10090
J . Phys. Chem. 1991, 95, 10090-I0097
Hydrogen Bonding in Fluids: An Equatiokof-State Approach C. Panayiotout and I. C. Sanchez* Department of Chemical Engineering and Center for Polymer Research, The University of Texas at Austin, Austin, Texas 78712- 1062 (Received: April 1, 1991)
This work attempts to develop a unified statistical thermodynamic model for hydrogen-bonded fluids and their mixtures. Its basic assumption is the division of intermolecular forces into physical (van der Waals) and chemical (hydrogen bonding) forces. A general formalism is presented valid for multicomponent systems of molecules having any number of hydrogen (proton) donor and acceptor groups. The model is applied to typical cases such as self-associated solvents, polymersolvent solutions, and polymer-polymer mixtures. Because of its equation-of-state character, the model may be used to describe the thermodynamic behavior of these systems over an extended range of external conditions that includes the critical and supercritical region. Its relation to the widely used chemical association models and the quasi-chemical models is discussed.
I. Introduction Hydrogen bonding is a subject of remarkable diversity that has attracted over time the intensive interest of chemists, physicists, biologists, and engineers.'-5 Protein folding and association, physical networks and gels, nonionic surfactants, ordering and association in microemulsions, bioseprations in aqueous two-phase systems, selective extraction of fermentation products, watersoluble polymers, thermodynamic properties of gasohols, polymer compatibilization, microphase separation and interfacial phenomena in block copolymers, solubility in the critical or supercritical region, and solvent effects in chemical equilibria are among many in a long list of areas where hydrogen bonding plays a vital role. Hydrogen bonding has received considerable attention in molecular modeling studies using quantum and molecular mechanics. However, there seems to be no consensus on how to describe hydrogen-bonding forces. Neither are there any strict bounds in the hydrogen-bonding energies. Values have been reported5q6 spanning the range from -5 to -155 kJ/mol. Hydrogen-bonding interactions, stronger and having a much longer lifetime than the ordinary van der Waals interaction^,^.^ are often referred to as chemical interactions. The latter implies the formation of association complexes (or clusters) or, simply, "associates". Because of the rather broad range of hydrogenbonding energies, the nature of associates and their very existence as distinguishable molecular entities remains a subject of controversy in the Nevertheless, the existence of these complexes has been invoked often in the past and forms the backbone of the various theories of associated solutions. Along with the association-complex approach another chemical approach exists in the literature. It has originated primarily from thermodynamic studies of water and aqueous solutions. In a seminal paper, Levine and Perramlo have presented a statistical mechanical treatment of hydrogen bonding in water. They point out that the focus should be on the correct counting of the number of arrangements of hydrogen bonds and not on the distribution of associates. In a similar context, Luck" has pointed out that the only equilibrium to be considered is that of the formation and rupture of hydrogen bonds and not of the equilibrium formation of the various associates. More recently and along these lines, VeytsmanI2 considers systems of molecules with one type of proton-donor group and one type of proton-acceptor group. In this lucid note, Veytsman proposes an approximate combinatorial expression for the number of ways of forming hydrogen bonds that does not invoke the existence of associates. A detailed comparison of these two chemical approaches (with associates and without associates) is not yet available. To these two versions of the chemical approach one should juxtapose the quasi-chemical approach.l*I7 In the quasi-chemical method associates are not invoked. Hydrogen bonds are not 'On leave of absence from the Department of Chemical Engineering, University of Thessaloniki, 54006 Thessaloniki, Greece. 0022-3654/91/2095-10090$02.50/0
distinguished from all other van der Waals or physical bonds. Through their interaction strength, hydrogen bonds impart a certain degree of nonrandomness in the molecular structure of the liquid. This approach has found limited success in systems with strong specific interactions. However, these successes sometimes have been overshadowed by an excessive amoung of ad hoc empiricism. In overwhelming majority, current thermodynamic theories of hydrogen bonding are confined to a limited range of external conditions. In a series of recent papers1E-20we have developed the LFAS (lattice-fluid associated solutions) model. This is an equation-of-state model for hydrogen-bonded fluid systems valid over an extended range of external conditions from the ordinary liquid state up to the critical or supercritical state. It is based on the lattice-fluid theory of f l ~ i d s ~and ' - ~invokes ~ the existence of associates. The LFAS model has proved successful in describing the thermodynamic behavior of systems of self-associated and cross-associated