Consider a binary lattice mixture in which every lattice site (volume ) holds exactly…

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Soft condensed matter · Soft condensed matter

Soft Matter at the Thermal Scale, Ch Polymer solutions and blends

Problem

Consider a binary lattice mixture in which every lattice site (volume v0v_0) holds exactly one chain segment: chains of species A have NAN_A segments each and chains of species B have NBN_B segments each. Let ϕ\phi denote the volume fraction of B, so the volume fraction of A is 1−ϕ1-\phi. ((a)) Using the lattice placement argument for whole chains (how many more places a chain can be put down in the mixture than in its own pure melt), derive the entropy of mixing per lattice site, Δsmix≡ΔSmix/(nkB)\Delta s_{\rm mix} \equiv \Delta S_{\rm mix}/(nk_B), where nn is the total number of sites, as a function of ϕ\phi, NAN_A and NBN_B. ((b)) Show that this result reduces to the ideal entropy of mixing of two small molecules when NA=NB=1N_A = N_B = 1. ((c)) Evaluate Δsmix\Delta s_{\rm mix}, in units of kBk_B, for a solution of a polymer with NB=250N_B = 250 segments per chain in a small-molecule solvent (NA=1N_A = 1) at a polymer volume fraction ϕ=0.40\phi = 0.40.

Browse the problem library