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Two-dimensional Monte Carlo simulations of a colloidal dispersion composed of polydisperse ferromagnetic particles in an applied magnetic field
Journal of Colloid and Interface Science 288 (2005) 475–488 www.elsevier.com/locate/jcis
Two-dimensional Monte Carlo simulations of a colloidal dispersion composed of polydisperse ferromagnetic particles in an applied magnetic field Masayuki Aoshima ∗ , Akira Satoh Department of Machine Intelligence and System Engineering, Faculty of System Science and Technology, Akita Prefectural University, 84-4 Ebinokuchi, Tsuchiya-aza, Honjyo, Akita 015-0055, Japan Received 18 October 2004; accepted 26 February 2005 Available online 16 April 2005
Abstract We have investigated aggregation phenomena in a polydisperse colloidal dispersion of ferromagnetic particles simulated by employing the cluster-moving Monte Carlo method in an applied magnetic field. The influence of both particle–particle and particle–field interactions on the aggregate structures is analyzed in terms of a pair correlation function. The results obtained in this study are summarized as follows: Under a strong magnetic field, chainlike clusters are formed along the magnetic field direction, and they become thickly clustered with an increase in the strength of the external magnetic field. Moreover, the thickly clustered chains are formed for a polydisperse system that has a large standard deviation of particle diameters. In contrast, for a very weak magnetic field, the strong interaction between the larger particles gives rise to the formation of various shapes in the chainlike clusters, including bending, looping, and branching. With an increase in the external magnetic field, these structures reorganize to form straight chainlike clusters. Furthermore, the thickness of the chainlike clusters for the polydisperse system is found to depend on the standard deviation of the particle-size distribution but is found to be independent of the magnetic field strength. 2005 Elsevier Inc. All rights reserved. Keywords: Magnetic fluid; Ferromagnetic colloidal dispersion; Particle-size distribution; Chain-like cluster; Monte Carlo simulation
1. Introduction A ferrofluid [1] is classed as a functional fluid because it exhibits attractive functional properties in an applied magnetic field. Another example of a functional fluid is an electrorheological (ER) fluid [2], which responds to an external electric field. These functional fluids are generally produced by dispersing functional particles in a liquid medium and only show the desired functional characteristics under certain conditions. Under the influence of a strong external magnetic field, the ferromagnetic particles in a ferrofluid tend to aggregate to form chain-like clusters [3,4]. These clusters give rise to a large resistance in a flow field. There* Corresponding author. Fax: +81 184 27 2191.
E-mail address: [email protected] (M. Aoshima). 0021-9797/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.02.093
fore, several fluid engineering applications using this phenomenon are under development: for example, mechanical dampers and actuators. In practice, a ferrofluid is a polydisperse colloidal dispersion. Aggregation phenomena arise during the application of an external magnetic field, and this has a significant influence on the physical properties of the fluid, particularly on the viscosity. A microscopic simulation approach such as the Monte Carlo method [5,6] is a powerful technique used to clarify the particle aggregation phenomena. Hence, several molecular simulations using this method have already been conducted on ferromagnetic colloidal dispersions [7–14]. Also, our previous studies clearly show that the aggregate formation is highly dependent on the strength of the magnetic particle–particle interactions for the case under a relatively strong magnetic field [15,16]. This is due to the
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fact that the magnetic moments of particles are strongly oriented toward the magnetic field direction. These previous investigations have generally tended to use an idealized dispersion comprising identical spherical particles with the same diameter. Using this simplified dispersion model, a large number of simulations have been conducted employing various methods of Stokesian dynamics [17,18], and Brownian dynamics [19–23] to investigate aggregation phenomena and rheological properties in a flow field. On the other hand, investigations on polydispersity of the aggregate structures are indispensable for clarifying aggregate structures and magnetic properties of a real ferrofluid. There are a number of studies on polydisperse ferromagnetic particle systems. For example, the log normal [24] and the normal [25] size distributions of particles are assumed for the polydispersity to carry out the MC simulations. Recently, Kristof and Szalai [26] studied the polydisperse system using gamma-distribution in the particle size. Kruse et al. [27] also carried out MC simulations using realistic particle-size distributions obtained from the experimental SAXS data of real ferrofluids. Furthermore, a bidisperse model system, which is supposed to consist of two fractions of magnetic particles with significant size differences, is assumed to describe the polydispersity by means of both the molecular dynamics (MD) simulation [28] and the theoretical consideration [29]. However, these studies mentioned above have mainly investigated the system for relatively weak magnetic interaction between particles. That is, they have studied the short chainlike clusters in detail. If the particle–particle interaction increases, a number of complicated microstructures are expected to be detected. In our previous study [30], we investigated a normal distributed polydisperse system of a ferromagnetic dispersion and clarified the influence of the particle-size distribution on the internal structure of aggregates in the absence of an external magnetic field. We used the cluster-moving Monte Carlo method [31], which is indispensable for simulating a strong interacting system. It was observed that clusters formed possess a complicated internal structure described as a network. In the present study we investigate the aggregation phenomena in a polydisperse system of ferromagnetic particles under conditions of an applied magnetic field. In fact, the cluster-moving Monte Carlo simulation has been conducted for a two-dimensional system. The results of the internal structures of aggregates are analyzed using the pair correlation function.
2. Polydisperse system of ferromagnetic colloidal particles In this study, we consider a two-dimensional system comprising of N spherical particles with a distribution of particle diameters, d, and a magnetic dipole at their center. We also
take into consideration the fact that the particles are coated with a steric layer of thickness δ. In the present study, we assume that the system possess a normal particle-size distribution with an average particle diameter d 0 and a standard deviation σ . If the magnetic moment of particle i and the uniform magnetic field strength are denoted by mi (mi = |mi |) and H (H = |H|), respectively, the energy of interactions be(H ) tween particle i and the magnetic field, ui , the energy of (m) magnetic dipole–dipole interaction, uij , and the interaction (v)
energy due to the overlapping of the steric layers, uij , are expressed as follows [6,30]: (H )
ui
= −kT ξi ni · H/H,
(m)
uij = kT λij
u(v) ij =
(1)
d03 ni · nj − 3(ni · tj i )(nj · tj i ) , 3 rj i
Ci − 1 2Ci λvi kT tδi + 1 −2 2− ln 2 tδi Ci tδi λvj kT tδi + dj /di 2Cj + ln 2− 2 tδi Cj Cj − dj /di , −2 tδi
(2)
(3)
in which Ci = Cj =
(1 + tδi )2 − (dj /di + tδi )2 + 4rj2i /di2 4rj i /di (dj /di + tδi )2 − (1 + tδi )2 + 4rj2i /di2 4rj i /di
, ,
(4)
where k is Boltzmann’s constant, T is the absolute temperature of the fluid, rj i is the magnitude of the vector rj i drawn from particles i to j , and ni and tj i denote the unit vectors given by ni = mi /mi (mi = |mi |) and tj i = rj i /rj i . The parameter tδi in Eqs. (3) and (4) is the ratio of the thickness of the steric layer δ to the radius of the solid portions of particle i, and is expressed as 2δ/di . The dimensionless parameters λvi and λvj in Eq. (3) are written as 2 2 πd02 ns dj di , λv , λvj = λv , λv = λvi = (5) d0 d0 2 in which λv is a dimensionless parameter representing the strength of steric interactions between two particles with the same average ensemble diameter d 0 , and ns is the number of surfactant molecules per unit area on the particle surface. If we assume that the magnitude of the magnetic moment of particles, m(= |m|), is in proportion to its particle volume πd 3 /6, then the dimensionless parameters ξi and λij in Eqs. (1) and (2) can be written as 3 di dj 3 di ξi = (6) ξ0 , λij = λ0 , d0 d02
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in which ξ 0 and λ0 are the dimensionless parameters for the particles with the mean diameter d 0 , representing the strength of particle–field and particle–particle interactions, relative to the thermal energy. These parameters are written as ξ0 = µ0 m0 H /kT and λ0 = µ0 m20 /4πd03 kT . 3. Monte Carlo simulations 3.1. Monte Carlo method In the present study, Monte Carlo (MC) simulations were conducted to investigate the aggregate structures of ferromagnetic colloidal dispersion. Our MC method is based on Metropolis’s algorithm [32], a powerful technique for generating microscopic thermodynamic equilibrium states. In this method a new trial configuration is generated by a random movement of a particle, and its acceptability is then determined by a probability in proportion to the Boltzmann factor exp(−U/kT ), where U is the change in the total energy of the system before and after the movement. However, the conventional Metropolis’s algorithm cannot reproduce physically reasonable aggregate structures for the case of a system with a strong particle–particle interaction. This is due to the fact that a particle move rarely overcomes particle interactions to escape from its cluster to form larger clusters. The cluster-moving MC algorithm [31] was developed for use in the simulation of a strongly interacting system by allowing a cluster to move as a composite single particle. In order to analyze the internal structures of the aggregates, snapshots of particle configuration are used for qualitative discussion. The number density of clusters and the pair correlation function g (2) (r) [15] are evaluated for quantitative analysis. In practice, the correlation function for direc(2) (r) and tions parallel and normal to the magnetic field, gpara (2) gnorm (r), are evaluated in the simulations. 3.2. The simulation parameters Two-dimensional MC simulations have been conducted under the following conditions: The particle number N and the area fraction of the particles to the system region φ a are taken as N = 900 and φa = 0.1, respectively. The particlesize distribution is assumed to be Gaussian, and we have considered two different cases of standard deviation, σ = 0.2 and 0.35, with the same average diameter d0 = 1.0. In this case, the standard deviation of σ ≈ 0.35 is almost maximum value because the minimum diameter of particles is approximately zero. The smooth Gaussian distribution curve is discretized into increments such as d = 0.02d0 and 0.035d0 for specifying the diameter of each particle, as shown in Fig. 1. According to the discretized Gaussian distribution, the number of particles with a specific diameter can be determined. The side length of the square simulation box, L, is taken as L = 85.62d0 and 89.12d0 for the polydisperse systems of σ = 0.2 and 0.35, and L = 84.07d0 for a monodis-
(a)
(b) Fig. 1. Particle-size distributions for σ = (a) 0.2 and (b) 0.35.
