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by Pietro Cicuta

Statistical Mechanics and Soft Condensed Matter Fluctuating membranes. by Pietro Cicuta. Slide 1: The thermally driven roughness of membranes can be analysed statistically. Reprinted with permission from Dr Markus Deserno, Carnegie Mellon University. Tangent vectors along x and y: where

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by Pietro Cicuta

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  1. Statistical Mechanics and Soft Condensed Matter Fluctuating membranes by Pietro Cicuta

  2. Slide 1: The thermally driven roughness of membranes can be analysed statistically.Reprinted with permission from Dr Markus Deserno, Carnegie Mellon University

  3. Tangent vectors along x and y: where Plane tangent to the surface at (x, y, h (x, y)): Position vector: s = (x, y,h (x, y)) Slide 2: Monge representation of a deformed membrane.

  4. Element of area dA: for small h: Surface metric g: = g dx dy Slide 3: Monge representation continued.

  5. 2D surface embedded in 3D space. Principal radii of curvature R1 and R2. Mean curvature Extrinsic curvature K=2H Gaussian curvature H and K are positive if the surfactant tails point towards the centre of curvature and negative if they point away from the centre. H > 0 H < 0 Slide 4: Curvature.

  6. Curvature where s is the arc length In one dimension: Non-trivial extension to two dimensions: Slide 5: Curvature of membranes.

  7. K = 2H • Work δE required to deform the membrane against tension and bending: Slide 6: Curvature and energy.

  8. The function h (x, y) can be decomposed into discrete Fourier modes or written in terms of its Fourier transform: Substituting into the expression for the fluctuation energy, we get: Slide 7: Fourier transform.

  9. Integrating over dx and dy generates a delta function, hence a simplified equation: • From equipartition of energy: • Spectrum for the mean square amplitude of fluctuations: Note the strong dependence on q, particularly in connection with the bending modulus. Slide 8: Fluctuation spectrum.

  10. Mean amplitude: qmin = 2π/Lqmax = 2π/dd ~ bilayer thickness Typically, bending stiffness is hence Slide 9: Mean amplitude of fluctuations.

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