Article pubs.acs.org/JPCB
Configurational Entropy in Thermoset Polymers Martin Jensen† and Johnny Jakobsen*,‡ †
Department of Chemistry and Bioscience, Aalborg University, 9220 Aalborg, Denmark Department of Mechanical and Manufacturing Engineering, Aalborg University, 9220 Aalborg, Denmark
‡
ABSTRACT: The configurational entropy describes the atomic structure in a material and controls several material properties. Often the configurational entropy is determined through dielectric or calorimetric measurements where the difference between the entropies of the crystalline state and the amorphous state is determined. Many amorphous materials such as thermoset polymers have a high crystallization barrier, greatly limiting the applicability of the existing methods for determining the configurational entropy. In this work, a novel differential scanning calorimetry (DSC) method, based on measurement of the glass transition temperature at different heating rates, for determination of the configurational entropy is introduced. The theory behind the method has a universal character for amorphous materials, as it solely involves measurement of the glass transition temperature. The temperature dependency of the configurational entropy is determined for epoxy resins and PMMA (poly(methyl methacrylate)) to demonstrate the versatility of the method. On the basis of the findings of the introduced method, the influence of the degree of cross-linking and the chemical structure of the network is discussed.
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INTRODUCTION Thermoset polymers are a widely used material class among others due to the low processing temperature and the mechanical properties. Thermoset polymers are used in highly different areas such as electronics,1−3 microfluidic devices,4,5 coatings,6,7 and matrix material in composite materials.8−11 Owing to their three-dimensional network, thermoset polymers are in most cases amorphous and crystallization through post processing is for most thermosets practically infeasible due to the associated time scale. Unlike the crystalline state, where the atomic configuration is fixed, a large number of atomic configurations can be achieved in the disordered amorphous state at a given temperature.12 The entropy in crystalline materials is solely constituted by the vibrational component, whereas the configurational term needs to be incorporated into the entropy description of amorphous matter. Besides the importance of the configurational entropy (Sc) in the characterization of the polymer network, the configurational entropy has been found to affect the properties of the amorphous material.12−15 The determination of Sc is complicated, and the chemical physics behind its determination has been debated.16 The configurational entropy of a material can be determined from the difference in heat capacity between the material in its amorphous and crystalline states, as this difference is equivalent to the configurational heat capacity (Cp,conf). Integration of Cp,conf/T with respect to temperature yields Sc.17 For amorphous materials with a high crystallization barrier, physical aging around the glass transition temperature (Tg) can be performed, as this lowers the energy state of the material to a level feasible of being infinitely close to that of the crystalline state. In order to mimic the energy state of the crystalline material, aging must be performed at low temperature, but the long structural relaxation time at low temperature extends the © 2015 American Chemical Society
aging duration to months or years. Impedance measurement directly on the pristine amorphous material is another feasible approach for determination of Cp,conf, thus avoiding the aging procedure of the calorimetry method, but it is restricted to strong network formers.18 Thus, there is a need for a fast and versatile method for quantifying the configurational entropy and its temperature dependence inamorphous materials. In a ground breaking paper by Adam and Gibbs, a relation between the thermodynamic property Sc and the dynamic parameter viscosity was introduced.12 Prediction of the temperature dependence of Sc requires a viscosity model usable over a large temperature range, but the most popular viscosity model, the VFT model, predicts zero entropy at a finite temperature highlighting a fundamental imbalance in the model.19 The novel MYEGA viscosity model employs the Angell fragility index (m),20 Tg, and the viscosity at the limit of infinite temperature, which has been shown to be universal for a material class19,21,22 as the system is dominated by kinetic energy at the limit of infinite temperature.19 By linking the viscosity determination of the MYEGA model to the relationship between viscosity and configurational entropy found by Adam and Gibbs, we present a universal calorimetric method for the determination of the configurational entropy and its temperature dependence in amorphous substances by using epoxy polymers and poly(methyl methacrylate) as examples.
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THEORETICAL CALCULATIONS The Adam−Gibbs equation relates the viscosity (η) and the configurational entropy as follows12 Received: October 29, 2014 Revised: March 16, 2015 Published: April 6, 2015 5645
DOI: 10.1021/jp510836y J. Phys. Chem. B 2015, 119, 5645−5649
Article
The Journal of Physical Chemistry B
Table 1. Glass Transition Temperature (Tg) in K Measured at Different Heating Rates for the Two Epoxy Systems and the PMMA Samplea 2 K/min system 1 (one-stage cure) system 1 (two-stage cure) system 2 (one-stage cure) PMMA a
345.5 344.9 348.2 379.9
± ± ± ±
0.7 0.9 0.7 0.4
5 K/min 348.5 347.7 351.3 381.3
± ± ± ±
10 K/min
1.0 1.1 0.9 0.4
349.5 348.9 353.0 383.6
± ± ± ±
1.2 0.8 1.0 0.7
20 K/min 351.4 350.9 356.1 385.5
± ± ± ±
1.0 1.0 1.0 0.3
40 K/min 353.5 353.3 360.2 388.1
± ± ± ±
1.1 1.1 0.8 1.1
The average value and the standard deviation are given.
log(η) = log(η∞) +
B (x ) TSc
B(x) = Tg*(log(η(Tg*)) − log(η(Tg* + 10))) (1)
⎛ ⎞−1 ⎜ ⎟ ⎜ 1 ⎟ Tg* ⎜ ⎟ − S ⎛ ⎞⎟ ⎜ Sc(Tg*) d( B(cx) ) ⎜⎜ Tg*⎜⎜Sc(Tg*) + dT ·10⎟⎟ ⎟⎟ ⎝ ⎠⎠ ⎝
where B(x) is a temperature independent material parameter depending on the chemical composition (x) and for thermosets the degree of cure and log(η∞) is the viscosity at infinite temperature. The viscosity input to the Adam−Gibbs equation is obtained through the MYEGA model19 log(η) = log(η∞) + (12 − log(η∞))
Tg*
T ⎡⎛ ⎞⎤ ⎞⎛ Tg* m − 1⎟⎟⎜⎜ − 1⎟⎟⎥ exp⎢⎜⎜ ⎢⎣⎝ 12 − log(η∞) ⎠⎝ T ⎠⎥⎦
Insertion of eq 5 into eq 1 gives Sc(Tg*) Sc(Tg*) +
(2)
MATERIALS AND METHODS Two epoxy systems have been studied in this work. Both consist of a DGEBA base with reactive diluents. For epoxy system 1, the curing agent contains isophorone diamine and Jeffamine D-230, whereas for epoxy system 2 the curing agent in addition to the two already mentioned amines also contains tertiary amines enabling homopolymerization. Epoxy system 1 was mixed in its stoichiometric ratio, and epoxy system 2 was mixed in the ratio recommended by the supplier. Epoxy system 1 was cured by means of two cure cycles: For the first cure cycle (one-stage), the mixed resin was cured directly at 353 K for 5 h after mixing. In the other curing cycle (two-stage), the mixed resin was left at room temperature overnight (∼18 h) and then cured in a preheated oven at 353 K for 5 h. Despite the difference in the curing cycles, the degree of cure is similar between the epoxies cured by means of the two cure cycles. Epoxy system 2 was only cured by the one-stage cure cycle. The poly(methyl methacrylate) (PMMA) was purchased from Vink and was delivered as a sheet of 1 mm thickness. The Tg of the samples was measured on a Mettler Toledo Star 1 DSC at varying heating rates (q) to tailor the thermal history of the samples. The samples were heated to 383 K (above Tg to erase the thermal history) and cooled to room temperature at the same rate as the heating. Afterward, the sample underwent another heating to 383 K at the same heating rate as the first scan and Tg was determined as the midpoint of the glass transition during the second heating. Rates of 2, 5, 10, 20, and 40 K/min were used, and five measurements were performed for each rate. Each DSC spectrum was checked for post cure through exothermic events during the first upscan, as this would interfere with the Tg
To eliminate the material constant B(x) in eq 1, the viscosity at Tg* and Tg* + 10 K is considered. Thus, on the basis of eq 1, an expression for the viscosity at Tg* and Tg* + 10 K is introduced. A temperature slightly above T*g is chosen for the elimination of B(x) as the MYEGA model uses data around Tg (m and Tg*) and the viscosity at the limit of infinitely high temperature. Since the configurational entropy increases with temperature as more atomic configurations become available, the Sc increases due to a temperature elevation of 10 K (dSc/ dT) and must be accounted for. This may be done by expressing the Taylor series for the configurational entropy at T = T*g + 10 K: (Sc(T*g + 10) = Sc(T*g ) + (dSc/dT)(T*g )·10 K + O); higher order terms have been omitted. Thus, from eq 1, we can write B (x ) Tg*Sc(Tg*)
= log(η∞) − log(η(Tg* + 10))
dT
⎞ ·10⎟ ⎠
dT
■
(3)
d(Sc)(Tg*)
Sc B(x)
To determine the configurational entropy at other temperatures, the temperature dependency from eq 3 is applied.
