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Hydrogen storage in a series of Zn-based IRMOFs studied by Sanchez–Lacombe equation of state
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 2 1 0 7 6 e2 1 0 8 2
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Hydrogen storage in a series of Zn-based IRMOFs studied by SanchezeLacombe equation of state Sajjad Javidi Alesaadi, Fatemeh Sabzi* Department of Chemical Engineering, Shiraz University of Technology, Shiraz 71555-313, Iran
article info
abstract
Article history:
In this work, the adsorptive behaviors of molecular hydrogen in 10 different isoreticular
Received 27 August 2014
metal-organic frameworks, namely, IRMOF-1, -2, -3, -6, -8, -10, -11, -13, -18 and -20 have
Received in revised form
been studied using SanchezeLacombe equation of state. SanchezeLacombe equation of
7 October 2014
state has three characteristic parameters: characteristic density, characteristic pressure,
Accepted 14 October 2014
and characteristic temperature which have been calculated from group contribution pro-
Available online 7 November 2014
cedure. The amount of hydrogen uptake in IRMOFs has been obtained through gasadsorbents phase equilibrium calculations at temperature 77 K and various pressures up
Keywords:
to 80 105 Pa. Finally, the results have been compared with the experimental data to show
IRMOF
the precision of SL equation of state in predicting the hydrogen sorption data.
Equation of state
Copyright © 2014, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
SanchezeLacombe Sorption Hydrogen
Introduction The rapid consumption of petroleum deposits and severe environmental impacts caused by burning fuel cells have driven the global research community to look for an alternative power source that is clean and sustainable [1,2]. Among many emerging new sources in the future, hydrogen is probably the best candidate [3e5]. However, the main bottleneck of hydrogen fuel cells is developing efficient, economic and safe hydrogen storing systems [6e8]. Without the new hydrogen storage system, it is difficult to switch to the hydrogen fuel cell engine because the existing hydrogen storage systems including compression orcryogenic storage are not economically viable and difficult to implement [9]. There is a third potential solution for hydrogen storage such as hydrogen adsorption in a new class of microporous adsorbents named
Metal Organic Frameworks (MOFs). Application of these functional materials is now a focus of considerable research due to their greater surface area and pore volumes [10e13]. Among all MOFs, Isoreticular Metal Organic Frameworks (IRMOFs) are a branch of MOFs which reticulated through the connection of octahedral Zn4O(O2Ce)6 secondary building units, resulting in frameworks based on the same primitive cubic topology. The alteration of linear ditopic carboxylate ligands has been shown to improve the hydrogen uptake level through the controlling specific surface area and pore sizes in the framework. In this study, IRMOF-1 prepared from benzene-1,4-dicarboxylic acid (BDC) and its bromo-, amino- and dihydrocyclobuta-derivatives i.e. IRMOF-2, -3 and -6 are viewed [14]. Other designs such as IRMOF-8 with naphthalene2,6-dicarboxylate (NDC) organic linker can also be wielded by the extension of the length between the metal centers leading to large void regions [15,16]. When the length of the linker
* Corresponding author. E-mail addresses: [email protected], [email protected] (F. Sabzi). http://dx.doi.org/10.1016/j.ijhydene.2014.10.064 0360-3199/Copyright © 2014, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
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i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 2 1 0 7 6 e2 1 0 8 2
reaches two phenyl rings as biphenyl-dicarboxylate (BPTC) in IRMOF-9 [14], interpenetration i.e. filling the void in one net by another net passing through it, is observed. The degree of interpenetration can be as high as fourfold when the linker contains a linear quarter-phenyl chain as 4,5,9,10tetrahydropyrenee2,7-dicarboxylate (HPDC) in IRMOF-11 [15,16] and pyrenee2,7-dicarboxylate (PDC) in IRMOF-13 [14]. An organic ligand with four methyl functional groups, 2,3,5,6-tetramethylbenzene-1,4-dicarboxylate (TMBDC), is considered in IRMOF-18 to study the effect of pore volume occupation [16]. Heteroatoms can be also incorporated in the aromatic backbone of the organic links, by using thieno[3,2-b] thiophene-2,5-dicarboxylic acide (TTDC) as the precursor IRMOF-20 [14]. Calculating the thermodynamic properties of gaseous processes such as exergy analysis has been performed for air, nitrogen and oxygen using three kinds of cubic equations of state [17]. Investigation on the solubility of gases in organic frameworks through the equation of state (EoS) was started in our group by publishing the work on the sorption of CO2, C2H2 and C2H4 in hydrogen-bonded organic framework (HOF-1a) using Perturbed Hard Sphere Chain Equation of State (PHSC EoS) [18]. This model has been also adopted to predict the hydrogen [19] and methane [20] storage in five adsorbents including MOF-5, MOF-177, MOF-200, MOF-205 and MOF-210. In this work, the SanchezeLacombe (SL) equation of state has been employed to examine its capability for interpreting hydrogen sorption data in above-mentioned macromolecules. The well-known lattice-fluid model of SL has been previously applied to describe properties of binary or multi-component ordinary compounds and polymeric systems [21e24]. For gas solubility prediction, the knowledge of three characteristic parameters, i.e. the characteristic density, r*, the characteristic pressure, P*, and the characteristic temperature, T*, is required. These parameters have been calculated for both hydrogen molecules and IRMOFs in order to use as scaling constants to correlate sorption data. Comparison of experimental and theoretical coexistence curves shows that SL EoS works very well in predicting hydrogen sorption behavior.
