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Predicting phase equilibrium for polymer solutions using COSMO-SAC
Accepted Manuscript Predicting phase equilibrium for polymer solutions using COSMO-SAC Paula B. Staudt, Renata L. Simões, Leonardo Jacques, Nilo S.M. Cardozo, Rafael de P. Soares PII:
S0378-3812(18)30183-3
DOI:
10.1016/j.fluid.2018.05.003
Reference:
FLUID 11825
To appear in:
Fluid Phase Equilibria
Received Date: 9 August 2017 Revised Date:
20 April 2018
Accepted Date: 1 May 2018
Please cite this article as: P.B. Staudt, R.L. Simões, L. Jacques, N.S.M. Cardozo, R.d.P. Soares, Predicting phase equilibrium for polymer solutions using COSMO-SAC, Fluid Phase Equilibria (2018), doi: 10.1016/j.fluid.2018.05.003. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Predicting phase equilibrium for polymer solutions using COSMO-SAC Paula B. Staudt, Renata L. Sim˜oes, Leonardo Jacques, Nilo. S. M. Cardozo, Rafael de P. Soares∗
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Laborat´orio Virtual de Predi¸ca˜ o de Propriedades - LVPP, Departamento de Engenharia Qu´ımica, Escola de Engenharia, Universidade Federal do Rio Grande do Sul, Rua Ramiro Barcelos, 2777, Bairro Santana, CEP 90035-007, Porto Alegre, RS, Brazil
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Abstract
With the aim of predicting mixture behavior based on information of pure components only, COSMO-based models have emerged as a promising alternative. Several works in the literature are available attesting the good performance of COSMO-RS variants in several different applications. However, the extension of COSMO calculations for polymers and other macromolecules was not extensively explored. In this work, a new procedure to evaluate the parameters of the polymers repeating unit for COSMO-
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based models is proposed. The idea of a large polymer molecule consisting of many repeating units is employed, however, a more comprehensive analysis using oligomers of different sizes is suggested. The COSMO-SAC model with parameters from the literature was used to test the new methodology. Infinite dilution activity coefficient (IDAC)
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and vapor-liquid equilibrium (VLE) calculations with homopolymer and copolymer systems were performed to verify the predictive capability of the approach. The good results obtained showed the adequacy of the proposed methodology to represent IDAC
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and VLE data of solvent-polymer and -copylymer systems. Keywords: COSMO-based models, polymer solution, VLE equilibrium, IDAC data
author. Tel.:+55 51 33083528; fax: +55 51 33083277 Email address: [email protected] (Rafael de P. Soares)
∗ Corresponding
Preprint submitted to Fluid Phase Equilibria
May 4, 2018
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1. Introduction Today, predictive models to calculate phase equilibrium of multicomponent sys-
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tems are gathering progressively more attention in chemical engineering research. In
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this context, group contribution models are important tools. For instance, the UNI-
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FAC model (Fredenslund et al., 1975) and its variants are widely used in research and
6
practical applications, mainly because once the interaction parameters are available, no
7
additional information is needed for phase equilibrium calculations. However, even for
8
groups with existing parameters, problems may be encountered. Further, the number
9
of groups with parameters available is limited and unreliable predictions may be expe-
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rienced for molecules with several functional groups or when functional groups appear
11
in an unusual way (Kehiaian, 1983; Wu and Sandler, 1991; Abildskov et al., 1996,
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1999; Soares and Gerber, 2013). Further, UNIFAC models are not entirely predictive
13
models, since the energetic interaction parameters (for each group pair) need to be
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optimized with experimental data (such as liquid-liquid, vapor-liquid and solid-liquid
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phase equilibrium, excess enthalpy, infinite dilution activity coefficients, etc.).
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In the aim of predicting mixture behavior based on information of pure components
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only, the COSMO-based models come as a promising alternative. The COSMO-RS
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model, developed by Klamt and co-workers (Klamt, 1995; Klamt et al., 1998; Klamt
19
and Eckert, 2000), was the first method to combine quantum chemical calculations with
20
statistical thermodynamics to estimate the activity coefficients for species in a mixture.
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Based on COSMO-RS, Lin and Sandler (2002) proposed the COSMO-SAC model
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that included the Staverman-Guggenheim expression to improve the combinatorial
23
contribution in the activity prediction. Many other works based on COSMO-RS and
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COSMO-SAC formulations could be cited regarding refinement or model parametriza-
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tion (Grensemann and Gmehling, 2005; Mu et al., 2007; Wang et al., 2007; Gerber and
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Soares, 2010; Hsieh et al., 2010; Soares, 2011).
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Good description of different kinds of systems are reported in the literature when
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applying COSMO-type models for thermodynamic calculations, ranging from solu-
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tions of small molecules at high pressure (Shimoyama et al., 2006; Constantinescu
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et al., 2005) to aqueous micellar solutions (Mokrushina et al., 2012), pharmaceutics
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and drug-like molecules solubility (Tung et al., 2008; Bouillot et al., 2013), and ionic
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liquids (Banerjee et al., 2006; Freire et al., 2007).
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Regarding polymer solutions, the literature about application of COSMO methods
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is relatively scarce (Yang et al., 2010; Goss, 2011; Reinisch et al., 2011; D´ıaz et al.,
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2013; Panayiotou, 2013; Shah and Yadav, 2013; Kuo et al., 2013). As the quantum
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chemical calculations for very large molecules like polymers are nearly impossible with
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the computational resources available today, a few papers presented specific method-
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ologies to perform the COSMO calculations for polymer molecules (Delley, 2006;
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Yang et al., 2010; Goss, 2011; Kuo et al., 2013; D´ıaz et al., 2013). The basic idea is
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that each polymer molecule consists of many repeating units and the model parameters
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(charge-density profile, cavity volume and cavity surface area) of the entire molecule
42
are calculated by summing up the values of the repeating units. The difference be-
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tween each procedure is how to identify and evaluate the repeating unit parameters. In
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the work of Delley (2006), the COSMO method was generalized to periodic boundary
45
conditions, allowing quantum calculations of infinite systems. Thus, by simply defin-
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ing the repeating unit, the molecular calculations are assumed to be valid to polymers
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and solid surfaces. In the research of Yang et al. (2010), several multimers containing
48
from 1 to 10 repeating units were constructed with the addition of head groups to com-
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plement the unsaturated chemical bonds. The most used head groups were the hydro-
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gen atom and the methyl group. After geometry optimization and energy calculation of
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the multimers, the mean value of the properties differences between two neighboring
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multimers correspond to the repeating unit properties. The disadvantage of considering
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the difference between consecutive multimers is the appearance of unrealistic negative
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values in the charge distribution profile. In Goss (2011), the author studied equilibrium
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sorption constants of chemicals in different polymers. The polymers were represented
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either by very small oligomers or monomers, end-capped with CH3 groups. In the fi-
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nal application the surface of the end-groups are disregarded. Kuo et al. (2013) used
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trimers ended capped with hydrogen atoms or the methyl groups to evaluate the sur-
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face area and charge of the homopolymer repeating unit. For copolymers, tetramers
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are used.
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Boucher (2015) studying the charge characteristics of 48 binary poly(3-hexylthiophene)– 3
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solvent mixtures with COSMO-RS model, used the 3-hexylthiophene monomer to rep-
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resent the polymer σ-profile. Parnis et al. (2016) modelled the properties of polyurethane
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foam (PUF) as an air sampling medium using COSMO-RS theory. To represent PUF
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polymer, different condensed isocyanate and polyol monomeric units were combined.
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Best results were obtained using 2,4-toluene diisocyante and glycerol combination.
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Also searching for polymer applications as passive air sampling, Okeme et al. (2016)
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studied the polydimethylsiloxane (PDMS)-air partition ratios for semi-volatile organic
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compounds. Hexamethyldisiloxane was employed as a chemical surrogate for poly-
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meric PDMS. The authors applied the Goss (2011) approach and reported deviations
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around 1% between the use of the monomer structure and the correction discussed by
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Goss (2011).
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In this work, a new procedure to evaluate the repeating unit parameters necessary
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for COSMO-based models is proposed. The idea of a large polymer molecule con-
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sisting of many repeating units is still employed, but a more comprehensive analysis
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of different oligomers when defining the appropriate repeating unit is suggested. The
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COSMO-SAC was used to test the new methodology with all model parameters taken
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from the literature. Infinite dilution activity coefficient (IDAC) and vapor-liquid equi-
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librium (VLE) calculations with homopolymer and copolymer systems were performed
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to verify the predictive capability of the approach and the results obtained here were
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compared to those from Kuo et al. (2013).
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2. The COSMO-SAC model
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In COSMO-based models, as in many group contribution methods, the activity coefficient is defined as a result of two contributions:
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ln γi = ln γires + ln γicomb
(1)
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The residual part is defined as the difference between the free energies of restoring
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the charges around the solute molecule in solution S and restoring the charges in a pure
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liquid i:
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ln γires =
∗res (∆G∗res i/s − ∆G i/i )
(2)
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The restoring free energies are computed by the COSMO-RS surface interacting theory where, for each contact between segments m and n, the interaction energy is given by: ! α0 HB (σm + σn )2 + Em,n (3) ∆Wm,n = 2
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where α0 = fpol 0.3a3/2 eff /0 is the constant for the misfit energy; 0 is the permittivity of
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2 vacuum; fpol is the polarization factor; aeff = πreff is the effective segment area with
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its respective radius reff ; σm and σn are the apparent surface charge densities of the
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contact segments m and n, respectively. In this work the values of reff = 1.52Å and
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HB fpol = 0.6917 were used, according to Wang et al. (2007). Em,n should account for
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hydrogen bond formation:
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HB Em,n = cHB max[0, σacc − σHB ]min[0, σdon + σHB ]
(4)
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where σacc and σdon are the larger and smaller σm and σn values. The constant for the
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hydrogen-bonding interaction cHB = 85580 kcal/mol Å4 /e2 and the cutoff σHB = 0.0084 e/Å2
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were taken from Lin and Sandler (2002).
Following the formulation presented in Gerber and Soares (2010), the resulting
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contacts in solution are computed by a statistical thermodynamics treatment as a func-
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tion of the composition and apparent surface charges distribution, the so-called σ-
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profiles. These profiles are given by the probability of finding an element with a charge
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density σ in a pure substance i: pi (σ) =
Ai (σ) Ai
(5)
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where Ai is the total cavity surface area, and Ai (σ) is the total surface area of all of the
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segments with a particular charge density σ. The calculation to obtain the σ-profile of a molecule starts with the computation
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of the ideal screening charge density distribution over the molecular surface using
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COSMO. The result is a surface divided into segments, each one with its own area and
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charge density. Finally, the three-dimensional geometric charge density distribution is
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averaged using another standard radius, ravg and then projected onto a histogram, which
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is known as the σ-profile, pi (σ). In this work the solvent σ-profiles were obtained as
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described by Gerber and Soares (2013), with the averaging radius ravg = 1.52Å (Wang
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et al., 2007). 5
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The combinatorial contribution ln γicomb is usually smaller than the residual one
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for small molecules. For polymer solutions this term becomes very important since
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it is responsible for taking into account differences in size and shape. The relevance
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of free-volume effects for polymer systems is widely discussed in the literature (Pat-
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terson, 1969; Oishi and Prausnitz, 1978; Elbro et al., 1990; Kontogeorgis et al., 1993,
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1994; Voutsas et al., 1995; Zhong et al., 1996; Kouskoumvekaki et al., 2002; Radfarnia
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et al., 2007; Loschen and Klamt, 2014). According to the authors, the inclusion of
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free volume effects in the combinatorial expression significantly improves the model
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accuracy in both VLE and LLE predictions. In order to evaluate the free-volume con-
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tribution, the polymer molar volume is necessary along with its van der Waals or hard
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core volume.
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In this work, the formulation recommended in Soares (2011) for the COSMO-SAC method is considered:
! ! φi φi z ln γicomb = ln φ0i + 1 − φ0i − qi ln +1− 2 θi θi
(6)
Eq. 6 consists of the well known Staverman-Guggenheim combinatorial contribution P with an exponent modification in the Flory-Huggins term, where φi = ri / j r j x j and P P φ0i = riR xi / j rRj x j are the normalized volume fraction; θi = qi / j q j x j is the normal-
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ized surface-area fraction; z is the coordination number, taken as 10; xi is the mole
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fraction; ri = Vi /r and qi = Ai /q are the normalized volume and surface-area, respec-
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tively; Ai is the cavity surface area and Vi is the cavity volume; q and r are universal
139
parameters of the model. Actually, since the model is independent of r, only the quo-
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tient z/q is relevant (Gerber and Soares, 2010). In the present work q is assumed as
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124 Å2 , as previosly reported by Soares (2011).
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For the exponent R, an expression similar to the one proposed by Voutsas et al.
(1995) was considered, for symmetric and asymmetric mixtures: ! Vi,small R= p 1− Vi,large
(7)
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where, in this work, Vi,small and Vi,large are the solvent and polymer cavity volume and
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p was taken simply as 1.
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It is worth noting that no free-volume effect is taken into account in this paper. The
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main reason to disregard such contribution is the fact that accurate experimental infor-
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mation about polymer density (or volume) is frequently unavailable. Although Loschen
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and Klamt (2014) recommended in their work the use of the free volume contribution
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of Elbro et al. (1990), the authors pointed out that this term is considerably sensitive
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to small changes in the density and a sufficiently accurate density of the polymer is
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needed. Additionally, a more extensive understanding of the proposed methodology
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can be performed with a classical model formulation with no use of extra information.
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It is likely that better results would be possible with a more sophisticated formulation
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in the combinatorial term.
