Accepted Manuscript. Toward a constitutive model for cure dependent modulus of a high temperature epoxy during the cure

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1 Acceted Manuscrit Toward a constitutive model for cure deendent modulus of a high temerature eoxy during the cure M Zarrelli, A.A Skordos, I.K Partridge PII: S (10) DOI: /j.eurolymj Reference: EPJ 5329 To aear in: Euroean Polymer Journal Received Date: 22 January 2010 Revised Date: 20 Aril 2010 Acceted Date: 7 June 2010 Please cite this article as: Zarrelli, M., Skordos, A.A, Partridge, I.K, Toward a constitutive model for cure deendent modulus of a high temerature eoxy during the cure, Euroean Polymer Journal (2010), doi: /j.eurolymj This is a PDF file of an unedited manuscrit that has been acceted for ublication. As a service to our customers we are roviding this early version of the manuscrit. The manuscrit will undergo coyediting, tyesetting, and review of the resulting roof before it is ublished in its final form. Please note that during the roduction rocess errors may be discovered which could affect the content, and all legal disclaimers that aly to the journal ertain.

2 Toward a constitutive model for cure deendent modulus of a high temerature eoxy during the cure M Zarrelli 1*, A A Skordos 2 and I K Partridge 2 1 Institute of Comosite and Biomedical Materials (IMCB) Research National Council (CNR), P.le E. Fermi, Portici 8055 (Nales) Italy 2 Comosites Centre, Cranfield University, Cranfield, Bedford, MK43 0AL, UK Abstract A constitutive model, based on Kohlrausch-Williams-Watts (KWW) equations, was develoed to simulate the evolution of the dynamic relaxation modulus during the cure of a high temerature eoxy. The basic assumtion of the modelling methodology roosed is the equivalence of the mechanisms underlying the evolution of the glass transition temerature and the relaxation time shift during the cure, leading to the use of a common otential function. This assumtion is verified by the comarison of normalised glass transition data and rincial relaxation times, which have been found to follow a single master curve. Results show satisfactory agreement between exerimental data and model rediction over the range of chemical conversion considered. Keywords: eoxy, comosites, relaxation times, master curve, modelling *Corresonding author: Dr Mauro Zarrelli CNR-Research National Council IMCB-Institute of Comosite and Biomedical Materials P E Fermi, Granatello, Portici (Nales), Italy mauro.zarrelli@imcb.cnr.it 1

3 1 Background During the olymerization of any synthetic thermosetting olymer the material undergoes chemical and hysical transformations which lead to the hardening of the material, with a corresonding transition from a viscous liquid to an elastic solid. Under normal service conditions the material is usually far below its glass transition temerature and follows linear elastic behaviour. However, during the manufacturing rocess, the olymer matrix exhibits viscoelastic behaviour which affects the develoment of residual stresses. If the resin is used as the matrix in a continuous fibre reinforced comosite, such residual stresses can lead to significant distortion in the final art. An accurate model for the develoment of residual stresses and the monitoring of deformations during the manufacturing stage requires a quantitative sub-model of the gradual evolution of the matrix mechanical roerties. Overestimation of the residual stress induced during the rocess can be eliminated only by considering the actual value of viscoelastic modulus evaluated at intermediate degrees of chemical conversion (degree of cure). Modelling of the modulus evolution is necessary for the analysis of both mechanical and non-mechanical loads develoed during the manufacturing rocess, due to interactions between the tooling and the often comlex material arrangement. The deendence of the viscoelastic roerties of thermosetting systems on the degree of cure has been the subject of only very few investigations, both in terms of exerimental results and modelling aroach. Suzuki et al.[1] resented relaxation data for eoxies cured following various cure cycles. The cure state of these samles was not determined and the relationshi between mechanical roerties and cure evolution could not be exlored. In a work by Kim et al. [2], a fixed value is assumed for the elastic modulus of the uncured and the fully cured systems, while a linear relationshi is assumed between the elastic modulus and the degree of conversion, in the entire region between gelled resin and its glassy state. White et al.[3] used a viscoelastic formulation to 2

4 study warage and residual stresses in asymmetric comosite laminates during cure, assuming a time- and cure- deendent model to simulate transverse modulus. Investigations of the behaviour of neat resin at various ost-gelation degrees of conversion under relaxation mode for various temeratures [4] have led to the develoment of a model of the relaxation modulus at lower degree of cure based on the Adams-Gibbs [5] formulation for the volume relaxation in glasses. Yi et al. [6] included viscoelastic effects in the analysis of residual stress develoment in laminated lates using a chemo-thermo-viscoelastic constitutive equation and considering a discontinuous model for the mechanical roerties with a ste at the gel oint. O`Brien et al. [7] have reorted a series of exeriments on the unreacted, artially reacted and ost-gelled resin. Simon et al. [8] have resented a study on the cure-deendent storage modulus of a commercial toughened eoxy resin, giving a general methodology to model the time-temerature-conversion effects in viscoelastic of thermosets. The resent study focuses on modelling the evolution of resin viscoelastic modulus during the cure. Exerimental results reorted in a comlementary study [9] are analysed in the context of KWW models which is modified to account for the deendence of mechanical roerties on conversion. The considered model uses the concet of rincial relaxation time for each degree of conversion with rincial relaxation times normalized with resect to the value for the fully cured state in order to find a suitable correlation with structural changes. Assuming that the same mechanism of structural evolution governs both the develoment of the glass transition temerature and of the mechanical modulus, an identical otential function can be adoted. This assumtion leads to an excellent agreement of the exerimental data with the model redictions. These results are suorted by dielectric relaxation measurements resented in the literature [10]. 3

5 2 Materials and Methods The synthetic resin system used in the resent work was a re-commercial formulation based on a tetrafunctional eoxy, cured with an amine hardener and containing an aromatic thermolastic comonent. Whilst the exact comosition of the resin is unimortant, the blend can be considered to be reresentative of the class of resins known as high temerature aerosace grade toughened eoxies. Partial curing of laques of the resin was erformed in a cavity formed by two glass lates and an aluminium sacer. Each of the lates was covered with a layer of PTFE/glass release film as well as liquid release agent (FREEKOTE 700). Five different laques were reared following isothermal curing schedules with a dwell temerature in the o C range. The final degree of conversion of the five lates as calculated using cure kinetics information on the resin system [11] was 0.68, 0.80, 0.87, 090 and Dynamical mechanical tests using a TA Instruments DMA, tye 2940, were erformed on rectangular stris taken from each of the artially cured late. A threeoint bending test mode configuration was emloyed. Each samle was equilibrated for about 5 minutes before a test segment starting at each temerature, to rovide a uniform temerature distribution inside the material. Three secimens were tested for each degree of conversion. Stress relaxation data were catured during each segment. More details of standard exerimental rocedures and data treatment for DSC and DMA analysis can be found in reference [9]. 3 Modeling Results 3.1 Cure-deendent ultimate relaxation modulus The variations of ultimate modulus were found to be significant and an aroriate model is develoed to exress the deendence on the degree of cure. Within the gelation region, where the 4

6 three-dimensional network starts to form, the modulus rises from a very low value of 1.38 MPa to a value of 60.9 MPa at temerature above the corresonding glass transition temerature.. Ultimate Equilibrium Modulus (GPa) Exerimental data Fitting function Degree of conversion Figure 1 - Exerimental oints and fitting curve for ultimate relaxed modulus The cure deendence of the equilibrium relaxation modulus is modelled using an emirical function roosed by O`Brien et al.[7], which exresses the value of relaxed modulus as a function of the degree of cure assuming a very low modulus value before gelation: [ E ( α )] D E log = C + Eq. 1 E α gel α FE 1+ e where C E, E D and F E are fitting arameters and α gel is the conversion at gelation (0.66 for the resin system of this study). It should be noted that this model is valid for degrees of cure over the gelation conversion. The values of the arameters obtained from the fitting of the exerimental results are reorted in table 1. Figure 1 shows a comarison between fitting results and exerimental data for the ultimate relaxation modulus indicating a close agreement. 5

7 3.2 Shift Factor: results and henomenological model For viscoelastic materials, the mechanical resonse is history-deendent and involves the use of a reduced time which can be exressed using a temerature-cure deendent shift factor as follows: t ξ = Eq. 2 a τ ( α, T ) where ξ is the reduced time, t is the actual exerimental time and a T ( α,t ) is the degree of cure deendent temerature-conversion shift factor. Figure 2-3D lot of exerimental shift factors vs. temerature and degree of cure For olymeric materials, shift factors are generally functions of temerature; in the case of a reactive system the deendence on fractional conversion needs to be taken into account in imlementing the model. Figure 2 shows the horizontal shift factors used to generate the master curves of stress relaxation modulus assuming as reference temerature the value of 302 K as indicated by the initial value of each shift factor curves in fig. 2 and 3. 6

8 Figure 3 - Shift factors for artially cured samles ( ) fitted with WLF and linear models As in the works of O Brien et al. [7] and Simon et al. [8], shift factor curves cannot be modelled using a WLF tye equation. The Williams-Landel-Ferry model can be alied only to the first art of each curve; for the remaining art, a linear shift factor model shows better agreement with the exerimental oints T-Tg ( C) ex 0.96 model 0.90 ex 0.90 model ex 0.80 model 0.87 ex 0.87 model 0.68 ex 0.68 model log(aτ) (a) 7

9 T-Tg ( C) (b) log(aτ) Figure 4 (a) Exerimental shift factor data vs. (T-Tg) and model fitting at each conversion (b) Model redictions over the full degree of cure range The shift factor curves for degrees of cure equal to 0.80, 0.87, 0.90 and 0.96 are illustrated in figure 3, alongside a linear and a WLF model fit in the high and low temerature region resectively. Both arts of each curve show very good agreement. Comarisons of these curves with the tan δ curves, obtained by torsional rheometry resented in [10] indicate that the intersection oint of the WLF and the linear models corresonds to the onset of the ost cure reaction. Presenting exerimental shift factors as a function of the differencet Tg, where T is the test temerature and T g the glass transition temerature for the corresonding level of conversion, allows the use of the following exression to model the shift factors uon the degree of cure and temerature: log C3 α ( C a ) = C α + C + ( C α + C )( T T ) + Eq.3 9 τ C + C g 5 α C4 aτ The fit of the model resented in eq. 3 to exerimental data, as obtained using the generalised reduced gradient nonlinear otimisation method imlemented in the solver of Microsoft Excel, is illustrated in figure 4.a. The values of the nine arameters of the model are given in table 2. The 8

10 redictions of the model described by eq. 3 over the whole degree of cure range are shown in figure 4.b. It has to be noticed that at this stage, Eq. 3, was build without any hysical meaning associated with the conversion deendent modulus. 3.3 Kohlrausch-Williams-Watts Model (KWW) The time-deendent roerties (mechanical, dielectric, thermal) of glass-forming materials (organic, mineral and metallic) follow the stretched exonential or so-called Kohlrausch-Williams-Watts [12-15] function (KWW). For a generic olymer system the modulus can be written as follows: ( t) β τ t E = E e Eq. 4 where β and τ, are two constants characteristic of the material and E is the fully relaxed modulus. The hysical meaning of the two arameters can be identified by considering the characteristic bell shae of the relaxation time distribution associated with the curve of relaxation modulus. The value of τ (called the characteristic relaxation time) reresents the central oint of the time distribution function, whereas β (called the non-exonential arameter) is inversely related to the breadth of the relaxation time distribution. The non-exonential arameter, ranges from 0 to 1, with β = 1 corresonding to a single-relaxation time Debye resonse. Lower values of β tyically reflect increased intermolecular co-oerativity as influenced by the chemical structure of the olymer, as well as otential constraints owing to the resence of crystallinity or crosslinks [16]. Fitting of the exerimental master curves was carried out using the Levenberg-Marquardt non-linear least squares fitting algorithm imlemented in Origin v6.0; the values of the KWW arameters for each degree of cure are reorted in Table 3. A high correlation coefficient of about has been obtained. Artificial oints have been added in order to force the fitting within the glassy state, adoting an average value of 3.43 GPa for the glassy relaxation modulus. 9

11 Eq. 4 imlies that different materials are characterised by distinct values of the arameters β and τ. The same assumtion can be made for the resin during the cure: at each degree of cure a air of KWW arameters ( τ ) β, can be evaluated. Consequently, a functional deendence for the values of these arameters on the degree of cure can be established. For a given reference temerature, T, and by using different values of the arameters ( τ ) ref general form of degree a cure deendent KWW model is obtained: E ( t T, α ) = E ( α ) ξ ( t, T, α ) τ ( α ) β ( α ) β, for each degree of cure, the following, e Eq. 5 The deendence of arameters β and τ on the degree of cure has not been investigated widely. However, Mijovic et al. [10] have obtained satisfactory results using a simle linear model for the non-exonential arameter β, as follows: ( α ) c 1β α + c β = Eq. 6 β 2 where c 1β and 2β c are fitting arameters. Figure 5 shows the linear fit for the values reorted in Table 3, with arameters c 1β, 2β c equal to and resectively. The increase in the breadth of the relaxation time distribution imlied by this deendence is related to the increase in the comlexity of the olymer network during cure which leads to a reduction in the mobility of the olymer molecules. 10

12 0.50 Beta arameter, Degree of conversion Figure 5 - Linear fitting of the beta arameter values A direct consequence of the increased network comlexity and resence of entanglements is a reduction in relaxation rate. For a generic olymer system, stress relaxation time decreases as the temerature increases, with a shar reduction in the modulus value over its glass transition temerature. In the case of an eoxy resin under stress, relaxation times increase with increasing degree of cure. Assuming that the molecular mobility influences the glass transition through the same mechanism that controls the stress relaxation function, the behaviour of the glass transition temerature at a fixed degree of cure can be normalized with resect to its value for a given reference conversion. This aroach leads to the definition of a otential function, which is identical to that obtained by normalizing the characteristic relaxation times with resect to the relaxation time at the same conversion. Thus: T T g g ( α ) ( α ) ref T g ( α ) = g Eq. 7 11

13 where ( α ) T is the glass transition temerature as a function of the degree of cure, and ( ) g T α is the reference conversion. If the same mechanism drives the change in normalised relaxation time then: ( α ) τ log = g ( α ) τ ( α ) τ Eq. 8 ref with ( α ) g ( α ) g T = Eq. 9 g τ where ( ) α τ is the relaxation time exressed as a function of the degree of cure, and ( ) value at the reference conversion α ref. g α ref ref τ is its Figure 6 - Normalized values of single relaxation time vs conversion with normalized values of glass transition temerature obtained exerimentally by thermal analysis and torsional rheometry alongside the Di Benedetto model (solid line) 12

14 Relaxation Modulus E(t) [GPa] KWW Model Predictions E E E E E+18 Log[ξ(t,T)] (min) (a) Relaxation Modulus E() [GPa] Fully cured E E E E E+19 Log[ξ(t,T)] (min) (b) Figure 7 (a) Model redictions and exerimental stress relaxation modulus curves. (b) Prediction for the relaxation modulus over the full degree of cure range. 13

15 Figure 6 reorts the normalised values of the single relaxation time τ and the glass transition temerature values taken from the DSC tests and tanδ curves; the best fitting obtained using the Di Benedetto [17-18] equation, commonly used to model the evolution of the glass transition temerature, is also suerimosed. The mentioned Di Benedetto model is a single fitting arameter model already used by the author for the same material to model the evolutionary rogression of glass transition temerature with degree of cure, and it can be suitable imlemented in comutational routine. The results show very good agreement for the two otential functions, thereby suorting the assumtion that the assumtion of a common effective mechanism that governs the change in glass transition temerature with the rogress of the reaction and the change in stress relaxation times during cure. It is imortant to note that measurements of the glass transition temerature made using different exerimental techniques will yield different results, due to the inherent time scales, relaxation mechanisms and secimen sizes associated with secific methods [19-20]. In order to aly correctly the assumtions underlying eq. 9, comarisons need to be based on congruent measurements. For this reason, values of the glass transition temerature were taken as the temerature of the eak in tanδ obtained by solid torsional rheometry. The stress relaxation modulus has been evaluated for all the levels of conversion using the system of eqs. 3 and 5-7 (or 8, deending on the exerimental data, glass transition temerature or rincial relaxation times, used to comute the otential function). Figure 7.a comares master curves based on exerimental stress relaxation data and on the redictions of the KWW model, indicating a very satisfactory agreement. The rediction over the whole range of conversions is reorted in Figure 7.b; the curves show a gradual shift towards higher values of reduced time as exected and a constant value for the glassy modulus. 14

16 t, T, α α log T α C3 α ( C a ) = C α + C + ( C α + C )( T T ) 9 + τ C + C α 6 1+ C4 aτ t ξ = a τ a τ t ( α, T) g log [ E ( α) ] τ log τ α ( α) = C ( α ) ref E = + g τ D 1+ e ( α) E α gel α FE α ( α) = c 1β α + c β β 2 ξ β τ E (, T, α) = E ( α) E t e ξ ( t, T, α ) τ ( α ) β ( α ) E Figure 8 Sub-model schematic. The comutational rocedure required for the imlementation of this model as a sub-model of rocess simulations is illustrated in Figure 8. The sub-model accets a time, temerature, degree of cure oint form the rocess simulation and returns a value for the cure and temerature deendent viscoelastic modulus of the thermosetting matrix. 4 Conclusions The cure deendent viscoelastic modulus of an eoxy system can be described by a modification of the general Kohlrausch-Williams-Watts equation. The KWW model has been imlemented by introducing a otential function to account for the structural evolution of the system from a liquid 15

17 olymer like to a solid vitrified material. The mechanism of relaxation is assumed directly related to the density of cross-links because of the inherent constraints imosed on molecular mobility as the network is formed. The otential function has been roduced by normalizing the rincial times with resect to the time of reference conversion, to account for the effect on the rincial relaxation times of the olymer structural evolution. The model shows a very good agreement with exerimental data for the conversion exerimentally examined. 5 Acknowledgements This work was funded by the Engineering and Physical Sciences Research Council (GR/M89454/01). 16

18 References [1] Suzuki K, Miyano Y and Kunio T. Change of viscoelastic roerties of eoxy resin in the curing rocess J. Al. Pol. Sci.1997; 21: [2] Kim K S and Hahn H.T. Residual stress develoment during rocessing of grahite/eoxy comosites J. Com. Sci. and Techn. 1989; 36: [3] White S R and Hahn H.T. Process modelling of comosite materials: residual stress develoment during cure. Part II: exerimental validation J. Com. Mat. 1992; 26: [4] White S R and Hahn H. T. Process modelling of comosite materials: residual stress develoment during cure. Part I: Model Formulation J. Com. Mat. 1992; 26: [5] Gibbs J H and Di Marzio E. A. Nature of the glass transition and the glassy state J. Chem. Phys. 1958; 28/3: [6] Yi S, Hilton H H and Ahmad M.F. Finite element analysis of thick thermosetting matrix comosite curing rocesses Proc. ASME Int. Mech. Eng. Congress and Ex. 1995; 65: [7] O`Brien D J, Mather P T and White S. R. Viscoelastic roerties of an eoxy resin during cure J. Com. Mat. 2001; 35: [8] SimonS L, MecKenna G B and Sindt O. A thermo-viscoelastic model for the mechanical roerties of an eoxy during cure J. of Al. Pol. Sci. 2000; 76/4: [9] Zarrelli M, Skordos A A and Partridge I.K. Thermo-mechanical analysis of a toughened thermosetting system J. Mech. of Com. Mat. 2008; 44/2: [10] Mijovi, J; Han Y; Sun M. Normal and segmental mode dynamics of end-functionalized oly(royleneoxide) by dielectric relaxation sectroscoy and dynamic mechanical sectroscoy Macrom. 2002; 35,

19 [11] Zarrelli M, Skordos A A and Partridge I.K. Investigation of cure induced shrinkage in unreinforced eoxy resin J. Plast. Rub. Com.2002; 31/9: [12] Ngai K L and Plazek D.J. Identification of different modes of molecular motion in olymers that causes thermorheological comlexity J. Rubber Chem. and Techn.1995; 68: [13] Kohlrausch L R 1863 Pogg. Ann. Phys. 1995; 119: 352. [14] Shlesinger M.F. Williams-watts dielectric relaxation: A fractal time stochastic rocess J. of Statis. Phy. 1994; 36/5: 6 [15] Hodge I M Adam-Gibbs Formulation of Enthaly Relaxation Near the Glass Transition. in J. Res. of the Nat. Inst. of Stand. and Techn. 1997, 195: 102 [16] Ngai K L Roland CM. Chemical structure and intermolecular cooerativity: dielectric relaxation results Macrom 1993; 26: [17] A.T. DiBenedetto, Prediction of the glass transition temerature of olymers: A model based on the rincile of corresonding states Journal of Alied Polymer and Science Part B Polym. Phys., 1987; 25; [18] Arturo Hale, Christoher W. Macosko, Harvey E. Bair Glass transition temerature as a function of conversion in thermosetting olymers Macrom. 1991; 24 (9): [19] Rieger J The glass transition temerature Tg of olymers-comarison of the values from differential thermal analysis (DTA, DSC) and dynamic mechanical measurements (torsion endulum) J. Pol. Testing, 2001; 20/2, [20] Mohammad S R, Al-Marhubi I M and Al-Mahrouqi A Measurement of glass transition temerature by mechanical (DMTA), thermal (DSC and MDSC), water diffusion and density methods: A comarison study J. Chem. Phy. Let. 2007; 440/ 4-6:

20 Figure Cations Figure 5 - Exerimental oints and fitting curve for ultimate relaxed modulus Figure 6-3D lot of exerimental shift factors vs. temerature and degree of cure Figure 7 - Shift factors for artially cured samles ( ) fitted with WLF and linear models Figure 8 (a) Exerimental shift factors vs. (T-T g ) and model at each conversion (b) Model redictions over the full degree of cure range Figure 5 - Linear fitting of the beta arameter values Figure 6 - Normalized values of single relaxation time vs conversion with normalized values of glass transition temerature obtained exerimentally by thermal analysis and torsional rheometry alongside the DiBenedetto model (solid line) Figure 7 (a) Model redictions and exerimental stress relaxation modulus curves. (b) Prediction for the relaxation modulus over the full degree of cure range. Figure 8 Sub-model schematic. 19

21 Tables C E D E F E log(pa) 4.10 log(pa) Table 1 Values of fitting arameter for eq. 1 C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 Log(C 9 ) Table 2 Values of the coefficients of the shift factor model (eq.3) Samle/conversion E (GPa) β τ E 0 (GPa) 3.43 Table 3 Values of the KWW arameters at different degrees of cure 20

22 Grahical Abstract Toward a constitutive model for cure deendent modulus of a high temerature eoxy during the cure M Zarrelli 1*, A A Skordos 2 and I K Partridge 2 21