Gradient-related forces

Saffman Force and Other Gradient-Related Forces... 

We have now reviewed most of the theory necessary for the evaluation of transport coefficients of liquid crystals. We are going to start by showing how the thermal conductivity can be calculated. In a uniaxially symmetric system this transport coefficient is a second rank tensor with two independent components. The component An n relates temperature gradients and heat flows in the direction parallel to the director. The component Aj j relates forces and fluxes perpendicular to the director. The generalised Fourier s law reads... 

The setting up of the constitutive relation for a binary system is a relatively easy task because, as pointed out earlier, there is only one independent diffusion flux, only one independent composition gradient (driving force) and, therefore, only one independent constant of proportionality (diffusion coefficient). The situation gets quite a bit more complicated when we turn our attention to systems containing more than two components. The simplest multicomponent mixture is one containing three components, a ternary mixture. In a three component mixture the molecules of species 1 collide, not only with the molecules of species 2, but also with the molecules of species 3. The result is that species 1 transfers momentum to species 2 in 1-2 collisions and to species 3 in 1-3 collisions as well. We already know how much momentum is transferred in the 1-2 collisions and all we have to do to complete the force-momentum balance is to add on a term for the transfer of momentum in the 1-3 collisions. Thus,... 

Some of the phenomenological coefficients relating forces and fluxes are already familiar from less general treatments of the subject. For example, the phenomenological coefficient relating a concentration gradient and a mass transfer flux is the diffusion coefficient. Other phenomenological coefficients are related to the ionic mobility, the coefficient of thermal conductivity, and the solvent viscosity. These are discussed in more detail later in this chapter. 

Convection is related to hydrodynamic transport. Generally fluid flow occurs because of natural convection (convection caused by density gradients) and forced convection, and may be characterized by stagnant regions, laminar flow, and turbulent flow. 

Since the membrane can swell freely, a function is needed to find the thickness of the membrane after swelling. This thickness is important in simulating transport in membranes because it directly affects many of the gradient driving forces. Due to a slight anisotropy in the membrane, the thickness can be related to the water content by [39]... 

Two features of this definition are worth noting. One is that EPH is defined as the heat of a reversible reaction, which essentially eliminates the various uncertainties arising from the irreversible factors such as overvoltage. Joule heat, thermal conductivity, concentration gradient and forced transfer of various particles like ions and electrons in electrical field, and makes the physical quantity more definite and comparable. This indicates that EPH is a characteristic measure of a cell reaction, because the term 8 (AG)/8T) p is an amoimt independent on reaction process, and only related to changes in the function of state. That is to say, EPH is determined only by the initial and the final states of the substances taking part in the reaction that occurs on the electrode-electrolyte interfaces, although other heats due to irreversible factors are accompanied. EPH is, unlike the heat of dissipation (Joule heat and the heats due to irreversibility of electrode processes and transfer processes), one of the fundamental characteristics of the electrode process. 

Convection is the movement of species owing to mechanical forces such as by stirring or hydrodynamic transport. The fluid flow can occur because of natural convection (because of density gradient) and forced convection and is characterized by stagnation regions, laminar flow, and turbulent flow. Convection usually can be eliminated on a short time scale. The convection mass flux (J j) is directly related to mass concentration and the solution velocity in the direction of electrode, v(x), and is given as... 

At low solvent density, where isolated binary collisions prevail, the radial distribution fiinction g(r) is simply related to the pair potential u(r) via g ir) = exp[-n(r)//r7]. Correspondingly, at higher density one defines a fiinction w r) = -kT a[g r). It can be shown that the gradient of this fiinction is equivalent to the mean force between two particles obtamed by holding them at fixed distance r and averaging over the remaining N -2 particles of the system. Hence w r) is called the potential of mean force. Choosing the low-density system as a reference state one has the relation... 

Nonporous Dense Membranes. Nonporous, dense membranes consist of a dense film through which permeants are transported by diffusion under the driving force of a pressure, concentration, or electrical potential gradient. The separation of various components of a solution is related directiy to their relative transport rate within the membrane, which is determined by their diffusivity and solubiUty ia the membrane material. An important property of nonporous, dense membranes is that even permeants of similar size may be separated when their concentration ia the membrane material (ie, their solubiUty) differs significantly. Most gas separation, pervaporation, and reverse osmosis membranes use dense membranes to perform the separation. However, these membranes usually have an asymmetric stmcture to improve the flux. 

The function W(X) is called the potential of mean force (PMF). The fundamental concept of the PMF was first introduced by Kirkwood [4] to describe the average structure of liquids. It is a simple matter to show that the gradient of W(X) in Cartesian coordinates is related to the average force. 

All criteria proposed here are constructed such that if absolutely no gradient of a particular type exists, then the value of the corresponding criterion is zero. For fast catalytic processes this is not reasonable to expect and therefore a value judgment must be made for how much deviation from zero can be ignored. For the dimensionless expressions the Damkdhler numbers are used as these are applied to each particular condition. The approach is that the Damkdhler numbers can be calculated from known system values, which are related to the unknown driving forces for the transport processes. 

If a diffusion potential occurs inside the membrane, the relation between mass transport and electrochemical potential gradient — as the driving force for the diffusion of ions — has to be examined in more detail. This can be done by three different approaches ... 

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