Autocorrelation functions involve concepts that one sometimes reads about in analyses of transport in liquids and are therefore concepts with which the student should be familiar. These are mathematical devices used in some discussions of the theory of transport in liquids, e.g., in treatment of viscous flow and diffusion. It is possible, for example, to represent the diffusion coefficient D in terms of the autocorrelation function of a particle's velocity. When a particle in solution collides with another (say, an ion colliding with a solvent molecule), the collision is tantamount to a force operating for a very short time on the particle. This time is much shorter than the time needed for the normal so-called "Stokes' resistance" (see Section 4.4.7), which slows down the particle that has "received a knock" to the average velocity (e.g., in flow). Now consider dealing with diffusion coefficients in terms of molecular dynamics (see a description of this technique in Section 2.3.2). This method is used to determine how the dynamics of motion of a given particle will be affected by its collision with other particles. Thus, it is important to know the relation between the velocity of a particle at the beginning (t = 0), that is, just after collision, and that at some later time t.
This is where one brings in the concept of the velocity autocorrelation/unction. This indeed concerns the velocity at t=0, v0, and the velocity at some subsequent time, vr To what extent does the subsequent velocity depend upon the velocity at time t = 0? That is the sort of information given by the autocorrelation function.
There are several technical details in a rigorous definition of the autocorrelation function for velocity. First, one has to remember the vectorial character of velocity, because clearly the direction in which the particle is knocked is important to its subsequent dynamic history. Then, according to the way it is defined, one has to take the product of the velocity at t = 0, v0, and that at the later chosen time, vr However, it is not as simple as just multiplying together the two vectors, v0 and v(. One has to allow for the distribution of positions and momenta of the particle in the system at the beginning, that is, at t = 0. To allow for this, one can introduce symbolically a probability distribution coefficient, g0. Therefore, the expression for the autocorrelation function will involve the product g0vovt.
Thereafter, there is only one step left, but it is a vital one. One has to carry out an averaging process for the entire liquid (or solution) concerned. Such averaging processes can be carried out in more than one way. One of these involves an integration with respect to time. One ends up by writing down the full-blown expression for the autocorrelation function as a function of an expression dependent on time, A(t). Then, in a general way, an autocorrelation function would run where the brackets represent "the average value of," as defined in Eq. (4.20).
How can this concept be used to calculate diffusion coefficients in ionic solutions? First one has to remember that for diffusion in one direction,
However, the displacement of the particle x is in reality a function of time and therefore can also be expressed in terms of an autocorrelation function similar to that presented in Eq. (4.115). One advantage of this procedure is that the autocorrelation function will depend only on a time interval, r -f, and not on the time itself. Through the use of Eq. (4.116) and some mechanics produced earlier in the century by Langevin, one finds that the diffusion coefficient D can be expressed by the time integral of the velocity autocorrelation function and eventually obtains the useful equation where fc is the frictional coefficient.
Does this concern ions in solution and electrochemistry? It does indeed concern some approaches to diffusion and hence the related properties of conduction and viscous flow. It has been found that the autocorrelation function for the velocity of an ion diffusing in solution decays to zero very quickly, i.e., in about the same time as that of the random force due to collisions between the ion and the solvent. This is awkward because it is not consistent with one of the approximations used to derive analytical expressions for the autocorrelation function.18 The result of this is that instead of an analytical expression, one has to deal with molecular dynamics simulations.
One of the simplest examples of this type of calculation involves the study of a system of rare-gas atoms, as in, e.g., calculations carried out on liquid argon. The relaxation time after a collision was found to be on the order of 10"13 s, which is about the same time as that for rather large ions (e.g., of 500 pm). Thus, much of what one learns from the MD study of molecular motion in liquid argon should be applicable to ionic diffusion.
Figure 4.40 shows the velocity autocorrelation function for liquid argon as calculated by Levesque et al. Looking now at this figure, one can see at first the fast exponential decay of the autocorrelation function. The function then becomes negative (reversal of velocity), indicating a scattering collision with another molecule. At longer times it trails off to zero, as expected, for eventually the argon atom's motion becomes unconnected to the original collision. The time for this to happen is relatively long, about 10"12 s.
In summary, then, autocorrelation functions are useful mathematical devices which, when applied to velocities, tell us to what degree the motion of a particle at a given instant is related to the impelling force of the last collision. Their usefulness is mainly in molecular dynamics, the principal computer-oriented method by which systems are increasingly being analyzed (Section 2.17).
18Here it is assumed that the influence of the collision lasts a lot longer than the force due to the collision itself.
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