Article pubs.acs.org/IECR
Superposition of Liquid−Liquid and Solid−Liquid Equilibria of Linear and Branched Molecules: Ternary Systems Thomas Goetsch,† Patrick Zimmermann,‡ Rebecca van den Bongard,† Sabine Enders,‡ and Tim Zeiner*,§ †
Department of Biochemical and Chemical Engineering, Laboratory of Fluid Separations, TU Dortmund University, Emil-Figge-Straße 70, D-44227 Dortmund, Germany ‡ Institute of Technical Thermodynamics and Refrigeration Engineering, Karlsruhe Institute of Technology, Engler-Bunte-Ring 21, D-76131 Karlsruhe, Germany § Institute of Chemical Engineering and Environmental Technology, Technical University Graz, Inffeldgasse 25 C, A-8010Graz, Austria S Supporting Information *
ABSTRACT: Oiling-out is an unwanted phenomenon during crystallization processes since it influences the product properties negatively and should, therefore, be avoided. To reduce the time of process development, thermodynamic modeling is usually applied. In the course of fitting model parameters, thermodynamic data of the present molecules are required. In case of branched molecules these thermodynamic data are often not available. To overcome this limitation, a methodology, which allows for the prediction of liquid−liquid equilibria (LLE) of binary systems containing branched molecules was developed recently. The developed methodology was applied in this contribution in order to predict the superposition of ternary LLE and solid−liquid equilibria (SLE) of the system n-hexadecane + 2,2,4,4,6,8,8heptamethylnonane + ethanol. To consider the influence of the molecular architecture on phase equilibria, the lattice cluster theory in combination with the chemical association lattice model was applied. The prediction of the ternary phase equilibria was based on the binary subsystems. It could be shown that the ternary LLE and the ternary SLE can be predicted in very good agreement with experimental data using the same set of model parameters. All model parameters were fitted using only binary LLE data of linear alkanes dissolved in ethanol. Neither binary experimental data of the branched alkane nor ternary ones were used for parameter fitting.
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INTRODUCTION Crystallization from solution is a reliable unit operation for the separation of isomers, because branched molecules have got much lower melting temperatures than their linear analogues leading to different solubilitites in a solvent. It is superior to conventional unit operations like distillation or extraction because of low selectivities and high purity demands.1 A need for the separation of isomers arises after nonregioselective reactions. One of these reactions is the hydroesterification of long-chain unsaturated esters of fatty acids. Besides the desired linear diester a number of branched isomers are produced simultaneously.2 The challenge during the crystallization from solution is the appearance of a superposition of a liquid−liquid (LLE) and solid−liquid equilibrium (SLE) called oiling-out. According to Kiesow et al.3,4 these phenomena can affect the product properties like crystal size and crystal shape. Furthermore, Yang and Rasmuson5,6 reported that the crystals are much smaller and more agglomeration can be observed due to the formation of a LLE. Beyond that, the particle size distribution is wider in comparison to crystallization without liquid−liquid demixing. Hence, it is essential to know the phase © 2016 American Chemical Society
equilibria in order to design and control crystallization processes. Since measuring the superposition of LLE and SLE is a complex task, it is reasonable to reduce the experimental effort by means of thermodynamic modeling. There are only a few investigations in literature that are dealing with the molecular architecture’s effect of isomers on LLE. The contributions of Hofman et al.7 as well as Reda et al.8 focused on binary LLE of isomeric C8 aliphatic monoethers dissolved in nitromethane and acetonitrile, respectively. It was shown that small differences in molecular architecture have a significant influence on the LLE. They tried to model this influence applying the modified UNIFAC9 model as well as the COSMO-SAC10 model. The UNIFAC model is not able to give quantitative predictions of the binary LLE because it cannot differentiate between the investigated isomers. Within COSMO-SAC every Received: Revised: Accepted: Published: 417
November 2, 2016 December 15, 2016 December 15, 2016 December 15, 2016 DOI: 10.1021/acs.iecr.6b04253 Ind. Eng. Chem. Res. 2017, 56, 417−423
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Industrial & Engineering Chemistry Research
