Two fractions of polystyrene in cyclohexane have measured critical (phase-separation)…

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

Two fractions of polystyrene in cyclohexane have measured critical (phase-separation) temperatures. In the Flory-Huggins count each chain occupies

N

lattice sites, one site being the volume of one cyclohexane molecule. For its coil size the same chain is instead

N_K

Kuhn segments, each of molar mass

m_K = 720\,\mathrm{g\,mol^{-1}}

and Kuhn length

b = 1.8\,\mathrm{nm}

. ((a)) Fraction 1:

N = 4700

T_c = 303.4\,\mathrm{K}

; Fraction 2:

N = 19\,000

T_c = 305.6\,\mathrm{K}

. Using the linear relation

1/T_c = A + Bx

with

x = N^{-1/2}+(2N)^{-1}

, find

A

and

B

from these two points and hence the extrapolated theta temperature

\Theta = 1/A

. Compare it with the theta temperature

307.7\,\mathrm{K}

(that is,

34.5^\circ\mathrm{C}

) found for this system from chain-dimension measurements. ((b)) An osmometer on fraction 2, held at

307.7\,\mathrm{K}

(where the second virial term vanishes), reads

\Pi = 1.18\,\mathrm{Pa}

at a mass concentration

c = 1.00\,\mathrm{g\,L^{-1}}

. Find the molar mass

M

from

\Pi = cRT/M

, and compare it with

M = N_Km_K

for fraction 2, which has

N_K = 3000

Kuhn segments. ((c)) At the theta temperature the chain is an ideal random walk with

R_g = b\sqrt{N_K/6}

. Estimate the overlap concentration

c^* = 3M/(4\pi N_{\rm Av}R_g^3)

for fraction 2, using the molar mass found in part (b).