# Membrane heterogeneity alters our interpretation of effective energy
barriers to transport

Nathanael S. Schwindt,1 Mor Avidar,2 Razi Epsztein,2 Anthony P. Straub,3
Michael R. Shirts1∗

1Department of Chemical & Biological Engineering, University of Colorado
Boulder,
Boulder, CO, USA 80309
2Department of Civil and Environmental Engineering, Technion - Israel
Institute of Technology,
Haifa, Israel
3Department of Civil, Environmental, and Architectural Engineering, University
of Colorado Boulder,
Boulder, CO, USA 80309

∗To whom correspondence should be addressed; E-mail:
<EMAIL_ADDRESS>
> Major efforts in recent years have been directed towards understanding
> molecular transport in polymeric membranes. Transition-state theory is an
> increasingly common approach to explore mechanisms of transmembrane
> permeation with molecular details, but most applications treat all free
> energy barriers to transport as equal. This assumption neglects the inherent
> structural and chemical heterogeneity in polymeric membranes. In this work,
> we expand the transition-state theory framework to include distributions of
> membrane free energy barriers. We show that the highest free energy barriers
> along the most permeable paths, rather than typical paths, provide the
> largest contributions to the experimentally-observed effective free energy
> barrier. We show that even moderate, random heterogeneity in molecular
> barriers will significantly impact how we interpret the mechanisms of
> transport through membranes. Naive interpretations of experimentally
> measured barriers can lead to incorrect assumptions about the underlying
> mechanisms governing transport and miss the mechanisms most relevant to the
> overall permeability.

## Introduction

Understanding the molecular-level mechanisms that govern transport and
selectivity in salt-rejecting membranes, such as those used in nanofiltration
(NF) and reverse osmosis (RO), are imperative for the development of next-
generation desalination technologies (?, ?, ?). Numerous models have been
proposed over the years to explain the observed transport and selectivity
trends in these membranes. However, these models are still insufficient to
provide the full picture of the interactions and phenomena that occur under
the extreme confinement imposed by the nanometer and sub-nanometer membrane
pores (?, ?, ?, ?). Developing new theoretical frameworks and approaches is
therefore crucial to improve our understanding of molecular transport in
polymeric membranes and design future membranes that can address specific
requirements (?, ?, ?).

Since the early stages of NF and RO model development, the solution-diffusion
model has been the most widely used description for transport through
polymeric membranes (?). In the solution-diffusion model, solutes and solvent
must dissolve into the membrane and diffuse through it along a concentration
gradient (?). Recent work has shown that the solution-diffusion model is
insufficient (and in some cases inappropriate) for explaining the molecular
mechanisms governing RO and NF membrane transport (?, ?, ?, ?). As a result, a
number of studies of RO and NF membranes have examined the utility of
measuring energy barriers to membrane permeability based on either the
Arrhenius framework (?, ?, ?, ?, ?) or the similar but more rigorous
transition-state theory framework (?, ?, ?, ?, ?, ?) in order to elucidate
details of the molecular mechanisms of transport.

At the simplest level, the Arrhenius activation energy model can be used to
understand the energetics of molecular barriers. The Arrhenius equation
relates the rate constant $k$ to a pre-exponential factor $A$ and the
reaction’s activation energy $E_{a}$, as shown in Eq. 1 where $R$ and $T$ are
the gas constant and temperature, respectively.

$\displaystyle k$ $\displaystyle=$ $\displaystyle
A\exp{\left(\frac{-E_{a}}{RT}\right)}$ (1)

Based on this framework, the activation parameters (i.e., the energy barrier
and the pre-exponential factor) are often measured since they can be directly
extracted from the slope and intercept of the linearized Arrhenius equation.
By assuming that membrane permeability is a direct function of some molecular-
level energy barrier to transport, the Arrhenius equation can be used to
estimate these energy barriers. Permeability ($P$), instead of $k$, is treated
as an Arrhenius rate in order to relate it to the activation parameters:

$\ln P=\ln A-\frac{E_{a}}{RT}$ (2)

These temperature dependent activation parameters could in theory
differentiate between mechanisms that are indistinguishable with the solution-
diffusion parameters because they are expected to correspond to molecular-
level phenomena, such as molecular rearrangement or ion dehydration (?).

A more thermodynamically rigorous model was proposed by Zwolinski, Eyring and
Reese, who described membrane transport using transition-state theory in 1949
(?), directly connecting permeability to enthalpic and entropic barriers. They
adopted Eyring’s original theory of reaction rates to describe membrane
transport in order to probe how free energy barriers govern permeability.
Instead of a quasi-equilibrium between the reactants and the activated
complexes, they considered a quasi-equilibrium between molecular jumps through
the membrane. They demonstrated the applicability of their framework with
biological membranes in a simple solution-membrane-solution framework. Fig. 1
depicts this framework applied to a polymeric membrane. A major shortcoming of
this representation of permeability is that it relies on the assumption that
membrane diffusion can be described as a series of molecular jumps over equal
free energy barriers, or equivalently as a single dominant free energy barrier
(?, ?, ?). Assuming the barriers within a membrane are either equal or
dominated by a single barrier does not isolate individual mechanisms and
ignores the inherent heterogeneity within polymeric membranes. Most notably,
such assumption hinders our understanding of the experimentally measured
effective free energy barriers and their associated enthalpic and entropic
barriers (?, ?, ?).

Figure 1: Schematic for the membrane model presented by Zwolinski and
coworkers (?). $\lambda_{M}$ is the jump length within the membrane,
$\lambda_{sm}$ is the jump length for the solution-membrane interface,
$\lambda_{ms}$ is the jump length for the membrane-solution interface, $k_{s}$
is the rate constant for the solution jumps, $k_{sm}$ is the rate constant for
the solution-membrane interfacial jump, $k_{ms}$ is the rate constant for the
membrane-solution interfacial jump, and $k_{M}$ is the rate constant for the
membrane jump. $m$ is the number of jumps along the transport coordinate, and
$\delta$ is the membrane thickness.

In this study, we extend the early work by Eyring and coworkers (?, ?),
presenting an interpretation of the transition-state theory approach that
accounts for distributions of free energy barriers that will exist within any
realistic membrane. Statistical mechanics tells us that to connect molecular
phenomena like jumps between voids in a membrane to macroscopic quantities
like permeability we must consider the probability distributions associated
with those molecular phenomena. We adapt theories for parallel arrays of pores
(?, ?) to molecular pathways through polymeric membranes, developing a novel
expression for membrane permeability in terms of molecular jumps along the
transport coordinates of many independent pathways. This expression for
permeability expands previous derivations (?, ?) to generalized membrane
barrier distributions. We relate the observable, effective membrane free
energy barrier and its enthalpic and entropic components to distributions of
energy barriers for individual molecular jumps. An effective free energy
barrier that averages molecular events has not been developed for this
theoretical framework, despite its growing application. Finally, we discuss
how our results change how researchers should interpret experimentally
observed free energy barriers in membranes, and how heterogeneity even at the
molecular level has a significant impact on membrane transport.

### Proposed Theoretical Framework

To construct our framework, we relax two of the main assumptions presented by
Zwolinski et al. (?), by allowing for distributions of membrane barriers and
jump lengths. We apply their equation for flux to a membrane with solution on
either side as in Fig. 1. We treat all solution jump rates $k_{s}$ as equal
and membrane jump rates $k_{M,j}$ as unequal. Similarly, we treat all solution
jump lengths $\lambda_{s}$ as equal and membrane jump lengths $\lambda_{M,j}$
as unequal. The full derivation is provided in the Supplementary Materials.
The permeability in terms of the free energy barriers for a single molecular
pathway becomes:

$P=\frac{\displaystyle\left(\frac{\lambda_{sm}}{\lambda_{ms}}\right)\left(\frac{k_{B}T}{h}\right)\exp{\left(\frac{-\left(\Delta
G_{sm}^{\ddagger}-\Delta
G_{ms}^{\ddagger}\right)}{RT}\right)}}{\displaystyle\sum_{j=1}^{m}\left(\frac{1}{\lambda_{M,j}}\right)\exp{\left(\frac{-\Delta
G_{M,j}^{\ddagger}}{RT}\right)}}$ (3)

where $P$ is permeability, and $k_{B}$, $T$, $h$, and $R$ are Boltzmann’s
constant, temperature, Planck’s constant, and the gas constant. $\lambda_{sm}$
and $\lambda_{ms}$ are the jump lengths from solution to membrane and membrane
to solution, respectively. Similarly, $\Delta G_{sm}^{\ddagger}$ and $\Delta
G_{ms}^{\ddagger}$ are the free energy barriers for the solution-to-membrane
jumps and membrane-to-solution jumps. $\Delta G_{M,j}^{\ddagger}$ is the free
energy barrier for membrane jump $j$.

