A concentration-dependent diffusion coefficient model for water sorption in composite

Accepted Manuscript A concentration-dependent diffusion coefficient model for water sorption in composite S. Joannès, L. Mazé, A.R. Bunsell PII: DOI: Reference:

S0263-8223(13)00454-6 http://dx.doi.org/10.1016/j.compstruct.2013.09.007 COST 5339

To appear in:

Composite Structures

Please cite this article as: Joannès, S., Mazé, L., Bunsell, A.R., A concentration-dependent diffusion coefficient model for water sorption in composite, Composite Structures (2013), doi: http://dx.doi.org/10.1016/j.compstruct. 2013.09.007

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

A concentration-dependent diffusion coefficient model for water sorption in compositeI S. Joann`esa,∗, L. Maz´ea , A.R. Bunsella a Centre

des Mat´ eriaux, Mines-ParisTech, CNRS UMR 7633 BP 87, 91003 Evry Cedex, France

Abstract The phenomenon of diffusion coefficients depending on concentration has been observed and studied experimentally for many years. This is particularly the case for the interdiffusion in metals [1] but has also been observed on various polymeric systems [2]. This study presents a concentration-dependent diffusion coefficient model for water sorption in a polymer matrix composite. It has been developped by studying the water uptake of a polyphthalamide short glass fibre reinforced material. According to experimental data, we postulate that the diffusion coefficient follows a modified Arrhenius equation with a concentration dependence of the pre-exponential factor but also of the activation energy. This choice is related to the evolution of the glass transition temperature during sorption, which has a profound influence on transport properties. The coupled partial differential equation of the model is solved for some applied examples by using an easy-to-implement finite difference scheme. Keywords: molecular diffusion, concentration-dependent diffusivity, activation energy, polyphthalamides

1. Introduction For several decades, polymeric materials have been increasingly used in many industrial fields and amongst others, water distribution equipment is an example [3]. Indeed, ease of implementation, low production costs and adapted mechanical properties are assets that now allow polymers to replace traditional metallic materials such as copper alloys. For domestic “hot”1 water supply and structural component under water pressure, some of high-performance polyamides such as reinforced polyphthalamides (PPA) have proved to be efficient [4]. The sustainability of hydraulic equipment is nevertheless a crucial issue for the industry. Most materials evolve by interaction with their hydro-thermal2 environment [5] and to expand polymeric material use, designers need I Application

to a polyphthalamide short glass fibre reinforced composite. author Email address: [email protected] (S. Joann` es) 1 We consider here that hot is less than 70◦ C. 2 The term “hydro-thermal” implies a simultaneous exposure to relative humidity or water immersion and to elevated temperatures. Preprint submitted to Composite Structures September 13, 2013 ∗ Corresponding

predictive models of age-dependent properties [6, 7, 8, 9]. The present paper is an initial response to this objective since it proposes a model based on physical processes able to give the water concentration over time and anywhere in the material. All results reported in this paper relate to a polyphthalamide matrix reinforced with short glass fibers. The reasoning of this paper rests on the fact that the diffusion coefficient, which will be described in the following paragraph, depends not only on the temperature but also on the concentration of water itself. 1.1. The diffusivity, fondamentals on transport phenomena In 1855, based on the experiments of T. Graham [10], A. Fick was the first to propose a conceptualization of molecular diffusion process by studying salt movement in liquids [11, 12]. He drew considerable inspiration from the analytic theory of heat conduction published by J. Fourier thirty-three years earlier [13]. Diffusion is the phenomenon governing the transport of material just as conduction is the mechanism of heat transfer. In the case of molecular diffusion, molecules are transported from a higher concentration region to one of lower concentration; it is a spontaneous phenomenon. For a transient state diffusion (t denoting time), to know the rate at which concentration C is changing at any given point in space, we have to use the well known Fick’s second law (1)3 ; which can be written as (2) and (3) for a one dimensionnal problem, x denotes the space variable. D is the diffusion coefficient or diffusivity 4 in dimensions of [length2 .time-1 ]. Z Z ∂ C dΩ = ∇.J dΩ ∂t Ω Ω ∂C = −∇.J = ∇. (D ∇C) ∂t

− ⇒

∂C ∂t ∂C ∂t

= =

µ ¶ ∂ ∂C D ∂x ∂x ∂2C D if D is constant ∂x2

(1)

(2) (3)

An early criticism of Fick’s work was related to its assumption of a constant diffusivity (3) and numerous situations are known where this assumption is not valid. If the temperature-variance of diffusion coefficients was soon considered, especially by S. Arrhenius and J.H. van’t Hoff, it took a few years for a concentration dependency to be considered. For metal alloys, C. Matano was one of the first to prove that, “the diffusion rate of A atoms into a B atom crystal lattice is a function of the amount of A atoms already in the B lattice” [1]. From the 1960s, similar observations were made for the water sorption in polymers and composites [14, 15, 16, 2]. In the first section 3 ∇ denotes here the divergence operator and J the diffusive flux vector field, a molecular transport mechanism which tends to homogenize the intensive quantity even in the absence of macroscopic motion of the medium. 4 Which can be compared to the thermal diffusivity, i.e. the thermal conductivity divided by density and specific heat capacity at constant pressure.

2

of this paper, we focus on the experimental water sorption kinetics for a short glass fibres reinforced PPA matrix. The water uptake for various thermal conditions have been recorded between 40◦ C and 70◦ C, coupled with immersion or relative humidities comprised between 40% RH and 100% RH. We then show the sorption kinetics depend on a temperature and water concentration. 1.2. A molecular based model to account for the diffusivity concentration dependency As we have just pointed out, the effects of water and temperature can be followed by sorption measurements. Experimentally, the water uptake can be plotted versus time5 and different types of absorption mechanisms might be seen [17]. The simplest one is described by a Fickian law and generally applies to glassy or rubbery polymers outside the glass transition temperature (Tg) zone. Since the Tg plays a key role in the diffusion kinetics [18, 19], some authors observe a “concentration dependency” of the transport properties in its vicinity [20]. It is now widely accepted that this non-Fickian behaviour results from the coupling between the diffusion, the viscoelastic relaxation of the polymer and chain motions [21]. Over the last few decades, models that have been developed to describe diffusion in polymers have fallen into two main categories: (i) Free volume models that consider the molecules diffuse through free voids [14]. As the temperature or the concentration is changed, the free volume and the occupied volume will both evolve leading to the sought dependency. (ii) Molecular models that analyse the polymer chain motions due to the penetrant by considering molecular forces. A penetrant molecule may exist in a hole of sufficient size and can “jump” into a neighbouring hole once it acquires a sufficient energy. It is a thermally activated process and is linked to the apparent activation energy of diffusion entering into an Arrhenius law. We adopt below this second approach for which one of the pioneers was P. Meares [22]. Having experimentally highlighted the concentration dependency in the first section of this paper, we then postulate that the diffusion coefficient follows a modified Arrhenius equation with a concentration dependence of the pre-exponential factor but also of the activation energy (§ 3.2). In the last section, we use a finite difference scheme for solving the diffusion transient equation obtained and we show some applied examples. 2. PPA sorption kinetics Polyphthalamides are thermoplastic synthetic polymers already used in the automotive industry where their chemical resistance and temperature stability are sought; one example could be the turbine water pump or water valves [23]. The semi-aromatic nature of PPA gives them higher mechanical properties than their aliphatic polyamides counterparts, especially at high temperatures associated with humid conditions. Indeed, the presence of aromatic rings has many advantages such as an increase in the glass transition temperature but also results in better resistance to hydrolysis, and better dimensional stability. Both amorphous and crystalline grades are available and they could be reinforced by glass fibres, as is the case in this study. 5 Preferably

the square root of time.

3

To obtain such materials, it is necessary to replace a certain amount of the adipic acid of PA66 by a terephthalic acid (TPA) or isophthalic acid (IPA)6 . The PPA considered in this study is a block copolymer PA6T-PA6I for which all the adipic entering the composition of PA66 has been replaced by terephthalic and isophthalic acids. To further increase its dimensional stability, this copolymer is then reinforced by short glass fibres (50% in mass). 2.1. Water sorption experimental observations The water uptake in the material can be characterized by two physical quantities: the diffusion coefficient related to the kinetics of absorption and the saturated concentration related to the maximum amount of water that the material can absorb. These two parameters can be identified on thin plates with the sorption curves which connect the exposure time to the hydro-thermal environment. The samples which have been used in this study were one millimeter thick injection molded plates (100x100 mm2 )7 . Before ageing, all samples were dried over silica gel in an oven at 40◦ C and stayed at this temperature until stabilization of the mass. In this way the reference dry level for measuring the water uptake could be identified. Mass gains as a result of water uptake, were then recorded by removing the sample from its ageing environment and by weighing it periodically on a precision balance8 . During the sorption test, the evolution of the water concentration C in the composite can be followed as a function of time t by evaluating the ratio between the mass uptake ∆M = M (t) − M0 and the initial reference mass of the specimen M0 . It is then possible to know the sorption kinetics and if an equilibrium occurs, determine the saturated concentration value C∞ . Regarding the composite hydro-thermal environment, ageing mechanisms can occur at various levels: water and temperature affect the constituents properties but can also affect fibre/matrix interface. The latter is by far the most feared because it quickly leads to the loss of the the composite material “nature”. Water penetration can cause swelling and plastification of the matrix and the main effect is to lower the glass transition temperature by several degrees. Sorption kinetics are presented in Figure 3 with an analytical estimation of the diffusivity which will be described in § 2.2. Except for Figure 1, experimental data points reported in this paper correspond to the average of three samples with a corrected standard deviation of less than 3% for the largest variability. Below 70◦ C and whatever moisture conditions studied, no significant mass loss of the material was observed. It was also seen that in those hydro-thermal conditions the diffusion effects seems to be reversible: at the top of Figure 1, sorption data points are displayed for three samples under isothermal 70◦ C and 70% relative humidity conditions; the samples were subjected to a first sorption (black symbols), desorption (drying at 70◦ C not shown) and a new sorption (open symbols) of which results are superimposed to the first obtained. ´ [4] pointed out that, even if the Tg was modified during sorption, it Moreover, L. Maze recovered after desorption (Figure 1, bottom). From a microstructure point of view, no 6 ASTM-D5336-03,

Standard Specification for Polyphthalamide (PPA) Injection Molding Materials. sample size allow a mono-dimensional diffusion through the thickness to be considered, thus eliminating the difficulties associated with the injection induced anisotropy [4, 24]. However, it was possible to extrapolate the results to larger thicknesses, as long as the diffusion was kept mono-dimensional [4]. 8 Mettler AT250 with 0.01 mg readability to 50 grams and 0.1 mg readability to 200 grams capacity. 7 This

4

significant damage, i.e. fibre/matrix debonding, was observed in the saturated material at the microscopic level (SEM in Figure 2). Thus, we do not consider here the case of chemical ageing (chemical permanent modification of the polymer matrix) or microstructure irreversible changes and we can assume that the material showed “pseudo-Fickean” behaviour.

Water uptake

C

(%)

1,5

1,0

70°C RH70% First sorption

0,5

Second sorption after desorption

0,0 2

0,15

4

6

8

10

12

tan delta

Time (days

14

16

18

20

1/2

/h)

0,10

Reference 70°C RH70% After desorption

0,05

0,00 0

20

40

60

80

100

120

140

160

180

Temperature (°C)

Figure 1: At the top, first and second water uptake measurements for three samples showing a small experimental scatter and supporting the hypothesis of reversible sorption effects. At the bottom, tan delta curves obtained for unaged, aged and rejuvenated samples. The good superposition of curves 1 and 3 is also a sign of the water sorption reversibility.

2.2. Highlighting the concentration dependency Considering initially each hydro-thermal condition separately, assuming uniform initial concentration distribution, negligible mass transfer resistance and constant specimen thickness, it becomes possible to obtain a first estimation of the diffusivities and the saturated concentrations by fitting the equation (3) to the experimental curves. Therefore, we use a simplified analytical expression described in the following paragraph. For the partial differential equation (3), the evolution of concentration with time is proportional to the second derivative of concentration with respect to distance. Solutions of this partial differential equation are obtained when specific boundary and initial conditions are applied. Based on trigonometrical series9 , J. Crank developped simplified mathematical solutions [25] for various geometries10 . For an infinitely large plate of 9 The following solution technique was first proposed by J. Fourier for the heat equation in his treatise “Th´ eorie analytique de la chaleur”, published in 1822 [13]. 10 Infinite and semi-infinite medium, plane sheets, cylinders and spheres.

5

Figure 2: SEM observations of the fracture surface of a dried reference specimen (left) and a water saturated one (70◦ C immersed condition at the right). No significant difference between dry and wet specimens can be seen and the fibre/matrix interface seems not to be affected.

thickness h (membrane), the amount of water absorbed C (t) can be computed11 at any time t by the expression (4). ∞ C (t) 8 X 1 =1− 2 exp C∞ π n=0 (2n + 1)2

Ã

! 2 −Dπ 2 (2n + 1) t h2

(4)

The two parameters of a Fickean diffusion can be clearly seen to appear in equation (4): D, which is here independent of time, of space and of the concentration of the diffusing species, and C∞ which is the water uptake at saturation. For a long-term absorption, when t > τ /15 (with τ = h2 /D the characteristic diffusion time), we can approximate the Fickean kinetic described in (4) by relation (5). This simplified expression allows a first estimation of the diffusivities and saturated concentrations to be obtained12 . Fitted curves are shown in Figure 3 that will be discussed below. C (t) 8 ≈ 1 − 2 exp C∞ π

µ

−Dπ 2 t h2

¶ (5)

11 Considering that the membrane is initially at a zero and uniform concentration and the surfaces are kept at a constant concentration C∞ . 12 Calculated D and C ∞ values are reported in Table A.1 at the end of this paper.

6

First, consider the results obtained for isothermal conditions 40◦ C but with varied humidities or water immersion (Figure 3, top). Considering each ageing condition separately, a good fit is obtained between the analytical simulations and experimental points and clearly demonstrate the water effect on the saturated concentration: the more the surrounding environment is rich in “water”, the more the saturated concentration is high. This is a classical result often observed with polyamide matrix composites [26].

Water uptake

C

(%)

3,0 2,5 2,0 1,5 1,0 0,5 0,0

Water uptake

C

(%)

3,0 2,5 2,0

40°C Immersed

1,5 1,0 0,5

40°C Immersed

40°C RH100%

60°C Immersed

40°C RH70%

70°C Immersed

40°C RH40%

Crank identification

0,0 0

2

4

6

8

10

Time (days

12

14

16

18

20

1/2

/h)

Figure 3: Experimental data points (symbols) obtained from the average value of three samples for each hydro-thermal ageing condition: isothermal (top) and immersed (bottom) conditions. The fitted (solid) curves corresponds to the analytical identification of the diffusion parameters D and C∞ using (5).

It seems at first sight that only the saturated concentration is affected by the water environmental content. Nevertheless, normalizing and comparing the isothermal simulations of Figure 3 shows a manifest variation between diffusivities (Figure 4) since the curves do not overlap: diffusivity increases with the water content. If now, the case of water immersion is considered (Figure 3, bottom) at different temperatures, two effects can be seen: • The diffusion kinetics are thermally activated, i.e. the higher the temperature, the larger is the diffusivity. • The saturated concentration drops as the temperature increases. Whilst the first result is expected, the second at the bottom of Figure 3 is much less obvious and the temperature dependence of the equilibrium moisture content is not very well established. Under immersed conditions, many authors have reported that the maximum water uptake is insensitive to the temperature [15, 27], whilst others observed either positive or negative13 temperature dependences [28, 29, 30]. This phenomenon is 13 Sometimes

called the reverse thermal effect.

7

1,1

1,0

Normalized water uptake

/

C C

0,9

0,8

0,7

0,6

Crank simulations

0,5

0,4

40°C RH40% 40°C RH70%

0,3

40°C RH100% 0,2

40°C Immersed

0,1

0,0 0

2

4

6

8

10

12

14

16

18

20

1/2

Time (days

/h)

Figure 4: Normalized isothermal sorption kinetic simulations highlighting the concentration dependency: the curves do not overlap.

nevertheless not in the scope of this paper. Also, it is assumed in this study that the saturated concentration is a known and imposed value which can be obtained from phenomenological considerations14 [4] and we focus on the diffusivity dependency, on temperature, but also especially on water content. To better visualize this phenomenon highlighted in Figure 4, Figure 5 shows the D values previously identified on a log scale, as a function of the inverse of temperature T . Beyond the evident thermal effect, we find that for similar concentrations15 experimental points seems to be aligned and the diffusivity increases with the water environmental content. It now remains to define the form and to quantify this hydro-thermal dependence of the diffusivity; this is the purpose of the following sections.

b 14 In the literature, a power law C ∞ = a (RH%) is often proposed to relate the equilibrium moisture uptake to the relative humidity environment [15, 5]. To take into account the thermal effect, an extended expression has been used in this study C∞ = (a0 + a1 T ) (RH%)b where a0 , a1 and b are model parameters. 15 The size and color of the bubbles (D, T ) are proportional to the saturated concentration.

8

70°C

Water uptake

C

3,794E-7

(%)

60°C 3,0

2,5

2,0

40°C

2

(mm /s)

5,4573E-8

1,5

6,9526E-8

D

1,0

0,5

8,1676E-8

0 2

D (mm /s) from experimental data 2

Arrhenius interpolation D =5504,8 (mm /s) & E =64,8 (kJ/mol) 0

a

2

Arrhenius interpolation D = 158,3 (mm /s) & E =56,3 (kJ/mol) 0

7,6502E-8

Arrhenius interpolation D = 0

0,0029

a

2

9,6 (mm /s) & E =49,5 (kJ/mol) a

0,0030

0,0031

0,0032

-1

1/T (K )

Figure 5: The diffusivity D versus 1/T : for similar concentrations, experimental points seems to be aligned. Bubble size and color are associated to the equilibrium concentration and the diffusivity increases with the water environmental content.

3. The hydro-thermal diffusivity dependency 3.1. The thermal effect, an Arrhenius-type rate theory Most reactions are faster when the temperature rises. This is essence of the argument advanced by the Swedish scientist S. Arrhenius (1859-1927) which he gained the Nobel Prize in Chemistry in 1903, by proposing an empirical law16 that reflects this phenomenon observed experimentally in many cases. He wrote his paper of 1889 by describing reaction rates of several published chemical reactions [32]. Assuming that the activation energy Ea does not depend on the temperature, which is a only reasonable assumption over a limited temperature range, the Arrhenius law for the diffusivity17 fits equation (6) where D0 represents the pre-exponential factor, R denotes the ideal gas constant (R=8,314 J.mol-1 .K-1 ) and T the temperature in Kelvin. µ D = D0 exp

−Ea RT

¶ (6)

Because of the alignment of experimental points obtained by plotting on a log scale D versus 1/T (Figure 5), we postulate that, in the range of temperatures and concentrations considered, the water diffusion in the PPA reinforced composite studied follows an Arrhenius rate type relation. In Figure 5, a network of lines is used to “schematize” this 16 Building

on the earlier work of the Dutch scientist J.H. van’t Hoff (1852-1921) [31], the first Nobel laureate in Chemistry in 1901. 17 It took almost thirty years and the work of S. Dushman and I. Langmuir [33] to consider that the temperature dependence of the diffusivity in solids obeys an Arrhenius law.

9

relation but also shows the assumptions that we adopt for the modelling: we assume a dependence of D0 to the water concentration which is clearly visible and we consider that the activation energy Ea , given by the slope of the lines, is also affected by the water concentration. DMA tests have shown that water uptake affects the alpha18 transition temperature of the composite studied and that this phenomenon is reversible (Figure 1, bottom and [4]); this justifies an additional degree of freedom on Ea (see § 1.2). Having laid the foundations of our assumptions, we can now build the concentration-dependent diffusion coefficient model by following a modified Arrhenius relation. 3.2. Modelling the diffusivity dependency, a modified Arrhenius relation To take into account the above observations, the diffusivity will be described by a modified Arrhenius law in which both the pre-exponential factor and the activation energy depend on the water concentration. It is then necessary to choose two simple expressions for D0 : C 7→ D0 (C) and Ea : C 7→ Ea (C). We limit this model to a temperature range from room temperature up to 70◦ C (no irreversible phenomenon seems to happen). The model is not relevant above 70◦ C, it is nevertheless possible to extend its lower bound up to a few degrees only, by extrapolation. It can be seen from Figure 5 that the diffusivity converges with a decreasing temperature and it is assumed (i) that this convergence attained at a temperature below the lower temperature bound of the model, 0◦ C (or 3.66 10-3 K-1 ) for example. This gives first constraint to respect. Moreover, at a fixed temperature greater than this convergent point, γ it is postulated (ii) that D follows a power law α + β (C) depending on the concentra3 tion; α, β and γ are model parameters ((α, β, γ) ∈ R+ ). This second condition ensures that, on the considered hydro-thermal range scale, the diffusivity is strictly increasing with concentration and tend towards an asymptote when the concentration tends to zero. Based on these two conditions, it is easy to write the relationship between Ea and D0 . Indeed, for two different temperatures Ta and Tb between the convergent point and 70◦ C, the activation energy evolution is given by equation (7).

Ea =

1 Tb

R −

(ln Da − ln Db )

1 Ta

(7)

By imposing a constant Db and for example Tb =0◦ C (273.15 K) (according to (i)) and γ Da = α + β (C) for a given Ta > Tb (according to (ii)), we get equations (8) and (9), where Ta , Tb , Db , α, β and γ are the model parameters.

Ea (C) D0 (C)

=

1 Tb

R −

γ

1 Ta

= Db exp

µ

(ln (α + β (C) ) − ln Db ) Ea (C) RTb

(8)

¶ (9)

18 The T transition corresponds to the glass transition and occurs when large segments of the polymer α start to move. This transition represents a major transition for many polymers, as physical properties change drastically as the material goes from a hard glassy to a rubbery state.

10

The model parameters are then identified by the use of a non-linear least squares algorithm, thus leading to the fitted surface represented in Figure 6. Activation energies obtained from about 45 kJ/mol to 65 kJ/mol, are within a range usually reported in the literature for similar materials: short glass fiber reinforced PA66 was found to have an activation energy of 52.3 kJ/mol [26] and 58.65 kJ/mol [34].

Figure 6: Surface representation of the diffusivity D versus temperature and water uptake. The experimental data points correspond to the discrete symbols.

4. Practical validation through field testing Based on physical considerations, the model described above is a “representation” of the phenomena ocurring within the material. To be fully useful, the model needs to be evaluated and validated by the use of simulation. Since no simple analytical solution seems to exist in such a case of dependency, the development of accurate numerical approximations are essential to obtain quantitative evaluation of the transient equation solution. To solve the dependent coefficient diffusion equation, we have chosen to use the finite difference method. It is one of several techniques for obtaining numerical solutions to equation (1) or (2) and offers a convenient and an easy-to-implement way to obtain good approximations of the solution. Without going into details, for that purpose described in a separate paper [35], a Crank-Nicolson algorithm has been implemented and is used to provide all the results presented below. 4.1. Discussion about the model accuracy Using the set of coefficients previously identified (in Figure 6), a first validation step of the model can be obtained by simulating the experimental data points at 40◦ C, 60◦ C and 70◦ C. This first step aims at validating the proposed model in its “mathematical 11

form” and is to ensure that we are able to recover the experimental sorption curves and to confirm the activation energy effect. Figure 7 shows these simulations superimposed on the data points.

50°C immersed &

Finite dif. prediction

Water uptake

C

(%)

3,0 2,5 2,0 1,5 1,0 0,5 0,0

Water uptake

C

(%)

3,0 2,5 2,0

40°C Immersed

1,5 1,0 0,5

40°C Immersed

40°C RH100%

60°C Immersed

40°C RH70%

70°C Immersed

40°C RH40%

Finite dif. simulation

0,0 0

2

4

6

8

10

Time (days

12

14

16

18

20

1/2

/h)

Figure 7: Experimental data points (symbols) obtained from the average value of three samples for each hydro-thermal ageing condition: isothermal (top) and immersed (bottom) conditions. The fitted (solid) curves corresponds to the finite difference simulation using the set of coefficient previously identified in Figure 6.

We notice a good correlation between experimental points and simulated curves but we also find that the correlation is even better when we move away from the glass transition temperature zone19 . Although we slightly undervalue the water uptake on the first portion of the sorption curves, the activation energy effect is nevertheless respected: diffusivity increases with the water environmental content by a temperature-dependent factor. To continue the discussion about the accuracy of the model, we can say that times to saturation can be readily predicted; this is one of the model objectives that we have stated as industrial goals. A second validation step is performed to simulate an intermediate immersed ageing condition (50◦ C at the bottom of Figure 7) which has not been used for identification purposes. As in previous simulations, the finite difference prediction is quite accurate and indicates that the proposed model, with its set of coefficients, is appropriate to simulate water sorption kinetics below 70◦ C. Indeed, one of the model objectives is to use it as a predictive tool, mainly when it is long and difficult to measure the water uptake. For example, it is the case to estimate the time 19 The material investigated is a block copolymer and the start of the glass transition zone changes from one hundred degrees for a dry composite to about 60◦ C when saturated by water (Figure 1 and [4])

12

required to reach saturation with an evolving and “real” hydro-thermal environment, this is what we present in the following paragraph. 4.2. A predictive tool The simple and efficient tool implemented for solving the concentration dependency of the diffusion transient equation has been succesfully used to simulate cyclic hydro-thermal conditions [4, 35]. In this additional work, a hot water supply component is considered: it is a fictive four millimeters thick wall equipment installed in Geneva for didactic reasons. This equipment is thus subjected to the Geneva weather20 ingress on the one side and to liquid water (50◦ C) on the other side. Under those hydro-thermal conditions, it is shown that more than four years are needed to reach the sorption equilibrium. Due to the inertia of the sorption kinetic highlighted, we can wonder if it is still necessary to use such detailed and daily climate data? We can also wonder how temperature and humidity affect the time to saturation? All those questions related to the model sensitivity are adressed below. Also, as in [4, 35], let’s consider a fictive four millimeters thick wall water distribution component, which is now installed in Amsterdam-Schiphol in north Holland (wetter than Geneva) and in Malaga in southern Spain (mediterranean climate with dry hot summers and mild winters). Considering the average daily temperature20 , there is a difference of more than 10◦ C between those two cities and this is a difference of about 20% for the average daily relative humidity20 . The lower part of Figure 8 shows a day by day temperature and relative humidity profile for Amsterdam and Malaga. We can then wonder about the different operating conditions of this equipment, particularly towards the saturation time. The saturation time is, in itself, not conclusive but is instructive in estimating the period during which the material will evolve. Indeed, knowing the temperature and the water concentration of the material may allow the evolution of the mechanical properties of the equipment to be predicted and help to set the right thickness. By imposing the hydro-thermal boundary conditions presented in Figure 8 on the one side and a constant 50◦ C temperature on the other side (immersed conditions), we get the water uptake versus time plotted at the top of Figure 8. As for Geneva, it takes more than four years to saturate a four millimeters thick product. From this figure and under these environmental conditions, we can obtain some interesting information: (i) Due to the inertia of the sorption kinetic, the results of a day by day simulation are very closed to those obtained with annual average values; there is thus no need to get daily climate data to obtain good approximations of the water uptake under any hydro-thermal condition. (ii) Even if the weather is very different, the sorption kinetics are quite similar although we can notice a faster water uptake in the case of Spain which will be discussed later. The effect of the temperature inside is preponderant in relation to changes in external environmental conditions. Just as easily, the simulation allow the water diffusion within the material to be determined ([4, 35]), which is difficult to obtain experimentally. We can then plot the water profiles throughout the thickness and versus time. Such graphs are shown in Figure 9 for 20 Meteorological

data come from the European Climate Assessment & Dataset project (http://eca.knmi.nl) which provides daily hydro-thermal measurements for more than 7500 places throughout Europe and the Mediterranean.

13

2,5

Water uptake

C

(%)

3,0

2,0

Atacama prediction

1,5 1,0

Malaga

daily &

Schiphol

0,5

average sim.

daily &

average sim.

30 RH

Temperature (°C)

Temperature

25

90

(Schiphol)

(Malaga)

20

80

15 70

10 RH

5

Temperature

(Malaga)

60

(Schiphol)

Relative humidity (%)

0,0

0 0

200

400

600

800

1000

1200

1400

Time (days)

Figure 8: Above: comparison between a day by day sorption simulation applying hydro-thermal boundary conditions (described below for two cities) and the water uptake obtained with temperature and relative humidity average values.

Amsterdam and Malaga. In those figures, environmental conditions are applied on the left side and the constant immersed temperature condition on the opposite side. Along the vertical axis, the seasonal effect is particularly visible. By comparing the water profiles obtained for Malaga and Amsterdam-Schiphol, we find that the water uptake is indeed faster in the case of Spain although hydro-thermal conditions seem less severe. This reflects the key role of the temperature in the sorption kinetic, a key role that appears in the sampling-based sensitivity analysis by scatterplots (Figure 10). Although the water content is lower in Spain, the high temperature contributes to the diffusion in the first order. Finally, we can wonder about the effect of a very low humidity on the sorption kinetics. Situated on the western coast of South America, the Atacama desert is one of the world’s driest place, with an annual average relative humidity21 of about 36%. The annual average temperature21 (around 12◦ C) is nevertheless quite similar to that of Geneva or Amsterdam. In this extreme and latter case, it is still interesting to see at the top of Figure 9 that time to saturation remains roughly the same.

21 World

Meteorological Organization, http://www.wmo.int

14

1400

3,5

Schiphol

1200

Malaga airport

(Amsterdam Airport)

Spain

3,0

(%)

Netherlands

Water uptake

Time (days)

2,5

C

1000

800

2,0 600

1,5 400

1,0

200

0,50

0 0

1

2

3

4

0

Thickness (mm)

1

2

3

4

Thickness (mm)

Figure 9: Water profiles throughout the thickness of a four millimeters thick composite subjected to evolving environmental conditions on the left side and constant immersed temperature condition on the opposite side for two cities: Amsterdam-Schiphol (left) and Malaga (right). We can see the seasonal effect along the vertical axis and the water uptake is faster in the case of Spain compared to Holland.

Malaga

-8

2,5x10

Amsterdam-Schiphol 2,0x10

-8

1,5x10

-8

1,0x10

D

2

(mm /s)

-8

-9

5,0x10

1,0

1,5

2,0

Amount of water

-8

2,5x10

C

2,5

3,0

(%)

-8

1,5x10

-8

1,0x10

D

2

(mm /s)

-8

2,0x10

-9

5,0x10

270

275

280

285

290

295

300

305

Temperature (K)

Figure 10: Scatter plots of the output variable D against individual input variables C and T which gives a direct visual indication of the model sensitivity.

15

5. Conclusion This paper is focussed on the water diffusion in a polyphthalamide matrix composite reinforced by short glass fibres. Recently, this high performance material was introduced to replace traditional metallic equipments dedicated to domestic water supply and subjected to evolving hydro-thermal environments. It is obvious that the reliability of such composite structures, the behaviour of which is a function of both temperature and humidity and thus time, will require accurate models to predict their long-term evolution. It has been shown that the diffusivity is doubly activated: thermally on the one hand and water content on the other hand. Having experimentally highlighted this diffusivity dependency, we propose a physically based model that is able to provide the water uptake at any time and anywhere in the structure. This model is based on a modified Arrhenius rate dependent equation with a concentration dependence of the pre-exponential factor but also of the activation energy. We justify these molecular energy considerations by the evolution of the glass transition temperature which control the sensitivity of the time-dependent behaviour to both temperature and water concentration. Finally, some applied examples provide interesting industrial results about the water sorption kinetics in two European cities. The temperature is of the first order for this composite and without temperature, high humidity will have little influence on the water uptake. Acknowledgements The authors would like to warmly thank S. Vago for sharing his insights and field experience which served as the foundation for this study. Appendix A. Analytical identification results Ageing conditions

D (mm2 /s)

C∞ (%)

40◦ C 40◦ C 40◦ C 40◦ C 60◦ C 60◦ C 70◦ C 70◦ C 70◦ C

8.1676 × 10−8 7.6502 × 10−8 6.9526 × 10−8 5.4573 × 10−8 3.794 × 10−7 2.9657 × 10−7 8.221 × 10−7 6.1253 × 10−7 4.4067 × 10−7

2.8149 2.6974 1.3861 0.5488 2.4459 2.3437 2.3539 2.2556 1.1173

Immersed RH100% RH70% RH40% Immersed RH100% Immersed RH100% RH70%

Table A.1: Diffusivities D and saturated concentrations C∞ obtained for each condition by the analytical solution of the Fick’s equation for moisture sorption with a constant diffusivity.

16

References [1] C. Matano, On the relation between the diffusion-coefficients and concentrations of solid metals (the nickel-copper system, Japanese Journal of Physics 8 (3) (1933) 109–113. [2] S. George, S. Thomas, Transport phenomena through polymeric systems, Progress in Polymer Science 26 (6) (2001) 985–1017. [3] L. Laiarinandrasana, E. Gaudichet, S. Oberti, C. Devilliers, Effects of aging on the creep behaviour and residual lifetime assessment of polyvinyl chloride (PVC) pipes., International Journal of Pressure Vessels and Piping 88 (2–3) (2011) 99–108. ´, Etude et mod´ [4] L. Maze elisation du vieillissement hygrothermique de polyphthalamides renforc´ es fibres de verre courtes, Ph.D. thesis, Mines-ParisTech, France (2012). [5] A. Bunsell, J. Renard, Fundamentals of fibre reinforced composite materials, Taylor & Francis, 2005. [6] R. Guedes, J. Morais, A. Marques, A. Cardon, Prediction of long-term behaviour of composite material, Computers and Structures 76 (1–3) (2000) 183–194. [7] I. Sevostianov, V. Verijenko, B. Verjenko, Evaluation of microstructure and properties deterioration in short fiber reinforced thermoplastics subjected to hydrothermal aging, Composite Structures 62 (3–4) (2003) 409–415. [8] B. Oliveira, G. Creus, An analyticalnumerical framework for the study of ageing in fibre reinforced polymer composites, Composite Structures 65 (3–4) (2004) 443–457. [9] J. Mercier, A. Bunsell, P. Castaing, J. Renard, Characterisation and modelling of aging of composites, in: Proceedings of the “Mat´ eriaux 2006” conference, Dijon, 2006. [10] T. Graham, On the diffusion of liquids, Philosophical Magazine and Journal of Science 37 (251) (1850) 341–349. [11] A. Fick, Ueber Diffusion, Annalen der Physik und Chemie 170 (1) (1855) 59–86. [12] A. Fick, On liquid diffusion, Philosophical Magazine and Journal of Science 10 (63) (1855) 31–39. [13] J. Fourier, Th´ eorie Analytique de la chaleur, Firmin Didot, Paris, 1822. [14] J. Crank, G. Park, Diffusion in Polymers, Academic Press, London, 1968. [15] C. Shen, G. Springer, Moisture absorption and desorption of composites materials, Journal of Composite Materials 10 (1) (1976) 2–20. [16] S. Roy, W. Xu, S. Park, K. Liechti, Anomalous moisture diffusion in viscoelastic polymers: modeling and testing, Journal of Applied Mechanics 67 (2) (2000) 391–396. [17] Y. Weistman, Effects of fluctuating moisture and temperature on the mechanical response of resin plate, Journal of Applied Mechanics 44 (4) (1977) 571–576. [18] R. Kander, R. J. Clark, M. Craven, Nylon 66/Poly(Vinyl Pyrrolidone) Reinforced Composites: II. Bulk mechanical properties and moisture effects, Composites Part A: Applied Science and Manufacturing 30 (1) (1999) 37–48. [19] J. Vrentas, C. Vrentas, Effect of glass transition on the concentration dependence of self-diffusion coefficients, Journal of Applied Polymer Science 89 (6) (2003) 1682–1684. [20] Tech. rep. [21] S. Lustig, J. Caruthers, N. Peppas, Continuum thermodynamics and transport theory for polymer-fluid mixtures, Chemical Engineering Science 47 (12) (1992) 3037–3057. [22] P. Meares, Polymers: Structure and Bulk Properties, Van Nostrand, London, 1965. [23] Tech. rep. [24] B. Boukhoulda, E. Adda-Bedia, K. Madani, The effect of fiber orientation angle in composite materials on moisture absorption and material degradation after hygrothermal ageing, Composite Structures 74 (4) (2006) 406–418. [25] J. Crank, The Mathematics of Diffusion, Oxford University Press, London, 1956. [26] D. Valentin, F. Paray, B. Guetta, The hygrothermal behaviour of glass fibre reinforced PA66 composites: a study of the effect of water absorption on their mechanical properties, Journal of Materials Science 22 (1) (1987) 46–56. [27] E. McKague, Jr., J. Reynolds, J. Halkias, Moisture diffusion in fiber reinforced plastics, Journal of Engineering Materials and Technology 98 (1) (1976) 92–95. [28] L. El-Sa’ad, M. Darby, B. Yates, Moisture absorption by epoxy resins: The reverse thermal effect, Journal of Materials Science 25 (8) (1990) 3577–3582. [29] A. Chaplin, I. Hamerton, H. Herman, A. Mudhar, S. Shaw, Studying water uptake effects in resins based on cyanate ester/bismaleimide blends, Polymer 41 (11) (2000) 3945–3956. [30] S. Karad, F. Jones, D. Attwood, Moisture absorption by cyanate ester modified epoxy resin matrices. Part II. The reverse thermal effect, Polymer 43 (21) (2002) 5643–5649.

17

[31] J. van’t Hoff, Etudes de dynamique chimique, Frederik Muller, Amsterdam, 1884. [32] S. Arrhenius, On the reaction velocity of the inversion of cane sugar by acids, Zeitschrift fur Physikalische Chemie 4 (2) (1889) 226–248. [33] S. Dushman, I. Langmuir, The diffusion coefficient in solids and its temperature coefficient, Physical Review 20 (1) (1922) 113. [34] Z. Mohd Ishak, J. Berry, Effect of moisture absorption on the dynamic mechanical properties of short carbon fiber reinforced Nylon 6,6, Polymer Composites 15 (3) (1994) 223–230. `s, L. Maze ´, A. Bunsell, A simple method for modelling the concentration-dependent [35] S. Joanne water sorption in reinforced polymeric materials, Composites Part B: Engineeringin preparation.

18