Coalescence of Upper and Lower Miscibility Gaps in Systems with

J . Phys. Chem. 1992, 96, 4056-4068

4056

the requirement of protein saturation by the ligand (eq 4) is approximately fulfilled. In the cases of 2’-AMP and 2’-CMP, only the values calculated at the three highest and two highest ligand concentrations, respectively, were averaged to give the figures listed in Table IV. The percent saturation of the protein with 8 mM 2’-CMP a t tl12was only about 60%. The values for AHba t 55.1 O C calculated according to eq 5 (column 4, Table IV) agree well with the titration values calculated from 25 O C using the values for A G given in Table I1 (column

5 , Table IV) and thus strengthen these values for A G .

Acknowledgment. We thank Professor C. Nick Pace of Texas A&M University for the supplies of purified RNase T, used in this research. This work was funded by grants from the National Institutes of Health (GM-04725) and National Science Foundation (DMB-8810329). Registry No. RNase TI, 9026-12-4; 2’-GMP, 130-50-7; 2’-CMP, 85-94-9; 3’-GMP, 117-68-0; 5’-GMP, 85-32-5; 2’-AMP, 130-49-4.

Coalescence of Upper and Lower Miscibility Gaps in Systems with Concentration-Dependent Interactionst K. &IC* Michigan Molecular Institute, Midland, Michigan 48640

and R. Koningsveld Polymer Institute XI, Waldfeuchtstraat 13, 6132 HH Sittard, The Netherlands (Received: October 9, 1991)

The merging of upper and lower critical solution temperature miscibility gaps is investigated theoretically for binary systems whose interaction parameter g depends strongly on concentration. The process is called forth by increasing the chain lengths m, and m2 of both components while keeping their ratio mz/mlconstant. The principal mechanisms of merging, referred to as sideways coalescence, seem to be different for systems symmetric (mz = m , ) and asymmetric (mz = 2mJ in molecular size. However, both displayed pattern sequences should be representative of small-molecule mixtures as well as polymer blends. Literature data on miscibility g a p and heats of mixing in the system chlorinated polyethylenelpoly(methy1methacrylate) point to the relevance of the theoretical considerations. Furthermore, criteria for various landmark situations are derived and temperature ( r ) derivatives of the interaction function g(cp2,T). For instance, depending in terms of concentration (e) on g(e,r), the boundary between the LCST and UCST behavior in Tvs plane may be not only horizontal (as customary) but also vertical, or, in general, inclined. The binodal slope, dT/dq2, can be rigorously related to the spinodal function, and the result utilized for pinpointing the moment of sideways coalescence between the two gaps. Also, the condition is derived for the singular point marking the transition between two types of binodal patterns.

In a previous report1 we discussed the upper and lower miscibility gaps merging by coincidence of either critical points (CPs) or precipitation thresholds. These phase diagrams were generated for Flory-Huggins systems with a concentration-independent interaction parameter g. The coexistence of the upper and lower critical solution temperatures (UCST and LCST) was here strictly due to a minimum, located at T = To, on the temperature dependence of g. Such phase diagrams show a kind of symmetry around To: and the observed USCT and LCST regions have been traditionally associated with the sign of enthalpy of mixing AH which, at any concentration, should be positive for T < To and negative for T > To. Polydispersity of the dissolved polymer can make diagrams quite complex1 but does not destroy their quasisymmetrical form.2 In fact, the quasisymmetry survives even some types of concentration dependence of g as long as there exist continuous sets of temperatures T I and T2, T I < TO< T2,such that g ( e , T , ) = g(cp2,Tz). For instance, this will happen if g can be written as a sum or a product of concentration and temperature functions, g(cpz,T) de)+ gdT), or tdcp,?,T)= gkcp,?)gdT), and g,(T) displays a minimum at To. It is conceivable, however, that if the UCST and LCST miscibility gaps were positioned next to each other at two different concentrations, rather than strictly above each other at the same concentration, they could merge sideways by a different mechanism altogether. Interpreting the above sign rule locally, such sideways coalescence should then be expected, for instance, in systems where, at constant temperature, AH switches its sign with ‘Dedicated to Professor Marshall Fixman on the occasion of his 60th birthday.

changing concentration. While the rigorous general criterion for the binodal curvature at a critical point involves the sign of the second concentration derivative of the enthalpy of mixing, ( @ A H / ~ M rather ~ ) ~ than , ~ ~that ~ of AH itself, it is apparent that this fact does not invalidate the above qualitative argument. Such systems with exo- and endothermic behavior at different mixture compositions have indeed been observedS and could be modeled by a strong, at least quadratic, concentration dependence of the interaction parameter combined with appropriate temperature dependence. Note that the existence of two miscibility gaps of the same type (e.g., both of the UCST type) in binary systems also requires at least quadratic function g(~pz).~,’ This expectation is confirmed below by some computer-generated examples demonstrating sideways coalescence as a mechanism alternative to the traditional head-on coalescence. More specifically, it is shown that the binodal contact may occur in any (1) solc, K.; Stockmayer, W.H.; Lipson, J. E. G.; Koningsveld, R. In Conremporary Topics in Polymer Science; Culbertson, B. M., Ed.; Plenum Press: New York, 1989; Vol. 6, ‘Multiphase Macromolecular Systems”, p 5.

(2) The discounting qualifier should emphasize that the symmetry is not necessarily complete. One can identify the same pairs of coexisting phases at temperatures TI and T2below and above To;in general, however, TI and T2 are not symmetrically positioned around To,i.e., To - TI # T2- To. (3) Rehage, G. 2. Narurforsch. 1955, loa, 301; Discuss. Faraday SOC. 1970, No. 49, 176. (4) Kiepen, F. Ph.D. Thesis, Duisburg, 1988. ( 5 ) Holleman, Th. Physica 1963, 29, 585. (6) Koningsveld, R.; Kleintjens, L. A. Pure Appl. Chem., Macromol. .. Chem. j973.8, 197. (7) Solc, K.; Kleintjens, L. A,; Koningsveld, R. Macromolecules 1984, 17, 573.

0022-365419212096-4056%03.00/0 0 1992 American Chemical Society

The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 4057

Upper and Lower Miscibility Gaps of three possible ways: between two CPs (traditional e c contact), between a C P and a noncritical point ( e n ) , or between two noncritical points (n-n). Criteria are derived for various special cases in terms of concentration and temperature derivatives of the interaction function g. The present treatment is restricted to strictly binary systems where both polymers are uniform with respect to molecular structure and molecular weight. The interaction function g(cp2,T)is defined through the free energy of mixing per lattice site given as usual: AG/NRT = (cpi/mi) In PI + ( ~ / m z )In M + gViM (1) where cpi and mi are the volume fraction and relative chain length (number of lattice sites occupied by a molecule) of component i, N is the total number of lattice sites, and R and T a r e the gas constant and temperature. At phase equilibrium, the difference in chemical potentials between the two phases has to be zero for both components, Le. A(Apl/mlRT) a ml-lA(ln cpl)

where

s2,= 2[(mlcpI3)-' + (m2e3)-lI - mZ0 + 4g30(W - M) + g40cpIM (7b)

and the ith temperature derivative of the spinodal is SOi

E

-2goi + 2gli(cpi - M) + gZiVlM

(7c)

The numerator SzOis recognized as the 'stability" criterion for critical points (cf. eq 29 of ref 7): Szo= 0 for a heterogeneous double CP, Szo> 0 for CPs of the first kind as defined by Korteweg'O (stable and metastable), and SzO< 0 for CPs of the second kind (unstable). On the other hand, the heat of mixing is

