ISSN:0301-5696    

DOI:10.1039/FS98217FX001

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Syrnp. Chem. SOC.,1982 17 7-10 Hydrophobic Interactions-a Historical Perspective BY FELIX FRANKS Department of Botany University of Cambridge Cambridge CB3 2EA Received 1st February 1983 To the best of my knowledge this is the first ever conference of physical chemists devoted exclusively to the important subject of the hydrophobic interaction. Its importance is amply demonstrated by its prolific literature as shown in fig. 1. Hydrophobic effects play a dominant role in a variety of technological processes in the stabilization of supermolecular biological structures and in the mechanisms associated with biosynthesis metabolism and physiological function. In order to put this Symposium in its historical context it is appropriate to trace the development of our understanding of hydrophobic effects from the thirties on- wards.The foundations appear to have been laid by Hartley who during his com- prehensive studies of association colloids realized that the formation of micelles in aqueous surfactant solutions occurs primarily as a result of an increase in the entropy of the solution.' Incidentally Hartley also suggested that water penetrates to some extent into the interior of the micelle a finding that is just now in the process of being rediscovered. Vapour-pressure measurements on alcohol + water mixtures together with the realization by Butler that in terms of their thermodynamics such solutions closely resemble aqueous solutions of rare gases and hydrocarbons led Eley to suggest that chemically inert molecules dissolve by the formation of cavities in the aqueous s01vent.~ These ideas were developed in considerable detail by Frank and Evans in a publication which still remains a landmark in aqueous-solution ~hemistry,~ a fact which is strikingly illustrated in fig.1. The authors discussed the origin of enthalpy- entropy compensation and the peculiar nature of water as an associated solvent which responds to the introduction of chemically inert species by spatial and orientationai rearrangements. This important realization lay dormant for many years during which time attempts were made to treat the eccentric properties of aqueous solutions of hydrocarbons and alkyl derivatives by more conventional method^.^,^ The term "hydrophobic bond " as a description of alkyl-group interactions in aqueous solution was coined by Kauzmann in his classic discussion of the basis of globular protein ~tability.~ This review acted as catalyst for much of the ensuing activity (see fig.l) especially among biochemists concerned with the interactions involved in protein folding and unfolding.8 In the meantime Nemethy and Scheraga published the first attempt to quantify the hydrophobic interaction in molecular terms,9 and Glew drew attention to the thermodynamic similarities between the crystalline gas hydrates and aqueous solutions of hydrocarbons.1° The structural and thermodynamic peculiarities of alcohol 4-water mixtures were reviewed and analysed in 1966,'l and since then such solutions have been subjected to ever more detailed scrutiny.The n.m.r. relaxation studies by Hertz Zeidler and their colleagues deserve special emphasis ;12*l3 they provided unambiguous evidence that hydrophobic solute molecules (ethylene oxide tetrahydrofuran acetone alcohols etc.) in dilute aqueous solutions are much more mobile than the smaller water molecules thus giving credence to the cavity solution model. Beginning in 1971 Ben-Naim published a series of studies of gas solubilities in various mixed aqueous solvents on the basis of HYDR OP HOBIC INTERACTI0NS-A HISTORI CA L PERSPECTIVE v) - 100Y .-Y 2000 -50 1000 -0 “hydrophobic:’--~ titles 1960 1965 1970‘ 1975 1980 year FIG.1.-Left-hand ordinate number of titles containing the word “ hydrophobic ” taken from Science Citation Permuterm Zizdex ; right-hand ordinate number of citations of key references.which he derived an expression for the potential of mean force WAA(r),between two hydrophobic molecules in water.14 Following on mainly from the writings of Ka~zmann,~ Nemethy and Scheraga and Tanford,ls the so-called hydrophobic bond was at that time pictured as a net attraction between alkyl groups in water leading to a partial reversal of the thermo- dynamically unfavourable introduction of apolar molecules into water. Contacts between apolar groups would thus replace the contacts between an apolar group and the water molecules making up the hydration cavity. Water molecules would then be able to relax into the normal unperturbed structure as shown in fig.2. The positive AS characteristic of the hydrophobic interaction could thus be accounted for. The thermodynamic functions associated with the hydrophobic pair interaction would thus be expected to have the opposite signs from those relating to hydrophobic hydration (the transfer of an isolated apolar molecule or residue from the ideal gas phase or from a hydrocarbon solvent to water). During the early seventies it became evident that this simple if appealing model could not be reconciled with the physical properties of dilute solutions of hydrophobic molecules. In particular the comprehensive thermodynamic studies of Desnoyers Jolicoeur and their associate~,’~~’~ but also light-scattering and small-angle X-ray scattering data led to the suggestion that the pairwise interactions of hydrophobic molecules or residues were more compatible with the clathrate hydrate geometry in which guest molecules occupy separate cages,lg i.e.the minimum in WAA(r)does not correspond to the sum of the solute van der Waals radii see fig. 2. The seventies also witnessed the application of several theoretical concepts to the calculation of so-called hydrophobicity values for proteins ; these were based mainly on measured free energies of transfer of amino acids from organic solvents to water,15J0 or calculated from interfacial free energies between hydrocarbons and water.21 Such treatments as well as others e.g. the scaled particle theory are able to model the free energy of solution of hydrocarbons in water and the free energy of the native protein relative to that of the unfolded state but they fail to account for the signs let alone the magnitudes of the temperature and pressure derivatives of the free energy.22 An important conclusion from the recent studies is that WA,(r) is not of a short-range nature.This is well demonstrated by the statistical-mechanical treatment of Pratt and Chandler,23 which also suggests that WAA(r)is an oscillating function. The long- F. FRANKS FIG.2.