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Decrease in the configurational entropy during a melt’s polymerization
Chemical Physics 305 (2004) 231–236 www.elsevier.com/locate/chemphys
Decrease in the configurational entropy during a meltÕs polymerization G.P. Johari
*
Department of Materials Science and Engineering, McMaster University, 1280 Main St. West Hamilton, Ont., Canada L8S 4L7 Received 25 August 2003; accepted 2 June 2004 Available online 30 July 2004
Abstract The manner in which the configurational entropy, Sconf, of a polymerizing melt decreases has been examined in context with a recent conclusion [S. Corezzi, D. Fioretto, P. Rolla, Nature, 420 (2002) 653] that Sconf varies as (1a/a0), where a is the measured extent of polymerization and a0(61) is a parameter calculated from the functionality of the meltÕs reacting molecules. It is shown that for Sconf to vary as (1a/a0): (i) Sconf of a melt should vanish when a becomes equal to a0, irrespective of the temperature and pressure, and the meltÕs relaxation time, s, becomes infinite; (ii) polymerization should terminate at a = a0. Experiments have shown the opposite, namely that: (i) Sconf of a fully polymerized state (a = 1 P a0) is large, s is as low as 1 ls and both Sconf and s vary with temperature and pressure; (ii) polymerization continues to occur beyond a0. Therefore, Sconf does not vary as (1a/a0). The relation, s(a) = s0exp[B/(1a/a0)], used for testing this variation requires that B and s0 be independent of a, which is not the case. 2004 Elsevier B.V. All rights reserved.
1. Introduction There is a general interest in determining the irreversible change in thermodynamic properties of a liquid in which chemical reactions lead to a macromoleculeÕs growth. Since such a growth irreversibly increases a liquidÕs viscosity at a fixed polymerization temperature, Tpoly, there is also a special interest in determining how molecular kinetics is controlled by irreversible chemical reactions. The phenomenon has been described in terms of a negative feedback between molecular diffusion, which allows a chemical reaction to occur, and chemical reaction which slows the diffusion [1]. Studies have also shown that changes in the thermodynamic and molecular kinetic properties during the course of a macromoleculeÕs growth in a polymerizing melt at a fixed Tpoly are analogous to the changes observed in these two properties on cooling a liquid of a fixed chem*
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0301-0104/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2004.06.056
ical structure [1]. For example, the entropy, volume and enthalpy decrease and viscosity, g, and relaxation time increase during a meltÕs polymerization, as also occurs during the cooling of the melt. Because thermodynamic and kinetic properties change together, one expects that g and the relaxation time of a liquid would be intrinsically related to a thermodynamic property. The change in thermodynamic properties due to the change in phonon frequency and anharmonic effects is finite but usually small when van der WaalsÕ interactions are replaced by covalent bonds during a macromoleculeÕs growth. Cooling a liquid of fixed chemical structure has a similarly small effect. In seeking a thermodynamic basis for vitrification, therefore, only the configurational entropy, Sconf, is currently used for describing the rate of molecular kinetics of a chemically stable liquid. Since each infinitesmal stage in the process of polymerization during which a fixed structure exists corresponds to a liquid of different chemical structure, and configurational restriction increases as the structure become increasingly covalently bonded, the observed
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increase in g and in the dielectric relaxation time, s, during a meltÕs polymerization can be related to the decrease in Sconf. For liquids in general, even Sconf has been difficult to estimate, because the vibrational component of the entropy of a liquid, which should be subtracted from the experimentally determined entropy, cannot be easily determined. Nevertheless, Sconf of polymers has been accurately calculated by using chain statistics and following FloryÕs original calculations based on a lattice hole model [2–4]. These calculations have been further used to determine the manner in which Sconf decreases on cooling a polymer melt, and then finally relating Sconf to the vitrification of a polymer [5–10]. Several of these studies have concluded that Sconf of a polymer would vanish at a T much higher than 0 K [5,6]. Others have concluded that Sconf would vanish only at 0 K [7–10]. Wang and Johari [10] have calculated Sconf of a melt during the course of polymerization, i.e, as a function of the extent of polymerization, a, at a fixed polymerizing temperature, Tpoly, by using polymer chain statistics in FloryÕs lattice hole model. The calculation showed that Sconf and the configurational heat capacity, Cp,conf, of a melt would remain finite at its full polymerization, i.e., at a = 1. It also led to two predictions, (i) Sconf and Cp,conf of an equilibrium melt would approach zero only at 0 K, which is contrary to the extrapolation of the excess entropy of a liquid over its crystal phase to zero at a temperature [11] of 20–30% below Tg (see also [12]), and (ii) Cp,conf would initially increase with increase in a and then decrease, thus showing a local maximum in the Cp against a plot at a fixed Tpoly. The first prediction has been experimentally verified by Pyda and Wunderlich [13], and the second prediction has been borne out by a recent study [14]. It should be stressed that vitrification of a liquid itself does not require that its excess entropy over the crystal phase value become zero at T > 0 K. Concurrently, Corezzi et al. [15] assumed that, in contrast to the prediction from the Flory model as extended by Wang and Johari [10] to the polymerization process, Sconf is inversely proportional to the number average degree of polymerization, xn, which in turn is defined as the average number of monomers per molecule [15]. Corezzi et al. recognized that this is a crude approximation and then used it to obtain a relation between s and a. To elaborate, their basic premise is that for a polymerizing melt, Sconf(a) = Sconf(0)/xn, where Sconf(0) is the configurational entropy of a melt consisting of only monomers at a = 0. After stating that the relation, Sconf(a) = Sconf(0)/xn, is ‘‘. . .not cruder than the entropy theory itself:’’ [15], they combined this relation with the configurational entropy theoryÕs Adam–Gibbs equation [16] and fitted the resulting equation to the available s data [15]. The apparently good fit to the data was seen as proof of the validity of this premise. They ultimately concluded that use of the
entropy relation, Sconf(a) = Sconf(0)/xn, in future would benefit materials processing, help in understanding a liquidsÕ structural fluctuation in terms of the potential energy landscape model at a fundamental level, as well as the phenomena observed in micro-emulsions, disordered structures, physical gels, percolation, etc. Both, the basic premise that Sconf (a) 1/xn and the apparent fit of the resulting equation to s data are intriguing. Moreover, Corezzi et al.Õs findings [15] conflict with the precepts of polymer chains statistics [7–10], which they overlooked. We examine the validity of this premise and their findings on fundamental grounds, and determine whether the relation, Sconf(a) 1/xn is physically justifiable. We also examine the merits of their ultimate sa correlation and the manner of its fitting by using experimental data, and show that the apparent success of this correlation is due to use of a limited range of data.
