A quantitative relationship between T_gs and chain segment structures of polystyrenes

Abstract

The glass transition temperature (T_g) is a fundamental characteristic of an amorphous polymer. A quantitative structure-property relationship (QSPR) based on error back-propagation artificial neural network (ANN) was constructed to predict T_gs of 107 polystyrenes. Stepwise multiple linear regression (MLR) analysis was adopted to select an optimal subset of molecular descriptors. The chain segments (or motion units) of polymer backbones with 20 carbons in length (10 repeating units) were used to calculate these molecular descriptors reflecting polymer structures. The relative optimal conditions of ANN were obtained by adjusting various network paramters by trial-and-error. Compared to the model already published in the literature, the optimal ANN model with [4-7-1] network structure in this paper is accurate and acceptable, although our model has more samples in the test set. The results demonstrate the feasibility and powerful ability of the chain segment structures as representative of polymers for developing T_g models of polystyrenes.

Keywords:
chain segments; glass transition temperature; polystyrenes; structure-property relationship

1. Introduction

The glass transition temperature (T_g) is known as the glass temperature or the transition temperature between glass and rubber states of amorphous materials. T_g is a fundamental characteristic and is taken as the most crucial property of amorphous polymeric materials^[1]. The nature of the theory in the glass and glass transition is unsolved, however, is taken as the deepest and most interesting problem in solid stated theory. Though T_g can be determined experimentally, the discrepancies in reported T_g values in the literature may be quite large, because (1) the transition happens over a comparatively wide temperature range, and (2) many factors affect T_g values, which include the structural, constitutional and conformational features of polymers, molecular weight, and experimental conditions such as the measuring method, duration of the experiment, and pressure during the measurement^[2]. In addition, experimental determination of T_gs cannot apply to those polymers that are not yet synthesized. Hence, it is necessary to develop theoretical methods for the prediction of T_gs.

Quantitative structure-property relationship (QSPR) models can be used to predict T_g values of polymers. This approach is based on the assumption that the variation of physicochemical properties of the compounds is dependent on changes of molecular structure, which can be characterized with descriptors. A major goal of QSPR approach is to develop a mathematical relationship between the property of interest and structural features^[3].

Some researchers have predicted T_gs of polymers with QSPR models. Van Krevelen^[4] predicted T_gs by using the group additive property theory. This method is only applicable to polymers whose contribution values are known. Bicerano^[2] developed a more universally QSPR model with R² (the square of the correlation coefficient R) being 0.95 and standard error (s) being 24.65 K for a data set of 320 polymers. The T_g model was based on the solubility parameter and the weighted sum of 13 topological bond connectivity parameters of the monomer structures. But the model is not validated with the test set. Joyce et al.^[5] built models for T_g prediction based on the monomer structures of 360 polymers. The model predicted the T_g values for a test set of polymers with a root mean square (rms) error of 35 K. Katritzky et al.^[6] introduced a four-parameter model with R² 0.928 for 21 medium molecular weight polymers and copolymers based on their repeat units. On a larger data set, Katritzky et al.^[7] developed a QSPR for the molar glass transition temperature (T_g/M) of 88 uncross-linked linear homopolymers. The model has five molecular descriptors and the s for T_g is 32.9 K. On the same data, Cao and Lin^[8] developed a QSPR model (R² = 0.9056) by using five molecular descriptors that focus on the influence of chain stiffness and intermolecular forces. Yu et al.^[9] developed stepwise multiple linear regression (MLR) for 107 polystyrenes and generated a QSPR model (R = 0.959 and s = 15.20 K) from the training set of 96 polystyrenes. The MLR model produced a rms error of 20.5 K for the test set comprising 11 polystyrenes. Recently, some quantum chemical descriptors calculated from repeating units or monomers were used to develop QSPR models for T_gs of polymers^[10-12].

Due to the large and variable size of polymer molecules, the QSPR models stated above, together with QSPR models of other polymer properties, are modeled by extrapolation from monomer structures or repeating units^[1]. These methods fail to account for the influences from neighboring repeating units. Especially for the T_g, the glass transition is resulted from Brownian motion of chain segments subjected to freezing or thawing. In this work, the chain segments (localized units or motion units) with 20 carbons (10 repeated units) in length were used to calculate descriptors for their corresponding polystyrenes and to develop QSPR models for their T_gs.

