Diffuso-Kinetics and Diffuso-Mechanics of Carbon Dioxide / Polyvinylidene Fluoride System under Explosive Gas Decompression: Identification of Key Diffuso-Elastic Couplings by Numerical and Experimental Confrontation
Oil & Gas Science and Technology - Rev. IFP Energies nouvelles, Vol.
This paper is a part of the hereunder thematic dossier published in OGST Journal, Vol. 70, No. 2, pp. 215-390 and available online here
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DOSSIER Edited by/Sous la direction de : B. Dewimille Perméation de gaz dans le polyéthylène semi-cristallin par simulation moléculaire et modèle élastique Renforcement des propriétés barrière aux gaz de matrices polyéthylène et polyamide par l'approche nanocomposite : facteurs clés et limitations Cinétique de diffusion et comportement diffuso-mécanique du système dioxide de carbone / polyfluorure de vinylidène sous décompression explosive de gaz : identification des couplages diffuso-élastiques majeurs par confrontation numérique et expérimentale > Development of Innovating Materials for Distributing Mixtures of Hydrogen and Natural Gas. Study of the Barrier Properties and Durability of Polymer Pipes > Effects of Thermal Treatment and Physical Aging on the Gas Transport Properties in Matrimid® Séparation de mélanges binaires de propylène et de propane par transport au travers des membranes de poly(éther-blocamide) incorporant de l'argent
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Perméabilité d’un matériau barrière EVOH utilisé dans des applications automobiles : développement métrologique pour des mélanges modèles de carburants
J. Zhao, C. Kanaan, R. Clément, B. Brulé, H. Lenda and A. Jonquières
Les effets du traitement thermique et du vieillissement physique sur les
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Fluids-Polymers Interactions: Permeability, Durability
Interactions fluides polymères : perméabilité, durabilité
Diffuso-Kinetics and Diffuso-Mechanics of Carbon
Dioxide / Polyvinylidene Fluoride System under
Explosive Gas Decompression: Identification of Key
Diffuso-Elastic Couplings by Numerical and
1 Institut Pprime UPR 3346 / CNRS - ISAE-ENSMA, 1 Av. Clément Ader, 86961 Futuroscope - France
2 IFP Energies nouvelles, 1-4 avenue de Bois-Préau, 92852 Rueil-Malmaison - France
* Corresponding author
Abstract — The work aims at identifying the key diffuso-elastic couplings which characterize a
numerical tool developed to simulate the irreversible ‘Explosive Decompression Failure’ (XDF) in
semi-crystalline polymer. The model proposes to predict the evolution of the gas concentration
and of the stress field in the polymer during the gas desorption [DOI:
10.10161j.compositesa.2005.05.021]. Main difficulty is to couple thermal, mechanical and diffusive effects that occur
simultaneously during the gas desorption. The couplings are splitting into two families:
– indirect coupling (i.e., phenomenology) that is state variables (gas concentration, temperature,
and pressure) dependent;
– direct coupling, (i.e., diffuso-elastic coupling) as polymer volume changes because of gas
The numerical prediction of the diffusion kinetics and of the volume strain (swelling) of PVF2
(polyvinylidene fluoride) under CO2 (carbon dioxide) environment is concerned. The prediction is carried
out by studying selected combinations of couplings for a broad range of CO2 pressures. The modeling
relevance is evaluated by a comparison with experimental transport parameters analytically identify
from solubility tests.
A pertinent result of the present study is to have demonstrated the non-uniqueness of the coefficients
of diffusion (D) and solubility (Sg) between the diffuso-elastic coupling (direct coupling) and
indirect coupling. Main conclusion is that it is necessary to consider concomitantly the two types
of couplings, the indirect and the direct couplings.
Re´ sume´ — Cine´ tique de diffusion et comportement diffuso-me´ canique du syste` me dioxide de carbone
/ polyfluorure de vinylide` ne sous de´ compression explosive de gaz : identification des couplages
diffuso- e´lastiques majeurs par confrontation nume´ rique et exp e´rimentale — Cette e´ tude a pour
objectif d’identifier les couplages diffuso-e´ lastiques majeurs qui entrent en jeu dans une
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
mode´ lisation des me´ canismes irre´ versibles « d’Endommagement par De´ compression Explosive de
gaz » (EDE) des mate´ riaux polyme` res semi-cristallins. Le mod e`le pre´ dit l’e´ volution de la
concentration en gaz et les champs de contraintes dans le mate´ riau au cours de la d e´sorption
de gaz [DOI: 10.10161j.compositesa.2005.05.021]. La difficulte´ principale est de coupler les
effets thermiques, les champs de contraintes et la diffusion du gaz qui se produisent
simultane´ ment au cours de la de´ sorption. Les couplages sont class e´s selon deux familles :
– couplage indirect (phe´ nome´ nologie) qui de´ pend des variables d’e´ tat (concentration en gaz,
tempe´ rature et pression) ;
– couplage direct (ou couplage diffuso-e´ lastique) pour lequel le volume du polyme` re e´ volue du
fait de la diffusion du gaz.
La pre´ diction nume´ rique des cine´ tiques de diffusion et des contraintes volumiques (gonflement)
du PVF2 (polyfluorure de vinylide` ne) sous environnement de CO2 (dioxyde de carbone) est
e´ tudie´ e a` travers une combinaison des couplages dans une large gamme de pressions en CO2.
La pertinence du mode` le est e´ value´ e par confrontation avec les parame` tres de transport qui
ont e´te´ e´value´ s selon la proce´ dure ‘analytique conventionnelle’ a` partir des mesures
exp e´rimentales de solubilite´ s.