2. Entropy and the extent of polymerization It is well recognized that polymerization decreases the entropy and increases the relaxation time of a melt, i.e., as molecular weight of the polymer increases, its entropy decreases and relaxation time increases. In a polymerization process, say by addition reactions between a monamine and digycidyl ether of bisphenol-A (a diepoxide) in a stoichiometric liquid mixture, the –NH2 group chemically reacts with two epoxide groups in two stages involving consecutive reactions, and produces a linear chain. This step-addition reaction occurs in the following sequence: One H-atom of the –NH2 group of the monoamine combines with the –O– group of the epoxide thus forming an –OH group and opening the terminal epoxide ring, and the N atom becomes covalently bonded to the terminal C atom of the diepoxide molecule. Then the second H-atom of the now sterically hindered –NH group in the R–NH–(CH2)n–CH3 molecule, where R denotes the partly reacted diepoxide molecule, chemically reacts similarly with the terminal epoxide group of another diepoxide molecule. Thus by losing its two amine protons, one amine molecule links two diepoxide molecules by its one N atom, leading to a structure of the type containing an –ABABA– sequence of the reacted molecules of diepoxide, A, and of monoamine, B. A similar reaction with a diamine produces a network structure when H atoms of both amine groups have reacted with the four terminal atoms epoxide groups of four diepoxide molecules. As these chemical reactions occur and the macromolecule grows, the viscosity of the liquid increases at a progressively rapid rate and the liquid (or elastomer) vitrifies before reaching complete polymerization when the polymerization temperature, Tpoly, is less than the vitrification temperature, Tg, of the fully polymerized structure. But when Tpoly is
G.P. Johari / Chemical Physics 305 (2004) 231–236
higher than this Tg, the fully polymerized state remains a viscous liquid if polymerization leads to linear chain and an elastomer if it leads to a molecular network structure. It seems necessary to first describe the manner by which Corezzi et al. [15] had obtained the relation, Sconf (a) 1/xn, and thereafter its correlation between the dielectric relaxation time, s, and a. Without providing a physical or molecular justification, they wrote [15] that it is realistic to expect that Wconf(a), the number of configurations available to a polymerized state characterized by a, is given by W conf ðaÞ ¼ ½W conf ð0Þ1=xn ;
ð1Þ
where Wconf(0), a constant is the number of configurations available to the monomer (unpolymerized) state at a = 0. In Eq. (1), as xn decreases from one to zero, a increases from zero to one. Since S = kBlnW, Eq. (1) becomes, S conf ðaÞ ¼
kB ln½W conf ð0Þ; xn
ð2Þ
where kB is the Boltzmann constant and xn is the number average degree of polymerization (or the average number of monomers per multimer). Since ln[Wconf(0)] is a constant at a fixed Tpoly, S conf ðaÞ /
1 : xn
ð2aÞ
The quantity xn was identified as the number average degree of polymerization, and was written as [15], xn = (1a/a0)1, where a0 is taken as a parameter determined from the average molecular weight. Macosko and Miller [17] had provided expressions for the average molecular weight and the magnitude of a0 for various types of addition reactions in terms of molecular ratio and functionality of the two reacting components of a polymerizing melt. Accordingly, a0 is determined only by the number of covalent bonds that can form between the reacting groups of molecules involved in polymerization, and not by the chemical structure of the molecules themselves or of the reacting groups. To briefly describe the definition of a0, we recall Macosko and MillerÕs [17] calculation of the average molecular weight for nonlinear polymers formed when Afi reacts with Bgj, and of the number average degree of polymerization, and use their Eqs. (56)–(59) to derive P P Bgj i Afi þ j P aA;0 ¼ ; ð3Þ i fiAfi where Afi represents the moles of component Afi, and Bgj the moles of component Bgj in the melt, and fi is the number of functional groups in Afi. According to Eq. (3), when one mole of a bifunctional component A reacts with one mole of component B, aA,0 is equal to one; when four moles of bifunctional A react with three
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moles of B, aA,0 = 0.875; and when two moles of bifunctional A react with one mole of B, aA,0 = 0.75. It is to be noted that polymerization is incomplete when aA,0 < a. The variation of xn with aA, i.e., the mole fraction of A groups that have reacted in the component A, has been defined as [15], xn(aA) = 1/(1ÆfAæaA), where 1=hfA i atheor , where atheor is taken as equal to Mocasko 0 0 and MillerÕs [17] aA,0 in Eq. (3) here. By expressing a as the extent of ‘‘conversion of the epoxy groups’’ [15], and substituting in Eq. (2) a S conf ðaÞ ¼ S conf ð0Þ 1 a0 k B ln½W conf ð0Þ ða0 aÞ: ð4Þ ¼ a0 Now Eq. (4) has several important consequences for thermodynamics of polymerization. Firstly, it means that Sconf would vanish when a = a0. In molecular terms it means that at a given Tpoly, which is higher than 0 K, there is only one configuration possible for the polymer structure formed by, say, addition reaction of two moles of bifunctional A and one mole of B at the extent a = a0 = 0.75. (This is reminiscent of the Kauzmann extrapolation [11], which had suggested that the excess entropy of a liquid over the crystal phase, which has been seen as equal to Sconf, would become zero on cooling the equilibrium liquid to a temperature 20–30% below Tg see also [12,13] for discussion of this extrapolation). Secondly, according to Eq. (4), Sconf of all polymers, which have been polymerized to the extent of a0, would be zero, irrespective of the value of their Sconf(0). This means that only one molecular arrangement of the polymer structure exists even when the experimentally observed state of the polymer at a = a0 is a viscous liquid or an elastomer at a high polymerization temperature. Thirdly, irrespective of the value of Sconf (0), Sconf for all T and P conditions of polymerization of all types of melts would become zero once its a is equal to a0, and, since Eq. (4) does not contain the terms T and P, a change in T or P at a = a0 would have no effect on Sconf. We now consider how Eq. (4) has been related to s in [15]. According to the Adam–GibbsÕ configurational entropy theory [16] for a chemically stable liquid, or for a polymerizing melt at a fixed value of a CðaÞ sðaÞ ¼ s0 ðaÞ exp ; ð5Þ TS conf ðaÞ where C ¼ Dlsc and s0 corresponds to the phonon time scale, i.e., 1013 to 1014 s. The term Dl is the free energy of activation per monomer segment in the co-operatively rearranging region, and sc is the critical entropy. The latter has been usually taken to be a constant for all liquids. The molecular basis of Eq. (5) is that as T is decreased, there is an increasing requirement for co-operative rearrangement of molecules in the liquid and
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therefore the size of the co-operatively rearranging region increases and Sconf decreases. Hence, TSconf for a fixed value of a decreases on cooling and s increases rapidly according to Eq. (4). When a increases on polymerization, the chemical structure of the liquid changes and hence the values of C and s0 change. To elaborate, in the Adam–Gibbs formalism [16], the quantity, Dl = DHa TDSa, where DHa is the enthalpy and DSa the entropy of activation. Since both DHa and DSa change as a polymerizing meltÕs chemical structure changes, C would change as a increases. As C appears in an exponent in Eq. (5), its small variation with a would have a relatively large effect on s, as recognized in the earlier studies of polymerization [18–20]. By substituting Eq. (4) in Eq. (5), Corezzi et al. [15] obtained, ln sðaÞ ¼ ln s0 þ
B Ba0 ¼ ln s0 þ ; ½1 ða=a0 Þ ½a0 a
ð6Þ
where B = C(a)/TSconf(0). The value of B is expected to vary with a because C varies with a. In the fitting of the dielectric relaxation time data, Corezzi et al. [15] have overlooked the strong effect of the variation of C with a, and have used fixed values of both s0 and B, although s0, which is related to phonon frequency is expected to change as Van der WaalsÕ interactions are replaced by covalent bonds during the growth of a macromolecule. One consequence of Eq. (6) is that s diverges at a0 generally for all polymerization processes, but there is no physical or molecular basis for it. It appears that this divergence is an artifact of the neglect in the variation of s0 and B with a, as discussed above. By substituting for C(a) in B = C(a)/TSconf(0), we may write, B ¼ sc ½DH a ðaÞ T DS a ðaÞ=TS conf ð0Þ. Since sc =TS conf ð0Þ is taken as a constant, B becomes proportional to [DHa(a)TDSa(a)] and its value varies with a. Thus, as more covalent bonds form and the melt densifies on polymerization, the growing hindrance to molecular structural rearrangement not only reduces Sconf but also changes the empirical quantity B through a change in DHa and DSa with increase in a.
3. A scrutiny of the features of the entropy-extent of polymerization relation The suggested success in fitting Eq. (6) to the s data in [15] has serious consequences for the validity of the calculation of thermodynamic properties based on polymer chain statistics [2–10]. Moreover, since it has been expected [15] that use of Eq. (2) would benefit materials processing, and help in understanding the picture of the potential energy landscape, and the phenomena of microemulsions, physical gels, disordered systems and percolation, it seems necessary that validity of the basic premise of Eq. (6) and thereafter of its fitting to the s
data already available in the literature should be carefully examined. This is done as follows: First, we consider a relatively simple feature of Eq. (4): Since Sconf(0) is a constant, differentiating Eq. (4) with respect to T yields, Cp,conf(a) = T(oSconf(0)/oT)P. This means that Cp,conf of a polymerizing melt at a fixed Tpoly would not change as a increases from zero to one, because (oSconf(0)/oT) is a constant. But the heat capacity of polymer melts and elastomers has been known to change with increase in the extent of polymerization or a polymerÕs molecular weight. Similarly, differentiating Eq. (4) with respect to pressure, P, yields, bconfVconf [= (oSconf(0)/oP)T], where bconf is the configurational contribution to the thermal expansion coefficient of the melt, and Vconf of that to the volume. This means that as a macromolecule grows, or a melt polymerizes, the product bconfVconf would not change. Therefore, in order for Eq. (4) to be valid, the observed change in Cp, b and V of a polymerizing melt should be due only to an increase in the phonon frequencies and decrease in the anharmonic effects. This is contrary to the well known observations that changes in Cp, b and V resulting from a meltÕs polymerization are generally much more than those resulting from changes in the phonon frequency and anharmonic forces, and that there is a considerable configurational contribution to the Cp, b and V of polymers as long as Tpoly > Tg. Second, according to Eq. (4) a plot of Sconf against a will be a straight line with an intercept Sconf(0), or kBln[Wconf(0)], and a slope of kBln[Wconf(0)]/a0] or equivalently of [Sconf(0)/a0]. The intercept would vary with T, and P only in as much as [Wconf (0)] would vary, and the slope would vary additionally with a0 of the melt. Moreover, such plots for different T and P would terminate at the same point on the axis where a = a0. This feature has two consequences: (i) Sconf of a melt polymerized to an extent a0 would be zero at all T and/or P conditions, i.e., the measured S of the melt would be only vibrational, since only one configuration would be available to the polymer structure, and (ii) polymerization would terminate at a = a0, because any further increase in a would make Sconf negative, i.e., a can not exceed a0. We examine these consequences below by a comparison against experiments. The Sconf of polymers whose a P a0 has been found to be generally greater than zero, and their state has been found to remain fluid when Tpoly is high. Also, for states at a P a0, their g and s have been found to increase on cooling and/or compression, implying that their Sconf decreases. For example, s of the fully polymerized state of stoichiometric composition (molar ratio 1:1) of diglycidylether of bisphenol-A cyclo-hexylamine melt, for which a0 = 1 (1.03 and 1.05 used in [15]), increases from 106 s at 393.2 K to 103.5 s at 367 K [19]. It is also common knowledge that fully polymerized
G.P. Johari / Chemical Physics 305 (2004) 231–236
4. Nature of the entropy equation and the fitting of the relaxation time data In view of the above-given observations, Eq. (6) provided by Corezzi et al. [15] has to be seen as an empirical relation of the form, ln s = A + [Ba0/(a0a)], which is similar to the equation, ln s = Ap + [Bp/(P0P)], used for fitting the variation of s with P, where Ap and Bp are adjustable parameters [24]. However, it is worth noting that there is also a different form of empirical relation, sðaÞ ¼ sða ¼ 0Þ exp½Sap ;
ð7Þ