−1 ⎛ T* ⎞⎤⎞ g ⎜ ⎟⎥⎟ ⎜ T − 1⎟⎥⎟ ⎝ ⎠⎦⎠
B (x ) ⎛ (Tg* + 10)⎜Sc(Tg*) + ⎝
( ) ·10
(6)
⎛ ⎡⎛ ⎞ Sc m = ⎜Tg*(12 − log(η∞)) exp⎢⎜⎜ − 1⎟⎟ ⎢⎣⎝ 12 − log(η∞) B(x) ⎜⎝ ⎠
+
d
⎛ log(η(Tg*)) − log(η(Tg* + 10)) ⎞ Tg* + 10 ⎟ = ⎜⎜1 − ⎟ T* log(η(Tg*)) − log(η∞) ⎝ ⎠ g
where T*g is the standard glass transition temperature of the material and is the Tg value measured by differential scanning calorimetry at a rate of 10 K/min.23 Combination of eqs 1 and 2 gives an expression for Sc:
log(η∞) − log(η(Tg*)) +
(5)
(4)
which reduces to 5646
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cure cycle where the epoxy has been allowed to set at room temperature followed by post cure at 353 K and a one-stage cure where the epoxy is cured directly at 353 K. As seen from Table 1, the two cure cycles result in similar Tg values and calculation of the m value shows that it is 58 and 56 for the onestage and two-stage samples, respectively. Since the difference in m value between the two cure cycles is within the experimental uncertainty, it is inferred that the fragility of the cured epoxy network is independent of the cure cycle and therefore solely related to the chemical composition and the degree of cross-linking of the epoxy network. In order to calculate the viscosity and the configurational entropy of the polymer, the viscosity at the limit of infinitely high temperature must be known (eq 3). However, similar to other thermosets, direct measurement of this parameter is infeasible, as the material decomposes prior to any melting point. Earlier viscosity studies on a wide range of glass formers have revealed that the viscosity universally approaches 1 mPa·s when the temperature is increased toward infinity,19,21 i.e., log(η ∞) is set to −3. To obtain a measure for the configurational entropy, eqs 3 and 4 are applied to determine the viscosity development and the relative configurational entropy in the glass transition range (Figure 1).
measurement during the second upscan. In order to obtain the Angell fragility index (m), the activation energy (ETg) for the glass transition is calculated from Tg at the different heating rates24 ETg = −
d(ln q) ·R ⎛1⎞ d⎜ T ⎟ ⎝ g⎠
(7)
where R is the universal gas constant. The fragility index is calculated from the activation energy as follows:25 m=
■
ETg RTg* ln 10
(8)
RESULTS AND DISCUSSION The glass transition temperature (Tg) of the two tested epoxy systems and the PMMA as a function of heating rate is summarized in Table 1. Tg of the samples is found to increase with the heating rate in an Arrhenian manner. Through eqs 7 and 8, the Angell fragility index (m) is calculated to 58 and 56 for the one-stage and two-stage cure cycles, respectively, which is characterizing a moderately fragile network.26 Comparison to other DGEBA based epoxies is tedious due to the strong dependency of fragility on the curing agent structure and the degree of cure.27,28 The DGEBA used in the epoxy base of both systems is bifunctional, whereas both amines in the curing agent are tetrafunctional. In this work, the fragility of another DGEBA epoxy system with the same two amines as in epoxy system 1 is explored, but in addition to these two amines, the curing agent contains tertiary amines, leading to a certain degree of homopolymerization in epoxy system 2. Hence, epoxy system 2 polymerizes by means of both step-growth polymerization and homopolymerization. This implies that epoxy system 2 has a lower degree of cross-linking than system 1 but a greater short-range rigidity due to an increased content of aromatic units and therefore comparison between fragility of epoxy systems 1 and 2 can shed light on the relative importance of the degree of cross-linking and the chemical backbone of the polymer on the fragility. With an m value of 38, epoxy system 2 exhibits a stronger behavior than epoxy system 1, showing the fragility of the epoxy network is more sensitive to changes in the chemical backbone than the cross-linking degree. Therefore, increased functionalities of the epoxy part, implying an increased degree of cross-linking (lower m), but a higher concentration of the more flexible aliphatic and cycloaliphatic amine groups (higher m) are overall expected to increase the m value of the cured epoxy. If the functionality of the amine is increased without altering the backbone of the amine molecule, the degree of cross-linking increases and the curing agent content relative to the epoxy content diminishes. The degree of cross-linking leads to a stronger network on a long-range scale, and the relative increase in the rigid aromatic group content present in the epoxy part enhances the shortrange rigidity. Both of the factors are expected to enhance the Arrhenian behavior, i.e., lower m. The structure of the epoxy network could be affected by the cure cycle, since a cure where a major part of the curing process is carried out slowly at a low temperature could allow for a more optimum position of each monomer unit in the growing network and thereby a cure cycle impact on the configurational entropy. Epoxy system 1 has been exerted to both a two-stage
Figure 1. Viscosity (black curve, left axis) and the ratio between the configurational entropy and the material constant B (dashed curve, right axis) as a function of temperature around the glass transition region for epoxy system 1 cured in a one-stage cure cycle.