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3. Extension to polymers
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As already mentioned, the available methodologies to perform COSMO calcula-
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tions for polymers are based on the structure of the repeating unit of the material. This
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is mainly because it is currently impractical (if not impossible) to accomplish the quan-
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tum mechanical calculations for actual very large macromolecules. To assemble the
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multimer structures and to accomplish a preliminary structure optimization the Jmol
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application (web: http://www.jmol.org/) and Avogadro software (Hanwell et al., 2012)
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version 1.0.3-5 were used. The COSMO calculations were performed with MOPAC
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(MOPAC2009) as described by Gerber and Soares (2013).
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In this work, the identification of a repeating unit is the initial step for the construc-
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tion of the σ-profiles of polymers. With the structure of the repeating unit defined,
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several oligomers were assembled for each polymer studied, always with an odd num-
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ber of units. To complement the unsaturated chemical bonds of the molecule, hydrogen
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atoms are added as end-cap groups. Each oligomer structure is optimized and then the
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apparent surface charges are calculated as in usual molecules. Examples of oligomer
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molecules are depicted in Figure 1 for linear polyethylene (LPE), poly(dimethyl silox-
173
ane) (PDMS), poly(caprolactone) (PCL), and poly(vinyl methyl ether) (PVME).
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After the whole molecule apparent surface charge is evaluated, only the surface
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of the central unit of the oligomer is kept, being the remaining surface portion disre-
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garded, as represented in Figure 1. By doing this, the σ-profile of the repeating unit
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(b) PDMS
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(a) LPE
(c) PVME
(d) PCL
Figure 1: Surface charge distribution of the central unit for oligomer structures: LPE with 9 repeating units,PDMS with 11 repeating units, PVME with 9 repeating units, and PCL with 3 repeating units.
alone is obtained. Naturally, this procedure is only an approximation and, in principle,
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the larger the oligomer the better because of the effect of the end groups in the central
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unit. In order to evaluate the minimum number of units necessary to obtain the ac-
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tual profile of a repeating unit, this procedure is repeated with an increasing number of
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units until constant responses are obtained. The properties investigated in this step are
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the activity coefficient at infinite dilution (IDAC) for different repeating unit-solvent
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pairs; the σ-profiles of the repeating unit; and the total charge of the unit, which must
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tend to its minimum. Due to computational difficulties, the largest oligomer studied
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consisted in 25 repeating units. For most polymers the procedure could be stopped
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with smaller molecules. The area and volume parameters for the repeating unit are
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determined from the difference in total volume and area of the two largest consecutive
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oligomers computed. Finally, the parameters of the polymer are obtained by multiply-
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ing the repeating unit parameters by the number of basic units of the macromolecule,
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given by the number-average molar weight of the polymer divided by the molecular
191
weight of the repeating unit. If the number average molecular weight is not available,
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the weight average molecular weight would be used instead.
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When dealing with copolymers, the copolymer unit is built with information of the
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monomers (homopolymer) repeating units. The number of repeating units of monomer
195
A and monomer B are estimated from the reported weight fractions of A and B in
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the material and the copolymer number everage molecular mass. Consequently, the
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number of repeating units of the copolymer is the sum of repeating units of A and
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B. The copolymer σ-profile is assembled by adding the profiles of monomers A and
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B weighted by their molar fraction in the copolymer into one single histogram. For
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the area and volume parameters of the copolymer base unit, the area and volume pa-
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rameters of the homopolymers are weighted by the molar fraction of A and B in the
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copolymer structure. Finally, the total copolymer parameters are evaluated as for ho-
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mopolymers, by multiplying the repeating unit parameters by the number of units of
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the macromolecule. It is important to notice that with this procedure, no differentia-
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tion between different copolymer arrangements is possible (random, diblock, triblock
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copolymer and so on).
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4. Results and Discussion
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4.1. Polymer sigma-profiles
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The following materials were studied in this work: linear polyethylene (LPE),
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poly(dimethyl siloxane) (PDMS), polyisobutylene (PIB), polybutadiene (PBD), poly(methyl
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metacrylate) (PMMA), polystyrene (PS), poly(vinyl acetate) (PVAc), poly(vinyl al-
212
cohol) (PVA), poly(caprolactone) (PCL), poly(ethylene oxide) (PEO), poly(propylene
213
oxide) (PPO), Poly(vinyl chloride) (PVC), poly(vinyl methyl ether) (PVME), and poly(acrilonitrile)
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(PAN). As described in section 3, the evaluation of the σ-profile for polymers was
215
based on the behavior of the central repeating unit inside oligomers of different sizes.
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The variables analyzed were the IDAC, σ-profile, and unit total charge. For LPE and
217
PDMS, the results are explored here and the analysis for the other materials is available
218
as supplementary material. The same criteria were adopted for all 14 polymers used
219
in this study. The IDAC values of different solvents diluted in the PDMS and LPE re-
220
peating units obtained from various oligomers are shown in Figure 2. Only the residual
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contribution was taken into account in this step.
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0.4
0.25 0.3
0.1 0.05
ln IDAC
dodecane octane 2-nitropropane
0.15
0.2
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ln IDAC
0.2
pentane benzene dichloroethane
0.1
0
0 -0.05
10
5
15
-0.1
(a) LPE
10
5
15
20
25
number of repeating units in the oligomer
number of repeating units in the oligomer
(b) PDMS
for different solvents.
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Figure 2: Variation in ln IDAC for the central repeating unit of LPE and PDMS inside different oligomers
As can be seen in Figure 2(a), a small variation in the ln-IDAC response for LPE
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was verified up to 11 repeating units, with no significant variation for larger oligomers
224
for the solvents tested. In Figure 2(b), similar results were obtained for PDMS with
225
n-pentane and benzene up to 19 repeating units. With dichloroethane an oscillation
226
was observed around the value of 0.3. It is expected that for PDMS more repeating
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units would be necessary to stabilize the central unit, probably because of the large
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number of atoms in the fundamental structure. It is also worth noting that the results
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confirm the assumption that the longer is the chain the weaker is the influence of the
230
head groups artificially added. For both polymers, similar results were also obtained
231
when analyzing the total charge of the repeating unit.
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In figures 3 and 4, the σ-profiles of the repeating units inside chains of different
sizes are depicted for LPE and PDMS, respectively. As can be seen in figures 3 and 4, for both polymers, the σ-profiles converge to a
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Figure 3: σ-profiles of the central repeating unit inside oligomers with 1, 7, 13 and 17 basic units for LPE.
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Figure 4: σ-profiles of the central repeating unit inside oligomers with 1, 9, 17, and 23 basic units for PDMS.
constant response as the oligomer chain grows. As expected, for PDMS the influence of
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the end group with the increase of the repeating unit number is more pronounced. The
237
analysis of activity coefficient values, total charge and σ-profiles in different oligomers
238
leads to the conclusion that each polymer needs an individual study to define the final
239
σ-profile of the representative repeating unit. While for LPE, the final σ-profile was
240
obtained using 13 units, for PDMS a molecule with 19 units was necessary. Some limitations of the proposed methodology should be noted. The most impor-
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tant is the fact that the polymer is considered as a single chain with the number of
243
repeating unit determined by the total molecular mass of the material. For this reason,
244
no effects of the entanglement of different chains, cross linking or polymer swelling
245
are taken into account. Actually, this limitation applies to most of the existing meth-
246
ods proposed so far. An alternative to overcome such limitation is the use of different
247
expressions for the combinatorial contribution, as proposed in Shah and Yadav (2013)
248
which added a correction term based on the Flory–Rehner equation and introduction of
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one additional empirical parameter. All structures considered in this work are freely availabe in the LVPP-Sigma profile
251
database at https://github.com/lvpp/sigma.
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4.2. Infilite dilution activity coefficient predictions
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After the σ-profiles defined, IDAC data of 61 different solvents diluted in PCL,
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PDMS, LPE, PEO, PS, PIB, PMMA, PVC, and PVAc were computed. The IDAC
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database used was assembled with data from literature (Hao et al., 1992) and contains
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1966 experimental points. The IDAC predictions can be seen in Figure 5.
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Logarithm of model IDAC
0
-2
-4
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-8
-6
-4
-2
0
Logarithm of experimental IDAC
Figure 5: Experimental IDAC logarithm versus COSMO-SAC IDAC logarithm for a total of 1966 points.
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Dashed lines represent the deviation of one logarithm unit.
As shown in Figure 5, very good predictions for IDAC data were possible. A small value of 0.25 for the ln AAD (absolute average deviation for the IDAC logarithm) was
259
observed for this dataset and the coefficient of determination obtained was R2 = 0.96.
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For the same dataset, predictions with the modified UNIFAC (Do) Jakob et al. (2006)
261
model would generate a ln AAD of 1.39.
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4.3. Vapor-Liquid equilibrium for homopolymer systems
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To verify the prediction strength of the COSMO-SAC model with the proposed
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methodology, VLE calculations were also performed. In all results presented, com12
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ponent 1 is the solvent and component 2 represents the polymer. Negligible polymer
266
vapor pressure was assumed (pure solvent vapor, y1 = 1) and only low pressure tests
267
were chosen, enabling the use of the modified Raoult’s Law:
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Py1 = P = x1 γ1 Psat 1
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(8)
where x1 is the molar fraction of the solvent in the liquid phase, γ1 the activity coef-
270
ficient of the solvent computed with the proposed method, and Psat 1 is the saturation
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pressure of the solvent.
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Although all computations are in molar basis (Eq. 8), better visualization of VLE
273
results is obtained in mass basis (w1 ). The deviations in VLE predictions were evaluated
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through the difference between the bubble pressure calculated using the COSMO-SAC
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exp
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model by Eq. (8) and the experimental value (Pi ): exp NP − Pi 1 X Pcalc i × 100 ∆P(%) = exp NP i Pi
(9)
where NP is the number of experimental points.
These results are grouped and shown in Appendix A for homopolymer systems and
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in Appendix B for copolymer-solvent mixtures. Additionally, a comparison to the work
280
of Kuo et al. (2013) is presented. In their work, the COSMO-SAC model was used to
281
calculate phase equilibrium of polymeric systems. They performed DFT/COSMO cal-
282
culations on trimers and used the middle unit charge and surface area. They also used
283
the free volume model of Elbro et al. (1990)
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According to Elbro et al. (1990), PDMS solutions with organic solvents do not
285
need free-volume corrections since the components have similar free volume percent-
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ages, and the results of Table A.1 are in agreement with this statement. Good results
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were obtained for these systems with an overall deviation of 8.2 %. Higher errors were
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observed for mixtures of PDMS with n-hexane and benzene. Although 27 % of devi-
289
ation was obtained for PDMS(3350) and hexane, still good response was possible as
290
can be seen in Figure 6(a). The same tendency was shown in the work of Kuo et al.
291
(2013), which presented a total deviation of 23.7 % for PDMS mixtures.
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In Table A.2, the results of polyethylene with different solvents are listed. In all
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tests the model parameters of LPE where used to represent the polymer, even to eval13
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22.5
Pressure [bar]
0.150 0.125 0.100 0.075
17.5 15.0 12.5 10.0 7. 5
0.050
5. 0
0.025
2. 5
0.000 0 .0
0 .1
0.2
0.3
SC
Pressure [bar]
0.175
Exp. 425.65 K Exp. 474.15 K COSMO-SAC
20.0
Exp. 303.0 K COSMO-SAC
RI PT
0.200
0. 0 0 .0 0. 1 0.2 0. 3 0.4 0.5 0 .6 0.7 0.8 0.9
0 .4
w1
w1
(b)
M AN U
(a)
Figure 6: VLE predictions for (a) n-hexane-PDMS(3350) at 303 K and (b) cyclopentane-LDPE(76000) at 425 K and 474 K. Experimental data from Hao et al. (1992) and Surana (1997).
uate the low density polyethylene (LDPE) behavior. In this case, the information of
295
free volume with the polymer density could be useful to distinguish between linear and
296
branched chains alhough the results obtained by Kuo et al. (2013) do not confirm this
297
hypothesis. Maybe the high pressure of the systems can be an issue and the assump-
298
tion of ideal gas phase may be not the best choice. For LDPE with cyclopentane, the
299
results are also depicted in Figure 6(b). The overall deviation for PE systems using the
300
proposed methodology is 8.8 % while with the COSMO-SAC used in Kuo et al. (2013)
301
is 31 %.
TE D
294
In Table A.3, VLE results for PEO and PIB with many solvents are listed and in Ta-
303
ble A.4 the results for PS mixtures. Very good results were obtained for PEO/benzene
304
with low and high polymer molecular mass with total deviation of 0.9 % compared to
AC C
EP
302
305
the experimental data. For the systems with PIB the overall error obtained was 6.7 %
306
similar to the mixtures of PS, 8.9 %. VLE diagrams for systems of PEO, PIB and PS
307
are presented in Figure 7.
