General Correlation for the Prediction of Theta (Lower Critical Solution

Jan 6, 2005 - General Correlation for the Prediction of Theta (Lower Critical. Solution Temperature) in Polymer Solutions. Hongwei Liu and Chongli Zho...
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Ind. Eng. Chem. Res. 2005, 44, 634-638

General Correlation for the Prediction of Theta (Lower Critical Solution Temperature) in Polymer Solutions Hongwei Liu and Chongli Zhong* Department of Chemical Engineering, Key Laboratory of Bioprocess of Beijing, Beijing University of Chemical Technology, Beijing 100029, China

In this work, a correlation for the estimation of the θ temperature (lower critical solution temperature, LCST) in polymer solutions is proposed in which both polymer and solvent molecules are characterized by the molecular connectivity index. A database of 169 data points, including 12 polymers and 69 solvents, was adopted to validate the new correlation. The results show that the correlation gives good estimations of θ(LCST), with an r2 value of 0.77 for the training set of 112 systems and an r2 value of 0.75 for the test set of 57 systems. Because one correlation was applied to all kinds of polymer solutions and the correlation requires only the molecular connectivity indices of the solvents and polymers in the calculations, the new correlation is a generalized predictive model that is easier to apply and has better predictive capability than the existing methods and models. Introduction Partially miscible polymer solutions often exhibit two coexistence curves: one at high temperature and one at low temperature.1 For strictly binary polymer solutions, the upper critical solution temperature (UCST) is located at the top of the coexistence curve, and the lower critical solution temperature (LCST) is located at the bottom of a coexistence curve at high temperature. For real systems, both the UCST and LCST deviate from the extrema because of the polydispersity of polymers. It is commonly known that both the UCST and LCST depend on the molar mass and pressure; in contrast, the θ temperatures, which are the critical solution temperatures at infinite chain length, are not affected by polymer molar mass. Although θ(LCST), the lower critical solution temperature at infinite chain length, is often regarded as less important than θ(UCST), it is an important property of polymer solutions because it can serve as an upper temperature limit for polymer processing. Currently, a great deal of experimental data are available, but estimation models are still required for the purpose of process development, design, and optimization. Although many models and methods have been proposed for modeling LCSTs, including θ(LCST), they can be mainly divided into two groups.1 The first group of models are those methods that have a solid theoretical basis2-10 but that require vapor-liquid or liquid-liquid experimental data to fit the adjustable parameters, resulting in limited predictive capability. The second group of models are empirical correlations relating θ(LCST) to other physiochemical properties, such as critical properties, density, or solubility parameters.1,11-13 These correlations are simple and have reasonable accuracy, but they cannot be applied to systems for which the required physicochemical properties are not available. An example of this kind of model is the recent work of Imre et al.,1 who related θ(LCST) to the critical temperature and density of the solvent. They proposed * To whom correspondence should be addressed. Tel. : +8610-64419862. E-mail: [email protected].

polymer-dependent correlations for polystyrene, polyethylene, and polypropylene. Our previous work14 showed that these correlations could not be applied to many systems because of the lack of the required critical properties of the solvents. To solve this problem, a method not belonging to the above two groups was proposed in our previous work,14 in which θ(LCST) in polymer solutions was related to the connectivity indices of the solvent concerned. Because molecular connectivity indices can be calculated once the molecular structure of the solvent is known, models based on them are predictive and do not require any experimental information once the models have been developed using limited experimental data points (training set). Therefore, models based on molecular connectivity indices are easier to apply and have better predictive capabilities than models from the other two groups mentioned above. Because of the complexity of polymer solutions, polymer-dependent correlations for eight polymers were proposed in our previous work.14 Although the correlations are predictive, they cannot be applied to other polymer solutions, limiting their applications to practical processes. To solve this problem, the method developed in our previous work is extended in this work, and a general correlation applicable to all kinds of polymer solutions is developed. Molecular Connectivity Index Molecular connectivity indices have been widely used as structural descriptors in the fields of pharmaceutics, biochemisty, environmental science and technology, and chemistry.15,16 The method has been extended to the field of polymers,17 and our previous works have shown that it is a very useful technique in quantitative structure-property relationship (QSPR) research for polymers and polymer solutions.14,18-21 Details of the definitions and the calculation methods for molecular connectivity indices can be found elsewhere,15,16 and the reader can also refer to a recent review of the development of the connectivity index written by Randi.22

10.1021/ie049367t CCC: $30.25 © 2005 American Chemical Society Published on Web 01/06/2005