which has been used to parameterize the s data by using the polymerization temperature-dependent parameters S and p [18–20,25]. For this fitting, s(a = 0) could be determined from the dielectric spectra measured at GHz frequencies [26,27]. Thus it appears that both Eqs. (6) and (7) can be fitted to the same data, and this ambiguity makes it necessary to examine the character of these two equations in detail. According to Eq. (6), a plot of ln s(a) against [1(a/ a0)]1 is a straight line, and according to Eq. (7) this plot is a stretched sigmoid shape curve which becomes pro-
gressively more stretched as the quantity [1(a/a0)]1 increases or a approaches a0. The straight line can partially overlap the approximately linear region of the stretched sigmoid shape curve, particularly around the region of its inflexion point. The plot of the measured values of log s against [1(a/a0)]1 for stoichiometric mixture of diglycidyl ether of bisphenol-A and cyclohexylamine in the molar ratio 1:1 polymerizing at 300.2 K is shown by circles in Fig. 1 here. These data had been reported in [19], but were not analysed. The straight line in Fig. 1 shows the plot of s calculated from Eq. (6) using a0 = 1.03, B = 8.8 and s0 = 1012.7 as given in [15]. It is evident that the measured s values lie on a curved line of a stretched sigmoid shape curve as expected from Eq. (7) and the deviations are maximum at the curveÕs extremes. When the polymerization temperature is increased, the range of overlap of the curves obtained from Eqs. (6) and (7) would increase and the slope of the plot would decrease, because the height of the stretched
-3
DGEBA-CHA melt polymerizing at 300.2 K
-4
log10 (τ)
materials become soft and fluid on heating, and hard and rigid on cooling and/or compression. Therefore, in contrast to the prediction of Eq. (4), Sconf of a polymerizing melt does not vanish and the measured entropy remains much higher than the vibrational entropy at a = a0. Melts have also been known to polymerize to an extent greater than a0. For example, Gillham and coworkers [21,22] have polymerized liquid diepoxides–diamine mixtures in the molar ratio 2:1 (a0 = 0.75) to an extent close to unity, i.e., to a = 1. Also, in a recent study aimed at determining experimental methods for studying the gradual transition of reaction kinetics from a mass-controlled to a diffusion-controlled mechanism during the polymerization process, a 2:1 molar mixture of diglycidyl ether of bisphenol-A with a diamine has been polymerized to a values as high as 0.89 [23], despite the fact that its a0 is 0.75. Perhaps, the most remarkable example of such an occurrence is the viscous liquid formed by polymerization of diglycidyl ether of 1,4 butanediol and hexamethylene1-6,diamine in 2:1 molar ratio, for which a0 is 0.75. It has been polymerized to the full extent of a = 1 at 326.1 K [19]. The dielectric relaxation time, s, of its fully polymerized state has been found to be less than 1 ls [19]. It is also noteworthy that in order to prevent change in their properties with time as a result of further chemical reactions, polymer processing and post-curing technologies are often based on achieving a closer to one, particularly for melts of compositions with a0 < 1.
235
-5
-6
-7
-8 1.4
1.6
1.8
2.0
2.2
2.4
2.6
-1
[1-(α/α0)]
Fig. 1. The logarithmic plots of the dielectric relaxation time against [1(a/a0)]1 for a stoichiometric composition of diglycidyl ether of bisphenol-A and cyclo-hexylamine in the molar ratio 1:1 polymerized at 300.2 K. Circles are the measured data points reported in [19], but not used for this purpose before, and the straight line is the calculated curve from Eq. (6) by using a0 = 1.03, B = 8.8 and s0 = 1012.7 s from Table I in [15]. Note that the experimental value of s lies along a stretched part of a sigmoid-shape curve as expected from Eq. (7) in the limiting range of [1(a/a0)]1. The s plot calculated by using a0 = 1.03, B = 8.8 and s0 = 1012.9 s (instead of s0 = 1012.7 s), will partly overlap the measured s plot, but the curvature will not be matched.
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G.P. Johari / Chemical Physics 305 (2004) 231–236
sigmoid curve decreases with increase in this temperature and the sigmoid-shape curves tend to gradually straighten out. (For example increase in s from a = 0 to a = 1 at high polymerizing temperature is less than that at low polymerizing temperature.) This occurrence is evident in Fig. 1 of [15], which shows that the slope of the plot of log s against [1(a/a0)]1 decreases as Tpoly is increased. For the sake of argument, one may overlook the neglect of the a-dependence of C and s0 and consider whether or not the values of the parameters used in the fitting of Eq. (6) in [15] are plausible. For a chemically stable state at a certain fixed a, Eq. (6) becomes, ln s = ln s0 + B/, where / = [a0/(a0a)], which is a constant. This is equivalent to Eq. (5) for a liquid of fixed chemical structure at the same fixed T as Tpoly, namely, ln s = ln s0 + Cr, where r = 1/TSconf. Since s0 is the same in both equations, it should correspond to the time scale of phonons, i.e., 1013 to 1014 s [28,29]. However, the s0 values used for fitting Eq. (6) are in the range of 107.8–109.5 s for 12 diepoxide-based polymers and are 1012.3 and 1012.7 s for two (Table 1 of [15]). These s0 values for 12 polymers seem too high to represent the phonon characteristics.
5. Conclusions Experiments have shown that: (i) Sconf of a fully polymerized state remains sufficiently high and s and g remain sufficiently low; (ii) polymerization continues to the full extent beyond the value of a0, (iii) s does not diverge at a0 as the melt polymerizes, and, (iv) at a P a0, the magnitude of Sconf, s and g of a fully polymerized state remain finite and their magnitudes vary with T and P. These observations remove the basis for Sconf being inversely proportional to the number average degree of polymerization, or that Sconf (1a/a0). A connection between the relaxation time and a by using the Adam–Gibbs equation can be made only by allowing: (i) the pre-exponential term to remain close to the phonon time scale; (ii) Dl, and therefore, the quantity C to change with the chemical structure of the liquid, and (iii) s to remain non-divergent at a = a0. (For this third case, it should be stressed that according to the Adam–Gibbs entropy formalism [16], s does not diverge on cooling a melt of fixed chemical structure.) We conclude that calculations based on FloryÕs chain statistics is appropriate [2–10], and a relation that allows Sconf and relaxation time of a fully polymer-
ized state to remain finite and to depend upon T and P, as experiments have shown, is preferable.