fluids. Recently it has been extended to polymer solutions in which the polymer can cross-associate with a selfassociating solvent.24 In general, hydrogen bonding in polymer solutions remains a challenging problem if modeled as a solution of associates. Extension of the LFAS model to polymer solutions in which the polymer can self-associate, or to any system forming networklike associates, is a formidable combinatorial problem that has resisted our attempts at solution. Other attempts toward solving this problem25,26use an unorthodox approach that is difficult to assess. (1) Vinogradov, S.; Linnell, R. Hydrogen Bonding, Van Nostrand Reinhold Co.: New York, 1971. (2) Walter, H., Brooks, D., Fisher, D., Eds. Partitioning in Aqueous Two-Phase Systems; Academic Press: New York, 1985. (3) Burchard, W., Ross-Murphy, S.B., Eds. Physical Networks, Polymers, and Gels; Elsevier Applied Science: London, 1990. (4) Bourrel, M., Schechter, R. S.,Eds. Microemuhiom and Relafed Systems; Marcel Dekker, Inc.: New York, 1988. (5) Reichardt, C. Solvent and Solvent Effects in Organic Chemistry; VCH Verlagsgesellschaft mbH: Weinheim, FRG, 1988. (6) Acree, W. E. Thermodynamic Properties of Nonelectrolyte Solutiom; Academic Press: New York, 1984. (7) Huyskens, P. L.; Howlait-Pinon, M. C.; Siegel. G. G.; Kapuku, F. J . Phys. Chem. 1988, 92, 6841. (8) Kohler, F. In Structure of Water and Aqueous Solutiom; Luck, W. A. P., Ed.; Verlag Chemie: Weinheim, FRG, 1974. (9) Marsh, K.; Kohler, F. J . Mol. Liquids 1985, 30, 13. (IO) Levine, S.;Perram, J. W . In Hydrogen-Bonded Solvent Systems; Covington, A. K., Jones, P., Eds.; Taylor and Francis: London, 1968. ( 1 1) Luck, W. A. P. Angew. Chem. 1980, 92, 29. (12) Veytsman, B. A. J. Phys. Chem. 1990, 94, 8499. (13) Guggenheim, E. A. Mixtures; Clarendon Press: Oxford, 1952. (14) Barker, J. A. J . Chem. Phys. 1952, 20, 1526; 1954, 22 375. (15) Prausnitz, J. M.; Lichtenthaler. R. N.; de Azevedo, E. G.Molecular Thermodynamics of Fluid Phase Equilibria, 2nd ed.; Prentice-Hall: New York, 1986. (16) Kehiaian, H. V.; Grolier, J. P. E.; Benson, G. C. J. Chim. Phys. 1978, 75, 1031. (17) Panayiotou, C.; Vera, J. H. Fluid Phase Equilib. 1980, 5, 35. (18) Panayiotou, C. J . Phys. Chem. 1988, 92, 2960. ( 1 9) Panayiotou, C. Fluid Phase Equilib. 1990, 56, 17 1. (20) Panayiotou, C . J . Solution Chem. 1991, 20, 97.
0 1991 American Chemical Society
Hydrogen Bonding in Fluids
The Journal of Physical Chemistry, Vol. 95, No. 24, 1991
It appears to be a hybrid between the aforementioned associates and nonassociates approaches. It is well-known that compressibility or equation-of-state effects may give rise to LCST (lower critical solution temperature) behavior in nonpolar polymer solutions. Traditionally, LCST behavior has been associated with hydrogen bonding or other strong interactions. The best suited models for describing the thermodynamic behavior of hydrogen-bonded polymer solutions are those combining the equation-of-state approach with the chemical association approach. In this paper we present an equation-of-state formalism for hydrogen-bonded systems valid for both the liquid and the gaseous state, for mixtures of ordinary solvents, as well as mixtures of polymers. In its most general version the model is valid for systems of fluids having any number and type of hydrogen-bonding groups. It includes the important class of systems (self-associating polymers) that the LFAS model does not include. Intermolecular forces are assumed to be divided into physical and chemical. As beforekkmJ4physical interactions are treated with the lattice-fluid mode1.21-23 In the spirit of the Levine and Perram paper,I0 we have extended the combinatorial expression of VeytsmanI2 to systems of molecules with any number and type of hydrogenbonding groups. The chemical contribution formalism does not invoke the existence of associates but instead emphasizes the enumeration of pair interactions between various hydrogenbonding acceptor and donor groups. In the next section we develop the formalism for the most general case of a multicomponent, multigroup system. In the section on applications the model is applied to pure self-associated fluids and their mixtures with inert solvents, to polymersolvent systems, and to polymer-polymer mixtures. The working equations are given in each case that yield examples of how to use the generalized formalism in specific cases. The very structure of the model facilitates its comparison with the parallel LFAS model. 11. Theory Consider a system of N I molecules of type 1, N2 molecules of type 2, ...,and Nl molecules of type r at temperature T and external pressure P. In the general case, there are m types of proton-donor groups and n types of proton-acceptor groups (free electron pairs) distributed in the molecules of the system. Let d! be the number of proton-donor groups of type i ( i = 1, m ) in each molecule of type k (k = 1, r ) and u: be the number of proton-acceptor groups of type j (j = 1, n) in the molecule of type k. The total number of donor groups of type i in the system ( N i ) is given by
Similarly, the total number of acceptor groups of type j in the system (N:) is given by
10091
