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Prediction of dielectric constants and glass transition temperatures of polymers by quantitative structure property relationships
EUROPEAN POLYMER JOURNAL
European Polymer Journal 43 (2007) 989–995
www.elsevier.com/locate/europolj
Prediction of dielectric constants and glass transition temperatures of polymers by quantitative structure property relationships Aihong Liu, Xueye Wang *, Ling Wang, Hanlu Wang, Hengliang Wang Xiangtan University, College of Chemistry, Xiangtan, Hunan 411105, China Received 23 July 2006; received in revised form 12 December 2006; accepted 13 December 2006 Available online 27 December 2006
Abstract The use of quantitative structure property relationships (QSPRs) is proposed for the calculation of the dielectric constants (e) and glass transition temperatures (Tg) for polymer. The descriptors involved in these models were calculated from the structures of the repeat units by the density functional theory (DFT) method. Subsets of these descriptors are used to build models in an attempt to find the best possible correlation between chemical structural parameter and dielectric constants or glass transition temperatures. Model with three quantum-chemical descriptors was selected for the prediction of e (R2 = 0.9086, s = 0.00103), and model with two quantum-chemical descriptors was selected for Tg(R2 = 0.9212, s = 21.7378). These results encourage the further application of QSPR methods to other classes of polymer. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Dielectric constants; Glass transition temperatures; Polymer; QSPR; Quantum-chemical descriptor
1. Introduction QSPR models of polymers are the theoretical basis for polymeric molecular designs and material designs [1–4], and have the ability to survey a list of possible candidates and exclude ones that do not fall into the desired property range for the application. In particular, no time is wasted on synthesis and testing of new materials that are deemed inappropriate by QSPR models.
* Corresponding author. Tel.: +86 0732 8292206; fax: +86 0732 8292477. E-mail address: [email protected] (X. Wang).
The electrical properties of polymers are important in many applications. The most widespread electrical application of polymers is the insulation of cables. In recent years, high-performance polymers have become important in the electronics industry, as encapsulants for electronic components, interlayer dielectrics, and printed wiring board materials. The dielectric constant, e, also called the relative permittivity, is an important fundamental molecular property, which can be a useful predictor of other electrical properties of polymers [5–7]. The primary importance of the dielectric constant lies in the fact that it is a measure of the ability of a substance to maintain a charge separation. The dielectric constant is a measure of polarization of the
0014-3057/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.eurpolymj.2006.12.029
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A. Liu et al. / European Polymer Journal 43 (2007) 989–995
medium between two charges when this medium is subjected to an electric field. A larger value of dielectric constant implies greater polarization of the medium between the two charges. Thus the dielectric constant represents the ability of a substance to separate charge and/or to orient it molecular dipoles in external electric field. The ability to predict dielectric constants theoretically is valuable in the molecular design of new materials. But the dielectric constant of a polymer is a function of so many parameters, such as temperature; rate (frequency) of measurement, structure and composition of polymer; morphology of specimens; impurities, fillers, plasticizers, other additives, and moisture (water molecules) in the polymer, and so on. As we all known, most of the previous works were about e for small organic molecules [8–10], we have not seen any reports to predict e values for polymer on quantitative structure property relationships (QSPRs). The aim of this study was to obtain a robust QSPR model of dielectric constants for a broadly representative group of high molecular weight polymers. Glass transition temperature, Tg, is also known as glass temperature or glass–rubber transition temperature. Tg of amorphous polymer is the most important property [5,11,12], which affects many other polymer properties such as heat capacity, coefficient of thermal expansion, and viscosity. Even if in the molecular design of the polymer, it is also necessary to predict the glass transition temperature of the designed polymer for evaluating it’s application. But it is difficult to determine experimentally, because the transition takes place over a comparatively wide temperature range and dependents on experimental conditions, such as the method of measurement, duration of the experiment, and pressure [13]. For this reason, the discrepancies in reported values of Tg in the literature can be quite large. There have been numerous attempts to predict Tgs for polymers on the basis of quantitative structure–property relationships. According to the view of Katrizky, there are two kinds of approaches, the empirical method and the theoretical estimation [14]. Empirical methods correlate the target property with other physical or chemical properties of polymers, for example: group additive property (GAP) [15]. The most widely referenced models of the theoretical estimations has been produced by Bicerano [5]. Bicerano produced a regression model
with a correlation coefficient (R) of 0.9749 and a standard error (s) of 24.65 K to relate Tg with the solubility parameter and the weighted sum of 13 structural parameters for the data set of 320 polymers. But he did not use external data set compounds to validate this model. Katritzky introduced a model with R2 of 0.928 for 22 medium molecular weight polymers using four parameters [16]. Katritzky applied the CODESSA method to predict the molar glass transition temperature (Tg/M) for 88 linear homopolymers, including polyethylenes, polyacrylates, polymethylacrylates, polystyrenes, polyethers and polyoxides as the data set, they used five parameters and generated a QSPR model with a standard error of 32.9 K for Tg values [14]. Cao and Lin tested the same set of 88 polymers using five descriptors in an attempt to derive a more physically meaningful QSPR with a coefficient of determination of R2 = 0.9056 and a standard error of 20.86 K [17]. Mattioni and Jurs developed a 10-descriptor model and an 11-descriptor model, which were used to predict Tg values for two diverse sets of polymers, respectively [18]. The kinds of polyacrylamides and polymethacrylamides are less than other, we want to produce a robust QSPR model that could predict Tg values for these polymers. These new molecular descriptors developed in this paper are calculated from the repeat unit structures with the hydrogen end-cap. 2. Materials and methods One of the steps necessary for this research was the assembly of a data set. A number of structurally heterogeneous addition linear polymers were selected. The reported experimental data have been taken from Refs. [19] and [20]. The experimented values of e for polymers were obtained by the Van Krevelen method at the condition: temperature (T) is 298.15 K and the frequency of measurement is 1 MHz. It is impossible to calculate descriptors directly for an entire molecule [18,21,22] because all the polymers have wide distribution of molecular weight and possess high molecular weight. As we know, if the molecular weight is high enough, the terminal groups hold only a very small proportion in a polymer and its effect on the properties can be ignored. Molecular descriptors calculated directly from the structure of the repeat units can be used for the study of QSPRs for polymers [23,24], since all the properties depends on the chemical structure of the
A. Liu et al. / European Polymer Journal 43 (2007) 989–995
a
H
R3
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H
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O
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H
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Calculated model R5
H H
n
R6
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H
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O R7
H R6 N4 R7
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Fig. 1. (a) The calculated models and structures for the polyalkenes, (b) the calculated models and structures for the polyacrylamides.