r ¼ P* M RT* r*
(3)
Note that for a high molecular weight polymer the value of r can be considered to be infinite, then (1/r) can be neglected. T*, P*, r* and V*are characteristic parameters which defined as [23]: T* ¼
ε* ; R
P* ¼
V* ¼ N rn* ;
ε* ; n*
r* ¼
M rn*
(4)
Where ε* and n* are the characteristic interaction energy per mer and the close-packed volume of a mer, respectively. N is the number of molecules. For a polymer-gas mixture it is necessary to use a mixing rule for the calculation of n*mix , ε*mix and rmix based on the corresponding values of the pure component parameters. There are different types of mixing rules for calculating mixture properties [23]. In this research work, the van der Waals mixing rule has been used for calculating the mixture properties as follows: n*mix ¼
Nc X Nc X i¼1
fi fj n*ij
(5)
j¼1
where n*ij ¼
n*ii þ n*jj 2
1 nij
(6)
with the parameter nij which accounts for the possible deviation ofn*ij from the arithmetic mean of the corresponding values,n*ii and n*jj , of the pure components. In this work, the value of the interaction parameter nij was assumed to be equal to zero. Accordingly, the values ofε*mix and rmix of the mixture will be obtained by the following equations: ε*mix
¼
r1 mix ¼
Nc X Nc 1 X
n*mix
i¼1
0 1
0:5
* * 1 kij An*ij fi fj @ εii εjj
(7)
j¼1
Nc
X fj rj
(8)
j¼1
Theory SanchezeLacombe EoS is based on lattice-fluid theory that treats the polymer chains as a set of interacting beads in a lattice. Following the original developments of Sanchez and Lacombe the general SL EoS can be written as [25,26]: 1 r2 þ P þ T ln 1 r þ 1 r ¼ 0 r
(1)
where r is the reduced density, P the reduced pressure and T the reduced temperature of a pure component. The parameters are reduced by characteristic constants as follows: r¼
r 1 V* ¼ ¼ ; * r n V
P¼
P ; P*
T¼
T T*
(2)
r is a size parameter that represents the number of lattice sites occupied by a molecule and can be related to the molecular weight according to the following equation:
where kij is a binary interaction parameter which represents the interaction energy between the ith and jth species in the mixture. fi is the volume fraction of the ith component in the mixture that can be expressed in terms of the mass fraction ui, the characteristic densityr*i and the characteristic volume n*i of the pure components: 2 !31 Nc uj 5 ui 4 X fi ¼ * * ri ni j¼1 r*j n*j
(9)
The chemical potential of each component in the two phases, one phase is gas and the other one consists of polymer and the sorbed gaseous molecules, must be set equal at the equilibrium state, i.e. mGi ¼ mPi . Following the developments of McHugh and Krukonis [27], the chemical potential of the ith component in a multi-component system can be expressed as:
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ri mi ¼RT ln fi þ 1 rmix 2 0 1 33 2 Nc Nc X X 2 fj n*ij ε*ij ε*mix fj n*ij A þ ε*mix 55 þ ri 4 r4 * @ nmix j¼1 j¼1 3 2 r RTn ð1 rÞlnð1 rÞ þ lnr ri 7 6 7 6 1 7 þ ri 6 7 6 Nc P * 5 4 * A þPn 2 fj n n j¼1
ij
Table 2 e Adjusted binary interaction parameters and statistical errors. Pairs (10)
mix
Results and discussions The exact estimation of the characteristic density, r*, the characteristic pressure, P*, and the characteristic temperature, T*, is of high importance for SL EoS to behave accurately. The regression of the PVT data of pure components is a simple method to find the characteristic parameters, but PVT data is not available for metal organic frameworks. Then, the group contribution method handed in by Boudouris et al. [22] has been chosen for extracting these scaling constants. From the knowledge of only the molecular structure of IRMOFs' repeating unit cells, first order and second order groups have been identified. The characteristic properties have been estimated by summing the contribution of each group in one specific property. Table 1 shows the calculated characteristic parameters for all the individual compounds of interest. Having in hand the three parameters characteristic to each component in the hydrogen and IRMOF systems, the solubility of gas can be calculated. Then, for each IRMOF/hydrogen pair the model contains only one parameter, which characterizes interactions between gas and macromolecules. This binary interaction parameter has been adjusted by fitting the parameter Kij to experimental hydrogen sorption data at 77 K. Regression of data points has been done by making use of the following objective function [24]: NdP exp X Pcal Pi i OF ¼ exp Pi i¼1
!2
Hydrogen-IRMOF-1 (adsorption) Hydrogen-IRMOF-1 (desorption) Hydrogen-IRMOF-2 (adsorption) Hydrogen-IRMOF-2 (desorption) Hydrogen-IRMOF-3 (adsorption) Hydrogen-IRMOF-3 (desorption) Hydrogen-IRMOF-6 (adsorption) Hydrogen-IRMOF-6 (desorption) Hydrogen-IRMOF-8 (adsorption) Hydrogen-IRMOF-8 (desorption) Hydrogen-IRMOF-9 (adsorption) Hydrogen-IRMOF-9 (desorption) Hydrogen-IRMOF-11 (adsorption) Hydrogen-IRMOF-11 (desorption) Hydrogen-IRMOF-13 (adsorption) Hydrogen-IRMOF-13 (desorption) Hydrogen-IRMOF-18 (adsorption) Hydrogen-IRMOF-18 (desorption) Hydrogen-IRMOF-20 (adsorption) Hydrogen-IRMOF-20 (desorption)
Kij
AAD%
AAD% (Kij ¼ 0)
0.36 0.42 0.76 0.22 0.71 0.72 0.82 0.80 0.69 0.67 0.78 0.81 0.79 0.72 0.71 0.82 0.73 0.79 0.27 0.32
0.69 2.15 0.92 1.19 0.58 1.12 0.75 1.02 0.66 2.07 1.06 1.56 0.93 2.18 1.10 1.43 0.96 1.88 0.99 1.48
0.89 2.70 1.52 2.02 1.27 2.02 1.33 1.75 1.28 3.16 1.61 2.35 1.63 3.14 1.66 2.40 1.67 3.42 1.31 2.04
has been performed to obtain the value of each binary parameter by the external convergence algorithm. The optimized binary interaction parameters have been brought in Table 2. In addition, this table reports the percentage of average absolute deviation (AAD%) expressed by the following formula: NdP cal exp 100 X Pi Pi AAD% ¼ NP i¼1 Pexp i
(12)
In order to show the predictive feature of SL equation of state, hydrogen sorption isotherms have been calculated based only on IRMOFs and hydrogen characteristic properties, and this is the so-called zero-parameter model, i.e. Kij ¼ 0. Satisfactory results have been also obtained with no adjustable parameter. Actually, with taking Kij ¼ 0 the interaction
(11)
exp
and Pi are calculated and experimental hydrogen wherePcal i gas pressures and Ndp is the number of equilibrium data points. A bubble point calculation at constant temperature
Table 1 e SanchezeLacombe characteristic parameters. Component H2 [28] Zn [29] IRMOF-1 IRMOF-2 IRMOF-3 IRMOF-6 IRMOF-8 IRMOF-9 IRMOF-11 IRMOF-13 IRMOF-18 IRMOF-20
T*(K)
P*(MPa)
r*(Kg/m3)
45.89 7128.00 551.06 534.27 610.79 576.19 669.70 664.26 831.26 733.89 704.48 547.46
100.00 8602.79 505.82 487.85 324.56 485.27 559.91 540.58 514.02 577.55 606.57 812.93
152.66 6110.00 975.37 950.08 1081.98 971.80 1064.90 1046.72 1203.18 1100.38 1100.60 896.56
Fig. 1 e Prediction of H2 sorption and desorption isotherms in IRMOF-1, Experimental data[16].
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 2 1 0 7 6 e2 1 0 8 2
Fig. 2 e Prediction of H2 sorption and desorption isotherms in IRMOF-2, Experimental data[14].
energy between hydrogen molecule and IRMOF has been ignored. Therefore, the calculated %AAD for all studied systems is larger when Kij ¼ 0, but the SL equation of state still works satisfactorily. In this study, hydrogen adsorption on and desorption from IRMOFs -1, -2, -3, -6, -8, -10, -11, -13, -18 and -20 has been predicted at 77 K by GC-LS EoS establishing the local equilibrium in both sides of membrane, feed and permeate. The experimental sorption amount which is in terms of the weight of hydrogen gas sorbed in the mass unit of IRMOF has been converted to volume fraction in order to use in equilibrium calculations. The results of calculations have been explained in more details as bellow: Zn-based IRMOFs are in principle reticulated through the connection of Zinc metal ions by polycoordinating ligands, resulting in porous frameworks with the same structurally primitive topology which is cubic. The simplest member of IRMOFs series is IRMOF-1 first introduced in 2003 by Yaghi's research group [30] known as MOF-5. Rowsell et al. measured
Fig. 3 e Prediction of H2 sorption and desorption isotherms in IRMOF-3, Experimental data[14].
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Fig. 4 e Prediction of H2 sorption and desorption isotherms in IRMOF-6, Experimental data[14].
the amount of hydrogen adsorbed on and desorbed from IRMOF-1 at 77 K and gas pressures up to 80 105 Pa [16]. In this study, hydrogen sorption on IRMOF-1 has been modeled through GC-SL EoS. Fig. 1 shows the calculated and experimental hydrogen uptake in IRMOF-1. The comparison of theoretical results, either with using Kij or without using it, and experimental coexistence curves reveals that the proposed model represents the sorption data reasonably well. One structural feature that may affect the amount of hydrogen adsorbed on an IRMOF is the electronic character of the linking organic units. These structures contain phenylene cores with available positions for chemical functional groups as demonstrated in IRMOF-2, -3 and -6 with bromo-, aminoand dihydrocyclobuta-derivatives, respectively. The hydrogen sorption isotherms measured by Rowsell et al. [14] at 77 K and gas pressures up to 80 105 Pa have been shown in Figs. 2e4. In this work, the amount of hydrogen adsorption on and
Fig. 5 e Prediction of H2 sorption and desorption isotherms in IRMOF-8, Experimental data[16].
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Fig. 6 e Prediction of H2 sorption and desorption isotherms in IRMOF-9, Experimental data[14].
Fig. 8 e Prediction of H2 sorption and desorption isotherms in IRMOF-13, Experimental data[14].
desorption from the above-mentioned IRMOFs has been calculated using SL EoS with the help of GC procedure. In comparison with the previously studied system, i.e. IRMOF-1, the amount of hydrogen sorption is lower in IRMOF-2 whereas in IRMOF-3 and -6 larger uptake has been observed. It seems the interaction energy of dihydrogen with the organic links of these materials is different in magnitude. For IRMOF-2, -3 and -6, this may be explained by the occupation of the small pores by the pendant groups. It seems that decreasing the interaction distances in the small pores by adorning the links with pendant groups slightly raise the average interaction potential and improve the uptake. However, the increased uptake by this strategy is compensated by the presence of heavier groups such as bromo-instead result in decreased sorption capacity. Referring to Table 2 for the first four macromolecules, it is evident that from IRMOF-1 to IRMOF-6 the amount of Kij is going towards the more negative values. Decreasing
trend in the amount of binary interaction parameter is a sign of the increasing interaction between H2 molecules and the apertures bounded by the edges of inorganic clusters. One of the important parameters influencing the sorption of gases in crystal molecules is the amount of accessible surface area, which can be enhanced by increasing the organic linker length. Rowsell et al. [16] reported larger void regions in IRMOF-8 by considering naphthalene-2,6-dicarboxylate (NDC) as an organic linker. Fig. 5 shows the calculated and experimental hydrogen uptake values for IRMOF-8. Compared with IRMOF-6, IRMOF-8 exhibits larger hydrogen uptake. An alternative method for manipulating the pore dimension in IRMOFs is catenation. Catenation means the interpenetration of frameworks in which the void in one net is occupied by another net passing through it occurring due to increasing the length of the ligand. Fig. 6 represents the modeled and experimental hydrogen sorption isotherms for
Fig. 7 e Prediction of H2 sorption and desorption isotherms in IRMOF-11, Experimental data[15,16].
Fig. 9 e Prediction of H2 sorption and desorption isotherms in IRMOF-18, Experimental data[16].
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 2 1 0 7 6 e2 1 0 8 2
Fig. 10 e Prediction of H2 sorption and desorption isotherms in IRMOF-20, Experimental data[14].
catenated IRMOF-9 [14] which are in good agreement. The amount of hydrogen uptake decreases because of lower structure porosity. The experimental amount of H2 uptake in IRMOF-11 [15,16] and -13 [14] along with calculated isotherms have been brought in Figs. 7 and 8. In spite of the reduction in pore size of these systems, the higher uptake has been observed. IRMOF11 and -13 are interwoven somehow there is minimal separation between the inorganic Zn4O units which causes the enhancement of the overlap of the opposite walls attractive potential. A comparison of the structure of IRMOF-11 and -13 reveals that the organic ligand in IRMOF-13, i.e. pyrene, is completely aromatic which results in higher interaction and higher sorption of H2. As shown in Fig. 9, and among all IRMOFs considered in this work, IRMOF-18 has the lowest amount of hydrogen sorption. It is clear that the introduction of too many functional groups on organic backbone, leads to occupation and reduction of void size which in turn limits the uptake of H2. Again the SL EoS has predicted the hydrogen sorption experimental data [16] very well. The same as previously studied systems, there is a good agreement between the calculated and experimental [14] hydrogen sorption isotherms in Fig. 10 for IRMOF-20. The length of the linker in IRMOF-20 is longer and narrower than other above-mentioned macromolecules. Polarity has been also enhanced due to the presence of a heteroatom, i.e. sulfur, in organic backbone. Therefore with respect to other linkers which show less or no polarity such as IRMOF-1 and -18, the amount of hydrogen sorption increases in this macromolecule.
Conclusions In this study, hydrogen adsorption on and desorption from IRMOF-1, -2, -3, -6, -8, -9, -11, -13, -18 and -20 have been investigated by applying the SanchezeLacombe equation of
21081
state at temperature 77 K and a wide range of pressure 0e80 105 Pa with reasonable precision. It is the advantage of SL EoS that only needs three molecular characteristic parameters, namely, the characteristic density, r*, the characteristic pressure, P*, and the characteristic temperature, T*, which have been calculated using group contribution method. There is only one adjustable binary interaction parameter, Kij which is calculated from the fitting of isothermal hydrogen sorption experimental data. The average absolute deviation is about 0.14 with Kij s 0.0 and 0.37 with Kij ¼ 0.0. These results indicate that neglecting binary interaction parameter has no effect on the precision of the modeling in prediction of H2 sorption isotherms. Among all aforementioned IRMOFs, IRMOF-18 and IRMOF13 show the lowest and the highest hydrogen storage capacity. In spite of having the same cubic topology, but IRMOFs show various sorption behaviors because of the quality or quantity of functional groups adding on organic linker, the presence of a heteroatom or aromatic structure in the network and catenation phenomena which cause reduction in pore size and enhancement of interaction energy.
Acknowledgments The authors wish to thank the computer facilities provided by Shiraz University of Technology.
Nomenclature AAD P T P T P* T* rmix OF Kij nij Ndp Nc ui
average absolute deviation pressure, MPa absolute temperature, K reduced pressure, SanchezeLacombe model, dimensionless reduced temperature, SanchezeLacombe model, dimensionless characteristic parameter, SanchezeLacombe model, MPa characteristic parameter, SanchezeLacombe model, K mixture parameter, SanchezeLacombe model, dimensionless objective function SanchezeLacombe binary interaction parameter between component i and j interaction parameter defined in Equation (6) number of equilibrium data number of components in the mixture mass fraction of the component i in the mixture
Greek symbols interaction energy of the mixture, ε*mix SanchezeLacombe model, J/mol close-packed molar volume of a mer of the mixture, n*mix m3/mol r absolute density, kg/m3 r reduced density, SanchezeLacombe model, dimensionless
21082 r* m ε* εij n* P f
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 2 1 0 7 6 e2 1 0 8 2
characteristic close-packed mass density, SanchezeLacombe model, kg/m3 chemical potential meremer interaction energy, SanchezeLacombe model, J/mol potential energy, J/mol closed-packed molar volume of a mer, m3/mol summation volume fraction
Subscripts and superscripts cal calculated exp experiment G gas i,j type of component p polymer
references
[1] Leaf D, Verlomec HJH, Hunt WF. Overview of regulatory/ policy/economic issues related to carbon dioxide. Environ Int 2003;29:303e10. [2] Thomas JM, Raja R. Designing catalysts for clean technology, green chemistry, and sustainable development. Ann Rev Mater Res 2005;35:315e50. [3] Wang QK, Zhu CC, Liu WH, Wu T. Hydrogen storage by carbon nanotube and their films under ambient pressure. Int J Hydrogen Energy 2002;27:497e500. [4] Bououdina M, Grant D, Walker G. Review on hydrogen absorbing materials-structure, microstructure, and thermodynamic properties. Int J Hydrogen Energy 2006;31:177e82. [5] Kruk M, Jaroniec M. Gas adsorption characterization of ordered organic-inorganic nanocomposite materials. Chem Mater 2001;13:3169e83. n G, Valde s-Solı´s T. Towards the hydrogen economy? [6] Marba Int J Hydrogen Energy 2007;32:1625e37. [7] Rivarolo M, Bogarin J, Magistri L, Massardo AF. Timedependent optimization of a large hydrogen generation plant using ‘spilled’ water at Itaipu 14 GW hydraulic plant. Int J Hydrogen Energy 2012;37:5434e43. [8] Durbin DJ, Malardier-Jugroot C. Review of hydrogen storage techniques for on board vehicle applications. Int J Hydrogen Energy 2013;38:14595e617. [9] DOE hydrogen, fuel cells & infrastructure technologies program multi-year research, development, and demonstration plan, hydrogen storage technical plan. 2007. http://www1.eere.energy.gov/hydrogenandfuelcells/mypp/. [10] Rowsell JLC, Yaghi OM. Metal-organic frameworks: a new class of porous materials. Microporous Mesoporous Mater 2004;73:3e14. [11] Rowsell JLC, Yaghi OM. Strategies for hydrogen storage in metal organic frameworks. Angew Chem Int Ed 2005;44:4670e9. [12] Rowsell JLC, Millward AR, Park KS, Yaghi OM. Hydrogen sorption in functionalized metal-organic frameworks. J Am Chem Soc 2004;126:5666e7.