2. Materials and Methods

2.1 Data set

Table 1 shows the experimental T_g data for 107 polystyrenes, which are taken from Brandrup et al.^[13]. The entire set contains a T_g value range of 208-490 K. The pendant groups presented in the benzene ring include halides, carbonyls, ethers, hydrocarbon chains, hydroxyl, hydroxyimino, aromatic rings, and other functional groups. These polystyrenes were randomly divided into a training set (70 polystyrenes) and a test set (37 polystyrenes). The training set was used to build a QSPR model, and the test set was adopted to evaluate the model.

Table 1
Molecular descriptors and T_g data of 107 polystyrenes.

No.	Polystyrene	ChiA_B(e)	SpMax_EA(bo)	H7s	DLS_01	Tg(K)a	Tg(K)b
	Training set
1	poly(4-dodecylstyrene)	0.288	3.734	8.818	0.50	221	234
2	poly(4-(octyloxymethyl)styrene)	0.286	3.734	8.417	0.50	231	239
3	poly[4-(1-ethlhexyloxymethyl)styrene]	0.282	3.768	9.048	0.50	250	283
4	poly[4-(4-hydroxybutoxymethyl)styrene]	0.281	3.734	10.155	0.00	293	305
5	poly(2-hexyloxycarbonylstyrene)	0.277	4.010	10.629	0.25	318	335
6	poly(5-bromo-2-butoxystyrene)	0.275	3.982	8.121	0.50	320	335
7	poly(2-pentyloxymethlstyrene)	0.283	3.903	6.016	0.50	320	335
8	poly(4-octanoylstyrene)	0.278	3.820	9.644	0.25	323	329
9	poly[4-(1-hydroxy-3-piperidinopropyl)styrene]	0.275	3.765	10.082	0.00	327	356
10	poly(4-butyrylstyrene)	0.270	3.820	8.314	0.50	347	348
11	poly(2-methoxystyrene)	0.271	3.901	5.874	0.50	348	371
12	poly(4-methoxymethylstyrene)	0.274	3.734	6.070	0.50	350	334
13	poly(5-bromo-2-methoxystyrene)	0.267	3.980	6.733	0.50	359	376
14	poly(2-methoxymethylstyrene)	0.274	3.903	6.404	0.50	362	356
15	poly(4-propoxycarbonylstyrene)	0.271	3.820	9.101	0.25	365	364
16	poly(4-ethoxycarbonylstyrene)	0.268	3.820	8.878	0.25	367	375
17	poly(4-isopropoxycarbonylstyrene)	0.265	3.822	9.488	0.25	368	373
18	poly(4-phenoxystyrene)	0.261	3.752	9.224	0.50	373	363
19	poly(4-diethylcarbamoylstyrene)	0.270	3.841	8.503	0.25	375	376
20	poly(2-ethylstyrene)	0.273	3.901	5.356	0.50	376	372
21	poly(3, 5-dimethylstyrene)	0.265	3.879	5.377	0.50	377	398
22	poly(2, 5- dichlorostyrene)	0.262	3.963	6.754	0.50	379	395
23	poly(4-methylstyrene)	0.268	3.720	5.042	0.50	382	392
24	poly(3, 4-dimethylstyrene)	0.265	3.919	5.207	0.50	384	402
25	poly(4-[(1-hydroxyimino)-2phenethyl]styrene)	0.260	3.857	11.994	0.00	384	398
26	poly(4-methoxycarbonylstyrene)	0.265	3.820	7.214	0.25	386	414
27	poly(4-acetylstyrene)	0.263	3.812	5.501	0.50	389	403
28	poly(2-ethoxycarbonylstyrene)	0.268	4.010	8.940	0.25	391	379
