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Journal of Membrane Science 280 (2006) 517–529

A molecular dynamics simulation study of surface effects ongas permeation in free-standing polyimide membranes

Sylvie Neyertz ∗, Anthony Douanne, David Brown

Laboratoire Mat´ eriaux Organiques a Propri´ et´ es Sp´ ecifiques (LMOPS), UMR CNRS 5041, Universit´ e de Savoie,

Bˆ at IUT, 73376 Le Bourget du Lac Cedex, France

Received 16 December 2005; received in revised form 7 February 2006; accepted 9 February 2006

Available online 6 March 2006

Abstract

A 141100-atom model of a glassy ODPA–ODA polyimide free-standing membrane, corresponding to a thickness of two average radii of

gyration for the 40-mers chains, has been studied using molecular dynamics (MD) simulations. Due to the large-scale of the fully atomistic model,

a parallelized particle-mesh technique using an iterative solution of the Poisson equation had to be implemented for the efficient evaluation of the

electrostatic interactions. With flattened-chain configurations, the density was found to adjust itself naturally in the middle of the membrane to

∼95% of the ODPA–ODA experimental value. At the free-standing surfaces, the density profile became sigmo ıdal, indicating surface roughness.

For comparison, two isotropic bulk models, one at the “normal” density as obtained for ODPA–ODA under ambient conditions and the other one at

95% of the normal-density, were built. Small gas probes were inserted into all three models in order to investigate whether the interfacial structure

of the glassy free-standing membrane can influence penetrant transport. Gas diffusion in the low-density part of the interface was found to be very

fast. The limiting value for the gas diffusion coefficient Dmembrane is only attained when the probes enter more dense regions in the membrane.

Indeed, Dmembrane compares well with Dbulk obtained for the 95%-density bulk system, i.e. about twice that in the normal-density bulk. Solubility is

larger in the membrane than in both bulk models, thus suggesting an effect of chain flattening in addition to the density. Adsorption is particularly

high at the free-standing interfaces.

© 2006 Elsevier B.V. All rights reserved.

Keywords: Molecular dynamics simulations; Large-scale fully atomistic polyimide; Glassy; Free-standing surfaces; Gas permeation

1. Introduction

Polyimides are high-performance macromolecules which are

often used in thin dense membrane applications [1–3], such as

gas-separation of oxygen or nitrogen from air or the purifica-

tion of natural gas [1]. However, their permeation properties are

strongly affected by the considerable number of parameters that

can influence diffusion, selectivity or other physical properties

over the entire processing stage [4–8]. Polyimide membranes

are often prepared by the so-called “solvent casting method”, in

which they are cast on a substrate from solutions prepared by

dissolution of the polyimide into a solvent such as m-cresol or

N -methylpyrrolidone [9]. Different casting surfaces will poten-

tially lead to different properties. It is known for example that

membranes which are cast on glass result in stiffer, stronger and

∗ Corresponding author. Tel.: +33 4 79 75 86 97; fax: +33 4 79 75 86 14.

E-mail address: [email protected](S. Neyertz).

more oriented films than when they are cast on liquids, such as

mercury [8].

We have recently carried out a comparative study of gas per-

meation in two fully atomistic models of a glassy polyimide

using molecular dynamics (MD) simulations [10] in order to

test for possible “skin-effects” [3,11–13] related to the presence

of interfaces with specific features. The 40-mer ODPA–ODA

homopolyimide (Fig. 1) was chosen as a test case since exper-

imental data for both gas permeability and diffusion in this

specific polyimide are available [14,15]. In addition,the prepara-

tion procedure for atomistic models of ODPA–ODA in the pure

bulk had been earlier optimized and validated with respect to

other experimental properties [16–18]. The first system studied

[10] was an isotropic 27-chain 56025-atom bulk model of the

ODPA–ODA amorphous phase [10,18], which was periodic in

three dimensions. The second system was a 68-chain 141100-

atom model of an actual membrane, which was created using

an original procedure based on the experimental solvent cast-

ing process [10]. The membrane had a width of 13 nm, i.e. two

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518 S. Neyertz et al. / Journal of Membrane Science 280 (2006) 517–529

Fig. 1. The (ODPA–ODA) polyimide chain.

average radii of gyration for the ODPA–ODA molecules. It was

periodic in the x - and y-directions, and confined between two

solid walls in the z-direction in order to be consistent with the

presence of the glass surface. Despite changes in configurations

and high-density interfaces in the vicinity of the walls, the per-

meability of small gas probes was found to be similar in both

bulk and membrane systems.

In the former case [10], calculations were speeded-up by a

factor of 70 using a soft-repulsive Weeks–Chandler–Andersen

(WCA) potential for the non-bonded interactions. While this

allowed for diffusive equilibrium and good statistics to be

attained in both bulk and membrane models, the polymer mem-

brane had to remain constrained between both walls in order tocompensate for the absence of attractive forces, and thus only

the presence of a high-density interface was tested [10]. In the

present work, we include more realistic Lennard–Jones (LJ) and

Coulombicinteractions for non-bondedinteractions. Thisallows

us to remove both walls around the membrane, so that the den-

sity at the interface and the interior will adjust itself naturally.

Such a model is likely to be more representative of reality where

the film is peeled off the glass plate and local rearrangement,

particularly at the surface, can occur.

Several examples of atomistic simulations of free-standing

polymer membranes can be found in the literature. Here, we

restrict ourselves to polymer/vacuum and polymer/small pen-etrants interfaces, and do not consider, for example, poly-

mer/polymer interfaces where the key features are adhesion and

miscibility [19–23]. Simulations of polymer/vacuum in the melt

[24–26] or in the glassy state [27,28] have shown that, although

densities decrease monotonically across the polymer–vacuum

interface, the chain structures are qualitatively similar to the flat-

tened configurations which develop near a solid wall [29], albeit

of more moderate amplitude. Usually, the interfacial width is

increased in a free-standing film and flexible polymer chain ends

have a tendency to segregate [25,30], unless they are sufficiently

attractive [31]. Mobility is also affected and the apparent glass

transition, which is dependenton thethickness of thefilm, is usu-

ally lower than in the bulk [32–34]. Crystallization can even beinitiated in the surface region and propagate into the interior of

the thin film [35,36]. MD simulations have been used to charac-

terizethe surface interactionsof thin films of randomcopolymers

based on styrene, butadiene and acrylonitrile with solvents [37],

as well as self-associating polymers [38]. However, in actually

very few cases up to now, and to our knowledge only in rubbery

matrices [39,40], have fully atomistic free-standing membrane

models been used to study gas transport.

