Thermodynamic minimum free-energy point

A number of statistical thermodynamic theories for the domain formation in block and graft copolymers have been formulated on the basis of this idea. The pioneering work in this area was done by Meier (43). In his original work, however, he assumed that the boundary between the two phases is sharp. Leary and Williams (43,44) were the first to recognize that the interphase must be diffuse and has finite thickness. Kawai and co-workers (31) treated the problem from the point of view of micelle formation. As the solvent evaporates from a block copolymer solution, a critical micelle concentration is reached. At this point, the domains are formed and are assumed to undergo no further change with continued solvent evaporation. Minimum free energies for an AB-type block copolymer were computed this way. 

Because of the thermodynamic imperative to attain a state of minimum free energy for the system as a whole, surface units are subjected to a net inward attraction normal to the surface. Geometrically, that can be equivalent to saying that the surface is in a state of net lateral tension defined as a force acting tangent to the surface at each point on it. It is this apparent tangential force that leads to the concept of a surface tension. The units of surface tension and of the excess surface free energy are dimensionally equivalent and, for pure liquids in equilibrium with their... 

It is customary [129] to define u0 independent of T, whilst r0 = a0t. In thermodynamic equilibrium the free energy must be a minimum as a function of m and m is assumed to have the mean-field value of m = /V. By this assumption the internal field acting at each point in the domain is the same,... 

To carry out the proof, we note that the LMA equations are equivalent to imposing an extremum on some function of the concentrations—the free energy F for v = const or the thermodynamic potential for p = const. It is in precisely this way, as is well known, that the LMA may be derived from general thermodynamic principles. We will solve the problem if we prove that the surface F or, under the conditions imposed on the concentration and for constant v or p, has one and only one minimum, and does not have either maxima or any other critical points (so-called minimax, or saddle points). 

Monovacancy. Statistical thermodynamics requires that if a vacancy is formed by removing an atom from the crystal and depositing it on the surface, then the free energy of the crystal must decrease as the number of created vacancies increases until a minimum in this free energy is reached. Because a minimum in the free energy exists for a certain vacancy concentration in the crystal, the vacancy is a stable point defect. The following facts about vacancies have been obtained experimentally (12). 

Step 4 is a frequency calculation on the geometry from step 3, again using the CASSCF(2,2)/6-31G method. The program might allow this step to automatically follow the optimization. In most cases the frequency calculation is desirable, to characterize the nature of the optimized structure as a minimum or some kind of saddle point, and to obtain thermodynamic data like zero point energy and enthalpy and free energy (Sections 2.5 and 5.5.2.1b). 

Beyond the thermodynamic control, the kinetics of chain rearrangement can dramatically influence the phase behavior leading to kinetically trapped structures, which do not necessarily correspond to an absolute free energy minimum of the system. Thus, the formation of block copolymer vesicles, from a kinetic point of view, can be a result of a transition from rod-like aggregates via flat, nonclosed lamellar structures. The kinetics of such transitions has been explored in [8], The transition steps are represented as follows ... 

The change in the free energy as a function of the reaction coordinate should have approximately the appearance of a plateau (Fig. 2). Note that the free energy profile shown in Fig..2 does not contradict the Jahn-Teller theorem [33]. This theorem affirms that the adiabatic potential has no minimum near the degeneration point, but this is not the case for the thermodynamic potential, in particular for free energy. However, the absence of the minimum of the adiabatic... 

You will note that when it is possible to plot as a function of G, the Gibbs free energy, a minimum occurs which is the number of defects at the given temperature for the given solid. Note also that the enthalpy of defects increases continuously as a function of Nd while both of the entropy-contributions decrease. The net effect is a minimum value for Gd, at which point we can get the number of intrinsic defects present. Thus, we have shown again that there is a specific number of intrinsic defects present in the soUd at any given temperature, this time by Thermodynamics. 

The first condition ensures a point of extremum (minimum or maximum), and the second condition ensures that the extremum is a minimum. The Gibbs free energy plays the role of a thermodynamic potential function at fixed temperature and pressure. By analogy to the mechanical system, the mathematical conditions that define a stable equilibrium state at constant pressure and temperature are ... 

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