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Cite This: ACS Appl. Polym. Mater. XXXX, XXX, XXX−XXX
Cross-Linking Reaction under a Stress and Temperature Field: Effect on Time-Dependent Rheological Behavior during Thermosetting Polymer Processing Sandeep Kumar,†,‡ N. Eswara Prasad,‡ and Yogesh M. Joshi*,† †
Department of Chemical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India Defence Materials & Stores Research & Development Establishment (DMSRDE), Kanpur 208013, India
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ABSTRACT: In most equipment that processes thermosetting polymers, a material is subjected to a large stress field while the cross-linking reaction is taking place. It is therefore vital to understand how the time-dependent rheology of a cross-linking thermosetting system gets affected by the stress field. In this work, an effect of the creep flow field on a model thermosetting system of epoxy resin undergoing chemical cross-linking reaction has been studied at different temperatures. It has been observed that creep curves obtained at different reaction times and stresses do not follow the Boltzmann superposition principle in the realtime domain. However, when transformed from the real-time domain to the effective time domain, the cross-linking system is observed to validate the Boltzmann superposition principle leading to time−reaction time−temperature− stress superposition. In the effective time domain, the real-time is suitably normalized by the time-dependent relaxation time, which shows a power law dependence, so that the material properties become invariant of the effective time. The corresponding power law exponent is observed to decrease with an increase in stress, suggesting stress to have a retarding effect on the evolution of relaxation time during the cross-linking reaction. Instead, the power law coefficient increases with an increase in temperature, indicating an enhanced time rate of evolution of the relaxation time at higher temperatures. This behavior also indicates that while stress impedes or temperature accelerates the evolution of mean retardation/relaxation time; it does not affect the shape of retardation/relaxation time spectra of the reacting system in the pregel regime. This work provides important insights into how the time-dependent rheology of thermosetting systems is affected by the stress and temperature fields and, therefore, may help in designing better processing equipment. KEYWORDS: Boltzmann superposition principle, effective time theory, effect of stress on cross-linking reaction, time−stress superposition, time−temperature superposition, creep, thermosetting materials the strong deformation/stress field poses challenges. Recently Kaushal and Joshi7 successfully extended the effective time domain theory to chemically cross-linked systems, wherein the principles of linear viscoelasticity were transformed to the effective time domain by appropriately normalizing the realtime by the time-dependent relaxation time.8−13 In this paper, we extend this work by studying the effect of stress and temperature on an epoxy resin system undergoing a crosslinking reaction. In the literature, two rheological approaches have been proposed to address the issues arising from the evolution of properties due to a system undergoing a cross-linking reaction. The first approach relates to carrying out the experiments fast enough so that the material does not undergo much change throughout the measurement. Such experiments are carried out either by using time-resolved rheometry or Fourier transform
I. INTRODUCTION In a cross-linking process, monomeric units or precursors having a functionality greater than two react with each other, which eventually leads to a three-dimensional network of the chemical bonds.1,2 Consequently, as the reaction proceeds, free-flowing liquid gets transformed into a solid state.3 During the reaction, the cross-linking density of a system continuously increases, causing viscoelastic properties of the cross-linking system to evolve as a function of time.4 As a result, the material does not obey the time-translational invariance (TTI) that requires its properties to be invariant of time.5 Consequently, the fundamental principle of viscoelasticitythe Boltzmann superposition principle (BSP)does not apply to the chemically cross-linked systems.6 Many chemically reacting polymeric systems, particularly the thermosetting polymers, are often subjected to the deformation/stress field in the processing equipment while their properties are continuously changing as a function of time.3 Since principles of viscoelasticity do not apply to time-dependent systems, analysis of the time-dependent rheological behavior under © XXXX American Chemical Society
Received: May 7, 2019 Accepted: August 2, 2019 Published: August 2, 2019 A
DOI: 10.1021/acsapm.9b00432 ACS Appl. Polym. Mater. XXXX, XXX, XXX−XXX
Article
ACS Applied Polymer Materials rheology.14−19 However, this approach does not help in solving BSP as material properties do change from time to time, and TTI still gets violated.20 In the second approach, the crosslinking reaction is completely terminated by the addition of inhibitor at different extents of reaction.21−24 In another complementary approach, but in the opposite direction, highly cross-linked material is exposed to strong radiation for a certain duration to cause scission of the cross-links.25 In these cases, since the time dependence ceases to exist, TTI is validated, and BSP can indeed be applied to the system.4 In the literature, it has been observed that such reaction terminated systems show time−temperature superposition.24 The termination of reaction at different times after starting the reaction leads to systems with different levels of cure. Carrying out rheological experiments on such reaction terminated systems lead to time−cure superposition.23,26−28 While TTI indeed gets validated due to termination of the reaction, this approach is still not useful as it does not give any information regarding how principles of viscoelasticity can be applied while the crosslinking reaction is taking place. In the conventional form, the Boltzmann superposition principle (BSP) is given by6
carried out validation of various forms of BSP for different materials that undergo time-dependent structural reorganization such as clay suspensions, emulsion paints, hair gel, etc.8,11−13,29,31 In time-dependent materials to apply eq 2 that is necessary for transformation to the effective time domain, a dependence of relaxation time on waiting time given by τ = τ(t) is required.7,13 For colloidal glasses, molecular (including polymeric) glasses, and spin glasses, the relaxation time has been observed to show a power−law dependence on waiting μ time given by τ = Aτ1−μ m tw, where A is a constant, τm is the characteristic time associated with the aging process and μ is the power-law exponent.8,32 The parameters A and τm are the properties of the material and are invariant of the deformation field.8−11 Various groups have also attempted the transformation of the creep data to the effective time-domain at different stresses and temperatures.10,11,33,34 It has been observed that in a limit of small stresses, wherein stress field does not influence the aging process, μ achieves the maximum value associated with the quiescent conditions. However, as stress increases μ have been observed to decrease. As the applied stress approaches the yield stress, μ tends to zero, suggesting complete rejuvenation of a material.11,34,35 Furthermore, the creep curves obtained at different aging times as well as stresses show self-similar curvature leading to time−aging time−stress superposition.11,35 For the aging soft glassy materials, the creep experiments at constant stress have also been carried out at different temperatures. It has been observed that μ increases with increase in temperature and the corresponding creep curves also show self-similar curvature leading to time−aging time−temperature superposition.10 In many processing units such as those for rotational, hot and cold press, transfer, reaction injection, pultrusion, compression molding, etc., a cross-linking reaction takes place under application of a strong deformation/stress field. Such a type of thermosetting processing, therefore, involves two aspects, the time-dependent rheology of the cross-linking material and effect of stress on the time dependency. While there have been many studies on how stress affects the timedependent rheology of glassy polymers, studies on the effect of deformation/stress field on the cross-linking reaction are very scarce. We believe that a comprehensive understanding of how the time-dependent rheology of thermosetting systems gets affected by the deformation field will prove vital in designing better equipment for the processing of the same. In this work, we investigate the effect of stress and temperature on the time evolution of the rheological behavior of thermosetting polymeric systems undergoing cross-linking reaction. We apply the effective time-domain theory to the cross-linking system and compare the results with the other time-dependent systems.