Acknowledgements It is a pleasure to thank Jingsong Wang of my research group for discussion on this subject.
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Decrease in the configurational entropy during a meltÕs polymerization G.P. Johari
*
Department of Materials Science and Engineering, McMaster University, 1280 Main St. West Hamilton, Ont., Canada L8S 4L7 Received 25 August 2003; accepted 2 June 2004 Available online 30 July 2004
Abstract The manner in which the configurational entropy, Sconf, of a polymerizing melt decreases has been examined in context with a recent conclusion [S. Corezzi, D. Fioretto, P. Rolla, Nature, 420 (2002) 653] that Sconf varies as (1a/a0), where a is the measured extent of polymerization and a0(61) is a parameter calculated from the functionality of the meltÕs reacting molecules. It is shown that for Sconf to vary as (1a/a0): (i) Sconf of a melt should vanish when a becomes equal to a0, irrespective of the temperature and pressure, and the meltÕs relaxation time, s, becomes infinite; (ii) polymerization should terminate at a = a0. Experiments have shown the opposite, namely that: (i) Sconf of a fully polymerized state (a = 1 P a0) is large, s is as low as 1 ls and both Sconf and s vary with temperature and pressure; (ii) polymerization continues to occur beyond a0. Therefore, Sconf does not vary as (1a/a0). The relation, s(a) = s0exp[B/(1a/a0)], used for testing this variation requires that B and s0 be independent of a, which is not the case. 2004 Elsevier B.V. All rights reserved.
1. Introduction There is a general interest in determining the irreversible change in thermodynamic properties of a liquid in which chemical reactions lead to a macromoleculeÕs growth. Since such a growth irreversibly increases a liquidÕs viscosity at a fixed polymerization temperature, Tpoly, there is also a special interest in determining how molecular kinetics is controlled by irreversible chemical reactions. The phenomenon has been described in terms of a negative feedback between molecular diffusion, which allows a chemical reaction to occur, and chemical reaction which slows the diffusion [1]. Studies have also shown that changes in the thermodynamic and molecular kinetic properties during the course of a macromoleculeÕs growth in a polymerizing melt at a fixed Tpoly are analogous to the changes observed in these two properties on cooling a liquid of a fixed chem*
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0301-0104/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2004.06.056
ical structure [1]. For example, the entropy, volume and enthalpy decrease and viscosity, g, and relaxation time increase during a meltÕs polymerization, as also occurs during the cooling of the melt. Because thermodynamic and kinetic properties change together, one expects that g and the relaxation time of a liquid would be intrinsically related to a thermodynamic property. The change in thermodynamic properties due to the change in phonon frequency and anharmonic effects is finite but usually small when van der WaalsÕ interactions are replaced by covalent bonds during a macromoleculeÕs growth. Cooling a liquid of fixed chemical structure has a similarly small effect. In seeking a thermodynamic basis for vitrification, therefore, only the configurational entropy, Sconf, is currently used for describing the rate of molecular kinetics of a chemically stable liquid. Since each infinitesmal stage in the process of polymerization during which a fixed structure exists corresponds to a liquid of different chemical structure, and configurational restriction increases as the structure become increasingly covalently bonded, the observed
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increase in g and in the dielectric relaxation time, s, during a meltÕs polymerization can be related to the decrease in Sconf. For liquids in general, even Sconf has been difficult to estimate, because the vibrational component of the entropy of a liquid, which should be subtracted from the experimentally determined entropy, cannot be easily determined. Nevertheless, Sconf of polymers has been accurately calculated by using chain statistics and following FloryÕs original calculations based on a lattice hole model [2–4]. These calculations have been further used to determine the manner in which Sconf decreases on cooling a polymer melt, and then finally relating Sconf to the vitrification of a polymer [5–10]. Several of these studies have concluded that Sconf of a polymer would vanish at a T much higher than 0 K [5,6]. Others have concluded that Sconf would vanish only at 0 K [7–10]. Wang and Johari [10] have calculated Sconf of a melt during the course of polymerization, i.e, as a function of the extent of polymerization, a, at a fixed polymerizing temperature, Tpoly, by using polymer chain statistics in FloryÕs lattice hole model. The calculation showed that Sconf and the configurational heat capacity, Cp,conf, of a melt would remain finite at its full polymerization, i.e., at a = 1. It also led to two predictions, (i) Sconf and Cp,conf of an equilibrium melt would approach zero only at 0 K, which is contrary to the extrapolation of the excess entropy of a liquid over its crystal phase to zero at a temperature [11] of 20–30% below Tg (see also [12]), and (ii) Cp,conf would initially increase with increase in a and then decrease, thus showing a local maximum in the Cp against a plot at a fixed Tpoly. The first prediction has been experimentally verified by Pyda and Wunderlich [13], and the second prediction has been borne out by a recent study [14]. It should be stressed that vitrification of a liquid itself does not require that its excess entropy over the crystal phase value become zero at T > 0 K. Concurrently, Corezzi et al. [15] assumed that, in contrast to the prediction from the Flory model as extended by Wang and Johari [10] to the polymerization process, Sconf is inversely proportional to the number average degree of polymerization, xn, which in turn is defined as the average number of monomers per molecule [15]. Corezzi et al. recognized that this is a crude approximation and then used it to obtain a relation between s and a. To elaborate, their basic premise is that for a polymerizing melt, Sconf(a) = Sconf(0)/xn, where Sconf(0) is the configurational entropy of a melt consisting of only monomers at a = 0. After stating that the relation, Sconf(a) = Sconf(0)/xn, is ‘‘. . .not cruder than the entropy theory itself:’’ [15], they combined this relation with the configurational entropy theoryÕs Adam–Gibbs equation [16] and fitted the resulting equation to the available s data [15]. The apparently good fit to the data was seen as proof of the validity of this premise. They ultimately concluded that use of the
entropy relation, Sconf(a) = Sconf(0)/xn, in future would benefit materials processing, help in understanding a liquidsÕ structural fluctuation in terms of the potential energy landscape model at a fundamental level, as well as the phenomena observed in micro-emulsions, disordered structures, physical gels, percolation, etc. Both, the basic premise that Sconf (a) 1/xn and the apparent fit of the resulting equation to s data are intriguing. Moreover, Corezzi et al.Õs findings [15] conflict with the precepts of polymer chains statistics [7–10], which they overlooked. We examine the validity of this premise and their findings on fundamental grounds, and determine whether the relation, Sconf(a) 1/xn is physically justifiable. We also examine the merits of their ultimate sa correlation and the manner of its fitting by using experimental data, and show that the apparent success of this correlation is due to use of a limited range of data.
