Modeling and Predicting the Glass Transition Temperature of Polymethacrylates Based on Quantum Chemical Descriptors by Using Hybrid PSO-SVR

Abstract

Based on six quantum chemical descriptors (|L-1.356|, E_total, qC6, α, q⁻, and E_therm), the hybrid PSO-SVR is proposed to establish a model for predicting the glass transition temperature (T_g) of 37 polymethacrylates. The prediction performance of SVR was compared with those of reported MLR and ANN models. The results show that the RMSE, MAPE, and R² calculated by SVR are superior to those achieved by MLR or ANN model for the identical training set and test set. This investigation reveals that the SVR model is more suitable to be used for prediction of the T_g values for unknown polymethacrylates possessing similar structure than the conventional MLR or ANN model, and provides a clue that the method proposed in this study may be useful in computer-aided design of new polymethacrylates with desired T_g.

1. Introduction

When heating a polymeric material over a certain temperature, the material would begin transfer from a glassy state to a rubbery state. This transition is referred to as the glass transition, which occurs over a range of a few degrees of temperature. The mid-point or the point of inflection of the transition is called the glass transition temperature (T_g). The T_g affects many other polymer properties such as thermal expansion coefficient, modulus, heat capacity and hardness, etc. These physical properties may change drastically below and above the T_g such that the T_g index is usually employed to determine the application scope of a polymeric material. In general, it is difficult to measure the T_g experimentally because the transition takes place over a comparatively wide temperature range and depends on experimental conditions, such as the method of measurement, duration of the experiment and pressure, etc.1 Moreover, T_g is quite dependent on the polymeric structural and constitutional features.2, 3 Thus, the development of theoretical modeling methods for prediction of the T_g is necessary.

The mechanical properties of thermoplastic polymers, particularly methacrylates polymers, play a decisive role in practical application. Many researchers have ever attempted to develop quantitative structure property relationship (QSPR) models for prediction of the T_g of polymethacrylates. For example, Camelio et al.4 chosen QSPR method involving the EVM (energy, volume, mass) model to predict the T_g of acrylate and methacrylate polymers with R² = 0.96. Dai et al.5 developed a QSPR model with R² = 0.980 and s = 8.3 K in terms of three parameters to calculate the T_g of 13 polyacrylates and 9 polymethacrylates. Liu et al.6 and Liu et al.7 used artificial neural network (ANN) to estimate the T_g for 113 polymers via four descriptors and 32 polymers via six descriptors, the root mean square error (RMSE) to the validation set achieved 17.53 and 8.35 K, respectively. Yu and Yi8 applied a QSPR model constructed by back-propagation neural network to regress the T_g of 192 vinyl polymers, including 95 polystyrenes, 73 polyacrylates, and 24 polymethacrylates, giving a validation RMSE of 20.195K. Yu and Wang9 constructed a QSPR model to correlate the T_g of aromatic heterocyclic polymers with the mean relative errors of 3.299% (R = 0.963, n = 28) for the training set and 3.064% (R = 0.959, n = 23) for the test set. Xu et al.10 used QSPR for the prediction of T_g of polystyrenes based on a set of 107 polystyrenes using ANN combined with genetic function approximation. In addition, other researchers11–13 utilized different QSPR models to predict the T_g values of polymers based on diverse parameters.

All the models stated above were based on constitutional (or topological) descriptors and quantum chemical descriptors, and developed with multiple linear/nonlinear regression (MLR/MNR) method and/or the ANN method. However research using the support vector regression (SVR) theory combined with particle swarm optimization (PSO) to model and predict the T_g of methacrylate polymers has not been reported previously.

In this study, motivated by Liu et al.'s previous work,7 we introduce a novel machine learning method, SVR hybrided with PSO, to model and predict the T_g of polymethacrylates based on six quantum chemical descriptors (|L-1.356|, E_total, qC6, α, q⁻, and E_therm), and its predictive errors were also compared with those of the MLR model and ANN model.

