J. Phys. Chem. 1992, 96,4052-4056
4052
bounds on the diffusion constant. These up r and lower bounds are given by DIU) = ( 0 . 1 0 9 ) k B T ( R / a ) z ~ q J Vand a DiL) = (0.040)kBT(R/a)2/3/qsNa, respectively. Note that these bounds are consistent with the scaling prediction Dll ( R / a ) 2 / previously 3 reported.’* Our estimates for DIldo indeed fall between these bounds (cf. Figure 4). However, their R dependence does not precisely follow the scaling of ref 12. Instead, DI!grows slightly slower than R2/3;we find DII R‘.61. This behavior is due to the R dependence of the transverse correlation length 6 found in our variational calculation. Although 6 is an increasing function of R , the ratio t / R is a slowly decreasing function. Thus, with increasing R, the cross-sectional concentration profile (cf. Figure 3) slowly approaches the constant density profile associated with the lower bound on the diffusion constant.26 The net R dependence of D,,comes from the product of the R2/3scaling term and the average over the cross-sectional monomer distribution, which
-
-
( 2 6 ) It must be emphasized, however, that R may not be increased indefinitely without also increasing N , in order to maintain the strongly confined conditions we assume in this paper.
is a slowly decreasing function of R. We should emphasize that this deviation from the scaling behavior is quite small, contributing less than 20% deviation over almost two decades of R . Furthermore, the precise form of this deviation may depend on the choice of trial function employed in our variational calculation. The results we have presented for narrow capillaries may be easily extended to other strongly confining geometries, such as closely spaced parallel plates. In this paper, we have restricted our attention to the case of flexible, neutral polymers, for which a Kirkwood approach is feasible. However, the transport properties of semiflexible or charged macromolecules are of particular interest also and have many important technological applications. Future work on transport in confined geometries should focus on such problems. Acknowledgment. This work was supported by the Grant-inAid for Scientific Research administered by the Ministry of Education, Science and Culture of Japan. J.L.H. was supported by a postdoctoral fellowship from the Japan Society for the Promotion of Science.
Thermodynamics of Binding of Mononucleotides to Ribonuclease T, Cui-QingHu and Julian M. Sturtevant* Departments of Chemistry and Molecular Biophysics and Biochemistry, Yale University, New Haven, Connecticut 0651 1 (Received: October 2, 1991)
The binding of mononucleotides to ribonucleases has received much attention in recent years, in part because of the possibility that the different s@icities of the ribonucleases might be. of interest in connection with specific protein-nucleic acid interactions. Among the most widely studied ribonucleases are ribonuclease A (RNase A), which is specific for pyrimidine nucleotides, specific for guanosine nucleotides (Egami, E.; Oshima, T.; Uchida, RNase U2,specific for purine nucleotides, and RNase TI, T. Mol. Biol. Biochem. Biophys. 1980,32, 250-277). In this paper we report determinations of the binding constants and enthalpies of binding of several mononucleotides to RNase TI,using both isothermal titration calorimetry (ITC) and differential scanning calorimetry (DSC). ITC was performed at several temperatures to yield values for the heat capacity changes. The DSC results extended the temperature range of validity of many of the thermodynamic data. The binding constants at 25 OC ranged from 5.8 X 10’ M-’ for 2‘-GMP to 7.7 X lo2 M-’ for 2’-CMP. The enthalpy variation was much smaller, ranging from -9.09 kcal mol-’ for 2’-GMP to -8.22 kcal mol-’ for 2’-CMP.
