Introduction to Statistical Thermodynamics

1
Introduction to Statistical Thermodynamics of
Soft and Biological Matter Lecture 4
Diffusion

• Random walk.
• Diffusion. Einstein relation.
• Diffusion equation.
• Random walks and conformations of polymer
  molecules.

Interactions I

• Osmotic pressure.
• Depletion force.
• Hydrophobic interactions.
• Electrostatic interactions. Debye screening.
• van der Waals attraction.

2
Diffusion
Albert Einstein
Robert Brown 1828
Water molecules (0.3 nm)
Pollen grain (1000 nm)
3
Universal properties of random walk
One-dimensional random walk
L
(step-size of random walk)
x
0
- random number (determines direction of i-th
step)
4
Diffusion coefficient
Number of random steps N corresponds to time t
From dimensional analysis
5
Diffusion coefficient and dissipation
Einstein relation
Friction coefficient
Viscosity
Particle size
- velocity
Force
6
Diffusion in two and three dimensions
One-dimensional (1D) random walk
Two-dimensional (2D) random walk
Three-dimensional (3D) random walk
7
Conformations of polymer molecules
(fully stretched polymer)
(coiled)
N number of segments
L length of elementary segment

• Universal properties of random walk describe
  conformations
• of polymer molecules.

Excluded volume effects and interactions may
change law!
8
Why power law is important?????
Ideal coil
Self-avoiding coil
9
More about diffusion Diffusion equation
Flux
10
Solution of diffusion equation
Concentration profile spreads out with time
11
Osmotic pressure
12
Depletion force
R
13
Hydrophobic interactions

• Amphiphiles (lipids) polar head-group and
  hydrophobic tail

Lipid molecule
Self assembly
14
Hydration repulsion
At small separations (lt1 nm), there is a
repulsion between surfaces in water due to
disruption of water molecular ordering (layering)
at the surfaces.
Hydration repulsion constitutes energetic barrier
for membrane fusion.
15
Electrostatic interactions
Two charges in medium with dielectric constant
Two charges in salt solution with dielectric
constant
16
Debye screening
17
van der Waals attraction

• Always present between molecules
• - Usually attractive between same species
• - Long range (power law)

van der Waals attraction between two atoms
Hamaker constant
vdW attraction is due to fluctuations of electron
clouds in atoms
18
Phase separation
Interactions can lead to phase separation