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The hydrophobic heat-capacity anomaly
Physica A 298 (2001) 229–236
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The hydrophobic heat-capacity anomaly P. Bhimalapuram, B. Widom ∗ Department of Chemistry, Baker Laboratory, Cornell University Ithaca, NY 14853-1301, USA
Abstract The excess heat capacity of a hydrophobic solute in water is calculated for four di)erent models, all with closed-loop solubility curves: (i) a phenomenological model based on a modi,ed regular-solution-theory equation of state; (ii) a decorated lattice-gas model equivalent to an underlying Ising model; and (iii) and (iv) van der Waals-like solution models incorporating c 2001 Elsevier Science B.V. All rights reserved. accurate hard-sphere equations of state. PACS: 05.20.Jj; 05.70.Ce; 82.60.Lf Keywords: Hydrophobic e)ect; Heat-capacity anomaly; Closed-loop coexistence curves
It is with great pleasure that we dedicate this account of our recent work to Dick Bedeaux, who has been an inspiration to all of us working in statistical mechanics and thermodynamics. Our theme is the “hydrophobic heat-capacity anomaly”, which is the experimental observation that the partial molar heat capacity of a hydrophobic solute is characteristically large and positive, in contrast with that of a non-hydrophobic solute [1–12]. We illustrate this e)ect in four models, all with closed-loop solubility curves: (i) a phenomenological model based on a modi,ed regular-solution-theory equation of state; (ii) Wheeler’s decorated lattice-gas model [13]; (iii) a van der Waals solution model with a Carnahan–Starling hard core [14]; and (iv) a van der Waals solution model with a Sanchez–van Rensburg hard core [15]. The reason for concentrating on models with closed-loop solubility curves, besides their intrinsic interest, is that it has been noted
Submitted for the special issue of Physica A in honor of Professor Dick Bedeaux on the occasion of his 60th birthday. ∗ Corresponding author. Tel.: +1-607-255-3363; fax: +1-607-255-4137. E-mail address: [email protected] (B. Widom). c 2001 Elsevier Science B.V. All rights reserved. 0378-4371/01/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 2 2 0 - 5
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Fig. 1. The curve is a parabola, and the ,gure then de,nes the temperature-dependent energy parameter w, with the three adjustable parameters TL (lower critical solution temperature), TU (upper critical solution temperature), and (0 ¡ ¡ 14 ). Here Tm = 12 (TL + TU ).
that their lower critical solution points are a manifestation of the hydrophobic e)ect [16]. They also entail a minimum in the solubility as a function of the temperature, which is also a signature of the hydrophobic e)ect [17]. The phenomenological model we treat assumes for the Gibbs free energy G(T; p; NA ; NB ) as a function of temperature T , pressure p, and numbers NA and NB of molecules of species A and B, G(T; p; NA ; NB ) = NA (Ao + kT ln xA + wxB2 ) + NB (Bo + kT ln xB + wxA2 ) ;
(1)
where Ao (T; p) and Bo (T; p) are the chemical potentials of pure A and B, where xA = NA =(NA +NB ) and xB = NB =(NA +NB ) are the mole (molecule) fractions, and where w is a phenomenological energy parameter, generalized from that in the regular-solution theory and here taken temperature dependent, with a temperature dependence as de,ned in Fig. 1. This form of temperature-dependent w, replacing what in the underlying regular-solution theory is a constant energy parameter, is in imitation of the relation of decorated lattice-gas models to underlying one-component lattice-gas or Ising models [13]. We hold p ,xed throughout. The coexistence curve with the choice of parameters TL (lower critical solution temperature) = 300 K, TU (upper critical solution temperature) = 425 K, and = 0:15, is shown in Fig. 2. With these values of the parameters the mutual solubility at the solubility minimum at Tm = 12 (TL + TU ) is only 4.6%. The partial molar(molecular) constant-pressure heat capacities (@Cp =@NA )T; NB and (@Cp =@NB )T; NA
P. Bhimalapuram, B. Widom / Physica A 298 (2001) 229–236
231
Fig. 2. Coexistence curve for the model de,ned by Eq. (1) and Fig. 1, with TL = 300 K, TU = 425 K, and = 0:15.