perse system, respectively. The initial configuration of particles and the initial directions of the magnetic moments are taken randomly. The thickness of the steric layer is taken as δ = 0.15d0 , and the dimensionless parameter, λv , representing the strength of the steric interactions between the two particles, as λv = 150. If the distance between the surfaces of the steric layer of the two particles is smaller than 0.2d 0 , then these particles are assumed to form a cluster. The cutoff radius for the particle–particle interactions, r coff , is 10d 0 . The maximum displacements of a particle and a cluster in one trial move are 0.5di and 0.5d 0 , respectively. The maximum value for a change in the particle orientation angle is 10◦ . The parameters r and θ for evaluating the pair correlation function are taken as r = d0 /20 and θ = 10◦ , respectively. Under these parametric conditions, we performed the MC simulations for various cases of the dimensionless parameters λ0 and ξ 0 such as λ0 = 2, 4, 6, 8 and ξ0 = 1, 2, 4, 30. Each simulation was executed up to approximately 450,000 MC steps, in which the cluster-moving algo-
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rithm was employed after every 80 MC steps up to the first 270,000 MC steps, and it was ensured that the system attains equilibrium. In the subsequent 180,000 MC steps, the conventional Metropolis algorithm was conducted instead of the cluster-moving procedure until the end of each run. The mean values were evaluated using the data of the last 180,000 MC steps. In order to estimate the finite size effect of the system, we compared the pair correlation functions and the number density of clusters for both systems of N = 900 and 1800 under the strong magnetic interaction between particles (λ0 = 8). It was observed that the pair correlation functions for shorter distance (r/d < 6) and the number density of clusters for a relatively small value of c coincide well with each other. Here, c is the number of particles in a cluster. However, the peaks at a larger value of c (c ≈ 100) depend on the system size. Therefore, we avoid discussing the accuracy in the size of the larger aggregates due to the uncertainty of peak positions of the number density distributions of clusters for a larger value of c.
4. Results and discussion 4.1. Influence of the interaction between particles on aggregate structures In the following sections, we consider the aggregate structures under a strong external magnetic field of ξ0 = 10. The influence of magnetic interactions between the particles on snapshot configurations and pair correlation functions for both small and large standard deviations of the particle-size distribution are compared. 4.1.1. Snapshots of aggregate structures First, we consider the influence of magnetic interactions between particles on snapshots of aggregate structures for the small standard deviation of the particle-size distribution (σ = 0.2). Fig. 2 shows the snapshots of the aggregate structures in equilibrium for λ0 = 2, 4, 6, and 8. Fig. 2a shows that for a relatively weak particle–particle interaction such as λ0 = 2, particles do not aggregate but move almost indepen-
Fig. 2. Influence of interactions between particles on aggregate structures for a polydisperse system with σ = 0.2 and ξ0 = 10: λ0 = (a) 2, (b) 4, (c) 6, (d) 8.
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Fig. 3. Influence of interactions between particles on aggregate structures for a polydisperse system with σ = 0.35 and ξ0 = 10: λ0 = (a) 2, (b) 4, (c) 6, (d) 8.
dently. In the case of λ0 = 4, shown in Fig. 2b, the particles aggregate to form numerous short chainlike clusters. In the case of λ0 = 6, shown in Fig. 2(c), several long chainlike clusters are observed and thick chainlike clusters can also be recognized. In the case of the relatively strong particle– particle interactions such as λ0 = 8, shown in Fig. 2d, long and thick chainlike clusters are formed. Next, we consider the results for the large standard deviation (σ = 0.35) shown in Fig. 3. Although the magnetic interaction between the particles is sufficiently weak for λ0 = 2, several chainlike clusters are observed in Fig. 3a, which is a significant contrast to the previous image in Fig. 2a. Fig. 3a clearly shows that particles with a large diameter are the nuclei for particle aggregation. It is observed that such chainlike clusters become thicker and larger with an increase in the particle–particle interaction. A comparison between Figs. 3d and 2d clarifies that chainlike clusters with a large standard deviation (σ = 0.35) possess more compact internal structures than those with a smaller standard deviation (σ = 0.2).
4.1.2. Number density distributions of clusters In order to quantitatively investigate the influence of the particle–particle interaction on the cluster formation, the number density distributions of clusters are analyzed and plotted on a logarithmic graph as shown in Fig. 4. The ordinates of the figures indicate the number of particles in a cluster c. Moreover, the abscissa of the figures shows the ratio of the number density of clusters n∗c (= nc /d02 ) to the number density of particles in the whole system n∗ (= d02 N/L2 ). The curves for noninteracting polydispersed particles for λ0 = 0 and ξ0 = 0 are also shown in Fig. 4 for comparison. Fig. 4a shows the number density of clusters for σ = 0.2. The curve for λ0 = 2 exhibits a monotonous decrease. Comparing the maximum values of c for λ0 = 2 and 0, we observe that the maximum number of particles in a cluster, c, for λ0 = 2 is slightly larger than that for λ0 = 0. This indicates the formation of a small amount of small clusters. As λ0 increases, the number density of clusters increases and finally the curve shows a distinguishable peak approximately at c ≈ 100 for λ0 = 8. This peak corresponds to the
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(a)
(b) Fig. 4. Influence of interactions between particles on number density distributions of clusters for σ = (a) 0.2, (b) 0.35.
thick chainlike clusters along the magnetic field direction, as shown in Fig. 2d. Fig. 4b shows the number density of clusters for σ = 0.35. For λ0 = 4, the peak indicating the formation of large clusters appears approximately at c ≈ 50, which is in contrast to the case of σ = 0.2 (Fig. 4a). For strong magnetic interaction between particles, such as λ0 = 6 and 8, several peaks corresponding to the formation of large clusters appear approximately at c ≈ 100; the monotonous decreasing curve at the lower region of c simultaneously shifts close to the curve for noninteracting dispersed particles (λ0 = 0, ξ0 = 0). These suggest that the larger particles form thick chainlike clusters and also that the smaller particles disperse as single particles throughout the space among the clusters. 4.1.3. Pair correlation functions of aggregate structures Fig. 5 shows the results of the pair correlation functions, (2) (2) gpara (r) and gnorm (r), in which Figs. 5a and 5b represent σ = 0.2, and Figs. 5c and 5d represent σ = 0.35. Each figure
depicts the curves for different magnetic particle–particle interactions such as λ0 = 2, 4, 6, and 8. Several sharp peaks are observed in the case when the pair correlation function is parallel to the magnetic field, shown in Figs. 5a and 5c, and these peaks became more significant with the larger values of magnetic interactions, λ0 . When λ0 = 4, the first, second, and third peaks appear at r/d0 ≈ 1.5, 3.0, and 4.5, respectively. The positions of the peaks are integer multiples of the average diameter of the particles including the steric layer (δ/d0 = 0.15). Hence, this fact clearly shows that the particles with average diameter of approximately (1.2d 0 ) constitute short stable chainlike clusters. This result agrees well with the results of other researcher’s studies assuming a bidisperse model system [28,29]. Thus, we conclude that the larger particles tend to form chainlike clusters along the magnetic field direction for relatively weak particle–particle interaction. As the value of λ0 increases, the positions of those peaks shift toward the shorter distances, and the peaks become sharper. This suggests that, as the magnetic interactions between the particles increase, the short clusters associate with smaller particles to form stable longer chainlike clusters along the magnetic field direction. The peaks for σ = 0.35 appear at longer distance positions than for σ = 0.2, which implies that the larger particles are contributing to the cluster formation. Comparing these pair correlation functions parallel to the magnetic field for polydisperse systems with the results for a monodisperse system [15], the peaks for a polydisperse system are much broader than that for a monodisperse system. This also supports the fact that the chainlike clusters for a polydisperse system consist of the particles with a size distribution. In the case when the pair correlation function is normal to the magnetic field, shown in Figs. 4b and 4d, the curves have gaslike characteristics and only the first peak occurs in both cases of σ = 0.2 and 0.35. As determined earlier, thick chainlike cluster for σ = 0.35 possess a more compact internal structure (not loose as in Fig. 2) than for σ = 0.2, so that the peaks for σ = 0.35 are much higher than for σ = 0.2. In the case of σ = 0.35, the curve for λ0 = 4 suggests that the clusters are observed even in the curve of a very small value of λ0 , which is in good agreement with the snapshot results shown in Fig. 3b. The results obtained above are summarized as follows: As the value of λ0 is increased, the particles form chainlike clusters along the magnetic field direction. In the case of a small standard deviation in the particle-size distribution (σ = 0.2), straight chainlike clusters extend along the magnetic field direction with an increase in the values of λ0 . This is due to the fact that single-moving particles combine with short chainlike clusters. In contrast with the above results, the snapshots for the large standard deviation (σ = 0.35) show the formation of thick chainlike clusters. These results can be explained by the fact that the larger particles form clumplike clusters, and they extend along the magnetic field direction with an increase in the values of λ0 .
