Sequence Distribution Effects on Glass Transition Temperatures of

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J. Phys. Chem. B 2008, 112, 93-99

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Sequence Distribution Effects on Glass Transition Temperatures of Copolymers: An Extended Gibbs-DiMarzio Equation in View of Bond Rotation Flexibility Guodong Liu,*,† Zhenying Meng,‡ Wei Wang,§ Yuelian Zhou,† and Liucheng Zhang† Institute of Polymer Science and Engineering, Hebei UniVersity of Technology, Tianjin, 300130, China, State Key Laboratory of Multiphase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, Beijing, 100080, China, and Institute of Polymer Chemistry, Nankai UniVersity, Tianjin, 300071, China ReceiVed: August 24, 2007; In Final Form: October 17, 2007

In view of bond rotation flexibility and the additivity of stiff energy, a new equation containing mole fractions of sequences and glass transition temperatures (Tg’s) of periodic copolymers was proposed. Meanwhile, the effect of stereochemical sequences was also taken into account. Even for the case of all substitutions on primary (C1), second (C2), and third (C3) carbon atoms adjoining to the given bond (triad sequences effect) to be considered, no intractable parameters exist in the new equation if the periodic copolymers of poly[AB], poly[AAB], and poly[BBA] could be acquired. This may facilitate its convenient application in comparison with the Ham equation (Ham, G. E. J. Macromol. Sci. Chem. 1975, A9, 461), which also concerned triad contributions. The new equation was applied to methyl methacrylate-styrene, ethylene-methyl methacrylate, and ethylene-vinyl acetate copolymers, and excellent fittings were obtained. The parameters obtained may predict Tg values of periodic copolymers that have not been acquired as well as provide useful information on the bond rotation flexibility.

1. Introduction The glass transition temperature, Tg, is one of the most important intrinsic characteristics of polymers that reflect their material properties and determine their potential applications. It is well-known that Tg’s of copolymers always deviate from linear relations as predicted by the Gordon-Taylor equation or Gibbs-DiMarzio (G-D) equation.1-2 It has been found that the sequence distribution effect must be taken into consideration when the composition dependence of copolymer Tg was studied.3-4 Assuming volume additivity of the repeating units in copolymers, Gordon and Taylor1 gave the resulting “Gordon-Taylor” equation:

Tg )

w1Tg1 + Kw2Tg2 w1 + Kw2

(1)

where Tg is the glass transition temperature of a copolymer, and wi and Tgi are the weight fraction and glass transition temperature of the component i, respectively. K is a parameter specified by the model. In the assumption of the Simha-Boyer rule,5 the Gordon-Taylor equation can be reformulated as the well-known Fox relation:6

w1 w2 1 ) + Tg Tg1 Tg2

(2)

To allow for diad sequence contribution to the Tg of a copolymer, Johnston3 extended the Fox equation as * Corresponding author. Tel: +86-22-60201998. Fax: +86-22-60202421. E-mail: [email protected]. † Hebei University of Technology. ‡ Chinese Academy of Sciences. § Nankai University.

1 w1P11 w2P22 w1P12 + w2P21 ) + + Tg Tg1 Tg2 Tg12

(3)

where Pij is the probability of forming the respective diad, and Tgij is the corresponding glass transition temperature contribution. On the basis of the assumption that the chain stiffness energy of a copolymer is additive, Gibbs and DiMarzio2 proposed an equation with a similar form to the Gordon-Taylor equation but containing coefficients with quite different physical significance:

nA(Tg - TgA) + nB(Tg - TgB) ) 0

(4)

where ni is the fraction of a rotatable bond of type i in the copolymer chain, and Tg and Tgi are glass transition temperatures of the copolymer and homopolymer, respectively. Barton4 added an extra term to the G-D equation in order to indicate the influence of diad sequences AB and BA:

Tg ) nAATgAA + nBBTgBB + nABTgAB + nBATgBA

(5)

where nij and Tgij are the mole fraction of a rotatable bond (or group) contained in the copolymer chain and the additive temperature associated with diad sequence ij, respectively. TgAB is the Tg of the alternating copolymer poly[AB] and is equal to TgBA. Suzuki derived a new expression of the Barton equation with a description of remarkable characteristics of the equation.7 His work allows us to estimate, with minimum data, the whole behavior of the Tg of a copolymer with varying degrees of sequence distribution. However, these expressions that consider diad sequence effect cannot be adapted to describe the asymmetrical and even S-shaped Tg versus composition relation. In most cases, the

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agreement obtained by assuming the additivity of triad sequences is substantially improved over that of diad sequences.8,9 In a previous paper,10 the Johnston equation was extended (see eq 6) to a formula including three adjustable parameters after considering the triad sequence distribution effect. This extended equation has been used to describe the Tg’s of methyl methacrylate-styrene (MMA-St) random copolymers. Although an excellent fitting was achieved, the predicted Tgalt ) 371.9 K showed large deviation from the experimental result of 364.2 K reported by Hirooka.11

1 w1P111 w2P222 w1P212 + w2P121 w1P112 ) + + + + Tg Tg1 Tg2 Tgalt Tg112 w2P221 (6) Tg221

Figure 1. Different bonds in a diad AB in a binary copolymer with consideration of C1 substitution.

Schneider extended the Gordon-Taylor relation to a cubic equation that has been used to fit the experimental curves between Tg and composition. The result is better than that obtained using the diad equations. Unfortunately, no relationship between the fitting parameters and physical characters of the copolymer components was available.12 Ham and Uematsu extended the Barton equation to describe the triad sequence effect as

Tg ) nAAATgAAA + nBBBTgBBB + nAABTgAAB + nBAATgBAA + nABBTgABB + nBBATgBBA + nABATgABA + nBABTgBAB (7) where nx and Tgx are, respectively, the mole fraction and Tg of the sequence specified by the suffix x. TgAAA and TgBBB could be determined experimentally. TgAAB ) TgBAA and TgABB ) TgBBA were supposed because of the symmetry of the triads. Unfortunately, there are still four intractable Tg values in the above equation. This means that it is difficult to utilize this equation at this stage because the polymer containing only one unique triad composition (e.g., AAB) does not exist. Ham8 reduced all eight Tg contributions to four: two homopolymer Tg values, TgAAB (which is assumed equal to TgBAA), and the fourth, coded as TgAB (assuming the equality between TgABA, TgBAB, TgABB, and TgBBA). Obviously, his supposition is difficult to understand. The equality between TgABA and TgABB is merely an unverifiable assumption. Uematsu and Honda9 also tried to reduce the eight different triad sequence Tg contributions to three groups (TgAA, TgBB, and TgAB) in four manners. In a certain manner of reduction (case D in their Table 1), they found the best agreement between theory and experiment. However, their manner of reduction is also unreasonable: triad BAB is related to TgAB, whereas ABA is related to TgBB instead of TgAB. In this paper, the effect of sequence distribution on copolymer Tg is investigated in view of bond rotation flexibility and additivity of stiff energy. The new equation proposed also concerns triad sequence influence, providing a reasonable method of reducing the number of intractable Tg contributions. 2. Theoretical Analysis Additivity of Bond Stiff Energy. The basement of the G-D and Barton equations is the additivity of the stiff energy of rotatable bonds. They proposed equations that were constructed on the basis of the additivity of contributions of different monomers or sequences. Undoubtedly, there is an obvious relationship between the content of the bond and that of the sequence; actually, differences exist. To simplify the complexion, only head-to-tail polymers composed of monosubstituted

Figure 2. Illustration of different bonds in a binary copolymer with consideration of both C1 and C2 substitutions.

