A novel method for predictions of the gas permeation parameters of polymers on the basis of their chemical structure

Journal of Membrane Science 487 (2015) 189–198

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Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci

A novel method for predictions of the gas permeation parameters of polymers on the basis of their chemical structure V. Ryzhikh a, D. Tsarev b, A. Alentiev a, Yu. Yampolskii a,n a b

A.V. Topchiev Institute of Petrochemical Synthesis, Russian Academy of Sciences, Leninskii Prospect, 29, Moscow, Russia OOO NII “Mitoingeniry”, Moscow State University, Russia

art ic l e i nf o

a b s t r a c t

Article history: Received 29 December 2014 Received in revised form 20 March 2015 Accepted 22 March 2015 Available online 4 April 2015

This work presents the results of further development of incremental (group contribution) methods for prediction of the gas transport parameters (permeability (P) and diffusion (D) coefficients) of amorphous glassy polymers. The source of information was a big Database that comprises the P and D values for about 900 polymers. The method of the modified atomic contributions was revisited using much enlarged set of polymeric structures. In addition, a novel one, the bond contribution method, was employed for the first time for predictions of the P and D values of a standard set of gases (He, H2, O2, N2, CO2 and CH4). It was shown that both approaches allow the accurate (improved) predictions of the gas permeation parameters but the latter is more precise (correlation factors R2 40.9). It was also demonstrated that found group contributions permit predictions of the permeability coefficients of random copolymers of different classes. & 2015 Elsevier B.V. All rights reserved.

Keywords: Permeability coefficients Diffusion coefficients Methods of predictions of gas permeation parameters Amorphous glassy polymers Random copolymers

1. Introduction A search for relationship between chemical structure and properties of organic compounds is a traditional problem of physical chemistry. When such quantitative relationship is found, it forms a basis for prediction of properties of unknown compounds. Among the properties successfully predicted one can mention enthalpy of formation of hydrocarbons [1], critical temperature and pressure of hydrocarbons [2] and other thermodynamic properties of low molecular weight organic compounds [3]. An important contribution in the chemical kinetics was a development of the predictions of the rate and equilibrium constants made by S.W. Benson and his school [4,5]. Predictions of the properties of polymers were started somewhat later. Here, the most important contribution was made in the book by Van Krevelen [6]. Another approach for predictions of the various properties of polymers was proposed by Askadskii [7]. A relevant parameter of polymers related to their applications in gas separation membranes is their permeability coefficients Pi. The first attempts to predict the Pi values on the basis of chemical structure of polymers using the group contribution method were made by Salame [8,9]. However, the disadvantage of these early

n

Corresponding author. E-mail address: [email protected] (Yu. Yampolskii).

http://dx.doi.org/10.1016/j.memsci.2015.03.055 0376-7388/& 2015 Elsevier B.V. All rights reserved.

works was an absence of detailed information on gas permeability of polymers with different chemical structure. Various observations prompted that there exists relationship between the chemical structure of polymers and their gas permeation parameters. Thus, introduction of Si(CH3)3 groups into various main chains resulted in increases in permeability and diffusion coefficients of polymers [10]. Similar effects though not so strong are exerted by appearance of CH3 or CF3 groups in aromatic backbones of polymers [11]. Extensive accumulation of the information of this kind started in the middle of 1980s [12], so a possibility appeared for predictions of the gas permeation parameters of polymers with much more widely varying structure. Jia and Xu [13] using regression analysis treated the permeability data of about 60 polymers for 6 gases. An interesting approach was proposed by Park and Paul [14]. In the works by Robeson et al. [15,16] the group contribution method was applied for the determination of the increments for relatively large structure elements of 65 amorphous glassy polymers with aromatic backbones studied in a single laboratory. A larger set of glassy polymers was used by Yampolskii et al. [17] who considered about 300 of amorphous glassy polymers that belonged to different chemical classes. In this work atomic contributions were sought for and the different contributions were ascribed for the same atoms in the main chain and side groups. Polyimides formed a significant part of the analyzed polymer structures, so it was possible to consider this chemical class of membrane materials separately. Since polyimides can be formally

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considered as alternative copolymers of diamine and dianhydride units, Alentiev et al. [18] found and used the contributions for these structural elements with the accuracy of predictions better than that given by the method of modified atomic contributions. Other attempts were made to predict gas permeability of gases in polymers [19]. During the subsequent years much more information on gas permeation parameters (permeability and diffusion coefficients) was accumulated. In order to take this information into account, a Database [20,21] was created in A.V. Topchiev Institute of Petrochemical Synthesis. The Database contains now the information on almost 900 amorphous glassy polymers. The following parameters are tabulated: permeability coefficients (Р), diffusion coefficients (D), solubility coefficients (S), activation energies of permeability (EP), and activation energies of diffusion (ED), some other physicochemical properties of polymers where available. The tabulated values of Р and D are cited for the temperature at which the measurement was performed according to the original publication; however, they are reduced to a standard temperature of 308 K using the values of ЕР and ED (original or obtained with the aid of specially developed correlations [22]). This is necessary for a comparison of different polymers and used for the predictions of transport parameters. Several correlations of EP and ED were considered in this work: the correlations with the fractional free volume, with activation energies found via the group contribution method, and the correlations of activation energies with the permeability coefficients at certain reference temperature (a consequence of kinetic compensation effect). The latter procedure was found to be the most accurate and subsequently was used for investigation of the temperature dependence of P and D in polymers. The most complete information is given for permeability coefficients: about 5500 records are available for Р. In the present work attempts are made to address the two problems: (i) to examine to what extent an expansion of the size of basic set of polymer structures affects the accuracy of the prediction of the gas permeation parameters; (ii) to investigate a novel approach for predictions of the permeability and diffusion coefficients, the one based not on atomic group contributions but on the increments characteristic for certain bonds within polymeric repeat units.

2. Principles of calculations For all the group contribution method, it is characteristic to represent the structure formulas of repeat units as a molecular graph. All the atoms of the repeat units except hydrogen are considered as vertices while bonds connecting those atoms are considered as edges. In relation to polymers, structural formula of the repeat unit is presented as a sum of fragments (in the simplest case it might be atoms or bonds), and a group contribution or increment is assigned to every such fragment. Summing up of these contributions with account for a number of corresponding fragments in the repeat unit can allow making a number estimation of necessary physicochemical parameter. A way of splitting of polymer structure into such fragments is a key point that determines the accuracy of the predictions. In the method of the modified atomic contributions, the same one that has been used previously [17], the polymer structure is split into the main chain and side groups. Different increments were assigned to the same atom when it appears either in the main chain or in side groups. It is assumed that the increments are constant for the whole set of complex structures, i.e. do not

depend on a mutual arrangement of the groups or their possible interactions. It is considered that the vertex of the molecular graph belongs to the main chain if there is at least one way from “head to tail” that includes this vertex by such traveling along the molecular graph when it is allowed to “visit” each vertex only once. In the opposite cases it is considered that this vertex belongs to side groups. Thus, while proceeding from polymer's structural formula one can obtain a unique set of fragments. The Database [20] was used in the calculations. It contains information for about 900 different amorphous glassy polymers that belong to different classes: Polyimides (315 structures), Polyacetylenes (107), Polyesters (86), Polyetheres (67), Polynorbornenes (53), Polysulfons (51) and other polymer classes. The values of P and D are tabulated in the Database for 26 various gases but the most extensive information is available for He, H2, O2, N2, CO2 and CH4. The whole set of the included polymers can be described using 16 fragments that belong to the main chain and 19 fragments of the side groups. These fragments and corresponding increments are given in Table 1S (see Supporting Information). In this way, a overdetermined system of equation was prepared. The prerequisite for such system is the inequality kcn

ð1Þ

where n is the number of different groups that are used to draw the structure of the repeat units of all k polymers. Such system is presented below: P mi nni1 8 lg X 1 ¼ > M nI i > < P mi n1ni2 lg X 2 ¼ M 2 nI i > > : lg X k ¼ P mi nnik nI i

ð2Þ

Mk

where X is the physicochemical parameter, mi is atomic mass of the i-th atom, ni is the number of atoms of i-th type, M is the molecular mass of the repeat unit, Ii is the increment (contribution) that corresponds to i-th type of atom. The solution of this overestimated system was performed using SVD method (Singular Value Decomposition) [23] and it gave optimal values of the increments for every fragment. These parameters can be used eventually for predictions of the properties of polymers comprised of the same fragments. An example of the splitting of a model polymer structure using this approach is given in Fig. 1 and Table 1. In the bond contribution method each edge of the molecular graph is classified as follows: single, double triple bonds and aromatic bonds. The bonds are considered separately if they connect different atoms. Thus, a certain property (type of the bond) is ascribed to each edge. The vertices of the molecular graph that are located at the ends of edges are considered in the same way as in the method of modified atomic contributions. Hence, the fragments of the structure are differed by the type of bond and end atoms. Because of this, the number of corresponding increments is significantly greater than in the method of modified atomic contributions. The corresponding increments are given in Table 2S (see Supporting Information). Accordingly, the following system of

Fig. 1. Example of splitting of a model polymeric structure in the method of modified atomic contributions.