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Fig. 5. Influence of interactions between particles on pair correlation functions for ξ0 = 10: (a) and (c) for the direction parallel to the field direction; (b) and (d) for the direction normal to the field direction; (a) and (b) for σ = 0.2; (c) and (d) for σ = 0.35.
4.2. Influence of magnetic field on aggregate structures In this section, we consider the aggregate structures for the strong magnetic interactions between particles such as λ0 = 8. The influence of the magnetic field on snapshot configurations and pair correlation functions for a monodisperse system and both small and large standard deviations of the particle-size distribution are compared. 4.2.1. Snapshots of aggregate structures The influence of magnetic field strength on the aggregate structure was investigated using snapshots of the aggregate structures in equilibrium for the magnetic fields ξ0 = 1, 2, 4, 30, as shown in Figs. 6, 7, and 8. First, we consider the influence of the magnetic field on snapshots of aggregate structures for the monodisperse system shown in Fig. 6. It is observed from Fig. 6a that moderately bent chainlike clusters are formed for a very weak magnetic field such as
ξ0 = 1. There exist several branchlike structures in the chainlike clusters. When ξ0 = 2 and 4 as shown in Figs. 6b and 6c, the chainlike clusters are not so extensively bent but the branchlike structures are still observed in the clusters. For a stronger magnetic field such as ξ0 = 30 (Fig. 6d), straight chainlike clusters with very little branchlike structures are formed along the magnetic field direction. Next, we consider the influence of interaction between a particle and the magnetic field on snapshots of aggregate structures for the small standard deviation of the particle-size distribution (σ = 0.2), as shown in Fig. 7. For all cases of particle–field interactions, the chainlike clusters are thicker than that for the monodisperse system (Fig. 6). In the case of a very weak particle–field interaction such as ξ0 = 1 and 2, shown in Figs. 7a and 7b, the structures of moderately bent chainlike clusters are almost similar to the snapshot for the monodisperse system (Figs. 6a and 6b). However, we can
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Fig. 6. Influence of magnetic field strength on aggregate structures for the monodisperse system with λ0 = 8: ξ 0 = (a) 1, (b) 2, (c) 4, (d) 30.
clearly observe the difference between the snapshots for the monodisperse and polydisperse systems; i.e., small looplike structures are included in the thick chainlike clusters in the polydisperse system. As the strength of particle–field interaction increases, the thick chainlike clusters with looplike structures become straight along the magnetic field direction. Finally, we consider the influence of interaction between a particle and the magnetic field on snapshots of aggregate structures for the large standard deviation of the particle-size distribution (σ = 0.35). Fig. 8 shows the snapshots of the aggregate structures in equilibrium for ξ0 = 1, 2, 4, and 30. In the case of a very weak particle–field interaction such as ξ0 = 1, shown in Fig. 8a, thick and bent chainlike clusters, which have several branchlike and small looplike structures, are formed. As the magnetic field increases for ξ0 = 2 and 4 (Figs. 8b and 8c), the thick chainlike clusters become aligned along the magnetic field direction. These clusters have compact internal structures without the small looplike structure. In the case of a significantly strong magnetic field such as
ξ0 = 30 (Fig. 8d), the straight chainlike clusters are seen to be thicker. Since the clusters are composed of well-aligned particles along the magnetic field direction, the branchlike structures are much less evident. 4.2.2. Number density distributions of clusters In order to quantitatively investigate the influence of particle–field interaction on the cluster formation, the number density distributions of clusters for the monodisperse and polydisperse systems are analyzed and plotted on logarithmic graphs as shown in Fig. 9. Also, the curve for noninteracting particles for λ0 = 0 and ξ0 = 0 are added to the graphs for comparison. Almost all curves for the monodisperse and polydisperse systems depend weakly on the strength of the particle–field interaction. Fig. 9a shows continuous, convex curves of the number density of clusters for the monodisperse system. As the particle–field interaction increase, the maximum number of particles in a cluster increases from approximately c ≈ 100 to 200, whereas the number density of smaller clusters
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Fig. 7. Influence of magnetic field strength on aggregate structures for a polydisperse system with σ = 0.2 and λ0 = 8: ξ 0 = (a) 1, (b) 2, (c) 4, (d) 30.
(c < 7) decreases. This shows the fact that smaller clusters aggregate to form larger clusters, and also that the sizes of the clusters are distributed continuously. On the other hand, the curves of the number density distributions of clusters for polydisperse systems are much more different than that for monodisperse systems. Fig. 9b shows the number density distributions of clusters for a small standard deviation (σ = 0.2). The curves for σ = 0.2 become linear at values lower than c ≈ 50. That is, the number densities of middle-sized clusters are lower than that in the case of the monodisperse system, whereas the densities of small clusters (c < 5), including single particles, are higher than that in the case of the monodisperse system. Moreover, the significant peak, which appears at approximately 100, corresponds to the formation of stable, thick chainlike clusters. When the contribution of the polydispersity is sufficiently large (σ = 0.35), as shown in Fig. 9c, the number density curves are split into two regions: linear curves at lower c and several peaks at higher c. The linear curves are slightly lower than the curve for noninteracting particles. This implies that
the particles that do not belong to the clusters are separated clearly from the particles in clusters. 4.2.3. Pair correlation functions of aggregate structures We now quantitatively investigate the effect of the magnetic field strength on the internal structure of the agglomerates by employing the pair correlation functions. Figs. 10 and 11 show the pair correlation functions of the monodisperse and the polydisperse (σ = 0.2 and 0.35) systems, respectively. First, we consider the pair correlation functions of the monodisperse system. Figs. 10a and 10b indicate the internal structures for the direction parallel to the magnetic field and for the direction normal to the magnetic field, respectively. In the case when the direction is parallel to the magnetic field, as shown in Fig. 10a, several of the peaks gradually increase with an increase in the value of ξ 0 . This implies that the particles in the chainlike clusters become well aligned along the magnetic field direction. Fig. 10b shows the pair correlation functions for the direction normal to the magnetic
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Fig. 8. Influence of magnetic field strength on aggregate structures for a polydisperse system with σ = 0.35 and λ0 = 8: ξ 0 = (a) 1, (b) 2, (c) 4, (d) 30.
field. In the case of a very weak particle–field interaction such as ξ0 = 1, shown in Fig. 10b, the pair correlation functions exhibit two clear, sharp peaks at r/d0 = 1.3 and 2.6, which are integer multiples of the diameter of a particle, including thickness of the steric layer. These peaks result from the bent chainlike clusters. With an increase in the particle– field interaction, these peaks decrease gradually. When the magnetic field increases significantly (ξ0 = 30), all these peaks finally disappeared because the thin straight chainlike clusters along the magnetic field are formed due to the application of a strong magnetic field. Next, we consider the pair correlation functions for the small standard deviation (σ = 0.2) shown in Figs. 11a and 11b. In the case when the direction is parallel to the magnetic field, as shown in Fig. 11a, some peaks gradually increase with an increase in the value of ξ 0 . This im-
plies that the particles in the chainlike clusters become well aligned along the magnetic field direction. Fig. 11b shows the pair correlation functions for the direction normal to the magnetic field. The intensity of the first peak of the functions is seen to be independent of the magnetic field. This implies that the thickness of the chainlike clusters remains unchanged with increasing strengths of the magnetic field. On the other hand, the second peak shown for ξ0 = 1 disappears with an increase in the magnetic field. This implies that the branchlike or bent structures of chainlike clusters for ξ0 = 1 disappear because thick, straight, chainlike clusters are formed due to the application of a strong magnetic field. Finally, we consider the pair correlation functions for the large standard deviation (σ = 0.35) shown in Figs. 11c and 11d. Fig. 11c shows the case for the correlation function
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Fig. 9. Influence of magnetic field strength on number density distributions of clusters: (a) for the monodisperse system; for σ = (b) 0.2 and (c) 0.35.
(a)
(b)
Fig. 10. Influence of magnetic field strength on pair correlation functions for the monodisperse system and λ0 = 8: (a) for the direction parallel to the field direction; (b) for the direction normal to the field direction.
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Fig. 11. Influence of magnetic field strength on pair correlation functions for λ0 = 8: (a) and (c) for the direction parallel to the field direction; (b) and (d) for the direction normal to the field direction; (a) and (b) for σ = 0.2; (c) and (d) for σ = 0.35.
parallel to the magnetic field direction. Several peaks that indicate the chainlike cluster formation gradually increase less than is the case for σ = 0.2, as shown in Fig. 11a. This implies that the particles are less aligned along the magnetic field direction even in the case of a strong magnetic field. Fig. 11d shows the pair correlation functions for the direction normal to the magnetic field direction. For ξ0 = 1, the pair correlation functions have both the first and second peaks. The small second peak originates from the branchlike structures or the bent chainlike clusters. As the strength of the magnetic field increases (ξ0 = 2 and 4), the second peak decreases slightly, while the first peak remains unchanged. This decrease in the second peak suggests that the branchlike structures or the bent chainlike clusters disappear. The first peak, which is independent of the magnetic field, indicates that the thickness of the chainlike clusters remains unchanged with an increase in the strength of the magnetic
field. In the case of a significantly strong magnetic field such as ξ0 = 30, the slight increase of the first peak implies the formation of thick chainlike clusters. The results obtained above are summarized as follows: In the case of the monodisperse system, thin, bent chainlike clusters become straight with an increase in the strength of the magnetic field. In the case of the small standard deviation of σ = 0.2, the branchlike, looplike, and bent chainlike clusters decrease with an increase in the strength of the magnetic field. In the case of a significantly strong magnetic field, particles align strongly along the magnetic field direction and form thick, straight, chainlike clusters. The reason for these characteristics may be interpreted as follows: In the case of a weak influence of the magnetic field, the particles that aggregate around the large particles prefer to form branchlike or clumplike structures rather than form straight chainlike clusters. This is because the aggregation is dominated by
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the influence of the interaction between the particles. A detailed discussion regarding the formation of the branchlike or the clumplike structures will be given in Section 4.3. As the strength of the magnetic field increases, the relative influence of the interaction between the particles decreases. This leads to the formation of straight chainlike clusters along the magnetic field direction. In the case of a large standard deviation of σ = 0.35, large clumplike clusters are formed in the initial stages of the chainlike cluster formation during the simulation. This is because there are several large particles that possess a particle– particle interaction strong enough to form the clumplike clusters. However, as the influence of the magnetic field dominates, the particles align along the magnetic field direction to form the straight chainlike clusters. This leads to the formation of the rather thick straight chainlike clusters, whereas the number of branchlike and bent structures found in the chainlike clusters decrease. 4.3. Influence of particle size on the interaction energy between three particles in a cluster One of the characteristics of the particle aggregation in the polydisperse system is the appearance of various interesting structures, such as the clumplike and the branchlike structures, due to the strong interaction of the larger particles. In order to investigate the formation of these structures, we consider a case in which the three particles are in contact with each other at the surface of their steric layers. The magnetic moments of these three particles is assumed to be aligned with the magnetic field direction. One of the three particles is denoted by i and the others are denoted by j . We also assume that two particles j have the same diameters dj . We then propose two representative configurations of these three particles, linear and triangular, as illustrated in Fig. 12. In order to investigate the influence of the diameter di of the particle i on potential energies of the linear and the (m) (m) triangular configurations, ulin and utri , we have derived the following equations from Eqs. (2) and (6) using a geometrical consideration, d03 di dj 3 d03 (m) + ulin /kT λ0 = −2 (rij + rjj )3 d02 rij3 2 3 dj d03 −2 2 (7) , 3 d0 rjj u(m) tri /kT λ0 = 2 +
di dj
3
d03 rij3
d02 2 3 dj d03 d02
3 rjj
1−3
,
2 − rjj 4rij2
4rij2
Fig. 12. Influence of particle diameter di /d0 on cluster formation energy for dj /d0 = 0.6.
i.e., rij = di /2 + dj /2 + 2δ, and rjj is the distance between the centers of the two particles j when they are in contact with each other, i.e., rjj = dj + 2δ. The graph of the potential energy changes, which are calculated from Eqs. (7) and (8) with the nondimensional particle diameter di /d0 for the case of dj /d0 = 0.6, are shown in Fig. 8. It can be seen that the potential energy curves of the linear and the triangular configurations intersect at di /d0 ≈ 1.26. This implies that the triangular configuration is more stable than the linear configuration when the value of di /d0 is larger than 1.26. This is because the influence of the attractive interaction between particles i and j dominates over the influence of the repulsive interaction between the two particles j with an increase in the particle diameter. The formation of the triangular configuration for the polydisperse system enhances the formation of the branchlike or the clumplike structures. If the system is three-dimensional, the chainlike clusters possibly branches off in more than two directions because another fundamental configuration of clusters may be formed, such as a pyramidal cluster. This may lead to more complicated network microstructures.
5. Conclusions
(8)
in which, rij is the distance between the centers of the two particles i and j when they are in contact with each other,
We have investigated the aggregation phenomena occurring in a polydisperse colloidal dispersion composed of ferromagnetic particles, simulated by means of the clustermoving Monte Carlo method, in an applied magnetic field. Influences of both particle–particle and particle–field interactions on the internal structures of the aggregate structures are analyzed in terms of a pair correlation function. The
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results obtained in this study are summarized as follows: Under a strong magnetic field, chainlike clusters are formed along the magnetic field direction and they become thick clusters with an increase in the strength of external magnetic field. Moreover, the thick chainlike clusters are formed more readily for a polydisperse system that has a large standard deviation of particle size. In contrast, for a relatively weak magnetic field, various shapes of the chainlike clusters, such as bending, small looplike and branchlike structures are formed due to a strong interaction between the larger particles. With an increase in the external magnetic field, these structures reorganize to form the straight chainlike clusters. Furthermore, the thickness of the chainlike clusters for the polydisperse system depends on the standard deviation of a particle-size distribution but is independent of the magnetic field strength.
Acknowledgment This work was partly supported by a Grant-in-Aid for Scientific Research (C) from the Japanese Ministry of Education, Sports, Science and Technology.
References [1] R.E. Rosensweig, Ferrohydrodynamics, Cambridge Univ. Press, Cambridge, UK, 1985. [2] W.A. Bullough (Ed.), Electro-Rheological Fluids, Magneto-Rheological Suspensions and Associated Technology, World Scientific, Singapore, 1996. [3] C.F. Hayes, J. Colloid Interface Sci. 52 (1975) 239. [4] B. Jeyadevan, I. Nakatani, J. Magn. Magn. Mater. 201 (1999) 62. [5] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1987.
