1. A Dynamical Model of Molecular Monolayers: Why Tethers Don’t Snap?
Lu Zou,* Violeta Beleva,* Andrew J. Bernoff,# James C. Alexander,+ J. Adin Mann Jr.! Elizabeth K. Mann*
*Dept. of Physics, Kent State University
# Dept. of Mathematics, Harvey Mudd College
+ Dept of Mathematics, Case Western Reserve University
! Dept of Chemical Engineering, Case Western Reserve University
Good morning, everyone!
The title of my presentation is “……..”

2. Why Don’t Tethers Snap? Relaxation of 8CB on Water/Air Interface
(First, show the tethers-relaxation video. ) (8CB bilayer on monolayer on the water/air interface)
First of all, I like to show you a small video of relaxation on the water/air interface.
The bright dumb-bell-shaped domains are 8CB layers. You can see there is a very thin tether connecting two round ends. Although there is a dust on it, there are still a lot of interesting things happen. So, I want to show you this video.
We saw the thin tethers get stronger and stronger, and the bola-shaped domains relax towards circles under the influence of line tension.
Seeing this relaxation, one may ask, why don’t tethers snap?
In this presentation, I will introduce a simplified dynamic model of molecular monolayers, which will give a possible answer.
Why Don’t Tethers Snap?

3. Introduction on Rayleigh instability (3D) and Hele-Shaw flow (2D)
OVERVIEW
Introduction on Rayleigh instability (3D) and Hele-Shaw flow (2D)
A dynamic model of molecular monolayers (2D)
Simulation and experimental results
Conclusion and prospects
At first of my presentation, I will introduce Rayleigh instability and Hele-Shaw Flow. In these two cases, the tether is unstable and will break up.
Then, I will present our model. It is also a 2D model. Compared with Hele-Shaw, you will find that our model is closer to the real experimental conditions. And the biggest difference is that, in our model, the tether is not snapping.
After that, you will see some simulation and experimental results.

4. Rayleigh Instability [1878]
Pure, cylindrical 3D fluid
Varicose mode fluctuations
Decrease area/surface energy
Break into droplets
Now, let’s start from Rayleigh Instability, which was raised by Rayleigh in 1878.
If we have pure, cylindrical 3D fluid, there will be varicose mode fluctuations on the surface. when there is a ‘neck’ structure like that, surface tension acts to reduce the surface area as well as to reduce the surface energy. Therefore, the neck becomes thinner and thinner until the neck is pinched off. Finally, the fluid will break into some droplets. A lot of experiments have proved this instability.
The surface tension here plays an important role. Similarly, in 2D case, one may predict that the line tension will keep the domain edge straight in order to reduce the energy. So, the tether should be stable. But this is not true. It is more complicated.
Now, let’s see what happens in 2D, in Hele-Shaw flow.

5. Hele-Shaw Cell constrains Height of gap
Before talking about Hele-Shaw flow, I want to introduce Hele-Shaw Cell first.
A Hele-Shaw cell is comprised of two parallel, rigid, transparent plates with spacers between them.
The green part is a liquid drop in the cell.
The height of the gap is much smaller than the dimensions of the plates. Because of this, Hele-Shaw cell is usually applied to approximate the 2D fluid in a lot of experiments.
However, this kind of approximate 2D fluid has some extra constrains on the upper and lower boundaries due to the two plates.
This is its major difference from the REAL 2D fluid cases. This is also the reason why we didn’t choose Hele-Shaw flow.
constrains
Height of gap

6. Evolution of a long, narrow bubble
This is the simulation of the evolution of a long narrow liquid bubble in a Hele-Shaw cell. The figure shows the central plane of the Hele-Shaw cell.
We can see some thin necks appear and then are pinched off. Finally, the long, narrow bubble becomes into several circles.
This result was reported by Glasner in It shows that the thin tether will break up.
This is similar as the 3D Rayleigh instability case. However, we still don’t see any experiment support it because it is hard to get the initial state in Hele-Shaw cell.
These are some previous work on thin tether relaxation. Now, I will present our model.
(Sharp interface model encounters a singularity at the point of breakup, it is hard to accomplish the continuation after this point. Diffuse interface model is insensitive to it, provide a unique way to continuing the solution past the breakup point.)
Ref: Glasner, Karl
A diffuse interface approach to Hele-Shaw flow NONLINEARITY 16 (1): JAN 2003

