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Ind. Eng. Chem. Res. 2008, 47, 3540-3550
Polarographic Method for Measuring Oxygen Diffusivity and Solubility in Water-Saturated Polymer Films: Application to Hypertransmissible Soft Contact Lenses Mahendra Chhabra,† John M. Prausnitz,†,‡ and Clayton J. Radke*,†,§ Department of Chemical Engineering, UniVersity of California, Berkeley, California 94720, Chemical Sciences DiVision, Lawrence Berkeley National Laboratory, Berkeley, California 94720, and Vision Science Group, UniVersity of California, Berkeley, California 94720
An electrochemical-polarographic method is described for measuring the diffusivity, D, and solubility, k, of oxygen in aqueous-saturated polymer films. While the apparatus and procedure are general for such films, it is here applied to determine D and k for oxygen in hypertransmissible soft contact lenses. Usually, only oxygen permeability, P, the product of D and k, is measured because P gauges the steady flux of oxygen through hydrogel membranes. However, we utilize the polarographic technique in the unsteady state and, hence, obtain D and k separately. Determination of each of these properties is critical for designing better lens materials that ensure sufficient oxygen supply to the cornea. We have measured oxygen diffusivities and solubilities for nine commercial soft contact lenses. Our data indicate that oxygen diffusivity is primarily responsible for the range of oxygen permeability observed for hypertransmissible soft contact lenses. For 2-hydroxyethyl methacrylate (HEMA)-based lenses, measured solubilities suggest that over 90% of the dissolved oxygen partitions to the polymer phase. 1. Introduction Polymer films are useful for a variety of technical applications including, for example, membrane-separation processes, food packaging, and contact lenses.1-5 In many of these applications, the polymer film is water saturated. For design of such films, it is essential to know oxygen permeability, P ) Dk, where D is oxygen diffusivity and k, a measure of solubility, is the inverse of Henry’s constant for oxygen in a polymer film.6 Diffusivity is sensitive to the detailed path(s) by which oxygen migrates through the membrane and is, therefore, sensitive to microstructure. Solubility, however, is sensitive primarily to the amounts of various material components and not primarily on how they are distributed in space. For optimum design of new materials, it is useful to know how D and k separately depend on the polymer-film composition and structure. We describe here a polarographic method that provides both D and k. While this method is general for aqueous-saturated polymer films, we here focus on measurement of D and k for oxygen in hypertransmissible soft contact lenses (SCLs). Essentially, the lens is placed on an electrode and submerged in nitrogen-saturated water to remove all oxygen in the system. Nitrogen-saturated water is then suddenly resaturated by air, and the current generated by oxygen consumption at the electrode is recorded as a function of time. The transient rise in current determines oxygen diffusivity, D, whereas the steadystate current determines oxygen permeability, P. Oxygen solubility, k, follows from the ratio of oxygen permeability to diffusivity. High lens oxygen diffusivities and solubilities are critical for normal functioning of the cornea,5,7 an avascular tissue that * To whom correspondence should be addressed. E-mail: radke@ berkeley.edu. Tel.: +1 510 642 5204. Fax: +1 510 642 4778. † Department of Chemical Engineering. ‡ Chemical Sciences Division, Lawrence Berkeley National Laboratory. § Vision Science Group.
depends on external oxygen sources to carry out cellular respiration. The cornea receives oxygen mainly from air during waking hours and from the palpebral conjunctiva (the back portion of the upper eyelid) during sleep.8,9 Contact-lens wear obstructs the flow of oxygen from the external environment to the cornea. If the obstruction is large enough, the result is insufficient oxygen supply. Corneal hypoxia initiates many unwarranted complications such as corneal edema, corneal acidosis, loss of transparency, epithelial keratitis, endothelial polymegethism, limbal hyperemia, neovascularization, and changes in myopia.5,10-14 Thus, it is imperative to ensure maximum transport of oxygen through a contact lens worn on the eye. The amount of oxygen available from the palpebral conjunctiva is about one-third compared to that from air.15 Thus, corneal oxygen supply is even more important for overnight lens wear. Transport of oxygen to the cornea can be maximized by designing lenses with both high oxygen diffusivity and solubility. We focus attention on measurement of oxygen diffusivity and solubility in SCLs, especially in hypertransmissible lenses, i.e., those with permeabilities >100 barrer. Lens materials supplied by the manufacturer, but not produced in the commercial molding process, may not necessarily be the same as that of the commercial lens.16,17 Hence, in this work, only commercially available soft contact lenses are investigated. Some previous measurements of oxygen diffusivity and solubility have been carried out for low-Dk hydrogel lenses18,19 or for various membranes.20-24 However, to our best knowledge, D and k values are not available for current high-Dk SCLs. Measurement of D and k for high-Dk lenses is difficult compared to that for low-Dk lenses because of faster response time (small lens thickness and large diffusivity), edge effects,17,25 significant boundary-layer resistances,18,19,22,24,26 and lens desiccation.26,27 On the basis of our earlier work on high-Dk measurements,28 we here overcome the difficulties encountered in measuring oxygen diffusivity and solubility for hypertransmissible SCLs. An improved electrochemistry-based polarographic method
10.1021/ie071071a CCC: $40.75 © 2008 American Chemical Society Published on Web 02/27/2008
Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3541 Table 1. Properties of Commercial Soft Contact Lenses Used in This Study brand name
material
manufacturer
base curve radiusa (mm)
hydration (wt % H2O)
L (µm)
Biomedics 38 Biomedics 55 Acuvue 2 Acuvue Advance Acuvue Oasys O2 Optix PureVision Biofinity Focus Night & Day
Polymacon Ocufilcon D Etafilcon A Galyfilcon A Senofilcon A Lotrafilcon B Balafilcon A Comfilcon A Lotrafilcon A
CooperVision CooperVision Johnson & Johnson Johnson & Johnson Johnson & Johnson CIBA Vision Bausch & Lomb CooperVision CIBA Vision
8.6 8.6 8.7 8.7 8.4 8.6 8.6 8.6 8.6
38 55 58 47 38 33 36 48 24
46 ( 1 81 ( 1 88 ( 1 82 ( 1 79 ( 3 86 ( 1 104 ( 1 89 ( 1 92 ( 5
a
Base-curve radius of various lenses was maintained as close as possible to 8.6 mm electrode radius to minimize curvature differences.
Figure 1. Schematic of the apparatus for measurement of oxygen diffusivity and solubility in soft contact lenses.
reports the transient current response of the lens to a step change in oxygen concentration, thereby enabling measurement of D and k. 2. Experimental Section 2.1. Materials. Except for Biofinity, commercial SCLs were obtained from Con-cise Contact Lens Co. (San Leandro, CA). CooperVision Inc. (Pleasanton, CA) supplied the Biofinity SCLs. The thickness of a commercial lens varies radially depending on the optical corrective power, typically measured in Diopter. A harmonic-mean thickness best gauges the average thickness within the central area of an axially symmetric lens.29 The harmonic-mean thickness of each lens was measured using an ET-3 electronic thickness gauge (Rehder Development Company, Castro Valley, CA) to an accuracy of (2 µm. Table 1 gives lens properties including brand name, material, manufacturer, base-curve radius, saturated-water content, and wet thickness, L. The optical power of all our commercial contact lenses used was -3.0 Diopters. 2.2. Apparatus. An improved polarographic apparatus is used to measure oxygen diffusivity and solubility for hypertransmissible SCLs. Details of the cell design and operation principles are available elsewhere.28 Figure 1 provides an overall schematic of the apparatus. In separate lines, nitrogen or air is supplied to the cell through a 3/8 in. filter (4ZK84, Grainger, Inc., Berkeley, CA) with the flow rate measured by rotameter (15-078-116, 15-078-137, Fischer Scientific, Santa Clara, CA). The exit stream from the flowmeter is water saturated using a sparged humidifier containing phosphate-buffer saline (PBS). Following humidification, gas is preheated by flowing through a copper
coil immersed in the same external heating bath as that used to maintain constant temperature in the cell (35 °C). Also, 3/8 in. Tygon tubing, coming from and returning to the bath, is tightly coiled around the outside of the central section of the cell and provides the heating source for temperature control. Foam insulation surrounding the cell minimizes heat losses. For our apparatus, the temperature of the external bath is maintained at 62 °C. Figure 2 shows the modified polarographic cell. The cell electrode assembly consists of a 14.3 mm diameter cylinder capped by a 8.6 mm radius spherical segment and supported by a larger diameter Lucite cylindrical base. The capping segment consists of a 4 mm diameter gold cathode (working electrode) surrounded by a concentric silver anode (counterelectrode). The two electrodes are isolated by epoxy resin to prevent short-circuiting. Once the lens is placed over the electrode assembly, a lens holder of 8.7 mm radius of curvature secures it in place. To avoid edge effects, the circular opening in the lens holder has the same diameter as that of the cathode.17,25,28 Preheated gas enters the cell through a 1/4 in. coarse cylindrical-fritted-glass sparger. The central region of the sparger is offset by 1 cm to enable proximity to the propeller; the bottom tip of the sparger is located 1 cm from the bottom of the stirrer shaft. A draft tube is mounted from the cell top and circumscribes both the stirrer shaft and sparger. The purpose of the draft tube is not only to provide well-defined flow profiles inside the cell but also to decrease the residence time of air in the cell. A small residence time ensures that fresh oxygen is rapidly available at the cathode. Accordingly, the transient current response is due only to the diffusion of oxygen to the cathode
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Figure 3. Current versus time for Biofinity commercial soft contact lens at 35 °C. The steady-state current is Imax, and the steady background current is IB. Voltage is applied at t ) 0 (indicated by left arrow). Air is bubbled (indicated by right arrow) when IB is reached.
Figure 2. Schematic of the polarographic cell. Reprinted with permission from ref 28. Copyright 2007 Elsevier.
and is not affected by mixing of air into the liquid. The propeller and draft tube are designed to drive oxygen in the air-saturated buffer directly toward the cathode for consumption. Air residence time is also minimized by increasing the air flow rate into the cell, as discussed later. A small unfilled gap of 2.5 cm is left at the cell top to allow bubbling of nitrogen/air at higher flow rates without buffer spillage. Bubbled gases exit the cell through an opening at the cell top fitted with glass-wool filter to minimize loss of any entrained buffer. Cables from the electrode assembly connect to a FAS1 femtostat (# 992-00005, Gamry Instruments, PA), installed as an external measurement module, and a controller card in the computer. Unsteady current data are recorded using Chronoamperometry application of physical electrochemistry software (PHE200, Gamry Instruments, PA). 2.3. Experimental Procedure. To provide precise assessment of D and k, the lens must initially be in a completely oxygenfree state. Therefore, the contact lens is initially soaked in a nitrogen-saturated phosphate buffer saline (PBS) solution [8.3 g/L of NaCl (roasted overnight at 450 °C to remove organic impurities), 0.467 g/L of NaH2PO4‚H2O, 4.486 g/L of Na2HPO4‚ 7H2O, pH ) 7.40 ( 0.1]30 in a separate vial for at least 24 h before use. Then, the lens is again soaked for at least 1 h in a beaker with nitrogen-saturated PBS while continuously bubbling pure nitrogen. The deoxygenated lens is carefully placed atop the electrode and fastened by the lens holder. A few drops of nitrogen-saturated PBS buffer are placed on the lens to maintain aqueous saturation. Also, filtered nitrogen is continuously fed through the cell during assembly. Once all parts of the cell are assembled, preheated nitrogen-saturated PBS solution at 35 °C (human-eye temperature) is poured into the cell from the top opening. Because the contact lens is exposed to air for a few minutes during cell assembly, an unknown amount of oxygen enters the lens during this period. Oxygen in the central region of the lens is easily purged because this region is in contact with the nitrogen-saturated solution before the start of the experiment. However, it is difficult to remove oxygen in the extended lens region covered by the lens holder. Therefore, after
lens assembly, nitrogen is continuously bubbled at a flow rate of 15 L/min through the cell for at least an additional 2 h to remove residual oxygen from the wings of the lens. The theory underlying this effect is discussed in Appendix A. Once the lens is nearly free of oxygen and thermal equilibrium is reached, the cell is operated at limiting-current condition for a few minutes to remove any additional residual oxygen. Applied voltage at limiting current is determined from an experimental current-voltage relationship as -0.9 V applied to the cathode (relative to the anode).28 Under this condition, oxygen molecules are immediately consumed as they reach the surface of the cathode. Consequently, oxygen concentration is nearly zero on the cathode surface. Once background current is reached, signifying purging of all oxygen, preheated and humidified air is bubbled at 15 L/min, establishing a step change in oxygen concentration from zero (when nitrogen is bubbled) to ∼20.9% in air using a four-way valve. Transient current increase is observed due to influx of oxygen through the lens onto the cathode. Air flow rate must be at least 10 L/min to ensure that the response of the system is due to diffusion of oxygen to the cathode and is not influenced by the time required to reach air saturation in the cell. This flow rate was determined by independent transient experiments at different air flow rates of 0.4, 2, 10, and 15 L/min. Nitrogen and air flow rates are maintained constant at 15 L/min to ensure that the thermal equilibrium established during the purging period is maintained even after the step change in oxygen concentration. During the transient experiment, the contact lens is kept water saturated by continual exposure to the aqueous PBS solution so as to mimic conditions corresponding to on-eye lens wear. The cell is stirred continuously at 1200 rpm for all experiments. High stirring speed is used to minimize the mass-transfer boundary-layer resistance.28 Figure 3 shows typical data for current as a function of time for a Biofinity commercial SCL. Initially, nitrogen is bubbled for 2 h (data not recorded during this time). Then, at t ) 0, -0.9 V voltage is applied to remove any residual oxygen. Once a steady background current, IB, is reached, a step change in oxygen concentration from zero to the concentration of oxygen in air is achieved by bubbling air into the aqueous buffer. As shown in Figure 3, transient current rises monotonically to a steady maximum current, Imax. For high-Dk lenses, we found that the current decreases a few minutes after reaching the
Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3543
linearly across the stagnant layer to the cathode surface where the oxygen concentration is c0. This concentration is nearly zero because of the limiting-current condition at the cathode. The total mass-transfer resistance, R, for oxygen to transport from the bulk solution to the cathode is given by28
R ) RBL +
Figure 4. Steady-state concentration profiles across the lens and surroundings, neglecting small differences in curvatures of the different layers.
maximum current, probably because of cathode passivation due to reduction of buffer constituent(s) (or a derived compound)31 or deposition of impurities. In such cases, only the current data up to the maximum current were analyzed. To remove any prior passivation, the electrode is lightly polished with soft tissue and simichrome polish (Stained Glass Garden, Berkeley, CA) before each experiment.
O2 + 2H2O + 4e- f 4OH-
(1)
Concomitantly, chloride ions in the saline buffer solution migrate to the surface of the counterelectrode (anode)33 and react according to32
Ag + Cl- f AgCl + e-
(2)
Figure 4 diagrams the steady-state oxygen concentration profile across the lens and surroundings. A mass-transfer boundary layer of thickness LBL resides at the top surface of the contact lens facing the bulk solution. LBL is set by stirring speed.28 The central layer is the soft contact lens of thickness L, and the bottom layer is the stagnant buffer layer of thickness LSL between the contact lens and the electrode. Oxygen concentration decreases nonlinearly across the boundary layer from cw∞ in the bulk solution to cwL at the boundary-layer/lens interface due to both convection and diffusion. Local phase equilibrium at the boundary-layer/lens interface relates cwL to the volume average oxygen concentration in the lens at the anterior surface, , through the overall partition coefficient, K, defined as
) KcwL
(3)
K is also the ratio of oxygen solubility in the contact lens (k) to that in water (kw):
K)
k kw
(4)
Because the oxygen concentration in the lens is small, the steady-state oxygen concentration decreases linearly through the lens to reach 〈cSL〉 at the lens/stagnant-layer interface due to molecular diffusion; 〈cSL〉 is again related by phase equilibrium to the oxygen concentration in the stagnant layer at this interface, cwSL, through the partition coefficient, K. Likewise, because of molecular diffusion, the oxygen concentration profile decreases
(5)
where RBL is the mass-transfer boundary-layer resistance in the aqueous solution, L is the harmonic-mean thickness of that portion of the contact lens exposed to bulk solution, Dk is the oxygen permeability of the contact lens, and RSL is the stagnantlayer resistance due to the stagnant buffer layer between the posterior surface of the lens and the cathode.28 A fabric net, which presses against the anterior surface of the SCL, minimizes the water cushion between the lens posterior surface and the curved electrode to decrease the undesired stagnant-layer resistance.28 3.2. Unsteady State. Unsteady diffusion of oxygen through the lens is described by Fick’s second law in one dimension
3. Theory 3.1. Steady State. The polarographic technique measures the current due to the flux of oxygen to a working electrode (here, the cathode). Reduction of oxygen molecules reaching the electrode surface occurs according to32
L + RSL Dk
∂2 ∂ )D ∂t ∂z2
(6)
where is local-volume-average oxygen concentration in the contact lens, t is time, z is the direction normal to the surface of the lens, directed toward the bulk solution, and D is the lens oxygen diffusivity. The radial direction is not considered because the length scale in this direction is greater than one order of magnitude larger than that in the normal direction. Consequently, radial concentration gradients are insignificant. Oxygen is totally purged from the cell to set an initial zero oxygen concentration everywhere, as discussed in the Experimental Section
) 0
(7)
The boundary condition at the surface of the electrode is zero oxygen concentration set by the limiting-current condition in the cell
) 0
(8)
The boundary condition at the top surface of the lens is related to the oxygen flux through the boundary layer
D
∂ ) km(cw∞ - cwL) ∂z
(9)
where km is the mass-transfer coefficient characterizing oxygen transport through the mass-transfer boundary layer and Leq is an equivalent lens thickness. Because the thickness of the stagnant buffer layer, LSL, is very small (5-10 µm),28 and because there is no oxygen consumption in the lens or in the stagnant layer, the stagnant buffer layer and the contact lens are assumed to be a single layer of equivalent resistance thickness, Leq. This approximation greatly simplifies the analytical solution. Leq is determined by taking into account both lens and stagnant-layer resistances from eq 5
Leq L ) + RSL Dk Dk
(10)
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Figure 5. Nondimensional excess current versus nondimensional time for R ) 1 and R ) 10 to compare the two-dimensional (2D) numerical model (dashed lines) with the one-dimensional (1D) analytical model (solid lines).
The stagnant-layer resistance, RSL, measured by independent experiments, is 0.1 µm/barrer.28 The mass-transfer boundarylayer resistance in eq 5 is rigorously accounted for in the analysis by eq 9. Transient flux of oxygen at a specific time, J(t), is established from the measured current according to Faraday’s law
J(t) )
I(t) - IB nFA
(11)
where I(t) is the measured transient current, IB is the background current in the absence of oxygen, n is the number of electrons consumed in the reduction of 1 molecule of oxygen (n ) 4), F is Faraday’s constant, and A is the cathode area. Solution of eqs 6-11, presented in eqs B6 and B7, is expressed in nondimensional form as
I(t) - IB ) f(τ;R) I∞
(12)
where I∞ is the current in the absence of mass-transfer boundarylayer resistance, f is the function expressing the dependence of current on τ and R (see eqs B6 and B7), τ is nondimensional time, τ ) Dt/Leq2, and R is a nondimensional parameter defined by
R)
kmLeqkw Dk
(13)
Parameter R represents the ratio of the sum of diffusional resistance of the contact lens and the stagnant layer to that of the mass-transfer boundary layer. To validate the one-dimensional approach, a two-dimensional diffusion process was modeled numerically using COMSOL Multiphysics 3.3a software. Details are given in Appendix C. By considering two extremes of oxygen permeability, i.e., 10 and 200 barrer, R values at these extremes are about 10 and 1, respectively. As shown in Figure 5, the nondimensional current over background is calculated at the cathode from the 2D analysis (dashed lines) and compared with that obtained from the 1D analysis in Appendix B (solid lines) for these extremes. The maximum current for R ) 1, which corresponds to highDk lenses, is lower compared to that for R ) 10, indicating that the mass-transfer boundary-layer resistance is significant for hypertransmissible lenses. We conclude that the one-
Figure 6. Inverse excess current versus inverse two-thirds power of the stirring speed for two commercial soft contact lenses. From the slopes, the average mass- transfer coefficient is km ) (6.8 ( 0.8) × 10-3 cm/s. This average km value is used for all soft contact lenses.
dimensional model provides an excellent approximation for unsteady-state diffusion of oxygen through the lens. We prefer the one-dimensional model because an analytical solution provides helpful insight into the physical significance of the parameters. To determine both D and k, three unknown parameters must be obtained from the measured current: mass-transfer coefficient, oxygen permeability, and oxygen diffusivity. First, steady-state current, Imax, is related to two of the unknown parameters, km and Dk, as
1 1 1 + ) Imax - IB kmcw∞nFA I∞
(14)
where I∞ is defined by
I∞ )
nFADkcw∞ kwLeq
(15)
The mass-transfer coefficient is directly proportional to the twothirds power of the stirring speed;28 from eq 14, km is determined from the slope of a plot of the inverse steady-state excess current as a function of the inverse two-thirds power of the stirring speed. Figure 6 shows such plots for PureVision and Focus Night & Day SCLs. Choice of high-Dk lenses in quantifying km is important because the boundary-layer resistance is most significant for these lenses, yielding a more reliable masstransfer coefficient. In Figure 6, stirring speed varies from 400 to 1200 rpm in intervals of 200 rpm. For all stirring speeds, the air flow rate is 15 L/min, the same as that for all the transient experiments. Since the cell hydrodynamics is maintained identical for all lenses, the slopes in Figure 6 should be the same for both lenses; km ) (6.8 ( 0.8) × 10-3 cm/s is determined from the average of these two slopes. This km is then used for data analysis of all subsequent SCLs. Air flow rate, stirrer and sparger location, and stirring speed are maintained constant to establish identical hydrodynamic conditions for all lenses. Oxygen permeability is sensitive to the particular choice of km. Our chosen km, determined from Figure 6, yields oxygen permeabilities consistent with those previously reported, obtained by extrapolation to zero mass-transfer boundary-layer resistance.28 Since km is now known from independent steady-state experiments, the second unknown parameter, oxygen permeability (Dk), is obtained from eq 14 by using Imax from the transient experiments conducted at 1200 rpm (e.g., Figure 7).
Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3545
Figure 7. Fitting of transient experimental data at 35 °C (dots) for Biofinity lens to eqs B6 and B7 (solid line). Table 2. Physical Constants parameter
value
n F Aa cw∞b kwc a Rd Dow e
4 96 487 °C/mol 0.13 cm2 2.2 × 10-7 mol/cm3 3.3 × 10-5 2 mm 9 mm 2.9 × 10-5 cm2/s
a Curved surface area of the cathode with 4 mm cathode diameter and 8.6 mm radius of curvature calculated using a formula from Section 9.1 of ISO 9913-1.43 b On the basis of solubility of oxygen in pure water.44 c mL of O2 (STP)/(mL‚mmHg). d Curved lens radius based on typical 15 mm lens diameter and 8.6 mm base-curve radius. e On the basis of the correlation by Wilke and Chang.45-47
Table 3. Oxygen Diffusivity, D, and Solubility, k, for Nine Water-Saturated Commercial Soft Contact Lenses brand name
sat. H2O (wt %)
Da (10-7 cm2/s)
ka,b
Dka (barrerc)
Biomedics 38 Biomedics 55 Acuvue 2 Focus Night & Day O2 Optix PureVision Acuvue Oasys Acuvue Advance Biofinity
38 55 58 24 33 36 38 47 48
0.7 ( 0.1 2.5 ( 0.2 3.2 ( 0.2 8.4 ( 0.7 5.8 ( 0.5 6.0 ( 0.6 4.5 ( 0.6 4.2 ( 0.1 6.1 ( 1.1
1.7 ( 0.2 0.7 ( 0.1 0.8 ( 0.1 2.3 ( 0.2 1.9 ( 0.1 1.5 ( 0.2 2.3 ( 0.1 1.5 ( 0.1 2.0 ( 0.2
11 ( 1 16 ( 1 26 ( 3 195 ( 4 112 ( 9 90 ( 4 103 ( 7 64 ( 2 123 ( 8
a Each value is the average of three separate experiments. The ( values refer to precision, not accuracy. b 10-3 mL of O2 (STP)/(mL‚mmHg). c 1 barrer ) 10-11 (cm2/s)(mL of O2 (STP))/(mL‚mmHg).
Oxygen diffusivity (D) is the final unknown parameter; it is determined by fitting the measured transient current to eqs B6 and B7, as outlined in Appendix B. Oxygen solubility (k) is determined from the ratio of oxygen permeability (P ) Dk) to oxygen diffusivity. Figure 7 shows a fit of theory to data for a Biofinity lens using the physical constants in Table 2. Satisfactory fits to theory are found for all lenses studied. Some amount of fitting discrepancy can be attributed to a nonuniform current density over the cathode due to the proximity and concentric placement of the electrodes. 4. Results and Discussion Table 3 shows experimental results for oxygen diffusivity and solubility for nine commercial SCLs. Each value corresponds to the average of three separate transient experiments.
Figure 8. Two-state model for oxygen distribution in HEMA-based hydrogel contact lenses: (1) oxygen solubilized in water phase (empty circles) and (2) oxygen adsorbed on polymer chains (filled circles). Solid lines represent polymer chains, and the space between them represents water phase. Note: not to scale.
Some uncertainty in data can be attributed to lens off-centering during manual placement on the electrode or due to small thickness changes in stagnant aqueous layer from one experiment to another.28 We observe that oxygen solubility in all studied silicone-hydrogel lenses (bottom six rows of Table 3) is approximately (2.0 ( 0.5) × 10-3 mL of O2 (STP)/(mL‚ mmHg), ∼2 orders of magnitude higher than that in water (3.3 × 10-5 mL of O2 (STP)/(mL‚mmHg), Table 2). Since oxygen solubilities in silicone-hydrogel lenses do not vary significantly from each other, we conclude that oxygen diffusivity is mainly responsible for the observed range of oxygen permeabilities for silicone-hydrogel lenses (60-200 barrer). The silicone backbone in a silicone hydrogel is flexible and, thus, more randomly packed. Apparently, this increased backbone mobility results in high diffusivity of oxygen in silicone-hydrogel lenses;33 perhaps the degree of polymer-chain mobility determines the range of oxygen permeability. Our results indicate that there is no correlation of oxygen diffusivity and solubility with water content for the highly oxygen-permeable silicone-hydrogel lenses. Oxygen diffusivities for the HEMA-based lenses (Biomedics 38, Biomedics 55, and Acuvue 2) from Table 3 are lower than those of Compan˜ and co-workers,18,19 possibly because we take separate account of oxygen diffusion through each of the various boundary layers. Accordingly, our oxygen diffusivities are not weighted by the high oxygen diffusivities of the aqueous boundary layers. However, oxygen solubility for the HEMAbased lenses is unexpectedly high, almost approaching that of the silicone-hydrogel lenses for Biomedics 38 lens. To a first approximation, oxygen solubility in a homogeneous hydrogel should be related to water uptake by k ) φwkw, where φw is the volume fraction of water and kw is the atmospheric solubility of oxygen in water (3.25 × 10-5 mL of O2 (STP)/ (mL‚mmHg), Table 2). Since the density of HEMA polymer is 1.3 g/cm3,34 the lens water weight fraction may be replaced closely by the volume fraction. Hence, oxygen solubility in HEMA-based lenses should exhibit values near to 10-5 mL of O2 (STP)/(mL‚mmHg), and k should increase with increasing water content. However, the k values for HEMA-based lenses are 2 orders of magnitude larger and k decreases with water content (Table 3). A possible explanation for the observed high oxygen solubilities in the measured HEMA material is illustrated in Figure 8.
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We propose that oxygen solubilized in a HEMA-based material is in two states: (1) dissolved in the water phase and (2) associated with the cross-linked polymer matrix.35 Empty circles in Figure 8 illustrate oxygen in the water-filled volume, while filled circles represent oxygen adsorbed with the polymer component. Since it is unlikely that HEMA in the highly aqueous swollen gel is fibrillar, oxygen is most likely adsorbed along individual polymer strands. To estimate how much oxygen is present in each state, we write the overall solubility35-39
k ) φwkw + φpkp
(16)
where φp is the polymer volume fraction and kp is the oxygen solubility attributed to oxygen adsorbed onto the polymer phase per unit volume. Equation 16 can explain the large observed solubilities of HEMA-based lenses provided kp is large. Unfortunately, direct use of eq 16 to quantify kp is not possible because the underlying two-state model demands a change in the value of the diffusion coefficients reported in Column 3 of Table 3. In the dual-state picture of oxygen partitioning in a hydrogel, the total or volume-average concentration becomes 〈c〉 ) φwc + φpcp, where c is the concentration of oxygen per unit volume of water phase and cp is the concentration of oxygen per unit volume of polymer phase. Substitution of this expression into eq 6, along with the assumptions that oxygen in the water and oxygen in the polymer are in local equilibrium and that oxygen diffuses only through the water phase, leads to a modified Fick’s law37-39
(φw + φpKp)
∂c ∂2c ) φ wD w 2 ∂t ∂z
(17)
where Dw is the oxygen diffusion coefficient in the water phase of the hydrogel and Kp ) cp/c ) kp/kw is the partition coefficient for oxygen in the polymer portion of hydrogel. Similar analyses have been carried out to model sorption kinetics of gases in silicone rubber filled with adsorptive molecular sieves and in glassy polymers.38,39 The second term in the bracket (φpKp) corresponds to a capacitive slowing of oxygen diffusion through the hydrogel. Equation 17 is an important result because, during unsteady uptake into or depletion of oxygen from the hydrogel, diffusion is characterized by an effective diffusion coefficient37-39
D)
φ wD w
(18)
(φw + φpKp)
that is reduced relative to the steady-state diffusion coefficient, φwDw, by transient oxygen partitioning to the polymer phase as gauged by Kp. According to the two-state model, the experimental transient current rise, such as that in Figure 7, is quantified by the effective diffusion coefficient in eq 18 (i.e., eqs B6 and B7) as listed in column 3 of Table 3. Conversely, at steady state, the left side of eq 17 is zero; steady diffusion is characterized by the diffusion coefficient Dw. Since the diffusion path length is not the same as the distance scale in Fick’s law, it is customary to express Dw as Dow/τ2, where Dow is the molecular diffusion coefficient of oxygen in bulk water (2.9 × 10-5 cm2/s, Table 2) and τ is the tortuosity.37 Accordingly, the permeability of the hydrogel, which is determined at steady state, is
Dk ) φwDwk )
φwDow τ2
k
(19)
Table 4. Two-State Model Parameters for HEMA-Based Lensesa brand name
Kp
K
kpb
φwDw (10-6 cm2/s)
kb
τ
Biomedics 38 Biomedics 55 Acuvue 2
10.6 8.6 10.5
7 4.4 5
3.5 2.8 3.4
0.5 1.1 1.6
2.3 1.4 1.6
4.8 3.8 3.3
a Each value is the average of three separate experiments. b 10-4 mL of O2 (STP)/(mL‚mmHg).
Since Dk is measured from the steady current and D is measured during the transient current rise, eqs 18 and 19 permit calculation of Kp and Dw or, equivalently, calculation of Kp and τ. Table 4 shows the results of this calculation. Kp values in column 2 of Table 4 reveal that, in the twostate model, the polymer chains adsorb ∼10 times more oxygen than is dissolved in the aqueous phase of the HEMA-based lenses. The overall partition coefficient, K ) φw + φpKp, is ∼5 for all three HEMA hydrogels. In the two-state model, oxygen solubility, k, is now smaller by ∼1 order of magnitude compared to that from the homogeneous-gel model in Table 3. This difference is because the diffusion coefficient, φwDw, is much higher than that in Table 3, at ∼10-6 cm2/s, due to correction for the capacitance effect. Equivalently, tortuosity values are not unreasonable at ∼4.40 Further, oxygen diffusivities (φwDw) increase with rising water content due to a larger proportion of high-diffusivity water phase. However, oxygen solubility (k) decreases with water content due to smaller proportion of the low-solubility water phase (relative to the high-solubility polymer phase). Nevertheless, oxygen permeabilities for HEMAbased lenses from Table 3 increase with water content, as reported earlier.18,41,42 Thus, we conclude that the increase in oxygen diffusivities with water content overcompensates for the observed decrease in oxygen solubilities. The proposed dualstate picture of oxygen partitioning in HEMA-based lenses provides a reasonable explanation for the measured diffusivities and solubilities in Table 3, whereas a homogeneous distribution of oxygen solubilized in the lenses does not. A similar analysis of the D and k results for the silicone hydrogels is currently not possible because neither the volume fraction of silicone and HEMA-like phases in the hydrogels nor their spatial distribution is known. Nevertheless, we expect an analogous inhomogeneous distribution of oxygen in the silicone hydrogels of Table 3. We regard the D and k results for the silicone hydrogels in Table 3 as apparent because the capacitive effect has not been accounted for in silicone hydrogels. However, it is likely that the capacity effect for the siliconehydrogel materials is less important due to the high oxygen diffusivity in the silicone phase. 5. Conclusions Determination of oxygen diffusivities and solubilities in lens materials is useful for design of next-generation soft contact lenses. This work describes a polarographic apparatus to measure separately oxygen diffusivity, D, and solubility, k, in aqueoussaturated polymer films, with special attention to hypertransmissible soft contact lenses. From unsteady polarographic data, we determine D from the transient current rise, Dk from the steady-state current, and k from the ratio of oxygen permeability to oxygen diffusivity. Oxygen solubility in silicone-hydrogel lenses is ∼2 orders of magnitude higher compared to that in pure water. Oxygen diffusivity mainly controls the range of oxygen permeabilities for hypertransmissible soft contact lenses. Oxygen diffusivity and permeability increase, and oxygen solubility decreases, with water content in conventional HEMAbased soft contact lenses. For HEMA-based lenses, oxygen
Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3547
partitioning onto the cross-linked polymer chains is ∼10 times more than that in water. However, there is no obvious relation between oxygen diffusivity or solubility with water content for silicone-hydrogel soft contact lenses. Acknowledgment The authors are grateful to CooperVision Inc. for partial financial support; to Professors Elton J. Cairns and John S. Newman for helpful guidance concerning electrochemistry; and for assistance with the experiments to Wouter J. Appelmans, Gary R. Bishop, Patrick Chen, Wei T. B. Gan, Henry Lawrence, Joe C. Liang, Wei Lu, and Michael W. Tsiang. J.M.P. acknowledges support from the Basic Sciences Division of the U.S. Department of Energy. Appendix A Estimate of Purging Time. During cell assembly, some oxygen re-enters the contact lens. To obtain quantitative values for D and k, a uniform zero-oxygen-concentration initial state is required in the lens. Hence, nitrogen purging is needed. We desire an estimate of the necessary purging time. During purging, part of the lens top surface is exposed to the solution bubbled with nitrogen. However, the remaining surface of the contact lens is covered by the lens holder and is not exposed to the solution. Thus, any oxygen present in this wing region of the lens must diffuse toward the central region before it is purged. To quantify the time to purge all oxygen, we adopt the simplified model shown in Figure A1. Because there is no sink for oxygen on the face of the lens covered by the lens holder, we consider only radial diffusion toward the central region. Therefore, the governing nondimensional equation is
(
)
∂c*(τ,r*) ∂c*(τ,r*) 1 ∂ r* ) ∂τ r* ∂r* ∂r*