-(a) Diagrammatic representation of the hydrophobic hydration of a non-polar particle. The dark cross-hatching represents the hydration cavity. (6) Hydrophobic interaction (Kauzmann- Nemethy and Scheraga-Tanford model) according to which the interaction is a partial reversal of process (a).(c) Solvent-separated hydrophobic pair interaction according to which E(r) is of a longer-range nature. This model is more consistent with the physical properties of dilute solutions and is consistent with recent computer calculations. It also has important implications for the role of hydrophobic interactions in the stabilization of proteins. [Figure reproduced from ref. (19) with permission.] range nature of the potential of mean force is sure to be of particular significance to the role of the hydrophobic interaction in the stabilization (or destabilization) of globular proteins and multisubunit ~tructures.~~ Very recent developments include high-precision vapour-pressure measurements on aqueous solutions of hydrocarbons 25 and computer studies of the energetics molecular distributions and dynamics of apolar species in water.26-28 The most important conclusions are (1) the solute-water distribution function gAw(r) suggests that the solute occupies cavities in a hydrogen-bonded water framework which bear a resemblance to those found in crystalline clathrate hydrates; (2) the water molecules making up the cavities are subject to stronger hydrogen bonds than those in bulk water and they also suffer a reduction in their diffusion rates; (3) WAA(r)is an oscillating function and has an effective range of several water molecular diameters; at ordinary temperatures the equilibrium separation of two apolar molecules is considerably greater than the sum of the van der Waals radii.Although the computer simulations have not produced any new insights they are in complete harmony with the previously advanced speculations l9 based on the thermodynamic and dynamic properties of dilute aqueous solutions. It is becoming increasingly clear that in a majority of globular proteins a considerable proportion of apolar amino-acid residues is exposed to the aqueous medium and that in protein crystals quite unorthodox water molecular distributions exist in the neighbourhood of such apolar "patches ".29 The question then is how does the solvent affect the net stability of native biomacromolecules ? Computer experiments on isolated protein molecules in the absence of the solvent however interesting in themselves will hardly provide the desired information.Nor can it be very fruitful to persist with treatments that are based on the summation of individual amino-acid hydrophobicity values. HYDROPHOBIC INTERACTIONS-A HISTORICAL PERSPECTIVE On the other hand it is to be hoped that the recent theoretical and experimental developments on so-called simple systems will eventually help to provide a better appreciation of the interplay of weak possibly long-range forces which control the structure and dynamics of biological macromolecules. The original aim of the organizing committee was to stimulate discussion of problems surrounding the inter- actions of one and two spherical molecules in aqueous solution i.e. descriptions of WAw(r)and WAA(r)and their associated distribution functions.However in the event scme of the processes which lead to multiple association in the forms of micelles lipid bil iyers and protein subunit aggregates are also included mainly to demonstrate the continuity between the dilute and the concentrated solution and the similarity between the rare-gas atom and the macromolecule with large numbers of hydrophobic sites in aqueous media. This brief historical outline is of necessity a personal view of the main develop- ments over a period of almost fifty years. As demonstrated in fig. 1 the literature covering the subject is vast so that the references included in this introduction to the Symposium should be regarded as signposts which pointed the way to subsequent developments. G. S. Hartley Aqueous Solutions of Parafin Chain Salts; a Study in Micelle Formation (Actual-ites Scientifiques et Industriels Paris 1936) p.387. I. M. Barclay and J. A. V. Butler Trans. Faraday SOC.,1938 34 1445. D. D. Eley Trans. Faraday SOC., 1939 35 1242 1281. €1. S. Frank and M. W. Evans J. Chem. Phys. 1945 13 507. ’J. H. Hildebrand and R. L. Scott The Solubility of Non-electrolytes (Dover New York 3rd edn 1964). J. S. Rowlinson Liquids and Liquid Mixtures (Plenum Press New York 2nd edn 1969). W. Kauzmann Adv. Protein Chem. 1959 14 1. J. F. Brandts J. Am. Chem. SOC.,1964,86,4291,4302. G. Nemethy and H. A. Scheraga J. Chem. Phys. 1962,36 3382 3401. lo D. N. Glew J. Phys. Chem. 1962 66 605. F. Franks and D. J. G. Ives Quart. Rev. Chem. SOC. 1966,20 1. E. v. Goldammer and M.D. Zeidler Ber. Bunsenges. Phys. Chem. 1969 73 4. l3 E. v. Goldammer and H. G. Hertz J. Phys. Chem. 1970,74 3734. l4 A. Ben-Naim Hydrophobic Interactions (Plenum Press New York 1980). l5 C. Tanford The Hydrophobic Efect-Formation of Micelles and Biological Membranes (Wiley-Interscience New York 1973). l6 J. E. Desnoyers L. Avedikian G. Perron and J-P. Morel J. Solution Chem. 1976,5 631. l7 C. Jolicoeur and H. L. Friedman Ber. Bunsenges. Phys. Chem. 1971,75,248; J. Phys. Chem. 1971 75 165. K. Iwasaki and T. Fujiyama J. Phys. Chem. 1977,81 1908; 1979 83,463. l9 F. Franks in Water-A Comprehensive Treatise ed. F. Franks (Plenum Press New York 1975) vol. 4 p. 1. 2o R. Wolfenden L. Anderson P. M. Cullis and C. C. B. Southgate Biochemistry 1981 20 849.21 C. Chothia J. Mol. Biol. 1976 105 1. 22 D. Y. C. Chan D. J. Mitchell B. W. Ninham and B. A. Pailthorpe in Water-A Comprehensive Treatise ed. F. Franks (Plenum Press New York 1979) vol. 6 p. 239. 23 L. Pratt and D. Chandler J. Chem. Phys. 1977 67 3683; 1980 73 3434. 24 J. L. Finney in Biophysicsof Water ed. F. Franks and S. F. Mathias (John Wiley Chichester 1982) p. 55. 25 E. E. Tucker and S. D. Christian J. Phys. Chem. 1979 73 426. A. Geiger A. Rahman and F. H. Stillinger J. Chem. Phys. 1979 70 263. 27 C. S. Pangali M. Rao and B. J. Berne J. Chem. Phys. 1979,71 2975. 28 S. Swaminathan and D. L. Beveridge J. Am. Chem. SOC., 1979 101 5832. 29 J. L. Finney in Wafer-A Comprehensive Treatise ed. F. Franks (Plenum Press New York 1979) vol. 6 p. 47.

