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Statistical Analysis of Thermally Driven Membrane Roughness in Soft Condensed Matter

This presentation delves into the statistical mechanics of thermally induced roughness in fluctuating membranes. We explore the Monge representation of deformed membranes and the concepts of curvature, including mean and Gaussian curvatures. The significance of surfactant orientations and their effect on curvature is addressed. Furthermore, we analyze the fluctuation energy through Fourier transforms, demonstrating how changes in bending stiffness influence the mean square amplitude of fluctuations. This work highlights the intricate relationship between curvature, energy, and material properties in soft condensed matter systems.

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Statistical Analysis of Thermally Driven Membrane Roughness in Soft Condensed Matter

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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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