Solute Diffusion under Centrifugation

Model the Lamm equation in a sector of a disk

This mass transport model describes the sedimentation and diffusion of a solute under ultracentrifugation in a disk sector cell.

The governing equation for describing this phenomenon is called the Lamm equation, which can be modeled with a 1D axisymmetric conservative mass transport equation.

Set up the mass transport model variable

vars

In[1]:=

vars={c[t,r],t,{r}};

Set up a line region

where

[cm]

denotes the radial position of the air-solution meniscus, which is formed during the centrifugation process, and

[cm]

the radial position of the cell bottom:

In[2]:=

=5.8;

=7.2;Ω=Line[{{

},{

}}];

Specify the model parameter's species diffusivity

[

/s]

and fluid flow velocity

, where

is the sedimentation coefficient in seconds (

) and

the angular velocity in

rad/s

In[3]:=

pars="DiffusionCoefficient"{d},"RegionSymmetry""Axisymmetric","MassConvectionVelocity"{scω^2r},"ModelForm""Conservative";

Set up the parameters with an initial unity concentration:

In[4]:=

parameters={c01,sc1.562*^-12,ω5236,d1.279*^-7};

Set up the initial conditions:

In[5]:=

ic={c[0,r]c0};

The flux is zero at both boundaries of the domain, so the boundary condition used is the

MassImpermeableBoundaryValue

that is applied at

.This specific boundary is the implicit default boundary condition for the conservative model.

Set up the equation:

In[6]:=

pde={MassTransportPDEComponent[vars,pars]0,ic}/.parameters

Out[6]=

0.0000428233c[t,r]+-

1.279×

-7

.

∇

{r}

c[t,r]+

∇

{r}

·{{-1.279×

-7

}}.

∇

{r}

c[t,r]+

∇

{r}

·{0.0000428233rc[t,r]}+

(1,0)

[t,r]0,{c[0,r]1}

Solve the PDE:

In[7]:=

Cfun=NDSolveValue[pde,c,{t,0,2525},r∈NDSolve`FEM`ToGradedMesh[Ω,"Alignment""Right","ElementCount"700]]

Out[7]=

InterpolatingFunction



Domain: {{0.,2530.},{5.8,7.2}}

Output: scalar

Data not saved. Save now



Visualize the concentrations at various points in time:

In[8]:=

Plot[{Cfun[0,r],Cfun[101,r],Cfun[404,r],Cfun[1010,r],Cfun[2525,r]},{r,

},PlotRange{0,1.5},AxesLabel{r,"C"},PlotLegends{"t=0 s","t=101 s","t=404 s","t=1010 s","t=2525 s"}]

Out[8]=

	t=0 s
	t=101 s
	t=404 s
	t=1010 s
	t=2525 s

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Solute Diffusion under Centrifugation

See Also

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Solute Diffusion under Centrifugation

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