Ind. Eng. Chem. Res., Vol. 44, No. 3, 2005 635

The general expression for the mth-order connectivity index is nm m+1

mχ t

)

∑∏(δ )

-0.5

i j

(1)

j)1 i)1

where m is the order of the connectivity index and t denotes a contiguous path type of fragment, which is divided into paths (P), clusters (C), path/clusters (PC), and chains (cycles) (CH). nm is the number of relevant paths, and δi is the atomic connectivity index, equal to the number of non-hydrogen atoms to which the ith nonhydrogen atom is bonded. If δi is replaced by δvi , the atomic valence connectivity index, we can obtain the expression for the mth-order valence connectivity index, m χvt , as follows nm m+1

mχv t

)

∑∏(δ )

v -0.5 i j

(2)

j)1 i)1

The above molecular connectivity indices can be calculated easily by hand as long as the molecular structure of the substance concerned is known; therefore, models based on them are predictive and easy to apply. Development of the New Correlation and Results and Discussion In our previous work,14 polymer-dependent correlations were developed for θ(LCST) in which only the connectivity indices of solvents were included. To develop a general correlation for θ(LCST) applicable to all kinds of polymer solutions, the connectivity indices of both solvents and polymers should be included in the correlation. To do this, a large database is necessary. A total of 169 experimental θ(LCST) data were collected from the literature,23-36 comprehending 12 polymers and 69 solvents. The polymers considered in this work are as follows: polyethylene (PE), polypropylene (PP), polybutene-1 (PB1), polyisobutylene (PIB), polypentene-1 (PP1), poly(4-methylpentene-1) (P4MP1), poly(cis-1,4-butadiene) (PBD), polystyrene (PS), poly(Rmethylstyrene) (PMS), poly(p-chlorostyrene) (PPCS), poly(dimethyl siloxane) (PDMS), and poly(isotactic methyl methacrylate) (PMMA). The data collected were randomly divided into two groups, a training set of 112 systems and a test set of 57 systems. On the basis of as analysis of the training set data, the following correlation was obtained

θ(LCST) ) 349.11-46.900χV(poly) + 155.903χVBB(poly) - 17.473χC(poly) + 153.991χSG(poly) - 208.663χSG(poly) + 49.493χP(sol) + 45.834χP(sol) + 44.165χP(sol) (3) r2 ) 0.77, Q2 ) 0.77, F ) 44.76, s ) 34.47, n ) 112 In eq 3, the unit for θ(LCST) is the kelvin. In total, eight structural descriptors were adopted in the new correlation, with five for characterizing the polymer and three for the solvent. The five molecular connectivity indices for polymers were calculated with the method proposed in our previous work.20 These parameters are

Table 1. Connectivity Indices of the Polymers Used in This Work polymer

0χV

3χVBB

polyethylene polypropylene polybutene-1 polyisobutylene polypentene-1 poly(4-methylpentene-1) poly(cis-1,4-butadiene) polystyrene poly(R-methylstyrene) poly(p-chlorostyrene) poly(dimethyl siloxane) poly(isotactic methyl methacrylate)

1.4142 2.2845 2.9916 3.2071 3.6787 4.5689 2.5689 4.6712 5.9390 5.7273 3.9083 4.5236

0.5000 0.3333 0.3333 0.0833 0.3333 0.3333 0.6667 0.3330 0.2500 0.2500 0.7501 0.2500

3χ

C

0.0000 0.2887 0.2887 1.2071 0.2887 0.6124 0.0000 0.3333 0.9469 0.6220 1.2071 1.0067

1χSG

3χSG

0.0000 0.5774 1.1154 1.0000 1.6154 1.9712 0.0000 3.1498 3.6052 3.5437 1.0000 2.4814

0.0000 0.0000 0.0000 0.0000 0.2887 0.4714 0.0000 1.6498 1.6498 2.0605 0.0000 0.6124