2. Entropy and the extent of polymerization It is well recognized that polymerization decreases the entropy and increases the relaxation time of a melt, i.e., as molecular weight of the polymer increases, its entropy decreases and relaxation time increases. In a polymerization process, say by addition reactions between a monamine and digycidyl ether of bisphenol-A (a diepoxide) in a stoichiometric liquid mixture, the –NH2 group chemically reacts with two epoxide groups in two stages involving consecutive reactions, and produces a linear chain. This step-addition reaction occurs in the following sequence: One H-atom of the –NH2 group of the monoamine combines with the –O– group of the epoxide thus forming an –OH group and opening the terminal epoxide ring, and the N atom becomes covalently bonded to the terminal C atom of the diepoxide molecule. Then the second H-atom of the now sterically hindered –NH group in the R–NH–(CH2)n–CH3 molecule, where R denotes the partly reacted diepoxide molecule, chemically reacts similarly with the terminal epoxide group of another diepoxide molecule. Thus by losing its two amine protons, one amine molecule links two diepoxide molecules by its one N atom, leading to a structure of the type containing an –ABABA– sequence of the reacted molecules of diepoxide, A, and of monoamine, B. A similar reaction with a diamine produces a network structure when H atoms of both amine groups have reacted with the four terminal atoms epoxide groups of four diepoxide molecules. As these chemical reactions occur and the macromolecule grows, the viscosity of the liquid increases at a progressively rapid rate and the liquid (or elastomer) vitrifies before reaching complete polymerization when the polymerization temperature, Tpoly, is less than the vitrification temperature, Tg, of the fully polymerized structure. But when Tpoly is
G.P. Johari / Chemical Physics 305 (2004) 231–236
higher than this Tg, the fully polymerized state remains a viscous liquid if polymerization leads to linear chain and an elastomer if it leads to a molecular network structure. It seems necessary to first describe the manner by which Corezzi et al. [15] had obtained the relation, Sconf (a) 1/xn, and thereafter its correlation between the dielectric relaxation time, s, and a. Without providing a physical or molecular justification, they wrote [15] that it is realistic to expect that Wconf(a), the number of configurations available to a polymerized state characterized by a, is given by W conf ðaÞ ¼ ½W conf ð0Þ1=xn ;
ð1Þ
where Wconf(0), a constant is the number of configurations available to the monomer (unpolymerized) state at a = 0. In Eq. (1), as xn decreases from one to zero, a increases from zero to one. Since S = kBlnW, Eq. (1) becomes, S conf ðaÞ ¼
kB ln½W conf ð0Þ; xn
ð2Þ
where kB is the Boltzmann constant and xn is the number average degree of polymerization (or the average number of monomers per multimer). Since ln[Wconf(0)] is a constant at a fixed Tpoly, S conf ðaÞ /
1 : xn
ð2aÞ
The quantity xn was identified as the number average degree of polymerization, and was written as [15], xn = (1a/a0)1, where a0 is taken as a parameter determined from the average molecular weight. Macosko and Miller [17] had provided expressions for the average molecular weight and the magnitude of a0 for various types of addition reactions in terms of molecular ratio and functionality of the two reacting components of a polymerizing melt. Accordingly, a0 is determined only by the number of covalent bonds that can form between the reacting groups of molecules involved in polymerization, and not by the chemical structure of the molecules themselves or of the reacting groups. To briefly describe the definition of a0, we recall Macosko and MillerÕs [17] calculation of the average molecular weight for nonlinear polymers formed when Afi reacts with Bgj, and of the number average degree of polymerization, and use their Eqs. (56)–(59) to derive P P Bgj i Afi þ j P aA;0 ¼ ; ð3Þ i fiAfi where Afi represents the moles of component Afi, and Bgj the moles of component Bgj in the melt, and fi is the number of functional groups in Afi. According to Eq. (3), when one mole of a bifunctional component A reacts with one mole of component B, aA,0 is equal to one; when four moles of bifunctional A react with three
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moles of B, aA,0 = 0.875; and when two moles of bifunctional A react with one mole of B, aA,0 = 0.75. It is to be noted that polymerization is incomplete when aA,0 < a. The variation of xn with aA, i.e., the mole fraction of A groups that have reacted in the component A, has been defined as [15], xn(aA) = 1/(1ÆfAæaA), where 1=hfA i atheor , where atheor is taken as equal to Mocasko 0 0 and MillerÕs [17] aA,0 in Eq. (3) here. By expressing a as the extent of ‘‘conversion of the epoxy groups’’ [15], and substituting in Eq. (2) a S conf ðaÞ ¼ S conf ð0Þ 1 a0 k B ln½W conf ð0Þ ða0 aÞ: ð4Þ ¼ a0 Now Eq. (4) has several important consequences for thermodynamics of polymerization. Firstly, it means that Sconf would vanish when a = a0. In molecular terms it means that at a given Tpoly, which is higher than 0 K, there is only one configuration possible for the polymer structure formed by, say, addition reaction of two moles of bifunctional A and one mole of B at the extent a = a0 = 0.75. (This is reminiscent of the Kauzmann extrapolation [11], which had suggested that the excess entropy of a liquid over the crystal phase, which has been seen as equal to Sconf, would become zero on cooling the equilibrium liquid to a temperature 20–30% below Tg see also [12,13] for discussion of this extrapolation). Secondly, according to Eq. (4), Sconf of all polymers, which have been polymerized to the extent of a0, would be zero, irrespective of the value of their Sconf(0). This means that only one molecular arrangement of the polymer structure exists even when the experimentally observed state of the polymer at a = a0 is a viscous liquid or an elastomer at a high polymerization temperature. Thirdly, irrespective of the value of Sconf (0), Sconf for all T and P conditions of polymerization of all types of melts would become zero once its a is equal to a0, and, since Eq. (4) does not contain the terms T and P, a change in T or P at a = a0 would have no effect on Sconf. We now consider how Eq. (4) has been related to s in [15]. According to the Adam–GibbsÕ configurational entropy theory [16] for a chemically stable liquid, or for a polymerizing melt at a fixed value of a CðaÞ sðaÞ ¼ s0 ðaÞ exp ; ð5Þ TS conf ðaÞ where C ¼ Dlsc and s0 corresponds to the phonon time scale, i.e., 1013 to 1014 s. The term Dl is the free energy of activation per monomer segment in the co-operatively rearranging region, and sc is the critical entropy. The latter has been usually taken to be a constant for all liquids. The molecular basis of Eq. (5) is that as T is decreased, there is an increasing requirement for co-operative rearrangement of molecules in the liquid and