Introduction Ribonuclease TI (RNase T I ) is a compact, globular protein containing a single chain of 104 amino acid residues and having a molecular mass of 11 085 Da. It contains two disulfide bridges, one of which links residues 6 and 103 to form an unusually large closed loop.’ RNase T, is an acidic protein with isoelectric pH equal to 3.8. RNase TI catalyzes the hydrolysis of singlestranded ribonucleic acid (RNA), causing cleavage with high specificity at the 3’-side of guanosine residues in the RNA. It has been suggested*-’ that the binding of mononucleotides to RNase TImight serve as a useful model system for the study of the biologically important process of specific protein-nucleic acid recognition. Several determinations of the affinities of mononucleotides to RNase TI have been made by various methods. A summar)“ of the results of these determinations shows binding constants at pH 5.Ch5.6 and 25 O C ranging, in the case of 2‘-guanosine monophosphate (2’-GMP), from 1.3 X 104 to 1.4 X l@M-I. Saenger et al.>’ have determined (1) Takahashi, K.J . Eiol. Chem. 1965, 240,PC4117. (2) Takahashi, K.;Moore, S.Enzymes 1982, IS, 435. (3) Heinemann, V.;Saenger, W. Nuture 1982, 299, 27. (4) Egami, F.; Oshima, T.; Uchida, T. Mol. Eiol.Eiochem. Eiophys. 1980, 32. 250. (5) Sergio, S.;Amisaki, T.; Ohnishi, H.; Tomita, K.; Heinemann, U.; Saenger, W. FEES Lett. 1985, 181, 129. ~I
~~
the three-dimensional structure of RNase TI and that of its 2’G M P complex by X-ray crystallography to a resolution of 1.9 A. Aside from binding constants at a single temperature, no thermodynamic data for the interaction of mononucleotides with RNase TI have been reported. We have employed isothermal titration calorimetry (ITC) and differential scanning calorimetry (DSC) to obtain thermodynamic data over a range of temperatures a t pH 5.5 for 2’-, 3’-, and 5’-GMP and 2’-adenosine monophosphate (2’-AMP) and 2’-cytosine monophosphate (2’-CMP).
Experimental Section RNase TI(clone), isolated from Escherichia coli and purified as described by Shirley et al.,* was kindly supplied by C. Nick Pace of Texas A&M University. Prior to use, the enzyme was dialyzed against 50 mM sodium acetate buffer, pH 5.5, and then filtered through a 0.22-pm Millipore filter. The concentration of the protein was determined using an absorbance of 1.67 at 278 nm for a 1 mg mL-’ solution (Pace, personal communication). The disodium salts of 2‘-, 3’-, and 5‘-GMP and 2‘-CMP and the acid form of 2’-AMP were purchased from Sigma and used ( 6 ) Sergio, S.; Oka, K.-I.; Ohnishi, H.; Tomita, K.; Saenger, W. FEES Lett. 1985,183, 115. (7) Ami, R.;Heinemann, U.; Tokuoka, R.; Saenger, W. J. Eiol. Chem. 1988, 263, 15358. (8) Shirley, B.A.; Stanssens, P.; Steyaert, J.; Pace, C. N. J . Eiol. Chem. 1989, 264, 11621.
0022-3654192 l2096-4052S03 .OO I O 0 1992 American Chemical Societv
The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 4053
Mononucleotide Binding to Ribonuclease TI
7
1.5
-
1.0
-
0
Y
a
\ Y
0
8 U
Mdl 0
I
mo
4000
8000
Tim (#.e)
0.5
-
:I
" 0.0
n 1
;-0 5
L
I
4s
I5
I 75
65
T e m D e r l t U r e / 'C
p
ea1
-100
-j50
j
1
Figure 2. Typical DSC curve for the thermal denaturation of RNase TI in the presence of 3'-GMP protein concentration, 2.025 mg mL-l; ligand concentration, 1.00 mM. Solid curve, observed data; dashed curves, best fit to a modified two-state model and calculated baseline; dotted lines, least squares pre- and posttransition baselines. Best fit parameters: tl,* = 60.5 O C ; AH'- = 127.3 kcal mol-'; = 0.99; = 1.96 kcal K-'mol-I; SD = 0.78% of maximal excess specific heat.
e,.,/& AG
1
-2004
1 -250!
I
0
.
I
6
.
r
.