2. Materials and Methods

2.1. Support Vector Regression

The support vector machine (SVM), proposed by Vapnik et al.,14 is a statistical learning approach based on the structure risk minimization principle. In recent years, the SVM has been regarded as the state-of-the-art technique for solving a variety of learning, classification, regression and prediction problems. SVR, as a regression version of SVM, is capable of solving non-linear regression problems using kernel functions, which is generally superior to other pattern recognition and regression approaches. SVR has been proved to exhibit a number of significant advantages such as excellent learning performance of small samples, good generalization ability, small errors, high calculation accuracy, etc.15 At present, it has become a focus in machine learning research and is extensively employed in a wide range of real-world problems.16–30

Suppose a sample is described by (x, y), where x represents the independent variable and y the dependent response. In general, the relationship between y and x is quite complex and highly nonlinear. For SVR, the basic idea is to map x from the original space X into a higher dimensional feature space F via a nonlinear mapping function ϕ(x), and then to conduct a linear regression in F space. Therefore, the purpose of SVR is to find a linear Equation (1) based on a given training dataset {(x₁,y₁), …, (x_n,y_n)}:

• ((1))

where w is a vector for regression coefficient, b is a bias. They are estimated by minimizing the regularized risk function R(C), namely:

• ((2))

• ((3))

where n is the number of training samples, C is a regularization factor, ε is a prescribed parameter controlling the tolerance to error, and (1/2)||w||² is used as a measurement of function flatness. The second term, , is the so-called empirical risk and measured by the ε-insensitive loss function L_ε(f(x_i)–y_i), which indicates that it does not penalize errors below ε.

In order to control function complexity and regression errors according to the desired precision, the slack variables ξ and ξ* are introduced to deal with the data points that do not satisfy Equation (3). Equation (2) and (3) can be transformed into the primal problem :

• 

• ((4))

In Equation (4), the first term increases the smoothness of the regression function to improve its generalization ability and the second term reduces errors. The regularization factor C is a positive constant, determining the tradeoff between the training error and the model flatness.

In order to obtain w and b in Equation (1), the Lagrange equation is built:

• ((5))

where α_i and α_i* are Lagrange multipliers to be solved. Only the nonzero values of the Lagrange multipliers are useful in the regression, and their corresponding samples are known as support vectors (SVs). The Equation (5) is obtained by setting the partial differential coefficients for w, b, ξ_i, and equal to zero:

• ((6))

Substituting Equation (6) into Equation (5), the dual optimization problem can be written as:

• 

• ((7))

Therefore, the function regression problem on SVR may come down to a quadratic programming problem. By minimizing Q, the array w can be written in terms of the Lagrange multipliers and training samples as:

• $${\bf w} = \sum\limits_{i = 1}^{l} {{\rm (}\alpha _{i} } - \alpha _{i}^{*} {\rm )}\phi {\rm (}{\bf x}_{i} {\rm )}$$, ((8))

where l is the number of SVs. Finally, the linear Equation (1) has the following explicit form:

• ((9))

In Equation (9), k(x,x_i) = ϕ(x)•ϕ(x_i) is a kernel function. There exist several types of commonly used kernel functions, such as linear kernel, radial basis kernel, polynomial kernel, sigmoid kernel, etc. In this study, the sigmoid kernel function (10) is used as the kernel function of the SVR because it tends to achieve better performance.

• ((10))

2.2. Choosing of SVR Parameters with PSO

The PSO method was proposed by Kennedy and Eberhart,31 being motivated by the social behavior of organisms such as bird flocking and fish schooling. It is an optimization technique. The generalization ability of SVR relies entirely on four parameters, i.e., ε of the ε-insensitive loss function, the regularization constant C, and the kernel function parameters α and β. Therefore, it is a key step to search for the optimal parameters (ε, C, α, β) for SVR. In this study, PSO was utilized to search for the optimal parameters (ε, C, α, β) of SVR, and the mean absolute percentage error (MAPE), which directly reflects the modeling performance of SVR, was chosen as the fitness function:

• ((11))

where n denotes the number of training samples, y_i represents the actual measured value and is the estimated value for the ith training sample.

