THERMODYNAMICS OF POLYMER SOLUBILITY IN POLAR AND NONPOLAR SYSTEMS R . F. BLANKS' AND J .
M.
P R A U S N I T Z
Department of Chemical EnginPering, C h i w r s i t y of Caiifornza, Rerheley. Calif.
An approximate descripiion of the thermodynamic properties of polymer solutions i s given b y the FloryHuggins equation. This equation includes a parameter which depends upon the intermolecular forces acting between the molecules in the solution. The theoretical critical value of i s approximately 0.5; for a given polymer.solvent pair must be less than 0.5 if the polymer i s to be appreciably soluble in the solvent.
x
x
x
x
Semiempirical itechniquss are prescribed for calculating from pure-component properties of the solvent and polymer; these techniquzs are based on an extension of the Hildebrand-Scatchard theory of solutions and on the theory of intermolecular forces. Polar and nonpolar solubility parameters are presented for a variety of solvents and for several polymers. The calculaiions are useful fot nonpolar systems, polar polymers or polar solvents, lor both, but are not reliable for systems where specific interactions (such as hydrogen bonding) are important. Brief consideration i s given to the solubility of polymers in mixed solvents.
THIS paper
is concerned ivith the solubility of polymers: given a certain polymer. what solvent \vi11 dissolve it? Much empirical knou 1ed::e is available to ans\ver this question, but it is still very difficult to interpret this knowledge in any systematic manner. There are two fundamental reasons for this difficulty. First. a completely satisfactory statistical mechanical description of polymer solutions is not yet available; second, the molecular interaction parameters which any such description must contain cannot. in general, be predicted from pure-component properties. However. for engineering purposes, an approximate treatment suggested by Flory (76) and Huggins ( 2 6 ) is usrful. especially for semiquantitative applications. 'Therefore, the present work is based on the Flory-Huggins treatment of polyner solutions and suggests a rapid and simple semiempirical technique for estimating whether or not a particular solvent \vi11 dissolve a particular polymer. .A similar technique for nonpolar solutions had been suggested by Burrell ( 7 7 ) . This work extends that of Burrell to systems which contain either polar polymers or polar solventu. or both. but it is not applicable to solutions that exhibit specific interactions such as hydrogen bonding. Flory-Huggins Equation
T h e thermodynamic properties of polymer solutions are frequently described in terms of the Flory-Huggins equation, which \vas first proposed more then 20 years ago; its use for data rrduction \vas pointed out by Gee (2,?).Scott ( 3 7 ) . and numerous other workers (78). This equation gives the free energy chang-e \vhich occurs \\.hen a noncrystalline polymer is mixed Xvith a solvent; for any given polymer-solvent pair it contains a characteristic parameter, X. callrd the Flory interaction parameter. \vhich reflects the intermolecular forces between the molecules in the solution. 'The Flory-Hupgins equation i s : AGrn ~~
~
R 7'
= n , In
@I
+ ri! In
+
x a l @ 2(n,
+ mnz)
(1)
Presrnt addrrss. Yniorr Carbide Plastics Co.. Round Brook, N. J.
I n Equation 1. AGm is the free energy of mixing the liquidlike. disoriented polymer u i t h the solvent; n l and n , refer to the number of moles of solvent and polymer. respectively; @I and are the volume fractions of solvent and polymer, defined bv @, E
nl
~~
ni
+ mn2 and
@
'
n,
mn2 mn2
+
where m is the ratio of molar volumes of polymer and solvent, and x, the Flory interaction parameter, is a dimensionless quantity ivhich is a function of the interaction energy characteristic of a given solvent-solute pair. If the free energy of mixing is negative, the polymer and solvent may mix spontaneously to form a solution. If it is positive. t\vo phases form and the polymer does not dissolve appreciably in the solvent. T h e first t\vo terms of Equation 1 account for the configurational entropy of mixing and are always negative. Therefore. for the polymer to be soluble in a solvent, the third term in Equation 1 must be small (if pocitive) or it must also be negative. For polymer solubility, therefore. the Flory interaction parameter must be either negative or a small positive number. Stability analysis sholvs that there is a maximum critical value of the Flory parameter, xC, below Lvhich the polymer and solvent are miscible over thr entire composition range. 0 < @ I < 1.0. and above which two phasrs form. T h e critical value of x is found to be ( 7 8 ) :
xc = 1 ~
(1
+ $-m)?