Le r e´sultat pertinent est d’avoir de´ montr e´ la non-e´ quivalence des coefficients de diffusion (D) et
de solubilite´ (Sg) entre le couplage diffuso-e´ lastique (couplage direct) et le couplage indirect. Une
bonne mod e´lisation du comportement diffuso-e´ lastique du polyme` re ne´ cessite la conside´ ration
concomitante des deux familles de couplages, direct et indirect.
Thanks to their good barrier properties, polymer
materials are encountered in the pressure sheath of flexible
oilcarrying pipes. With respect to the transported
hydrocarbons, the permeability to corrosive gases, such as
hydrogen sulphide (H2S) or carbon dioxide (CO2), must
be sufficiently low to minimize the corrosion of the metal
reinforcements; the metal reinforcements ensure the
mechanical resistance of the pipes structure. However,
the polymer materials are not fully impermeable to gases
]. The loss of barrier task is accentuated in the case of
sharp-pressure variations generating thus irreversible
‘explosive’ deterioration of the polymeric structures.
The phenomenon is called ‘Explosive Decompression
Failure’ (XDF) [
]. Various forms of damage appear,
i.e., cracks, blisters or structure-like foams within the
polymer. The irreversible damage appears because of
strong thermo-diffuso-mechanical couplings.
By considering especially the transport properties of
carbon dioxide (CO2) into polyvinylidene fluoride
(PVF2), the gas diffusion and solubility coefficients in
the petroleum exploitation sheath cannot be determined
through experiments because of the size of the sheath. As
a result, it becomes of prime importance to use
representative specimens and to propose modeling tools allowing
the prediction of polymer behavior under real conditions
of use. Several studies relating to the transport
phenomena of gas into polymer have made it possible to
establish the gas diffusion law and to evaluate the
solubility with the swelling of various gas/polymer systems
for various pressures and temperatures. Adapted
accordingly, a non-exhaustive bibliographic review is
presented below on CO2/PVF2 systems.
The mathematical theory of diffusion [
] for an
isotropic system is based on the assumption of
proportionality between the flux ~J of diffusing substance and
the concentration gradient r~C through an elementary
surface (C in ppm). The first Fick’s law is expressed in
Equation (1). D is called the diffusion coefficient (D in
This first law is applicable for example in the steady
state reached when the gas concentration in a membrane
does not vary with time and the flux is constant. In
transient state, the matter transfer by diffusion is different
from zero and the penetrant concentration is a function
of position and time. The second Fick’s law of
diffusion describes this non-steady state as expressed in
The diffusion parameter D is a non constant
characteristic of the gas/polymer system. When the interactions
between the polymer chains and the gas molecules are
high (i.e., CO2 diffusion through fluoride polymers) the
diffusion coefficient depends on the gas concentration
inside the material. Within the general framework of
gas/polymer systems, a relation of the type connects
the diffusion coefficient to the thermodynamic variables
(T, temperature; p, gas pressure) of the system as given in
D ¼ DðC; T ; pÞ
These dependencies of D state the various chemical
affinities between gas and polymer. These
phenomenological couplings were named indirect couplings [
For systems in which the solubility essentially obeys to
the Henry’s law (i.e., hydrocarbons in elastomers), the
dependence of the diffusion coefficient on the sorbed
penetrant concentration has been empirically
represented, for a given temperature, by linear Equation (4)
and exponential Equation (5) [
ð Þ ¼ D0ð1 þ bCÞ
Linear model ð4Þ
ð Þ ¼ D0 exp ðbCÞ
Exponential model ð5Þ
Equation (5) is used when the dependence on
concentration becomes important. D0 is the limit of D when the
concentration C tends to zero (Dlong). b is a constant
parameter and is temperature dependent. If b is zero,
the model corresponds to Fick’s laws, Equations (1, 2),
where D stays constant. These models have been
considered in the works of Benjelloun-Dabaghi [
For systems in which the solubility does not exactly
obey to the Henry’s law but rather than an isotherm of
Flory-Huggins type (i.e., highly soluble gases in rubbery
polymers), the concentration dependence of D can be
represented consistently with Equation (6) [
ð Þ ¼ Dð0Þ exp
1 þ rC
At a given temperature, r is a constant related to
penetrant/polymer interactions. Fujita et al. [
proposed a similar relationship from considerations
based on the free volume theory.
The effect of pressure on gas diffusion through rubbery
polymers has been studied to develop membranes useful
of service for gases separation. Stern et al. [
concluded that the pressure influence could be explained
as the result of two opposite phenomena. In the first
effect, the hydrostatic pressure leads to an increase of
the polymer density and thus to a reduction of polymer
free volume. In the second effect, the increase in pressure
induces a concentration enhance of molecules diffusing
within the matrix. The diffusion in turn plasticizes the
macromolecular chains involving a larger free volume.
Naito et al. [
] have studied the permeability of a
series of pure gases into polymers in the rubbery state,
such as polyethylene, polypropylene, and
polybutadienes. The effect of gas pressure was carried out up to
10 MPa. To discriminate the two antagonistic effects,
i.e., hydrostatic constraint and plasticization, for a given
temperature the dependency of the diffusion coefficient
with pressure has been proposed as expressed in
DðC; pÞ ¼ Dð0; 0Þ exp ðbhp
Dð0; 0Þ is the diffusion coefficient at atmospheric
pressure and for zero penetrating gas concentration. The
term exp ðbhpÞ is the hydrostatic pressure effect. bh is a
negative coefficient expressing the reduction diffusion
abilities of the material under pressure. The term
exp ðaCÞ indicates the number of molecules increase
dissolved into the polymer and giving rise to plasticization.