2.3. Dataset and Modeling Method

The training set and test set used in this study was generated by Liu et al.7, which was originally taken from ref. 32 and is composed of 32 polymethacrylates. Table 1 tabulates the T_g values and the indices of related six quantum chemical descriptors (|L-1.356|, E_total, qC6, α, q⁻ and E_therm) for 25 training samples and seven test samples. Here, L is side-chain length, referring to the distance from the ester O⁵ to its furthest atom on the side chain R⁶ (see Figure 1), here R⁶ = C_nH_2n+1 (n is the atomic number of carbon of the alkyl on the side chain of polymethacrylates, n = 1, 2, 3, 4, 6, 8, 10, 12, 14, 16). After analyzing the relationship between the side-chain length L and T_g, Liu et al.7 found that when L = 1.356 nm (n = 10), the T_g would decline to the lowest point where the free volume and the intermolecular attraction strike a balance. Thus taking L = 1.356 nm as the length of the side chain at which T_g is lowest, irrespective of whether the side chain is longer or shorter than 1.356 nm.7 E_total is the total energy of the macromolecular. qC6 is the net charge of carbon atom C6 connected directly to ester O⁵. α is the molecular average polarizability. q⁻ is the net charge of the most negative atom and E_therm is the thermal energy of the polymethacrylates. All the quantum chemical descriptors were calculated directly from the structure of the monomer with the Gaussian 03 program, at the DFT/B3LYP/6-31G(d) level with the keywords OPT, POLAR, FREQ, and the optimized structure was characterized as true local energy minima on the potential energy surface, without imaginary frequencies. In this study, the polymethacrylates were represented by their repeating units end-capped by hydrogen atoms to calculate molecular descriptors. The detailed definition and calculation for above quantum chemical descriptors are also given in ref. 7. In addition, other five independent samples as tabulated in Table 3 were taken from the literatures.2, 8

Table 1. Quantum chemical descriptors and glass transition temperature T_g for 32 polymethacrylates.7

No.	Polymer	|L-1.356| [nm]	Etotal [au]	qC6 [au]	q^− [au]	α [au]	Etherm [kJ  ·  mol^−1]	Exp.Tg [K]
a) Test sample.
1	Poly(4-cyanopheny1-methacylate)	0.68	−631	0.3604	−0.5274	125.33	555.98	428
2	Poly(phenyl methacrylate)	0.83	−538.75	0.3562	−0.5336	104.34	550.12	407
3a)	Poly(tert-butyl methacrylate)	1.02	−464.97	0.3118	−0.4964	89.6	642.01	380
4	Poly(4-methoxycarbonyphenyl-methacrylate)	0.54	−766.4	0.3561	−0.5273	139.68	680.28	379
5	Poly(methyl methacrylate)	1.15	−347.02	−0.2169	−0.4805	58.09	410.71	373
6	Poly(4-tertbutylphenyl methacrylate)	0.6	−696.01	0.3444	−0.5318	152.5	866.35	371
7	Poly(2-chloroehtyl-methacrylate)	0.85	−845.93	−0.028	−0.4787	79.26	466.24	365
8a)	Poly(2-hydroxyehtyl-methacrylate)	0.96	−461.54	−0.0555	−0.609	72.93	502.01	358
9	Poly(cycHexyl-methacrylate)	0.93	−542.39	0.1318	−0.4913	108.94	742.84	356
10	Poly(1,1,1-trifluoro-2-propyl-methacrylate)	1	−723.37	−0.0032	−0.4758	79.55	510.31	354
11a)	Poly(isoproyl methacrylate)	1.02	−425.65	0.1303	−0.4859	79.49	565.94	354
12	Poly(2-hydroxypropyl-methacrylate)	0.9	−500.86	−0.0369	−0.6099	83.17	581.1	349
13	Poly(2-cyanoethyl-methacrylate)	0.88	−478.57	−0.0328	−0.4765	81.35	487.92	347
14	Poly(ethyl methacrylate)	1.02	−386.33	−0.0315	−0.4832	69.08	488.81	338
15a)	Poly(sec-butyl methacrylate)	0.96	−464.97	0.1328	−0.4856	89.9	642.15	333
16	Poly(phenethyl-methacrylate)	0.6	−617.39	−0.0266	−0.4806	126.9	714.2	329
17	Poly(benzyl methacrylate)	0.75	−577.99	−0.1442	−0.485	114.84	636.01	327
18	Poly(isobutyl methacrylate)	1	−464.96	−0.0233	−0.4828	90.42	644.9	326
19	Poly(2-bromoehtyl-methacrylate)	0.94	−2957.44	−0.0336	−0.4794	86.46	467.36	325
20a)	Poly(3,3-dimethylbutyl-methacrylate)	0.96	−543.59	−0.0422	−0.4844	111.59	800.59	318
21	Poly(neopenyl methacrylate)	0.9	−504.28	−0.0207	−0.4826	100.58	719.13	312
22	Poly(proyl methacrylate)	0.9	−425.65	−0.0258	−0.483	80.01	564.88	308
23	Poly(2-methoxyethyl-methacrylate)	0.79	−500.85	−0.0631	−0.4787	83.89	582.85	297
24a)	Poly(butyl methacrylate)	0.77	−464.96	−0.0315	−0.4835	90.99	643.75	293
25	Poly(hexdecyl methacrylate)	0.77	−936.73	−0.0312	−0.4837	224.75	1590.82	288
26	Poly(pentyl methacrylate)	0.64	−504.28	−0.045	−0.4841	101.65	725.07	283
27	Poly(Hexyl methacrylate)	0.51	−543.59	−0.0312	−0.4835	113.14	803.64	273
28	Poly(2-ethylhexyl-methacrylate)	0.52	−622.21	−0.0487	−0.489	134.51	959.99	263
29	Poly(octyl methacrylate)	0.26	−622.22	−0.0311	−0.4836	135.37	961.05	253
30	Poly(Tetradecyl-methacrylate)	0.51	−858.1	−0.0312	−0.4837	202.37	1433.38	233
31	Poly(dodecyl methacrylate)	0.26	−779.47	−0.0312	−0.4837	180	1275.94	208
32a)	Poly(decyl-methacrylate)	0	−700.85	−0.044	−0.4847	157.51	1118.66	203

