Diffusion-constrained systems

The theory of Brownian motion for a constrained system is more subtle than that for an unconstrained system of pointlike particles, and has given rise to a substantial, and sometimes confusing, literamre. Some aspects of the problem, involving equilibrium statistical mechanics and the diffusion equation, have been understood for decades [1-8]. Other aspects, particularly those involving the relationships among various possible interpretations of the corresponding stochastic differential equations [9-13], remain less thoroughly understood. This chapter attempts to provide a self-contained account of the entire theory. 

In this section, we use simple phenomenological arguments to construct a diffusion equation for constrained systems, in a notation that is common to both... 

Brownian motion of a constrained system of N point particles may also be described by an equivalent Markov process of the Cartesian bead positions R (f),..., R (f). The constrained diffusion of the Cartesian coordinates may be characterized by a Cartesian drift velocity vector and diffusivity tensor... 

The connection between a diffusion equation and a corresponding Markov diffusion process may be established through expressions for drift velocities and diffusitivies. The drift velocity for both unconstrained and constrained systems may be expressed in an arbitrary system of coordinates in the generic form... 

An approximate resolution of the system above leads to an equation relating the substrate concentration at the active sites, (CA) [=0, to its value at the boundary between the constrained diffusion layer and the linear diffusion layer, (Ca)x=2Rq ... 

The content of diffusion equation (2.175) for such a model is, moreover, independent of our choice of a system of 3,N coordinates for the unconstrained space. Constrained Brownian motion may thus be described by a model with a mobility and an effective potential /eff in any system of 3N coordinates for... 

Throughout this section, we will use the notation X (t),..., X t) to denote a unspecified set of L Markov diffusion processes when discussing mathematical properties that are unrelated to the physics of constrained Brownian motion, or that are not specific to a particular set of variables. The variables refer specifically to soft coordinates, generalized coordinates for a system of N point particles, and Cartesian particle positions, respectively. The generic variables X, ..., X will be indexed by integer variables a, p,... = 1,...,L. 

In many cases, changes in one extensive quantity are coupled to changes in others. This occurs in the important case of substitutional components in a crystal devoid of sources or sinks for atoms, such as dislocations, as explained in Section 11.1. Here the components are constrained to lie on a fixed network of sites (i.e., the crystal structure), where each site is always occupied by one of the components of the system. Whenever one component leaves a site, it must be replaced. This is called a network constraint [1]. For example, in the case of substitutional diffusion by a vacancy-atom exchange mechanism (discussed in Section 8.1.2), the vacancies are one of the components of the system every time a vacancy leaves a site, it is replaced by an atom. As a result of this replacement constraint, the fluxes of components are not independent of one another. 

The substitutional binary alloy diffusion illustrated in Fig. 3.3 is discussed in a treatment pioneered by Darken [6] (see also Crank s book [7]). The system has three components, species 1, species 2, and vacancies, and is assumed to be at constant pressure and temperature with sites that can only be created or destroyed at sources (i.e., the system is network constrained except at dislocations or interfaces). The fluxes are obtained from Eqs. 2.21 and 2.32 ... 

Another system obeying Fick s law is one involving the diffusion of small interstitial solute atoms (component 1) among the interstices of a host crystal in the presence of an interstitial-atom concentration gradient. The large solvent atoms (component 2) essentially remain in their substitutional sites and diffuse much more slowly than do the highly mobile solute atoms, which diffuse by the interstitial diffusion mechanism (described in Section 8.1.4). The solvent atoms may therefore be considered to be immobile. The system is isothermal, the diffusion is not network constrained, and a local C-frame coordinate system can be employed as in Section 3.1.3. Equation 2.21 then reduces to... 

The superscript 0 on the diffusivities listed above refers to the fact that these are for dilute solutions. In a concentrated system the rate of rotation will be slowed down considerably because of steric hinderance from nearest neighbors. The nature of the entanglements from other rods onto a test rod is such that the translational motion perpendicular to the rod axis becomes highly constrained. The translation along the chain axis, on the other hand, is for the most part unaffected. The steric interactions imposed by the neighboring rods on a single test rod can be modeled by placing such a rod within a tube of radius ac... 

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