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} $

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-DelbrÃ¼ck 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

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

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 $

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!)

“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

Possible advantages

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}
$
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