J. Phys. Chem. 1992,96,9479-9487 (9)Pitzcr, K.S.Activity Coefficients in Electrolyte Solutions, 2nd ed.; Pitzer, K. S., Ed.;CRC Press: Boca Raton, FL, 1991;pp 75-154. (10)Pitzer, K.S.J. Solut. Chem. 1975,4, 249-265. (1 1) Roy, R. N.; Gibbons, J. J.; Peiper, J. C.; Pi-, K. S. J. Phys. Chem. 1983,87,2365-2369. (12)Staples, B. R.; Nuttall, R. L. J . Phys. Chem. Ref. Data 1977,6, 385-407. (13) Linke, W. F. Solubilities of Inorganic and Metal Organic Compounds; American Chemical Society: Washington, 1965;Vol. I. (14)Phutela, R. C.; Pitzcr, K. S.J . Solut. Chem. 1983, 12, 201-207. (15) Broul, M.; Nyvlt, J.; Sohnel, 0. Solubility in Inorganic Two-Component Systems; Elsevier: Amsterdam, 1981. (16)Wu, Y. C.; Rush, R. M.; Scatchard, G. J. Phys. Chem. 1968,72, 40484053. (17)Robinson, R. A.; Covington, A. K. J. Res. Natl. Bur. Stand. 1W, 72A,239-245. (18) Harvie, C. E.; Weare, J. H. Geochim. Cosmochim. Acta 1980,44, 981-997. (19)Harvie, C. E.; Eugster, H. P.; Weare, J. H. Geochim. Cosmochim. Acfa 1982,46,1603-1618. (20)Pitzer, K.S.Ber. Bunsen-Ges. Phys. Chem. 1981,85,952-959. (21)Pitzer, K. S.J . Am. Chem. Soc. 1980,102,2902-2906. (22)Goldberg. R. N.; Nuttall, R. L. J . Phys. Chem. Ref. Data 1978,7, 263-3 1 1. (23)Quill, L. L.; Robey. R. F. J. Am. Chem. Soc. 1937,59,2591-2595. (24)Mironov, K.E.; Popov, A. P.; Khripin, L. A. Russ. J. Inorg. Chem. l!h%,II, 1499-1503. (25)Silcock, H. L. Solubilities of Inorganic and Organic Compounds; Pergamon: Oxford, 1979;Vol. 3. (26)Robinson, R. A. J . Am. Chem. Soc. 1940,62,313(F-3131.
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Phase Diagrams of "Living" Polymers at Fixed Initiator Concentrations L.Reo6 Corrales* and John C . Wheeler Department of Chemistry, 0340, University of California-Son Diego, La Jolla, California 92093 (Received: May 8, 1992)
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The equilibrium polymerization of 'living" polymers can be described by the n 0 limit of an annealed dilute n-vector model of magnetism. Phase diagrams for fixed magnetic field, corresponding to fixed initiator activity, show very flat coexistence curves in the vicinity of the nonsymmetric tricritical point found in the annealed dilute n 0 vector model. However, in an experiment it is often the initiator concentration rather than the activity that can be held fmed. Fixing the initiator concentration instead of the initiator activity produces phase boundaries analogous to dew-bubble curves, where the maxima are not necessarily the critical points of the system. In this paper, interesting new phase diagrams are predicted that do not have flat coexistence curves in the vicinity of the nonsymmetric tricritical point. However, these types of phase diagrams do produce apparently flat coexistence curves away from the nonsymmetric tricritidal point. The details of the coexistence surfaces that reveal threephase equilibrium and other features are shown in density, field spaces corresponding to the initiator concentration, solvent concentration, reduced temperature,and critical solvent temperature. In addition, an analogous phase diagram for a particular sulfur solution is presented and contrasted to phase diagrams of previous work.
I. Introductioa Solutions in which polymerization can occur under conditions of chemical equilibrium between monomer and its various polymers display interesting and unusual phase diagrams. Solutions of elemental sulfur in various organic solvents illustrate the variety of phase diagrams possible.' Solutions of the "living polymers" of SwamZ provide another potential example currently under experimental in~estigation.~ An elegant chemical equilibrium theory to explain the sulfur phase diagrams was given by Scott: based on ideas of F l ~ r y Gee: , ~ and Tobolsky and Eisenberg.' Wheeler and PfeutyS showed that a lattice model of polymerization in sulfur and sulfur solutions was equivalent to an n-vector model of an annealed dilute magnet in the limit n 0 and that Scott's theory was equivalent to the mean-field approximation to this dilute magnet. Subsequently, Kennedy and Wheeler9 proposed that the study of living polymers was amenable to similar treatment. They showed that a lattice model for living polymer
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Address correspondence to this author at the Molecular Science Rcscarch Center, Pacific Northwest Laboratory, Richland, WA 99352.