+ (ml-' - m2-l)A(ez + A.(xIM~) =0

A(Ap2/m2RT) E m2-'A(ln 'p2)

+ (m2-] - ml-')Acpl +

which also implies that (cf. eq 7c)

Nx~vI') = 0 (2) where xI = g - glocpl,x2 = g + g 1 0 ~the r 8operator A stands for the difference between the two phases, and the doubly subscripted fij denotes a mixed partial derivative of g relative to concentration and temperature:

(3) 1. Criteria for UCST and LCST A simple way of deriving the condition determining the type of a critical point (CP) is by analyzing the spinodal. In binary systems, a C P is located either at a maximum (UCST) or a minimum (LCST) of the spinodal. Hence, its type is revealed by the sign of the second derivative d2T/de21sp,Cptaken along the spinodal at the CP. The spinodal equation for a system with concentration-dependent parameter g is8v9 1 S(R,T) I -

miQi

1 +- 2g + 2glo(cpI - M) + g20cp1M = 0 mzM (4)

The differentials d e and d T along the spinodal have to satisfy the equation

d S a SI0 d e

+ Sol d T = 0

where, as above for g in eq 3, the subscripts denote the partial derivatives. Thus the slope of the spinodal is dT/dMlsp = -Slo/Sol

(5)

and the CP is characterized by Slo= 0, i.e.'v9

Sl0p

1 -1 6gIo + 3g20(cp1- M) + g30'PI'h = 0

mlcp12

m2M2

(6) The second derivative is obtained by differentiating eq 5:

!?I

=

d e 2 SP

+ Soz[dT/d~l~p) - Soi(S20 + S ~ l [ d T / W s p )- SIO(SII (7) Sol2

For the C P (Slo= 0) it is simplified to (8) Koningsveld, R.; Staverman, A. J. J. Polym. Sci., Part A 2 1968.6, 325.

(9) Koningsveld, R. Ph.D. Thesis, Leiden, 1967.

The above equations confirm that the often quoted simple LCST/UCST criterion in terms of the sign of AH is applicable ifg is independent of concentration or if, at least, only the constant term of g depends on temperature, Le., gll = gZl= 0. Then, with the sign of the denominator Solof eq 7a being the same as that of AH (cf. eqs 7c and 8), and its numerator Szopositive, eq 7a indeed predicts a maximum for endothermic mixing, and a minimum for exothermic one, regardless of concentration. In the general case, however, the miscibility gap type is determined by the sign of the entire denominator Sol,i.e., by the sign of AHzo. Note that Rehage's conclusion3 of an LCST gap requiring AHH, > 0 (i.e., So, < 0), and a UCST gap calling for AH20< 0 (i.e., Sol> 0), is valid only for stable and metastable CPs; in the case of unstable CPs the reverse is true. 1.1. Boundary between the LCST and UCST Regions. The above relations also offer a general definition of the boundary between LCST and UCST (i.e., of the locus for c-c coalescence), and identification of its two principal causes. Evidently, the boundary is characterized by Sol= 0, i.e., AH20= 0, which may be realized in several different ways: (i) Traditionally, this boundary has been thought of as a particular critical temperature To separating UCST from LCST behavior (Le., a horizontal in a T vs (p2 diagram). From the definition of Sol(eq 7c), however, it is seen that this simple concept is valid only under the above-mentioned special conditions glI = gZl= 0, when the boundary To is strictly due to the minimum of the g(T) function at To where gol(To)= 0. This type of boundary normally does not occur in systems with monotonous temperature dependence of g. (ii) Another special instance involves systems with monotonous temperature dependence of g, where the boundary arises strictly due to g's appropriate concentration dependence. Classical examples are cases modeled in section 2 where g is given as a quadratic function: g = goo = go + gl'h + gzMZ having the concentration derivatives8

( 1Oa)

g10 = g1 + 2g2M g20 = 2gz (lob) where each coefficient gk is allowed to depend on Tin the usual manner: gk = sk + hk/T

(1h)

(10) Korteweg, D. J. Sitzungsber. Kais. Akad. Wiss., Math.-Natunviss. Klasse 1889, 98, 2 Abt. A, 1154.

solc and Koningsveld

4058 The Journal of Physical Chemistry, Vol. 96, No. 10, 1992

0.4

0.2

0.0

I .o

0.8

0.6 42

-

-

Figure 1. Example of a system with UCST/LCST boundary (- - -) increasing from 300 K for (p20 0 to 400 K for 'p20 1. Interaction parameter is defined in the text. Chain lengths for coalescing binodals (thick lines) are as follows: 1, m l N 1.4745, m 2 = 26.672; 2, m 1 = 6.1465, m2N 2.6006; 3, m lN 26.771, m2N 1.4819. For LCST/UCST and hourglass binodals (thin lines) they are as follows: 4, m, N 5.9796, m 2 = 2.53; 5, m l N 6.3814, m 2 = 2.7.

Surprisingly, here the temperature T can be factored out of eq 7c, and the condition Sol= 0 written in terms of cp2 only as

ho + h d 2 e - vI) + 3 h 2 * ( e - v1) = 0

(114

leading to the uncommon concept of a particular critical concentration cp,, as the boundary separating UCST and LCST, independent of T. Evidently, in a T vs diagram such a boundary would be displayed as a vertical line located at eowhich depends only on the ratios of his. If h2 = 0, the solution is simply 'p20 =

( h -~ h0)/3h1

otherwise it is (Mo)1,2 =

f

[(

;)2

+

-1 KO

(1lb)

- Kl

112

(llc)

where K~ = (h2- h0)/h2,K~ = (h2 - h l ) / h 2 ,with physical roots restricted to the interval 0 C ~0 C 1. More generally, this behavior should occur in all systems where g(cp2)is a power series whose coefficients follow an identical 'simple" function of temperature. Note that the result 11 would be the same if, e.&, the second term of (1Oc) were logarithmic, hk In T; however, gk cannot have both a reciprocal and a logarithmic term. (iii) In general case, both of the above causes may be involved and the boundary between UCST and LCST can be of any orientation. In analogy to eq 5 , its slope is dT/d'&lbnd

= -SIl/s02

(12a)

It is apparent that this equation describes correctly the previous special instances i and ii where the slope acquires values of 0 (Sl = 0) and (So, = 0), respectively. The general case is illustrated in Figure 1 for a system where g is linear in concentration, with the coefficients gk of eq loa containing also a logarithmic term (see Appendix): k = 0, 1 gk = S k + hk/T + CI; In T

-

-

For example, if the critical boundary line is desired to approach 300 K for cp20 0 and 400 K for cp20 1, then for arbitrarily selected values hl = 100 and c1 = 0.05, the condition Sol= 0 calls for ho = 820 and co = 2.45. Furthermore, if the depicted systems are to behave at composition limits as truly high polymers dissolved in the other monomer (i.e., ml 1 and m2 m for a0 0, and m2 1 and m I m for p20 I), the spinodal condition (4) determines the remaining unknowns as so N -16.168 797 and sI -0.579 719. With the interaction function g thus settled, the ml, m 2pairs yielding the coalescing patterns at various critical concentrations can be determined as usual from the spinodal and