2 Since m is a very large number for polymers. the criterion for polymer solubility is essentially givrn by x 5 0.5. I t has been found experimentally (72) that the molecular tveight of the polymer may have an effect on polymer solubility and on the value of x uced in Equation 1 , 'I-his effect \vas not taken into account in the approximatr relationships ivhich are discussed in this \verb If Equation 1 is used to describe the thermodynamic propertiesof polymer solutions. then the problemof predicting pol>-mer VOL. 3
NO. 1
FEBRUARY 1964
1
solubility is equivalent to that of predicting reasonably accurate values of the Flory interaction parameter. I n Equation 1 x is a free energy parameter, and therefore it includes both entropy and enthalpy contributions:
x=
Xs
+ XH
(3)
most general form this theory- is not limited to nonpolar species but it considers only configurational contributions and neglects contributions from i:iternal degrees offreedom such as rotation and vibration. T h e Hildebrand-Scatchard regular solution theory equation for the heat of mixing may be written
Huggins' derivation for the free energy of mixing predicts that the entropy parameter X , = 1,z, \\.here z is the lattice coordination number (26) ; however. many workers have found X, empirically to have a value of approximately 0.3 to
0.4 (38). T h e parameter X, is related to the heat of mixing of the polymer with the solvent. If it is assumed that the heat of mixing in this case is given by the Hildebrand-Scatchard regular solution theory for nonpolar substances, it is found that
AH,,, = v1Al? (nl
very small, and therefore the solubility parameters of the polymer and solvent must have very nearly the same value. T h e approximate equality of the solubility parameters of polymer and solvent is a useful a priori test of the suitability of a given solvent for a given polymer. This criterion has been extensively applied by Burrell ( 7 7 ) ; however. Equation 4 applies only to nonpolar systems and is not valid for systems containing polar molecules or for those Lvhere specific interactions are present. Qualitatively. hydrogen bonding may have important effects in determining polymer solubility ( 7 7 , 38). Recently, a very rough attempt was made to quantify this effect (27),but only association of the solvent was considered ; solvation between solvent and polymer was neglected. Several 'solution theories for polar molecules have been proposed in recent years (2. 28-30, 40) and new theoretical work on polymer solutions also appears to be promising (72: 73). However. none of these newer theoretical efforts seem a t this time to be developed to a sufficient degree to permit their use in engi-.eering work for obtaining a fairly simple method of polymer solubility estimation. No doubt the newer theoretical research Mill in due time be reduced to engineering application, but for the present, Equation 1 is still a useful point of departure for illdustrial calculations. Although a number of assumptions not applicable to polar solutions discussed thoroughly by T o m p a (47) were made in deriving Equation 1 it has met with reasonable success in many applications. This is probably due to the well-known fact that an oversimplified model often introduces serious errors in calculating separately either the heat or entrap). of mixing. which frequently tend to cancel each other when the free energy of mixing is calculated (23). Therefore, this work aims to provide useful applications of Equation 1 by developing a more general method of estimating the Flory interaction parameter. T h e estimation method developed below appears to give good results for polymer-solvent systems which contain either polar or nonpolar species, or both. provided that hydrogen bonding and other specific interactions are absent. Calculation of Flory Interaction Parameter
T h e enthalpy contribution.
xH: for
systems containing polar
molecules can be calculated by a suitable modification of the Hildebrand-Scatchard theory of solutions (24. 33, 42). I n its 2
l&EC FUNDAMENTALS
(5)
where A12 is the interchange energy density for the solventsolute pair. T h e use of Equation 5, together with the FloryHuggins expression for the athermal configurational entropy of mixing. gives
T h e interchange energy density. A I ? . is given by '412
where ~1 is the solvent molar volume, and where 61 and 62 are the solubility parameters of solvent and polymer. respectively. T h e use of Equation 4 allows only positive values for x,. Thus. if x is to have a value less than 0.5: X, must be
+ mnr)@i@n
=
(Cii
+
-
C P ~
(7)
2612)
where the til's characterize the intermolecular forces acting between molecules (or molecular segments) i and j. For the pure components. c l 1 and C Z Y are the cohesive energy densities of components 1 and 2 :
I n Equation 8, AEiuis the energy of vaporization of substance X i and 7 1 are defined, respectively, as the nonpolar and the polar solubility parameters of substance z. T h e quantity c12 represents the intermolecular forces acting between solvent molecules 1 a n d molecular segments of polymer 2 in solution. It may include dispersion forces, dipoledipole forces. dipole-induced dipole forces. and specific interactions (solvation) between solute and solvent: z to a gas a t zero pressure;
612
= Fdigp(X1rX~)
+
Fdi-di('l.Tr)
+
Find(X1:71rx2rT2)
+
where the F's stand for unspecified functions. Equation 9 have the following significance:
F s p e c int
(9)
T h e terms in
Fdia,,(Xi: A,) represents the (nonpolar) dispersion forces acting between dissimilar molecules. O n the ,basis of the London theory it may be represented by the geometric mean of the nonpolar forces present in the pure components: F d ~ s , ( X ~ X2) 3
= hlX2
Fdi-,i,(r ~2l ), represents the interactions between permanent dipoles in the solute and solvent. If no specific forces (such as hydrogen bonding) are present, it is a function of the purecomponent polar solubility parameters, r , and 7 2 . For polar molecules or molecular segments which may be represented by spherical force fields with small ideal dipoles a t their center (25)
Fdl-dl(71, 7 2 ) = 7172
T h e induction contribution. Flnd(XI, 7 2 . XP, T I ) ? represents dipole-induced dipole interactions between solute and solvent and may be represented by \ I I ( T ~ X ? . i 2 X 1 ) , where \k is a n unspecified function. represents the contribution to the pair interaction F,,,, energy from any solvation which may occur in solution. Very little theoretical knowledge is available to characterize this interaction. For the special case where neither solvent nor solute is polar, Equation 7 reduces to the familiar result A12
=
(61
- 62)'
where = Xi: since 7 i= 0. .4ny specific interactionsoccurring between molecules 1 a n d 2 add a positive term to c12 and correspondingly decrease '412. This means that X , also decreases and therefore mutual solubility of the two species is enhanced. This qualitative result is intuitively correct! since any specific interaction, such as hydrogen bonding. between a solvent and a polymer helps to "pull" the polymer into solution. T h e above development strictly only applies to liquidlike polymers, I n order to dissolve partially crystalline polymers, the free energy necessary to melt the crystals must be supplied. This is often a large additional contribution to the free energy of mixing. particularly if the polymers are much below their melting points. 'Iherefol-e. crystalline polymers a t a temperature significantly below their melting point are usually soluble only in those solvents with which they have some specific interaction \vhich results in a negative value of A,?. Cohesive Energy Density and Polar and Nonpolar Solubility Parameters
According to Iiildebrand's definition: the solubility parameter is the square root of the cohesive energy density which. in turn. is defined as the energy of (complete) vaporization divided by the condensed molar volume. When considering fluids whose molecules possess permanent dipoles. it has been found useful to separate the cohesive energy densities iiito polar and nonpolar parts. Splitting the energy of vaporization of a polar fluid into nonpolar and polar parts permits the follobving definition of solubility parameters to b r made.
Polar solubility paramrter
T~
z
(
AE;;(p))' ~
*
(1 1)
where A E D l ( n pand ) A E C L ( pare , the nonpolar and the polar contributions to the energy of vaporization of fluid 2-that
is.
For nonpolar molecules . l E F l ( p = ) 0,
-
A E c i ( n p aj :n d X i =
ai.
T~
=
0, LEGi( t o t s l )
Bondi and Sinikin (7:' have shown how the energies of vaporization of polar liquids can be divided into polar and nonpolar contributions by using the homomorph idea of Brown ( 9 ) . ( T h e homornorph of a polar molecule is a nonpolar molecule having very nearly the same size a n d shape as those of the polar molecule.) This concept has been used in interpretations of the second virial coefficients of polar gases (5)and in Lvork related to the solvent selectivity of polar solvents (33). T h e homomorph concept is rrlatively easy to apply. T h e polar energy of vaporization is simply the difference betrveen the experimentally determined total energy of vaporization and the energy of vaporization of the homomorph a t the same reduced temperature. If the polar molecule has a straight-chain structure. a normal paraffin may be used as a homomorph. I n that ca?e, the energy of vaporization of the homomorph (which is equal to the nonpolar energy of vaporization of the polar molecule) may be estimated from Figure 1. T h e size of the homomorph may be considered as a continuous. rather than a discontinuous, variable. This mak.es it convenient to use the molar volume of the polar liquid as the independent variable as
MOLAR V O L U M E V,
Figure 1. carbons
, c c . /g-mole
Energy of vaporization for straight-chain hydro-
shown in Figure 1. This step follo\\s from the basic idea that the molar volume of the polar component is equal to that of its homomorph as long as the comparison is made a t the same reduced temperature. I n principle it would be possible to construct plots similar to Figure 1 for homomorphs of other shapes. such as disks? globules. etc. I n practice. however. there are insufficient data available to make smooth plots of this type. Therefore. for a polar molecule whose homomorph is not a linear hydrocarbon. the homomorph is chosen as the nonpolar molecule most closely resembling the polar molecule in shape and molar volume kvhen compared a t the same reduced temperature. For example, the homomorph of chlorobenzene might be toluene, the homomorph of methyl isobutyrate is 2-methylbutane, and so on. O n e must remember to find the energy of vaporization of the homomorph a t the same reduced temperature as that of the polar molecule, and to compare molar volumes a t this temperature also. For most solvents of interest. the molar volumes are readily accessible in standard reference books. Reliable values for energies of vaporization, ho\vever. are not found as easily. Bondi has developed a method for calculating energies of vaporization from group contributions when experimental values are not available (6). Tables I and I1 give polar and nonpolar solubility parameters for a number of industrially important solvents. all calculated a t 25' C. Values for the energy of vaporization were taken from the literature ( 7 , 8, 75. 24, 37, 34. 39. 43) or calculated by Bondi's method. If the required critical temperatures could not be found, they were calculated using the method of Lydersen as outlined by Reid and Sherwood (35). T h e temperature dependence of the energies of vaporization \vas found with an empirical equation given by \Vatson ( 4 4 ) . T h e calculation of polar and nonpolar polymer solubility parameters for polymers is based on the method of Small (35). This method had to be modified in order to calculate both polar and nonpolar solubility parameters, since the original procedure did not differentiate between polar and nonpolar forces. A few polymer solubility parameters calculated with the new procedure are given in Table 111. With Small's method, solubility parameters are constructed from a table of additive group constants called molar-attraction constants. For nonpolar polymers
VOL. 3
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FEBRUARY 1964
3
T h e polymer is divided into repeating structural units: ZF refers to the sum of the molar-attraction constants for all of the groups in the structural unit; MIITis the molecular weight of the structural unit, and d is the polymer density. For polar polymers Small's method was modified in such a way that both X and T may be calculated for the polymer. According to Small,
where AE'(total) refers to the pseudo-energy of vaporization of the structural unit of the polymer. Then for the polymer, as for the monomeric liquids: ~
AEti(total) - AE"(,P) - -~
s
+ -AEti(V) - ,
X'polymer
+
~
~
(15) ~
S e x t . X2 and T~ are found for the monomeric liquid most closely resembling the structural unit of the polymer. For example, the monomeric liquid most closely resembling the repeating unit of poly(viny1 acetate),
would be ethyl acetate :
CH 3-CH2
I
0-C-CH3
/I
0 T h e energy of vaporization of the monomeric liquid is then divided into polar and nonpolar parts using the homomorph concept previously described. To obtain X and T for the polymer it is assumed that the nonpolar and the polar contributions to the total cohesive energy density are the same for the monomer as for the polymer structural unit. Thus
Table l.