The permeability in Eq. 3 only describes transport along a single molecular
pathway. The observed permeability is a combination of all accessible
molecular paths, similar to the parallel array of pores described by Wendt et
al. (?). We apply this relationship to our expression for permeability to
obtain an area-weighted permeability across many parallel paths. Therefore,
the overall permeability for $n$ paths per unit area each with $m_{i}$
barriers is:

$P=\sum_{i=1}^{n}\left[\frac{\displaystyle\left(\frac{A_{i}}{A_{0}}\right)\left(\frac{\lambda_{sm}}{\lambda_{ms}}\right)\left(\frac{k_{B}T}{h}\right)\exp{\left(\frac{-\left(\Delta
G_{sm}^{\ddagger}-\Delta
G_{ms}^{\ddagger}\right)}{RT}\right)}}{\displaystyle\sum_{j=1}^{m_{i}}\left(\frac{1}{\lambda_{M,i,j}}\right)\exp{\left(\frac{-\Delta
G_{M,i,j}^{\ddagger}}{RT}\right)}}\right]$ (4)

where $A_{i}$ is the cross-sectional area for path $i$, $A_{0}$ is the
membrane unit area being considered, and $m_{i}$ is the number of jumps for
path $i$. Similar to Eq. 3, $\lambda_{M,i,j}$ and $\Delta
G_{M,i,j}^{\ddagger}$ are the jump length and free energy barrier for the
$j^{th}$ membrane jump on path $i$, respectively.

We express the effective free energy barrier from Eyring’s original derivation
(Eq. S5) in terms of distributions of membrane free energy barriers and jump
lengths across many parallel paths with different numbers of jumps. To do
this, we equate Eq. S5 to Eq. 4 and solve for $\Delta G_{eff}^{\ddagger}$. Eq.
5 gives the resulting analytical expression for the overall effective free
energy barrier, the main theoretical result of this paper.

$\Delta
G_{eff}^{\ddagger}=-RT\ln{\left[\sum_{i=1}^{n}\frac{\displaystyle\left(\frac{A_{i}}{A_{0}}\right)\left(\frac{\delta}{\lambda_{avg}^{2}}\right)\left(\frac{\lambda_{sm}}{\lambda_{ms}}\right)}{\displaystyle\sum_{j=1}^{m_{i}}\left(\frac{1}{\lambda_{M,i,j}}\right)\exp{\left(\frac{\Delta
G_{M,i,j}^{\ddagger}}{RT}\right)}}\right]}+\left(\Delta
G_{sm}^{\ddagger}-\Delta G_{ms}^{\ddagger}\right)$ (5)

We can decompose this effective free energy barrier into enthalpic and
entropic terms. Under the same assumptions as the original expression by
Zwolinski et al. but expanding to include parallel paths, in addition to the
terms related to averaging the free energy barriers, we find an effective
entropic penalty resulting from the fraction of membrane area accessible to
permeation. The permeability only depends on the path areas that are
accessible to transport. The accessible area to transport is not necessarily
the total membrane area, as shown in Eq. S7. This result is consistent with
experimental barriers calculated for ions in NF membranes, where the entropy
was attributed to geometric constraints on the void volumes (?, ?). Eq. 6 more
clearly shows this “entropic” penalty if we additionally assume all paths are
identical. Zwolinski et al. implicitly assumed that the entire area is
accessible to transport, or equivalently that $\sum_{i=1}^{n}A_{i}=A_{0}$ such
that the entropic penalty is 0. The area fraction accessible to transport,
because it is not temperature dependent, would manifest as part of the overall
effective entropy. When we expand the scenario presented by Zwolinski et al.
to parallel paths, the overall equation becomes:

$\Delta G_{eff}^{\ddagger}=\left[\Delta H_{M}^{\ddagger}+\Delta
H_{sm}^{\ddagger}-\Delta H_{ms}^{\ddagger}\right]-T\left[\Delta
S_{M}^{\ddagger}+\Delta S_{sm}^{\ddagger}-\Delta
S_{ms}^{\ddagger}+R\ln{\left(\sum_{i=1}^{n}\frac{A_{i}}{A_{0}}\right)}\right]$
(6)

$\Delta H_{M}^{\ddagger}$ and $\Delta S_{M}^{\ddagger}$ are the enthalpic and
entropic barriers within the membrane, and similarly, $\Delta
H_{sm}^{\ddagger}$, $\Delta S_{sm}^{\ddagger}$, $\Delta H_{ms}^{\ddagger}$,
and $\Delta S_{ms}^{\ddagger}$ are the enthalpic and entropic barriers at the
solution-membrane ($sm$) and membrane-solution ($ms$) interfaces.

In this study, we focus on the scenario where transport is primarily hindered
by diffusion through the membrane, not membrane entry or exit (?, ?). As a
result, we treat the jumps across the solution-membrane and membrane-solution
interfaces as fast and their associated free energy barriers as negligible and
constant across all paths through the membrane. In this case, $\Delta
G_{sm}^{\ddagger}$ and $\Delta G_{ms}^{\ddagger}$ are constant for all $i$
paths and small compared to $\Delta G_{M,i,j}^{\ddagger}$.

In order to examine the consequences of our proposed framework, we need to
make choices about the distributions of membrane barriers. We select two
common distributions. First, we assume a fixed mean and normally distributed
barrier heights around this mean. Physically, this distribution would model
membranes with uniform pore structure or homogeneous polymer density at the
molecular level. Most molecular pathways would thus have similar environments
and jump mechanisms, such that the barriers would be similar for all pathways.
Second, we choose exponentially distributed barrier heights to represent
membranes with a large amount of heterogeneity. All pores would have regions
of unfavorable mechanisms with a few high barriers, as well as regions of low-
barrier mechanisms more similar to free diffusion. Further details on the
choice of these distributions and their parameters are given in the Materials
and Methods section.

## Results

### The single path effective free energy barrier is highly dependent on the
heterogeneity of the individual barrier distributions

We find that the single path effective free energy barrier is slightly below
the maximum free energy barrier of the underlying distributions, and
significantly larger than the mean free energy barrier. The single path
effective free energy barrier can be related to the distribution of membrane
free energy barriers $\Delta G_{M,j}^{\ddagger}$ by assuming a single path $i$
where the entire area is accessible to transport. Fig. 2A shows where the
single path effective free energy barrier would lie for one realization of the
barrier distribution, assuming the underlying distributions for the enthalpic
and entropic barriers follow normal distributions and exponential
distributions. Fig. 2B shows free energy profiles for single pathways through
the membrane with barrier distributions like Fig. 2A.

The single path effective free energy barrier is most affected by the largest
individual barriers, consistent with Giddings and Eyring’s “kT-cutoff” (?). We
numerically verify the “kT-cutoff” model by showing that the largest barriers
(i.e. those within $k_{B}T$ of the maximum barrier) determine the single path
effective barrier, as shown in Fig. 2A. Larger variance in the underlying
barrier distributions introduces high outliers that significantly increase the
single path effective free energy barrier, as demonstrated with the
exponentially distributed barriers. We generalize Scheuplein’s conclusions
that the locations of barriers along the transport coordinate do not matter
and that the overall rate constant can be expressed as a probability-weighted
sum of rates to incorporate arbitrary distributions of membrane barriers (?).
We verify these conclusions numerically in Fig. 2. In it, we see that the
single path effective barrier is slightly below the maximum barrier and does
not depend on the locations or orders of the barriers.

Figure 2: A realization of distributions of membrane barriers along a single
path. (A) For both realizations considered, the effective free energy barrier
for a single path lies within the high tail of the distribution, near the
maximum barrier. Larger variance in the distribution results in a
significantly larger effective barrier. The effective free energy barrier is
shown as a dashed vertical line. The mean free energy barrier for both
distributions is 12.5 kcal/mol with further discussion in Materials and
Methods. We use 200 jumps of 2 Å each through a single path at 300 K. (B) The
effective free energy barrier along a single path is not dependent on the
locations of the barriers. The effective free energy barriers for each path
are shown as dashed horizontal lines. We use 100 jumps of 2 Å each. Enthalpic
and entropic barriers are each drawn independently from the specified
distributions and combined to calculate the free energy barrier.

### The overall effective free energy barrier is determined by the highest
barriers in the most permeable paths

Figure 3: Effective free energy barriers depend on the largest underlying
barriers. The overall effective free energy barrier is near the maximum of the
individual free energy barriers and near the minimium of the single path
effective free energy barriers. We show distributions of membrane barriers
with the same mean at 12.5 kcal/mol. The underlying enthalpic and entropic
barriers are all equal (A), normally distributed (B), and exponentially
distributed (C). In this figure, the entropic penalty from the accessible area
for transport is assumed to be 0. The effective free energy barriers are shown
as dashed vertical lines. The overall effective free energy barriers are
calculated by Eq. 5. We use 200 jumps of 2 Å each for all 2000 paths.

Expanding the model to a membrane comprising many parallel paths, we find the
overall effective free energy barrier through the membrane from Eq. 5 lies
within the high tail of the underlying barrier distributions and the low tail
of the single path effective barriers. In Fig. 3, we show the overall
effective free energy barriers for 2000 paths compared to the distributions of
individual free energy barriers and the distributions of single path effective
barriers. If all paths have equal individual barriers as in the original
Zwolinski et al. derivation, the overall effective free energy barrier
collapses to be identical to an individual membrane barrier, as shown in Fig.
3A.

Fig. 3B and Fig. 3C show that the overall effective barrier for the membrane
lies near the maximum individual barrier. Equivalently, the overall effective
barrier lies near the lowest single path effective barrier. Therefore, when we
consider distributions of free energy barriers across many pathways, the
overall effective free energy barrier to permeability is not the difference in
free energy between the species in solution and the species at the top of the
highest potential energy barrier. Rather, it is heavily dependent on the
highest barriers within the paths with the lowest single path effective
barriers. These single path effective barriers are most dependent on the
highest individual barriers along the path. We demonstrate that the overall
effective barrier is typically determined by the paths whose highest barriers
are relatively low in Fig. 4A and Fig. 4B, where we plot each path’s maximum
barrier. The overall effective barrier is near the lowest maximum barriers,
which is in turn is near, but not at, the top of the distribution of
individual barrier heights.