Figure 3. Different bonds in an alternating copolymer with consideration of both C1 and C2 substitutions.

vinyl monomers are considered in this work. Assuming that monomer A has a substituent X and monomer B has a substituent Y, the case of a diad AB in a binary copolymer is shown in Figure 1. It could be found that there are two kinds of bonds existing (marked by bA and bB) in diad AB, even when only C1 substitution is considered. So it may be improper to distribute only one Tg contribution, TgAB, to diad AB when a theoretical equation is based on the additivity of bond stiff energy, . According to the G-D theory,13 the conformation entropy is a unique function of /kT, and the Tg of a copolymer will be the sum of the Tg contributions of bonds bA and bB, named as TgbA and TgbB. To notice that the homopolymer of A is composed of bond bA, TgbA should be equal to TgA used in the G-D equation. Similarly, TgbB should be equal to TgB. Then we obtain

Tg ) nbATgbA + nbBTgbB ) nATgA + nBTgB

(8)

where nbi and ni are the mole fractions of bond bi and unit i, and Tgbi and Tgi are the Tg’s of a fictitious polymer composed of bond bi uniquely (namely, Tgi) and a homopolymer of unit i. This equation is formally the same as the G-D equation, but with quite different essence.

Sequence Distribution Effect on Tg’s of Copolymers

J. Phys. Chem. B, Vol. 112, No. 1, 2008 95

[B] rB [A] nBB ) nBPBB ) [A] [B] rA + rB +2 [B] [A] nAB ) nAPAB )

nBA ) nBPBA )

1 [A] [B] rA + rB +2 [B] [A]

1 ) nAB [A] [B] rA + rB +2 [B] [A]

(12)

(13)

(14)

Then we have

Figure 4. Illustration of different bonds in a binary copolymer with consideration of C1, C2, and C3 substitutions.

Effect of C2 Substitution on Rotation Flexibility of Bonds. Torsions about bonds in nearly all real polymers, and especially vinyl polymers, are interdependent. Obviously, the substituent on C2 will affect the rotation flexibility of a given bond greatly. Taking this case into account, there are four kinds of bonds in a binary copolymer, as shown in Figure 2. They will have different stiff energies and, in turn, different Tg contributions to the copolymer. We code the four different kinds of bonds as bAA, bBB, bAB, and bBA, respectively, where bij means a bond with C1 and C2 substituted by substituents of monomer i and j, respectively. Then the Tg of a binary copolymer will be

Tg ) nbAATgbAA + nbBBTgbBB + nbABTgbAB + nbBATgbBA

(9)

where nbij and Tgbij are the mole fraction of bond bij and the corresponding Tg contribution, respectively. Accordingly, the following techniques can be employed to calculate nbij. From Figure 2, we know that the mole fraction of a particular bond is related to that of the diad sequence closely. So, the mole fractions of different diads can be calculated from the monomer feed proportions. Copolymer composition is related to the monomer feed ratio by the following well-known equation:

[A] +1 rA [B] A ) B [B] rB +1 [A]

(10)

where A and B are the mole fractions of units A and B in a copolymer, [A] and [B] are the mole fractions or mole concentrations of the feed monomers, and ri is the reactivity ratio of monomer i. The mole fractions of different diads are as follows:8

[A] rA [B] nAA ) nAPAA ) [A] [B] rA + rB +2 [B] [A]

(11)

1 1 nbAA ) nAA + nAA ) nAA 2 2

(15)

1 1 nbBB ) nBB + nBB ) nBB 2 2

(16)

1 1 nbAB ) nAB + nBA ) nAB 2 2

(17)

1 1 nbBA ) nBA + nAB ) nAB 2 2

(18)

nbAA + nbBB + nbAB + nbBA ) 1

(19)

1/2 was added because there are two bonds in one monomer unit. Since nbAB ) nbBA ) nAB, eq 9 can be changed as

Tg ) nAATgbAA + nBBTgbBB + nAB(TgbAB + TgbBA) (20) Noticing that an alternating copolymer of A with B, poly[AB], contains two kinds of bonds, bAB and bBA, in equivalent content (shown in Figure 3), the Tg of poly[AB] should be

1 Tg[AB] ) (TgbAB + TgbBA) 2

(21)

Then eq 20 can be changed to

Tg ) nAATgA + nBBTgB + 2nABTg[AB]

(22)

