CHAPTER 3 HARVESTING OXYGEN: THE OXYTRANSPORTER
3.3 A NANOSCALE FILLER MATERIAL
3.3.2 VALIDATING THE NANOPOROUS ANTI-CONDENSATION FILLER
A treated Vycor sample were then tested for oxygen permeability (Figure 3.15). The sample was placed in a heated water bath. To reduce axial oxygen transport out of the sample, it was tightly jacketed by a polyolefin heat shrink tube. The top of the sample was left above the water bath and
exposed to a step function of gas which goes from atmosphere to 100% humidified oxygen. The oxygen was humidified by passing it through a 20ºC bubbler between the oxygen source and the sample. An AL-300 AP-coat Fluorometrics Instruments oxygen probe and NeoFox reader were used to measure the response curve. This experiment was compared to an oxygen transport simulation in COMSOL.
Figure 3.15: Oxygen permeability of hydrophobic Vycor 2930. (a) Diagram of test setup.
Humidified oxygen is blown over silanized Vycor 2930. A polyolefin sheath surrounds the Vycor to prevent radial oxygen diffusion. The Vycor is placed inside a vial of water with ambient oxygen content. That vial is placed in a water bath on a hotplate which is agitated with a magnetic stirrer to maintain a constant temperature. Doing so assures an accurate temperature without introducing convective transport to the experiment. An AP coated oxygen probe (Fluorometrics Instruments) is placed against the Vycor’s base to measure oxygen. This was run at several temperatures. (b) The resultant curve at 37ºC was used to fit for the permeation constant of the material, which was found to be 3×104 Barrer, with a reduced-χ2 of 1.14.
Simulation began with a time dependent transport equation without convection. Both the tank of water and sample begin at ambient concentration. The boundary condition for the top of the sample is a constant oxygen concentration at the top surface with 760mmHg of oxygen tension. The boundary with water is calculated such that the flux is continuous, ∇c1|x1=x = ∇c2|x2=x, and the partial pressures are continuous across the boundary. Concentration is converted to partial pressure by Henry’s law, c = Hp, where H is the solubility constant. The permeability is the product of the diffusivity and solubility. Parameters for the model can be found in Table 3.5. The solubility of Vycor is assumed to be the product of air, 1/RgasT and its porosity, 0.28 [3.29]. The diffusivity is approximated by scaling the diffusivity of air by a factor, a [3.30]:
𝐷𝑜𝑥,𝑣𝑦𝑐 = 𝑎𝐷𝑜𝑥,𝑣𝑦𝑐 = 𝑎 (1.13 × 10−9[m2 s ] ( 𝑇
1[K])
1.724
) (3.9)
This accounts for the change in permeability over a wide range of temperatures. The exact division of solubility and diffusivity does not change the result of the simulation, only their product, the permeability, is relevant. The permeability of the material was swept until the curve matched the experiment at 37ºC (Figure 3.15c), which was 3x104 Barrer at 37ºC.
Table 3.5: Parameters used in the COMSOL simulation of Vycor. Most parameters are sourced from literature. The diffusion coefficient for Vycor was scaled from the diffusion coefficient from air. The scaling factor, a, was determined by the simulation to the experiment at 37ºC. The solubility coefficient was taken from the equivalency for air to Henry’s law, 𝑝 = 𝑐/𝑅𝑔𝑎𝑠𝑇, where it was scaled by the open pore percentage. Since Fick’s law can be written in terms of permeation,
𝜕𝑝
𝜕𝑡= 𝑃𝛻2𝑝, the exact division between the solubility coefficient and diffusivity is unimportant;
only their product affects the simulation.
PARAMETER VALUE
POROSITY OF VYCOR 2930 [3.29]
0.28 DIFFUSION COEFFICIENT OF
O2 IN AIR [3.30] 1.13 × 10−9[m2 s ] ( 𝑇
1[K])
1.724
DIFFUSION COEFFICIENT OF
O2 IN VYCOR 𝑎 (1.13 × 10−9[m2 s ] ( 𝑇
1[K])
1.724
) DIFFUSION COEFFICIENT OF
O2 IN WATER [3.31]
3.33×105 [cm2/s]
SOLUBILITY OF O2 IN WATER [3.14]
1.3×10-3 [mol∙L-1∙atm-1]
SOLUBILITY OF O2 IN VYCOR 0.28 𝑅𝑔𝑎𝑠𝑇 VYCOR 2930 MEAN PORE SIZE
[3.26]
44 [Å]
VYCOR 2930 STANDARD DEV.