Fig. 3. Mcp as a function of T, with the coexistence curve as in Fig. 2.
for an A–B mixture are calculated from (1). We de,ne @Cp Mcp; = lim − cp ; N →0 @N T; N
(2)
where cp is the constant-pressure heat capacity per molecule of pure ‘’. Since this phenomenological model is A–B symmetric, Mcp; A = Mcp; B = Mcp . Fig. 3 shows the
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Fig. 4. Coexistence curve for Wheeler’s decorated lattice-mixture model [13], with his parameters U1 =k = 1000 K, U2 =k = − 170 K, q = w = 6.
variation of Mcp with T . We see a very high maximum of about 150k in Mcp at T 368 K, just a little above Tm (= 362:5 K). It is positive over an extended temperature range, approximately 325 K ¡ T ¡ 415 K, only a little shorter than the temperature range of the coexistence curve itself. This is a very strong hydrophobic heat-capacity anomaly. Those found in the next three models are much weaker. We have calculated the analogous quantities for Wheeler’s decorated lattice-gas model [13]. The corresponding heat capacities are now those at ,xed volume and composition, and the analog of Mcp; in (2) we now call Mcxv; . We choose for illustration the parameter values (in the notation of Ref. [13]) U1 =k = 1000 K, U2 =k = − 170 K, q = ! = 6. Like the phenomenological model discussed above, the Wheeler model is A– B symmetric, which ensures Mcxv; A = Mcxv; B = Mcxv . The resulting coexistence curve and Mcxv are displayed in Figs. 4 and 5.The very large peak in Mcxv is not the hydrophobic heat-capacity anomaly we seek but is associated with the antiferromagnetic phase transition in the underlying Ising model [13]. The positive Mcxv at higher temperatures is the heat-capacity anomaly that is the object of our present study, although it is seen to be markedly weaker than that in Fig. 3. The general form of the van der Waals-like equation of state for mixtures [15] is p=
RT a f( ) − 2 ; Vm Vm
(3)
where Vm is the molar volume and f( ) is the hard-sphere compression factor in terms of the packing fraction ( = b=(4Vm )). Both the covolume parameter b and the attraction parameter a are modeled as second order in the mole fraction, so that
P. Bhimalapuram, B. Widom / Physica A 298 (2001) 229–236
233
Fig. 5. Mcxv as a function of T for the decorated lattice-mixture model, with the coexistence curve as in Fig. 4. The high peak at the very low temperature is not the hydrophobic heat-capacity anomaly but a reOection of an underlying antiferromagnetic phase transition in the model.
a = i; j aij xi xj and b = i; j bij xi xj . For a mixture of equal sized molecules (bAA = bBB = bAB ), b is independent of the mixture composition. We considered two closed-form hard-sphere compression factors, namely, the Carnahan–Starling (CS) and the Sanchez– van Rensburg (SvR), with f given by fCS ( ) =
1 + + 2 − 3 ; (1 − )3
(4)
fSvR ( ) =
1 + 1:024385 + 1:10437 2 − 0:4611472 3 − 0:743038 4 ; 1 − 2:975615 + 3:007000 2 − 1:097758 4
(5)
respectively. For a particular choice of dimensionless parameters % and & (following the notation in Ref. [15]) which result in closed-loop coexistence curves, we calculated the Mcp; for Carnahan–Starling (with & = 0:39, % = − 0:01) and Sanchez–van Rensburg (with & = 0:39; % = − 0:001) compression factors. In Figs. 6 –9 are the coexistence curves and corresponding Mcp; for these two models. (T ∗ = kT=(aAA =b) is the scaled temperature in these ,gures.) In these models, too, the positive Mcp; is the hydrophobic heat-capacity anomaly, again obviously much weaker than that in the ,rst model, Fig. 3. It is seen that the quantitative results are strongly model dependent, but qualitatively all lead to positive Mcp or Mcxv . Since positive Mcp and low solubility are both manifestations of hydrophobicity one may anticipate a quantitative connection between them. When the solubility of the hydrophobe A in the solvent B is so low that Henry’s law is satis,ed in the saturated solution, and when the reference phase is pure A, the
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Fig. 6. Coexistence curve as a function of the scaled temperature T ∗ (= kT=(aAA =b)) for the van der Waals-like equation of state with Carnahan–Starling hard-sphere compression factor and for the dimensionless global parameters (notation as in Ref. [15]) & = 0:39 and % = − 0:01.
