Faraday Symp. Chem. SOC.,1982 17,93-108 Geometric Relaxation in Water Its Role in Hydrophobic Hydration BY RUFUS LUMRY AND EZIOBATTISTEL Chemistry Department University of Minnesota Minneapolis Minnesota 55455 U.S.A. AND CARMEL JOLICOEUR Chemistry Department UniversitC de Sherbrooke Sherbrooke Quebec J1K 2R1 Canada Received 17th September 1982 The thermodynamic quantities AGO AH",AS"and AC; associated with the solvation of argon in water and aqueous mixtures are reinterpreted on the basis of two contributions. The first is related to the hydrogen-bonding connectivity of water and is assumed to be approximately represented by corresponding thermodynamic quantities in solvents such as hydrazine and ethylene glycol; following a proposal by Lumry and Frank [R.Lumry and H. S. Frank Proc. 6th Int. Biophys. Congr. (1978) vol. 7 p. 554; R. Lumry in Bioenergetics and Thermodynamics; Model Systems ed. Y. Braibanti (Reidel Dordrecht 1980) p. 4051 this contribution determines the free energy of hydrophobic hydration and is dominated by a positive enthalpy. The second contribution which is responsible for marked enthalpy-entropy compensation and large heat-capacity effects in argon hydration is assigned to the characteristic fluctuation behaviour of liquid water. This assignment is substantiated by comparisons of the bulk properties of water and various other liquids and a model is suggested to rationalize the uncommon thermodynamic properties of water aqueous mixtures and solutions of hydrophobic solutes. The phenomenological representation proposed is an overlay of a randomly connected H-bond network and a local fluctuation process defined in terms of a minimum cooperative unit.This process labelled "geometric relaxation" is pictured as a cooperative H-bond rearrange- ment involving a water molecule coordinated by four neighbours. The limiting microstates of the cooperative units are "short-bond " forms (with short stiff near-linear bonds) and " long-bond " forms (with long weak bent hydrogen bonds). The first is dominated by the low enthalpy of short hydrogen bonds and the second by the high entropy resulting from motions of the water molecules on flexible hydrogen bonds into the free volume not available to the " short-bond " form. The thermodynamic quantities associated with the introduction of hydrophobic compounds in water exhibit a number of uncommon features.Near room temper- ature the transfer of hydrophobic solutes (e.g.rare gases small hydrocarbons) from a hydrocarbon environment to water typically occurs with the following thermody- namic changes a positive Gibbs energy a negative enthalpy a large negative entropy and a large positive heat capacity (at constant pressure). At the time of these early observation^,^ -3 and for many years thereafter such thermodynamic features appeared remarkably unique to inert solutes in aqueous solutions and were thus logically correlated with the unusual properties of liquid water. Our main objective in the present paper is to determine from thermodynamic arguments to what extent and at what level the phenomenon of "hydrophobic hydration " depends on the unique properties of water.In performing this analysis the path which appeared most convenient may be outlined as follows. First we compare several bulk and thermodynamic properties of water and other GEOMETRIC RELAXATION IN WATER liquids emphasizing distinctions between water and other fluids having high cohesive- energy density due to extensive hydrogen bonding. The inferences derived from such comparisons are carried over to the examination of properties of selected mixtures of water with hydrogen-bonded liquids. This is followed by a discussion of one par- ticular process which has attracted long-standing interest the dissolution of argon in water and in several other liquids and mixtures.These combined data provide arguments which will be used to suggest a molecular description through which the characteristic properties of liquid water as well as the thermodynamic attributes of the hydrophobic hydration can be understood. BULK PROPERTIES OF WATER AND OTHER LIQUIDS It has long been recognized that the uncommon behaviour of liquid water is at least in part a consequence of the near-tetrahedral configuration and 1 1 ratio of its donor to acceptor groups. These rare molecular properties greatly favour the formation of three-dimensional H-bonding arrays which are highly adaptable to externally imposed constraints as evidenced by the many structural forms of ice and the variety of organic and inorganic hydrates.The presence of a highly connected H-bonding network in liquid water must have a marked influence on its properties and on the properties of aqueous solutions and to some extent the same should be true for any liquid which maintains an extensive H-bond connectivity. This then suggests an obvious question already raised in numerous different ways and which we reformu- late as follows for any given property what is the relative importance of effects originating in random H-bond connectivity to those which are due to short-range molecular phenomena dominated by the unique properties of the water molecule (assuming at this point that such contributions can indeed by isolated)? A qualitative answer to such questions may be sought from comparisons of the properties of water with those of other liquids chosen such that their molecular properties will bear as much resemblance to water as is chemically possible.Numerous investigations have followed such a comparative approach and it is interesting to recall that Bernal and Fowler in their discussion of the specific features of liquid water remarked that " the properties of liquid hydrazine would repay more careful investigation". As additional experimental data become available