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Brownian MotionFluctuations, Dynamics, and Applications$

Robert M. Mazo

Print publication date: 2008

Print ISBN-13: 9780199556441

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780199556441.001.0001

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(p.268) Appendix F Mutual Diffusion and Self-Diffusion

(p.268) Appendix F Mutual Diffusion and Self-Diffusion

Source:
Brownian Motion
Publisher:
Oxford University Press

In this appendix we want to discuss the difference between mutual diffusion and self-diffusion from the point of view of the thermodynamics of irreversible processes. A fuller discussion of this topic can be found in Tyrrell and Harris (1984).

F.1 Mutual Diffusion

We first consider mutual diffusion in a two component mixture. Component 0 denotes the solute and component 1 denotes the solvent. Let Ji denote the diffusion flux of component i in the barycentric coordinate system. That is, the flux is measured relative to the local center of mass motion of the fluid. The basic description of diffusion is Fick’s law

(F.1.1) Appendix F Mutual Diffusion and Self-Diffusion

Because we are in the barycentric coordinate system, J0 + J1 = 0 and only one of the fluxes is independent. It then follows that L 00 = -L 01. The relation between the flux J0 and the thermodynamic forces, the gradients of chemical potential, is then

(F.1.2) Appendix F Mutual Diffusion and Self-Diffusion

Using the Gibbs-Duhem relation c 0∇µ0 + c1∇µ 1 = 0, eqn (F.2) can be written

(F.1.3) Appendix F Mutual Diffusion and Self-Diffusion

where x 1 is the mole fraction of component 1. Consequently the relationship between the phenomenological coefficient L 00 and the mutual diffusion coefficient, is

(F.1.4) Appendix F Mutual Diffusion and Self-Diffusion

(p.269) F.2 Self-Diffusion

Now we consider the same system except that now some of the solute molecules (component 0) are tagged somehow. The tagged mo;ecules are denoted as component α and the untagged solute molecules as component β. The fractions of tagged and untagged molecules are yα and yβ respectively. cα + cβ = c 0. It is assumed that molecules of type α and of type β are identical except for tagging.

The fluxes of the solute species can be written as

(F.2.1) Appendix F Mutual Diffusion and Self-Diffusion

In other words, the tagged system is treated as a three component mixture. The l coefficients are the phenomenological coefficients appropriate to the discussion of the three component system. The chemical potentials of the tagged species are

(F.2.2) Appendix F Mutual Diffusion and Self-Diffusion

Here, µ 0 is the chemical potential of component zero. It is not a standard state chemical potential; there is no implication that the solution is ideal.

In a self diffusion experiment, the concentrations of species 0 and 1 are uniform. Thus µ 0 = µ 1 = 0. Since yα + yβ = 1, = -∇. Consequently

(F.2.3) Appendix F Mutual Diffusion and Self-Diffusion

Writing Jβ = -Ds as the definition of the self-diffusion coefficient, Ds we find that

(F.2.4) Appendix F Mutual Diffusion and Self-Diffusion

F.3 Relation between Dm and D s

Now we return to the general situation where components 0 and 1 are not necessarily uniform in space. Since J0 = Jα + Jβ, we have

Appendix F Mutual Diffusion and Self-Diffusion
(p.270)
(F.3.1) Appendix F Mutual Diffusion and Self-Diffusion

where we have used the Onsager relation lαβ = lβα. For this equation to be compatible with eqn (F.1.3) the coefficient of ▽µ 0 must be L 00/x 1 and the coefficient of ▽ must vanish. These conditions yield

(F.3.2) Appendix F Mutual Diffusion and Self-Diffusion

where we have again used the Onsager relations. The second of eqns (F.3.2) can be used to eliminate lαβ from the first, and we find

(F.3.3) Appendix F Mutual Diffusion and Self-Diffusion

The equation for Ds becomes

(F.3.4) Appendix F Mutual Diffusion and Self-Diffusion

To go any further it is necessary to know more about the y dependence of the l coefficients. Irreversible thermodynamics is silent on the subject of the dependence of phenomenological coefficients on thermodynamic state. It just assumes that the coefficients are independent of gradients so that the theory is linear. It is clear, however, that there is no a priori reason why eqns (F.3.3) and (F.3.4) should be the same. First of all, (F.3.3) contains the thermodynamic derivative (∂µ 0/∂c 0)T, P as a factor while (F.3.4) does not. Secondly, the remaining factors are not necessarily the same. In the limit c 0 → 0, however, the thermodynamic derivative does approach RT/c 0. Nevertheless, non thermodynamic arguments are needed to conclude that DmDs as c 0→ 0, that is, at infinite dilution of solute.