In the present paper, we make a comparative study of gas

permeation in bulk models of the glassy ODPA–ODA polyimide

with that in the corresponding free-standing membrane model.

The latter originates from the earlier 141100-atom membrane

model of the same polymer confined between two walls [10]

and the relevant computational details are given in Section 2.

Removing the walls results in the relaxation of the polyimide

matrix and the high-density interfaces; these effects are anal-

ysed in Section 3.1. In Section 3.2, diffusion and solubility

results for highlymobile gasprobes in the bulk andfree-standing

membrane models are discussed. It should be noted that the

free-standing polyimide model is over 20–50 times larger than

the typical sizes used for these glassy systems [17,41], and that

consequently these calculations with full excluded-volume and

electrostatic interactions have proven extremely expensive in

terms of computational time (

∼26,000 h monoprocessor on an

IBM Power 4 for 1.1 ns of simulation). Within this context,relatively slow penetrants such as oxygen [15] could not be con-

sidered, but the use of small gas probes allowed for significant

diffusion to be attained through the highly rigid membrane on

the limited timescale accessible to the fully atomistic MD sim-

ulations.

2. Computational details

2.1. Force-field

The chemical structure of an (ODPA–ODA) chain is shown

in Fig. 1 [9,17,18,42]. Each chain has a total of 40 monomersand 2075 atoms. The gas probe was modeled as a single atom.

All calculations were performed using the MD code of the gmq

package [43] in its parallel form, ddgmq [44], on the French

IDRIS (Orsay) and CINES (Montpellier) supercomputing cen-

tres as well as on local resources at the University of Savoie.

The force-field and the parameters for the polyimide are

the same as described before [10,17,18], with “bonded” angle-

bending, torsional and out-of-plane interactions arising from

near-neighbour connections in the structure. The so-called

“non-bonded” excluded-volume and electrostatic potentials are

applied to all atom pairs separatedby more than twobondson the

same molecule or belonging to different molecules. The model

gas probe is also identical to that used previously [10]. Its smallsize, σ = 1.88 A, allows it to be highly mobile in the membrane.

Standard combination rules are used for all cross terms [18],

except for gas–gas interaction parameters which are set to zero

to represent an ideal gas.

As explained in Section 1, “non-bonded” interactions in the

gas permeation study of the confined membrane were described

solely by the purely repulsive WCA potential [10], which is a

reasonable approximation to the more realistic excluded-volume

Lennard–Jones and electrostatic forms when weakly interacting

gases and rigid matrices are being investigated [18]. The low

computational cost of the WCA form allowed a steady state

uptake of gas to be achieved in these MD simulations [10], but



S. Neyertz et al. / Journal of Membrane Science 280 (2006) 517–529 519

thepolymer membrane hadto be constrained between rigid walls

to maintain its density. Removing the walls and allowing for

the polymer to relax in a realistic way means that we have to

revert to the much slower LJ (Eq. (1)) and electrostatic (Eq.

(2)) combination of non-bonded potentials in order to maintain

cohesion in the membrane:

U LJ(rij ) = 4ε

σ rij

12 −

σ rij

6

(1)

U coul(rij ) =qiqj

4πε0rij (2)

where r ij is the distance between atoms i and j, ε the well-depth

and σ the distance at which the LJ potential is zero, qi and q j

the partial charges on atoms i and j and ε0 is the vacuum per-

mittivity. U LJ(r ij) and U coul(r ij) are the corresponding energies.

Considering the size of the polymer membrane, this amounts

to a single integration time-step of 10−15 s taking about 85 s

monoprocessor CPU time on an IBM Power 4.

The Ewald summation method, which is routinely used toevaluate the electrostatic interactions in reasonably small mod-

els [45,46], was applied to the bulk models under study. On the

other hand, a parallelized particle-mesh technique using an iter-

ative solution of the Poisson equation was implemented for the

membrane since it is much more efficient than the Ewald sum for

large systems and it is also naturally applicable to models which

are periodic in two dimensions [47]. Using this technique, both

the short-range and the long-range contribution of the electro-

static potential are calculated in real space. The former is a sum

of pair interactions, while the latter is handled with a smooth

projection of discrete point charges to a grid [48] f ollowed by a

multigrid-approach to resolving Poisson’s equation in order-Noperations [49].

2.2. The free-standing membrane and the bulk models

The free-standing membrane originates from the procedure

described previously [10]. Theaverageradiusof gyration S 21/2

for an ODPA–ODA chain was shown to be equal to ∼58.4 A

at 700 K by using a well-documented “hybrid Pivot Monte

Carlo-Molecular Dynamics (PMC/MD) single-chain sampling”

approach in the melt [10,16–18,50–55]. Since S 21/2 is usu-

ally quoted as being the minimum distance for the influence of

the interface [28,29,56–61], the size of the model was based

on a length of ∼2S 21/2 for the cell, i.e. a total of 141100atoms in 68 chains. The membrane was then created using an

original approach designed to mimick loosely the experimental

solvent casting process [10]. The polymer chains were gener-

ated with the hybrid PMC/MD single-chain sampling method in

the melt and placed at a density corresponding to a 10% (w/w)

solution. Following the progressive introduction of excluded-

volume (using the WCA soft-repulsive form for the non-bonded

potentials), an impenetrable wall was added on either side of the

polymer and the membrane was compressed in the z-direction

until the density in its middle reached the experimental value

of 1368kgm−3 [14], while the basis vectors in the x - and y-

direction remained constant. Internal stresses were then released

with a constant-volume (NVT) run of 3000 ps at 700 K. At that

stage, the Lennard–Jones and Coulombic non-bonded poten-

tials were introduced at 700 K for 250 ps. Finally, the system

was cooled down to 300 K and allowed to settle for 100 ps, the

drift in coulombic energy being only 0.03% over the last 50 ps.

In all cases, the temperature was maintained through loose-

coupling to a heat bath [62] with a coupling constant equal

to 0.1 ps. The final size of the confined membrane model was

112 A× 113 A× 158 A.

In the free-standing membrane, the part of the wall which was

interacting directly to constrain the polymer, but was invisible to

the gas probe, was removed. Only the middle four layers of the

initial 10-layer tetrahedral diamond lattice arrangement [10], i.e.

a total of 1680 atoms linked through flexible bonds and bends,

were retained in order to prevent gas probes from escaping into

thevacuum.However, the wall atoms were moved away from the

polymer at such a distance that thecut-offof 10 A effectivelypre-

vented them from interacting. The total width of the remaining

wallamountedto

∼14 A. At that stage,the parameters associated

with the lattice diffusion multigrid procedure for the calculationof electrostatic interactions [47] were set to 0.24 A−1 for the β

coefficient and to h x = 0.700 A, h y = 0.707 Aand h z = 0.702 Afor

the mesh spacing in the x -, y- and z-directions, respectively. Par-

ticle charges were interpolated onto the 64 closest grid points

of the mesh using a Gaussian charge spreading function. The

optimized value for the charge-spreading diffusion coefficient

was D = 0.0578 A2 per iteration and the number of steps for the

diffusion scheme was N t =72.

Upon removal of the wall constraint, the polymer was found

torelaxin the z-direction from a total length of ∼130 to∼140 A.

The density of the free-standing membrane relaxed within less

than 100 ps and the final size of the membrane+ wall model wasadjusted to 112 A× 113 A× 180 A. A schematic representation

of its left interface is shown in Fig. 2a. For comparison, the left

interface of the confined membrane [10] is presented in Fig. 2b.

It is clear that in the free-standing membrane, the interface has

expanded significantly and has become a lot rougher than its

confined counterpart.

The ODPA–ODA bulk simulations are based on the 6225-

atom model also used previously [18], which had a density

of 1377kg m−3, i.e. within 0.7% of the experimental value of

1368kgm−3 [14]. The Ewald summation method was used to

evaluate electrostatic interactions with the same parameters as

before [18], the only difference with our former calculations

being the gas probe interaction parameters. This model will bereferred to afterwards as the “normal-density” bulk.

In order to make comparisons with the lower density of the

free-standing polymer membrane, the “normal-density” bulk

was driven smoothly from a side length of ∼41.4 to ∼42.1 A,

and was allowed to relax for 100 ps in the pure state. This second

model will be referred to as the “95%-density” bulk with a final

value of 1301kg m−3.

2.3. Insertion of gas probes

The number of gas probes np to insert [18,63] into the bulk

models was set to 25. This is sufficient to give good statistics




Fig. 2. Compared close-ups of the free-standing ODPA–ODA membrane surface (a) on the left side and that of the corresponding confined membrane surface (b)

[10]. Both snapshots havebeen taken at theend of their respectiveproductionruns andshow theleft interfacial zone from∼50 A fromthe polyimide center-of-masses.

They have dimensions of ∼30 A along the y-axis and the color code is the following: cyan, C; red, O; blue, N and white, H. These schematic representations are

displayed using the VMD 1.8.2 software [90]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the

article.)

whilst implying reasonable external pressures [18]. The number

of gas probes absorbed into the polymer, at a temperature T , is

related to the external pressure P applied and is directly linked to

the probability of insertion for the gas into the polymer, pip. For

small penetrants, this probability can be obtained from Widom’stest-particle insertion method [18,64,65],

pip =

exp

−U kBT

(3a)

where U is the change in potential energy associated with a

trial insertion of the gas probe into the polymer matrix. pip is

directly related to the excess chemical potential of the gas in the

polymer, µex by:

µex = −RT ln

exp

−U kBT

= −RT ln(pip) (3b)

Considering the volume of the polymer V p, the external pres-sure is thus:

P ≈ npkBT

pipV p(4)

As explained before [18], in a simulation using 3D peri-

odic boundary conditions, the external pressure and the gas

concentration can be (and often are) set independently. In our

calculations, we aim to have a consistency between these vari-

ables by using Eq. (4).

In a membrane model, the external pressure and the gas

concentration are linked as in reality. One thus has to also con-

sider the number of gas probes in the gas phase ng as well as

the volume of the gas phase, V g. Assuming a probability of

insertion of 1 into the gas phase, i.e. ideality, it can be shown

that:

P ≈(n

p +n

g)k

BT

V g + pipV p(5)

Widom’s test particle insertion method was used on all the

stored configurations of each production run for the pure poly-

mer systems with 500,000 trials per configuration using the

parameters associated with the specific gas probe of this work.

pip was found to be 0.14 for the “normal-density” bulk model,

which corresponds to an external pressure of ∼110× 105 Pa

(∼110 bars) when 25 gas atoms are included. The 95%-density

bulk was run under constant-volume conditions, so no exter-

nal pressure was necessary to maintain the density. For the

membrane model,

pip

was obtained as a function of z for bin

widths of 1 A and revealed a much more complicated profile(see Fig. 11 later). Each slab of volume 112 A× 113 A× 1 A

was thus multiplied by its respective pip and the total con-

tribution summed. From the reverse use of Eq. (5) and con-

sidering the same P of ∼110× 105 Pa (∼110 bars), this led

to a total of np + ng = 3280 gas probes to be inserted into the

free-standing membrane. Since it has been shown that intro-

ducing penetrants (which do not interact with each other) on

one side or on both sides of a membrane model leads to anal-

ogous permeation curves [40], 1640 probes were randomly

inserted into the free space on one side and the remaining

1640 into the free space on the other side of the polymer

membrane.




It is important to note that the traditional experimental set-

up uses a pressure gradient in order for the permeation rate to

be measured [66]. However, this is not necessary in the case

of an atomistic simulation, as the gas atoms are continually

entering, leaving and diffusing through the membrane even

in the absence of an actual pressure gradient; indeed this is

the fundamental spontaneous process that is probed by apply-

ing the relatively small pressure gradients used in real exper-

iments. As we record the positions of all atoms as a function

of time, it is straightforward to see how the density distri-

bution of a particular subset of the gas atoms changes with

time.Consequently, no additional external pressure gradient was

applied.

Following equilibration, the production run for each bulk

model was carried out. The “normal-density” + gas model was

run for 3000 ps under NPT conditions with an applied pres-

sure P of110× 105 Pa (110 bars),whilethe “95%-density” + gas

model was run for 1000 ps under NVT conditions. The much

more expensive membrane + gas model was simulated for a total

period of 1100 ps. In all cases, configurations were stored at 1or 5 ps intervals, and thermodynamic and conformational data

every 1ps for post-analysis.

3. Results and discussion

The free-standing membrane model is obviously very simi-

lar in terms of chain structure and configurations to the confined

membrane it originates from [10]. It is clear that the influence

of the preparation procedure for such a glassy fully atomistic

film is a key factor, but, to date, relatively few techniques

have been described in the literature. One common approachis that of extending the z-axis in order to eliminate interac-

tions of the parent chains with their images in one direction

[22,25,26,37,38]. However, it is very unlikely that rigid chains,

such as polyimides, would relax much at ambient temperatures

in the timescale of a few nanoseconds. This also applies to the

so-called healing method where a snapshot from an equilibrated

bulk film is duplicated in the z-direction and the two films are

merged, thus increasing the thickness of the film [25]. A pro-

cedure related to ours has been used for example by Kikuchi

et al. [67,68] who compressed a bulk polyisoprene chain with a

repulsive wall, albeit starting from a box big enough to contain a

single-chain under the periodic boundary conditions. The future

in the field certainly lies in multiscale approaches, which involvereverse mapping from equilibrated coarse-grained models using

dynamic MC [21,25,69] or MD [70,71] simulations. However,

they require well-parameterized coarse-grained models of the

polymers in question, which are well beyond the scope of this

work.

In the present paper, we investigate if the interfacial struc-

ture of a free-standing membrane can influence gas permeation.

Section 3.1 will point outthe actualdifferencesbetween thefree-

standing membrane and its confined counterpart [10]. The gas

probe solubility and diffusion through the free-standing model

will then be compared to those of the corresponding bulk models

in Section 3.2.

Fig. 3. Mass density distribution for the polyimide as a function of the distance

z from the center-of-mass of the membrane. The curve has been symmetrized

about the membrane COM and normalized with respect to the density of the

reference bulk system, ρexp =1377kgm−3. The slab width is 1 A. The black

circles are fits of the sigmoidal part of ρ( z) to the form of Eq. (6) (see text for

details).

3.1. Characterization of the free-standing polyimide

membrane

3.1.1. Densities and interfacial thickness

The previously prepared [18] 6625-atom “normal-density”

bulk model has a density of ρbulk =1377kgm−3. The confined

membrane was prepared with a similar density at thecenter of the

membrane but had a higher-density (up to 1840 kg m−3) layer of

∼6 A thickness in the vicinity of each planar surface [10]. Upon

removal of thewalls, thepolymer matrixlosesthesehigh-density

surface layers and the middle part of the model relaxes alsotowards a lower average value of 1301 kg m−3. This decrease is

a direct resultof the flatteningof thechains during themembrane

construction phase [10], and results in densities approximately

5% lower than the experimental bulk value of 1368 kg m−3 [14].

For this reason, the “95%-density” bulk model with a density of

1301kgm−3 was also prepared and examined.

Fig. 3 shows the mass density distribution, ρ( z), of the poly-

mer chains in the membrane model, relative to the bulk density

ρbulk , as a function of the distance, z, from the middle of a given

slab of width 1 A to the center-of-mass (COM) of the membrane.

The flattening of the chain configurations leads to more apparent

fluctuations, of

∼30kgm−3, as a function of z than for the con-

fined membrane, but the ρ( z) remain fairly linear up to ∼60 Afrom the COM. No smooth oscillations of the density profile are

evident and, consequently, there is no ordered chain layering in

the core of the membrane.

As seen before [22,25–28,30,31,33,34,38,40,67,69,72], the

density profile at the free-standing surface is sigmoıdal. Fig. 3

exemplifies the difficulties associated with precisely determin-

ing the boundaries of the membrane, and there are several pos-

sible definitions for the interfacial thickness. For example, it has

been defined as being the distance over which mass density falls

from its bulk value to zero [27,28]. Despite its very large mag-

nitude, our membrane does not attain true bulk behaviour at the

centre due to the method of preparation, the large S 2

1/2

of the




ODPA–ODA molecules, and the rigidity of the chains. If we use

instead the average relaxed density of 1301 kg m−3 for the mid-

dle of the membrane as our reference point, then ρ( z) smoothly

decrease from 60.5 to 72 A, thus leading to an interfacial thick-

ness of 12.5 A. As reported elsewhere [27], it is significantly

smaller than the overall chain dimensions. Unlike rubbery poly-

mers [10,25–27,33,34,69], no tendency for certain species, such

as chain ends or special groups, to migrate to the surface was

identified in the case of our glassy model.

Another way to identify the interfacial thickness

[30,31,38,72] is to fit the sigmoıdal part of the mass den-

sity curve ρ( z) to the following hyperbolic equation [73,74]:

ρ(z) = ρmiddle1 − tanh[2(z− h)/w]

2(6)

where ρmiddle is the density of the middle region of the film, h

the location of the interface and w is the interfacial width. ρ( z)

does fit very well to the form of Eq. (6) and the resulting curve is

displayedas filled circles in Fig.3. The optimized parametersare

ρmiddle =1306kgm−3, h = 66.9 Aforthelocationoftheinterfaceandw=4.5 A for the interfacial width. The value of ρmiddle is in

good agreement with the actual average of 1301 kg m−3.A third

way to locate the interface [30] is to determine the point where

dρ( z)/d z is at its minimum value near the free surface. Here, the

minimum is situated at h = 65.5 A. In all cases, the width of the

free-standing interface is indeed larger than that of the confined

interface [10,30].

3.1.2. Structure

To characterize chain alignment, the second order Legen-

dre polynomial functions, P2(cos θ α), were calculated where the

angle θ α for a triplet of atoms {i, j, k } is that between the α-axis and the vector between atoms i and k [75]. This analysis

was carried out for all so-called “pivot angles” which includes

(see Fig. 1) the flexible C–O–C ether bridges of the ODA and

ODPA moieties and the rigid C–N–C ODA–ODPA linkages

[17]. The P2 function is especially interesting since its limit-

ing values, with respect to the normal to the interface, are −1/2

for a perfectly perpendicular alignment, 1 for a perfectly parallel

alignment and 0 for a random alignment of the vectors defining

θ α. Fig. 4 shows that the free-standing membrane retains the

features of the confined system, i.e. a tendency towards a par-

allel alignment of the chains with respect to the surface in the

z-direction originating from the flattening induced in the chains

during the compression step of the model preparation. However,in the free-standing membrane, there is not the same precipitous

fall in P2(cos θ α) towards −1/2 at the interface as found in the

confined model (see Fig. 4 of our previous paper [10]) where

the presence of the wall forces a near perfect parallel alignment

of the chains to it. While a pronounced alignment of the chains

in the vicinity of the interface has been extensively described in

the literature for confined systems [10,27,29,56,60,61], it is also

known to be present but attenuated in a free-standing membrane

[22,25–27,31,33], as shown in Fig. 4. Indeed, the tendency for

parallel orientation has been shown to decrease with increased

roughness [72], which can clearly be related to Fig. 2. No spe-

cific features such as end-beads which orient perpendicular to

Fig. 4. Average P2(cos θ z) for pivot angles situated in slabs of width 2 A. They

are displayed as a function of the z distance from the position of the pivot-angle

middle atom to the membrane COM. Ether bridges and ODPA–ODA links are

COCand CNCangles,respectively. Thecurveshave beensymmetrizedaboutthe

membrane COM. The standard errors are smaller than the size of the symbols.

the surface [25] or become almost random [26] in flexible poly-

mers were identified in the glassy polyimide. Unlike P2(cos θ z),

the P2(cos θ x ) and P2(cos θ y) functions were superimposable,

thus showing that there was no preference with respect to either

x or to y in the membrane model.

As far as polymer chain configurations are concerned, the

average mean-square radius of gyration in the free-standing

membrane, S 2=2750 A2, is practically the same as that found

in the confined membrane, S 2= 2760 A2 [10], indicating that

the relaxation that takes place is not due to changes in the con-

formations of these stiff chains. It should be noted that, in both

cases, the component of the average mean-square radius of gyra-

tion perpendicular to the interface, S 2⊥, represents only ∼1%of the total of S 2, which confirms the flattened nature of the

polyimide chains.

3.1.3. Dynamics

In their investigation on free surfaces of glassy atactic

polypropylene [28], Mansfield and Theodorou had reported

enhancements in the small mean-square displacements (MSD)

of atoms in the vicinity of the interface. While it is out of the

question to follow long-range atomic polyimide diffusion given

the timescale of our MD simulations, there is indeed a consistent

trend in thelocalmobility dependingon thedistanceof theatoms

to the interface. This is displayed in Fig. 5, where the MSDs,(r i(t + t 0)− r i(t 0))2,arerecordedfor t =10,50and100psforall

polymer atoms situated, at t 0, in slabs of width 10 A averaged

over all possible time origins. For comparison, the polyimide

density profile is also given in Fig. 5. While the absolute values

for the polyimide MSDs are very small, there is undoubtedly

an enhancement of atomic mobility close to the interface, which

Mansfield and Theodorou attributed to the lower density prevail-

ing there [28]. However, similar to their and other work [75],

this dynamical feature extends further into the polymer than

the mere perturbation of density, and what has been called the

“dynamical interfacial thickness” is indeed about twice as large

as the thickness obtained from the mass density profile. Such a




Fig. 5. Left axis: polyimide mean-square displacements (MSD) as a function of

the z distance to the membrane COM for all atoms situated in a slab of width

10 A at time t 0. Three time-intervals are shown, i.e. t = 10, 50 and 100 ps. The

MSDshave beenaveraged over all polyimide atoms and all possible timeorigins

of the production run. Standard errors are smaller than the size of the symbols.

Right axis: polyimide mass density ρ( z) as a function of z with a slab width of

1 A. All curves have been symmetrized about the membrane COM.

transmission of enhanced mobility has been attributed to molec-

ular connectivity [28] and the increase of local diffusivity at the

surface has been reported in several other studies of both rub-

bery and glassy polymers [30,32,34]. The concept of a fluidlike

interfacial region has even been used to explain free-standing

film behaviour at the vicinity of the glass transition temperature

[33], but in the present work, the temperature is much too low

to see such phenomena.

3.2. Permeation of small gas probes

3.2.1. Trajectories

Typical trajectories for individual gas probes traversing the

free-standing membrane are displayed in Fig. 6. They are quite

characteristic of small mobile penetrants in glassy matrices

[16,18] with the gas probes oscillating within available free vol-

umes and jumping between different voids when the limited

fluctuations of the glassy matrix allow for the temporary open-

ing of channels [65,76,77]. Although the gas probe dimensions

are smaller, Fig. 6 is qualitatively similar to the motion of helium

Fig. 6. Typical trajectories of gas probes along the z-direction of the free-

standing membrane model.

in polyimides [18], albeit with shorter times of residence in the

holes [10]. As in the confined membrane, the z-component of

the gas probe MSDs for penetrants situated in the middle of

the membrane is found to be lower by ∼20% than the x - and

y-components. This was attributed to the anisotropic configura-

tions of the membrane and to a slight channeling effect along the

x – y plane [10]. The x -, y- and z-components of the correspond-

ing MSDs in the bulk systems are lower for the normal-density

bulk whereas they are slightly higher in the 95%-density bulk

with respect to the membrane. However, both bulk systems do

display an isotropic diffusion since the three components are the

same for a given bulk model.

3.2.2. Diffusion coefficients in the bulk

Gas diffusion coefficients in the bulk models were obtained

by averaging the gas MSDs over all penetrants and over all pos-

sible time origins of the production runs, t 0. The self-diffusion

coefficient D is obtained from Einstein’s equation [18], and is

a good approximation for the true diffusivity [65] as long as

there are no interactions between the gas probes [78,79]. In thepresent case, the crossover from the anomalous to the Fickian

diffusion regime occurs in less than 100 ps, as shown by the fact

that the slope of log (MSD) versus log (t ) goes to one [18]. The

resulting diffusion coefficients are Dbulk =1.0× 10−4 cm2 s−1

for the normal-density bulk and Dbulk =2.2× 10−4 cm2 s−1 for

the 95%-density bulk. These values were further confirmed by

using the probability density distribution of displacement vector

components, which can be fitted by a single Gaussian curve in

the Fickian regime [18,80], as well as by calculating the self

part of the van Hove correlation function, Gs(r , t ) [18,81,82].

As expected, both approaches give the same Dbulk than those

evaluated using the MSD versus time curves.

3.2.3. Diffusion coefficient in the free-standing membrane

Considering the set-up of the membrane model, the gas con-

centration, c, is expected to vary as a function of time and of

the z-position; it will thus be referred to as c( z, t ). However, the

simplest and most common way of solving the one-dimensional

Fick’s law diffusion equation is through the identification of a

regime in whichthe diffusioncoefficient D is assumed to be inde-

pendent of both z and concentration [79,83]. In terms of time t ,

this is indicated by a linear relationship between the weight gain

of the polymer and t 1/2.

The number uptake of gas probes in the membrane over the

whole simulation run is presented as a function of t 1/2 in Fig. 7.As before, such an analysis requires the definition of the width of

the membrane L [10]. Since several possible definitions for the

membrane thickness have been put forward (Section 3.1.1) and

the surface of the membrane is highly heterogeneous (Fig. 2), we

considered L /2 = 60.5 A corresponding to the start of the dense

part of the membrane, L /2 = 66.9 A obtained from Eq. (6) and

L /2 = 65.5 A referring to the point where dρ( z)/d z passes through

a minimum at the interface. In addition, the number uptake was

studied under the hypothetical condition that L /2 = 50 A, which

would be situated well inside the dense part of the model. The

uptake curves differ at short times but the limiting slopes are

very similar, thus showing that the definition of the location of




Fig. 7. The number uptake of gas probes by the membrane as a function of the

square root of time, t . Due to the uncertainty in the definition of the membrane

thickness, L, severalvalues related to the interfacehave beenconsidered (see text

andlegendfor details),in additionto an hypothetical L /2 = 50 A whichis situated

well into the dense part of the model. The actual uptake curves are displayed

with symbols and lines whereas the straight lines are the corresponding linear

fit to t 1/2.

the interface does not fundamentally affect the analyses. When

L /2 is defined as being 60.5, 65.5 or 66.9 A, the curves do not

scale linearly as t 1/2 at short times (t < 225 ps) with the critical

differences being at very short times (t < 20 ps). This is related

to the initial filling-up of the interface by the gas where the

diffusion coefficient varies with z. As the definition of L /2 gets

closer to the dense part of the interface, this non-t 1/2 dependence

of diffusion disappears, as can be seen from the uptake curve

of the hypothetical L /2 = 50 A. This indicates that the diffusion

coefficient settles to a constant value in the dense part of the

membrane. The progressive tendency of the L /2 > 60.5 A uptakecurves to become linear to t 1/2 means that the transition from a

large value for the diffusion coefficient in the gas phase to a low

constant value inside the membrane occurs over a certain range.

It is clear that, despite the choice of a fast-diffusing model gas

probe, the available simulation time did not allow for the steady

state regime to be reached. Using results from a numerical pro-

cedure (see later), this can be estimated at 3.5 ns. However, the

equilibrium and non-equilibriumuptake curves have beenshown

to be superimposable in the case of the confined membrane [10],

thus suggesting that the non-equilibrium experiment is sufficient

for our purpose.

If D is independent of the concentration and of the z-positionwithin the membrane, and diffusion in the gas phase is consider-

ably higher, thediffusion coefficient canbe evaluated from fits of

the time-dependent density distributions to the following solu-

tion of the one-dimensional diffusion equation in a semi-infinite

system [79,83]:

c(z, t ) = c0erfc

z√ 4Dt

(7)

In Eq. (7), z is the coordinate in a reference system where

z = 0 is the left edge of a medium which extends to z = +∞. In

the case of a finite-length membrane of length L, this solution

can only be used up until gas probes start to exit from the far

Fig. 8. Mass density distributions of labeled gas probes as a function of the

z-position in the membrane at time-intervals of 50, 100, 200 and 300 ps. The

actual profiles (displayed with symbols) have been averaged over all possible

time origins by labeling gasprobes which areinside or outside of themembrane,

defined by L /2 = 66.9 A, at each time origin. They are shown as a function of

z = z + ( L /2). The slab width is 1˚

A and the left and right profiles have beensymmetrized. The lines are fits to the form of Eq. (7) with a constant diffusion

coefficient of D =1.9× 10−4 cm2 s−1.

side of the membrane. With the parameters for the gas probe

used in this work and the actual thickness of the membrane,

this restricts the analyses to time-intervals shorter than 500 ps.

In addition, the following boundary conditions are required:

c( z≥ 0, t =0)=0; c( z = 0, t > 0 ) = c0. Gas concentration versus

time curves were obtained by artificially labeling gas probes as

being on the “left”, zi(t 0) <− L /2, or on the “right”, zi(t 0) > L /2 at

any particular time origin t 0 and then subsequently following the

evolution of the distribution of these t 0-labeled atoms. In orderto improve the resolution, all possible time origins were used

and the distributions for “left” and “right” molecules were sub-

sequently symmetrized. They are displayed in Fig. 8, where the

largest of the possible definitions of L /2 of 66.9 A has been used,

along with fits to the form of Eq. (7) f or each given time-interval

under study. While all the fits were obtained with a constant dif-

fusion coefficient of D = Dmembrane =1.9× 10−4 cm2 s−1, they

are only valid from about z =6.5 A onwards, which corresponds

to z =±60.5 A. As noted before, this is actually the region where

the density approaches that of the middle region of the film. The

erfc fits are thus not valid in the smoothly decreasingρ( z) partof

the interfacial region. Fig. 9 characterizes diffusion in this low-

density part of thepolyimide by selectingthosegas probeswhichare initially in the pure gas phase, i.e. situated at | zi(t 0)|> 72 A

at time origin t 0. Their concentration versus time profile is aver-

aged over all time origins separated by 1 ps on a total production

time of 500 ps. Within 1–2 ps, the interfacial structure is clearly

almost entirely filled up with gas, which means that, despite

an intermediate behaviour between diffusion in the pure gas

phase and that through the dense membrane, it is a lot closer

to the former case. Indeed, the polyimide has a very irregular

structure in the immediate vicinity of the interface (see Fig. 2)

with sufficiently large and connected holes for the gas probes to

move easily. This is progressively reduced as the polymer den-

sity increases but still remains a lot faster than transport in both




Fig. 9. The evolution of the average gas density distribution as a function of the

z-position in the low-density interfacial part of the membrane at time-intervals

of 1, 2 and 5 ps. All curves have been averaged over as many time origins as

possible and over both sides of the membrane. The 0 ps reference labels those

gas probes which are situated at z <−72 A or at z > 72 A at a given time origin.

bulk models ( Dbulk =1–2 A2 ps−1). This confirms that D( z) in

the low-density part of the interface is close to that of the ideal

gas and only drops to a limiting lower value in the more dense

interior regions of the membrane.

As before [10], we can see in Fig. 8 that the c0 coefficient

in Eq. (7) slightly decreases with increasing t . This has been

linked to the fact that the stationary source boundary condition,

c( z = 0, t > 0 ) = c0, is notfullyconsistent with theconditions used

here, i.e. the partial pressure of the necessarily limited number

of labeled gas probes in the (fixed volume) reservoir gradually

diminishes as gas diffuses into the membrane. However, this

was found to be of no consequence on the value of D for theconfined membrane [10]. In order to check that it is also the

case here andto avoid the problem of the inappropriate boundary

conditions, a second approach [10] in which the solution of the

one-dimensionaldiffusionequation is obtained numerically [83]

was attempted. The range of the membrane in the z dimension

was divided into equal intervals of δ z = 1 A and the range in t

into intervals of δt = 0.1 ps. With finite difference methods, it is

possible to obtain the concentration at the middle of a slab z and

at time t + δt , c( z, t + δt ) with good precision from the following

numerical integration algorithm:

c(z, t

+δt

) =c

(z, t

)+Dδt

δz2 (c

(z+δz, t

)

+ c(z− δz,t ) − 2c(z, t )) (8)

provided that ( D δt / δ z2)<1/2 [83]. As has been found with the

erfc analysis, the left “edge” of the membrane has to be situated

at zmin =−60.5 A from the COM in order to have a description

of the concentration versus time profiles consistent with a con-

stant D( z) in the membrane. It can also be seen from Fig. 8 that

a good choice for the initial gas concentrations at the edges of

the membrane, c( zmin, 0), is ∼10kgm−3, which corresponds

to the inflexion point for all the concentration curves when the

probes enter the membrane. For the numerical solution, the ini-

tial amount of gas situated on the left side of the membrane,

QL(0), was determined from the integral of the real average gas

distribution symmetrized over z <−60.5 A and z >60.5 A. This

region includes both the true gas reservoir and the low-density

part of the interface. At each δt , the amount of gas either remain-

ing in the original left region, QL(t ), having diffused into the

membrane, QM(t ), or having exited through the opposite right

interface ( zmax = 60.5 A), QR(t ), was evaluated. Since diffusion

in the interfacial part of the membrane is a lot faster than that in

the membrane, the concentrations at the edges of the membrane

could be safely reset at each integration loop using:

c(zmin, t ) = c(zmin, 0)QL(t )

QL(0)(9a)

c(zmax, t ) = c(zmin, 0)QR(t )

QL(0)(9b)

Numerical solutions to the form of Eq. (8) obtained using

the diffusion coefficient evaluated from the erfc fits, i.e.

D = Dmembrane =1.9× 10−4 cm2 s−1, are displayed in Fig. 10 as

a function of the z-position and at different time-intervals, t .For convenience, the concentrations have been converted to z-

dependent mass densities so as to be compared with the actual

average mass density distributions. They confirm that the value

given by Eq. (7) is a rather good approximation (within sta-

tistical uncertainties) and that the non-constant c0 of Fig. 8

does not affect the diffusion coefficient. In order to further

check the parameters used, the numerical integration procedure

was carried out up to 1.1 ns and the predicted uptake of gas

probes was found to be superimposable to the actual uptake

with L /2 = 60.5 A shown in Fig. 7. Furthermore, as mentioned

before, carrying on the numerical integration procedure up to

10 ns with the same parameters revealed that the steady state

will probably not be reached before 3.5 ns under the conditionsof the present model.

A third way to estimate the diffusion coefficient in the mem-

brane would have been to perform a time-lag analysis [66,84] by

following the quantity of penetrants that have crossed the entire

membrane as a function of time. Although this has been done for

the confined membrane [10], the basic assumption behind this

Fig. 10. Same as Fig. 8 except that the lines are now from numerical solutions

using the algorithm given in Eq. (8) with D =1.9× 10−4 cm2 s−1 and the mass

density distribution is displayed as a function of z.




approach is that statistics are accumulated in the steady state

of the permeation process. Consequently, we cannot apply it to

the current set of non-equilibrium data. However, this former

analysis only confirmed the results of both the analytical erfc

and numerical approaches in the confined membrane [10], thus

suggesting that the value of Dmembrane =1.9× 10−4 cm2 s−1 for

the gas probe diffusion coefficient in the free-standing poly-

imide model is reliable. Dmembrane is equal to twice Dbulk for

the normal-density bulk system, which shows that the pres-

ence of a free-standing interface and flattened configurations

in the vicinity of a glassy polyimide film will not act as a

retardant for diffusion. This is primarily governed by lower den-

sity since Dmembrane is of a similar order of magnitude than

Dbulk =2.2× 10−4 cm2 s−1 obtained for the 95%-density bulk

model.

3.2.4. Solubility in the free-standing membrane

For small penetrants, such as the gas probe under study, sol-

ubility in a polyimide membrane is known to follow Henry’s

law in that it is linearly proportional to the pressure, S = S cP,where S c is the solubility coefficient expressed in appropriate

units [18,65]. It is, of course, directly linked to the average

probability of insertion for the gas into the polymer, pip, which

has been given in Eq. (3a). Using Widom’s test insertion tech-

nique [64,65] and 1 million attempted trials per configuration,

pip was found for the gas probe used here to be equal to

0.16± 0.01 in the normal-density bulk + gas and 0.20± 0.01

in the 95%-density bulk + gas systems. Similar analyses on the

pure bulk systems gave 0.14± 0.01 and 0.19± 0.01, respec-

tively. Although the size of the probe is not realistic, the

bulk+gas pip would correspond roughly to 0.14× 10−5 cm3

(STP) cm−3

Pa−1

(0.14 cm

3

(STP) cm−3

bar−1

) for the normal-density model and0.18× 10−5 cm3 (STP) cm−3 Pa−1 (0.18 cm3

(STP) cm−3 bar−1) for the 95%-density model. The related

excess chemical potentials µex of the gas in the polyimide (Eq.

(3b)) are thusof the order of4.6kJ mol−1 for the normal-density

model and 4.0 kJ mol−1 for the 95%-density model. As for the

confined membrane [10], Widom’s insertion technique was also

applied to the free-standing membrane model as a function of

the z-position of the inserted probe with a resolution of 1 A and

a total of 10 million insertions.

The results presented in Fig. 11 display the symmetrized

pip( z) for the membrane along with the pip for both bulk

models. Superimposing the polyimide mass density curve shows

that the pip( z), albeit higher than both bulk values, are fairlyconstant from the membrane COM up to the start of the low-

density interface. The lower density and the flattening of the

chains associated with the compression step in the membrane

preparation procedure lead to a general increase of solubility,

which is twice that of the normal-density polyimide bulk and 1.5

time higher than the 95%-density bulk. The associated excess

chemical potentialµex decreases to∼3.1kJmol−1. However, as

the polyimide density decreases, the solubility then increases in

a sigmoıdal way, reaching even values where pip( z)>1. The

latter refers to a further decrease of the excess chemical poten-

tial and the presence of very favourable interactions between the

penetrants and the polymer chains at the interface. Such inter-

Fig. 11. The average probability of insertion for the gas probes into the free-

standingmembrane as a functionof z, pip( z), obtainedusing Widominsertion’s

technique[64,65] (squares with a maximum standard error of 0.02). pip in both

the normal-density (medium dash) and the 95%-density bulk systems (short

dash) are shown for comparison, as well as the average slab mass density for the

polyimide (line). All curves have been symmetrized about the membrane COM

and the slab width is 1 A.

actions get progressively less easy as the gas progresses into

the denser parts of the matrix, which illustrates the fact that

higher density leads to less space available for weakly inter-

acting penetrants, i.e. lower solubility [85]. Thus in addition

to faster diffusion, the low-density interface is also associated

with higher solubility, which will even increase permeation. It

should be noted that results presented here are for a specific

model gas probe which has a higher solubility into the poly-

imide than, for example, helium [14]. However, this increased

diffusion + solubility combination allowed for significant pene-

tration into the glassy rigid membrane model to occur in about

∼1ns.

4. Conclusions

The purpose of this study was to allow a very large-scale

initially confined glassy polyimide fully atomistic membrane

model [10] to adjust itself to its natural density by removing

the walls which held it in place, and test whether this relax-

ation had an effect on gas permeation properties. To stabilize the

free-standing surfaces, realistic Lennard–Jones and Coulombic

interactions had to be considered in place of the purely repul-

sive non-bonded potential used in the confined membrane [10].

This required the implementation of a parallelized particle-mesh

technique using an iterative real-space solution of the Poissonequation since it is a lot more efficient for large systems than the

commonly used Ewald sum [45,46] and can be applied to peri-

odicity in two dimensions [47]. Despite this improvement, the

timescale available to the MD simulation remained necessarily

limited due to the considerable size of the membrane model.

As expected, the free-standing membrane retained in a more

diffuse manner the main features of its confined counterpart

which had been created using a strategy designed to mimick

the experimental solvent casting process [10]. Chains were still

flattened and stacked parallel to the interfaces but the density

in the middle of the membrane adjusted itself to ∼95% of the

ODPA–ODA experimental value. At the free-standing surfaces,




the density profile was found to fall smoothly to zero, and this

was associated with surface roughness, such as that displayed in

Fig. 2a, and a relative increase in atomic mobility. Within this

context, the definition of interfacial thickness was not obvious

and could be estimated in several ways, but this had no effect on

the permeation properties.

Results for the diffusion and solubility of the gas probe in the

membrane model show clearly that the low-density polyimide

interface is highly permeable with both very high diffusion and

high solubility. Transport is only slowed down as the gas probes

enter the dense parts of the ODPA–ODA glassy membrane. In

that case, permeation is mostly governed by the lower density

prevailing due to the flattened chain configurations. However,

although Dmembrane≈ Dbulk for the 95%-density bulk model, the

anisotropy of the membrane leads to a slight increase in solubil-

ity, thus suggesting once again a faster permeability coefficient

in such structures.

There are still some specific features in this polyimide mem-

brane which need to be addressed at a later stage. Although

it was designed with respect to the experimental solvent cast-ing process, the preparation procedure of such a fully atomistic

large-scale glassy modelremains non-trivial. In addition, solvent

evaporation was not modeled explicitly and this could influence

the chain configurations. Another open question is the size of

the gas probe. As explained in the text, it was kept very small

in an attempt to follow gas motion through the membrane with

sufficient statistics. Of course, it can be argued that larger and

more realistic probes, such as O2 or N2, would maybe modify to

some small extent the structure of the rigid matrix [86–89]. With

our current computational resources, this is presently a difficult

problem to tackle if full electrostatic and excluded-volume are to

be considered in a system of this size. A last point for discussionis that, with hindsight, considering the way in which the density

and the chain alignment (Figs. 3 and 4) rise up to their limiting

values within a relatively narrow range, it is tempting to think

that the central ∼60 A of the model film could have been dis-

carded. This would certainly represent a saving of about a factor

of 2 in terms of CPU time but it is not entirely obvious whether

the results would have been the same. Such a question could

only be answered by a study of a smaller system. In addition,

with a thinner system alone, there would have been lingering

doubts as to whether the true plateau had been reached. It is also

worth pointing out that the actual membrane model length of

∼140 A remains small with respect to the typical experimental

dimensions of a few microns for those thin-films [14].Nevertheless, this work does answer the questions related

to the effect of confinement and to the use of a purely repulsive

excluded-volume potential in our initially confined ODPA–ODA

polyimide membrane [10].

Acknowledgments

The IDRIS (Orsay, France) and CINES (Montpellier, France)

supercomputing centres and the University of Savoie are

acknowledged for computer time. The Rhone-Alpes region is

thanked for funds dedicated to computer hardware and a doc-

toral grant for AD.

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