Mean field lattice equations of state

ELSEVIER

Fluid Phase Equilibria 99 (1994) 121- 133

Mean field lattice equations of state VI. Prediction of volume and surface parameters via Bondi’s group contribution theory E.J. Beckman a,*, T.A. Hoefling a, L. Van Opstal b, R. Koningsveld

b, R.S. Porter ’

a Chemical Engineering Department, University of Pittsburgh, Pittsburgh, PA, USA b Chemistry Department, University of Antwerp, Wilrijk, Belgium c Polymer Science and Engineering Department, University of Massachusetts, Amherst, MA, USA Received 22 July 1993; accepted 14 March 1994

Abstract

In preceding publications in this series, the Mean Field Lattice Gas (MFLG) model has been shown to produce a useful description of the p VT behavior of both polymers and low molar mass materials. Here we introduce a method, referred to as use of the Bondi constraint, by which the number of adjustable parameters required to apply the model can be reduced from five to three. This is accomplished by taking advantage of the general observations that (1) the parameter which represents the number of segments per molecule (or polymer repeat unit), m, , varies linearly with the van der Waals volume as calculated by Bondi, and (2) the ratio of the molecular surface areas of a compound and a reference fluid, calculated from the MFLG parameters m, and y,, can be equated to the analogous ratio of surface areas calculated using Bondi’s group contribution methodology. We show that this predictive method for derivation of the parameters m, and y, , produces a p VT description for ethane and ethylene which is comparable to that produced using the standard five-parameter fit. Thus the Bondi constraint is a useful means of generating material parameters that are necessary to apply the model to compounds for which few volumetric data are available. Keywords: Theory; Equation

of state; Group contribution;

Volume; Surface parameters; Mean Field Lattice Gas

1. Introduction

The effect of changes in volume on the free energy in polymer solutions has been modeled using one of two general strategies: that of the cell approach (Prigogine, 1957; Flory et al., 1964), and that of the lattice gas formulation, which is the focus of this work. The lattice gas approach, which has been in use since the 1920s (Schottky et al., 1929), is one in which vacant * Corresponding

author.

037%3812/94/%07.00 0 1994 - Elsevier Science B.V. All rights reserved SSD10378-3812(94)02512-Y

122

E.J. Beckman et al. 1 Fluid Phase Equilibria 99 (1994) 121-133

sites, or holes, are introduced into the lattice. A pure component is therefore assumed to be a pseudo-binary mixture, where changes in volume with pressure or temperature are modeled by changes in the concentration of the holes. In the first approximation the distribution of holes on the lattice is assumed to be random, although this can be readily modified to include the presence of specific interactions between segments. The apparent concentration dependence of the interaction parameter in conventional lattice models can be at least partially explained via the introduction of the concept of segmental contact surface areas during the formulation of the internal energy of mixing of holes and segments (Staverman, 1937). Segments and holes are permitted distinct coordination numbers, which are assumed to be proportional to their respective surface areas. The total number of nearest neighbor contacts of segments or holes is thus proportional to the contact surface area. The change in the internal energy upon mixing holes and segments, A& is assumed to be equal to the change in the number of the various contact pairs on mixing, multiplied by their respective interaction energies, where it is assumed that contact with a hole involves zero interaction energy. The number of segment-segment contacts is thus calculated using Regular Solution Theory (i.e. a mean field approach) and surface fractions, instead of volume fractions. Ultimately, AU is derived as

(1) where

y1 =

1 -ol/oo

N4 =no+nlm,

no, nl are the number of holes and segments, co, crl are the contact surface areas of holes and segments, and wll is the segment-segment interaction energy. Note that the effective interaction parameter, (AU/4,& N4RT), is dependent upon concentration without the use of purely empirical parameters. (Note: in recent applications of the Mean Field Lattice Gas (MFLG) model by Van Opstal and Koningsveld (1993a, b), the interaction parameter g,, is defined as ( - wl, 01/2R).This combines the factor (1 - yl) with g,, , y et does not affect the essential structure of the model.) If the change in the entropy on mixing holes and segments is assumed to follow the simple Flory-Huggins-Staverman expression, the Helmholtz free energy of mixing, AA, becomes: AA/N+RT=40