ISSN:0301-5696    

DOI:10.1039/FS9821700007

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC., 1982 17 11-24 Precise Vapour-pressure Measurements of the Solubilization of Benzene by Aqueous Sodium Octylsulphate Solutions BY EDWINE. TUCKER AND SHERRILD. CHRISTIAN Department of Chemistry The University of Oklahoma Norman Oklahoma 73019 U.S.A. Received 11th August 1982 Accurate vapour-pressure-solubility results have been obtained for aqueous solutions of sodium octylsulphate(SOS) and benzene at 15,25,35 and 45 "C. Thermodynamic constants are reported for reactions of benzene with SOS monomers and micelles ; hydrophobic effects are important in stabiliz- ing the aggregates formed between monomeric benzene and SOS species. The observed non-linearity of degree of solubilization with benzene activity indicates that benzene molecules interact cooperatively within the micelles.A mass-action model employing a form of the Poisson distribution equations modified to account for cooperativity provides an excellent fit of all the solubilization data at each temperature. Activity coefficients are reported for benzene in the micellar species as functions of temperature and micellar composition. Among the salient properties of aqueous micellar systems is the ability of these solutions to dissolve large concentrations of hydrocarbons. There have been numerous studies of the extent of solubilization of hydrocarbons by aqueous surfac- tants and many attempts to characterize the solubilized components by physical methods. -3 From a thermodynamic point of view little is known about the dependence of the free energy energy or entropy of solubilized hydrocarbon molecules on either concentration or temperature.Except for recent reports from this laborat~ry,~~~ only a few publications provide any information about the variation of the extent of solubilization with hydrocarbon activity or fugacity. It has generally been assumed either explicitly or implicitly that the solubilization of hydrocarbons occurs as predicted by Henry's law; that is that the concentration of the solubilized species varies linearly with solute activity.6 The large majority of solubilization studies have in fact been restricted to investigations of solutions prepared by allowing the aqueous surfactant solutions to equilibrate directly with pure organic liquids or liquid mixtures.The purpose of the present research has been to provide detailed information about the thermodynamics of solubilization of a hydrocarbon benzene by an aqueous surfactant sodium octylsulphate (SOS) at concentrations below and above the critical micelle concentration (c.m.c.). A high-precision automated vapour-pressure apparatus '** has been used to investigate the formation of discrete complexes between benzene and SOS. Below the c.m.c. the adducts which must be considered include the 1 1 complex and other small molecular species; above the c.m.c. solubiliz- ation occurs primarily through formation of aggregates containing one or more benzene molecules and n surfactant anions. A detailed mass-action model based on an extension of the Poisson distribution equations is employed to correlate the many hundreds of sets of equilibrium vapour-pressure data obtained at benzene fugacities varying from 0 to 70% of saturation at temperatures from 15 to 45 "C.SOLUBILIZATION OF BENZENE BY sos EXPERIMENTAL VAPOUR-PRESSURE MEASUREMENTS An automated vapour-pressure apparatus and techniques described previously '9' were used to measure equilibrium pressures of benzene above aqueous solutions contained in a thermostatted central reservoir. In each series of measurements a known volume of SOS solution was introduced into the central reservoir which had a total volume of 510.9 cm3 and the system was evacuated to the equilibrium vapour pressure of the aqueous solution. The small amount of water volatilized from the sample during the evacuation step was collected and accurately weighed.Under microcomputer control successive samples of benzene stored in an external reservoir at 50 "C,were volatilized into the central reservoir through a 6-port high-pressure chromatography valve; the quantity of benzene added per increment is highly reproducible equalling (2.135 rf 0.002) x mol in the present series of experiments. The total vapour pressure was monitored continuously and after each sample addition comparisons of successive pressure values were made at regular intervals. Approximately 30 to 45 min were required to reach equilibrium which was judged to have been attained when the pressure varied by less than 3 mTorr over 5 min. The primary data consisted of several series of measured equilibrium vapour-pressure values (p) at constant temperature (T),for equal increments of benzene added to a system of known liquid and vapour volume and total composition.Table 1 lists the entire collection of 588 vapour-pressure values obtained at 4temperatures (15,25,35and 45 "C)for accurately measured volumes of solutions having known initial concentrations of SOS. In the initial processing of data it is necessary to convert the measured total vapour pressures to fugacities of benzene at known total concentrations of benzene and SOS in the aqueous solution. This requires information about the virial coefficients of benzene in the vapour phase,' as well as partial molar volume data for benzene in the aqueous phase. The volume of benzene added is in each case small compared with the volume of the condensed phase; consequently the partial molar volume of benzene need not be known with great accuracy.A complete table of the derived solubilization results available from the authors lists values of the total aqueous concentrations of SOS and of benzene the fugacity of benzene the concentration of benzene monomer in the aqueous phase (inferred by assuming that monomeric benzene obeys Henry's law on the molarity basis and using a Setchenow coefficient to account for the salting-out effect of added Na+ on the solubility of monomeric benzene) and the absolute temperature for all of the samples investigated. CHEMICALS Sodium octylsulphate (Eastman Kodak Reagent Grade) was purified as described pre- vio~sly.~ Benzene (Mallinckrodt A.R.Grade) was fractionally distilled and carefully dried before use. RESULTS INITIAL EXAMINATION OF SOLUBILIZATION Before developing a detailed mathematical model for the solubilization data we shall consider some general features of the vapour-pressure-solubility results. Fig. 1 shows some of the data obtained at 25 "C for pure water for [SOS]z 0.15 mol dm-3 and for [SOS]z0.29 mol dm-3. In the absence of dissolved SOS the partial pressure of benzene in equilibrium with its aqueous solutions increases nearly linearly with benzene concentration because only a small fraction of the dissolved benzene exists as self-associated species.8 However in the SOS solutions the partial pressure of benzene increases much less rapidly as the concentration of benzene [B] increases.Qualitatively this seems to imply that there are cooperative interactions between E. E. TUCKER AND S. D. CHRISTIAN 75 60 45 h c, .I 0 M 2 30 15 0.02 0.04 0.06 benzene molarity FIG.1.-Benzene fugacity plotted against molarity for solutions at 25 "Cin HzO(-) SOS (X) and 0.29 mol dm-3 SOS (0). 0.15 mol dm-3 4.0 3.0 2 .o d 1.0 0 FIG.2.-Ratios of increments in solubilized benzene concentration to increments in benzene monomer concentration at 15 "Cas functions of benzene monomer concentration for varying SOS concentra-tions curve A 0.3 rnol dm-3 SOS; curve B 0.21 mol dm-3 SOS; curve C 0.15 rnol dm-3 SOS; curve D 0.1 mol dm-3 SOS.c- P TABLE 1.-PARTIAL PRESSURES OF BENZENE ABOVE AQUEOUS SOLUTIONS OF SODIUM OCTYLSULPHATE a -~~ ~=25.00OC T=l5.00 "C Tz25.00 OC T=25.00 "C Ts35.00 "C T=45.00 "C T=l5.00 "C T=25.00 "c T=25.00 "c T=35.00 "c Tz45.00 "c n-w=20.925 n-w=l1.691 n-wz12.869 n-w=11.650 n-w=l2.489 n-w=11.957 n-w=7.9318 n-w=8.7455 n-w=7.8929 n-w=8.4696 n-w=8.1083 2.1700€+00 2.1620~90 2.955M40 3.09SOE*00 4.027OE+OO 5.1 51 OE40 2.3140EMO 3.24SDE90 3.1540E90 4,26mE+00 5.lc6oE+oo 4.3360E90 4.3180E90 5.88mE90 4.1 870E90 8.0380E90 1.0261E41 4.59loE.90 6.4620E90 6.287OE+00 6.4260~*00 1.0270€+01 6.4970E90 6.4660E+00 8.81 70E90 9.2730E90 1.2030E91 1.5355E91 6.8240E90 9.6240€+00 9.376oE+Oo 1.2544E91 1-556CE91 8.6580E+00 8.61 60E90 1.1757E91 1.2362E91 1. S998E91 2.0457E91 9.0471 E90 1.2783E91 1.2437E91 1.6626€+01 2.0421 E91 1.0805E91 1.0761E91 1.C684E41 1.541X91 1.W77E91 2.5552E91 1.1236E91 1.5883E+01 1.5461E91 2.0696E 91 2.548X+01 1.2966E91 1.2907E91 1,7603E91 1.0469E91 2.3965E91 3.0649E91 1.%03E91 1.8WOE+O1 1.8474E91 2.473% *01 3.0507E41 1.5121E+01 1.5045E+Ol 2.051 3E91 2.1539€91 2.7947E*01 3.5737E91 1.5547EtOl 2.2034E91 2.1 63X+O1 2.8757E91 3.5506€+01 1.7272E91 1.71 94E91 2.342SE91 2.4581E91 3.1 9lZE91 4.0854E91 1.7652E91 2.5045E91 2.4370E91 3.271 4E91 4.0464EMl 1.941 7E91 1.9333E91 2.6342E91 2.7631E91 3.5867E91 4.5933E91 1.9743E41 2.8042E91 2.7282E91 3.665X91 4.543QE91 2.1 571 E91 2.1 467E 41 2.9263E91 3.0677E91 3.9828E91 S.lWSEt01 2 .1782E91 3.1024E91 3.01 SEE91 4.0584E41 S.0329€*01 2.3723E91 2.361 lE41 3.2148Ei-01 3.3714E91 4.3774E91 5.6076E41 2.381 1EM1 3.5WE41 3.2994E91 C,647OE91 5.522XdJl 2.5882E91 2.5742E41 3.5051E91 3.6755E91 4.773X91 6.1 132E+O1 2.5807E91 3.6855EMl 3.57QIE91 C.832K91 4.0092E*01 2.8026E91 2.7857E91 3.7962E41 3.9785691 5.1685E*01 6.6177E91 2.7756E91 3.9734E91 3.8561E91 5.21 42E41 4.4924E91 3.0179€91 2.W67E91 4.0854E91 4.2816E91 5.5620E91 7.1228E91 2.9701 €91 4.2583E91 4.1298E91 5.5937E91 6.9702E91 3.233OE91 3.M81 EN9 4.374SE91 4.5856E91 5.95?7E91 7.6247E41 3.1604E91 L539lE+Ol 4.4012E91 5.9707E+01 ?.cc43€91 3.4469E41 3.41 85E+01 4.6655E+01 4.8835E91 6.3506E91 8.1 24SE41 3.3469E91 4.8172E91 4.67W91 6.3448E41 7.9l71E91 3.6618E91 3.4293E91 4. %S2E91 5.1822E91 6.7460E91 8.6296E91 3.531 5E91 5 .OW 4EM1 4.9355E91 6.71 40OE91 8.3807E91 3.8765E91 3.8397E91 5.2436641 5.48Qk91 7.1381 EM1 9.1328E91 3.7121 E91 5.3613E91 5.1 961 €91 7.0809€91 8.8565E91 4.0897E91 4.0486E91 5.5317E41 5.7768E91 7.5M2E 91 9.6339E91 3.8898E91 5.4284E91 5.4546€+01 7.4452E91 9.321 5E91 4.3041 €91 4.2566E91 5.8202E91 6.071 9E91 7.9231E91 1-01 36E92 4.064% 91 5.8937E91 5.7096E91 7.8048E91 9.783SE*01 4.51 89E91 4.4641E91 6.1 046E41 6.3655E91 8.31 38E91 1.0635EN2 4.2347E91 6.1551 E41 5.9602E#1 8.1 627E91 1.024&*02 4.7334E91 6.3930E41 6.65 7% 91 8.7040E41 1.t 131 €92 4,4035E91 6.4112E41 6.2076EtOl 8.51 SOEM1 1.0701E92 4.947X91 6.6794E91 4.9485E41 9.0924E91 1.1630E42 4.5668€+01 6.6645E91 4.451 OEM1 8.8646E91 1.1154E42 5.161 R91 4.9636E91 9.4804E91 1.21 28E92 6.9l62E91 6.6931E91 9.2098691 1.1 60R42 5.3756EMl 9.8680E91 1.2622E92 7.161 1EN1 4.9292Eal 9.5524€91 1.MSlE+02 5.588x91 1.0253E42 1.31 17E92 9.8906E91 1.2494EM2 5.8D27€*01 1.0636E92 1.5609E92 1.0225E92 1.293CEtO2 6.01 74~91 1.41 OOE42 1.0554E92 1-3371 €92 6.231 M91 1.3803E92 6.443X91 6.6574E91 TABLE1.-cont.T= 15.00 "C T= 25.00 "C T= 25.00 "C T= 35.00 "C T= 45.00 "C T= 15.00 "C T= 25.00 "C T= 25.00 "C T= 35.00 "C T=45.00 "C n-w= 5.8398 n-wc6.0523 n-w= 5.81 81 n-w=5.9480 n-w= 5.8475 n-w=4.0130 n-w=4.3478 n-w= 3.9947 n-w=4.2473 n-w=4.0347 2.1 2W)EaO 2.9520Et00 2.WWE+00 3.8530E90 4.7980E+00 1.9490E90 2.7510E90 2.6270E90 3.6060E90 4.491oE+00 4.1 9WE90 5.8700E90 5.7990E90 7.6580E90 9.5920E90 3.8760E90 5.4770E90 5.3020E90 7.179OE90 8.9720E90 6.2680E90 8.76&+00 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of the given moles of water (n-w) and 0.024 15 moles of sodium octylsulphate in a volume of 510.9 cm3.c-r Each successive pressure value represents the addition of 2.135E-04 moles of benzene to the enclosed system.vl SOLUBILIZATION OF BENZENE BY sos benzene molecules dissolved in the micelles although an additional factor to consider is that the concentration of free or monomeric SOS must decrease as benzene-SOS solubilizates are formed. Fig. 2 illustrates the tendency toward greater solubilization at the higher benzene concentrations; the figure shows the dependence of A[B],,,/Ac on C at nearly constant [SOS] (where A[B],, is the change in the concentration of benzene solubilized by SOS corresponding to a change in benzene monomer con- centration AcB). At concentrations of SOS less than the c.m.c. (approximately 0.136 rnol dm-3) the AIB],,l/AcB values are quite small and these values increase relatively little as C increases.Above the c.m.c. a very large percentage increase in A[B],,,/ AcB occurs as C varies from 0 to 0.016 mol dmA3. Note that derivative plots like that shown in fig. 2 indicate the high quality of the data; no solubilization data of comparable precision are available in the literature. Each of the data sets in table I consists of 20-35 measurements of total pressure for solutions prepared by adding equal increments of benzene to an aqueous solution having a nearly constant SOS concentration. For each point in the separate data sets values of the total concentration of benzene [B] and the benzene monoLver concentration c, were inferred in the initial processing of data (vide supra). It is possible to fit the individual data sets {[B] c,> to quartic equations [B] = acg + bck + CC~,+ dci (1) nearly to the precision of the data (ca.5 x lov6mol dm-3 in [B]). The collection of empirical parameters (a byc and d) obtained by fitting the separate sets may then be used to interpolate the data or to arrange them in useful forms for plotting. 0.0 7 I 1 0 0.08 0.16 0.24 0.32 SOS molarity FIG.3.-Curves of aqueous benzene solubility at fixed monomer concentrations of benzene (0.002 0.004 . . . 