-

-

--

-

-

critical conditions. Three such binodals are displayed in Figure 1 with critical concentrations selected as 0.2, 0.6, and 0.8. The m l , m2 values required for that are given in the legend. The boundary line computed from the condition Sol= 0 appears to be almost linear, with the largest deviation of AT 1.5 K found a t cpzO N 0.5. The agreement between independently computed binodals and the boundary line is excellent, attesting to the validity of employed relations. The middle coalescing binodal is complemented by binodals of two other systems with identical m2/ml ratios but increased or decreased mi values, to produce the familar hourglass and UCST/LCST patterns. Note that the switch in the sign of dZT/de21sp,CP, announced by the condition Sol= 0 in eq 7a, can also have another physical expression than the LCST/UCST coalescence; it may signify the instant where a closed-loop miscibility gap disappears and the system becomes miscible in the entire composition range. The two phenomena can be distinguished by the effect of perturbation in T and e,carried out a t To and eo"across" the boundary line, on the sign of Sol.From eq 12a, the slope of such perpendicular perturbation is

(6T/6e)' = S 0 2 / S l l

( 12b)

and its effect on Sol is 6Sol' = S116e'

+ So26T'

=

+ So22)6TL/S~2= (S1I2+ S022)~eL/S11 (12c)

(SlI2

[Two equivalent expressions on the right-hand side of eq 12c are given for convenience. Note that, e.g., for case i with the horizontal boundary line and SII = 0, the lattermost expression is unsuitable, which is understandable since in this case it is the temperature that has to be perturbed.] The outcome thus depends on the sign of So, and/or Sll. Say, for (meta)stable CPs (S20> 0) with So, negative, a positive perturbation 6T results in a negative Sol(eq 12c), i.e., in a positive d2T/dq221s,cp (eq 7a) producing an LCST gap, that points to an LCST/U&T coalescence pattern. By the same token, a positive So, signifies a closed-loop pattern. If the concentration-dependent terms of eq 7c can be disregarded (e.g., if g is independent of conantration), this criterion again reproduces the well-known rule: from eqs 7c and 8 we then have sign (So2)= s i g n (go,) = sign (aAH/aT). For instance, enthalpy of mixing dropping with growing temperature from positive to negative values makes So, C 0 and thus should generate an hourglass type diagram with coalescing LCST/UCST gaps, which it indeed does. Obviously in the special case ii with the vertical boundary, the relevant question becomes which type of gap (LCST or UCST) occupies a given concentration range. The answer is found by analyzing the lattermost expression in eq 12c, the term with variation in q2: For (meta)stable CPs, SllC 0 yields UCST at lower concentrations (cp < cpzO) and LCST at higher ones (cp > cp20), whereas a positive SI1 has an opposite effect. All of the above quoted results are reversed for unstable CPs with S20 C 0. In conclusion, it is interesting to note that the locus alone for LCST/UCST (Le., c,c type) coalescence can be defined in terms of the interaction function g(e,T) only, independently of ml and m2. It is not until an actual binodal, coalescing at certain concentration (or temperature), is computed that particular chain lengths have to be specified. 2. Sideways Coalescence As the title of this section suggests, its focus is on sideways coalescence (n-n and e n ) . One can see below, however, that this process is often accompanied by the head-on type coalescence (c-c) as well, even though it might not occur in the stable region. For demonstration, we choose for the interaction function g a quadratic concentration dependence and a reciprocal temperature dependence, as defined by eqs 10a and 1Oc. Thus, each gjl of eqs 7c and 8 is identical in form to gp of eq 10a,b, only with gk replaced by -hk/P. The function (1Oc) is known to have a limited applicability and represents only the most primitive approach to


Upper and Lower Miscibility Gaps TABLE I: Interaction Parameters Used for Producing Phase Diagrams in Figures 2 and 4 m Z / m 2= 2b m2/ml = 1‘ i Si hilK Si hilK 0 1 2

0.51887 -0.65955 0.26722

-63.378 126.76 0

0.17989 -0.55813 0.328 13

9.6964 136.33 -41.261

Figure 2. Figure 4. g(?31.899 However, it is sufficient for the present illustrative purposes. The parameters Xk of eq 2 are then given by relations XI = go - gl(cpI - cpz) - gzcpz(2cp1 - cpz) ( 1Oe) x2 = go + 2glcpz + 3 m 2 It is apparent that with these interactions, the boundary between LCST and UCST should be vertical (cf. eq 11). Phase diagrams can now be constructed with eqs 2-4,6, and 10, provided the values for various parameters are available. We choose a set of values that forces the equations to describe systems of present interest. With small-molecule systems ( m l N 1, m2 N 1) we would need the pressure as a variable to follow the merging process. With chain molecules we shall utilize varying chain length for this purpose. To fix the sk and hk values of eq lOc, we set the two CPs for ml = 10 at T = 350 K, cp2 = 0.75, and at T = 300 K , ‘p2 = 0.25. This selection, together with additional spinodal points (T; cp2) at (350; 0.13) and (300; 0.87), forces the system to exhibit both the UCST and LCST behavior for the two cases discussed below. With the two CPs, eqs 4 and 6 provide altogether four relations, while the two spinodal points substituted into eq 4 supply two more. The total set of six linear equations then fully determines six unknowns, S k and hk, k = 0, 1, 2. The calculation of binodals proceeds by a trial-and-error method as follows: For a selected concentration of one phase, p2,a value is tried for the conjugate phase, e*, and the temperatures T I and T2are computed from the relations for A(Accl) and A(Ap,) of eq 2, respectively. The process is repeated for a modified cp2* until T I is sufficiently close to T2,resulting in the equilibrium values of cpz* and T for the selected cpz. Another option is to employ for computation the equilibrium functions F, G, and B introduced in section 3. Both methods yield identical results. 2.1. Symmetric Systems ( m , = m2). First we investigate a sequence of systems where both chain lengths are equal ( m l = m2). The results are displayed in Figure 2. Perhaps it is not surprising that for this kind of symmetry, eq 1l b together with the parameter constants listed for m 2 / m l= 1 in Table I predicts the LCST/UCST coalescence in the middle at cpzO = Furthermore, (meta)stable LCST CPs should occupy the range cp2 < eo,whereas their counterparts, UCST CPs, should be limited to the range cpz > eo.This prediction is consistent with all binodals computed for this system. Figure 2h shows that for ml = 10 the above set of parameters generates a diagram with merged miscibility gaps, although the two spinodals are still separate. The CPs are both metastable, and the stable portion of the binodal does not have the usual hourglass shape but looks rather like a distorted tube. The shape of the phase diagram appears to be surprisingly sensitive to the perturbation of conditions. With all other parameters kept constant, a minute reduction of both chain lengths to m1 = 9.3 (Figure 2a) produces a trivial form with two separate gaps. However, the process of disengagement (or merging) is anything but trivial and will be described in detail below. As the chain lengths increase and the gaps come close to rubbing shoulders (at m , = 9.9 in Figure 2b), the AG(cp,,T) surface develops around two of its spinodal points two additional plaits located within the two plaits corresponding to the original LCST and UCST gaps already present. The projections of these new plaits on the plane of the T(cp2)phase diagram have the shape of two oppositely oriented crescents which, at this point, are not stable yet. The schematic Figure 2c with tie-lines clearly shows that the crescents are mutually conjugated. Note in particular