Molar Volumes and Solubility Parameters of Nonpolar Solvents at 25' C. u, Cc./G. Mole
Solsent
Paraffins Methane Ethane Propane Butane 2-Methylpropane (isobutane) Pentane 2-Methylbutane (isopentane) 2.2-Dimethylpropane (neopentane) Hexane Heptane 2,2,3-Trimethylbutane Octane 2.2,4-Trimethylpentane (iso-octane) ~ Nonane i > ~ ~ ~ Decane Dodecane Tetradecane Hexadecane Octadecane Eicosane Olefins Ethylene Propene Butene cis-2-Bu tene trans-2-Butene 2-Methylpropene (isobutene) 1,3-Butadiene Isoprene Cyclic hydrocarbons Cyclopentane Cyclohexane Methylc yclohexane Aromatics Benzene Toluene Ethylbenzene o-Xylene rn-Xylene $-Xylene Propyl benzene Mesitylene Styrene Tetralin Carbon tetrachloride a
53a 69a 8ja 101.43 105.48 116.104 117.383 123.31 131.598 147.456 146.087 163.530 166.079 179.670 195.905 228.579 261.312 294.083 326.93 359.83
4.7a 5.69 6.2a 6.618 6.250 7 021 6.747 6.117 7 242 7.423 6.942 7.554 6.849 7 648 7.722 7.841 7.927 7.993 8.04 8.08
630 79a 95.285 91 .17 93.76 95.43 87.96 100.780
5.5a 6.1a 6.7 7.2 7.0 6.7 7.1 7.45
94 713 108.744 128.332
8.10 8.182 7.816
89.399 106 847 123,064 121.193 123.456 123 919 140.110 140 115 559
9.147 8.907 8.80 9 00 8 80 8 75 8.65 8.80 9.3 9.5 8.58
97.10
2
Estimated from gas solubility data ( 3 4 ) .
tions for the correlation of these parameters are Equations 6, 7: 8, and 9. Discussion of x parameters is divided into three parts which discuss the experimental results for nonpolarnonpolar systems, polar-nonpolar systems. and polar-polar systems, respectively. Nonpolar-Nonpolar Systems. In nonpolar-nonpolar systems only dispersion forces are to be expected. I n this case:
and
A12 = ( 6 1
Application of Theory to Correlations of Experimental Data
Reliable experimental data on the thermodynamic properties of polymer solutions are not plentiful and, therefore, if a correlation is to be made, it is necessary to use swelling data and viscosity data in addition to thermodynamic information such as vapor pressure measurements. Techniques for calculating Flory interaction parameters from experimental measurements are discussed by Flory et ai.(77-22). T h e agreement between experimental x values for the same systems obtained from these three sources is far from perfect. Ho\vever. the agreement is \vel1 ivithin the accuracy of both the experimental measurements and the theoretical limitations of the Flory-Huggins equation. T h e basic equa4
6:
( Cal./Cc. )I(
l&EC FUNDAMENTALS
-
6*)2
(1 8)
T h e entropy parameter, x,? would be expected to depend on the size and shape of the solvent molecules and also on those of the polymer groups. Both of the nonpolar solubility parameters appearing on the right-hand side of Equation 18 are known. Values of x, were calculated from Equations 6 and 18, using experimental values of x for three nonpolar polymers in a large number of nonpolar solvents. The mean value of X, from 23 calculations was 0.34; the standard deviation from the mean was 0.08.\vhile the largest deviation from the mean was 0.15. It appeared that there might be a regular increase in X, with chain length for normal paraffins. Lvhile the branched paraffins possess considerably lower values for
xS,
Table II.
Molar Volumes and Nonpolar and Polar Solubility Parameters of Polar Solvents at 25' C.
AE?,Q> AEUnOlai, h T Kcal, /G. M o l e Kcal. / G . M o l e ( Cal. / C c . ) l i 2 ( Cal,/Cc.
L'.
Solvent
Cc./G. >Mole
T,,' K .
Halogen compounds 80.673 536 Chloroform 73 460.2 Ethyl chloride 1,2-Dichloroethane (ethyl79.44 563 ene chloride) 84.8 522. 4La 1,l-Dichloroethane 100.4 536.5L 1.1.1-Trichloroethane 90,2 554L Trichloroethylene 87 650 1,2-Dibromoethane 102.1 632.4 Chlorobenzene Ketones 73.987 Acetone 508.7 550L 90.164 Methyl ethyl ketone 107.46 565L Methyl propyl ketone 577L 125.81 Methyl isobutyl ketone 140.779 610L Methyl amyl ketone 140.69 602L Dipropyl ketone 157.45 631 L Hexamethyl ketone 115.6 614L Mesityl oxide Esters 98.5 523.3 Ethyl acetate 115.7 549.2 Propyl acetate 132.5 579 Butyl acetate 591L 148.9 Amyl acetate 115.5 546.1 Ethyl propionate Ethers 104.748 467 Diethyl ether 103.81 467 Methyl isopropyl ether 85.7 573L Dioxane Miscellaneous 90,35 60 6 Nitropropane 52.9 547.9 Acetonitrile 70.3 564.2 Propionitrile L refers to T , calculated with Lydersen's method ( 3 5 ) .