7. A dynamic model of molecular monolayersZ Ω
Z = 0
Subphase fluid
Fundamental Hydrodynamic Equations
Instead of two boundaries on the upper and lower of the domain in Hele-Shaw cell, in our model, the monolayer is floating on a subphase fluid.
the depth of the subphase fluid is much bigger than the thickness of the domain on the surface.
The green domain stands for the monolayer.
z=0 is located at the interface of the subphase fluid and the monolayer.
In our experiments, the subphase fluid is pure deionized water and the size of the monolayer is about 100 microns.
The same as other models, our model also starts from these two fundametal hydrodynamic equations– Stokes equation and Continuity equation.
We applied these two equations on both bulk and surf. This makes the problem very complicated to solve. To our knowledge, no one can solve this problem explicitly. So, we need some assumptions to simplify the problem.
Stokes Equation
Continuity Equation

8. Assumptions on the subphase fluid
Horizontal flow
Boundary condition
Bulk viscosity ηbulk [Ref]
For the subphase fluid, we assume that it is a horizontal flow, that is, there is no z-component of the flow velocity u.
And at the boundary of the bulk fluid, we assume there is no flow.
In this paper, almost every case was observed and they find the bulk viscosity is dominated.
Therefore, we don’t ignore the effect due to the bulk viscosity. In fact, much stronger than this, we even assume the viscosity of the surface phase is negligible. This is one of our assumption on the surface. Now, let’s see the other assumptions on the surface phase.
Ref: Elizabeth K. Mann
Hydrodynamics of Domain Relaxation in a Polymer Monolayer PRE 51 (6): JUN 1995

9. Assumptions on the surface
2D Fluid (η and KG)
One component [Ref1]:
Elasticity KG [Ref1]:
Surface pressure Π
Surface Viscosities [Ref2]:
Electrostatic forces
gas Ω
liquid
First assumption, the domains behave as two-dimensional fluid, which can be characterized by viscosity η and Gibbs elasticity KG. This is analogous with the 3D case.
In most experimental conditions, there is only one component on the water/air interface. So, we assume that inside of domain Ω, it is 2-dimensional liquid and outside of Ω, it is 2-dimensional gas. And at the boundary of Ω, the line tension λ balances with the difference of inside and outside surface pressures, so, this equation is satisfied. Here, κ is the curvature of the local boundary.
Because the gas phase is very dilute, we assume Kgas goes to 0. On the contrary, the liquid phase goes to infinity. In the other words, we assume the liquid phase is incompressible.
This is a major different assumption from others’ work. For example, in Stone and McConnell’s work, they assume that the inside and outside of Ω have the same viscosity and elasticity. Obviously, our assumption is much nearer to the real experimental conditions.
As I said just now, the viscosity of the subphase fluid is dominated, therefore, here we assume that the viscosities of the two surface phases are both ignorable.
Besides these, we still consider the electrostatic forces, including Van De Waals. But we don’t consider the dipole-dipole repulsion in our model because it is not easy to apply. So, at present, we just leave it aside.
Ref1: H. A. Stone; H. M. McConnell; Proc. R. Soc. Lond. A 448:
Ref2: Elizabeth K. Mann; PRE 51 (6): JUN 1995

10. Result on Small Distortion Limit For 2D
Now, let’s have a look at some results with this model.
First, let’s see the case of small distortion limit. The domain boundary is supposed to be a little bit off a circle and follow this function. In most experiments, n=2. That is, the domain is ellips.
From Stone and McConnell’s work, when the surface viscosity is negligible compared with the subphase fluid viscosity, for the case of n=2, we can get L/w-1 is exponential decay to relaxation time t. Here Tc is a function of line tention (lambda), bulk viscosity(eta), domain size (Ro) and mode number n.
Our experiments show the same exponential decay relationship. (Domain A and B are of different thicknesses.)
Using our dynamic model, we can obtain the same conclusion. So, our model works well on the small distortion limit. L w
Ref: H. A. Stone; H. M. McConnell
Hydrodynamics of quantized shape transitions of lipid domains Proc. R. Soc. Lond. A 448:

