by Pietro Cicuta

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.
4
Surface metric g

• Element of area dA
• for small h

?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 K2H
• 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 gt 0
H lt 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 dE 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 2p/L qmax 2p/d d bilayer thickness
Typically, bending stiffness is hence
Slide 9 Mean amplitude of fluctuations.