Figure 4: The effective free energy barrier and flux are dominated by paths
with low maximum barriers. (A,B) The effective free energy barrier in the case
of many pathways is primarily determined by the paths with the smallest
maximum barriers. For normally (A) and exponentially (B) distributed enthalpic
and entropic barriers, the overall effective free energy barrier is shown as a
dashed line. The entropic penalty from the accessible area for transport is
assumed to be 0 to highlight the effect of the barrier heights. (C) More
heterogeneous free energy landscapes create a few highly permeable paths that
dominate the flux, as shown by the fraction of the flux through the most
permeable paths. The fluxes are calculated in the case of normally distributed
enthalpic and entropic barriers with increasing variance as given by the
standard deviation ($\sigma$) of $\Delta G_{M,i,j}^{\ddagger}$. The standard
deviation for the enthalpic barriers ranged from $1\text{\times}{10}^{-4}$ to
10 kcal/mol, and the standard deviation for the entropic barriers ranged from
$3.3\text{\times}{10}^{-7}$ to $3.3\text{\times}{10}^{-2}$ kcal/mol$\cdot$K.

### Heterogeneity in molecular pathways dictates membrane flux

Intuitively, the overall flux is most determined by the paths with the highest
permeability, and Fig. 4C confirms this trend in the transition-state theory
model for molecular pathways through a membrane. Importantly, this is true not
only for macroscopic defects, but also for mechanistic molecular barriers. If
all individual membrane barriers are equal, the flux is evenly distributed
across all parallel paths as shown in the straight, dark blue line. As more
heterogeneity is introduced from the distributions of membrane barriers, the
flux is skewed towards highly permeable paths. Distributions of free energy
barriers within the membrane create more favorable paths through the membrane.
Physically, paths through easily traversed pores will contribute most to the
total permeability, and paths that require energetically unfavorable
rearrangement and hopping will contribute least to the permeability.

For a real membrane, all the molecular pathways will have variance in their
energy barriers, jumps, and total path length, and therefore outlier pathways
with high permeability will contribute the most to observable energy barriers.
Fig. 5 gives one realization of the model where the underlying barriers, jump
lengths, and the number of jumps are each normally distributed, and we
highlight two important free energy profiles through the membrane—the most
permeable path (blue) and the path with the smallest maximum barrier (red).
The highest permeability paths have low maximum barriers and fewer jumps. In
the model, fewer jumps corresponds to fewer opportunities for high outliers in
the membrane barrier distribution. These paths might correspond to large pores
or defects in the membrane, where molecules can easily take large jumps. A low
maximum barrier may represent a pore with a single small constriction, but
otherwise relatively open.

In contrast to the substantial effect from distributions in the barrier
thermodynamics, we find the overall effective free energy barrier does not
vary much, on the order of 0.5 kcal/mol, with distributions of number of jumps
or jump lengths assuming all barrier heights are equal. This finding has a
caveat that the overall effective barrier can be decreased moderately when a
non-negligible number paths contain only a few (single digit numbers) jumps,
where the effects might be as large as 1.5 kcal/mol. However, this is still a
small contributing factor compared to the effects from variation in barrier
heights. In the case of varying both barrier heights and numbers of jumps, we
expect a larger difference in the overall effective barrier caused by paths
with both small numbers of jumps and no high energy barriers among those
jumps. We do not explore this regime quantitatively in this study due to the
large number of possible variables. We present an in-depth discussion of the
effects of jump lengths and number of jumps in the Supplementary Materials
(Figs. S2 and S3).

Figure 5: Smallest maximum barrier path and the most permeable path through a
membrane with normal barriers, jump lengths, and jump numbers. Pathways with
low maximum barriers and a few large jumps contribute most to the overall
effective barrier. Over a realization of 2000 paths with normally distributed
enthalpic and entropic barriers, jump lengths, and number of jumps, the path
with the smallest maximum barrier is shown in red, and the path with the
highest permeability is shown in blue. While the most permeable path does not
have the smallest maximum barrier, its maximum barrier is low and it requires
fewer jumps. The overall effective free energy barrier for 2000 paths, shifted
by the effective entropic penalty from parallel paths to better show the
direct connection of effective barriers to the barrier height distribution, is
shown as a dashed line.

In Fig. 5, we show a realization where the most permeable path does not have
the smallest maximum barrier. While its maximum barrier is comparatively
small, it is not the smallest maximum barrier. We tested how frequently the
smallest maximum barrier path is also the most permeable path for both
normally distributed and exponentially distributed underlying barriers. For
1000 realizations of normally distributed membrane barriers, the smallest
maximum barrier path is the most permeable path in 60.8% of the realizations.
That percentage jumps to 90.0% for exponentially distributed barriers with the
same mean. Of the realizations where the smallest maximum barrier path is not
the most permeable path, the maximum barrier in the most permeable path is
similar to the smallest maximum barrier 95.9% of the time for normally
distributed barriers and 97.0% of the time for exponentially distributed
barriers. Barriers are considered similar if they are within $k_{B}T$, as
defined by Giddings and Eyring’s “kT-cutoff” (?).

### Overall effective enthalpic and entropic barriers are larger than the
typical barriers experienced by molecules in the membrane

Figure 6: Distributions of enthalpic, entropic, and free energy barriers and
their overall effective barriers. Effective entropic and enthalpic barriers
are found towards the tails of their underlying distributions. In these
examples, the effective entropic barriers have a larger relative shift
compared to the effective enthalpic barriers due to the accessible area
entropic penalty. We calculate the overall effective barriers for 22,000 paths
from the linearized permeability vs. temperature. Standard errors for the
effective barriers and means are shown as lightly shaded regions. Error for
the means and normally distributed barriers are too small to be visible.

Zwolinski and coworkers’ expression for permeability (given in our Eq. S5) has
been used to estimate the overall effective enthalpic and entropic barriers to
membrane transport. Typically, this equation is linearized so the slope is
$-\frac{\Delta H_{eff}^{\ddagger}}{R}$ and the intercept is $\frac{\Delta
S_{eff}^{\ddagger}}{R}$. Therefore, the enthalpic and entropic contributions
to the permeability can be estimated by simply measuring permeability at a
range of temperatures (?). We follow this approach using permeabilities from
our numerical model evaluated at a range of temperatures. Individual barriers
at each temperature are drawn from random distributions with the same
parameters.

As with free energies, the overall effective enthalpy and entropy calculated
from the linearized fit lie in the high magnitude tail of the underlying
distributions of enthalpies and entropies. Fig. 6 demonstrates how the
measured enthalpic and entropic barriers are larger in magnitude than their
respective average barriers in the membrane. These distributions are for
22,000 paths with 200 jumps each for temperatures at 10 K increments between
250 K and 350 K. We simulate 22,000 paths (which would be approximately
equivalent to 1 , following the same procedure provided in the Supplementary
Materials) to reduce the error in the effective barriers for the exponential
distributions, since the exponential distributions have higher variance.

The relative shift in the entropic barrier is larger than that for the
enthalpic barrier because the entropic shift also includes the contribution
from transport-accessible area, as that contribution is temperature
independent and would be interpreted as entropy. Additionally, in Fig. 6, the
mean entropic contribution $\left<-T\Delta S_{M,i,j}^{\ddagger}\right>$ is
larger than the enthalpic contribution, further exaggerating the difference in
effective barriers. Increasing the variance in the membrane barrier
distribution increases the magnitudes of the overall effective enthalpic and
entropic barriers, as shown by the higher variance exponential distributions
in Fig. 6. Higher variance introduces higher maximum barriers along single
paths, which heavily influences the overall barrier to transport. We find that
observable entropic and enthalpic barriers are, again, not representative of
the typical or mean mechanisms in the membrane but rather, of the rate-
limiting mechanisms along only the most permeable paths.

### Implications for experimental study of effective energy barriers in RO and
NF membranes

Figure 7: Experimental linearized transition-state theory plots and the
resulting overall effective enthalpic barriers. Overall effective barriers
measured experimentally are similar across different salts and membranes,
indicating that highest barriers in the most permeable paths are also similar.
(A) Linearized transition-state theory plots for the permeability of NaCl and
NaF in the NF270 membrane and NaCl in the SW30 RO membrane. The least squares
fit is shown as a line for each system. (B) Overall effective enthalpic
barriers calculated from the slopes in (A) and Fig. S4. The errors shown are
the propagated errors from the linear regression. Experimental conditions
during filtration: a single-salt solution of NaCl or NaF at 5 mM, 10-40 °C, pH
7, 33 bar, and crossflow velocity of 2.13 m/s.

In experimental studies of molecular transport in polymeric membranes, the
measured barrier is considered an overall effective parameter that represents
the transport of a given solute, and the physical meaning has not been fully
established. Based on the current study, we can better analyze and understand
effective energy barriers in the context of many individual energy barriers in
parallel and series.

To demonstrate the implications our analysis has on the experimental study of
effective energy barriers, we extracted effective transition state barriers
from permeabilities of a selection of salts in a selection of membranes.
Specifically, we experimentally measured the permeability of sodium chloride
(NaCl) in a NF membrane at six temperatures to extract the effective enthalpic
barrier for the salt transport from the slope of the linearized transition-
state theory plot (Fig. 7A). We also performed a similar measurement for the
transport of sodium fluoride (NaF) in the same NF membrane and for NaCl in a
RO membrane (Fig. 7A). Finally, we measured the increase of NaCl conductivity
with temperature in water and constructed its corresponding linearized
transition-state theory plot (Fig. S4). The effective enthalpic barriers
measured for the four cases are shown in Fig. 7B.

Fig. 7B does show an increasing effective enthalpic barrier to transport with
a denser medium (water $<$ NF $<$ RO) or a larger and more strongly hydrated
species (NaCl $<$ NaF). These trends are intuitive as a denser medium or
larger species may require higher molecular adjustments and arrangements
during diffusion jumps. The effective enthalpic barriers measured for membrane
permeability are slightly higher than typical barrier values reported for
water or ion diffusion in water, (?, ?) indicating a hindered diffusion
compared to free diffusion in water. However, the statistical uncertainty is
large, indicating that any differences in the enthalpic barrier heights are
likely small and do not appear to be significantly affected by the substantial
change in ion size, nor by membrane density. This observation supports the
picture that the average ion environment only loosely affects the transport
along the most important paths. For example, both the NF and RO membranes
likely have larger voids that dominate the flux and result in similar
effective barriers, despite the significant difference in the chemistry of the
membranes.

Additionally, the effective barriers to ion transport measured in the
membranes are much lower than typical activation energies reported for
diffusion in denser polymers, such as Nafion, polyimide, and
perfluorosulfonate (?, ?, ?). The barriers in these polymers would likely be
associated with high energy pathways through dense networks. While in the
membranes, the barriers would likely be associated with larger pores (or
voids) where molecules can pass through while being relatively unaffected by
the polymer environment.

Our finding that the overall effective energy barrier is dictated by the
highest barrier in the most permeable path is also supported by prior
experimental data examining the heterogeneity of polyamide RO membranes (?).
Culp et al. identified water diffusion pathways in polyamide RO membranes and
estimated the local flux along those pathways. They found that the average
diffusion coefficient in the polymer was unable to predict membrane water
permeability, rather that the nanoscale heterogeneity controlled membrane
permeability. They identified the same two levels of heterogeneity that we
explore, namely heterogeneity across parallel paths and heterogeneity within a
single path in the direction of transport.

Interestingly, the membrane samples studied had pathways through the membrane
with significantly more heterogeneity along pathways than between pathways,
i.e. each of the parallel pathways had highly correlated flux as a function of
distance through the membrane, rather than being random, as in our model,
because of inhomogeneities at the mesoscale. For such paths, sections along
the transport coordinate with very low local flux (high outlier barriers)
significantly limited the total flux along all paths. On the other hand, the
high permeability membranes had narrower distributions of local flux along the
transport coordinate. Consequently, the most permeable membranes had barrier
distributions with low variance, such that there were not many outliers and
the highest barriers were relatively low.

## Discussion

In this work, we find that even moderate, statistically random heterogeneity
in energy barriers will significantly impact how we interpret the mechanisms
of transport through the membrane. In polymeric membranes used for RO and NF,
structural and chemical heterogeneity, such as non-uniform pores or charged
functional groups, introduce a wide variety of free energy barriers to
permeability (?, ?). For a cross-section of membrane, the overall effective
barrier is most dependent on the highest barriers in the most permeable paths
with smaller contributions from the other parallel paths. The enthalpic and
entropic components, and thus the overall free energy barrier increase with
increasing heterogeneity in the membrane.

To design membranes with desired separation capabilities, we must control the
highest barriers to transport in the most permeable paths. Defects or voids in
the membrane along the transport coordinate will significantly increase the
permeability by decreasing the highest barriers to transport through
individual paths, leading to flux hot spots (?). Designing membranes with more
uniform energy barriers will distribute flow through more paths.
Experimentally, increasing the homogeneity of the polyamide films has been
shown to increase water flux and permselectivity.

There are a number of limitations to this study due to the approximations
required to perform numerical experiments, but the main conclusions of the
work are unaffected since the analysis is generalized to be broadly
applicable. We explored two physically motivated distributions of barrier
heights and jumps; in virtually all cases, these distributions are unknown,
and we may not have explored the parameter ranges that are most physically
relevant. We do not consider the coupled diffusion of multiple species, which
can constrain transport via requirements of electroneutrality for ions. We
also do not include external driving forces in our analysis, but previous work
has developed this theory, which effectively scales the energy barriers (?,
?). The magnitudes of the barriers would change, but our interpretation of
effective energy barriers would not. Additionally, we treat the interfacial
barriers as negligible in our analysis, but some studies have suggested these
may be significant factors in modeling membrane transport (?). To include the
effect of the interfaces, there are two significant scenarios to address. In
the first case, interfacial barriers dominate the transport. Only the
interfacial barriers and their heterogeneity across parallel paths would need
to be considered. In the second case, interfacial barriers are of similar
magnitude to barriers within the membrane. But because the order of the
individual barriers does not change the interpretation of the single path
effective free energy barrier, these barriers can be included in the overall
framework with corresponding distributions. A remaining important limitation
is that we assumed independent and separate pathways through the membrane, but
in reality, the molecular pathways may merge and interconnect. Incorporating
this additional heterogeneity is beyond the scope of the current study, but
can be easily extended to introduce correlated barrier distributions between
paths and consideration of topological effects, as for example in the work of
Culp et al. (?). It would be difficult to determine a priori appropriate
covariances for any given polymeric membrane. However, by treating the
molecular jumps as resistances, many different topologies could be explored
with parallel-series circuit models. This circuit model theory is well-
developed for interconnected pathway flows and can be readily expanded to
include varying barriers (?, ?, ?, ?).

Our work shows that the conventional theoretical framework for transition-
state theory energy barriers leads to incorrect interpretations of
experimental effective free energy, enthalpic, and entropic barriers. Even
analysis of experimental results based on transition-state theory will produce
effective barriers that are not easily related to mechanistic details at the
atomistic level. In particular, the observed effective enthalpies and
entropies do not necessarily correspond to either the most frequent or the
highest mechanistic barriers occurring in the system. For example, attempts to
match barriers to specific enthalpies of ion dehydration within the membrane
are unlikely to be successful, as the free energy barrier of an individual
mechanistic event may be several kcal/mol different from the measured
effective free energy barrier, and thus chemical design attempts may focus on
the wrong interactions. Additionally, attempts to understand membrane barriers
by looking at typical events in the membrane via simulation may focus on the
wrong events, as the typical free energy barrier is not necessarily relevant
in the overall permeability. Similarly, the highest barriers encountered in
membrane pores may not be relevant, as it is only the highest barriers on the
most permeable paths that primarily contribute to the experimentally
observable barrier.

## Materials and Methods

### Numerical methods

We numerically evaluate our expanded transition-state theory model for
membrane permeability by drawing heights for each of the individual enthalpic
and entropic barriers from independent random distributions. In this paper, we
use two commonly occurring probability distributions to represent the
distribution of energy barriers, normal (Gaussian) and exponential
distributions, for both enthalpic and entropic barriers. Normal and
exponential distributions occur in many natural phenomena, and represent
extremes of possible behavior. Normal distributions serve as well-behaved two-
sided distributions around a set mean, whereas exponential distributions are
one-sided with highest probability near zero and long tails.

We draw heights of the enthalpic barriers from distributions with mean 3.5
kcal/mol, which corresponds to the observed effective enthalpic barrier for
chloride within NF membranes at 300 K (?), and we draw entropic barriers from
distributions with mean -0.03 kcal/mol$\cdot$K, which corresponds to the
observed effective entropic barrier for chloride under the same conditions
(?). This combination results in an effective free energy of $\Delta
G_{eff}^{{\ddagger}}=\Delta H_{eff}^{{\ddagger}}-T\Delta S_{eff}^{{\ddagger}}$
= 12.5 kcal/mol, at 300 K. Unless otherwise specified, the standard deviation
for the normally distributed enthalpic barriers is 1.17 kcal/mol, and the
standard deviation for the normally distributed entropic barriers is 0.01
kcal/mol$\cdot$K. These standard deviations ensure the normally distributed
barriers represent membranes with less heterogeneity than the exponentially
distributed barriers. Exponential distributions are defined by a single
parameter, so specifying their mean is enough to fully define them.

In simulating membranes with multiple paths, we use $2\text{\times}{10}^{-4}$
as an estimate for the number of paths per $\text{\AA}^{2}$. This estimate is
an order of magnitude smaller than the estimated packing density of single-
walled carbon nanotubes with a diameter of 0.5 nm (more information provided
in the Supplementary Materials), to account for the heterogeneity of polymer
membranes. Here, we show trends for 2000 paths through the membrane unless
otherwise stated, which roughly corresponds to a unit area of 0.1 , enough to
converge average results across a distribution of paths. See Fig. S1 for
determination of the number of paths needed for convergence. The code
implementation for our numerical analysis is on Github at
https://github.com/shirtsgroup/eyring_model.

### Crossflow filtration experiments

Filtration experiments were performed with two types of flat-sheet commercial
membranes in a crossflow mode – a loose polyamide NF membrane (NF270, Dow
FilmTec) and a tight polyamide RO membrane (SW30, Dow FilmTec). Single-salt
solutions of NaCl and NaF at 5 mM were used as feed solutions. The filtration
experiments were carried out at pH 7, with an applied pressure of 33 bar and a
crossflow velocity of 2.13 m/s. In order to calculate transition-state theory
barriers, the salt flux was measured at 6 temperatures from 10 °C to 40 °C,
from which the permeability at these temperatures was calculated using

$P=\frac{J_{s}}{C_{m}-C_{p}}$ (7)

where $J_{s}$ is the salt flux and $C_{m}$ and $C_{p}$ are the salt
concentrations on the membrane surface in the feed side and in the permeate
solution, respectively. Concentration polarization on the membrane surface and
$C_{m}$ were evaluated using previously reported methods given also in the
Supplementary Materials (?). The enthalpic and entropic barriers were
extracted from the slope and intercept of the linearized Eyring plot, as shown
previously (?). These experiments were replicated 3 times for each
temperature.

### Measurement of barriers to transport for salt in water

Energy barriers to transport of salt in water were calculated by measuring the
conductivity (Eutech Instruments, CON2700) of 5 mM sodium chloride solutions
at 4 different temperatures between 25 °C and 45 °C. The barriers for the
conductivity were then extracted using the same method applied to calculate
the barriers of the permeability. The resulting transition-state theory plot
for conductivity in water is provided in the Supplementary Materials.

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## Acknowledgments

Funding: This material is based upon work supported by the National Science
Foundation under Grant No. CBET-2136835 and the United States-Israel
Binational Science Foundation (BSF), Jerusalem, Israel (grant No. 2021615).

Author contributions: NSS and MRS designed the study. NSS performed all
numerical experiments. MRS supervised the numerical research. MA collected the
experimental data. MA, NSS, and MRS analyzed the experimental data. NSS, RE,
APS and MRS revised and edited the manuscript. RE supervised the experimental
research.

Competing interests: The authors declare that they have no competing
interests.

Data and materials availability: All data needed to evaluate the conclusions
in the paper are present in the paper and/or the Supplementary Materials. All
code to generate and analyze the data and figures can be found at
https://github.com/shirtsgroup/eyring_model.

Supplementary Materials for
Membrane heterogeneity alters our interpretation of effective energy barriers
to transport
Nathanael S. Schwindt et al.
∗Corresponding author. Email<EMAIL_ADDRESS>
This PDF file includes:

  * Supplementary Text

  * Figs. S1 to S5

  * Table S1 to S2

  * References (46 to 50)

## Supplementary Text

Previous theoretical framework
The original derivation by Zwolinski and coworkers (?) modeled membrane flux
in terms of point-to-point jumps of molecules governed by rate constants.
Thus, the net flux ($Q$) between equilibrium positions within the membrane
becomes the difference in the forward and backward molecular jump rates
through a cross-sectional area. A single barrier with equal forward and
backward rate constants ($k$) and jump lengths ($\lambda$) leads to Fick’s
first law of diffusion (Eq. S1) with diffusion coefficient $D=k\lambda^{2}$.

$Q=-D\frac{\text{d}C}{\text{d}x}$ (S1)

At steady state, the flux is a set of rate equations relating all local
equilibrium positions along the direction of transport. Assuming a constant
flux across the membrane and eliminating all the local concentrations gives an
expression for the flux in terms of the local rate constants $k_{i}$, jump
lengths $\lambda_{i}$, and initial $C_{0}$ and final $C_{m+1}$ concentrations
shown in Eq. S2, where $n$ is the total number of jumps along the transport
coordinate.

$Q=\frac{\displaystyle
k_{0}\lambda_{0}C_{0}-\prod_{i=1}^{n}\left(\frac{k_{i}^{\prime}\lambda_{i}^{\prime}}{k_{i}\lambda_{i}}\right)k_{n+1}^{\prime}\lambda_{n+1}^{\prime}C_{n+1}}{\displaystyle
1+\sum_{r=1}^{n}\prod_{i=1}^{r}\left(\frac{k_{i}^{\prime}\lambda_{i}^{\prime}}{k_{i}\lambda_{i}}\right)}$
(S2)

Under transition state theory, the individual rate constants $k_{i}$ can be
related to free energy barriers $\Delta G^{\ddagger}_{i}$ by

$k_{i}=\kappa_{i}\frac{k_{B}T}{h}\exp{\left(\frac{-\Delta
G^{\ddagger}_{i}}{RT}\right)}$ (S3)

$\kappa_{i}$ is the transmission coefficient (generally assumed to be unity
for membrane processes), and $k_{B}$, $T$, and $h$ are Boltzmann’s constant,
temperature, and Planck’s constant, respectively. Zwolinski et al. (?) and
later del Castillo et al. (?) expanded the expression for flux in terms of
free energy barriers to include external forces. Here, we explore the model
without external forces, as external forces will only increase or decrease the
free energy barriers without impacting the behavior of the model.

Zwolinksi and coworkers verified their model on biological membranes using a
simple setup with four distinct rate constants for the solution $k_{s}$, the
solution-membrane interface $k_{sm}$, the membrane $k_{M}$, and the membrane-
solution interface $k_{ms}$. The authors evaluated Eq. S2 for the solution-
membrane-solution scenario under the assumptions that all jump lengths are
equal, all free energy barriers within the membrane are equal, and diffusion
within the membrane is the dominating step. They arrived at the following
equation for membrane permeability ($P$)

$P=\frac{k_{sm}k_{M}\lambda}{mk_{ms}}$ (S4)

As a result, they expressed membrane permeability in terms of a single,
effective free energy barrier that includes the solution-membrane, membrane,
and membrane-solution barriers. They claimed that this effective free energy
barrier represents the difference in free energy between the species in
solution and the species at the top of the highest potential energy barrier
within the membrane. They extracted the enthalpic ($\Delta
H_{eff}^{\ddagger}$) and entropic ($\Delta S_{eff}^{\ddagger}$) contributions
to permeability from the Gibbs-Helmholtz relation.

$\displaystyle P$ $\displaystyle=$
$\displaystyle\left(\frac{\lambda^{2}}{\delta}\right)\left(\frac{k_{B}T}{h}\right)\exp{\left(\frac{-\Delta
G_{eff}^{\ddagger}}{RT}\right)}$ (S5) $\displaystyle=$
$\displaystyle\left(\frac{\lambda^{2}}{\delta}\right)\left(\frac{k_{B}T}{h}\right)\exp{\left(\frac{\Delta
S_{eff}^{\ddagger}}{R}\right)}\exp{\left(\frac{-\Delta
H_{eff}^{\ddagger}}{RT}\right)}$

$\delta$ in Eq. S5 is the membrane thickness, defined as $\delta=m\lambda$.
This expression has been applied to both biological and polymeric membrane
systems as a way to explore the molecular mechanisms governing membrane
permeability (?, ?, ?).

Giddings and Eyring also explored barrier kinetics primarily through the lens
of nucleation (?). Starting from Eq. S2, the authors represented the effective
free energy barrier for flux in terms of the individual point-to-point rate
constants. While they did not explicitly state the similarity, the effective
free energy barrier is in the form of multiple parallel resistances (see
Equation 7 in reference (?)). They developed a “kT-cutoff model” to identify
the non-negligible barriers (i.e. those within $k_{B}T$ of the maximum
barrier). They concluded that for a series of jumps over unequal free energy
barriers, the highest barrier does not define the overall flux but rather
contributes the most to a sum of non-negligible barriers. Furthermore, they
showed that the effective barrier depends only on the magnitude of the
contributing barriers, not on their order.

Scheuplein further explored the idea that position does not matter in his
analysis of Gidding and Eyring’s multibarrier kinetics model more specifically
applied to membrane permeability (?). Scheuplein grouped membrane barriers of
similar size and represented membrane transport across many unequal groups as
transport across a series of membranes with equal barriers. For barrier groups
$\alpha,\beta,...,\omega$, the permeability becomes

$\frac{1}{P}=\left(\frac{m}{\lambda}\right)\sum_{i=\alpha}^{\omega}\left(\frac{p_{i}}{K_{si}k_{i}}\right)$
(S6)

$m$ is the total number of barriers, $K_{si}$ is the partition coefficient
from the solution to the $i^{th}$ minimum in the membrane, and $p_{i}$ is the
probability of occurrence of the $i^{th}$ kind of barrier. This representation
shows that the permeability is dependent on the individual probabilities and
rate constants within the membrane. Therefore, the permeability is most
affected by the highest and the most probable membrane barriers.

This equation leads to interpreting membrane permeability as combining
parallel resistances. Wendt et al. derived a similar interpretation of
permeability for pores in series (?), and del Castillo et al. explicitly
showed how the multibarrier kinetic model can be thought of under this context
(?). Wendt and coworkers’ primary assumptions were that transport can be
treated as one-dimensional and that there is no internal concentration
polarization within the membrane. From these assumptions, they showed that the
overall flux in a series of non-sieving pores is equivalent to flux through a
single pore with an overall permeability in the form of parallel resistances.
Expanding to an array of pores, they showed that the overall flux in parallel
pores is a sum of the individual pore fluxes. The overall permeability for the
parallel array of pores is the area-weighted sum of the $n$ individual pore
permeabilities ($P_{i}$) as shown in Eq. S7.

$P=\sum_{i=1}^{n}\frac{A_{i}}{A_{0}}P_{i}$ (S7)

where $A_{i}$ is the individual pore area and $A_{0}$ is the total membrane
area considered, generally assumed to be a unit area. The individual pore
areas are not required to sum to the total area. As a result, the overall
permeability $P$ describes transport through the accessible area. If the pore
areas do sum to the total area, the overall permeability becomes a weighted
average, and the entire membrane area is accessible for transport. del
Castillo et al. also explored these permeability expressions under arbitrary
external forces, arguing that the overall flux depends on the distribution of
parallel permeabilities, but in most cases, it will be near the pure diffusion
limit. Additionally, they provided a weak constraint on the applicability of
the multibarrier kinetic model for membrane transport.

Derivation of the permeability with distributions of barriers, jumps, and
paths
To construct our framework, we start with the main assumptions of Eyring’s
multibarrier kinetic model, and then relax some of these assumptions. Their
assumptions were:

  1. 1.

steady state flux can be represented by point-to-point molecular jumps between
locally equilibrated states,

  2. 2.

membrane transport is one-dimensional in the solution-membrane-solution
framework,

  3. 3.

an aqueous solution is diffusing through the membrane, and membrane diffusion
is the primary hindrance to transport,

  4. 4.

the transmission coefficient is one for all rate constants,

  5. 5.

the free energy barriers within the membrane are a series of equal free energy
barriers, and

  6. 6.

the jump lengths between local barriers are equal.

We start with Eq. S8 (?) and relax the assumptions that the free energy
barriers within the membrane are a series of equal free energy barriers and
the jump lengths between local barriers are equal.

$Q=\frac{\displaystyle
k_{0}\lambda_{0}C_{0}-\prod_{i=1}^{n}\left(\frac{k_{i}^{\prime}\lambda_{i}^{\prime}}{k_{i}\lambda_{i}}\right)k_{n+1}^{\prime}\lambda_{n+1}^{\prime}C_{n+1}}{\displaystyle
1+\sum_{r=1}^{n}\prod_{i=1}^{r}\left(\frac{k_{i}^{\prime}\lambda_{i}^{\prime}}{k_{i}\lambda_{i}}\right)}$
(S8)

As a result, the numerator expands to

$\displaystyle
k_{0}\lambda_{0}C_{0}-\prod_{i=1}^{n}\left(\frac{k_{i}^{\prime}\lambda_{i}^{\prime}}{k_{i}\lambda_{i}}\right)k_{n+1}^{\prime}\lambda_{n+1}^{\prime}\lambda
C_{n+1}$
$\displaystyle=k_{s}\lambda_{s}C_{0}-\left[\left(\frac{k_{s}\lambda_{s}}{k_{s}\lambda_{s}}\right)...\left(\frac{k_{s}\lambda_{s}}{k_{s}\lambda_{s}}\right)\left(\frac{k_{s}\lambda_{s}}{k_{sm}\lambda_{sm}}\right)\left(\frac{k_{ms}\lambda_{ms}}{k_{M,1}\lambda_{M,1}}\right)\left(\frac{k_{M,1}\lambda_{M,1}}{k_{M,2}\lambda_{M,2}}\right)...\right.$
$\displaystyle\hskip
60.22499pt\left.\left(\frac{k_{M,m-1}\lambda_{M,m-1}}{k_{M,m}\lambda_{M,m}}\right)\left(\frac{k_{M,m}\lambda_{M,m}}{k_{ms}\lambda_{ms}}\right)\left(\frac{k_{sm}\lambda_{sm}}{k_{s}\lambda_{s}}\right)\left(\frac{k_{s}\lambda_{s}}{k_{s}\lambda_{s}}\right)...\left(\frac{k_{s}\lambda_{s}}{k_{s}\lambda_{s}}\right)\right]k_{s}\lambda_{s}C_{n+1}$
$\displaystyle=k_{s}\lambda_{s}C_{0}-\left[\left(\frac{k_{s}\lambda_{s}}{k_{s}\lambda_{s}}\right)^{s-1}\left(\frac{k_{s}\lambda_{s}}{k_{sm}\lambda_{sm}}\right)\left(\frac{k_{ms}\lambda_{ms}}{k_{M,1}\lambda_{M,1}}\right)\left(\frac{k_{M,1}\lambda_{M,1}}{k_{M,2}\lambda_{M,2}}\right)...\right.$
$\displaystyle\hskip
60.22499pt\left.\left(\frac{k_{M,m-1}\lambda_{M,m-1}}{k_{M,m}\lambda_{M,m}}\right)\left(\frac{k_{M,m}\lambda_{M,m}}{k_{ms}\lambda_{ms}}\right)\left(\frac{k_{sm}\lambda_{sm}}{k_{s}\lambda_{s}}\right)\left(\frac{k_{s}\lambda_{s}}{k_{s}\lambda_{s}}\right)^{s^{\prime}-1}\right]k_{s}\lambda_{s}C_{n+1}$
$\displaystyle=k_{s}\lambda_{s}C_{0}-\left(1\right)k_{s}\lambda_{s}C_{n+1}$

The denominator expands to

$\displaystyle
1+\sum_{r=1}^{n}\prod_{i=1}^{r}\left(\frac{k_{i}^{\prime}\lambda_{i}^{\prime}}{k_{i}\lambda_{i}}\right)$
$\displaystyle=1+\left(s-1\right)\left[\left(\frac{k_{s}\lambda_{s}}{k_{s}\lambda_{s}}\right)^{s-1}\right]_{s-1}+\left[\left(\frac{k_{s}\lambda_{s}}{k_{s}\lambda_{s}}\right)^{s-1}\left(\frac{k_{s}\lambda_{s}}{k_{sm\lambda_{sm}}}\right)\right]_{s}$
$\displaystyle\hskip
20.075pt+\left[\left(\frac{k_{s}\lambda_{s}}{k_{s}\lambda_{s}}\right)^{s}\left(\frac{k_{s}\lambda_{s}}{k_{sm}\lambda_{sm}}\right)\left(\frac{k_{ms}\lambda_{ms}}{k_{M,1}\lambda_{M,1}}\right)\right]_{s+1}$
$\displaystyle\hskip
20.075pt+\left[\left(\frac{k_{s}\lambda_{s}}{k_{s}\lambda_{s}}\right)^{s-1}\left(\frac{k_{s}\lambda_{s}}{k_{sm}\lambda_{sm}}\right)\left(\frac{k_{ms}\lambda_{ms}}{k_{M,1}\lambda_{M,1}}\right)\left(\frac{k_{M,1}\lambda_{M,1}}{k_{M,2}\lambda_{M,2}}\right)\right]_{s+2}$
$\displaystyle\hskip
20.075pt+...+\left[\left(\frac{k_{s}\lambda_{s}}{k_{s}\lambda_{s}}\right)^{s-1}\left(\frac{k_{s}\lambda_{s}}{k_{sm}\lambda_{sm}}\right)\left(\frac{k_{ms}\lambda_{ms}}{k_{M,1}\lambda_{M,1}}\right)\left(\frac{k_{M,1}\lambda_{M,1}}{k_{M,2}\lambda_{M,2}}\right)...\right.$
$\displaystyle\hskip
65.24374pt\left.\left(\frac{k_{M,m-1}\lambda_{M,m-1}}{k_{M,m}\lambda_{M,m}}\right)\left(\frac{k_{M,m}\lambda_{M,m}}{k_{ms}\lambda_{ms}}\right)\right]_{s+m+1}$
$\displaystyle\hskip
20.075pt+\left[\left(\frac{k_{s}\lambda_{s}}{k_{s}\lambda_{s}}\right)^{s-1}\left(\frac{k_{s}\lambda_{s}}{k_{sm}\lambda_{sm}}\right)\left(\frac{k_{ms}\lambda_{ms}}{k_{M,1}\lambda_{M,1}}\right)\left(\frac{k_{M,1}\lambda_{M,1}}{k_{M,2}\lambda_{M,2}}\right)...\right.$
$\displaystyle\hskip
40.15pt\left.\left(\frac{k_{M,m-1}\lambda_{M,m-1}}{k_{M,m}\lambda_{M,m}}\right)\left(\frac{k_{M,m}\lambda_{M,m}}{k_{ms}\lambda_{ms}}\right)\left(\frac{k_{sm}\lambda_{sm}}{k_{s}\lambda_{s}}\right)\right]_{s+m+2}$
$\displaystyle\hskip
20.075pt+\left(s^{\prime}-1\right)\left[\left(\frac{k_{s}\lambda_{s}}{k_{s}\lambda_{s}}\right)^{s-1}\left(\frac{k_{s}\lambda_{s}}{k_{sm}\lambda_{sm}}\right)\left(\frac{k_{ms}\lambda_{ms}}{k_{M,1}\lambda_{M,1}}\right)\left(\frac{k_{M,1}\lambda_{M,1}}{k_{M,2}\lambda_{M,2}}\right)...\right.$
$\displaystyle\hskip
80.29999pt\left.\left(\frac{k_{M,m-1}\lambda_{M,m-1}}{k_{M,m}\lambda_{M,m}}\right)\left(\frac{k_{M,m}\lambda_{M,m}}{k_{ms}\lambda_{ms}}\right)\left(\frac{k_{sm}\lambda_{sm}}{k_{s}\lambda_{s}}\right)\left(\frac{k_{s}\lambda_{s}}{k_{s}\lambda_{s}}\right)^{s^{\prime}-1}\right]_{s+m+s^{\prime}+1}$
$\displaystyle=1+\left(s-1\right)+\left(\frac{k_{s}\lambda_{s}}{k_{sm}\lambda_{sm}}\right)+\left(\frac{k_{s}\lambda_{s}k_{ms}\lambda_{ms}}{k_{sm}\lambda_{sm}k_{M,1}\lambda_{M,1}}\right)+\left(\frac{k_{s}\lambda_{s}k_{ms}\lambda_{ms}}{k_{sm}\lambda_{sm}k_{M,2}\lambda_{M,2}}\right)+...$
$\displaystyle\hskip
20.075pt+\left(\frac{k_{s}\lambda_{s}k_{ms}\lambda_{ms}}{k_{sm}\lambda_{sm}k_{M,m}\lambda_{M,m}}\right)+\left(\frac{k_{s}\lambda_{s}}{k_{sm}\lambda_{sm}}\right)+1+\left(s^{\prime}-1\right)$
$\displaystyle=s+s^{\prime}+2\left(\frac{k_{s}\lambda_{s}}{k_{sm}\lambda_{sm}}\right)+\left(\frac{k_{s}\lambda_{s}k_{ms}\lambda_{ms}}{k_{sm}\lambda_{sm}}\right)\sum_{j=1}^{m}\left(\frac{1}{k_{M,j}\lambda_{M,j}}\right)$

where the subscripts on bracketed terms track the sum over all jumps. Eq. S8
simplifies to

$Q=\frac{k_{s}\lambda_{s}\left(C_{0}-C_{n+1}\right)}{\displaystyle
s+s^{\prime}+2\left(\frac{k_{s}\lambda_{s}}{k_{sm}\lambda_{sm}}\right)+\left(\frac{k_{s}\lambda_{s}k_{ms}\lambda_{ms}}{k_{sm}\lambda_{sm}}\right)\sum_{j=1}^{m}\left(\frac{1}{k_{M,j}\lambda_{M,j}}\right)}$
(S9)

Therefore, the permeability becomes

$P=\frac{1}{\displaystyle\left(\frac{s}{k_{s}\lambda_{s}}\right)+\left(\frac{s^{\prime}}{k_{s}\lambda_{s}}\right)+\left(\frac{2}{k_{sm}\lambda_{sm}}\right)+\left(\frac{k_{ms}\lambda_{ms}}{k_{sm}\lambda_{sm}}\right)\sum_{j=1}^{m}\left(\frac{1}{k_{M,j}\lambda_{M,j}}\right)}$
(S10)

In membrane transport, the jump rates through solution ($k_{s}$) are
significantly larger than those through the membrane interface and the bulk
membrane. As a result, the permeability can be expressed only in terms of the
interfacial and membrane rate constants as shown in Eq. S11.

$\frac{1}{P}=\left(\frac{2}{k_{sm}\lambda_{sm}}\right)+\left(\frac{k_{ms}\lambda_{ms}}{k_{sm}\lambda_{sm}}\right)\sum_{j=1}^{m}\left(\frac{1}{k_{M,j}\lambda_{M,j}}\right)$
(S11)

The first term in Eq. S11 is associated with diffusion through the solution-
membrane interface, and the second term is associated with diffusion through
the membrane. For most polymeric membranes, the rate-determining step is
diffusion through the membrane, so Eq. S11 can be approximated with only the
second term. The resulting expression for permeability in terms of the rate
constants for transport is shown in Eq. S12.

$P=\frac{k_{sm}\lambda_{sm}}{\displaystyle
k_{ms}\lambda_{ms}\sum_{j=1}^{m}\left(\frac{1}{k_{M,j}\lambda_{M,j}}\right)}$
(S12)

Under transition state theory, the individual rate constants $k_{i}$ can be
related to free energy barriers $\Delta G^{\ddagger}_{i}$ by

$k_{i}=\kappa_{i}\frac{k_{B}T}{h}\exp{\left(\frac{-\Delta
G^{\ddagger}_{i}}{k_{B}T}\right)}$ (S13)

Relating Eq. S12 to the associated free energy barriers with Eq. S13 yields
Eq. S14 for permeability across a series of unequal membrane barriers in terms
of the free energy barriers at transition $\Delta G_{j}^{\ddagger}$.

$P=\frac{\displaystyle\left(\frac{\lambda_{sm}}{\lambda_{ms}}\right)\left(\frac{k_{B}T}{h}\right)\exp{\left(\frac{-\left(\Delta
G_{sm}^{\ddagger}-\Delta
G_{ms}^{\ddagger}\right)}{RT}\right)}}{\displaystyle\sum_{j=1}^{m}\left(\frac{1}{\lambda_{M,j}\exp{\left(\frac{-\Delta
G_{M,j}^{\ddagger}}{RT}\right)}}\right)}$ (S14)

Derivation of the effective free energy barrier
Zwolinski and coworkers express the effective free energy barrier to
permeability as

$\displaystyle P$ $\displaystyle=$
$\displaystyle\left(\frac{\lambda^{2}}{\delta}\right)\left(\frac{k_{B}T}{h}\right)\exp{\left(\frac{-\Delta
G_{eff}^{\ddagger}}{RT}\right)}$ (S15) $\displaystyle=$
$\displaystyle\left(\frac{\lambda^{2}}{\delta}\right)\left(\frac{k_{B}T}{h}\right)\exp{\left(\frac{\Delta
S_{eff}^{\ddagger}}{R}\right)}\exp{\left(\frac{-\Delta
H_{eff}^{\ddagger}}{RT}\right)}$

We incorporate parallel molecular pathways and distributions of membrane jumps
and barriers into the transition-state theory model for membrane permeability,
as shown in Eq. S16.

$P=\sum_{i=1}^{n}\left[\frac{\displaystyle\left(\frac{A_{i}}{A_{0}}\right)\left(\frac{\lambda_{sm}}{\lambda_{ms}}\right)\left(\frac{k_{B}T}{h}\right)\exp{\left(\frac{-\left(\Delta
G_{sm}^{\ddagger}-\Delta
G_{ms}^{\ddagger}\right)}{RT}\right)}}{\displaystyle\sum_{j=1}^{m_{i}}\left(\frac{1}{\lambda_{M,i,j}\exp{\left(\frac{-\Delta
G_{M,i,j}^{\ddagger}}{RT}\right)}}\right)}\right]$ (S16)

We equate these expressions for permeability and solve for the effective free
energy barrier in terms of the distributions of membrane barriers and parallel
paths.

$\Delta
G_{eff}^{\ddagger}=-RT\ln{\left[\sum_{i=1}^{n}\frac{\displaystyle\left(\frac{A_{i}}{A_{0}}\right)\left(\frac{\delta}{\lambda_{avg}^{2}}\right)\left(\frac{\lambda_{sm}}{\lambda_{ms}}\right)}{\displaystyle\sum_{j=1}^{m_{i}}\left(\frac{1}{\lambda_{M,i,j}}\right)\exp{\left(\frac{\Delta
G_{M,i,j}^{\ddagger}}{RT}\right)}}\right]}+\left(\Delta
G_{sm}^{\ddagger}-\Delta G_{ms}^{\ddagger}\right)$ (S17)

Estimating the number of paths per unit area
We estimate the number of paths per unit area for the polyamide membrane to be
an order of magnitude fewer than what is expected for single-walled carbon
nanotubes (SWCNT). SWCNT with diameter 1.7 nm have been reported to pack with
density $1.9\text{\times}{10}^{12}$ paths per (?). If all the area is occupied
by circular nanotubes with diameter 1.7 nm and negligible thickness, the
theoretical packing density is $4.4\text{\times}{10}^{13}$ paths per . We use
this ratio of actual packing density to theoretical packing density to
approximate the actual packing density of SWCNT with diameter 0.5 nm, the
reported average pore size for polyamide membranes (?, ?).

$\left(\frac{\text{actual, }d=1.7\unit{}}{\text{theoretical,
}d=1.7\unit{}}\right)\times\left(\text{theoretical,
}d=0.5\unit{}\right)=\left(\text{actual, }d=0.5\unit{}\right)$ (S18)

We estimate the actual packing density of SWCNT with diameter 0.5 nm to be
$2.2\text{\times}{10}^{13}$ paths per or 0.22 paths per . Therefore, we
estimate the actual number of paths per unit area for polyamide membranes to
be 0.022 paths per . The results we present consider a total unit area of 0.1
, or $1.0\text{\times}{10}^{7}$ $\text{\AA}^{2}$, and a single path area of
$\pi(5\text{\AA})^{2}=19.635~{}\text{\AA}^{2}$. These areas correspond to 2196
paths per 0.1 .

Effect of jump distributions on the effective free energy barrier
Given a fixed membrane thickness, the distribution of number of jumps and the
length of jumps are directly related. Thus, we can examine the effects of only
the distribution in the number of jumps for a given membrane thickness. We
choose a physically realistic membrane thickness and hold all membrane free
energy barriers equal. We draw the number of jumps from a (truncated) normal
distribution because we can change the variance while maintaining a physically
relevant mean. For this analysis, we model 5000 paths through a membrane of
thickness 400 Å. We set the mean number of jumps to be 100, and we adjust the
jump length to ensure the membrane thickness remains constant. We vary the
standard deviation in the number of jumps between 5 and 200. Because this can
result in a negative number of jumps, we redraw each negative draw from a
normal distribution until no paths have a non-positive number of jumps. This
results in a nearly normal distribution for large variances, but a truncated
distribution at $N=1$ and below for larger variances.

We find that the effective free energy barrier decreases with increasing
variance in the number of jumps, but the change is significantly less than the
effects from distributions of barrier heights in all physical scenarios. Fig.
S2 shows the relationship between the effective free energy barrier and the
standard deviation in number of jumps.

The effective barrier decreases negligibly when the variance is small. When
the variance becomes larger, a small number of paths have very few jumps,
which results in moderately decreased effective barrier, up to 1.5 kcal/mol.
The barrier decreases negligibly again for larger standard deviations of the
truncated distribution, when the number of paths with $N=1$ barriers
predominates. In contrast, modest variance in the barrier height distribution,
as shown by the normally distributed barrier heights in Fig. 3B in the main
text, changes the effective barrier by 5.3 kcal/mol.

At large standard deviation, the effective barrier becomes dominated by paths
with only a few jumps. Fig. S3A confirms this trend by showing the percentage
of the total permeability through each path for the highest variance
distribution (standard deviation 200). Conversely, when all paths have nearly
the same number of jumps, the permeability is evenly distributed across the
paths, as shown in Fig. S3B. The standard deviation in the number of jumps for
Fig. S3B is 5.

Physically, larger jumps along the transport coordinate with the approximately
the same membrane thickness reduce the number of barriers the molecules must
cross. Jump lengths affect the single path permeability as a sum of reciprocal
jump lengths, so small jumps contribute more than large jumps. The
distributions of jump lengths introduce some smaller jumps that drive the
permeability lower and the effective free energy barrier higher. Individual
jump lengths are likely to be correlated with their associated free energy
barrier. However, the exponential contribution of the free energy barriers
will dominate the contribution from the jump lengths. For membranes with
heterogeneity in their free energy barrier distributions, the variability of
the smallest maximum barrier contributes significantly more than variability
in the number and length of jumps through the membrane, and we thus focus
primarily on the distribution of barrier heights in this study.

Accounting for concentration polarization in the membrane
Previously reported barriers for NF and RO membranes range from 0 to $\sim$17
kcal/mol with most values lie between $\sim$4 and $\sim$8 kcal/mol (?).
However, most of the reported values in the literature are likely an
overestimation of the real barriers, as these values were measured without
accounting for the increasing concentration polarization of the transported
solutes with temperature. This phenomenon leads to higher concentration
gradient over the membrane (and therefore higher driving force) with
temperature, resulting in an increased solute flux that is not related to
intrinsic activation (i.e., a permeability increase with temperature). Our
measurements rigorously accounted for concentration polarization and therefore
reflect more reliably the intrinsic barriers.

In brief, to account for concentration polarization during the measurement of
the permeability at the different temperatures, evaluation of the salt
concentration on the membrane surface, $C_{m}$, was performed at each
temperature by retrieving the mass transfer coefficient in the boundary layer,
$k$, using the following correlation for the Sherwood number based on laminar
(Eq. S19) and turbulent (Eq. S20) flows in a rectangular channel without a
spacer (?):

$\text{Sh}=1.85\left(\text{Re}\text{Sc}\frac{d_{h}}{L}\right)^{0.33}$ (S19)
$\text{Sh}=0.04\text{Re}^{0.75}\text{Sc}^{0.33}$ (S20)

where Sh is the Sherwood number
$\left(\text{Sh}=\left(\frac{kd_{h}}{D}\right)\right)$, Re is the Reynolds
number ($\sim$3295 in our system), Sc is the Schmidt number
$\left(\text{Sc}=\frac{\nu}{D}\right)$, where $D$ is the diffusion coefficient
and $\nu$ is the kinematic viscosity), $d_{h}$ is the hydraulic radius
($1.55\text{\times}{10}^{-3}$ m in our system), and $L$ is the cell length
(0.06 m in our system). The height and width of the flow channel in our system
were 0.8 mm and 25 mm, respectively. Because Re was in the borderline of
laminar and turbulent flow in our system, we examined both the laminar and
turbulent correlations. The diffusion coefficients of the different ions at
the tested temperatures were calculated with the Stokes-Einstein equation
using Stokes radii (Table S2). For each salt, the diffusion coefficient of the
slower ion was used for the calculations of the Sherwood number. The evaluated
$k$ values were then used in the film theory equation (Eq. S21) to measure
$C_{m}$.

$\frac{C_{m}-C_{p}}{C_{f}-C_{p}}=\exp{\left(\frac{J_{w}}{k}\right)}$ (S21)

$C_{p}$ and $C_{f}$ are the salt concentrations in the permeate and the feed
solution, respectively, $J_{w}$ is the permeate flux (L m-2 h-1), and $k$ is
the mass transfer coefficient (m s-1).

Comparing the Arrhenius plots and transition-state theory plots
Energy barriers to permeability are often measured as Arrhenius barriers, and
the effective parameters are determined as the slope and intercept of $\ln(P)$
vs $1/T$. However, this form neglects the temperature dependence of the
prefactor that is explicitly stated in transition-state theory. The difficulty
with the transition-state theory approach is the need for additional
parameters, namely average jump length $\lambda$ and membrane thickness
$\delta$ in Eq. S22, which are challenging to measure.

$P=\left(\frac{\lambda^{2}}{\delta}\right)\left(\frac{k_{B}T}{h}\right)\exp{\left(\frac{-\Delta
G_{eff}^{\ddagger}}{RT}\right)}$ (S22)

We perform both linear fits, and we determine the goodness of fit is not
significantly different between the models. The $R^{2}$ is 0.642 for the
Arrhenius treatment and 0.569 for the transition-state theory treatment. In
Fig. S5, we show the linear fits for $\ln(P)$ and $\ln(P/T)$ for NaCl in the
NF270 membrane. Table S1 shows the effective enthalpic barriers for each
linearization. The errors shown are the standard errors in the slope parameter
for the linear fit propagated to the effective enthalpic barrier. All the
enthalpic barriers from transition state theory are within the mutual margin
of error of the effective activation energies from Arrhenius theory.

More importantly, the trends are completely preserved between the two
approaches. Effective enthalpic barriers are 0.5 to 0.6 lower in the
transition state theory versus Arrhenius theory in all cases, with the
variation within 0.1, much smaller than the error.

(a) Normally distributed barriers

(b) Exponentially distributed barriers

Figure S1: Convergence testing to determine necessary number of paths. The
effective free energy barrier converges for both normally and exponentially
distributed barriers for the distribution variances used in this paper
converges within summation over 2000 pathways. Barrier distributions with
higher variance will take more pathways to converge. Black points are single
realizations of calculated effective free energy barriers, and red points are
effective free energy barriers averaged over all realizations. We calculate
100 realizations for each number of paths.

Figure S2: Effective barrier decreases with increasing variance in the number
of jumps. We show the overall effective free energy barrier as a function of
the standard deviation for normally distributed numbers of jumps with mean
100. The jump length is adjusted to maintain a constant membrane thickness of
40 Å. For each standard deviation, we calculate the overall effective free
energy barrier over 5000 paths.

Figure S3: Paths with only a few jumps contribute most to the permeability. We
calculate the percentage of the total permeability through paths with
different numbers of jumps. The jump length is adjusted to maintain a constant
membrane thickness of 40 Å, and we calculate the overall effective free energy
barrier over 5000 paths. (A) The standard deviation in the normally
distributed number of jumps is 5, and the mean is 100. The permeability is
evenly spread across paths, since all paths have similar number of jumps. (B)
The standard deviation in the normally distributed number of jumps is 200, and
the mean is 100. The permeability is dominated by paths with only a few jumps.

Figure S4: Effective enthalpy for NaCl in water. Linearized transition-state
theory plot for the conductivity of NaCl in water, which corresponds to free
diffusion of the ions. The least squares fit is shown as a line.

Figure S5: Comparison of transition-state theory and Arrhenius plots
Estimation of the effective enthalpic barriers from the transition-state
theory model and the Arrhenius model are indistinguishable within error.
Scatter points are the experimental data linearized to fit the corresponding
model. The least squares fits are shown as lines. A 95% confidence interval
(shaded region) is provided with each least squares fit, determined by a
nonparametric bootstrap over 1000 bootstraps.

System | Linearization |  $E_{a}$ or $\Delta H_{eff}^{\ddagger}$ (kcal/mol)
---|---|---
NaCl (NF270) | $\ln(P)$ | $4.2\pm 0.9$
$\ln(P/T)$ | $3.6\pm 0.9$
NaF (NF270) | $\ln(P)$ | $4.5\pm 1.1$
$\ln(P/T)$ | $3.9\pm 1.1$
NaCl (RO) | $\ln(P)$ | $5.0\pm 1.3$
$\ln(P/T)$ | $4.5\pm 1.3$
Table S1: Effective barriers from Arrhenius and transition-state theory
models. The activation parameter $E_{a}$ and the effective enthalpic barrier
$\Delta H_{eff}^{\ddagger}$ are within their mutual margin of error for all
systems.

Species | Stokes radius (nm) (?)
---|---
Sodium (Na+) | 0.184
Fluoride (F-) | 0.166
Chloride (Cl-) | 0.121
Table S2: Stokes radii for the ions tested in the experimental filtration
measurements. All data is from reference (?)