The form of eq 22 is similar to the Barton equation, but the parameters have different meanings. If the Tg of alternating copolymer poly[AB] had been obtained, eq 22 could be used to investigate the Tg’s of binary copolymers conveniently. The conformational entropy associated with a given bond in a vinyl polymer can also be strongly affected by the stereochemical sequence. It depends on the chain tacticity, i.e., isotactic or syndiotactic. There may be eight kinds of bonds in a binary copolymer when concerning tacticity at the same time. Therefore, eq 9 may be described as

Tg ) nbAAiTgbAAi + nbAAsTgbAAs + nbBBiTgbBBi + nbBBsTgbBBs + nbABiTgbABi + nbABsTgbABs + nbBAiTgbBAi + nbBAsTgbBAs (23) where the subscripts i and s represent isotactic and syndiotactic, respectively. Although eq 23 may be more precise theoretically, the application may be abrogated because of its overmany intractable parameters. However, eq 23 can be simplified to eq 22 if one assumes that the tacticity of a diad in a random copolymer is the same as that in a homopolymer or an alternating copolymer.

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Effect of C1, C2, and C3 Substitutions on the Rotation Flexibility of Bonds. As mentioned above, in many cases the agreement of assuming the individual contributions of various triad sequences is substantially improved, which rises superior to the one obtained with equations assuming additivity of the contributions of diad sequences. The rotation flexibility of a given bond will be also affected by C3 substitution. Consequently, there are eight kinds of bonds in a binary copolymer, as shown in Figure 4. They are named bAAA, bAAB, bABA, bABB, bBBB, bBBA, bBAB, and bBAA, where bijk means a bond with C1, C2, and C3 substituted by groups in monomer i, j and k, respectively. Similarly, they will have different stiff energies and Tg contributions. Then, the Tg of a binary copolymer will be

Tg ) nbAAATgbAAA + nbAABTgbAAB + nbABATgbABA + nbABBTgbABB + nbBBBTgbBBB + nbBBATgbBBA + nbBABTgbBAB + nbBAATgbBAA (24) where nbijk and Tgbijk are the mole fraction of bond bijk and the corresponding Tg contribution, respectively. Also, the mole fraction of every kind of bond could be obtained from the content of the corresponding triad sequence. The mole fractions of different triads are as follows:8

(25)

[A] rA [B] 1 ‚ nAAB ) nAPAAPAB ) [A] [B] [A] rA + rB + 2 rA +1 [B] [A] [B]

(26)

1 1 ‚ [A] [B] [B] rA + rB + 2 rB +1 [B] [A] [A]

[B] rB [A] nABB ) nAPABPBB ) ‚ [A] [B] [B] rA + rB + 2 rB +1 [B] [A] [A] 1

[B] [B] rB rB [A] [A] nBBB ) nBPBBPBB ) ‚ [A] [B] [B] rA + rB + 2 rB +1 [B] [A] [A] nBBA ) nBPBBPBA ) nABB

nBAB ) nBPBAPAB )

1 1 ‚ [A] [B] [A] rA + rB + 2 rA +1 [B] [A] [B]

(32)

Then the mole fractions of different bonds could be calculated using eqs 25-32, referring to Figure 4, and eq 24 will be changed as

Tg ) nAAATgbAAA + nAAB(TgbAAB + TgbABA) + nBABTgbABB + nBBBTgbBBB + nABB(TgbBBA + TgbBAB) + nABATgbBAA (33) Obviously, eq 33 could be used to interpret most Tgcomposition curves of binary copolymers because it has four adjustable parameters: (TgbAAB + TgbABA), TgbABB, (TgbBBA + TgbBAB) and TgbBAA. Unfortunately, these parameters could not be obtained experimentally because the polymer composed of the corresponding single or binate bonds does not exist. Tg values obtained by data regressing will have lower reliability. This fatally depresses the application of the equation. Most alternating copolymers could be synthesized by complex copolymerization.14 An alternating copolymer containing units A and B is composed of two kinds of bonds, bABB and bBAA, in equal mole fraction, i.e.,

1 Tg[AB] ) (TgbABB + TgbBAA) 2

(34)

It should also be noted that

nABB + nABA ) nAAB + nBAB

[A] [A] rA rA [B] [B] ‚ nAAA ) nAPAAPAA ) [A] [B] [A] rA + rB + 2 rA +1 [B] [A] [B]

nABA ) nAPABPBA )

nBAA ) nBPBAPAA ) nAAB

(35)