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therefore the size of the co-operatively rearranging region increases and Sconf decreases. Hence, TSconf for a fixed value of a decreases on cooling and s increases rapidly according to Eq. (4). When a increases on polymerization, the chemical structure of the liquid changes and hence the values of C and s0 change. To elaborate, in the Adam–Gibbs formalism [16], the quantity, Dl = DHa TDSa, where DHa is the enthalpy and DSa the entropy of activation. Since both DHa and DSa change as a polymerizing meltÕs chemical structure changes, C would change as a increases. As C appears in an exponent in Eq. (5), its small variation with a would have a relatively large effect on s, as recognized in the earlier studies of polymerization [18–20]. By substituting Eq. (4) in Eq. (5), Corezzi et al. [15] obtained, ln sðaÞ ¼ ln s0 þ
B Ba0 ¼ ln s0 þ ; ½1 ða=a0 Þ ½a0 a
ð6Þ
where B = C(a)/TSconf(0). The value of B is expected to vary with a because C varies with a. In the fitting of the dielectric relaxation time data, Corezzi et al. [15] have overlooked the strong effect of the variation of C with a, and have used fixed values of both s0 and B, although s0, which is related to phonon frequency is expected to change as Van der WaalsÕ interactions are replaced by covalent bonds during the growth of a macromolecule. One consequence of Eq. (6) is that s diverges at a0 generally for all polymerization processes, but there is no physical or molecular basis for it. It appears that this divergence is an artifact of the neglect in the variation of s0 and B with a, as discussed above. By substituting for C(a) in B = C(a)/TSconf(0), we may write, B ¼ sc ½DH a ðaÞ T DS a ðaÞ=TS conf ð0Þ. Since sc =TS conf ð0Þ is taken as a constant, B becomes proportional to [DHa(a)TDSa(a)] and its value varies with a. Thus, as more covalent bonds form and the melt densifies on polymerization, the growing hindrance to molecular structural rearrangement not only reduces Sconf but also changes the empirical quantity B through a change in DHa and DSa with increase in a.
3. A scrutiny of the features of the entropy-extent of polymerization relation The suggested success in fitting Eq. (6) to the s data in [15] has serious consequences for the validity of the calculation of thermodynamic properties based on polymer chain statistics [2–10]. Moreover, since it has been expected [15] that use of Eq. (2) would benefit materials processing, and help in understanding the picture of the potential energy landscape, and the phenomena of microemulsions, physical gels, disordered systems and percolation, it seems necessary that validity of the basic premise of Eq. (6) and thereafter of its fitting to the s
data already available in the literature should be carefully examined. This is done as follows: First, we consider a relatively simple feature of Eq. (4): Since Sconf(0) is a constant, differentiating Eq. (4) with respect to T yields, Cp,conf(a) = T(oSconf(0)/oT)P. This means that Cp,conf of a polymerizing melt at a fixed Tpoly would not change as a increases from zero to one, because (oSconf(0)/oT) is a constant. But the heat capacity of polymer melts and elastomers has been known to change with increase in the extent of polymerization or a polymerÕs molecular weight. Similarly, differentiating Eq. (4) with respect to pressure, P, yields, bconfVconf [= (oSconf(0)/oP)T], where bconf is the configurational contribution to the thermal expansion coefficient of the melt, and Vconf of that to the volume. This means that as a macromolecule grows, or a melt polymerizes, the product bconfVconf would not change. Therefore, in order for Eq. (4) to be valid, the observed change in Cp, b and V of a polymerizing melt should be due only to an increase in the phonon frequencies and decrease in the anharmonic effects. This is contrary to the well known observations that changes in Cp, b and V resulting from a meltÕs polymerization are generally much more than those resulting from changes in the phonon frequency and anharmonic forces, and that there is a considerable configurational contribution to the Cp, b and V of polymers as long as Tpoly > Tg. Second, according to Eq. (4) a plot of Sconf against a will be a straight line with an intercept Sconf(0), or kBln[Wconf(0)], and a slope of kBln[Wconf(0)]/a0] or equivalently of [Sconf(0)/a0]. The intercept would vary with T, and P only in as much as [Wconf (0)] would vary, and the slope would vary additionally with a0 of the melt. Moreover, such plots for different T and P would terminate at the same point on the axis where a = a0. This feature has two consequences: (i) Sconf of a melt polymerized to an extent a0 would be zero at all T and/or P conditions, i.e., the measured S of the melt would be only vibrational, since only one configuration would be available to the polymer structure, and (ii) polymerization would terminate at a = a0, because any further increase in a would make Sconf negative, i.e., a can not exceed a0. We examine these consequences below by a comparison against experiments. The Sconf of polymers whose a P a0 has been found to be generally greater than zero, and their state has been found to remain fluid when Tpoly is high. Also, for states at a P a0, their g and s have been found to increase on cooling and/or compression, implying that their Sconf decreases. For example, s of the fully polymerized state of stoichiometric composition (molar ratio 1:1) of diglycidylether of bisphenol-A cyclo-hexylamine melt, for which a0 = 1 (1.03 and 1.05 used in [15]), increases from 106 s at 393.2 K to 103.5 s at 367 K [19]. It is also common knowledge that fully polymerized
G.P. Johari / Chemical Physics 305 (2004) 231–236
4. Nature of the entropy equation and the fitting of the relaxation time data In view of the above-given observations, Eq. (6) provided by Corezzi et al. [15] has to be seen as an empirical relation of the form, ln s = A + [Ba0/(a0a)], which is similar to the equation, ln s = Ap + [Bp/(P0P)], used for fitting the variation of s with P, where Ap and Bp are adjustable parameters [24]. However, it is worth noting that there is also a different form of empirical relation, sðaÞ ¼ sða ¼ 0Þ exp½Sap ;
ð7Þ