Figure 1. The calculated models of polymethacrylate.

In this study, in order to directly compare the MLR, ANN, and SVR models' modeling, prediction and generalization performance, the training and test procedures were carried out in terms of the same training and test samples as previously conducted by Liu et al.7. Therefore, as indicated in Table 1, 25 samples were employed as a training dataset in the training process to develop a model, the other seven samples acted as the test set. The five independent samples acted as the independent set to further validate the generalization ability of the established SVR model.

The SVR model was self-adjusted to make the training error (MAPE) as small as possible via learning with the training samples and optimizing the parameters through continuous training adjustment for 10 000 times. Finally, an optimal SVR model was obtained via a sigmoid kernel with the penalty factor C of 56700186162.908470, insensitivity factor ε of 4.011811 and kernel function parameters (α, β) of (0.150243, −4.425184).

2.4. Generalization Performance Evaluation

The T_g values of the seven test samples and five independent samples were calculated by using the constructed SVR model. Besides the index of MAPE, three other indices, i.e., mean absolute error (MAE), RMSE, and correlation coefficient (R²), were used for performance evaluation, as defined by Equation (12–14), respectively:

• ((12))

• ((13))

• ((14))

where m denotes the number of test/independent samples, y_j and stand for the targets (measured) and predicted values of the jth test/independent sample respectively, is the mean experimental value as well as is the mean predicted value for all test/independent samples.

3. Results and Discussion

Table 2 gives a list of the T_g values calculated by MLR7 (see Equation (15)), ANN7 and SVR for 25 training samples and seven test samples. Table 3 lists five independent samples for further assessment of the established SVR model. Table 4 summarizes the statistical RMSE, MAE, MAPE, and R² for the calculations by using the MLR, ANN, and SVR models for the training set, test set and independent set, respectively. Figure 2 depicts a pair wise comparison of the experimental values versus estimated values calculated via SVR.

• ((15))

Table 2. Comparison the experimental and calculated T_g of 32 polymers among different regression methods.