defined as follows: 0χV(poly) is the polymer zero-order valence connectivity index, 3χVBB(poly) is the polymer third-order valence connectivity index contributed by the chain backbone,3χC(poly) is the polymer third-order cluster connectivity index, 1χSG(poly) is the polymer firstorder connectivity index contributed by the side groups, and 3χSG(poly) is the polymer third-order connectivity index contributed by the side groups. The use of both chain backbone and side group indices can distinguish the local structures of polymers in more detail. For solvents, on the other hand, three connectivity indices were adopted to describe their structures, namely, the third- to fifth-order connectivity indices. The values of the connectivity indices for the polymers and solvents used in this work are listed in Tables 1 and 2, respectively. The calculation procedure for the connectivity indices of 2-methyl hexane and P4MP1 is given as an example in the Supporting Information. The statistical parameters of the new correlation for the training set systems are reported below eq 3, and the calculated results are depicted in Figure 1 (the tabulated results are provided as Supporting Information). The results shown in Figure 1 demonstrate that the new model provides a good representation of θ(LCST) for most systems, with an overall average absolute relative deviation (AARD) of 5.44% for the 112 systems tested. Among the 112 systems considered, 18 systems have AARDs larger than 10%, and three have AARDs larger than 20%, namely, PIB-2-methylbutane (21.07%), PPCS-ethyl carbitol (27.23%), and PPCS-n-butyl carbitol (34.54%). It can be concluded that the new correlation does not work well for PPCS-carbitol solutions, but it is difficult to obtain a definite conclusion regarding the large deviation for the PIB-2-methylbutane system given that a total of 21 PIB-hydrocarbon systems were considered in this work and the correlation provided very good results for most of them, with only three solvents showing AARDs larger than 10%. Because the experimental data used in this work were taken from various sources and different researchers might have used different measuring methods, the reported data have different measurement accuracies and systematic errors. In addition, θ(LCST) is not easy to measure, and no real polymer can have an infinite chain length. As a result, the experimental uncertainties in θ(LCST) can be large, depending on the measurement methods, polymer samples used, researchers’ techniques, etc. Considering these factors, it seems that the new correlation works well for the representation of θ(LCST) of polymer solutions, using only the molecular connectivity indices of the polymers and solvents as input parameters.

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Table 2. Connectivity Indices of the Solvents Used in This Work solvent

3χ P

4χ P

n-butane n-pentane n-hexane n-heptane n-octane n-nonane n-decane n-undecane n-dodecane n-tridecane n-cetane 2-methylbutane 2,2-dimethylbutane 2,3-dimethylbutane 2-methylpentane 3-methylpentane 2,4-dimethylpentane 2,3-dimethylpentane 2,2-dimethylpentane 3,3-dimethylpentane 2,2,3-trimethylbutane 2-methylhexane 3-methylhexane 2,2,4-trimethylpentane 2-methylheptane 3-methylheptane 2,2-dimethylhexane 2,4-dimethylhexane 2,5-dimethylhexane 3,4-dimethylhexane 3-ethylpentane 2,2,4,4-tetramethylpentane 2,3,4-trimethylhexane cyclopentane cyclohexane cycloheptane cyclooctane methylcyclopentane methylcyclohexane ethylcyclopentane n-propylcyclopentane benzene toluene methyl acetate ethyl acetate n-propyl acetate i-popyl acetate n-butyl acetate isobutyl acetate sec-butyl acetate tert-butyl acetate n-pentyl acetate i-amyl acetate n-hexyl acetate ethyl n-butyrate methyl ethyl ketone diethyl ketone ethyl propyl ketone dipropyl ketone diethyl ether diethyl malonate 1-octanol ethyl carbitol n-butyl carbitol propylene oxide butyl chloride tetrahydrofuran (THF)

0.5000 0.7071 0.9571 1.2071 1.4571 1.7071 1.9571 2.2071 2.4571 2.7071 3.4571 0.8165 1.0607 1.3333 0.8660 1.3938 0.9428 1.7820 1.0000 1.9142 1.7321 1.1350 1.4784 1.0206 1.3850 1.7474 1.2803 1.5707 1.3214 2.2593 1.7321 1.0607 2.5931 1.2500 1.5000 1.7500 2.0000 1.6438 1.8938 2.0521 2.1717 1.5000 1.8938 0.8165 0.8660 1.1350 0.9428 1.3850 1.3214 1.5707 1.0206 1.6350 1.5629 1.8850 1.5629 0.8165 1.3938 1.4784 1.5629 0.7071 2.3677 1.7071 1.7071 2.2071 0.5774 0.7071 1.2500

0.0000 0.3536 0.5000 0.6768 0.8536 1.0303 1.2071 1.3839 1.5607 1.7375 2.2678 0.0000 0.0000 0.0000 0.5774 0.2887 0.9428 0.4714 0.7500 0.2500 0.0000 0.6124 0.6969 1.2247 0.8026 0.7567 0.7071 0.9714 0.6667 0.8047 0.8660 1.5910 1.0289 0.8839 1.0607 1.2374 1.4142 1.1299 1.3067 0.6969 1.5629 1.0607 1.3067 0.0000 0.5774 0.6124 0.9428 0.8026 0.6667 0.9714 1.2247 0.9794 0.9343 1.1561 1.1299 0.0000 0.2887 0.6969 1.1299 0.4082 1.9082 1.0303 1.0303 1.3838 0.0000 0.3536 0.8839