29	poly(4-cyanostyrene)	0.262	3.767	6.441	0.50	393	394
30	poly(3-hydroxymethystyrene)	0.271	3.767	6.319	0.25	398	400
31	poly(2, 4-dichlorostyrene)	0.262	3.958	7.169	0.50	406	388
32	poly(2,4,5-trimethylstyrene)	0.263	4.085	6.131	0.50	409	421
33	Poly[4-(bis(trimethylstanny)methyl)styrene]	0.261	3.867	5.232	0.50	413	414
34	poly(2, 5-dimethylstyrene)	0.265	3.963	5.744	0.50	416	398
35	poly(4-tert-butylstyrene)	0.263	3.871	4.717	0.50	422	415
36	poly(2,4,6-trimethylstyrene)	0.263	4.135	6.020	0.50	435	432
37	poly(2-carboxystyrene)	0.261	4.003	7.687	0.00	450	466
38	Poly(4-benzoylstyrene)	0.257	3.934	8.698	0.50	371	385
39	Poly(4-phenylacetylstyrene)	0.259	3.829	10.145	0.50	351	357
40	Poly(2-phenylaminocarbonylstyrene)	0.258	4.016	8.500	0.00	464	465
41	Poly(4-phenylstyrene)	0.260	3.914	6.531	0.50	434	401
42	Poly(4-piperidinocarbonylstyrene)	0.269	3.841	8.483	0.25	387	379
43	Poly[4-(3-piperidinopropionyl)styrene]	0.272	3.821	11.064	0.25	311	336
44	Poly(4-propoxysulfonylstyrene)	0.261	4.697	9.650	0.25	490	484
45	Poly(4-p-toluoylstyrene)	0.255	3.951	10.707	0.50	372	370
46	Poly{3-[bis(trimethylsiloxy)boryl]styrene}	0.264	3.811	15.282	0.25	308	314
47	Poly (4- [bis(trimethylstannyl)methyl] styrene)	0.261	3.867	5.136	0.50	413	415
48	poly(4-hexylstyrene)	0.283	3.734	7.830	0.50	246	252
49	poly[4-(1-hydroxy-1-methylbutyl)styrene]	0.269	3.885	6.338	0.25	403	416
50	poly[4-(1-hydroxy-1-methylhexyl)styrene]	0.273	3.885	10.191	0.25	364	344
51	poly[4-(1-hydroxy-1-methylpropyl)styrene]	0.266	3.884	5.442	0.25	459	441
52	poly(4-ethylstyrene)	0.273	3.733	4.802	0.50	350	355
53	poly(4-nonylstyrene)	0.287	3.734	8.706	0.50	220	236
54	poly(4-tetradecylstyrene)	0.289	3.734	9.289	0.50	237	232
55	poly[4-(2-hydroxybutoxymethyl)styrene]	0.276	3.734	11.064	0.00	319	309
56	poly(5-bromo-2-pentyoxystyrene)	0.277	3.982	7.429	0.50	322	337
57	poly(4-isopentyloxystyrene)	0.275	3.734	8.632	0.50	330	316
58	poly(2-butoxymethylstyrene)	0.281	3.903	6.267	0.50	340	339
59	poly(4-valerylstyrene)	0.273	3.820	8.735	0.50	343	337
60	poly(4-butoxycarbonylstyrene)	0.273	3.820	10.176	0.25	349	344
61	poly(4-methoxy-2-methylstyrene)	0.268	3.965	6.787	0.50	358	371
62	poly(2-isopropoxymethylstyrene)	0.273	3.903	7.998	0.50	361	341
63	poly(4-isobutoxycarbonylstyrene)	0.267	3.821	9.820	0.25	363	363
64	poly(4-fluorostyrene)	0.265	3.720	7.357	0.50	368	376
65	poly(styrene)	0.271	3.622	4.565	0.50	364	356
66	poly(4-propionylstyrene)	0.267	3.820	6.429	0.50	375	377
67	poly(2,3,4,5,6,-pentafluorostyrene)	0.249	4.413	17.543	0.50	378	377
