Date post: | 21-Dec-2015 |
Category: |
Documents |
View: | 239 times |
Download: | 4 times |
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
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
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
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