308
In Table A.5 the results for VLE of PCL, PVC, PPO, PVME and PVA are shown. A
309
very good agreement with the experimental data was achieved, mainly with PPO. For
310
PVME bigger deviations were expected since no further investigation was accomplish
311
about the polymer structure and in its free-volume contribution. A comparison between
14
ACCEPTED MANUSCRIPT
Pressure [bar]
0.5 0.4 0.3 0.2
Exp. 323.45 K Exp. 343.15 K COSMO-SAC
0.1
0. 0 0.1 0. 2 0. 3 0.4 0.5 0.6 0.7 0. 8 0.9 1 .0
0.0 0.0
0.1
0.2
0.3
0.4
w1
w1
(a)
(b) 0.7
0.5 0.4 0.3 0.2 0.1 0.0 0.0
0.6
Exp. 298.15 K Exp. 343.15 K COSMO-SAC
M AN U
Pressure [bar]
0.6
0.5
SC
Pressure [bar]
0.6
RI PT
Exp. 298.15 K Exp. 313.15 K Exp. 338.15 K COSMO-SAC
0.7 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15
0.1
0.2
0. 3
0.4
0.5
0.6
w1
TE D
(c)
Figure 7: VLE predictions for (a) benzene-PEO(600000) at 323 K and 343 K, (b) n-hexane-PIB(50000) at 298 K, 313 K, 338 K Kand (c)2-butanone-PS(290000) at 298 K and 343 K. Experimental data taken from Hao et al. (1992) and Bawn et al. (1950).
trichloromethane activity results in PVME using polymer trimers with different tatici-
313
ties was made by Kuo et al. (2013). The authors found deviations of 25.6 % for isotatic
314
PVME and 6.5 % for syndiotatic PVME when compared with experimental data, illus-
AC C
EP
312
315
trating the importance of this analysis for some materials. However, polymer tacticities
316
are not reported in most of the literature VLE data, making this study a difficult task.
317
In Figure 8 are shown the VLE predictions PCL-carbon tetrachloride (8(a)), for
318
PVA-water (8(b)) and PBD-cyclohexane (8(c)). The deviations in pressure were 4.7 %,
319
7.0 %, and 6.2 % respectively, using experimental data from Hao et al. (1992) for PCL,
320
from Palamara et al. (2004) for PVA, and from Gupta and Prausnitz (1995) for PBD.
321
Based on model VLE predictions, the proposed methodology to extend the use
15
0. 5
Pressure [bar]
0. 4 0. 3 0. 2
Exp. 338.15 K COSMO-SAC
0. 1
0. 0 0. 1 0.2 0. 3 0. 4 0. 5 0.6 0. 7 0.8 0.9 1.0
1 .2 1 .1 1 .0 0 .9 0 .8 0 .7 0 .6 0 .5 0 .4 0 .3 0 .2 0 .1 0 .0
Exp. 373.15 K Exp. 383.15 K Exp. 363.15 K COSMO-SAC 0 .1
0 .2
0 .3
0 .4
0 .5
w1
w1
(a)
(b)
0.35
0.25 0.20 0.15 0.10 0.05
Exp. 333.15 K COSMO-SAC
M AN U
Pressure [bar]
0.30
0 .6
SC
Pressure [bar]
0. 6
RI PT
ACCEPTED MANUSCRIPT
0.1
0.2
0.3
0.4
0.5
0.6
0.7
w1
TE D
(c) Figure 8: VLE predictions for (a) carbon tetrachloride-PCL(33000) at 338 K, (b) water-PVA(116000) at 363 K, 373 K and 383 K and (c) cyclohexane-PBD(250000) at 333 K. Experimental data from Hao et al. (1992), Palamara et al. (2004), and Gupta and Prausnitz (1995).
of COSMO-based models to polymer systems shows promising results for apolar and
323
associating mixtures. However, as it considers the polymer as a liquid of independent
324
repeating units, further developments are necessary in order to enlarge its application to
EP
322
crystalline materials. Also, the inclusion of free-volume contributions in the COSMO-
326
SAC combinatorial formulation must be verified.
327
4.4. Vapor-Liquid equilibrium for copolymer systems
AC C 325
328
329
330
The VLE of copolymer-solvent mixtures predictions were compared to experimen-
tal data from Wohlfarth (2001) and Gupta and Prausnitz (1995). In Table B.6, results for ethylene-vinyl acetate copolymer with 39.4 wt% of vinyl
16
ACCEPTED MANUSCRIPT
acetate (E-VA/39w), 41.4 wt% vinyl acetate (E-VA/41w), and 70 wt% vinyl acetate (E-
332
VA/70w) are shown. Both copolymer have low molecular weight and no information
333
of Mw was available, only Mn . Good results with low ∆P(%) values were obtained
334
with an overall deviation of 7.9 %. For the systems studied, Kuo et al. (2013) had an
335
error of 10 %.
RI PT
331
In Table B.7 the VLE results for poly(styrene-acrylonitrile) with 25 wt% acryloni-
337
trile (S-AN/25w) and 28 wt% acrylonitrile (S-AN/28w) are listed. Very low deviations
338
were obtained with S-AN/28w while erros around 9 % were observed for the system
339
S-AN/25w with toluene. S-AN/25w and S-AN/28w are both random copolymers with
340
similar amount of acrylonitrile, so their equilibrium with toluene should result almost
341
the same. In this sense the errors of 1 % and 9 % presented in Table B.7 for S-AN/25w
342
and S-AN/28w toluene systems are not expected. A plausible explanation is the qual-
343
ity of the experimental data for S-AN/25w-toluene system. The VLE diagrams with
M AN U
SC
336
toluene for both polymers can be verified in Figure 9. As can be seen in the S-AN/25w0.200
0 .7
0.175
TE D
Pressure [bar]
0.150 0.125 0.100 0.075 0.050
EP
0.025
Exp. 313.5 K Exp. 323.15 K Exp. 333.15 K COSMO-SAC
0.000 0.00
0.25
0.50
0.75
Pressure [bar]
0 .6
1.00
Exp. 343.15 K Exp. 373.15 K COSMO-SAC
0 .5 0 .4 0 .3 0 .2 0 .1 0 .0 0 .0 0.1 0.2 0 .3 0 .4 0 .5 0. 6 0.7 0 .8 0. 9 1 .0
w1
w1
(b)
AC C
(a)
Figure 9: VLE predictions for styrene-acrylonitrile copolymer with toluene: (a) 25 wt of acrylonitrile (SAN/25w) and (b) 28 wt of acrylonitrile (S-AN/28w). Experimental data from Wohlfarth (2001).
344 345
toluene diagram (Figure 9(a)), a sudden change in the equilibrium pressure is observed
346
for a small change in the composition for w1 around 0.5. This indicates a possible
347
problem in the measurements shown in this figure. The COSMO-SAC model agree
348
with the data above this mole fraction and appears to do the same in the low solvent
17
ACCEPTED MANUSCRIPT
concentration region. None of the experimental data used here for this copolymer was
350
analysed in the work of Kuo et al. (2013), just S-AN with higher content of acriloni-
351
trile. The authors reported an overall deviation of 9.8 % for solvent/S-AN systems.
352
With the proposed methodology a total error of 4.9 % was achieved.
353
RI PT
349
Finally, the VLE results for systems containing the block copolymer of poly(styrenebutadiene) with 30 wt% styrene (S-b-BR/30w); the copolymer styrene-butadiene (with
355
no specification) containing 41 wt% styrene, (S-BR/41w), 45 wt% styrene (S-BR/45w),
356
and 77 wt% styrene (S-BR/77w) are shown in Table B.8. For the ethylbenze solvent,
357
lower deviations in equilibrium pressure conditions are achieved as the styrene content
358
increases, disregarding the effect of the molecular mass of polymer. The structure of
359
the block copolymer S-b-BR/30w is not taken into account in this methodology, which
360
can also be an important factor. With other solvents, good results were obtained.
361
5. Conclusions and further works
M AN U
SC
354
A new alternative to assemble the σ-profiles for polymers was proposed in this
363
work. The procedure is also based on a repeating unit structure that represents the ma-
364
terial of interest, as the other alternatives available in the literature. In this work a wider
365
analysis is made to extract the repeating unit information from different oligomers, and
366
an σ-profile with a much reduced end group effect is obtained. The σ-profiles of all
367
polymers studied in this work along with the ones for around 1500 solvents were in-
368
cluded in the freely available LVPP-Sigma database, available at https://github.
369
com/lvpp/sigma. The COSMO-SAC model with parameters from the literature was
370
used to predict infinite dilution activy coefficient and vapor-liquid equilibrium of sol-
EP
TE D
362
vent/polymer and solvent/copolymer systems. Comparing to a similar methodology
372
available in the literature, smaller deviations were achieved and good results were pos-
373
sible for apolar and associating mixtures. Improvements on the combinatorial contri-
374
bution are probably necessary for the representation of liquid-liquid equilibrium data.
375
Appendix A. VLE results for homopolymer-solvent systems
AC C 371
376
In Table A.1, the results of PDMS with different solvents are presented. 18
ACCEPTED MANUSCRIPT
Table A.1: VLE results for systems with organic solvents and PDMS.
Polymer
T (K)
NP
∆P(%)
n-hexane
PDMS(3350)
303
8
27.7
PDMS(6650)
303
8
21.6
PDMS(15650)
303
8
17.3
PDMS(26000)
303
8
15.3
313
8
9.1
303
8
13.1
PDMS(26000)
313
7
7.2
b toluene
PDMS(1540)
298
12
1.7
313
14
4.7
308
10
0.3
n-heptane
PDMS(140000)
chloroform
PDMS(89000)
2,2,4-trimethylpentane
PDMS(958)
(Hao et al., 1992)
(Hwang et al., 1998)
8
18.8
(Hao et al., 1992)
(Hwang et al., 1998) (Hao et al., 1992) (Hao et al., 1992)
(Hao et al., 1992)
7
0.2
(Hao et al., 1992)
4.8
(Hao et al., 1992)
298
12
8.9
298
12
5.2
313
17
5.1
(Hao et al., 1992)
303
3
5.1
(Hao et al., 1992)
303
3
3.6
(Hao et al., 1992)
303
3
3.7
(Hao et al., 1992)
303
3
3.0
(Hao et al., 1992)
PDMS(61000)
303
3
3.7
(Hao et al., 1992)
PDMS(220000)
303
3
3.1
PDMS(1540)
298
16
3.3
16
24
(Hao et al., 1992)
PDMS(4170)
298
11
7.8
11
24.3
(Hao et al., 1992)
PDMS(6650)
303
8
13.2
8
33.5
(Hao et al., 1992)
PDMS(26000)
303
8
17.6
8
34.7
(Hao et al., 1992)
PDMS(4600)
12
5.2
(Hao et al., 1992) (Hao et al., 1992)
(Hao et al., 1992)
8.2
23.4
EP
TE D
PDMS(55000)
377
(Hao et al., 1992)
8
PDMS(40000)
overall
(Hao et al., 1992)
323
PDMS(16000)
benzene
Ref.
(Hao et al., 1992)
303
PDMS(4170)
2-butanone
∆P(%)
M AN U
PDMS(89000) n-pentane
NP
RI PT
Solvent
Kuo et al. (2013)
SC
This work
In Table A.2, the results of polyethylene with different solvents are depicted. In all tests the model parameters of LPE where used to represent the polymer, even to
379
evaluate the low density polyethylene (LDPE) behavior.
AC C 378
380
381
382
383
In Table A.3, VLE results for PEO and PIB with many solvents are depicted and in
Table A.4 the results for PS mixtures. In Table A.5 the results for VLE of polymers with no IDAC data available are
shown.
19
ACCEPTED MANUSCRIPT
This work Solvent
Polymer
T (K)
NP
∆P(%)
NP
∆P(%)
1-pentene
LDPE(76000)
423
5
6.8
5
49.4
3-pentanone
LDPE(76000)
423
6
15.1
6
15.1
478
11
7.1
6
34.1
6
1.9
6
27.6
6
53.6
cyclopentane
LDPE(76000)
425 474
6
10.4
n-pentane
LDPE(76000)
423
6
12.4
propyl acetate
LDPE(76000)
426
5
8.9
5
474
5
7.4
5
386
387
388
(Surana, 1997)
(Surana, 1997)
(Surana, 1997)
(Surana, 1997)
SC
(Surana, 1997)
13.9
(Surana, 1997)
24
(Surana, 1997)
31.1
Appendix B. VLE results for copolymer-solvent systems
In Table B.6, results for copolymer poly(ethylene-vinyl acetate) with 39.4 wt% of vinyl acetate (E-VA/39w) and 41.4 wt% vinyl acetate (E-VA/41w) are presented. In Table B.7 are shown the VLE results for poly(styrene-acrylonitrile) with 25 wt% acrylonitrile (S-AN/25w) and 28 wt% acrylonitrile (S-AN/28w).
TE D
385
8.7
Ref.
(Surana, 1997)
M AN U
overall
384
Kuo et al. (2013)
RI PT
Table A.2: VLE results for systems with organic solvents and polyethylene.
In Table B.8 the VLE predictions of block copolymer of poly(styrene-butadiene)
390
with 30 wt% styrene (S-b-BR/30w); the copolymer styrene-butadiene (with no spec-
391
ification) containing 41 wt% styrene, (S-BR/41w), 45 wt% styrene (S-BR/45w), and
392
77 wt% styrene (S-BR/77w), are shown.
AC C
EP
389
20
ACCEPTED MANUSCRIPT
This work Solvent
Polymer
benzene
PEO(5700)
PEO(600000)
T (K)
NP
∆P(%)
NP
∆P(%)
318
5
1.0
5
3.6
(Hao et al., 1992)
343
9
0.5
9
6.0
(Hao et al., 1992)
323
5
0.6
5
2.8
(Hao et al., 1992)
343
8
1.5
8
3.8
(Hao et al., 1992)
298
6
7.0
308
6
6.4
318
5
5.9
328
5
5.8
PIB(1350)
298
PIB(2250000)
298
PIB(50000)
298
308 n-hexane
313 338
(Hao et al., 1992) (Hao et al., 1992) (Hao et al., 1992) (Hao et al., 1992)
10
2.7
12
5.9
12
(Hao et al., 1992)
5.7
(Hao et al., 1992)
4
13.1
4
12.2
(Hao et al., 1992)
8
5.5
8
10.3
(Hao et al., 1992)
9
7.6
9
9.9
(Hao et al., 1992)
7
7.4
7
7.9
(Hao et al., 1992)
6.7
TE D
overall
4.0
SC
0.9 PIB(1170)
Ref.