68	poly(4-chlorostyrene)	0.266	3.720	5.821	0.50	383	392
69	poly(2, 4-dimethylstyrene)	0.265	3.958	5.417	0.50	385	402
70	poly(4-bromostyrene)	0.267	3.720	5.249	0.50	391	395
	Test set
71	poly(4-chloro-3-fluorostyrene)	0.261	3.919	7.155	0.50	395	388
72	poly(2-isobutoxycarbonylstyrene)	0.268	4.010	7.650	0.25	400	401
73	poly(4-hydroxymethystyrene)	0.271	3.733	5.854	0.25	413	394
74	poly[4-(1-hydroxy-1-methylethyl)styrene]	0.261	3.871	5.408	0.25	438	454
75	poly(4-hexyloxymethystyrene)	0.284	3.734	8.839	0.50	253	249
76	poly(4-propoxymethylstyrene)	0.279	3.734	8.429	0.50	295	284
77	Poly[4-(sec-butoxymethyl)styrene]	0.276	3.734	8.869	0.50	313	309
78	poly(5-bromo-2-propoxystyrene)	0.273	3.982	7.680	0.50	327	345
79	poly(2-butoxycarbonylstyrene)	0.273	4.010	10.353	0.25	339	345
80	poly[2-(2-dimethylaminoethoxycarbonyl)styrene]	0.269	4.010	10.534	0.25	342	352
81	poly(2-ethoxymethylstyrene)	0.277	3.903	7.002	0.50	347	342
82	poly(2-isopentyloxymethylstyrene)	0.277	3.903	6.615	0.50	351	346
83	poly(4-sec-butylstyrene)	0.273	3.764	5.284	0.50	359	365
84	poly(4-methoxystyrene)	0.270	3.733	5.556	0.50	362	374
85	poly(2-pentyloxycarbonylstyrene)	0.275	4.010	9.383	0.25	365	354
86	poly(3-methylstyrene)	0.268	3.752	5.205	0.50	370	394
87	poly(2,5-difluorostyrene)	0.260	3.963	9.602	0.50	374	365
88	poly(2-propoxycarbonylstyrene)	0.271	4.010	8.667	0.25	381	374
89	poly(2-fluoro-5-methylstyrene)	0.263	3.963	8.028	0.50	384	373
90	poly(4-chloro-3-methylstyrene)	0.264	3.919	6.218	0.50	387	390
91	poly(2-chlorostyrene)	0.267	3.880	5.996	0.50	392	382
92	poly(4-dimethylaminocarbonylstyrene)	0.263	3.837	6.944	0.25	398	423
93	poly(2-methoxycarbonylstyrene)	0.265	4.010	7.907	0.25	403	408
94	poly(2-methylstyrene)	0.268	3.880	5.080	0.50	409	392
95	poly(4-hydroxystyrene)	0.266	3.720	6.714	0.25	433	416
96	poly(4-decylstyrene)	0.287	3.734	8.641	0.50	208	236
97	poly(2-isopentyloxycarbonylstyrene)	0.270	4.010	10.984	0.25	341	344
98	poly(5-bromo-2-ethoxystyrene)	0.270	3.982	7.632	0.50	353	355
99	poly(2-hydroxymethystyrene)	0.271	3.901	6.795	0.25	433	402
100	poly(2-octyloxystyrene)	0.285	3.903	6.698	0.50	286	322
101	poly(4-octylstyrene)	0.286	3.734	8.727	0.50	228	239
102	poly(3, 4-dichlorostyrene)	0.262	3.919	6.395	0.50	401	395
103	poly(4-hexanoylstyrene)	0.275	3.820	10.423	0.50	339	321
104	poly(2-phenoxycarbonylstyrene)	0.258	4.016	8.596	0.25	397	427
105	poly(2-methaminocarbonylstyrene)	0.266	4.010	7.803	0.00	462	456
106	poly(4-p-anisoylstyrene)	0.257	3.953	10.577	0.25	376	387
107	Poly(2-phenethyloxymethylstyrene)	0.267	3.903	6.696	0.50	336	372