4.85 3.85
1,921 1.424
7.74 7.26
4.88 4.42
Fig. 1 Fig. 1
4.90 4.89 5.46 5.76 5.90 8.70
2.687 2.139 1.899 2.00 3.510 0.878
7.87 7.59 7.37 7.99 8.24 9.23
5.83 5.02 4.35 4.71 6.34 2.93
Fig. 1 Isobutane Neopentane Isopentane Fig. 1 Toluene
4.25 5.45 6.40 7.05 8.50 8.40 9.45 8.12
2.778 2.308 2.148 2.168 2.508 2.591 2.658 1,678
7.58 7.77 7.72 7.49 7.77 7 73 7.75 8.38
6.13 5.06 4.47 4.15 4.22 4.29 4.11 3.81
Fig. 1 Fig. 1
5.45 6.70 7 75 8 65 6 70
2.658 2.127 2 40 2 652 2 338
7.44 7.61 7.65 7 62 7 62
5.19 4.29 4 26 4 22 4 50
5.20 5.12 6.27
0,537 0.75 1.853
7.05 7.02 8.55
2.27 2.69 4.65
Fig. 1 Isobutane Cyclopentane
5.85 3.30 4.50
3.928 4.084 3.68
8.02 7.90 7.96
6.57 8.79 7.20
Fig. 1 Fig. 1 Fig. 1
However. there u e r e inwfficient data available to obtain a good correlation of X, with size and shape of the molecules under consideration.
Therefore, the mean value,
x,
=
Homomorph
0.34,
was adopted for all systems.
Polar-Nonpolar Systems. I n polar-nonpolar systems the forces between solute and solvent include not only dispersion forces but also dipole-induced dipole forces and possibly specific interactions between the molecules. For polar-nonpolar systems with z being either 1 or 2, depending upon which component is the polar one,
T h e temperature T , is that a t which the polar solubility parametel. 7 , was obtained. T h e ratio ( T , / T ) is suggested by the Keesom formula (25) a n d accounts in a n approximate way for the temperature dependence of the polar-polar interaction of molecule i. T h e parameter is a n empirical term which accounts for inductive effects; specific interactions are not considered. I n order to obtain values of the parameter P from experinien-
Isohexane Fin. 1 Fig. 1 Fig. 1 2-Methylpentene Fig. Fie. Fig. Fig. Fig.
tal d a t a it is necessary to specify the value of 2,.
1 1 1 1 1
T h e entropy
interaction parameter, xS, is a function of the size and shape of the molecules in solution. T h e size and shape of a polar molecule are essentially the same as those of its nonpolar homomorph a n d therefore X, was assigned the value of 0.34, the mean for
x,
from the nonpolar-nonpolar calculations.
From experimental values of
x
for a vaiiety of systems.
xH
is
found from Equation 3. With this information, together with Equations 6 and 19 and the polar and nonpolar solubility parameters of the solute and solvents. parameter P was evaluated. T h e results of this calculation are shown in Figure 2 . A good correlation of P with X n P r pwas obtained in accordance with theoretical expectations. Polar-Polar Systems. I n systems where both components are polar. dispersion forces, induction forces. dipole-dipole forces, a n d specific interactions may all be important in deter-
*
Table 111.
*
RUBBER IN POLAR SOLVENTS
o
POLAR POLYMERS IN NOM-POLAR SOLVENTS
Nonpolar and Polar Solubility Parameters of Some Polymers at 25' C.
x Polymer
(Cal./Cc.)'#*
Rubber Polyisobuty1en.e Polystyrene Polybutadiene Polyeth ylrne Poly(vi1iy1 chloride) Poly(viny1 acetate) Pol)-imethyl methacrylate) Poly( propylene oxide)
8.15 7.70 9.10 8.32 8.10 8.16 7.72 7.69 7.02
T
(Cal./Cc.)' 0 0 0 0 0
4.96 5 36 5.15 2.70
*
0
5 IO 15 2 0 2 5 30 35 INDUCTION ENERGY DENSITY FUNCTION A,,
40 45 50 r P ,coIortes/cc.
55
Figure 2. Contribution of dipole-induced dipole interactions to interchange energy density at 25" C. VOL. 3
NO.
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FEBRUARY
1964
5
mining the interchange energy density. I n these solutions, where both molecules possess permanent dipole moments, induction forces are small in comparison to dipole-dipole interactions and can be ncglected. It is then convenient to classify polar-polar solutions according to \vhether specific interactions betiveen the different species in solution are expected to be present or not. Specific interactions which may occur in the polar-polar systems ark hydrogen bonds. Pimentel and McClellan (32) have given a good discussion of molecular groups which foim hydrogen bonds, and although new evidence for hydrogen bonding is continually being reported, their discussion is a guide in classifying polar-polar solutions. For polar-polar systems where hydrogen bonding is not expected to occur between solute and solvent:
A12 = (XI
-
A?)* f
(71
-
(20)
72)'
Equation 20, however. is valid only when all of the quantities are evaluated at the same temperature. T h e temperature dependence of A 1 2 should be accounted for when utilizing experimental Flory inreraction parameters which have been obtained a t temperatures other than the temperature where the solubility parameteis were evaluated. LVhile it is true that the nonpolar solubility parameters, X. are temperaturedependent, it is also true that in regular-solution theory the heat of mixing is independent of temperature for nonpolar systems; this suggests that the group v l ( A 1 - An)* is also independent of temperature. However. the heat of mixing in systems containing polar species is a function of temperature since the average potential energy between two small permanent dipoles varies inversely as the absolute temperature. as shown by Keesom (25). The Keesom formula suggests that the square of the polar solubility parameter is inversely proportional to T . For small temperature intervals?
rT*
(Ti)T2 =
(TTI)T**
where T, is the temperature a t \vhich T~ \vas evaluated (298' K . in this work). Equation 20 then becomes 9 1 2
(Ai -
X2)'
f
T* T
- (71 - 7
~
)
~
(21)
Table IV. Calculated and Experimental Flory Interaction Parameters for Several Polymer-Polar Solvent Systems
T, Solvent
OK.