No.	Polymer	Tg [K]
		Exp.	MLR		ANN		SVR
			Cal.	Error [%]	Cal.	Error [%]	Cal.	Error [%]
a) Test sample.
1	Poly(4-cyanopheny1-methacylate)	428	420	−1.87	418	−2.34	424	−0.93
2	Poly(phenylmethacrylate)	407	405	−0.49	415	1.97	408	0.25
3	Poly(tert-butyl methacrylate)	380	368	−3.16	369a)	−2.89	380a)	0.00
4	Poly(4-methoxycarbonyphenyl-methacrylate)	379	391	3.17	382	0.79	379	0.00
5	Poly(methyl methacrylate)	373	347	−6.97	359	−3.75	378	1.34
6	Poly(4-tertbutylphenyl methacrylate)	371	370	−0.27	365	−1.62	368	−0.81
7	Poly(2-chloroehtyl-methacrylate)	365	339	−7.12	346	−5.21	360	−1.37
8	Poly(2-hydroxyethyl-methacrylate)	358	359	0.28	347a)	−3.07	364a)	1.68
9	Poly(cycloHexyl-methacrylate)	356	346	−2.81	342	−3.93	353	−0.84
10	Poly(1,1,1-trifluoro-2-propyl-methacrylate)	354	350	−1.13	358	1.13	359	1.41
11	Poly(isopropyl methacrylate)	354	355	0.28	353a)	−0.28	347a)	−1.98
12	Poly(2-hydroxypropyl-methacrylate)	349	348	−0.29	350	0.29	344	−1.43
13	Poly(2-cyanoethyl-methacrylate)	347	346	−0.29	333	−4.03	343	−1.15
14	Poly(ethyl methacrylate)	338	344	1.78	337	−0.30	335	−0.89
15	Poly(sec-butyl methacrylate)	333	344	3.30	332a)	−0.30	338a)	1.50
16	Poly(phenethyl-methacrylate)	329	328	−0.30	315	−4.26	331	0.61
17	Poly(benzyl methacrylate)	327	338	3.36	334	2.14	333	1.83
18	Poly(isobutyl methacrylate)	326	335	2.76	328	0.61	324	−0.61
19	Poly(2-bromoehtyl-methacrylate)	325	329	1.23	322	−0.92	331	1.85
20	Poly(3,3-dimethylbutyl-methacrylate)	318	322	1.26	317a)	−0.31	316a)	−0.63
21	Poly(neopentyl methacrylate)	312	319	2.24	308	−1.28	308	−1.28
22	Poly(propyl methacrylate)	308	327	6.17	309	0.32	309	0.32
23	Poly(2-methoxyethyl-methacrylate)	297	311	4.71	300	1.01	298	0.34
24	Poly(butyl methacrylate)	293	307	4.78	290a)	−1.02	291a)	−0.68
25	Poly(hexdecyl methacrylate)	288	272	−5.56	280	−2.78	283	−1.74
26	Poly(pentyl methacrylate)	283	286	1.06	280	−1.06	278	−1.77
27	Poly(Hexyl methacrylate)	273	269	−1.47	276	1.10	271	−0.73
28	Poly(2-ethylhexyl-methacrylate)	263	263	0.00	272	3.42	266	1.14
29	Poly(octyl methacrylate)	253	232	−8.30	245	−3.16	249	−1.58
30	Poly(Tetradecyl-methacrylate)	233	246	5.58	240	3.00	236	1.29
31	Poly(dodecyl methacrylate)	208	221	6.25	211	1.44	211	1.44
32	Poly(decyl-methacrylate)	203	194	−4.43	210a)	3.45	205a)	0.99

Table 3. Parameters, experimental and calculated T_gs by SVR model for five independent polymethacrylates.

No.	Polymer	|L–1.356| [nm]	Etotal [au]	qC6 [au]	q^− [au]	α [au]	Etherm [kJ · mol^−1]	Exp. Tg [K]	Cal. Tg [K]
1	Poly(2-ethylbutyl methacrylate)	0.81	−543.59	−0.0395	−0.4834	111.64	802.69	284	294
2	Poly(cyclopentyl methacrylate)	0.89	−503.07	0.1413	−0.4851	97.1	665.9	348	338
3	Poly(cyclobutyl methcarylate)	0.94	−463.72	0.1543	−0.4836	87.106	584.07	351	349
4	Poly(3,3,5-trimethylhexyl methacrylate)	0.53	−661.52	−0.0562	−0.4854	143.69	1036.37	274	262
5	Poly(cyclodecyl methacrylate)	0.75	−699.61	0.1367	−0.4872	149.68	1061.9	331	331

Table 4. Prediction performance comparison among different regression methods.