(van der Waals) and chemical (hydrogen bonding) forces. As already mentioned, this is the basic assumption for the numerous associated solution models in the literature. On the basis of this assumption we can express the canonical partition function as
Q
QPQH
(5)
Formally, the system configurational energy can be separated into a nonpolar contribution and a polar contribution. The latter would include possible hydrogen-bonding interactions. The approximation inherent in eq 5 is equivalent to approximating the classical phase integral as the product of two phase integrals. In one the polar interactions are turned off, while in the other the nonpolar interactions are turned off. This “decoupling” of the interactions is clearly an approximation. A. Physical Partition Function. Since we wish to develop a model valid for both ordinary solvents and polymers, liquids as well as gases, we will adopt for the physical part the lattice-fluid partition f u n c t i ~ n . ~ ’This - ~ ~will facilitate the comparison of the present model with the parallel chain-association LFAS model.18-20924According to the lattice-fluid theory the molecules are arranged on a quasi-lattice of N , sites, No of which are empty. Each molecule of type k is divided into rk segments of clw-packed volume u*k in the pure state. Thus the total number of molecular segments in the system is I
I
where N is the total number of molecules ( N = Z i N k ) and xk is the mole fraction of component k in the mixture. The total number of lattice sites is then N , = rN
+ No
(7)
The average interaction energy per segment of molecule k is given by23
where ekk is the interaction energy per k-k contact and sk is the average number of contacts per segment k (a surface-to-volume ratio characteristic of molecule k). The hydrogen-bonding groups, as parts of their molecules, are also characterized by the same L F parameters as any other group in the molecule. The hydrogen-bonding energy is in excess of this physical or van der Waals interaction. For the mixture, and in the one-fluid approach, the following combining and mixing rules are assumed I
v* = Z4P*f i
(9)
I
N,‘ = XajkNk k
(2)
The total number of donors ( N d )and acceptors (N,) is given by where the segment fractions 4f are defined by
m
Nd = C N J
(3)
N , = ?NJ
(4)
i
j
4i = riNi/rN = xiri/r
(1 1)
and the surface fractions Bi by the equation
Our objective is to write down a tractable approximate partition function for this system. To do this we assume that the configurational partition function of our system can be factored. One factor disregards the existence of hydrogen bonds and considers only the physical intermolecular interactions, while a second factor takes into account the existence of hydrogen bonds in the system. This is equivalent to dividing the intermolecular forces into physical
If desired, a quadratic dependence of u* on composition can be invoked to secure more quantitative agreement with experimental volumes of A Berthelot-type combining rule is adopted for the cij, namely
(21) Sanchez, 1. C.; Lacombe, R. J . Phys. Chem. 1976,80, 2352; 1976, 80, 2568. (22) Sanchez, I. C.; Lacombe, R. Macromolecules 1978, / I , 1145. (23) Panayiotou, C. Macromolecules 1987, 20, 861.
(24) Panayiotou, C.; Sanchez, I. C. Macromolecules, in press. (25) Painter, P. C.; Park, Y.; Coleman, M. M. Macromolecules 1989, 22, 570, 580, 586. (26) Painter, P. C.; Graf, J.; Coleman, M. M. J . Chem. Phys. 1990, 92, 6166.
10092 The Journal of Physical Chemistry, Vol. 95, No. 24, 1991
The dimensionless parameter tijis expected to have values close to unity (equal to one in Berthelot's rule). For binary mixtures, eq IO is cast to the more convenient form e*
= dJlf*l
+ 42t*2 - 4JIfl2Xl2
(14)
where t*1
XI2 =
+ (s1/s*)c*2
- 2(sI/s2)%*12 RT
Panayiotou and Sanchez index refers to the acceptor. The total number of hydrogen bonds is
What is now required is the number of ways Q of distributing the Nu bonds among the functional groups of the system. By extending the recent arguments of VeytsmanI2 to the general case of multigroup molecules, we have
(15)
and
or Equation 15 indicates that, in the application of the theory, the absolute values of sI)s are not needed. It is the ratio si/sj which is needed as in Flory equation-of-state theory.*' In many applications it is sufficient to assume that all the sI)s are equal. The total lattice-fluid volume of the system is given by VLF = N p * = rNv*ij = P 8 (17) where P is the close-packed volume of the system and 8 = 1/p is the reduced volume ( p being the reduced density) which is obtained from the equation of state to be developed later. Similarly, the total potential energy of the system arising from physical intermolecular interactions is given by21-23
= rNpc* (18) Segment fractions, &, should be distinguished from the site fractions fk defined by -ELF
and for the site fraction of empty sites No N , - r N fo = - = p N, N, On the basis of these definitions and nomenclature, the canonical partition function for the physical interactions is given by QdT*NO*INkl)= (I/fo)No$(wk/fk)N' ~XP(-ELF/RT)
(21a)
or
where Nio is the number of unbonded donors of type i and Noj is the number of unbonded acceptors of type j : n
Nio = N i - CNij
(25)
I
m
Noj = N,'
- XNij i
(26)