M AN U
overall n-pentane
Kuo et al. (2013)
RI PT
Table A.3: VLE results for systems with PEO and PIB.
9.2
Table A.4: VLE results for different systems with PS. This work
Kuo et al. (2013)
T (K)
NP
∆P(%)
NP
∆P(%)
Ref.
PS(290000)
298
11
14.6
11
11.1
(Bawn et al., 1950)
343
6
11.5
6
11.3
(Bawn et al., 1950)
carbon tetrachloride
PS(500000)
293
15
11.6
14
7.5
(Hao et al., 1992)
3-pentanone
PS(200000)
293
12
15.7
12
22.7
(Hao et al., 1992)
PS(500000)
293
12
13.2
(Hao et al., 1992)
PS(63000)
288
8
10.1
(Hao et al., 1992)
303
8
12.4
(Hao et al., 1992)
333
7
9.3
(Hao et al., 1992)
303
11
2.5
(Hao et al., 1992)
318
11
2.0
338
11
1.6
(Hao et al., 1992)
298
11
7.9
(Hao et al., 1992)
318
11
6.6
(Hao et al., 1992)
338
11
5.8
Solvent
Polymer
EP
2-butanone
AC C
benzene
cyclohexane
toluene
PS(154000)
PS(154000)
overall
8.9
21
11
1.3
(Hao et al., 1992)
(Hao et al., 1992) 10.8
ACCEPTED MANUSCRIPT
Table A.5: VLE results for solvent/polymer systems of PCL, PPO and PVME.
Polymer
carbon tetrachloride
PCL(33000)
T (K)
NP
∆P(%)
338
9
4.7
NP
PVC(77300)
338
8
13.4
2-butanone
PMMA (19770)
322
8
6.1
methanol
PPO(1120)
248
4
0.3
263
4
0.2
273
4
7.0
298
4
0.2
320
11
1.1
11
333
7
0.9
7
343
6
1.0
6
347
14
1.2
(Hao et al., 1992)
8
12.1
(Hao et al., 1992)
(Hao et al., 1992)
(Hao et al., 1992) (Hao et al., 1992) (Hao et al., 1992)
2.5
(Hao et al., 1992)
3.7
(Hao et al., 1992)
6.2
(Hao et al., 1992)
13.9
(Hao et al., 1992)
M AN U
PPO(500000)
Ref.
(Hao et al., 1992)
carbon tetrachloride
benzene
∆P(%)
SC
Solvent
Kuo et al. (2013)
RI PT
This work
PVME(14000)
298
13
17.3
water
PVA(116000)
363
10
9.2
(Palamara et al., 2004)
373
4
5.6
(Palamara et al., 2004)
383
4
6.3
overall
13
(Hao et al., 1992)
benzene
4.9
(Palamara et al., 2004)
7.7
Solvent benzene
TE D
Table B.6: VLE results for solvent/copolymer E-VA systems.
Copolymer
Mn (g/mol)
T (K)
NP
∆P(%)
E-VA/39w
3650
303
13
9.1
323
14
7.6
E-VA/41w
EP
butyl acetate
AC C
propyl acetate
ethyl acetate
methyl acetate
This work
E-VA/41w
E-VA/41w
E-VA/41w
4620
4620
4620
4620
Kuo et al. (2013) NP
323
8
16.0
8
14.6
343
8
16.4
8
15.9
363
8
16.9
8
16.6
303
8
4.4
8
6.0
323
8
7.3
8
8.8
343
8
10.0
8
8.7
363
8
11.0
8
12.5
303
7
1.5
7
2.2
323
7
1.9
7
2.9
343
7
2.8
7
2.8
303
8
1.7
8
6.0
323
8
1.6
8
6.6 25.2
cyclohexane
E-VA/70w
50000
353
7
5.7
9
chloroform
E-VA/70w
50000
333
8
13.4
8
overall
7.9
22
∆P(%)
12.3 10.0
RI PT
ACCEPTED MANUSCRIPT
Table B.7: VLE results for solvent/copolymer S-AN systems with the proposed methodology. ∆P(%)
Solvent
Copolymer
Mn (g/mol)
T (K)
NP
toluene
S-AN/25w
90000
313
12
9.5
323
12
10.3
12
10.7
S-AN/28w
46000
343
11
1.1
m-xylene
S-AN/28w
46000
398
8
2.8
p-xylene
S-AN/28w
46000
398
8
1.0
toluene
S-AN/28w
46000
343
overall
7
1.4
11
2.2
M AN U
373
SC
333 benzene
4.9
Solvent
TE D
Table B.8: VLE results for solvent/copolymer S-BR systems.
ethylbenzene
Kuo et al. (2013)
Copolymer
Mn (g/mol)
T (K)
NP
∆P(%)
NP
S-b-BR/30w
108000
373
10
13.8
9
15
403
20
13.5
19
14.5
EP
S-BR/41w
S-BR/45w
S-BR/77w
AC C
This work
86300
130430
117650
373
9
7.2
398
7
5.4
373
11
11.7
403
22
11.6
373
13
3.7
403
15
4.7
∆P(%)
cyclohexane
S-BR/41w
86300
343
8
9.4
8
2.8
benzene
S-BR/41w
86300
343
9
1.2
9
1.7
toluene
S-BR/41w
86300
343
8
1.3
8
0.8
373
8
1.7
8
1.6
overall
7.1
23
6.0
ACCEPTED MANUSCRIPT
394
Acknowledgement This work was partially supported by CNPq-Brazil under grants no. 454557/2014-0
RI PT
393
and 304046/2016-7.
396
References
397
Abildskov, J., Constantinou, L., Gani, R., 1996. Fluid Phase Equilib. 118 (1), 1–12.
398
Abildskov, J., Gani, R., Rasmussen, P., O’Connell, J., 1999. Fluid Phase Equilib. 158160, 349–356.
M AN U
399
SC
395
400
Banerjee, T., Singh, M. K., Khanna, A., 2006. Ind. Eng. Chem. Res. 45 (9), 3207–3219.
401
Bawn, C. E. H., Freeman, R. F. J., Kamaliddin, A. R., 1950. Trans. Faraday Soc. 46,
402
677–684.
Boucher, D. S., 2015. Colloids Surf., A 487, 207–213.
404
Bouillot, B., Teychen´e, S., Biscans, B., 2013. Ind. Eng. Chem. Res. 52 (26), 9276–
405
406
407
9284.
TE D
403
Constantinescu, D., Klamt, A., Geana, D., 2005. Fluid Phase Equilib. 231 (2), 231– 238.
Delley, B., 2006. Mol. Simul. 32 (2), 117–123.
409
D´ıaz, I., D´ıez, E., Camacho, J., Le´on, S., Ovejero, G., 2013. Fluid Phase Equilib. 337, 6–10.
AC C
410
EP
408
411
412
Elbro, H. S., Fredenslund, A., Rasmussen, P., Oct. 1990. Macromolecules 23 (21), 4707–4714.
413
Fredenslund, A. A., Jones, R. L., Prausnitz, J. M., 1975. AIChE J. 21 (6), 1086–1099.
414
Freire, M. G., Santos, L. M., Marrucho, I. M., a.P. Coutinho, J. a., 2007. Fluid Phase
415
416
Equilib. 255 (2), 167–178. Gerber, R. P., Soares, R. D. P., 2010. Ind. Eng. Chem. Res. 49 (16), 7488–7496. 24
ACCEPTED MANUSCRIPT
Gerber, R. P., Soares, R. P., 2013. Braz. J. Chem. Eng. 30 (1), 1–11.
418
Goss, K.-U., 2011. Anal. Chem. 83 (13), 5304–5308.
419
Grensemann, H., Gmehling, J., 2005. Ind. Eng. Chem. Res. 44 (5), 1610–1624.
420
Gupta, R. B., Prausnitz, J. M., Jul. 1995. J. Chem. Eng. Data 40 (4), 784–791.
421
Hanwell, M., Curtis, D., Lonie, D., Vandermeersch, T., Zurek, E., Hutchison, G., 2012.
424
SC
423
J. Cheminformatic 4, 17.
Hao, W., Elbro, H. S., Alessi, P., 1992. Polymer Solution Data Collection. DECHEMA Chemistry Data Series. Frankfurt.
M AN U
422
RI PT
417
425
Hsieh, C.-M., Sandler, S. I., Lin, S.-T., 2010. Fluid Phase Equilib. 297 (1), 90–97.
426
Hwang, S., Kim, J., Yoo, K.-P., Jul. 1998. J. Chem. Eng. Data 43 (4), 614–616.
427
Jakob, A., Grensemann, H., Lohmann, J., Gmehling, J., nov 2006. Ind. Eng. Chem.
428
Res. 45 (23), 7924–7933.
Kehiaian, H., Jan. 1983. Fluid Phase Equilib. 13, 243–252.
430
Klamt, A., 1995. J. Phys. Chem. 99 (7), 2224–2235.
431
Klamt, A., Eckert, F., 2000. Fluid Phase Equilib. 172 (1), 43–72.
432
Klamt, A., Jonas, V., Burger, T., Lohrenz, J. C. W., 1998. J. Phys. Chem. A 102 (26), 5074–5085.
Kontogeorgis, G. K., Coutsikos, P., Tassios, D., Fredenslund, A., 1994. Fluid Phase
AC C
434
EP
433
TE D
429
435
436
437
438
439
440
Equilib. 92, 35–66.
Kontogeorgis, G. M., Fredenslund, A., Tassios, D. P., Feb. 1993. Ind. Eng. Chem. Res. 32 (2), 362–372.
Kouskoumvekaki, I. A., Michelsen, M. L., Kontogeorgis, G. M., 2002. Fluid Phase Equilib. 202 (2), 325–335. Kuo, Y.-C., Hsu, C.-C., Lin, S.-T., 2013. Ind. Eng. Chem. Res. 52 (37), 13505–13515. 25
ACCEPTED MANUSCRIPT
Lin, S. T., Sandler, S. I., 2002. Ind. Eng. Chem. Res. 41 (5), 899–913.
442
Loschen, C., Klamt, A., 2014. Ind. Eng. Chem. Res. 53 (28), 11478–11487.
443
Mokrushina, L., Yamin, P., Sponsel, E., Arlt, W., 2012. Fluid Phase Equilibria 334,
444
RI PT
441
37–42.
Mu, T., Rarey, J., Gmehling, J., 2007. Ind. Eng. Chem. Res. 46 (20), 6612–6629.
446
Oishi, T., Prausnitz, J. M., Jul. 1978. Ind. Eng. Chem. Process Des. Dev. 17 (3), 333–
448
449
450
451
339.
Okeme, J. O., Parnis, J. M., Poole, J., Diamond, M. L., Jantunen, L. M., 2016. Chemosphere 156, 204–211.
M AN U
447
SC
445
Palamara, J. E., Zielinski, J. M., Hamedi, M., Duda, J. L., Danner, R. P., 2004. Macromolecules 37 (16), 6189–6196.
Panayiotou, C., 2013. Polymer 54 (6), 1621–1638.
453
Parnis, J. M., Mackay, D., Harner, T., 2016. Atmos. Pollut. Res. 7 (1), 155–161.
454
Patterson, D., Nov. 1969. Macromolecules 2 (6), 672–677.
455
Radfarnia, H. R., Kontogeorgis, G. M., Ghotbi, C., Taghikhani, V., 2007. Fluid Phase
457
458
Equilib. 257 (1), 63–69.
Reinisch, J., Klamt, A., Eckert, F., Diedenhofen, M., 2011. Fluid Phase Equilib. 310 (1-
EP
456
TE D
452
2), 7–10.
Shah, M. R., Yadav, G. D., 2013. J. Membr. Sci. 427, 108–117.
460
Shimoyama, Y., Iwai, Y., Takada, S., Arai, Y., Tsuji, T., Hiaki, T., 2006. Fluid Phase
AC C
459
461
Equilib. 243 (1-2), 183–192.
462
Soares, R. D. P., 2011. Ind. Eng. Chem. Res. 50 (5), 3060–3063.
463
Soares, R. D. P., Gerber, R. P., Apr. 2013. Ind. Eng. Chem. Res., 11159–11171.
464
Surana, R. K.; Danner, R. P. H. A. B. B. N., 1997. Fluid Phase Equilib. 139, 361–370. 26
ACCEPTED MANUSCRIPT
466
Tung, H.-H., Tabora, J., Variankaval, N., Bakken, D., Chen, C.-C., 2008. J. Pharm. Sci. 97 (5), 1813–20.
RI PT
465
467
Voutsas, E., Kalospiros, N. S., Tassios, D., 1995. Fluid Phase Equilib. 109 (1), 1–15.
468
Wang, S., Sandler, S. I., Chen, C.-C., 2007. Ind. Eng. Chem. Res. 46 (22), 7275–7288.
469
Wohlfarth, C., 2001. CRC Handbook of Thermodynamic Data of Copolymer Solutions. CRC Press: Boca Raton, FL.
SC
470
Wu, H. S., Sandler, S. I., 1991. Ind. Eng. Chem. Res. 30 (5), 881–889.
472
Yang, L., Xu, X., Peng, C., Liu, H., Hu, Y., 2010. AIChE J. 56 (10), 2687–2698.
473
Zhong, C., Sato, Y., Masuoka, H., Chen, X., 1996. Fluid Phase Equilib. 123 (1-2), 97–106.
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S0378-3812(18)30183-3
DOI:
10.1016/j.fluid.2018.05.003
Reference:
FLUID 11825
To appear in:
Fluid Phase Equilibria
Received Date: 9 August 2017 Revised Date:
20 April 2018
Accepted Date: 1 May 2018
Please cite this article as: P.B. Staudt, R.L. Simões, L. Jacques, N.S.M. Cardozo, R.d.P. Soares, Predicting phase equilibrium for polymer solutions using COSMO-SAC, Fluid Phase Equilibria (2018), doi: 10.1016/j.fluid.2018.05.003. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Predicting phase equilibrium for polymer solutions using COSMO-SAC Paula B. Staudt, Renata L. Sim˜oes, Leonardo Jacques, Nilo. S. M. Cardozo, Rafael de P. Soares∗
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Laborat´orio Virtual de Predi¸ca˜ o de Propriedades - LVPP, Departamento de Engenharia Qu´ımica, Escola de Engenharia, Universidade Federal do Rio Grande do Sul, Rua Ramiro Barcelos, 2777, Bairro Santana, CEP 90035-007, Porto Alegre, RS, Brazil
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Abstract
With the aim of predicting mixture behavior based on information of pure components only, COSMO-based models have emerged as a promising alternative. Several works in the literature are available attesting the good performance of COSMO-RS variants in several different applications. However, the extension of COSMO calculations for polymers and other macromolecules was not extensively explored. In this work, a new procedure to evaluate the parameters of the polymers repeating unit for COSMO-
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based models is proposed. The idea of a large polymer molecule consisting of many repeating units is employed, however, a more comprehensive analysis using oligomers of different sizes is suggested. The COSMO-SAC model with parameters from the literature was used to test the new methodology. Infinite dilution activity coefficient (IDAC)
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and vapor-liquid equilibrium (VLE) calculations with homopolymer and copolymer systems were performed to verify the predictive capability of the approach. The good results obtained showed the adequacy of the proposed methodology to represent IDAC
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and VLE data of solvent-polymer and -copylymer systems. Keywords: COSMO-based models, polymer solution, VLE equilibrium, IDAC data
author. Tel.:+55 51 33083528; fax: +55 51 33083277 Email address: [email protected] (Rafael de P. Soares)
∗ Corresponding
Preprint submitted to Fluid Phase Equilibria
May 4, 2018
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1. Introduction Today, predictive models to calculate phase equilibrium of multicomponent sys-
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tems are gathering progressively more attention in chemical engineering research. In
4
this context, group contribution models are important tools. For instance, the UNI-
5
FAC model (Fredenslund et al., 1975) and its variants are widely used in research and
6
practical applications, mainly because once the interaction parameters are available, no
7
additional information is needed for phase equilibrium calculations. However, even for
8
groups with existing parameters, problems may be encountered. Further, the number
9
of groups with parameters available is limited and unreliable predictions may be expe-
10
rienced for molecules with several functional groups or when functional groups appear
11
in an unusual way (Kehiaian, 1983; Wu and Sandler, 1991; Abildskov et al., 1996,
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1999; Soares and Gerber, 2013). Further, UNIFAC models are not entirely predictive
13
models, since the energetic interaction parameters (for each group pair) need to be
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optimized with experimental data (such as liquid-liquid, vapor-liquid and solid-liquid
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phase equilibrium, excess enthalpy, infinite dilution activity coefficients, etc.).
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In the aim of predicting mixture behavior based on information of pure components
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only, the COSMO-based models come as a promising alternative. The COSMO-RS
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model, developed by Klamt and co-workers (Klamt, 1995; Klamt et al., 1998; Klamt
19
and Eckert, 2000), was the first method to combine quantum chemical calculations with
20
statistical thermodynamics to estimate the activity coefficients for species in a mixture.
21
Based on COSMO-RS, Lin and Sandler (2002) proposed the COSMO-SAC model
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that included the Staverman-Guggenheim expression to improve the combinatorial
23
contribution in the activity prediction. Many other works based on COSMO-RS and
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COSMO-SAC formulations could be cited regarding refinement or model parametriza-
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tion (Grensemann and Gmehling, 2005; Mu et al., 2007; Wang et al., 2007; Gerber and
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Soares, 2010; Hsieh et al., 2010; Soares, 2011).
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Good description of different kinds of systems are reported in the literature when
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applying COSMO-type models for thermodynamic calculations, ranging from solu-
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tions of small molecules at high pressure (Shimoyama et al., 2006; Constantinescu
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et al., 2005) to aqueous micellar solutions (Mokrushina et al., 2012), pharmaceutics
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and drug-like molecules solubility (Tung et al., 2008; Bouillot et al., 2013), and ionic
32
liquids (Banerjee et al., 2006; Freire et al., 2007).
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Regarding polymer solutions, the literature about application of COSMO methods
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is relatively scarce (Yang et al., 2010; Goss, 2011; Reinisch et al., 2011; D´ıaz et al.,
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2013; Panayiotou, 2013; Shah and Yadav, 2013; Kuo et al., 2013). As the quantum
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chemical calculations for very large molecules like polymers are nearly impossible with
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the computational resources available today, a few papers presented specific method-
38
ologies to perform the COSMO calculations for polymer molecules (Delley, 2006;
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Yang et al., 2010; Goss, 2011; Kuo et al., 2013; D´ıaz et al., 2013). The basic idea is
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that each polymer molecule consists of many repeating units and the model parameters
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(charge-density profile, cavity volume and cavity surface area) of the entire molecule
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are calculated by summing up the values of the repeating units. The difference be-
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tween each procedure is how to identify and evaluate the repeating unit parameters. In
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the work of Delley (2006), the COSMO method was generalized to periodic boundary
45
conditions, allowing quantum calculations of infinite systems. Thus, by simply defin-
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ing the repeating unit, the molecular calculations are assumed to be valid to polymers
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and solid surfaces. In the research of Yang et al. (2010), several multimers containing
48
from 1 to 10 repeating units were constructed with the addition of head groups to com-
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plement the unsaturated chemical bonds. The most used head groups were the hydro-
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gen atom and the methyl group. After geometry optimization and energy calculation of
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the multimers, the mean value of the properties differences between two neighboring
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multimers correspond to the repeating unit properties. The disadvantage of considering
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the difference between consecutive multimers is the appearance of unrealistic negative
54
values in the charge distribution profile. In Goss (2011), the author studied equilibrium
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sorption constants of chemicals in different polymers. The polymers were represented
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either by very small oligomers or monomers, end-capped with CH3 groups. In the fi-
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nal application the surface of the end-groups are disregarded. Kuo et al. (2013) used
58
trimers ended capped with hydrogen atoms or the methyl groups to evaluate the sur-
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face area and charge of the homopolymer repeating unit. For copolymers, tetramers
60
are used.
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Boucher (2015) studying the charge characteristics of 48 binary poly(3-hexylthiophene)– 3
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solvent mixtures with COSMO-RS model, used the 3-hexylthiophene monomer to rep-
63
resent the polymer σ-profile. Parnis et al. (2016) modelled the properties of polyurethane
64
foam (PUF) as an air sampling medium using COSMO-RS theory. To represent PUF
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polymer, different condensed isocyanate and polyol monomeric units were combined.
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Best results were obtained using 2,4-toluene diisocyante and glycerol combination.
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Also searching for polymer applications as passive air sampling, Okeme et al. (2016)
68
studied the polydimethylsiloxane (PDMS)-air partition ratios for semi-volatile organic
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compounds. Hexamethyldisiloxane was employed as a chemical surrogate for poly-
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meric PDMS. The authors applied the Goss (2011) approach and reported deviations
71
around 1% between the use of the monomer structure and the correction discussed by
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Goss (2011).
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In this work, a new procedure to evaluate the repeating unit parameters necessary
74
for COSMO-based models is proposed. The idea of a large polymer molecule con-
75
sisting of many repeating units is still employed, but a more comprehensive analysis
76
of different oligomers when defining the appropriate repeating unit is suggested. The
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COSMO-SAC was used to test the new methodology with all model parameters taken
78
from the literature. Infinite dilution activity coefficient (IDAC) and vapor-liquid equi-
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librium (VLE) calculations with homopolymer and copolymer systems were performed
80
to verify the predictive capability of the approach and the results obtained here were
81
compared to those from Kuo et al. (2013).
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2. The COSMO-SAC model
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In COSMO-based models, as in many group contribution methods, the activity coefficient is defined as a result of two contributions:
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ln γi = ln γires + ln γicomb
(1)
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The residual part is defined as the difference between the free energies of restoring
87
the charges around the solute molecule in solution S and restoring the charges in a pure
88
liquid i:
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ln γires =
∗res (∆G∗res i/s − ∆G i/i )
(2)
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The restoring free energies are computed by the COSMO-RS surface interacting theory where, for each contact between segments m and n, the interaction energy is given by: ! α0 HB (σm + σn )2 + Em,n (3) ∆Wm,n = 2
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where α0 = fpol 0.3a3/2 eff /0 is the constant for the misfit energy; 0 is the permittivity of
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2 vacuum; fpol is the polarization factor; aeff = πreff is the effective segment area with
95
its respective radius reff ; σm and σn are the apparent surface charge densities of the
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contact segments m and n, respectively. In this work the values of reff = 1.52Å and
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HB fpol = 0.6917 were used, according to Wang et al. (2007). Em,n should account for
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hydrogen bond formation:
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HB Em,n = cHB max[0, σacc − σHB ]min[0, σdon + σHB ]
(4)
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where σacc and σdon are the larger and smaller σm and σn values. The constant for the
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hydrogen-bonding interaction cHB = 85580 kcal/mol Å4 /e2 and the cutoff σHB = 0.0084 e/Å2
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were taken from Lin and Sandler (2002).
Following the formulation presented in Gerber and Soares (2010), the resulting
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contacts in solution are computed by a statistical thermodynamics treatment as a func-
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tion of the composition and apparent surface charges distribution, the so-called σ-
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profiles. These profiles are given by the probability of finding an element with a charge
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density σ in a pure substance i: pi (σ) =
Ai (σ) Ai
(5)
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where Ai is the total cavity surface area, and Ai (σ) is the total surface area of all of the
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segments with a particular charge density σ. The calculation to obtain the σ-profile of a molecule starts with the computation
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of the ideal screening charge density distribution over the molecular surface using
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COSMO. The result is a surface divided into segments, each one with its own area and
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charge density. Finally, the three-dimensional geometric charge density distribution is
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averaged using another standard radius, ravg and then projected onto a histogram, which
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is known as the σ-profile, pi (σ). In this work the solvent σ-profiles were obtained as
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described by Gerber and Soares (2013), with the averaging radius ravg = 1.52Å (Wang
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et al., 2007). 5
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The combinatorial contribution ln γicomb is usually smaller than the residual one
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for small molecules. For polymer solutions this term becomes very important since
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it is responsible for taking into account differences in size and shape. The relevance
122
of free-volume effects for polymer systems is widely discussed in the literature (Pat-
123
terson, 1969; Oishi and Prausnitz, 1978; Elbro et al., 1990; Kontogeorgis et al., 1993,
124
1994; Voutsas et al., 1995; Zhong et al., 1996; Kouskoumvekaki et al., 2002; Radfarnia
125
et al., 2007; Loschen and Klamt, 2014). According to the authors, the inclusion of
126
free volume effects in the combinatorial expression significantly improves the model
127
accuracy in both VLE and LLE predictions. In order to evaluate the free-volume con-
128
tribution, the polymer molar volume is necessary along with its van der Waals or hard
129
core volume.
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In this work, the formulation recommended in Soares (2011) for the COSMO-SAC method is considered:
! ! φi φi z ln γicomb = ln φ0i + 1 − φ0i − qi ln +1− 2 θi θi
(6)
Eq. 6 consists of the well known Staverman-Guggenheim combinatorial contribution P with an exponent modification in the Flory-Huggins term, where φi = ri / j r j x j and P P φ0i = riR xi / j rRj x j are the normalized volume fraction; θi = qi / j q j x j is the normal-
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ized surface-area fraction; z is the coordination number, taken as 10; xi is the mole
137
fraction; ri = Vi /r and qi = Ai /q are the normalized volume and surface-area, respec-
138
tively; Ai is the cavity surface area and Vi is the cavity volume; q and r are universal
139
parameters of the model. Actually, since the model is independent of r, only the quo-
140
tient z/q is relevant (Gerber and Soares, 2010). In the present work q is assumed as
141
124 Å2 , as previosly reported by Soares (2011).
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For the exponent R, an expression similar to the one proposed by Voutsas et al.
(1995) was considered, for symmetric and asymmetric mixtures: ! Vi,small R= p 1− Vi,large
(7)
145
where, in this work, Vi,small and Vi,large are the solvent and polymer cavity volume and
146
p was taken simply as 1.
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It is worth noting that no free-volume effect is taken into account in this paper. The
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main reason to disregard such contribution is the fact that accurate experimental infor-
149
mation about polymer density (or volume) is frequently unavailable. Although Loschen
150
and Klamt (2014) recommended in their work the use of the free volume contribution
151
of Elbro et al. (1990), the authors pointed out that this term is considerably sensitive
152
to small changes in the density and a sufficiently accurate density of the polymer is
153
needed. Additionally, a more extensive understanding of the proposed methodology
154
can be performed with a classical model formulation with no use of extra information.
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It is likely that better results would be possible with a more sophisticated formulation
156
in the combinatorial term.
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3. Extension to polymers
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As already mentioned, the available methodologies to perform COSMO calcula-
159
tions for polymers are based on the structure of the repeating unit of the material. This
160
is mainly because it is currently impractical (if not impossible) to accomplish the quan-
161
tum mechanical calculations for actual very large macromolecules. To assemble the
162
multimer structures and to accomplish a preliminary structure optimization the Jmol
163
application (web: http://www.jmol.org/) and Avogadro software (Hanwell et al., 2012)
164
version 1.0.3-5 were used. The COSMO calculations were performed with MOPAC
165
(MOPAC2009) as described by Gerber and Soares (2013).