Fig. 7. Mcp; as a function of the scaled temperature T ∗ (= kT=(aAA =b)) for the van der Waals-like equation of state with the coexistence curve in Fig. 6.
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Fig. 8. Coexistence curve as a function of the scaled temperature T ∗ (= kT=(aAA =b)) for the van der Waals-like equation of state with Sanchez–van Rensburg hard-sphere compression factor and for the dimensionless global parameters (notation as in Ref. [15]) & = 0:39 and % = − 0:001.
Fig. 9. Mcp; as a function of the scaled temperature T ∗ (= kT=(aAA =b)) for the van der Waals-like equation of state with the coexistence curve in Fig. 8.
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relation is Mcp = kT
d2 d d2 [T ln y(T )] = kT 2 ln y(T ) ; ln y(T ) + T dT dT 2 dT 2
(6)
where y(T ) is the solubility of A in B expressed as the mole ratio NA =NB at saturation. When the solubility has a minimum as a function of the temperature, as is typical [17], the term d ln y(T )=dT vanishes at the minimum while d 2 ln y(T )=dT 2 is positive there, so necessarily Mcp ¿ 0 in the neighborhood of the temperature of minimum solubility. Acknowledgements We gratefully acknowledge help and advice from Dr. Volker Weiss and helpful conversations with Dr. Luis-Miguel Trejo. This work has been supported by the US National Science Foundation and the Cornell Center for Materials Research. References [1] J.T. Edsall, J. Am. Chem. Soc. 57 (1935) 1506. [2] A. Ben-Naim, Water and Aqueous Solutions: Introduction to a Molecular Theory, Plenum Press, New York, 1974, p. 318. [3] S.J. Gill, N.F. Nichols, I. WadsTo, J. Chem. Thermodynamics 8 (1976) 445. [4] A. Ben-Naim, Hydrophobic Interactions, Plenum Press, New York, 1980, pp. 203–204. [5] C. Tanford, The Hydrophobic E)ect: Formation of Micelles and Biological Membranes, 2nd Edition, Wiley, New York, 1980, pp. 21–28. [6] R.L. Baldwin, Proc. Natl. Acad. Sci. USA 83 (1986) 8069. [7] G.I. Makhatadze, P.L. Privalov, J. Chem. Thermodynamics 20 (1988) 405. [8] B. Madan, K. Sharp, J. Phys. Chem. 100 (1996) 7713. [9] P.M. Wiggins, Physica A 238 (1997) 113. [10] K.A.T. Silverstein, A.D.J. Haymet, K.A. Dill, J. Chem. Phys. 111 (1999) 8000. [11] N.T. Southall, K.A. Dill, J. Phys. Chem. B 104 (2000) 1326. [12] S.W. Rick, J. Phys. Chem. B 104 (2000) 6884. [13] J.C. Wheeler, J. Chem. Phys. 62 (1975) 433. [14] L.V. Yelash, T. Kraska, Ber. Bunsenges. Phys. Chem. 102 (1998) 213. [15] R.L. Scott, Phys. Chem. Phys. 1 (1999) 4225. [16] M.E. Paulaitis, S. Garde, H.S. Ashbaugh, Curr. Opin. Colloid Interface Sci. 1 (1996) 376. [17] B. Guillot, Y. Guissani, J. Chem. Phys. 99 (1993) 8075.
