Thermodynamic potential parameter volume dependence

The competitive nucleation and growth of two intermediate phases in volumes of nanometric size is investigated, taking into account the mole fraction depletion of the parent phase. The problem has been solved in the framework of the classical method by representation of the thermodynamic potential in the mole fraction-size space. It is shown that, depending on the particle size and thermodynamic parameters of parent and intermediate phases, there exist the following possible situations (i) total inhibition of separation, (ii) formation and total stabilization of the metastable phase instead of a stable one, (iii) relative stabilization of the metastable phase with the temporary delay of its transformation into the stable phase, (iv) formation and growth of a stable phase when the metastable phase does not appear at all, (v) formation and growth of the stable phase via the metastable phase. This was applied to the coherent precipitation of metastable AhLi ordered phase in supersaturated solid solution Al(Li). 

The dimensionless parameters that define the thermodynamic state of this system are the total volume fraction ( ) = (nl6)n (with n being the total number concentration, n = n, + n, the relative concentrations = njn, and the potential parameters K2, and z. The free-diffusion coefficients D° are also assumed identical for both species, that is, D = D = D°. Explicit values of the parameters a and D are not needed, since the dimensionless dynamic properties, such as t), only depend on the dimensionless parameters specified above, when expressed in terms of the scaled variables ka and t/to, where to = a /D . Besides solving the SCGLE scheme, in reference [21] Brownian dynamics simulations were generated for the static and dynamic properties of the system above. 

The first term on the right is the formula for the chemical potential of component a at density pa = na/V in an ideal gas, as would be the case if interactions between molecules were negligible, fee is Boltzmann s constant, and V is the volume of the solution. The other parameters in that ideal contribution are properties of the isolated molecule of type a, and depend on the thermodynamic state only through T. Specifically, V/A is the translational contribution to the partition function of single a molecule at temperature T in a volume V... 

Although the kinetics of liquid uptake to attain gel-saturation is history-dependent, the composition at the true end-state (i.e. thermodynamic equilibrium in excess liquid) is not therefore the observed end-state is usually reproducible [19]. Gel-saturation is attained when the restraining force (per unit area) of the polymeric crosslinked network becomes equal and opposite to the osmotic pressure that causes the system to swell [20], In other words saturation is achieved when the chemical potential of swelling liquid, p1 in the swollen network is equal to the chemical potential of the excess pure liquid, p , outside the network. It was logical to anticipate that the volume of liquid sorbed per gram of polymer, at this state of thermodynamic equilibrium with excess liquid, would correlate with the molecular structure of the liquid. In fact two parameters already exist which relate the sorption affinity to the molecular structure, namely the solubility parameter, 8, first proposed by Hildebrand [21], and the interaction parameter, %, introduced by Flory [22] and Huggins [23-26],... 

To frame this point, we give simple estimates of temperature and pressure derivatives assuming that the thermodynamic state dependence of the radii may be neglected. We will consider a simple ion and the Born formula (Pettitt, 2000) the interaction contribution to the chemical potential of such a solute is charge on the ion and R is its Born radius see Section4.2. We assume that these radius parameters are independent of the thermodynamic state. Considering the partial molar volume first, we have... 

Here e and a are parameters dependent on the type of atoms, e is the well depth of the Lennard-Jones potential achieved at ry = ro = The r attractive term in the Lennard-Jones (LJ) potential finds theoretical justification in the 1930 quantum mechanical perturbation theory calculation of London. Recent modeling work has shown that an exponential term, A exp(—kr), for excluded volume interactions provides a more satisfactory representation of thermodynamic properties than the term of the LJ potential. Such an exponential term for the repulsions also has better theoretical justification. 

The two-parameter formalism makes no stipulations concerning the temperature dependence of the radius of gyration and of the second virial coefficient, except that both quantities must reflect any temperature dependence of the excluded volume integral P through the variable z. Without an explicit theory of the behavior of j , we can only assert general thermodynamic requirements. We recall that the osmotic pressure is given by —(RT/Vi) In, with the activity of the solvent and Fj its molar volume. Then using equation (63), we can write the chemical potential of the solvent as a... 

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