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Table 1 Corresponding increments.

Fig. 2. Example of splitting of a model polymeric structure in the bond contribution method.

equation is formed and should be solved: P ni1 8 lg X 1 ¼ > N nI i > < P ni21 lg X 2 ¼ N nI i > P nik2 > : lg X k ¼ nI i

ð3Þ

Nk

Here X is the physicochemical property of the polymer, ni is the number of the bonds of i-th type in the repeat unit, N is the total number of the bonds in the repeat unit, Ii is the increment characteristic for each bond of i-th type. For the solution of this overestimated system the program RIADA (Rapid Incremental Algorithm Development Application) was used. An example of the splitting of a model polymer structure using the bond contribution method is given in Fig. 2 and Table 2.

3. Results and discussion 3.1. The method of the modified atomic contributions The same approach as used earlier [17] was employed in search for the predicted permeability coefficients using the method of the modified atomic contributions. As has been mentioned above, the P values at 308 K were used in the calculations. They were either reported at this temperature in the original works or recalculated

to 308 K using the observed activation energy of permeation found using the correlations described in Ref. [22]. No correction was made to take into account the pressure effects on the P values because all the calculations were made for light gases for which permeability coefficients only weakly depend on pressure. Fig. 3 shows the correlations between Pexp and Ppred values. It is seen that the clouds of the data points are located symmetrically around the median lines that correspond the complete agreement between these two parameters. In this regard, there is a difference with previous results of similar calculations [17] where the clouds of the data pointes were skewed in relation to the medians. As is seen from Table 3 the scatter of the points corresponds to the standard deviation less than one order. The number of considered structures was the greatest in the case of oxygen and nitrogen, for helium and hydrogen it was smaller though in all the cases the number of used increments (32–35) was much smaller than the number of considered structures, so the system of the Eq. (2) was always strongly overestimated. A similar procedure was conducted in calculation of the predicted values of the diffusion coefficients. Here helium and hydrogen were excluded from the set of gases because of lack of needed information for these light gases. Fig. 4 shows the correlations between experimental and predicted diffusion coefficients for O2 and N2. In the case of the diffusion coefficients the correlation coefficients are somewhat smaller than those for the

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Table 2 Corresponding increments.

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Fig. 3. Comparison of experimental and calculated permeability coefficients (method of the modified atomic contributions) for various gases using the whole set of the structures from the Database. 1 Barrer¼10  10 cm3 (STP) cm cm  2 s  1 (cm Hg)  1 ¼ 3.36 10  16 mol s  1 m  1 Pa  1.

permeability coefficients. The clouds of the data points still are located symmetrically around the medians of the plots. It can be noted that in our previous publication [17] no procedure was described for prediction of the diffusion coefficients of glassy polymers with arbitrary structure. It is of interest to consider the effects of the number of the polymer structures included into correlations. This can be made by comparing the correlations for nitrogen shown in Fig. 3 and Fig. 5. It was taken from the thesis by A.Alentiev [24]. It is seen that the scatter of the data points is markedly larger what is reflected in smaller value of the correlation coefficient R2.

Table 3 Standard deviation σ of the correlations Pexp and Ppred obtained using the method of the modified atomic contributions and correlation coefficients R2 for various gases.

3.2. The bond contribution method

atoms as has been illustrated in Fig. 2 and Table 2. In this method the number of the increments for the polymers of Database is greater than one hundred and they are collected in Table 2S. The correlations between Ppred found using this method and Pexp are shown in Fig. 6. On the basis of the predictions for all the gases

The bond contribution method has not been used so far. In this method, the bonds are considered to be different if they are of different type (e.g. single or double) and connect different

Gas

σ (log[P, Barrer])

R2

H2 He O2 N2 CO2 CH4

0.81 0.72 0.79 0.83 0.87 0.83

0.88 0.88 0.84 0.86 0.83 0.85

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Fig. 4. Comparison of experimental Dexp and predicted Dpred values (method of the modified atomic contributions) for oxygen and nitrogen for the whole set of polymers from the Database.