41 ln~o+~ln~,+

4041(1 - Y1) (g#) 1 -Y*&

Eq. (2) is essentially the free energy expression derived by Kanig ( 1963). During the modeling of various gas-gas, gas-liquid, and polymer-liquid systems, Kleintjens (1979) and Koningsveld (Kleintjens and Koningsveld, 1980, 1982) found that addition of two empirical parameters, CI]and glo, to Eq. (2) greatly improved the model description of the pVT behavior of gases and low molar mass liquids. The free energy expression of this MFLG model is as follows:

E.J. Beckman et al. 1 Fluid Phase Equilibria 99 (1994) 121-133

123

(3) The need for these empirical parameters was initially assumed to be due to deviations from the simple entropy of mixing expression used in the construction of the model. The Kanig model is thus a specific case (CC,,g,, = 0) of the MFLG model. The equation of state is derived from the free energy (Eq. 3) in the usual way: (4) therefore: -pvo/RT

-d2

+k,o+gdT)U

= In c$~+ &

e’

1

where Q = 1 - y1&. Although the equation of state is sufhcient to define the p VT behavior of high polymers, additional constraints, such as equations defining vapor-liquid equilibrium (VLE) and the gas-liquid critical point, are required to completely describe the behavior of gases and low molar mass liquids. VLE is defined by equating the chemical potential of the segments and the holes (the latter is the same as equating the pressure) in the saturated liquid and vapor phases: /C=/C

T,nj

i=o,

Following the analysis of Kleintjens expressed via Eqs. (7) and (8):

1

(1979)

(6) the critial condition

for the MFLG

model is

(8) The complete set of equations resulting from combinations of Eqs. (3) and (6) -(8) can be found in Kleintjens (1979). In the MFLG model the volume per lattice site, vo, is fixed for all substances. Frenkel (1946) has asserted that v. for lattice models should approximate the size of small atoms and, as such, lie between 5 and 25 cm3 mol-‘. In the MFLG model, v. is presumed to be a scaling parameter and is initially set to 25 cm3 mol-’ for all substances. As shown by Beckman, et al. (1987) if varied in the 5-35 cm3 mol-’ range, the value of v. changes the absolute value of the material parameters but does not significantly affect the quality of the model description of p VT data.

2. Origin of material parameters in the MFLG model As defined above, the material parameters ml and yl in the MFLG should correlate well to the size and surface areas of various functional groups, if the model provides a good representation

E.J. Beckman et al. / Fluid Phase Equilibria 99 (1994) 121- 133

124

of behavior at the molecular level. However, because the absolute values of ml and y1 for a material are based on the value of u. (here chosen as 25 cm3 mol-‘), unless the choice of v. is extremely fortuitous, we would not expect to be able to a priori calculate ml and yl. Consequently we have investigated the degree of correlation between values of ml and yl, which have been determined from computer fitting of pVT data to the MFLG model equations to segmental surface areas and volumes calculated using Bondi’s group contribution values. 2.1. The Parameter Estimation Program (PEP) Determination of the adjustable parameters for the various compounds examined in this study was carried out using the Parameter Estimation Program PEP developed at Dutch State Mines (DSM) Research (Geleen, Netherlands). The PEP finds the values of the parameters, &, which are solutions to the minimization problem (Hillegers, 1986) as follows. Minimize i

(Xi -

xywyxi - xp>

i=l

(9)

subject to the constraints

(10) where Xi are the calculated data points, xy are the true values, and 51is the error variance matrix. The constraints, fi, are the appropriate equations (equation of state, VLE, critical point expressions) in the implicit form, as shown in Eq. (10). All variables are therefore assumed to have been measured with some error. The errors, or tolerances, which comprise 0, are assumed to be random in nature and have been set as follows: temperature 50.25 K; T, + 0.01 K; pressure +0.5 bar; PC +O.l bar; density &OS%; critical density f 0.1%. The errors can be assigned either absolute or relative (%) values. The best values for the parameters, therefore, are those which fit the surface represented by the constraints (Eq. 10) to the data x7, such that the sum of the squares of the distance between x7 values and their projections onto the surface is minimal. The distances, Xi - xy, are weighted by the corresponding tolerances. The problem is solved iteratively for the & value using a Gauss-Newton type of algorithm.