polymer molecule, and all this structure was conditioned by the repeat unit structure. Therefore, we adopt this method and focus on the following model (Fig. 1) to calculate molecular quantum-chemical descriptors. The structures for polymers were endcapped with methyl and hydrogen, respectively. For example, the structure used to calculate descriptors for polyethylene is the butane molecule. The structure of the repeat unit for each polymer was drawn and preoptimized with the chemoffice 3D program. For obtain the necessary quantum-chemical descriptors for the further calculations, the preoptimized structures were then optimized with a semiempirical AM1 [25] method. Semi empirical quantum-chemical methods (such as AM1) include empirical parameters, which come from a special system [26]. Thus these methods are not universality for others system. DFT method does not include any empirical parameters and belongs to the family of ab initio methods. In the DFT method, the exact exchange term used in the Hartree–Fock method is replaced by a more general expression:the exchange correlation functional. The simplicity of the DFT stems from the fact is that it uses a functional of electron density to model exchange and correlation. Further, DFT energy includes, besides the exchange contributions, also some portion of the correlation energy. In addition, DFT requires shorter computer times when compared with conventional ab initio methods with
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inclusion of electron correlation [27]. Care should be taken, however, when applying DFT to molecular clusters, where the dominant part of the stabilization energy comes from the dispersion energy [28]. The reason is that DFT method does not include the dispersion energy. In this paper, the calculated models are repeat units of polymers. The dispersion energy is very smaller than the total energy. Therefore, we adopt DFT method to optimize and calculate the models by the GAUSSIAN 03 program [29], at the B3LYP/6-31G (d) level [30–32] and obtain six quantum-chemical descriptors: ELUMO, EHOMO, q, Ethermal, S and EHF. The descriptor ELUMO and EHOMO are molecular orbital energy of the lowest unoccupied molecular orbital and the highest occupied molecular orbital, respectively. The descriptor q is the most negative net atomic charge in a molecule. Ethermal and S are the thermal energy and the entropy of a system at the condition T = 298.150 K and P = 1.0000 atm, respectively; and all are the contributions from translational, vibrational, and rotational movement of a molecule. The last descriptor EHF is the total energy of the whole system. 3. Results and discussion 3.1. Dielectric Constants By carrying out the correlation between the e values of 22 polyalkenes and the six descriptors (ELUMO, EHOMO, q, Ethermal, S and EHF) using multiple linear stepwise regression analysis with SPSS 12.0 program, the following optimal equation has been obtained: e ¼ 2:22646 þ 0:3259ELUMO þ 0:2236q þ 0:00006S
ð1Þ
R = 0.9532, R2 = 0.9086, s = 0.00103, F = 59.6483, n = 22. The QSPR models shows good correlations (R = 0.9532) between the e values and the three descriptors: ELUMO, q, S, which are listed in Table 1. Fig. 2a shows the plot of experimental values versus calculated values with Eq. (1). The standard deviation (s) value is 0.00103, which is satisfied. The characteristics of the three descriptors are given in Table 2. According to the statistical theory, if there is VIFj (variance inflation factor) P 10 in a model, the descriptor j is strongly correlated with the others
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Table 1 Parameters ELUMO, q, S and e values for 22 polyalkenes No.
Polymer name
ELUMO (a.u.)
q (c)
S (cal/mol Æ K)
ea
eb
ec
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
Poly(butene-1) Poly(decylethylene) Poly(3,3-dimethylbutylethylene) Poly(1,1-dimethylethylene) Poly(4,4-dimethylpentylethylene) Poly(1,1-dimethyltrimethylene) Poly(dodecylethylene) Poly(ethylene) Poly(ethyl-2-propylene) Poly(1-heptene) Poly(heptylethylene) Poly(1-hexene) Poly(hexylethylene) Poly(isobutylethylene) Poly(isopentylethylene) Poly(isopropylethylene) Poly(1-methyloctamethylene) Poly(nonylethylene) Poly(propylene) Poly(propyl-2-propylene) Poly(1-octene) Poly(1,1,2-trimethyltrimethylene)
0.08845 0.08845 0.07141 0.07664 0.07459 0.07624 0.07965 0.09504 0.07679 0.08651 0.08624 0.08686 0.08639 0.08058 0.08470 0.07803 0.08726 0.08857 0.08130 0.07966 0.08893 0.07724
0.452563 0.452509 0.451482 0.453985 0.450942 0.447269 0.450809 0.441482 0.454115 0.452597 0.452718 0.452440 0.452605 0.454428 0.453025 0.455289 0.444510 0.452504 0.449261 0.455140 0.452415 0.466527
86.446 147.892 112.854 84.659 120.674 92.389 161.196 73.085 91.204 109.02 124.149 101.687 117.066 101.002 108.38 91.849 124.155 139.969 79.336 100.358 117.597 97.805
2.159 2.163 2.157 2.155 2.158 2.157 2.163 2.165 2.157 2.162 2.162 2.161 2.162 2.157 2.158 2.155 2.162 2.163 2.157 2.158 2.162 2.154
2.160 2.164 2.156 2.156 2.158 2.157 2.162 2.164 2.156 2.161 2.161 2.160 2.161 2.158 2.160 2.156 2.164 2.163 2.158 2.157 2.162 2.154
2.160 2.164 2.156 2.156 2.158 2.157 2.162 2.162 2.156 2.161 2.161 2.160 2.161 2.158 2.160 2.156 2.164 2.163 2.158 2.157 2.162 2.153
a b c
Experimental values from Ref. [19]. Calculated values from the Eq. (1). Predicted values from LOO method.
