Brian A Camley & Frank LH Brown

Dynamic scaling in phase separation kinetics for quasi-two-dimensional membranes

The Journal of chemical physics 135 (22) 225106 (2011)

Amphiphiles meeting UW 2016

Daniel Duque
CEHINAV, UPM, Spain
daniel.duque@upm.es

### Resources

• Lately, I have been doing hydrodynamics (Naval Eng)
• So, it seemed natural to do research on membrane dynamics while at UW

## Some simulations

pFEM method

### Conserved dynamics

We have conserved dynamics, since $\int \phi = \langle\phi\rangle$ is constant. "Model B" procedure begins with a free energy functional
$F=\int d\mathbf{r} \, f(\phi,\nabla\phi)$

With the chemical potential
$\mu = \frac{\delta F }{\delta \phi}$

Recipe:
$\frac{\partial \phi}{\partial t} = \gamma \nabla^2 \mu$

It's easy to show that $f=\frac a2 \phi^2$ usual diff results, $\frac{\partial \phi}{\partial t} = D \nabla^2 \phi$

### Cahn - Hilliard theory

For phase separation, vdW-G-L:
$f(\phi) = - \frac a2 \phi^2 + \frac b4 \phi^4 + \frac c2 (\nabla\phi)^2$

Whichs yields the chemical potential
$\mu = \frac{\delta F }{\delta \phi} = - a \phi + b \phi^3 - c \nabla^2\phi$

Our conserved dynamics is
$\frac{\partial \phi}{\partial t} = \gamma \nabla^2 \mu ,$
the celebrated Cahn-Hilliard equation

### Convection and hydrodynamics

Convection of $\phi$ due to velocity field $\mathbf{u}$:
$\frac{\partial \phi}{\partial t} \rightarrow \frac{d \phi}{d t}$           $\frac{d \phi }{d t} = \frac{\partial \phi}{\partial t} + \mathbf{u} \cdot \nabla \phi$

Then, convection - diffusion is "simply"
$\frac{d \phi}{d t} = D \nabla^2 \phi$

and convected CH is "simply"
$\frac{d \phi}{d t} = \gamma \nabla^2 \mu ,$

### The momentum equation

Applying convection to velocity itself results in the Navier-Stokes equation:

$\rho \frac{d \mathbf{u} }{d t} = \eta \nabla^2 \mathbf{u} - \nabla p + \mathbf{f}$

Similar to Newton's 2nd law (yet so different!):
$m \frac{d \mathbf{u} }{d t} = - \alpha \mathbf{u} + \mathbf{f}$

### The two equations

With the chemical potential
$\mu = - a \phi + b \phi^3 - c \nabla^2\phi$

$\frac{d \phi}{d t} = \gamma \nabla^2 \mu,$

$\rho \frac{d \mathbf{u} }{d t} = \eta \nabla^2 \mathbf{u} - \nabla p - \phi \nabla \mu$

There's $a$, $b$, and $c$, which can be related to: $\sigma$, $\xi$, and $\langle\phi\rangle$.

There's also $D$, $\eta$, $\rho$, length and time scales $\ldots$
But I can show that everything may be reduced to two parameters!
$\mu = - \phi + \phi^3 - \nabla^2\phi$
$\frac{d \phi}{d t} = D^* \nabla^2 \mu,$
$\mathrm{Re} \frac{d \mathbf{u} }{d t} = \nabla^2 \mathbf{u} - \nabla p - \phi \nabla \mu$

### Low Reynolds hydrodynamics

The Reynolds number is very low in membranes, so we end up with the creeping flow (Stokes) equation:
$\nabla^2 \mathbf{u} - \nabla p - \phi \nabla \mu = 0$

Which can be solved for any force:
$\nabla^2 \mathbf{u} - \nabla p + f = 0$

### Quasi-2D hydrodynamics

There is indeed an additional solvent-mediated force !
$\nabla^2 \mathbf{u} - \nabla p - \phi \nabla \mu - \frac{\xi}{L_\mathrm{SD}} \int d^2\mathbf{r'} K(\mathbf{r}-\mathbf{r'}) \mathbf{u} (\mathbf{r'}) = 0 ,$

whose range is given by the Saffman-Delbrück length:
$L_\mathrm{SD} = \frac{\eta_\mathrm{m} }{ 2 \eta_\mathrm{f} }$

this term, I still don't know how to model ...