[6] A. Satoh, Introduction to Molecular-Microsimulation of Colloidal Dispersions, Elsevier, Amsterdam, 2003. [7] R.W. Chantrell, A. Bradbury, J. Popplewell, S.W. Charles, J. Phys. D 13 (1980) L119. [8] R.W. Chantrell, A. Bradbury, J. Popplewell, S.W. Charles, J. Appl. Phys. 53 (1982) 2742. [9] J.J. Weis, D. Levesque, Phys. Rev. E 48 (1993) 48. [10] J.J. Weis, D. Levesque, Phys. Rev. Lett. 71 (1993) 2729. [11] D. Levesque, J.J. Weis, Phys. Rev. E 49 (1994) 5131. [12] R.W. Chantrell, G.N. Coverdale, M.E. Hilo, K. O’Grady, J. Magn. Magn. Mater. 157/158 (1996) 250. [13] P.J. Camp, G.N. Patey, Phys. Rev. E 62 (2000) 5403. [14] T. Ukai, T. Maekawa, H. Morimoto, J. Mod. Phys. B 16 (2002) 2352. [15] A. Satoh, R.W. Chantrell, S. Kamiyama, G.N. Coverdale, J. Colloid Interface Sci. 178 (1996) 620. [16] A. Satoh, R.W. Chantrell, S. Kamiyama, G.N. Coverdale, J. Colloid Interface Sci. 181 (1996) 422. [17] A. Satoh, G.N. Coverdale, R.W. Chantrell, J. Colloid Interface Sci. 231 (2000) 238. [18] A. Satoh, J. Colloid Interface Sci. 243 (2001) 342. [19] A. Satoh, R.W. Chantrell, G.N. Coverdale, J. Colloid Interface Sci. 209 (1999) 44. [20] H. Morimoto, T. Maekawa, J. Phys. A Math. Gen. 33 (2000) 247. [21] S. Melle, O.G. Calderon, G.G. Fuller, M.A. Rubio, J. Colloid Interface Sci. 247 (2002) 200. [22] A. Satoh, G.N. Coverdale, R.W. Chantrell, J. Colloid Interface Sci. 231 (2000) 238. [23] P. Ilg, M. Kroger, S. Hess, A.Y. Zubarev, Phys. Rev. E 67 (2003) 061401. [24] P. Bhatt, R.V. Mehta, S.P. Bhatnagar, R.V. Upadhyay, D. Srinivas, Indian J. Eng. Mater. Sci. 5 (1998) 356. [25] J. Garcia-Otero, M. Porto, J. Rivas, A. Bunde, Phys. Rev. Lett. 84 (2000) 167. [26] T. Kristof, I. Szalai, Phys. Rev. E 68 (2003) 041109. [27] T. Kruse, A. Spanoudaki, R. Pelster, Phys. Rev. B 68 (2003) 054208. [28] Z. Wang, C. Holm, Phys. Rev. E 68 (2003) 041401. [29] A.O. Ivanov, S.S. Kantorovich, Colloid J. 65 (2003) 166. [30] M. Aoshima, A. Satoh, J. Colloid Interface Sci. 280 (2004) 83. [31] A. Satoh, J. Colloid Interface Sci. 150 (1992) 461. [32] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, J. Chem. Phys. 21 (1953) 1087.
Two-dimensional Monte Carlo simulations of a colloidal dispersion composed of polydisperse ferromagnetic particles in an applied magnetic field Masayuki Aoshima ∗ , Akira Satoh Department of Machine Intelligence and System Engineering, Faculty of System Science and Technology, Akita Prefectural University, 84-4 Ebinokuchi, Tsuchiya-aza, Honjyo, Akita 015-0055, Japan Received 18 October 2004; accepted 26 February 2005 Available online 16 April 2005
Abstract We have investigated aggregation phenomena in a polydisperse colloidal dispersion of ferromagnetic particles simulated by employing the cluster-moving Monte Carlo method in an applied magnetic field. The influence of both particle–particle and particle–field interactions on the aggregate structures is analyzed in terms of a pair correlation function. The results obtained in this study are summarized as follows: Under a strong magnetic field, chainlike clusters are formed along the magnetic field direction, and they become thickly clustered with an increase in the strength of the external magnetic field. Moreover, the thickly clustered chains are formed for a polydisperse system that has a large standard deviation of particle diameters. In contrast, for a very weak magnetic field, the strong interaction between the larger particles gives rise to the formation of various shapes in the chainlike clusters, including bending, looping, and branching. With an increase in the external magnetic field, these structures reorganize to form straight chainlike clusters. Furthermore, the thickness of the chainlike clusters for the polydisperse system is found to depend on the standard deviation of the particle-size distribution but is found to be independent of the magnetic field strength. 2005 Elsevier Inc. All rights reserved. Keywords: Magnetic fluid; Ferromagnetic colloidal dispersion; Particle-size distribution; Chain-like cluster; Monte Carlo simulation
1. Introduction A ferrofluid [1] is classed as a functional fluid because it exhibits attractive functional properties in an applied magnetic field. Another example of a functional fluid is an electrorheological (ER) fluid [2], which responds to an external electric field. These functional fluids are generally produced by dispersing functional particles in a liquid medium and only show the desired functional characteristics under certain conditions. Under the influence of a strong external magnetic field, the ferromagnetic particles in a ferrofluid tend to aggregate to form chain-like clusters [3,4]. These clusters give rise to a large resistance in a flow field. There* Corresponding author. Fax: +81 184 27 2191.
E-mail address: [email protected] (M. Aoshima). 0021-9797/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.02.093
fore, several fluid engineering applications using this phenomenon are under development: for example, mechanical dampers and actuators. In practice, a ferrofluid is a polydisperse colloidal dispersion. Aggregation phenomena arise during the application of an external magnetic field, and this has a significant influence on the physical properties of the fluid, particularly on the viscosity. A microscopic simulation approach such as the Monte Carlo method [5,6] is a powerful technique used to clarify the particle aggregation phenomena. Hence, several molecular simulations using this method have already been conducted on ferromagnetic colloidal dispersions [7–14]. Also, our previous studies clearly show that the aggregate formation is highly dependent on the strength of the magnetic particle–particle interactions for the case under a relatively strong magnetic field [15,16]. This is due to the
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fact that the magnetic moments of particles are strongly oriented toward the magnetic field direction. These previous investigations have generally tended to use an idealized dispersion comprising identical spherical particles with the same diameter. Using this simplified dispersion model, a large number of simulations have been conducted employing various methods of Stokesian dynamics [17,18], and Brownian dynamics [19–23] to investigate aggregation phenomena and rheological properties in a flow field. On the other hand, investigations on polydispersity of the aggregate structures are indispensable for clarifying aggregate structures and magnetic properties of a real ferrofluid. There are a number of studies on polydisperse ferromagnetic particle systems. For example, the log normal [24] and the normal [25] size distributions of particles are assumed for the polydispersity to carry out the MC simulations. Recently, Kristof and Szalai [26] studied the polydisperse system using gamma-distribution in the particle size. Kruse et al. [27] also carried out MC simulations using realistic particle-size distributions obtained from the experimental SAXS data of real ferrofluids. Furthermore, a bidisperse model system, which is supposed to consist of two fractions of magnetic particles with significant size differences, is assumed to describe the polydispersity by means of both the molecular dynamics (MD) simulation [28] and the theoretical consideration [29]. However, these studies mentioned above have mainly investigated the system for relatively weak magnetic interaction between particles. That is, they have studied the short chainlike clusters in detail. If the particle–particle interaction increases, a number of complicated microstructures are expected to be detected. In our previous study [30], we investigated a normal distributed polydisperse system of a ferromagnetic dispersion and clarified the influence of the particle-size distribution on the internal structure of aggregates in the absence of an external magnetic field. We used the cluster-moving Monte Carlo method [31], which is indispensable for simulating a strong interacting system. It was observed that clusters formed possess a complicated internal structure described as a network. In the present study we investigate the aggregation phenomena in a polydisperse system of ferromagnetic particles under conditions of an applied magnetic field. In fact, the cluster-moving Monte Carlo simulation has been conducted for a two-dimensional system. The results of the internal structures of aggregates are analyzed using the pair correlation function.