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solutions is identical with the same annealed dilute n 0 vector model of magnetism and that the mean-field approximation to the magnetic model is useful in predicting phase diagrams of the living polymer system. References 8 and 9 together demonstrate that the global phase diagrams for model sulfur solutions, model living polymer solutions, and a model dilute magnet similar to the Blume, Emery, and Griffiths'O model of magnetism are all the same. The rather different phase diagrams that result in the three cases arise from differences in the way that the "working curve" of a given model cuts through the global phase diagram (see, for example, Figure 1 of ref 8 and Figure 6 of ref 9). The chemical equilibrium model for living polymers contains a parameter K1,which serves effectively as a combined activity of initiators and equilibrium constant for the reaction of an initiator with a monomer to initiate polymerization. There is also an equilibrium constant Kpfor propagating a chain of 1 monomer units to a chain of 1 1. Kennedy and Wheeler studied the model in the limit of small initiator concentration, corresponding to the limit K 1 0. They found interesting phase diagrams that contained a tricritical point analogous to that found in He3-He4
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0022-3654/92/2096-9479$03.00/00 1992 American Chemical Society
Corrales and Wheeler
9480 The Journal of Physical Chemistry, Vol. 96, No. 23, 1992
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Figure 1. Coexistence curve for the case of constant K 1 = KINTcp= (1/2h2)NTCP = 0.001 1 and T l / T p= (Tl/Tp)NTcp= 0.9299. The curve scala with an exponent = (see text). It shows the enhanced flatness of the coexistence curve near a higher-order critical point. The NTCP sits centered on the maximum of the curve. The two phasts in cquilibrium are determined by eq 2.5, where the left and right branches are given by Ixw’, xp)] and (xmlr”,x,”), respectively. In contrast, Figure 2 shows the phase boundary for fixed xp.
mixtures at very low temperatures and in models of sulfur solutionsEand dilute ferromagnets.lO*llThe correspondence of the polymer model and the dilute magnet model shows that there is a symmetric tricritical point in which three phases, two of which are related by a mirror symmetry, are in critical equilibrium at zero field. More recently Wheeler12 discovered that at larger K I the model contained a nonsymmetric (generic) tricritical point (NTCP) analogous to that found in three- and morecomponent mixtures13and studied the model in the vicinity of the nonsymmetric tricritical point for fixed KI(or h, the magnetic field in the magnetic language). At the tricritical value of K1, the coexistence curve in the (T, x h ) plane, where xwlvis the fraction of solvent, is described by the critical exponent = whereas for the symmetric tricritical point the effective critical exponent is /3 = 1. Thus one sees,at fmed Kl, a dramatically flat coexistence curve near the nonsymmetric tricritical point (as shown in Figure 1). There is, however, an artificiality in comparing phase diagrams with Kl held fixed with at least some experimental diagrams. Holding K1fixed corresponds to holding the activity of the polymers constant. In certain cases of ionic polymerization, an equilibrium between an inactive ion pair (or other complex) and active free ions may exist14 that makes the approximation of constant K1 appropriate, but in other cases, it is the concentration of polymers, x rather than the activity, K1,that should be held constant. Witkn the magnetic analogy, this corresponds to holding the product hm rather than the magnetic field, h, constant, where m is the magnetization per spin. In this paper we examine the phase diagrams of the dilute n 0 vector model of magnetism as applied to systems of living polymers in nonzero field (Le., Kl >O). In particular, we examine the model for living polymers with parameters corresponding to polytetrahydrofuran. We also show and discuss briefly a phase diagram for a given sulfur solution with fmed x pwhen only chain polymers are allowed (corresponding to this model). In section I1 we briefly present the results of the mean-field approximation of the annealed dilute n 0 vector model with some slight modifications in notation from those of previous publications. The phase diagrams containing the NTCP for fixed K1and fixed x p are compared in section 111 for various values of the solvent parameter, TI,which is the critical temperature of the monomer with the solvent. The phase diagrams at fixed x p are explained in detail. Further examples are presented that show three-phase equilibrium in the proximity of the NTCP and a phase diagram that appears to be flat that is well away from the NTCP. The details of the coexistence surface for these values of the parameters are examined in the (T,xp)and (xp x ~planes. ) A phase diagram for sulfur at fixed initiator concentration with parameter values away from the nonsymmetric trinitical point is found to have a high-temperature miscibility gap with phase boundaries similar to those found for the low-temperature miscibility gap in poly-
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tetrahydrofuran. “Initiators” in this case can be small amounts of solvent or hydrogen pregcnt in the system that react with the sulfur and raise the effective value of K1to much larger values than occur in pure sulfur. Preliminary studies indicate that very flat phase diagrams also result for model sulfur solutions and can be obtained by optimizing the values of T I and x,. The two- and threephase regions for sulfur solutions are more complex and are not presented here. The parameters that lead to a flat phase boundary reveal that rather small amounts of initiators arc needed in order to see phase diagrams like those for sulfur in a mixture of cis- and tram-decalin. This provides yet another possible mechanism for the observation of dramatically flat phase diagram seen for sulfur systems.12*1s In saction IV we discuss our results.
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II. Correspondence of Living Polymers a d the D 0 film The details of the dilute n 0 vector model of magnetism as applied to living polymers have been presented el~ewhere.~ We will summarize the mappings between the polymer and magnetic models and state the results of the mean-field approximation. The correspondence between polymers and magnets is as follows
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= 3e + y2hm, x p = Y2hm, xmIv= 1 - x, (2.1) where, for living polymers, KI is the equilibrium constant for initiation, K,,’is the equilibrium constant for the reaction of a polymer end with a monomer in a specific adjacent cell, & is the fraction of lattice cells filled with monomers that are incorporated in chain polymers, xp is the number of polymers divided by the number of cells and is also equal to the initiator concentration in the limit that all initiators form polymers and where, for the magnet, h = H/(kT), 3 = J/(kT) and Hand J are the magnetic field and (dimensionleas) spin coupling constant, m and e are the magnetization and spin coupling energy per site of the corresponding dilute magnet, and xmlV= 1 x, is the fraction of sites occupied by solvent, where x, is the number of sites containing spins. The notation to be used here will be soh = solvent and o = unreacted monomer, which is consistent with the more recent york of ref 15 and is different from that of refs 8 , 9 , and 12. In the latter references 4swas the fraction of cells filled with monomers in any form rather than solvent, here it is replaced with 1 - xdv, which is the fraction of sites not occupied by solvent. In all the subsequent equations the appropriate substitutions have been made for con!istency. In addition there is a chemical potential difference A = (po - h V ) / ( k T )between monomer and solvent and a microscopic mixing energy t = K/(kT)that plays the same role in both models. The following resutls are obtained by applying the molecular field approximation to the dilute n 0 vector model: 4 = Y2qk(l- x,lv)2 + Y2qJm2+ In xWlv (2.2a)
K I= V2h2, K,,’= 3,
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e = Y2qm2
xmlV= (1 + exp(h
(2.2b)
+ qk(1 - xwlV))[l+ ‘/2(qSm + h)2]l-’ (2.2c)
m = ( q j m + h)x,l, exp(h
+ q&1
- x,lv))
m
I
(2.2d) where 4 is the thermodynamic potential natural to the independent variables 3, k,h, and h and where q is the coordination number of the underlying lattice.- Here, 4,e, x h , and in are considered to be functions of 3, k,A, and h. The above equations can be rearranged to give m 1 - x,lv = + f/zm(qjm+ h) (2.3a) qjm + h
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A=
- xWlv)+ In
(q3.I + h)X,l”
(2.3b)
The representation of xWlvas a function of m, 3, and h greatly simplifies the equation for criticality and phase equilibrium. For a fmed t, 3, and h it is then possible to find two-phase equilibrium
The Journal of Physical Chemistry, Vol. 96, No.23, 1992 9481
Phase Diagrams of Living Polymers for two values of m, say "and m': that satisfy B(J, k, h, m? = k,h, m")
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