Soft Matter Phases

Instructor: Cao Xin
中文版：软物质相
Back to: Soft Matter Physics Course Notes

Colloids and Thermal Scale

A colloid is a dispersed phase suspended in a continuous phase, typically (10,\mathrm{nm}\sim10,\mu\mathrm{m}) in size. Interactions are often measured in units of k_B T.

thermal=interaction comparable to $k_BT$ is reversible

diffusion=$D=\frac{k_B T}{6\pi\eta R}$

MSD=$\langle\Delta r^2\rangle=2dDt$

van der Waals Attraction and Electric Double Layer

atoms=$U(r)=-C/r^6$

plates=$W(h)=-\frac{H}{12\pi h^2}$

spheres=$U_{vdW}(h)\approx-\frac{HR}{12h}$

wall=$U_{vdW}(h)\approx-\frac{HR}{6h}$

Charged surfaces form a Stern layer and a diffuse layer. Higher salt concentration gives shorter Debye length and shorter-ranged repulsion.

Debye=$\lambda_D=\sqrt{\frac{\epsilon_0\epsilon_r k_BT}{\sum_i q_i^2c_i}}$

screening=$\phi(h)=\phi_0e^{-h/\lambda_D}$

Yukawa=$\phi(r)\propto e^{-r/\lambda_D}/r$

zeta=$U=\epsilon_0\epsilon_r\zeta E/\eta$

DLVO Theory

DLVO adds electric double-layer repulsion and van der Waals attraction. High salt lowers the repulsive barrier and promotes aggregation.

DLVO=$U_{DLVO}(h)=U_e(h)+U_{vdW}(h)$

repulsion=$U_e(h)=Ae^{-h/\lambda_D}$

attraction=$U_{vdW}(h)\approx-\frac{HR}{12h}$

salt=higher c gives smaller lambda_D and lower barrier

Depletion Force

Small particles are excluded from the depletion layer around large particles. When two depletion layers overlap, available volume for small particles increases, creating entropy-driven attraction.

osmotic=$p=nk_BT$

depletion=$U(r)=-p\Delta V(r)$

AO=$\Delta V(r)=\frac{\pi}{6}(a+b-r)^2(a+b+r/2)$

range=range approx small-particle diameter b

Hard Spheres and Glass

Hard spheres have no attraction but can crystallize by entropy. At high volume fraction, particles become caged by neighbors and form a colloidal glass.

hard sphere=$u(r)=\infty\;(r<\sigma),\;0\;(r\ge\sigma)$

phi=$\phi=Nb/V,\;b=\pi\sigma^3/6$

CS=$Z=\frac{1+\phi+\phi^2-\phi^3}{(1-\phi)^3}$

coexist=$\phi_f=0.494,\;\phi_s=0.545$

packing=$\phi_0=\frac{\pi}{3\sqrt2}\approx0.74$

KTHNY theory describes 2D melting as solid to hexatic to liquid through defect unbinding. Active colloids may undergo motility-induced phase separation.

Exam Cue

Identify the interaction: vdW, electrostatic, depletion, hard sphere, or activity. For phases, start from free energy A = U - TS. Hard spheres and depletion force are especially entropy-driven.