t
γ (t ) =
∫−∞ J(t − tw) ddtσ dtw w
(1)
where γ(t) is the strain induced in a material at the present time t, σ = σ(tw) is the stress applied to a material at the past time tw such that tw varies from −∞ to t, and J(t − tw) is the creep compliance that depends only on time elapsed since the application of stress (t − tw). In the time-dependent materials, properties including creep compliance depend not only on time elapsed since application of the stress field (t − tw) but also the time at which stress field is applied (tw), J = J(t − tw, tw).9,20,29 Consequently, eq 1 cannot be applied to the timedependent materials including systems undergoing chemical cross-linking reaction. Kaushal and Joshi7 reported that principles of viscoelasticity could be applied to the systems undergoing cross-linking reaction by using effective timedomain theory. In this theory, the effective time (ξ) is defined as29,30 ξ(t ) = τξ
∫0
t
dt ′/τ(t ′)
(2)
where τ is the average relaxation time that depends on the realtime (t), while τξ is the constant relaxation time. In the effective time domain, owing to the mentioned normalization, the material clock gets readjusted in such a fashion that relaxation time remains constant at a value of τξ in the effective time domain.8 Kaushal and Joshi7 showed that the transformation of the BSP from the real-time domain to the effective time domain is given by ξ
γ (ξ ) =
∫−∞ J(ξ − ξw) ddξσ dξw w
II. MATERIAL AND EXPERIMENTAL PROCEDURE (3)
In this work, we use epoxy resin and curing agent tri ethylene tetramine as a model thermosetting cross-linking polymeric system. Epoxy resin belongs to the “epoxide” family. It is widely used as a matrix medium for the fabrication of composites. The primary reason for using this system is the feasibility to carry out the rheological tests at room temperature.36 The general properties of the curing system are given in Table 1. The curing mechanism of the epoxy system by the cross-linking agent is mentioned in Figure 1. It can be seen that a lone pair of polyamine attacks the electron-deficient epoxide ring.37 Consequently, the epoxy ring on the monomer opens leading to a
where ξw = ξ(tw). Kaushal and Joshi7 carried out experiments on PDMS (Sylgard 184 supplied by the Dow Chemical Company) undergoing a gelation reaction. They verified BSP in the stress-controlled mode (as shown by eq 3) as well as in the strain-controlled mode and, finally, validated the convolution relation [ξ = ∫ ξ0 G(ζ)J(ξ − ζ) dζ] in the effective time domain relating the creep compliance and the stress relaxation modulus. Interestingly Joshi and co-workers also B
DOI: 10.1021/acsapm.9b00432 ACS Appl. Polym. Mater. XXXX, XXX, XXX−XXX
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ACS Applied Polymer Materials Table 1. General Properties of the Epoxy System properties
key data
Epoxy Resin (Trade Name Araldite LY556, Huntsman) aspect (visual) clear, viscous, pale yellow liquid epoxy content (ISO 3000) 5.30−5.45 [equiv/kg] viscosity at 25 °C (ISO 12058-1) 10 000−12 000 [mPa s] density at 25 °C (ISO 165) 1.15−120 [g/cm3] molecular weight 340.42 [g/mol] Curing Agent (Trade Name Aradur HY951, Huntsman) aspect (visual) clear liquid viscosity at 25 °C (ISO 12058-1) 50−100 [mPa s] density at 25 °C (ISO 165) 1.15−120 [g/cm3]
Figure 2. Experimental protocol adopted in the present work.
(G″) moduli show a power-law dependence on frequency (ω) given by4,38 G′ = G″cot(nπ /2) =
formation of free hydroxyl group along with a network formed by a tertiary amine linkage. Before blending with a curing agent, to remove any volatile materials, epoxy LY556 is kept in a vacuum oven at 100 °C for an hour. Subsequently and after cooling back to the room temperature, a predetermined amount of epoxy LY556 (90.91 mass %) is added to a beaker along with the equivalent amount of HY951 (9.09 mass %). After quick but vigorous manual mixing, the sample is placed in the shear cell of the rheometer, and the experiments are started. In the present study, we carry out the oscillatory shear experiments at different frequencies and the creep experiments at different stresses. In the oscillatory shear experiment, the sample is subjected to a succession of frequency sweeps (0.1−30 rad/s) at stress magnitude of 1 Pa. We carry out the oscillatory experiments on Anton Paar MCR 501 rheometer with concentric cylinder geometry (cup diameter 28.915 mm and bob diameter 26.65 mm). Unlike oscillatory shear experiments, in the creep experiments, the stress field was applied after waiting for a certain time (tw) after mixing the epoxy resin and the curing agent, as shown in Figure 2. This time is termed as waiting time (tw). In this work, we carry out creep experiments at different tw, different stresses, and three temperatures. For the creep experiments, we use AR-G2 rheometer with parallel plate geometry (disposable parallel plate geometry with aluminum as a material of construction having 25 mm diameter and gap on 1 mm).