2. Polydisperse system of ferromagnetic colloidal particles In this study, we consider a two-dimensional system comprising of N spherical particles with a distribution of particle diameters, d, and a magnetic dipole at their center. We also
take into consideration the fact that the particles are coated with a steric layer of thickness δ. In the present study, we assume that the system possess a normal particle-size distribution with an average particle diameter d 0 and a standard deviation σ . If the magnetic moment of particle i and the uniform magnetic field strength are denoted by mi (mi = |mi |) and H (H = |H|), respectively, the energy of interactions be(H ) tween particle i and the magnetic field, ui , the energy of (m) magnetic dipole–dipole interaction, uij , and the interaction (v)
energy due to the overlapping of the steric layers, uij , are expressed as follows [6,30]: (H )
ui
= −kT ξi ni · H/H,
(m)
uij = kT λij
u(v) ij =
(1)
d03 ni · nj − 3(ni · tj i )(nj · tj i ) , 3 rj i
Ci − 1 2Ci λvi kT tδi + 1 −2 2− ln 2 tδi Ci tδi λvj kT tδi + dj /di 2Cj + ln 2− 2 tδi Cj Cj − dj /di , −2 tδi
(2)
(3)
in which Ci = Cj =
(1 + tδi )2 − (dj /di + tδi )2 + 4rj2i /di2 4rj i /di (dj /di + tδi )2 − (1 + tδi )2 + 4rj2i /di2 4rj i /di
, ,
(4)
where k is Boltzmann’s constant, T is the absolute temperature of the fluid, rj i is the magnitude of the vector rj i drawn from particles i to j , and ni and tj i denote the unit vectors given by ni = mi /mi (mi = |mi |) and tj i = rj i /rj i . The parameter tδi in Eqs. (3) and (4) is the ratio of the thickness of the steric layer δ to the radius of the solid portions of particle i, and is expressed as 2δ/di . The dimensionless parameters λvi and λvj in Eq. (3) are written as 2 2 πd02 ns dj di , λv , λvj = λv , λv = λvi = (5) d0 d0 2 in which λv is a dimensionless parameter representing the strength of steric interactions between two particles with the same average ensemble diameter d 0 , and ns is the number of surfactant molecules per unit area on the particle surface. If we assume that the magnitude of the magnetic moment of particles, m(= |m|), is in proportion to its particle volume πd 3 /6, then the dimensionless parameters ξi and λij in Eqs. (1) and (2) can be written as 3 di dj 3 di ξi = (6) ξ0 , λij = λ0 , d0 d02
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in which ξ 0 and λ0 are the dimensionless parameters for the particles with the mean diameter d 0 , representing the strength of particle–field and particle–particle interactions, relative to the thermal energy. These parameters are written as ξ0 = µ0 m0 H /kT and λ0 = µ0 m20 /4πd03 kT . 3. Monte Carlo simulations 3.1. Monte Carlo method In the present study, Monte Carlo (MC) simulations were conducted to investigate the aggregate structures of ferromagnetic colloidal dispersion. Our MC method is based on Metropolis’s algorithm [32], a powerful technique for generating microscopic thermodynamic equilibrium states. In this method a new trial configuration is generated by a random movement of a particle, and its acceptability is then determined by a probability in proportion to the Boltzmann factor exp(−U/kT ), where U is the change in the total energy of the system before and after the movement. However, the conventional Metropolis’s algorithm cannot reproduce physically reasonable aggregate structures for the case of a system with a strong particle–particle interaction. This is due to the fact that a particle move rarely overcomes particle interactions to escape from its cluster to form larger clusters. The cluster-moving MC algorithm [31] was developed for use in the simulation of a strongly interacting system by allowing a cluster to move as a composite single particle. In order to analyze the internal structures of the aggregates, snapshots of particle configuration are used for qualitative discussion. The number density of clusters and the pair correlation function g (2) (r) [15] are evaluated for quantitative analysis. In practice, the correlation function for direc(2) (r) and tions parallel and normal to the magnetic field, gpara (2) gnorm (r), are evaluated in the simulations. 3.2. The simulation parameters Two-dimensional MC simulations have been conducted under the following conditions: The particle number N and the area fraction of the particles to the system region φ a are taken as N = 900 and φa = 0.1, respectively. The particlesize distribution is assumed to be Gaussian, and we have considered two different cases of standard deviation, σ = 0.2 and 0.35, with the same average diameter d0 = 1.0. In this case, the standard deviation of σ ≈ 0.35 is almost maximum value because the minimum diameter of particles is approximately zero. The smooth Gaussian distribution curve is discretized into increments such as d = 0.02d0 and 0.035d0 for specifying the diameter of each particle, as shown in Fig. 1. According to the discretized Gaussian distribution, the number of particles with a specific diameter can be determined. The side length of the square simulation box, L, is taken as L = 85.62d0 and 89.12d0 for the polydisperse systems of σ = 0.2 and 0.35, and L = 84.07d0 for a monodis-
(a)
(b) Fig. 1. Particle-size distributions for σ = (a) 0.2 and (b) 0.35.
perse system, respectively. The initial configuration of particles and the initial directions of the magnetic moments are taken randomly. The thickness of the steric layer is taken as δ = 0.15d0 , and the dimensionless parameter, λv , representing the strength of the steric interactions between the two particles, as λv = 150. If the distance between the surfaces of the steric layer of the two particles is smaller than 0.2d 0 , then these particles are assumed to form a cluster. The cutoff radius for the particle–particle interactions, r coff , is 10d 0 . The maximum displacements of a particle and a cluster in one trial move are 0.5di and 0.5d 0 , respectively. The maximum value for a change in the particle orientation angle is 10◦ . The parameters r and θ for evaluating the pair correlation function are taken as r = d0 /20 and θ = 10◦ , respectively. Under these parametric conditions, we performed the MC simulations for various cases of the dimensionless parameters λ0 and ξ 0 such as λ0 = 2, 4, 6, 8 and ξ0 = 1, 2, 4, 30. Each simulation was executed up to approximately 450,000 MC steps, in which the cluster-moving algo-
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rithm was employed after every 80 MC steps up to the first 270,000 MC steps, and it was ensured that the system attains equilibrium. In the subsequent 180,000 MC steps, the conventional Metropolis algorithm was conducted instead of the cluster-moving procedure until the end of each run. The mean values were evaluated using the data of the last 180,000 MC steps. In order to estimate the finite size effect of the system, we compared the pair correlation functions and the number density of clusters for both systems of N = 900 and 1800 under the strong magnetic interaction between particles (λ0 = 8). It was observed that the pair correlation functions for shorter distance (r/d < 6) and the number density of clusters for a relatively small value of c coincide well with each other. Here, c is the number of particles in a cluster. However, the peaks at a larger value of c (c ≈ 100) depend on the system size. Therefore, we avoid discussing the accuracy in the size of the larger aggregates due to the uncertainty of peak positions of the number density distributions of clusters for a larger value of c.
4. Results and discussion 4.1. Influence of the interaction between particles on aggregate structures In the following sections, we consider the aggregate structures under a strong external magnetic field of ξ0 = 10. The influence of magnetic interactions between the particles on snapshot configurations and pair correlation functions for both small and large standard deviations of the particle-size distribution are compared. 4.1.1. Snapshots of aggregate structures First, we consider the influence of magnetic interactions between particles on snapshots of aggregate structures for the small standard deviation of the particle-size distribution (σ = 0.2). Fig. 2 shows the snapshots of the aggregate structures in equilibrium for λ0 = 2, 4, 6, and 8. Fig. 2a shows that for a relatively weak particle–particle interaction such as λ0 = 2, particles do not aggregate but move almost indepen-
Fig. 2. Influence of interactions between particles on aggregate structures for a polydisperse system with σ = 0.2 and ξ0 = 10: λ0 = (a) 2, (b) 4, (c) 6, (d) 8.
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Fig. 3. Influence of interactions between particles on aggregate structures for a polydisperse system with σ = 0.35 and ξ0 = 10: λ0 = (a) 2, (b) 4, (c) 6, (d) 8.
dently. In the case of λ0 = 4, shown in Fig. 2b, the particles aggregate to form numerous short chainlike clusters. In the case of λ0 = 6, shown in Fig. 2(c), several long chainlike clusters are observed and thick chainlike clusters can also be recognized. In the case of the relatively strong particle– particle interactions such as λ0 = 8, shown in Fig. 2d, long and thick chainlike clusters are formed. Next, we consider the results for the large standard deviation (σ = 0.35) shown in Fig. 3. Although the magnetic interaction between the particles is sufficiently weak for λ0 = 2, several chainlike clusters are observed in Fig. 3a, which is a significant contrast to the previous image in Fig. 2a. Fig. 3a clearly shows that particles with a large diameter are the nuclei for particle aggregation. It is observed that such chainlike clusters become thicker and larger with an increase in the particle–particle interaction. A comparison between Figs. 3d and 2d clarifies that chainlike clusters with a large standard deviation (σ = 0.35) possess more compact internal structures than those with a smaller standard deviation (σ = 0.2).
4.1.2. Number density distributions of clusters In order to quantitatively investigate the influence of the particle–particle interaction on the cluster formation, the number density distributions of clusters are analyzed and plotted on a logarithmic graph as shown in Fig. 4. The ordinates of the figures indicate the number of particles in a cluster c. Moreover, the abscissa of the figures shows the ratio of the number density of clusters n∗c (= nc /d02 ) to the number density of particles in the whole system n∗ (= d02 N/L2 ). The curves for noninteracting polydispersed particles for λ0 = 0 and ξ0 = 0 are also shown in Fig. 4 for comparison. Fig. 4a shows the number density of clusters for σ = 0.2. The curve for λ0 = 2 exhibits a monotonous decrease. Comparing the maximum values of c for λ0 = 2 and 0, we observe that the maximum number of particles in a cluster, c, for λ0 = 2 is slightly larger than that for λ0 = 0. This indicates the formation of a small amount of small clusters. As λ0 increases, the number density of clusters increases and finally the curve shows a distinguishable peak approximately at c ≈ 100 for λ0 = 8. This peak corresponds to the
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(a)
(b) Fig. 4. Influence of interactions between particles on number density distributions of clusters for σ = (a) 0.2, (b) 0.35.