Fig. 2. (a) Plot of experimental e versus calculated e, (b) plot of experimental e versus predicted e.
Table 2 The characteristics of descriptors ELUMO, q and S in Eq. (1)
Constants ELUMO q S
Coefficients Std. error
t-test
Sig.
2.22646 0.3259 0.2236 0.00006
87.89669 7.97912 4.22781 6.35138
0.00000 0.00000 1.18871 0.00046 1.17589 0.00000 1.01519
0.02533 0.04084 0.05283 0.00001
VIF
and is not significant to explain the model, which is not reliable. In this paper, all the values of VIF are less than 1.2, which shows that these descriptors are weakly correlated with each other. Validation is a crucial aspect of any QSAR/ QSPR modeling [33]. One of the most popular validation criteria is leave-one-out [34,35] cross-validated method. For this reason, in order to assess the predictability of the model found, the internal
A. Liu et al. / European Polymer Journal 43 (2007) 989–995
validation was carried out. The properties are predicted with the leave-one-out method. For leaveone-out method, the n samples is split into two sets, one set is one data as testing set, the other is n 1 samples as training set. The QSPR model obtained from training set is used to predict the testing set. This is done in turn until each sample is left out once. Using this approach, the model had a crossvalidation q2cv value is 0.847. This value of q2cv can be considered as proof of the high predictive ability of the model (q2cv > 0:6). For that reason, the QSPR model obtained in this paper is suitable for making accurate predictions for polyalkenes. The predicted values were also shown in Table 1. The plots of predicted values versus experimental values are shown in Fig. 2b. In fact, Fig. 2b is very similar to Fig. 2a, i.e. predicted values are very close to calculated values. A conclusion can be drawn that these QSPR models are reliable. On the one hand, According to Refs. [2] and [36], a larger ELUMO or q denotes a smaller intermolecular force, which makes the side group to move easily, and results in a larger e value; On the other hand, a larger S expresses a larger molecular polarizability, and indicates a very larger e value. Therefore, the dielectric constants e are positive correlation with ELUMO, q and S (see Eq. (1)).
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3.2. Glass transition temperatures Correlating the glass transition temperatures (Tg) values of 20 polyacrylamides to the six descriptors with multiple linear stepwise regression analysis, the QSPR equation with two descriptors (in Table 3) is obtained. Fig. 3a shows the plot of experimental values versus calculated values with Eq. (2). In contrast to previous works [5,14–18], the model with only two parameters (Ethermal and EHF) still has a smaller standard error. The characteristics of the two descriptors are given in Table 4, which suggests that the two descriptors all are significant descriptors from the P-value test, and the two descriptors are weakly correlated with each other from the VIFvalue test. T g ¼ 450:3612 1:37657Ethermal 0:29043EHF
ð2Þ
R = 0.9598, R2 = 0.9212, s = 21.7378, F = 99.337, n = 20. The properties are predicted with the leave-oneout method for the same experimental set in Table 3. The plots of predicted values versus experimental values are shown in Fig. 3b. The predicted values are very close to the calculated values, which can be seen from Table 3. The cross-validation q2 value
Table 3 Parameters Ethermal, EHF, and Tg values for 20 polyacrylamides No.
Polymer name
Ethermal (Kcal/mol)
EHF (Hartree)
Tg (K)a
Tg (K)b
Tg (K)c
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Poly(acrylamide) Poly(N-butylacrylamide) Poly(N-sec-butylacrylamide) Poly(N-tert-butylacrylamide) Poly(N,N-dibutylacrylamide) Poly(N,N-diisopropylacrylamide) Poly(N-dodecylacrylamide) Poly(isodecylacrylamide) Poly(isohexylacrylamide) Poly(isononylacrylamide) Poly(isoctylacrylamide) Poly(N-isopropylacrylamide) Poly(N-(1-methylbutyl)acrylamide) Poly(N-methyl-N-phenylacrylamide) Poly(morpholylacrylamide) Poly(N-octadecylacrylamide) Poly(piperidylacrylamide) Poly(4-carboxyphenylmethacrylamide) Poly(4-ethoxycarbonylphenylmethacrylamide) Poly(4-methoxycarbonylphenylmethacrylamide)
51.995 165.917 128.273 127.661 203.570 165.501 279.182 241.080 165.917 222.241 233.488 108.808 147.126 125.519 118.205 390.601 126.302 136.658 174.077 155.357
247.288 483.171 404.546 404.546 561.789 483.155 719.046 640.426 483.171 601.112 561.798 365.230 443.857 517.642 478.534 954.935 458.669 706.215 784.833 745.513
438 319 390 401 333 393 280 313 344 325 339 403 380 453 420 162 381 473 441 453
451 362 391 392 333 363 275 304 362 319 292 407 377 428 427 190 410 467 439 453
454 365 391 391 333 361 274 304 364 318 285 407 376 426 427 219 412 465 438 453
a b c
Experimental values from Ref. [20]. Calculated values from the Eq. (2). Predicted values from LOO method.