3. The Bondi constant For low molar mass materials such as C02, n-alkanes, and cyclohexane, a significant body of VLE, supercritical pVT, and critical point data are readily available, allowing an unambiguous determination of the five material parameters of the MFLG model (ci = (ml/M), yl, tll, glo, gl i, where A4 is the molecular weight). Although VLE and critical point data are obviously not available for macromolecules, pVT data at several molar masses exist for a limited number of polymers, including polystyrene (PS) and poly( dimethylsiloxane) (PDMS). However, for most polymeric materials, sufficient p VT data are not available to provide unambiguous values of the five parameters. In such cases, the y parameter for these materials is dependent on M, y1 and ml for PS, and the surface areas of the repeat unit in question and that of PS are calculated using Bondi’s method (1968a, b):

E.J. Beckman et al. / Fluid Phase Equilibria 99 (1994) 121-133

c&N1 - Yl~/~PS~Ml-Yw) = WSPS

125

(11)

where Mf and M& are the repeat unit molar masses for the polymer in question, and PS, and S, and S,, are the Bondi-calculated repeat unit surface areas. Consequently, for the majority of the polymers examined, the number of adjustable parameters is reduced to four. Subsequently, we will refer to this procedure, by which the number of adjustable parameters may be reduced, as the use of the ‘Bondi Constraint’. While PS is used as the reference material in Eq. (11), other substances for which a large amount of experimental data are available can be employed. For example, in the determination of the MFLG parameters for the n-alkanes, pentane has been used as the reference material during the fitting procedure. Using the procedure outlined above, the MFLG model parameters were derived for a number of polymers and low molar mass fluids. A linear correlation was then regressed between the MFLG ml parameter and the molecular volume of the molecule (or repeat unit) as calculated using Bondi’s group contribution method. This correlation, plus Eq. (ll), were then used to predict values of y1 and ml for both ethane and ethylene. A comparison was drawn between the quality of the MFLG model description using the predicted values of ml and y1 (three free parameters; glo, gll , a,) and the description found when all five parameters were allowed to vary.

4. Results and discussion 4.1. Low molar mass materials Using literature critical point, pVT, and VLE data, MFLG parameters have been determined for a series of n-alkanes, as well as C02, SOZ, CF3H, CC&, CC&, cyclohexane, and benzene. The parameter values for n-nonane are determined from the VLE data above, whereas the values of the other alkanes are derived using VLE, critical state, and isothermal PV data. Consequently, while y1 generally varies smoothly with ml (Van Opstal, 1990) for the n-alkane series, the value for n-nonane deviates from the overall trend. It is interesting to note that the derived values for g,, , the parameter which is defined to account for the segment-segment interaction energy, are all generally within 10% of 500 for the n-alkanes. In-depth comparisons between literature data and the model fit can be found in references by Van Opstal (1990) and Beckman (1988). The MFLG parameters for these materials are shown in Table 1. 4.2. Polymers Using the Bondi Constraint (Eq. 1l), with the exception of PS and PDMS, the MFLG equation of state is fit to pVT data for a variety of polymers at temperatures above Tg because the MFLG is based on an equilibrium distribution of holes and segments, it is not designed to describe the properties of glassy materials). The model parameters for the polymers are shown in Table 2. The ml parameter (number of segments per molecule, or per repeat unit for polymers) is then plotted vs. the molecular volume as calculated using Bondi’s group contribution method. Regressing these data leads to ml = 0.1985 + 0.0584 V,

