pore_size_from_hydrodynamic_pressure: pore_size_from_hydrodynamic_pressure

In tbgitoo/textureAnalyzerGels: Reading and analyzing texture analyzer data

pore_size_from_hydrodynamic_pressure

Description

Order of magnitude estimation of the pore size from stress hysteresis. The idea is to estimate approximately an effective pore size that could explain higher stress by the pressure build-up due to lag in pore fluid evacuation as opposed to underpressure when relaxing the same gel. In general, such differences are large with micro- or nanoporous gels, but very small with macroporous gels and thus difficult to measure at all but very high compression rates in macroporous gels.

Usage

pore_size_from_hydrodynamic_pressure(P_up=500,P_down=1000,gel_thickness=1e-3,v_compression=1e-5,viscosity=1e-3,disk_radius=2e-3)

Arguments

P_up	Pressure (stress) observed at a given strain during relaxation ("up" movement of the chuck)
P_down	Pressure (stress) observed during compression ("down" movement of the chuck), this is generally larger than P_up
gel_thickness	Height of the gel, in meters
v_compression	Speed of the chuck, identical except for direction in down and up movement. Units: m/s
viscosity	Viscosity of the pore fluid, in Pa*s
disk_radius	Radius of the disk sample, in meters

Details

This function is based on an order of magnitude estimation. For a single pore, Poiseuille's law of laminar flow resistance in a cylindrical tube gives:
Qsingle_pore = deltaP*pi*r^4/8/viscosity/disk_radius
where deltaP=P_down-P_up, and r is the characteristic pore radius we are looking for. There are many more or less warranted assumptions, such as all pores having the same length of disk_radius, all pores being cylindrical and of identical radius r and so forth, but one has to keep in mind that this is an order-of-magnitude estimation only.

For N pores in parallel, the flow rate is N times higher; it must at the same time match the pore fluid evacuation rate given by the volume displacement of the chuck:
Qtotal = v_compression*pi*disk_radius^2 = N*Qsingle_pore
The number of parallel pores can roughly be obtained from the cross section surface, roughly at half the disk radius:
N = disk_radius*pi*gel_thickness/pi/r^2
Assembling, we get: Qtotal = disk_radius*pi*gel_thickness/pi/r^2 * deltaP*pi*r^4/8/viscosity/disk_radius = disk_radius/disk_radius * pi/pi*pi * gel_thickness * r^4/r^2 * deltaP/8/viscosity =
Qtotal = pi*gel_thickness*r^2*deltaP/8/viscosity = v_compression*pi*disk_radius^2

The final expression, used in this function, for the pore radius estimation is thus:
=> r = disk_radius*sqrt(v_compression *8 *viscosity /gel_thickness/deltaP)

Value

Numerical estimation of the pore radius

Author(s)

Thomas Braschler

Examples

pore_size_from_hydrodynamic_pressure(P_up=500,P_down=1000,gel_thickness=1e-3,v_compression=1e-3,viscosity=1e-3,disk_radius=2e-3)

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