Sailing the surfactant sea:Dynamics of rigid and flexible bodies in

interfaces and membranes

Alex J. LevineDepartment of Chemistry and Biochemistry

University of California, Los Angeles

University of Colorado, BoulderAugust 2006

P.G. Saffman and M. Delbrück, Brownian Motion in Biological Membranes, Proc. Nat. Acad. Sci. 72, 3111 (1975).

Collaborators (Theory)

• F.C. Mackintosh

• T.B. Liverpool

• M. Henle

Collaborators (Experiment)

• A.D. Dinsmore, R. McGorty

• M. Dennin

• V. Prasad, S. Koehler, and E. Weeks

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).

Hydrodynamics in membranes and on monolayers:The importance of looking below the surface

Mobilities of particles in a membrane

How to determine the particle mobilities?

• Microrheology in membranes/interfaces: Both translational and rotational

(E. Weeks) Two-particle microrheology on interfaces.

•Dynamics of phases separation in multi-component membranes

Lipid raft formation as 2d phase separation.

Transmembrane protein aggregation kinetics

•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

Why consider membrane hydrodynamics?

Rapid respreading of lung surfactant is important for minimizing the work of inhalation

Needle viscometry

Understanding the physical properties of lung surfactant

Describing the dynamics of a membrane or interface:

Boundary Conditions:

Displacement fieldon the globally flat

interface:

Flow of the Newtonian

Sub/super-phase:

Velocity decays into theinfinite surrounding fluid.

No slip

Vertical Displacementof the interface:

Force Balance in the Membrane

Hydrodynamic stress from the sub- and super-phase

Externally applied forces

For the surrounding Newtonian fluids

viscoelastic inplane forces

Bending forces

Shear

T. Chou et al. (1995); D.K. Lubensky and R.E. Goldstein (1996);H.A. Stone and A. Ajdari (1998).

Compression

Calculating the response function:

Determine the displacement of the bead (radius a).

Putting a force on a particle:

compression shear

Summing over the modes excited by this force:

10 20 30 40 50

-1

-0.5

0.5

1

The single particle response function

CompressionShear

The in—plane response:

The exponential screening of shearwaves in an elastic medium coupled to a viscous fluid.

For a viscous membrane

The Saffman-Delbrück result for transmembrane proteins.

Max Delbrück(From the CalTech archives)

In contrast with three-dimensional objects,the diffusivity of transmembrane proteins is only weakly dependent on their size.

The Saffman-Delbrückin the membrane

The Stokes-Einsteinresult in three dimensions

Hydrodynamic interactions:

Specializing to a viscous membrane and in-plane forces

Microrheology on an interface

PS beads, a=0.85 m, spread at interface

20 X objective, N.A=0.5, frame rate=30 frames/s

Human Serum Albuminat air-water interface (bulk c0.03-0.45 mg/ml)

( , ) ( ) ( )r t r t r t

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

R

ra

rb

r

rr

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

Dragging a Rod: An example of extended objects in the membrane

The Kirkwood Approximation

Top View: Viscous membrane

Aspect Ratio

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, butis a number of order one.

Note: Hydrodynamic Cooperativity:

The mobility of a rod in the membrane

What is the difference between parallel and perpendicular drag?

Only parallel drag has the log term The ratio is now length dependent

Perpendicular drag is larger.

But, in 3d:

Ans: Losing HydrodynamicCooperativity

Why are parallel and perpendicular drag different?

Parallel flow consistent with 3d flow field.

Perpendicular flow implies no short paths around the rod.

Two consequences of the free-draining case:

Purely algebraic rotationaldrag

For flexible rods…

Where:

Correlation Functions:

Note the cross-over from 2d Lennon-Brochard to free draining [F. Brochard and J.F. Lennon J. Phys. (France) 36, 1035 (1979).

Small LargeSmall Large

Colloids at an Interface

4.3 nm diameter CdSe at water/toluene interface:

l0 ≈ 48 μm

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).

Y. Lin, H. Skaff, T. Emrick, A.D. Dinsmore, and T.P. Russell, Nanoparticle Assembly and Transport at Liquid-Liquid Interfaces, Science 299, 226 (2003).

• 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

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

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

l0:

40

20

13.3

5

2

Value of l0 from MSD: 13.3 µm

Droplet diameter: 52 µm

Rod length: 7.0 µm

Studying the velocity field in more detail…

Hydrodynamics in curved space?

McGorty, Levine, Dinsmore unpublished (2006)

How does the curvature ofthe sphere affect the surface

flows?

Hydrodynamics on curved surfaces

But, how to find the shear stresses from the surrounding fluids?

Ans. Apply results from Sir Horace Lamb

where: and

Note the combined effects of Geometry and Viscosity

Specialize to an incompressible, viscous membrane:

Mapping the velocity field on the sphere

Highviscosity surface

orSmall

Sphere

Lowviscosity surface

orLargeSphere

Symmetric Case

Mark Henle & AJL

(Vectors x 2)

Mapping the velocity field on the pinned sphere

Highviscosity surface

orSmall

Sphere

Lowviscosity surface

orLargeSphere

Calculating the mobility of a point particle on the sphere

R

Removing the uniform rotation of the sphere by transforming toa co-rotating frame so that the total angular momentum of the sphere

and its contents vanishes

R

a

The mobility can be calculated for a sphere with a fixed point atthe south pole as well.

Mobility on a Pinned Sphere

Henle & Levine unpublished (2006)

The mobility on a sphere can be larger or smallerthan the flat case depending on whether the smallerviscosity is inside or outside.

The velocity field around the rod

R

Summary:

(i) 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: k=2/ln(AL/a).

(iii) On spheres, geometry (radius of curvature) controls particle modifies particle mobility at fixed viscosities.

The cause: