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Factors governing water absorption by composite matrices
Composites Science and Technology 62 (2002) 487–492 www.elsevier.com/locate/compscitech
Factors governing water absorption by composite matrices I. Merdas*, F. Thominette, A. Tcharkhtchi, J. Verdu Ensam, 151 bd de L’Hopital, 75013 Paris, France Received 17 November 2000; accepted 19 July 2001
Abstract The water transport mechanisms in organic matrices were reconsidered in two ways. Attention is focused on solubility rather than concentration in the analysis of equilibrium properties and a model in which polymer/water interactions govern, at least partially, diffusion is proposed. This model allows us to explain a certain number of experimental results, for instance: the fact that the number of sorbed water molecules is lower than the number of polar groups and that it increases pseudo parabolically with this latter; and the facts that diffusivity is globally a decreasing function of hydrophilicity; that its apparent activation energy is generally close to that for solubility, and that it tends to vary in the same sense with structure. # 2002 Published by Elsevier Science Ltd. Keywords: Organic matrices
1. Introduction Humid ageing is widely recognized as one of the main causes of long-term failure of organic composites with an organic matrix exposed to the atmosphere or in contact with aqueous media. There are several recognised modes of humid ageing: by plasticization of the matrix; differential swelling related to concentration gradients; embrittlement linked to the degradation of the macromolecular skeleton by hydrolysis; osmotic cracking; hygrothermic shock with change of water state; damage localised at the matrix/fibre interface. In all of these cases, two characteristics play a significant roˆle: the affinity of the matrix for water (hydrophilicity), and the penetration rate of water in the material (diffusivity). The first of these factors has been extensively studied in the literature since the 1970s, but no consensus was reached. A brief summary follows. Many authors have suggested that the quantity of absorbed water at equilibrium is determined by the available free volume [1–2]. It is easy to demonstrate the inconsistency of this theory: liquids (rich in free volume) such as hydrocarbons or low glass transition temperature rubbers such as poly(dimethylsiloxane) are hydrophobic. It should be admitted, therefore, that the * Corresponding author. E-mail address: [email protected] (I. Merdas).
hydrophilicity is related to the existence of specific interactions between the water molecules and polar groups of the polymer. The hypothesis that the equilibrium water concentration is a molar additive function can lead to relationships having some predictive quality [3,4] if one considers only rather narrow families of materials. In larger structural series, however, it appears that the molar contribution of a given hydrophilic group, for example an alcohol, is a pseudo-parabolic function of the concentration of this group in the polymer [5]. Tendencies for variation of the hydrophilicity with the structure, for example the fact that epoxy-amine networks are generally more hydrophilic than unsaturated polyesters crosslinked by styrene, are well understood but we do not have a theory allowing to predict the equilibrium water concentration with reasonable precision. The equilibrium water concentration can be an increasing (e.g. in polyesters [6]), decreasing (e.g. in certain polyimides [7]) or independent (e.g. in certain epoxy-amine networks [8]) functions of temperature. No consistent explanation has been given, to our knowledge, for these trends. Concerning the mechanisms of diffusion and the structure/diffusivity relationships, they have been relatively little studied. The water diffusivity was linked to morphology [9] (the internodular areas, not very compact, would be preferential paths for the diffusion); to
0266-3538/02/$ - see front matter # 2002 Published by Elsevier Science Ltd. PII: S0266-3538(01)00138-5
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local molecular mobility [10] or free volume [11]. However, these theories do not seem to lead to a clear vision of the problem under study. It appeared interesting to us to reconsider the structure/properties relationships in this field, by privileging the following ideas: (i) Reconsider the hydrophilicity from thermodynamic point of view than and to rather study the water solubility in the matrix than its equilibrium concentration. (ii) Try to find a concept of the hydrophilic site (at the molecular level), which is compatible with the whole experimental data available. (iii) Not dissociate the concepts of solubility and diffusivity, in other words consider that the water/ polymer interactions play a significant role in the diffusion.
2. Experimental The essential of the experimental results have already seen reported [3–6,8,12]. Isothermal tests are carried out at temperatures ranging between 20 and 100 C. The test is retained when the mass gain is stabilized, in other words when the sorption equilibrium is established. Then, if M is the mass gain for a sample with initial mass Mo, one determines: M The relative mass gain m ¼ and its equilibrium Mo value is: m!m1 when t!1 The equilibrium water mass fraction in the material: m1 w1 ¼ 1 þ m1 The equilibrium water concentration in the material: w1 ðmol m3 ) C1 ¼ 18 103 where is the density of material (wet) in kg m3. Diffusivity is analyzed in the frame of the hypothesis of a Fickian mechanism. The relative mass gain is plotted against the square root of time. If the initial part of the curve (typically until m1/2) is linear, one determines p its slope dm/d t and one deduces coefficient of dif the L dm 2 pffiffi where L is the fusion D by the relation: D ¼ 4d t sample thickness. An example of sorption curves obtained with polyetherimide [12] is shown in Fig. 1. 3. Results 3.1. Plasticizing effect of water The water plasticizes the matrices by inducing decrease of their glass transition temperature Tg. This
Fig. 1. Water diffusion curves in PEI at different temperatures.
decrease is generally compatible with the free volume theory whose simplified version can be written: 1 1 ¼ þ Av Tg Tgp
ð1Þ
is the water volume fraction in the w material of density , the density of water being w. Tgp is the glass transition temperature of dry polymer and A a constant in theory given by: 1 1 A¼ ð2Þ Tgw Tgp where v ¼ w
where Tgw is Tg of water, close to 120 K. A is of the order of (6.0 0.5) 103 K1 for the majority of the composites matrices. By deriving Tg=f (w), one obtains: dTg ¼ AT2gp w dw
1 1 þ ATgp w w
2
ð3Þ
For moderate values of w (typically w40.07), one can use in a first approach: dTg ¼ AT2gp dw
ð4Þ
dTg 670 K For example, for Tgp 350 K one obtains: dw dTg for Tgp 500 K one obtains: 1580 K dw In other words, the decrease of Tg is of the order of 6.7 K per percent of absorbed water for a polymer of relatively weak Tgp (350 K), and more than twice for a material of relatively high Tgp (500 K). The tendency of variation of plasticizer effect with the structure (increase with the glassy transition temperature of polymer), and the order of magnitude of this effect ( 10 K by percent) are in excellent agreement with the experimental results [5,12,13]. This is a strong argument against theories according to which water occupies the free volume. Indeed, in order to achieve a plasticizer effect, it is necessary that the water dissolves
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itself in the matrix and forms a single phase with this latter. The industrial matrices under study seldom absorb more than 7% of water, which means that their Tg does not decrease more than 47 K if Tgp 350 K and of 110 K if Tgp 500 K. In a general way, under the use conditions, the material will remain in the glassy state, even if it is saturated by water. 3.2. Solubility, effect of the temperature Since in the studied cases, the concentration remains weak (w40.07) and the polymer does not change its state, one can in a first approach, consider that the equilibrium water concentration C obeys the Henry’s law: C ¼ Sp
ð5Þ
where S is the coefficient of solubility and p the water vapor pressure. S varies in theory with the temperature according to Arrhenius law: Hs S ¼ So exp with Hs : heat of dissolution: ð6Þ RT However, the sorption tests at various temperatures are made out at saturated vapor pressure. Between 20 and 100 C, the vapor pressure obeys also Arrhenius law: Hp ð7Þ p ¼ po exp with Hp ¼ 43 kJ mol1 RT The combination of Eqs. (5)–(7) gives: Hc C ¼ Co exp with Co ¼ So po and Hc RT ¼ ðHs þ Hp Þ
489
to Van Amerongen [14], the heat of dissolution would be related to the Lennard–Jones temperature TLJ of the diffusant by the following relationship: Hs
ð500 10TLJ Þ 1200 ðJ mol1 Þ R
ð10Þ
Water is characterized by an especially high cohesive energy density, leading to an especially high TLJ value (TLJ=809 K), which in turn leads, according to Eq. (10), to an especially low Hs value (Hs 63 kJ mol1). Experimental values of Hs are not so low, Hs450 kJ mol1), but they agree with the trend indicated by Eq. (10). Our results show that Hs increases in (absolute value) with the polymer polarity (and the cohesivity). This dependence must be taken into account in a possible structure/heat of dissolution relationship, which is not the case of the Van Amerongen relation. This latter could be modified as follows: Hs ¼ Tp 10TLJ R
ð11Þ
where Tp (which has the dimension of a temperature) would be a decreasing function of the polarity of polymer: typically Tp 4700 K for a polymer not very polar, like PET and Tp 2500 K for a polar polymer such a polyimide or an epoxy-amine rich in OH groups. The results of Van Amerongen [14] suggest a compensation effect, in other words the existence of a linear link between the logarithm of the preexponentiel coefficient So and the heat of dissolution Hs since: Ln So ¼ 6:65 0:05 TLJ
ð8Þ
All the experimental results can be explained if Hs is negative and close to Hp in absolute value. Thus, C will increase or decrease with temperature according to whether Hs is lower or higher than Hp in absolute value. Since Hs is not very different from Hp, one can generally write: 2 1 C ð9Þ Hp Hs RT C T This relation allows a rapid determination of Hs from slope C/T. The fact that the dissolution is very exothermic (negative Hs with a high absolute value), can be predicted by the structure/property relationships available on polymer/solvent interactions. For example, according
Our own results (Table 1), if they reveal a tendency of So to increase with Hs, do not confirm, however, the existence of a compensation phenomenon. In other words, So and Hs are certainly not controlled by the same structural factors. One can write: Ln So ¼ Ap 0:05 TLJ
ð12Þ
where Ap, from the data of Table 1, would increase with the polymer polarity (and hydrophilicity). Using Eqs. (11) and (12), one obtains: Hs ¼ Tp þ 200Ap þ 200LnSo R
ð13Þ
If Tp+200 Ap were constant, one would have a compensation effect, which a priori would be a coincidence. Thus we know that the characteristics of solubility (So, Hs) are qualitatively in agreement with the tendencies already observed in studies of polymer/solvent
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Table 1 Pre-exponentiel coefficient and apparent energy of solubility and diffusivity Polymer
Hs Do So (mol cm3 Pa1) (kJ mol1) (m2 s1)
PET 1.591012 PC 2.31014 UP 8.81014 PSU 2.41014 PMMA 6.91014 PA11 5.51014 PEI 1.41014 PPI 5.31014 DGEBA/DDA 3.51014 DGEBD/Etha 1.81014 PES 7.51016 DGEBA/Etha 3.41015 IP 960 6.01015
28 37 37 38 37 39 42 44 43 45 45 46 47
1.6105 2.3106 5.2107 8.6106 3.3105 1.4103 1.5105 4.5106 3.6 8.9103 1.7106 9.2106 3.4105
ED (kJ mol1) 42 32 33 33 43 54 43 42 80 61 32 39 43
. The distribution of the distances between close polar groups is Gaussian, the average distance is of course a decreasing function of the concentration but it is assumed that the standard deviation is constant. . A hydrophilic site consisting of two polar groups, is able to bind a water molecule by two hydrogen bonds.
To a hydrophilic site, a given pair of polar groups must be sufficiently close e.g. the distance between both groups must be lower or equal to a critical value dc. The probability to have a hydrophilic site is thus given by: ð dc p¼ o
interactions: dissolution is strongly exothermic when the polymer is strongly hydrophilic. This is still an argument against the theories according to which the water occupies free volume. Indeed, in this last case, dissolution should be athermic (Hs=0). 3.3. Nature of hydrophilic sites If one gives to So and Hs their usual physical meaning, one expects that So is dependent primarily on the concentration in hydrophilic sites and their configuration whereas Hs is rather related to the force of the interactions between water and these sites. This leads us to ask us about the nature of this latter. The general tendencies of the structure/hydrophilicity relationships can establish a hierarchy of the molar contributions of the various groups present in polymers,
H¼
M:m1 ¼ SHi 18
ð14Þ
where M is the molar mass of the monomer unit, H the number of water moles absorbed by a monomer unit and Hi the contribution of the ith elementary group present in the monomer unit. Eq. (14) applies well to relatively broad families of networks epoxy-amine [3,4]. However, as soon as one tries to apply it to wider families, one observes that the molar contribution of a given hydrophilic group, for example alcohol (HOH), is a pseudo-parabolic function of the concentration of this group [5]. Moreover, HOH is generally quite lower than unity, so we have to explain why some groups OH fix a water molecule and others not. On the basis of the observations made by the spectroscopists [15], according to which water is always doubly or perhaps triply bound, the following assumptions can be made:
1 ð d dc Þ 2 pffiffiffiffiffiffi exp dd 2 2 2
ð15Þ
and C ¼ p CH
ð16Þ
where C is the water equilibrium concentration and CH is the hydrophilic groups concentration. For epoxyamine networks, this model give excellent results with = 0.25 nm [5]. Is it possible to extend this model to polymers of low or moderate hydrophilicity and which is the influence of structure on dc and ? The research field is still largely open in this domain. 3.4. Diffusion By considering the sorption tests results, one can think that the diffusion is rather strongly thermoactived, whereas the equilibrium properties are not very dependent on the temperature. Since one considers the coefficient of solubility S, one is obliged to change point of view since Hs is of the order of magnitude (but of opposite sign) of the activation energy of the diffusion ED (Table 1). Is this a coincidence? To try to answer this question, we must take into account two significant observations: . When D is measured on both sides of Tg in some cases of epoxy-amine networks, one does not observe a discontinuous variation of D at Tg [5]. In other words, D does not appear to depend closely on segmental mobility into the polymer. . In relatively homogeneous structural series, like the epoxy-amine networks [4], diffusivity is a decreasing function of the water equilibrium concentration: the higher is the hydrophilicity, the slower is the diffusion (Fig. 2).
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The diffusion thus behaves as a Fickian process with an apparent Dapp such as: 2 ð18Þ Dapp ¼ D þ
Fig. 2. Dapp variation with 1/m1 for epoxy-amines systems.
These results indicate clearly that the water/polymer interactions play a role in the diffusion. The elementary jump of a water molecule from a site P1 to a site P2 could be tentatively schematized by the following sequence involving three distinct steps:
The results of sorption tests, for instance in the case of polyetherimide (Fig. 2) are generally interpreted in terms of Fickian model, with Dapp values. The relation (18) enables us to understand why Dapp is a decreasing function of the equilibrium concentration. Indeed the latter should be an increasing function of the ratio probability of trapping/probability of release, in other 2 Þ . words a decreasing function of ½ð þ Numeric simulations (studies in progress in our laboratory), confirm well that the curves of sorption can be followed by the Langmuir mechanism but it is still too early to outline the structure/properties relationships in this field.
4. Conclusions I.
[P1. . .W]
!
II.
W
!
III.
W+P2
!
P1+W
Dissociation of the polymer/water complex Jump from P1 to P2
[P2. . .W]
Formation of the polymer/water complex
The dissociation of the polymer/water complex, the water molecule migration from one site to another (governed by a traditional diffusion mechanism) and its capture by a hydrophilic site to reconstitute the complex. Schematically, we have to compare two rate constants k1 and k2 respectively linked to processes I and II: k1 is a decreasing function of polymer/water bond energy; and k2=D/d2 where D is the true diffusivity of water in the matrix and d is the distance between two close sites. For D high or d weak, the diffusion is controlled by the dissociation of the polymer/water complex. There is a formalism perfectly adapted to the analysis of this problem: the Langmuir model of diffusion, applied to the composites until now. We will not reconsider the detailed description of this exposed model because it is well described in the literature [16]. Let us recall that it involves three parameters: D the diffusion coefficient of free water molecules; the probability (by time unit) for a free molecule, to be trapped on a hydrophilic site and the probability (by time unit), for a trapped molecule, to be released. The curve of sorption obeys Fick in its initial part, where it pffiffi is linear in t. m 4 1 pffiffiffiffiffiffi One can show that : ¼ pffiffiffi Dt m1 þ L
ð17Þ
In the study of the structure/hydrophilicity relationships of polymers, the most pertinent parameter is probably solubility S rather than the water equilibrium concentration C1 and its variation with the temperature: if the heat of dissolution Hs < 43 kJ mol1, C1 increases with T, if Hs > 43 kJ mol1, C1 decreases with T. For many moderately polar polymers, Hs is close to 43 kJ mol1, which explains the fact that the water equilibrium concentration is not very dependent on the temperature. The relationships between solubility and polymer structure appear complex. One of the reasons of this complexity is probably the nature of the hydrophilic sites which one supposes that they consist of two groups enough close to form a complex with the water molecule. This explains the fact that only part of the polar groups is able to fix a water molecule and that the number of water molecules per polar group increases pseudo parabolically with the concentration of the polar groups. Several experimental observations, in particular the fact that (apparent) diffusivity seems to decrease when hydrophilicity increases, show that the polymer/water interactions influence the diffusion rate. This is not surprising when one knows that water establishes relatively strong hydrogen bonds with the polar groups. In such a case, the Langmuir diffusion model appears more adapted than the Fickian model to describe the water transport in polymer. This model appears compatible with the fact that apparent diffusivity is a decreasing function of hydrophilicity. These results can open new prospects for analysis of structure/property relationships in this field. They can lead us to consider the concept of water solubility in polymer with a high
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activation energy of dissolution, and to reject the free volume theory.
References [1] [2] [3] [4] [5] [6]
Adamson MJ. J Mater Sci 1980;15:1736–45. Johncok P, Tudgey GT. Br Polym J 1986;18:242. Morel E, Bellenger V, Verdu J. polymer 1985;26:1719–24. Bellenger V, Verdu J, Morel E. J Mater Sci 1989;24:63–8. Tcharkhtchi A, Bronnec PY, Verdu J. Polymer 2000;41:5777–85. Bellenger V, Mortaigne B, Verdu J. J Appl Polym, Sci 1989; 41:1225–33.
[7] Sacher E, Susko JR. J Appl Polym Sci 1979;23:2355–64. [8] See, for example: Zhou J, Lucas JP. Polymer 1999;40:5505–12. [9] Gupta VB, Drzal LT, Rich MJ. J Appl Polym Sci 1985; 30:4467. [10] Cukierman S, Halary JL, Monnerie L. Polym Eng Sci 1992; 35:534. [11] Vrentas JS, Duda JL, Ling HC. Macromolecules 1988;21:1470–5. [12] Merdas I, Thominette F, Verdu J. J Appl Polym Sci 2000; 77:1439–44. [13] Ellis TS, Karasz FE. Polym Eng Sci 1984;25:664. [14] Van Amerongen GJ. Rubb Chem Technol 1964;37:1069. [15] Fuller RT, Fornes RE, Memory J. J Appl Polym Sci 1979; 23:1871. [16] Carter HG, Kibler KG. J Compos Mat 1978;12:118–31.
Factors governing water absorption by composite matrices I. Merdas*, F. Thominette, A. Tcharkhtchi, J. Verdu Ensam, 151 bd de L’Hopital, 75013 Paris, France Received 17 November 2000; accepted 19 July 2001
Abstract The water transport mechanisms in organic matrices were reconsidered in two ways. Attention is focused on solubility rather than concentration in the analysis of equilibrium properties and a model in which polymer/water interactions govern, at least partially, diffusion is proposed. This model allows us to explain a certain number of experimental results, for instance: the fact that the number of sorbed water molecules is lower than the number of polar groups and that it increases pseudo parabolically with this latter; and the facts that diffusivity is globally a decreasing function of hydrophilicity; that its apparent activation energy is generally close to that for solubility, and that it tends to vary in the same sense with structure. # 2002 Published by Elsevier Science Ltd. Keywords: Organic matrices
1. Introduction Humid ageing is widely recognized as one of the main causes of long-term failure of organic composites with an organic matrix exposed to the atmosphere or in contact with aqueous media. There are several recognised modes of humid ageing: by plasticization of the matrix; differential swelling related to concentration gradients; embrittlement linked to the degradation of the macromolecular skeleton by hydrolysis; osmotic cracking; hygrothermic shock with change of water state; damage localised at the matrix/fibre interface. In all of these cases, two characteristics play a significant roˆle: the affinity of the matrix for water (hydrophilicity), and the penetration rate of water in the material (diffusivity). The first of these factors has been extensively studied in the literature since the 1970s, but no consensus was reached. A brief summary follows. Many authors have suggested that the quantity of absorbed water at equilibrium is determined by the available free volume [1–2]. It is easy to demonstrate the inconsistency of this theory: liquids (rich in free volume) such as hydrocarbons or low glass transition temperature rubbers such as poly(dimethylsiloxane) are hydrophobic. It should be admitted, therefore, that the * Corresponding author. E-mail address: [email protected] (I. Merdas).
hydrophilicity is related to the existence of specific interactions between the water molecules and polar groups of the polymer. The hypothesis that the equilibrium water concentration is a molar additive function can lead to relationships having some predictive quality [3,4] if one considers only rather narrow families of materials. In larger structural series, however, it appears that the molar contribution of a given hydrophilic group, for example an alcohol, is a pseudo-parabolic function of the concentration of this group in the polymer [5]. Tendencies for variation of the hydrophilicity with the structure, for example the fact that epoxy-amine networks are generally more hydrophilic than unsaturated polyesters crosslinked by styrene, are well understood but we do not have a theory allowing to predict the equilibrium water concentration with reasonable precision. The equilibrium water concentration can be an increasing (e.g. in polyesters [6]), decreasing (e.g. in certain polyimides [7]) or independent (e.g. in certain epoxy-amine networks [8]) functions of temperature. No consistent explanation has been given, to our knowledge, for these trends. Concerning the mechanisms of diffusion and the structure/diffusivity relationships, they have been relatively little studied. The water diffusivity was linked to morphology [9] (the internodular areas, not very compact, would be preferential paths for the diffusion); to
0266-3538/02/$ - see front matter # 2002 Published by Elsevier Science Ltd. PII: S0266-3538(01)00138-5
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local molecular mobility [10] or free volume [11]. However, these theories do not seem to lead to a clear vision of the problem under study. It appeared interesting to us to reconsider the structure/properties relationships in this field, by privileging the following ideas: (i) Reconsider the hydrophilicity from thermodynamic point of view than and to rather study the water solubility in the matrix than its equilibrium concentration. (ii) Try to find a concept of the hydrophilic site (at the molecular level), which is compatible with the whole experimental data available. (iii) Not dissociate the concepts of solubility and diffusivity, in other words consider that the water/ polymer interactions play a significant role in the diffusion.
2. Experimental The essential of the experimental results have already seen reported [3–6,8,12]. Isothermal tests are carried out at temperatures ranging between 20 and 100 C. The test is retained when the mass gain is stabilized, in other words when the sorption equilibrium is established. Then, if M is the mass gain for a sample with initial mass Mo, one determines: M The relative mass gain m ¼ and its equilibrium Mo value is: m!m1 when t!1 The equilibrium water mass fraction in the material: m1 w1 ¼ 1 þ m1 The equilibrium water concentration in the material: w1 ðmol m3 ) C1 ¼ 18 103 where is the density of material (wet) in kg m3. Diffusivity is analyzed in the frame of the hypothesis of a Fickian mechanism. The relative mass gain is plotted against the square root of time. If the initial part of the curve (typically until m1/2) is linear, one determines p its slope dm/d t and one deduces coefficient of dif the L dm 2 pffiffi where L is the fusion D by the relation: D ¼ 4d t sample thickness. An example of sorption curves obtained with polyetherimide [12] is shown in Fig. 1. 3. Results 3.1. Plasticizing effect of water The water plasticizes the matrices by inducing decrease of their glass transition temperature Tg. This
Fig. 1. Water diffusion curves in PEI at different temperatures.
decrease is generally compatible with the free volume theory whose simplified version can be written: 1 1 ¼ þ Av Tg Tgp
ð1Þ
is the water volume fraction in the w material of density , the density of water being w. Tgp is the glass transition temperature of dry polymer and A a constant in theory given by: 1 1 A¼ ð2Þ Tgw Tgp where v ¼ w
where Tgw is Tg of water, close to 120 K. A is of the order of (6.0 0.5) 103 K1 for the majority of the composites matrices. By deriving Tg=f (w), one obtains: dTg ¼ AT2gp w dw
1 1 þ ATgp w w
2
ð3Þ
For moderate values of w (typically w40.07), one can use in a first approach: dTg ¼ AT2gp dw
ð4Þ
dTg 670 K For example, for Tgp 350 K one obtains: dw dTg for Tgp 500 K one obtains: 1580 K dw In other words, the decrease of Tg is of the order of 6.7 K per percent of absorbed water for a polymer of relatively weak Tgp (350 K), and more than twice for a material of relatively high Tgp (500 K). The tendency of variation of plasticizer effect with the structure (increase with the glassy transition temperature of polymer), and the order of magnitude of this effect ( 10 K by percent) are in excellent agreement with the experimental results [5,12,13]. This is a strong argument against theories according to which water occupies the free volume. Indeed, in order to achieve a plasticizer effect, it is necessary that the water dissolves
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itself in the matrix and forms a single phase with this latter. The industrial matrices under study seldom absorb more than 7% of water, which means that their Tg does not decrease more than 47 K if Tgp 350 K and of 110 K if Tgp 500 K. In a general way, under the use conditions, the material will remain in the glassy state, even if it is saturated by water. 3.2. Solubility, effect of the temperature Since in the studied cases, the concentration remains weak (w40.07) and the polymer does not change its state, one can in a first approach, consider that the equilibrium water concentration C obeys the Henry’s law: C ¼ Sp
ð5Þ
where S is the coefficient of solubility and p the water vapor pressure. S varies in theory with the temperature according to Arrhenius law: Hs S ¼ So exp with Hs : heat of dissolution: ð6Þ RT However, the sorption tests at various temperatures are made out at saturated vapor pressure. Between 20 and 100 C, the vapor pressure obeys also Arrhenius law: Hp ð7Þ p ¼ po exp with Hp ¼ 43 kJ mol1 RT The combination of Eqs. (5)–(7) gives: Hc C ¼ Co exp with Co ¼ So po and Hc RT ¼ ðHs þ Hp Þ
489
to Van Amerongen [14], the heat of dissolution would be related to the Lennard–Jones temperature TLJ of the diffusant by the following relationship: Hs
ð500 10TLJ Þ 1200 ðJ mol1 Þ R
ð10Þ
Water is characterized by an especially high cohesive energy density, leading to an especially high TLJ value (TLJ=809 K), which in turn leads, according to Eq. (10), to an especially low Hs value (Hs 63 kJ mol1). Experimental values of Hs are not so low, Hs450 kJ mol1), but they agree with the trend indicated by Eq. (10). Our results show that Hs increases in (absolute value) with the polymer polarity (and the cohesivity). This dependence must be taken into account in a possible structure/heat of dissolution relationship, which is not the case of the Van Amerongen relation. This latter could be modified as follows: Hs ¼ Tp 10TLJ R
ð11Þ
where Tp (which has the dimension of a temperature) would be a decreasing function of the polarity of polymer: typically Tp 4700 K for a polymer not very polar, like PET and Tp 2500 K for a polar polymer such a polyimide or an epoxy-amine rich in OH groups. The results of Van Amerongen [14] suggest a compensation effect, in other words the existence of a linear link between the logarithm of the preexponentiel coefficient So and the heat of dissolution Hs since: Ln So ¼ 6:65 0:05 TLJ
ð8Þ
All the experimental results can be explained if Hs is negative and close to Hp in absolute value. Thus, C will increase or decrease with temperature according to whether Hs is lower or higher than Hp in absolute value. Since Hs is not very different from Hp, one can generally write: 2 1 C ð9Þ Hp Hs RT C T This relation allows a rapid determination of Hs from slope C/T. The fact that the dissolution is very exothermic (negative Hs with a high absolute value), can be predicted by the structure/property relationships available on polymer/solvent interactions. For example, according
Our own results (Table 1), if they reveal a tendency of So to increase with Hs, do not confirm, however, the existence of a compensation phenomenon. In other words, So and Hs are certainly not controlled by the same structural factors. One can write: Ln So ¼ Ap 0:05 TLJ
ð12Þ
where Ap, from the data of Table 1, would increase with the polymer polarity (and hydrophilicity). Using Eqs. (11) and (12), one obtains: Hs ¼ Tp þ 200Ap þ 200LnSo R
ð13Þ
If Tp+200 Ap were constant, one would have a compensation effect, which a priori would be a coincidence. Thus we know that the characteristics of solubility (So, Hs) are qualitatively in agreement with the tendencies already observed in studies of polymer/solvent
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Table 1 Pre-exponentiel coefficient and apparent energy of solubility and diffusivity Polymer
Hs Do So (mol cm3 Pa1) (kJ mol1) (m2 s1)
PET 1.591012 PC 2.31014 UP 8.81014 PSU 2.41014 PMMA 6.91014 PA11 5.51014 PEI 1.41014 PPI 5.31014 DGEBA/DDA 3.51014 DGEBD/Etha 1.81014 PES 7.51016 DGEBA/Etha 3.41015 IP 960 6.01015
28 37 37 38 37 39 42 44 43 45 45 46 47
1.6105 2.3106 5.2107 8.6106 3.3105 1.4103 1.5105 4.5106 3.6 8.9103 1.7106 9.2106 3.4105
ED (kJ mol1) 42 32 33 33 43 54 43 42 80 61 32 39 43
. The distribution of the distances between close polar groups is Gaussian, the average distance is of course a decreasing function of the concentration but it is assumed that the standard deviation is constant. . A hydrophilic site consisting of two polar groups, is able to bind a water molecule by two hydrogen bonds.
To a hydrophilic site, a given pair of polar groups must be sufficiently close e.g. the distance between both groups must be lower or equal to a critical value dc. The probability to have a hydrophilic site is thus given by: ð dc p¼ o
interactions: dissolution is strongly exothermic when the polymer is strongly hydrophilic. This is still an argument against the theories according to which the water occupies free volume. Indeed, in this last case, dissolution should be athermic (Hs=0). 3.3. Nature of hydrophilic sites If one gives to So and Hs their usual physical meaning, one expects that So is dependent primarily on the concentration in hydrophilic sites and their configuration whereas Hs is rather related to the force of the interactions between water and these sites. This leads us to ask us about the nature of this latter. The general tendencies of the structure/hydrophilicity relationships can establish a hierarchy of the molar contributions of the various groups present in polymers,
H¼
M:m1 ¼ SHi 18
ð14Þ
where M is the molar mass of the monomer unit, H the number of water moles absorbed by a monomer unit and Hi the contribution of the ith elementary group present in the monomer unit. Eq. (14) applies well to relatively broad families of networks epoxy-amine [3,4]. However, as soon as one tries to apply it to wider families, one observes that the molar contribution of a given hydrophilic group, for example alcohol (HOH), is a pseudo-parabolic function of the concentration of this group [5]. Moreover, HOH is generally quite lower than unity, so we have to explain why some groups OH fix a water molecule and others not. On the basis of the observations made by the spectroscopists [15], according to which water is always doubly or perhaps triply bound, the following assumptions can be made:
1 ð d dc Þ 2 pffiffiffiffiffiffi exp dd 2 2 2
ð15Þ
and C ¼ p CH
ð16Þ
where C is the water equilibrium concentration and CH is the hydrophilic groups concentration. For epoxyamine networks, this model give excellent results with = 0.25 nm [5]. Is it possible to extend this model to polymers of low or moderate hydrophilicity and which is the influence of structure on dc and ? The research field is still largely open in this domain. 3.4. Diffusion By considering the sorption tests results, one can think that the diffusion is rather strongly thermoactived, whereas the equilibrium properties are not very dependent on the temperature. Since one considers the coefficient of solubility S, one is obliged to change point of view since Hs is of the order of magnitude (but of opposite sign) of the activation energy of the diffusion ED (Table 1). Is this a coincidence? To try to answer this question, we must take into account two significant observations: . When D is measured on both sides of Tg in some cases of epoxy-amine networks, one does not observe a discontinuous variation of D at Tg [5]. In other words, D does not appear to depend closely on segmental mobility into the polymer. . In relatively homogeneous structural series, like the epoxy-amine networks [4], diffusivity is a decreasing function of the water equilibrium concentration: the higher is the hydrophilicity, the slower is the diffusion (Fig. 2).
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The diffusion thus behaves as a Fickian process with an apparent Dapp such as: 2 ð18Þ Dapp ¼ D þ
Fig. 2. Dapp variation with 1/m1 for epoxy-amines systems.
These results indicate clearly that the water/polymer interactions play a role in the diffusion. The elementary jump of a water molecule from a site P1 to a site P2 could be tentatively schematized by the following sequence involving three distinct steps:
The results of sorption tests, for instance in the case of polyetherimide (Fig. 2) are generally interpreted in terms of Fickian model, with Dapp values. The relation (18) enables us to understand why Dapp is a decreasing function of the equilibrium concentration. Indeed the latter should be an increasing function of the ratio probability of trapping/probability of release, in other 2 Þ . words a decreasing function of ½ð þ Numeric simulations (studies in progress in our laboratory), confirm well that the curves of sorption can be followed by the Langmuir mechanism but it is still too early to outline the structure/properties relationships in this field.
4. Conclusions I.
[P1. . .W]
!
II.
W
!
III.
W+P2
!
P1+W
Dissociation of the polymer/water complex Jump from P1 to P2
[P2. . .W]
Formation of the polymer/water complex
The dissociation of the polymer/water complex, the water molecule migration from one site to another (governed by a traditional diffusion mechanism) and its capture by a hydrophilic site to reconstitute the complex. Schematically, we have to compare two rate constants k1 and k2 respectively linked to processes I and II: k1 is a decreasing function of polymer/water bond energy; and k2=D/d2 where D is the true diffusivity of water in the matrix and d is the distance between two close sites. For D high or d weak, the diffusion is controlled by the dissociation of the polymer/water complex. There is a formalism perfectly adapted to the analysis of this problem: the Langmuir model of diffusion, applied to the composites until now. We will not reconsider the detailed description of this exposed model because it is well described in the literature [16]. Let us recall that it involves three parameters: D the diffusion coefficient of free water molecules; the probability (by time unit) for a free molecule, to be trapped on a hydrophilic site and the probability (by time unit), for a trapped molecule, to be released. The curve of sorption obeys Fick in its initial part, where it pffiffi is linear in t. m 4 1 pffiffiffiffiffiffi One can show that : ¼ pffiffiffi Dt m1 þ L
ð17Þ
In the study of the structure/hydrophilicity relationships of polymers, the most pertinent parameter is probably solubility S rather than the water equilibrium concentration C1 and its variation with the temperature: if the heat of dissolution Hs < 43 kJ mol1, C1 increases with T, if Hs > 43 kJ mol1, C1 decreases with T. For many moderately polar polymers, Hs is close to 43 kJ mol1, which explains the fact that the water equilibrium concentration is not very dependent on the temperature. The relationships between solubility and polymer structure appear complex. One of the reasons of this complexity is probably the nature of the hydrophilic sites which one supposes that they consist of two groups enough close to form a complex with the water molecule. This explains the fact that only part of the polar groups is able to fix a water molecule and that the number of water molecules per polar group increases pseudo parabolically with the concentration of the polar groups. Several experimental observations, in particular the fact that (apparent) diffusivity seems to decrease when hydrophilicity increases, show that the polymer/water interactions influence the diffusion rate. This is not surprising when one knows that water establishes relatively strong hydrogen bonds with the polar groups. In such a case, the Langmuir diffusion model appears more adapted than the Fickian model to describe the water transport in polymer. This model appears compatible with the fact that apparent diffusivity is a decreasing function of hydrophilicity. These results can open new prospects for analysis of structure/property relationships in this field. They can lead us to consider the concept of water solubility in polymer with a high
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activation energy of dissolution, and to reject the free volume theory.
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