Minimization of the free energy

The first term comes from the entropy and takes implicitly into account the fact that we deal with a chain (linear memory) the second term comes from the energy and is derived from the properties of a monomer solution. With Lifshitz, we grant that the equilibrium concentration is given by the functional derivative 

Finally, by taking eqns (14.4.35) and (14.4.32) into account, we can write the above equation in the explicit form 

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While the surface tension of the adhesive, is easily measured in the laboratory, the other terms in Wa, by themselves, are not. A second easily measurable property associated with the solid-liquid-air system, however, is the contact angle, 9, the angle, drawn in the liquid, between the solid-liquid and the liquid-air interfaces, drawn in the plane perpendicular to the three-phase interline, as shown in Fig. 4. Minimization of the free energy of the solid-liquid-air... 

The shape of a droplet or of the front end of a film can be determined from the surface energies and interaction forces between the interfaces. These also determine the equilibrium thickness of a liquid film that completely wets a surface. The calculation is done by minimization of the free energy of the total system. In a two-dimensional case the free energy of a cylindrical droplet can be expressed as [5] ... 

The obvious disadvantage of this simple LG model is the necessity to cut off the infinite expansion (26) at some order, while no rigorous justification of doing that can be found. In addition, evaluation of the vertex function for all possible zero combinations of the reciprocal wave vectors becomes very awkward for low symmetries. Instead of evaluating the partition function in the saddle point, the minimization of the free energy can be done within the self-consistent field theory (SCFT) [38 -1]. Using the integral representation of the delta functionals, the total partition function, Z [Eq. (22)], can be written as... 

In practice, the deformation parameters e/ (l > 2) are determined from the minimization of the free-energy of the system in full analogy to the total momentum Q of a Cooper pair. And they can be determined in a volume conserv-... 

The actual numerical evaluation of the cell partition functions, and the final minimization of the free energy are extremely involved, and several approximations were introduced by WR to simplify the calculations. For details we refer the reader to the original paper 64> and the elegant review by Stillinger 3>. The principle results are ... 

In actuality, the successive magnetic layers are not strictly antiparallel under zero external field, nor are they strictly parallel when under a saturating external field. The orientation of magnetic domains is dictated by minimization of the free energy, which... 

As is well known (see, for instance, Van Beijeren and Nolden, 1987), the equilibrium crystal shape is the shape that minimizes the total surface free energy at a given fixed volume. From the minimization of the free energy calculated above we can construct the equilibrium shape of the crystal around the (110) facet. This shape depends crucially... 

Minimization of the free energy of this arbitrary state of a system containing the reactants, subject to the condition of equality of the two free energies, yields an expression for the free energy of the reactants in this centered distribution and, thereby, for AF. The functional form of the equation for AF is given by Equation 3, and that for AF v, the free energy of formation of the centered distribution from the product, is given by Equation 4 (8). 

The total energy is expressed in terms of kinetic energy, potential energy including nuclear attraction, Coulomb repulsion and LDA exchange, and correlation energy. Minimization of the free energy F = E — TS, with respect... 

The minimization of the free energy with respect to ip leads to the equation... 

The equilibrium state is determined by a minimization of the free energy. The total interaction in colloidal systems is mainly determined by electrostatic interactions and can be divided in three components. The first component occurs at interactions between net charged molecules or molecules with asymmetric charged distribution. These charge distributions can often be described by multipole expansions, i.e., a combination of monopole, dipole, quadrupole, etc., and is a fruitful approach if each multipole expansion can be described by one or two terms. The interaction is given by the sum of interactions between the terms, where the first contribution is the interaction between ions and is given by Coulomb s law and is the main contribution in systems with... 

Statistical mechanical manipulations of the functional integral representation of Q are necessary for inhomogeneous systems (Helfand, 1975c Hong and Noolandi, 1981). Minimization of the free energy fixes the equilibrium spatial distribution of polymer and solvent. Edwards random field technique (1965) leads to... 

The minimization of the free energy in accordance with White-Johnson-Dantzig allows the determination of the composition of the detonation products in equilibrium state. The thermodynamic parameters (enthalpy, entropy, free enthalpy, free energy) of the detonation products can be calculated using functions based on the enthalpy //°-//°. The coefficients c0,. .. c4 can be found in literature J. Chem. Phys 1958, 28, 751 755 Prop., Expl. Pyrotech. 1985,10, 47-52] ... 

The observed interactions between molecules in a crystal are a consequence of the minimization of the free energy for the entire atomic... 

In such cases a local-equilibrium structure may be obtained theoretically by minimization of the free energy of the system under the constraint of a fixed alloy composition in the surface region [8,17-24]. Although this approach is very similar to the one used for bulk systems, it should be modified due to the specific features introduced by the surface. First of all, since the structure of the underlying bulk system is fixed, it acts as the source of an external field for the surface alloy, creating, for instance, epitaxial strain. Secondly, since the surface is an open system, it allows the formation of a great variety of different structures, which may not have any connection at all to the crystal structure of the substrate. Finally, the surface is a spatially inhomogeneous system, and thus different alloy components have their own... 

This matrix formulation may be used in the iterative procedure by replacing the inner cycle with the solution of linear equation system of eq.(51) (Coitino et al., 1995a). However, this approach could be too cumbersome a more interesting application is the direct minimization of the free energy functional. We need to make a digression here. 

Here f C plays the role of an energy and S C of an entropy. Lifshitz s method consists in finding proper approximate expressions for S C and E C, and in determining the mean concentration C(r) by minimization of the free energy... 

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