Sailing the surfactant sea:

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University of Colorado, Boulder August 2006
Sailing the surfactant sea Dynamics of rigid and
flexible bodies in interfaces and membranes
P.G. Saffman and M. Delbrück, Brownian Motion in
Biological Membranes, Proc. Nat. Acad. Sci. 72,
3111 (1975).
Alex J. Levine Department of Chemistry and
Biochemistry University of California, Los Angeles
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Collaborators (Theory)
Collaborators (Experiment)

• A.D. Dinsmore, R. McGorty
• M. Dennin
• V. Prasad, S. Koehler,
• and E. Weeks

• F.C. Mackintosh
• T.B. Liverpool
• M. Henle

Papers

• A.J. Levine and F.C MacKintosh Dynamics of
  viscoelastic membranes PRE 66, 061606 (2002)
• A.J. Levine, T.B. Liverpool, and F.C. MacKintosh
  Mobility of extended bodies in viscous
• films and monolayers PRE 69, 021503 (2004).
• A.J. Levine, T.B. Liverpool, and F.C. MacKintosh
  Dynamics of rigid and flexible extended
• bodies in viscous films and membranes PRL 93,
  038102 (2004).

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Hydrodynamics in membranes and on monolayers The
importance of looking below the surface
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Mobilities of particles in a membrane
How to determine the particle mobilities?
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Why consider membrane hydrodynamics?

• Microrheology in membranes/interfaces Both
  translational and rotational
• (E. Weeks) Two-particle microrheology on
  interfaces.

• Dynamics of rigid or semiflexible rods in
  membranes/interfaces
• (J. Zasadzinski) Needle viscometry
• (M. Dennin) Actin dynamics on a monolayer.
• (A.D. Dinsmore) Rod mobilities on the surface of
  spherical droplets

• Dynamics of phases separation in multi-component
  membranes
• Lipid raft formation as 2d phase separation.
• Transmembrane protein aggregation kinetics

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Understanding the physical properties of lung
surfactant
Rapid respreading of lung surfactant is important
for minimizing the work of inhalation
Needle viscometry
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Describing the dynamics of a membrane or
interface
Vertical Displacement of the interface
Displacement field on the globally
flat interface
Flow of the Newtonian Sub/super-phase
Boundary Conditions
No slip
Velocity decays into the infinite surrounding
fluid.
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Force Balance in the Membrane
viscoelastic inplane forces
Bending forces
Hydrodynamic stress from the sub- and super-phase
Externally applied forces
For the surrounding Newtonian fluids
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Shear
T. Chou et al. (1995) D.K. Lubensky and R.E.
Goldstein (1996)H.A. Stone and A. Ajdari (1998).
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Compression
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Calculating the response function
Putting a force on a particle
Summing over the modes excited by this force
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The single particle response function
Compression
Shear
The inplane response
For a viscous membrane
The exponential screening of shear waves in an
elastic medium coupled to a viscous fluid.
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The Saffman-Delbrück result for transmembrane
proteins.
The Saffman-Delbrück in the membrane
In contrast with three-dimensional objects, the
diffusivity of transmembrane proteins is only
weakly dependent on their size.
The Stokes-Einstein result in three dimensions
Max Delbrück (From the CalTech archives)
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Hydrodynamic interactions
Specializing to a viscous membrane and in-plane
forces
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Microrheology on an interface
?r?
PS beads, a0.85 ?m, spread at interface 20 X
objective, N.A0.5, frame rate30 frames/s Human
Serum Albumin at air-water interface (bulk
c?0.03-0.45 mg/ml)
?rr

• Measure vector displacements of particles ?r for
  200 frames
• Determine lt ?r2(?) gt (1-particle MSD)
• Determine Drr(R,?) and D??(R,?) from
  displacements for different R, ?