that neither of the two extrema of each crescent is a CP although they are located on the spinodal; rather each extremum is at equilibrium with a cusp point of the other crescent (see section 3.1. for a proof). Similar behavior has been observed in binary systems before’ and will be seen repeatedly in this work (see, e.g., Figure 2r). In fact, these relations even survive introduction of polydispersity: conjugation between a cusp and a spinodal point of the binodal has been also observed in isothermal ternary diagrams,” and the cusps of cloud-point curves for solutions of polydisperse polymers are known to be in equilibrium with extrema of shadow curves.12 The fact that tie-lines sketched in Figure 2c cross the narrow onephase range between the gaps (Figure 2b) is not as inconsistent as it might seem. The equilibria of Figure 2c are less stable than those of the two gaps and occur at a higher level of AG in an isothermal AG(‘p2) graph. Figure 3 illustrates this point for T = 339.3 K, the maximum of the right-hand crescent in Figure 2b marked by I. Figure 3 shows that the narrow onephase channel is really the more stable state. With the chain lengths growing from ml = 9.9 up, the system necessarily passes through a state where the gaps make a tangential contact (sharing not just a point but also the slope). At that moment the middle phase, represented by the point of contact, can be at equilibrium with either one of the two other phases on the outer sides of the touching binodals; this formally constitutes a nonvariant three-phase eq~i1ibrium.l~Note that the outer two phases also represent points of tangential contact between the stable binodals and the metastable parts of the crescentlike binodals approaching from inside. As the chain lengths further increase to m1 = 9.95, the two gaps penetrate each other (Figure 2e) and, simultaneously, the two new crescent binodals become partly stable as they protrude beyond the envelope of the original LCST and UCST gaps (arc M N in Figure 2d[A],e). The above single nonvariant three-phase equilibrium splits into two, at two different temperatures, separated by a regular two-phase region. Next comes the transformation of twin crescent binodals into the complex pattern shown as a dashed line in Figure 2g. The transition occurs for m l = m l t as each crescent binodal makes a contact also with the metastable part of the “other” original binodal, this time from the outside (see Figure 2e,f). It is a “point” contact between the two, Le., it is a singular point where the binodal slope dT/dcp21cPCabruptly switches from a south-north direction to an east-west. Hence, the slope here is not defined uniquely, rather only in the sense of two different limits for m1 mlt(+)and m1 mlt(-).This fact can be utilized for derivation of a criterion for this point, and for proving it has to be located on the spinodal originally separating the two binodals (see section 3.2). For instance, for symmetric systems discussed in this section, the left crescent contacts the left branch of the UCST gap for m I t N 9.9613 at p‘: 0.287 and T‘ N 303.1. This event is illustrated in Figure 2f displaying two types of curves: (1) the LCST gap (IC) and the spinodal (s) drawn for a single value of m1 = 9.961 (since neither IC nor s is too sensitive to changing m l ) , and (2) the relevant portions of other, fast changing, binodals plotted for several close values of m , . Note that out of the latter curves, only one binodal retains the original unaltered pattern of Figure 2e [the crescent (cr) and the UCST (uc) portions of the binodal for m I = 9.9610 in Figure 2fl. A very slight increase in

-

-

(11) Tompa, H. Trans. Faraday SOC.1949, 1142. (12) &IC, K. Macromolecules 1977, 10, 1101. ( 1 3) It should be noted that this three-phase equilibrium is fundamentally different from the one commonly encountered around heterogeneous double critical points in ternary and higher systems where the separation is due to the greatly differing chain lengths.1’.12In the latter case, all three phases can actually coexist (and have been indeed visually observed as three layers) in an entire range of concentrations and temperatures. On the other hand, in the present case all three phases can never physically coexist; even if Tcould be accurately adjusted to the value characterizing the point of contact between the two gaps, one could only bring to equilibrium either the left pair or the right pair of phases, depending on the mixture composition, but never both pairs (Le., all three phases) at the same time. See also: Koningsveld, R.; Stockmayer, W. H.; Nies, E. Polymer Phase Diagrams; Oxford University Press, in press.

4060 The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 rfi

Solc and Koningsveld T/K

m, = 9.3

m,

0.95

360

320

280 0.5

0.5

1

m ,= 9.9

360

320

280

1

0.5

.e--.,

I!

\. 1

\I

I

PI

[AI

d

[CI

Figure 2. Phase diagrams calculated with the set of parameters listed in Table I for m 2 = m , ( m ,values indicated in the figures). Critical points, 0;stable bincdals, -; unstable and metastable binodals, - - -; spinodals, tie lines, 0 ; three-phase lines, A. Parts c, d, and f' are schematic; part f is a magnified detail. - a * ;

Upper and Lower Miscibility Gaps


5

1

m , = 10.6

360

320

280 I

0.5

Figure 2.

1

The Journal of Physical Chemistry, Vol. 96, No. 10, 1992

4062

,

AGINRT

Figure 3. Concentration profile of free energy of mixing for M , = m2 = 9.9 at T = 339.3 K. Arrows indicate conjugated stable phases. I is the spinodal point (also displayed in Figure 2b), conjugated with the cusp.

m , makes the cr and uc binodals coalesce at the singular point S, and change fundamentally the pattern afterward, producing new minima and maxima spreading from S along the spinodal s (pairs of curves d, e, and f for m , = 9.9614,9.9615, and 9.9620, respectively). Furthermore, by necessity, a contact has also to occur between the curves conjugated with the above two. It is a tangential contact at et* 0.9264 between the inner right arc of the right crescent and the right branch of the UCST gap (Figure 2d[B]), which upon further growth of m l breaks into a pair of cusps (Figur 2d[C]). Both events change dramatically the continuity of binodals, and the new pattern makes possible a gracious exit of threephase states from the scene: The protruding portion (p in Figure 20, originally for m , < m l t belonging to the crescentlike binodal (cr), now continuously ties via a pair of extrema (I and A on the marked branches of d-f) into the descending left-hand branch of the UCST gap (uc). With further growth in m1 both of these extrema ‘ride down” the spinodal and merge into a point of inflection as they reach the LCST critical point LC (see also Figure 2g for m1 = 9.964603). It is apparent that at LC (qZcN 0.24965, T, N 300.8993) also the left two phases of the three-phase equilibria pictured in Figure 2f (A)become identical, and the three-phase equilibrium is thus reduced to a regular two-phase one. Simultaneously, the associated pair of cusps on the right-hand branch of the UCST binodal is reduced to zero size, and the dicontinuity in its slope consequently disappears. Thus LC is a critical end point. Analogous metamorphosis happens with the right crescent, resulting in the symmetric form of Figure 2g. The phase coexistence in the region of nonvariant threephase equilibria of Figure 2f is shown in detail in the schematic Figure 2f’. The sigmoidal binodal with a minimum and a maximum on the left side is associated with three branches, tied together by two cusps, on the right. Further growth in ml leaves both the LCST and UCST CPs metastable (Figure 2h for m , = IO), and the experimentally observable phase diagram assumes very simple shape of a distorted tube. This is a good example, however, of how deceiving can be the innocent looks of the stable envelope, in fact hiding inside a ballet of metastable and unstable binodals with rich past (see above) as well as future (see below). A hint of the latter is also raised by the above prediction of an LCSTIUCST boundary at cpz = 0.5 and, particularly, of a c-c coalescence that should occur for m , N 10.5767. This indeed happens: With m , growing above 10, the upper and lower sigmoidal critical portions move apart while the left and right crescentlike branches move together, creating a narrow neck (Figure 2i for m , = 10.575) which eventually collapses for m1 N 10.5767 into another singular point contact at cpZc = 0.5 (LUC’ in Figure 2j). Figure 2j represents again a transition state between two distinct binodal and spinodal patterns. Upon closer look LUC’ is rec-