12
I
18
-
1
'
24
Injection Nuder
Figure 1. Titration curve of 3'-GMP binding to RNase TI at 15 OC: RNase TI concentration, 0.280 mM; 3'-GMP concentration, 6.04 mM. Top panel: recording of 20 injections of 3'-GMP of 5 WL each into 1.3 mL of RNase T I . Bottom panel: filled circles, integrated heats of injections after deduction of small dilution heats; solid curve, best fit to the data using the model of a single set of binding sites with equal binding constants. Best fit parameters: Kb = 1.37 X lo5 M-I; AHb = -7.26 kcal mol-'; n = 0.99.
without further purification except for dehydration to constant weight at 70 O C in a vacuum oven. In each case, the weight loss on dehydration corresponded to the stated content of water of crystallization. The ligands were dissolved in the same buffer as was the protein, the pH being readjusted as necessary. The ligand concentrations were determined by weight. Calorimetric titrations were performed in the Omega microcalorimeter manufactured by Microcal, Inc., Northampton, MA. A description of this instrument has been published by Wiseman et al? In our experiments, 1.3 mL of 0.2-0.4 mM protein solution was titrated with 20-26 computer-controlled injections of 4-5 1 L each of ligand a t 20-30 times the protein concentration. The Omega includes electrical compensation so that return to the baseline after each injection is very rapid. The data for a typical titration experiment are reproduced in Figure 1 . 3'-GMP at a concentration of 6.04 mM was added to 0.280 mM RNase TI at 15 OC in 20 injections of 5 pL each. The upper panel in the figure is a recording of the actual injections, and the lower panel shows the integrated enthalpy value for each injection (filled circles), after deduction of very small heats of dilution, and the best fit curve based on the model of a single set of binding sites with equal binding constants. The curve fitting, which is carried out by software supplied with the calorimeter, leads to values for &, the binding constant, AHb,the molar enthalpy change, and n, the number of binding sites per molecule. Wiseman et al.9 pointed out that for optimal results the product, C, of the total molarity of substrate, [PI,, times Kb(M-I) should be between 1 and 1000, preferably between 10 and 100. The value of C for the experiment reported in Figure 1 was 38. Differential scanning calorimetric (DSC) experiments utilized the MC-2 instrument, also manufactured by Microcal. A scan rate of 1 K m i d was employed, it having been ascertained by experiments at lower scan rate that calorimetric lag did not sig(9) Wiseman, T.; Williston, S.;Brandts, J. F.; Lin, L.-N. Anal. Biochem. 1989, 179, 131.
nificantly affect the data. The DSC scans were fully reversible under all conditions employed in this work. The DSC curves of excess apparent specific heat as a function of temperature, after deduction of the instrumental baseline, were evaluated by a least squares fit of the observed data to the theoretical curve for a single two-state process modified to permit the van't Hoff enthalpy, AHVH, which controls the progress of the transition with increasing temperature according to the van't Hoff equation d In Kd -dT
--AHH,,
(1)
RP
where & is the equilibrium constant for the process, to differ from the calorimetric, or true, enthalpy, A&.'' A typical DSC curve is shown in Figure 2. Here the solid curve is the observed data, the dashed curves are the theoretical curve and the calculated chemical baseline, and the dotted lines are the linear least squares observed pre- and posttransition baselines. The calculated baseline is obtained by proceeding from the pretransition baseline to the posttransition baseline in proportion to the extent of the reaction, due allowance being made for A G , the permanent change in heat capacity accompanying the process.I0 The curve fitting leads to values for tlI2,the temperature of half-completion of the transition, AHd and AHvH. Brandts and Lin" have recently shown how binding constants and binding enthalpies can be evaluated from DSC data obtained in the presence of ligands, provided the ligand binding constant is sufficiently high for the native state and practically zero for the denatured state. This opens up the possibility of a considerably expanded temperature range for thermodynamic data for binding processes since the unfolding transition for most proteins occurs well above room temperature. It is interesting that this technique can be applied for binding constants that are far beyond the range of applicability of ordinary isothermal procedures, including noncalorimetric procedures. For the situation of a protein having a single reversible two-state unfolding transition in the presence of an effectively saturating concentration of a ligand which binds to the native protein at a single site, the equilibria involved are native (N) + ligand (L) e NL
N
F!
denatured (D)
Kb = INL1 [Nl [LI
Kd =
[Dl [NI
(2)
(3)
At tIl2,if the concentration of free N can be neglected, [NL]1/2 (10) Sturtevant, J. M. Annu. Rev. Phys. Chem. 1987, 38, 463. ( 1 1 ) Brandts, J. F.; Lin, L.-N. Biochemistry 1990, 29, 6927.
4054 The Journal of Physical Chemistry, Vol. 96, No. 10, 1992
Hu and Sturtevant
TABLE I: Thermodynamic Parameters for the Binding of Mononucleotides to RNase TI As Determined by Titration Calorimetry at Various Temperatures temp, OC no. of titrations K b , M-' -Affb, kcal mol-' - A G , cal K" mol-' n 2'-GMP 15.0 25.0 35.0
3 3 3
(9.04 f 0.27) X lo5 (5.78 f 0.14) X IO5 (3.35 0.13) x 105
15.0 25.0 35.0
3 3 3
(1.33 f 0.09) (8.44 f 0.16) (5.48 f 0.04)
15.0 25.0 35.0
13 4 11
(1.24 f 0.04) X IO4 (9.63 f 0.33) X lo3 (6.69 f 0.12) x 103
5.0 15.0 25.0
3 3 3
(1.25 f 0.01) x 104 (7.43 f 0.44) X IO' (4.82 f 0.19) X IO'
5.0 15.0 25.0
3 3 3
(1.73 f 0.13) X 10' (1.13 f 0.03) X lo3 (7.66 f 0.26) X IO2
*
8.67 f 0.12 9.09 f 0.09 9.38 f 0.07
9 = 0.99
7.49 f 0.21 7.64 f 0.02 8.52 f 0.04
9 = 0.86
1.02 f 0.02 1.10 f 0.02 1.11 f 0.05
35.5 f 1.7
3'-GMP X X X
lo5 lo4 lo4
51.5 f 9.3
1.00 f 0.02 1.12 f 0.02 1.08 f 0.05
5'-GMP 8.08 f 0.14 8.42 f 0.16 9.30 f 0.13 2'-AMP
*
7.59 0.12 8.04 f 0.35 8.58 f 0.27
61.0 f 6.9 9 = 0.94
1.18 f 0.03 1.10 f 0.01 1.12 f 0.02
49.5 f 2.6 ? = 1.00
1.12 f 0.06 1.08 f 0.07 0.96 f 0.04
35.5 f 0.3
1.O' 1,v 1.O'
2'-CMP
" n was held equal
7.51 f 0.23 7.86 f 0.16 8.22 f 0.63
9 = 1.00
to 1 during the curve-fitting process.
TABLE 11: Thermodynamic Properties for the Binding of Mononucleotides to Ribonuclease TI at 25 O C ligand 2'-GMP 3'-GMP 5'-GMP 2'-AMP 2'-CMP
Kb, 5.50 X 8.34 x 9.11 x 4.80 X 7.68 X
IO5 104 103 10' lo2
-A(?., kcal mol-' 7.83 6.71 5.40 5.02 3.94
-AfCHt
kcal mol-' 8.73 7.81 5.41 7.35 6.35
= [D]1/2so that Kb/Kd = 1/[L]1/2. Since Kd = 1 at to, the temperature of half-completion of the transition in the absence of ligand, the Gibbs-Helmholtz equation, can be employed to evaluate Kb[LlIl2at t l j z :
+
where and Toare respectively t 1 / 2+ 273.15 and to 273.15 and [LlIl2 = [L], - [P],/2, with [L], and [PI, being the total ligand and protein concentrations. AC", is evaluated by least squaring M d as a function of T (calculations not shown), the temperature of denaturation being varied by varying the pH. If values for A H b , the enthalpy change in reaction 2, and A G , the corresponding heat capacity change, have been obtained by titration or some other form of isothermal mixing calorimetry, the Gibbs-Helmholtz equation can again be employed to evaluate Kb at some other temperature such as To. The enthalpy of binding can be calculated from DSC data, again provided the folded protein is essentially saturated with and AH, are respectively ligand, by means of eq 5. Here
e,2
m
b
(at to) = M I 2 - N
o
- (tl/2
- to)(ACd, + A($)
(5)
the enthalpies of unfolding in the presence of ligand at t I l 2and in the absence of ligand at to.