The viscosity of epoxy system 1 exhibits a decrease of 8 orders of magnitude in the 60 K temperature range in Figure 1. The slope of the viscosity curve increases as the temperature is reduced, since the material is moderately fragile. The viscosity predicted in this work is the equilibrium viscosity, and the actual viscosity of the polymer (iso-structural viscosity) might deviate from the former depending on the thermal history of the sample, in particular for more fragile networks.29 Furthermore, the determined viscosity is the steady state viscosity (low shear rate), which can deviate from the dynamic viscosity at higher shear rates, in particular at low conversion degrees.30 The viscosity temperature profile determined by the means of this method is the viscosity of the thermoset at the given degree of cure. For undercured thermosets, post cure is likely to initiate at temperatures around or above Tg and consequently for those polymers the actual viscosity above Tg continuously changes as a result of the post cure, leading to a discrepancy between the actual viscosity and the viscosity determined by the means of the method presented in this work. In Figure 1, the configurational entropy increases with 5647
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temperatures, such as cryogenic temperature, the configurational entropy is controlled by the network structure (degree of cross-linking), whereas at room temperature and above it is controlled by the molecular structure. For the PMMA sample, which is a thermoplastic material with a high crystallization barrier, the configurational entropy as a function of temperature gives reasonable results, suggesting that the method is applicable to both thermosets and thermoplastic materials. On the basis of the agreement between the determined configurational entropy and the principles of physics, it is inferred that the determined entropy values and the temperature dependency of the configurational entropy are correct. Therefore, the method presented in this work is considered to be simple and versatile for determination of the configurational entropy of amorphous materials.
temperature as the number of atomic configurations available grows. Since the viscosity, and hence the Sc prediction, is based on Tg* and m, two parameters characterizing the viscosity around the glass transition region, and the viscosity at the limit of infinite temperature, trends below the glass transition region are solely based on extrapolation of higher temperature data. To explore the versatility of the applied approach, the configurational entropy from absolute zero temperature to a temperature above the glass transition temperature is considered (Figure 2). At absolute zero temperature, the
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CONCLUSIONS In this work, a novel differential scanning calorimetry method for determination of the configurational entropy in amorphous materials is presented. Through measurement of the glass transition temperature at different heating rates, the viscosity− temperature relationship is established, and through application of the Adam−Gibbs equation, the configurational entropy and its temperature dependency can be determined. The theory behind the method has a universal and fundamental character, which makes the method applicable to amorphous materials in general. Determination of the configurational entropy of two epoxies with similar chemical compositions finds that the network fragility is more governed by the chemical composition rather than the functionality of the monomers. The configurational entropy is at low temperatures controlled by the degree of cross-linking and by the molecular rigidity at higher temperatures.
Figure 2. Configurational entropy as a function of absolute temperature determined by eqs 3 and 6 for epoxy system 1 (solid black curve), epoxy system 2 (dashed black curve), and PMMA (dashed gray curve).
configurational entropy should vanish, although some authors31−34 have reported a Kauzmann entropy catastrophe35 where the configurational entropy reaches 0 at a finite temperature. For all samples, the configurational entropy exhibits a linear dependency around the glass transition region and at low temperature asymptotically approaches zero as the absolute temperature goes toward zero. The verification of this principle in chemical physics serves as a confirmation for the validity of the applied approach for determination of the configurational entropy. Since the determination of the configurational entropy as a function temperature relies on the viscosity predictions by the MYEGA model, it is concluded that the physical foundation of the model reflects the changes in atomic configuration during both heating and cooling. The solid physical foundation of the model can account for the extrapolation stability from the glass transition temperature to absolute zero temperature. Comparison of the configurational entropy of the two epoxy systems provides the background for elucidating the effect of chemical composition and degree of cross-linking on the configurational entropy. The entropy starts to increase at a low temperature for the relatively low cross-linked system 2, suggesting that, at very low temperatures, the configurational entropy is governed by the degree of cross-linking. At higher temperature, a reverse trend is seen; i.e., the low cross-linked system with a large content of rigid aromatic groups exhibits a weaker temperature dependency with respect to configurational entropy, implying that, starting from somewhat below Tg, the strength of the cross-links is inadequate to compensate for the increased molecular mobility and therefore the molecular structure becomes more important. Consequently, at low
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AUTHOR INFORMATION
Corresponding Author
*Phone: + 45 23367236. E-mail: [email protected]. Author Contributions
The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS F. Christensen is acknowledged for experimental assistance. REFERENCES
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(27) Andelic, S.; Fitz, B.; Mijovic, J. Reorientational Dynamics and Intermolecular Cooperativity in Reactive Polymers. 2. Multifunctional Epoxy-Amine Systems. Macromolecules 1997, 30, 5239−5248. (28) Mijovic, J.; Sy, J. W. Segmental and Normal Mode Dynamics During Network Formation. J. Non-Cryst. Solids 2002, 307−310, 679− 687. (29) Gupta, P. K.; Heurer, A. Physics of the iso-structural viscosity. J. Non-Cryst. Solids 2012, 358, 3551−3558. (30) Halley, P. J.; Mackay, M. E. Chemorheology of Thermosets − An Overview. Polym. Eng. Sci. 1996, 36, 593−609. (31) Gibbs, J. H.; DiMarzio, E. A. Nature of the Glass Transition and the Glassy State. J. Chem. Phys. 1958, 28, 373−383. (32) Sciortino, F.; Kob, W.; Tartaglia, P. Inherent Structure Entropy of Supercooled Liquids. Phys. Rev. Lett. 1999, 83, 3214−3217. (33) Scala, A.; Starr, F. W.; La Nave, E.; Sciortino, F.; Stanley, H. E. Configurational Entropy and Diffusivity of Supercooled Water. Nature 2000, 406, 166−169. (34) Sastry, S. The Relationship Between Fragility, Configurational Entropy and the Potential Energy Landscape of Glass Forming Liquids. Nature 2001, 409, 164−167. (35) Kauzmann, W. The Nature of the Glassy State and the Behavior of Liquids at Low Temperature. Chem. Rev. 1948, 43, 219−256.
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Configurational Entropy in Thermoset Polymers Martin Jensen† and Johnny Jakobsen*,‡ †
Department of Chemistry and Bioscience, Aalborg University, 9220 Aalborg, Denmark Department of Mechanical and Manufacturing Engineering, Aalborg University, 9220 Aalborg, Denmark
‡
ABSTRACT: The configurational entropy describes the atomic structure in a material and controls several material properties. Often the configurational entropy is determined through dielectric or calorimetric measurements where the difference between the entropies of the crystalline state and the amorphous state is determined. Many amorphous materials such as thermoset polymers have a high crystallization barrier, greatly limiting the applicability of the existing methods for determining the configurational entropy. In this work, a novel differential scanning calorimetry (DSC) method, based on measurement of the glass transition temperature at different heating rates, for determination of the configurational entropy is introduced. The theory behind the method has a universal character for amorphous materials, as it solely involves measurement of the glass transition temperature. The temperature dependency of the configurational entropy is determined for epoxy resins and PMMA (poly(methyl methacrylate)) to demonstrate the versatility of the method. On the basis of the findings of the introduced method, the influence of the degree of cross-linking and the chemical structure of the network is discussed.
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INTRODUCTION Thermoset polymers are a widely used material class among others due to the low processing temperature and the mechanical properties. Thermoset polymers are used in highly different areas such as electronics,1−3 microfluidic devices,4,5 coatings,6,7 and matrix material in composite materials.8−11 Owing to their three-dimensional network, thermoset polymers are in most cases amorphous and crystallization through post processing is for most thermosets practically infeasible due to the associated time scale. Unlike the crystalline state, where the atomic configuration is fixed, a large number of atomic configurations can be achieved in the disordered amorphous state at a given temperature.12 The entropy in crystalline materials is solely constituted by the vibrational component, whereas the configurational term needs to be incorporated into the entropy description of amorphous matter. Besides the importance of the configurational entropy (Sc) in the characterization of the polymer network, the configurational entropy has been found to affect the properties of the amorphous material.12−15 The determination of Sc is complicated, and the chemical physics behind its determination has been debated.16 The configurational entropy of a material can be determined from the difference in heat capacity between the material in its amorphous and crystalline states, as this difference is equivalent to the configurational heat capacity (Cp,conf). Integration of Cp,conf/T with respect to temperature yields Sc.17 For amorphous materials with a high crystallization barrier, physical aging around the glass transition temperature (Tg) can be performed, as this lowers the energy state of the material to a level feasible of being infinitely close to that of the crystalline state. In order to mimic the energy state of the crystalline material, aging must be performed at low temperature, but the long structural relaxation time at low temperature extends the © 2015 American Chemical Society
aging duration to months or years. Impedance measurement directly on the pristine amorphous material is another feasible approach for determination of Cp,conf, thus avoiding the aging procedure of the calorimetry method, but it is restricted to strong network formers.18 Thus, there is a need for a fast and versatile method for quantifying the configurational entropy and its temperature dependence inamorphous materials. In a ground breaking paper by Adam and Gibbs, a relation between the thermodynamic property Sc and the dynamic parameter viscosity was introduced.12 Prediction of the temperature dependence of Sc requires a viscosity model usable over a large temperature range, but the most popular viscosity model, the VFT model, predicts zero entropy at a finite temperature highlighting a fundamental imbalance in the model.19 The novel MYEGA viscosity model employs the Angell fragility index (m),20 Tg, and the viscosity at the limit of infinite temperature, which has been shown to be universal for a material class19,21,22 as the system is dominated by kinetic energy at the limit of infinite temperature.19 By linking the viscosity determination of the MYEGA model to the relationship between viscosity and configurational entropy found by Adam and Gibbs, we present a universal calorimetric method for the determination of the configurational entropy and its temperature dependence in amorphous substances by using epoxy polymers and poly(methyl methacrylate) as examples.