thick chainlike clusters along the magnetic field direction, as shown in Fig. 2d. Fig. 4b shows the number density of clusters for σ = 0.35. For λ0 = 4, the peak indicating the formation of large clusters appears approximately at c ≈ 50, which is in contrast to the case of σ = 0.2 (Fig. 4a). For strong magnetic interaction between particles, such as λ0 = 6 and 8, several peaks corresponding to the formation of large clusters appear approximately at c ≈ 100; the monotonous decreasing curve at the lower region of c simultaneously shifts close to the curve for noninteracting dispersed particles (λ0 = 0, ξ0 = 0). These suggest that the larger particles form thick chainlike clusters and also that the smaller particles disperse as single particles throughout the space among the clusters. 4.1.3. Pair correlation functions of aggregate structures Fig. 5 shows the results of the pair correlation functions, (2) (2) gpara (r) and gnorm (r), in which Figs. 5a and 5b represent σ = 0.2, and Figs. 5c and 5d represent σ = 0.35. Each figure
depicts the curves for different magnetic particle–particle interactions such as λ0 = 2, 4, 6, and 8. Several sharp peaks are observed in the case when the pair correlation function is parallel to the magnetic field, shown in Figs. 5a and 5c, and these peaks became more significant with the larger values of magnetic interactions, λ0 . When λ0 = 4, the first, second, and third peaks appear at r/d0 ≈ 1.5, 3.0, and 4.5, respectively. The positions of the peaks are integer multiples of the average diameter of the particles including the steric layer (δ/d0 = 0.15). Hence, this fact clearly shows that the particles with average diameter of approximately (1.2d 0 ) constitute short stable chainlike clusters. This result agrees well with the results of other researcher’s studies assuming a bidisperse model system [28,29]. Thus, we conclude that the larger particles tend to form chainlike clusters along the magnetic field direction for relatively weak particle–particle interaction. As the value of λ0 increases, the positions of those peaks shift toward the shorter distances, and the peaks become sharper. This suggests that, as the magnetic interactions between the particles increase, the short clusters associate with smaller particles to form stable longer chainlike clusters along the magnetic field direction. The peaks for σ = 0.35 appear at longer distance positions than for σ = 0.2, which implies that the larger particles are contributing to the cluster formation. Comparing these pair correlation functions parallel to the magnetic field for polydisperse systems with the results for a monodisperse system [15], the peaks for a polydisperse system are much broader than that for a monodisperse system. This also supports the fact that the chainlike clusters for a polydisperse system consist of the particles with a size distribution. In the case when the pair correlation function is normal to the magnetic field, shown in Figs. 4b and 4d, the curves have gaslike characteristics and only the first peak occurs in both cases of σ = 0.2 and 0.35. As determined earlier, thick chainlike cluster for σ = 0.35 possess a more compact internal structure (not loose as in Fig. 2) than for σ = 0.2, so that the peaks for σ = 0.35 are much higher than for σ = 0.2. In the case of σ = 0.35, the curve for λ0 = 4 suggests that the clusters are observed even in the curve of a very small value of λ0 , which is in good agreement with the snapshot results shown in Fig. 3b. The results obtained above are summarized as follows: As the value of λ0 is increased, the particles form chainlike clusters along the magnetic field direction. In the case of a small standard deviation in the particle-size distribution (σ = 0.2), straight chainlike clusters extend along the magnetic field direction with an increase in the values of λ0 . This is due to the fact that single-moving particles combine with short chainlike clusters. In contrast with the above results, the snapshots for the large standard deviation (σ = 0.35) show the formation of thick chainlike clusters. These results can be explained by the fact that the larger particles form clumplike clusters, and they extend along the magnetic field direction with an increase in the values of λ0 .
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Fig. 5. Influence of interactions between particles on pair correlation functions for ξ0 = 10: (a) and (c) for the direction parallel to the field direction; (b) and (d) for the direction normal to the field direction; (a) and (b) for σ = 0.2; (c) and (d) for σ = 0.35.
4.2. Influence of magnetic field on aggregate structures In this section, we consider the aggregate structures for the strong magnetic interactions between particles such as λ0 = 8. The influence of the magnetic field on snapshot configurations and pair correlation functions for a monodisperse system and both small and large standard deviations of the particle-size distribution are compared. 4.2.1. Snapshots of aggregate structures The influence of magnetic field strength on the aggregate structure was investigated using snapshots of the aggregate structures in equilibrium for the magnetic fields ξ0 = 1, 2, 4, 30, as shown in Figs. 6, 7, and 8. First, we consider the influence of the magnetic field on snapshots of aggregate structures for the monodisperse system shown in Fig. 6. It is observed from Fig. 6a that moderately bent chainlike clusters are formed for a very weak magnetic field such as
ξ0 = 1. There exist several branchlike structures in the chainlike clusters. When ξ0 = 2 and 4 as shown in Figs. 6b and 6c, the chainlike clusters are not so extensively bent but the branchlike structures are still observed in the clusters. For a stronger magnetic field such as ξ0 = 30 (Fig. 6d), straight chainlike clusters with very little branchlike structures are formed along the magnetic field direction. Next, we consider the influence of interaction between a particle and the magnetic field on snapshots of aggregate structures for the small standard deviation of the particle-size distribution (σ = 0.2), as shown in Fig. 7. For all cases of particle–field interactions, the chainlike clusters are thicker than that for the monodisperse system (Fig. 6). In the case of a very weak particle–field interaction such as ξ0 = 1 and 2, shown in Figs. 7a and 7b, the structures of moderately bent chainlike clusters are almost similar to the snapshot for the monodisperse system (Figs. 6a and 6b). However, we can
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Fig. 6. Influence of magnetic field strength on aggregate structures for the monodisperse system with λ0 = 8: ξ 0 = (a) 1, (b) 2, (c) 4, (d) 30.
clearly observe the difference between the snapshots for the monodisperse and polydisperse systems; i.e., small looplike structures are included in the thick chainlike clusters in the polydisperse system. As the strength of particle–field interaction increases, the thick chainlike clusters with looplike structures become straight along the magnetic field direction. Finally, we consider the influence of interaction between a particle and the magnetic field on snapshots of aggregate structures for the large standard deviation of the particle-size distribution (σ = 0.35). Fig. 8 shows the snapshots of the aggregate structures in equilibrium for ξ0 = 1, 2, 4, and 30. In the case of a very weak particle–field interaction such as ξ0 = 1, shown in Fig. 8a, thick and bent chainlike clusters, which have several branchlike and small looplike structures, are formed. As the magnetic field increases for ξ0 = 2 and 4 (Figs. 8b and 8c), the thick chainlike clusters become aligned along the magnetic field direction. These clusters have compact internal structures without the small looplike structure. In the case of a significantly strong magnetic field such as
ξ0 = 30 (Fig. 8d), the straight chainlike clusters are seen to be thicker. Since the clusters are composed of well-aligned particles along the magnetic field direction, the branchlike structures are much less evident. 4.2.2. Number density distributions of clusters In order to quantitatively investigate the influence of particle–field interaction on the cluster formation, the number density distributions of clusters for the monodisperse and polydisperse systems are analyzed and plotted on logarithmic graphs as shown in Fig. 9. Also, the curve for noninteracting particles for λ0 = 0 and ξ0 = 0 are added to the graphs for comparison. Almost all curves for the monodisperse and polydisperse systems depend weakly on the strength of the particle–field interaction. Fig. 9a shows continuous, convex curves of the number density of clusters for the monodisperse system. As the particle–field interaction increase, the maximum number of particles in a cluster increases from approximately c ≈ 100 to 200, whereas the number density of smaller clusters
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Fig. 7. Influence of magnetic field strength on aggregate structures for a polydisperse system with σ = 0.2 and λ0 = 8: ξ 0 = (a) 1, (b) 2, (c) 4, (d) 30.