on the physical properties of H-bonded liquids the classical approach which may have been exhausted at the time of Bernal and Fowler may now be worth further pursuit. A detailed comparison among several representative liquids (hydrocarbon alcohol polyol water) will be presented elsewhere and the results confirm that the discriminating ability of thermodynamic properties of liquids increases with the order of differentiation with respect to T or p.For example values of the temperature dependence of the heat capacity at constant pressure dC,/dT categorize liquids more sharply than the C,,values themselves. The same is observed for the thermal expan- sion coefficient (a) and its temperature derivative. Focusing on the latter property we illustrate in fig. 1 a values for homologous alkanes alcohols and a,w-diols at 25 "C. Consistent with expectations a decreases as the cohesive energy of the liquids increases (e.g.with increasing chain length in a homologous series or with increasing number density of H-bonding groups). The a values included for liquid hydrazine (N2H4) and hydrogen peroxide (H202)are approximately those expected on the basis of the number density of H-bonding groups (i.e.H202 behaves quite accurately as the zeroth member of the homologous a,co-diols). The value of a for water at room temperature is only half of the lowest value shown in fig. 1 and as is well known becomes negative at temperatures below 4 "C. The behaviour of the heat capacity of liquids [expressed per unit volume (ci) to be consistent with a] shows a similar qualitative distinction R. LUMRY E. BATTISTEL AND C. JOLICOEUR between water and other H-bonding liquids,6 and so does the compressibility.'' The fact that such distinctions appear with increasing contrast as we examine higher T or p derivatives would immediately indicate some fundamental difference in the thermo- dynamic fluctuation behaviour of water as compared with other liquids.Addressing the fluctuation behaviour of water essentially amounts to defining a I I I I I I I 1 1 1 * I Yo 2 . 8 0 CX,W -DIOLS 0 0 1 3 5 7 9 11 no. of carbon atoms FIG.1.-Thermal expansion coefficients of liquids in several homologous series at 25 "C. The error bars indicate maximum departure of the data from the curves shown. Data sources alkane al- kanols and diols ref. (6); NzH4,ref. (7); Hz02,ref. (8); HzO,ref. (9). set of microstates with comparable free energies although with different values of H S V etc. and much of the written history of the water puzzle deals with the definition of micro- or macro-states in terms of structural models.The approach followed here is mainly concerned with the identification of the main characteristic features of these fluctuations and the mechanisms through which they can occur. As may be recalled the mean-square fluctuations in Hand S (a;) or (a:) which are also the second moments [ proportional to (~idth)~] of the H or S probability density distribution functions are given by (a;} =kT2Cpand (a;} =kC,. Higher derivatives of Cpwith respect to Twill be related successively to the higher moments of the H or Sdistribution functions the third moment defining their assymmetry and so on. Provided accurate temperature derivatives of Cpare available standard mathe- matical formalism will enable one to approximate with increasing accuracy the Hand -1 HZ02 GEOMETRIC RELAXATION IN WATER Sprobability distributions from the various moments.l' The same of course applies to other thermodynamic quantities related to fluctuations the heat capacity at constant volume (C,) specifying internal energy fluctuations the isothermal compressibility (p) defining volume fluctuations and the thermal expansion coefficient reflecting the correlation in energy-volume fluctuations (covariance).At least two aspects of the thermodynamic fluctuation behaviour of water are worth emphasis at this point. First the low correlation of the internal energy-volume fluctuations (which becomes negative at lower temperatures and increasingly negative in supercooled water) cannot be explained from the type of H-bond connectivity found in other H-bonded liquids studied thus far.Secondly a remarkable analysis carried out by Stey et aZ.I2using C, (and C,) and their temperature derivatives for water and several other liquids yielded a bimodal enthalpy probability density distri- bution function only for water. With the above observations these two points warrant the conclusion that some feature of the fluctuation behaviour of liquid water cannot be understood from " extrapolations " based on the properties of other liquids using H-bond-connectivity or cohesive-energy (number density of H-bonds) arguments. The thermodynamic properties of liquid water are thus envisaged here as due to three contributions (1) an intramolecular part (2) a contribution from random H-bonding connectivity and (3) a characteristic fluctuation contribution from a process to be specified.The magnitude of each contribution will vary for different properties but the observations quoted above indicate that the relative importance of the third contribution will increase as higher T or p derivative properties are examined. The above dissection of thermodynamic properties is identical to that offered by Oguni and Angel1 l3 in interpreting heat capacities of water and aqueous mixtures although stated from a different point of view. This deconvolution is artificial in the sense that whatever is specific to the water molecule and to interactions between water molecules will show up in the H-bonding connectivity as well as in other local molecular effects which may be present.Such separation is not required for ab initio theoretical treatment based on a correct wavefunction but it appears to be a useful concept for understanding hydrophobic