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THEORETICAL CALCULATIONS The Adam−Gibbs equation relates the viscosity (η) and the configurational entropy as follows12 Received: October 29, 2014 Revised: March 16, 2015 Published: April 6, 2015 5645
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Table 1. Glass Transition Temperature (Tg) in K Measured at Different Heating Rates for the Two Epoxy Systems and the PMMA Samplea 2 K/min system 1 (one-stage cure) system 1 (two-stage cure) system 2 (one-stage cure) PMMA a
345.5 344.9 348.2 379.9
± ± ± ±
0.7 0.9 0.7 0.4
5 K/min 348.5 347.7 351.3 381.3
± ± ± ±
10 K/min
1.0 1.1 0.9 0.4
349.5 348.9 353.0 383.6
± ± ± ±
1.2 0.8 1.0 0.7
20 K/min 351.4 350.9 356.1 385.5
± ± ± ±
1.0 1.0 1.0 0.3
40 K/min 353.5 353.3 360.2 388.1
± ± ± ±
1.1 1.1 0.8 1.1
The average value and the standard deviation are given.
log(η) = log(η∞) +
B (x ) TSc
B(x) = Tg*(log(η(Tg*)) − log(η(Tg* + 10))) (1)
⎛ ⎞−1 ⎜ ⎟ ⎜ 1 ⎟ Tg* ⎜ ⎟ − S ⎛ ⎞⎟ ⎜ Sc(Tg*) d( B(cx) ) ⎜⎜ Tg*⎜⎜Sc(Tg*) + dT ·10⎟⎟ ⎟⎟ ⎝ ⎠⎠ ⎝
where B(x) is a temperature independent material parameter depending on the chemical composition (x) and for thermosets the degree of cure and log(η∞) is the viscosity at infinite temperature. The viscosity input to the Adam−Gibbs equation is obtained through the MYEGA model19 log(η) = log(η∞) + (12 − log(η∞))
Tg*
T ⎡⎛ ⎞⎤ ⎞⎛ Tg* m − 1⎟⎟⎜⎜ − 1⎟⎟⎥ exp⎢⎜⎜ ⎢⎣⎝ 12 − log(η∞) ⎠⎝ T ⎠⎥⎦
Insertion of eq 5 into eq 1 gives Sc(Tg*) Sc(Tg*) +
(2)
MATERIALS AND METHODS Two epoxy systems have been studied in this work. Both consist of a DGEBA base with reactive diluents. For epoxy system 1, the curing agent contains isophorone diamine and Jeffamine D-230, whereas for epoxy system 2 the curing agent in addition to the two already mentioned amines also contains tertiary amines enabling homopolymerization. Epoxy system 1 was mixed in its stoichiometric ratio, and epoxy system 2 was mixed in the ratio recommended by the supplier. Epoxy system 1 was cured by means of two cure cycles: For the first cure cycle (one-stage), the mixed resin was cured directly at 353 K for 5 h after mixing. In the other curing cycle (two-stage), the mixed resin was left at room temperature overnight (∼18 h) and then cured in a preheated oven at 353 K for 5 h. Despite the difference in the curing cycles, the degree of cure is similar between the epoxies cured by means of the two cure cycles. Epoxy system 2 was only cured by the one-stage cure cycle. The poly(methyl methacrylate) (PMMA) was purchased from Vink and was delivered as a sheet of 1 mm thickness. The Tg of the samples was measured on a Mettler Toledo Star 1 DSC at varying heating rates (q) to tailor the thermal history of the samples. The samples were heated to 383 K (above Tg to erase the thermal history) and cooled to room temperature at the same rate as the heating. Afterward, the sample underwent another heating to 383 K at the same heating rate as the first scan and Tg was determined as the midpoint of the glass transition during the second heating. Rates of 2, 5, 10, 20, and 40 K/min were used, and five measurements were performed for each rate. Each DSC spectrum was checked for post cure through exothermic events during the first upscan, as this would interfere with the Tg
To eliminate the material constant B(x) in eq 1, the viscosity at Tg* and Tg* + 10 K is considered. Thus, on the basis of eq 1, an expression for the viscosity at Tg* and Tg* + 10 K is introduced. A temperature slightly above T*g is chosen for the elimination of B(x) as the MYEGA model uses data around Tg (m and Tg*) and the viscosity at the limit of infinitely high temperature. Since the configurational entropy increases with temperature as more atomic configurations become available, the Sc increases due to a temperature elevation of 10 K (dSc/ dT) and must be accounted for. This may be done by expressing the Taylor series for the configurational entropy at T = T*g + 10 K: (Sc(T*g + 10) = Sc(T*g ) + (dSc/dT)(T*g )·10 K + O); higher order terms have been omitted. Thus, from eq 1, we can write B (x ) Tg*Sc(Tg*)
= log(η∞) − log(η(Tg* + 10))
dT
⎞ ·10⎟ ⎠
dT
■
(3)
d(Sc)(Tg*)
Sc B(x)
To determine the configurational entropy at other temperatures, the temperature dependency from eq 3 is applied.