AI?.
Q. 21
xcaicda
Yerptl
Poly(viny1 Chloride) Ethylene dichloride 326 0.8995 0.454 0.46b 349 0.8474 0.446 0.43b 1,2-Dibromoethane 298 2.075 0.645 0,468" Nitropropane 326 2.612 0 709 0.44b Poly(viny1 Acetate) 0 514 0 . 407d Dioxane 298 1.2023 0 . 437d Acetone 298 0.6054 0,416 0 . 42!Id Methyl ethyl ketone 298 0.0946 0.354 Poly( methyl Methacrylate) 0 475c 0 351 Ethyl acetate 298 0.0636 n-Propyl acetate 298 0.7454 0 486 0 487c 0.519 0 50OC n-Butyl acetate 298 0 7988 0 556 0 507e n-Amyl acetate 298 0 8654 298 0.2937 0 397 Methyl isobutyrate 0 482e 0.461 0 481e .Acetone 298 0.9718 Methyl ethyl ketone 298 0,0154 0.342 0 469e Methyl propyl ketone 298 0.4593 0.423 0 475e 0.558 0 4966 Methyl isobutyl ketone 298 I ,0340 0 493e 0.545 Methyl amyl ketone 298 0.8665 0.514 0 502e Diisopropyl ketone 298 0 7360 0 518" 298 1 0833 0.626 Hexamethyl ketone " x.7 ass7frned constant at 0.34. h ( 7 4 ) . c (8). d (70). e (27).
6
I&EC FUNDAMENTALS
SOLUBLE R E G I O N
''
0
0.2 0.4 0.6 08 1.0 VOLUME FRACTION ACETONE ' [SOLUTE-FREE EASISI
Figure 3. Predicted solubility of poly(methy1 methacrylate) in acetone-hexane mixtures
As before, the entropy parameter.
x,.
from the nonpolar
correlation was used for the polar-polar system. T h e parameter x may then be calculated from Equations 6 and 21 using the polar and nonpolar solubility parameters for the polymer and solvent in question. For systems where there is no hydrogen bonding (or other specific interaction) between polymer and solvent: the agreement between experimental and calculated x values is very good considering the accuracy of the x values themselves. For 19 polar-polar systems, the average difference between calculated and experimental x values was 0.07. These systems included the polymers poly(viny1 chloride) poly(viny1 acetate), and poly(methy1 methacrylate) ; the solvents included ketones, esters: dioxane, nitroparaffins? and halogenated hydrocarbons. Details of the calculations are given in Table I\'. For polar-polar systems in which specific interactions take place betkveen components 1 and 2. Equation 21 may not be used to predict x with any degree of accuracy or reliability. This is because strong specific interactions between molecules in solution cause an increase in cI2 and a corresponding decrease in A12 which is not taken into account in Equation 21. ~
Mixed Solvents
The ternary system. polymer-liquid 1-liquid 3, is of considerable practical interest to the polymer industry. Phase equilibria in ternary systems of this kind have been extensively discussed by Scott (37) and also by Flory (78). For a ternary system, making essentially the same assumptions that were made in deriving Equation 5> Scatchard (36) has shown that per mole of mixture, Ahm
= ~~[~412@1f @ 2 A13QlrP3
+
i423@2@3]
(22)
According to Equation 22, the polymer solubility in a mixed solvent is related to the binary interchange energy densities, A reasonable estimate of the Flory interaction parameter between a polymer and a mixed (two-component) solvent can therefore be made as follows. (Subscripts: first solvent. 1 ; second solvent, 3 ; polymer. 2 ; 0 refers to mixed solvent.)
xS
= 0.34
(23)
3. Poly(methy1 Methacrylate) Solubility in Mixtures of Hexane-Acetone at 25" C. (Subscript 1 refers to hexane, subscript 3 to acetone.) where uo =
@,VI
I
X,
-1@ 3 u 3 (Solute-free basis)
= 0.34
and
+ cz - 2602)
A02
=
(CO
co
=
@ 1 2 c ~ ~ @32c33 f
co2
=
@lclz
+ 2@1@'3613(solute-free basis) + a3cz3(solute-free basis)
To illustrate the estimation of polymer solubility in a binary solvent mixture, the solubility of poly(methy1 methacrylate) in acetone-hexane mixture!; is considered below. Calculations, as well as experimental sNslubility studies. indicate that acetone is a solvent for poly(methy1 methacrylate) but that hexane is not. I t is frequently dtsirable in practice to dilute a prime solvent, acetone in this c;ise, with a relatively inexpensive nonsolvent such as hexane. I t is therefore of interest to consider the following questions: Even though hexane is not a solvent for poly(methy1 methauylate) I kill mixtures of hexane a n d acetone dissolve the polymer? If so. how much hexane can the sulvent mixture tolerate before it becomes a nonsolvent? T h e sample calculation below attempts to answer these questions.