PORE SIZE [3.26]
4 [Å]
CONDUCTIVITY OF VYCOR 2930 [3.32]
1.4 [W∙m-1∙K-1] CONDUCTIVITY OF
POLYOLEFIN [3.33]
0.2 [W∙m-1∙K-1] POLYOLEFIN THICKNESS 350 [µm]
VYCOR 2930 SHAFT RADIUS 1 [mm]
VYCOR 2930 SHAFT LENGTH 4 [mm]
As the water bath is heated, the water vapor inside the Vycor sample will surpass the saturation pressure predicted by the Kelvin equation, at which point water would condense inside the pores (Figure 3.16). The time constant was determined by fitting the function:
𝑝 = 𝐴𝑒−𝑡/𝜏+ 𝐵 (3.10)
where τ represents the time constant. A smaller time constant is indicative of greater permeability as τ ∝ 1/P. As the water bath temperature increases the time constant is expected to drop, the pore saturation pressure is reached, at which point some pores will become occluded.
A multi-part COMSOL simulation calculated temperature, water vapor partial pressure, bulk saturation pressure, and p/psat,bulk. If p/psat,bulk surpasses the ratio determined by the Kelvin equation, the diffusivity of the region is switched from that of hydrophobic Vycor to that of water.
Heat transport is given by the equation:
∇ ∙ (k∇T) = 0, (3.11)
where k is the thermal conductivity of the material. The water bath is at uniform temperature. Contact between the Vycor and polyolefin coating is assumed to be continuous in temperature and thermal flux. Parameters for the model can be found in Table 3.5. Water vapor transport is calculated similar to oxygen transport with the boundaries in contact with water and fixed at concentrations given by
the saturation partial pressure of water vapor at their respective temperatures [3.21]. The results of these simulations are used to calculate p/psat,bulk.
Using the Kelvin equation, the critical pore radius above which condensation would occur can be found by:
rpore = 2γVm
RgasT log(p/pbulk) (3.12) Note that the pores of Vycor are not all uniform in size, and tend to be normally distributed with a mean diameter, μpore, of 44Å and a standard deviation, σ, of 4Å [3.26]. Since any section of the FEM mesh includes a large number of pores, the cumulative function for a normal distribution can be used to determine the percentage of pores above or below the critical pore diameter. The diffusion constant of un-occluded Vycor is scaled to the percent of pores that remain below the critical diameter. The percent of pores above this critical diameter is multiplied by the diffusivity of water.
Therefore, for every mesh tetrahedral, there is a diffusion coefficient given by:
Dvyc= {
Dox,vyc(T), p/psat,bulk< 1 Dox,vyc(T)
2 [1 + erf (2rpore− μpore
σ√2 )] +DH2O
2 [1 − erf (2rpore− μpore
σ√2 )] , p/psat,bulk≥ 1
(3.13) This diffusivity was then used on a simulation of the oxygen permeability of the Vycor sample. The time constant was found by fitting equation (3.10) to the simulated data (Figure 3.16). The experimental time constant increases sooner than the simulated one; suggesting the pore size may be 10Å to 20Å larger than expected from literature. Regardless, experiment and simulation agree that the nanoporous glass is resistant to condensation under typical physiological conditions.
Figure 3.16: Hydrophobic Vycor’s resistance to occlusion under high humidity. (a) The water bath was heated to various temperatures, and a step function of 100% oxygen was applied. The oxygen tension at the probe (Figure 3.15a) was measured over time. The time constant of the resultant curve, similar to Figure 3.15c, was fitted. The same procedure was done for the simulation. (b) The oxygen tension at 100s in 87ºC water bath, and the ratio of pore equilibrium partial pressure to bulk partial pressure in 87ºC water bath are shown. These time constants were then plotted. Below the equilibrium partial pressure of the pores, no condensation would be expected. Therefore, as temperature increases the time constant decreases or remains relatively flat. As the pores being to condense, fewer high permeation paths through the material are left open and the net permeation decreases, thereby increasing the time constant. Both simulation and experiment demonstrate that hydrophobic Vycor is robust in physiological conditions.