Interacting Brownian particles

Having seen the basic tools for describing Brownian motion, we now consider more realistic situations. Suppose that a collection of spherical Brownian particles, all having equal size, are suspended in a fluid and interacting with each other. Such systems are often found in colloidal suspensions. As we shall discuss in the next section, the study of this 

To obtain the Smoluchowski equation for such a system, we first calculate the mobility matrix. Let Rt,Ri, . , i Ar = i be the positions of the spheres and Fi, J, . Fv be the forces acting on them. We assume that there eire no external torques acting on the particles. Then the velocities of the particles are written ast 

In a very dilute suspension, the velocity of a particle is determined only by the force acting on it, and the mobility matrix becomes 

To calculate the particle velocities V (n = l,2. N) we have to know the fluid velocity v(r) created by the external forces acting on the particles. In the usual condition of Brownian motion, the relevant hydrodynamic equation of motion is that of the low Reynolds number 

The hydrodynamic interaction. The force acting on the particle m creates a velocity field and causes the motion of other particles. 

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When one is interested in slow modes of motion of the system, each macromolecule of the system can be schematically described in a coarse-grained way as consisting of N + 1 linearly-coupled Brownian particles, and we shall be able to look at the system as a suspension of n(N + 1) interacting Brownian particles. An anticipated result for dynamic equation of the chains in equilibrium situation can be presented as a system of stochastic non-Markovian equations... 

As was demonstrated by Pyshnograi (1994), the last term in (6.7) can be written in symmetric form, if the continuum of Brownian particles is considered incompressible. In equation (6.7), the sum is evaluated over the particles in a given macromolecule. The monomolecular approximation ensures that the stress tensor of the system is the sum of the contributions of all the macromolecules. In this form, the expression for the stresses is valid for any dynamics of the chain. One can consider the system to be a dilute polymer solution or a concentrated solution and melt of polymers. In any case the system is considered as a suspension of interacting Brownian particles. 

In equation (6.33), the stresses in the moving viscoelastic liquid (6.31) are added to the stresses in the continuum of Brownian particles. When the equations of motion are formulated, we have to take into account the presence of the two interacting and interpenetrating continuous media formed by the viscoelastic liquid carrier and the interacting Brownian particles that model the macromolecules. However, the contribution of the carrier in the case of a concentrated solution is slight, and we shall ignore it henceforth. 

MCT considers interacting Brownian particles, predicts a purely kinetic glass transition, and describes it using only equilibrium structural input, namely the equilibrium structure factor Sq [3,46] measuring thermal density fluctuations. MCT-ITT extends this statistical mechanics, particle based many-body approach to dispersions in steady flow assuming a linear solvent velocity profile, but neglecting the solvent otlrerwlse. 

Without doing detailed quantitative analysis of the data, it can be stated that the polyion diffusion can be qualitatively described by two theoretical concepts. The first concept capable of qualitative explanation of the polyion diffusion is the concept based on considering polyions as interacting Brownian particles with direct interactions between polyions and hydrodynamic interactions. The short-time collective diffusion coefficient for a system of interacting Brownian particles treated by statistical mechanics is calculated from the first cumulant F of the dynamic structure factor S(q, t) as [15-17]... 

Pusey PN. The dynamics of interacting Brownian particles. J Phys A Math Gen 1975 8 1433-1440. 

Ackerson BJ. Correlations for interacting Brownian particles. J Chem Phys 1976 64 242-246. 

Klein R. Interacting Brownian particles the dynamics of colloidal suspensions. In Mallamace F, Stanley HE, eds. The Physics of Complete Systems. Amsterdam IOS Press, 1997 301-345. 

A starting point for developing a DDFT for N interacting Brownian particles is the Fokker-Planck (or Smoluchowsky) equation for the probability density W( r,, - = i. . for finding particle 1 at position ri, particle 2 at position V2,. .., particle i at position r and finally particle N at position Tat. For pairwise interactions among the particles, we have... 

Archer AJ, Rauscher M (2004) Dynamical density functional theory for interacting Brownian particles stochastic or deterministic J Phys A Math Gen... 

This is the basic equation describing the interacting Brownian particles. 

The value of these approaches to the problem is that they demonstrate the possibility of evaluating the memory function through the intermolecular correlation functions and structural dynamic factor of the system of interacting Brownian particles. Theoretical evaluations of the memory function are based on some approximations which, apparently, do not allow the calculation of quantities for the limiting case c M, but they are good in the cross-over region from Rouse to entanglement dynamics. 

So, one can consider the parameters r, Xi and Me to be equivalent. One of these parameters is introduced in each theory of polymer dynamics. Note that the correlation time is expressed through the two-particle correlation function and the dynamic structure factor of the system of the interacting Brownian particles in many-chain theories [54, 91]. 

Kikuchi et al. demonstrated, however, that MMC is not restricted to the calculation of equilibrium properties, but can also be used to study dynamic properties. Specifically, they applied the MMC method to the study of Brownian motion of a harmonically bound particle [19]. The same authors further extended the method to study interacting Brownian particles including the effects of hydrodynamic interactions [20]. 

Cichocki, B. 1988. Linear kinetic theory of a suspension of interacting Brownian particles. 1. Enskog... 

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