Sample class	Method	RMSE [K]	MAE [K]	MAPE [%]	R^2
Training samples	MLR	12.01	9.28	3.01	0.9468
	ANN	8.35	6.76	2.07	0.9776
	SVR	3.79	3.43	1.09	0.9948
Test samples	MLR	8.94	7.42	2.50	0.9748
	ANN	6.58	5.00	1.62	0.9958
	SVR	4.04	3.28	1.01	0.9946
Independent samples	SVR	8.34	6.8	2.26	0.9417

Figure 2. Plot of experimental T_g versus calculated T_g by SVR.

From Table 2, it can be viewed that the percentage errors of most samples calculated by SVR model are smaller than those calculated by the MLR model and ANN model. It can be found by simple statistics that, there are 15 samples with a small absolute percentage error (APE) in the range of 0–1% achieved by SVR model, while there are only eight and nine samples whose APEs are in this range calculated by the MLR and ANN models, respectively. From Table 2, it also can be seen that the maximum APE 1.98% predicted by SVR for sample #11, is smaller than those (8.30%, #29) achieved by MLR model and (5.21%, #7) calculated by ANN model.

The maximum APE (|(262-274)/274 × 100%| = 4.38%, #4) calculated by the established SVR model in Table 3 is also smaller than that (5.21%) achieved by the ANN model. By comparison the calculated T_g values with the measured T_g indices among MLR, ANN and SVR in Table 2, it reflects that the T_g values predicted by SVR model are in quite better agreement with the experimental values than those calculated by the MLR model or ANN model.

From Table 4, it can be found that the MAPE of the SVR model for training samples, test samples were 1.09% and 1.01%, which were smaller than those (3.01%, 2.5%) achieved by MLR model and (2.07%, 1.62%) by ANN model, respectively. The RMSEs calculated by MLR and ANN models for both training set and test set were >5 K, while the RMSE predicted by the SVR model for the training set and test set were 3.79 and 4.04 K, which were both lower than those of MLR and ANN models. These mean that the prediction result of the SVR model is more accurate than both of the MLR and ANN models. Meanwhile, the MAE of T_g calculated by the established SVR model for the independent set was 6.8 K. It has been reported that there exists measurement error of about 2–4 degrees,33, 34 or even over 10 degrees35 measured by different experimental techniques. These imply that the established SVR model is acceptable in practice to satisfactorily predict the T_g of an unknown polymethacrylate with similar structure outside of the training set.

The correlation coefficient R² between the measured T_gs and SVR regressed T_gs for the training set, test set were 0.9948, 0.9946, both were greater than those (0.9468, 0.9748) and (0.9776, 0.9958) for MLR model and the ANN model. Again illustrate that the regressed T_gs for the training samples or the predicted T_gs for the test samples by SVR model are in more agreement with the measured T_gs than those calculated by the MLR model or the ANN model. From Figure 2, it also can be seen that there is strong agreement among the training results, the test results and the evaluation results, indicating no over-fitting issues with SVR. These results strongly demonstrate that the SVR model is more suitable for accurately making T_g prediction.

Moreover, in order to further convince the statistical results of SVR model, the whole 37 samples used in this study is also partitioned into the training, test and independent sets in the ratio 50%, 25%, and 25% by using k-Nearest Neighbor (kNN). Therefore, in this typical split the training, test and independent sets are comprised of 19, 9, and 9 samples, respectively. For this partition, the modeling and prediction results of SVR are tabulated in Table 5, and the related statistical RMSE, MAE, MAPE, and R² are listed in Table 6. It can be viewed from Table 6 that the MAEs of T_g calculated by the SVR model for the test set and independent set are 7.44 and 8.0 K, respectively, which is also acceptable in practice. From Tables 4 and 6, a phenomenon can be found by comparison the prediction results for the test and independent samples that, in general, the more the number of training samples is, the greater the prediction and generalization ability of SVR would be.

Table 5. The 19 training samples, 9 test sample and 9 independent samples along with their measured T_gs and calculated T_gs by SVR.

