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Pure and Applied Chemistry

The Scientific Journal of IUPAC

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Volume 88, Issue 3

Issues

Shedding light on the hydrophobicity puzzle

Giuseppe Graziano
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  • Dipartimento di Scienze e Tecnologie, Università del Sannio, Via Port’Arsa 11 – 82100 Benevento, Italy
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Published Online: 2016-02-08 | DOI: https://doi.org/10.1515/pac-2015-1003

Abstract

A general theory of hydrophobic hydration and pairwise hydrophobic interaction has been developed in the last years. The main ingredient is the recognition that: (a) cavity creation (necessary to insert a solute molecule into water) causes a solvent-excluded volume effect that leads to a loss in the translational entropy of water molecules; (b) the merging of two cavities (necessary to form the contact minimum configuration of two nonpolar molecules) causes a decrease in the solvent-excluded volume effect and so an increase in the translational entropy of water molecules. The performance of the theoretical approach is tested by reproducing both the hydration thermodynamics of xenon and the thermodynamics associated with the formation of the contact minimum configuration of two xenon atoms, over a large temperature range.

Keywords: cavity creation; entropy; hydrophobic hydration; hydrophobicity; ICSC-34; pairwise hydrophobic interaction; small size of water molecules; solvation; solvent-excluded volume effect; water

Article note:

A collection of invited papers based on presentations at the 34th International Conference on Solution Chemistry (ICSC-34), Prague, Czech Republic, 30th August–3rd September 2015.

Introduction

The hydrophobic effect is still considered to be the main driving force and stabilizing interaction of protein folding, formation of micelles and double-layer membranes and molecular recognition processes [1–3]. It is usually divided into two arms: the so-called hydrophobic hydration and hydrophobic interaction [1, 4]. Hydrophobic hydration refers to the transfer of a nonpolar solute from a fixed position in the gas phase to a fixed position in water [5], and has peculiar thermodynamic features: the most famous of which is the large and negative entropy change that leads to a large and positive Gibbs energy change (i.e. to the poor solubility of nonpolar species in water). Moreover, a large and positive heat capacity change affects both the hydration enthalpy and entropy changes, but has little effect on the hydration Gibbs energy change due to enthalpy–entropy compensation [1, 2]. In any case, the hydration Gibbs energy change increases over the 0–100°C temperature range, at 1 atm. On the other hand, hydrophobic interaction (HI) refers to the process in which several nonpolar molecules cluster together in water to avoid the contact with water molecules and should occur spontaneously, characterized by a negative Gibbs energy change (note that HI cannot be investigated from the experimental point of view for purely nonpolar solutes in view of their poor solubility in water) [4]. In particular, theoretical approaches and computer simulations have been used to study the association of two small nonpolar molecules, such as methane and neopentane, in water (i.e. pairwise HI) [4, 6–12]. It emerged that pairwise HI is characterized by a potential of mean force (i.e. the trend of the Gibbs energy as a function of the distance between the two molecules) that presents a minimum at the contact configuration, a maximum at the desolvation barrier and a second minimum at the solvent-separated configuration [6–12]. It is widely recognized that the formation of the contact minimum, (cm) configuration is characterized by a positive entropy change that drives the association. The latter proves to be increasingly favored on increasing the temperature [10, 11].

All these features can be rationalized by means of a single statistical mechanical approach [13–20]. The fundamental ingredient is the solvent-excluded volume effect associated with cavity creation, whose magnitude is amplified in water by the small size of water molecules in comparison to that of the other common liquids [13–15]. The solvent-excluded volume effect leads to a decrease in the accessible configurational space, and so to a large decrease in the translational entropy of water molecules. This is the molecular origin of the large and negative entropy change associated with the hydration of nonpolar solutes. It is a basic geometric fact that the solvent-excluded volume effect is correlated to the water accessible surface area (WASA), of the solute molecule [18, 21]. This geometric evidence allows a simple rationalization of HI. The association of two or more nonpolar molecules is entropically favored around room temperature due to the large WASA decrease associated with “complex” formation, that leads to a gain of translational entropy for water molecules [18–20].

To test the suitability of the devised theoretical approach, I have selected: (a) the hydration of xenon over the 0–100°C temperature range, at 1 atm, because experimental data of very good precision are available [22] (see Table 1 and Fig. 1); (b) the pairwise HI of xenon over the 0–100°C temperature range, because a careful computational investigation was performed by Paschek [11], and can be used as benchmark. In both cases the thermodynamic quantities calculated by means of the devised theoretical approach are in line with the “experimental” ones, over the whole considered temperature range. The “take home message” is that an entropy gain or loss in water (and in a liquid in general) should not be automatically considered the signature of a decrease or an increase, respectively, of structural order. They should be associated with an increase or decrease in the accessible configurational space for water molecules, as a consequence of the solvent-excluded volume effect.

Table 1

Experimental hydration thermodynamic functions of Xe, according to the Ben–Naim standard (i.e. transfer from a fixed position in the gas phase to a fixed position in water), at 1 atm, over the 0–100°C temperature range, from the study of Krause and Benson [22].

Trend of the experimental thermodynamic functions associated with Xe hydration, at 1 atm, over the 0–100°C temperature range, according to the Ben–Naim standard [22].
Fig. 1:

Trend of the experimental thermodynamic functions associated with Xe hydration, at 1 atm, over the 0–100°C temperature range, according to the Ben–Naim standard [22].

Theoretical approach

Hydrophobic hydration

The process of transferring a solute molecule from a fixed position in the gas phase to a fixed position in water (i.e. hydration according to the Ben–Naim standard [5]) is dissected into two sub-processes [13–16]: (a) creation in water of a cavity suitable to host the solute molecule; (b) insertion of the solute molecule into the cavity and turning on the solute–water attractive potential. When the latter is weak in comparison to the water–water H-bonds (a condition satisfied by nonpolar solutes in water), the reorganization of water–water H-bonds is characterized by an almost complete enthalpy–entropy compensation and does not affect the overall Gibbs energy change [13, 14, 23]. Thus, the Ben–Naim standard hydration Gibbs energy change, ΔG˙, is given by:

ΔG=ΔGc+Ea (1)(1)

where ΔGc is the reversible work of cavity creation, and Ea is the solute–water attractive potential energy. The two terms, ΔGc and Ea, represent, respectively, the direct entropic and energetic perturbations of water caused by solute insertion.

Cavity creation, at constant temperature and pressure, even though it leads to an increase of the liquid volume equal to the cavity partial molar volume, produces a geometric constraint for the liquid molecules. The centers of the latter cannot enter the spherical shell between the van der Waals surface of the cavity and its solvent accessible surface area (WASA in water) [18]. This geometric constraint produces a solvent-excluded volume that leads to a significant decrease in the accessible configurational space and so to a translational entropy loss for liquid molecules. Therefore, ΔGc is a purely entropic quantity given by [24]:

ΔGc=TΔSx (2)(2)

where ΔSx is the entropy change due to the solvent-excluded volume effect. In response to the direct perturbation, there is a reorganization of water–water H-bonds that produces both enthalpy ΔHr and entropy ΔSnx contributions. Thus, the total hydration enthalpy and entropy changes are given by:

ΔH=Ea+ΔHr (3)(3)

ΔS=ΔSx+ΔSnx (4)(4)

The notation ΔSnx emphasizes that this is the non-excluded volume entropy contribution, originating from the water–water H-bond reorganization. The latter is characterized by an almost complete enthalpy–entropy compensation [13–16, 23], and so:

TΔSnx=ΔHr (5)(5)

In addition, the weakness of the solute–water attractive potential implies that the water–water H-bond reorganization is mainly due to cavity creation, so that ΔHr≈ΔHc and ΔS˙≈ΔSc, as originally devised by Pierotti [25] (note that ΔSc is made up of both ΔSx and ΔSnx contributions; for a complete statistical mechanical demonstration see [26]).

Estimates for ΔGc and ΔHc are calculated by means of the classic scaled particle theory [25–28], SPT, formulas to create a spherical cavity in a liquid (neglecting the pressure-volume term for its smallness at P=1 atm):

ΔGc=RT{ln(1ξ)+[3ξ/(1ξ)]x+[3ξ/(1ξ)]x2+[9ξ2/2(1ξ)2]x2} (6)(6)

ΔHc=[RT2ξαP/(1ξ)3][(1ξ)2+3(1ξ)x+3(1+2ξ)x2] (7)(7)

where R is the gas constant, αP is the isobaric thermal expansion coefficient of the solvent, ξ is the volume packing density of the solvent, which is defined as the ratio of the physical volume of a mole of solvent molecules over the molar volume of the solvent, v1 (i.e. ξ=π·σ13·NAv/6·v1); x21, and σ1 is the hard-sphere diameter of the solvent molecules; σ2 is the hard-sphere diameter of the solute molecule and corresponds to the cavity diameter, defined as the diameter of the spherical region from which any part of the solvent molecules is excluded. Calculations have been performed using the experimental values of the density and αP of water at 1 atm [29] (see Table 2 for a compilation of several physical properties of water; the surface tension values come from [30]), and the following hard-sphere diameters: σ(H2O)=2.8 Å, close to the location of the first peak in the oxygen-oxygen radial distribution function of water [31, 32]; σ(Xe)=4.0 Å, as fixed by Pierotti [25]. Both are assumed to be temperature-independent. The cavity entropy change is readily calculated by means of ΔSc=(ΔHc–ΔGc)/T.

Table 2

Experimental values of the molar volume, isobaric thermal expansion coefficient, isothermal compressibility [29], and liquid–vapor surface tension [30], at 1 atm, over the 0–100°C temperature range, for water (part A), and carbon tetrachloride (part B).

The Ea value for the Xe–water attractive potential energy, at 20°C and 1 atm, –20.5 kJ mol−1, has been fixed using Eq. (1), the experimental ΔG˙ number and the classic SPT-ΔGc estimate. This value is considered to be temperature-independent, and is close to both the number, –21.5 kJ mol−1, obtained by means of MD simulations [33] in the SPC/E water model [34], and to the number, –20.2 kJ mol−1, calculated by means of the Pierotti’s formula [25] at 20°C, using the above diameters and ε/k=120 K for water, and 230 K for Xe [35]. Note that, according to the Pierotti’s formula, Ea(100°C)=Ea(20°C)·ξ (100°C)/ξ (20°C)=–19.4 kJ mol−1, with a very small decrease in magnitude; this supports the assumption to consider Ea temperature-independent in water.

Pairwise hydrophobic interaction

Bringing two nonpolar solutes, such as two Xe atoms, from a fixed position at infinite separation to a fixed position at contact distance in water, keeping constant temperature and pressure, is called pairwise HI [4]. The associated Gibbs energy change is given by:

ΔG(HI)=E(cm)+δG(HI) (8)(8)

where E(cm) is the Xe–Xe van der Waals interaction energy in the contact-minimum, cm, configuration, and should not depend on the presence of the solvent and its nature; δG(HI) is the indirect part of the reversible work to do the process, and accounts for the specific features of the solvent in which pairwise HI occurs. A general relationship exists between δG(HI) and the Ben–Naim standard hydration Gibbs energy of the Xe–Xe cm configuration and of the Xe atom, respectively [4, 19, 20]:

δG(HI)=ΔG(XeXe)2ΔG(Xe) (9)(9)

Use of Eq. (1) in the definition of δG(HI) leads to:

δG(HI)=[ΔGc[XeXe]2ΔGc(Xe)]+[Ea(XeXe)2Ea(Xe)]=δGc(HI)+δEa(HI) (10)(10)

Clearly, δG(HI) is not affected by the water–water H-bond reorganization because the latter is a compensating process [4]. The δG(HI) value could be obtained by calculating: (a) ΔGc to create in water a cavity suitable to host a couple of Xe atoms in the cm configuration, and a cavity suitable to host a single Xe atom; (b) Ea to turn on the attractive interactions between a couple of Xe atoms in the cm configuration and all the surrounding water molecules, and between a single Xe atom and all the surrounding water molecules. Actually, I have devised a different procedure.

A decrease in solvent-excluded volume occurs in the cm configuration of two spheres, and it can be accounted for by the corresponding WASA decrease. The WASA relevance is confirmed by the occurrence of a robust linear relationship between the Gibbs energy of the cm configuration for the pairwise HI of 13 alkane pairs and the average amount of buried WASA [36]. Moreover, both classic SPT calculations [18], and computer simulations using detailed water models [37, 38] indicate that: (a) by keeping the cavity VvdW fixed, the ΔGc magnitude depends upon cavity shape, and proves to be proportional to the cavity WASA; (b) the value of the ΔGc/WASA ratio calculated for spherical cavities can be used, to a good approximation, also for non-spherical cavities. The WASA decrease due to pairwise HI is given by:

ΔWASA=2WASA(Xe)fWASA (11)(11)

where WASA(Xe) is the WASA of an Xe atom and fWASA is the fraction of WASA of a single cavity that becomes not accessible to water molecules when the two spherical cavities approach each other; fWASA depends upon their center-to-center distance according to this analytical formula [19, 20]:

fWASA=2π(1cosα)/4π (12)(12)

where cosα=d(center-to-center)/[σ(sphere)+σ(H2O)], σ(sphere)=σ(Xe)=4.0 Å, and σ(H2O)=2.8 Å; both diameters are considered to be temperature-independent). Thus, WASA(Xe)=4π·(2.0+1.4)2=145 Å2, fWASA=0.206 at the cm configuration [i.e. d(center-to-center)=4.0 Å], and ΔWASA=–59.7 Å2. On these grounds, the δGc(HI) quantity is given by:

δGc(HI)=[ΔGc(Xe)/WASA(Xe)]ΔWASA=ΔGc(Xe)(1cosα) (13)(13)

To reproduce the Gibbs energy of the cm configuration of some alkanes and fullerene at 1 atm and room temperature, I assumed that also δEa(HI) scales linearly with ΔWASA [19, 39, 40]. However, to achieve a close agreement with computer simulation results, ΔWASA had to be multiplied by a factor equal to 1.2 [19, 20]. This 20% increase should be due to the fact that two solute molecules in the cm configuration interact with a lower number of water molecules due to the decrease in the value of the cavity contact correlation function for water molecules contacting the surface of the cavity hosting two solute molecules [41, 42]. Thus, the δEa(HI) quantity is given by:

δEa(HI)=[Ea(Xe)/WASA(Xe)]1.2ΔWASA=1.2Ea(Xe)(1cosα) (14)(14)

It is worth noting that δGc(HI) provides a negative Gibbs energy change favoring pairwise association, whereas δEa(HI) provides a positive Gibbs energy change contrasting pairwise association. The rationale is that bringing two Xe atoms from a fixed position at infinite distance to a fixed position at contact distance in water causes a WASA decrease that leads to both: (1) a translational entropy gain for water molecules; (b) a loss of favorable Xe–water energetic interactions. Thus, δG(HI) is the balance of these two contrasting contributions.

The enthalpy change associated with the formation of the cm configuration of two Xe atoms in water has to account also for the water–water H-bond reorganization and is given by [43]:

ΔH(HI)=E(cm)+δEa(HI)+δHr(HI) (15)(15)

where the first two terms on the right-hand-side have already been explained, and δHr(HI) is the enthalpy change due to the overlap of the hydration shells of the two separated atoms upon formation of the cm configuration and the consequent release of some water molecules to the bulk. According to the present approach, δHr(HI) is given by:

δHr(HI)=ΔHr(Xe)(1cosα) (16)(16)

where ΔHr(Xe) is the enthalpy change due to the water–water H-bond reorganization upon the insertion of an Xe atom (see the numbers listed in third column of Table 4). The entropy change associated with the formation of the cm configuration of two Xe atoms in water is readily calculated by means of ΔS(HI)=[ΔH(HI)–ΔG(HI)]/T.

Results and discussion

Hydrophobic hydration of xenon

Classic SPT calculations have been performed, over the 0–100°C temperature range, at 1 atm, to obtain ΔGc estimates for the creation in water of a cavity suitable to host an Xe atom, using the experimental density of water at the various temperatures (see the numbers in the second column of Table 2), and the following hard-sphere diameters: σ(H2O)=2.8 Å and σ(Xe)=4.0 Å, both considered to be temperature-independent. The ΔGc estimates, listed in the second column of Table 3, are positive and increase with temperature: ΔGc(in kJ mol−1)=23.9 at 0°C, 25.6 at 20°C, and 29.9 at 100°C, in line with MD simulation results in the SPC/E water model [33]. As σ(H2O) is constant and the volume packing density ξ decreases slightly with temperature (see the numbers in the last column of Table 2), the ΔGc increase is due to the RT factor present in Eq. (6) and in all the theoretical relationships for ΔGc [24–28]; a factor that is related to the average kinetic energy of the solvent molecules bombarding the cavity surface. The estimate of the Xe–water interaction energy is Ea=–20.5 kJ mol−1, and is considered to be temperature-independent because the average distance among the solute and surrounding water molecules changes very little with temperature in view of the almost constancy of the water density [29]. The ΔGc+Ea numbers, listed in the fourth columns of Table 3, prove to be practically identical to the experimental ΔG˙ values over the whole 0–100°C temperature range (see Fig. 2). This agreement means that: (1) the theoretical approach is able to reproduce experimental data over a large temperature range; (2) classic SPT works well in water; (3) there is no need to take into account a contribution from the reorganization of water–water H-bonds (i.e. the latter process should be characterized by an almost complete enthalpy–entropy compensation, as emerged from the theoretical approach).

Table 3

Estimates of the Gibbs energy change, ΔGc, associated with the creation in water of a cavity suitable to host Xe, calculated by means of Eq. (6), at 1 atm, over the 0–100°C temperature range; estimate of the Xe–water interaction energy, Ea, considered to be temperature-independent; comparison between the ΔGc+Ea, numbers and the experimental ΔG˙ values.

Comparison between the (ΔGc+Ea), (Ea+ΔHc) and T·ΔSc functions (the full lines), calculated by means of classic SPT formulas, and the experimental values (the points) of ΔG˙, ΔH˙ and T·ΔS˙, respectively, for the hydration of Xe, over the 0–100°C temperature range, at 1 atm.
Fig. 2:

Comparison between the (ΔGc+Ea), (Ea+ΔHc) and T·ΔSc functions (the full lines), calculated by means of classic SPT formulas, and the experimental values (the points) of ΔG˙, ΔH˙ and T·ΔS˙, respectively, for the hydration of Xe, over the 0–100°C temperature range, at 1 atm.

The classic SPT-ΔHc estimates, listed in the second column of Table 4, show a marked increase with temperature: ΔHc(in kJ mol−1)=–1.0 at 0°C, 3.3 at 20°C, and 17.4 at 100°C. This temperature dependence corresponds to that of αP of water, whose values are listed in the third column of Table 2. In fact, according to Eq. (7), ΔHc is directly proportional to αP, which represents the ensemble correlation between volume fluctuations and enthalpy fluctuations [24, 26], and the latter in water are attributed to the transient H-bond reorganization. The important point is that the ΔHc numbers are close, over the whole considered temperature range, to the ΔHr=ΔH˙–Ea values, listed in the third column of Table 4 (see also Fig. 2). This means that the water–water H-bond reorganization upon insertion of an Xe atom is satisfactorily described by the reorganization associated with the process of cavity creation [13, 25]. The latter proves to be an endothermic process, except at 0°C, suggesting that there is no increase in structural order for the water molecules surrounding the cavity (i.e. the Xe atom).

Table 4

Estimates of the enthalpy change, ΔHc, and entropy change, ΔSc, associated with the creation in water of a cavity suitable to host Xe, calculated by means of classic SPT relationships, over the 0–100°C temperature range, at 1 atm, to be compared with the values of enthalpy change due to the water–water H-bond reorganization upon Xe insertion, ΔHr=ΔH˙ – Ea, and the experimental ΔS˙ ones, respectively.

The classic SPT-ΔSc estimates, listed in the fourth column of Table 4, are large negative and increase with temperature: ΔSc(in J K−1mol−1)=–91.1 at 0°C, –76.6 at 20°C, and –33.5 at 100°C. The important point is that such ΔSc estimates are close to the experimental ΔS˙ values over the whole considered temperature range (see also Fig. 2). This means that the large and negative hydration entropy change is mainly due to the process of cavity creation and that, classic SPT, notwithstanding its simplicity, works well. This statement holds for the hydration of a very large set of polar and nonpolar molecules [44].

However, it is necessary to deepen the analysis. The ΔSc quantity is made up of two parts [26]: (a) the entropy decrease due to the solvent-excluded volume effect for cavity creation, ΔSx=–ΔGc/T, see Eq. (2); (b) the entropy change due to the water–water H-bond reorganization upon cavity creation, ΔSnx=ΔHc/T, see Eq. (5). The ΔSx numbers are calculated from the ΔGc estimates and are listed in the sixth column of Table 4. They: (a) are large negative and decrease very slightly in magnitude over the 0–100°C temperature range because the density of water decreases by only 4% over the same temperature range; (b) represent the loss in translational entropy of water molecules for the solvent-excluded volume effect, and rule the hydration process. On the other hand, the ΔSnx numbers, listed in the last column of Table 4, show a marked increase with temperature because they are linked to the ΔHc ones. They are positive, except at 0°C, indicating an entropy gain due to the water–water H-bond reorganization. However, the latter process is characterized by a complete enthalpy–entropy compensation, and does not affect the hydration Gibbs energy change [13, 14, 23]. It is clear that this structural reorganization is also characterized by a large and positive heat capacity change that can be well described by taking into account a two-state equilibrium between intact and broken H-bonds in bulk water and in the hydration shell (i.e. the so-called Muller’s model [45]). Application of Muller’s model confirms that there is no increase in structural order for the water molecules in the hydration shell of nonpolar solutes [46, 47]. This result is strongly supported by both direct structural measurements, such as neutron scattering and NMR [48, 49], and computer simulations using detailed water models [50, 51].

Cavity thermodynamics

To better clarify the special features of water, the thermodynamic functions associated with the creation of a cavity of 4 Å diameter in H2O and CCl4, calculated by means of classic SPT formulas, Eqs. (6) and (7), are shown in Fig. 3 (the calculations have been performed in the same way for the two liquids, using the experimental data reported in Table 2). The comparison emphasizes that: (a) ΔGc(H2O)>>ΔGc(CCl4) over the whole considered temperature range because the small size of water molecules enlarges the magnitude of the solvent-excluded volume effect, even though the volume packing density of water is significantly smaller than that of CCl4 (compare the numbers in the last column of Table 2) [13, 15, 17]; (b) ΔHc(H2O)<<ΔHc(CCl4) simply because αP(CCl4) is significantly larger than αP(H2O) (see the experimental values listed in the third column of Table 2), indicating that the structural reorganization of solvent molecules occurs to a much larger extent in the organic solvent due to the weakness of the van der Waals interactions existing among CCl4 molecules in comparison to the strength of water–water H-bonds [13, 26]; (c) T·ΔSc(H2O) is a large and negative function over the whole 0–100°C temperature range, whereas T·ΔSc(CCl4) is small negative at low temperatures, and becomes small positive at high temperatures. The trend of T·ΔSc(CCl4) can be understood by recognizing that there are two entropy contributions (see above), one due to the solvent-excluded volume effect always negative, and the other due to the structural reorganization of solvent molecules always positive. The latter has no actual effect on ΔGc because it is exactly compensated by ΔHc in all liquids. Water is special because its molecules are the smallest among the liquid substances at room temperature and 1 atm, and its H-bonds are sufficiently strong to keep almost constant the density and to remain intact upon the creation of molecular-sized cavities [13]. This conclusion emerged also from the analysis of hydrophobicity in suitably modified water models [52, 53].

Thermodynamic functions associated with the creation of a cavity of 4 Å diameter, calculated by means of classic SPT formulas, over the 0–100°C temperature range, at 1 atm, in water (panel A) and in carbon tetrachloride (panel B).
Fig. 3:

Thermodynamic functions associated with the creation of a cavity of 4 Å diameter, calculated by means of classic SPT formulas, over the 0–100°C temperature range, at 1 atm, in water (panel A) and in carbon tetrachloride (panel B).

The experimental values of the liquid–vapor surface tension, γ, of H2O and CCl4 are listed in the fifth column of Table 2 [30]. It is well known that γ of water is significantly larger than that of CCl4 and all organic liquids. Therefore, there has been the claim that cavity creation is more costly in water than in other liquids simply because γ of water is larger [3, 54, 55]. In this respect, it should be pointed out that γ is a macroscopic thermodynamic quantity that cannot provide a molecular level rationalization of hydrophobic hydration. In particular, it is not clear how γ of water should be connected with the large and negative entropy changes that are the fundamental feature of hydrophobic hydration [26]. Moreover, the γ values of water decrease on increasing the temperature, whereas the ΔGc values in water rise on increasing the temperature (a similar discrepancy occurs also in the case of CCl4).

Pairwise hydrophobic interaction of xenon

To calculate the Gibbs energy change associated with the formation of the cm configuration of two Xe atoms, it is necessary to calculate the quantities in Eq. (8). The Xe–Xe van der Waals interaction energy in the cm configuration is E(cm)=–1.9 kJ mol−1, corresponding to ε/k=230 K [35], and is in line with the calculations by Paschek [11]. The E(cm) term is considered to be temperature-independent (see the second column of Table 5). The δGc(HI) and δEa(HI) numbers, calculated by means of Eqs. (13) and (14), are listed in the third and fourth columns of Table 5 [note that δEa(HI) is considered to be temperature-independent just as Ea(Xe)]. The δGc(HI) numbers are negative and increase in magnitude with temperature, favoring pairwise HI because they measure the gain in translational entropy of water molecules due to the WASA decrease associated with the formation of the cm configuration. On the other hand, the δEa(HI) numbers are positive, contrasting pairwise HI because they account for the loss of Xe–water energetic interaction due to the WASA decrease associated with the formation of the cm configuration. The δG(HI) values, representing the water contribution to cm formation and listed in the fifth column of Table 5, are negative (except at 0°C) and not large, in view of the balance between the contrasting δGc(HI) and δEa(HI) contributions. This should be a general feature of pairwise HI [56]. However, as the two energetic terms in Eqs. (8) and (10) have opposite signs, when the two nonpolar objects are not spherical (i.e. two plates) [18, 57], E(cm) and δEa(HI) should balance each other to a large extent, so that the δGc(HI) term should dominate pairwise HI.

Table 5

Estimates of the quantities characterizing the cm configuration of the pairwise HI of Xe, at 1 atm, over the 0–100°C temperature range, calculated by means of Eqs. (9), (13) and (14): (a) ΔG(HI)=E(cm)+δG(HI); (b) δG(HI)=δGc(HI)+δEa(HI).

The ΔG(HI) numbers, obtained by means of Eq. (8) and listed in the sixth column of Table 5, prove to be negative and increase in magnitude with temperature. The formation of the cm configuration of two Xe atoms is thermodynamically favored over the 0–100°C temperature range. Moreover, these ΔG(HI) estimates are close to the values obtained by Paschek [11], performing MD simulations using the SPC/E water model over the same temperature range (see the values reported in the last column of Table 5 and Fig. 4). It is worth underscoring that there is no fitting, the theoretical approach allows the calculation of all the relevant quantities in a direct way.

Temperature dependence of the functions describing the formation of the contact-minimum configuration of two Xe atoms (see the numbers listed in Tables 5 and 6). The points correspond to the ΔG(HI) values calculated by Paschek, by means of MD simulations in the SPC/E water model [11].
Fig. 4:

Temperature dependence of the functions describing the formation of the contact-minimum configuration of two Xe atoms (see the numbers listed in Tables 5 and 6). The points correspond to the ΔG(HI) values calculated by Paschek, by means of MD simulations in the SPC/E water model [11].

The δHr(HI) numbers, calculated by means of Eq. (16) and listed in the second column of Table 6, prove to be negative, except at 0°C. The reorganization of water–water H-bonds associated with the formation of the cm configuration is exothermic because the release of some water molecules from the hydration shell of the two Xe atoms to the bulk leads to an energy gain (i.e. note that, according to the Muller’s model [45–47], the H-bonds in the hydration shell of nonpolar solutes are slightly more broken than those in bulk water). In any case, the ΔH(HI) numbers, calculated by means of Eq. (15) and listed in the third column of Table 6, are positive over the whole considered temperature range, and prove to be in line with those obtained by Paschek [11] (note that Paschek did not list numerical values, but showed plots from which it is not simple to extract reliable numbers). Thus, the formation of the cm configuration of two Xe atoms is endothermic and entropy-driven. In fact, the TΔS(HI) numbers, listed in the fourth column of Table 6, are positive (see also Fig. 4). Their main contribution comes from the gain in translational entropy of water molecules for the WASA decrease associated with the formation of the cm configuration.

Table 6

Estimates of the enthalpy and entropy changes characterizing the cm configuration of the pairwise HI of Xe, at 1 atm, over the 0–100°C temperature range, calculated as: (a) ΔH(HI)=E(cm)+δEa(HI)+δHr(HI); (b) T·ΔS(HI)=ΔH(HI)–ΔG(HI).

Conclusion

The devised theoretical approach is independent of SPT, the latter is used because it provides analytical formulas to calculate reliable estimates of the ΔGc quantities. On the other hand, SPT calculations take into account the specific energetic and geometric features of water–water H-bonds, because the experimental water density values are used as input data. It is well known that SPT results are strongly dependent on the hard-sphere diameter assigned to water molecules [58]. However, the selected σ(H2O)=2.8 Å diameter is physically sound, corresponding to the location of the first peak in the oxygen-oxygen radial distribution function of water [32].

The aim of the present work is to show that a single theoretical approach can account, in a coherent way, for most of the peculiar features of hydrophobic hydration and pairwise HI. Actually, it has also been extended to rationalize the conformational stability of globular proteins [59–61]. The physical ground of the approach is the recognition that the solvent-excluded volume effect plays a fundamental role in such processes and can be “measured” by the WASA of the cavity (molecule) to be inserted in water, or by the WASA lost when solute molecules form a contact configuration or when a polypeptide chain folds. A decrease in solvent-excluded volume leads to a gain of translational entropy for water molecules, driving the association of nonpolar molecules and the folding of globular proteins.

References

About the article

Corresponding author: Giuseppe Graziano, Dipartimento di Scienze e Tecnologie, Università del Sannio, Via Port’Arsa 11 – 82100 Benevento, Italy, Phone: +39/0824/305133, Fax: +39/0824/23013, E-mail:


Published Online: 2016-02-08

Published in Print: 2016-03-01


Citation Information: Pure and Applied Chemistry, Volume 88, Issue 3, Pages 177–188, ISSN (Online) 1365-3075, ISSN (Print) 0033-4545, DOI: https://doi.org/10.1515/pac-2015-1003.

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