solc and Koningsveld ognized as the above predicted LCSTIUCST homogeneous double critical point (c-c coalescence) which, with another increase in m , , splits “vertically” into a pair, UC‘ and LC’, of unstable CPs (of the second kindlo) marking extrema of newly developed binodals. As shown in Figure 2k for m1 = 10.6, each of the new CPs occupies together with one of the original metastable LCST/UCST CPs a double-sigmoidal self-contained binodal. Such bisigmoids are well-known from other ~ t u d i e s .Typically, ~ they are transient forms that, with suitably changing conditions, keep shrinking until they disappear in a heterogeneous double CP located at the point of inflexion of the spinodal. This happens in the present case, too, specifically for ml N 11.805 at cpz N 0.30612, T 265.02, and at cpz N 0.69388, T N 413.63; at these two points, S, SI,, and S20 of eqs 4, 6, and 7b assume zero values. Beyond m l = 11.805, the phase diagram has no internal structure. It consists merely of a distorted tubelike binodal and a similarly shaped spinodal with no critical points (Figure 21 for ml = 12). The principles of these peculiarities have long ago been laid down by Gibbs,I4 van der Waals and Kohnstaqm,15 and SchreinemakersI6and have recently been treated by Solc et al.7 and by Feix et al.I7 The AG expression used here obeys classical rules, as it should. These rules require, inter alia, that a horizontal three-phase line in a binary isobaric T(cpz)plot is bordered by one twephase region on one side and two two-phase ranges on the other side.13 The intermediate state of “touching shoulders” seems to be in contradiction in this respect. However, we have a limiting case in hand in which the two-phase range between the two three-phase lines in Figure 2e has just vanished into a single line. 2.2. Asymmetric Systems ( m 2= 2m1).We turn now to the practically more probable case of a system asymmetric in chain lengths. We set m2 = 2ml and derive the parameter values listed in Table I in the same manner as before, basing them on the system m, = 10, m2 = 20. Note that the LCST/UCST boundary here stays vertical, but is shifted to ‘pz0N 0.2722 (cf. eq 1 IC). Again, for ml = 10 we have overshot our target in that the selected parameter values already produce merged gaps (Figure 4c), and one has to go down to m 1 / m 2= 9.5119 to split the single gap into two (Figure 4a). Here the merging process seems to differ from the previous symmetric case. Within the accuracy of the displayed plots, the prospective point of contact on the LCST gap is a CP which, upon approaching the UCST binodal, induces in the latter a marked shoulder (Figure 4a). Evidently, the point of contact is a critical end point constituting the beginning of the three-phase region.13 The UCST binodal, on the other hand, develops here a horizontally oriented point of inflexion (Figure 4b for m I = 9-82), that later decomposes into two noncritical extrema (Figure 4c for m , = 10) coexisting with a pair of cusps grown on its right branch. Recall that similar behavior has been observed in the past on UCST gaps for other binary systems.‘ Also, note the similarity with the Figure 2f’. In any case, this process should be classified as c-n type. During further increase of the chain lengths, the patterns of the spinodal and binodal change drastically. The shoulder on the UCST spinodal also develops a horizontal point of inflexion; this time, however, (i) it happens before the LCST CP contacts the UCST spinodal, and (ii) the point of inflexion is critical. It is in fact a heterogeneous double CP with coordinates ‘p2 N 0.29346, T = 295.025, and m l 10.01 18, as determined from the condition S = Slo= SzO= 0 (eqs 4, 6 , and 7b). This double C P further splits into metastable and unstable critical points positioned at the maximum and minimum of the spinodal and of a new bisigmoidal binodal, respectively (Figure 4d for m , = 10.0135; (14) Gibbs, J. W. Collected Works; Dover Reprint, 1961. (15) van der Waals, J. D.; Kohnstamm, Ph. Lehrbuch der Thermodynamik; Leipzig, 1912. (16) Schreinemakers, F. A. H. In Die hererogenen Gleichgewichre uom Standpunkre der Phasenlehre; Bakhuis Roozeboom, H. W., Ed.; Braunschweig, 1913. (17) Feix, G. Z. Phys. Chem. 1983, 264, 369. Feix, G.; Muller, E.; Bittrich, H.-J. Wiss. Z. TH Merseb. 1987, 29, 424.

Upper and Lower Miscibility Gaps cf. also Figure 8 of ref 7). As m, is increased to 10.01365 (Figure 4e), the upper metastable C P merges with the CP of the LCST binodal at cp2 N 0.2722 (as predicted above for c-c coalescence), forming a homogeneous double CP. Simultaneously, one of the spinodal branches becomes vertical as required under present circumstances: Because of the chosen temperature dependence for g (cf. eq lOc), the condition for c-c coalescence by two CPs, So, = 0, makes the temperature-dependent terms of the spinodal, eq 4, cancel each other. Hence, a t the 'p2 root of the remaining equation 1 1 S(M) = - -~ I Q I mzM 2[s0 + ~ ~ ( -2d~ +2 ~ ~ Z M-( dM l = 0 ( 4 4

+

the spinodal condition is satisfied independent of temperature. Note that the same phenomenon appeared in Figure 2j. As expected, further growth of m, transforms the spinodal and binodal around the singular homogeneous double C P to an hourglass form which, together with the cusps below, produces a bootlike binodal for m, = 10.0137 (Figure 4 0 . Its "heel" gets higher and higher (Figure 4g-h) until the maximum of the "sole" (which is a noncritical point) makes a tangential n-n contact with the noncritical sigmoidal part of the UCST binodal. [Remarkable on this state is that the contact is being made by maxima of the two curves, yet it is of n-n type since the maxima are noncritical, each conjugated with a distant cusp.] Simultaneously, the "toe" of the boot has extended and connected to the upper cusp of the cusp pair on the right UCST branch, first noticeable in Figure 4g (although present since Figure 4c). All this occurs just beyond m, = 10.3 and constitutes a topologically symmetrical situation for inner binodals. A slight further increase of ml reshapes the pattern again: The left noncritical maximum breaks up, sealing off the left cusp pair, while the contact between the two cusps on the right-hand side "thickens" to generate a new protruding bisigmoid (Figure 4i-k) oriented oppositely to the original left one in Figure 4d. In the process, the left gap above the three-phase line keeps growing while the right one keeps shrinking until, at m1 = 15, it disappears in a critical end point (Figure 41). The final form of the stable binodal is again very simple: a somewhat deformed tree-trunk type diagram (see Figure 4m for m, = 17.5). Unlike in the symmetric case, however, here the internal bisigmoid with a metastable and an unstable CP stays and never disappears, as is apparent from the lack of solutions with ml > 17.5 for the set of equations S = Slo= Sz0 = 0. 2.3. Summary. In both cases examined above, merging of UCST and LCST miscibility gaps appears to be a much more complex process than one would expect from deceptively simple stable binodal envelopes. The internal structures include crescentand bisigmoid-shaped binodals with pairs of c u s p and double CPs of heterogeneous as well as homogeneous type. The lattermost point represents in fact a c-c contact, i.e., a head-on coalescence, between two metastable or two unstable binodals that seems to be always accompanying the more obvious sideways coalescence of the external stable parts. Commonly present features are also nonvariant three-phase equilibria and critical end points, although their development for the two above cases differs in detail: In the symmetric system, after being generated by a sideways noncritical n-n contact for 9.9 < m, < 9.95, the three-phase equilibria appear in pairs a t two different temperatures and vanish correspondingly via a pair of critical end points for m , N 9.9646. In the m2 = 2ml case, on the other hand, there is only a single three-phase equilibrium for each m,, sandwiched between two critical end points at m , N 9.82 and m, N 15. Figure 5 displays the concentration dependence of the heat of mixing, calculated with eq 8 for the two cases discussed above. We note that our model involves a AH independent of temperature and chain length, and we have a unique curve for each case. This is an obvious limitation of the model that will have to be amended to fit actual systems. For the present it suffices to observe that the sign of AHzo,which is determinative of lower or upper critical miscibility, varies with p2in accordance with the earlier considerations. Particularly instructive is the asymmetric system where