Results and Discussion Titration Calorimetry. The data obtained by titration calorimetry are summarized in Table I. Three or more titrations were performed at each temperature with each nucleotide, and the uncertainties listed in the table are standard errors of the mean.
-a,
-Affbi
-AG,
kcal mol-'
cal K-' mol-'
cal K-' mol-I
9.05 7.88 8.60 8.57 8.22
4.09 3.92 10.73 11.9 13.2
35.5 51.5 61.0 49.5 35.5
AC
The values for were obtained by least squaring the enthalpy-temperature data, with the indicated coefficients of determination, 9.It is interesting that, although the nucleotide binding is enthalpy driven, the binding constants at 25 "C cover a 750-fold range while the enthalpy values differ at most by 20%. The deviation from unity of the values for n for the first four nucleotides appears to be greater than the experimental uncertainty of the titrations. We cannot explain the discrepancy. n was set equal to unity in the curve fitting for 2'-CMP because the small size of Kb precluded allowing n to float. Since A@ is well determined by the enthalpy-temperature data, we have evaPuated the van't Hoff enthalpies, A&, by linear least squaring of the function R In K - AC(ln T + 298.15/T) against 1/ T. The smoothed thermodynamic data listed in Table I1 were calculated from the results of this least squaring and that of A&, vs T . The values for Kb in the literature, as summarized by Egami et a1.: range from 1.3 X lo4 to 1.5 X lo5 M-' for 2'-GMP at 25 OC and pH 5.0-5.6. The range is 2.9 X lo3 to 1.3 X lo5 M-' for 3'-GMP, and 1 X l@ to 3.3 X 104 M-' for 5'-GMP. The present value for 2'-GMP is almost 4 times the highest literature value, and those for 3'- and 5'-GMP lie within the very wide range of literature values. The one value listed for 2'-AMP, 2.3 X 102 M-I, is only one-twentieth of the value reported here. These very large discrepancies are difficult to understand. Hirono and KollmanI2 have recently employed a thermodynamic perturbation method implemented with molecular dynamics to estimate the difference in the free energy of binding, AAG, between 2'-GMP and 2'-AMP. Their result, 2.76 kcal mol-', agrees fortuitously well with the calorimetric value at 25 OC, 2.81 kcal mol-'. According to their calculations, the active site of RNase Ti electrostaticallyfavors the binding of 2'-GMP over that of 2'-AMP by 44 kcal mol-', but this large difference is reduced by other interactions favoring 2'-AMP, including the 7 kcal mol-' (12) Hirono, S.; Kollman, P.A. J. Mol. Biol. 1990, 212, 197.
The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 4055
Mononucleotide Binding to Ribonuclease TI
TABLE I11 Least Squarecl Coefficients Showing the Variation of t , / 2with In [Ll,a d the Variation of AHLwith t l l l for the Thermal D e ~ t u n t i o nof RNase TIin the Presence of Mononucleotides“ hand
range of [L],, mM
no. of expts
A
2’-GMP 3’-GMP 5‘-GMP 2’-AMP 2’-CMP
0.4-7.2 0.4-8.0 0.6-10.0 1.0-8.0 1.0-8.0
6 6 7 5 5
180.2 188.4 185.2 229.9 309.9
“In [L], = A
+ 1000B/T,/2.
B -60.75 -62.86 -61.32 -75.79 -101.94
AG,
‘9
kcal mol-’
kcal K-l mol-I
rz
0.998 0.999 0.997 0.993 0.966
58.91 8 1.36 6.07 -7.02 42.64
1.181 0.746 2.014 2.076 1.143
0.926 0.764 0.974 0.998 0.987
+ AGtI/~.