Figure A2. Nondimensional lateral diffusion current as a function of time for various equivalent lens thicknesses. A high-thickness lens requires more time to reach 100 h to expunge all oxygen present in the wings of the 300 µm thick lens. Without purging, we found experimentally that, for the same lens material, oxygen diffusivity increased with increasing lens thickness. However, when sufficient nitrogen purging was performed, the lens-thickness effect disappeared. Thus, it is critical to purge essentially all oxygen from the lens to obtain reliable oxygen diffusivity and solubility, especially for highthickness and high-permeable lenses. As shown in Figure A2, it takes ∼2 h to reach 1% of the steady current, I∞, for a thin lens (∼100 µm). Because the commercial lenses used in this study are e100 µm in thickness, the cell is bubbled with N2 for 2 h before initiating the step change from nitrogen to air. Appendix B One-Dimensional Model. Equations 3, 4, and 6-9 are solved using separation of variables to obtain the transient oxygen profile in the central region of lens as ∞
) βz +
2
()
(B1)
eq
where is the volume-average oxygen concentration, β is defined by
β)
kmKcw∞ DK + kmLeq
and bi are defined by
bi ) -
(
(B2)
)
2(R + 1)ALeq sin(λi) λi λi - sin(λi) cos(λi)
(B3)
Here, λi are the nondimensional eigenvalues of the eigenequation
tan(λi) ) -
λi R
(B4)
and the nondimensional parameter, R, is defined in eq 13. The magnitude of transient oxygen flux, J, at the surface of the cathode is determined from the oxygen-concentration profile (eq B5):
|
∂ J)D ∂z z)0
(B5)
Oxygen flux is directly related to the measured transient current by Faraday’s law to obtain the general solution:
I(t) - IB
)
( )[ R
1 - 2(R + 1)
[
(
∞
(-1)j-1 eR(aτ+j) erfc R xτ + ∑ j)1
j
)]
2xτ
(B7)
To fit the current data for oxygen diffusivity, we use a golden section least-square fitting (fminbnd function, MATLAB 6.0.0.88 Release 12) using the first term (j ) 1) of the short-time solution eq B7 until I(t) ) 0.2(Imax - IB). Thereafter, the first three terms (i ) 1-3) of the general solution, eq B6, are employed until steady-state current is reached. Increasing the number of terms in the short-time and general solutions does not result in any significant changes in data fitting. The transition factor of 0.2 is determined from short-time and general solutions. For each R, there is an overlap region of these solutions. On the basis of the common overlap region for extremes, R ) 1 and 10, an optimized transition factor of 0.2 from the short-time solution to the general solution is obtained. The function f(τ;R) in eq 12 represents the combined form of eqs B6 and B7 with the transition factor of 0.2. This function is plotted as the one-dimensional analytical model in Figure 5. Appendix C Axisymmetric Numerical Model. To determine whether the one-dimensional description in eqs B6 and B7 is quantitative, a two-dimensional cylindrical lens diffusion process was modeled using COMSOL Multiphysics 3.3a. Curvature of the lens is neglected; the contact lens is approximated as a flat disk. For simplicity, the thickness of the stagnant layer is incorporated into the equivalent lens thickness, Leq, as defined in eq 10. This approximation should not result in loss of generality. The governing equation in nondimensional form is
(
)
1 ∂ ∂c* ∂2c* ∂c* ) r* + ∂τ r* ∂r* ∂r* ∂z*2
(
)
(C1)
where z* ) z/Leq. The initial condition is
c*(0,z*,r*) ) 0
(C2)
The boundary condition at the posterior lens surface adjacent to the cathode surface due to the limiting-current condition is
c*(τ,0,r* e a/Leq) ) 0
(C3)
Because oxygen is not consumed by the anode, the boundary condition on the posterior lens surface surrounding the cathode is
∂c*(τ,0,r* > a/Leq) )0 ∂z*
(C4)
Because of mass-transfer resistance, the boundary condition at the anterior lens surface exposed to the bulk solution is
I∞
R+1
I∞
) 2R
λi z
bi e-λ τ sin ∑ L i)1 i
I(t) - IB
∞
∑ i)1
(
sin(λi)
λi - sin(λi) cos(λi)
) ]
e-λi τ (B6) 2
Unfortunately, for short times, eq B6 requires an inordinate number of terms to fit the measured current. Hence, a shorttime solution is obtained from a Laplace transformation of the original differential equation and boundary conditions. The system is then solved in the transformed space (s being the Laplace variable), expanded asymptotically for large s (or t close to zero), and finally inverted to give
∂c*(τ,1,r* e a/Leq) ) R[1 - c*(τ,1,r* e a/Leq)] (C5) ∂z* Oxygen transport through the lens holder is assumed to be insignificant; thus, the boundary condition at the anterior lens surface covered by the lens holder is
∂c*(τ,1,r* > a/Leq) )0 ∂z*
(C6)
Because of symmetry, the boundary condition at the center of the lens is
Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3549
∂c*(τ,z*,0) )0 ∂r*
(C7)
Since the outer edge of the lens is not a source/sink of oxygen, oxygen flux at this boundary is assumed to be zero
∂c*(τ,z*,r* ) R/Leq) )0 ∂r*
(C8)
The magnitude of average nondimensional oxygen flux on the surface of the cathode, J* ) J/J∞ where J∞ is the oxygen flux in the absence of mass-transfer boundary-layer resistance, is determined by integrating the local oxygen flux as
J* )
2Leq2 a
2
∫0a/L
eq
r*
∂c* dr* ∂z*
(C9)
The average oxygen flux is directly related to the transient current by Faraday’s law as
I - IB ) J* I∞
(C10)
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ReceiVed for reView August 6, 2007 ReVised manuscript receiVed December 3, 2007 Accepted December 4, 2007 IE071071A