• ^a
  T_g data were taken from Brandrup et al.^[13^];
• ^b
  T_g data were calculated with the ANN model.

2.2 Descriptor computation

A polymeric material consists of a mixture of giant molecules. Therefore, it is impossible to calculate descriptors directly from molecular structures of the polymeric material. Two approaches have been adopted to resolve this problem. One is using the repeating unit to calculate descriptors for the corresponding polymer. The other is using the monomer as representative of the corresponding polymer^[1].

T_g is a temperature point used to express transition region, where polymer chain segments can move from frozen to movement (or vice versa). Below the glass transition region, 1-4 chain atoms are involved in motion. Further, these motions are largely restricted to vibrations and short-range rotational motions. During the glass transition region, 10-50 chain atoms attain sufficient thermal energy to move in a coordinated manner. In T_g region, these chain atoms (motion units) are first mobilized before the whole molecule starts moving. On further heating, the increased energy allotted to the chains permits them reptate out through entanglements rapidly and flow as individual molecules^[14,15].

The structures of polymer chain segment have an effect on its glass transition and are correlated to T_gs. According to above theory of glass transition, descriptors calculated from the chain segments are more accurate in describing structures affecting polymer T_gs than that from repeating units and monomers. From a theoretical point of view, the chain segment used to calculate descriptors is longer, the descriptors are more accurate in characterizing polymers. The motion units related to glass transition of polymers usually contain 10-50 carbons in length. In addition, a too long segment taken into account may cause difficulty in calculating descriptors, and a too short segment cannot sufficiently represent the structure of motion units. Thus, chain segments with 20 carbons in chain length were used to calculate molecular descriptors for the corresponding polymers.

Polymeric chain segments containing 20 main chain carbons of polystyrenes were first sketched using ChemBioDraw Ultra 11.0 in ChemBioOffice 2008 program. For example, the structure model consisting of 10 repeating units end-capped by two hydrogens (see Figure 1) was adopted as the representative structure of poly(styrene) (No. 65 in Table 1) to calculate the descriptors.

Figure 1
The calculated models of poly(styrene).

Subsequently, the sketched 2D molecular structures were converted to 3D structures and optimized using a molecular mechanics (MM2 force field) in ChemBio3D Ultra 11.0 with the convergence criterion of minimum rms of gradient value being 0.01 kcal/molÅ. The optimized molecules were saved in Sybyl mol2 (.mol2) format as the input files for Dragon software^[16]. Lastly, 4885 descriptors were calculated for each energy-minimized motion unit with Dragon software. Descriptors with constant or near constant values and with pair correlation greater than or equal to 0.90 were removed in order to reduce redundant and non-useful information. After excluding redundant and non-useful variables, 551 descriptors were remained to undergo descriptor selection. A relative optimal subset of descriptors was obtained by applying MLR analysis in IBM SPSS Statistics 19.

2.3 Artificial neural network

The optimal descriptors subset was fed to artificial neural network (ANN) as input vectors. ANNs are computational models, which simulate the human brain behavior. The common networks consist of an input layer, some number of hidden layers (intermediate layers) and an output layer. Each layer includes a number of processing nodes, called neurons or units. Each node in the network is influenced by those nodes to which it is connected in a highly complex and parallel way. The degree of influence is dictated by the values of the links or connections. Through a training algorithm, the overall behavior of ANNs can be modified by adjusting the weights (or the values of the links or connections). After learning from the input dataset, ANNs acquire knowledge and can be applied on test set data not present in the training set. The output layer produces the prediction values of properties interested. One of the most popular algorithms applied in the training phase is the error back-propagation (BP) algorithm. The number of neurons in the hidden layer shouled be optimized by trial and validation until no obvious improvement was seen for that model^[17].

3. Results and Discussions

By analyzing the correlation between the 551 descriptors and T_gs of 70 polystyrenes in the training set with stepwise MLR analysis in IBM SPSS Statistics 19, Equation 1 and the corresponding statistical results were obtained.

$$T_g=1346.200-4546.871ChiA\_B(e)+100.47SpMax\_EA(bo)-12.653H7s-122.231DLS\_01$$ (1)

$$n=\mathrm{70,}R=\mathrm{0.955,}{R}^2=\mathrm{0.912,}s=\mathrm{16.717,}F=167.652$$

where n is the number of samples from the training set; s is the standard error of estimate; R is the correlation coefficient; F is the Fischer ratio.

The four molecular descriptors, ChiA_B(e), SpMax_EA(bo), H7s and DLS_01 appearing in above MLR model and the corresponding descriptor values are shown in Table 1. Their descriptor characteristics are listed in Table 2; and their definitions^[18] are shown in Table 3. Calculated results with Equation 1 are depicted in Figure 2A. The rms errors of T_gs of the training and test sets are 16.1 and 22.4 K, respectively.

Table 2
Characteristics of descriptors appearing in MLR model.