πS ωn 2Γ(n)sin(nπ /2)
(4)
where n is the power-law index (0 ≤ n ≤ 1), Γ(n) is the Euler gamma function, and S is the gel strength. The loss tangent (tan δ = G″/G′) corresponding to the critical gel state given by tan δ = tan(nπ/2) is independent of the frequency. Winter and co-workers4,38 proposed that in the pregel state, tan δ decreases with increase in frequency while, in the postgel state, it shows an increase with frequency. At the point of the critical gel state, tan δ becomes independent of the frequency. To study the evolution of viscoelastic properties of the studied epoxy resin while it undergoes cross-linking reaction, we subject the system to the cyclic frequency sweep experiments and obtain the time evolution of G′ and G″ at different frequencies. In Figure 3 we plot tan δ at various times after the cross-linking reaction is started as a function of frequency. It should be noted that the values of tan δ at different frequencies are obtained by applying a succession of frequency sweeps to the chemical cross-linking system. Therefore, strictly, every point in Figure 3 has been measured at different times. However, to simplify the analysis, we associate time at which the highest frequency measurement was carried out to the corresponding series. At the beginning of the experiment, the value of G′ (not shown) is too small, but that of G″ is sufficiently large. Consequently, tan δ up to the time of 100 min shows significant fluctuations, and therefore, we get meaningful data only beyond 100 min. It can be seen that up to 180 min, tan δ decreases with increase in frequency, which, according to the Winter criterion,4,38 is suggestive of a pregel state. Furthermore, due the very fact that tan δ is plotted on a logarithmic scale, the decrease in tan δ at lower times is far more severe, which becomes weaker as time increases. At
III. RESULTS AND DISCUSSION A polymeric material undergoing a cross-linking reaction is known to undergo the sol−gel transition. In a seminal contribution, Winter and co-workers4,38 proposed that, while undergoing the sol−gel transition, the system passes through the critical gel state, where the cross-linked percolated network spans the entire three-dimensional space. They also proposed that at the critical gel transition the elastic (G′) and viscous
Figure 1. (a) Curing mechanism37 of epoxy system Araldite LY556 with a cross-linking agent HY-951. (b) Chemical expression for the R group. C
DOI: 10.1021/acsapm.9b00432 ACS Appl. Polym. Mater. XXXX, XXX, XXX−XXX
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ACS Applied Polymer Materials
Figure 3. Loss tangent (tan δ) plotted as a function of frequency at different times after the cross-linking reaction is started. The region where tan δ decreases with frequency is a pregel state while the region where tan δ increases with time is a postgel state. At the point of critical gel transition (≈200 min), tan δ becomes independent of frequency.