which has been used to parameterize the s data by using the polymerization temperature-dependent parameters S and p [18–20,25]. For this fitting, s(a = 0) could be determined from the dielectric spectra measured at GHz frequencies [26,27]. Thus it appears that both Eqs. (6) and (7) can be fitted to the same data, and this ambiguity makes it necessary to examine the character of these two equations in detail. According to Eq. (6), a plot of ln s(a) against [1(a/ a0)]1 is a straight line, and according to Eq. (7) this plot is a stretched sigmoid shape curve which becomes pro-
gressively more stretched as the quantity [1(a/a0)]1 increases or a approaches a0. The straight line can partially overlap the approximately linear region of the stretched sigmoid shape curve, particularly around the region of its inflexion point. The plot of the measured values of log s against [1(a/a0)]1 for stoichiometric mixture of diglycidyl ether of bisphenol-A and cyclohexylamine in the molar ratio 1:1 polymerizing at 300.2 K is shown by circles in Fig. 1 here. These data had been reported in [19], but were not analysed. The straight line in Fig. 1 shows the plot of s calculated from Eq. (6) using a0 = 1.03, B = 8.8 and s0 = 1012.7 as given in [15]. It is evident that the measured s values lie on a curved line of a stretched sigmoid shape curve as expected from Eq. (7) and the deviations are maximum at the curveÕs extremes. When the polymerization temperature is increased, the range of overlap of the curves obtained from Eqs. (6) and (7) would increase and the slope of the plot would decrease, because the height of the stretched
-3
DGEBA-CHA melt polymerizing at 300.2 K
-4
log10 (τ)
materials become soft and fluid on heating, and hard and rigid on cooling and/or compression. Therefore, in contrast to the prediction of Eq. (4), Sconf of a polymerizing melt does not vanish and the measured entropy remains much higher than the vibrational entropy at a = a0. Melts have also been known to polymerize to an extent greater than a0. For example, Gillham and coworkers [21,22] have polymerized liquid diepoxides–diamine mixtures in the molar ratio 2:1 (a0 = 0.75) to an extent close to unity, i.e., to a = 1. Also, in a recent study aimed at determining experimental methods for studying the gradual transition of reaction kinetics from a mass-controlled to a diffusion-controlled mechanism during the polymerization process, a 2:1 molar mixture of diglycidyl ether of bisphenol-A with a diamine has been polymerized to a values as high as 0.89 [23], despite the fact that its a0 is 0.75. Perhaps, the most remarkable example of such an occurrence is the viscous liquid formed by polymerization of diglycidyl ether of 1,4 butanediol and hexamethylene1-6,diamine in 2:1 molar ratio, for which a0 is 0.75. It has been polymerized to the full extent of a = 1 at 326.1 K [19]. The dielectric relaxation time, s, of its fully polymerized state has been found to be less than 1 ls [19]. It is also noteworthy that in order to prevent change in their properties with time as a result of further chemical reactions, polymer processing and post-curing technologies are often based on achieving a closer to one, particularly for melts of compositions with a0 < 1.
235
-5
-6
-7
-8 1.4
1.6
1.8
2.0
2.2
2.4
2.6
-1
[1-(α/α0)]
Fig. 1. The logarithmic plots of the dielectric relaxation time against [1(a/a0)]1 for a stoichiometric composition of diglycidyl ether of bisphenol-A and cyclo-hexylamine in the molar ratio 1:1 polymerized at 300.2 K. Circles are the measured data points reported in [19], but not used for this purpose before, and the straight line is the calculated curve from Eq. (6) by using a0 = 1.03, B = 8.8 and s0 = 1012.7 s from Table I in [15]. Note that the experimental value of s lies along a stretched part of a sigmoid-shape curve as expected from Eq. (7) in the limiting range of [1(a/a0)]1. The s plot calculated by using a0 = 1.03, B = 8.8 and s0 = 1012.9 s (instead of s0 = 1012.7 s), will partly overlap the measured s plot, but the curvature will not be matched.
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G.P. Johari / Chemical Physics 305 (2004) 231–236
sigmoid curve decreases with increase in this temperature and the sigmoid-shape curves tend to gradually straighten out. (For example increase in s from a = 0 to a = 1 at high polymerizing temperature is less than that at low polymerizing temperature.) This occurrence is evident in Fig. 1 of [15], which shows that the slope of the plot of log s against [1(a/a0)]1 decreases as Tpoly is increased. For the sake of argument, one may overlook the neglect of the a-dependence of C and s0 and consider whether or not the values of the parameters used in the fitting of Eq. (6) in [15] are plausible. For a chemically stable state at a certain fixed a, Eq. (6) becomes, ln s = ln s0 + B/, where / = [a0/(a0a)], which is a constant. This is equivalent to Eq. (5) for a liquid of fixed chemical structure at the same fixed T as Tpoly, namely, ln s = ln s0 + Cr, where r = 1/TSconf. Since s0 is the same in both equations, it should correspond to the time scale of phonons, i.e., 1013 to 1014 s [28,29]. However, the s0 values used for fitting Eq. (6) are in the range of 107.8–109.5 s for 12 diepoxide-based polymers and are 1012.3 and 1012.7 s for two (Table 1 of [15]). These s0 values for 12 polymers seem too high to represent the phonon characteristics.
5. Conclusions Experiments have shown that: (i) Sconf of a fully polymerized state remains sufficiently high and s and g remain sufficiently low; (ii) polymerization continues to the full extent beyond the value of a0, (iii) s does not diverge at a0 as the melt polymerizes, and, (iv) at a P a0, the magnitude of Sconf, s and g of a fully polymerized state remain finite and their magnitudes vary with T and P. These observations remove the basis for Sconf being inversely proportional to the number average degree of polymerization, or that Sconf (1a/a0). A connection between the relaxation time and a by using the Adam–Gibbs equation can be made only by allowing: (i) the pre-exponential term to remain close to the phonon time scale; (ii) Dl, and therefore, the quantity C to change with the chemical structure of the liquid, and (iii) s to remain non-divergent at a = a0. (For this third case, it should be stressed that according to the Adam–Gibbs entropy formalism [16], s does not diverge on cooling a melt of fixed chemical structure.) We conclude that calculations based on FloryÕs chain statistics is appropriate [2–10], and a relation that allows Sconf and relaxation time of a fully polymer-
ized state to remain finite and to depend upon T and P, as experiments have shown, is preferable.
Acknowledgements It is a pleasure to thank Jingsong Wang of my research group for discussion on this subject.
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