Descriptors	Unstandardized Coefficients	Std. Error	Standardized Coefficients	t	Sig.	VIF
Constant	1346.200	110.153		12.221
ChiA_B(e)	-4546.871	273.430	-0.685	-16.629	0.000	1.249
SpMax_EA(bo)	100.047	14.650	0.288	6.829	0.000	1.309
H7s	-12.653	0.959	-0.556	-13.200	0.000	1.307
DLS_01	-122.231	13.833	-0.364	-8.836	0.000	1.247

Table 3
The symbol, class and definition for descriptors appearing in MLR models.

Symbol	Class	Definition
ChiA_B(e)	2D matrix-based descriptors	Average randic-like index from burden matrix weighted by Sanderson electronegativity
SpMax_EA(bo)	Edge adjacency indices	Leading eigenvalue from edge adjacency matrix weighted by bond order
H7s	GETAWAY descriptors	H autocorrelation of lag 7 / weighted by I-state
DLS_01	Drug-like indices	Modified drug-like score from Lipinski (4 rules)

Figure 2
Plots of calculated vs. experimental T_g values of polystyrenes: (A) for MLR model; (B) for ANN model.

The four descriptors are then fed to ANN as input vectors. The optimal condition of the neural network was obtained by adjusting various parameters by trial-and-error. The architecture of the final optimum BP neural network is [4-7-1], with the number of hidden layer being 1, the nodes in hidden layer being 7, the permission error being 0.00001, the momentum being 0.6, and the sigmoid parameter being 0.9. The results from ANN method are listed in Table 1 and depicted in Figure 2B, which indicate that the predicted T_g values are close to the experimental ones. The rms error of training set is 13.6 K (R = 0.939). The test set rms error is 17.1 K (R = 0.902) which is less than the errors from the test set in previous model (20.5 K)^[9]. The mean relative error for the 107 polystyrenes in Table 1 is 3.4%, less than that from the model of Yu et al.^[9] (3.7%). Furthermore, it should be noted that the test set in this paper possesses 37 polystyrenes, more than the number of samples (11 polystyrenes)^[9]. And it is much easier to obtain better results on small test set of polymers. In comparison to previous model on T_gs of polystyrenes^[9], the statistic qualities of our model is accurate and acceptable. Therefore, it is feasible to calculate molecular descriptors from the chain segments of polymer backbones comprising 10 repeating units for developing T_g model of polystyrenes.

Table 2 shows that each descriptor in Equation 1 has a Sig.-value near to 0, and less than the default level of 0.05, which suggest that these descriptors are significant for T_gs. Moreover, all variance inflation factor (VIF) values are less than 2, far less than the default value of 10. Thus these descriptors are “pure” without “mixing” or contamination from other descriptors, and each descriptor reflects some particular molecular structures affecting T_gs.

According to the t-test, the most significant descriptor in the MLR model is ChiA_B(e) (2D matrix-based descriptors)^[16]. ChiA_B(e) denotes the average randic-like index from burden matrix weighted by Sanderson electronegativity and is defined as follow:

$$ChiA\_B(e)=\frac{Chi\_M(e)}{nBO}$$ (2)

Where nBO is the number of graph edges. Chi_M(e) is the Randic-like index calculated by applying Sanderson electronegativity as the vertex weighting scheme and a H-depleted molecular graph as a square matrix:

$$Chi\_M(e)={{\displaystyle \sum _{i=1}^{nSK-1}{\displaystyle \sum _{j=i+1}^{nSK}{\alpha }_{ij}\cdot \left[V{S}_i(M;e)\cdot V{S}_j(M;e)\right]}}}^{-1/2}$$ (3)

Here nSK means the number of graph vertices; VS_i(M) is the ith matrix row sum; α_ij are the elements of the adjacency matrix, which are equal to one for pairs of adjacent vertices, and zero otherwise. ChiA_B(e) reflects information about interatomic distances, bond distances, ring types, planar and non-planar systems and atom types^[16]. A small ChiA_B(e) indicates a small interatomic distances, which results in a low degree of freedom for rotation and leads to high T_g.