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Fig. 3. (a) Plot of experimental Tg versus calculated Tg, (b) Plot of experimental Tg versus predicted Tg.
Table 4 The characteristics of descriptors Ethermal and EHF in Eq. (2) Constants Ethermal EHF
Coefficients
Std. error
t-test
Sig.
VIF
450.3612 1.37657 0.29043
17.31536 0.10865 0.04796
26.00675 12.6697 6.05541
0.00000 0.00000 0.00001
2.57295 2.57295
is 0.882, which is satisfied. The results are indicated that these QSPR models are reliable. On the basis of the t-test (in Table 4), the most significant descriptor appearing in regression Eq. (2) is the descriptor Ethermal that relates to the number of atoms in a molecule. The descriptor Ethermal decreases Tg values. This phenomenon may be explained by that a molecule with a larger Ethermal possesses more atoms lying in the side groups, which increases the space of the polymeric chains, leads to a larger free volume and results in a smaller Tg value. As seen from Eq. (2), the second significant descriptor is EHF. The more negative EHF stands a larger molecular polarizability, cause a stronger intermolecular force, enhance stiffness of the polymeric chains, and result in a larger Tg. 4. Conclusions Two successful general QSPR models for the prediction of the properties e and Tg of a variety of high molecular weight polymers are proved to be accurate and reliable. The quantum-chemical descriptors can be obtained quickly and accurately from the repeat units of polymers and the QSPR model is easy for predicting. These results encourage the further application of QSPR methods to other classes
of polymer, such as copolymers, cross-linked polymers, and biopolymers. Acknowledgements The author wish to acknowledge the financial supports from the Scientific Research Fund of Hunan Provincial Education Department (No. 05A002), the National Natural Science Foundation of China (50675185) for the research work. References [1] Ulmer II Charles W, Smith Douglas A, Sumpter Bobby G, Noid Donald I. Computational neural networks and the rational design of polymeric materials: the next generation polycarbonates. Comput Theor Polym Sci 1998;8:311. [2] Yu XL, Wang XY, Gao JW, Li XB, Wang HL. QSPR studies of polyvinyls by density functional theory. Polymer 2005;46:9443. [3] Karelson M, Lobanov VS, Katritzky AR. Quantum-chemical descriptors in QSAR/QSPR studies. Chem Rev 1996;96: 1027. [4] Afantitis A, Melagraki G, Sarimveis H, Koutentis PA, Markopoulos J, Igglessi-Markopoulou O. Prediction of intrinsic viscosity in polymer–solvent combinations using a QSPR model. Polymer 2006;47:3240. [5] Bicerano J. Prediction of Polymers Properties. 2nd ed. New York: Marcel Dekker; 1996.
A. Liu et al. / European Polymer Journal 43 (2007) 989–995 [6] Gareth Hougham, Guiliana Tesoro, Alfred Viehbeck. Influence of free volume change on the relative permittivity and refractive index in fluoropolyimides. Macromolecules 1996; 29:3453. [7] Schweitzer RC, Morris JB. Improved quantitative structure property relationships for the prediction of dielectric constants for a set of diverse compounds by subsetting of the data set. J Chem Inf Comput Sci 2000;40:1253. [8] Cocchi M, De Benedetti PG, Seeber R, Tassi L, Ulrici A. Development of quantitative structure-property relationships using calculated descriptors for the prediction of the physicochemical properties (nD, q, bp, e, g) of a series of organic solvents. J Chem Inf Comput Sci 1999;39:1190. [9] Schweitzer RC, Morris JB. The development of a quantitative structure property relationship (QSPR) for the prediction of dielectric constants using neural networks. Anal Chim Acta 1999;384:285. [10] Sild S, Karelson M. A general QSPR treatment for dielectric constants of organic compounds. J Chem Inf Comput Sci 2002;42:360. [11] Afantitis A, Melagraki G, Makridima K, Alexandridis A, Sarimveis H, Iglessi-Markopoulou O. Prediction of high weight polymers glass transition temperature using RBF neural networks. J Mol Struct (Theochem) 2005;716:193. [12] Camelio P, Cypcar CC, Lazzeri V, Waegell B. A novel approach toward the prediction of the glass transition temperature:application of the EVM model, a designer QSPR equation for the prediction of acrylate and methacrylate polymers. J Polym Sci Part A: Polym Chem 1997; 35(13):2579. [13] Krause S, Gormley JJ, Roman N, Shetter JA, Watanabe WH. Glass temperatures of some acrylic polymers. J Polym Sci Part A: General Papers 1965;3(10):3573. [14] Katritzky AR, Sild S, Lobanov V, Karelson M. Quantitative structure-property relationship (QSPR) correlation of glass transition temperatures of high molecular weight polymers. J Chem Inf Comput Sci 1998;38:300. [15] Van Krevelen DW. Properties of Polymers, Their estimation and correlation with chemical structure. 2nd ed. Amsterdam: Elsevier; 1976. [16] Katritzky AR, Rachwal P, Law KW, Karelson M, Lobanov VS. Prediction of polymer glass transition temperatures using a general quantitative structure-property relationship treatment. J Chem Inf Comput Sci 1996;36:879. [17] Cao CZ, Lin YB. Correlation between the glass transition temperatures and repeating unit structure for high molecular weight polymers. J Chem Inf Comput Sci 2003;43:643. [18] Mattioni BE, Jurs PC. Prediction of glass transition temperatures from monomer and repeat unit structure using computational neural networks. J Chem Inf Comput Sci 2002;42:232. [19] National Institure for Materials Science, Polymer Database: http://polymer.nims.go.jp/PoLyInfo/. [20] Brandrup J, Immergut EH, Grulke EA. Polymer handbook. 4th ed. New York: John Wiley and Sons Inc.; 1999.