truncated at second order because terms of higher order do not improve the results significantly. In this contribution a ternary system containing the solvent ethanol (A), the branched alkane 2,2,4,4,6,8,8-heptamethylnonane (B), and the linear alkane nhexadecane (C) was investigated. The calculation of the Helmholtz free energy of the ternary system is based on the binary subsystems. For every binary subsystem the interaction energy ε is defined as
isomer has a different enthalpic contribution to the Helmholtz free energy. However, it is also not able to give quantitative predictions of the binary LLE.8 Applying other thermodynamic models, pure component parameters have to be adjusted to experimental data. This common procedure cannot be applied when experimental data of the pure components are missing, which is often the case for branched molecules. To overcome this shortcoming Goetsch et al.11 developed a methodology, which allows for the prediction of binary LLE of systems containing branched molecules. They investigated binary systems of linear and branched alkanes dissolved in an alcohol. Within their methodology only phase equilibria data of linear molecules are used for the fitting of the necessary model parameters. The model parameters are then combined with architectural information on the branched molecules in order to predict phase equilibria of these isomers. Goetsch et al.11 applied the lattice cluster theory (LCT) developed by Freed and coworkers.12−20 The LCT calculates the Helmholtz free energy under direct consideration of the molecular architecture where architecture parameters are determined by the chemical structure. Since alcohols are highly polar components the LCT was combined with the chemical association lattice model (CALM)21,22 in order to take the self-association of the alcohol into account. Besides CALM, the LCT was already successfully combined with Wertheim’s theory.23−26 The predicted binary LLE of Goetsch et al.11 showed a very good agreement (average relative deviations between 1.79% and 3.93%) to experimental data. In addition to predict binary LLE of systems containing branched molecules they also investigated the superposition of binary LLE and SLE. For two binary systems containing a long-chain alkane and an alcohol it was shown that both phase equilibria can be calculated in very good agreement to experimental data applying only one set of model parameters. The same finding was reported for the superposition of LLE and SLE of hyperbranched polymers dissolved in an alcohol.27 The aim of this work is the prediction of the superposition of ternary LLE and SLE based on the binary subsystems. As a ternary model system, the long-chain alkane n-hexadecane, its branched isomer 2,2,4,4,6,8,8-heptamethylnonane, and the solvent ethanol were investigated. The binary subsystem containing 2,2,4,4,6,8,8-heptamethylnonane and ethanol was predicted using the methodology of Goetsch et al.11 As thermodynamic model an incompressible version of the LCT was applied. It is combined with CALM in order to describe the self-association of ethanol. The predicted phase equilibria are compared with experimental data, which were measured for this contribution.
ε = εii + εjj − 2εij
(1)
where εii, εjj, and εij represent the interaction energies between two segments of type ii, jj, and ij, respectively. The LCT is defined in terms of segment fractions, which can be written as
ϕi =
niNi NL
(2)
In eq 2 ni is the number of molecules of component i, Ni is the number of segments a molecule of component i is divided into, and NL is the total number of lattice sites. The investigated ternary phase equilibria are assumed to depend not on the pressure. Hence, the Helmholtz free energy is equal to the Gibbs free energy (segment-molar), which can be written as follows:18,26 ΔGsLCT = RT
∑ i
ϕi Ni
ln(ϕi) −
ΔE1 ΔE2 ΔS − − R RT RT
(3)
In eq 3 the first term equals the entropic part of the Flory− Huggins theory (FH).29 The other three terms contain corrections to the FH mean field, where ΔS represent the entropic corrections and ΔE1 and ΔE2 represent the enthalpic corrections of first and second order, respectively. ΔS, ΔE1, and ΔE2 can be derived using the tables I, II, and III reported by Dudowicz and Freed18 and the corrections made by Dudowicz et al.30 The exact expressions can be found in literature.26 To take the molecular architecture into consideration the LCT offers architecture parameter. Originally six architecture parameters were supposed to be necessary to define the molecular architecture properly. Later, it could be shown by Langenbach et al.28 that some of the architecture parameters are connected to each other so that a reformulation of the LCT with only three architecture parameters is possible. These three architecture parameters are the number of bonds N1,i, the number of two consecutive bonds N2,i, and the number of three consecutive bonds N3,i. In this contribution the architecture parameters listed in Table 1 were estimated according to the chemical formula for the molecules of interest. Since the LCT in the applied version is not able to describe interactions of strongly polar components, it is combined with CALM in order to take the self-association of ethanol into account. Chemical Association Lattice Model. Browarzik21 developed CALM to calculate phase equilibria of binary and
