Time-Dependent Friction Function Exact Expression

1 Time-Dependent Friction Function Exact Expression 

Let us go back to Equation 1.1, but now imagine that the Brownian particle diffuses in a colloidal dispersion formed by other N particles in the volume V with which it interacts by means of pairwise direct (i.e., conservative) interactions but in the absence of hydrodynamic interactions. The pairwise force that this tracer particle (T) exerts on particle / is given by Fj- = -V M( r - rj. ), so that Equation 

We now need a relaxation equation that couples the time derivative of the variable 8 (r, t) with this variable and with y it). This must be a linear version of a generalized diffusion equation, whose structure is dictated by the rigid format of the GLE of Equation 1.8 applied to the vector 8a(t) = [vt-(0, 8n (0] which leads to [24] 

Equations 1.11 and 1.12 provide an exact description of the Brownian motion of the tracer particle coupled to the fluctuations of the local concentration n (r, t) in terms of the components of the vector 8a(0 = [vy(0. 5 (f)l. If one is interested only in the tracer s velocity, one wonld have to eliminate 8n ( ) from this description, and the corresponding process is referred to as a contraction of the description. In the present case, this is achieved by formally solving Eqnation 1.12 for 8n (r, t) and substituting the solution in Equation 1.11, which then becomes 

The function F (k, t) just defined is the intermediate scattering function, except for the asterisk, which indicates that the position vectors r (0 and r,(0) have their origin in the center of the tracer particle. Denoting by the position of the tracer particle referred to a laboratory-fixed reference frame, we may re-write F (k, t) as 

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Let us now apply the general concepts above to the description of the Brownian motion of a tracer particle that interacts with the other particles of a colloidal dispersion [23,55]. In this manner we will derive an exact result for the time-dependent friction function A (t), which is later given a useful approximate expression. 

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