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In this work, the identification of a repeating unit is the initial step for the construc-
167
tion of the σ-profiles of polymers. With the structure of the repeating unit defined,
168
several oligomers were assembled for each polymer studied, always with an odd num-
169
ber of units. To complement the unsaturated chemical bonds of the molecule, hydrogen
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atoms are added as end-cap groups. Each oligomer structure is optimized and then the
171
apparent surface charges are calculated as in usual molecules. Examples of oligomer
172
molecules are depicted in Figure 1 for linear polyethylene (LPE), poly(dimethyl silox-
173
ane) (PDMS), poly(caprolactone) (PCL), and poly(vinyl methyl ether) (PVME).
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After the whole molecule apparent surface charge is evaluated, only the surface
175
of the central unit of the oligomer is kept, being the remaining surface portion disre-
176
garded, as represented in Figure 1. By doing this, the σ-profile of the repeating unit
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(b) PDMS
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(a) LPE
(c) PVME
(d) PCL
Figure 1: Surface charge distribution of the central unit for oligomer structures: LPE with 9 repeating units,PDMS with 11 repeating units, PVME with 9 repeating units, and PCL with 3 repeating units.
alone is obtained. Naturally, this procedure is only an approximation and, in principle,
178
the larger the oligomer the better because of the effect of the end groups in the central
179
unit. In order to evaluate the minimum number of units necessary to obtain the ac-
180
tual profile of a repeating unit, this procedure is repeated with an increasing number of
181
units until constant responses are obtained. The properties investigated in this step are
182
the activity coefficient at infinite dilution (IDAC) for different repeating unit-solvent
183
pairs; the σ-profiles of the repeating unit; and the total charge of the unit, which must
184
tend to its minimum. Due to computational difficulties, the largest oligomer studied
185
consisted in 25 repeating units. For most polymers the procedure could be stopped
186
with smaller molecules. The area and volume parameters for the repeating unit are
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determined from the difference in total volume and area of the two largest consecutive
188
oligomers computed. Finally, the parameters of the polymer are obtained by multiply-
189
ing the repeating unit parameters by the number of basic units of the macromolecule,
190
given by the number-average molar weight of the polymer divided by the molecular
191
weight of the repeating unit. If the number average molecular weight is not available,
192
the weight average molecular weight would be used instead.
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When dealing with copolymers, the copolymer unit is built with information of the
194
monomers (homopolymer) repeating units. The number of repeating units of monomer
195
A and monomer B are estimated from the reported weight fractions of A and B in
196
the material and the copolymer number everage molecular mass. Consequently, the
197
number of repeating units of the copolymer is the sum of repeating units of A and
198
B. The copolymer σ-profile is assembled by adding the profiles of monomers A and
199
B weighted by their molar fraction in the copolymer into one single histogram. For
200
the area and volume parameters of the copolymer base unit, the area and volume pa-
201
rameters of the homopolymers are weighted by the molar fraction of A and B in the
202
copolymer structure. Finally, the total copolymer parameters are evaluated as for ho-
203
mopolymers, by multiplying the repeating unit parameters by the number of units of
204
the macromolecule. It is important to notice that with this procedure, no differentia-
205
tion between different copolymer arrangements is possible (random, diblock, triblock
206
copolymer and so on).
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4. Results and Discussion
208
4.1. Polymer sigma-profiles
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The following materials were studied in this work: linear polyethylene (LPE),
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poly(dimethyl siloxane) (PDMS), polyisobutylene (PIB), polybutadiene (PBD), poly(methyl
211
metacrylate) (PMMA), polystyrene (PS), poly(vinyl acetate) (PVAc), poly(vinyl al-
212
cohol) (PVA), poly(caprolactone) (PCL), poly(ethylene oxide) (PEO), poly(propylene
213
oxide) (PPO), Poly(vinyl chloride) (PVC), poly(vinyl methyl ether) (PVME), and poly(acrilonitrile)
214
(PAN). As described in section 3, the evaluation of the σ-profile for polymers was
215
based on the behavior of the central repeating unit inside oligomers of different sizes.
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The variables analyzed were the IDAC, σ-profile, and unit total charge. For LPE and
217
PDMS, the results are explored here and the analysis for the other materials is available
218
as supplementary material. The same criteria were adopted for all 14 polymers used
219
in this study. The IDAC values of different solvents diluted in the PDMS and LPE re-
220
peating units obtained from various oligomers are shown in Figure 2. Only the residual
221
contribution was taken into account in this step.
RI PT
216
SC
0.4
0.25 0.3
0.1 0.05
ln IDAC
dodecane octane 2-nitropropane
0.15
0.2
M AN U
ln IDAC
0.2
pentane benzene dichloroethane
0.1
0
0 -0.05
10
5
15
-0.1
(a) LPE
10
5
15
20
25
number of repeating units in the oligomer
number of repeating units in the oligomer
(b) PDMS
for different solvents.
TE D
Figure 2: Variation in ln IDAC for the central repeating unit of LPE and PDMS inside different oligomers
As can be seen in Figure 2(a), a small variation in the ln-IDAC response for LPE
223
was verified up to 11 repeating units, with no significant variation for larger oligomers
224
for the solvents tested. In Figure 2(b), similar results were obtained for PDMS with
225
n-pentane and benzene up to 19 repeating units. With dichloroethane an oscillation
226
was observed around the value of 0.3. It is expected that for PDMS more repeating
227
units would be necessary to stabilize the central unit, probably because of the large
AC C
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222
228
number of atoms in the fundamental structure. It is also worth noting that the results
229
confirm the assumption that the longer is the chain the weaker is the influence of the
230
head groups artificially added. For both polymers, similar results were also obtained
231
when analyzing the total charge of the repeating unit.
232
233
234
In figures 3 and 4, the σ-profiles of the repeating units inside chains of different
sizes are depicted for LPE and PDMS, respectively. As can be seen in figures 3 and 4, for both polymers, the σ-profiles converge to a
10
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ACCEPTED MANUSCRIPT
M AN U
SC
Figure 3: σ-profiles of the central repeating unit inside oligomers with 1, 7, 13 and 17 basic units for LPE.
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Figure 4: σ-profiles of the central repeating unit inside oligomers with 1, 9, 17, and 23 basic units for PDMS.
constant response as the oligomer chain grows. As expected, for PDMS the influence of
236
the end group with the increase of the repeating unit number is more pronounced. The
237
analysis of activity coefficient values, total charge and σ-profiles in different oligomers
238
leads to the conclusion that each polymer needs an individual study to define the final
239
σ-profile of the representative repeating unit. While for LPE, the final σ-profile was
240
obtained using 13 units, for PDMS a molecule with 19 units was necessary. Some limitations of the proposed methodology should be noted. The most impor-
AC C
241
EP
235
242
tant is the fact that the polymer is considered as a single chain with the number of
243
repeating unit determined by the total molecular mass of the material. For this reason,
244
no effects of the entanglement of different chains, cross linking or polymer swelling
245
are taken into account. Actually, this limitation applies to most of the existing meth-
246
ods proposed so far. An alternative to overcome such limitation is the use of different
247
expressions for the combinatorial contribution, as proposed in Shah and Yadav (2013)
248
which added a correction term based on the Flory–Rehner equation and introduction of
11
ACCEPTED MANUSCRIPT
250
one additional empirical parameter. All structures considered in this work are freely availabe in the LVPP-Sigma profile
251
database at https://github.com/lvpp/sigma.
252
4.2. Infilite dilution activity coefficient predictions
RI PT
249
After the σ-profiles defined, IDAC data of 61 different solvents diluted in PCL,
254
PDMS, LPE, PEO, PS, PIB, PMMA, PVC, and PVAc were computed. The IDAC
255
database used was assembled with data from literature (Hao et al., 1992) and contains
256
1966 experimental points. The IDAC predictions can be seen in Figure 5.
SC
253
M AN U
Logarithm of model IDAC
0
-2
-4
-6
TE D
-8
-8
-6
-4
-2
0
Logarithm of experimental IDAC
Figure 5: Experimental IDAC logarithm versus COSMO-SAC IDAC logarithm for a total of 1966 points.
257
EP
Dashed lines represent the deviation of one logarithm unit.
As shown in Figure 5, very good predictions for IDAC data were possible. A small value of 0.25 for the ln AAD (absolute average deviation for the IDAC logarithm) was
259
observed for this dataset and the coefficient of determination obtained was R2 = 0.96.
260
For the same dataset, predictions with the modified UNIFAC (Do) Jakob et al. (2006)
261
model would generate a ln AAD of 1.39.
262
4.3. Vapor-Liquid equilibrium for homopolymer systems
AC C 258
263
To verify the prediction strength of the COSMO-SAC model with the proposed
264
methodology, VLE calculations were also performed. In all results presented, com12
ACCEPTED MANUSCRIPT
ponent 1 is the solvent and component 2 represents the polymer. Negligible polymer
266
vapor pressure was assumed (pure solvent vapor, y1 = 1) and only low pressure tests
267
were chosen, enabling the use of the modified Raoult’s Law:
268
Py1 = P = x1 γ1 Psat 1
RI PT
265
(8)
where x1 is the molar fraction of the solvent in the liquid phase, γ1 the activity coef-
270
ficient of the solvent computed with the proposed method, and Psat 1 is the saturation
271
pressure of the solvent.
SC
269
Although all computations are in molar basis (Eq. 8), better visualization of VLE
273
results is obtained in mass basis (w1 ). The deviations in VLE predictions were evaluated
274
through the difference between the bubble pressure calculated using the COSMO-SAC
M AN U
272
exp
275
276
277
model by Eq. (8) and the experimental value (Pi ): exp NP − Pi 1 X Pcalc i × 100 ∆P(%) = exp NP i Pi
(9)
where NP is the number of experimental points.
These results are grouped and shown in Appendix A for homopolymer systems and
279
in Appendix B for copolymer-solvent mixtures. Additionally, a comparison to the work
280
of Kuo et al. (2013) is presented. In their work, the COSMO-SAC model was used to
281
calculate phase equilibrium of polymeric systems. They performed DFT/COSMO cal-
282
culations on trimers and used the middle unit charge and surface area. They also used
283
the free volume model of Elbro et al. (1990)
EP
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278
According to Elbro et al. (1990), PDMS solutions with organic solvents do not
285
need free-volume corrections since the components have similar free volume percent-
286
ages, and the results of Table A.1 are in agreement with this statement. Good results
AC C
284
287
were obtained for these systems with an overall deviation of 8.2 %. Higher errors were
288
observed for mixtures of PDMS with n-hexane and benzene. Although 27 % of devi-
289
ation was obtained for PDMS(3350) and hexane, still good response was possible as
290
can be seen in Figure 6(a). The same tendency was shown in the work of Kuo et al.
291
(2013), which presented a total deviation of 23.7 % for PDMS mixtures.
292
In Table A.2, the results of polyethylene with different solvents are listed. In all
293
tests the model parameters of LPE where used to represent the polymer, even to eval13
ACCEPTED MANUSCRIPT
0.225
22.5
Pressure [bar]
0.150 0.125 0.100 0.075
17.5 15.0 12.5 10.0 7. 5
0.050
5. 0
0.025
2. 5
0.000 0 .0
0 .1
0.2
0.3
SC
Pressure [bar]
0.175
Exp. 425.65 K Exp. 474.15 K COSMO-SAC
20.0
Exp. 303.0 K COSMO-SAC
RI PT
0.200
0. 0 0 .0 0. 1 0.2 0. 3 0.4 0.5 0 .6 0.7 0.8 0.9
0 .4
w1
w1
(b)
M AN U
(a)
Figure 6: VLE predictions for (a) n-hexane-PDMS(3350) at 303 K and (b) cyclopentane-LDPE(76000) at 425 K and 474 K. Experimental data from Hao et al. (1992) and Surana (1997).
uate the low density polyethylene (LDPE) behavior. In this case, the information of
295
free volume with the polymer density could be useful to distinguish between linear and
296
branched chains alhough the results obtained by Kuo et al. (2013) do not confirm this
297
hypothesis. Maybe the high pressure of the systems can be an issue and the assump-
298
tion of ideal gas phase may be not the best choice. For LDPE with cyclopentane, the
299
results are also depicted in Figure 6(b). The overall deviation for PE systems using the
300
proposed methodology is 8.8 % while with the COSMO-SAC used in Kuo et al. (2013)
301
is 31 %.
TE D
294
In Table A.3, VLE results for PEO and PIB with many solvents are listed and in Ta-
303
ble A.4 the results for PS mixtures. Very good results were obtained for PEO/benzene
304
with low and high polymer molecular mass with total deviation of 0.9 % compared to
AC C
EP
302
305
the experimental data. For the systems with PIB the overall error obtained was 6.7 %
306
similar to the mixtures of PS, 8.9 %. VLE diagrams for systems of PEO, PIB and PS
307
are presented in Figure 7.