For higher pressures, beyond 20 MPa, Equation (7) is
extended as concentration, temperature, and pressure
]. The diffusion coefficient D is then
expressed in Equation (8):
DðC; T ; pÞ ¼ D0ðT ; pÞ exp ðbCÞ
In Equation (8), only the diffusion through the
thickness of a membrane is considered. The parameters D0
and b have been identified for various level of pressure
for a given temperature. But most probably, to correctly
evaluate this model, it would have been preferable to
identify the parameters on one test and to validate them
on the others.
The models of gas diffusion in the open literature are
greatly established on couplings focused on the
dependencies of D on concentration, pressure and
temperature. However, solubility tests carried out on CO2/
PVF2 system show the existence of a high
diffusomechanical coupling. This coupling results in an
important swelling of the material during the sorption stage,
40% for a pressure of 100 MPa, as it will be described
further in the experimental investigation (Sect. 2.1).
As a result, the polymer elastic contribution (i.e.,
mechanical response dues to volume change) to the
diffusion phenomena becomes of the utmost importance
]. Moreover, considerable hydrostatic, and so
mechanical, pressure induces complex stress fields within the
polymeric material especially under the high pressures
constraints. This additional complexity must be added
in the analysis of the couplings between mechanical and
To date, a body of models tentatively describes the
existence of strong couplings between mechanical,
thermal and diffusive phenomena occurring in polymeric
materials in fluid transport, with the additional difficulty
in modeling the state of the fluid within the polymer.
Theoretical descriptions of these coupling phenomena
are based on empirical or thermodynamic
interpretations, as given in the selected open literature [
As an attempt, Rambert et al. [
] described the
effect of gas on the mechanical behavior of material
through diffuso-mechanical coupling, as it will be
explained in Equation (10). The diffuso-mechanical
coupling is classically defined by an isotropic expansion
coefficient linked to gas concentration. In this modeling,
the CO2/PVF2 system has been considered as a mixture
for which the molecular diffusion obeys to the laws’
Fick, Equations (1, 2). The originality of Rambert
et al. [
] approach consists in the addition of a
coupling term, due to the gradient of the elastic volume
strain, to the divergence of the substance diffusing flux as
was given in Equation (2). The additional terms in
Equations (10, 12), presented in Section 2, are named direct
couplings. The chosen method has been to systematically
eliminate the indirect couplings, as described previously,
in order to evaluate the effect of the direct couplings.
This model has been applied to a structure subjected to
an explosive decompression, and a parametric study
showed the implication of the direct couplings during
stages of transition.
From a purely mechanical point of view, one can see
that two types of couplings appear, direct and indirect.
The indirect couplings were validated in a majority of
system having diffusion kinetics at the mercy of pressure,
concentration and/or temperature. Nevertheless, they
are not enough to describe the volume strain induced
by the sorption of a gas into a polymer. The direct
couplings can be used to describe concomitantly the
diffusion kinetics and the mechanical behavior of a polymer
subjected to a gas diffusion. The indirect couplings
outline the dependency of the transport parameters on the
state variables; indirect means that these couplings can
be substituted into the direct couplings, which involve
the transport parameters. All these couplings must be
evaluated, in particular to improve the analysis of the
experimental tests and for a more accurate modeling of
the basic phenomena involved during the sorption and
the diffusion of gas into polymer.
Consequently, the present work aims at studying the
relevance and the limits of these couplings by using the
formalism of the thermodynamics on the example of
CO2/PVF2 system. A first discussion was introduced in
our previous work [
]. In the diffusion problem, see
the following Equations (11, 12), the formalism of the
thermodynamics is described within two parameters, kl
(a coupling parameter related to the effect of the
chemical potential gradient on the gas mass flux) and ac (the
isotropic expansion coefficient related to mass
transport). A new identification of the diffusion problem,
dependent on the polymer volume variation and gas
concentration in the polymer, is proposed by numerically
quantifying the parameters kl and ac of the direct
This article is organized into three parts. In Section 1,
a brief presentation of the indirect couplings proposed
by Rambert et al. [
] is presented. Section 2
describes the experimental solubility tests. These tests
permit to determine the transport parameters of CO2
in PVF2. The mass and the volume of (gas+polymer)
mixture are measured during gas desorption at a given
temperature and various pressures levels. For any level
of pressure, desorption can be decomposed into two
phases: a first phase of decompression during which it
can be supposed that temperature fluctuations do not
notably influence diffusion when compare to the effect
of concentration, and a second phase at atmospheric
pressure and constant ambient temperature for which
measurements have been done. The couplings with
thermal phenomena are neglected; only the
diffusomechanical couplings during gas desorption are studied.
Two pressures ranges are of interest. At low pressures
tests (0.5, 1.5 and 4 MPa), only the mass have been
measured after gas decompression. At high pressures tests
(25, 50 and 100 MPa), the mass and the volume have
been measured. In Section 3, the ability of the various
types of couplings is studied to predict the diffusion
kinetics and the volume strain of CO2/PVF2 system.
Experimental measurements and computed values are
compared. In a first attempt, the study of indirect
couplings is examined. We estimate the effect of pressure
and gas concentration in the polymer with linear and
exponential dependencies of the diffusion coefficient,
Equations (4, 5); the volume change is considered
explicitly as well as the gradients effects of the volume
deformation due to the sorption stage. In a second attempt,
the study of diffuso-elastic direct coupling is carried
out, Equation (12). In both the attempts, the high
pressures tests are studied because the polymer volume
evolution is known for such tests. By using a constant
diffusion coefficient adapted at each level of pressure
of the solubility test, the expansion coefficient related
to the diffusion is determined in order to obtain the
maximum swelling of the specimen at equilibrium. The effect
of the hydrostatic compression of the polymer is
compared to the plasticizing effect of gas, Equation (7).
Finally, the coupling with the elastic volume strain in the
diffusion law is evaluated and compared with the effect
of the gas concentration.
– Diffusion problem:
1 DIFFUSO-MECHANICAL MODEL: DIRECT COUPLINGS
The direct couplings are described according to the
thermo-diffuso-elastic-linear approach of Rambert
et al. [
]. The thermal couplings are neglected
and only the diffuso-elastic couplings are retained.
This model has been developed in the framework of
the generalized standard media, and therefore follows
a classical approach. In this thermodynamic frame,
the Elementary Representative Volume (ERV)
describes a bi-component, continuous, homogeneous
medium: polymer and gas, at a macroscopic scale.
The introduction of a first potential (specific free
energy) defines the laws connecting the thermodynamic
forces (stress, chemical potential) with dual variables
(strain ee, gas concentration C). With the assumption
of small perturbations, i.e., small strain and small
difference of gas concentration, the Taylor expansion of
this potential up to the second order leads to linear
coupled constitutive equations.
The dissipations, which are only associated with the
diffusion phenomenon in the elastic case, are obtained
by using a second potential, i.e., dissipation potential.
The dissipation potential depends on the coupling
phenomena which are taken into account. Once these are
defined, under the assumption of uncoupling between
dissipative phenomena, a complementary evolution law
is obtained, Equations (11, 12). It is to be noted that
the model was developed by prescribing constant values
for all the material parameters.
By considering the previous assumptions, the
diffusoelastic model is finally defined via a mechanical problem
and a problem of diffusion [
] under the following forms:
– Mechanical problem:
d~iv r þ q~f ¼ 0
r ¼ r0 þ k tree I þ 2lee
ð3k þ 2lÞac ðC
r is the total Cauchy stress tensor (Pa) applied to the
CO2/PVF2 mixture. q is the average mixture density
(kg.m 3). ~f is the body force per unit mass (N.kg 1)
at any point within the ERV. r0 and C0 represent the
initial stress (Pa) and gas concentration (cm3 (STP).
cm 3), respectively. k and l are the Lame´ elastic
coefficients (Pa). ac is the isotropic expansion coefficient
related to mass transport. I is the identity matrix;
kl is a coupling parameter related to the effect of the
chemical potential gradient on the gas mass flux
(kg.s.m 3). ee is the elastic volume strain and tr is the
Equations (11, 12) convey couplings between
diffusion and elastic phenomena. The first terms in the right
hand side of Equation (10) are of the Hooke’s law types.
The last term corresponds to the effect of the gas
diffusion on the mechanical behavior of the material. In a
reciprocal way, in the diffusion law, Equation (12), the
product ðkl acÞ reflects the effect of the elastic volume
strain (tree) on the evolution of the gas concentration in
Added in the flux Equation (12), the
phenomenological laws expressing the effects of concentration and
pressure on the diffusion coefficients, see Equations (4, 5) or
(8), have complemented this model. Consequently, the
two types of couplings are present at various levels in
the diffusion Equation (12).
For the various CO2 Saturation Vapor pressures
(SVps) considered in the present work, the sorption
curve of the CO2/PVF2 system is supposed to follow
the Henry’s law [
]. The Henry’s law expresses the
relationship between the applied pressure p (Pa) and
the gas concentration C through a solubility coefficient
Sg (cm3 (STP).cm 3.Pa 1), as given in Equation (13):
C ¼ Sgp
2 EXPERIMENTAL INVESTIGATION: SOLUBILITY TEST
AND ANALYTICAL ANALYSIS OF DIFFUSION
In the conduction of solubility tests, the evolutions of the
mass and volume of CO2-saturated PVF2 during
desorption are concerned. Mass measurements permit to define
two transport parameters, the coefficients of solubility
Sg and of diffusion D. Only the parameter D is studied.
This parameter can be analytically calculated at the
beginning and the end of desorption curves accordantly
to two methods. The analytical analysis gives rise to two
constant diffusion coefficients, namely Dshort and Dlong,
related to the desorption curve. Discussion on both the
diffusion coefficients is established.
2.1 Description of the Solubility Test – Experimental
The poly(vinyli-dene fluoride) PVF2 investigated is a
main component as efficient barrier against light fluids
for making on-duty pipe-lines seals [
]. The polymer
characteristics are found in [
]. It is a Kynar 50HD,
a polymer without additives like plastifiants or
elastomers. Melting temperature and cristallinity degree are
168 C and 48%, respectively. Purity of carbon dioxide
CO2 is given at 99.5% and used without further
purification. Physical properties of CO2 can be found in
The solubility tests have been performed at IFP
Energies nouvelles (IFPEN). They can be conducted
according to two stages: sorption and desorption. A
polymeric specimen is placed in a pressure cell [
thermally controlled. The pressure cell is connected to
high pressure valves allowing the circulation of the gas.
During the sorption stage, temperature and gas pressure
are gradually increased, and then maintained constant
until the specimen is fully saturated with gas. At
lowpressure tests (0.5, 1.5 and 4 MPa), the sorption
temperature is 313 K. At high-pressure tests (25, 50 and
100 MPa), the sorption temperature is 403 K. Heating
the pressure cell reduces the saturation period in
sorption phase. Indeed, for a given pressure, the diffusion
coefficient is generally related to the temperature by an
Arrhenius’s law . After saturation, the pressure cell,
in which the polymer seats, is quenched with water until
the temperature drops to 298 K by maintaining the
pressure constant. During the desorption phase, a fast
decompression to the atmospheric pressure is first
achieved. Then, the sample is left out the pressure cell
and the specimen volume is measured at regular time
intervals under atmospheric pressure and at ambient
temperature (the first measurement is carried out about
15 min after decompression).
The measurements of mass m(t), volume V(t) and
density q of the specimen during desorption have been made
by double weighing, in the ambient air and in immersion
into ethanol. The uncertainty on measurements is 0.01 g.
Archimedes’ principle is used to calculate the volumes of
the specimen during desorption. The immersion time
(< 30 s) into ethanol is supposed to be sufficiently small
so that the sample does not absorb it. The mass and
the volume variations are calculated according to
M ð%Þ ¼ 100 mðtÞ
DV ð%Þ ¼ 100 V ðtÞ
m(i) and V(i) are the initial mass and volume of the
The solubility tests are conducted on a membrane
with a square base (m(i) = 4.44 g, V(i) = 2.56 cm3,
thickness of 1.1 mm, and base length of 47 mm) for
low pressures and on a ring shaped membrane (m(i) =
7.62 g, V(i) = 4.37 cm3, thickness of 4.8 mm, and
external and interior radius of respectively 25 and 10.5 mm)
for high pressures [
]. The density of native PVF2 is
1 745 kg.m 3.
The mass evolutions during desorption for the six
applied pressures are illustrated in Figure 1. Because of
the high pressure amplitude between tests, the mass
variation is plotted for the low pressure tests in a separate
Figure 2. The first point of each curve, in grey,
corresponds to the extrapolation at zero time of the
desorption curve. Extrapolation is carried out by using a
polynomial function of degree 4 on the first four points
of each curve. This kind of representation is preferred
to the graphs plotted classically in literature by
normalizing the mass because it makes more easily to represent
a broad range of pressures. However, it can also loose
some information. For example, at the highest pressures,
after approximately nine hours of desorption, a
convergence point appears in the mass variation curves,
contrary to the normalized mass curves. For the test
investigated at 4 MPa, it seems that a residual CO2 mass
exists in the membrane (Fig. 2). Indeed, for the
lowpressure tests at 0.5 and 1.5 MPa, the sample recovers
its initial mass at the end of desorption, whereas at
4 MPa it remains approximatively 0.13% of gas in the
membrane (approximatively 6 mg of gas). For the
high-pressure tests (Fig. 1), we can think that a residual
3) 1 750
.g 1 700
its 1 650
)2 1 600
+P 1 550
(C 1 500
quantity of gas of approximately 0.8% remains in the
ring (approximatively 60 mg of gas).
The volume evolution of the ring for the high-pressure
tests is plotted in Figure 3. In the same way as a residual
quantity of gas seems to stay in the sample, a residual
volume strain of approximatively 4% is noticed. At the
end of measurements, the density of the samples is
always lower than the initial density. This remark can
be correlated with the macroscopic observations carried
out during desorption which suggest that sudden blisters
appear within the material.
Contrary to mass desorption kinetics, a not clear
convergence point appears in volume variation kinetics
(Fig. 3); but the mass and volume evolution curves
during desorption follow the same tendency. Indeed,
Experimental 25 MPa
Experimental 50 MPa
Experimental 100 MPa
2.2 Analytical Evaluation of the Diffusion Coefficient
When a membrane of polymer is immersed in a
penetrant, the concentration C in the material is given, at
time t, by the solution of the second Fick’s law, Equation
(2), with a constant diffusion parameter, assuming that
during desorption the gas diffuses only in the thickness
direction, i.e., in one-dimension (1-D) [
Dð2n þ 1Þ2p2t!
2l denotes the membrane thickness. The sample centre is
located at x = 0. Cmax, the gas concentration in contact
with the membrane faces, is considered as immediately
prescribed. This equation does not take into account
the diffusion towards the edges of the membrane, as
far as, in this system, the membrane thickness is very
small compared to the other dimensions. In the Fick’s
law the volume change is not taking into account.
Equation (15) can be integrated to obtain the mass
of penetrant, M, absorbed by the sample at time t,
assuming that temperature and pressure stay constant
Based on the desorption equation, Equation (16), two
‘conventional’ analytical methods, called ‘short time
method’ and ‘long time method’, are used to determine
the diffusion coefficients at the beginning and at the
end of the desorption [
]. The analytical values of Dshort
and Dlong are given in Table 1.
2.2.1 Analytical Determination of the So Called `Short Time
Diffusion Coefficient' Dshort
Dshort predicts the beginning of the experimental
desorption curve. Its identification makes use of the
extrapolated point as previously defined by the polynomial
function of degree 4. The first part of the experimental
curve assumes to obey to a semi-infinite diffusion law.
Moreover, presuming that the diffusion corresponds to
a Fickian behavior, the curve M/Mmax versus t1/2/l is
linear between 1 and 0.5. In that case, the value of t/l2
for which the ratio M/Mmax equals ½ is expressed by
Equation (17) and Dshort is given by Equation (18) [
l2 1 ¼
]. Mmax represents the mass of the diffusing species
absorbed by the material at equilibrium, l stands for the
membrane thickness [
n¼0 ð2n þ 1Þp2 exp
Dð2n þ 1Þ2p2t!
2.2.2 Analytical Determination of the `Long Time Diffusion
Dlong predicts the end of the experimental desorption
curve. When M/Mmax is lower than 0.4, the desorption
equation can be written as Equation (19). K is a constant
By plotting ln(M/Mmax) as a function of time t, the
end of this curve is nearly a straight line whose slope
can be used to calculate the parameter Dlong as given in
The diffusion coefficient Dshort highly depends on the
first point of the semi-infinite desorption curve. This
point is estimated by extrapolating the desorption curve
since the decompression starts. Consequently, if this
point is overestimated, the coefficient Dshort
overestimates the beginning of the desorption kinetics.
Conversely, if this point is underestimated, the coefficient
Dshort underestimates the beginning of the desorption
kinetics. Even, the accuracy of the coefficient Dlong is
low because the assessment of the slope in Equation
(20) depends on the number of points experimentally
Dshort and Dlong coefficients increase when the applied
pressure rises. So, when the applied pressure is high, the
slope of desorption curve, at a given time, increases
(Fig. 1). Moreover, for a given level of pressure, the
necessity to use two diffusion coefficients in order to
describe at best the desorption kinetics means that the
parameter D depends on the gas concentration in the
It assesses the necessity to study the influence of gas
pressure and gas concentration on the diffusion
coefficients D, Equations (4, 5) and (8). This analysis can be
carried out by using optimizing methods [
] or by
numerically simulating the test of solubility. This later
numerical alternative consists in carrying out a
parametric study of D by numerically simulating the solubility
test with the Fick’s law. In Section 3, the comparison
of the results of these two analytical and numerical
methods makes it possible to conclude on their abilities to
evaluate the diffusion parameters at the beginning and
the end of desorption.
3 NUMERICAL SIMULATION OF THE BEHAVIOR OF PVF2 WITH CO2: IDENTIFICATION OF KEY COUPLINGS
To numerically simulate the diffuso-mechanical
behavior of the CO2/PVF2 system, the two types of couplings
are considered, namely the direct couplings, Equations
(9-12), and the indirect couplings, Equations (4, 5).
These laws have been implemented into the engineering
finite element code AbaqusTM by means of an User
ELement (UEL) subroutine, as applied in [
simulate the couplings, it is used 9 000 finite elements
defined in the UEL with 20 nodes and 27 integration
points. Details can be found in [
This tool gives the volume evolution of the sample and
the gas concentration in the sample during the solubility
test. The following Equation (21) expresses the mass of
the gas in terms of its gas concentration C, m(i)
represents the initial mass of the specimen:
mg ¼ 1 10 6C
To optimize the computing time, only a quarter of the
membrane with a square base and one section of the ring
have been modeled by taking into account the
appropriate symmetries. As regards the loading and the boundary
conditions, they have been defined so as to reproduce the
experimental test conditions, i.e., a sample placed on a
porous support in a pressure cell and subjected to a
variation of pressure and temperature. The contact between
the sample and the porous support has not been modeled
in a refined way during the simulation in order to
minimize the difficulty of the study.
Properties of the native PVF2 at 21 C are a Poisson’s
ratio of 0.38 and a Young modulus of 1 743 MPa [
The value chosen for the solubility coefficient is that
obtained experimentally at IFPEN (Tab. 2). In the
AbaqusTM code, the solubility parameter is expressed in
In the parametric study, the three unspecified
parameters are the diffusion coefficient D, and the direct
coupling parameters, ac and kl. The concentration and
pressure dependencies, Equations (4, 5) and (8), are
first studied neglecting thus the direct couplings (i.e., ac
= kl = 0). Then, the effect of the diffuso-elastic
couplings on the desorption kinetics and on the mechanical
behavior of the material is investigated by using a
constant diffusion coefficient adapted at each level of
pressure. Finally, this study compares these two kinds of
coupling and concludes on the type of coupling that
must be used so as to characterize as well as possible
the behavior of the PVF2 subjected to a CO2 pressure
3.1 Parametric Evaluation of the Indirect Couplings –
Influence of Pressure and Concentration on the
The indirect couplings describe the mass evolution of a
specimen during a solubility test, but not explicitly the
evolution of specimen volume. So, only the diffusion
kinetics is considered in this paragraph.
First of all, due to the uncertainty of the analytical
methods to precisely determine the diffusion coefficients
Dshort and Dlong as previously discussed, a numerical
identification of the two parameters is performed at each
level of pressure of the solubility tests. The simulations
are achieved to minimize the experience/model
difference and to identify the diffusion coefficients without
hypothesis. The minimization solves the problem of
diffusion in three-dimension (3-D, AbaqusTM) without
imposing any direction on diffusion flux. It uses the
Experimental 0.5 MPa
Experimental 1.5 MPa
Experimental 4 MPa
Dshort 1.45 10-13 0.5 MPa
Dshort 3.00 10-13 1.5 MPa
Dshort 2.42 10-12 4 MPa
Experimental 25 MPa
Experimental 50 MPa
Experimental 100 MPa
Dshort 90 10-12 25 MPa
Dshort 180 10-12 50 MPa
Dshort 360 10-12 100 MPa
classical diffusion element defined in AbaqusTM by means
of the Fick’s law. This parametric method permits to
reproduce as well as possible the beginning and the end
of the desorption kinetics by searching appropriate values
of Dshort (Fig. 5, 6) and Dlong (Fig. 7, 8). The new diffusion
coefficients can be compared with those previously
obtained thanks to the analytical methods (Tab. 1).
The analytical values with the numerical values
remain in the same order of magnitude (10 12 m2.s 1).
For low-pressure tests, lower than 1.5 MPa, Dshort has
certainly been overestimated by the analytical method
whereas for the other pressures it has been
underestimated. These discrepancies notably stem from the rough
extrapolation at the initial time of the experimental
desorption curve. So, even if the analytical method can
be used to quickly obtain the diffusion coefficients at
the beginning and the end of desorption, the numerical
simulation seems to be more suitable to determine these
parameters in a more precise way.