5χ

P

0.0000 0.0000 0.2500 0.3536 0.4786 0.6036 0.7286 0.8536 0.9786 1.1036 1.4786 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.4082 0.2041 0.0000 0.4330 0.4928 0.5303 0.3333 0.6667 0.1667 0.0000 0.0000 0.2722 0.0000 0.7500 0.8750 1.0000 0.2887 0.9010 0.5761 0.7803 0.7500 0.9010 0.0000 0.0000 0.4082 0.4082 0.4330 0.6667 0.3333 0.0000 0.5675 0.4714 0.6925 0.2887 0.0000 0.0000 0.2041 0.2887 0.0000 0.6381 0.6036 0.6036 0.8536 0.0000 0.0000 0.0000

A comparison with existing models was not made in this work because no common comparison basis could be found. The existing theoretical models require systemdependent parameters that have to be obtained by fitting experimental VLE or LLE data. The empirical correlations, on the other hand, are polymer-dependent,

Figure 1. Experimental versus calculated θ(LCST) for the training set of 112 polymer solution systems.

Figure 2. Experimental versus predicted θ(LCST) for the test set of 57 polymer solution systems.

that is, different correlations or coefficients are required for different polymers. The new correlation, in contrast, is a general correlation that can be applied to all kinds of polymer solutions and that requires only the molecular connectivity indices of polymers and solvents in question. To validate its predictive capability, the new correlation was used to predict θ(LCST) values for the systems in the test set. The predicted results for the 57 test set systems are shown in Figure 2 (the tabulated results are given in the Supporting Information). The new model gives an r2 value of 0.75, comparable to that of 0.77 for the training set systems. Figure 2 shows that the new correlation gives good predictive results for most systems concerned, with an overall AARD of 5.63%. Among the 57 systems, 10 have AARDs of larger than 10%, and only one system, PS-tert-butyl acetate, has an AARD of larger than 20%. Considering the complexity of polymer-solvent systems and the simplicity of the new correlation, the predictive results can be considered satisfactory. The new correlation should be useful for solvent selection and process design and optimization, as it can provide at least a crude estimate even before the solvent and/or polymer being synthesized have been tested. Conclusion The general correlation proposed in this work can be applied to all kinds of polymer solutions, as it requires

Ind. Eng. Chem. Res., Vol. 44, No. 3, 2005 637

only the molecular connectivity indices of the polymers and solvents in the calculations. Because molecular connectivity indices can be calculated once the molecular structure of the compound in question is known, the new correlation is predictive and very easy to apply. The calculated results obtained in this work, for systems in both the training set and the test set, show that the new correlation can provide satisfactory estimations of θ(LCST) for most systems, leading to the conclusion that it is useful for practical purposes, in that it can provide a crude estimate even before the solvent and/or polymer being synthesized have been tested. This work also demonstrates that topological indices are useful structural descriptors for polymer-containing systems. Acknowledgment The financial support of the Natural Science Foundation of China (Contract 20476003), TRAPOYT, the Trans-Century Training Programme Foundation for the Talents by the Education Ministry of China, P.R.C., and the Beijing Committee of Science and Technology (Contract 9558101100) is greatly appreciated. Supporting Information Available: Calculation procedure for the connectivity indices of 2-methyl hexane and P4MP1 as an example and tabulated results for the application of the correlation to the training and test set data. This material is available free of charge via the Internet at http://pubs.acs.org. Nomenclature F ) Fischer ratio n ) number of systems nm ) number of relevant paths Q2 ) leave-one-out cross-validated correlation coefficient r2 ) correlation coefficient s ) standard deviation Greek Letters mχ ) mth-order connectivity index t m χv ) mth-order valence connectivity t

δ ) simple connectivity index δv ) valence connectivity index

index

Subscripts C ) cluster i ) atom i P ) path t ) contiguous path type of fragment Superscripts BB ) contribution of the chain backbone calc ) calculated value exp ) experimental value pred ) predicted value SG ) contribution of the side groups

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Received for review July 19, 2004 Revised manuscript received November 4, 2004 Accepted November 18, 2004 IE049367T