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THEORY Lattice Cluster Theory. In this work, we used an incompressible version of the LCT applicable for a multicomponent system.26,28 The LCT was originally derived by Freed and co-workers12−20 in order to take the molecular architecture directly into account to calculate the Helmholtz free energy. They defined a partition function for structured molecules on a lattice, which is formally exact but not solvable analytically. Therefore, the partition function of the LCT was expanded in a double series of 1/z and ε/kBT where z is the lattice coordination number and ε/kBT represents the dimensionless interaction energy of nearest neighbor segments. According to Dudowicz and Freed,18 the double series can be
Table 1. Architecture Parameters of the LCT for Ethanol, nHexadecane, and 2,2,4,4,6,8,8-Heptamethylnonane ethanol n-hexadecane HMNa a
418
Ni
N1,i
N2,i
N3,i
3 16 16
2 15 15
1 14 24
0 13 16
2,2,4,4,6,8,8-Heptamethylnonane. DOI: 10.1021/acs.iecr.6b04253 Ind. Eng. Chem. Res. 2017, 56, 417−423
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Industrial & Engineering Chemistry Research
defined for a binary system: the interaction energy ε/kB and the two association parameters K0A and ΔhasA . The two association parameters describe the self-association of the solvent. Therefore, they are not dependent on the solute and remain constant for any pair of solvent and solute. Hence, these two parameters can be fitted to binary LLE data of a linear molecule and afterward used for the prediction of the binary LLE of the branched molecule. This procedure is not possible for the interaction energy ε/kB because it refers to a pair of solute and solvent. It could be shown in the literature11,31 that the interaction energy can be written as a linear function of the chain length within a certain range. This finding was used by Goetsch et al.11 for estimating the interaction energy between a branched molecule and a solvent. They showed that the interaction energy can be approximated by the length of the backbone of the branched molecule. For three binary systems they predicted the binary LLE with a maximal average relative deviation of 3.93%. Applying the described procedure it is possible to predict binary LLE of systems containing branched molecules without having used experimental data of these molecules. For the prediction of the ternary phase equilibria in this contribution five model parameters have to be known in total: the interaction energy ε/kB for every binary subsystem (where the interaction energy between n-hexadecane and 2,2,4,4,6,8,8heptamethylnonane is zero by definition) and the two association parameters K0A and ΔhasA in order to describe the self-association of ethanol. All parameters were already determined by Goetsch et al.11 Values of ΔhasA = −30685.5 J mol−1 and K0A = exp(−9.35) were found for the two association parameters, the interaction energy between ethanol and hexadecane was fitted to ε/kB = 11.29 K, and the interaction energy between ethanol and 2,2,4,4,6,8,8-heptamethylnonane was estimated to ε/kB = 4.48 K.
ternary systems with one associating component. It is an association model that offers a straightforward way of calculating the excess Gibbs energy without any numerical iteration. CALM treats the self-association of solvent (A) as chemical equilibrium between two associating chains with degrees r and r′ and one associating chain of degree r + r′: Ar + Ar′ ⇄ Ar+r′
(4)
This procedure of describing the self-association of solvent (A) leads to a distribution of association chain lengths, which can range from 1 to ∞. To take the effect of temperature on the association into consideration, Browarzik21 introduced the equilibrium association constant KA: ⎛ Δhas ⎞ KA = K 0A exp⎜ − A ⎟ ⎝ RT ⎠
(5)
ΔhasA
where is the association enthalpy. Applying the mentioned equations, the contribution of CALM to the segment-molar Gibbs energy can be written as ϕ ⎛ ΔGsas 1 ⎞ ⎟⎟ln(1 + KÃ ϕA ) = − A ⎜⎜1 + RT NA ⎝ KÃ ϕA ⎠ +
ϕA ⎛ 1 ⎞ ⎟ln(1 + KÃ ); ⎜1 + NA ⎝ KÃ ⎠
KÃ = KA exp(1)
(6)
Both contributions, the contribution of the LCT as well as the contribution of CALM, are aggregated to the segmentmolar Gibbs free energy of the system: ΔGs = ΔGsLCT + ΔGsas
(7)