[13] Wong-Foy AG, Matzger AJ, Yaghi OM. Exceptional H2 saturation in microporous metal-organic frameworks. J Am Chem Soc 2006;128:3494e5. [14] Rowsell JLC, Yaghi OM. Effects of functionalization, catenation and variation of the metal oxide and organic linking units on the low-pressure hydrogen adsorption properties of metal-organic frameworks. J Am Chem Soc 2006;128:1304e15. [15] Eddaoudi M, Kim J, Rosi N, Vodak D, Wachter J, O'Keeffe M, et al. Systematic design of pore size and functionality in isoreticular MOFs and their application in methane storage. Science 2002;295:469e72. [16] Rowsell JLC, Millward AR, Park KS, Yaghi OM. Hydrogen sorption in functionslized metal-organic frameworks. J Am Chem Soc 2004;126:5666e7. [17] Rashidi MM, Rahimzadeh N, Laraqi N. Evaluation of the equations of state for air, nitrogen and oxygen on throttle reduction efficiency by using exergy analysis. Int J Exergy 2011;9:297e318. [18] Sabzi F, Molaei H. Sorption of CO2, C2H2 and C2H4in HOF-1a studied by PHSC equationof state. Fluid Phase Equilibria 2013;360:23e8. [19] Javidi Alesaadi S, Sabzi F. Hydrogen storage in a series of Znbased MOFs studied by PHSC equation of state. Int J Hydrogen Energy 2014;39:14851e7. [20] Tahmooresi M, Sabzi F. Sorption of methane in a series of Zn-based MOFs studied by PHSC equation of state. Fluid Phase Equilibria 2014;381:83e9. [21] De Angelis MG, Merkel TC, Bondar VI, Freeman BD, Dighiri F, Sarti GC. Hydrocarbon and fluorocarbon solubility and dilation in poly(dimethylsiloxane): comparison of experimental data with predictions of the Sanchez-Lacombe equation of state. J Polym Sci Part B Polym Phys 1999;37:3011e26. [22] Boudoris D, Constantinou L, Panayiotou C. Prediction of volumetric behavior and glass transition temperature of polymers: a group contribution approach. Fluid Phase Equilibria 2000;167:1e19. [23] Neau E. A consistent method for phase equilibrium calculation using the SanchezeLacombe latticeefluid equation-of-state. Fluid Phase Equilibria 2002;203:133e40. € la € J, Kokko E, [24] Bashir MA, Ali MA, Kanellopoulos V, Seppa Vijay S. The effect of pure component characteristic parameters on Sanchez-Lacombe equation of state predictive capabilities. Macromol React Eng 2013;7:193e204. [25] Sanchez IC, Lacombe RH. An elementary molecular theory of classical fluids. Pure fluids. J Chem Phys 1976;80:2352e62. [26] Sanchez IC, Lacombe RH. Statistical thermodynamics of polymer solutions. Macromolecules 1978;11:1145e56. [27] McHugh M, Krukonis V. Supercritical fluid extraction. 2nd ed. Stoneham, MA: Butterworth-Heinemann; 1994. [28] Kanellopoulos V, Mouratides D, Pladis P, Kiparissides C. Prediction of solubility of a-olefins in polyolefins using a combined equation of state molecular dynamics approach. Ind Eng Chem Res 2006;45:5870e8. [29] von Bergen Y, von Bergen R. A lattice fluid approach to liquid metals: equation of state parameters and periodic properties. Revista Latinoamericana de Metalurgia y Materiales 2000;20:79e86. [30] Rosi NL, Eckert J, Eddaoudi M, Vodak DT, Kim J, O'Keeffe M, et al. Hydrogen storage in microporous metal-organic frameworks science 2003;300:1127e9.
Available online at www.sciencedirect.com
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Hydrogen storage in a series of Zn-based IRMOFs studied by SanchezeLacombe equation of state Sajjad Javidi Alesaadi, Fatemeh Sabzi* Department of Chemical Engineering, Shiraz University of Technology, Shiraz 71555-313, Iran
article info
abstract
Article history:
In this work, the adsorptive behaviors of molecular hydrogen in 10 different isoreticular
Received 27 August 2014
metal-organic frameworks, namely, IRMOF-1, -2, -3, -6, -8, -10, -11, -13, -18 and -20 have
Received in revised form
been studied using SanchezeLacombe equation of state. SanchezeLacombe equation of
7 October 2014
state has three characteristic parameters: characteristic density, characteristic pressure,
Accepted 14 October 2014
and characteristic temperature which have been calculated from group contribution pro-
Available online 7 November 2014
cedure. The amount of hydrogen uptake in IRMOFs has been obtained through gasadsorbents phase equilibrium calculations at temperature 77 K and various pressures up
Keywords:
to 80 105 Pa. Finally, the results have been compared with the experimental data to show
IRMOF
the precision of SL equation of state in predicting the hydrogen sorption data.
Equation of state
Copyright © 2014, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
SanchezeLacombe Sorption Hydrogen
Introduction The rapid consumption of petroleum deposits and severe environmental impacts caused by burning fuel cells have driven the global research community to look for an alternative power source that is clean and sustainable [1,2]. Among many emerging new sources in the future, hydrogen is probably the best candidate [3e5]. However, the main bottleneck of hydrogen fuel cells is developing efficient, economic and safe hydrogen storing systems [6e8]. Without the new hydrogen storage system, it is difficult to switch to the hydrogen fuel cell engine because the existing hydrogen storage systems including compression orcryogenic storage are not economically viable and difficult to implement [9]. There is a third potential solution for hydrogen storage such as hydrogen adsorption in a new class of microporous adsorbents named
Metal Organic Frameworks (MOFs). Application of these functional materials is now a focus of considerable research due to their greater surface area and pore volumes [10e13]. Among all MOFs, Isoreticular Metal Organic Frameworks (IRMOFs) are a branch of MOFs which reticulated through the connection of octahedral Zn4O(O2Ce)6 secondary building units, resulting in frameworks based on the same primitive cubic topology. The alteration of linear ditopic carboxylate ligands has been shown to improve the hydrogen uptake level through the controlling specific surface area and pore sizes in the framework. In this study, IRMOF-1 prepared from benzene-1,4-dicarboxylic acid (BDC) and its bromo-, amino- and dihydrocyclobuta-derivatives i.e. IRMOF-2, -3 and -6 are viewed [14]. Other designs such as IRMOF-8 with naphthalene2,6-dicarboxylate (NDC) organic linker can also be wielded by the extension of the length between the metal centers leading to large void regions [15,16]. When the length of the linker