and the probability Pij is defined in the following paragraph. The derivation for 52 follows from a generalization of the following example: Consider a system containing D donors and A acceptors that forms H hydrogen bonds. The number of ways of choosing H donors from the donor population D is just the binomial coefficient D ! / ( D- H)!H!.Similarly, the number of ways of choosing H acceptors from the population A is the binomial coefficient A ! / ( A - H)!H!. Now a specific donor can hydrogen bond with any of H acceptors, a second donor can hydrogen bond with any of the remaining H - 1 acceptors, and so on. The number of ways that H bonds can be formed between H donors and H acceptors is H!. Thus, the total number of ways that H bonds can form between D donors and A acceptors is the product of the binomial coefficients times H!or Q = D ! A ! / [ ( D - H ) ! ( A - H)!H!]. However, this is a gross overestimation of the number of bonds because we have failed to account for the bonding requirement that donor and acceptor be in close spatial proximity. The mean-field probability P that a specific acceptor will be proximate to a given donor is proportional to the volume of the acceptor group divided by the total system volume; i.e., P 1/V p/rN. Even spatial proximity does not guarantee that a bond will form. Bond formation requires that donor and acceptor adopt a unique spatial orientation with respect to one another. Formation of the bond is also accompanied by a loss of rotational degrees of freedom. Steric considerations will also come into play in bond formation. In general, for a donor i-acceptor j pair, the probability is given by
-
N
Qp(T,No,{Nkl) = ( 1 - p)-Nop-NfI(wk/4k)Nkexp(rNpc*/RT) k
(21b) where a k is the number of configurations available to an rk-mer in the close-packed pure state; in this work wk is treated as a constant characteristic of molecules or type k and it will cancel out in all applications of our interest. B. Hydrogen-BondingPartition Function. It should be stressed, once again, that the interaction energies giving rise to the hydrogen-bonding contribution to the partition function are in excess of the physical interactions of the preceding paragraph. They are pertinent only to proton-donor and acceptor groups of the molecules which interact with both physical and chemical forces. Let EiY be the favorable energy change upon hydrogen bond formation between a donor group i and an acceptor groupj. (This energy is in excess of any physical interaction energy.) There are Nijsuch bonds in the system and, thus, the total hydrogen-bonding energy of the system is
Pij = &?/Rp/rN (27) where Sip is the entropy loss (intrinsically negative) associated with hydrogen bond formation of an (id) pair. Thus, in our simple example above, we correct our overestimation of D-A pairs by multiplying by the mean-field product PH. The canonical partition function for hydrogen bonding can now be written as QH(
T,NO,INkl) = C QH(T,No,{NkI,VijI) lNvl
m n
EH = CCNijEj,O i l
(22)
In general E:, # Ej: and Nij # Nji; throughout we adopt the convention that the inner index refers to the donor and the outer (27) Flory,
P.J . Discuss. Faraday Soc. 1970, 7, 49.
where and the sum is over all allowable values of Nip
(28a)
C. Complete Partition Function. The Gibbs partition function 9 is given by *(T,p,lNk\) =
5 QdT,No,INkI)
NpO
QH(T,NovINkI,INijI)exP(-PV/RT)
(30)
IN0
or n
m
[ v i - XkU i k ] [ v i - F Y k j ] p exp(-Gij”/RT)
vij
m
In
m n
+ EENijVi,O i j
(43b)
which are a set of ( m X n) simultaneous quadratic equations in the uil. Substitution of eq 43a into (37) yields the compact form
where the system volume is given by V = rNijo*
10093
The Journal of Physical Chemistry, Vol. 95, No. 24, 1991
Hydrogen Bonding in Fluids
(31)
vi0 + Eui In -
ud
(44a)
j
and the V,,”s are the volume changes accompanying hydrogen bond formation between an ( i j ) pair. The Gibbs free energy of this system is G = -kT In \k (32)
or
The Gibbs partition function is evaluated in the usual way by finding the maximum term. This is equivalent to equating the Gibbs free energy to the genertic term in the partition function and then minimizing with respect to b and each of the Nil. The minimization conditions are ( a G / a a )T,P,(NtJ,(N,jI = (33)
Notice that the mn 1 order parameters, the set {vi]} b, are coupled and require the simultaneous solution of mn + 1 equations, eqs 39 and 43. D. Basic Thermodynamic Quantities. For each basic thermodynamic quantity of the real mixture there is one contribution arising from physical interactions and one contribution arising from hydrogen bonding. Thus, the heat of mixing at ordinary pressures is given by
=
(ac/aNij)T,PP,IN~l,lN,,l
(34)
+
+
m n
I
HE
where G is given by G = GLF
+ GH
(35)
I
-p/p
(45) where uiJk = NiJk/rNand NiJkis the number of donor i-acceptor
The lattice-fluid contribution is given by G L F / k T= rN
I
+ xp&kek* + z:CEij”[vij - CUiJk]} k i i k
EE = rN(-pt*
j hydrogen bonds in pure component k. Similarly, the volume
+ P b / P + (5 - 1 ) In (1 - p ) +
of mixing is given by
= rN(ib* - ifi&Pk* k
m n
+ XEvi:[vij i J
- i v i j k ] I (46) k
The equation for the chemical potential of component k(&) is obtained as follows:
and the chemical contribution by
.(47aj but from eqs 33 and 34 this simplifies to
where uij
5
Nij/rN
u,,,
N,/rN
. udl
Ndi rN
etc
(38a)
and Gi,O = :F
+ PK? = Eij” + PVij” - TS,O
(38b)
pkJRT = In ($&)
Equation 33 yields p2
+ P + F[ln (1 - p ) + p ( 1 - I/?)]
