Polymers

Instructor: Xu Heng
中文版：聚合物物理
Back to: Soft Matter Physics Course Notes

Core Picture

A polymer is a long chain made of many monomers. The key questions are how entropy, bending rigidity, excluded volume, and solvent quality determine chain conformation.

Ideal Chain

An ideal chain neglects excluded volume and long-range interactions, so it behaves like a random walk.

contour=$L=Nb$

end-to-end=$\langle R^2\rangle=Nb^2$

size=$R\approx bN^{1/2}$

Rg=$R_g^2=\langle R^2\rangle/6$

The end-to-end distribution is approximately Gaussian, giving entropic elasticity.

Gaussian=$P(R)\propto e^{-3R^2/(2Nb^2)}$

entropy F=$F(R)=\frac{3k_BTR^2}{2Nb^2}$

force=$f=\frac{3k_BTR}{Nb^2}$

Wormlike Chain

Semiflexible chains require bending rigidity.

bending=$E_b=\frac{\kappa}{2}\int\left|\frac{d\mathbf u}{ds}\right|^2ds$

persistence=$l_p=\kappa/(k_BT)$

correlation=$\langle\mathbf u(s)\cdot\mathbf u(0)\rangle=e^{-s/l_p}$

WLC R2=$\langle R^2\rangle=2l_pL\left[1-\frac{l_p}{L}(1-e^{-L/l_p})\right]$

For L much smaller than l_p, the chain behaves like a rod. For L much larger than l_p, it behaves like a flexible chain with Kuhn length about 2l_p.

Non-Ideal Chain and Flory Theory

Real chains cannot pass through themselves. Excluded volume swells the chain in a good solvent.

Flory F=$F/k_BT\approx\frac{R^2}{Nb^2}+\frac{vN^2}{R^d}$

Flory nu=$\nu=3/(d+2)$

3D good=$R\approx bN^{3/5}$

theta=$R\approx bN^{1/2}$

poor=$R\approx bN^{1/3}$

Polymer Solution

Flory-Huggins theory combines mixing entropy and interaction energy. Polymer mixing entropy is small because the whole chain moves as one object.

FH=$f/k_BT=\frac{\phi}{N}\ln\phi+(1-\phi)\ln(1-\phi)+\chi\phi(1-\phi)$

excluded=$v=b^3(1-2\chi)$

theta=$\chi=1/2$

overlap=$\phi^*\approx Nb^3/R^3$

spinodal=$\partial_\phi^2f=0$

Exam Cue

First identify the model: ideal chain, wormlike chain, self-avoiding chain, or polymer solution. The most important size scalings are N^(1/2), N^(3/5), and N^(1/3).