Fig. 5. Comparison of experimental and calculated permeability coefficients of nitrogen (method of the modified atomic contributions) based on a smaller data set [24].

the standard deviations and correlation coefficients are given in Table 4. It indicates that in all the cases the standard deviations are smaller than one order. Both methods for predictions gave approximately the same level of the standard deviation, though the correlation coefficients are somewhat greater for the bond contribution method. A similar treatment of the data on the diffusion coefficients using the bond contribution method resulted in obtaining the correlations shown in Fig. 7. A comparison of the correlation coefficients R2 of the two method shows that in the case of bond contribution method the observed R2 values are higher. 3.3. Analysis of individual polymer classes Some classes of glassy polymers are presented in the Database by sufficiently great number of individual structures. Hence, inequality (1) for them is fulfilled, so it is possible to examine the correlations between experimental and predicted permeability coefficients within a certain class of polymers. It can be assumed that in such case the accuracy of the predictions within a certain polymer class might be better than for the correlations within the whole set of glassy polymers. Such work was made for polyacetylenes, polyesters and polyethers which are presented with rather big number of chemical structures (see Section 2). The predictions can be made only by the method of the modified atomic contributions, because for the bond contribution method the number of possible increment is too big for fulfilling of the inequality (1). The results of such analysis are presented here for combined group of polyesters and polyethers (Fig. 8) and for polyacetylenes (Fig. 9). In

preparation of these correlations the increments were sought for only for this massive of polymers. It is seen that a quite good correlation is observed between Pexp and Pcalc. The correlation coefficient R2 ¼ 0.87 is higher than corresponding correlation coefficient given in Fig. 3. Entirely different results were obtained for polyacetylenes as Fig. 9 shows. A much bigger scatter of the data points is obvious, and the correlation coefficient is much lower than that for the whole set of glassy polymers (Fig. 3). It indicates that the formal chemical structure of the repeat unit in this class does not allow to predict accurately the permeation rate, and other factors must be taken into account. In the case of polyactylenes possible effects of cis/trans configuration of double bonds in the main chain can be responsible for such behavior. Indeed, it has been shown [25] that the samples of poly(trimethylsilyl propyne) (PTMSP) with low content of cis-isomer show much higher permeability coefficients than the samples with prevailing trans-isomer: P(O2)¼11300 Barrer at 20% of cis-isomer and 6900 Barrer at 65% of cis-isomer. The PTMSP samples obtained using diffeent catalysts and with variation of cis-content are characterized by different chain rigidity (Kuhn segments). They show helical secondary structure with different steps of the helix [26,27]. Another factor that is important for highly permeable polyacetylenes but can be neglected for the polymers of lower permeability is well documented fast aging of these polymers (reduction of permeability coefficients and free volume) [28–30]. An opposite process, swelling in non-solvents (alcohols) results in strong increases in gas permeability and free volume of PTMSP and other high free volume polyacetylenes [31]. All these phenomena are likely responsible for a wide variation of the observed permeability coefficient of PTMSP as is shown by arrow in Fig. 9 (variation of P(O2) from 2300 to 19500 Barrer). In other words, for these polymers the primary structure as presented by chemical formula of the repeat unit is a poor descriptor of gas permeation properties and secondary structures should be considered.

3.4. Tests of predicting ability The improved quality of the predictions of the gas permeation parameters on the basis of extended set of polymers and using a novel approach, the bond contribution method, is obvious from Figs. 3–7. However, there are other ways to assess the quality of the predictions made. Fig. 10 shows “synthetic” permeability – permselectivity diagram, where the data points were obtained by the method of bond contribution and the line is the Upper bound of 2008 [32]. It is

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Fig. 6. Comparison of experimental and calculated permeability coefficients (bond contribution method) for various gases using the whole set of the structures from the Database.

Table 4 Standard deviation σ of the correlations Pexp and Ppred obtained using the method of the bond contributions and correlation coefficients R2 for various gases. Gas

σ (log[P, Barrer])

R2

H2 He O2 N2 CO2 CH4

0.77 0.82 0.81 0.85 0.78 0.78

0.94 0.94 0.91 0.91 0.90 0.91

seen that the diagram is rather similar to the original diagram presented by L.M.Robeson [32]. It is also possible to consider the correlation of the separation factors αij ¼ Pi/Pj. An example of such correlation is shown in Fig. 11.

The accuracy of the prediction of αij is worse than in the case of the permeability coefficients because of the effects of the scatter of the permeability coefficients in the numerator and denominator. So it is less suitable for accurate predictions of this parameter though provides an appropriate trend of the separation factors. Synthetic Robeson diagram and correlation between experimental and calculated separation factors for method of modified atomic contributions are given in Figs. 3S and 4S respectively. Another approach for assessment of the accuracy of the predictions was also tested. For a number of polymers their transport parameters were not used in the search for the increments of both employed methods. Therefore, their predicted permeability coefficients can be considered as completely independent of their experimental P values. These works [33–49] reported permeability coefficients of polyimides, polyamides, polyethers, polynorbornenes and some other polymers. The comparison of the experimental permeability coefficients P(O2) for this group of polymers and the values found via the bond contribution method is shown in Fig. 12. Similar correlation for the method of the

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Fig. 7. Comparison of experimental Dexp and predicted Dpred values (method of bond contributions) for oxygen and nitrogen for the whole set of polymers from the Database.

