Bulk free energy

Bulk free energy of crystallization per unit volume... 

Transport term , i.e. the rate at which molecules arrive at the surface Apportioning factor proportion of the bulk free energy released during stem deposition Lateral growth rate of a sector Strain surface free energy... 

We now discuss the effects of finite chain length. The difficulties arise from the definition of a bulk free energy term, when the very nature of the chains constrains the crystal thickness to be finite. There are two different approaches to this problem the first to be considered is due to Hoffman et al. [31] and is a simple modification of the infinite chain case, but is somewhat lacking in theoretical justification the second, due to Buckley and Kovacs [23], aims to correct this deficiency and suggests that the interpretation of experimental data given by Hoffman s approach is misleading. 

Hoffman takes Eq. (2.13) as the definition of bulk free energy per unit volume at the melting point, neglecting the fact that a surface contribution is also included. 

Fig. 3.2. Variation of free energy of a cluster of molecules on a surface as a function of the number of molecules, at a fixed supercooling. N is the number of molecules at which the free energy is a maximum. Any cluster larger than Ns is stable. The dashed curves show the contributions from the increase in surface area and the decrease in bulk free energy. Increasing the supercooling shifts all curves towards the origin and decreases the height of the maximum...

The most difficult part of the theory lies in obtaining actual values for AF and v. For a large cluster of N molecules the extra surface tension due to the incremental surface area, edA, contributes an increase to the total free energy, whilst the bulk free energy per volume summed over the incremental volume, AF dV, gives a decrease to the total free energy. Hence, AF can be estimated as the maximum value of ad A — AF dV as a function of N. It is found that AF is proportional... 

For h close to hmin the main variation of S arises from 1 — (kl/k + )h l m ". Using kl/k+ = exp ( — AfjkT), where A/ is the bulk free energy difference for a unit we finally arrive at ... 

Where the particle of B contains m molecules, AGB is the bulk free energy change per molecule, a is the shape factor (4irr2 for a spherical interface) and y is the strain energy per unit area of interface. For a spherical nucleus, where vm is the volume of product per molecule ,... 

The manner in which a film is formed on a surface by CVD is still a matter of controversy and several theories have been advanced to describe the phenomena. ] A thermodynamic theory proposes that a solid nucleus is formed from supersaturated vapor as a result of the difference between the surface free energy and the bulk free energy of the nucleus. Another and newer theory is based on atomistic nucle-ation and combines chemical bonding of solid surfaces and statistical mechanics. These theories are certainly valuable in themselves but considered outside the scope of this book. 

Here scalar order parameter, has the interpretation of a normalized difference between the oil and water concentrations go is the strength of surfactant and /o is the parameter describing the stability of the microemulsion and is proportional to the chemical potential of the surfactant. The constant go is solely responsible for the creation of internal surfaces in the model. The microemulsion or the lamellar phase forms only when go is negative. The function/(<))) is the bulk free energy and describes the coexistence of the pure water phase (4> = —1), pure oil phase (4> = 1), and microemulsion (< ) = 0), provided that/o = 0 (in the mean-held approximation). One can easily calculate the correlation function (4>(r)(0)) — (4>(r) (4>(0)) in various bulk homogeneous phases. In the microemulsion this function oscillates, indicating local correlations between water-rich and oil-rich domains. In the pure water or oil phases it should decay monotonically to zero. This does occur, provided that g2 > 4 /TT/o — go- Because of the < ), —<(> (oil-water) symmetry of the model, the interface between the oil-rich and water-rich domains is given by... 

This mechanism may account for the stability, in the absence of any external stabilising agent, of amphiphilic homopolymers in the fully collapsed/glo-bular state. The total free energy of a collapsed macromolecule includes a surface energy contribution in addition to the bulk free energy. Obviously, to form a stable particle, the outer shell of the particle should be hydrophilic enough. 

However, this choice of t / implies that surfaces of stems need to be formed first before any gain in the bulk free energy of the stems. 

Recrystallization occurs when a crystalline material is plastically deformed at a relatively low temperature and then heated [1]. The as-deformed material possesses excess bulk free energy resulting from a high density of dislocations and point-defect debris produced by the plastic... 

The critical radius given by Eq. 19.92 is equal to the critical radius for homogeneous nudeation in the bulk liquid. This is the expected result because 7LM = ySM (so that the liquid/solid interface makes an angle of 90° with the mold) and the inward pressure on the interface due to curvature, AP = 2-yLS/R (Eq. 12.4), is then exactly balanced by the change in bulk free energy across the interface, jjphase trans — gB (Eq. 12.1). Substitution of Eq. 19.92 into Eq. 19.91 yields the critical free energy for nudeation ... 

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