An introduction to statistical physics (1982) by William Geraint V. Rosser
By William Geraint V. Rosser
Advent to Statistical Physics (Mathematics and Its functions) [Paperback
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Denoting the corresponding number of such bonds by N(1), N(2), N(3) and N(4), respectively, then N(1) ϭ Nzρ(1 Ϫ ρ) N(2) ϭ Nzρ2 N(3) ϭ Nz(1 Ϫ ρ)2 N(4) ϭ Nzρ(1 Ϫ ρ) where, N equals half the total number of atoms in this equiatomic system, z is the coordination number (number of nearest neighbors, equal to 8 in this case), and ρ is the fraction of A(B) atoms on the I(II) sub-lattice. ]}2 Using Stirling’s approximation for the factorial of large numbers, then S ϭ Ϫ2 Nk(ρ ln ρ ϩ (1 Ϫ ρ)ln(1 Ϫ ρ)) Since the energy of the superlattice is given by NzEAB and the entropy of configuration of the superlattice is zero then the disordering energy and entropy are ΔE ϭ Ϫ2Nzρ(1 Ϫ ρ)[EAB Ϫ (EAA ϩ EBB)/2] ΔS ϭ Ϫ2Nk(ρ ln ρ ϩ (1 Ϫ ρ)ln(1 Ϫ ρ)) Minimizing the free energy ΔF ϭ ΔE Ϫ TΔS yields the equilibrium value of the occupation probability, ρ, and by its relation to the long-range order parameter η, the equilibrium value of the latter.
At very low densities (the area per molecule is then much larger than the cross-sectional area a*, of an isolated molecule) an adsorbed monolayer behaves as a 2D gas. In this state, an amphiphilic molecule can exhibit its conformational entropy without interference from neighboring molecules. Compression of this layer then produces a first-order phase transition to a “liquid expanded” (LE) condensed phase. The energy of interaction between the molecules acts to stabilize the LE phase although in this transformation there is little change in configurational entropy.
The crumpling process in this model corresponds to an increase in the number of folds along the chain as the ends of the polymer chain come towards each other. Experimental results that distinguish between the two models are not available at this writing. For both models, however, the phase transition produced by simulations is second order as long as the attractive driving force is between monomers alone. In the above phase transitions of an isolated homopolymer, as the temperature decreases the entropy associated with each succeeding phase is less than that of the phase stable at the higher temperature.