No.	Polymer	Exp. Tg [K]	Cal. Tg [K]
a) Test samples; b) Independent samples.
1	Poly(4-cyanopheny1-methacylate)	428	428
2	Poly(phenyl methacrylate)	407	407
3	Poly(tert-butyl methacrylate)	380	380
4	Poly(4-methoxycarbonyphenyl-methacrylate)	379	379
5a)	Poly(methyl methacrylate)	373	363
6	Poly(4-tertbutylphenyl methacrylate)	371	371
7b)	Poly(2-chloroehtyl-methacrylate)	365	334
8a)	Poly(2-hydroxyehtyl-methacrylate)	358	366
9	Poly(cycHexyl-methacrylate)	356	356
10b)	Poly(1,1,1-trifluoro-2-propyl-methacrylate)	354	350
11	Poly(isopropyl methacrylate)	354	354
12	Poly(2-hydroxypropyl-methacrylate)	349	349
13b)	Poly(2-cyanoethyl-methacrylate)	347	337
14	Poly(ethyl methacrylate)	338	338
15	Poly(sec-butyl methacrylate)	333	343
16	Poly(phenethyl-methacrylate)	329	329
17	Poly(benzyl methacrylate)	327	327
18a)	Poly(isobutyl methacrylate)	326	328
19a)	Poly(2-bromoehtyl-methacrylate)	325	350
20	Poly(3,3-dimethylbutyl-methacrylate)	318	318
21b)	Poly(neopenyl methacrylate)	312	310
22	Poly(proyl methacrylate)	308	313
23	Poly(2-methoxyethyl-methacrylate)	297	297
24	Poly(butyl methacrylate)	293	293
25b)	Poly(hexdecyl methacrylate)	288	285
26b)	Poly(pentyl methacrylate)	283	278
27a)	Poly(Hexyl methacrylate)	273	272
28	Poly(2-ethylhexyl-methacrylate)	263	263
29	Poly(octyl methacrylate)	253	252
30a)	Poly(Tetradecyl-methacrylate)	233	233
31a)	Poly(dodecyl methacrylate)	208	204
32	Poly(decyl-methacrylate)	203	203
33b)	Poly(2-ethylbutyl methacrylate)	284	294
34b)	Poly(cyclopentyl methacrylate)	348	341
35a)	Poly(cyclobutyl methcarylate)	351	353
36a)	Poly(3,3,5-trimethylhexyl methacrylate)	274	259
37b)	Poly(cyclodecyl methacrylate)	331	331

Table 6. Prediction performance of SVR for 19 training samples, 9 test samples and 9 independent samples

Sample class	RMSE [K]	MAE [K]	MAPE [%]	R^2
Training samples	2.58	0.84	0.26	0.9978
Test samples	10.74	7.44	2.40	0.9681
Independent samples	11.85	8.0	2.39	0.9034

4. Conclusion

In this paper, the SVR combined with PSO was proposed to estimate the glass transition temperature of polymethacrylates based on six quantum chemical descriptors (|L-1.356|, E_total, qC6, α, q⁻, and E_therm). The modeling performance of the proposed hybrid PSO-SVR method was compared with those of MLR and ANN methods, the comparison results revealed that the calculated errors of the SVR model were smaller than those of MLR model or ANN model for the identical training set and test set. The generalization ability of the SVR model was also superior to those of the MLR and ANN models. The evaluation results via independent samples confirmed that the established SVR model is acceptable in practice. This investigation demonstrates that the methodology introduced in this study can successfully rationalize the T_g of polymethacrylates, and the derived SVR model is more suitable to be used for prediction of the T_g values for unknown polymethacrylates possessing similar structure than the conventional multivariable linear regression model or ANN.

Acknowledgements

This work was supported by the Fundamental Research Funds for the Central Universities (CDJXS10101107), the Innovative Talent Funds for Project 985 (WLYJSBJRCTD201102), the Scientific Research Foundation for the Returned Overseas Chinese Scholars of Ministry of Education, China (2008101-1), the Natural Science Foundation of Chongqing, China (CSTC2006BB5240) and the Program for New Century Excellent Talents in Universities of China (NCET-07-0903).