hydration and the properties of aqueous mixtures. PROPERTIES OF SELECTED AQUEOUS MIXTURES In the line of the foregoing remarks we now inquire about how the characteristic fluctuation behaviour of water will be affected by addition of hydrophilic cosolvents which can in their pure form support extensive H-bonding connectivity i.e. as manifested by their high enthalpy and entropy of vaporization high viscosity and high specific heat. Interesting candidates for such investigations on mixtures are liquids such as N2H4and H202as recently studied by Oguni and Angell,13 or ethylene glycol (EG) investigated by the present authors.The general trends in the thermo- dynamic results on aqueous mixtures of these liquids may be summarized as follows. When high-derivative properties (such as a ci da/dT and dci/dT) of the mixtures are plotted against mole fraction of cosolvent X(cos) the curves obtained exhibit a common three-region pattern a region MI corresponding to 0 < X(cos) < 0.3 where the properties change rapidly with X(cos); a region MIIIin the range 0.4 < X(cos) < 1.0 where the properties of the mixture depend only weakly on X(cos); and a cross-over region between MI and MIIIdesignated M,I. This is illustrated in fig. 2 with a and da/dT of the EG water mixtures; as could be anticipated from dis- cussion of the properties of liquids the distinction of MI and MllIregions becomes sharper as higher-derivative properties are examined.R. LUMRY E. BATTISTEL AND C. JOLICOEUR The similar shape of curves describing the properties of different mixtures in the MI region suggests a statistical nature for the cosolvent effect on the characteristic fluc- tuations of water. The high cosolvent mole fraction required to eliminate the charac- teristic water behaviour further suggests that the disruptive cosolvent effect operates at short range. The improved qualitative distinction between the MI and MIIIregions obtained with higher T derivatives (e.g. fig. 2) confirms that such distinction can be adequately discussed in terms of changes in fluctuation behaviour with X(cos).A closer examination of thermodynamic properties of mixtures for example in terms of partial molar properties leads to other interesting clues on the origin of the characteristic fluctuation contributions in liquid water. For hydrazine + water mixtures a plot of cpof each component against X(H,O) again shows the MI-MIII division [with respect to X(cos)] as illustrated in fig. 3. For example cp(H20)varies only FIG.2.-Thermal expansion coefficient (m) and its temperature dependence in ethylene glycol + water mixtures at 25 "C. Data from ref. (5). slightly in the concentration range 0 < X(H,O) < 0.6 and increases sharply as X(H,O) -+ 1. The order of this Cpincrease with respect to water concentration is readily estimated from a plot of the cp(H20) increments against [X(H,O]".These cp(H,O) increments were taken as [cp(H,O) -cp(H20*)], where c,(H,O*) is the value of cp(H,O) extrapolated from the near linear region 0 < X(H,O) < 0.5 to X(H,O) = 1. We chose the view here that this extrapolation yields the partial-molar heat capacity of water in a hypothetical liquid state where its characteristic fluctuations have been eliminated. Within the combined uncertainties of the experimental data and extrapolation we find n = 5. The difference between the molar heat capacity of liquid water and C,(H,O*) is ca. 2 cal K-' mol-' as also estimated earlier by Oguni and Angel1.l3 If this difference is attributed to the characteristic fluctuations of liquid water then the above crude estimate would suggest that pentameric water units are responsible for this contribution.Clearly such inferences must be taken with due 98 GEOMETRIC RELAXATION IN WATER I I I 1 ____________---___ I I 20 15 0 -5 t \- I I I I I I I 0 0.2 0.4 0.6 0.8 1.0 FIG.3.-Partial molar heat capacity of hydrazine cp(N2H4), and water Cp(H20),in hydrazine +water mixtures at 17 "C. Data from ref. (13). caution since quantitative differences appear with different cosolvents; e.g. ethylene glycol exhibits characteristic hydrophobic behaviour below X(cos) = 0.OfL5 None-theless these results emphasize the short-range statistical nature of cosolvent effects on the characteristic fluctuations of water.THERMODYNAMICS OF Ax- SOLVATION HYDROPHOBIC HYDRATION The analysis of the thermodynamic quantities associated with hydrophobic hydration follows essentially the route outlined above in discussing the characteristic features of liquid water. Using argon as hydrophobic solute we again concen-trate on a dissection of the hydrophobic-hydration process and corresponding thermodynamic properties into contributions arising respectively from the H-bond connectivity (cohesive energy density) and from the characteristic fluctuations of liquid water. In achieving this separation we make further use of similarities and differences between water and other H-bonded liquids namely N2H and EG. In discussing bulk properties of liquid water the separation of contributions is still an intellectual exercise but it becomes necessary in discussing hydrophobic hydration specificially in understanding the origin of Gibbs-energy changes.This follows from the work of Benzinger l4 (later expanded by Lumry and Frank 15*16) who showed that there is no rigorous connection between AG(T)and AS(T) for processes at constant p and 7',although (2AG/2T),and AS are of course related. The main implications of these analyses are given here only as required for subsequent discussion; detailed derivations can be found el~ewhere.~.'~ Using only fundamental thermodynamic principles it can be shown that the experi- mental quantities AHo(T)and ASo(T)for a reaction each contain two contributions which have been labelled " motive " (m) and " compensation " (c):~,'~ AHo = AH +-AH," AS' = AS:+ AS:.R. LUMRY E. BATTISTEL AND C. JOLICOEUR As implied by their designation the compensation contributions cancel out exactly in AGo(T),so the latter is determined exclusively by the " motive " contributions AGO = AH -TAS:. Except for reactions for which thermodynamic data are available at very low temper- atures (to allow evaluation of the heat integral and the " compensation "contribution) there is no rigorous way of identifying the " motive " contributions which are the relevant parts for understanding AGo(T). Several examples have now been examined and show that the magnitude of AH,"(T)and AS:@) can be significant even in simple chemical reactions or systems and becomes very important in cooperative processes such as reactions involving proteins and ma~r0rn01e~~le~.~~ For example the motive entropy of copper metal at 300 K is only half of its total standard molar entropy." On the other hand the motive contribution to the unfolding enthalpy of ribonuclease- A was estimated by Benzinger and Hammer l8 as ca.6% of the total enthalpy! The compensation of AH by TASO,in AGO can be called " second-law " compen-sation behaviour. Since it is exact at all temperatures its manifestation for example in a series of related processes (e.g. reactions involving homologous series systematic pH changes etc.) would yield a linear relationship between consecutive AH; and AS; values with a slope equal to the experimental temperature.This may be written in the form AHP(T) = A + T,ASY(T) (1) where i represents values for each reaction in a series of closely related reactions and T is the compensation temperature. The criterion T = (T,,,,) for two or more values of (T,,,,) allows unambiguous identification of " second-law compensation". In addition to second-law compensation ad hoc compensation can occur between the motive contributions AH:, and AS;,,. Such behaviour is very common but has no rigorous thermodynamic basis. It can also lead to a linear correlation between AH; and AS of the type'given by eqn (I) although with T # (Texpt). The experi- mental distinction between " second-law " and " motive " compensation is thus readily achieved by determining the dependence of T on the experimental temperature over a sufficient temperature range.Returning to the thermodynamics of argon solvation we examine several transfer quantities collected in fig. 4 (due to Frank and Lumry Is) in which we want to consider the following transfer processes at 25 "C. process I Ar(cyc1ohexane) -+Ar(water) process 2 Ar(cyc1ohexane) stephAr(hydrazine) Zh Ar(water). AGO for process 1 is positive and very close to that observed in the first step of process 2. AHoand ASo for process 1 are both large and negative (as is typical of hydrophobic hydration defined by such transfer processes) and of similar magnitude to the corres- ponding quantities for the second step of process 2. In fact the first step of process 2 exhibits AGO > 0 and ASoz 0.ACpO is large for process 1 and this shows up almost exclusively in the second step of process 2. On the basis of AGO values only we would thus conclude that hydrophobic hydra- tion is in no way special since it has its counterpart in hydrazine and ethylene glycol (fig. 4). If we accept the view that liquid hydrazine is a good "model " for liquid GEOMETRIC RELAXATION IN WATER water without its characteristic fluctuation behaviour we would conclude that the hydration of Ar in such a solvent (and its low solubility) is dominated by a positive enthalpy rather than by a negative entropy. In fact the transfer of Ar from cyclohexane to hydrazine occurs with AS" close to zero in spite of the high connectivity of hydrazine. The large negative AH" and AS" values observed in the hydrazine -f water transfer nearly cancel each other at 25 "C.They are thus good approximations to those parts AGekcal AH kcal ASo=cal K" / AH -2.68 AS: 17.3 I I CY c L o H EXAN E] \ AHk9 5 AG'2.84 AH'= 2.27 A S'=-l. 9 IETHYLENE GLYCOL^ FIG.4.-Thermodynamic functions for the transfer of one mole of argon among various solvents at 25 "C. The standard state used for argon is unit mole fraction. Data sources ethylene glycol ref. (19); cyclohexane and water ref. (20); hydrazine ref. (21). which do not contribute to AGO and which could be attributed to the characteristic fluctuation behaviour of liquid water. The quantitative degree to which hydrazine is a good reference liquid for the analysis performed above is still open to question (it certainly will not be reliable at all levels of comparison) but the separation it allows is conceptually important for a complete rationalization of the thermodynamics of hydrophobic hydration.Similar deductions had also been inferred by Shinoda and Fujihara 22 from an analysis of the temperature dependence of hydrocarbon solubilities in water (e.g. assuming that water at ca. 200 "C has lost its characteristic fluctuation contribution). Using excess thermodynamic functions and order-disorder effects in alkanes Barbe and Patterson 23 provided strong support for the Shinoda-Fujihara interpretation of thermodynamic properties of hydrocarbons in water. The above partitioning of the thermodynamic quantities of hydrophobic hydration is also of importance for the interpretation of interactions among hydrophobic solutes (" hydrophobic effect " in water).The formation of surfactant micelles a partial case of hydrophobic interactions in water at ca. 160 "C and in hydrazine at 25 "C has AGmiCvalues comparable to those observed in water at room temperat~re.~~ HYDROPHOBIC HYDRATION IN AQUEOUS MIXTURES As a last step in this thermodynamic analysis we examine the changes in AH" and AT for dissolution of argon in several aqueous mixtures using the data of Ben-Naim and coworkers. Fig. 5 shows AH" values for the transfer of Ar from water to water+ EG mixtures of increasing EG mole fraction plotted against the corresponding AS" for the same process.1g (The analysis would preferably be carried out with water + R.LUMRY E. BATTISTEL AND C. JOLICOEUR hydrazine mixtures but no such data are yet available.) As observed earlier with bulk properties of mixtures the AH"-AS" data of fig. 5 exhibit three distinct com- position regions. In the MI region 0 < X(EG) < 0.3 AH" and AS"show accurate compensation with T z 295 K; the latter is independent of temperature in the range 5-25 "Cso the compensation behaviour is of motive origin. In the MllIregion 0.4 < X(EG) < 1.0 motive compensation is also observed although with a very different I I .I LO00 Al.1 /A " 1.o !-3000 E ..( I * c1 .2 s + 'ji 2000 B 6 a 1000 0 5 10 15 ASow-,EG + w/~l K-' mol-' FIG.5.-Conventional enthalpy-entropy compensatlion plot for the transfer of one mole of argon from water to mixtures of water with ethylene glycol.Temperatures A,5; 0 15; 0 25 "C. The line has a slope of 295 K. Deviations from the line appear at X(EG) = 0.3 as indicated. The solid dot at the top is for transfer from water to pure hydrazine. The insert shows ACg for the transfer of argon from water to ethylene glycol + water mixtures. Data from ref. (19). slope T,x 80 K. The difference of behaviour in the MI and Mill regions is demonstrated more clearly and more reliably in a plot of AGO against AH",as suggested by Krug et aZ.,25although the diagram shown in fig. 5 is fully adequate for present purposes. Similar data also reported by Ben-Naim and coworkers on aqueous GEOMETRIC RELAXATION IN WATER mixtures with other cosolvents (dioxan,26 methanol 27 and ethanol 28) yield linear AHo-ASo compensation lines in the MI region which are virtually superimposable on that in fig.5. Some chemical specificity towards cosolvent is evident in the MI,and MIIIregions but in the MI region T is constant within errors of a few degrees for this set of cosolvents. This is a key observation indicating that the motive com- pensation pattern with T,FZ 295 K can be assigned to a characteristic process of water. The effect of added cosolvents on this process again appears largely statistical. These conclusions are supported by the transfer heat-capacity data (water -+ EG+ water mixtures) as shown in fig.5 (insert). The large positive heat capacity (Ci)of Ar in water is virtually eliminated by addition of EG at X(EG) FZ 0.25 and similar effects are also found with the other cosolvents studied. The absence of significant heat-capacity change for solution of Ar in any other solvent including N,H4 supports the proposal that the characteristic fluctuations of liquid water are responsible for this unique Cpbehaviour. Returning to the AHo -ASo data in fig. 5 we note that the AHo and ASo values for the transfer of Ar from water to hydrazine lie very close to the projection of the H-S compensation line from the MI region. This point may be viewed as the ultimate AHo and ASO values which would be reached if EH and the other cosolvents were able to maintain the high connectivity of water in the mixtures of high X(cos).Having stated some of the distinguishing features of water and hydrophobic hydration we present below a qualitative model for their explanation. The model is constructed largely from concepts already in the literature reassembled in a way which seemed most consistent with current knowledge of thermodynamic properties of water and aqueous systems. THE GEOMETRIC-RELAXATION MODEL The uncommon properties of water and aqueous solutions appear to originate in fluctuations which differ sharply from those resulting from H-bond perturbation or breakage in other highly connected H-bonded liquids. Such important H-S com-pensation and Cpeffects as those found for the hydrophobic hydration of argon must be associated with a process allowing large H-S fluctuations at near constant G.The phenomenology of a model should thus provide a basis for understanding such fluctuations and as well an explanation for the covariance of internal energy and volume which is unusually small in water at ordinary temperatures and becomes negative at low temperatures. The model we propose consists of a random network of essentially pairwise hydrogen bonds with nearly continuously distributed energies angles and donor- acceptor distances similar to that described by Sceats and Rice.29 However we postulate that embedded in this connectivity system are clusters having appropriate geometry to allow cooperative electronic and nuclear rearrangements into short-lived but well structured tetrahedral H-bonded units.The minimum cluster size should be the symmetrical pentamer or a tetrameric fragment either of which allows simultaneous bond contraction with increased bond strength in the bonds to the " central " water molecule. Such cluster fluctuation processes between the long-bond and short-bond forms we call " geometric relaxation". It is illustrated for a pentameric cluster in fig. 6. Only the two limiting forms are proposed to make major contributions to the partition function; these are described as follows. Short-bondform The five water molecules are linked by short stiff linear hydrogen bonds. Bending as well as torsional and rotational motions are severely hindered. This form has low enthalpy and low entropy but through its rigid geometry increases the free volume and thus to a smaller degree the total volume which is otherwise deter- R.LUMRY E. BATTISTEL AND C. JOLICOEUR mined by the connectivity system. Much of the free volume associated with this form is available to solutes. Long-bond form With increasing hydrogen-bond length the tetrahedral constraint diminishes so that cluster cooperativity is replaced by the simple pairwise hydrogen- bonding interactions of the connectivity system Bending librational and rotational freedom increase and a wide variety of hydrogen-bond-breaking processes becomes available. The weakened and broken hydrogen bonds are stabilized by the resulting entropy increases. The local free volume is utilized by water and becomes less available to solutes.Hydrogen-bond bending and better packing produce a reduction in total volume. From our discussion of enthalpy-entropy compensation in a process characteristic 0 FIG. 6.-Structural representation of the geometric relaxation of a pentameric cluster neglecting effects of neighbouring molecules. of water the enthalpy required to stretch the hydrogen bonds in the short- to long- bond fluctuation in clusters surrounding an argon atom would be exactly cancelled by the gain in entropy at ca. 295 K. For larger hydrophobic solutes T is likely to be different. Before proceeding to further characterization we note that the overlay of a ran- dom H-bonded network and a fluctuation process as proposed here represents an intermediate situation between the " bond-lattice " model of Angel1 30 and the " flickering-cluster " model of Frank and Wen.31 The former consists of a totally H-bonded network (quasi-lattice ground state) in which the breaking of hydrogen bonds is progressively allowed (e.g.with increasing T). The latter is described essentially as an equilibrium mixture of a bulky species with more hydrogen bonding and an interstitial (dense) species with less hydrogen bonding rather than longer hydrogen bonds; these two forms can interchange roles rapidly in cooperative thermal fluc- tuations (" flickering "). Our proposal retains features of both of these models. At 25 "C aside from de-emphasizing H-bond breaking as a critical event it more closely resembles the interstitial model as detailed by Frank and Cha~.~~ In super- cooled water short-bond forms dominate and the extreme fluctuational behaviour is attributed to short- to long-bond soliton waves in larger clusters rather than to hydrogen-bond breaking.At no temperature does our proposal require the identific- ation of structures other than the requirement for almost perfect tetrahedral symmetry about central water molecules in short-bond forms. Multiple association of pent-amers or clusters with average pentagonal-dodecahedra1 geometry satisfies this single requirement but may have low probability in the absence of an appropriate solute. In any event flickering is so fast that large clusters have little structural identity except around the smallest units e.g. pentamers or tetramers.Large clusters nevertheless exist for times longer than small-solute rotational times. SOME HISTORICAL FOUNDATION The cooperative nature of geometric relaxation is postulated on the basis of several properties of hydrogen bonds the distinction of classes of hydrogen bonds in solids GEOMETRIC RELAXATION IN WATER a significant degree of hydrogen-bond coop-ativity in clusters above some minimum size and a high degree of enthalpy-entropy compensation behaviour in the fluctuation of these clusters. Early discussions (e.g. by Coulson 33) turned attention to the covalent component of hydrogen bonds and to the complex behaviour of hydrogen- bonding interactions with variations in bond lengths and angles. Ab initio quantum-mechanical treatments have more recently provided information about these matters 34 for networks dominated by pairwise bonding but the full complexity of the behaviour of hydrogen bonds in condensed systems included polymolecular cooperativity and polarization problems which remain largely ~nresolved.~~ On the other hand a number of attempts have been made to classify hydrogen bonds in solids where geo- metries are constrained.Comprehensive analyses of data reported by Pimentel and M~Clellan,~~ and more recently by Brown,37 indicate that hydrogen bonds in solids tend to group into two categories distinguishing " symmetrical " and '' unsymmetri-cal " '' strong " and " weak ",or " short '' and " long " hydrogen bonds. There is now ample spectroscopic evidence showing that the majority of hydrogen bonds in liquid water cannot be sharply divided into two categoties i.e.as sharply as is required for true two-state models. However in the H-bonded network of water despite the broad variety of instantaneous configurations it is not unreasonable to assume that the hydrogen bonds can be assigned to two limiting groups corresponding to short-bond and long-bond forms of the clusters. The long-bond forms are characteristic of the pairwise-bonded connectivity system which predominates above 4 "C. The cooperative behaviour of hydrogen bonds manifested by a reduction in total energy in a many-body interaction relative to the sum of pairwise interactions is well supported. Extensive ab initio 34 and molecular-dynamics 38 results on water clusters of various sizes as well as experimental data for dilute water in weakly interacting solvents,39 consistently show the cooperative nature of interactions among water molecules.(It should be mentioned here that the charge rearrangement implied in the stabilization of larger aggregates has been found to increase with solvent polarity in several H-bonding adducts investigated by Fritsch and ZundeL40) Frank and Wen emphasized the importance of this cooperativity and its dependence on the changing covalent character of the hydrogen bond in the " flickerjng-cluster " model.31 Geo-metric relaxation is based on the same argument namely that the electronic rearrange- ment subsequent to the formation of a first hydrogen bond between a pair of water molecules favours the formation of a second hydrogen bond with a third neighbour and so on.However we identify the tetrahedral pentameric cluster as the basic cooper- ative unit largely because the order (or index) of cooperativity in electron rearrange- ments would be expected to be larger in these many-body interactions than in sequen- tial formations of hydrogen bonds in a random chain. The balance of enthalpy-against-entropy changes associated with H-bond form- ation is also of central importance to our discussion of fluctuation behaviour. H-S compensation effects in hydrogen bonds was demonstrated in some detail by Pimentel and McClellan in 1971 and earlier in their 1960 discu~sion.~~ From a large body of data on reactions yielding H-bonded pairs with phenol as proton donor they showed that systematic variations in the acceptor produced AHo-ASo compensation patterns with T values of 320 K or greater.Their interpretation of these findings was that in weakly interacting solvents the higher the bond energy the greater the rotational librational and bonding freedom and thus the greater the entropy. We propose that such H-S compensation also operates between the limiting forms of " short " and '' long " hydrogen bonds in liquid water the enthalpy required to increase the 0-H --0 distances being compensated by an entropy gain from H-bond bending and torsional librational rotational and translational modes of the water molecules. R. LUMRY E. BATTISTEL AND C. JOLICOEUR The occurrence of this compensation behaviour in a process involving a minimum number of water molecules (chosen here as 5) is due to the requirement for a large entropy change to balance changes in energy of strong hydrogen bonds.Without this cooperativity T would lie far above room temperature. CHARACTERIZATION AND CONSEQUENCES We have determined from a crude estimate based on Cr,data that the minimum " cluster" undergoing geometric relaxation in bulk water consists of ca. 5 water molecules. The enthalpy and entropy of the relaxation process cannot be evaluated in a direct fashion but most analyses of data which exhibit apparent two-state behaviour yield AH" and AS"values of ca. 2.5 kcal mol-I and ca. 8 cal K-' m01-l.~~ Since geometric relaxation can account for the apparent two-state behaviour of some observables (i.e.those which are sensitive to H-bond lengths and energies) we tem- porarily adopt the enthalpy and entropy values from two-state analysis. This will allow further first-order characterization of the phenomena involved. We noted in comparisons involving water hydrazine and their mixtures that the difference between the molar heat capacity of liquid water (C;)and the heat capacity of a hypothetical water free of its characteristic fluctuations [C,(H,O*)] is ca.2 cal K-' mol-I at 290 K. Generally the molar relaxation contribution Ci of a given two-state process to Cpof a system is calculated from C,' = AH2X1XJRT2 where AH is the enthalpy of the process and X,and X2represent the mole fractions of each form.At the compensation temperature assumed to be 295 K the populations of the long-and short-bond forms of the pentameric units are identical A' = X2. Using these and AH = 2.5 kcal mo1-l we calculate that ca. 20% of the water molecules are involved in geometric relaxation i.e. comparing the above 2 cal K-' mol-I with the maximum possible contribution from this process when X = X2 = 0.5. Con-sequently only ca. 10% of the molecules are in clusters of the short-bond form; the remaining 90% are all considered as indistinguishable parts of the connectivity system. The fraction of molecules involved in geometric relaxation will depend on the extent of H-bond connzctivity. Hence changes in temperature (or addition of hydrophilic cosolvents) will affect thk process mainly through modifications in the H-bond connectivity.Another interesting aspect of the geometric relaxation process is the lifetime of the short- and long-bond forms assumed here. Sound absorption 42 and depolarized Rayleigh scattering 43 indicate the presence of two relaxation effects with characteristic times of the order of 0.5 and 3 ps. The shorter of these appears to exhibit a two-state behaviour consistent with the geometric relaxation process. Montrose et al.43using depolarized Rayleigh scattering found a Lorentzian peak with Arrhenius behaviour corresponding to a relaxation time of 0.6 ps at 20 "C; the activation enthalpy and entropy were found respectively as 2.4 kcal mol-I and 6 cal K-' mol-l. These authors and more recent investigations of depolarized Rayleigh scattering 44 assign this process to H-bond rupture.However the Rayleigh scattering reflects changes in proton polarizabilities and these could likely be attributed to H-bond stretching as well as to H-bond breaking. The actual occurrence of broken hydrogen bonds is of course not to be discarded but such events are considered here as limiting situations in the connectivity system. At present there is considerable uncertainty about the extent of H-bond breaking even at 100 "C. With regard to the hydrophobic hydration comparing the properties of argon in hydrazine we noted earlier that for hydrazine +water AGO = -0.3 kcal mol-I AH" = 106 GEOMETRIC RELAXATION IN WATER -4.8 kcal mol-' and AT = -14.5 cal K-' mol-'. Using the AH" and AS"values from two-state analysis (2.5 kcal mol-I and 8 cal K-l mol-') we would estimate that with this minor change in Gibbs energy one mole of argon shifts (on the average) two moles of clusters into the short-bond form.This rough estimate is consistent with the high heat capacity of Ar in water the maximum relaxational heat capacity from two moles of clusters would be ca. 35 cal K-' mol-' while ci (Ar) in water lies between 40 and 50 cal K-' mol-' (determined from the temperature dependence of its ~olubility'~). A possible likely alternative is that argon increases the probability of an average pentagonal-dodecahedron structure in which it is enclosed. One simple mechanism through which Ar can impose a pronounced bias on the geometric relaxation process is believed to be through the occupation of free volume available with the short-bond form which favours the increase in the average popul- ation and cluster size of the short-bond forms responsible for the heat capacities of solution.The hydrophobic hydration of Ar in aqueous mixtures in the MI region then appears best described by a loss of C; associated with geometric relaxation because of statistical disruption of the water H-bond connectivity. As the mole fraction of co- solvent is increased the probability of finding a minimal size cluster of proper geo- metry decreases and essentially vanishes as the cosolvent :water molar ratio exceeds 1 :4. The statistical aspect is consistent with the analysis of Frank and Franks 45 for hydrocarbon solubility in urea +water mixtures with the added feature of geo- metric relaxation.Finally it is appropriate to outline the relationship of the present interpretation of the Ar thermodynamic data to the original explanation offered by Ben-Naim."~~~ Asin the above interpretation Ben-Naim separated the standard enthalpy and entropy of solution into two parts labelled " static " and " relaxational ". The static parts were viewed as the quantities which would be measured if the two-state equilibrium of liquid water was " frozen-in ". The " relaxation " contribution was assigned to the effect resulting from a shift in this two-state equilibrium for which exact entropy- enthalpy compensation was explicitly assumed (second-law compensation). The magnitude of the latter contribution was related to the size of the clusters and the effect of cosolvents was suggested to decrease the size of these clusters.Apart from obvious differences in the representations used for liquid water departure of our interpretation from the above may be seen primarily in the implication of" motive " rather than " second-law " compensation and in the emphasis on cooperative fluctu- ations among microstates of a single macrostate rather than changes in equilibrium between structural macrostates. Clearly our data-base for the interpretation of hydrophobic hydration is largely that provided by Ben-Naim and coworkers; our main reasons for suggesting an alternate explanation stem from explicit considerations of some reference liquids such as hydrazine and ethylene glycol to identify the source variety and magnitude of the special fluctuational behaviour in bulk water and hydrophobic hydration.CONCLUDING REMARKS We conclude by summarizing the main thermodynamic features which can be con- sistently understood on the basis of a qualitative representation of water which in- cludes both random H-bond connectivity and a geometric-relaxation process specific to water. While a large portion of the properties of water and aqueous solutions are accounted for by random H-bonding the added geometric-relaxation feature offers a basis for explaining large H-S fluctuation and compensation behaviour as well as apparent two-state behaviour of numerous observables. It also provides an R.LUMRY E. BATTISTEL AND C. JOLICOEUR explanation for the bimodal probability distribution function of the enthalpy des- cribed by Stey et aZ.12 Of equal importance is the fact that the low (and sometimes negative) correlation in energy-volume fluctuations is a natural consequence of the properties postulated for short-bond forms of clusters. The model also provides a consistent approach to the explanation of thermodynamic properties of hydrophobic solutes in water and in aqueous mixtures without extensive use of structural concepts. In the light of the achievements of theoretical and simulation methods applied to liquid water 47 and to hydrophobic hydrati~n,~~ the proposal of a new phenomenologi- cal representation may be viewed as excessive complication.Our reasons for pur- suing such an approach are quite straightforward. The description of fluctuation behaviour involves high derivatives of the Gibbs energy with respect to T and p and such derivatives become increasingly difficult to generate accurately in analytical or simulation approaches. For example Cp and its temperature derivatives involve second and higher derivatives of weak potential-energy functions. Clearly this is an exceedingly demanding situation in any theory and requires either a high degree of parametrization or the extensive use of input from experimental data (e.g. radial distribution functions mode frequencies etc.). Likewise cooperativity understand- ably remains a difficult matter to incorporate in current theoretical and simulation calculations.Thermodynamics has played less of a role in guiding the constructing of models which have the proper fluctuational characteristics than its potential warrants. The qualitative model proposed here is basically a working hypothesis sufficiently flexible to explain a variety of thermodynamic effects manifested by water and different types of aqueous solutions. It is on the other hand sufficiently precise to be fully testable in the spirit of a comment by Frank so on the flexibility of models “ a model which cannot be wrong can expect very little credit for being right ”. This work was supported by the U.S.P.H.S. through the National Heart Lung and Blood Institute and by the National Science Foundation (R. L.) and by the Natural Science and Engineering Research Council of Canada (C.J.).R. L. acknow- ledges with pleasure his great indebtedness to Prof. Henry Frank whose many valuable suggestions include the use of hydrazine as “ inhibited’’ water. The authors are grateful to Prof. A. Angel1 for providing detailed experimental data and to Dr. F. Etzler for calculations of numerous transfer quantities of argon. 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