Then eq 33 could be changed to

Tg ) nAAATgA + nBBBTgB + 2(nAAB + nBAB)Tg[AB] + nAAB (TgbAAB + TgbABA - TgbABB) + nABB(TgbBBA + TgbBAB - TgbBAA) (36) Noticing that a periodic copolymer, poly[AAB], is composed of bonds bAAB, bABA, and bBAA in equal mole fraction, i.e.,

1 Tg[AAB] ) (TgbAAB + TgbABA + TgbBAA) 3 (27)

(37)

Similarly, for the case of poly[BBA],

1 Tg[BBA] ) (TgbBBA + TgbBAB + TgbABB) 3

(38)

Introducing eqs 35, 37, and 38 into 33, we then have

(28)

(29)

(30)

(31)

Tg ) nAAATgA + nBBBTgB + 2(nABA - nAAB)Tg[AB] + 3nAABTg[AAB] + 3nBBATg[BBA] (39) where Tg[AB], Tg[AAB], and Tg[BBA] are the Tg’s of periodic copolymers poly[AB], poly[AAB], and poly[BBA], respectively. (Normally poly[AB] is called an alternating copolymer.) So, if all three periodic copolymers could be acquired,15 all the Tg contributions could be determined experimentally. Meanwhile, the mole fractions of triads could be calculated from the reactivity ratios and feed compositions of the comonomers. Therefore, eq 39 has no intractable parameter. So, eq 39 is quite suitable for the prediction of copolymer Tg’s. If the periodic copolymers poly[AAB] and poly[BBA] cannot be acquired, there are only two intractable parameters in eq 39. Even if the alternating copolymer poly[AB] also cannot be acquired, there are only three adjustable parameters, far lower than the original six in eq 24 as well as in the Ham equation. This indicates that eq 39 is more suitable for predicting the Tg’s of copolymers

Sequence Distribution Effect on Tg’s of Copolymers

Figure 5. Plots according to eq 22 for MMA-St random copolymers. Dashed line: eq 22 with Tg[MS] ) 364.2 K. Solid line: eq 22 with Tg[MS] ) 369.0 K.

than eqs 24 and the Ham equation. Compared with that of Ham8 and Uematsu,9 the reduction method of intractable parameters is reasonable. In this case, there may be as many as 32 kinds of bonds in a binary copolymer if the stereochemical structure is also taken into consideration. It will give a result similar to that of eq 39 if one assumes that the tacticity of a triad in statistical copolymers is the same as it in homopolymers or periodic copolymers. 3. Results and Discussion Copolymers of Methyl Methacrylate with Styrene (MMASt). In a previous paper we prepared random MMA-St copolymers by emulsion polymerization10 and found an asymmetric relation curve between Tg and composition. The Johnston equation, which only considers the contribution of diad sequences, is not suitable for description of the Tg-composition relation. According to eq 22, the relation between Tg - nMMTgM nSSTgS and 2nMS will be a straight line with a slope of Tg[MS] and a zero intercept. The experimental data of the Tg’s of poly(methyl methacrylate) (PMMA) and polystyrene (PS) (385.8 K and 375.7 K, respectively) were used in the data analysis. The plot in Figure 5 shows a good result. Tg[MS] is obtained to be 369.0 K (rMMA ) 0.46 and rSt ) 0.52 are used16), which is close to the 364.2 K of an alternating copolymer synthesized by Hirooka.11 Seemingly, only considering the effect of C1 and C2 substitutions is enough, whereas the predicted Tg’s give large deviations from the experimental data using both 369.0 K (fitting result in Figure 5) and 364.2 K (experimental value by Hirooka) for Tg[MS], as shown in Figure 6. While it can be seen that the fitting result obtained using eq 39 and Tg[MS] ) 364.2 K is excellent. This indicates that the substituent on C3 will strongly affect the rotation flexibility of the given bond. The fitting results of Tg[MMS] ) 380.4 K and Tg[SSM] ) 367.3 K indicate that the periodic copolymer poly[MMA-MMA-St] may have a Tg of 380.4 K, which is 5.4 K lower than that of PMMA, while poly[St-St-MMA] has Tg of 367.3 K, which is 8.4 K lower than that of PS. One may notice that there is a little difference between the data of Hirooka and our results, especially at FSt > 0.6. This may come from the different synthesis techniques, which may result in different stereochemical structures, as well as the different Tg determination conditions and data-acquiring meth-

