Title: Introduction to Statistical Thermodynamics
1Introduction 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.
2Diffusion
Albert Einstein
Robert Brown 1828
Water molecules (0.3 nm)
Pollen grain (1000 nm)
3Universal properties of random walk
One-dimensional random walk
L
(step-size of random walk)
x
0
- random number (determines direction of i-th
step)
4Diffusion coefficient
Number of random steps N corresponds to time t
From dimensional analysis
5Diffusion coefficient and dissipation
Einstein relation
Friction coefficient
Viscosity
Particle size
- velocity
Force
6Diffusion in two and three dimensions
One-dimensional (1D) random walk
Two-dimensional (2D) random walk
Three-dimensional (3D) random walk
7Conformations 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!
8Why power law is important?????
Ideal coil
Self-avoiding coil
9More about diffusion Diffusion equation
Flux
10Solution of diffusion equation
Concentration profile spreads out with time
11Osmotic pressure
12Depletion force
R
13Hydrophobic interactions
- Amphiphiles (lipids) polar head-group and
hydrophobic tail
Lipid molecule
Self assembly
14Hydration 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.
15Electrostatic interactions
Two charges in medium with dielectric constant
Two charges in salt solution with dielectric
constant
16Debye screening
17van 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
18Phase separation
Interactions can lead to phase separation