200 min, tan δ becomes almost constant with respect to frequency, indicating the critical gel state of this system to be around this time as per the Winter criterion. At the critical gel state, tan δ = 1, suggesting that G′ = G″ marks the critical gel state for this system (this is just a coincidence, the critical gel state need not be at tan δ = 1 or G′ = G″; the critical gel state occurs when tan δ becomes independent of frequency4). At the higher times tan δ can be seen to be increasing with an increase in frequency, which represents a postgel state. At further high times, however, both G′ as well as G″ become so high due to the progressive increase in the cross-link density of the gel that the rheometer is no longer able to monitor change in the dynamic moduli. After the material is incorporated in the shear cell, and it has attained the thermal equilibrium, the cross-linking system is subjected to constant stress at different waiting times (tw) after the reaction has started. In Figure 4a the strain induced during creep while in Figure 4b the corresponding strain rate is plotted as a function of time at different waiting times. It can be seen that strain increases monotonically and eventually reaches a plateau while the corresponding strain rate is almost constant up to a point and then decreases. Furthermore, with an increase in tw, the strain, as well as strain rate induced in the material, can be seen to be decreasing when compared at the same creep time. The time at which strain attains a plateau (or strain rate starts decreasing) can also be seen to be decreasing with the increase in waiting time. Overall, it can be seen that the cross-linking system attains a plateau as the total reaction time (sum of waiting time and creep time) approaches around 100 min. This point is of the same order of magnitude as the time required to reach the critical gel point (≈200 min) at which the cross-linking system forms a space spanning percolated network.4 The state of a cross-linking system before this critical gel point is termed as the pregel state, while the state of a cross-linking system after the critical gel point is represented as the postgel state. In this work, we restrict ourselves to the pregel state of the system as creep flow is possible only before the gel point. The behavior shown in Figure 2 clearly suggests that the strain (and therefore compliance) is not just a function of time elapsed since the application of stress but also depends on
Figure 4. (a) Strain induced in the material and (b) corresponding shear rate plotted as a function of creep time for the experiments started at different waiting times (from top to bottom tw = 1800, 2700, 3000, 3300, 3600, 4500 s) for an applied constant stress of 100 Pa. It can be seen that strain increases and eventually attains a plateau.
waiting time at which it is applied: γ = γ(t − tw, tw). Such additional dependence on tw does not allow application of the BSP in its conventional form to the time-dependent systems.20,39 As discussed in the Introduction, the BSP can be applied by transforming the material behavior from the realtime domain to the effective time domain. However, according to eq 2 dependence of relaxation time on time τ(t) is needed to express the effective time domain. For the creep data shown in Figure 2, we obtain τ(t) by using a standard procedure that involves the horizontal shifting of the creep data in a limit of t − tw ≪ tw as discussed in detail elsewhere.7,13,30 We observe that the obtained τ(t) shows a power-law dependence on the time given by τ = Aτm1− μtwμ
(5)
Incorporation of eq 5 into eq 2 leads to expression of the effective time domain for the observed relaxation time dependence, which upon adjustment gives [ξ(t ) − ξ(tw )]A τξτmμ − 1
=
t 1 − μ − tw1− μ (1 − μ)
(6)
In Figure 5, we plot compliance as a function of (t − (1 − μ) for the data (at 100 Pa) mentioned in Figure 4. We also plot the compliance data obtained at 25 °C as a function of (t1−μ − t1−μ w )/(1 − μ) at the other constant values of stresses as well. The effective time-domain theory is applicable only when a material preserves the shape of its relaxation time spectrum while the mean relaxation time undergoes time evolution.7,30,39 It has been observed that the spectrum of 1−μ
D
t1−μ w )/
DOI: 10.1021/acsapm.9b00432 ACS Appl. Polym. Mater. XXXX, XXX, XXX−XXX
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ACS Applied Polymer Materials
onslows down the evolution of relaxation time. Interestingly all the superpositions shown in Figure 5 have a similar curvature on a double logarithmic plot suggesting that the shape of retardation time spectrum is not just independent of tw but also that of stress (the retardation time spectra can be considered to be proportional to the time derivative of the creep compliance and hence related to the curvature of the creep curve40). As a result, suitable horizontal shifting of the individual superpositions leads to a comprehensive time− reaction time−stress superposition, as shown in Figure 7. It
Figure 5. Time−reaction time superposition in the effective time domain for reactions carried out under different creep stresses (squares 30 min, circles 45 min, up triangles 50 min, down triangles 55 min, left triangle 60 min). The superposition associated with 100 Pa has been shifted vertically (ζ = 10) for better visibility, while for all the other stresses no vertical shifting has been applied (ζ = 1).