=
2.0
1.0
-c
0.0
-1.0
\ -2.0
.93
I
I
I
I
2.95
2.07
2.99
3.01
1000/T
,,~
3.03
3.05
140
1
4
0
$
W
7
0
8
0
9
0
Temperature 1 ‘c
Figure 3. DSC curves of the thermal denaturation of RNase TIin the presence of various concentrations of 3’-GMP: protein concentration, 2.025 mg mL-’; concentration of ligand, (A) 0.0, (B) 0.20, (C) 0.40, (D) 0.60, (E) 1.00, (F) 1.80, (G) 4.00, (H) 8.00 mM. difference in desolvation free energies also favoring 2’-AMP. Differential Scanning Calorimetry. Figure 3 illustrates the variation of tl12 with nucleotide concentration for the case of 3’-GMP. As expected for a ligand which dissociates on protein unfolding, increases with increasing ligand concentration, thus giving the appearance of stabilizing the protein. Table 111gives the results of least squaring the van? Hoff plots of In [L], vs 1/TlI2, and the linear plots of AHL, the total denaturational enthalpy in the presence of ligand, vs tl12,and the plots are illustrated in Figure 4. In Table 111, the constants A and B given in the fourth and fifth columns permit the evaluation of [L], in millimolar, by means of the equation [L], = exp(A + 1000B/T1/2) (6) The constants & and listed in the seventh and eighth columns give A@, in kilocalories per mole, according to the equation
AHL=
+Aqtlp
%
(7)
The sixth and ninth columns list the coefficients of determination for the two least squarings. It can be shown13that if the protein is essentially saturated with ligand before denaturation, the slope B in Table I11 should equal -AHvHInR. Thus, for the case of 2’-GMP, the mean value of obtained by curve fitting of the DSC data obtained in the presence of 2’-GMP and calculated to 55.1 OC, assuming to have the same temperature variation as is 116.6 kcal mol-’; the value of -BR is 120.7 kcal mol-] giving 0.97 for n. The values
eH
e,
I
I
I
60
65
70
Tamparature / ‘C
F i e 4. (A) Plots of In [L], vs 1000/Tl for RNase TI in the presence of mononucleotides: 0, 2’-GMP; A, 3’-dMP 0,5’-GMP a, 2’-AMP 0,2’-CMP. (B) Plots of AHd vs temperature: 0,2’-GMP; 0 , 3’-GMP 0,5’-GMP M, 2’-AMP A, 2’-CMP.
TABLE I V Tbennodynrmics of the Binding of Mononucleotides to Ribonuclease TIat 55.1 O C As Evaluated from DSC Data
Kb, M-I ligand ” 2’-GMP 3’-GMP 5‘-GMP 2’-AMP 2’-CMP
DSC (1.44 i 0.05) X (1.94 i 0.07) X (4.22 f 0.06 X (1.44 i 0.03) X 6.50 X lo2
A& kcal mol-’ titration DSC titration 10.1 lo’ 1.25 X lo5 10.6 i 0.7 lo4 2.19 X 10‘ 8.9 f 1.2 9.4 lo3 2.10 X lo3 11.8 i 0.3 11.1 10’ 1.14 X 10’ 8.6 2.0 x 102 9.3
n 0.97 1.02 1.04
of n for the three nucleotides having the largest binding constants are listed in the last column of Table IV and confirm the existence of one binding site per molecule of RNase TI. The results of evaluating Kb from the DSC data by the procedure given by Brandts and Lin” are given in the second column of Table IV and may be compared with the values in the third column calculated by means of the Gibbs-Helmholtz equation from the titration data at 25 OC given in Table 111. The mean titration values for A H b and A G listed in Table I were used in converting the values for K b at t l l zobserved in the DSC experiments to t, (55.1 “C). Good agreement between the DSC and titration values is only to be expected and is found for 2’- and 3’-GMP, which have sufficiently large binding constants so that
aH
(13) Fukada, H.;Sturtevant, J. M.; Quiocho, F. A. J . B o / . Chem. 1983, 258, 13193.
55
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 t l 1 2was 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 m z / m lconstant. 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
