Supplemental Information

Predictive simulation of non-steady-state transport of gases through

a polymer membrane

Marielle Soniat,a,b Meron Tesfayec,d, Daniel Brooks,e Boris Merinov,e William A. Goddard, III,e

Adam Z. Weber a,c and Frances A. Houle*a,b

a Joint Center for Artificial Photosynthesis, Lawrence Berkeley National Laboratory, Berkeley,

CA 94720

b Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720

c Energy Storage and Distributed Resources Division, Lawrence Berkeley National Laboratory,

Berkeley, CA 94720

d Department of Chemical and Biomolecular Engineering, University of California, Berkeley, CA

94720

e Materials and Process Simulation Center (MSC), Beckman Institute, California Institute of

Technology, Pasadena, CA, 91125

* Author to whom correspondence should be addressed. fahoule@lbl.gov, (510) 495-8135.

Contents1. Molecular dynamics simulations methods........................................................................3

2. PDMS experimental sample preparation..........................................................................4

3. Experimental permeation measurement............................................................................4

4. Sensitivity of lifetime to initial excited state dye concentration.......................................5

5. Sensitivity of lifetime to quenching rate constant.............................................................6

6. Determination of the excitation rate constant...................................................................7

7. Sensitivities of the sorption and desorption simulations...................................................8

8. Additional experimental data..........................................................................................10

9. Method for simulating the experimental gas leak...........................................................12

10. Method for simulating the absorption site concentration increase within the polymer during a pressure or concentration rise...................................................................................14

11. Additional simulations of time-dependent permeation...............................................15

12. Predictive Simulations of Mixtures of Gases..............................................................17

13. Predictive Simulation of diurnal cycles.......................................................................19

14. References...................................................................................................................21

Nomenclature for symbols used only within the SI. Other symbols are defined in the main text.

Symbol Meaning Unit

d Gaussian shell width Å

f Gaussian density Å-3

r⃗i position Å

rpup ratio of upstream pressure to its final value --

α fraction of successful encounters --

1. Molecular dynamics simulations methods

The surface is described using the method of Willard and Chandler.1 Each polymer atom, at

location r⃗i, is assigned a Gaussian shell with width of d = 3.0 Å. This creates a “coarse-grained

density,” f(r⃗ ), with units of Å-3,

f (r⃗ )=∑⃗ri

1d3 ¿¿

¿ (S1)

Where this coarse-grained density reaches half its bulk value, f = 0.035, a surface is constructed

on a 1-Å grid. The parameters for Gaussian width and grid fineness control the smoothness of

the surface. Several combinations are tested to determine the sensitivity of sticking coefficient

results to these parameters. The Willard and Chandler surface definition is selected over other

commonly used methods, such as the “10-90” definition or the Gibbs dividing surface, because it

provides information about the instantaneous, local interface.

A change of ±0.01 in the f at which the surface is constructed results in a ≈1.2 Å shift in the

surface. This magnitude of shift in the surface location has a minimal effect on the sticking

coefficient. Since most desorbed molecules are far from the surface, the choice of surface region

primarily affects the distinction between absorbed and adsorbed molecules. Thus, upper bound

for the sticking coefficient, the fraction of absorbed and adsorbed molecules, is insensitive to the

choice of surface.

2. PDMS experimental sample preparation

Poly(dimethyl siloxane) (PDMS) base and a proprietary crosslinker (Sylgard 184, Dow

Corning Corp., Auburn, MI) are mixed in a 10:1 ratio by weight. The solution is stirred with

wooden spatula for 2 minutes. The solution is degassed under vacuum until there are neither

visible bubbles nor gas pockets (> 1 hr). The manufacturer-provided viscosity for PDMS

prepared in this way is 3.5 Pa∙s.2 The structure of PDMS is shown in Figure 1 of the main body.

Silicon wafers (6-inch, Silicon 100, p-type, Pure Wafer, San Jose, CA) are prepared by

rinsing with deionized water, followed by blow drying with N2, two rinses with isopropyl

alcohol, and finally drying with N2. Each wafer is placed on a hot plate at 353 K for a few

minutes to remove any residual alcohol. The wafer is then placed in a vacuum desiccator and

exposed to a vapor of trichloro(1,1,2,2-perfluoocytl)silane (Sigma Aldrich, Saint Louis, MO).

This is done to facilitate later removal of PDMS film from the wafer.3

The PDMS solution is then spun cast onto the silane-treated wafer. Samples are spun for

20 sec at 200 revolutions per minute (rpm) followed by 90 sec at 300 rpm using a Laurell spin

coater (Laurell Technologies Corp., North Wales, PA). The polymer is then dried in a vacuum

oven for 19 hours at 323 K. All samples are cut into small squares and slowly peeled off the

silicon wafer under water. Samples are stored in deionized water until placed in the testing

chamber.

3. Experimental permeation measurement

Prior to the permeation experiment, the PDMS film samples are dried on a flat vacuum plate

at ambient temperature for at least an hour. Samples are then placed in the permeation assembly,

backed by a filter paper and sandwiched between two flat aluminum supports. The metal support

allows for transport through a defined active area but does not alter the measured permeability.

The active area ranges from 0.186 to 0.203 cm2 for this study. The sample assembly is then

placed in the permeation cell for measurement. The sample exposed to vacuum of 3 kPa or less

for at least 10 hours to remove any residual water or gas pockets. Initially, the downstream valve

connecting the permeation cell to the vacuum pump is closed, and slow pressure rise in the

downstream volume, (dpds/dt)leak, is monitored to test for any defects in the experimental

apparatus. The sample is then exposed to dry gas at the pressure of interest on the upstream side.

The upstream pressure is recorded so that it can be included in the simulation. As gas permeates

through the membrane, the pressure rise in the closed downstream volume, (dpds/dt)SS, is

monitored. Once steady state, signaled by a linear rise in downstream pressure over time, is

reached, the permeability is calculated. The thicknesses used in this study are outside of the

range in which PDMS displays thickness-dependent permeability.4

As a test for swelling, the thicknesses of a PDMS film before and after sorption of CO2 under

external pressure of 820 kPa were measured using ellipsometry. No change in thickness was

detected. Other studies indicate that swelling of PDMS under these conditions should be 1% or

less.5 Among the gases in this study, CO2 is the most likely to cause swelling, indicating that for

all the conditions studied herein, swelling is negligible.

4. Sensitivity of lifetime to initial excited state dye concentration

The lifetime of PtOEP, i.e. the time for intensity to decay to 1/e of its initial value, in PS

depends upon the initial concentration of excited state dye. The initial excited state dye

concentration calculated from Equation 5 in the main text with the reported experimental

variables is 9.96 × 10-3 M. However, simulation shows that this value is too large, and that

[3PtOEP]init = 5.85 × 10-4 M gives a better fit to the experimental phosphorescence decay data.

The simulation is sensitive to the value of [3PtOEP]init, since agreement with experiment

deteriorates if this value changes by more than -10% or +20%.

Figure S1. Simulated and experimental phosphorescence decay in air. The experimental data points from Ref. 6 are marked with black x’s, and the gray region indicates the local standard deviation (over 11 data points). The dashed line indicates the lower detection limit of the instrument. The blue markers are for the simulation data using the reported value of [ 3PtOEP]init

= 9.96 × 10-3 M. The red markers are for a simulation using a reduced value of [3PtOEP]init = 5.85 × 10-4 M. Variation greater than +20% (cyan) or -10% (magenta) from the best fit value of [3PtOEP]init no longer lies within experimental error bars.

5. Sensitivity of lifetime to quenching rate constant

In the presence of O2, the lifetime of PtOEP is reduced due to an additional pathway for

decay from the excited state. The sensitivity of lifetime, τ, to the quenching rate constant, kq, is

shown in Figure S2. In addition to the scenarios shown in the figure, changes in the quantum

yield do not change the lifetime, nor do changes of ±50% in the TTA rate constant, probably

because the concentration of excited-state dye is so low.

Oftentimes, reactions in condensed phase are thought to be diffusion-limited. The

Smoluchowski equation is then used to calculate a rate constant from the diffusion coefficient in

the following manner,

k q=α kd=α ( 4 π N Av DAB RAB ) (S2)

where kd is the diffusion-limited rate constant, NAv is Avogadro’s number, and RAB is the

encounter distance. RAB is assumed throughout this work and in References 7 and 8 to be 10 Å for

O2 with PtOEP. DAB is the mutual diffusion coefficient, equal to the sum of self-diffusion

coefficients of species A and B, and α is the fraction of encounters that result in successful

quenching reactions, assumed to be 1. It should be noted that for triplet-triplet quenching, α = 1/9

is also reasonable, based on quantum considerations.9-10 Here, A and B are the oxygen and dye

species, and the dye diffusivity is assumed to be negligible compared to that of oxygen. The

value of kq = 7.57 × 107 M-1 s-1 calculated from the Smoluchowski equation is ≈4 times higher

than the experimental value. Simulation using the Smoluchowski value results in

phosphorescence decay clearly outside the experimental error bound, as shown in Figure S2.

Such a large discrepancy may indicate that few of the encounters results in successful reactions

or that the reaction also plays an important role in limiting the overall quenching rate.

Figure S2. Sensitivity of lifetime in air to kq for PtOEP in PS. Simulation of lifetime using the kq

from experiment in Reference 6 (black) agrees well with experiment. Changes in kq of ±50% (cyan, magenta) start to deviate from the experiment. The kq calculated from the Smoluchowski equation (red) is clearly outside of experimental error bounds.

6. Determination of the excitation rate constant

In oxygen sorption and desorption experiments, the dye in the polymer is continuously

excited by incoming light. The initial concentration of excited-state PtOEP is not reported and is

different from that in the pulsed laser experiment, due to the difference in energy densities of the

laser beams. However, starting from any concentration of excited-state PtOEP, the system

rapidly equilibrates (within 0.001 seconds) to a constant concentration of both ground-state and

excited-state PtOEP under conditions of continuous excitation. The intensity ratio B = I0/Ieq -1 is

the same whether a single well-mixed compartment or a multi-compartment model, with each

compartment well-mixed, is used. The dependence of B on the excitation rate constant is shown

in Figure S3. The maximum value of B from simulation is lower than the experimentally

reported value of 74 ± 6. The difference may be due to variability in experimental parameters

leading to a slightly higher concentration of O2 being present within the polymer than expected,

or a lower photon detection efficiency.

Figure S3. The ratio of intensities of the phosphorescent emission without oxygen to with oxygen, B + 1 = I0/Ieq, calculated from kex (see main text).

It is observed that the emitted intensity increases when the dye has increased quantum yield,

increased kex, increased total dye concentration, or decreased (effective) quenching rate constant.

However, these effects are operant both with and without oxygen present, and the ratio of those

two intensities is insensitive to them. Increasing the oxygen concentration increases the ratio

without substantially changing the I(t) curve (see Figure S4 below).

7. Sensitivities of the sorption and desorption simulations

Sensitivities of simulation results to properties of the polymer are investigated thoroughly.

Figure S4 shows that changes in D and S alter the oxygen sorption profiles but that many

different O2 profiles give similar I(t) profiles. Figures S5 and S6 shows the sensitivity to the

description of absorption and desorption, respectively, varying each property independently.

Properties of the interfacial region can be varied over orders of magnitude without affecting the

agreement between simulated and experimental I(t). Though not shown here, a variety of other

poly(n-alkyl(amino) thionylphosphazenes) were studied in Ref. 7 and simulated by us. We see

similar trends in terms of insensitivity of I(t) to surface properties, slight sensitivity to bulk

properties, and disagreement between calculated and simulated amounts of oxygen within the

membrane.

(a) (b)

(c) (d)

Figure S4. Sensitivity of I(t) (calculated from [O2(p)]) to diffusivity, solubility, and permeability: (a) I(t) during sorption, (b) amount of O2 in the polymer during sorption, (c) I(t) during desorption, and (d) amount of O2 in the polymer during desorption. The orange region shows the range of I(t) and amount of O2 simulated using ±10% changes in the experimental permeability Pm. The cyan region shows the variation if D or S is varied by 50% but adjusting the other value to maintain the experimental Pm.

(a) (b) (c)

Figure S5. Sensitivity of I(t) (calculated from [O2(p)]) to components of the surface adsorption process: (a) the orange region shows I(t) for a sticking coefficient from 10-16 to 1.0 but is such a

narrow distribution that it is hidden behind the black line; the black line is for a value of 0.1; and the cyan curve is for 10-17; (b) the orange region shows I(t) for a concentration of surface sites from 0.05 M to 1.66 M; the black line is for a surface site concentration of 1.66 M; the cyan line is for a surface site concentration of 0.017 M; (c) the orange region shows I(t) for diffusivity in the surface region from 1 × 10-15 to 4 × 10-10 m2/s; the black line is for a diffusion coefficient of 4 × 10-10 m2/s; the cyan line is for a diffusion coefficient of 4 × 10-16 m2/s.

(a) (b) (c)

Figure S6. Sensitivity of I(t) (calculated from [O2(p)]) to surface components of desorption: (a) the orange region, hidden behind the black curve, shows I(t) for desorption rate constants from 1.4 × 104 to 1.4 × 1020 s-1; the black line is for a desorption rate constant of 1.4 x 1011 s-1; (b) the orange region, again mostly hidden behind the black curve, shows I(t) for variation in surface site concentration from 1.66 × 10-4 to 1.66 M; the black line is for a surface site concentration of 1.66 M; (c) the orange region shows I(t) for diffusivity in the surface region from 4 × 10 -11 to 4 × 10-8

m2/s; the black line if for a diffusion coefficient of 4 × 10 -10 m2/s; the cyan curve shows I(t) for D = 1 × 10-11 m2/s.

8. Additional experimental data

A selection of additional experimental permeation curves is shown in Figure S7. Data for

all additional permeation curves are available in a separate SI spreadsheet. The entire set of

experimental data is summarized in Table S1. For all experiments, the downstream collection

volume is Vds = 41.30 ± 0.07 cm3, and the temperature is 308 K.

(a) (b)

(c) (d)

Figure S7. Experimental permeation curves for (a) CH4 through L5S3, (b) CO2 through L5S2, (c) N2 through L8S3, and (d) O2 through L8S2. Each curves is labeled with the upstream pressure in kPa. The shaded regions represent an experimental sample-to-sample standard deviation of 10%.

gas sample

ID

l * A error

in A

pup

μm cm2 cm2 kPa

CH4 L4S3 163 0.190 0.001 127, 210, 313, 414

L5S3 165 0.193 0.001 128, 214, 311, 416, 620

L8S3 163 0.203 0.004 130, 126, 214, 212, 313, 313, 418, 419, 621

CO2 L4S2 163 0.203 0.004 123, 126, 212, 211, 312, 321, 417

L5S2 165 0.200 0.002 123, 123, 207, 209, 311, 312, 416, 414

L8S2 163 0.186 0.002 124, 126, 213, 209, 312, 312, 417

N2 L4S2 163 0.203 0.004 125, 123, 208, 209, 312, 421, 416

L4S3 163 0.190 0.001 127, 214, 313, 413

L5S1 165 0.187 0.003 141, 122, 119, 139, 138, 138, 203, 205, 210, 204,

204, 208, 204, 218, 204, 309, 309, 317, 309, 309,

308, 414, 417, 414, 617, 618, 620, 827, 828

L5S2 165 0.200 0.002 125, 210, 311, 415

L5S3 165 0.193 0.001 128, 207, 312, 420

L8S2 163 0.186 0.002 124, 209, 211, 312, 312, 415, 418

L8S3 163 0.203 0.004 130, 207, 311, 417

O2 L4S2 163 0.203 0.004 126, 123, 207, 312, 312, 416, 416

L5S2 165 0.200 0.002 124, 128, 208, 213, 312, 324, 418, 418

L8S2 163 0.186 0.002 123, 126, 212, 207, 312, 312, 417, 417

* The error in the thickness measurements is ±1 μm for all the samples.

Table S1. Summary of all experiments.

9. Method for simulating the experimental gas leak

In the experiments, it is possible for gas to leak into the apparatus from outside because the

downstream volume is at reduced pressure. To properly account for this, we include the

following zeroth order reaction in the collection volume compartment only:

leak k leak→

gas+leak(S4)

where kleak is determined from the experimental leak rate (dp1/dt)leak. Because the experimental

leak rate is small in all cases, it could be neglected in the simulation without introducing

significant error. We include it in all simulations for the sake of completeness. The values of kleak

used in the simulations are reported in Table S2.

For each simulation, the initial residual gas concentration in the downstream receiver

volume is set equal to the value measured in the experiment. Neglect of this quantity results in

the simulated permeation curve being offset (too low) compared to experiment at the start time

but does not change the shape of the curve.

Gas Figure Sample ID pup kact kleak initial [gas]ds

kPa M-1s-1 M-1s-1 M

N2 9 L4S2 122 0.0015 1 × 10-12 2.08 × 10-7

206 0.0016 1 × 10-12 1.56 × 10-7

416 0.0024 1 × 10-12 1.56 × 10-7

CO2 10 L4S2 122 0.02 5.44 × 10-11 1.04 × 10-7

206 0.017 5.17 × 10-11 1.04 × 10-7

311 0.015 4.72 × 10-11 1.04 × 10-7

CH4 S9a L8S3 122 0.011 1.14 × 10-11 5.20 × 10-8

122 0.011 5.30 × 10-12 0

206 0.011 7.39 × 10-12 5.20 × 10-8

311 0.011 5.21 × 10-12 0

311 0.012 4.82 × 10-12 0

416 0.01 6.49 × 10-12 5.20 × 10-8

620 0.013 4.13 × 10-12 0

O2 S9b L4S2 122 0.003 6.01 × 10-11 1.04 × 10-7

311 0.004 6.23 × 10-11 1.04 × 10-7

416 0.004 5.73 × 10-11 5.20 × 10-8

N2 S9c L8S3 122 0.002 1.36 × 10-11 0

206 0.0015 3.54 × 10-12 0

311 0.002 7.33 × 10-12 0

416 0.0015 6.70 × 10-12 5.20 × 10-8

L5S2 416 0.0005 1 × 10-12 1.04 × 10-7

Table S2. Values for kact and kleak used in simulations.

10. Method for simulating the absorption site concentration increase within the polymer during a pressure or concentration rise

In the experiment, the upstream pressure does not instantly jump from zero to the desired

pressure, but increases over a finite period of time, typically 5 to 20 seconds. As the external

pressure increases, the maximum concentration of gas within the polymer will also increase, in

accordance with Henry’s law. To capture this effect in a kinetic scheme, we implement the

reaction

latent k act'

❑→

site (S3)

in each compartment, where k’act = kact × rpup where kact is the 0th-order rate constant for activation

of a latent internal absorption site and rpup is ratio of the current upstream pressure to its final

value. Such a scheme assumes that equilibration between the external pressure and the internal

available sites is instantaneous over the time-scales of interest. The rate constant is

phenomenological, with a best value that leads to complete conversion of all “latent” species to

“site” species and takes the entire time for the experimental pressure rise to do so. The values of

kact used in all the simulations are listed in Table S2.

11. Additional simulations of time-dependent permeation

The simulations of CH4 and O2 permeation are shown in Figure S8a and S8b,

respectively, in order to show the general applicability of the multi-scale simulations to a variety

of gas types. Also shown in Figure S8c is N2 permeation of a different PDMS sample from that

shown in the main text, in order to show agreement for a different membrane thickness and area.

The literature value of solubility, the upstream pressure, and the diffusivity, calculated in the

manner described in the main text, used in the multi-scale simulations are listed in Table S3.

(a) (b)

(c)

Figure S8. Comparison of simulation and experimental permeation curves. The simulation results are shown in red. The shaded regions represent an experimental sample-to-sample standard deviation of 10%. The curves are labeled with the upstream pressure in kPa. (a) CH4 for experimental sample L8S3, (b) O2 for experimental sample L4S2, (c) N2 for experimental sample L8S3.

Gas S (a) Sample ID pup D

M/Pa kPa m2/s

CH4 1.85 × 10-7 L8S3 122 1.86 × 10-9

122 1.86 × 10-9

206 1.86 × 10-9

311 1.83 × 10-9

311 1.84 × 10-9

416 1.88 × 10-9

620 1.84 × 10-9

N2 3.96 × 10-8 L8S3 122 2.84 × 10-9

206 2.83 × 10-9

311 2.80 × 10-9

416 2.84 × 10-9

L5S2 416 3.66 × 10-9

O2 7.93 × 10-8 L4S2 122 3.57 × 10-9

311 3.59 × 10-9

416 3.54 × 10-9

(a) Reference 11

Table S3. Input values for additional multi-scale simulations. The thickness and area of each sample are listed in Table S1.

12. Predictive Simulations of Mixtures of Gases

The surface description is modified slightly for the description of mixed gas absorption. In

single gas permeation, no desorption from the upstream interface is allowed. This assumption

introduces minimal error; e.g., a test with CO2 shows that 1.64 M of the adsorption sites will be

occupied of the 1.66 M available for a single gas. However, without desorption at the upstream

interface for mixtures of gases, the gas with the greatest collision frequency will occupy all of

the adsorption sites immediately, and no other gas will have a chance to adsorb. Allowing

reversible adsorption at the upstream interface slows the simulation down considerably because

it is a rapidly established equilibrium that is expensive to model. To overcome this challenge, a

short simulation (<1 sec. of simulation time) can be run with reversible absorption to determine

the equilibrium concentration of adsorption sites occupied by each gas. The equilibrium

concentration of surface sites occupied by each gas is then used in the full simulation with

adsorption only. The concentration of surface sites for each gas in each simulation is listed in

Table S4. As long as the concentration of surface sites is higher than the concentration of binding

sites within the polymer, this method works.

A test with a N2-O2 mixture at equal upstream partial pressures agrees with the experimental

selectivity11 of 2 when this method is used. The result is shown in Figure S9a. In Figure S9b the

downstream gas mixture is shown for a mixture that is 80 kPa N2 and 21 kPa O2 upstream,

representative of the composition of air. It can be seen that, though the permeability of O2 is

greater, the higher upstream concentration of N2 allows it to dominate the downstream gas

mixture. Table S4 presents the input values for simulations of mixtures of gases and also for the

simulations in the main text that include Ar and CO2.

(a) (b)

Figure S9. Permeation of Mixtures of N2 and O2. (a) The upstream pressure of both gases is 101 kPa. (b) The upstream pressure is consistent with the composition of air (80 kPa N2, 21 kPa O2).

Figure 11a 11b 11c S10a S10bN2 (a) pup kPa 101 80 80 101 80

[asite] M 0.379 1.30 1.30 0.858 1.32kads s-1 1.46 × 1022 1.14 × 1022 1.14 × 1022 1.46 × 1022 1.14 × 1022

S M/Pa 3.96 × 10-8

D m2/s 3.40 × 10-9

O2 (a) pup kPa 101 21 21 101 21[asite] M 0.368 0.326 0.326 0.802 0.327kads s-1 1.36 × 1022 2.86 × 1021 2.86 × 1021 1.36 × 1022 2.86 × 1021

S M/Pa 7.93 × 10-8

D m2/s 3.40 × 10-9

Ar (b) pup kPa 101 0.95 0.95[asite] M 0.823 0.0340 0.0340kads s-1 3.06 × 1022 2.86 × 1020 2.86 × 1020

S M/Pa 1.41 × 10-7

D m2/s 1.70 × 10-9

CO2 pup kPa 101 0.041 0.041

[asite] M 0.0769 1.2 × 10-4 1.2 × 10-4

kads s-1 2.97 × 1021 8.90 × 1017 8.90 × 1017

S M/Pa 5.68 × 10-7

(a)5.68 × 10-7 5.68 × 10-6

D m2/s 2.20 × 10-9

(a)2.20 × 10-9 2.20 × 10-8

(a) S and D from Ref. 11

(b) S and D from Ref. 12

Table S4. Input values for multi-scale simulations of mixtures of gases, the results of which are shown in Figure 11 of the main text and Figure S10.

13. Predictive Simulation of diurnal cycles

The distribution of CH4 within various polymer membranes during a diurnal cycle is shown

in Figure S10. The increase from 0 to the steady-state concentration is most clearly visible for

Hyp1 (Fig. S10b). The linear drop in concentration from the upstream to the downstream side,

which is typical of steady state, can be seen for most of the operating time for all polymers. The

two hypothetical polymers with reduced permeability also maintain this steady-state

concentration profile after CH4 production has ceased, due to accumulation of CH4 in the

upstream catholyte compartment. In contrast, the high permeability of PDMS allows all

accumulated CH4 to permeate and returns to zero concentration within an hour of the cessation

CH4 production.

(a) (b)

(c)

Figure S10. Concentration profile of CH4 during a diurnal cycle through polymer membranes (a) PDMS, (b) Hyp1, and (c) Hyp2. In all the contour plots, darker colors represent lower concentration, and lighter colors represent higher concentration. Position 0 is defined as the catholyte-polymer interface. The concentration is highest at the catholyte-polymer interface and falls linearly through the membrane once steady state is reached.

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