(12)

E.J. Beckman et al. / Fluid Phase Equilibria 99 (1994) 121-133

126

Table 1 MFLG model parameters for various low molar mass liquids gll

Reference

-0.84339

539.29

- 0.73033 0.78639 -0.73650 -0.69360 - 0.74650 -0.48863

480.55 500.32 498.29 491.22 508.15 489.12

Younglove and Ely, 1987 Kay, 1940, Gehrig and Lentz, 1979 Kay, 1940, Stewart et al., 1954 Kay, 1940, Nichols et al., 1955 Kay, 1940, Felsing and Watson, 1942 Das and Kuloor, 1967a, Carmichael et al., 1953 Das and Kuloor, 1967b, Gehrig and Lentz, 1983

ml

Yl

al

g10

3.39 3.95

-0.38797

0.64077 0.70117 0.77117 0.74697 0.73454 0.89840 0.40365

- 1.1446 -1.1162 - 0.98264 - 0.6663 -0.61682 -0.53524

519.81 695.79 438.72 684.17 452.22 369.02

n-alkanes Butane Pentane Hexane Heptane Octane Nonane Decane

4.51 5.05 5.59 6.52 6.40

-0.57436 -0.60392 -0.63231 -0.65716 -0.570 - 0.76369

Others CQZ SQZ CF,H CC&H ccl, Cyclohexane

1.31 1.57 1.69 2.53 3.26 3.56

- 1.2010 - 1.2619 - 1.2983 -0.8127 - 1.2060 - 1.4931

0.91499 0.80814 0.76250 0.31539 0.84540 0.91134

Benzene

3.36

-0.3163

0.25690 -0.68958

802.47

Michels, 1936a, b, Perry and Chiltem, 1973 Perry and Chiltem, 1973, Kang et al., 1961 Perry and Chiltern, 1973, Hou and Martin, 1959 Campbell and Chaterjee, 1968 Campbell and Chaterjee, 1969 Kerns et al., 1974, Hales and Townsend, 1972, Washburn, 1928, Timmermans, 1965, Young, 1910 Kudchadker et al., 1968, Gehrig and Lentz, 1977, Vargaftik, 1975

Table 2 MFLG parameters for various polymers ml

a

Yl

a1

&?I0

gll

References Olabisi and Simha, 1975 Beret and Prausnitz, 1975 Beret and Pausnitz, 1975 Venneman et al., 1987 Olabisi and Simha, 1975 McKinney and Goldstein, 1974 Zoller, 1978 Hellwege et al., 1961 Oels, 1977, Ueberreiter and Otto-Laupenmuhlen, 1953 Zoller et al., 1976, Quach and Simha, 1971 Olabisi and Simha, 1975 Olabishi and Simha, 1975 Quach and Simha, 1971 Zoller, 1978 Zoller, 1978 Zoller and Hoehn, 1982

LDPE PIB PDMS PEO PBMA PVAc PH PVC Pst

1.24 2.29 2.68 1.46 5.08 2.705 9.395 1.69 3.70

- 0.6988 - 1.4053 - 0.7788 - 1.1800 - 1.2948 -1.2711 - 1.0274 - 1.1695 -0.9290

-4.6367 - 6.0670 - 5.9422 - 7.9597 - 4.9260 -8.4356 -6.3637 - 11.626 - 7.5649

2.7835 4.2930 4.1607 4.7596 2.8330 5.2265 3.7139 8.1917 5.5625

1159.23 957.47 759.50 1439.72 1124.93 1348.07 1580.25 1316.59 1118.87

PMMA PcHMA PoMSt PC PS PPO

3.14 5.70 4.325 8.18 13.71 4.23

- 1.4452 - 1.2194 -0.9777 - 0.9590 -0.7608 - 1.0364

- 8.3661 - 5.2486 -5.3182 - 3.9033 -4.6538 - 1.0639

5.4736 3.3402 3.8093 1.9627 2.7383 -0.0553

1356.04 1145.21 1076.81 1490.04 1567.18 1202.32

a Segments per repeat unit.

127

E.J. Beckman et al. 1Fluid Phase Equilibria 99 (1994) 12I- 133

0

50

150 200 100 Vg (ems/MOLE )

250

Fig. 1. The MFLG number of segments, m,, vs. the van der Waals volume according to Bondi (Bondi, 1968a, b), where u0 = 25 cm3 mol-‘. (CRU = chemical repeat unit.)

with a correlation coefficient of 0.996. Fig. 1 shows a plot of ml vs. the van der Waal’s volume according to Bondi, and the fitted curve (Eq. 12). Derivation of the molecular volumes and surfaces for use in Eqs. ( 11) and (12) can be accomplished using Bondi’s group contribution methodology or that of Slonimskii et al. (1970). In general, the latter approach produces values which agree well with those of Bondi, except in the case of oxygen, for which Slonimskii et al. derive a value significantly larger than that of Bondi (5.8 vs. 3.7 cm3 mol-‘). 4.3. Predictions

using the Bondi group contributions and the MFLG model

Given the excellent correlation method, and those derived from the the Bondi constraint (Eq. 11) to correlations for the cases of ethane

between the MFLG data calculate y, and ethylene.

molecular volumes calculated using Bondi’s fit, as well as the general success found in using we evaluated the predictive power of these First, literature data (critical point, VLE, and

Table 3 MFLG parameters for ethane and ethylene: all parameters adjustable

Ethane Ethylene

ml

Yl

al

g10

gll

1.97 1.79

- 0.88423 -0.75154

0.72905 0.62349

-0.81965 -0.88998

397.0 427.57

Table 4 MFLG parameters for ethane and ethylene using Eq. (12)

Ethane Ethylene

Vu

ml

cl (=m,lM)

27.34 23.88

1.795 1.593

0.05984 0.05690

128

E.J. Beckman et al. / Fluid Phase Equilibria 99 (1994) 121-133

Table 5 Prediction of y for ethane and ethylene using Eqs. ( 11) and (12)

4.25 3.72

Ethane Ethylene

-0.77265 - 0.75246

Table 6 MFLG parameters for ethane and ethylene with m, ,

Ethane Ethylene

y,

predicted by Eqs. (11) and (12)

al

ET10

g11

0.1999 0.03614

-0.71541 -0.73920

479.55 499.53

o--0 =Literature values -=3 Free Parameters =5 Free Parameters

350 325 300 275 250 225 200

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

Density (g/cc)

Fig. 2. Comparison of the fit of the five-parameter MFLG model with that of the three-parameter y, predicted using Eqs. 11 and 12) for the VLE data of ethylene.

version (m, and

supercritical p VT) (Jahangiri et al., 1986; Younglove and Ely, 1987) were fit to the MFLG model equations, providing the five adjustable parameters (Table 3). Next, Eq. (12) was used to predict the cl parameter for each fluid (see Table 4). These values, plus the ml and y1 values for pentane, were used in the Bondi constraint (Eq. 11) to predict y1 for ethane and ethylene (Table 5). Finally, these predicted values for m 1 and y1 were fixed for each compound, and the remaining three MFLG parameters were derived by a fit to the literature volumetric data (values shown in Table 6). A comparison between the quality of this three-parameter fit vs. the five-parameter fit is shown graphically in Figs. 2-5. The model performs quite well at low to medium density and deviates from the experimental values in the compressed liquid range. Here the five-parameter MFLG model offers a better description than the three-parameter Bondi constraint version. The difference between the two descriptions decreases with increasing temperature, as seen in the VLE (Figs. 2 and 4) as well as the near critical isotherms (Figs. 3(a) -3(c) and 5(a) -5(c)).