The second significant descriptor is the GETAWAY (GEometry, Topology, and Atom-Weights AssemblY) descriptor, H7s (H autocorrelation of lag 7 / weighted by I-state). The descriptor H7s encodes information on structural fragments, such as the effective position of substituents and fragments in the molecular space, and accounts information on molecular size and shape as well as for specific atomic properties^[16]. A large H7s suggests that a polymer has a large side group, which decreases the volume ratio of phenyl ring to other substituent groups. While the aromatic or cyclic structure in bulky side groups increases rotational barrier for backbone chain and leads to high T_g. Therefore, a polymer with large H7s may have a low T_g.

The next significant descriptor is the Drug-like indice DLS_01. The descriptor DLS_01, being modified drug-like score from Lipinski (4 rules), is calculated as 1 minus Lipinski Alert Index (LAI), while LAI is defined as the ratio between the number of satisfied conditions over the total number of conditions, i.e., (1) there are more than 5 H-bond donors; (2) there are more than 10 H-bond acceptors (N and O atoms); (3) molecular weight (MW) is over 500; and (4) Moriguchi's logP (MLogP) is over 4.15^[16]. DLS_01 is related to the number of intermolecular hydrogen bonds, which increase intermolecular force and determine the magnitude of molecular aggregates. Polymer molecules with small DLS_01 hold together more strongly due to intermolecular hydrogen bonds and are unable to mover that easily, and possess high T_gs.

According to the t-test, the last significant descriptor in the MLR model is SpMax_EA(bo). Edge adjacency index, SpMax_EA(bo), is derived from the H-depleted molecular graph and encodes the connectivity between graph edges. It is leading eigenvalue from edge adjacency matrix weighted by bond order. SpMax_EA(bo) reflects molecular shape and implies the substituent position in the phenyl ring for styrenes^[16]. Compared to styrenes with substituents lying in p-or m-positions of the phenyl ring, a styrene with a substituent lying in o-positions usually has a larger SpMax_EA(bo), which can be seen from Table 1. The substituents in o-positions will enhance rotational barrier for backbone chain, increase rigidity of polymer chains and result in higher T_gs^[9].

Despite a variety of factors affecting the T_g values of polymeric materials, intermolecular forces and molecular flexibility (or rigidity) are two important factors related to T_gs. The descriptor DLS_01 reflects the intermolecular forces, while descriptors ChiA_B(e), SpMax_EA(bo) and H7s indicate the stiffness of polymer. Therefore, the four descriptors can predict T_gs sufficiently.

Figure 3 (Williams plot) was obtained to visualize the applicability domain of the ANN model in this paper. According to Williams plot based on standardized residuals vs. leverages, predictions for only those samples that fall into this domain may be considered reliable^[19,20]. Figure 3 shows that only the two samples No. 44, poly(4-propoxysulfonylstyrene) and No. 67, poly(2,3,4,5,6,-pentafluorostyrene) in the training set have larger leverage h values (0.448 and 0.456, respectively), greater than the warning leverage h* (= 0.214). But their standardized residual values (0.430 and 0.075, respectively) are less than 3. Thus the two samples, poly(4-propoxysulfonylstyrene) and poly(2,3,4,5,6,-pentafluorostyrene), can stabilize the ANN model of polystyrenes and make it more accurate.

Figure 3
Williams plot for polystyrenes with a warning leverage of 0.214.

4. Conclusions

Four molecular descriptors calculated from the chain segments of main chains comprising 10 repeating units were adopted for developing QSPR model of T_gs for polystyrenes. MLR analysis was used to select the optimal subset of descriptors after molecular descriptor generation for each chain segment. The developed ANN model was proved to be accurate and acceptable, with the absolute mean errors for the whole data set is 3.4%, which is less than that of the model published in the literature, although our model possesses more samples for the test set. Therefore, it is feasible calculating molecular descriptors from the chain segments comprising 10 repeating units in length to develop ANN model of T_gs for polystyrenes.

5. Acknowledgements

The project was supported by the National Natural Science Foundation of China (No. 21472040) and Scientific Research Fund of the Hunan Provincial Education Department (No. 16A047).

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Publication Dates

• Publication in this collection
  Jan-Mar 2017

History

• Received
  20 Feb 2016
• Reviewed
  02 May 2016
• Accepted
  24 May 2016