Consider a molecule that has N elements. Every element can be either in the stat

Question

Consider a molecule that has N elements. Every element can be either in the state or state . In the former case the element has length a and energy a, while in the latter, length b and energy b. At the equilibrium state with temperature we are interested to find the relation between the length of the molecule L and the tension force f applied to its ends.

(a) Construct partition function Z in the case of a canonical ensemble and perform an exact summation with the help of binomial formula. Note: since molecule is under the tension with force f one has to add corresponding potential energy -fL(Na,Nb) in addition to (Na,Nb) in the exponential when computing the partition function.

(b) Find now the desired relation between the average length of the molecule and applied tension force

(c) Introduce =(a-b)/ and find the asymptotic form for <L> to the leading non-vanishing order in f assuming that {fa,fb}<<.

(d) Make a plot of an exact expression for <L>/Nb versus fa/ for =-10 and b/a=5/4.

***Note: Part D is not necessary for a LIFE-SAVER rating.

Explanation / Answer

the effects of solvent selectivity and temperature on phase behavior and transition of diblock copolymer solution, self-consistent field theory is modified to incorporate the short-range interaction and non-local effects. Inhomogeneous free-energy density is shown to be dependent on solvent selectivity, temperature and copolymer concentration. Enthalpic quantity and entropic contributions are crucial to phase diagrams of diblock copolymer solution. Three selective strengths of solvent --weak, moderate and strong-- are chosen for comparison. For a weakly selective solvent, theoretical and experimental results illustrate the same variation tendency in the phase boundary of the order-disorder transition for a symmetric diblock of polystyrene and polyisoprene. Self-consistent field equations can be used to calculate the exact FCC-BCC structural phase transition temperatures in moderately and strongly selective solvents. Detailed comparison with the experimental phase diagrams including lamellar, cylindrical and spherical structures is presented.

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