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

Why this article

Figure 1

Why this article

  • 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 $

Parameter overload!


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...
Lengths
case $\xi$ (nm) $L$ ($\mu$m) $L_\mathrm{SD}$ (nm)
Fig 1 $40$ $30$ $2500$
Fig 2 $20$ $20$ $25$ (!)
Figs 3 , 6 & 12 $5$ $5$ $6.2$ (!)
Fig 7 $40$ $30$ $\infty$

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

Figure 1


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.

Figure 1

Figure 1

Figures 3&4

Figure 3 Figure 4
$R_n \sim t^{1/2} $
The article really concerns dynamic scaling, not so much actual systems

Figures 3&5

Figure 3 Figure 5
$R \sim t^{1/2} $

Figures 6 & 7

Figure 6 Figure 7
No scaling observed

Figures 8 & 9

Figure 8 Figure 9
$R \sim t^{1/3} $

Figures 10 & 11

Figure 10 Figure 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 $
Times
case $t_0$ ($\mu$s) #steps = $t_0 / \Delta t$
Fig 1 $2\times 10^{-3}$ $10^{-4}$ !
Fig 2 $5\times 10^{-3}$ $5\times 10^{-5}$ !!
Fig 3 $3\times 10^{-2}$ $8\times 10^{-3}$ !
Fig 7 $2$ $0.2$

Diffusion times

The diffusion time scale is set by $t_0 /D^* $. Some cases look better in this regard (not Figure 1!)
Times
case $t_0/D^*$ ($\mu$s) #steps = $t_0/D / \Delta t$
Fig 1 $0.05$ $5\times 10^{-3} $ !
Fig 2 $61$ $0.62$
Fig 3 $0.8$ $0.8$
Fig 7 $50$ $5$

Some results

Thanks




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


Created with reveal.js, the HTML Presentation Framework

Figure 2

Figure 2

Figure 12

Figure 12

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
Motivation picture
Starting with particles + velocity values
Motivation picture
Let's project onto a fixed mesh
Motivation picture
Using some triangulation (pFEM style)
Motivation picture
The problem is on the mesh only
Motivation picture
We compute new velocities (and pressure)
Motivation picture
We project back onto the particles ...
Motivation picture
Particles are finally moved, Lagrangian-ly

Possible advantages

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} $
with a quadratic 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)^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

Errors for Taylor-Green

Velocity err vs $\Delta t$

Errors for Taylor-Green

Pressure err vs $\Delta t$

Errors for Taylor-Green

Velocity err vs CPU time


Errors for Taylor-Green

Pressure err vs CPU time


Rayleigh-Taylor instability


pFEM
projFEMq
projFEM6
OpenFOAM

Extra: Kelvin-Helmholtz instability




pFEM
projFEMq

Conclusions

Thank you for your attention

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;
}