308
In Table A.5 the results for VLE of PCL, PVC, PPO, PVME and PVA are shown. A
309
very good agreement with the experimental data was achieved, mainly with PPO. For
310
PVME bigger deviations were expected since no further investigation was accomplish
311
about the polymer structure and in its free-volume contribution. A comparison between
14
ACCEPTED MANUSCRIPT
Pressure [bar]
0.5 0.4 0.3 0.2
Exp. 323.45 K Exp. 343.15 K COSMO-SAC
0.1
0. 0 0.1 0. 2 0. 3 0.4 0.5 0.6 0.7 0. 8 0.9 1 .0
0.0 0.0
0.1
0.2
0.3
0.4
w1
w1
(a)
(b) 0.7
0.5 0.4 0.3 0.2 0.1 0.0 0.0
0.6
Exp. 298.15 K Exp. 343.15 K COSMO-SAC
M AN U
Pressure [bar]
0.6
0.5
SC
Pressure [bar]
0.6
RI PT
Exp. 298.15 K Exp. 313.15 K Exp. 338.15 K COSMO-SAC
0.7 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15
0.1
0.2
0. 3
0.4
0.5
0.6
w1
TE D
(c)
Figure 7: VLE predictions for (a) benzene-PEO(600000) at 323 K and 343 K, (b) n-hexane-PIB(50000) at 298 K, 313 K, 338 K Kand (c)2-butanone-PS(290000) at 298 K and 343 K. Experimental data taken from Hao et al. (1992) and Bawn et al. (1950).
trichloromethane activity results in PVME using polymer trimers with different tatici-
313
ties was made by Kuo et al. (2013). The authors found deviations of 25.6 % for isotatic
314
PVME and 6.5 % for syndiotatic PVME when compared with experimental data, illus-
AC C
EP
312
315
trating the importance of this analysis for some materials. However, polymer tacticities
316
are not reported in most of the literature VLE data, making this study a difficult task.
317
In Figure 8 are shown the VLE predictions PCL-carbon tetrachloride (8(a)), for
318
PVA-water (8(b)) and PBD-cyclohexane (8(c)). The deviations in pressure were 4.7 %,
319
7.0 %, and 6.2 % respectively, using experimental data from Hao et al. (1992) for PCL,
320
from Palamara et al. (2004) for PVA, and from Gupta and Prausnitz (1995) for PBD.
321
Based on model VLE predictions, the proposed methodology to extend the use
15
0. 5
Pressure [bar]
0. 4 0. 3 0. 2
Exp. 338.15 K COSMO-SAC
0. 1
0. 0 0. 1 0.2 0. 3 0. 4 0. 5 0.6 0. 7 0.8 0.9 1.0
1 .2 1 .1 1 .0 0 .9 0 .8 0 .7 0 .6 0 .5 0 .4 0 .3 0 .2 0 .1 0 .0
Exp. 373.15 K Exp. 383.15 K Exp. 363.15 K COSMO-SAC 0 .1
0 .2
0 .3
0 .4
0 .5
w1
w1
(a)
(b)
0.35
0.25 0.20 0.15 0.10 0.05
Exp. 333.15 K COSMO-SAC
M AN U
Pressure [bar]
0.30
0 .6
SC
Pressure [bar]
0. 6
RI PT
ACCEPTED MANUSCRIPT
0.1
0.2
0.3
0.4
0.5
0.6
0.7
w1
TE D
(c) Figure 8: VLE predictions for (a) carbon tetrachloride-PCL(33000) at 338 K, (b) water-PVA(116000) at 363 K, 373 K and 383 K and (c) cyclohexane-PBD(250000) at 333 K. Experimental data from Hao et al. (1992), Palamara et al. (2004), and Gupta and Prausnitz (1995).
of COSMO-based models to polymer systems shows promising results for apolar and
323
associating mixtures. However, as it considers the polymer as a liquid of independent
324
repeating units, further developments are necessary in order to enlarge its application to
EP
322
crystalline materials. Also, the inclusion of free-volume contributions in the COSMO-
326
SAC combinatorial formulation must be verified.
327
4.4. Vapor-Liquid equilibrium for copolymer systems
AC C 325
328
329
330
The VLE of copolymer-solvent mixtures predictions were compared to experimen-
tal data from Wohlfarth (2001) and Gupta and Prausnitz (1995). In Table B.6, results for ethylene-vinyl acetate copolymer with 39.4 wt% of vinyl
16
ACCEPTED MANUSCRIPT
acetate (E-VA/39w), 41.4 wt% vinyl acetate (E-VA/41w), and 70 wt% vinyl acetate (E-
332
VA/70w) are shown. Both copolymer have low molecular weight and no information
333
of Mw was available, only Mn . Good results with low ∆P(%) values were obtained
334
with an overall deviation of 7.9 %. For the systems studied, Kuo et al. (2013) had an
335
error of 10 %.
RI PT
331
In Table B.7 the VLE results for poly(styrene-acrylonitrile) with 25 wt% acryloni-
337
trile (S-AN/25w) and 28 wt% acrylonitrile (S-AN/28w) are listed. Very low deviations
338
were obtained with S-AN/28w while erros around 9 % were observed for the system
339
S-AN/25w with toluene. S-AN/25w and S-AN/28w are both random copolymers with
340
similar amount of acrylonitrile, so their equilibrium with toluene should result almost
341
the same. In this sense the errors of 1 % and 9 % presented in Table B.7 for S-AN/25w
342
and S-AN/28w toluene systems are not expected. A plausible explanation is the qual-
343
ity of the experimental data for S-AN/25w-toluene system. The VLE diagrams with
M AN U
SC
336
toluene for both polymers can be verified in Figure 9. As can be seen in the S-AN/25w0.200
0 .7
0.175
TE D
Pressure [bar]
0.150 0.125 0.100 0.075 0.050
EP
0.025
Exp. 313.5 K Exp. 323.15 K Exp. 333.15 K COSMO-SAC
0.000 0.00
0.25
0.50
0.75
Pressure [bar]
0 .6
1.00
Exp. 343.15 K Exp. 373.15 K COSMO-SAC
0 .5 0 .4 0 .3 0 .2 0 .1 0 .0 0 .0 0.1 0.2 0 .3 0 .4 0 .5 0. 6 0.7 0 .8 0. 9 1 .0
w1
w1
(b)
AC C
(a)
Figure 9: VLE predictions for styrene-acrylonitrile copolymer with toluene: (a) 25 wt of acrylonitrile (SAN/25w) and (b) 28 wt of acrylonitrile (S-AN/28w). Experimental data from Wohlfarth (2001).
344 345
toluene diagram (Figure 9(a)), a sudden change in the equilibrium pressure is observed
346
for a small change in the composition for w1 around 0.5. This indicates a possible
347
problem in the measurements shown in this figure. The COSMO-SAC model agree
348
with the data above this mole fraction and appears to do the same in the low solvent
17
ACCEPTED MANUSCRIPT
concentration region. None of the experimental data used here for this copolymer was
350
analysed in the work of Kuo et al. (2013), just S-AN with higher content of acriloni-
351
trile. The authors reported an overall deviation of 9.8 % for solvent/S-AN systems.
352
With the proposed methodology a total error of 4.9 % was achieved.
353
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349
Finally, the VLE results for systems containing the block copolymer of poly(styrenebutadiene) with 30 wt% styrene (S-b-BR/30w); the copolymer styrene-butadiene (with
355
no specification) containing 41 wt% styrene, (S-BR/41w), 45 wt% styrene (S-BR/45w),
356
and 77 wt% styrene (S-BR/77w) are shown in Table B.8. For the ethylbenze solvent,
357
lower deviations in equilibrium pressure conditions are achieved as the styrene content
358
increases, disregarding the effect of the molecular mass of polymer. The structure of
359
the block copolymer S-b-BR/30w is not taken into account in this methodology, which
360
can also be an important factor. With other solvents, good results were obtained.
361
5. Conclusions and further works
M AN U
SC
354
A new alternative to assemble the σ-profiles for polymers was proposed in this
363
work. The procedure is also based on a repeating unit structure that represents the ma-
364
terial of interest, as the other alternatives available in the literature. In this work a wider
365
analysis is made to extract the repeating unit information from different oligomers, and
366
an σ-profile with a much reduced end group effect is obtained. The σ-profiles of all
367
polymers studied in this work along with the ones for around 1500 solvents were in-
368
cluded in the freely available LVPP-Sigma database, available at https://github.
369
com/lvpp/sigma. The COSMO-SAC model with parameters from the literature was
370
used to predict infinite dilution activy coefficient and vapor-liquid equilibrium of sol-
EP
TE D
362
vent/polymer and solvent/copolymer systems. Comparing to a similar methodology
372
available in the literature, smaller deviations were achieved and good results were pos-
373
sible for apolar and associating mixtures. Improvements on the combinatorial contri-
374
bution are probably necessary for the representation of liquid-liquid equilibrium data.
375
Appendix A. VLE results for homopolymer-solvent systems
AC C 371
376
In Table A.1, the results of PDMS with different solvents are presented. 18
ACCEPTED MANUSCRIPT
Table A.1: VLE results for systems with organic solvents and PDMS.
Polymer
T (K)
NP
∆P(%)
n-hexane
PDMS(3350)
303
8
27.7
PDMS(6650)
303
8
21.6
PDMS(15650)
303
8
17.3
PDMS(26000)
303
8
15.3
313
8
9.1
303
8
13.1
PDMS(26000)
313
7
7.2
b toluene
PDMS(1540)
298
12
1.7
313
14
4.7
308
10
0.3
n-heptane
PDMS(140000)
chloroform
PDMS(89000)
2,2,4-trimethylpentane
PDMS(958)
(Hao et al., 1992)
(Hwang et al., 1998)
8
18.8
(Hao et al., 1992)
(Hwang et al., 1998) (Hao et al., 1992) (Hao et al., 1992)
(Hao et al., 1992)
7
0.2
(Hao et al., 1992)
4.8
(Hao et al., 1992)
298
12
8.9
298
12
5.2
313
17
5.1
(Hao et al., 1992)
303
3
5.1
(Hao et al., 1992)
303
3
3.6
(Hao et al., 1992)
303
3
3.7
(Hao et al., 1992)
303
3
3.0
(Hao et al., 1992)
PDMS(61000)
303
3
3.7
(Hao et al., 1992)
PDMS(220000)
303
3
3.1
PDMS(1540)
298
16
3.3
16
24
(Hao et al., 1992)
PDMS(4170)
298
11
7.8
11
24.3
(Hao et al., 1992)
PDMS(6650)
303
8
13.2
8
33.5
(Hao et al., 1992)
PDMS(26000)
303
8
17.6
8
34.7
(Hao et al., 1992)
PDMS(4600)
12
5.2
(Hao et al., 1992) (Hao et al., 1992)
(Hao et al., 1992)
8.2
23.4
EP
TE D
PDMS(55000)
377
(Hao et al., 1992)
8
PDMS(40000)
overall
(Hao et al., 1992)
323
PDMS(16000)
benzene
Ref.
(Hao et al., 1992)
303
PDMS(4170)
2-butanone
∆P(%)
M AN U
PDMS(89000) n-pentane
NP
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Solvent
Kuo et al. (2013)
SC
This work
In Table A.2, the results of polyethylene with different solvents are depicted. In all tests the model parameters of LPE where used to represent the polymer, even to
379
evaluate the low density polyethylene (LDPE) behavior.
AC C 378
380
381
382
383
In Table A.3, VLE results for PEO and PIB with many solvents are depicted and in
Table A.4 the results for PS mixtures. In Table A.5 the results for VLE of polymers with no IDAC data available are
shown.
19
ACCEPTED MANUSCRIPT
This work Solvent
Polymer
T (K)
NP
∆P(%)
NP
∆P(%)
1-pentene
LDPE(76000)
423
5
6.8
5
49.4
3-pentanone
LDPE(76000)
423
6
15.1
6
15.1
478
11
7.1
6
34.1
6
1.9
6
27.6
6
53.6
cyclopentane
LDPE(76000)
425 474
6
10.4
n-pentane
LDPE(76000)
423
6
12.4
propyl acetate
LDPE(76000)
426
5
8.9
5
474
5
7.4
5
386
387
388
(Surana, 1997)
(Surana, 1997)
(Surana, 1997)
(Surana, 1997)
SC
(Surana, 1997)
13.9
(Surana, 1997)
24
(Surana, 1997)
31.1
Appendix B. VLE results for copolymer-solvent systems
In Table B.6, results for copolymer poly(ethylene-vinyl acetate) with 39.4 wt% of vinyl acetate (E-VA/39w) and 41.4 wt% vinyl acetate (E-VA/41w) are presented. In Table B.7 are shown the VLE results for poly(styrene-acrylonitrile) with 25 wt% acrylonitrile (S-AN/25w) and 28 wt% acrylonitrile (S-AN/28w).
TE D
385
8.7
Ref.
(Surana, 1997)
M AN U
overall
384
Kuo et al. (2013)
RI PT
Table A.2: VLE results for systems with organic solvents and polyethylene.
In Table B.8 the VLE predictions of block copolymer of poly(styrene-butadiene)
390
with 30 wt% styrene (S-b-BR/30w); the copolymer styrene-butadiene (with no spec-
391
ification) containing 41 wt% styrene, (S-BR/41w), 45 wt% styrene (S-BR/45w), and
392
77 wt% styrene (S-BR/77w), are shown.
AC C
EP
389
20
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This work Solvent
Polymer
benzene
PEO(5700)
PEO(600000)
T (K)
NP
∆P(%)
NP
∆P(%)
318
5
1.0
5
3.6
(Hao et al., 1992)
343
9
0.5
9
6.0
(Hao et al., 1992)
323
5
0.6
5
2.8
(Hao et al., 1992)
343
8
1.5
8
3.8
(Hao et al., 1992)
298
6
7.0
308
6
6.4
318
5
5.9
328
5
5.8
PIB(1350)
298
PIB(2250000)
298
PIB(50000)
298
308 n-hexane
313 338
(Hao et al., 1992) (Hao et al., 1992) (Hao et al., 1992) (Hao et al., 1992)
10
2.7
12
5.9
12
(Hao et al., 1992)
5.7
(Hao et al., 1992)
4
13.1
4
12.2
(Hao et al., 1992)
8
5.5
8
10.3
(Hao et al., 1992)
9
7.6
9
9.9
(Hao et al., 1992)
7
7.4
7
7.9
(Hao et al., 1992)
6.7
TE D
overall
4.0
SC
0.9 PIB(1170)
Ref.
