Fields, chemical potential

To understand fully the chemical reaction processes that take place in rock assemblages, it is necessary to introduce the concept of chemical potential Much the same as in a gravitational potential, in which an object tends to fall from a high to a low altitude, in a chemical potential field the reaction or flow direction of components always tends to proceed from a high to a low chemical potential region. 

This parameter has the magnitude of an energy (i.e., kJ/mole, kcal/mole) and represents the energy driving toward equilibrium in a chemical potential field— i.e., the higher A becomes, the more the solid phase of interest is unstable in solution. In a heterogeneous system at equilibrium, based on the principle of... 

Theory for the self- and tracer-diffusion of a diblock copolymer in a weakly ordered lamellar phase was developed by Fredrickson and Milner (1990). They modelled the interactions between the matrix chains and a labelled tracer molecule as a static, sinusoidal, chemical potential field and considered the Brownian dynamics of the tracer for small-amplitude fields. For a macroscopically-oriented lamellar phase, they were able to account for the anisotropy of the tracer diffusion observed experimentally. The diffusion parallel and perpendicular to the lamellae was found to be sensitive to the mechanism assumed for the Brownian dynamics of the tracer. If the tracer has sufficiently low molecular weight to be unentangled with the matrix, then its motion can be described by a Rouse model, with an added term representing the periodic potential (Fredrickson and Bates 1996) (see Fig. 2.50). In this case, motion parallel to the lamellae does not change the potential on the chains, and Dy is unaffected by... 

The inverse argument shows how a chemical potential field imposes a particular energy distribution on units of matter. 

The separation force in concentration-FFF is a chemical potential field established by a concentration gradient of a mixed solvent across the channel in order... 

The inverse argument shows how a chemical potential field imposes a particular distribution on units of matter. Consider energy levels at Hi and Hj in this field and assume these to be occupied by n and rij molecules respectively. These numbers can be thought of as representing the relative activities at the two levels. The balance between action and affinity is then described, as before, by the relationship... 

If one considers (Fig. 4.83) an infinitesimal length dx parallel to the direction of the electric- and chemical-potential fields, one can obtain the electric-potential difference dy/ and the chemical-potential difference dfi across the length dx... 

At this point, we have adapted the concept of chemical potential so as to apply it in situations where the stress field is nonhydrostatic as well as nonuniform through space. We have related a material s strain rate or change-of-dimension behavior to its chemical-potential field. But so far we have discussed only problems of continuum mechanics type—the same problems that we were able to discuss effectively in terms of stress. We have used chemical potential in describing continuum-mechanics behavior, the left-hand two boxes in Figure 17.1 but we have not yet used chemical potential in describing change of concentration of an atomic species, which is of course one of the concept s most natural and powerful uses. 

It is possible to transform the path integrals in eqn [43] to a functional integral over the chemical potential fields < asso-dated with the corresponding spedes ... 

Entropy generation is considered a key concept of nonequiHbrium thermodynamics. Let us view entropy generation for the following cases isolated systems, systems in a homogeneous thermostat, and systems in a nonhomogeneous environment (in the temperature gradient field, in chemical potential field, etc.). At that, let us divide systems into two types weakly nonequilibrium (linear) and far from equilibrium (nonlinear). 

The in-plane tensile stress in the direction normal to the hole and the inplane shear stress on the surface of the hole are zero because of the condition of zero applied traction. The strain energy density as a function of position along the surface of the hole is [/(ly) = (t /2E where E = E/ X — v ) is the plane strain elastic modulus. The chemical potential field over the surface of the hole is then... 

Next, suppose that j and x in (9.7) are the actual mass flux field and chemical potential field on S. Furthermore, suppose that q is any possible mass flux field on S, that is, it shares any properties of periodicity, smoothness and remote decay characteristics with j, but is otherwise arbitrary. Form the inner product of each side of (9.7) with q and then integrate each... 

For a specified shape of the surface S and a prescribed chemical potential field X over S, this functional is maximum when the trial velocity field n is in fact that actual velocity field Vn defined by (9.9). The variational approach in terms of surface flux was introduced in a numerical study of grain boundary transport and cavitation by Needleman and Rice (1980), for which the grain boundary chemical potential is given by (8.14). It was introduced in the way as described here by Suo (1997) and Zhang et al. (1999) as a basis for study of surface evolution. 

The local chemical potential field corresponding to a spatial distribution of component B in an substitutional solid solution was... 

Diffusion, or chemical diffusion, is a transport phenomenon that is driven by gradients of chemical potentials. Particles migrate in a system spontaneously from an initial state to a steady state with uniform chemical potential field. Based on a general relation, the average velocity for molecules of species i v, is written as ... 

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