11. Lubrication Theory H(x, t) X Ref: L. Zhornitskaya; A. L. Bertozzi
The next case is the thin tether. At the beginning of my presentation, the video shows that there is no rupture on the tether, even though it is very very thin. We wish our model could give the same result.
Now, we only consider the thin tether. The coordinate is set up on the surface. We assume the tether is symmetric about x axis. Y is the distance from x axis to the domain edge. Using the lubrication theory, we can get this 4th order diffusion equation. (click)
This is a more general form. For Hele-Shaw case, we can get the same equation with n=1. So, our model also covers Hele-Shaw flow.
Now, our problem is how to solve this equation. Luckily, mathematicians already did some great job.(click) Zhornitskaya and Bertozzi prove that the solution of this equation is positive all the time and also gives us a numerical scheme to solve this lubrication equation. With this scheme, we can simulate the evolution of the tether.
Ref: L. Zhornitskaya; A. L. Bertozzi
Positivity-preserving numerical schemes for lubrication-type equations SIAM J. NUMER. ANAL. 37(2):

12. Simulation result Initial state:
Let’s see what will happen, if we choose a cosine function as the initial state. (PLAY VIDEO)
In the initial state, the tether is almost pinched off.
You can see there is no rupture and the tether relaxes to a straight line, which is in some ways consistent with the first video I just showed you.
However, you can still see it is different from the real relaxation. That is because we give some specific constrains to the simulation.
Initial state:

13. Discussion on the Simulation
Periodic Boundary condition
No ends
What constrains should be applied at the ends of the tether?
Firstly, periodic boundary condition is applied to simulate the relaxation process. We also assume the ends of the tether are fixed. These assumptions are not the same as the real experimental condition. (click) It raises a question to ourselves – What constrains should be applied at the ends of the tether?
At present, we are still working on this problem. We wish to find out a better boundary condition to simulate the real case.

14. Hole Closing
The last case I want to show you is hole closing.
This is PDMS monolayer. If there is a hole on this monolayer, will it stay there or disappear? With our model, our collaborator, Dr. Bernoff predicted that it will disappear linearly. Here is our experiment result. (PLAY VIDEO)
Attention to this part.
The contrast is very small because the film is too thin.
This is some recent prediction from him. I still don’t know how he worked it out.
Poly(dimethyl)siloxane (PDMS) monolayer on water/air interface

15. Conclusion
A simplified model with assumptions close to the real experimental conditions
Prospect
Line tension determination
Entire range of the relaxation behavior
So, comparing our model with others’, as well as with experimental results, we know this is a simplified model with assumptions close to the real experimental conditions and it works for some cases.
One of the major goals of this work is to develop a theoretical framework to determine the experimental line tension. Ideally, the whole range of relaxation behavior could be used in the line tension determination.
We are still working towards this goal.

16. Dr. Elizabeth K. Mann (Kent State University)
Acknowledgement
Dr. Elizabeth K. Mann (Kent State University)
Dr. Andrew J. Bernoff (Harvey Mudd College)
Dr. James C. Alexander (Case Western Reserve University)
Dr. J. Adin Mann Jr. (Case Western Reserve University)
Ms. Violeta Beleva (Kent State University)
Ms. Ji Wang (Kent State University)
Supported by National Science Foundation under Grant No
At the end, I want to thank my supervisor, Dr. Elizabeth Mann, she raised this idea and her paper in 1995 is one of the start points of this work.
I also want to thank all of our collaborators.
Dr. Andrew Bernoff, he is the mathematician deriving and solving most of the equations.
Dr. James C. Alexander and Dr. J. Adin Mann Jr., they set up the origin of this model with Dr. Elizabeth Mann.
Miss Violeta Beleva made program to do the simulation.
Ms. Ji Wang helped me to perform all the experiments.
This project is supported by …..

17. Frequent Questions Brewster Angle Microscope (set-up)
Green Function  Hele Shaw
F(n=2)=5PI/16 (Stone); F(n=2)=5PI/12
Hole closing, linearly

18. Brewster Angle Microscope (set-up)
CCD
Water Surface L2 L1 A P B Ei

19. Hole Closing Linearly