* Corresponding author. E-mail addresses: [email protected], [email protected] (F. Sabzi). http://dx.doi.org/10.1016/j.ijhydene.2014.10.064 0360-3199/Copyright © 2014, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
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reaches two phenyl rings as biphenyl-dicarboxylate (BPTC) in IRMOF-9 [14], interpenetration i.e. filling the void in one net by another net passing through it, is observed. The degree of interpenetration can be as high as fourfold when the linker contains a linear quarter-phenyl chain as 4,5,9,10tetrahydropyrenee2,7-dicarboxylate (HPDC) in IRMOF-11 [15,16] and pyrenee2,7-dicarboxylate (PDC) in IRMOF-13 [14]. An organic ligand with four methyl functional groups, 2,3,5,6-tetramethylbenzene-1,4-dicarboxylate (TMBDC), is considered in IRMOF-18 to study the effect of pore volume occupation [16]. Heteroatoms can be also incorporated in the aromatic backbone of the organic links, by using thieno[3,2-b] thiophene-2,5-dicarboxylic acide (TTDC) as the precursor IRMOF-20 [14]. Calculating the thermodynamic properties of gaseous processes such as exergy analysis has been performed for air, nitrogen and oxygen using three kinds of cubic equations of state [17]. Investigation on the solubility of gases in organic frameworks through the equation of state (EoS) was started in our group by publishing the work on the sorption of CO2, C2H2 and C2H4 in hydrogen-bonded organic framework (HOF-1a) using Perturbed Hard Sphere Chain Equation of State (PHSC EoS) [18]. This model has been also adopted to predict the hydrogen [19] and methane [20] storage in five adsorbents including MOF-5, MOF-177, MOF-200, MOF-205 and MOF-210. In this work, the SanchezeLacombe (SL) equation of state has been employed to examine its capability for interpreting hydrogen sorption data in above-mentioned macromolecules. The well-known lattice-fluid model of SL has been previously applied to describe properties of binary or multi-component ordinary compounds and polymeric systems [21e24]. For gas solubility prediction, the knowledge of three characteristic parameters, i.e. the characteristic density, r*, the characteristic pressure, P*, and the characteristic temperature, T*, is required. These parameters have been calculated for both hydrogen molecules and IRMOFs in order to use as scaling constants to correlate sorption data. Comparison of experimental and theoretical coexistence curves shows that SL EoS works very well in predicting hydrogen sorption behavior.
r ¼ P* M RT* r*
(3)
Note that for a high molecular weight polymer the value of r can be considered to be infinite, then (1/r) can be neglected. T*, P*, r* and V*are characteristic parameters which defined as [23]: T* ¼
ε* ; R
P* ¼
V* ¼ N rn* ;
ε* ; n*
r* ¼
M rn*
(4)
Where ε* and n* are the characteristic interaction energy per mer and the close-packed volume of a mer, respectively. N is the number of molecules. For a polymer-gas mixture it is necessary to use a mixing rule for the calculation of n*mix , ε*mix and rmix based on the corresponding values of the pure component parameters. There are different types of mixing rules for calculating mixture properties [23]. In this research work, the van der Waals mixing rule has been used for calculating the mixture properties as follows: n*mix ¼
Nc X Nc X i¼1
fi fj n*ij
(5)
j¼1
where n*ij ¼
n*ii þ n*jj 2
1 nij
(6)
with the parameter nij which accounts for the possible deviation ofn*ij from the arithmetic mean of the corresponding values,n*ii and n*jj , of the pure components. In this work, the value of the interaction parameter nij was assumed to be equal to zero. Accordingly, the values ofε*mix and rmix of the mixture will be obtained by the following equations: ε*mix
¼
r1 mix ¼
Nc X Nc 1 X
n*mix
i¼1
0 1
0:5
* * 1 kij An*ij fi fj @ εii εjj
(7)
j¼1
Nc
X fj rj
(8)
j¼1
Theory SanchezeLacombe EoS is based on lattice-fluid theory that treats the polymer chains as a set of interacting beads in a lattice. Following the original developments of Sanchez and Lacombe the general SL EoS can be written as [25,26]: 1 r2 þ P þ T ln 1 r þ 1 r ¼ 0 r
(1)
where r is the reduced density, P the reduced pressure and T the reduced temperature of a pure component. The parameters are reduced by characteristic constants as follows: r¼
r 1 V* ¼ ¼ ; * r n V
P¼
P ; P*
T¼
T T*
(2)
r is a size parameter that represents the number of lattice sites occupied by a molecule and can be related to the molecular weight according to the following equation:
where kij is a binary interaction parameter which represents the interaction energy between the ith and jth species in the mixture. fi is the volume fraction of the ith component in the mixture that can be expressed in terms of the mass fraction ui, the characteristic densityr*i and the characteristic volume n*i of the pure components: 2 !31 Nc uj 5 ui 4 X fi ¼ * * ri ni j¼1 r*j n*j
(9)
The chemical potential of each component in the two phases, one phase is gas and the other one consists of polymer and the sorbed gaseous molecules, must be set equal at the equilibrium state, i.e. mGi ¼ mPi . Following the developments of McHugh and Krukonis [27], the chemical potential of the ith component in a multi-component system can be expressed as:
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ri mi ¼RT ln fi þ 1 rmix 2 0 1 33 2 Nc Nc X X 2 fj n*ij ε*ij ε*mix fj n*ij A þ ε*mix 55 þ ri 4 r4 * @ nmix j¼1 j¼1 3 2 r RTn ð1 rÞlnð1 rÞ þ lnr ri 7 6 7 6 1 7 þ ri 6 7 6 Nc P * 5 4 * A þPn 2 fj n n j¼1
ij
Table 2 e Adjusted binary interaction parameters and statistical errors. Pairs (10)
mix
Results and discussions The exact estimation of the characteristic density, r*, the characteristic pressure, P*, and the characteristic temperature, T*, is of high importance for SL EoS to behave accurately. The regression of the PVT data of pure components is a simple method to find the characteristic parameters, but PVT data is not available for metal organic frameworks. Then, the group contribution method handed in by Boudouris et al. [22] has been chosen for extracting these scaling constants. From the knowledge of only the molecular structure of IRMOFs' repeating unit cells, first order and second order groups have been identified. The characteristic properties have been estimated by summing the contribution of each group in one specific property. Table 1 shows the calculated characteristic parameters for all the individual compounds of interest. Having in hand the three parameters characteristic to each component in the hydrogen and IRMOF systems, the solubility of gas can be calculated. Then, for each IRMOF/hydrogen pair the model contains only one parameter, which characterizes interactions between gas and macromolecules. This binary interaction parameter has been adjusted by fitting the parameter Kij to experimental hydrogen sorption data at 77 K. Regression of data points has been done by making use of the following objective function [24]: NdP exp X Pcal Pi i OF ¼ exp Pi i¼1
!2
Hydrogen-IRMOF-1 (adsorption) Hydrogen-IRMOF-1 (desorption) Hydrogen-IRMOF-2 (adsorption) Hydrogen-IRMOF-2 (desorption) Hydrogen-IRMOF-3 (adsorption) Hydrogen-IRMOF-3 (desorption) Hydrogen-IRMOF-6 (adsorption) Hydrogen-IRMOF-6 (desorption) Hydrogen-IRMOF-8 (adsorption) Hydrogen-IRMOF-8 (desorption) Hydrogen-IRMOF-9 (adsorption) Hydrogen-IRMOF-9 (desorption) Hydrogen-IRMOF-11 (adsorption) Hydrogen-IRMOF-11 (desorption) Hydrogen-IRMOF-13 (adsorption) Hydrogen-IRMOF-13 (desorption) Hydrogen-IRMOF-18 (adsorption) Hydrogen-IRMOF-18 (desorption) Hydrogen-IRMOF-20 (adsorption) Hydrogen-IRMOF-20 (desorption)
Kij
AAD%
AAD% (Kij ¼ 0)
0.36 0.42 0.76 0.22 0.71 0.72 0.82 0.80 0.69 0.67 0.78 0.81 0.79 0.72 0.71 0.82 0.73 0.79 0.27 0.32
0.69 2.15 0.92 1.19 0.58 1.12 0.75 1.02 0.66 2.07 1.06 1.56 0.93 2.18 1.10 1.43 0.96 1.88 0.99 1.48
0.89 2.70 1.52 2.02 1.27 2.02 1.33 1.75 1.28 3.16 1.61 2.35 1.63 3.14 1.66 2.40 1.67 3.42 1.31 2.04
has been performed to obtain the value of each binary parameter by the external convergence algorithm. The optimized binary interaction parameters have been brought in Table 2. In addition, this table reports the percentage of average absolute deviation (AAD%) expressed by the following formula: NdP cal exp 100 X Pi Pi AAD% ¼ NP i¼1 Pexp i
(12)
In order to show the predictive feature of SL equation of state, hydrogen sorption isotherms have been calculated based only on IRMOFs and hydrogen characteristic properties, and this is the so-called zero-parameter model, i.e. Kij ¼ 0. Satisfactory results have been also obtained with no adjustable parameter. Actually, with taking Kij ¼ 0 the interaction
(11)
exp
and Pi are calculated and experimental hydrogen wherePcal i gas pressures and Ndp is the number of equilibrium data points. A bubble point calculation at constant temperature
Table 1 e SanchezeLacombe characteristic parameters. Component H2 [28] Zn [29] IRMOF-1 IRMOF-2 IRMOF-3 IRMOF-6 IRMOF-8 IRMOF-9 IRMOF-11 IRMOF-13 IRMOF-18 IRMOF-20
T*(K)
P*(MPa)
r*(Kg/m3)
45.89 7128.00 551.06 534.27 610.79 576.19 669.70 664.26 831.26 733.89 704.48 547.46
100.00 8602.79 505.82 487.85 324.56 485.27 559.91 540.58 514.02 577.55 606.57 812.93
152.66 6110.00 975.37 950.08 1081.98 971.80 1064.90 1046.72 1203.18 1100.38 1100.60 896.56
Fig. 1 e Prediction of H2 sorption and desorption isotherms in IRMOF-1, Experimental data[16].
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 2 1 0 7 6 e2 1 0 8 2
Fig. 2 e Prediction of H2 sorption and desorption isotherms in IRMOF-2, Experimental data[14].
energy between hydrogen molecule and IRMOF has been ignored. Therefore, the calculated %AAD for all studied systems is larger when Kij ¼ 0, but the SL equation of state still works satisfactorily. In this study, hydrogen adsorption on and desorption from IRMOFs -1, -2, -3, -6, -8, -10, -11, -13, -18 and -20 has been predicted at 77 K by GC-LS EoS establishing the local equilibrium in both sides of membrane, feed and permeate. The experimental sorption amount which is in terms of the weight of hydrogen gas sorbed in the mass unit of IRMOF has been converted to volume fraction in order to use in equilibrium calculations. The results of calculations have been explained in more details as bellow: Zn-based IRMOFs are in principle reticulated through the connection of Zinc metal ions by polycoordinating ligands, resulting in porous frameworks with the same structurally primitive topology which is cubic. The simplest member of IRMOFs series is IRMOF-1 first introduced in 2003 by Yaghi's research group [30] known as MOF-5. Rowsell et al. measured
Fig. 3 e Prediction of H2 sorption and desorption isotherms in IRMOF-3, Experimental data[14].
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Fig. 4 e Prediction of H2 sorption and desorption isotherms in IRMOF-6, Experimental data[14].
the amount of hydrogen adsorbed on and desorbed from IRMOF-1 at 77 K and gas pressures up to 80 105 Pa [16]. In this study, hydrogen sorption on IRMOF-1 has been modeled through GC-SL EoS. Fig. 1 shows the calculated and experimental hydrogen uptake in IRMOF-1. The comparison of theoretical results, either with using Kij or without using it, and experimental coexistence curves reveals that the proposed model represents the sorption data reasonably well. One structural feature that may affect the amount of hydrogen adsorbed on an IRMOF is the electronic character of the linking organic units. These structures contain phenylene cores with available positions for chemical functional groups as demonstrated in IRMOF-2, -3 and -6 with bromo-, aminoand dihydrocyclobuta-derivatives, respectively. The hydrogen sorption isotherms measured by Rowsell et al. [14] at 77 K and gas pressures up to 80 105 Pa have been shown in Figs. 2e4. In this work, the amount of hydrogen adsorption on and
Fig. 5 e Prediction of H2 sorption and desorption isotherms in IRMOF-8, Experimental data[16].
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Fig. 6 e Prediction of H2 sorption and desorption isotherms in IRMOF-9, Experimental data[14].
