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.
Diffusion
Albert Einstein
Robert Brown: 1828
Water molecules (0.3 nm):
Pollen grain (1000 nm)
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)
N-th step of random walk:
(N-1)-th step of random walk:
Verify!
Diffusion coefficient
Number of random steps N corresponds to time t:
From dimensional analysis:
Diffusion coefficient and dissipation
Einstein relation:
Friction coefficient:
Viscosity
Particle size
- velocity
Force
Diffusion in two and three dimensions
One-dimensional (1D) random walk:
Two-dimensional (2D) random walk:
Three-dimensional (3D) random walk:
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!
Why power law is important?????
(coiled)
Ideal coil:
L=0.3 nm
Self-avoiding coil:
More about diffusion… Diffusion equation
Surface area: A
x
Flux:
– concentration of particles
(depends on coordinate x and time t)
Solution of diffusion equation
– concentration of particles
verify this is the solution!
c(x,t) Concentration profile spreads out with time
x
Osmotic pressure
Free energy of ideal gas:
N – number of particles
V - volume
concentration:
Pressure:
Osmotic forces:
Protein solution
Concentration difference induces
osmotic pressure
Semi-permeable membrane
(only solvent can penetrate)
Depletion force
R
Free energy gain:
A – surface area of contact
R – small particle radius
- small particles concentration
Hydrophobic interactions
• Amphiphiles (lipids): polar head-group and hydrophobic tail
Lipid
molecule
Self assembly
Hydrophobic interaction
is due to disruption of entropy of hydrogen bonding of water
chain (tail) (hate water)
polar head (love water)
Hydration repulsion
At small separations (<1 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.
Electrostatic interactions
Two charges in medium with dielectric constant
R
Interaction energy:
Two charges in salt solution with dielectric constant
---
----+- - + - +
--- + R -
++ ++
+
+
+
+
++
Screened interactions:
Debye screening
----
----+- - + - +
--- + R -
++ ++
+
+
+
+
++
Screened interactions:
- Debye radius
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
Phase separation
Interactions can lead to phase separation: