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J. Phys. Chem. 1996, 100, 5172-5181
FEATURE ARTICLE Mixing Schemes in Aqueous Solutions of Nonelectrolytes: A Thermodynamic Approach Yoshikata Koga† Department of Chemistry, The UniVersity of British Columbia, 2036 Main Mall, VancouVer, BC V6T 1Z1 Canada ReceiVed: August 15, 1995; In Final Form: January 17, 1996X
Thermodynamic studies were carried out on aqueous solutions of some nonelectrolytes. The quantities proportional to the second derivatives of Gibbs energy were measured directly and in small increments in mole fraction or temperature. Therefore, we were able to differentiate once more with respect to mole fraction or temperature. Generally, the higher the order of the derivative, the more detailed the information it contains. Using these second and third derivatives, an attempt was made at elucidating the mixing schemes, the way in which solute and solvent H2O molecules mix with each other. For the following nonelectrolytes studied so far, 2-butoxyethanol, tert-butyl alcohol, 2-butanone, isobutyric acid, and dimethyl sulfoxide, there are separate regions in the single-phase domain of the temperature-mole fraction field, in each of which the mixing scheme is qualitatively different from those of the other regions. The details of each mixing scheme were elucidated from the behavior of the second and third derivatives of Gibbs energy. In the water-rich region (I), the effect of a solute was suggested to enhance the hydrogen bonds of H2O in the vicinity of the solute but to diminish the hydrogen bond probability of the bulk H2O away from the solute. When the hydrogen bond probability at a certain region of bulk water decreases to that of the percolation threshold, this mixing scheme is no longer operative, and mixing scheme II sets in. The transition from the mixing scheme of region I to that in region (II) was found to accompany anomalies in the third derivatives of Gibbs energy, in contrast to the normal phase transitions which are associated with anomalies in the second derivatives of Gibbs energy. In the intermediate region (II), it was suggested that the solution consists of two kinds of clusters rich in each component. In the solute-rich region (III), at least for the 2-butoxyethanol and dimethyl sulfoxide cases, the clusters of purely solute molecules exist and H2O molecules are “adsorbed” on the surfaces of such clusters.
I. Introduction Understanding microscopic details in aqueous solutions is important in a wide variety of fields. Indeed, a vast amount of work has been devoted to this subject and studies up to the early 1980s were comprehensively reviewed.1-3 It is generally agreed that a nonelectrolyte solute modifies the hydrogen bond network of solvent H2O. In this article is summarized our recent experimental attempt since 1986 at adding some more insight to the mixing schemes of aqueous solutions of nonelectrolytes. By the mixing scheme, it is meant the detailed manner (as microscopic as possible) in which solute and solvent molecules mix with each other. The word “mixing scheme” is preferred in this article over “solution structure”, which has been frequently used in the literature, since the latter contains a connotation of a solid-like nature. It is emphasized that our approach is purely thermodynamical. In contrast to the general feeling, we believe a thermodynamic approach is equally useful as other techniques (spectroscopies, diffraction techniques, etc.) in understanding the mixing scheme. A similar article has been published in Japanese.4 However, the present English version contains more recent findings and new discussions. In particular a new section (X) is added. Table 1 lists thermodynamic quantities in the (p, T, ni) variable system, where ni is the amount of the i-th component. All the thermodynamic quantities are some derivatives of Gibbs † X
Phone: (604) 822-3491. Fax: 822-2847. E-mail:
[email protected]. Abstract published in AdVance ACS Abstracts, March 1, 1996.
0022-3654/96/20100-5172$12.00/0
TABLE 1: Thermodynamic Quantities of the (p, T, ni) Variable Systema 0th 1st derivative derivative 2nd derivative 3rd derivative G
H;{T} S;{T} V;{p} µi;{ni}
Cp;{T,T} Hi;{T,ni} Si;{T,ni} Vi;{p,ni} κT;{p,p} Rp;{p,T} 〈(∆xi)2〉;{ni,ni}
4th derivative
∂Cp/∂T;{T,T,T} ∂Cp(i)/∂p;{T,T,ni,p} ∂Cp/∂p;{T,T,p} ∂2Hi/∂ni2;{T,ni,ni,ni} Cp(i);{T,T,ni} ∂2Si/∂ni2;{T,ni,ni,ni} ∂Hi/∂ni;{T,ni,ni} ∂Si/∂ni;{T,ni,ni} ∂Vi/∂ni;{p,ni,ni} κT(i);{p,p,ni} Rp(i);{p,T,ni}
a Data taken from ref 4. Reproduced by permission of The Crystallographic Society of Japan.
energy, G, with respect to a variable or a combination thereof. They are classified in the table by the order of the derivative. The variables in braces after the semicolons are the variables of differentiation. For example,
H ) -T2{∂(G/T)/∂T}, hence H;{T} Cp ) T(∂S/∂T) ) -T(∂2G/∂T2), hence Cp;{T,T} The Gibbs energy, G, the zeroth order derivative, dictates the fate of an equilibrium system. The first differentiation with respect to T separates the entropy and the enthalpy effects. Further differentiation with respect to T gives the heat capacity, Cp, which is related to the degree of entropy fluctuation of the © 1996 American Chemical Society
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system. It may therefore be safe to state that the higher the order of the derivative, the more detailed the information that it contains. Indeed, Hepler used (∂Cp/∂p);{T,T,p}, a third derivative, to discuss the degree of “structure” in H2O and D2O. He further discussed whether a solute i is “structure”-making or -breaking when dissolved in H2O by the sign of {∂Cp∞(i)/ ∂p};{T,T, ni,p}, where Cp∞(i) is the partial molar heat capacity of i at infinite dilution.5
2-butoxyethanol (BE),8 2-butanone (BUT),9 and dimethyl sulfoxide (DMSO)10 were determined in this manner. III. Second Derivatives
Consider a solution (nB, nW) consisting of nB moles of nonelectrolyte B and nW moles of W (H2O). The total thermodynamic function F can be written as
As shown in Table 1, there are seven second derivatives. The excess partial molar enthalpy of B, HBE, and volume, VBE, can be determined directly and accurately by calorimetry9-16 and dilatometry,17 respectively. Exactly as the definitions (eqs 1 and 2) the thermal or the volume change is directly measured on addition of a small enough amount of B. Namely, for the process {nB of B + nW of W} + {δnB of B} f {(nB + δnB) of B + nW of W}, the enthalpy change, δq, for example, is measured. It follows that
F ) nBFB° + nWFW° + FE
δq ) H(nB + δnB,nW) - H(nB,nW) - δnBHB°
II. Partial Molar Quantities/Many-Body Problem
where FB° and FW° are the molar quantities of pure B and W, respectively. The excess function, FE (or its molar quantity, FmE ) FE/(nB + nW)), is the manifestation of complicated manybody interactions among B and W molecules. Theoretically, the detail of such many-body interactions is not yet fully understood at present. Experimentally, on the other hand, the following process is used to evaluate the actual contribution of B toward the value of F of the entire system. Change the amount of nB infinitesimally, δnB, and measure the response of the entire system in terms of F, δF, toward such a perturbation. In this process, p, T, and nW are kept constant. In the limit of δnB f 0, the quotient (δF/δnB) is nothing but the partial molar quantity of B, FB,
FB ) (∂F/∂nB)p,TnW
(1)
Or the excess partial molar quantity is written as
FBE ) (∂FE/∂nB)p,T,nW
(2)
Thus, the partial molar quantity of B is the actual contribution of B toward the global value of F of the entire system. Or one may say it represents the actual situation in terms of F that a B molecule experiences under the influence of many-body interactions from other B and W molecules. For F ) G, eq 1 defines the chemical potential of B, µB, which can be written as
µB ) µB° + RT ln xB + µBE where xB is the mole fraction of B and µBE is the excess chemical potential (partial molar Gibbs energy) of B, defined by eq 2. Conventionally it is written with the activity coefficient γB as
µBE ) RT ln γB
and
(δq/δnB) ) (∂H/∂nB) - HB° ) HBE if δnB is small enough, which can be checked by changing the size of δnB. Exactly the same principle is applied for the excess partial molar volume, VBE, using a dilatometer. (From this point on, the partial differentiation is performed by keeping other variables in (p, T, ni) constant without showing them explicitly in the subscript.) The excess partial molar entropy of B, SBE, is then calculated from µBE and HBE.6-10 VBE can also be determined by a densimeter, if the density data are measured accurately and in small increments in mole fraction to enable differentiation.18,19 Various methods are available in determining the response functions, Cp, κT , and Rp. The concentration fluctuation, 〈(∆xi)2〉 , are measured by small angle X-ray scattering or light scattering. From theoretical treatment, on the other hand, fluctuations are rigorously related to second derivatives as20-23
〈(∆S)2〉 ) kT(∂S/∂T)
(4)
〈(∆V)2〉 ) -kT(∂V/∂p)
(5)
〈(∆S)(∆V)〉 ) kT(-∂S/∂p) ) kT(∂V/∂T)
(6)
∆ signifies the difference of the local and instantaneous value from the ensemble average of that quantity. In evaluating such quantities, it is convenient to divide the system into coarse grains consisting of an equal number of molecules and to determine the quantity within this grain. Since S and V are extensive quantities, the fluctuations calculated by the right hand sides of eqs 4-6 depend on the size of the coarse grain. To avoid this ambiguity, it was proposed to normalize the fluctuations by the volume of the coarse grain.24,25 Thus, eqs 4-6 are rewritten as
〈(∆S/V)2〉 ) (kT/V2)(∂S/∂T) ) RCp/Vm2
(7)
If the equilibrium partial pressure of B, pB, is known, µBE is calculated as
〈(∆V/V)2〉 ) -(kT/V2)(∂V/∂p) ) RTκT/Vm
(8)
µBE ) RT ln{pB/(xBpB°)} + cB
〈(∆S/V)(∆V/V)〉 ) (kT/V2)(∂V/∂T) ) RTRp/Vm
(9)
(3)
where pB° is the vapor pressure of pure B at the same temperature and cB is the virial correction due to nonideality of the gas phase mixture. The value of cB is normally negligibly small in comparison with the value of the first term of the right side of eq 3, particularly for aqueous solutions of nonelectrolytes. The values of µBE for B ) tert-butyl alcohol (TBA),6,7
The last equalities on the right were obtained by converting the Boltzmann constant, k, to the molar gas constant, R, and V and S to their molar values. The actual size of the coarse grain must be small enough for fluctuations to be detected but large enough for thermodynamic quantities to be defined. An estimate indicates about 105 molecules.23
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TABLE 2: Reference Numbers of the Data of Second Derivatives in Aqueous Solutions Available in the Literature solute HiE SiE ViE Cp κT Rp 〈(∆xi)2〉
tert-butyl alcohol 13*, 14* 6*, 7* 42 33 42 26*, 37, 39, 40
22isobutyric dimethyl butoxyethanol butanone acid sulfoxide 15*, 16* 8* 18*, 19* 27*, 28*, 32 31* 29* 36, 38, 41*
9* 9* 17* 34 31* 17*
30*
10* 10* 10* 35
For the concentration fluctuation, a remaining second derivative of G, it may be more convenient to define the coarse grain with a constant number of molecules of solvent.20,23 Hence
〈(∆nB)2〉nW ) (kT)/(∂G2/∂nB2)nW
(10)
is similarly converted to
〈(∆xB)2〉nW ) (kT)/{N(∂µB′/∂xB)nW} ) RT(1 - xB)/(∂µB/∂xB)nW (11) where nW determines the size of the coarse grain, N ) nB + nW , with nB being the ensemble average, µB′ the chemical potential of one molecule of B, and µB that per mole of B, as normally obtained experimentally. As discussed below, it is more convenient to use the right hand sides of eqs 7-9 and the response functions themselves separately in discussing the peculiar nature of H2O.24 Table 2 lists the second derivatives available now in the literature for aqueous solutions of some nonelectrolytes. They will be used in the subsequent discussion. The reference numbers with * refer to our own work. IV. Third Derivatives Consider now the following derivative of HBE with respect to nB
HB-B ≡ N(∂HBE/∂nB);{T,nB,nB} ) (1 - xB)(∂HBE/∂xB) (12) This third derivative signifies the effect of additional B molecules on the value of HBE. Since enthalpy has a stability criterion, δH e 0, the sign of the above derivative has some significance. If HB-B < 0, this implies that an additional B makes the enthalpic situation of existing B more favorable than before. Thus, we may state that the B-B interaction is “attractive” in terms of enthalpy. Conversely, if HB-B > 0, then the B-B interaction is “repulsive” in terms of enthalpy. At present, there is no theory that supports the above argument rigorously without resorting to a model system. Within the mean-field approximation, for example, since the interaction enthalpy has the leading term proportional to xB2, it is elementary to show consistency with the above argument. For a binary system (B,W), the Gibbs-Duhem relation dictates that HB-B and HW-W must have the same sign, while HB-W has the opposite sign. This means that if the B-B interaction is attractive, then so should be the W-W interaction, while the B-W interaction is repulsive. The same argument is applicable for the entropy analog, except that the sign is reversed, since the stability criterion is the opposite, δS g 0. Hence, the B-B interaction is “attractive” in terms of entropy, if SB-B > 0. Since the stability criterion exists only for G, H, and S, the above argument does not apply for the remaining third derivatives with respect to nB. In such
cases, those equivalent to eq 12 signify simply the effect of an additional B on the quantity. Taking a derivative inevitably magnifies uncertainty. In order to evaluate a third derivative and to make any meaningful use of it, it is essential that the second derivative must be determined accurately and in small increments in the variable in question. For example, the excess partial molar enthalpies, HBEE, were directly determined typically within 0.1% and in the mole fraction increment of 0.001. As a result, the values of HBE-BE were evaluated within several percent, which turned out to be sufficient for elucidating the mixing scheme, as will be shown below. V. LCST/UCST When the B-B interaction becomes strongly attractive, phase separation occurs. There are two crudely classified classes. Phase separation can occur at higher temperatures with the lower critical solution temperature (LCST) or at low temperatures with the upper critical solution temperature (UCST). This difference may be understood intuitively as follows: G ()H - TS) dictates the fate of an equilibrium system. At high temperatures the TS term becomes predominant, while at low temperatures the H term determines G. For the LCST to occur, therefore, the B-B interactions must be attractive in terms of entropy and repulsive in terms of enthalpy; i.e., SB-B > 0, and HB-B > 0. In this way, the entropic B-B attraction causes phase separation at high temperatures, while the enthalpic B-B repulsion (and hence the B-W attraction) makes the system well mixed at low temperatures. Conversely, SB-B < 0 and HB-B < 0 will lead to phase separation with the UCST. A more general argument relating the way in which the phase boundary slants in the temperature-mole fraction field to the signs of the above third derivatives was presented earlier.9 This covers not only LCST and UCST but also other cases including the closed loop boundary. It will not be repeated here. Another more elementary but less intuitive explanation leading to the same conclusion for LCST and UCST cases is given in the Appendix. Thus, the signs of HB-B and SB-B can serve for understanding the mixing scheme, if a phase separation with the LCST or the UCST is known to exist. Namely, if the signs of HB-B and SB-B are consistent with the existence of the LCST or the UCST, then the mixing scheme operating in the region must be such that the solution is preparing for phase separation. VI. Liquid H2O/Hydrogen Bond Network The peculiar properties of liquid H2O and the explanations in terms of hydrogen bonds have been well documented.43-45 An additional remark may be made, however, on the response functions, Cp, κT, and Rp of H2O in comparison with those of benzene, say, and the fluctuations calculated using them.24,25 The fact that κT for H2O is 4.5 × 10-5 bar-1 and 9.6 × 10-5 bar-1 for benzene suggests rigidity of H2O. The volume fluctuation calculated by eq 8, on the other hand, is 0.0621 for H2O and 0.0268 for benzene, suggesting the presence of hydrogen-bonded patches in H2O. The volume fluctuation for H2O diminishes sharply as temperature increases and at about 70 °C exhibits a similar increase to that of benzene. This suggests that icelike patches diminish as temperature increases and hence supports the so-called “mixture model” of liquid water. The value of Cp for H2O is 75.4 J K-1 mol-1, and that of benzene, 136.1 J K-1 mol-1, while the entropy fluctuation per volume (eq 7) gives the values of 1.92 J2 K-2 cm-6 for H2O and 0.14 J2 K-2 cm-6 for benzene. Hence, the degree of entropy fluctuation properly defined is enormous for H2O in comparison with that for benzene. This suggests that the “bent
Feature Article hydrogen bond or continuum model” may be more appropriate. Rp of 2.5 × 10-4 K-1 for H2O in comparison with 12.1 × 10-4 K-1 for benzene may be understood as the breakage of a hydrogen bond accompanied by volume decrease on temperature rise. The cross-fluctuation derived from Rp as eq 9 is 0.0351 J K-1 cm-3 for H2O while that for benzene is 0.0334 J K-1 cm-3 at 25 °C. The peculiarity of H2O in this cross-fluctuation appears in its sharp temperature dependence, which is consistent with the presence of hydrogen bonds with large volume.24 Noteworthy is, however, the suggestion by Stanley et al.46 that the hydrogen bond network in liquid H2O is “percolated” at normal temperatures. This means that at any instance the hydrogen bond network is connected throughout the entire bulk of the H2O. At the same time, the positions of four hydrogenbonded H2O molecules turned out to be correlated, not random, so that tiny patches of icelike regions are present. This provides a basis for estimating the hydrogen bond probability as a function of temperature, using the existing density data. The result indicated that the hydrogen bond probability decreases as temperature increases and at about 80 °C reaches 0.39, which is the bond percolation threshold for the ice Ih structure.46 If the hydrogen bond connectivity in liquid H2O is topologically the same as that in ice Ih, this implies that H2O is a giant molecule and should be written as (H2O)∞ below about 80 °C. Other estimates47-50 on the hydrogen bond probability agree on the fact that it decreases on temperature rise, while the value varies from 0.547 to 0.950 at 0 °C. Stanley’s estimate lies approximately in the middle of the other estimates. The starting point of Stanley et al.46 is basically that of the continuum model. By taking into account the spatial distribution of various hydrogen-bonded species, however, they noted site correlation among four-bonded species, providing some support for the mixture model as well. Another noteworthy development is a recent high-resolution inelastic neutron scattering study of ice by Li et al.51-53 They observed two well separated molecular optic bands in ice I. The following model reproduced the spectra very well. Namely, in ice I there are two kinds of hydrogen bond, strong and weak, in the force constant ratio of about 2:1 and the existence ratio of about 2:1, randomly distributed throughout the ice structure. The origin of this phenomenon was sought in the way in which other protons are distributed in the nearest neighbor of the hydrogen bond in question.51-53 Another explanation may be probable noting the fact that the H-O-H angle in ice I is not far from its gaseous value, 104.5°, while the OsOsO angle is that of diamond, 109°.43 Thus, at any instant if a hydrogen atom is aligned in the OsO direction for a proper (strong) hydrogen bond, the other hydrogen atom will have to be off aligned. Whether such two kinds of hydrogen bond do indeed exist in liquid H2O is not yet certain with the present resolution of inelastic neutron scattering.52 However, as will be discussed below, if a solute causes H2O molecules in its immediate vicinity to form strong hydrogen bonds, then this would weaken the hydrogen bonds of H2O away in the bulk. VII. Mixing Schemes of Aqueous 2-Butoxyethanol By far the most extensively studied compound is aqueous 2-butoxyethanol (BE), which will be mainly discussed here. Figure 1 shows the phase diagram for BE-H2O. One of the conclusions from our work8,15,16,18,19,24,27-29,31,41 is that two more lines (broken lines in Figure 1) are necessary to separate the single phase domain into three regions, I, II, and III, in each of which the mixing scheme is qualitatively different from those of others. It should be emphasized that these three regions do not represent new phases, but they are all within a single phase
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Figure 1. Phase diagram of aqueous 2-butoxyethanol (BE) and mixing scheme diagram: s, phase boundaries; - - -, mixing scheme boundaries. The region labeled as “Two Liquids” consists of two kinds of liquid droplets, the compositions of which are those at the two end points of the isothermal tie line. “Two Solids” represents a mixture of ice and the addition compound (xBE ) 0.026, see Figure 5). “Solid Solution” is a macroscopically homogeneous solution, the components of which are H2O and BE. Microscopically, however, the addition compound may exist as a molecular unit. The other molecular unit is not known, since the phase diagram for xBE > 0.45 has not yet been determined, since solutions do not freeze into a proper solid but form glass. Reproduced with permission from ref 4. Copyright 1995 The Crystallographic Society of Japan.
domain. What is different in these three regions is the mixing scheme, the detailed way in which BE and H2O molecules mix with each other, or the short to medium range order.16 Figure 2 shows HBEE;{T,nBE}15,16 and TSBEE,8 and Figure 3 shows HWE 15,16 and TSWE,8 against xBE. It is immediately obvious from the HBEE data in Figure 2b that the thermodynamic behavior and hence the mixing scheme is different in mole fraction regions I (xBE < 0.0175), II (0.0175 < xBE < 0.46), and III (xBE > 0.46) at 25 °C. The arrows in the figure indicate apparent boundaries. Such a realization was only possible because HBEE data were available. The earlier data of the excess (integral) molar enthalpy, HmE ) xBEHBEE + (1 - xBE)HWE, did not show the distinction of three regions conspicuously enough. Of course, HmE and HBEE are mathematically related as
HBEE ) (∂HE/∂nBE) ) HmE + (1 - xBE)(∂HmE/∂xBe)
(13)
In reality, however, with even high-quality data for HmE, graphical differentiation introduces a large error, particularly in the dilute region. Moreover, if there are indeed three regions of different thermodynamic behavior, a conventional practice of fitting HmE data with a single analytic function leads to erroneous results in HBEE when eq 13 is used, particularly at the boundary regions. As is evident in Figure 2b, HBEE is constant and zero, and TSBEE is also almost zero in region III, xBE > 0.46 at 25 °C. This suggests that the environment for BE is the same as that in pure BE in terms of enthalpy and entropy. Hence BE molecules seem to cluster together, perhaps in a micellar form. Thus, an incoming BE molecule settles within this cluster of purely BE. On the other hand, the dependence of HWE and SWE shown in Figure 3 suggests that H2O molecules are
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Figure 3. Excess partial molar enthalpy, HWE, and entropy (times temperature), TSWE, of H2O in BE-H2O at 25 °C: 9, HWE (ref 15); 2, TSWE (ref 8). Reproduced with permission from ref 4. Copyright 1995 The Crystallographic Society of Japan.
Figure 2. Excess partial molar enthalpy, HBEE, and entropy (times temperature), TSBEE, of BE in BE-H2O at 25 °C. The arrows indicate the apparent boundary at which thermodynamic behavior seems to change: 9, HBEE, (ref 15); 2, TSBEE (ref 8). Reproduced with permission from ref 4. Copyright 1995 The Crystallographic Society of Japan.
“adsorbed” on the surfaces of such BE clusters with a constant heat and a somewhat coverage dependent entropy of “adsorption”. Namely, H2O molecules are acting as a single molecule, not as molecules hydrogen bonded to each other in this region. As shown in Figure 1, the LCST is 50 °C in region II, which means that the B-B interaction must be attractive in terms of entropy but repulsive in terms of enthalpy. Indeed, Figure 2b indicates HBE-BE > 0, and SBE-BE > 0. In a sense, therefore, the solution is properly behaving in preparation for phase separation at higher temperatures. Thus, the solution may consist of two kinds of clusters, rich in each component, preparing for phase separation above the LCST. The percolation present in pure H2O no longer exists in this region. As the temperature and the composition approach the LCST, these clusters grow in size and become macroscopic at the LCST. Light scattering36,38 and small angle X-ray scattering (SAXS)54 indicated the forward scattering intensity increases sharply near the LCST. For tert-butyl alcohol (TBA)-H2O, the angle dependence of SAXS indicated the size of such a cluster is about 30 Å.39 It is also interesting to note that when the BE-H2O solution in this region is cooled toward the freezing point, the small angle X-ray scattering intensity also increases sharply.41 The mixing scheme operating in region I is consistent with
the commonly adopted explanation of “iceberg formation” or the enhancement of the hydrogen bond network.1,3,44,45 However, a small modification appears to be necessary, as discussed later. Figure 2a shows that the values of HBEE and TSBEE extrapolated to xBE ) 0 are -17 and -26 kJ mol -1. Namely, the first BE molecule enters H2O with a large enthalpy gain and a larger entropy (× temperature) loss. Such a large enthalpy gain and entropy loss can hardly be explained by the process by which a BE molecule breaks a hydrogen bond away from other BE molecules in the pure state and forms hydrogen bonds with a limited number of surrounding H2O molecules.13 The majority of this enthalpy gain and entropy loss must be coming from the effect of BE on the surrounding H2O. Namely, the hydrogen bonds among H2O in the vicinity of BE are enhanced. The second BE molecule, then, would settle far away from the first, so as to gain enthalpy equally. In terms of entropy, on the other hand, it would come close to the first to avoid a large entropy loss. If both effects balance out, i.e., |HBEE| ) |TSBEE|, BE molecules would distribute randomly in water. Since the entropy term is predominant, BE molecules tend to come closer than random distribution with hydrogen bond enhanced H2O in between them, completely consistent with “hydrophobic attraction”.1,3 The isotope effects on HBEE in this region also support the above view.55 Temporarily we use BEOH instead of BE as the abbreviation for 2-butoxyethanol. There was found no isotope effect when BEOH or BEOD was used. When D2O was used in place of H2O, the values of HBEE were about -1 kJ mol-1 more negative than those when BE was added to H2O. This supports the fact that HBEE is mainly determined by the nature of the solvent H2O (or D2O) but not by the OH (or OD) part of BE. Moreover, the fact that HBEE is more negative in D2O than in H2O is consistent with the enhancement of hydrogen bonds in solvent water, since the hydrogen bond strength is believed to be stronger in D2O than in H2O.44-46 Figure 4 shows the BE-BE interactions in terms of enthalpy and entropy at 25 °C. The fact that TSBE-BE > HBE-BE > 0 indicates the BE-BE interaction is attractive in terms of entropy but repulsive in terms of enthalpy and that the entropy attraction surpasses the enthalpy repulsion is completely consistent with the above arguments. The other important point to note is that the BE-BE interactions, both enthalpy repulsion and entropy attraction, become stronger as xBE increases and then suddenly
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Figure 4. BE-BE interactions in terms of enthalpy, ∂HBEE/ ∂nBE;{T,nBE,nBE}, and entropy, T∂SBEE/∂nBE at 25 °C: 9, ∂HBEE/ ∂nBE;{T,nBE,nBE} (ref 15); 2, T∂SBEE/∂nBE;{T,nBE,nBE} (ref 8). Reproduced with permission from ref 4. Copyright 1995 The Crystallographic Society of Japan.
collapse at xBE ) 0.0175 at 25 °C. This suggests that something drastic occurs at this point. We suggest that this anomaly marks the transition point separating regions I and II. These peaks in Figure 4 have a similar appearance to heat capacity anomalies associated with phase transitions. Desnoyers et al. observed that the partial molar heat capacity of BE, Cp(BE);{T,T,nBE}, showed a sharp peak at the same locus32 and noted that it resembles the phase transition.56 It should be emphasized, however, that the transition from mixing scheme I to mixing scheme II is not the phase transition, in which a long range order parameter changes. Rather, it is the transition of the mixing scheme, in which some short to medium range order may be what changes.16,27 As will be suggested later, this transition may involve loss of hydrogen bond percolation. Another important difference to note is that the phase transitions are associated with anomalies in the second derivatives, while the present transition of the mixing scheme appears to be accompanied by anomalies in the third derivatives. This point is further pursued in the next section. The boundary between regions II and III was found to be more subtle, associated with a step anomaly in the fourth derivative.8 VIII. Transition of Mixing Scheme from I to II The boundary between mixing schemes I and II seems to accompany peak anomalies in ∂HBEE/∂nBE;{T,nBE,nBE}, ∂SBEE/ ∂nBE;{T,nBE,nBE}, and CpE(BE);{T,T,nBE}, all of which are the third derivatives of G. It turned out that the following three third derivatives also showed peak anomalies at about the same locus: ∂VBEE/∂nBE;{p,nBE,nBE},19 ∂κT/∂nBE;{p,p,nBE},31 and ∂Rp/ ∂nBE;{p,T,nBE}.29 On the other hand, ∂Cp/∂T;{T,T,T} showed step anomalies.27, 28 In Figure 5 all the loci of anomalies in the third derivatives are collected. Also included are the loci at which various dynamic properties change. Thermal57 and ionic conductivities57,58 showed a sharp change in slope. Foam stability also made a sharp change.59 The hydrodynamic radii of diffusing species determined by ultra sonic relaxation60 and by dynamic light scattering61 were two orders of magnitude larger in region I than those in region II. Thus, there is little
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Figure 5. Boundary between mixing schemes I and II for BE-H2O. The loci of the anomalies in various third derivatives and other dynamic properties: circle, ∂HBEE/∂nBE (ref 16); square, estimates of ∂HBEE/∂nBE (ref 19); triangle, ∂HBEE/∂nBE (ref 15); upside down triangle, ∂SBEE/ ∂nBE (ref 8); diamond, ∂VBEE/∂nBE (ref 19); hexagon, ∂Rp/∂nBE (ref 29); circle with a vertical line through it, ∂Cp/∂T (ref 27); square with a vertical line through it, ∂κT/∂nBE (ref 31); triangle with a line through it, ∂Cp/∂T (ref 28); upside down triangle with a line through it, ∂Cp/∂T (ref 63); diamond with a line through it, ∂Cp/∂nBE (ref 32); hexagon with a line through it, ionic conductivity (ref 58); circle with a horizontal line through it, thermal conductivity (ref 57); square with a diagonal line through it, foam stability (ref 59).
doubt that the mixing scheme in BE-H2O changes at the boundary shown in Figure 5. The transition of the mixing scheme is not unique to BEH2O. For aqueous tert-butyl alcohol (TBA), the loci of peak anomalies in the following third derivatives are shown in Figure 6: ∂HTBAE/∂nTBA;{T,nTBA,nTBA},13,14 ∂STBAE/∂nTBA,6,7 ∂VTBAE/ ∂nTBA;{p,nTBA,nTBA},42 ∂Rp/∂nTBA;{p,T,nTBA},42 and ∂Cp/ ∂nTBA;{T,T,nTBA}.33 For aqueous isobutyric acid (IBA), there were step anomalies in ∂HIBAE/∂nIBA, as shown in Figure 7.30 In this case, ∂HIBAE/∂nIBA < 0 in region II, naturally leading to phase separation at the UCST, which indeed occurs as shown in Figure 8. In region I, on the other hand, ∂HIBAE/∂nIBA > 0, as in region I for the other solutes. A similar step anomaly was found at xBUT ) 0.033 between mixing schemes I and II for aqueous 2-butanone (BUT).9,17 For aqueous dimethyl sulfoxide (DMSO), ∂HDMSOE/∂nDMSO;{T,nDMSO,nDMSO} and the entropy analog showed a sort of step anomaly at xDMSO ) 0.28 at 25 °C,10 as shown in Figure 9. The type of anomalies mentioned so far are of peak, as in Figure 4, or of step type, as in Figures 7 and 9. If a third derivative in region I increases and that in region II takes a smaller value, then the transition appears as a peak. If, on the other hand, a third derivative stays constant in region I, then the anomaly would be of a step type shown in Figure 7, while if a third derivative decreases in region I, then a step anomaly looks like that shown in Figure 9. Thus, the type of anomaly depends on the detail of the behavior in region I and may not be important for the transition of the mixing scheme itself. Indeed, Figure 1062 indicates the anomaly in ∂HBEE/∂nBE changes its shape from peak to step as a third component, DMSO, is added to BE-H2O, while the locus of the anomaly stays almost constant. We may therefore generalize that the transition of
5178 J. Phys. Chem., Vol. 100, No. 13, 1996
Figure 6. Boundary between mixing schemes I and II for tert-butyl alcohol-H2O. The loci of peak anomalies of the following third derivatives: O, ∂HTBAE/∂nTBA and ∂STBAE/∂nTBA (refs 7 and 13; 4, ∂VTBAE/∂nTBA (estimated using data in ref 42); 0, ∂Rp/∂nTBA (estimated using data in ref 42); 3, ∂Cp/∂nBE (estimated using data in ref 33). Reproduced with permission from ref 4. Copyright 1995 The Crystallographic Society of Japan.
Koga
Figure 8. Phase and mixing scheme diagram for IBA-H2O: s, phase boundary; - - -, mixing scheme boundary (ref 30). Reproduced with permission from ref 4. Copyright 1995 The Crystallographic Society of Japan.
Figure 7. IBA-IBA interaction in terms of enthalpy, ∂HIBAE/∂nIBA, for isobutyric acid (IBA)-H2O at 25 °C. The arrow indicates the locus of the step anomaly.30 Reproduced with permission from ref 4. Copyright 1995 The Crystallographic Society of Japan.
the mixing scheme from I to II is accompanied by some form of anomalies in the third derivatives of the Gibbs energy. IX. Some Details of Mixing Scheme I in BE-H2O
Figure 9. DMSO-DMSO interaction in terms of enthalpy, ∂HDMSOE/ ∂nDMSO, for dimethyl sulfoxide (DMSO)-H2O at 25 °C. The arrows indicate the mixing scheme boundaries.10 Reproduced with permission from ref 4. Copyright 1995 The Crystallographic Society of Japan.
Two features are noteworthy in Figure 5. The mixing scheme boundary for BE-H2O ends at the incongruent melting point 269.5 K of an addition compound of xBEadd ) 0.0260,63 1:36 in terms of mole ratio. This value is approximately equal to the number of H2O molecules tightly surrounding a BE molecule, assuming both to be hard spheres with the values of molar volume in their respective pure states. The addition compound therefore may be a clathrate. This fact must have a close bearing on mixing scheme I, in which a BE molecule enhances the hydrogen bonds in its vicinity, or rather, one may say that the BE-H2O mixture is prepared in region I for the formation of this clathrate below freezing. The other end of the mixing scheme boundary seems to end at about 85 °C at xBE ) 0. For TBA-H2O also, the boundary seems to extrapolate to about 85 °C, as shown in Figure 6. The mixing scheme boundary for BE-H2O above 60 °C is only an estimate19 using the HBEE and Cp(BE) data below 60 °C.16
Nevertheless, this value of 85 °C is close to the percolation threshold in pure H2O, as mentioned above. Namely, at xBE ) 0 the hydrogen bond network is percolated below about 80 °C, while the percolation breaks down above that temperature. This hints that mixing scheme I may retain the percolation nature of pure H2O. As discussed above, the solution consists of two kinds of clusters in region II and is preparing for phase separation at higher temperatures. Thus there is no percolation in region II. The mixing scheme boundary between I and II may therefore be the percolation threshold. As shown in Figures 2 and 4, the BE-BE interaction in terms of enthalpy and entropy becomes stronger as xBE increases up to the mixing scheme boundary. This behavior may be explained by the local enthalpy and entropy profile in space around BE molecules as sketched in Figure 11. Note that the ordinate is in enthalpy gain and entropy loss. Prior to the
Feature Article
Figure 10. BE-BE interactions in terms of enthalpy, ∂HBEE/∂nBE, in BE-DMSO-H2O: b, xDMSO ) 0.02499; 9, xDMSO ) 0.05450; 2, xDMSO ) 0.08993.62 Reproduced with permission from ref 4. Copyright 1995 The Crystallographic Society of Japan.
Figure 11. Enthalpy gain and entropy loss profile due to the presence of solute BE. Reproduced with permission from ref 4. Copyright 1995 The Crystallographic Society of Japan.
presence of the first BE molecule, the enthalpy and entropy profiles take a global average independent of location in space, except for small fluctuations. The first BE molecule would alter the profile as shown in Figure 11. The tail of the profile extends to infinity via a percolated hydrogen bond network, as will be discussed in the next section. The second BE molecule, due to the entropy-enthalpy imbalance discussed above, comes closer to the first than in the random distribution. The degree of alteration of the profile due to the second BE is smaller than that caused by the first. Thus, the second BE has less enthalpy gain and less entropy loss; i.e., ∂HBEE/∂nBE > 0, and ∂SBEE/ ∂nBE > 0. This trend continues until xBE reaches the boundary. Earlier,9,17 the enthalpy and entropy profile shown in Figure 11 was simply taken to be proportional to the hydrogen bond probability profile. Namely, the ordinate of Figure 11 was labeled as “the hydrogen bond probability” in earlier papers.9,17 If such is the case, the bulk of H2O becomes more and more icelike as xBE increases, and eventually at the boundary, hydrogen bonding among H2O may be complete. This argu-
J. Phys. Chem., Vol. 100, No. 13, 1996 5179 ment, however, causes self-contradiction at xBE ) 0 on the boundary. In pure H2O, the hydrogen bond probability at 80 °C is 0.39, the percolation threshold, while at the boundary at xBE ) 0 + δ the hydrogen bond probability is 1. Another difficulty for this model is that the value of xBE at the boundary should increase as temperature increases, since the higher the temperature the lower the hydrogen bond probability in pure H2O and thus the more BE molecules needed to make the hydrogen bonds complete. The actual findings are the complete opposite, as shown in Figure 5. The boundary slants to smaller values of xBE as temperature increases. Furthermore, a recent study on the molar conductivity of H+OH- at infinite dilution58 suggested that the hydrogen bond probability of bulk H2O in fact decreases smoothly to the boundary as xBE increases. The entropy and volume fluctuations as written by eqs 7 and 8 showed a smooth decrease as xBE increased,24 the same trend as the temperature rise for pure H2O, suggesting also the hydrogen bond probability decreases. We note in passing that Cp data themselves increase32 up to about xBE ) 0.016, while κT decreases,31 indicating the contradicting effects of BE. When the respective fluctuations are properly calculated by eqs 7 and 8, however, both fluctuations point to the consistent picture.24,25 Namely, the solute BE reduces both fluctuations in aqueous solutions. These contradictions were originated from converting the ordinate of Figure 11 directly to the hydrogen bond probability. This was done under the assumption that there is only a single kind of hydrogen bond. If we allow some distribution in hydrogen bond strength inherent to the “continuum model” (or perhaps two-tiered as suggested for ice I51-53), the above difficulties may be reconciled. Namely, the enthalpy gain and entropy loss profile in Figure 11 is regarded as a product of the hydrogen bond probability profile times the hydrogen bond strength profile. If the hydrogen bonds in the vicinity of BE are stronger than those in the bulk away from BE, for example,64 then it is quite feasible that the hydrogen bond probability becomes less than that of pure H2O in the bulk away from the solute BE. As xBE increases, the hydrogen bond probability may eventually reach the value of the percolation threshold in such regions; whereupon, mixing scheme II, consisting of two kinds of clusters, sets in. If we recognize the boundary between I and II (Figure 5) to be the percolation threshold, then the observed manner in which the boundary slants can naturally be explained. Namely, at higher temperatures, at which the hydrogen bond probability is smaller already in pure H2O, a smaller amount of BE is required to bring the hydrogen bond probability down to the value of the percolation threshold. As pointed out above, the difference in shape of the third derivatives is essentially coming from the details of the solutesolute interactions operating in region I. This may be reworded in terms of the enthalpy gain and the entropy loss profile sketched in Figure 11. The essence of the sketch (Figure 11) is that the total amount that is altered by the second solute is less than that caused by the first. This feature is applicable to all the above cases. The rate of change in the amounts altered by successive addition of the solute molecules, however, may increase (for BE and TBA), stay constant (for IBA and BUT), or decrease (for the DMSO case). The first brings about a peak anomaly at the boundary, as shown in Figure 4, the second, a step anomaly, as shown in Figure 7, and the third, a kind of step, as shown in Figure 9. These differences come about from subtle differences in curvature of the enthalpy gain and entropy loss profile shown in Figure 11 and in turn of the hydrogen bond probability profile and the hydrogen bond strength profile. These differences must be caused by the differences in competi-
5180 J. Phys. Chem., Vol. 100, No. 13, 1996
Koga the interaction potential is infinitely small and infinite in range.67 Thus, it appears highly likely that there is no Henry’s law region in aqueous solutions. Such a long range interaction may only be possible via a “percolated” hydrogen bond network. XI. Conclusion As demonstrated above, the second and higher order derivatives of Gibbs energy are useful in elucidating the mixing schemes in aqueous solutions of nonelectrolytes. There is no reason why this methodology should not be applied for other solutions. Indeed, studies have been begun on aqueous solutions of biopolymers and electrolytes. In our work summarized above, p in the (p, T, ni) variable system has been inadvertently kept constant. Studies on the p-dependency of these quantities will no doubt add important information. The mixing schemes elucidated above purely by thermodynamic studies may be quite relevant to NMR work.64,68-70 Other spectroscopic works and diffraction studies (particularly neutron scattering) on aqueous solutions are awaited.
Figure 12. Dependence of HBE and TSBE as a function of xB for the UCST case. Reproduced with permission from ref 4. Copyright 1995 The Crystallographic Society of Japan.
Acknowledgment. I wish to thank all my co-workers: the late Aase Hvidt, Peter Westh, and the members of the Hvidt group, University of Copenhagen; Jim Davies and his students, The University College of the Cariboo; Tooru Atake and his students, Tokyo Institute of Technology; Keiko Nishikawa and her students, Yokohama National University; Sachio Murakami and Katsutoshi Tamura, Osaka City University; last but not least, my many students at the University of British Columbia. I have learned much from discussions with Loren G. Hepler, University of Alberta, Tadashi Kato, Tokyo Metropolitan University, and Sadao Takagi and Takayoshi Kimura, Kinki University. This research was supported by the NSERC of Canada, the NATO collaborative research program, and the Ministry of Education, Science and Culture, Japan. This paper is dedicated to Professor Emeritus of Tokyo University, Dr. Shoji Makishima. I learned the very essence of the many-body problem through his “Pattern Dynamics”.71 Appendix: Phase Separation The diffusional stability criterion dictates ∂µB/∂xB g 0.
Figure 13. Dependence of HBE and TSBE as a function of xB for the LCST case. Reproduced with permission from ref 4. Copyright 1995 The Crystallographic Society of Japan.
tive effects of the hydrophobic and hydrophilic ends of each solute. The details are yet to be elucidated. X. Henry’s Law Region in Aqueous Solutions? In a sufficiently dilute solution, it is expected that the solutesolute interactions are absent or can be ignored. Then the solution is said to be in the Henry’s law region. Thus, the excess chemical potential, µBE, and the excess partial molar enthalpy, HBE, and entropy, SBE, of B would stay constant in a dilute limit. For TBA-H2O, HBE did not show any sign of convergence to a constant value for xTBA > 1 × 10-3 .13 The vapor pressure measurements indicated that at xTBA ) 7 × 10 -5 the solution is definitely not yet in the Henry’s law region.65,66 The presence of a single molecule of TBA is felt by at least 15 000 molecules of H2O. The excess partial molar enthalpy measurements indicated that the H2O-D2O mixture behaves as a strictly regular solution, in spite of the fact that the hydrogen bonds are stronger and more abundant in D2O than H2O.11 This means that the intermolecular interactions operating in the H2O-D2O mixture are of a mean-field type, which is rigorously true when
µB ) µB° + RT ln xB + µBE
(A1)
µBE ) HBE - TSBE
(A2)
Note that µBE ) 0, HBE ) 0, and SBE ) 0 at xB ) 1. Differentiation of eq A1 with respect to xB gives
∂µB/∂xB ) RT/xB + ∂µBE/∂xB
(A3)
The first term of eq A3 is always positive. If the second term ∂µBE/∂xB < 0, ∂µB/∂xB may become negative for a certain range in xB. Phase separation then occurs in that range to keep ∂µB/ ∂xB non-negative. The two simplest cases for ∂µBE/∂xB < 0 are that HBE and TSBE change as a function of xB, as sketched in Figures 12 and 13. Case A1: As is evident from Figure 12, HB-B ≡ ∂HBE/∂xB < 0, and SB-B ≡ ∂SBE/∂xB < 0; the B-B interaction is attractive in terms of enthalpy and repulsive in terms of entropy. For ∂µBE/∂xB < 0, HBE must stay larger than TSBE, which may be violated at high enough temperatures. For if TSBE > HBE, then the sign of ∂µBE/∂xB will become positive, and no phase separation is needed to obey the diffusional stability criterion. Hence this case brings about the UCST.
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