www.elsevier.com/locate/physa
The hydrophobic heat-capacity anomaly P. Bhimalapuram, B. Widom ∗ Department of Chemistry, Baker Laboratory, Cornell University Ithaca, NY 14853-1301, USA
Abstract The excess heat capacity of a hydrophobic solute in water is calculated for four di)erent models, all with closed-loop solubility curves: (i) a phenomenological model based on a modi,ed regular-solution-theory equation of state; (ii) a decorated lattice-gas model equivalent to an underlying Ising model; and (iii) and (iv) van der Waals-like solution models incorporating c 2001 Elsevier Science B.V. All rights reserved. accurate hard-sphere equations of state. PACS: 05.20.Jj; 05.70.Ce; 82.60.Lf Keywords: Hydrophobic e)ect; Heat-capacity anomaly; Closed-loop coexistence curves
It is with great pleasure that we dedicate this account of our recent work to Dick Bedeaux, who has been an inspiration to all of us working in statistical mechanics and thermodynamics. Our theme is the “hydrophobic heat-capacity anomaly”, which is the experimental observation that the partial molar heat capacity of a hydrophobic solute is characteristically large and positive, in contrast with that of a non-hydrophobic solute [1–12]. We illustrate this e)ect in four models, all with closed-loop solubility curves: (i) a phenomenological model based on a modi,ed regular-solution-theory equation of state; (ii) Wheeler’s decorated lattice-gas model [13]; (iii) a van der Waals solution model with a Carnahan–Starling hard core [14]; and (iv) a van der Waals solution model with a Sanchez–van Rensburg hard core [15]. The reason for concentrating on models with closed-loop solubility curves, besides their intrinsic interest, is that it has been noted
Submitted for the special issue of Physica A in honor of Professor Dick Bedeaux on the occasion of his 60th birthday. ∗ Corresponding author. Tel.: +1-607-255-3363; fax: +1-607-255-4137. E-mail address: [email protected] (B. Widom). c 2001 Elsevier Science B.V. All rights reserved. 0378-4371/01/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 2 2 0 - 5
230
P. Bhimalapuram, B. Widom / Physica A 298 (2001) 229–236
Fig. 1. The curve is a parabola, and the ,gure then de,nes the temperature-dependent energy parameter w, with the three adjustable parameters TL (lower critical solution temperature), TU (upper critical solution temperature), and (0 ¡ ¡ 14 ). Here Tm = 12 (TL + TU ).
that their lower critical solution points are a manifestation of the hydrophobic e)ect [16]. They also entail a minimum in the solubility as a function of the temperature, which is also a signature of the hydrophobic e)ect [17]. The phenomenological model we treat assumes for the Gibbs free energy G(T; p; NA ; NB ) as a function of temperature T , pressure p, and numbers NA and NB of molecules of species A and B, G(T; p; NA ; NB ) = NA (Ao + kT ln xA + wxB2 ) + NB (Bo + kT ln xB + wxA2 ) ;
(1)
where Ao (T; p) and Bo (T; p) are the chemical potentials of pure A and B, where xA = NA =(NA +NB ) and xB = NB =(NA +NB ) are the mole (molecule) fractions, and where w is a phenomenological energy parameter, generalized from that in the regular-solution theory and here taken temperature dependent, with a temperature dependence as de,ned in Fig. 1. This form of temperature-dependent w, replacing what in the underlying regular-solution theory is a constant energy parameter, is in imitation of the relation of decorated lattice-gas models to underlying one-component lattice-gas or Ising models [13]. We hold p ,xed throughout. The coexistence curve with the choice of parameters TL (lower critical solution temperature) = 300 K, TU (upper critical solution temperature) = 425 K, and = 0:15, is shown in Fig. 2. With these values of the parameters the mutual solubility at the solubility minimum at Tm = 12 (TL + TU ) is only 4.6%. The partial molar(molecular) constant-pressure heat capacities (@Cp =@NA )T; NB and (@Cp =@NB )T; NA
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231
Fig. 2. Coexistence curve for the model de,ned by Eq. (1) and Fig. 1, with TL = 300 K, TU = 425 K, and = 0:15.
Fig. 3. Mcp as a function of T, with the coexistence curve as in Fig. 2.
for an A–B mixture are calculated from (1). We de,ne @Cp Mcp; = lim − cp ; N →0 @N T; N
(2)
where cp is the constant-pressure heat capacity per molecule of pure ‘’. Since this phenomenological model is A–B symmetric, Mcp; A = Mcp; B = Mcp . Fig. 3 shows the
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Fig. 4. Coexistence curve for Wheeler’s decorated lattice-mixture model [13], with his parameters U1 =k = 1000 K, U2 =k = − 170 K, q = w = 6.
variation of Mcp with T . We see a very high maximum of about 150k in Mcp at T 368 K, just a little above Tm (= 362:5 K). It is positive over an extended temperature range, approximately 325 K ¡ T ¡ 415 K, only a little shorter than the temperature range of the coexistence curve itself. This is a very strong hydrophobic heat-capacity anomaly. Those found in the next three models are much weaker. We have calculated the analogous quantities for Wheeler’s decorated lattice-gas model [13]. The corresponding heat capacities are now those at ,xed volume and composition, and the analog of Mcp; in (2) we now call Mcxv; . We choose for illustration the parameter values (in the notation of Ref. [13]) U1 =k = 1000 K, U2 =k = − 170 K, q = ! = 6. Like the phenomenological model discussed above, the Wheeler model is A– B symmetric, which ensures Mcxv; A = Mcxv; B = Mcxv . The resulting coexistence curve and Mcxv are displayed in Figs. 4 and 5.The very large peak in Mcxv is not the hydrophobic heat-capacity anomaly we seek but is associated with the antiferromagnetic phase transition in the underlying Ising model [13]. The positive Mcxv at higher temperatures is the heat-capacity anomaly that is the object of our present study, although it is seen to be markedly weaker than that in Fig. 3. The general form of the van der Waals-like equation of state for mixtures [15] is p=
RT a f( ) − 2 ; Vm Vm
(3)
where Vm is the molar volume and f( ) is the hard-sphere compression factor in terms of the packing fraction ( = b=(4Vm )). Both the covolume parameter b and the attraction parameter a are modeled as second order in the mole fraction, so that
P. Bhimalapuram, B. Widom / Physica A 298 (2001) 229–236
233
Fig. 5. Mcxv as a function of T for the decorated lattice-mixture model, with the coexistence curve as in Fig. 4. The high peak at the very low temperature is not the hydrophobic heat-capacity anomaly but a reOection of an underlying antiferromagnetic phase transition in the model.
a = i; j aij xi xj and b = i; j bij xi xj . For a mixture of equal sized molecules (bAA = bBB = bAB ), b is independent of the mixture composition. We considered two closed-form hard-sphere compression factors, namely, the Carnahan–Starling (CS) and the Sanchez– van Rensburg (SvR), with f given by fCS ( ) =
1 + + 2 − 3 ; (1 − )3
(4)
fSvR ( ) =
1 + 1:024385 + 1:10437 2 − 0:4611472 3 − 0:743038 4 ; 1 − 2:975615 + 3:007000 2 − 1:097758 4
(5)
respectively. For a particular choice of dimensionless parameters % and & (following the notation in Ref. [15]) which result in closed-loop coexistence curves, we calculated the Mcp; for Carnahan–Starling (with & = 0:39, % = − 0:01) and Sanchez–van Rensburg (with & = 0:39; % = − 0:001) compression factors. In Figs. 6 –9 are the coexistence curves and corresponding Mcp; for these two models. (T ∗ = kT=(aAA =b) is the scaled temperature in these ,gures.) In these models, too, the positive Mcp; is the hydrophobic heat-capacity anomaly, again obviously much weaker than that in the ,rst model, Fig. 3. It is seen that the quantitative results are strongly model dependent, but qualitatively all lead to positive Mcp or Mcxv . Since positive Mcp and low solubility are both manifestations of hydrophobicity one may anticipate a quantitative connection between them. When the solubility of the hydrophobe A in the solvent B is so low that Henry’s law is satis,ed in the saturated solution, and when the reference phase is pure A, the
234
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Fig. 6. Coexistence curve as a function of the scaled temperature T ∗ (= kT=(aAA =b)) for the van der Waals-like equation of state with Carnahan–Starling hard-sphere compression factor and for the dimensionless global parameters (notation as in Ref. [15]) & = 0:39 and % = − 0:01.