The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 4063 the heat of mixing stays positive in the entire composition range, yet it shows UCST/LCST pattern. At present it is not clear how general is the behavior displayed by the above two examined systems, i.e., how is it modified by choosing different ratios m2/ml and/or different interaction functions. However, it provides a picture that is not inconsistent with other observations. For instance, the much wider ml interval of three-phase equilibria observed for the asymmetric case, compared to the m, = m2 system, points to the molecular-size disparity as being conducive to three-phase separation. But the same conclusion has been drawn many years ago for ternary systems solvent-P(ml)-P(m2) with simple interactions, where the threephase separation demands that m2/ml ratio be at least lo.]' Reports of shoulders and two extrema on miscibility gaps for polymer blends are not rare, and in our opinion they are a sure observable sign of complicated stability conditions within the phase boundary. It would be desirable to know the conditions for the appearance of various phase diagram patterns displayed in this report. To this end we present in the next section an analysis for some of the situations encountered. 3. Criteria for Some Special Situations Following the procedures listed in ref 18, one can write for a bulk binary system the two phase-equilibrium functions generally as

= Ul(cpl + VI*) + Uz(M + M*) +

F(M9Ulr~2977

2(ml-' - mz-')AM + (VIM* + m*dAg(glo(PIM + glo*PI*M*)AM = 0 (13a)

=

G((PZP1,~2,77 0 2

-

UI

+A M n

- MI1

+ A(BI0cpIM) = 0 (1 3b)

where A stands for the difference between the incipient (*) and principal ( ) phases [e.g., A(gp2) = g*cpz* - gcp2], uk is the separation factor for the component k defined by the relation pk* = pk exp(ukmk)

(14)

and gijare the partial concentration and temperature derivatives of the interaction parameter g, eq 3. For reasons of symmetry (appreciated when calculating the derivatives) we employ both separation factors u1 and u2 as separate variables, although they are tied together by the identity B(ch,U,,Uz)

= PI* + e*- 1 = 0

(15)

With cp, = 1 - cpz and the relation 14 used for both incipient concentrations vk*, it is apparent that F, G, and B have variables as indicated. 3.1. Cusp Systems. A cusp point on the cloud-point curve (CPC) is characterized by the condition

where the derivatives are taken along the CPC. Figuratively speaking, eq 16 expresses the fact that a walker, moving along the CPC in time (uk)in a two-dimensional plane (T,cp2),has to stop at a cusp point, turn around, and start walking again in the opposite direction, while the time keeps flowing. Typically, cusp points appear in pairs referred to as cusp systems. Thanks to eq 16, for cusp points the T and terms in the total differentials of equilibrium functions can be dropped, so that, e.g. d F = F,, dol + Fn2du2 = 0 [(P~,TJ dB = Bo, dal

+ B,, do2 = 0

[a]

(17)

where the indexed functions denote partial derivatives, Le., B,, ( L U ~ / ~ U , )etc. , , ~ ,By ~ , combining the two equations in (17) and subsequent rearranging with the aid of relations (1 3)-( 15), the resulting cusp criterion is simply (18) solc,

K. Macromolecules

1986, 19, 1166.

4064


Solc and Koningsveld m , = 10.01365

4

a

I

0.5

296

f

02

.3

.25

m,

10.15

0.5

B

07

.5

Id

,

):,

Wi 3

Figure 4. Same as Figure 2 except for m 2 = 2 4 . Parts d-f show only magnified relevant portions of full-scale diagrams.

i

Upper and Lower Miscibility Gaps


.5

.5

1

1

.5

1

4

0.5

Figure 4. Continued

0.5

1

0.5

1

1


4066 1 -+--

miQi*

1 m2%*

2g*

+

2g10*(cpl*- e*) g20*(PI*M*= +

(18) which is recognized (cf. eq 4) as the spinodal function S((p2*,T) written, however, for the incipient phase. The composition coexisting with the principal-phase CUSP thus has to belong also to the spinodal, i.e., it has to be a point of intersection of the binodal with the spinodal, as shown many Years ago by Tompa for ternary systems with simple interactions." In fact, binodal cusps conjugated with the spinodal led van der Waals'' to coin the term "spinodal", the Latin word 'spinus" meaning thorn (Or CUSP). Moreover, since the CUSP is approached by two binodal arcs with identical slopes dT/dv, heading either both UP or both down, the incipient point has to be located at an extremum of the conjugate binodal as shown in Figure 2b,c and others. The statement may have even more significance when turned around as a warning: Not every point am"mn to the SPinodal and bhodal is necessarily a critical point, even if it is located a t a local temperature extremum of the binodal. It is a CP only if the two curves have a common slope. 3.2. slope of the ~Oud-PointCWe. Sideways ~ A e s c e n c eof two gaps in the ml = m2 case Occurs via two off-critical contact points, with the slopes of both binodals becoming identical (see section 2.1.). Accurate pinpointing of such a situation by inspecting numerical data on binodals is difficult. It would be desirable to have a well-defined criterion for it, based on the equality of both values and slopes for both binodals. The slope of the CPC is best obtained from the total differentials of the functions F, G, and B: d F 5 F,, dal F,, da2 + Fa d e + FT d T = 0

+

dG

G,, dal +

dB

5

s,

B,, dol

da2 + G, de + cTd T = 0

+ B,, daz + B,

(19)

d e =0

By eliminating da, and da2, one gets

FJBG - GJBF + B,YGF GTYBF- FTYBG

(20)

solc and Koningsveld 4b). The distinction between a critical and noncritical binodal can be made by perturbing the system (for instance, by changing r l ) : the latter one separates from the CP, whereas the former one retains it (cf. Figure 4c). (ii) If the spinodal slope is not zero, the binodal displays at the common point an extremum equilibrated with a distant cusp (cf. crescents in Figure 2b). (b) In rare cases, the denominator of eq 21a becomes zero at the common point of spinodal and binodal, making the binodal slope expression indeterminate, of the 0 / 0 type. Such event announces a drastic change in the binodal pattern, such as OCcurring, e.g., at the singular point S of Figure 2f. This statement has been confirmed numerically: at the point S the variables indeed satisfy, in addition to eqs 2 and 4, also the condition %(go1 - g 1 1 d A e + (p2*A(gol(PI)= 0

(21b)

(2) Conversely, one can say that for coalescence via a C P (whether the c-c or c-n type), the correct horizontal slope is guaranteed by eq 21a. (3) For n-n type coalescence, the equality of the s l o p for two binodals at their point of contact e , T (which can be equilibrated with either of the two phases A and B) demands that the righthand sides of eq 21a, written for A and for B, are equal. After dividing both sides by S(v2,T),the condition can be simplified to (22) (4) For systems with "simple" interactions (where only godepends on temperature), the partial derivative golis independent of concentration, and eq 22 is reduced to the trivial form (c2,** = v2.B'. This result indicates that here the n-n coalescence is not possible, leaving c-n and c--c types as the only acceptable options. Similarly, eq 21b shows that for these systems, the singularity discussed in (1 b) above can appear only at a CP, specifically at the c--c coalescence where the LCST/UCST pattern switches to the hourglass type. The binodal slope relation 21a remains legitimate; in fact it can be simplified under these conditions to

where

WC) YBF= B,,F,, - B,,F,,,

etc.