### Cases

There should be a table for simulation cases! All the details are given as figure captions...
 case Lengths

(The spacial resolution is always $L/1024$)

### Figure 1

This case seems to be the only one with actual, realistic parameters ! For the rest of the cases, the solvent's viscosity is either $400\times$ that of water, or zero.

### Figures 3&4

$R_n \sim t^{1/2}$
The article really concerns dynamic scaling, not so much actual systems

### Figures 3&5

$R \sim t^{1/2}$

### Figures 6 & 7

No scaling observed

### Figures 8 & 9

$R \sim t^{1/3}$

### Figures 10 & 11

$R \sim t^{1/2}$

### Criticism

• No dimensional study
• One set of reasonable parameters, the rest have no justification (why not vary the temperature, add detergent ... )
• I am not sure thermal noise should be included at all
• As mentioned, the spatial resolution does not really cover the interfacial zones, of width $\xi$
• I think the temporal scale is also too coarse

### Times

The velocity time scale is set by $t_0 = \eta \xi / \sigma$
 case Times

### Diffusion times

The diffusion time scale is set by $t_0 /D^*$. Some cases look better in this regard (not Figure 1!)
 case Times

# Thanks

This presentation is at: https://ddcampayo.github.io

Created with reveal.js, the HTML Presentation Framework

## Particle-in-cell idea

“You don't have to move all your particles, you may keep some fixed, as in the old Particle-in-Cell method”

J Monaghan, SPHeric 2015 at Parma

Starting with particles + velocity values

Let's project onto a fixed mesh

Using some triangulation (pFEM style)

The problem is on the mesh only

We compute new velocities (and pressure)

We project back onto the particles ...

Particles are finally moved, Lagrangian-ly

• From particles: linear algebra may be done once !!
• From particles: easier boundary conditions and FSI
• From mesh: avoidance of false diffusion through particles

## pFEM simulations

 p-FEM: particle + FEM proj-FEM: particle + ( mesh + FEM )

## The problem with projection

Unfortunately, projection may spoil convergence
$\frac{d A}{d t} = Q$      $\frac{d r }{d t} = u$

$E= T$ $(\Delta t)^2$ $\left( \ddot{Q} + \frac{Q'' u_0^2}{ \mathrm{Co}^2} \right)$ $+ \underbrace{T \color{red}{ (\Delta t)} A'' \frac{ u_0^2 }{ \mathrm{Co}_h^2} \left( 1+ \frac{1}{ m^2} \right) }_\mathrm{proj}$
But, this can be fixed!

## Improving projection

The former holds for linear projection !

$E= T \color{red}{ (\Delta t)^2 } \left( \ddot{Q} + \frac{Q'' u_0^2}{ \mathrm{Co}^2} \right) + \underbrace{T \color{red}{ (\Delta t) } A'' \frac{ u_0^2 }{ \mathrm{Co}_h^2} \left( 1+ \frac{1}{ m^2} \right) }_\mathrm{proj}$
$E= T \color{red}{ (\Delta t)^2 } \left( \ddot{Q} + \frac{Q'' u_0^2}{ \mathrm{Co}^2} \right) + \underbrace{T \color{red}{ (\Delta t)^2 } A''' u_0^3 \left( \frac{1}{ \mathrm{Co}_H^3} + \frac{1}{ \mathrm{Co}_h^3} \right) }_\mathrm{proj}$

... or the offending term can be made small by particle crowding

## Zalesak's disk

 Linear FE Quad mesh Quad mesh + parts Equal parts. mesh Part crowding

## Taylor-Green vortex sheet

 pFEM projFEMq projFEMq projFEM6

## Rayleigh-Taylor instability

 pFEM projFEMq projFEM6 OpenFOAM

## Extra: Kelvin-Helmholtz instability

 pFEM projFEMq

## Conclusions

• Linear projection is not a good idea ...
• Future: Other boundaries, free surface.
• Future: quadratic FEM + better time integrator.

## CGAL programming


void areas(Triangulation& T) {

for(F_f_it fc=T.finite_faces_begin();
fc!=T.finite_faces_end();
fc++) {

Periodic_triangle pt=T.periodic_triangle(fc);

Triangle t=T.triangle(pt);

fc->area=t.area();

}

return;
}