+
uo
=
Aoz
= (xo
\E13
= 13.0 cal./cc.
xo
=
70
= @IT1
@I71
-
@lX1
@3u3
id'
+ +
+
(To
-
.2)'
+
2(@1@3*13
-
@1 Q l Z )
@3X3
a373
1. Solubility of Poly(methy1 Methacrylate) in Acetone at 25 " C. This is a polar-polar system where hydrogen bond-
T h e results of this calculation (Figure 3) show that poly(methyl methacrylate) is soluble in mixtures of acetone and hexane as long as the volume fraction of acetone (solute-free basis) is greater than about 0.55, and that the "best" solvent mixture for the poly(methy1 methacrylate) corresponds to a volume fraction of 0.7 for acetone on a solute-free basis. These calculations are in favorable agreement ivith unpublished d a t a of Billmeyer ( 3 ) ,who reports: "For the system poly(methy1 methacrylate)-hexane-acetone a t 25' C., the point of initial precipitation was observed a t a volume fraction of acetone of 0.60 to 0.65 (solute-free basis). At a volume fraction of 0.55 about 55% of the polymer \vas precipitated in one case."
ing between solvent and polymer is not expected. 3 refers to acetone.)
Acknowledgment
x xs
=
'432
=
x,
(Subscript
+ X,{
T h e authors are grateful to Arthur K. Doolittle for encouragement; to the Kettering Foundation for financial support in the form of a g r a n t ; and to the it'oodrow \\'ikon Foundation for a fellowship.
= 0.34
-
0 3
A212
+
'432 =
0.972 cal./cc.
xH
=
0.1206
x
= 0.1206
(73
-
Nomenclature
72)'
Aij
-+ 0.34 = 0.46
(Xexpt, =
0.48)
Since x < 0.5! acetone is a solvent for poly(methy1 methacrylate) a t 25' C. This is confirmed experimentally.
2. Solubility of Poly(methy1 Methacrylate) in Hexane at 25" C. This is a polar-nonpolar systek.' (Subscript 1 refers to hexane.)
+ x,,
x
=
x,
xS
=
0.34
A12
=
(Al -
X?)2
+
=
c = d = AEu =
F
=
G, = AH,, Ah, MI+'
= = =
ni
=
R T T, TR
= =
u z
=
= = =
interchange energy density for components cal./cc. cohesive energy density, cal./cc. density, g./cc. energy of vaporization, ca!.;'mole molar attraction constant, (cal.-cc.)1'2 free energy of mixing heat of mixing heat of mixing per mole of solution molecular tveight number of moles of component i gas constant absolute temperature, ' K . critical temperature, ' K. reduced temperature molar volume: cc./g.-mole lattice coordination number
i and j
GREEKLETTERS 6 = solubility parameter 722
- 2Q
X
A 1 r 2 = 37.28 cal./cc. T
I
= 9.75 cal./cc. (Figure 2)
Ai2
=
(7.686 -- 7,242)' f (5.148)' - 2(9.75)
xH
=
1.599
x
=
1.94
Since > 0.6. the calculation indicates t h a t hexane is n o t a solvent for poly(methy1 methacry-late) a t 25' C., \vhich is verified experimentally.
@
$
for a nonpolar substance, (cal. /cc.)l!? = nonpolar solubility parameter of a polar substance, (cal./cc.) 1 ' 2 = polar solubility parameter of a polar substance, (cal. icc.)l'' = volume fraction = Flory interaction parameter = induction energy parameter (Figure 2)
SUBSCRIPTS i. j. 1, 2, 3. 0 = property of a substance or component of a
*
H 5'
mixture standard temperature enthalpy term = entropy term
= =
VOL. 3
NO. 1
FEBRUARY 1964
7
m
= = =
np
p
change of thermodynamic property upon mixing nonpolar portion of energy of vaporization polai. portion of energy of vaporization
literature Cited
American Petroleum Institute, Pittsburgh, Pa.: Project 44, ’Selected Values of Properties of Hydrocarbons and Related Compounds.” 1955. (2) Barker. J. A.. J . Chrm. Phps. 20, 1526 (1952). (3) Billmeyer, 1. \ V . > private communication. (4) Blanks. R. F., doctoral dissertation. University of California, Barkrlrv. Calif.. 1963 (5) Blanks: R. F.. Prausnitz. J. M., A.1.Ch.E. J . 8 , 86 (1962). ( 6 ) Bondi, A,. Z b i d . , 8, 610 (1962). (7) Bondi. A , . Simki’n: D. J., Ibid.. 3 , 473 (1957). (8) BristoLv. G.. \Vatson. \ V . F.. Trans. Faraday SOC.54, 1731. 1742 (1958). (9) Brown, H. C. Pt ai...J. A m . Chem. SOC.7 5 , 1 (1953). (10) Brolvning. G . V.? Ferry, J. D., J . C h m . Phys. 17, 1107 (1
~
\
~~~~~
,
i\ l- 9. 4 9.,.\
(11) Burrell, H.. Znterchem. Rei’. 14, 3; 31 (1955). (12) Conway. B. E.. Lakhanpal, M . L., J . Polymer Sci. 46, 75: 93, 111 (1960). (13) Delmas. G.. Patterson. D., Somcynsky, T., Zbid., 5 7 , 79 11062). (14) Doty. P. M.. Zablr. H. Z . . Zbid.: 1 , 90 (1946). (15) Dreisbach, R. R.; Adz,an. Chrni. Sei. No. 15 (1955). (16) Florv. P. J.. J . Chem. Phys. 10, 51 (1942). (17) 1 b d . : ’ 1 8 , 108 (1950). (18) Flory. P. J.. “Principles of Polymer Chemistry,” Cornell University Press, N e w York. 1953. (19) Flory. P. J.. Fox. T. G., Jr., J . Am. Chem. Soc. 7 3 , 1904 (1951). (20) Florv. P. J.. Rehner, J.: J . Chpm. Phys. 11, 521 (1943). (21) Fox.’T. G. et al.. Poiymer3, 71. 111 (1962). \ - -
--I