relaxation time preserves its shape before the critical gel point.7 Consequently, in the effective time domain, we consider the creep data only up to the critical gel point that is when it starts showing a plateau (around 80−100 min after the sample is placed in the shear cell although the actual critical point is attained at ≈200 min). It can be seen from Figure 5 that creep data for a given constant stress but, at different reaction times, shows excellent superposition that can be termed as time− reaction time superposition. It should be noted that the initial creep data, that is associated with the developing flow region, does not participate in the superposition as also observed for the aging colloidal systems.29,33 In Figure 6, the corresponding values of μ, for which the superpositions are obtained, are plotted as a function of creep
Figure 7. Time−curing time−stress superposition of compliance in the effective time-domain (squares 30 min, circles 45 min, up triangles 50 min, down triangles 55 min, left triangle 60 min). (inset) Variation of horizontal shift factor as a function of μ that leads to the estimation of characteristic time τm associated with the cross-linking reaction.
can be seen that, in Figure 5, the compliance is plotted as a μ−1 function of (t1−μ − t1−μ w )/(1 − μ) or [ξ(t) − ξ(tw)]A/(τξτm ), wherein only μ depends on stress, while A, τm, and τξ are independent of stress. In the effective time domain, we fix the dominant relaxation time of a material to a constant value of τξ by carrying out the transformation mentioned in eq 2. As a result, the very fact that there is a superposition suggests that the abscissa of Figure 7 must be given by [ξ(t) − ξ(tw)]/τξ. In Figure 7, the individual superpositions at various constant stresses have been shifted horizontally by a factor of α. Consequently, to represent the abscissa of Figure 7 by [ξ(t) − ξ(tw)]/τξ, the factor α is given by ln α = (μ(σ) − μ(σR))ln τm. In this expression μ(σR) is the power-law exponent associated with the reference stress (σR), which we consider to be 100 Pa in Figure 7. In the inset of Figure 7, we plot ln α as a function of μ(σ) − μ(σR) for all the explored stresses. It can be seen that the experimental data indeed demonstrates the behavior: ln α = (μ(σ) − μ(σR))ln τm, which on one hand suggests selfconsistency of the analysis, while, on the other hand, it leads to an estimation of characteristic time scale to be τm = 261 s by which relaxation time evolves as the reaction proceeds. The time−reaction time−stress superposition suggests that application of stress retards the cross-linking process (thereby affecting the mean relaxation time) in such a manner that it preserves the shape of the retardation time (and hence the relaxation time) spectrum. We also carry out the cross-linking reaction and the associated creep experiments at 100 Pa for higher temperatures 35 and 45 °C. Since the rate of cross-linking gets faster at higher temperatures, we employ shorter tw at which constant creep stress is applied. The corresponding creep curves have been plotted in Figure 8. It can be seen that the qualitative
Figure 6. Power law exponent μ plotted as a function of creep stress. The line represents a linear fit given by eq 7.
stress. It can be seen that the value of μ, which is around 6 for constant stress of 100 Pa, decreases linearly to around 4 for the constant stress of 1000 Pa. The dependence of μ on σ over the explored stresses can be represented as μ = 6.43 − 0.0024σ
(7)
Very interestingly this behavior of decrease in power-law coefficient μ with an increase in stress has been observed to be very similar to that reported for soft glassy materials11,34,35 and polymeric glasses.8 This behavior suggests that application of greater intensity of stresswhile cross-linking reaction is going E
DOI: 10.1021/acsapm.9b00432 ACS Appl. Polym. Mater. XXXX, XXX, XXX−XXX
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ACS Applied Polymer Materials
Figure 8. Evolution of strain plotted as a function of time for creep experiments started at different curing times as mentioned for applied constant stress of 100 Pa at temperatures (a) 35 and (b) 45 °C.
behavior of the creep curves at 35 and 45 °C is similar to that obtained at 25 °C plotted in Figure 4, wherein lesser strain has been observed to get induced in the material at higher values of tw. Furthermore, similar to that observed at 25 °C, the creep data at 35 and 45 °C also shows time−reaction time superposition. In Figure 9, we plot the corresponding
Figure 10. Creep time−reaction time−temperature−stress superposition for all the creep data in the effective time domain. The horizontal shift factor β is plotted in the inset as a function of T as discussed in the text.
