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

1. 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

2. 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).

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

4. Mobilities of particles in a membrane How to determine the particle mobilities?

5. 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

6. Understanding the physical properties of lung surfactant Rapid respreading of lung surfactant is important for minimizing the work of inhalation Needle viscometry

7. 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.

8. 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

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

10. Compression

11. Determine the displacement of the bead (radius a). compression shear Calculating the response function: Putting a force on a particle: Summing over the modes excited by this force:

12. The single particle response function Compression Shear The in—plane response: For a viscous membrane The exponential screening of shear waves in an elastic medium coupled to a viscous fluid.

13. 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)

14. Hydrodynamic interactions: Specializing to a viscous membrane and in-plane forces

15. ra R rb Microrheology on an interface r PS beads, a=0.85 m, spread at interface 20 X objective, N.A=0.5, frame rate=30 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 < r2()> (1-particle MSD) • Determine Drr(R,) and D(R,)from displacements for different R, 

16. 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

17. Dragging a Rod: An example of extended objects in the membrane Top View: Viscous membrane The Kirkwood Approximation Aspect Ratio

18. 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:

19. Perpendicular drag is larger. But, in 3d: 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

20. Why are parallel and perpendicular drag different? Parallel flow consistent with 3d flow field. Perpendicular flow implies no short paths around the rod.

21. Correlation Functions: 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

22. 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). 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

23. 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

24. 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

25. 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

26. Hydrodynamics in curved space? How does the curvature of the sphere affect the surface flows? McGorty, Levine, Dinsmore unpublished (2006)

27. Hydrodynamics on curved surfaces Specialize to an incompressible, viscous membrane: 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

28. 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

29. Mapping the velocity field on the pinned sphere High viscosity surface or Small Sphere Low viscosity surface or Large Sphere

30. Calculating the mobility of a point particle on the sphere  R a R 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.

31. 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)

32. The velocity field around the rod R

33. 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: k=2/ln(AL/a). • (iii) On spheres, geometry (radius of curvature) controls particle modifies particle mobility at fixed viscosities. The cause: