Restricted equilibrium

Notice that the number NA of atoms is usually small compared to the number NS of species, and hence the RAND algorithm is very effective in terms of computational effort. The rank of the atom matrix, however, must be equal to the number NA of atoms. At this point it is interesting to remark that instead of the atom matrix we can use a virtual atom matrix, i.e., the matrix of reaction invariant coefficients if the atom matrix is not available or we are interested in a restricted equilibrium. For details see Section 1.8.1. 

Fig. 16. Interaction potential energy between adsorbed layers vs. separation under conditions of restricted equilibrium as calculated by Scheutjens and Fleer (198S). Various adsorbed amounts are achieved for two chain lengths by varying bulk solution concentration when the surfaces are well separated the necessary q>b for each curve are given in the table. Other parameters include %,= L X = 0.5, and A0 = 0.5.

Under some conditions, though, an adsorbed layer may not be in full equilibrium with bulk solution. For restricted equilibrium, the adsorbed amount of polymer is held constant adsorbed chain configurations are assumed to be optimally distributed, but only solvent is free to move between the layer and bulk solution. Since ZA normalizes G(r, s)G(r, n — s)em integration of Eq. (79) with Eq. (69) over r gives c — (pi Jn where [Pg.182]

The SCF [Eq. (39)] minimizes the free energy of the interfacial system. The Helmholtz free energy per area of segment for restricted equilibrium is... 

Interaction potential energies for restricted equilibrium based on Ploehn and Russel s (1989) equations are given in dimensionless form as... 

Like the models of de Gennes (1982) and Scheutjens and Fleer (1985 Fleer and Scheutjens, 1986), the SCF model predicts monotonic attraction between adsorbed layers under conditions of full equilibrium. For constant total potential (curve C) displays an attractive minimum as well as a steep repulsive wall. Potentials for various different combinations of n and (pb (i.e., dosage at infinite separation). The reasons for this difference are not clear at this point. 

Fig. 23. Typical interaction potential energy vs. gap half-width (scaled on Rt) for restricted equilibrium (curve C) calculated by Ploehn (1988) using the Flory-Huggins SCF of Eq. (70). The components of the potential are the potential for full equilibrium, i.e., the surface tension, (curve A) and the chains chemical potential (curve B). Parameters are x = 1, x = 0.488, and n= 1129 the dosage

To obtain the probability for such a fluctuation we introduce another state, state t, which is a fictitious restricted equilibrium system in which (1) the charge distribution is Po and (2) the nuclear polarization is P e, that is same as in the equilibrium state P in which the charge density is pe. We want to calculate the free energy difference, AGo z, between the restricted equilibrium state t and the fully equilibrated state 0. This is the reversible work needed to go, at constant temperature and pressure, from state 0 to state t. 

These are the same relations as in Eq. (16.23), with 3 replacing 0 everywhere. Restricted equilibrium state t. 

Note that p, D, , and P are all functions of position. Two types of relationships appear in these equations First (terms with white background) there are those that stem from electrostatic definitions. Another type (terms with light-grey background) are constitutive linear response relationships that are assumed valid at equilibrium. In the equilibrium states described by Eqs (16.23) and (16.24) both are satisfied. The restricted equilibrium state described by Eq. (16.25) is characterized by the fact that the nuclear polarization is not allowed to relax to its equilibrium value for the given electric field, but instead restricted to the same value it would have in the equilibrium state 6 (last equation in (16.25)with dark-grey background). 

Another important issue which is not included in the dynamic modelling here is the type of kinetic rate equations used. This is an important problem that still requires extensive experimental and theoretical research. From chapter 3, we notice that most kinetic rate equations for catalytic reactions are based on restrictive steady state assumption or more often the even more restrictive equilibrium adsorption-desorption assumption. Therefore, these rate equations are not necessarily valid for the description of the catalytic process under unsteady state conditions. 

Several theoretical models describing surface interactions between irreversibly adsorbed flexible polymers have been developed. The most common approaches are based on scaling arguments (181) or on self-consistent mean field calculations (180). The irreversibility criterion implies that the polymer adsorption/desorption rate has to be slow when compared to the approach rate of the surfaces. Under such circumstances, the total amount of polymers on the surfaces is independent of the surface separation and the system is not in true equilibrium with the bulk solution. However, it is often the case that the speed of approach is sufficiently slow for the irreversibly adsorbed polymers to adopt the most favourable conformation for each surface separation. Hence, there is equilibrium within the layer. This situation is referred to as quasi-equilibrium, or restricted equilibrium. 

R., Unnikrishnan, G., and Thomas, S. (2004) Investigation on interfacial adhesion of short sisal/coir hybrid fiber reinforced natural rubber composites by restricted equilibrium swelling technique. Compos. Interfaces, 11, 489-513. 

Xo is simply the constant for the restricted equilibrium involving the molecules M and those molecules of MY which are wholly uncombined with X. It is independent of x, except insofar as activity coefficients vary with x. The denominator of Lj is given by a similar expression with Ko replaced by unity. The equilibrium constant L,- may therefore be written as ... 

Another important timescale is the contact time. We have to distinguish whether the approach and retraction of the two interacting surfaces is so slow that the system is in equilibrium at all stages or not. In the first case, we talk about full equilibrium, the second case is called restricted equilibrium [1404, 1405]. In... 

Whether full or restricted equilibrium is established does not only depend on the chemical nature of the polymer, the surfaces, and the solvent but also depends on the specific geometry. Small particles have only a small gap zone from which polymer can more easily diffuse in and out. They equilibrate faster than large particles. With respect to the measuring technique, full equilibrium is easier to reach in AFM experiments than with the SFA. In experiments, one manifestation of relaxation processes in polymer layers are differences in the approaching and retracting force curve. For very slow processes, even a difference between subsequent force curves is observed [1353, 1396]. 

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