1 1 1 1 1 1 2 kTV G = dG = d " " = = + + # ! ! !

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080312 Quiz 9 Polymer Properties

1) Fluctuations in composition exist above and below the cloud point for a polymer blend displaying an LCST.

a) Explain the difference between fluctuations above and below the cloud point based on the free energy diagram (a plot of free energy versus composition) and

b) explain the difference between fluctuations above and below the cloud point based on a Fourier transform of the fluctuations to amplitude versus wave vector (inverse

wavelength). You can use both a spinodal model as well as a bimodal model above the cloud point. (A real space sketch of the fluctuations is also useful for comparison).

c) If the magnitude (amplitude) of the fluctuation is φ, explain why <φ²> would be used to measure the strength or intensity of the thermal fluctuations for a mixture of polymer A and polymer B.

2) When we apply a stress field, σ, to a sample a strain, ε, results. We use a linear constitutive equation to relate stress and strain, ε = J σ. The energy expended in straining the material with this stress field is calculated from

!

G =

"

dG⁼

"

^#^d^$^.

a) Write a similar constitutive equation for fluctuations and an applied field (the chemical potential). Give an expression for the energy associated with a fluctuation at wave vector k.

b) Write the Boltzman probability function and the Gaussian Probability function then apply these to to the case of fluctuations in composition.

c) Show that <φ²> = αk kT.

3) The scattering structure factor for a polymer is given by S(k) = <φ²>/Vc = αk kT/Vc. For an athermal polymer blend (χ = 0) or polymer A and polymer B the collective response coefficient, αk0, is given by

!

1

"k

0 = 1

"k AA + 1

"k

BB where αkAA is the scattering coefficient of polymer A alone in a melt of polymer A.

a) Calculate the internal field for this mixture and explain what it is.

b) Use the internal field to calculate the expression for αk0 given above.

c) Show how this expression can be modified for χ ≠ 0 to yield

!

1

"_k⁰ ⁼ 1

"_k^AA ⁺ 1

"_k^BB ^# 2$kT

V₀ .

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ANSWERS: 080312 Quiz 9 Polymer Properties

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