−1 ⎛ T* ⎞⎤⎞ g ⎜ ⎟⎥⎟ ⎜ T − 1⎟⎥⎟ ⎝ ⎠⎦⎠
B (x ) ⎛ (Tg* + 10)⎜Sc(Tg*) + ⎝
( ) ·10
(6)
⎛ ⎡⎛ ⎞ Sc m = ⎜Tg*(12 − log(η∞)) exp⎢⎜⎜ − 1⎟⎟ ⎢⎣⎝ 12 − log(η∞) B(x) ⎜⎝ ⎠
+
d
⎛ log(η(Tg*)) − log(η(Tg* + 10)) ⎞ Tg* + 10 ⎟ = ⎜⎜1 − ⎟ T* log(η(Tg*)) − log(η∞) ⎝ ⎠ g
where T*g is the standard glass transition temperature of the material and is the Tg value measured by differential scanning calorimetry at a rate of 10 K/min.23 Combination of eqs 1 and 2 gives an expression for Sc:
log(η∞) − log(η(Tg*)) +
(5)
(4)
which reduces to 5646
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cure cycle where the epoxy has been allowed to set at room temperature followed by post cure at 353 K and a one-stage cure where the epoxy is cured directly at 353 K. As seen from Table 1, the two cure cycles result in similar Tg values and calculation of the m value shows that it is 58 and 56 for the onestage and two-stage samples, respectively. Since the difference in m value between the two cure cycles is within the experimental uncertainty, it is inferred that the fragility of the cured epoxy network is independent of the cure cycle and therefore solely related to the chemical composition and the degree of cross-linking of the epoxy network. In order to calculate the viscosity and the configurational entropy of the polymer, the viscosity at the limit of infinitely high temperature must be known (eq 3). However, similar to other thermosets, direct measurement of this parameter is infeasible, as the material decomposes prior to any melting point. Earlier viscosity studies on a wide range of glass formers have revealed that the viscosity universally approaches 1 mPa·s when the temperature is increased toward infinity,19,21 i.e., log(η ∞) is set to −3. To obtain a measure for the configurational entropy, eqs 3 and 4 are applied to determine the viscosity development and the relative configurational entropy in the glass transition range (Figure 1).
measurement during the second upscan. In order to obtain the Angell fragility index (m), the activation energy (ETg) for the glass transition is calculated from Tg at the different heating rates24 ETg = −
d(ln q) ·R ⎛1⎞ d⎜ T ⎟ ⎝ g⎠
(7)
where R is the universal gas constant. The fragility index is calculated from the activation energy as follows:25 m=
■
ETg RTg* ln 10
(8)
RESULTS AND DISCUSSION The glass transition temperature (Tg) of the two tested epoxy systems and the PMMA as a function of heating rate is summarized in Table 1. Tg of the samples is found to increase with the heating rate in an Arrhenian manner. Through eqs 7 and 8, the Angell fragility index (m) is calculated to 58 and 56 for the one-stage and two-stage cure cycles, respectively, which is characterizing a moderately fragile network.26 Comparison to other DGEBA based epoxies is tedious due to the strong dependency of fragility on the curing agent structure and the degree of cure.27,28 The DGEBA used in the epoxy base of both systems is bifunctional, whereas both amines in the curing agent are tetrafunctional. In this work, the fragility of another DGEBA epoxy system with the same two amines as in epoxy system 1 is explored, but in addition to these two amines, the curing agent contains tertiary amines, leading to a certain degree of homopolymerization in epoxy system 2. Hence, epoxy system 2 polymerizes by means of both step-growth polymerization and homopolymerization. This implies that epoxy system 2 has a lower degree of cross-linking than system 1 but a greater short-range rigidity due to an increased content of aromatic units and therefore comparison between fragility of epoxy systems 1 and 2 can shed light on the relative importance of the degree of cross-linking and the chemical backbone of the polymer on the fragility. With an m value of 38, epoxy system 2 exhibits a stronger behavior than epoxy system 1, showing the fragility of the epoxy network is more sensitive to changes in the chemical backbone than the cross-linking degree. Therefore, increased functionalities of the epoxy part, implying an increased degree of cross-linking (lower m), but a higher concentration of the more flexible aliphatic and cycloaliphatic amine groups (higher m) are overall expected to increase the m value of the cured epoxy. If the functionality of the amine is increased without altering the backbone of the amine molecule, the degree of cross-linking increases and the curing agent content relative to the epoxy content diminishes. The degree of cross-linking leads to a stronger network on a long-range scale, and the relative increase in the rigid aromatic group content present in the epoxy part enhances the shortrange rigidity. Both of the factors are expected to enhance the Arrhenian behavior, i.e., lower m. The structure of the epoxy network could be affected by the cure cycle, since a cure where a major part of the curing process is carried out slowly at a low temperature could allow for a more optimum position of each monomer unit in the growing network and thereby a cure cycle impact on the configurational entropy. Epoxy system 1 has been exerted to both a two-stage
Figure 1. Viscosity (black curve, left axis) and the ratio between the configurational entropy and the material constant B (dashed curve, right axis) as a function of temperature around the glass transition region for epoxy system 1 cured in a one-stage cure cycle.
