Introduction to Statistical Thermodynamics

of Soft and Biological MatterLecture 4Diffusion

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

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

Interactions I

Diffusion

Robert Brown: 1828

Albert Einstein

Pollen grain (1000 nm)

Water molecules (0.3 nm):

Universal properties of random walk

0

L (step-size of random walk)

- random number (determines direction of i-th step)

One-dimensional random walk:

N-th step of random walk:

(N-1)-th step of random walk:

Verify!

x

Diffusion coefficient

From dimensional analysis:

Number of random steps N corresponds to time t:

Friction coefficient:

Diffusion coefficient and dissipation

Viscosity Particle size

Einstein relation:

- 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

* Excluded volume effects and interactions may change law!

L – length of elementary segment

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

(fully stretched polymer)(coiled)

N – number of segments

Why power law is important?????

(coiled)

L=0.3 nm

Ideal coil:

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

verify this is the solution!

c(x,t)

x

Concentration profile spreads out with time

– concentration of particles

Osmotic pressure

Free energy of ideal gas:

concentration:N – number of particlesV - volume

Pressure:

Osmotic forces: Concentration difference inducesosmotic pressure

Semi-permeable membrane(only solvent can penetrate)

Protein solution

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

Self assembly Lipidmolecule

chain (tail) (hate water)

polar head (love water)

Hydrophobic interactionis due to disruption of entropy of hydrogen bonding of 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.

R

Electrostatic interactions

Two charges in medium with dielectric constant

Interaction energy:

Two charges in salt solution with dielectric constant

Screened interactions:

R +- +

++

+ ++

-

-- --

-

--

-

---

--

--

--

+ +++

+ +

+ -

Screened interactions:

R +- +

++

+ ++

-

-- --

-

--

-

---

--

--

--

+ +++

+ +

+ -

Debye screening

- 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: