Prediction of the Rheological Properties of a Curing Thermoset System

Article pubs.acs.org/Macromolecules

Prediction of the Rheological Properties of a Curing Thermoset System J. H. Daly, D. Hayward, and R. A. Pethrick* West CHEM, Department of Pure and Applied Chemistry, University of Strathclyde, 295 Cathedral Street, Glasgow G1 1XL, UK ABSTRACT: The variation of the viscosity as a function of the degree of conversion for an anhydride cured epoxy resin is modeled using a combination of a Monte Carlo statistical model of the polymerization process and constitutive equations describing the motion of the entities in the mix. The degree of conversion was monitored as a function of time using Fourier transform infrared spectroscopy over the temperature range 80−130 °C. The rate data are analyzed using the Kamal equation and activation energy of 57 and 64 kJ mol−1 obtained from the temperature dependence of the rate constants. The gelation and vitrification times for the cure process measured as a function of temperature were measured using a vibrating probe curometer. The variation of the viscosity measured experimentally and predicted are found to be similar when compared as a function of the degree of conversion. A time−temperature transformation was constructed and indicated that the gelation point occurred at approximately the same degree of conversion for all the temperatures investigated. The influence of the cure process on the final mechanical propeties of the resin is explored through dynamic mechanical thermal analysis. This paper indicates the possibility of predicting the variation of the viscosity prior to gelation based on the cure kinetics for reactive polymer systems.

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solutions and melts13−19 and demonstrated the influence of chain topography, molecular weight, and concentration on the rheological characteristics. For the accurate description of the rheological behavior of high molecular weight polymer systems, account has to be taken of the influence of the motion of one polymer system on that of its neighbors as exemplified in the tube model of Doi, Edwards, and de Gennes.18,19 Neighboring chains constrain the motion of an identified chain to within a tube, and diffusion occurs by curvilinear diffusion (“reptation”). The initial model has been refined to include the retraction of the arms of branched chains and the breathing motion of extended loops. These latter motions add to the complexity of the dynamic behavior of flow under various stress/strain rate conditions. For complex topographies, H-shapes,21−23 combs,24 and multiarm polymers25,26 require consideration of the process of the arm retraction prior to reptation, and it is envisaged that free ends retract back to the outermost layer of branch points; these become mobile, activating deeper retractions toward the second layer, and so on.27 The relaxation time of a given tube segment in an ensemble of branched polymers depends on the curvilinear distance to the free end that eventually retracts and disentangles it and on the dangling trees attached to this path that slow down the retraction process. This type of model together with Monte Carlo simulation has been successfully applied to the prediction of the viscoelastic characteristics of polydisperse branched polymers

INTRODUCTION In composite manufacture and many applications of thermoset resins, being able to predict the change of the viscosity with time is an important factor in the efficient production of adhesive bonds, cure of composites structures, and encapsulation of electronic circuits.1 Epoxy resins are widely used as both a single- and two-component curable resin system.2−4 In the case of two-component systems, the cure is achieved by the use of a multifunctional amine, whereas single-component cure is often achieved using an anhydride plus a catalysts.3 In composite manufacture, consolidation of the matrix is achieved by application of pressure to the structure being created.1 Premature application of pressure can lead to resin depleted regions, whereas to late application will produce a voided structure. Ideally, the pressure should be applied in the time window where the viscosity is beginning to increase prior to gelation of the resin. A number of techniques have been developed allowing the cure process to be monitored, either in terms of change in the chemical composition or by measurement of some bulk physical property.5−10 In this paper we explore the connection between the chemistry occurring in the resin and the changes in the viscosity. In a previous paper, the use of the DryAdd Monte Carlo simulation software to predict the rheological profile of a number of branched chain molecules was reported.11,12 A good correspondence was observed between the experimental data for the shear rate dependence of the viscosity and the predictions from theory. A number of papers have addressed the problem of accurately modeling the rheological behavior of polymer © XXXX American Chemical Society

Received: November 10, 2012 Revised: April 12, 2013

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Figure 1. Cure chemistry for an anhydride cured epoxy resin: initiation.

generated by “metallocene” catalysts.28 In the present case, the condensation polymerization will only at the point of gelation create a significant proportion of chains with molecular weight above the entanglement limit, and it will be assumed that the system will initially approximate to the simple Doi−Edwards model. The accurate description of the rheology between gelation and vitrification would require consideration of the more complete descriptions of the dynamic behavior but will not be considered here. In this paper, a comparison is made between the predictions of the viscosity obtained from theory and experimental for the anhydride cure of an epoxy resin in the region up to the point of gelation. In order to perform the calculations, the extent of conversion of the epoxy and anhydride during the cure process is required and Fourier transform infrared (FTIR) studies of the cure kinetics were under taken. This paper explores the possibility of using computer simulation of the kinetics of cure to generate the time dependence of the topography of the growing polymer chains and using of constitutive equations to predict its viscosity.

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plates, each 5 mm thick and 25 mm diameter. Scans were obtained between 4000 and 500 cm−1 at 1 cm−1 resolution, normally with data accumulated in 32 scans. The resulting collection time was ∼90 s. The NaCl plates and film were heated to the desired temperature in a thermostatically controlled brass holder. Temperature control was achieved using a control and readout controller with a Pt100 resistor as the sensing element. The temperature was stable to 0.1 °C and estimated to be accurate to better than 0.5 °C. Supplementary measurements on a pair of NaCl plates with an embedded thermocouple indicated that the plates reached within 1 °C of the cell temperature within 2 min at the lowest temperature (80 °C) and within 4 min at the highest temperature (150 °C). It took typically 10 min for the FTIR sample chamber atmosphere to reach its equilibrium value once the sample had been loaded. For each set of measurements, the backgrounds of the two plates were recorded, and an unreacted film of material placed between the plates at room temperature and compressed until the absorption fell below 0.6, ensuring that measurements were performed within the linear range of the detector. The assembly was then taken from the spectrometer and stored in a sealed bag until the thermostat reached equilibrium, where upon it was placed in the spectrometer and measurements made. The room temperature unreacted spectrum was used as the zero reference for each isothermal measurement. Rheometry. Dynamic viscosity measurements were obtained using the Strathclyde vibrating probe curometer. Measurements were made on ∼1 mL samples of the mixture using a blade of area ∼15 mm2 and oscillating at a frequency of 2 Hz with amplitude of ∼500 μm. The sample was thermostatically controlled in a preheated silicone bath. Supplementary measurements showed that it took 3 min to reach equilibrium at the highest temperature (125 °C). The real and imaginary components of the damping coefficient were used to calculate the complex viscosity as has been described previously.29 Dynamic Mechanical Thermal Analysis (DMTA). A plaque of the cured material was produced bars 15 mm long and 5 mm wide. DMTA measurements were performed using a Polymer Laboratory MkII DMTA fitted with a power head. The DMTA measures the complex elastic modulus and was operated at 1 Hz in a two point bending mode over a range of temperature form 0 to 180 °C at a scan rate of 3 °C/min. Care was taken to allow for effects of thermal

EXPERIMENTAL SECTION

Materials. The epoxy resin, based on the diglycidyl ether of bisphenol A, Huntsman epoxy (MY750), was cured with methyltetrahydrophthalic anhydride (HY917) and catalyzed by the addition of 2,4,6-tris(dimethylaminomethyl)phenol, accelerator (DY073), used in a ratio of 100:85:2 by weight. A master batch of the material was prepared and stored in a sealed container at ∼−20 °C until required. The infrared spectra indicate that no reaction has taken place during storage. Cure was achieved by heating the mixture to 90 °C for 3 h, followed by 1.5 h at 130 °C and 2.5 h at 150 °C. A sample was also produced by heating the mixture to 160 °C for 16 h. Fourier Transform Infrared Spectroscopy (FTIR). FTIR measurements were performed using a Nicolet Impact 410 FTIR fitted with a Whatman Purge Gas Generator. The purge gas was air with a nominal dew point of −73 °C and a CO2 content of ∼1 ppm. Measurements were made on films placed between two polished NaCl B

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Figure 2. Propagation and termination stage of reaction.

d[M*2 ] = −k1[M 2][M1*] − k 2[M1][M*2 ] dt

expansion when clamping the sample. The data output is in the form of the storage and loss modulus as a function of temperature.

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BACKGROUND THEORY OF EPOXY−ANHYDRIDE CURE The kinetics will be briefly reviewed in order to determine the correct for for analysis of the FTIR data. The cure kinetics of anhydride/epoxy systems are complex, and Khanna30 and Chandra31 in a series of papers on anhydride cure have proposed the mechanism shown in Figure 1. Initial Stage of Reaction. The rate of disappearance of M1 and M2 at the initial stages of the reaction can be expressed as d[M1] = −k 2[M1][M*2 ] dt d[M 2] = −k1[M 2][M1*] dt

d[M1*] = −k1[M 2][M1*] + k 2[M1][M*2 ] dt

(4)

For the fast initiation through the epoxy−tertiary amine reaction, the initial conditions can be written as [M*1 ] ≈ [C]0, [M*2 ] = 0, and [M1] ≈ [M*1 ]0 where [C]0 is the initial concentration of the tertiary amine catalysts. Dividing eq 4 by eq 1, one obtains d[M*2 ] k [M*][M 2] = 1 1 +1 d[M1] k 2[M*2 ][M1]

(5)

Since [M1]0 = [M2]0 and in the case of stepwise addition [M1] ≈ [M2] also each mole of M1* and M2* contains 1 mol of the tertiary amine

(1)

[M1*] + [M*2 ] = [C]0

(6)

Combining eqs 5 and 6 and rearranging, one obtains

(2)

1 ⎡ d[M*2 ] ⎤ ⎢ ⎥ = d[M1] k′[C]0 ⎣ K − 1/[M*2 ] ⎦

(3) C

(7)

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where k′ = k1/k2 and K = (1 + k′)/k′[C]0. Integration of eq 7 yields

tertiary amine with the epoxide group of species which are much larger than the monomer. It is further assumed that there is a steady state for the total concentration of active species, that is

1 ⎡ [M*2 ] 1 ln K ⎤ ⎢ + 2 ln(1 − [M*2 ]K ) − 2 ⎥ k′[C]0 ⎣ K K K ⎦ = [M1] + constant

d([M1*] + [M*2 ]) =0 dt

(8)

and the rates of interconversion of M1* and M2* are equal, that is

Since the acetylation reaction is generally faster than the carboxylate−epoxy reaction, then k1 ≫ k2 and [M*1 ]K can be approximated by [M*2 ]/[C]0. Expanding ln(1 − [M*2 ]K) and neglecting higher order terms, eq 8 then simplifies to [M*2 ]2 ln K − − = [M1] + constant 2k[C]0 k′[C]0 K 2

k1[M1*][M 2] = k 2[M*2 ][M1]

Using the initial conditions for eq 5 to evaluate the constant in eq 9 then becomes

[C]0 = [M1*] + [M*2 ] + [C]

(10)

where K′ = (2k1k2[C]0) . Defining epoxy conversion as α by α = ([M1]0 − [M1])/[M1]0, eq 10 can be rewritten as (11)

where K″ = K′[M1]01/2 = (2k1k2[C]0[M1]0)1/2. Equation 11 describes the initial stages of cure when initiated by the tertiary amine−epoxy reaction. If however initiation involves the tertiary amine−anhydride reaction, the initial conditions can be written as [M*1 ] = 0, [M*2 ] ≈ [C0], and [M1] ≈ [M1] ≈ [M1]0. Following the above arguments leads to [C]0 dα = K ′(1 − α) α[M1]0 + dt 2k′

d[M1*] = k i1′ [C][M1] − k1[M1*] + k 2[M*2 ][M1] dt

(20)

d[M*2 ] = k1[M1*][M 2] − k 2[M*2 ][M1] dt − k t′([M*2 ]2 + [M1*][M*2 ])

(21)

The application of eqs 17 and 19 leads to k i1′ [C][M1] = k t″([M*2 ]2 + [M1*][M*2 ])

(22)

Since the initial concentrations of M1 and M2 are equal and they are consumed at equal rates in alternating type of polymerization reactions, it follows that [M1] ≈ [M2] during the course of the reaction. Equation 18 thus yields

(12) −3

Since the catalysts concentration [C]0 is small (3.6 × 10 −1.7 × 10−2 m mol/g) and k′ = k1/k2, eq 12 can be simplified to dα = K ″ α(1 − α) dt

(19)

The steady state assumption of eq 17 therefore implies that [C] is constant. Moreover, since

1/2

dα = K ″ α(1 − α) dt

(18)

Equation 17 is quantitatively in accord with the reported observations that k1 ≫ k2, [M2*] ≫ [M1*], and [M1] ≈ [M2] for equimolar epoxy and anhydride in the feed. By mass balance on the catalysts we now have

(9)

d[M1] = −K ′[M1] [M1]0 − [M1] dt

(17)

[M1*] ≈ k 2[M*2 ]/k1

(23)

Substituting for M1 in eq 22, one obtains (13)

[M*2 ] =

which is identical to eq 11 and indicates that identical rate are predicted irrespective of whether the carboxylate or anhydride initiation is invoked. Intermediate Stage of Reaction. After the initial stage, when the concentration of the active species M1* and M2* build up, termination reactions involving these active species assume significance. The main termination reaction is assumed to involve the regeneration of the tertiary amine. Regeneration of the Amine. According to the mechanism of Tanaka and Kakiuchi32 for the case in which the initiation is through the tertiary amine−epoxy reaction, the regeneration steps can be written as shown in Figure 2. The curing in the intermediate stage is assumed to involve initiation, propagation, and termination, the rates of which are given by R i = k i1′ [C][M1]

(14)

R p = k1[M1*][M 2] + k 2[M*2 ][M1]

(15)

R t = k t′′[M*2 ]2 + k t′[M1*][M*2 ]

(16)

ki′1[C][M1] k t′(1 + k 2/k1)

(24)

A combination of eqs 14, 17, and 18 yields ⎛ d[M1] d[M 2] ⎞ R p = −⎜ + ⎟ = 2k 2[M*2 ][M1] ⎝ dt dt ⎠

(25)

Considering that M1 and M2 are consumed by alternating copolymerization, a substitution for [M2*] from eq 22 then yields −2

d[M1] = 2k 2[M1]1.5 dt

k i1′ [C] k t′(1 + k 2/k1)

(26)

Equation 26 can be written in terms of conversion α as dα = K ‴(1 − α)1.5 dt

(27)

where K‴ = k2{(k′i1[C][M1]0)/(k′t (1 + k2/k1))} . A similar set of equations can be created for the case of the amine− anhydride reaction. The rate of initiation, propagation, and termination being given by 1/2

where [C] is the concentration of the tertiary amine catalysts at any time t. It may be noted that ki1 in eq 14 is different from ki1 in the above scheme, since the former relates to reaction of the

R i = k i2′ [C][M 2] D

(28)

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Figure 3. Waterfall plots of spectra over the range 2000−700 cm−1 for anhydride−epoxy cured isothermally at 105 °C.

R p = k 2[M1][M*2 ] + k1[M 2][M1*]

(29)

R t = k t″[M1*]2 + k t′[M1*][M*2 ]

(30)

conditions at 80, 90, 105, 125, and 150 °C. The shortest practical time interval between measurements was 5 min, and for this time interval the number of scans was set at 16 to reduce the scan time to below 1 min. The time interval was increased as the reaction proceeded and the reaction rate decreased. At certain of the lower temperatures, an “equilibrium” spectrum was observed at less than 100% conversion and is consistent with the concept of diffusionlimited reaction rates when the viscosity is high. For these temperatures the cell temperature was raised to 150 °C to check that complete cure could be achieved. The assignment of the infrared spectra of the resin components has been previously reported by Antoon and Koenig33 and Garton and Daly.34 The changes in the spectra for the reaction mixture as a function of time for an isothermal cure at 105 °C are shown in Figure 3. The CO stretch frequencies for the anhydride at 1780 and 1858 cm−1 are well resolved from those of the ester which is formed at 1740 cm−1, allowing the progress of the cure to be followed in terms of the decrease of the anhydride and the growth of the ester. The consumption of the epoxy can be monitored by following the peak at 915 cm−1. A very small O− H peak is observed in the final spectrum of the cured material at 3500 cm−1 indicative of the presence of pendant OH in the final material but not observed in the initial starting materials. The lack of significant intensity to the OH stretch indicates that the C−O− generated by the catalytic action of the amine is almost exclusively consumed in further reactions. To allow for possible changes in the film thickness during the course of the reaction, the spectra were referenced to the ring absorption at 1510 cm−1. A Mathcad routine was developed based on the Levenberg− Marquardt minimization function which allowed area of the overlapping spectra for the anhydride at 1780 cm−1 and the ester band at 1740−1 to be analyzed (Figure 4). It was assumed that the peak shapes are invariant as the reaction proceeds. The

Proceeding in the same way as in the previous section, we obtain dα = K ⁗(1 − α)1.5 dt

(31)

where K⁗ = k1{ki2′ [C]/[kt″(1 + k1/k2)]}1/2. This leads to a similar relationship for the conversion with time equation. It is therefore not possible to differentiate from the initial analysis the mechanism whereby the reaction proceeds. The mechanism presented above does not include the possibility of nucleophilic attack of the pendant hydroxyl group on the epoxy ring, a process which is known to change to form of the kinetics in the latter stages of the cure process and is in part ascribed to autocatalytic reactions observed in many systems. The accepted method of fitting kinetic data for epoxy−amine reactions uses the form proposed by Kamal.30 dα = (k1 + k 2α m)(1 − α)n dt

(32)

where α is the extent of cure, k1 and k2 are rate constants with an Arrhenius form, and m and n are rate orders and the value of n is predicted to be 1.5. Equation 32 can be modified to allow for incomplete cure and has the form dα = (k1 + k 2α m)(αf − α)n dt

(33)

where αf is the final extent of cure at a particular temperature. This form of the equation will be used in the analysis of the kinetic data. This study investigates the cure of a simple epoxy with an anhydride catalyzed by the addition of a tertiary amine. FTIR Study of Anhydride−Epoxy Cure. FTIR spectra were recorded as a function of time for isothermal cure E

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Figure 6. Variation of the area of the epoxy peak at 910 cm−1 as a function of time for isothermal cures carried out at 80, 90, 105, 115, 125, and 150 °C.

Figure 4. Changes in spectra for the CO region as a function of cure.

Table 1 variation of the peak area as a function of time for cures carried out isothermally over the temperature range 80−150 °C are presented in Figure 5 for the variation of the anhydride peak at 1780 cm−1 and for the epoxy peak at 910 cm−1 in Figure 6.

temp (°C) 80 90 105 115 125 150 mean value activation energy (kJ mol−1)

k1 (min) −3

2.14 × 10 6.03 × 10−3 7.49 × 10−3 0.015 0.026 0.06 57

k2 (min)

m

n

0.022 0.058 0.189 0.152 0.215 1.07

2.69 3.31 2.31 1.57 1.67 2.275 2.30

0.85 1.21 1.60 1.16 1.07 1.79 1.28

64

Figure 5. Variation of the area of the anhydride peak at 1780 cm−1 as a function of time for isothermal cures carried out at 80, 90, 105, 115, 125, and 150 °C.

The kinetic data were fitted to eq 33 using a Mathcad routine which allowed calculation of the parameters k1, k2, m, and n at each temperature. The results are fitting of the curves are summarized in Table 1. The temperature variation of the rate constant k1 and k2 are presented in Figure 7. The respective activation energies are 57 and 64 kJ mol−1; however, the uncertainty in the values is ±4 kJ mol−1, making that values essentially identical. These values are similar to those reported in the literature for similar cure reactions.35−42 The fitting coefficient n and m were found to vary with the temperature but had mean values of respectively 1.28 and 2.30. The mechanism presented above would imply a value of n of 1.5 and a value of m of 2. The variation with temperature of these coefficients indicates that the detailed mechanism is

Figure 7. Temperature variation of the rate constant k1 and k2.

probably changing with temperature as the proportions of termination and auto catalytic processes change. Detailed studies of the cure behavior of DGEBA with the diglycidyl ether of poly(propylene glycol) DGEPPG cured with 2-ethyl-4-methylimidazole have shown that in the more viscous mixtures diffusion control influences the rates of the reaction close to vitrification.41 The values of n obtained from the analysis are comparable to those observed in this study; however, the values of m are generally smaller. The inclusion of diffusion control into eq 32 has been shown to allow analysis of the data to be extended beyond gelation and influences the value of m. Harsch et al. have reported values of 65.1 and 68.1 F

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kJ/mol for the reaction catalyzed with 1-methylimidazole.42 Activation energies of the order of 70−80 kJ mol−1 have been reported from the cure of DGEBA with methyltetrahydrophthalic anhydride using benzyldimethylamine as catalysts.41 Changing the anhydride to pyromellitic dianhydride and incorporation of carboxy telechelic poly(tetramethylene oxide) yields values of the activation energy in the range 28− 77 kJ mol−1 depending on the ratio of the additives.40 Using tetrabutylammonium bromide as catalysts, values of the activation energy of 48 and 54 kJ mol−1 have been reported,39 and a Raman study of cure of tetraglycidyl-4,4′-diaminodiphenylmethane with hexahydrophthalic anhydride yields values of 48 −50 kJ mol−1, which are comparable to those reported here. Rheometry. Isothermal cure data obtained using the Strathclyde curometer are presented in Figure 8.

Figure 9. DMTA of a sample cured using the standard cure schedule and a sample cured at 160 °C for 16 h.

is constructed on the basis of the degrees of conversion lines obtained by mapping the anhydride consumption as a function of time. The superimposed line connecting the points maps the variation of the time to reach the gelation point for the cure. It may be noted that this latter contour line coincides with a degree of conversion of ∼70%, indicating that the conditions for gelation are essentially independent of the cure temperature and is consistent with previous predictions concerned with network formation.45−47 Modeling of the Viscosity Profile. The DryAdd Monte Carlo simulation program was set up to follow the reaction scheme summarized above with the ratio of the rate constant being taken as those form the FTIR measurements. The program allows the consumption of the monomers to be calculated as a function of the degree of conversion of the anhydride α. At each stage the “connectivity” matrix for the species created during the reaction were calculated. The “connectivity” matrix calculated at each point during the simulated cure process contains information on the molecular weight of the individual species present and their topography. The calculation of the viscoelastic behavior of the total system therefore requires individual calculations for each of the species present and allowance for effects of chain topography. The prediction of the viscosity of the reaction mixture is based on the calculation of the dynamic viscosity for the entities which are being created as a function of time. The viscosity of the mixture can be described as the sum of components: the high-frequency limiting behavior which corresponds to that of the unreacted monomers (viscosity of the monomer blend) and the contributions from the motion of the polymer chains which are created as a consequence of the cure process. Initially, the chains will be relatively short and can be described by Rousetype behavior,48 but as the reaction proceeds the molecular weight will approach and exceed the critical entanglement value and reptation motion of the chain has to be considered18,19 Normal Mode Relaxation Processes. The co-operative relaxation of a polymer chain may be described either by a Rouse48 or Wang and Zimm model49 dependent upon the nature of the interaction of the surrounding media. A polymer melt will be best described by the Rouse model in which the polymer is made up of N freely jointed segments and has a molar mass below a critical value for entanglement, Mc. The frequency-dependent complex modulus can be described by

Figure 8. Variation of viscosity with time for isothermal measurements carried out between 80 and 130 °C for an anhydride−epoxy cure.

The time to reach a viscosity of 100 Pa·s was taken as indicative of the system approaching gelation and corresponds to the time at which a peak was observed in the imaginary component of the damping coefficient. At the highest temperature a kink in the viscosity time profile is observed at or about the point at which gelation occurs and is consistent with the changes in the n and m coefficients obtained form the FTIR analysis. At lower temperatures the viscosity plots show a smooth change with time, with slower changes indicative of the differences in rate constant of the processes at lower temperatures. DMTA. Dynamic mechanical thermal analysis was carried out on a test piece which had been subjected to the “standard” cure of heating at 90 °C for 3 h, followed by 1.5 h at 130 °C and 2.5 h at 150 °C, and another test piece was postcured for a further 16 h at 160 °C (Figure 9). The sample cured at 160 °C shows a single glass transition peak at 116 °C, but the more complex “standard” cure shows two peaks: one located at ∼85 °C and the higher at 35 °C. The staged cure is achieving a higher glass transition temperature, but the existence of the lower peak at 85 °C is evidence for the plasticization effects of lower molecular weight unreacted species being present in the matrix and consistent with the FTIR observation that the spectra reach “equilibrium” conditions with unreacted monomer. Time−Temperature Transformation Diagrams. Gillham43,44 has proposed that cure data may be rationalized using time−temperature transform diagram (Figure 10). The diagram G

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Figure 10. Time−temperature transformation diagram for anhydride−epoxy reaction. Key (right-hand column: degree of conversion).

cRT G*(ω)Rouse = M

N

∑ j=1

generated form the Monte Carlo simulations. The values of GN for calculation of eq 36 were scaled to appropriate values for polydimethylsiloxane.52 Total Viscoelastic Response. It was assumed that the modulus terms are additive at least in the limit of relatively low molecular weights for which the reptation contribution is small or absent. The total modulus for the system is then given by

iωτj 1 + ωτj

(34)

where c and M denote the concentration and molecular weight of the polymer, respectively, and R and T are the gas constant and absolute temperature and τj in eq 34 is given by τj = τ/j2, with ⎛ ξ N b2 ⎞ τ = ⎜ 20 a2 ⎟M2 ⎝ 6π m0 RT ⎠

G*(ω) = G*reptation (ω) + G*(ω)Rouse + G*Liquid (ω) (38)

(35)

The zero shear viscosity of the mixture is evaluated from eq 38 using the relation η0 = [G″/ω]w→0. For a system in which there is a molecular mass distribution, the calculation will then need to be performed for all the species present and summed over all the molecules present and over all frequencies. In a previous paper11 we considered the effects of chain branching on the normal mode relaxation and showed that through connecting the modes to the connectivity table generated by the DryAdd simulation software it was possible to predict the viscosity profile of amine cured resins systems. In this system, the cure process produces linear polymers, and the simple forms of eqs 36−38 were used to predict the relaxation times and the viscosity behavior. The change of the molecular weight and its distribution as a function of the degree of conversion α obtained from the DryAdd simulation are shown in Figure 11. The values of the weight- and number-averaged molecular weights Mw and Mn presented in Figure 11 include the unreacted monomer. The polymerization process involves a stepwise growth of the polymers, and initially the values calculated were largely dominated by the contribution from the monomers. Over the initial 20% conversion, the value of Mw has increased from 284 to 349, whereas Mn has increased from 195 to 225 and Mw/Mn has a value of 1.5. However, interrogation of the detail of the changes in molecular weight indicates that large polymer chains are not produced in a significant number until ∼57% conversion, whereupon there is a steady growth until at ∼80% conversion nearly all the monomers have disappeared, and the reactions involve joining up short polymer chains. At 66% conversion, the longest chain component has M = 244 780 (composed of 1421 units), but this component constitutes only 7% of the total polymer present and is diluted by lower

where ξ0 is the monomer friction coefficient, Na is Avogadro’s number, and m0 and b are the molecular weight and effective step length of the repeat unit. In the calculation the short chains (as well as long entangled chains explained below), monomer friction coefficient has been assumed to equate to the initial viscosity of the reactive system. For the case of a linear reactive polymer system the values of M are obtained directly from the DryAdd simulation of the polymerization process and allow prediction of the variation of the viscosity as a function of cure time.12 Above Mc it is necessary to include in the calculation the effects of chain entanglement or reptation. Terminal or Reptation Motion. The complex modulus for the reptation motion of an entangled polymer molecule may be described by the relationship18,19 G*(ω) = GN

∑ j = odd

iωTd /j 2 8 j 2 π 2 1 + iωTd /j 2

(36)

where GN is the entanglement plateau modulus and Td is the terminal relaxation time related to GN defined and the steady state flow viscosity η0: Td =

12η0 π 2GN

(37)

Equations 36 and 37 are controversial for actual entangled polymers; however, the above formulation describes accurately the observed behavior for linear narrow molecular weight polydimethylsiloxanes51,52 and will be used here. In the case of cure producing a branched chain polymer the approach outlined by Forsman50 would be adopted. The chain topography can be obtained form the connectivity table H

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conversion this system will become a gel and exhibit viscoelastic characteristics. Up to 20% conversion the increase in the viscosity is a consequence of the creation of a small number of polymer chains with molar mass ∼2000−3000 and corresponding to about 1% of the total increase of the viscosity. Above 20% conversion the proportion of such species increases as reflected in the steady growth in the viscosity between 20% and 60% conversion. At about 66% the longest chains approach Mc, and the theoretical predictions contain a significant contribution from reptation motion. The correspondence between experiment and theory is good over the region 20−70%, indicating that the simple approach adopted to the prediction of the viscosity is mirroring the experimental data.

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CONCLUSIONS This paper indicates that it is possible by combining the predictions of the structures created during polymerization using a Monte Carlo simulation program with a combination of Rouse and reptation modes calculations to predict the variation of the viscosity of the reaction mixture as a function of the degree of conversion. For this simple anhydride−epoxy cure up to the gelation point the agreement between experiment and theory is surprising good. The principles of being able to predict the viscosity variation as a function of the degree of conversion and hence time are illustrated for the cure of this anhydride−epoxy system. It is possible however that the addition of other model of chain motion will be required to accurately describe the viscoelastic properties between gelation and vitrification and for reactions which generate more complex topographies.

Figure 11. Molar mass (Mn, Mw) and molar mass distribution (Mw/ Mn) variation as a function of the degree of conversion.

molecular weight material. The value of Mw/Mn is now 5.8, indicating that a significant proportion of the polymer chains have a high value of Mn. The creation of the longer polymer chains will allow the viscosity to increase rapidly and entanglement to ensue. The proportion of long to short chains changes during the polymerization process and is reflected in the changing value of Mw/Mn. The computer simulation indicates that the proportion of longer chains grows significantly between 70% and 80% conversion as reflected in the marked increase in the values of Mw/Mn. Initially, below 20% conversion the reaction mixture is essentially a dispersion of a few short polymer chains in a solvent of the monomer. As the reaction proceeds, the length of the dominant polymer chains increases, until above 60% the system is a polymer melt where the components have a broad molecular weight distribution. For the conversion ≤60% in the calculation, the molecular weight is less than 12 000 even for the longest (and dilute) chain component, and hence the Rouse model is applicable in that range of concentration. Above 60% conversion, an increasing contribution to the viscosity will arise polymer chains with molecular weights greater than Mc, and this is reflected in an increasing contribution from reptation. The variation of the measured and calculated viscosity at a dynamic frequency corresponding to 2 Hz is shown in Figure 12. As the reaction proceeds, there is a steady increase in the viscosity with degree of conversion. At greater than 70%

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (R.A.P.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The support of EPSRC in funding D.H. and J.H.D. and the help of John Dunseith are gratefully acknowledged. REFERENCES

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Figure 12. Variation of the experimental viscosity and the predictions from the DryAdd theoretical modeling joined with the prediction of the viscosity. I

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