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
Position vector: s = (x, y, h (x, y))
Tangent vectors along x and y:
rx  (1, 0, hx ) ry  (0, 1, hy )
where
h
 hx
x
 h

 or  hy 
 y

Plane tangent to the surface at (x, y, h (x, y)):


r r 
 h ,h ,1
n

x
y
rx  ry
1  h
x
2
x
y
 hy2
Slide 2: Monge representation of a deformed membrane.

1/ 2
Surface metric g:
g  (1  hx2  hy2 )
Element of area dA:

dA  rx  ry dxdy  1  hx2  hy2
= g dx dy
for small h:
Slide 3: Monge representation continued.

1/ 2
dxdy
•
•
•
2D surface embedded in 3D space.
Principal radii of curvature R1 and R2.
Mean curvature
1 1
1 
H    
2  R1 R2 
•
•
Extrinsic curvature K=2H
Gaussian curvature
1
KG 
R1R2
•
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.
Curvature
  2r 
c  n. 2 
 s 
where s is the arc length
In one dimension:
Non-trivial extension to two dimensions:
Slide 5: Curvature of membranes.

F   dA K 2
2
K = 2H
•
Work δE required to deform the membrane against tension and bending:
Slide 6: Curvature and energy.
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.
•
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.
Mean amplitude:
qmin = 2π/L
qmax = 2π/d
Typically, bending stiffness is
hence
Slide 9: Mean amplitude of fluctuations.
d ~ bilayer thickness