0.016 mol dm-3) against SOS concentration at 25 "C. It is convenient to use the derived quartic equations to obtain values of [B] corresponding to chosen values of c, at nearly constant values of [SOS]. Fig. 3 displays the results at 25 "C; the smoothed curves passing through the interpolated points indicate the dependence of [B] on [SOS] at values of C varying from 0.002 to E.E. TUCKER AND S. D. CHRISTIAN 0.016 mol dm-3 in increments of 0.002 mol d~n-~. These curves show that small amounts of benzene are solubilized below the c.m.c. and that a substantial increase in solubilization occurs beyond the c.m.c. It is also evident that the value of the c.m.c. (estimated to be equal to the concentration of SOS at the intersection of the two nearly linear portions of each curve) decreases as cB increases. Such an effect has been observed previously,1° but the present results are unique in showing how the entire solubilization curve is displaced as the benzene activity gradually increases. In the detailed analysis of the solubilization data (vide infra) it is inferred that the concentration of monomeric octylsulphate anion c,- must decrease as [SOS] increases beyond the c.m.c.Estimates are made of values of cA-for SOS solutions in the limit of zero benzene activity. Using these cA-values one may calculate the concentration of micellar SOS [SOS],ic at c = 0 for each SOS solution for which the concentration exceeds the c.m.c. The derived values of the parameters for the quartic equations for each data set [see eqn (l)] may then be used to calculate the limiting equilibrium constant for each total concentration of SOS where [BImicis the total concentration of benzene in micellar species i.e. [B],, minus the concentration of benzene in non-micellar complexes with SOS.Kl may be interpreted as the equilibrium constant (divided by the micelle aggregation number n) for the reaction A, + B+A,B (3) where A, represents the SOS micelle containing n octylsulphate anions B represents monomeric benzene in dilute aqueous solution and A,B denotes the micellar species containing only one solubilized benzene molecule. Although the average value of the micelle aggregation number n cannot be obtained with accuracy from the present results Kl can be obtained (once c,-is known) directly from the solubilization data at each total concentration of SOS. Table 2 lists Kl values inferred for the solutions TABLE 2.-EQUILIBRIUM CONSTANTS FOR TRANSFERRING BENZENE INTO SODIUM OCTYLSULPHATE MICELLES temperature/"C Kl/dm3moI-l a 15.00 10.3 25.00 10.8 35.00 10.5 45.00 10.1 Kl represents the equilibrium constant (divided by n) for the reaction A +B+A,B in the limit as [benzene]+O.Standard states for the solutes (B = benzene and A = micelles of n octylsulphate anions) are unit molarity ideal dilute solution states. at the largest SOS concentrations at 15,25,35 and 45 "C; Kl appears to have a maxi- mum at ca. 27 "C. At 25 "C,AH = 0.29 & 0.14 kcal mo1-l and ACp = -95 & 17 cal rno1-l K-l for the reaction [eqn (3)] representing the transfer of a benzene mole- cule from dilute aqueous solution into the interior of the SOS micelle. Note that 27 "C is approximately the temperature at which the c.m.c. for sodium octylsulphate reaches its minimum value.'l At this temperature AH for transferring an octylsulphate anion from aqueous solution into the micelle is approximately zero SOLUBILIZATION OF BENZENE BY sos and ACp is ca.-70 cal mol-I K-I for this transfer.11J2 Thus the micellization process is characterized by enthalpy and heat capacity changes very similar to those pertaining to the transfer of one benzene molecule into the micelle. A MASS-ACTION MODEL FOR SOLUBILIZATION Previously we used mass-action equations similar to those derived from B.E.T. adsorption theory to fit solubilization data for cyclohexane with sodium octylsulphate micelles and for benzene and cyclohexane with sodium deoxycholate micelle~.~ One (or two) discrete equilibrium constants were used to represent the association of the first (or first and second) hydrocarbon molecules with the micelle binding site.A step-wise equilibrium constant was employed to account for the binding of each additional hydrocarbon molecule by micelles already containing solubilized hydro- carbon. In the case of the cyclohexane SOS system an equilibrium constant for the reaction A- + cyclohexane+A-• cyclohexane was adequate to account for the soh- bilization results in the premicellar region. An assumed micelle aggregation number of I5 or 16 provided the best fit of the data for a wide range of cyclohexane activities and SOS concentrations. When attempts were made to use similar simple association models with the benzene SOS solubilization data it became apparent that the size of the micelles could not be assumed to remain constant as the benzene activity varied.Successful modelling required the assumption that the average micelle size gradually increases as the fugacity of benzene is increased. There is also some evidence in the literature that the solubilization of aromatic compounds by alkylsulphate micelles does lead to increases in micelle size.I3 Incorporating a degree of polydispersity in the mass- action model is therefore not unreasonable but it does complicate the mathematical analysis of the data. In trying to develop a mass-action scheme that is both reasonable and economical in its use of adjustable parameters we decided to approach the solubilization problem from a different statistical point of view. The Poisson distribution seems a good zeroth-order model because it quite simply predicts relative concentrations of soh-bilizates containing 1 2 3 .. . i molecules of benzene in the limiting case where no interaction occurs between the bound benzenes.14 Thus for micelles A, the expres- sions for the total micellar SOS and bound benzene concentrations are [SOS]mic= ncAn[l + a + a2/2!+ a3/3! + . . .] = nca,,exp(a) and [BImic= cA,[a + a’ + d/2!+ a4/3! + . . .] = acA,,exp(a) (4) where a = nK1cB. (The constant Kl is the solubilization constant introduced in the previous section; note that Kl equals l/n times the equilibrium constant for the reaction A + B +A,B.) Eqn (4) predicts that the ratio of the concentration of benzene bound in the micelle to that of all micellar forms of SOS will increase linearly with cB or with benzene activity.However it is clear that any reasonable analysis of the solubilization data will require accounting for cooperativity in the binding of benzene. That is as cB increases the ratio [B],,c/[SOS],ic/cB increases (see for example fig. 2). Although the unmodified Poisson equations do not provide a quantitative fit of the data they do serve as a logical point of departure in developing a model that accounts for intramicellar benzene-benzene interactions. To represent micelles containing one benzene molecule only the a term is required in the expressions for bound B and micellar SOS [eqn (4)]. But in considering micelles containing two E. E. TUCKER AND S. D. CHRISTIAN benzene molecules we propose adding a multiplicative term eb to account for the free energy of interaction of the B .-. Bpair at the average distance of contact. If b > 0 the A,B2 species will form with a larger equilibrium constant than that predicted by the original Poisson distribution; but if b < 0 the A,B2 species will have a smaller formation constant than that given by the Poisson equations. In considering the micellar species A,B3 it seems reasonable to include the factor eb to the third power to account for an average of 3 times as many B -B interactions (presumably at the same average distance of contact) as in the A,B2 species. Similarly for the A,B species the term (eb)6 is used as a factor to enhance (or inhibit) formation of that solubilizate. Proceeding in this way we may derive the following expressions for the concentrations of micellar SOS and bound benzene [SOS],ic = mA,[l + a + a2eb/2!+ a3(eb)3/3!+ .. . + ai(eb)'(l-l)12/i!+ . . . ] and [B'Jmic (5) = cA,[~ + a2eb+ a3(eb)3/2!+ . . . + ai(eb)'('-l)l2/(i-I)! + . . .I. Eqn (5) have proved to be quite efficient in fitting solubilization data; for a given micelle A, they involve only the two unknowns Kl (needed in calculating a) and b the interaction parameter. Unfortunately we have been unable to derive closed-form expressions that will represent the series. Therefore in practice the partial sums are ob- tained numerically the series being terminated at values of i for which the ai(eb)'('- l)12/i! terms become insignificant. APPLICATIONS OF THE MASS-ACTION MODEL IN FITTING SOLUBILIZATION RESULTS In attempting to correlate solubilization data throughout wide ranges of SOS concentration and benzene activity we have encountered two major problems (1) It appears to be necessary to consider the existence of at least two different micelle sizes and to include separate solubilization constants for each type of micelle.(2) The thermodynamic equilibrium constant for forming each micelle pertains to reactions of the type pNa+ + qA-+Na,AqPd4. Therefore the activities of the monomeric ions and the micelle should be used (rather than concentrations) in any rigorous mass-action model for fitting the solubilization data. Our previous modelling of the cyclohexane SOS data led to a good correlation of results without considering the polydispersity of the micelles or the dependence of activity coefficients on total SOS c~ncentration.~ However considerably smaller amounts of cyclohexane are solubilized (compared with benzene) and the range of SOS concentrations covered is smaller than for the present system.We are able to account for the major effects of micelle polydispersity on solubiliz- ation by assuming that only two micelles exist the species containing 16 A-anions and the aggregate of 22 A-anions. Obviously other combinations of micelle sizes could be used in modelling but the chosen sizes seem reasonable and they lead to an adequate fit of all the data. Problems arising from the variation of ionic activity coefficients with SOS con-centration are not easily solved.However it seems reasonable to ignore activity- coefficient effects in fitting the separate series of solubilization data at nearly constant [SOS] because the addition of benzene alone should not cause significant variations in the ionic strength. We have been able to estimate that at [SOS] = 0.30moI dm-3 the concentration of the A-anion is 0.120mol dm-3. This value derives from our mass-action simulation of SOS activity data obtained from Kale and Evans in SOLUBILIZATION OF BENZENE BY sos advance of p~b1ication.l~ The simulation includes the effects of ionic strength on activity coefficients and it leads to the result that the degree of counterion binding (of Na+) is ca. 40%. In fitting the solubilization data at each temperaturq we have fixed the value of cA-at 0.120 mol dm-3 (in the absence of benzene) at the highest con- centrations of SOS.At other SOS concentrations we have not forced cA-to have any predetermined value; instead the cA-values are inferred from the complete mass-action correlation of the data. In the analysis the formation constants for the two micelles A16 and AZ2 are assumed to be independent parameters to be determined by least-squares optimization at each SOS concentration. In other words instead of attempting to account quantitatively for activity coefficient effects throughout the range of SOS concentrations we have allowed the individual micelle formation con- stants to vary with concentration. These equilibrium constants thus pertain to the reactions 16 A-+A16 and 22 A-+ A22 where concentrations of species may now be used in place of activities in the mass-action fitting of data.In fitting the data at SOS concentrations less than the c.m.c. it is necessary to introduce the equilibrium constant K,, for the reaction A- + B -+ A-.B plus one other formation constant to account for a species with several A- anions and at least two benzene molecules. The complex A6B2 is assumed to form with an equilibrium constant & and both K, and Kbt are inferred from the solubility data for [SOS] < 0.12 rnol dm-3. The derived values of these constants are incorporated into the com- posite model used to correlate the entire collection of data at each temperature. Given the assumptions and rationale in the preceding several paragraphs we may summarize the features of the actual model used in fitting data (I) The equilibrium constants K16 = cA16/ci!and K22= cA2Jc2! are parameters to be determined separ- ately for each SOS concentration greater than the c.m.c.[except for the largest SOS TABLE 3.-MASS-ACTIONPARAMETERS FOR SODIUM OCTYLSULPHATE-BENZENE COMPLEXES IN AQUEOUS SOLUTION 15.00 "C 25.00 "C 35.00 "C 45.00 "C Kll/dm3 mol-' a K62/105dm21 mol-' K for A16 micelles/ 0.45 i0.02 2.63 & 0.05 9.97 If 0.05 0.62 i0.01 5.72 * 0.10 10.10 f0.03 0.83 i0.02 5.16 i0.06 10.01 & 0.05 0.93 i0.02 5.12 i0.06 8.88 i0.06 dm3 rnol - Kl for AZ2micelles/ 11.19 i0.05 11.74 & 0.05 11.44 i0.05 11.29 f0.06 dm3 mol-l b for A16 micelles 0.036 & 0.002 0.027 i0.002 0.031 f0.002 0.028 i0.004 b for A22micelles 0.055 & 0.001 0.049 i0.001 0.050 5 0.001 0.042 f0.001 r.m.s.d.(mol dm-3) 9.82 x 10-6 2.71 x 10-5 7.97 x 10-6 1.04 x 10-5 a Equilibrium constant for A + B -+ AB (where A denotes octylsulphate and B denotes benzene); equilibrium constant for 6A $-2B -+ A&; equilibrium constant for A16 + B +A16.B divided by 16; * equilibrium constant for A22-f B +A22.B,divided by 22; cooperativity parameter in modified Poisson equations for B * B interactions in AI6 micelle (see text); cooperativity parameter in modified Poisson equations for B * * * B interactions in A22micelle; root-mean-square deviation in total benzene molarity. concentration see (3) below]. (2) At the highest SOS concentrations at each tem- perature cA-is taken to be 0.120 mol dm-3 in the absence of benzene.(3) Values of K16 and K22at the largest values of [SOS] are fixed to force the total concentrations of A- in these two forms to be equal consistent with cA-= 0.120 mol dm-3. (4) The micelles are individually assumed to solubilize benzene according to the modified E. E. TUCKER AND S. D. CHRISTIAN Poisson distribution including a cooperativity parameter b for each size of micelle. The constants Kl and b used in relating [B],, and [SOS],i to the benzene activity are parameters to be inferred in the least-squares analysis of all the data at a given temperature. (5) Formation constants for the 1 1 SOS-benzene complex and the solubilizate assumed to contain 6 A-anions and 2 benzene molecules are the only equilibrium constants introduced to account for the formation of non-micellar aggregates.The least-squares fitting of data requires as a first step developing expressions to relate [SOS] and the total benzene concentration to the concentrations of the mono- mers cB and cA-.4 The expression for [SOS] can be solved numerically for cA-using provisional values for all of the equilibrium constants including Kl and b in the modified Poisson equations for both types of micelle. The calculated value of cA-may then be used together with the same provisional values of all the equilibrium constants to predict a value of the total benzene concentration for each of the data sets. The non-linear least-squares program NLLSQ l6 is used to minimize the sum of squares of deviations between the calculated and experimental total benzene con- centrations and to infer values of the equilibrium constants and their standard devia- tions.Table 3 summarizes some of the extensive results obtained from the least- squares analysis of data at all four temperatures. DISCUSSION AND CONCLUSIONS The values of the root-mean-square deviation (r.m.s.d.) in total benzene concen- tration given in table 3 indicate the excellent goodness of fit achieved by utilizing the mass-action model to correlate all of the solubilization data. Except for the results at 25 "C,for which all 245 data sets are fitted to an r.m.s.d. of 2.71 x lo-' mol dm-3 the model in general yields r.m.s.d. values that are within a factor of two of the value corresponding to the extremely high precision of the measurements (ca.5 x mol dmM3). Considering the fact that the total concentration of benzene in each series of measurements attains values of the order of several hundredths mol dm-3 we conclude that the modelling quite successfully reproduces the major features of the solubilization phenomena. The values of the equilibrium constants for forming the 1 1 complex increase with temperature but at a decreasing rate as may be expected for a typical hydrophobic association complex. Using the K, values we may calculate that AH = 5.6 & 0.6 kcal mo1-I at 25 "C for the reaction A-+ B -+ A-B. We also obtain the value ACp = -170 & 80 cal K-l mol-' a more negative quantity than that obtained for similar complexes of hydrocarbons with non-ionic molecules.* The K62 values do not show any clear trend but they appear to pass through a maximum in the temperature range 15-35 "C.Of major interest are the values of the constants Kl and b needed to fit the solu- bilization data to the modified Poisson model [eqn (5)]. K for the A, micelle is consistently larger at each temperature than K for the AI6 species and the co- operativity parameter b is also considerably larger for A, than for A16. At each temperature the average of the Kl values for the AI6and micelles are equal to within 3% to the Kl values in table 2 which were inferred from the limiting solubiliz- ation data using eqn (2) without assuming anything about the size of the micelles. The b values (ca. 0.03 for the A16 micelles and 0.05 for the A2 micelles) correspond to free energies of pairwise interaction of ca.-20 cal mol-' for the smaller and -30 cal mol-1 for the larger micelles. The complete least-squares analysis provides many other results that may be used SOLUBILIZATION OF BENZENE BY sos to characterize the formation of solubilizates in the SOS benzene system; however extensive details cannot be given here because of lack of space. At each temperature the concentration of the octylsulphate anion reaches a maximum of ca. 0.14 mol dmm3 at [SOS] z 0.16 mol dm-3 in the absence of added benzene and cA-decreases uni- formly as [SOS] increases further reaching the assumed value of 0.120 mol dm-3 at ca. 0.30 mol dm-3. The simulation of activity data for aqueous SOS solutions described above leads to quite comparable values of c,- and recent analyses of activity data for aqueous solutions of sodium dodecylsulphate and sodium decanoate yield anion concentrations that vary similarly with the total surfactant concentra- tion."-I9 A decrease in c,-of ca.10% occurs for each series of results at constant [SOS] as the benzene fugacity varies from 0 to 70% saturation. There is a sub-stantial increase in the relative proportion of the A22solubilizates compared with the A, species as the benzene activity increases at constant [SOS]. There is also a pronounced increase in the relative proportion of A22micelles as [SOS] increases at constant benzene activity. The general view emerging from the rather complex analysis presented here is that as the benzene activity increases the benzene molecules are solubilized preferentially within the A22species reflecting the tendency of benzene to interact more strongly with the larger micelles and also with other benzene mole- cules within the micelle.A comparison may be made of the thermodynamic constants in table 2 with constants for the transfer reaction benzene (dilute aqueous solution) -+benzene (pure liquid).' For the latter reaction AH = -0.56 kcal mol-' at 25 "Cand ACp = -61 cal mo1-I K-l compared with the values AH = 0.29 kcal mo1-' and ACp = -95 cal mol-I IS-'obtained here for the reaction A, + B -+A,,B in dilute aqueous solution. Both reactions have large negative values of ACp,presumably owing to the abnormally large heat capacity of monomeric benzene dissolved in water.20 The transfer enthalpy for benzene (liquid) -+ benzene (in the micellar interior) is ca.0.85 kcal mol-l at 25 "C implying that the environments of the benzene molecules are quite similar in the two states. The present results do not seem to support the conclusion that arom- atic hydrocarbon molecules solubilize preferentially in the vicinity of the polar-ionic region of the micelle rather than in the hydrocarbon core.21 It may be mentioned that the enthalpy of transfer of benzene into the micelle does become more exothermic as the benzene activity increases reflecting the enhancement in benzene-benzene inter- actions within the micelle as the total intramicellar concentration of benzene increases.The availability of values of [SOS],, and [BImicfrom the analysis of the solubiliz- ~~~, ation data makes it possible to examine the dependence of Y~ the activity co- efficient of benzene in SOS micelles (on the pure-component standard-state basis) on XB,mic, the average mole fraction of benzene in all the micellar forms. Thus XB,mic = [Blmic/([Blmic + [sos]~ic) and YB,mic =fB/(XB.micfi> wherefB is the fugacity of benzene in equilibrium with the micellar solution andfi is the fugacity of pure liquid benzene at the same temperature. Fig. 4 is a plot of yB,,, against XB,,, for the solutions at [SOS]z 0.3 mol dm-3 at 15,25 35 and 45 "C. The activity coefficient values indicate a considerable degree of positive deviation from Raoult's law the deviation decreasing with increasing temperature.There is also a decrease in yB,micwith increasing XB,mic at each temperature reflecting cooperative interactions between benzene molecules within the micelles. Recent research from this laboratory has shown that high-precision vapour-pressure measurements can provide detailed information about the thermodynamic quantities for forming complexes between hydrocarbons and other aqueous solutes. The results 4.4 < 4.2 m ?-4.0 3.8 0 0.05 0.10 0.15 0.20 XB,rniC FIG.4.