Fig. 8 e Prediction of H2 sorption and desorption isotherms in IRMOF-13, Experimental data[14].
desorption from the above-mentioned IRMOFs has been calculated using SL EoS with the help of GC procedure. In comparison with the previously studied system, i.e. IRMOF-1, the amount of hydrogen sorption is lower in IRMOF-2 whereas in IRMOF-3 and -6 larger uptake has been observed. It seems the interaction energy of dihydrogen with the organic links of these materials is different in magnitude. For IRMOF-2, -3 and -6, this may be explained by the occupation of the small pores by the pendant groups. It seems that decreasing the interaction distances in the small pores by adorning the links with pendant groups slightly raise the average interaction potential and improve the uptake. However, the increased uptake by this strategy is compensated by the presence of heavier groups such as bromo-instead result in decreased sorption capacity. Referring to Table 2 for the first four macromolecules, it is evident that from IRMOF-1 to IRMOF-6 the amount of Kij is going towards the more negative values. Decreasing
trend in the amount of binary interaction parameter is a sign of the increasing interaction between H2 molecules and the apertures bounded by the edges of inorganic clusters. One of the important parameters influencing the sorption of gases in crystal molecules is the amount of accessible surface area, which can be enhanced by increasing the organic linker length. Rowsell et al. [16] reported larger void regions in IRMOF-8 by considering naphthalene-2,6-dicarboxylate (NDC) as an organic linker. Fig. 5 shows the calculated and experimental hydrogen uptake values for IRMOF-8. Compared with IRMOF-6, IRMOF-8 exhibits larger hydrogen uptake. An alternative method for manipulating the pore dimension in IRMOFs is catenation. Catenation means the interpenetration of frameworks in which the void in one net is occupied by another net passing through it occurring due to increasing the length of the ligand. Fig. 6 represents the modeled and experimental hydrogen sorption isotherms for
Fig. 7 e Prediction of H2 sorption and desorption isotherms in IRMOF-11, Experimental data[15,16].
Fig. 9 e Prediction of H2 sorption and desorption isotherms in IRMOF-18, Experimental data[16].
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 2 1 0 7 6 e2 1 0 8 2
Fig. 10 e Prediction of H2 sorption and desorption isotherms in IRMOF-20, Experimental data[14].
catenated IRMOF-9 [14] which are in good agreement. The amount of hydrogen uptake decreases because of lower structure porosity. The experimental amount of H2 uptake in IRMOF-11 [15,16] and -13 [14] along with calculated isotherms have been brought in Figs. 7 and 8. In spite of the reduction in pore size of these systems, the higher uptake has been observed. IRMOF11 and -13 are interwoven somehow there is minimal separation between the inorganic Zn4O units which causes the enhancement of the overlap of the opposite walls attractive potential. A comparison of the structure of IRMOF-11 and -13 reveals that the organic ligand in IRMOF-13, i.e. pyrene, is completely aromatic which results in higher interaction and higher sorption of H2. As shown in Fig. 9, and among all IRMOFs considered in this work, IRMOF-18 has the lowest amount of hydrogen sorption. It is clear that the introduction of too many functional groups on organic backbone, leads to occupation and reduction of void size which in turn limits the uptake of H2. Again the SL EoS has predicted the hydrogen sorption experimental data [16] very well. The same as previously studied systems, there is a good agreement between the calculated and experimental [14] hydrogen sorption isotherms in Fig. 10 for IRMOF-20. The length of the linker in IRMOF-20 is longer and narrower than other above-mentioned macromolecules. Polarity has been also enhanced due to the presence of a heteroatom, i.e. sulfur, in organic backbone. Therefore with respect to other linkers which show less or no polarity such as IRMOF-1 and -18, the amount of hydrogen sorption increases in this macromolecule.
Conclusions In this study, hydrogen adsorption on and desorption from IRMOF-1, -2, -3, -6, -8, -9, -11, -13, -18 and -20 have been investigated by applying the SanchezeLacombe equation of
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state at temperature 77 K and a wide range of pressure 0e80 105 Pa with reasonable precision. It is the advantage of SL EoS that only needs three molecular characteristic parameters, namely, the characteristic density, r*, the characteristic pressure, P*, and the characteristic temperature, T*, which have been calculated using group contribution method. There is only one adjustable binary interaction parameter, Kij which is calculated from the fitting of isothermal hydrogen sorption experimental data. The average absolute deviation is about 0.14 with Kij s 0.0 and 0.37 with Kij ¼ 0.0. These results indicate that neglecting binary interaction parameter has no effect on the precision of the modeling in prediction of H2 sorption isotherms. Among all aforementioned IRMOFs, IRMOF-18 and IRMOF13 show the lowest and the highest hydrogen storage capacity. In spite of having the same cubic topology, but IRMOFs show various sorption behaviors because of the quality or quantity of functional groups adding on organic linker, the presence of a heteroatom or aromatic structure in the network and catenation phenomena which cause reduction in pore size and enhancement of interaction energy.
Acknowledgments The authors wish to thank the computer facilities provided by Shiraz University of Technology.
Nomenclature AAD P T P T P* T* rmix OF Kij nij Ndp Nc ui
average absolute deviation pressure, MPa absolute temperature, K reduced pressure, SanchezeLacombe model, dimensionless reduced temperature, SanchezeLacombe model, dimensionless characteristic parameter, SanchezeLacombe model, MPa characteristic parameter, SanchezeLacombe model, K mixture parameter, SanchezeLacombe model, dimensionless objective function SanchezeLacombe binary interaction parameter between component i and j interaction parameter defined in Equation (6) number of equilibrium data number of components in the mixture mass fraction of the component i in the mixture
Greek symbols interaction energy of the mixture, ε*mix SanchezeLacombe model, J/mol close-packed molar volume of a mer of the mixture, n*mix m3/mol r absolute density, kg/m3 r reduced density, SanchezeLacombe model, dimensionless
21082 r* m ε* εij n* P f
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 2 1 0 7 6 e2 1 0 8 2
characteristic close-packed mass density, SanchezeLacombe model, kg/m3 chemical potential meremer interaction energy, SanchezeLacombe model, J/mol potential energy, J/mol closed-packed molar volume of a mer, m3/mol summation volume fraction
Subscripts and superscripts cal calculated exp experiment G gas i,j type of component p polymer
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