where the reduced pressure, P,is defined by P = P/P* = P v * / c * the reduced temperature, T, by = T / P = RT/c* and a modified average chain length, i , by
where the L F contribution is23
=0
+ 1-
rk
+
(39) (40) (41)
where uH is the fraction of hydrogen bonds in the system. Equation 39 is identical in form to the original lattice-fluid equation of state2’ except that 1 /? replaces the usual 1 / r term. Equation 34 yields (the derivative is taken holding all members of the set { N J constant except Nij) uij/uiouo, = 5 exp(-G,,”/RT) for all (ij) (43a)
and the hydrogen-bonding contribution is
where the fraction of hydrogen bonds uH is defined as in eq 42 and d? and a: are defined as in eqs 1 and 2. Notice that the sum ZNkpk k
yields eqs 36
+ 44, as it should.
10094 The Journal of Physical Chemistry, Vol. 95, No. 24, 1991 TABLE I: LF Scaling Parameters for Pure Fluids" fluid T*.K P*,MPa methanol 496 315 ethanol 464 328 I -propanol 418 3 20 I-butanol 494 3 20 I -pentanol 513 313 534 301 I-hexanol 1 -octanol 552 301 I -decanol 566 30I 416 298 n-hexane 487 309 n-heptane 499 416 chloroform 541 605 poly(ethy1ene oxide) poly(propy1ene oxide) 5 24 421 polystyrene 135 351 poly(viny1 methyl ether) 628 418
p*,
kg/m3 186 826 858 863 865 867 812 816 175 800 1109
1112 1093
0
1 IO5
1117
OThe characteristic density p * , the size parameter r , the closepacked segment volume v*, and the molecular weight M are related to one another by rv* = M/p*.
111. Applications In this section we will apply the model to three typical cases: self-associated solvent, polymer-solvent mixtures, and polymerpolymer mixtures. A. Self-Associated Solvents. As already mentioned, the present equation-of-state model may be used to describe the thermodynamic behavior of hydrogen-bonded systems over extended ranges of external conditions. In applying the model to pure fluids one may extract all needed information and determine the purecomponent characteristic equation-of-state parameters. The latter may be used subsequently in the description of the mixtures. There are a number of compilations of pure-component properties that can be used for this purpose. Two particularly useful compilations are those of Timmermans28 and of Daubert and D a r ~ n e r . ~In~ the case of self-associated fluids, the present model requires the determination of three lattice-fluid scaling constants (e*, u s , r ) and three hydrogen-bonding constants (E', So, P)for each type of hydrogen bond formed. These parameters have been determined for a number of fluids in our parallel LFAS mode1,18-20where the linear association complexes were considered as real molecular entities with their own physical properties. The immediate question is whether the pure-component parameters determined for the LFAS model may be used in the present model. To answer this question we have applied the present model to the case of pure 1-alkanols by using the LFAS pure-component properties, namely, the scaling constants reported in Table I and the hydrogen-bonding parameters '?l = -25.1 kJ/mol, S,O = -26.5 J/(K mol), and P = -5.6 cm3/mol. Pure 1 -alkanols have one proton donor and one acceptor per molecule. In this case there is only one type of hydrogen bond in the system and the minimization condition, eq 43, gives
( N - N I l ) 2= N l , ( r N 6 )exp(Gllo/RT)
Panayiotou and Sanchez
(50)
Figure 1. Vapor pressures of pure I-alkanols. Solid curves were calculated by the present model. Experimental data are from ref 29. Numbers near the curves indicate the number of carbon atoms in the corresponding I-alkanols.
1.32 V5P
("9
)
1.26
1.2 0
3
50
Figure 2. Specific volumes of I-alkanols. Solid curves were calculated by the present model. Experimental data are from ref 29: 0,methanol; 0,1-butanol; A, 1-octanol.
9 00 HE
iJ/mol)
600
3 OC
The physically meaningful solution of this quadratic equation is r u H = Nl -I = 1 - [ A ( A N
+ 4)]'/2 - A 2
(51)
where A = r6 exp(Gllo/RT)
(52)
From eqs 48 and 49, the chemical potential is
( 2 8 ) Timmermans, J. Physico-Chemical Constants of Pure Organic Compounds; Elsevier: New York, 1950 Vol. I, p 196;Vol. 11. (29)Daubert, T. E.,Danner, R. P.,Eds. Data Compilation Tables of Properties of Pure Compounds; AlChE Symp.Ser. No. 203;American Institute of Chemical Engineers: New York, 1985.
n u
00
05
x,
1.0
Figure 3. Heats of mixing for the system ethanol ( I ) + n-hexane (2). Experimental data are from ref 34. Solid line calculated by the present model: 0, 25 OC; A, 45 "C.
In Figure 1 are shown experimental and calculated vapor pressures for 1-alkanols, while in Figure 2 are shown the corresponding specific volumes. For all practical purposes the performance of the present model in describing the thermodynamic behavior of pure 1-alkanols, by using the previously determined LFAS is the same as that of LFAS itself. This is also true for the case of mixtures of self-associated 1-alkanols
The Journal of Physical Chemistry, Vol. 95, No. 24, I991 10095
Hydrogen Bonding in Fluids
v Figure 5. E~perimental'~ and calculated heats of mixing for the system chloroform (1) poly(ethy1ene oxide) (2) at 5.53 OC. Solid line calculated by the present model.
+
Figure 4. ExperimentalUand calculated heats of mixing for the system
1-octanol ( I ) model: 0, 25
+ n-heptane (2).
Solid lines calculated by the present
OC;A, 45 O C .
with inert solvents as can be seen in Figures 3 and 4. The experimental and calculated heats of mixing for the systems ethanol n-hexane and octanol + n-heptane, respectively, are compared. No binary or any other parameter has been adjusted here. B. Polymer-Solvent Systems. The results of the preceding paragraph suggest a possible equivalence between the present approach and the classical "associated solutions" approach of consecutive formation of association complexes. The equivalence refers in particular to the equality of equilibrium constants for the formation of hydrogen bonds. As a further test of the validity of this proposition we will apply the present model to two polymer-solvent systems for which on the one hand there are excellent experimental datag0s3' and on the other hand data have been recently described by the LFAS m0de1.2~ The systems are chloroform-poly(ethy1ene oxide) (PEO) and chloroform-poly(propylene oxide) (PPO). In both cases the solvent has one proton-donor group while the polymer has only one type of proton acceptor. Let "a" be the number of these equivalent proton-acceptor groups per polymer molecule. Since there is only one type of hydrogen bond in the system, there is only one minimization condition, namely
+
(NI- Nll)(aN2- NI1)= N l 1 W 8 )exp(GI1O/RT) (54) The physically meaningful solution of this quadratic equation is rvH
NII =N =
XI
+ 0x2 4- A - [(XI + U X +~ A)'
-4
~ ~ 1 ~ 2 ]
2 (55)
where A is given, in form, by eq 52. Subscript 1 refers to the solvent and 2 to the polymer; the xi are mole fractions. For the solvent, the two contributions to the chemical potential are
and K1.H XI = r l v H- In RT X I - run
(57)
(30) Malcolm, G. N.; Baird, C. E.; Bruce, G. R.;Cheyne, K. G.; Kershaw, R. W.; Pratl, M.C. 1.Polym. Sci. A-2 1969, 7, 1495. (31) Kershaw, R. W.; Malcolm, G.N. Tram. Faraday Soc. 1968,64,323.
For the polymer (component 2) the physical contribution to the chemical potential is given by interchanging the indices in eq 56, while the chemical contribution is given by pZ.H/RT = rZvH- a In 0x2 axz - run To compare with the LFAS modelz4we will use the same values for the number of proton-acceptor groups in the polymers, namely, a = 7 for PEO, which is the number of CHzCHzOrepeating units in the molecule, and a = 18 for PPO, which is half the CH(CH g ) C H 2 0repeating units in the molecule. The latter number takes into account the steric effect of the pendant CH3 group in the hydrogen bond formation as discussed previ~usly.~'Alternatively, we could use a = 36, but with a different value (more negative) of the bond entropy So. The hydrogen-bonding parameters for both systems suggested by the LFAS model24are E' = -1 1.44 kJ/mol, So = -9.74 J/(K mol), and = -0.85 cm3/mol. By using the same set of parameters, the present model reproduces quantitatively the vapor pressures in both systems and the heats of mixing and volumes of mixing in the system chloroform-PPO. The heats of mixing in the system chloroform-PEO are underestimated, on the average, by 4% while the calculated volumes of mixing are, as expected,24 only qualitatively correct (calculated minimum = -1.4 cm3/cm3,compared to the corresponding experimental = -0.7 cm3/cm3). Many sets of values for E" and 3' can reproduce simultaneously the heats of mixing and the vapor pressures for both systems. In all cases, however, the hydrogen-bonding free energy (E' - T p ) remains practically the same and equal to the value calculated by the above LFAS values (-8.72 kJ/mol at 5.53 "C). One such ~ / ~ set is .@ = -10.41 kJ/mol and 9 = -6.1 1 J/(K mol). In Figures 5-8 are compared the experimental data of Malcolm et al.30731 for the heats of mixing and vapor pressures of the two systems with the present model using the above set of hydrogen-bonding parameters at 5.53 "C. The procedure for obtaining the above set of hydrogen-bonding parameters was the same as the one used previously;24they were obtained from a simultaneous correlation of all experimental data for the system CHCIg-PEO only. The parameters, so obtained, were used to predict the mixing quantities of the system CHCIg-PPO. Once again, no lattice-fluid binary parameters were adjusted in these applications. The calculations by the present model are practically indistinguishable from the LFAS calculation^.^^ In the same Figures 5-8, the corresponding calculations of the original lattice-fluid model without association21-z3are also shown for comparison. C. Polymer-Polymer Systems. Polymer pairs are as a rule immiscible. Chemical modification through incorporation of complementary hydrogen-bonding groups in the two polymers can induce or enhance miscibility. This possibility is of particular interest in view of the technological importance of miscible (compatible) polymer blends. Associated with polymer-polymer miscibility is the universally observed phase separation with in-
10096 The Journal of Physical Chemistry, Vol. 95, No. 24, 1991
Panayiotou and Sanchez partially modified polystyrene by copolymerizing styrene with a relatively small number of vinyl phenyl hexafluoro dimethyl carbinol monomers. The OH group of the carbinol is a relatively strong proton donor while the -0- group in PVME is a proton acceptor. The enhancement of miscibility will be indicated by the relative abundance and strength of the two types of hydrogen bonds. Let "d" be the number of OH groups per modified PS molecule and -an be the number of -0- groups per PVME molecule. Note that an OH group acts as both a donor and an acceptor. For simplicity we consider monodisperse polymers. Since there are two types of hydrogen bonds in the system (self-association and cross-association) there will be two minimization conditions, namely
( ~ N-IN I I- N12)(dN1- Nil) = ( ~ N C I N Iexp(GiiO/RT) I (594 and
Figure 6. ExperimentalMand calculated vapor pressures for the system chloroform ( I ) + poly(ethy1ene oxide) (2) at 5.53 OC. Solid line calculated by the present model. Dashed line calculated by the original LF without association (6 = 1 for physical contribution).
( ~ N-IN I l - N12)(aN2- N I 2 )= (rNE)N12exp(GI2O/RT) (604 Subscript 1 stands for PS and 2 for PVME. Gllois the free energy of self-association and GI2Othe free energy of cross-association. This is a system of two coupled quadratic equations with two unknowns, NI1and N I 2 . They can easily be solved numerically by rewriting the equations as NI 1 = 2dxl + A l l - rv12 N [(2dxl + A l l - rv12)2 - 4dxl(dxl -
(59b)
N12 ax2 + A I 2- rull N [ ( d x , + ax2 + A12- rv11)2- 4ax2(dxl-
(60b)
rvll
and rv12 3 - = d x l Figure 7. Experimental3'and calculated heats of mixing for the system chloroform (1) poly(propy1ene oxide) (2) at 5.53 OC. Symbols as in Figure 5 .
+
80 P ("
Hg)
LO
0
where A l l = rij exp(GlIo/RT)
(32) Pcarce, E. M.;Kwei, T. K.; Min,B. Y. J . Mucromol. Sci.-Chem. 1984, A 2 / , 1181.
AI2=
rij exp(GI2O/RT)
(61)
One small, near zero, trial value is initially assumed for vI2 and used in eq 59b to make a first estimation of v l l . This value for u I I is then used in eq 60b to calculate the new value for vI2. This algorithm is stable and the solution, to the desired accuracy, is obtained after a few iterations. This iteration scheme must be combined with the one used to solve the LF equation of state, eq 39. The value of the reduced volume ii required in eqs 59 and 60 can initially be calculated by assuming that both v I 1 and uI 2 are zero. After the values of uII and vI2 are computed, as described above, then a new value of 1/i is computed:
Since r itself is large for this system of two polymers and the fraction of hydrogen bonds vH is small, the approximation 1/F = 0 is quite good in this application (effectively decoupling ii from u I I and u I 2 ) , but in general it is not. The equations for the lattice-fluid contributions to the chemical potential are identical in form to eq 56. The hydrogen-bonding contributions are given by the following equations:
Figure 8. Experimental" and calculated vapor pressures for the system chloroform ( I ) + poly(propy1ene oxide) ( 2 ) at 5.53 O C . Symbols as in Figure 6.
creasing temperature (LCST behavior). By introducing hydrogen-bonding interactions, one seeks to raise the LCST of these systems. The present model is particularly suited for these type of studies. As an example, we will apply the model to the mixture of polystyrene (PS) with poly(viny1 methyl ether) (PVME). Pearce et al.32have made an extensive study of compatibilization through hydrogen bonding in the system PS-PVME. They
+
dX1 _ -- r l v H - d In - d In PI,H
d x l - ruH
RT
dxl dx, - rvll
(63)
and
-P2.n- - r2vH - a In RT
ax2 ax2 - rv12
The L F parameters for the system PS-PVME are available from our previous work.23 As it was pointed out in ref 23, the phase behavior of this system is very sensitive to the molecular provide weights of the two polymers. Unfortunately Pearce et incomplete information on molecular weights of their samples.
The Journal of Physical Chemistry, Vol. 95, No. 24, 1991 10091
Hydrogen Bonding in Fluids O
170
I--
0.0
0’2
%Garb
:
0.5
Figure 9. Experimental3* and calculated (solid curve) LCST’s for the system poly(viny1 methyl ether) modified polystyrene versus the carbinol content (in mol %) of the modified polystyrene.
+
The molecular weight of the unmodified PS was set equal to 22 100 in order to match the observed LCST with that calculated by the L F parameters of ref 23. The molecular weight of PVME was retained equal to 5 1 500. For lack of any additional information, the hydrogen-bonding parameters for the OH group of the carbinol repeating units (self-association) was set equal to the corresponding parameters of the O H of 1-alkanols, namely, Ell0= -25.1 kJ/mol and Sllo= -26.5 J/(K mol). In the temperature range of interest the hydrogen-bonding entropy for the cross-association plays a secondary role and for lack of any additional relevant information it was set equal to that of the self-association entropy, namely, Sllo= SI?= -26.5 J/(K mol). At normal pressures the qF are unimportant for determining phase equilibria and were set equal to zero. We are left, then, with one unknown parameter, the cross-association energy E,?. This parameter can be determined either from one experimental LCST for a given carbinol content in the modified PS or from an overall best fit of the available experimental data. In Figure 9 we compare the experimental LCST’S~~ with theory by setting E,? = -23 kJ/mol. The calculated LCSTs are obtained as the minimum in the corresponding binodal curves. In view of the experimental uncertainties and the complexities of these systems, the results of Figure 9 are rather gratifying.
IV. Discussion and Conclusions In this paper we have developed and tested a general thermodynamic framework for the treatment of hydrogen-bonded fluids and their mixtures. Although in the model the intermolecular forces are assumed divided into physical and chemical, the existence of association complexes as real molecular entities is not
invoked. In this sense the present approach is different from the classical approach of the various “associated solution modelswin the literature. Is this then a quasi-chemical model? There is a formal resemblance. Both satisfy the law of mass action through equations such as (43a). However, this is the only resemblance between this model and Guggenheims’s quasi-chemical theory.I3 The equations that define the number of (id)pairs A’,,in the two models are quite different. The set of equations (43b) are much more tractable than those encountered in the quasi-chemical method. Although there is no obvious formal relation between the present approach and the chemical approach of “associated solutions” models (one typical representative is the LFAS mode118-20J4),the results presented in the Applications section indicate that the treatments of hydrogen bonding by the two approaches are essentially equivalent. The important advantage of the present approach over the LFAS approach is its straightforward extension to polymer mixtures and, in general, to systems forming multidimensional physical networks through hydrogen-bond association. The present model is, in particular, applicable to water and its solutions. Application of the model to aqueous polymer solutions will be the subject of a forthcoming publication. In the particular polymer-polymer mixture studied in the Applications section, the percentage of the styrene repeat units with OH groups was quite low (less than 0.5%). This permitted the safe assumption that the L F scaling constants of the modified PS are the same as in the unmodified PS. In the general case, where the percentage of the modifying monomer is not low, the L F constants should be obtained by the appropriate averaging procedure used for random copolymer^.^^ The present model properly accounts for the dependence of hydrogen bonding on density. Thus, it can be used over an extended range of external conditions up to the critical or supercritical state. Although it represents a significant improvement over the original L F t h e ~ r y , ~it’ -retains ~ ~ the LF’s approximate character and inherent drawbacks. Nevertheless, it presents a consistent thermodynamic framework for studying hydrogenbonded systems and as such it may form the basis for more refined treatments. Acknowledgment. Financial support for this work has been provided by the Air Force Office of Scientific Research and the National Science Foundation Division of Materials Research. (33)Panayiotou, C. Makromol. Chem. 1987,188, 2133. (34)Christensen. J. J.; Hanks, R. W.; Izatt, R. M. Handbook o f H e a f s of Mixing; Wiley: New York, 1982;supplement, 1986.