Fig. 7. Mcp; as a function of the scaled temperature T ∗ (= kT=(aAA =b)) for the van der Waals-like equation of state with the coexistence curve in Fig. 6.
P. Bhimalapuram, B. Widom / Physica A 298 (2001) 229–236
235
Fig. 8. Coexistence curve as a function of the scaled temperature T ∗ (= kT=(aAA =b)) for the van der Waals-like equation of state with Sanchez–van Rensburg hard-sphere compression factor and for the dimensionless global parameters (notation as in Ref. [15]) & = 0:39 and % = − 0:001.
Fig. 9. Mcp; as a function of the scaled temperature T ∗ (= kT=(aAA =b)) for the van der Waals-like equation of state with the coexistence curve in Fig. 8.
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relation is Mcp = kT
d2 d d2 [T ln y(T )] = kT 2 ln y(T ) ; ln y(T ) + T dT dT 2 dT 2
(6)
where y(T ) is the solubility of A in B expressed as the mole ratio NA =NB at saturation. When the solubility has a minimum as a function of the temperature, as is typical [17], the term d ln y(T )=dT vanishes at the minimum while d 2 ln y(T )=dT 2 is positive there, so necessarily Mcp ¿ 0 in the neighborhood of the temperature of minimum solubility. Acknowledgements We gratefully acknowledge help and advice from Dr. Volker Weiss and helpful conversations with Dr. Luis-Miguel Trejo. This work has been supported by the US National Science Foundation and the Cornell Center for Materials Research. References [1] J.T. Edsall, J. Am. Chem. Soc. 57 (1935) 1506. [2] A. Ben-Naim, Water and Aqueous Solutions: Introduction to a Molecular Theory, Plenum Press, New York, 1974, p. 318. [3] S.J. Gill, N.F. Nichols, I. WadsTo, J. Chem. Thermodynamics 8 (1976) 445. [4] A. Ben-Naim, Hydrophobic Interactions, Plenum Press, New York, 1980, pp. 203–204. [5] C. Tanford, The Hydrophobic E)ect: Formation of Micelles and Biological Membranes, 2nd Edition, Wiley, New York, 1980, pp. 21–28. [6] R.L. Baldwin, Proc. Natl. Acad. Sci. USA 83 (1986) 8069. [7] G.I. Makhatadze, P.L. Privalov, J. Chem. Thermodynamics 20 (1988) 405. [8] B. Madan, K. Sharp, J. Phys. Chem. 100 (1996) 7713. [9] P.M. Wiggins, Physica A 238 (1997) 113. [10] K.A.T. Silverstein, A.D.J. Haymet, K.A. Dill, J. Chem. Phys. 111 (1999) 8000. [11] N.T. Southall, K.A. Dill, J. Phys. Chem. B 104 (2000) 1326. [12] S.W. Rick, J. Phys. Chem. B 104 (2000) 6884. [13] J.C. Wheeler, J. Chem. Phys. 62 (1975) 433. [14] L.V. Yelash, T. Kraska, Ber. Bunsenges. Phys. Chem. 102 (1998) 213. [15] R.L. Scott, Phys. Chem. Phys. 1 (1999) 4225. [16] M.E. Paulaitis, S. Garde, H.S. Ashbaugh, Curr. Opin. Colloid Interface Sci. 1 (1996) 376. [17] B. Guillot, Y. Guissani, J. Chem. Phys. 99 (1993) 8075.