M AN U
overall n-pentane
Kuo et al. (2013)
RI PT
Table A.3: VLE results for systems with PEO and PIB.
9.2
Table A.4: VLE results for different systems with PS. This work
Kuo et al. (2013)
T (K)
NP
∆P(%)
NP
∆P(%)
Ref.
PS(290000)
298
11
14.6
11
11.1
(Bawn et al., 1950)
343
6
11.5
6
11.3
(Bawn et al., 1950)
carbon tetrachloride
PS(500000)
293
15
11.6
14
7.5
(Hao et al., 1992)
3-pentanone
PS(200000)
293
12
15.7
12
22.7
(Hao et al., 1992)
PS(500000)
293
12
13.2
(Hao et al., 1992)
PS(63000)
288
8
10.1
(Hao et al., 1992)
303
8
12.4
(Hao et al., 1992)
333
7
9.3
(Hao et al., 1992)
303
11
2.5
(Hao et al., 1992)
318
11
2.0
338
11
1.6
(Hao et al., 1992)
298
11
7.9
(Hao et al., 1992)
318
11
6.6
(Hao et al., 1992)
338
11
5.8
Solvent
Polymer
EP
2-butanone
AC C
benzene
cyclohexane
toluene
PS(154000)
PS(154000)
overall
8.9
21
11
1.3
(Hao et al., 1992)
(Hao et al., 1992) 10.8
ACCEPTED MANUSCRIPT
Table A.5: VLE results for solvent/polymer systems of PCL, PPO and PVME.
Polymer
carbon tetrachloride
PCL(33000)
T (K)
NP
∆P(%)
338
9
4.7
NP
PVC(77300)
338
8
13.4
2-butanone
PMMA (19770)
322
8
6.1
methanol
PPO(1120)
248
4
0.3
263
4
0.2
273
4
7.0
298
4
0.2
320
11
1.1
11
333
7
0.9
7
343
6
1.0
6
347
14
1.2
(Hao et al., 1992)
8
12.1
(Hao et al., 1992)
(Hao et al., 1992)
(Hao et al., 1992) (Hao et al., 1992) (Hao et al., 1992)
2.5
(Hao et al., 1992)
3.7
(Hao et al., 1992)
6.2
(Hao et al., 1992)
13.9
(Hao et al., 1992)
M AN U
PPO(500000)
Ref.
(Hao et al., 1992)
carbon tetrachloride
benzene
∆P(%)
SC
Solvent
Kuo et al. (2013)
RI PT
This work
PVME(14000)
298
13
17.3
water
PVA(116000)
363
10
9.2
(Palamara et al., 2004)
373
4
5.6
(Palamara et al., 2004)
383
4
6.3
overall
13
(Hao et al., 1992)
benzene
4.9
(Palamara et al., 2004)
7.7
Solvent benzene
TE D
Table B.6: VLE results for solvent/copolymer E-VA systems.
Copolymer
Mn (g/mol)
T (K)
NP
∆P(%)
E-VA/39w
3650
303
13
9.1
323
14
7.6
E-VA/41w
EP
butyl acetate
AC C
propyl acetate
ethyl acetate
methyl acetate
This work
E-VA/41w
E-VA/41w
E-VA/41w
4620
4620
4620
4620
Kuo et al. (2013) NP
323
8
16.0
8
14.6
343
8
16.4
8
15.9
363
8
16.9
8
16.6
303
8
4.4
8
6.0
323
8
7.3
8
8.8
343
8
10.0
8
8.7
363
8
11.0
8
12.5
303
7
1.5
7
2.2
323
7
1.9
7
2.9
343
7
2.8
7
2.8
303
8
1.7
8
6.0
323
8
1.6
8
6.6 25.2
cyclohexane
E-VA/70w
50000
353
7
5.7
9
chloroform
E-VA/70w
50000
333
8
13.4
8
overall
7.9
22
∆P(%)
12.3 10.0
RI PT
ACCEPTED MANUSCRIPT
Table B.7: VLE results for solvent/copolymer S-AN systems with the proposed methodology. ∆P(%)
Solvent
Copolymer
Mn (g/mol)
T (K)
NP
toluene
S-AN/25w
90000
313
12
9.5
323
12
10.3
12
10.7
S-AN/28w
46000
343
11
1.1
m-xylene
S-AN/28w
46000
398
8
2.8
p-xylene
S-AN/28w
46000
398
8
1.0
toluene
S-AN/28w
46000
343
overall
7
1.4
11
2.2
M AN U
373
SC
333 benzene
4.9
Solvent
TE D
Table B.8: VLE results for solvent/copolymer S-BR systems.
ethylbenzene
Kuo et al. (2013)
Copolymer
Mn (g/mol)
T (K)
NP
∆P(%)
NP
S-b-BR/30w
108000
373
10
13.8
9
15
403
20
13.5
19
14.5
EP
S-BR/41w
S-BR/45w
S-BR/77w
AC C
This work
86300
130430
117650
373
9
7.2
398
7
5.4
373
11
11.7
403
22
11.6
373
13
3.7
403
15
4.7
∆P(%)
cyclohexane
S-BR/41w
86300
343
8
9.4
8
2.8
benzene
S-BR/41w
86300
343
9
1.2
9
1.7
toluene
S-BR/41w
86300
343
8
1.3
8
0.8
373
8
1.7
8
1.6
overall
7.1
23
6.0
ACCEPTED MANUSCRIPT
394
Acknowledgement This work was partially supported by CNPq-Brazil under grants no. 454557/2014-0
RI PT
393
and 304046/2016-7.
396
References
397
Abildskov, J., Constantinou, L., Gani, R., 1996. Fluid Phase Equilib. 118 (1), 1–12.
398
Abildskov, J., Gani, R., Rasmussen, P., O’Connell, J., 1999. Fluid Phase Equilib. 158160, 349–356.
M AN U
399
SC
395
400
Banerjee, T., Singh, M. K., Khanna, A., 2006. Ind. Eng. Chem. Res. 45 (9), 3207–3219.
401
Bawn, C. E. H., Freeman, R. F. J., Kamaliddin, A. R., 1950. Trans. Faraday Soc. 46,
402
677–684.
Boucher, D. S., 2015. Colloids Surf., A 487, 207–213.
404
Bouillot, B., Teychen´e, S., Biscans, B., 2013. Ind. Eng. Chem. Res. 52 (26), 9276–
405
406
407
9284.
TE D
403
Constantinescu, D., Klamt, A., Geana, D., 2005. Fluid Phase Equilib. 231 (2), 231– 238.
Delley, B., 2006. Mol. Simul. 32 (2), 117–123.
409
D´ıaz, I., D´ıez, E., Camacho, J., Le´on, S., Ovejero, G., 2013. Fluid Phase Equilib. 337, 6–10.
AC C
410
EP
408
411
412
Elbro, H. S., Fredenslund, A., Rasmussen, P., Oct. 1990. Macromolecules 23 (21), 4707–4714.
413
Fredenslund, A. A., Jones, R. L., Prausnitz, J. M., 1975. AIChE J. 21 (6), 1086–1099.
414
Freire, M. G., Santos, L. M., Marrucho, I. M., a.P. Coutinho, J. a., 2007. Fluid Phase
415
416
Equilib. 255 (2), 167–178. Gerber, R. P., Soares, R. D. P., 2010. Ind. Eng. Chem. Res. 49 (16), 7488–7496. 24
ACCEPTED MANUSCRIPT
Gerber, R. P., Soares, R. P., 2013. Braz. J. Chem. Eng. 30 (1), 1–11.
418
Goss, K.-U., 2011. Anal. Chem. 83 (13), 5304–5308.
419
Grensemann, H., Gmehling, J., 2005. Ind. Eng. Chem. Res. 44 (5), 1610–1624.
420
Gupta, R. B., Prausnitz, J. M., Jul. 1995. J. Chem. Eng. Data 40 (4), 784–791.
421
Hanwell, M., Curtis, D., Lonie, D., Vandermeersch, T., Zurek, E., Hutchison, G., 2012.
424
SC
423
J. Cheminformatic 4, 17.
Hao, W., Elbro, H. S., Alessi, P., 1992. Polymer Solution Data Collection. DECHEMA Chemistry Data Series. Frankfurt.
M AN U
422
RI PT
417
425
Hsieh, C.-M., Sandler, S. I., Lin, S.-T., 2010. Fluid Phase Equilib. 297 (1), 90–97.
426
Hwang, S., Kim, J., Yoo, K.-P., Jul. 1998. J. Chem. Eng. Data 43 (4), 614–616.
427
Jakob, A., Grensemann, H., Lohmann, J., Gmehling, J., nov 2006. Ind. Eng. Chem.
428
Res. 45 (23), 7924–7933.
Kehiaian, H., Jan. 1983. Fluid Phase Equilib. 13, 243–252.
430
Klamt, A., 1995. J. Phys. Chem. 99 (7), 2224–2235.
431
Klamt, A., Eckert, F., 2000. Fluid Phase Equilib. 172 (1), 43–72.
432
Klamt, A., Jonas, V., Burger, T., Lohrenz, J. C. W., 1998. J. Phys. Chem. A 102 (26), 5074–5085.
Kontogeorgis, G. K., Coutsikos, P., Tassios, D., Fredenslund, A., 1994. Fluid Phase
AC C
434
EP
433
TE D
429
435
436
437
438
439
440
Equilib. 92, 35–66.
Kontogeorgis, G. M., Fredenslund, A., Tassios, D. P., Feb. 1993. Ind. Eng. Chem. Res. 32 (2), 362–372.
Kouskoumvekaki, I. A., Michelsen, M. L., Kontogeorgis, G. M., 2002. Fluid Phase Equilib. 202 (2), 325–335. Kuo, Y.-C., Hsu, C.-C., Lin, S.-T., 2013. Ind. Eng. Chem. Res. 52 (37), 13505–13515. 25
ACCEPTED MANUSCRIPT
Lin, S. T., Sandler, S. I., 2002. Ind. Eng. Chem. Res. 41 (5), 899–913.
442
Loschen, C., Klamt, A., 2014. Ind. Eng. Chem. Res. 53 (28), 11478–11487.
443
Mokrushina, L., Yamin, P., Sponsel, E., Arlt, W., 2012. Fluid Phase Equilibria 334,
444
RI PT
441
37–42.
Mu, T., Rarey, J., Gmehling, J., 2007. Ind. Eng. Chem. Res. 46 (20), 6612–6629.
446
Oishi, T., Prausnitz, J. M., Jul. 1978. Ind. Eng. Chem. Process Des. Dev. 17 (3), 333–
448
449
450
451
339.
Okeme, J. O., Parnis, J. M., Poole, J., Diamond, M. L., Jantunen, L. M., 2016. Chemosphere 156, 204–211.
M AN U
447
SC
445
Palamara, J. E., Zielinski, J. M., Hamedi, M., Duda, J. L., Danner, R. P., 2004. Macromolecules 37 (16), 6189–6196.
Panayiotou, C., 2013. Polymer 54 (6), 1621–1638.
453
Parnis, J. M., Mackay, D., Harner, T., 2016. Atmos. Pollut. Res. 7 (1), 155–161.
454
Patterson, D., Nov. 1969. Macromolecules 2 (6), 672–677.
455
Radfarnia, H. R., Kontogeorgis, G. M., Ghotbi, C., Taghikhani, V., 2007. Fluid Phase
457
458
Equilib. 257 (1), 63–69.
Reinisch, J., Klamt, A., Eckert, F., Diedenhofen, M., 2011. Fluid Phase Equilib. 310 (1-
EP
456
TE D
452
2), 7–10.
Shah, M. R., Yadav, G. D., 2013. J. Membr. Sci. 427, 108–117.
460
Shimoyama, Y., Iwai, Y., Takada, S., Arai, Y., Tsuji, T., Hiaki, T., 2006. Fluid Phase
AC C
459
461
Equilib. 243 (1-2), 183–192.
462
Soares, R. D. P., 2011. Ind. Eng. Chem. Res. 50 (5), 3060–3063.
463
Soares, R. D. P., Gerber, R. P., Apr. 2013. Ind. Eng. Chem. Res., 11159–11171.
464
Surana, R. K.; Danner, R. P. H. A. B. B. N., 1997. Fluid Phase Equilib. 139, 361–370. 26
ACCEPTED MANUSCRIPT
466
Tung, H.-H., Tabora, J., Variankaval, N., Bakken, D., Chen, C.-C., 2008. J. Pharm. Sci. 97 (5), 1813–20.
RI PT
465
467
Voutsas, E., Kalospiros, N. S., Tassios, D., 1995. Fluid Phase Equilib. 109 (1), 1–15.
468
Wang, S., Sandler, S. I., Chen, C.-C., 2007. Ind. Eng. Chem. Res. 46 (22), 7275–7288.
469
Wohlfarth, C., 2001. CRC Handbook of Thermodynamic Data of Copolymer Solutions. CRC Press: Boca Raton, FL.
SC
470
Wu, H. S., Sandler, S. I., 1991. Ind. Eng. Chem. Res. 30 (5), 881–889.
472
Yang, L., Xu, X., Peng, C., Liu, H., Hu, Y., 2010. AIChE J. 56 (10), 2687–2698.
473
Zhong, C., Sato, Y., Masuoka, H., Chen, X., 1996. Fluid Phase Equilib. 123 (1-2), 97–106.
AC C
EP
TE D
474
M AN U
471
27