient αi as

αi =

FIG. 2. Influence of cutoff distance of hydrodynamic interactions on averaged viscosities for the case of the additivity of forces.

chain-like clusters in the direction of a magnetic field, the relative velocity of a particle of a cluster to the shear flow becomes large as the particle is farther away from the mass center of the cluster. When the particle, away from the mass center of the cluster, is in a nearly particle–particle touching situation, it has a significant influence on the motion of the other particles of the cluster, which is clearly understandable from the characteristics of the resistance functions shown in Fig. 1. Consider this problem in more detail using Fig. 3. If particles 1 to 3 are in a particle–particle touching situation, the lubrication effect does not occur among these particles. If particles 3 and 4 are in a nearly touching situation, the significant influence due to the lubrication effect is directly exerted on particle 1 as if there were no particle 2 between particles 1 and 3. In other words, the screening effect of particle 2 is not taken into account. It is, therefore, quite understandable that the averaged viscosity is more significantly influenced as the value of the cutoff distance (hydro) becomes large since the above-mentioned situation occurs rcoff more frequently with increasing cutoff distance. Hence, we have to develop an approximate technique concerning the screening effect (since the exact treatment is known to be problem). In the present study, we have adopted the following approximate method. If the minimum distance of particle i to the other

FIG. 3.

Screening effect of a third particle.

 (min)∗ ∗  ri − rα1   ∗ ∗   rα2 − rα1     

1 0

 ¡ ∗ ¢ (min)∗ ∗  < rα2  rα1 < ri   ¢ ¡ (min)∗ . ∗ ≥ rα2 ri    ¢  ¡ (min)∗  ∗ ≤ rα1 ri

[19]

The coefficient αi is used as a weighting coefficient when hydrodynamic interactions are taken into account. That is, the hydrodynamic interactions of a particle of interest with particles far away from it are taken into account without the weighting coefficient when the particle does not belong to a stable cluster. On the other hand, the weighted hydrodynamic interactions with the coefficient αi are taken into consideration when the particle belongs to a stable cluster, which means that the screening effect of a third particle is taken into consideration indirectly. This approximate treatment has been used for the additivity of velocities as well. From the various simulations, we have had the conclu∗ ∗ ∗ ∗ and rα2 are taken as rα1 = 1.96 and rα2 = 2.00, sion that if rα1 the averaged values of viscosity are almost independent of the (hydro)∗ . Hence, all simulations in the values of the cutoff radius rcoff present study were obtained using these values. 4.2. Snapshots of Aggregate Structures in Equilibrium Figure 4 shows the snapshots of the aggregate structures in equilibrium for Rm = 20, in which Fig. 4a is for the additivity of forces (AF), Fig. 4b for the additivity of velocities (AV), and Fig. 4c for the approximation of ignoring hydrodynamic interactions between particles (AIHI). The figure on the left-hand side is viewed from an oblique angle and the figure on the right-hand side from the magnetic field. It is seen that particles aggregate to form a wall-like structure for each case and also that a significant difference in those structures is not qualitatively recognized. This wall-like structure is mainly due to the magnetic interactions between particles since the viscous shear forces are significantly dominated by the magnetic forces for Rm = 20 and also since a similar wall-like structure is obtained for AIHI, Fig. 4c. In the small system of N = 64, thick chain-like clusters are formed with difficulty and the predominantly thin chain-like clusters incline toward the shear flow direction, and such thin clusters are presumed to aggregate to form wall-like clusters due to the magnetic interactions, as shown in Fig. 4. Hence, it is expected that thicker wall-like aggregates are formed for a larger system, which will be clearly shown later for N = 512. The results of the pair correlation function are shown in Fig. 5 to see the quantitative difference in the aggregate structures between AF and AV. Figure 5a is for the former case and Fig. 5b for the latter case. Also, the results for AIHI are shown in Fig. 5c for comparison. All results are obtained for Rm = 20, in which the magnetic interactions dominate the viscous shear forces. The notation of θ in the figures refers to the angle measured from the magnetic field direction to the shear flow direction. In addition, to clarify the difference more clearly, the correlation function

ADDITIVITY OF VELOCITIES

FIG. 4. Snapshots of aggregate structures for Rm = 20: (a) additivity of forces; (b) additivity of velocities; (c) without hydrodynamic interactions.