Using the Gibbs free energy of the system ternary LLE and SLE are calculated applying standard thermodynamics. Concerning the calculation of the SLE we made the assumptions that the solid phase is composed of the pure substance and that the differences in heat capacities of solid and liquid phase are negligible. This leads to eq 8 for the calculation of the SLE. Δμi (T , ϕi)
=−
RT
ΔhiSL ⎛ T ⎞ ⎜1 − SL ⎟ RT ⎝ Ti ⎠
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MATERIALS AND METHODS Materials. In this contribution LLE and SLE of the ternary system n-hexadecane + 2,2,4,4,6,8,8-heptamethylnonane + ethanol were estimated. Table 2 shows all used chemicals Table 2. Chemicals Used in the Present Contribution
(8)
chemical name
All results are presented in mass fractions; thus the segment fractions have to be converted into mass fractions using eq 9. wi =
Mi ϕ Ni i Mj
∑j
ϕ
Nj j
(9)
Predicting Phase Equilibria of Branched Molecules. In the beginning of thermodynamic modeling it is mandatory to adjust model parameters to experimental data. Regarding branched molecules experimental data are scarce, which is the reason why thermodynamic modeling of systems containing branched molecules is often not possible. Recently, Goetsch et al.11 introduced a methodology to overcome this limitation, which will be shortly explained in this section. The basic idea is to combine architecture parameters that can be extracted from the molecular architecture with model parameters that were fitted to experimental data of linear molecules. On the one hand architecture parameters can be defined a-priori within the LCT. On the other hand model parameters have to be fitted to experimental data. In total, three model parameters have to be
a
CAS no.
source
purity
remark used for LLE experiments used for LLE experiments used for SLE experiments used for SLE experiments used for LLE and SLE experiments internal standard for GC
ethanol
64-17-5
VWR
100%
n-hexadecane
544-76-3
amresco
99%
ethanol
64-17-5
Merck
99%
n-hexadecane
544-76-3
99%
HMNa
4390-04-9
dibutyl ether
142-96-1
SigmaAldrich SigmaAldrich Alfa Aesar
98% 99%
2,2,4,4,6,8,8-Heptamethylnonane.
including their purities. To ensure the purity of ethanol it was stored together with molecular sieves by Merck having pore sizes of 3 Å. All other chemicals were used without further purification. 419
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EXPERIMENTS LLE Measurements. The ternary LLE of the system nhexadecane + 2,2,4,4,6,8,8-heptamethylnonane + ethanol was measured by titration experiments as well as by estimating tie lines. First, the binodal curve was determined by cloud-point titration. Therefore, binary mixtures were prepared and placed in a tempered water bath for 30 min, where the temperature can be adjusted with an accuracy of ±0.1 K. All mixtures were clear at this point. Afterward, 0.015 g of the third component was added to each mixture and it was checked after 2 min whether the mixtures became turbid or stayed clear. This procedure was repeated until all mixtures turned turbid, which indicates the liquid−liquid transition. Besides measuring the binodal curve by cloud-point titration, also tie lines of the ternary systems were determined. Therefore, ternary samples of known composition within the miscibility gap were prepared and placed in the tempered water bath where they were stirred for 1 h. After that, the samples remained another 24 h in the tempered water bath for equilibration. In this time the samples separated into two phases and the equilibrium between these two phases was achieved. The upper phase was separated from the lower phase using a syringe and both phases were weighed. This procedure is advisible for checking the phase ratio that is given by the gas chromatograph. Afterward, the composition of the two phases were analyzed by gas chromatography as described below. SLE Measurements. The ternary SLE was determined in a similar way than the determination of the binodal curve. At first, binary mixtures of 2,2,4,4,6,8,8-heptamethylnonane + ethanol were prepared. Because the SLE does not change within the miscibility gap, we made use of the already known ternary LLE such that only mixtures with very high and very low concentration of ethanol were prepared in order to measure the ternary SLE exclusively outside the miscibility gap. The mixtures were placed in the tempered water bath and stirred continuously. After 30 min, 0.015 g of n-hexadecane were added to the mixtures. After 5 min it was checked whether there were precipitates visible or not. If there were no precipitates apparent, the procedure was repeated until there were precipitates in every sample. To prove the quality of the gained results, the mixtures were heated up afterward. If the precipitates disappear already at small temperature increases, then the measured SLE has a good accuracy. In this work 0.2 K up to 0.4 K were required to make the precipitates disappear. Analytics. In order to determine the compositions of lower and upper phase of the tie lines a gas chromatograph from Shimadzu (type GC-14A) was applied. The gas chromatograph was equipped with a nonpolar column Innopeg-FFAP having a length of 25 m, an inner diameter of 0.32 mm and a film thickness of 0.48 μm. Helium with a velocity of 0.35 cm/s was used as carrier gas. At the end of the column, a flame ionization detector with a temperature of 573.15 K analyzed the contents of n-hexadecane and 2,2,4,4,6,8,8-heptamethylnonane. The content of ethanol was determined via a mass balance. Calibration curves with relative deviations of 1.29% and 0.91% were prepared for n-hexadecane and 2,2,4,4,6,8,8heptamethylnonane, respectively. Dibutyl ether was used as internal standard in order to determine the mass fractions of the alkanes in the samples. Every sample was analyzed three times. The average mass fraction out of these three measurements was used for the further evaluation. In order to separate the peaks of the present components, the following method was
applied: First, the column is heated up to a temperature of 353.15 K, which is held constant for 3 min. Afterward, the temperature is raised to a temperature of 503.15 K applying a heating rate of 30 K/min. After another 3 min, the column is cooled down to a temperature of 353.15 K. The method takes 11 min in total, typical retention times were 3.1 min for ethanol, 3.5 min for dibutyl ether, 7.2 min for 2,2,4,4,6,8,8heptamethylnonane and 9.3 min for n-hexadecane.
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RESULTS The aim of this contribution is the prediction of the superposition of LLE and SLE for the ternary system nhexadecane + 2,2,4,4,6,8,8-heptamethylnonane + ethanol based on the binary subsystems. This implies that no experimental data of the ternary system was used for adjusting model parameters. All necessary model parameters were determined as described in the Theory section. Before predicting the superposition of ternary LLE and SLE, it should be proved whether the model is capable of predicting the ternary phase equilibrium based only on the binary subsystems. Therefore, the binodal curve and three tie lines are predicted at a temperature of 298.15 K, which is higher than the melting temperature of n-hexadecane. In order to validate the ternary prediction, the binodal curve as well as tie lines were estimated (see Tables S1 and S2). All experimental data are shown with the corresponding uncertainties. These uncertainties are calculated by Gaussian error propagation based on the errors of the used scale and of the gas chromatograph. Figure 1 shows the comparison of calculated and measured data.
Figure 1. LLE of the ternary system n-hexadecane + 2,2,4,4,6,8,8heptamethylnonane + ethanol at a temperature of 298.15 K. Experimental points on the binodal curve are shown as gray diamonds; experimental tie lines are shown as gray stars connected by dashed lines. The binodal curve as well as the tie lines (white stars connected by solid lines) were calculated using the LCT in combination with CALM.
Regarding binodal curve and tie lines an excellent agreement of predicted and experimental data can be found for the compositions on the ethanol-rich side. Likewise, the slope of the tie lines is predicted in excellent agreement with the experimental data. Small deviations between predicted and calculated data arise on the ethanol-lean side. This finding is not surprising, since our model also show small deviations for the binary system n-hexadecane + ethanol.11 Nevertheless, it can be stated that the prediction of the ternary phase equilibria 420
DOI: 10.1021/acs.iecr.6b04253 Ind. Eng. Chem. Res. 2017, 56, 417−423
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Industrial & Engineering Chemistry Research based on the binary subsystems is in very good agreement to the experimental data. Having gained trust in the ternary prediction based on the binary subsystems, it can be checked whether the superposition of ternary LLE and SLE, which is essential for the design of crystallization processes, can be predicted correctly. The already fitted model parameters were used for the prediction of both phase equilibria. Regarding the calculation of the ternary SLE, additionally the melting temperature and enthalpy of fusion is needed. Values of TSL = 291.07 K and ΔhSL = 52433 J mol−1 were estimated in our recent work.11 It was assumed that the solid phase only contains n-hexadecane. Figure 2 shows the predicted super-
Figure 3. Superposition of LLE (diamonds) and SLE (triangles) of the ternary system n-hexadecane + 2,2,4,4,6,8,8-heptamethylnonane + ethanol at a temperature of 278.15 K. Lines were calculated using the LCT in combination with CALM. The dashed lines denote the metastable LLE.