After substituting the derivatives and making some rearrangements, it turns out that every one of the five products on the right-hand side of eq 20 contains as a multiplier the left-hand side of eq 18, the spinodal function S(**,T) evaluated at the incipient concentration. Thus the result is greatly simplified if both the numerator and denominator of eq 20 are divided by this common factor, producing the general relation S(e9T)Ae

e(g01 - gllCp1)Ae + e*A(g01(PJ

(2 a)

Here, as before, Acp, is the concentration difference between the two equilibrium phases (A* = e*- cpz), and goland gll are the partial derivatives of the interaction parameter (cf. eq 3). Particularly surprising of this result is the simple relation between the CPC's slope and the spinodal function S (this time evaluated a t the binodal point of interest). Several remarks are called for here: At any intersection with the spinodal (where s = 01, the numerator of the slope relation 21a is zero. There are two possible consequences of this fact: (a) Normally, the denominator of eq 21a is nor zero, and the binodal has to have here a horizontal slope. (i) If the spinodal slope ( 5 ) is zero as well, the common point is a critical point. Usually the binodal passing through it is a critical binodal (in fact, each C P has to have at least one such binodal) in which m e it displays here an extremum (cf. the LCST binodal in Figure 4b). However, it might also be a binodal which displays a point of inflexion, equilibrated with a distant budding pair of cusps (cf. the UCST binodal in Figure

The formula 21c is consistent with the result reported for dx/d(p2 previously (eq 8 of ref 19), but the present form is more attractive for its simplicity. ( 5 ) Note that, in the sense of a limit, eq 21a yields the correct answer even at notoriously ill-behaving cusps [where S(p2*,T) = 0, making the original expression 20 of the indeterminate 0/0 type]. The above statements are consistent with all of our numerical calculations carried out so far. 4. Discussion Though the calculations reported above were carried out for two rather Of oligomeric systems* we may expect the results to have a wider applicability. Multiplying both sides of

eq

by

m19

we

Obtain

m , A G / N R T = cpI In cpI

+ (*/m)

In

+ (gml)cpl*

(23)

where m = m 2 / m l .Since the scaling of AG by ml has no influence on the present results, we note that it is the asymmetry of the system* mv that governs its properties* the value of ml which may either be representative for a mixture Of l ) Or a polymer The ( m 1; m l appropriate value of the interaction parameter is gml, not g, and must be adapted to the system in hand* Thus, we may the present considerationsto have a general validity* The concentration dependence of g has to be quite strong to that obviously produce the the sideways malescence of miscibility gaps. It has been indicated that molecular (19) solc, K. J . Polym. Sei., Polym. Phys. Ed. 1974, 12, 1865.

-. .. ..., upper ana Lower MisciDiiiry craps .I

.I.

n


~

Figure 7. Heat of mixing measured for the system chlorinated alkane/oligomeric poly(methy1 methacrylate)23showing the sigmoid behavior going with U C and LC demixing. The two curves refer to alkane samples differing in chlorine content.

0.5

I

Figure 5. Heat of mixing, LW,calculated with eq 8 for mz = m, (top) and m2 = 2ml (bottom).

polymers may be treated as copolymers2sand that small changes in chemical composition affect thermodynamic stability in a sensitive manner similar to that of changes in chain length.26 A more direct piece of evidence is provided by measurements of heats of mixing of chlorinated paraffins with an oligomeric poly(methy1 m e t h a ~ r y l a t e ) . ~The ~ results of Walsh et al. are presented in Figure 7 and show the sigmoidal concentration dependence consistent with the simultaneous Occurrence of LC and UC demixing in one system. A factor detracting from the already meager experimental support is the inhomogeneity of Walsh et al.’s samples with respect to chain length and chemical composition, not having been included in the present analysis. Both aspects are subject of current study.

Acknowledgment. Partial support of one of us (K.S.) by the National Science Foundation, Grant No. DMR-8808294,is gratefully acknowledged.

Appendix: Temperature Dependence of the Interaction Parameter Numerical studies such as the present one usually assume the temperature dependence of the interaction parameter as

100

g = s + h/T

.s Figure 6. Cloud-point curves for the system chlorinated polyethylene/ poly(methy1 methacrylate)22showing a trend to sideways coalescence;w 2 = weight fraction of PMMA.

association may lead to expressions for g that produce, inter alia, bimodal miscibility gaps20or distorted “tubes”.21 The analytical form of g(*) then deviates from the simple series used here but, formally, results in a strong dependence of g on (p2. We are not aware of any direct experimental support for these theoretical results in the literature. Work by Walsh et a1.22v23 on mixtures of chlorinated polyethylene and poly(methy1 methacrylate) seems to provide some indications, however. Figure 6 shows LCST and UCST cloud-point curves, the shape and location of which could provide a case in point for sideways coalescence. Indeed, Walsh et al. found a merged gap if the chlorinated polyethylene was replaced by a sample with a different chlorine content. Another experimental indication is contained in work by MacKnight et al.24 It is known that chemically modified (20) Koningsveld, R. Paper presented at the IUPAC Symposium on Macromolecules, Montreal, 1990. (21) Coleman, M. M.; Graf, J. F.; Painter, P. C. Specific Inreructions arid the Miscibiliry of Polymer Blends; Technomic Publishing Co.: Lancaster, 1991. (22) Walsh, D. J.; Shi Lainghe; Chai Zhikuan Polymer 1981, 22, 1005. (23) Walsh, D. J.; Higgins, J. C.; Chai Zhikuan Polymer 1982, 23, 336. (24) Cong, G.; Huang, Y.; MacKnight, W. J.; Karasz, F. E. Mucromolecules 1986, 19, 2765.

(‘41)

where s and h are constants standing for entropic and enthalpic contributions to g. The form of eq A1 has been suggested long ago by a large number of experimental investigation^.^' It has also been known for a long time that the g(T) dependence expressed in (Al) does not suffice in the description of LC and UC miscibility behavior in the same system. An early theoretical study on the subject, based on Prigogine’s model of the liquid state, indicated that the addition of a linear term in T should be expected to be relevant in general.28 One might alternatively use a classic, partly empirical, argument to show that eq A1 cannot be anything but a first appro~imation.8,~ Remembering the Gibbs-Helmholtz equation for the heat of mixing:

we note that the assumption A1 introduced into eq 1 involves

AH/NR = h q l e

(A3)

the usual Van Laar-Bragg/Williams result, independent of T. If next we derive Acp, the change in heat capacity upon mixing, we obtain AC,

=(au/aq,,

=o

(A41

(25) Koningsveld, R.; MacKnight, W. J. Mukromol. Chem. 1989, 190, 419. (26) Koningsveld, R.; Kleintjens, L. A.; Schoffeleers,H. M. Pure Appl. Chem. 1974, 39, 1. (27) Rehage, G. Kunsrsroffe 1963, 53, 605. (28) Delmas, G.; Patterson, D.; Somcynski,T. J . Polym. Sci. 1%2,57, 59.

J. Phys. Chem. 1992, 96, 4068-4074

4068

a result that will usually not conform to the experimental evidence. Adding a linear as well as a logarithmic term to g g = s + h/t bT + c In T (~5)

+

we have

AcP = -~1(~2(26T + C)

('46) Thus we see that a linear dependence of Acp on T directly leads

to expression A5 for g. In practice such a Ac,( 7") behavior will represent a first approximation only, but it serves the present qualitative purpose. The g(T) function IOd,used in the text for illustration, thus involves b = 0 and makes Acp nonzero but independent of T . Registry NO. PMMA, 9011-14-7.

Dielectric Friction and Solvation Dynamics: A Molecular Dynamics Study Margaret Bruehlt and James T. Hynes* Department of Chemistry and Biochemistry, University of Colorado, Boulder, Colorado 80309-021 5 (Received: October 17, 1991)

The van der Zwan-Hynes relation connecting the solvation time and the dielectric friction for a solute in a polar solvent is tested via molecular dynamics computer simulation. For partially and fully ionic diatomic solute pairs in a model polar aprotic solvent, for which there is considerable dielectric friction, the relation is found to be satisfied to within a factor of 2 and to correctly follow the trends observed for different solute pairs. Deficiencies of the relation are also discussed. Pronounced solute rotational caging associated with strong electrostatic solutesolvent interactions is also observed.

1. Introduction

The concept of the dielectric friction' on a translating ion or a rotating dipole immersed in a polar solvent has a long history, beginning with the work of Born in 1920.* For example, the time lag of the surrounding polar solvent electric polarization for a steadily rotating dipole leads,3 in a simple dielectric continuum description, to a dielectric friction constant proportional to the longitudinal dielectric relaxation time4 of the solvent. Considerable effort has been made to experimentally test this and related predictions in the context of molecular reorientation An independent measure of the dynamics of a polar solvent in the presence of dipolar (and/or charged) solutes is the so-called 'solvation time" T ~available , from time-dependent-fluorescence (TDF) measurements.* For example, in a Franck-Condon electronic transition in which there is an excited-state dipole moment jie different from the ground-state value iig, there will be subsequent solvent relaxation from the equilibrium appropriate to ii,to that appropriate to jie; this relaxation can be related to the time evolution of the TDF Stokes shift.8-'0 A convenient molecular measure of the solvation time scale is the time area of the normalized equilibrium time correlation function (tcf)

should approximately hold, connecting the solvation time to the ratio of the dielectric friction constant {d to the initial time value

(1) For reviews see: Wolynes, P. G. Annu. Reu. Phys. Chem. 1980, 31, 345; Madden, P. A.; Kivelson, D. Adv. Chem. Phys. 1984, 56, 479. (2) Born, M. Z . Phys. 1920, 1 , 221. (3) Nee, T.; Zwanzig, R. J . Chem. Phys. 1970,52,6353. Hubbard, J. B.; Wolynes, P. G. J. Chem. Phys. 1978, 69, 998. (4) See,e.g.: Hubbard, J.; Onsager, J. J . Chem. Phys. 1977, 67, 4850; Hubbard, J. B. J . Chem. Phys. 1978, 68, 1649. ( 5 ) (a) Templeton, E. F. G.; Kenney-Wallace, G. A. J. Phys. Chem. 1986, 90, 2896, 5441. (b) Blanchard, G. J. J . Phys. Chem. 1988, 92, 6303. (c) Kivelson, D.; Spears, K. G. J. Phys. Chem. 1985, 89, 1999. (d) Phillips, L. A.; Webb, S.P.; Yeh, S. W.; Clark, J. H. J . Chem. Phys. 1985,89, 17. (e) Alavi, D. S.; Waldeck, D. H. J . Phys. Chem. 1991,95,4848. ( f ) Nakahara, M.; Ibuki, K. J . Chem. Phys. 1986,85,4654. (g) Ben-Amotz, D.; Scott, T. W. J. Chem. Phys. 1987,87, 3739. (6) Simon, J. D.; Thompson, P. A. J . Chem. Phys. 1990, 92,2891. (7) Alavi, D. S.; Hartman, R. S.; Waldeck, D. H. J . Chem. Phys. 1991, 94,4509. (8) For r a n t reviews, see, e.g.: (a) Barbara, P. F.; Jarzeba, W. Ado. Photochem. 1990, 15, 1. (b) Barbara, P. F.; Kang, T. J.; Jarzeba, W.; Fonseca, T. In Perspectives in Photosynthesis; Jortner, J., Pullman, B., Eds.; Kluwer: Dordrecht, 1990; p 273. (c) Simon, J. D. Acc. Chem. Res. 1988, 21, 128. (d) Kosower, E. M.; Huppert, D. Annu. Rev. Phys. Chem. 1986,37, 127. (e) Bagchi, B. Annu. Reu. Phys. Chem. 1989,40, 115. ( f ) Maroncelli, M.; MacInnis, J.; Fleming, G. R. Science 1989, 243, 1674. (g) Maroncelli, M. J . Mol. 139. 1991, in press. (9) Bagchi, B.; Oxtoby, D. W.; Fleming, G. R. Chem. Phys. 1984,86,257. (10) van der Zwan, G.; Hynes, J. T. J. Phys. Chem. 1985,89,4181. Note that the friction defined in this reference is equal to the product of the friction Here 6 A E is the fluctuation in the difference, for fixed solvent and the moment of inertia I in the present work. molecular coordinates, of the solute-solvent interaction energy (11) (a) Carter, E. A.; Hynes, J. T. J . Chem. Phys. 1991, 94, 5961. (b) for the excited and ground dipole The molecular See also: Hynes, J. T.; Carter, E. A.; Ciccotti, G.; Kim, H. J.; Zichi, D. A.; formula eq 1.1 can be identified with the experimental solvation Ferrario, M.; Kapral, R. In Perspecriues in Photosynthesis; Jortner, J., Pullman, B., Eds.; Kluwer: Dordrecht, 1990; p 133. time T~ mentioned above, upon the assumption of linear response: (12) Maroncelli, M. J . Chem. Phys. 1991, 94, 2084. T~ = T ~ There ~ can 'be some ~ deviations from a strict validity (13) Maroncelli, M.; Fleming, G. R. J. Chem. Phys. 1988, 89, 5044. of this a s s u m p t i ~ n , l ' - ~but ~ * 'these ~ will not directly concern us (14) Bader, J. S.; Chandler, D. Chem. Phys. Lett. 1989,157, 501. Levy, here. R. M.; Kitchen, D. B.; Blair, J. T.; Krogh-Jespersen, K. J . Phys. Chem. 1990, 94, 4470. These two solvent dynamical measures have been approximately (15) Fonseca, T.; Ladanyi, B. M. J . Phys. Chem. 1991, 95, 2116. related by van der Zwan and Hynes (vdZ-H),I0 in an effort (16) (a) van der Zwan, G.; Hynes, J. T. J . Chem. Phys. 1982, 76, 2993; designed to provide an experimental route to the independent J . Chem. Phys. 1983, 78, 4174; Chem. Phys. 1984, 90, 21. Zichi, D.A.; prediction of dielectric friction effects in chemical reactions inHynes, J. T. J. Chem. Phys. 1988,88,2513. (b) van der Zwan, G.; Hynes, volving nuclear dipolar realignment or charge di~p1acement.l~~'~ J. T. Chem. Phys. Left. 1983, 101, 367. (c) Ashcroft, J.; Besnard, M.; Aquada, A.; Jonas, J. Chem. Phys. Lett. 1984, 110,420. For the dipolar solute case, these authors showed that the relation (17) (a) Gertner, B. J.; Wilson, K. R.; Hynes, J. T. J. Chem. Phys. 1989, 90, 3537. (b) Bergsma, J. P.; Gertner, B. J.; Wilson, K. R.; Hynes, J. T. J . TS = cd/cd(t=O) (1.2) Chem. Phys. 1987.86, 1356. Keirstead, W.; Wilson, K. R.; Hynes, J. T. J . Chem. Phys. 1991, 95, 5256. Roux, B.; Karplus, M. J. Phys. Chem. 1991, 95, 4856. 'NSF Postdoctoral Fellow 1990-1992.

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Coalescence of Upper and Lower Miscibility Gaps in Systems with

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