superpositions have been plotted at the reference stress of 100 Pa with abscissa given by α(t1−μ − t1−μ w )/(1−μ) = [ξ(t) − ξ(tw)]A/(τξ[τm(T)]μ(σR)−1). The characteristic time scale associated with the cross-linking reaction τm, which is independent of stress, is dependent on temperature with an Arrhenius dependence given by10 τm = τm0exp(U /kBT )
(8)
where U is the energy barrier associated with the chemical reaction and τm0 is the fastest characteristic time scale associated with very high temperature. Therefore, equivalent to the process adopted for the shifting at different stresses, the temperature shift factor (β) can be related to temperaturedependent characteristic time scale as β ∼ [τm(T)]μ(T)−1/A. The very fact that the superposition at any temperature has been shifted to superposition associated with the reference temperature (25 °C) leads to
Figure 9. Creep time−reaction time superposition plotted in the effective time domain for the creep data at 100 Pa shown in Figure 8 for 35 and 45 °C. The creep time−reaction time−stress superposition with reference stress of 100 Pa plotted in Figure 7 is also plotted alongside. (inset) Change in μ as a function of temperature.
β = [τm(T )]μ(T ) − 1 /[τm(TR )]μ(TR ) − 1
superpositions along with the time−reaction time−stress superposition at 25 °C with reference stress of 100 Pa. In the inset of Figure 9, we plot μ required to obtain a superposition at the respective temperatures. It can be seen that the value of μ increases with increase in temperature. This behavior of μ essentially suggests that the logarithmic rate of growth of relaxation time with respect to time (μ = d ln τ/d ln tw) due to cross-linking reaction increases with temperature. Interestingly for a soft glassy material μ has been observed to decrease with increase in temperature. The origin of this behavior is discussed elsewhere.10 This observation essentially suggests that the cross-linking process is inherently different than the physical aging process observed in soft glassy materials. It can be seen that all the three superpositions plotted in Figure 9 show similar curvature suggesting a possibility of superposition with respect to temperature. In Figure 10, we plot creep time−reaction time−temperature−stress superposition by carrying out the horizontal shifting of the individual superpositions shown in Figure 9 using a factor β. The shifting of the creep data at different temperatures on the time axis essentially suggests normalization with respect to the temperature-dependent relaxation time. In Figure 9, individual
(9)
In the inset of Figure 10, we plot β as a function of T, which can be seen to be increasing with T. In Figures 5, 7, 9, and 10, the creep curves can be seen to be having a linear dependence on (t1−μ − t1−μ w ). To understand this behavior, let us consider a single-mode Maxwell model with constant modulus G and relaxation time showing a powerlaw dependence on time given by eq 5 so that the viscosity is μ given by η = Gτ = AGτ1−μ m tw. If we subject the Maxwell model to the creep flow field shown in Figure 1, with a magnitude of stress σ0, the strain induced in a material is given by41 G
1−μ − tw1− μ) γ 1 (t =1+ σ0 Aτm1− μ (1 − μ)
(10)
In the above equation, the first term on the right is due to the elasticity of the Maxwell model while the second term on the right is due to the viscous dashpot. Interestingly this second term is the same as the abscissa while term on the left side of eq 10 is proportional to the ordinate of Figures 5, 7, 9, and 10. This expression clearly shows that compliance is linear with respect to (t1−μ − t1−μ w ) because of the contribution of the Newtonian dashpot, which can be attributed to the pregel state of the present system. F
DOI: 10.1021/acsapm.9b00432 ACS Appl. Polym. Mater. XXXX, XXX, XXX−XXX
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ACS Applied Polymer Materials
down the evolution of relaxation time. This behavior has been observed to be similar to that of in physically aging materials wherein aging behavior also slows down by applied stress. The power-law coefficient, on the other hand, is observed to increase with increasing temperature suggesting that the crosslinking reaction is accelerated with an increase in temperature. Finally, the respective superpositions at different stresses and temperatures show a similar curvature on a double logarithmic scale suggesting invariance of the shape of retardation time spectra with respect to reaction time, stress, and temperature, although the mean retardation (or relaxation) time does get affected by the reaction time, stress, and temperature. As a result, suitable horizontal and vertical shifting of the data leads to time−reaction time−temperature−stress superposition. This work has important implications for the processing of thermosetting materials. Often while designing thermosetting processing equipment, the time dependence of the crosslinking reaction is considered to be independent of the deformation/stress field the material experiences. The present work shows that the rate of cross-linking reaction is influenced by the stress field in a nontrivial fashion and that the dependence is a strong function of temperature. We hope that the insights provided by the present work are useful in better designing of processing equipment for the thermosetting polymeric systems.
