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. 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 particles V - volume Pressure: Osmotic forces: Concentration difference induces osmotic 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 Lipid molecule chain (tail) (hate water) polar head (love water) Hydrophobic interaction is 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

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: