Gas Permeation in Semicrystalline Polyethylene as Studied by Molecular Simulation and Elastic Model
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Gas Permeation in Semicr ystalline Polyethylene
as Studied by Molecular Simulation and Elastic Model
P. Memari1,2, V. Lachet2∗ and B. Rousseau1
∗ Corresponding author
Résumé — Perméation de gaz dans le polyéthylène semi-cristallin par simulation moléculaire et
modèle élastique — Dans ce travail, nous utilisons la simulation moléculaire pour étudier la
perméation de deux gaz (CH4 et CO2) dans le polyéthylène. Ces simulations sont conduites à des températures
inférieures à la température de fusion du polymère. Bien que dans de telles conditions, le polyéthylène
soit à l’état semi-cristallin, des boîtes de simulation contenant exclusivement du polymère amorphe sont
utilisées. Dans de précédents travaux [Memari P., Lachet V., Rousseau B. (2010) Polymer 51, 4978],
nous avons montré que les effets de la morphologie complexe des matériaux semi-cristallins pouvaient
être pris en compte de manière implicite par une contrainte ad-hoc exercée sur la phase amorphe. Dans
le présent travail, nous montrons que cette approche peut être mise en oeuvre non seulement pour le
calcul de propriétés d’équilibre mais également pour le calcul de propriétés de transport comme les
coefficients de diffusion. De plus, en utilisant le modèle élastique de Michaels et Hausslein [Michaels
A.S., Hausslein R.W. (1965) J. Polymer Sci. : Part C 10, 61], cette contrainte ad-hoc peut être reliée à
la fraction de chaînes qui contribuent au terme d’énergie élastique dans le matériau. Nous constatons
que les propriétés de transport dans les régions amorphes sont fortement influencées par cette fraction
Abstract — Gas Permeation in Semicrystalline Polyethylene as Studied by Molecular Simulation
and Elastic Model — We have employed molecular simulation to study the permeation of two different
gases (CH4 and CO2) in polyethylene. The simulations have been performed at temperatures below
the polymer melting point. Although under such conditions, polyethylene is in a semicrystalline state,
we have used simulation boxes containing only a purely amorphous material. We showed in previous
works [Memari P., Lachet V., Rousseau B. (2010) Polymer 51, 4978] that the effects of the complex
morphology of semicrystalline materials on solubility can be implicitly taken into account by an
ad-hoc constraint exerted on the amorphous phase. Here, it has been shown that our method can be
applied not only for the calculation of equilibrium properties but also for transport properties like
diffusion coefficients. In addition, the ad-hoc constraint has been theoretically related to the fraction
of elastically effective chains in the material by making use of Michaels and Hausslein elastic model
[Michaels A.S., Hausslein R.W. (1965) J. Polymer Sci.: Part C 10, 61]. We observe that the transport
properties in amorphous regions are strongly governed by this fraction of elastically effective chains.
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In many areas of engineering, polymers are used as barriers
to protect materials from gas or liquid contamination. For
instance, polymers are used in food packaging, membrane
separation processes, biomedical devices, offshore oil and
gas production, etc. The relevant property for this process
is the penetrant permeability which quantifies the flux of
matter passing through the polymeric membrane.
Permeability, Pe, is the product of two terms: the solubility, S ,
describing the solute concentration into the polymer phase
and the Fickian diffusion, D, describing the solute mass
transport inside the polymer:
Pe = S × D
The knowledge of both solubility and Fickian diffusion
coefficient (thereafter, called diffusion coefficient) is required for
a complete description and understanding of permeability.
From an experimental point of view, there has been a
large amount of work on the solubility and diffusion of small
penetrants in polymers. Several techniques like
], permeation cell [
] and NMR [
] have been
adopted to measure solubility and diffusion coefficients.
From a theoretical point of view, these two quantities are
usually described and calculated through the concept of free
], the dual-mode transport model [
molecular theories [
Besides experiments and theory, molecular simulation
is an attractive tool to calculate such properties.
Molecular simulation, by describing the system at the
molecular level, may reveal the microscopic mechanisms that
play an important role in the barrier properties. Thanks
to the increase of computing power and to new
methodological developments, several authors have investigated
the modeling of gas solubility in polymers at a
molecular level. For example, Gusev and Suter [
the infinite dilution solubility of methane in polycarbonate.
] computed Henry’s constant of different
gases in amorphous atactic polypropylene by using a test
particle insertion method similar to Widom’s method [
Later, de Pablo et al. [
] employed Gibbs ensemble
Monte Carlo simulations in order to compute the
solubility of short chain alkanes in polyethylene melts beyond
Henry’s regime. Several studies have followed, considering
many different penetrant gases and more complex polymers
such as polystyrene [
], poly(styrene-alt-maleic anhydride)
copolymer, pols-(styrene-stat-butadiene) rubber and atactic
], polyimide [
], polyamide [
polyethylene terephthalate [
In the early 1990s, Gusev et al. [
], Müller-Plathe [
], Müller-Plathe et al. , Takeuchi [
], as well
as Pant and Boyd [
], used molecular simulation to
investigate the diffusion of small penetrants in
polymers. Pant and Boyd [
] studied the transport
properties of small penetrants in amorphous polyethylene and
polyisobutylene. Gusev et al. [
] and Müller-Plathe
et al. [
] used molecular dynamics method to show
that gases diffuse through polymers in a sequence of
activated jumps between neighboring locations (called hopping
mechanism). By using the same method, van der Vegt 
achieved to observe a transition temperature between
hopping-like and liquid-like diffusion.
Despite considerable work on solubility and diffusion
coefficients, computer simulation of both quantities beyond
infinite dilution limit and using the same molecular model
are rare. In this paper, we will compute simultaneously
solubility and diffusion coefficients in order to evaluate the
permeability. Furthermore, in most of the previous works
mentionned above, transport properties have been obtained
in melt or purely amorphous polymer or, in the case of
semicrystalline materials, within the assumptions that:
– the amorphous phase is the only permeable phase;
– the amorphous phase characteristics are not affected by
the presence of the crystalline regions.
The first assumption is strongly supported by
experimental evidences. Experiments by Michaels and Bixler [
on different polyethylene grades with different degrees of
crystallinity permitted to establish the following relationship
between the solubility, S , and the diffusion coefficient, D,
in the semicrystalline material versus the solubility and the
diffusion coefficient in the amorphous phase (denoted by
S = S a(1 − χ)
0 < χ < 1
τ, β > 1
where χ is the (volume or mass) fraction of crystalline
regions. In Equation (3), τ is a tortuosity factor accounting
for the increase of the diffusion path caused by the presence
of crystallites and β is a chain immobilization factor which
takes into account the reduction of the amorphous chain
segment mobility due to the proximity of crystallites.
The second assumption is not always valid. In
semicrystalline materials, the amorphous phase can be perturbed
by the presence of crystallites. Thereby, several authors
have reported large deviations between experimental data
and theoretical predictions when they assume that the
amorphous phase properties are not affected by the crystalline
regions. Van der Vegt et al. [
] showed that gas solubilities
in amorphous polyethylene, computed by molecular
simulation, are 1.4 to 3.5 times greater than experimental
values (S a) corrected from amorphous content (Eq. 2). Using
another molecular model, Nath and de Pablo [
] found that
the solubility of small molecules in amorphous
polyethylene is overestimated, especially for highly soluble gases.
Hu and Fried [
] also reported solubility coefficients of
small gas molecules in poly(organophosphazenes) five times
larger than what would be expected by extrapolating values
reported for semicrystalline samples to a 100% amorphous
The crystalline region effect on the amorphous phase
has been accounted for by theoretical approaches [
based on Flory and Rehner work [
]. This effect comes
from the fact that some polymer chains leaving a crystalline
region may be trapped in another crystalline region. The
network formed by these tie segments may constrain the
amorphous regions of the polymer. This effect, called the elastic
effect, has been invoked by several authors to correct the
predicted solubilities by equations of state or group contribution
] but was never accounted for in molecular
In a previous work [
], we were able to implicitly
account for the effect of crystalline regions upon gas
solubilities by using Monte Carlo simulations in the osmotic
ensemble in an original way. We reproduced experimental
solubilities by exerting an additional uniform constraint on
our purely amorphous simulation box. A single constraint
value emerged, independent of the gas nature,
characteristic of the semicrystalline material. We concluded that the
role of this ad-hoc constraint is to reproduce the effective
density of the permeable phase of the material. This method
has also shown its success toward the computation of gas
mixture solubilities in semicrystalline polyethylene [
The methodology mentioned above has been applied to
polyethylene (PE), because of its wide use in industrial
applications and its relative simplicity as a molecular model.
In the present work, we extend the proposed approach to
permeability calculations in semicrystalline PE. Two different
penetrants are investigated: carbon dioxide and methane,
mostly chosen because of their relevance in gas production.
The obtained results are then validated against experimental
This article is organized as follows. In the next
section, we rapidly review the methodological aspects of this
work. Section 2 discusses estimations of transport properties
in the amorphous region of semicrystalline polymer from
experimental data. In Section 3, we present our simulation
results on gas solubility, gas diffusion coefficient and gas
permeation. The increase of diffusion coefficient with gas
concentration in polymer is captured. From the study of
transport property temperature dependency, activation
energies are calculated and compared with experiments. A link
between the ad-hoc constraint used in our simulations and
the elastic effect is then provided in Section 4. Finally,
Section 5 summarizes our findings.
We have employed Monte Carlo (MC) simulations in the
osmotic ensemble to obtain the solubility of different gases
in polyethylene. The details of these MC simulations are
given elsewhere [
]. Molecular Dynamics (MD)
simulations were performed in the canonical ensemble to obtain
the diffusion coefficient of penetrants. Nosé-Hoover
thermostat with an explicit time reversible integrator [
been used. A timestep of 2 fs was used in our simulations.
The carbon-carbon bonds in polyethylene were constrained
to their equilibrium value (1.535 A˚) using the Rattle
Initial MD configurations are issued from the output of
the osmotic ensemble MC simulations with number of
penetrant molecules and volume equal to the average MC values.
Initial particle velocities are taken from a Gaussian
distribution. Self-diffusion coefficients, D∗a, are computed from
the mean square displacement msd(t) of the N penetrant
molecule center of mass positions, ricm(t):
using the Einstein equation [
For each system, the production run lasts 80 ns. During this
period, the mean displacement was larger than ten molecular
diameters and we checked that penetrant molecules reached
the diffusive mode (i.e. the linear regime of msd(t)).
The polymer studied in this work is a model of linear
polyethylene (PE) with 70 carbon atoms per chain. The
force field and the mixing rule are identical to the ones used
in our previous works [
2 ESTIMATION OF TRANSPORT PARAMETERS
IN THE AMORPHOUS PHASE FROM EXPERIMENTS
There are experimental measurements available for
solubility and diffusion coefficients of gas in semicrystalline PE.
However the simulations refer only to the amorphous part
of the material, thus comparison of simulation results with
experiments requires further elaboration.
Thanks to Equation (2), the conversion of measured
solubility coefficients S to solubility coefficients in amorphous
phase S a is straightforward. The only required
parameter is the crystallinity ratio χ which can be measured
independently by several techniques (Differential Scanning
Calorimetry (DSC), density measurement, Raman
spectroscopy, Transmission Electron Microscopy (TEM) and
Nuclear Magnetic Resonance (NMR) [
The determination of the diffusion coefficient in the
amorphous region, Da, from experimental value D, is not
straightforward. Parameters τ and β involved in Equation (3) are not
easily measured experimentally. Therefore, we could only
obtain estimates for Da from three different methods.
In the first method, Da is extrapolated from the diffusion
coefficient in polymer melts using the following equation:
Da = D exp (−ΔEd(1/T − 1/T ))
where T is a temperature above the polymer melting point,
D is the diffusion coefficient in polymer melt at T , ΔEd
is an activation energy for diffusion and T is an arbitrary
temperature below the melting point. We assumed that ΔEd
is constant over the whole temperature range.
The second method uses diffusion coefficient values in a
100% amorphous polymer grade. For polyethylene, there
isn’t a 100% amorphous grade at low temperature. Other
polymers, such as natural rubber, are usually considered as
its amorphous analogue [
CH4 (Sim σPE = 80 MPa)
CO2 (Sim σPE = 80 MPa)
In the present work, we compare our simulation results with
the values estimated from these three methods (Eq. 6-8).
3 SIMULATION RESULTS
In Figure 1, we present CO2 and CH4 concentrations in
the amorphous phase of polyethylene at 293 K and 298 K
respectively, obtained from Monte Carlo simulations in
the osmotic ensemble. During these simulations, an
additional isotropic constraint σPE of 80 MPa is imposed on
the simulation box to account for the effect of
crystallites by reproducing the average density of the permeable
]. As can be seen, a good agreement between
simulation and experiments is observed. It is also shown that the
gas concentration in polyethylene is linearly proportional to
the pressure in this pressure range. In such cases, the Fickian
diffusion coefficient (Da) is identical to the Maxwell-Stefan
diffusion coefficient [
]. In addition, if the polymer chain
mobility is neglected comparing to solute velocities, the
Maxwell-Stefan diffusion coefficient can itself be
approximated by the penetrant self-diffusion coefficient (D∗a). (For
Fig. 1, see also [
]). Therefore, diffusion coefficient
Da can be calculated from Equation (5). A msd(t) curve
and its time derivative for CO2 diffusion in PE is given in
Figure 2. The average diffusion coefficient is computed in
the region where the slope of msd(t) is constant.
In Figures 3 and 4, we present the simulation results for
CO2 diffusion at 293 K and CH4 diffusion at 298 K. (For
Fig. 3, see also [
6, 7, 59, 60
] and for Fig. 4, see [
]). It is
shown that the use of an additional constraint of 80 MPa on
the amorphous box leads to a decrease of the diffusion
coefficient. This observation is related to the increased density
of the amorphous region under such an additional constraint,
reducing gas mobility.
Experimental data are also reported in these graphs.
Laguna et al. [
] reported two different experimental values
Dissolved gas concentrations as a function of pressure for
CO2 and CH4, calculated by Monte Carlo simulations in the
osmotic ensemble (at 298 K for CH4 and 293 K for CO2).
Simulations are performed with an additional constraint of 80 MPa.
Solid lines represent the average experimental concentrations
from references [
] and dashed lines correspond to
standard deviations from this average.
106<D> = 1.306 ± 0.056 cm2/s
a) Mean square displacement and b) its time derivative
versus time. The plots correspond to the simulation of 3 CO2
molecules in 15 chains of nC70 at 293 K with an additional
constraint of 80 MPa. The region where the slope of msd(t) is
constant is in green.
for CO2 diffusion coefficient: a diffusion coefficient
measured by permeation cell technique and a self-diffusion
coefficient measured by NMR. The value obtained by NMR
is an order of magnitude higher than the one obtained by
permeation experiments. Except this NMR value and the
estimate from the extrapolation method, all experimental
values corrected either by Equation (7) or by Equation (8)
are close to the simulation results obtained with an
additional constraint of 80 MPa. This observation confirms the
need of an additional constraint to allow for a more realistic
description of the amorphous region. It is also concluded
that the extrapolation method is not reliable to estimate the
diffusion coefficients at low temperatures. Main reasons are:
– the diffusion activation energy is not constant with
– the effect of crystalline regions on the gas diffusion in the
amorphous phase is completely ignored by the
We also observe that the diffusion coefficient increases
with gas concentration in the polymer. This is related to
the plasticization of polymer chains due to the presence of
penetrant molecules. Vittoria [
] describes this behaviour
using an exponential function:
Da = Da∞ exp (γCa)
In this relationship, Da∞ refers to the diffusion coefficient at
infinite dilution which is related to the fraction of free
volume and γ is the concentration coefficient which is a
measure of the plasticization effect. Da∞ and γ values obtained
from our simulation results are given in Table 1.
A diffusion slightly faster is obtained for CO2
compared to CH4. Such behaviour has been already reported
in the literature with other polymers: polypropylene [
and polystyrene [
], and has been interpreted on the basis
of molecular structure considerations, attributing the faster
diffusion to faster axial motions in case of linear molecules.
Finally, permeabilities have been computed from our
molecular simulation diffusion and solubility coefficients.
Diffusion coefficient of methane versus concentration in the
amorphous phase of PE at 298 K. Two series of simulations
are presented: with (black) and without (red) an additional
constraint of 80 MPa. Dashed lines are exponential fits of
the simulation results (Eq. 9). Experimental data are from
Michaels and Bixler [
] and Lundberg [
a Estimated from Equation (6), b estimated from
Equation (7), c estimated from Equation (8).
Calculated values are presented in Table 2 and shown in
Figure 5 for the two studied gases. Data are in good
agreement with experimental results of Michaels and Bixler at
infinite dilution [
As the morphology of semicrystalline polyethylene
evolves with temperature, we showed in our previous
] that the additional constraint must decrease
with increasing temperature: from 80 ± 10 MPa at room
temperature (293-298 K) to 60 ± 10 MPa at 333 K and
40 ± 10 MPa at 353 K. We have thus computed CO2
and CH4 solubility and diffusion coefficients at these three
temperatures (see Tab. 2, 3) and we have then estimated
activation energies for solubility, diffusion and
permeability processes (see Tab. 4). Calculated activation energies
are compared with experimental values of Flaconnèche
et al.  and Michaels and Bixler [
]. The orders of
magnitude are well reproduced. However, the activation
energy for the diffusion process are slightly
underestimated by our simulations. This difference can be related
to the chain immobilization factor (β) whose contribution
4 INTERPRETATION OF THE ADDITIONAL CONSTRAINT
BASED ON THE ELASTIC MODEL
The non crystalline regions of semicrystalline polymer are
composed of elastically effective and elastically ineffective
chain segments. The elastically effective chain segments
consist exclusively of intercrystalline tie segments. The
network formed by these tie segments has been
considered as crosslinks constraining the amorphous phase of the
polymer. This network results in an additional elastic
]. Several authors have invoked this elastic
effect to correct their calculated gas solubility coefficients
in the amorphous phase of semicrystalline polymers. The
correction term is usually introduced as a contribution to the
activity coefficients of the different species. Some modified
equations of state are proposed that allow the calculation of
thermodynamics properties including the elastic effect
40, 44-47, 67
Different elastic models exist in the literature. The Flory
and Rehner model, which is dedicated to the study of
crosslinks rubbers [
40, 42, 43
], assumes a Gaussian
distribution of the tie segments. A few years later, this model
has been extended to semicrystalline polymers [
considering a Hookean behaviour of the tie segments.
Some authors have tried to compare these two elastic
]. Doong and Ho  showed that the model
to the activation energy is not taken into account in our
Solubility, diffusion and permeability coefficients in PE amorphous
phase for CO2 and CH4. The results are obtained by molecular
simulations with an additional constraint of 80 MPa applied
on the amorphous phase
proposed by Michaels and Hausslein [
], especially due
to its temperature dependency, is more relevant in
In the elastic model with Hookean segments, the energy
cost associated with the stretching of an elastically effective
tie segment is described by [
where r is the average distance between adjoining
crystallites, u the average number of methylene units in the tie
segment and K a spring constant. The force due to an elastically
effective tie segment, faelastic, is given by −∇(ΔUelastic). By
summing over all ν elastically effective segments, the total
force on the polymer amorphous phase can be obtained:
Felastic = −ν∇(ΔUelastic) = −ν
where rˆ is the unit vector in the r direction. The constraint
on the amorphous phase due to the elastically effective tie
segments is then:
σPE = − Va
where Va is the total volume of the amorphous regions.
Michaels and Hausslein [
] and more recently
Banasazak et al. [
], based on the thermodynamic
equilibrium between crystalline and amorphous phases, have
shown the following relationship (see Eq. 26 in [
where ΔH f is the polymer molar heat of fusion, vaP the
partial molar volume of polymer in the amorphous polymer
phase, φ the volume fraction of polymer in the amorphous
polymer phase, Tm the melting point of crystallites in the
presence of penetrant molecules and f the fraction of
elastically effective chains in the amorphous regions. Substituting
Equation (13) into Equation (12), we obtain the following
expression for the additional constraint:
ΔH f 1 1
R T − Tm
2 3fφ − 1
It is interesting to note that Equation (14) predicts a
nonzero constraint value even at zero penetrant concentration,
i.e. for pure polymer. In this case, φ = 1, vaP = 1/ρa where
ρa is the number density of the pure amorphous region and
Tm = Tm, the melting point of pure semicrystalline polymer.
Equation (14) can thus be rewritten:
σPE = 3ρaRT
ΔH f 1 1
R T − Tm
23f − 1
In the following, we compare the values of σPE
computed using Equation (15) with the values used in our
molecular simulations. We recall that our constraint
values were calibrated from comparison between experimental
CO2 solubiliy coefficients on PE samples with crystallinity
χ = 50% and simulated values computed in the osmotic
]. All quantities required in Equation (15),
except f , have been taken from experimental results of
Flaconnèche et al. obtained on a medium density
polyethylene sample [
]. Few authors have reported values for the
empirical factor f in order to reproduce solubility data of
penetrants in different PE samples [
39, 44, 45, 47
]. f values
are typically in the range 0.27-0.50. In Figure 6, we present
the additional constraint value, σPE, calculated from
Equation (15) as a function of f at 298 K, 333 K and 353 K. At
these temperatures, the additional constraint values used in
our simulations are respectively 80 ± 10 MPa, 60 ± 10 MPa
and 40 ± 10 MPa . Figure 6 shows that a value of f ,
around 0.65-0.75, can reproduce the additional constraints at
all temperatures. This fraction of elastically effective chains
is slightly larger than the values reported in the litterature.
Such a large value might be related to the use of rather
small chain lengths (70 carbon atoms) to model
polyethylene. The use of longer chains would indeed lead to lower
additional constraint values, and thus to lower fractions of
elastically effective chains.
We have shown in this work that our recently proposed
technique based on osmotic ensemble Monte Carlo
] provides an effective method for evaluation of
gas solubility in the amorphous phase of semicrystalline
polymer. The real material density is reproduced when
an ad-hoc constraint is applied to the polymer amorphous
phase. This density value can then be imposed in molecular
dynamics simulations without the use of a priori
experimental results. The resulting diffusion coefficients are in good
agreement with those obtained from experiments (Michaels
and Bixler [
] among others).
Our methodology permitted the computation of the
transport parameters (diffusion coefficient and permeability) of
polyethylene amorphous region toward carbon dioxide and
methane. The study of diffusion coefficient at different gas
concentrations allowed us to capture the plasticization effect
in polyethylene. In addition, we reproduced reasonable
Elastic model of Michaels and Hausslein [
seen to be able to justify the effective constraint values
employed in the simulations. It is shown that intercrystalline
tie segments exert an additional constraint on the polymer
amorphous phase, what should be accounted for in any
molecular simulation of the amorphous phase. Our findings
suggest that the fraction of elastically effective chains is
independent of the gas pressure (in the range of 0-8 MPa)
or nature (for CO2 and CH4) or temperature (in the range of
293-353 K). Rather, it is a characteristic of the polymer with
a given crystallinity and history. For this approach to be
fully predictive, an a priori knowledge is required for this
fraction of elastically effective chains.
PM would like to thank IFP Energies nouvelles for
financial support. Jean-Marie Teuler (Laboratoire de Chimie
Physique) is gratefully acknowledged for support in code
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