The present work provides many interesting insights in understanding how the stress and temperature field has a direct influence on the chemical cross-linking process in the pregel state. The observed behavior suggests that, while stress slows down or temperature speeds up the growth of mean retardation/relaxation time, it does not affect the shape of retardation/relaxation time spectra of the reacting system in the pregel regime. The results of the present work have significant consequences in designing equipment that processes the thermosetting materials. This work suggests that the regions, where the extent of deformation field or stress is greater, are expected to be weaker than the other regions experiencing the lesser stresses. We believe that this aspect will affect the amount of time that thermosetting material is needed to spend in processing equipment. Furthermore, in case a mold used for thermoset processing has narrow paths through which flow occurs while filling the cavity, the high stresses may get generated in the thin region. Consequently, that specific region is expected to be weaker than the other regions. If this thin part connects other thick parts in the final product, it is very likely that material will fail at this thin region if a priori precaution is not taken. Owing to limitations associated with the rotational rheometer the present work explores stresses only up to 1000 Pa, while in real processing equipment the stress level is expected to be significantly higher. However, we feel that the explored stress range is large enough to indicate what might happen at the larger stresses through the relationship between μ and σ given by eq 7 assuming that such a relation holds at the higher stresses as well. Second, in processing equipment, there indeed are regions wherein material gets subjected to stresses studied in this work, where the results could be directly useful. Moreover, given the highest achievable stress limit of a rotational rheometer, this is the best that can be achieved since in a capillary rheometer cross-linking reaction can occur inside the barrel, where stresses are too small. The important result of the present work that stress impedes the time evolution of relaxation time suggests that kinetics of chemical cross-linking reaction is affected by the applied stress field. However, how stress affects the kinetics of a chemical cross-linking reaction certainly needs further investigation and is beyond the scope of the present work.
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AUTHOR INFORMATION
Corresponding Author
*Email:
[email protected]. ORCID
Yogesh M. Joshi: 0000-0001-6692-858X Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We acknowledge financial support from the Science and Engineering Research Board (SERB), Department of Science and Technology, Government of India.
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IV. CONCLUSION In this work, we study the creep flow behavior of epoxy resin while it is undergoing the cross-linking reaction. We obtain the creep curves at different reaction times, stresses, and temperatures. Typically, a time-dependent system such as reacting system does not obey the principles of linear viscoelasticity represented by the Boltzmann superposition principle (BSP) due to the invalidation of the time-translational invariance. However, time-dependent materials have been observed to validate the BSP, when material behavior is transformed from the real-time to the effective time domain. We observe that creep curves at different reaction times, stresses, and temperatures indeed obey time−reaction time superposition in the effective time domain. Furthermore, the relaxation time of the studied cross-linking system is observed to show a power-law dependence on time irrespective of the reaction time, stress, and temperature in the pregel state. Interestingly the corresponding power-law exponent has been observed to decrease with an increase in stress at which the creep experiments have been carried out. This suggests that the applied stress does affect the reaction kinetics, thereby slowing
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