At the end of desorption, the diffusion coefficient
should be the same for the different pressures. The
results obtained with these two methods show that it
is necessary to use two diffusion coefficients to fit the
end of the desorption kinetics: one for the low-pressure
tests and another one for the high-pressure tests, higher
than the first. This result is confirmed as illustrated in
Figure 8 by simulating the solubility test at 25 MPa with
the same Dlong diffusion coefficient as that of the low
pressure tests (Tab. 1). In order to understand this
change in behavior, the following assumption is put
forth. The temperature during the sorption phase is
not the same for low-pressure tests (313 K) and for
Experimental 1.5 MPa
Experimental 4 MPa
D0 = Dlong (1.5 MPa) = 1.0 10-13, β = 3 10-4 (linear law)
D0 = Dlong (4 MPa) = 1.0 10-13, β = 3 10-4 (linear law)
high-pressure tests (403 K). A strong reorganization of
matter (i.e., molecular chains) is thus possible during
sorption at 403 K. However, decompression is preceded
by a quenching during which the polymer, perhaps,
does not have time to relax to its initial state of
equilibrium. Moreover, diffusion increases with temperature
]; it is thus possible that the polymer behavior after
quenching is equivalent to its behavior at 403 K,
explaining a faster desorption for high pressures than
for low pressures.
The diffusion parameter Dshort seems to increase
whatever the characterization method when the applied
pressure rises. For pressures higher than 0.5 MPa, this
dependency on pressure does not make it possible to
obtain the whole desorption kinetics (Fig. 5).
Undeniably, due to this dependency, for pressures higher than
1.5 MPa, only the beginning of the desorption curve with
a diffusion coefficient Dshort (Fig. 5, 6) or its end with a
diffusion coefficient Dlong (Fig. 7, 8) can be obtained.
The experimental desorption curve lies between the
two desorption curves simulated with the Fick’s law
and using the parameters Dshort and Dlong, respectively.
So, during desorption, the coefficient D decreases from
Dshort to Dlong. Consequently, beyond 0.5 MPa in which
Fick’s law is sufficient to describe the desorption phase
of a solubility test, the effect of concentration on the
diffusion parameter D becomes not negligible.
Linear and Exponential Diffusion Laws
In order to study the dependence of the diffusion
coefficient with gas concentration, linear Equation (4) and
exponential Equation (5) diffusion laws are incorporated
into the Rambert’s subroutine UEL [
For the pressures of 1.5 and 4 MPa, the solubility tests
are simulated with the linear law by using the diffusion
coefficient Dlong obtained at these pressures as a value
for the parameter D0, i.e., 1910 13 m2.s 1. A parametric
study with subroutine UEL has been performed to
determine the parameter b that induces the best prediction of
the beginning of the experimental desorption curve. This
parameter has the same value for the two pressures and
is taken as 3 9 10 4 (Fig. 9). With these parameters, only
the end of desorption curve at 4 MPa is not correctly
predicted, probably because of gas trapped in the material.
These results show that a linear dependence is sufficient
to correctly model the diffusion behavior of the PVF2 for
CO2 pressures up to 4 MPa.
The same type of simulation is carried out for the high
pressures tests by applying the linear and exponential
laws. The diffusion coefficient Dlong previously obtained
for these tests is chosen as a value for the parameter D0,
i.e., 1.5 9 10 12 m2.s 1. Numerical simulation results at
25 MPa are illustrated in Figure 10; the numerical results
are given for both linear and exponential laws. As well,
Fick’s law data previously calculated with a Dshort of
90 9 10 12 m2.s 1 are plotted (Fig. 6). The estimate of
the beginning of desorption kinetics is improved by
increasing the concentration dependency on mass
variation (or diffusion coefficient). However, even by
prescribing high dependencies on gas concentration, the
experimental desorption curve cannot be completely
Similar numerical analysis executed at 50 and
100 MPa permits to express the dependence of the
parameter b for each applied pressure by Equations
– for the Linear law:
b ¼ 6
– for the Exponential law:
These equations show that the gas concentration
influence on the diffusion coefficient in CO2/PVF2
system is predominant at high pressures. These results also
show that the classical laws of diffusion are insufficient
to predict the desorption kinetics for the high pressures
tests. To improve the evaluation of the diffusion of
CO2 in PVF2, the knowledge of the thermodynamic state
of the dissolved gas becomes point of interest [
raises a fundamental problem and would direct further
In the following part, a first approach is proposed
where the gas dissolved in polymer is described by a
simple chemical potential [
3.2 Evaluation of the Direct Couplings
The objectives of the section are to describe the
swelling of PVF2 and to evaluate the effect of the direct
couplings between the diffusion coefficient and gradient
of elastic volume strain (tree), Equation (12), on mass
Parametric Study of the ac Expansion Parameter
The volume deformation of the samples in the tests at
0.5, 1.5 and 4 MPa has not been experimentally
measured. Therefore, the ac parameter study is restricted to
high pressures. To obtain the ac expansion parameter
corresponding to the experimental volume strain
threshold at the end of sorption, that is to say the simulation of
the maximum swelling when the polymer is saturated by
gas, high pressure tests are numerically simulated by
uncoupling the diffusion phenomena Equation (12)
(kl = 0, ac 6¼ 0) and by using the diffusion coefficients
Dshort obtained previously with the numerical
simulations (Tab. 1). The unique value of the ac parameter
equals 0.6 (Fig. 11) [
By assuming that the membranes subjected to low
pressures have the same behavior than those subjected
to high pressures (i.e., ac is pressure independent (25 to
100 MPa)), their maximum volume strain can be
calculated (Tab. 3) by using the above parameter ac and by
inverting the mechanical constitutive Equation (10).
One must specify that in this model the diffusion
coefficient does not have any effect on the value of the ac
parameter; this assumption will have to be validated
during the experiments.
Parametric Study of the kl Coupling Parameter
For a constant diffusion coefficient, the diffusion
kinetics is now modified by the diffuso-mechanical coupling
which depends on the product of the kl and ac
parameters. By using the ac coefficient, as previously defined, the
effect of this coupling on desorption kinetics is then
explored. Figure 12 compares the numerical simulations
of the mass variation in the uncoupled case (i.e., Fick’s
law) and in the case of direct couplings (i.e., various
products (kl 9 ac).
The coupling induces a strong deceleration of the
desorption but is unable to account for the desorption
kinetics. In fact this coupling is strongly related to the
gradient of the elastic volume strain, which is perhaps
too small in the case of the solubility test studied here.
In order to have a significant effect, a high value of kl
Concerning the effect of the coupling on the volume
variations during desorption (Fig. 13), the remarks are
the same as for the mass evolution. The deceleration of
the volume strain can be correlated with that of the
desorption kinetics (Fig. 12).
Indeed, the increase in the sample size during sorption
involves an increase in distance to be crossed by the gas
molecules within the material. Thus, for a given diffusion
coefficient, these molecules must spend more time to
travel a given distance in the sample. The inverse effect
occurs when the material contracts during desorption.
In Figure 13, for a high coupling (kl = 3.45 9 10 14),
the sample volume continues to increase during
decompression whereas the quantity of gas starts to decrease.
It means that the contraction of the sample due to
desorption does not compensate for the volume
expansion related to the external pressure drop, because a high
coupling induces an important deceleration of the
diffusion of gas towards the outside. For example, at a
pressure of 25 MPa, during decompression the increase by
volume is 0.88%. This variation can be broken up into
two parts. One is induced by the hydrostatic pressure
drop which involves a volume increase by 1.03%, it is
a purely mechanical effect. The other one is related to
the contraction due to the desorption during the external
pressure drop, it is the effect of the coupling between
diffusion and mechanical phenomena. The latter is
obtained by the difference between the total increase
and that of mechanical origin, 0.15%. Then, as soon
as the external pressure does not vary any more, the
sample is subjected to the atmospheric pressure, and
desorption involves the contraction of the sample (Fig. 13).
We now propose to predict the beginning of the
desorption kinetics by using the parameters of direct
coupling (Fig. 14). The diffusion coefficient Dshort is
multiplied by a factor of two. The diffuso-elastic coupling
modifies the diffusion coefficient. So, even if this
coupling does not model correctly the kinetics desorption,
it must be taken into account in order to characterize
the diffusion coefficient of CO2 in PVF2.
3.3 Direct Couplings and Indirect Couplings
The discussion compares the identical parameters of the
direct couplings as defined in Figure 12. An indirect
coupling is incorporated into the diffusion coefficient law
Equation (5) by taking D0 = 1.5 9 10 12 m2.s 1 and
the law Equation (22) for the b parameter. The
diffuso-elastic coupling becomes negligible compared to
the influence of the gas concentration. At the end of gas
sorption, concentration in the material is high, inducing
a high diffusion coefficient (D0 = 3.0 9 10 12 m2.s 1)
due to the law describing the dependency on
Desorption kinetics of CO2, for high-pressure tests (25, 50
and 100 MPa), with constant Dshort (Fick’s law) multiplied
by a factor at each level of pressure and a diffuso-elastic
concentration. So, at the beginning of the decompression
the first term of Equation (12) is prevailing compared to
the diffuso-elastic coupling, the second term of
So, when investigating the behavior of PVF2 subjected
to a CO2 pressure fluctuation, it is necessary to
simultaneously consider the direct and indirect couplings. The
phenomenological laws, where D is a function of
pressure and concentration, can describe the non linear
behavior of CO2 diffusion in PVF2. However, the same
laws are unable to describe the mechanical behavior of
PVF2 during the sorption-desorption of CO2 in the
material. Concerning the direct couplings, they can
predict the change in volume of the material and the kinetics
of diffusion. Nevertheless, as soon as the influence of
pressure and concentration becomes important, the
introduction of the indirect couplings into the direct
couplings proves to be necessary in order to correctly
describe the diffusion stage.
The present study brings out some pertinent conclusions
about the type of diffuso-elastic couplings, that must be
considered in order to describe the behavior of PVF2
subjected to a CO2 pressure fluctuation.
The diffusion coefficient becomes pressure and gas
concentration dependent for pressures higher than
0.5 MPa. For the studied system, the effect of the
pressure on the diffusion coefficient is relatively
weak compared to that of the gas concentration.
The b parameter related to gas/polymer interactions is
a function of pressure. The prediction of desorption
kinetics is as better as the degree of the dependency on
concentration is high. Nevertheless for tests investigated
at high pressures ( 25 MPa), the phenomenological
laws, for which D is a function of pressure and gas
concentration, are insufficient to completely describe the
The direct diffuso-mechanical couplings from
Rambert et al. [
] give the volume strain threshold
of PVF2 at the end of the CO2 sorption with an
expansion coefficient connected to mass transport. This
expansion coefficient is independent of the applied pressure. In
this model, the coupling with the gradient of the volume
strain introduced into the diffusion equation is insufficient
to account for the highly non-linear behavior of the
diffusion kinetics (high pressures 25 MPa). This coupling can
be neglected compared to the effect of concentration on the
diffusion coefficient during a solubility test.
In order to simultaneously describe the volume
change of PVF2 and the diffusion kinetics of CO2 in
PVF2, it is necessary to consider concomitantly the
indirect couplings and the direct couplings.
For high CO2 pressure, desorption in PVF2 usually
takes place in a rubbery state and lasts several hundred
hours. It would be relevant in the future to consider a
diffuso-visco-elastic coupling rather than a
diffusoelastic one. Moreover, in order to take into account the
residual mass and volume strain in the material, it appears
necessary to develop a coupled model involving damage.
This research started by a project in polymer ‘blistering’
completed in co-operation with IFP Energies nouvelles,
Rueil-Malmaison, France [
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Manuscript accepted in October 2013 Published online in May 2014