J. Phys. Chem. B, Vol. 112, No. 1, 2008 97

Figure 6. Experimental and predicted Tg’s of MMA-St copolymers. Open squares denote statistical copolymers and homopolymers. Filled squares denote periodic and statistical copolymers synthesized by Hirooka. Dashed line: eq 22 with Tg[MS] ) 364.2 K. Dotted line: eq 22 with Tg[MS] ) 369.0 K. Solid line: eq 39 with Tg[MS] ) 364.2 K. Dash-dotted line: eq 39 with Tg[MS] ) 364.2 K and Tg[MMS] ) 375 K. FSt represents mole fractions of styrene in MMA-St copolymers.

ods. However, the deviations of the Tg data are still within the range of determination error. The data of Hirooka were also regressed, and the predicted values fit the experimental data exactly, as shown in Figure 6. The result gave 377.5 K for Tg[MMS] and 369.8 K for Tg[SSM]. It was only a little deviation from that obtained from our data. These values are reliable because large deviations could be created if different values were used. For example, the fitting results with Tg[MS] ) 364.2 K and Tg[MMS] ) 375 K in Figure 6 show a large deviation. Although the Tg contribution for a given bond could not be solved by this method, the fitting of MMASt copolymers using eq 36 gives 412.7 K for TgbMMS + TgbMSM - TgbMSS and 373.5 K for TgbSSM + TgbSMS - TgbSMM. This may give some information concerning the influence of the C2 and C3 substitutions on the rotation flexibility of a given bond. Copolymers of Ethylene with Methyl Methacrylate (EMMA). Experimental data of the Tg’s of E-MMA random copolymers at different compositions are shown in Figure 7. The homopolymers of polyethylene (PE) and PMMA have Tg’s of 192.2 K and 378.2 K, respectively. An excellent fitting result with eq 39 is also observed by using Tg[EM] ) 292.2 K, rE ) 0.34, and rMMA ) 2.5, as reported by Yokota.17 The fitting gives 269.2 K and 324.2 K for the Tg’s of the periodic copolymers poly[E-E-MMA] and poly[MMA-MMA-E], respectively. Interestingly, the fitting result of 269.2 K for Tg[EEM] is close to the experimental 267.2 K within the determination error, while the predicted Tg of a statistical copolymer with the same composition by Yokota using a quadratic regression equation is 5.3 K lower than 267.2 K. Copolymers of Ethylene with Vinyl Acetate (E-VAc). Yokota synthesized an alternating copolymer of ethylene with vinyl acetate, poly[E-VAc], which has a Tg of 233 K.18 It is 33 K lower than the 266 K predicted by the Fox equation, and 15.5 K lower than the 248.5 K predicted by the G-D equation (the Tg’s of PE and poly(VAc) are 192 K and 305 K, respectively). Obviously, the Tg predicted from eq 22 (or the Barton equation) and the Johnston equation will give negative deviations from the G-D or Fox equations. Nevertheless, the Tg’s of the random copolymers show positive deviations. By using Tg[EVA] ) 233 K, the theoretical predicted values (eq 22, rE ) 0.97, rVAc ) 1.0216) are shown in Figure 8 (dotted line)

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Liu et al. comparison with statistical copolymers. Yokota reported that the Tg of poly[MCA-Pr] increased from 56 to 76 °C when the propylene unit was substituted by more flexible units of EE (poly[MCA-E-E]).15 For statistical copolymers, it has also been reported that introducing a more flexible unit of vinylidene chloride could increase the Tg of poly(methyl acrylate) as much as 32.5 °C.11 Fitting by assuming poly[E-E-VAc] with a lower Tg has also been attempted. The result shows larger deviations. Further Discussion. When the Tg’s of a copolymer composed of multiple repeating units are investigated, eqs 9 and 24 will be extended as n

Tg )

∑ i)1

n

nbiiTgi +

n

Figure 7. Tg-composition relationship for E-MMA copolymers. Open squares denote statistical copolymers and homopolymers. Filled squares denote periodic copolymers. Dashed line: G-D equation. Solid line: eq 39 with Tg[EM] ) 292.2 K. FMMA: mole fractions of methyl methacrylate in E-MMA copolymers.

Tg )

∑ nbijTgbij

(40)

i,j)1 j*i n

nbiiiTgi + ∑ nbijkTgbijk ∑ i)1 i,j,k)1

(41)

k or j*i

where Tgi is the glass transition temperature of the homopolymer of i, and nijk and Tgbijk are mole fraction and Tg contribution of bijk, respectively. Unfortunately, the utilization of these equations, especially eq 41, will be hindered by the fact that they have too many intractable parameters that could not be determined experimentally but can only be solved by data regression. As discussed above, the Tg of a polymer may also be greatly affected by the tacticity, such as in the case of PMMA.19,20 When the stereochemistries of C1, C2, and C3 substituents are taken into account, there are four kinds of C-C bonds in the PMMA homopolymer. This may give a reasonable interpretation of the variational Tg of PMMA and will be discussed in the future. 4. Summary and Conclusions

Figure 8. Experimental and predicted Tg’s of E-VAc copolymers. Open squares denote statistical copolymers and homopolymers. Filled squares denote periodic copolymers. Dashed line: G-D equation. Dotted line: eq 22 with Tg[EVA] ) 233 K. Solid line: eq 39 with Tg[EVA] ) 233 K. FVAc represents mole fractions of vinyl acetate in E-VAc copolymers.

with negative deviations from the G-D equation, opposite to the experimental data. The predicted Tg’s with eq 39 taking Tg[EVA] ) 233 K are also plotted in the same figure with an excellent fitting result. This means that the effect of C3 substitution (in other words, the triad sequence) on the rotation flexibility or stiff energy of the given bond could not be neglected in E-VAc copolymers. This further means the variation of Tg contribution of the corresponding bond. Tg[EEVA] ) 236.0 K and Tg[VAVAE] ) 297.2 K were obtained, indicating that the Tg of the periodic copolymer poly[E-E-VAc] may be 44 K higher than that of PE, while the Tg of poly[VAc-VAc-E] is only 7.8 K lower than that of PVAc. It is the reason the Tg’s of E-VAc random copolymers have positive deviations from the G-D equation, a feature different from that of the alternating copolymer poly[E-VAc]. Outwardly, the Tg of poly[E-E-VAc] should be lower than that of poly[E-VAc] because poly[E-VAc] contains less flexible ethylene units. This comes into conflict with our predicted result. However, this may be the case because the periodic copolymers usually have quite different glass transition characters in

We proposed a new sequence distribution-copolymer Tg equation in view of bond rotation flexibility and additivity of the corresponding stiff energies. The new equation has no meaningless and intractable parameters. When the substituents on C1, C2, and C3 are all concerned, the equation is composed of the mole fractions of triads and the Tg’s of periodic copolymers poly[AB], poly[AAB], and poly[BBA]. These Tg contributions could be determined experimentally. Even if poly[AAB] and poly[BBA] could not be acquired, the equation has only two intractable parameters, and they can be solved reliably by data regression. The equation was applied to investigate the Tg’s of MMA-St, E-MMA, and E-VAc random copolymers. The fitting results are good, even for E-VAc copolymers, whose Tg’s show opposite deviation from the G-D equation compared with the alternating copolymer. The equation can also reliably provide some useful information, such as the effect of substitution on bond rotation flexibility, and the Tg’s of periodic copolymers have not been synthesized yet. Acknowledgment. This work is supported by the Natural Science Foundation of Hebei Province (B2007000019) and the Natural Science Foundation of Tianjin (05YFJMJC05800). References and Notes (1) (2) (3) (4) (5) (6)

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