A pair of colloidal spheres interacts through a hard core of diameter plus a square…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Interacting systems, real gases and phase equilibria

Soft Matter at the Thermal Scale, Ch Brownian particles, sedimentation and the hard-sphere fluid

Problem

A pair of colloidal spheres interacts through a hard core of diameter

\sigma

plus a square well:

U = \infty

for

r < \sigma

U = -u_0

for

\sigma \le r < \lambda_{\rm w}\sigma

and

U = 0

for

r \ge \lambda_{\rm w}\sigma

, with

u_0 = 1.5\,k_BT

and

\lambda_{\rm w} = 1.3

. ((a)) Write the Mayer function

f(r) = e^{-U(r)/k_BT} - 1

in each of the three ranges of

r

. ((b)) Evaluate

B_2 = -2\pi\int_0^\infty f(r)\,r^2\,\mathrm{d}r

and write the result as

B_2/B_2^{\rm HS}

, with

B_2^{\rm HS} = \tfrac{2\pi}{3}\sigma^3

. ((c)) Evaluate the ratio and state whether the pair interaction is net attractive or net repulsive.

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