347

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AKIRA SATOH

at r ∗ = 1.99 is shown as a function of θ in Fig. 6, in which Fig. 6a is for Rm = 5 and Fig. 6b for Rm = 20. First, we show the relationship between the aggregate structure and each peak in the correlation function using Fig. 5c. Since aggregates have a wall-like structure with thickness of a single-particle radius for Rm = 20, it is quite understandable that each peak in the correlation function corresponds to the correlation of particle 0 with the particle of the corresponding number. Next, consider the results shown in Figs. 5 and 6b. It is seen that there are quantitative differences among the three cases, although the results obtained by AF and AV are in qualitative agreement with those for AIHI. Also, we seen that the results by AV have medium characteristics between AF and AIHI. We discuss the results in Fig. 6 more carefully concerning this point. The discrepancy of the peak value relative to that for AF is as follows. The difference in the peak at θ = 30◦ is (21%, 17%) for Rm = (20, 5), respectively, for AV, and (32%, 31%) for AIHI. Also, the relative difference in the peak at θ = 40◦ is (5%, 23%) for (AV, AIHI), respectively. These results clearly show that the internal structure of wall-like aggregates approaches that for AF as the influence of the magnetic interactions decreases.

FIG. 5. Pair correlation functions for Rm = 20: (a) additivity of forces; (b) additivity of velocities; (c) without hydrodynamic interactions.

FIG. 6. Pair correlation functions at r ∗ = 1.99: (a) for Rm = 5; (b) for Rm = 20.

ADDITIVITY OF VELOCITIES

FIG. 7. Influence of shear rate on averaged viscosity.

4.3. Averaged Viscosities Figure 7 shows the influence of the shear rate on the averaged viscosities for AF, AV, and AIHI. As in Fig. 2, the maximum and minimum values of the subaveraged viscosity for 50,000 time steps are used as an error bar for Rm = 50 and 100. It is seen that the viscosity increases significantly with increasing values of Rm or with decreasing the shear rate, and such non-Newtonian properties almost coincide quantitatively among the three cases. We have already seen that particles aggregate to form a stable wall-like structure in equilibrium when the magnetic forces significantly dominate the viscous shear forces. In this situation, the lubrication effect seldom occurs so that it is quite understandable that there is no significant difference in the averaged viscosity among the three cases.

349

FIG. 8. Transient properties of subaveraged viscosity with time step for Rm = 20.

is thicker than that for N = 64 and is composed of several layers of particles. For the case of N = 512, thick chain-like clusters are expected to be formed along the magnetic field direction (8–10). Hence, we understand that such thick chain-like clusters associate to form thick wall-like aggregates, shown in Fig. 9,

4.4. Transient Properties from an Initial State To clarify the transient properties from an initial state, the evolution of the subaveraged viscosity for 500 time steps with time is shown in Fig. 8 for Rm = 20, in which the results are for the above-mentioned three cases. Since the magnetic forces significantly dominate the viscous shear forces for Rm = 20, the three curves agree well with each other at the initial stage. As the computation proceeds, the magnetostatic attraction causes particles to approach each other and eventually they nearly touch. At this stage the curves of AF and AV deviate significantly from the curve for AIHI. The result of AV is seen to be in good agreement with that of AF. 4.5. Aggregate Structures for a Relatively Large System Finally, we show a snapshot of the aggregate structures in equilibrium for a relatively large system, N = 512, in Fig. 9, which was obtained for AV for Rm = 50. It is seen from comparing the results with Fig. 4b that the wall-like aggregate structure

FIG. 9. Snapshot of aggregate structure for a relatively large system, N = 512, for Rm = 50.

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AKIRA SATOH

under the circumstances of the applied magnetic field and the shear flow. Further investigation will be required to clarify this behavior. 5. CONCLUSIONS

The characteristics and differences in the two approximations, i.e., the additivity of forces and the additivity of velocities, on which Stokesian dynamics methods are based, have been investigated. Stokesian dynamics simulations of a ferromagnetic colloidal dispersion have been carried out for a simple shear flow, and the aggregate structures and averaged viscosities have been evaluated. From the results of aggregate structures in equilibrium, the correlation functions obtained by the additivity of forces are quantitatively different from those by the approximation of ignoring hydrodynamic interactions, although a qualitative agreement is recognized. The results obtained by the additivity of velocities give medium characteristics between the two above approximations. The shape of the pair correlation function for the additivity of velocities approaches that for the additivity of forces as the influence of magnetic interactions decreases. From the results of transient characteristics from an initial state, the results by the additivity of velocities agree well with those by the additivity of forces. In contrast, the results without hydrodynamic interactions deviate from those of the two additivity approximations at a time step when a nearly particle–particle touching situation starts to appear. ACKNOWLEDGMENTS The author gratefully acknowledges Prof. R. W. Chantrell and Dr. G. N. Coverdale for their valuable advice on this work. This work was partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education and Science of Japan.

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