(c < 7) decreases. This shows the fact that smaller clusters aggregate to form larger clusters, and also that the sizes of the clusters are distributed continuously. On the other hand, the curves of the number density distributions of clusters for polydisperse systems are much more different than that for monodisperse systems. Fig. 9b shows the number density distributions of clusters for a small standard deviation (σ = 0.2). The curves for σ = 0.2 become linear at values lower than c ≈ 50. That is, the number densities of middle-sized clusters are lower than that in the case of the monodisperse system, whereas the densities of small clusters (c < 5), including single particles, are higher than that in the case of the monodisperse system. Moreover, the significant peak, which appears at approximately 100, corresponds to the formation of stable, thick chainlike clusters. When the contribution of the polydispersity is sufficiently large (σ = 0.35), as shown in Fig. 9c, the number density curves are split into two regions: linear curves at lower c and several peaks at higher c. The linear curves are slightly lower than the curve for noninteracting particles. This implies that
the particles that do not belong to the clusters are separated clearly from the particles in clusters. 4.2.3. Pair correlation functions of aggregate structures We now quantitatively investigate the effect of the magnetic field strength on the internal structure of the agglomerates by employing the pair correlation functions. Figs. 10 and 11 show the pair correlation functions of the monodisperse and the polydisperse (σ = 0.2 and 0.35) systems, respectively. First, we consider the pair correlation functions of the monodisperse system. Figs. 10a and 10b indicate the internal structures for the direction parallel to the magnetic field and for the direction normal to the magnetic field, respectively. In the case when the direction is parallel to the magnetic field, as shown in Fig. 10a, several of the peaks gradually increase with an increase in the value of ξ 0 . This implies that the particles in the chainlike clusters become well aligned along the magnetic field direction. Fig. 10b shows the pair correlation functions for the direction normal to the magnetic
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Fig. 8. Influence of magnetic field strength on aggregate structures for a polydisperse system with σ = 0.35 and λ0 = 8: ξ 0 = (a) 1, (b) 2, (c) 4, (d) 30.
field. In the case of a very weak particle–field interaction such as ξ0 = 1, shown in Fig. 10b, the pair correlation functions exhibit two clear, sharp peaks at r/d0 = 1.3 and 2.6, which are integer multiples of the diameter of a particle, including thickness of the steric layer. These peaks result from the bent chainlike clusters. With an increase in the particle– field interaction, these peaks decrease gradually. When the magnetic field increases significantly (ξ0 = 30), all these peaks finally disappeared because the thin straight chainlike clusters along the magnetic field are formed due to the application of a strong magnetic field. Next, we consider the pair correlation functions for the small standard deviation (σ = 0.2) shown in Figs. 11a and 11b. In the case when the direction is parallel to the magnetic field, as shown in Fig. 11a, some peaks gradually increase with an increase in the value of ξ 0 . This im-
plies that the particles in the chainlike clusters become well aligned along the magnetic field direction. Fig. 11b shows the pair correlation functions for the direction normal to the magnetic field. The intensity of the first peak of the functions is seen to be independent of the magnetic field. This implies that the thickness of the chainlike clusters remains unchanged with increasing strengths of the magnetic field. On the other hand, the second peak shown for ξ0 = 1 disappears with an increase in the magnetic field. This implies that the branchlike or bent structures of chainlike clusters for ξ0 = 1 disappear because thick, straight, chainlike clusters are formed due to the application of a strong magnetic field. Finally, we consider the pair correlation functions for the large standard deviation (σ = 0.35) shown in Figs. 11c and 11d. Fig. 11c shows the case for the correlation function
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Fig. 9. Influence of magnetic field strength on number density distributions of clusters: (a) for the monodisperse system; for σ = (b) 0.2 and (c) 0.35.
(a)
(b)
Fig. 10. Influence of magnetic field strength on pair correlation functions for the monodisperse system and λ0 = 8: (a) for the direction parallel to the field direction; (b) for the direction normal to the field direction.
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Fig. 11. Influence of magnetic field strength on pair correlation functions for λ0 = 8: (a) and (c) for the direction parallel to the field direction; (b) and (d) for the direction normal to the field direction; (a) and (b) for σ = 0.2; (c) and (d) for σ = 0.35.
parallel to the magnetic field direction. Several peaks that indicate the chainlike cluster formation gradually increase less than is the case for σ = 0.2, as shown in Fig. 11a. This implies that the particles are less aligned along the magnetic field direction even in the case of a strong magnetic field. Fig. 11d shows the pair correlation functions for the direction normal to the magnetic field direction. For ξ0 = 1, the pair correlation functions have both the first and second peaks. The small second peak originates from the branchlike structures or the bent chainlike clusters. As the strength of the magnetic field increases (ξ0 = 2 and 4), the second peak decreases slightly, while the first peak remains unchanged. This decrease in the second peak suggests that the branchlike structures or the bent chainlike clusters disappear. The first peak, which is independent of the magnetic field, indicates that the thickness of the chainlike clusters remains unchanged with an increase in the strength of the magnetic
field. In the case of a significantly strong magnetic field such as ξ0 = 30, the slight increase of the first peak implies the formation of thick chainlike clusters. The results obtained above are summarized as follows: In the case of the monodisperse system, thin, bent chainlike clusters become straight with an increase in the strength of the magnetic field. In the case of the small standard deviation of σ = 0.2, the branchlike, looplike, and bent chainlike clusters decrease with an increase in the strength of the magnetic field. In the case of a significantly strong magnetic field, particles align strongly along the magnetic field direction and form thick, straight, chainlike clusters. The reason for these characteristics may be interpreted as follows: In the case of a weak influence of the magnetic field, the particles that aggregate around the large particles prefer to form branchlike or clumplike structures rather than form straight chainlike clusters. This is because the aggregation is dominated by
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the influence of the interaction between the particles. A detailed discussion regarding the formation of the branchlike or the clumplike structures will be given in Section 4.3. As the strength of the magnetic field increases, the relative influence of the interaction between the particles decreases. This leads to the formation of straight chainlike clusters along the magnetic field direction. In the case of a large standard deviation of σ = 0.35, large clumplike clusters are formed in the initial stages of the chainlike cluster formation during the simulation. This is because there are several large particles that possess a particle– particle interaction strong enough to form the clumplike clusters. However, as the influence of the magnetic field dominates, the particles align along the magnetic field direction to form the straight chainlike clusters. This leads to the formation of the rather thick straight chainlike clusters, whereas the number of branchlike and bent structures found in the chainlike clusters decrease. 4.3. Influence of particle size on the interaction energy between three particles in a cluster One of the characteristics of the particle aggregation in the polydisperse system is the appearance of various interesting structures, such as the clumplike and the branchlike structures, due to the strong interaction of the larger particles. In order to investigate the formation of these structures, we consider a case in which the three particles are in contact with each other at the surface of their steric layers. The magnetic moments of these three particles is assumed to be aligned with the magnetic field direction. One of the three particles is denoted by i and the others are denoted by j . We also assume that two particles j have the same diameters dj . We then propose two representative configurations of these three particles, linear and triangular, as illustrated in Fig. 12. In order to investigate the influence of the diameter di of the particle i on potential energies of the linear and the (m) (m) triangular configurations, ulin and utri , we have derived the following equations from Eqs. (2) and (6) using a geometrical consideration, d03 di dj 3 d03 (m) + ulin /kT λ0 = −2 (rij + rjj )3 d02 rij3 2 3 dj d03 −2 2 (7) , 3 d0 rjj u(m) tri /kT λ0 = 2 +
di dj
3
d03 rij3
d02 2 3 dj d03 d02
3 rjj
1−3
,
2 − rjj 4rij2
4rij2
Fig. 12. Influence of particle diameter di /d0 on cluster formation energy for dj /d0 = 0.6.
i.e., rij = di /2 + dj /2 + 2δ, and rjj is the distance between the centers of the two particles j when they are in contact with each other, i.e., rjj = dj + 2δ. The graph of the potential energy changes, which are calculated from Eqs. (7) and (8) with the nondimensional particle diameter di /d0 for the case of dj /d0 = 0.6, are shown in Fig. 8. It can be seen that the potential energy curves of the linear and the triangular configurations intersect at di /d0 ≈ 1.26. This implies that the triangular configuration is more stable than the linear configuration when the value of di /d0 is larger than 1.26. This is because the influence of the attractive interaction between particles i and j dominates over the influence of the repulsive interaction between the two particles j with an increase in the particle diameter. The formation of the triangular configuration for the polydisperse system enhances the formation of the branchlike or the clumplike structures. If the system is three-dimensional, the chainlike clusters possibly branches off in more than two directions because another fundamental configuration of clusters may be formed, such as a pyramidal cluster. This may lead to more complicated network microstructures.
5. Conclusions
(8)
in which, rij is the distance between the centers of the two particles i and j when they are in contact with each other,
We have investigated the aggregation phenomena occurring in a polydisperse colloidal dispersion composed of ferromagnetic particles, simulated by means of the clustermoving Monte Carlo method, in an applied magnetic field. Influences of both particle–particle and particle–field interactions on the internal structures of the aggregate structures are analyzed in terms of a pair correlation function. The
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results obtained in this study are summarized as follows: Under a strong magnetic field, chainlike clusters are formed along the magnetic field direction and they become thick clusters with an increase in the strength of external magnetic field. Moreover, the thick chainlike clusters are formed more readily for a polydisperse system that has a large standard deviation of particle size. In contrast, for a relatively weak magnetic field, various shapes of the chainlike clusters, such as bending, small looplike and branchlike structures are formed due to a strong interaction between the larger particles. With an increase in the external magnetic field, these structures reorganize to form the straight chainlike clusters. Furthermore, the thickness of the chainlike clusters for the polydisperse system depends on the standard deviation of a particle-size distribution but is independent of the magnetic field strength.
Acknowledgment This work was partly supported by a Grant-in-Aid for Scientific Research (C) from the Japanese Ministry of Education, Sports, Science and Technology.
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