The viscosity of epoxy system 1 exhibits a decrease of 8 orders of magnitude in the 60 K temperature range in Figure 1. The slope of the viscosity curve increases as the temperature is reduced, since the material is moderately fragile. The viscosity predicted in this work is the equilibrium viscosity, and the actual viscosity of the polymer (iso-structural viscosity) might deviate from the former depending on the thermal history of the sample, in particular for more fragile networks.29 Furthermore, the determined viscosity is the steady state viscosity (low shear rate), which can deviate from the dynamic viscosity at higher shear rates, in particular at low conversion degrees.30 The viscosity temperature profile determined by the means of this method is the viscosity of the thermoset at the given degree of cure. For undercured thermosets, post cure is likely to initiate at temperatures around or above Tg and consequently for those polymers the actual viscosity above Tg continuously changes as a result of the post cure, leading to a discrepancy between the actual viscosity and the viscosity determined by the means of the method presented in this work. In Figure 1, the configurational entropy increases with 5647
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temperatures, such as cryogenic temperature, the configurational entropy is controlled by the network structure (degree of cross-linking), whereas at room temperature and above it is controlled by the molecular structure. For the PMMA sample, which is a thermoplastic material with a high crystallization barrier, the configurational entropy as a function of temperature gives reasonable results, suggesting that the method is applicable to both thermosets and thermoplastic materials. On the basis of the agreement between the determined configurational entropy and the principles of physics, it is inferred that the determined entropy values and the temperature dependency of the configurational entropy are correct. Therefore, the method presented in this work is considered to be simple and versatile for determination of the configurational entropy of amorphous materials.
temperature as the number of atomic configurations available grows. Since the viscosity, and hence the Sc prediction, is based on Tg* and m, two parameters characterizing the viscosity around the glass transition region, and the viscosity at the limit of infinite temperature, trends below the glass transition region are solely based on extrapolation of higher temperature data. To explore the versatility of the applied approach, the configurational entropy from absolute zero temperature to a temperature above the glass transition temperature is considered (Figure 2). At absolute zero temperature, the
■
CONCLUSIONS In this work, a novel differential scanning calorimetry method for determination of the configurational entropy in amorphous materials is presented. Through measurement of the glass transition temperature at different heating rates, the viscosity− temperature relationship is established, and through application of the Adam−Gibbs equation, the configurational entropy and its temperature dependency can be determined. The theory behind the method has a universal and fundamental character, which makes the method applicable to amorphous materials in general. Determination of the configurational entropy of two epoxies with similar chemical compositions finds that the network fragility is more governed by the chemical composition rather than the functionality of the monomers. The configurational entropy is at low temperatures controlled by the degree of cross-linking and by the molecular rigidity at higher temperatures.
Figure 2. Configurational entropy as a function of absolute temperature determined by eqs 3 and 6 for epoxy system 1 (solid black curve), epoxy system 2 (dashed black curve), and PMMA (dashed gray curve).
configurational entropy should vanish, although some authors31−34 have reported a Kauzmann entropy catastrophe35 where the configurational entropy reaches 0 at a finite temperature. For all samples, the configurational entropy exhibits a linear dependency around the glass transition region and at low temperature asymptotically approaches zero as the absolute temperature goes toward zero. The verification of this principle in chemical physics serves as a confirmation for the validity of the applied approach for determination of the configurational entropy. Since the determination of the configurational entropy as a function temperature relies on the viscosity predictions by the MYEGA model, it is concluded that the physical foundation of the model reflects the changes in atomic configuration during both heating and cooling. The solid physical foundation of the model can account for the extrapolation stability from the glass transition temperature to absolute zero temperature. Comparison of the configurational entropy of the two epoxy systems provides the background for elucidating the effect of chemical composition and degree of cross-linking on the configurational entropy. The entropy starts to increase at a low temperature for the relatively low cross-linked system 2, suggesting that, at very low temperatures, the configurational entropy is governed by the degree of cross-linking. At higher temperature, a reverse trend is seen; i.e., the low cross-linked system with a large content of rigid aromatic groups exhibits a weaker temperature dependency with respect to configurational entropy, implying that, starting from somewhat below Tg, the strength of the cross-links is inadequate to compensate for the increased molecular mobility and therefore the molecular structure becomes more important. Consequently, at low
■
AUTHOR INFORMATION
Corresponding Author
*Phone: + 45 23367236. E-mail: [email protected]. Author Contributions
The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes
The authors declare no competing financial interest.
■ ■
ACKNOWLEDGMENTS F. Christensen is acknowledged for experimental assistance. REFERENCES
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