Fluid membranes

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Fluid membranes
Elastic properties and fluctuations
A brief overview of a few basic concepts
Markus Deserno
AK Spiess Klausurtagung in Hirschegg, September
4th, 2003
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Aim of the upcoming introduction
Provide some simple insight into the elastic and
statistical properties of fluid membranes.
? Helfrich Hamiltonian will be our new best
friend!
Disclaimer
The following will be eclectic, incomplete,
biased, simplified, theoretical and not at all
authoritative!
Well, too bad for you . . .
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Todays menu
What are fluid membranes?
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Todays menu
What are fluid membranes?
Equilibrium shapes
Fluctuations
Differential geometry
Fourier decomposition
Helfrich Hamiltonian
Average height
Shape equation
Persistence length
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Fluid membranes
. . . are quasi-twodimensional elastic sheets
with a vanishing in-plane shear-modulus.
The typical example to keep in mind are lipid
bilayers.
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Todays menu
What are fluid membranes?
Equilibrium shapes
Fluctuations
Differential geometry
Fourier decomposition
Helfrich Hamiltonian
Average height
Shape equation
Persistence length
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Todays menu
What are fluid membranes?
Equilibrium shapes
Fluctuations
Differential geometry
Fourier decomposition
Helfrich Hamiltonian
Average height
Shape equation
Persistence length
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How to describe a curve in 3d
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How to describe a surface in 3d
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What could the energy depend on?
No, unless theres a field!
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What could the energy depend on?
No, unless theres a field!
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What could the energy depend on?
In principle yes . . . But not if we assume
inextensibility !
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What could the energy depend on?
Yes!
In harmonic approximation the bending energy then
will be proportional to the square of the
curvature !
? Wormlike chain model !
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Same game for membranes
. . . with one more possible deformation
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Hence . . .
The elastic energy of a fluid membrane depends on
its local curvature deformation!
However . . .
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Curvature of a surface
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Curvature of a surface
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Curvature of a surface
It turns out to be enough to know these two
principal curvatures in order to locally
understand the defor- mation of the surface!
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Why two curvatures are enough
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Mean and Gaussian curvature
Instead of working with principal curvatures, it
is customary to use instead the following two
alternative expressions
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Todays menu
What are fluid membranes?
Equilibrium shapes
Fluctuations
Differential geometry
Fourier decomposition
Helfrich Hamiltonian
Average height
Shape equation
Persistence length
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Curvature energy Helfrich Hamiltonian
In the spirit of harmonic theory, the energy
should depend on quadratic invariants of the
deformation!
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Spontaneous curvature
If the surface has a spontaneous curvature, this
can be taken into account in the following way
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Total energy of a surface
The total energy of a deformed surface is the
surface integral of the local bending energy
? Energy becomes a functional of the shape !
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Closed membranes Vesicles
Vesicles also have constraints on surface and
volume.
The equilibrium vesicle shape is found by
minimizing this energy. This leads to a
variational problem, and a corresponding
Euler-Lagrange-equation.
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Todays menu
What are fluid membranes?
Equilibrium shapes
Fluctuations
Differential geometry
Fourier decomposition
Helfrich Hamiltonian
Average height
Shape equation
Persistence length
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Shape equation for vesicles
This equation is outrageously complicated to
solve!
Mathematicians have been studying this equation
for more than 200 years . . .
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Example of a minimal surface
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Todays menu
What are fluid membranes?
Equilibrium shapes
Fluctuations
Differential geometry
Fourier decomposition
Helfrich Hamiltonian
Average height
Shape equation
Persistence length
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Fluctuations of an almost flat membranes
Membrane shape can be described by giving the
height h as a function of horizontal position x
and y.
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Energy functional
Instead of solving the shape equation in this
case (which is easy), we want to understand the
statistics of fluctuations.
Lets expand the shape in Fourier modes!
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Todays menu
What are fluid membranes?
Equilibrium shapes
Fluctuations
Differential geometry
Fourier decomposition
Helfrich Hamiltonian
Average height
Shape equation
Persistence length
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Fourier decomposition
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Todays menu
What are fluid membranes?
Equilibrium shapes
Fluctuations
Differential geometry
Fourier decomposition
Helfrich Hamiltonian
Average height
Shape equation
Persistence length
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Profile fluctuations
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What do these fluctuations look like?
Undulations of a fluid SOPC vesicle, imaged via
phase contrast microscopy
Courtesy of Jonas Henriksen, MEMPHYS, DTU,
Lyngby, Denmark
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Special case bending dominance
Fluctuations are scale invariant !
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Todays menu
What are fluid membranes?
Equilibrium shapes
Fluctuations
Differential geometry
Fourier decomposition
Helfrich Hamiltonian
Average height
Shape equation
Persistence length
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Final goodie persistence length
The following goes back to P. G. deGennes and C.
Taupin, J. Phys. Chem. 86, 2294 (1982).
n(0)
n(r)
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Final goodie persistence length
This is exponential in the bending stiffness !
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Thats all for today!
Thanks for the invitation and for your interest!