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[21] Yu XL, Wang XY, Gao JW, Li XB, Wang HL. Prediction of glass transition temperatures for polystyrenes by a fourdescriptors QSPR model. Macromol Theory Simul 2006;15:94. [22] Garcı´a-Domenech R, de Julia´n-Ortiz JV. Prediction of indices of refraction and glass transition temperatures of linear polymers by using graph theoretical indices. J Phys Chem B 2002;106:1501. [23] Sun Lili, Zhou Liping, Yu Yu, Lan Yukun, Li Zhiliang. QSPR study of polychlorinated diphenyl ethers by molecular electronegativity distance vector (MEDV-4). Chemosphere 2007;66:1039. [24] Marrero-Ponce Yovani. Total and local (atom and atom type) molecular quadratic indices:significance interpretation, comparison to other molecular descriptors, and QSPR/ QSAR applications. Bioorg Med Chem 2004;12:6351. [25] Dewar MJS, Zoebisch EG, Healy EF, Stewart JJP. AM1: a new general purpose quantum mechanical molecular model. J Am Chem Soc 1985;107:3902. [26] Wiff DR, Altieri MS, Goldfarb IJ. Predicting glass transition temperatures of linear polymers, random copolymers, and cured reactive oligomers from chemical structure. J Polym Sci Part B: Polym Phys 1985;23(6):1165. [27] Hobza P, Sponer J. Structure, energetics, and dynamics of the nucleic acid base pairs:nonempirical ab initio calculations. Chem Rev 1999;99:3247. [28] Katritzky AR, Sild S, Karelson M. Correlation and prediction of the refractive indices of polymers by QSPR. J Chem Inf Comput Sci 1998;38:1171. [29] Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, et al. Gaussian 03, Revision B.05; Gaussian. Pittsburgh: PA, 2003. [30] Becke AD. Density-functional thermochemistry. III. The role of exact exchange. J Chem Phys 1993;98:5648. [31] Lee C, Yang W, Parr RG. Development of the Colle– Salvetti correlation-energy formula into a functional of the electron density. Phys Rev B 1988;37:785. [32] Francl MM, Pietro WJ, Hehre WJ, Binkley JS, Gordon MS, DeFrees DJ. Self-consistent molecular orbital methods. XXIII. A polarization-type basis set for second-row elements. J Chem Phys 1982;77(7):3654. [33] Golbraikh A, Tropsha A. Beware of q2! J Mol Graph Model 2002;20:269. [34] Ovidiu Ivanciuc, Teodora Ivanciuc, Daniel Cabrol-Bass. QSAR for dihydrofolate reductase inhibitors with molecular graph structural descriptors. J Mol Struct (Theochem) 2002; 582:39. [35] Shansheng Yang, Wencong Lu, Nianyi Chen, Qiannan Hu. Support vector regression based QSPR for the prediction of some physicochemical properties of alkyl benzenes. J Mol Struct (Theochem) 2005;719:119. [36] Yu XL, Wang XY, Gao JW, Li XB, Wang HL. Quantitative structure–property relationship studies of polymeric cohesive energy by density functional theory. J Polym Sci Part B: Polym Phys 2006;44:409.
European Polymer Journal 43 (2007) 989–995
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Prediction of dielectric constants and glass transition temperatures of polymers by quantitative structure property relationships Aihong Liu, Xueye Wang *, Ling Wang, Hanlu Wang, Hengliang Wang Xiangtan University, College of Chemistry, Xiangtan, Hunan 411105, China Received 23 July 2006; received in revised form 12 December 2006; accepted 13 December 2006 Available online 27 December 2006
Abstract The use of quantitative structure property relationships (QSPRs) is proposed for the calculation of the dielectric constants (e) and glass transition temperatures (Tg) for polymer. The descriptors involved in these models were calculated from the structures of the repeat units by the density functional theory (DFT) method. Subsets of these descriptors are used to build models in an attempt to find the best possible correlation between chemical structural parameter and dielectric constants or glass transition temperatures. Model with three quantum-chemical descriptors was selected for the prediction of e (R2 = 0.9086, s = 0.00103), and model with two quantum-chemical descriptors was selected for Tg(R2 = 0.9212, s = 21.7378). These results encourage the further application of QSPR methods to other classes of polymer. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Dielectric constants; Glass transition temperatures; Polymer; QSPR; Quantum-chemical descriptor
1. Introduction QSPR models of polymers are the theoretical basis for polymeric molecular designs and material designs [1–4], and have the ability to survey a list of possible candidates and exclude ones that do not fall into the desired property range for the application. In particular, no time is wasted on synthesis and testing of new materials that are deemed inappropriate by QSPR models.
* Corresponding author. Tel.: +86 0732 8292206; fax: +86 0732 8292477. E-mail address: [email protected] (X. Wang).
The electrical properties of polymers are important in many applications. The most widespread electrical application of polymers is the insulation of cables. In recent years, high-performance polymers have become important in the electronics industry, as encapsulants for electronic components, interlayer dielectrics, and printed wiring board materials. The dielectric constant, e, also called the relative permittivity, is an important fundamental molecular property, which can be a useful predictor of other electrical properties of polymers [5–7]. The primary importance of the dielectric constant lies in the fact that it is a measure of the ability of a substance to maintain a charge separation. The dielectric constant is a measure of polarization of the
0014-3057/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.eurpolymj.2006.12.029
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A. Liu et al. / European Polymer Journal 43 (2007) 989–995
medium between two charges when this medium is subjected to an electric field. A larger value of dielectric constant implies greater polarization of the medium between the two charges. Thus the dielectric constant represents the ability of a substance to separate charge and/or to orient it molecular dipoles in external electric field. The ability to predict dielectric constants theoretically is valuable in the molecular design of new materials. But the dielectric constant of a polymer is a function of so many parameters, such as temperature; rate (frequency) of measurement, structure and composition of polymer; morphology of specimens; impurities, fillers, plasticizers, other additives, and moisture (water molecules) in the polymer, and so on. As we all known, most of the previous works were about e for small organic molecules [8–10], we have not seen any reports to predict e values for polymer on quantitative structure property relationships (QSPRs). The aim of this study was to obtain a robust QSPR model of dielectric constants for a broadly representative group of high molecular weight polymers. Glass transition temperature, Tg, is also known as glass temperature or glass–rubber transition temperature. Tg of amorphous polymer is the most important property [5,11,12], which affects many other polymer properties such as heat capacity, coefficient of thermal expansion, and viscosity. Even if in the molecular design of the polymer, it is also necessary to predict the glass transition temperature of the designed polymer for evaluating it’s application. But it is difficult to determine experimentally, because the transition takes place over a comparatively wide temperature range and dependents on experimental conditions, such as the method of measurement, duration of the experiment, and pressure [13]. For this reason, the discrepancies in reported values of Tg in the literature can be quite large. There have been numerous attempts to predict Tgs for polymers on the basis of quantitative structure–property relationships. According to the view of Katrizky, there are two kinds of approaches, the empirical method and the theoretical estimation [14]. Empirical methods correlate the target property with other physical or chemical properties of polymers, for example: group additive property (GAP) [15]. The most widely referenced models of the theoretical estimations has been produced by Bicerano [5]. Bicerano produced a regression model
with a correlation coefficient (R) of 0.9749 and a standard error (s) of 24.65 K to relate Tg with the solubility parameter and the weighted sum of 13 structural parameters for the data set of 320 polymers. But he did not use external data set compounds to validate this model. Katritzky introduced a model with R2 of 0.928 for 22 medium molecular weight polymers using four parameters [16]. Katritzky applied the CODESSA method to predict the molar glass transition temperature (Tg/M) for 88 linear homopolymers, including polyethylenes, polyacrylates, polymethylacrylates, polystyrenes, polyethers and polyoxides as the data set, they used five parameters and generated a QSPR model with a standard error of 32.9 K for Tg values [14]. Cao and Lin tested the same set of 88 polymers using five descriptors in an attempt to derive a more physically meaningful QSPR with a coefficient of determination of R2 = 0.9056 and a standard error of 20.86 K [17]. Mattioni and Jurs developed a 10-descriptor model and an 11-descriptor model, which were used to predict Tg values for two diverse sets of polymers, respectively [18]. The kinds of polyacrylamides and polymethacrylamides are less than other, we want to produce a robust QSPR model that could predict Tg values for these polymers. These new molecular descriptors developed in this paper are calculated from the repeat unit structures with the hydrogen end-cap. 2. Materials and methods One of the steps necessary for this research was the assembly of a data set. A number of structurally heterogeneous addition linear polymers were selected. The reported experimental data have been taken from Refs. [19] and [20]. The experimented values of e for polymers were obtained by the Van Krevelen method at the condition: temperature (T) is 298.15 K and the frequency of measurement is 1 MHz. It is impossible to calculate descriptors directly for an entire molecule [18,21,22] because all the polymers have wide distribution of molecular weight and possess high molecular weight. As we know, if the molecular weight is high enough, the terminal groups hold only a very small proportion in a polymer and its effect on the properties can be ignored. Molecular descriptors calculated directly from the structure of the repeat units can be used for the study of QSPRs for polymers [23,24], since all the properties depends on the chemical structure of the
A. Liu et al. / European Polymer Journal 43 (2007) 989–995
a
H
R3
C1
C2
H
R4
CH3
n
Polymer
b
H
R5
C1
C2
H
C3
O
Polymer
H
R3
C1
C2
H
R4
CH3
Calculated model R5
H H
n
R6
N4
C1
C2
H
C3
O R7
H R6 N4 R7
Calculated model
Fig. 1. (a) The calculated models and structures for the polyalkenes, (b) the calculated models and structures for the polyacrylamides.
polymer molecule, and all this structure was conditioned by the repeat unit structure. Therefore, we adopt this method and focus on the following model (Fig. 1) to calculate molecular quantum-chemical descriptors. The structures for polymers were endcapped with methyl and hydrogen, respectively. For example, the structure used to calculate descriptors for polyethylene is the butane molecule. The structure of the repeat unit for each polymer was drawn and preoptimized with the chemoffice 3D program. For obtain the necessary quantum-chemical descriptors for the further calculations, the preoptimized structures were then optimized with a semiempirical AM1 [25] method. Semi empirical quantum-chemical methods (such as AM1) include empirical parameters, which come from a special system [26]. Thus these methods are not universality for others system. DFT method does not include any empirical parameters and belongs to the family of ab initio methods. In the DFT method, the exact exchange term used in the Hartree–Fock method is replaced by a more general expression:the exchange correlation functional. The simplicity of the DFT stems from the fact is that it uses a functional of electron density to model exchange and correlation. Further, DFT energy includes, besides the exchange contributions, also some portion of the correlation energy. In addition, DFT requires shorter computer times when compared with conventional ab initio methods with
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inclusion of electron correlation [27]. Care should be taken, however, when applying DFT to molecular clusters, where the dominant part of the stabilization energy comes from the dispersion energy [28]. The reason is that DFT method does not include the dispersion energy. In this paper, the calculated models are repeat units of polymers. The dispersion energy is very smaller than the total energy. Therefore, we adopt DFT method to optimize and calculate the models by the GAUSSIAN 03 program [29], at the B3LYP/6-31G (d) level [30–32] and obtain six quantum-chemical descriptors: ELUMO, EHOMO, q, Ethermal, S and EHF. The descriptor ELUMO and EHOMO are molecular orbital energy of the lowest unoccupied molecular orbital and the highest occupied molecular orbital, respectively. The descriptor q is the most negative net atomic charge in a molecule. Ethermal and S are the thermal energy and the entropy of a system at the condition T = 298.150 K and P = 1.0000 atm, respectively; and all are the contributions from translational, vibrational, and rotational movement of a molecule. The last descriptor EHF is the total energy of the whole system. 3. Results and discussion 3.1. Dielectric Constants By carrying out the correlation between the e values of 22 polyalkenes and the six descriptors (ELUMO, EHOMO, q, Ethermal, S and EHF) using multiple linear stepwise regression analysis with SPSS 12.0 program, the following optimal equation has been obtained: e ¼ 2:22646 þ 0:3259ELUMO þ 0:2236q þ 0:00006S
ð1Þ
R = 0.9532, R2 = 0.9086, s = 0.00103, F = 59.6483, n = 22. The QSPR models shows good correlations (R = 0.9532) between the e values and the three descriptors: ELUMO, q, S, which are listed in Table 1. Fig. 2a shows the plot of experimental values versus calculated values with Eq. (1). The standard deviation (s) value is 0.00103, which is satisfied. The characteristics of the three descriptors are given in Table 2. According to the statistical theory, if there is VIFj (variance inflation factor) P 10 in a model, the descriptor j is strongly correlated with the others
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A. Liu et al. / European Polymer Journal 43 (2007) 989–995
Table 1 Parameters ELUMO, q, S and e values for 22 polyalkenes No.
Polymer name
ELUMO (a.u.)
q (c)
S (cal/mol Æ K)
ea
eb
ec
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
Poly(butene-1) Poly(decylethylene) Poly(3,3-dimethylbutylethylene) Poly(1,1-dimethylethylene) Poly(4,4-dimethylpentylethylene) Poly(1,1-dimethyltrimethylene) Poly(dodecylethylene) Poly(ethylene) Poly(ethyl-2-propylene) Poly(1-heptene) Poly(heptylethylene) Poly(1-hexene) Poly(hexylethylene) Poly(isobutylethylene) Poly(isopentylethylene) Poly(isopropylethylene) Poly(1-methyloctamethylene) Poly(nonylethylene) Poly(propylene) Poly(propyl-2-propylene) Poly(1-octene) Poly(1,1,2-trimethyltrimethylene)
0.08845 0.08845 0.07141 0.07664 0.07459 0.07624 0.07965 0.09504 0.07679 0.08651 0.08624 0.08686 0.08639 0.08058 0.08470 0.07803 0.08726 0.08857 0.08130 0.07966 0.08893 0.07724
0.452563 0.452509 0.451482 0.453985 0.450942 0.447269 0.450809 0.441482 0.454115 0.452597 0.452718 0.452440 0.452605 0.454428 0.453025 0.455289 0.444510 0.452504 0.449261 0.455140 0.452415 0.466527
86.446 147.892 112.854 84.659 120.674 92.389 161.196 73.085 91.204 109.02 124.149 101.687 117.066 101.002 108.38 91.849 124.155 139.969 79.336 100.358 117.597 97.805
2.159 2.163 2.157 2.155 2.158 2.157 2.163 2.165 2.157 2.162 2.162 2.161 2.162 2.157 2.158 2.155 2.162 2.163 2.157 2.158 2.162 2.154
2.160 2.164 2.156 2.156 2.158 2.157 2.162 2.164 2.156 2.161 2.161 2.160 2.161 2.158 2.160 2.156 2.164 2.163 2.158 2.157 2.162 2.154
2.160 2.164 2.156 2.156 2.158 2.157 2.162 2.162 2.156 2.161 2.161 2.160 2.161 2.158 2.160 2.156 2.164 2.163 2.158 2.157 2.162 2.153
a b c
Experimental values from Ref. [19]. Calculated values from the Eq. (1). Predicted values from LOO method.
Fig. 2. (a) Plot of experimental e versus calculated e, (b) plot of experimental e versus predicted e.
Table 2 The characteristics of descriptors ELUMO, q and S in Eq. (1)
Constants ELUMO q S
Coefficients Std. error
t-test
Sig.
2.22646 0.3259 0.2236 0.00006
87.89669 7.97912 4.22781 6.35138
0.00000 0.00000 1.18871 0.00046 1.17589 0.00000 1.01519
0.02533 0.04084 0.05283 0.00001
VIF
and is not significant to explain the model, which is not reliable. In this paper, all the values of VIF are less than 1.2, which shows that these descriptors are weakly correlated with each other. Validation is a crucial aspect of any QSAR/ QSPR modeling [33]. One of the most popular validation criteria is leave-one-out [34,35] cross-validated method. For this reason, in order to assess the predictability of the model found, the internal
A. Liu et al. / European Polymer Journal 43 (2007) 989–995
validation was carried out. The properties are predicted with the leave-one-out method. For leaveone-out method, the n samples is split into two sets, one set is one data as testing set, the other is n 1 samples as training set. The QSPR model obtained from training set is used to predict the testing set. This is done in turn until each sample is left out once. Using this approach, the model had a crossvalidation q2cv value is 0.847. This value of q2cv can be considered as proof of the high predictive ability of the model (q2cv > 0:6). For that reason, the QSPR model obtained in this paper is suitable for making accurate predictions for polyalkenes. The predicted values were also shown in Table 1. The plots of predicted values versus experimental values are shown in Fig. 2b. In fact, Fig. 2b is very similar to Fig. 2a, i.e. predicted values are very close to calculated values. A conclusion can be drawn that these QSPR models are reliable. On the one hand, According to Refs. [2] and [36], a larger ELUMO or q denotes a smaller intermolecular force, which makes the side group to move easily, and results in a larger e value; On the other hand, a larger S expresses a larger molecular polarizability, and indicates a very larger e value. Therefore, the dielectric constants e are positive correlation with ELUMO, q and S (see Eq. (1)).
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3.2. Glass transition temperatures Correlating the glass transition temperatures (Tg) values of 20 polyacrylamides to the six descriptors with multiple linear stepwise regression analysis, the QSPR equation with two descriptors (in Table 3) is obtained. Fig. 3a shows the plot of experimental values versus calculated values with Eq. (2). In contrast to previous works [5,14–18], the model with only two parameters (Ethermal and EHF) still has a smaller standard error. The characteristics of the two descriptors are given in Table 4, which suggests that the two descriptors all are significant descriptors from the P-value test, and the two descriptors are weakly correlated with each other from the VIFvalue test. T g ¼ 450:3612 1:37657Ethermal 0:29043EHF
ð2Þ
R = 0.9598, R2 = 0.9212, s = 21.7378, F = 99.337, n = 20. The properties are predicted with the leave-oneout method for the same experimental set in Table 3. The plots of predicted values versus experimental values are shown in Fig. 3b. The predicted values are very close to the calculated values, which can be seen from Table 3. The cross-validation q2 value
Table 3 Parameters Ethermal, EHF, and Tg values for 20 polyacrylamides No.
Polymer name
Ethermal (Kcal/mol)
EHF (Hartree)
Tg (K)a
Tg (K)b
Tg (K)c
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Poly(acrylamide) Poly(N-butylacrylamide) Poly(N-sec-butylacrylamide) Poly(N-tert-butylacrylamide) Poly(N,N-dibutylacrylamide) Poly(N,N-diisopropylacrylamide) Poly(N-dodecylacrylamide) Poly(isodecylacrylamide) Poly(isohexylacrylamide) Poly(isononylacrylamide) Poly(isoctylacrylamide) Poly(N-isopropylacrylamide) Poly(N-(1-methylbutyl)acrylamide) Poly(N-methyl-N-phenylacrylamide) Poly(morpholylacrylamide) Poly(N-octadecylacrylamide) Poly(piperidylacrylamide) Poly(4-carboxyphenylmethacrylamide) Poly(4-ethoxycarbonylphenylmethacrylamide) Poly(4-methoxycarbonylphenylmethacrylamide)
51.995 165.917 128.273 127.661 203.570 165.501 279.182 241.080 165.917 222.241 233.488 108.808 147.126 125.519 118.205 390.601 126.302 136.658 174.077 155.357
247.288 483.171 404.546 404.546 561.789 483.155 719.046 640.426 483.171 601.112 561.798 365.230 443.857 517.642 478.534 954.935 458.669 706.215 784.833 745.513
438 319 390 401 333 393 280 313 344 325 339 403 380 453 420 162 381 473 441 453
451 362 391 392 333 363 275 304 362 319 292 407 377 428 427 190 410 467 439 453
454 365 391 391 333 361 274 304 364 318 285 407 376 426 427 219 412 465 438 453
a b c
Experimental values from Ref. [20]. Calculated values from the Eq. (2). Predicted values from LOO method.
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Fig. 3. (a) Plot of experimental Tg versus calculated Tg, (b) Plot of experimental Tg versus predicted Tg.
Table 4 The characteristics of descriptors Ethermal and EHF in Eq. (2) Constants Ethermal EHF
Coefficients
Std. error
t-test
Sig.
VIF
450.3612 1.37657 0.29043
17.31536 0.10865 0.04796
26.00675 12.6697 6.05541
0.00000 0.00000 0.00001
2.57295 2.57295
is 0.882, which is satisfied. The results are indicated that these QSPR models are reliable. On the basis of the t-test (in Table 4), the most significant descriptor appearing in regression Eq. (2) is the descriptor Ethermal that relates to the number of atoms in a molecule. The descriptor Ethermal decreases Tg values. This phenomenon may be explained by that a molecule with a larger Ethermal possesses more atoms lying in the side groups, which increases the space of the polymeric chains, leads to a larger free volume and results in a smaller Tg value. As seen from Eq. (2), the second significant descriptor is EHF. The more negative EHF stands a larger molecular polarizability, cause a stronger intermolecular force, enhance stiffness of the polymeric chains, and result in a larger Tg. 4. Conclusions Two successful general QSPR models for the prediction of the properties e and Tg of a variety of high molecular weight polymers are proved to be accurate and reliable. The quantum-chemical descriptors can be obtained quickly and accurately from the repeat units of polymers and the QSPR model is easy for predicting. These results encourage the further application of QSPR methods to other classes
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