Effective free energy

The effective free energy of the system of interfaces takes the general fonn [19, 80 and 81]... 

Random interface models for ternary systems share the feature with the Widom model and the one-order-parameter Ginzburg-Landau theory (19) that the density of amphiphiles is not allowed to fluctuate independently, but is entirely determined by the distribution of oil and water. However, in contrast to the Ginzburg-Landau approach, they concentrate on the amphiphilic sheets. Self-assembly of amphiphiles into monolayers of given optimal density is premised, and the free energy of the system is reduced to effective free energies of its internal interfaces. In the same spirit, random interface models for binary systems postulate self-assembly into bilayers and intro-... 

Many solvents do not possess the simple structure that allows their effects to be modeled by the Langevin equation or generalized Langevin equation used earlier to calculate the TS trajectory [58, 111, 112]. Instead, they must be described in atomistic detail if their effects on the effective free energies (i.e., the time-independent properties) and the solvent response (i.e., the nonequilibrium or time-dependent properties) associated with the... 

Fig. 7. The effective free-energy potentials for retraction of the free end of arms in a mon-odisperse star polymer melt. The upper curve assumes no constraint-release, the lower two curves take the dynamic dilution approximation with the assumptions (Ball-...

For this problem already the simple mean field approximation becomes rather involved [197,213]. Therefore, we describe here only an approach, which is even more simplified, appropriate for wavenumbers q near the characteristic wavenumber q, but strictly correct neither for q—>0 nor for large q the spirit of our approach is similar to the long wavelength approximation encountered in the mean field theory of blends, Eq. (7). That is, we write the effective free energy functional as an expansion in powers of t t and include terms (Vv /)2 as well as (V2 /)2, as in the related problem of lamellar phases of microemulsions [232,233],namely [234]... 

Starting point is the description of the interface in terms of the drumhead model [273], i.e. we disregard for the moment the fact that the interface has a non-trivial intrinsic profile with width w0, and treat it like a sharp kink in terms of the local interfacial height h(x,y) only, x,y being coordinates along the lower surface (at z=0). Then the effective free energy functional Heff [h] associated with interfacial fluctuations can be written as [220]... 

Although the foregoing electron-transfer theory is preoccupied with describing the electron-transfer step itself, in order to understand the kinetics of overall reactions it is clearly also important to provide satisfactory models for the effective free energy of forming the precursor and successor states from the bulk reactant and product, wv and ws, respectively. As outlined in Sect. 2.2, it is convenient to describe the influence of the precursor and successor state stabilities upon the overall activation barrier using relations such as... 

The addition of tributyl phosphate (TBP) to the dodecane acts to reduce the SHG signal from an initial (partial) monolayer of PNP at the dodecane/water interface the dependence shown in Figure 1.5 can be fitted to a Langmuir-like equation (15) from which an effective free energy of adsorption for TBP, AadsG Bp. can be extracted [48]. 

The chemical potential therefore represents the contribution per mole of each component to the total free energy. It is the effective free energy per mole of each component in the mixture and is always less than the free energy of the pure substance. 

The chemical potential is the effective free energy per mole of each component of a solution. In general, the chemical potential of a component is identical in all phases of a system at equilibrium at a fixed temperamre and pressure. 

Because the recent experiments and simulations reviewed here concentrated on the universal aspects of the novel non-equilibrium transition, focus will be laid on the MCT-ITT approach. Reassuringly, however, many similarities between the MCT-ITT equations and the results by Miyazaki and Relchman exist, even though these authors used a different, field theoretic approach to derive their results. This supports the robustness of the mechanism of shear-advection in (7) entering the MCT vertices in (lid, 14), which were derived independently in [40, 41] and [43 5] from quite different theoretical routes. This mechanism had been known from earlier work on the dynamics of critical fluctuations in sheared systems close to phase transition points [61], on current fluctuations in simple liquids [62], and on incoherent density fluctuations in dilute solutions [63], Different possibilities also exist to include shear into MCT-inspired approaches, especially the one worked out by Schweizer and coworkers including strain into an effective free energy [42]. This approach does not recover the (idealized) MCT results reviewed below but starts from the extended MCT where no true glass transition exists and describes a crossover scenario without, e.g., a true dynamic yield stress as discussed below. 

By virtue of this free energy, the evolution equations for these two sets of fields are coupled and demand numerical solution. For the case to be discussed here, an interesting step in the analysis corresponds to the construction of a new effective free energy Fgjy[i/ (x)] in which explicit reference to the elastic fields has been removed. This step is effected at the price of calculating an auxiliary quantity which serves as a potential within the effective free energy. In particular, the new free energy functional is of the form... 

We simply have to take into account that three interfacial tensions compete / snvt for the solid-vapor interface, f t for the liquid-vapor interface, and /j 1, for the solid-liquid interface. We have surface melting if /im - (/ini + /,ni) = A/ > 0. The effective free energy replacing eq. (263) is, for short range forces, being the melting temperature... 

A similar quantity svc can be calculated for the cluster body as well. In accordance with the definition given in [6.11], Svc represents the complexity of the structure displaying the degree of the SS uncertainty for the system with a broken ergodidty. In addition to the free energy of the cluster, we shall also introduce the effective free energy... 

Solvent Effects Free Energy Calculations and the Real Solid/Liquid Interface An important question arising from DFT calculations of the oxide/vacuum interface performed at 0 K, is the extrapolation of the results to a more realistic solid-liquid... 

Parsons made an attempt to derive some theoretical values for the preexponential term for the above processes, based on statistical thermodynamics and classical absolute rate theory. He pointed out the effect of possible surface mobility of the loosely bonded activated complex on the rate of reaction, or rather that of the mobility of the adsorbate, since the latter implies a mobile activated complex. Low bonding strength of H in the electrochemical environment may well allow mobility, provided that the effective free energy of the adsorbed species is less than A further activation entropy effect... 

In this equation, y is the interaction strength, c(r) the crosslink concentration, the smectic order parameter, and Vz (r) the relative displacement of the rubber matrix. Witkowski and Terentjev [132] evaluated (15) for (r) = 1, which is valid deep in the smectic phase, i.e., far below the smectic-nematic transition. Using the so-called replica trick, they integrated out the rubbery matrix fluctuations and obtained an effective free-energy density that depends only on the layer displacements M(r). Under the restriction that wave vector components along the layer normal dominate over in-layer components, q q, and considering only long-... 

If there is an electrochemical potential gradient A/x driving the mobility of the polymer across the pore, then F(x) is directly related to A/x. In fact, all contributing factors to the interaction between the pore and the polymer and the electrochemical potential gradients across the pore can be combined into an effective free energy F(x). Substitution of Equation 6.64 into Equation 6.63 yields... 

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