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CONCLUSIONS In this contribution the superposition of LLE and SLE was predicted for the ternary system n-hexadecane + 2,2,4,4,6,8,8heptamethylnonane + ethanol applying the LCT in combination with CALM. The prediction was done based on the binary subsystems, which means that no ternary data was used for parameter fitting. Challenging for the fitting of model parameters is that experimental data of branched molecules are often not available. Therefore, we made use of our recently developed methodology for predicting LLE of binary systems containing branched molecules,11 where only phase equilibria data of linear molecules are used for parameter fitting. In this contribution we extended this methodology in order to predict ternary LLE. The LCT considers the molecular architecture directly within the calculation of the Helmholtz free energy by introducing architecture parameters. It is therefore able to describe the influence of molecular structure on the phase equilibria. In order to consider the self-association of the strongly polar solvent ethanol, the LCT was combined with CALM. Applying the fitted model parameters in combination with architecture parameters the superposition of ternary LLE and SLE was predicted for two different temperatures. The agreement of predicted and measured data was very good for the LLE, where tie lines and the binodal curve were investigated, as well as for the SLE. Thus, this contribution offers a convenient way of predicting the ternary map for the crystallization of branched molecules from solution.
Figure 2. Superposition of LLE (diamonds) and SLE (triangles) of the ternary system n-hexadecane + 2,2,4,4,6,8,8-heptamethylnonane + ethanol at a temperature of 283.15 K. Lines were calculated using the LCT in combination with CALM. The dashed lines denote the metastable LLE.
position of ternary LLE and SLE for a temperature of 283.15 K. The predicted phase equilibria are compared with experimental data listed in Table S3. Below the SLE curve the LLE is shown as a dashed line. This indicates that the LLE is only metastable in this region. The SLE was only determined outside the LLE, because a mixture within the miscibility gap would directly separate into two phases. Comparing predicted and measured SLE data an excellent agreement can be seen. Regarding the binodal curve, a very good agreement between predicted and measured data can be seen. Only the lowest point on the ethanol-lean side shows a small deviation to the predicted bimodal curve. In order to prove the capability of our model to consider the temperature dependency of the system the superposition of ternary LLE and SLE was also predicted for a temperature of 278.15 K (Figure 3) and compared with experimental data measured in this contribution (Table S4). Again, the agreement of predicted and measured data is excellent for the ternary LLE as well as for the ternary SLE. Therefore, the model is able to consider the influence of temperature on the superposition of ternary LLE and SLE. Comparing the experimental data at 283.15 and 278.15 K, it can be seen that the region of the ternary SLE is growing with lower temperatures. This leads to a bigger region where a crystallization process is possible. However, also the region where an oiling-out can occur is growing, which has to be considered within the process design.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.6b04253. Experimental data shown in Figures 1−3 (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Tim Zeiner: 0000-0001-7298-4828 421
DOI: 10.1021/acs.iecr.6b04253 Ind. Eng. Chem. Res. 2017, 56, 417−423
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Industrial & Engineering Chemistry Research Notes
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The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work is part of the Collaborative Research Centre “Integrated Chemical Processes in Liquid Multiphase Systems” coordinated by the Technische Universität Berlin. Financial support by the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged (TRR63). We gratefully acknowledge the valuable technical assistance of Karin Daniel and Katrin Kissing.
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NOMENCLATURE ΔGs = segment-molar Gibbs energy Δhasi = association enthalpy of component i ΔhSL i = enthalpy of fusion of component i KB = Boltzmann constant Ki = association constant K̃ i = Kiexp(1) K0i = pre-exponential factor defined by eq 5, Mi = molar mass of component i ni = number of molecules of component i Ni = segment number of component i NL = total number of lattice sites R = universal gas constant T = temperature TSL i = melting temperature of component i wi = mass fraction of component i z = coordination number
Greek Letters
ε = interaction energy defined by eq ϕ = segment fraction μ = chemical potential
1
Superscripts
as = association LCT = lattice cluster theory SL = solid−liquid phase transition
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REFERENCES
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DOI: 10.1021/acs.iecr.6b04253 Ind. Eng. Chem. Res. 2017, 56, 417−423
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DOI: 10.1021/acs.iecr.6b04253 Ind. Eng. Chem. Res. 2017, 56, 417−423
