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

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

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.
Self Assembly 5 5

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

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

8. Substituting into the expression for the fluctuation energy, we get:
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.
Brownian motion 8

9. From equipartition of energy:
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.
Brownian motion 9

10. qmin = 2π/L qmax = 2π/d d ~ bilayer thickness
Mean amplitude:
qmin = 2π/L qmax = 2π/d d ~ bilayer thickness
Typically, bending stiffness is
hence
Slide 9: Mean amplitude of fluctuations.
Brownian motion 10