Fig. 8. Comparison of experimental and calculated permeability coefficients for oxygen (method of the modified atomic contributions) in combined group of polyesters and polyethers from the Database.

Fig. 10. Synthetic Robeson diagram for oxygen/nitrogen pair. Data points are obtained by the method of bond contribution and the line is the Upper bound of 2008 [32].

Fig. 11. Correlation between experimental and calculated separation factors α(O2/ N2). The separation factors are found using the method of bond contribution. Fig. 9. Comparison of experimental and calculated permeability coefficients for oxygen (method of the modified atomic contributions) in polyacetylenes from the Database.

modified atomic contributions is presented in Supporting Information (Fig. 5S). It can be seen that both procedures allow relatively accurate prediction of permeability of independent (“novel”) group of amorphous polymers with modest permeability. There is a group of the data points that shows maximum deviation from these correlations on both plots, i.e. independently of the method of prediction. They correspond to the poly(arylene ether)s studied in [37]. It can be caused by a systematic error in determination or some unknown effect. In any case, this result requires additional corroboration.

3.5. Prediction of permeability of copolymers Most of the known attempts to predict gas permeation parameters of polymers were limited to glassy homopolymers [13–18]. Only the work [19] presented a single example of predicting the permeability of copolymers, in particular copolyimides. Here the results are presented on prediction of the permeability coefficients of random copolymers of different classes. For the permeability coefficient P in random copolymers the following linear equation can be found in the literature [50]:   lg P ¼ ϕn lg P A þ 1  ϕ n lg P B ð4Þ where PA is the predicted permeability coefficient of the homopolymer A, and PB is the predicted permeability coefficient of the homopolymer B, and ϕ is the volume fraction of comonomer A.

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Fig. 12. Correlation of the experimental [33–49] and predicted permeability coeffificents of oxygen in “novel” polymers (not used in calculating of the increments in the bond contribution method). Dotted line represents one order of magnitude deviation.

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Database two methods were employed for predictions of the P and D values of six gases. The method of the modified atomic contributions gives better results in predictions (correlation factors R2 of 0.85–0.88) when the number of the considered structures increased to 500–900 as compared with smaller sets of the structures (250) analyzed previously using the same method. Much better correlations between experimental and predicted P values were obtained when the bond contribution method was used that employs much larger set of different increments. The same conclusion can be made regarding the predictions of the diffusion coefficients: the observed R2 values increased from 0.80 to 0.92. Possible way for improvements of the accuracy of the prediction can comprise consideration of particular polymer classes that are represented in the Database by sufficiently large structures. Here the results turned out contradictory. For low permeable (highly selective) polymers that belong to combined class of polyesters and polyethers the results were satisfactory: the accuracy of predictions somewhat increased. But quite opposite result was obtained for polyacetylenes. Much greater scatter of the data points on the correlations can be apparently explained as follows: for this class of polymers not only the primary structure (chemical formula of repeat units) but also space organization of the chains (secondary structure) can strongly affect the observed permeability. At last, it was shown that using the found increments it is possible to predict accurately the permeability coefficients of random copolymers of various structures.

Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.memsci.2015.03.055.

Fig. 13. Correlation of experimental and calculated values of permeability coefficients of oxygen (modified atom contribution method) for copolymers (dotted lines correspond to71 order of magnitude deviation).

There are some problems with its application: the volume fraction ϕ is seldom reported in the original membrane works and its estimation is not easy. In addition, there are examples of deviations from the linear form of Eq.(4) [51]. So we used a similar equation with mole fraction m: lg P ¼ mn lg P A þ ð1  mÞn lg P B

ð5Þ

This or similar equations were successfully used for different random copolymers [19,52]. The existing literature on gas permeation properties of copolymers is very extensive. We used a limited set of data on random copolymers. It included a test massive for about 40–45 random copolymers of different classes, namely copolyimides [53–55], copolyesters [56], copolysulphones [57], and some other copolymers: polynorbornenes [39,58], functionalized copoly(vinyl chloride)s [59], perfluorinated copolymers [60]. Fig. 13 indicates that such simple procedure gives relatively good results: the data points are located symmetrically along the median that corresponds to full coincidence between predicted and experimental P values and the deviations from the median do not exceed one order. It makes reasonable to undertake more extensive predictions of the transport parameters of random copolymers.

4. Conclusions Using the extended set of polymeric chemical structures and corresponding gas permeation parameters (P and D) from the

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