E.J. Beckman et al. 1 Fluid Phase Equilibria 99 (1994) 121-133

129

0.900 0.800 -0.700

o---o =Literature values -=3 Free Parameters -=5 Free Parameters

--

‘;‘

0.600--

.z 4

0.500--

Ex

0.400--

z $

0.300

--

0.200

--

-I 30

130

230

(a)

330

430

530

Pressure

630

730

830

(bar)

0.800 0.700

o--o=Lkerature values -=3 Free Parameters =5 Free Parameters

0.600 F 2 x .z : $

0.500 0.400 0.300 0.200 0.100 0.000 30

130

230

330

430

Pressure

630

730

630

(bar)

o--o =Literature values -=3 Free Parameters =5 Free Parameters

0.600 < m

530

0.500 0.400

z z $

0.300 0.200 0.100

0.000

r 30

(4

130

230

330

430

Pressure

530

630

730

6 i0

(bar)

Fig. 3. Comparison of the fit of the five-parameter MFLG model with that of the three-parameter version (m, and y, predicted using Eqs. 11 and 12) for density-pressure data for ethylene at: (a) 290 K, (b) 320 K; (c) 350 K.

E.J. Beckman et al. 1 Fluid Phase Equilibria 99 (1994) 121-133

130

1

o--o=Literature values -=3 Free Parameters -=5 Free Parameters

325

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

Density (g/cc)

Fig. 4. Comparison of the fit of the five-parameter MFLG model with that of the three-parameter y, predicted using Eqs. 11 and 12) for the VLE data of ethane.

o--o=Literature values -=3 Free Parameters =5 Free Parameters

0.800 0.700

3

0.600

2

0.500

cx E 2

0.400 0.300 0.200 0.100 0.000 30

130

230

330

530

430

Pressure

(a)

630

730

8

(bar)

o--0 =Literature values -=3 Free Parameters -=5 Free Parameters 3 < s

0.600

.fi x : 0”

0.400

0.500

0.300 0.200 0.100

0.000 4 30

(b) Fig. 5. (a), (b) (Caption opposite.)

130

230

330

430

Pressure

530 (bar)

630

730

830

version (m, and

E.J. Beckman et al. 1 Fluid Phase Equilibria 99 (1994) 121-133

131

o--o=Literature values --=3 Free Parameters -=5 Free Parameters 0.600

‘;‘

4

0.500

x

0.400

t

0.300

22 Y

2

0.200 0.100 0.000 30

130

230

330

(cl

430

Pressure

530

630

730

6 i0

(bar)

Fig. 5. Comparison of the fit of the five-parameter MFLG model with that of the three-parameter version (m, and y, predicted using Eqs. 11 and 12) for density-pressure data for ethane at: (a) 315 K; (b) 335 K, (c) 370 K.

5. Conclusions The MFLG model provides a good description of the volumetric behavior of both polymers and low molecular weight liquids. As shown above, the MFLG material constants ml and yl, which represent the number of segments per molecule and the contact surface area per segment, respectively, correlate well to molecular volumes and surface areas derived using Bondi’s group contribution method. Because two material constants can therefore be calculated a priori, use of the model requires derivation of only three parameters for a given substance. This can be accomplished by solution of the three-model equations which describe the gas-liquid critical point, a regression of VLE data (low molecular weight liquids), or via regression of one or two isobars for polymers. Given these results, it would appear that the Bondi constraint, (Eqs. 11 and 12), in conjunction with the MFLG model, represents a useful means of generating volumetric predictions when few data are available. For example, after using Eqs. (11) and (12) to determine ml and y1 for a particular low molar mass compound, knowledge of the critical parameters (T,, V,, and PC) is all that is required to calculate the other three parameters. For a polymer, the Bondi constraint plus one or two isobars could be used to derive the parameters.

Acknowledgment L.V.O. wishes to thank DSM, Netherlands,

for support while at the University of Antwerp.

References Beckman, E.J., 1988. Ph.D. Thesis, Modelling the phase behavior of supercritical gas-polymer of Massachusetts, Amherst, MA.

mixtures. University

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