(22) Fox, T. G., Flory, P. J., J . Am. C h m . Soc. 73, 1909, 1915 (1951). (23) Gee, G.: Trans. Znst. Rubber Ind. 18, 266 (1943). (24) Hildebrand, J. H., Scott. R. L.. “Solubility of hTonelectrolytes,“ Reinhold. Nrw York: 1950. (25) Hirschfelder. J. 0 . .Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids.“ \Viley, New York: 1954. (26) Huggins, M. L.. Ann. .Y. I?, Acad. Scz. 43, 1 (1942): J . Am. Chem. SOC.64, 1712 (1942); 2nd. Eng. Chem. 35, 216 (1943). . ig.Fedetation SOC.Paint Techno). (27) Lieberman: E. P., O ~ C D 34, No. 444. 30 (1962). (28) Munster. A , . J . C/zim. Phys. 49, 128 (1952). ( 2 9 ) Munster. A , . Trans. Faraday Soc. 46, 165 (1950). (30) Munster. A . Z. Physik. Chem. Leijrig 196, 106 (3950). (31) Nickerson. .T. K., doctoral dissartation. Texas University, Austin. ?’ex., 1960. (32) Pimentel, G. C.. McClellan: A. L., ’.The Hydrogen Bond,” LV. H. Freeman and Co., San Francisco, 1960. (33) Prausnitz. J. M . , Anderson, K . , ‘4.I.Ch.B. .7. 7 , 96 (1961). (34) Prausnitz, J. M., Shair, F. H.. Ibtd., p. 682. (35) Reid, R. C . . Sherwood. T. K.. “Properties of Gases and Liquids.” McGraw-Hill. New York, 1958. (36) Scatchard, G., Trans. Faraday SOC.33, 160 (1937): Chem. Rem. 8, 321 (1931); J . .4m. Chem. SOC. 56, 995 (1934). (37) Scott. R. L.: Magat. M.. J . Chem. Phys. 13, 172, 178 (1945). (38) Small. P. A , J . Appl. Chem. 3 , 71 (1953). (39) Timmermans. J.: “Physico-Chemical Constants of Pure Organic Compounds.” Elsevier. Amsterdam. 1950. (40) Tompa. H.. J . Chem. Phys. 21, 250 (1953). (41! Tompa. H., ”Polymer Solutions,” Acadrmic Press. New l o r k . 1956. (42) Van Arkel. .A. E.. Trans. Faraday Soc. 42B, 81 (1946). (43) \Valkrr. E. E.. J . ‘4ppl. Chem. 2, 470 (1952). (44) [Vatson, K . M . , Ind. Eng. Chem. 35, 398 (1943). RECEIVED for re\iew May 31. 1963 ACCEPTED October 14. 1963
DEW AND BUBBLE ISOTHERM CALCULATIONAL METHOD FOR BINARY .
SYSTEM PHASE AND VOLUMETRIC BEHAVIOR N E W E L L C. R O D E W A L D , ’ J .
A . D A V I S , 2 A N D
F R E D
K U R A T A
CenfPr for Research zn Engineering Sctpnce. 1 *nzutrsit) of k a n s a s . Laurence. K a n
A method for calculation of liquid and vapor compositions together with saturated volumetric properties of binary systems was developed and successfully used on the helium-nitrogen system. These calculations require only quantities measured in the determination of the dew and bubble points. The phose rule
Y, specifies that for an isothermally univariant system, if the pressure i s fixed, the intensive properties-X, $, #-are set, regardless of the amount of the phases present. Obviously a numerical solution for these variables requires four simultaneous equations. Since it i s possible to write two equations from material balances on nitrogen and helium at each volume per cent liquid, two independent runs are required-a dew isotherm and a bubble isotherm, The quantities which must be measured are the amount of known composition gas metered to the cell and the volume per cent liquid at cell conditions. In the limit, these equations go to the dew and bubble points. HE measurement of the solubility of light gases in liquids T p r e s r n t s man)i problems. jvhich arise primarily from the phase behavior peculiarities of these systems-i.e.. thr lo\v concentration of the gas component in the liquid phase and the low concentration of the liquid constituent in the vapor phase. For many industrially important binary systems such as a light hydrocarbon-\yarer. helium-methane. or helium--nitrogen, the solubility of the light gas in the liquid is less than 1 mole % over conqiderable ranges of pressure and temperature.
Prescnt addrrss. Continental Oil Co., Ponca City: Okla. Present address. Denver Rrsearch Crntrr. Marathon Oil Co.. Littleton. Colo. 1
2
6
I&EC FUNDAMENTALS
Methods frequently used to study the phase behavior of theae systems include the flowing-condensation method. the circulation method. and the static method. In these methods: samplrs of both phases must be \vithdra\vn for analyses either from circulation lines or directly from the equilibrium chamber. ,411 of thrse methods have bern used to study the liquid-vapor behavior of the helium--nitrogen system ( 7 . 1. 1-6. 7 0 ) . Surprisingly large difFerences exist among the data of these investigators. ranging from a maximum of 2.5 to a minimum of 0 to Syc, 7‘he primary source of disagreement is probably sampling error. This error ma); be reduced by taking a representative sample, by getting a single-phase sample. by purging leads adequately beforr sampling. and by sampling ivithout upset-
