diffusion_from_disk: diffusion_from_disk

In tbgitoo/textureAnalyzerGels: Reading and analyzing texture analyzer data

diffusion_from_disk

Description

Remaing total fraction of solute in a originally homogeneously loaded disk, with the boundary condition of c=0 on the portour

Usage

diffusion_from_disk(r,D,t)

Arguments

r	Radius of the disk
D	Diffusion coefficient
t	Time since the beginning of the diffusion

Details

The general solution of the diffusion equation in cylindrical coordinates can be expressed in terms of Bessel functions. It can for instance be found in Keil 1971. For a disk with initially homogenous concentrations, the coefficients take a slightly particular form (e.g. https://mathworld.wolfram.com/HeatConductionEquationDisk.html) such that the integral giving the remaining solute fraction becomes:

B(tau)=4*SUM_n 1/jon^2 * exp(-jon^2*tau)

where tau=D*t/r^2 and where the jon terms indicate the n-th zero of the Bessel function of the first kind, order 0 (e.g. J0). The 4 arises due to the particular value of the infinite series for tau=0, where the exponential terms all evaluate to 1 (c.f. deLyra 2013).

The time-dependency of the remaining solute in a disk was used to evaluate hydrogel pore size from water loss under compressive load Braschler et al. (2015).

Value

Fraction of solute still remaining in the disk

Author(s)

Thomas Braschler

References

E. Keil, 13.12.1971, Solution of the diffusion equation in cylindrical coordinates and comparison with experiments, https://cds.cern.ch/record/1129859/files/CM-P00065889.pdf

Wolfram web ressources: https://mathworld.wolfram.com/HeatConductionEquationDisk.html

Jorge L. deLyra, On the Sums of Inverse Even Powers ofZeros of Regular Bessel Functions, https://arxiv.org/abs/1305.0228 Braschler T, Songmei W, Wildhaber F, Bencherif SA, Mooney DJ, "Soft nanofluidics governing minority ion exclusion in charged hydrogels", Soft Matter, Issue 20, 2015, Supplementary 2.

See Also

The function makes use of the bessel_zero_J0 function

Examples

t=seq(from=0,to=1,by=0.01)
remaining=diffusion_from_disk(1,1,t)
plot(remaining~t)

tbgitoo/textureAnalyzerGels documentation built on March 30, 2022, 4:53 a.m.