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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

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Diffusion Robert Brown: 1828 Albert Einstein Pollen grain (1000 nm) Water molecules (0.3 nm):

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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

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Diffusion coefficient From dimensional analysis: Number of random steps N corresponds to time t:

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Friction coefficient: Diffusion coefficient and dissipation Viscosity Particle size Einstein relation: - velocity Force

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Diffusion in two and three dimensions One-dimensional (1D) random walk: Two-dimensional (2D) random walk: Three-dimensional (3D) random walk:

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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

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Why power law is important????? (coiled) L= 0.3 nm Ideal coil: Self-avoiding coil:

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More about diffusion… Diffusion equation Surface area: A x Flux: – concentration of particles (depends on coordinate x and time t)

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Solution of diffusion equation verify this is the solution! c(x,t) x Concentration profile spreads out with time – concentration of particles

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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

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Depletion force R Free energy gain: A – surface area of contact R – small particle radius - small particles concentration

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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

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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.

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R Electrostatic interactions Two charges in medium with dielectric constant Interaction energy: Two charges in salt solution with dielectric constant Screened interactions: R + - + + + ++ + - - -- - - - - - - - - - - - - - - + + + + + + + -

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R + - + + + ++ + - - -- - - - - - - - - - - - - - - + + + + + + + - Debye screening - Debye radius

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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

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Phase separation Interactions can lead to phase separation:

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