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Master curve

• Fits are from theory - A.J. Levine and F.C.
  MacKintosh, Phys Rev E 66, 061606(2002)
• Characterizes flow/strain fields over different
  length scales

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Dragging a Rod An example of extended objects in
the membrane
Top View Viscous membrane
The Kirkwood Approximation
Aspect Ratio
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The mobility of a rod in the membrane
Recall the Stokes result in three dimensions
Drag on a rod of length L, radius a. The constant
A depends on details of the ends, but is a number
of order one.
Note Hydrodynamic Cooperativity
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What is the difference between parallel and
perpendicular drag?
Ans Losing Hydrodynamic Cooperativity
Only parallel drag has the log term
The ratio is now length dependent
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Why are parallel and perpendicular drag different?
Parallel flow consistent with 3d flow field.
Perpendicular flow implies no short paths around
the rod.
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Two consequences of the free-draining case
Purely algebraic rotational drag
For flexible rods
Where
Note the cross-over from 2d Lennon-Brochard to
free draining F. Brochard and J.F. Lennon J.
Phys. (France) 36, 1035 (1979).
Small
Large
Small
Large
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Colloids at an Interface

• Self-assembled nanoparticles at an interface
  could lead to materials with interesting optical,
  magnetic and electric properties
• Nanoparticles on droplets provide high surface
  area allows for efficient chemical processes on
  nanoparticles

Y. Lin, A. Boker, H. Skaff, D. Cookson, A.D.
Dinsmore, T. Emrick, and T.P. Russell,
Nanoparticle Assembly at Fluid Interfaces
Structure and Dynamics, Langmuir 21, 191 (2005).
4.3 nm diameter CdSe at water/toluene
interface l0 48 µm
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Data Collection

• Chain of paramagnetic beads is moved across the
  interface
• Move the chains by waving a magnet nearby
• 0.3 µm PMMA beads
• Water droplets in hexadecane

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Comparison to Theory

• Experimental and theoretical flow fields overlaid
• The value of l0 used for the theoretical flow
  field was obtained from the MSD plot (13.3 µm in
  this case)
• Experimental and theoretical rod is 7.0 µm long.
  Theoretical rod is 1.05 µm wide experimental is
  0.95 µm

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Studying the velocity field in more detail
l0 40 20 13.3 5 2
Value of l0 from MSD 13.3 µm Droplet diameter
52 µm Rod length 7.0 µm
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Hydrodynamics in curved space?
How does the curvature of the sphere affect the
surface flows?
McGorty, Levine, Dinsmore unpublished (2006)
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Hydrodynamics on curved surfaces
But, how to find the shear stresses from the
surrounding fluids?
Ans. Apply results from Sir Horace Lamb
and
where
Note the combined effects of Geometry and
Viscosity
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Mapping the velocity field on the sphere
Symmetric Case
High viscosity surface or Small Sphere
Low viscosity surface or Large Sphere
(Vectors x 2)
Mark Henle AJL
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Mapping the velocity field on the pinned sphere
High viscosity surface or Small Sphere
Low viscosity surface or Large Sphere
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Calculating the mobility of a point particle on
the sphere
Removing the uniform rotation of the sphere by
transforming to a co-rotating frame so that the
total angular momentum of the sphere and its
contents vanishes
The mobility can be calculated for a sphere with
a fixed point at the south pole as well.
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Mobility on a Pinned Sphere
The mobility on a sphere can be larger or
smaller than the flat case depending on whether
the smaller viscosity is inside or outside.
Henle Levine unpublished (2006)
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The velocity field around the rod
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Summary

• For small objects (specifically, for which L
  l0), the drag coefficients
• become independent of both the rod orientation
  and aspect ratio. In agreement with
• the Saffman/Delbrück result.
• (ii) For larger rods of large aspect ratio, ??
  Becomes purely linear in the rod length L
• For parallel drag ?k2??/ln(AL/a).
• (iii) On spheres, geometry (radius of curvature)
  controls particle modifies particle mobility at
  fixed viscosities.

The cause