-Activity coefficient of micellar benzene plotted against benzene mole fraction in sodium octylsulphate micelles. reported here are the most extensive ever obtained for an aqueous surfactant system. Hydrophobic association effects are examined quantitatively in a way not possible with other physical methods for studying solubilization.We are using the techniques and models described in the present article to investigate other surfactant systems; a report on the effect of added NaCl on properties of benzene-SOS solubilizates is in preparation. The research described here has been supported by the National Science Found- ation (grant CHE-8103084) the United States Department of Energy Bartlesville Energy Technology Center (contract DE-AT 1981 BC10476) the University of Oklahoma Energy Resources Center and the University of Oklahoma Mining and Mineral Resources Research Institute. M. E. L. McBain and E. Hutchinson Solubilization and Related Phenomena (Academic Press New York 1955).’P. H. Elworthy A. T. Florence and C. B. McFarlane Solubilization by Surface Actiue Agents and Its Application in Chemistry and the Biological Sciences (Chapman and Hall London 1968). P. Mukerjee in Solution Chemistry of Surfactants ed. K. L. Mittal (Plenum New York 1978) vol. 1. S. D. Christian E. E. Tucker and E. H. Lane J. Colloid Interface Sci. 1981 84 423. S. D. Christian L. S. Smith D. S. Bushong and E. E. Tucker J. Colloid Interface Sci. in press. A. Wishnia J. Phys. Chem. 1963 67 2079; I. B. C. Matheson and A. D. King J. Colloid Interface Sci. 1978,66,464; S. A. Simon R. V. McDaniel and T. J. McIntosh J. Phys. Chem. 1982,86 1449. ’E. E. Tucker and S. D. Christian J. Chem. Thermodyn. 1979 11 1137. E. E. Tucker E.H. Lane and S. D. Christian J. Solution Chem. 1981 10 1. M. B. King Phase Equilibrium in Mixtures (Pergamon Oxford 1969) p. 260. lo S. J. Rehfield J.Phys. Chem. 1967 71 738. l1 E. D. Goddard and G. C. Benson Can. J. Chem. 1957 35,986. l2 G. L. Musbally G. Perron and J. E. Desnoyers J. Colloid Interface Sci.,1974 48 494. SOLUBILIZATION OF BENZENE BY sos l3 H. W Offen D. R. Dawson and D. F. Nicoli J. Colloid Interface Sci. 1981 80 118; H. Coll J. Phys. Chem. 1970,74 520. l4 S. S. Atik M. Nam and L. A. Singer Chem. Phys. Lett. 1979 67 75. K. Kale and D. F. Evans unpublished work. l6 S. D. Christian and E. E. Tucker Am. Lab. 1982 14(9) 31. l7 T. Sasaki M. Hattori J. Sasaki and I(.Nukina Bull. Chem. Soc. Jpn 1975,48 1397. S. G. Cutler P.Meares and D. G. Hall J. Chern. Soc. Faraday Trans. 1 1978 74 1758. l9 E. Vikingstad J. Colloid Interface Sci. 1979 72 68. 'O F. Franks in Water-A Comprehensive Treatise ed. F. Franks (Plenum New York 1973) chap. 1. '' P. Mukerjee and J. R. Cardinal J. Phys. Chem. 1978 82 1978.

ISSN:0301-5696    

DOI:10.1039/FS9821700011

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Symp. Chem. Soc. 1982 17 25-30 Nuclear Magnetic Resonance Relaxation Investigation of Tetrahydrofuran and Methyl Iodide Clathrates AND MANFRED BY MARC JUNIO DIETER 0. LEICHT D. ZEIDLER Institut fur Physikalische Chemie der RWTH Aachen Templergraben 59 D-5 100 Aachen Federal Republic of Germany Received 13th August 1982 Proton and deuteron spin-lattice relaxation rates of the guest molecules have been measured as a function of temperature in methyl iodide and tetrahydrofuran clathrates. By use of the isotopic- dilution method in proton n.m.r. i.e. employing the compounds (CH31),.(CD31)l -;17D20 and (C4H80)x*(C4D80)1 -,*17D20 with varying mole fractions x intramolecular and intermolecular contributions to the proton relaxation rate may be separated.Deuteron relaxation rates have been measured in the compounds CD31*17H20 and C4D80*17H20. The maximum temperature range covered was 0 to-120 "C. The proton relaxation rates contain rather large intermolecular contributions ca. 45% for methyl iodide and 23% for tetrahydrofuran. The correlation times obtained from the intramolecular contributions lie in the range 1-12 and 0.5-1 ps for methyl iodide and tetrahydrofuran respectively which is the range typical for liquids. This confirms the well known fact that the enclathrated guest molecules perform very fast motion. The evaluation of the deuteron relaxation rates leads to correla-tion times which are comparable to those derived from the intramolecular proton rate for tetrahydro- furan but differ in the case of methyl iodide.This indicates anisotropic rotational motion of the methyl iodide molecule in the clathrate cage. A summary of n.m.r. investigations on clathrate hydrates has been given by Davids0n.l Here we mention only those papers which include relaxation time measurements methyl chloride,2 pr~pane,~ sulphur hexafluoride and tetrahydro- furan hydrate. To our knowledge no attempt has been made to separate relaxation contributions for example by using the isotopic-dilution method,6 and thus to obtain more detailed information on the molecular motion of the clathrate components. Clathrate hydrates form according to well known rules depending on the chemical nature of the guest molecule. In the present investigation tetrahydrofuran and methyl iodide are the guest molecules and they both form clathrates of the type M.17H20.The water molecules in these clathrates are arranged in pentagonal dodeca- FIG.1.-Cage structures of type M.17H20 clathrates. Only oxygen positions are drawn. N.M.R. INVESTIGATION OF CLATHRATES hedra and hexakaidecahedra (having 4 hexagons and 12 pentagons as faces) with cage diameters 4.8 and 6.9 A respectively (see fig. 1). Only the larger cages can be occupied by the tetrahydrofuran and methyl iodide molecules because of their size. Both hydrates are stable up to 4.3 "C at normal pressure. They can be further stabilized by also filling the smaller cages with suitable molecules for example nitro- gen. It is well known that the encaged molecules show large mobility even at tempera- tures far below the ice point.This motion must be largely rotational because the water cage prohibits any translational motion except within the cage. In order to obtain more details about this motion one can employ the well established n.m.r. relaxation methods often used in liquid-state studies. (a) If dipolar interactions dominate the relaxation intra- and inter-molecular contributions are separated by the isotopic-dilution technique. Intramolecular contributions are associated with rotational motion whereas intermolecular contributions are largely effected by translational motion. (6) If quadrupolar interactions dominate only rotational motion is probed. By comparison of rotational correlation times for different quadrupolar nuclei (or with the aforementioned rotational dipolar contribution) information on anisotropic rotation is obtained.In the present paper the proton and deuteron n.m.r. relaxations (showing dipolar and quadrupolar interactions respectively) of the guest molecules are studied and the isotopic-dilution method (using deuteron dilution) is employed in the proton n.m.r. This means that clathrates of the type (resonant nucleus underlined) (C,H,O) (C4D80)1-x-17D20,(CH,I);(CD,l) -;17D20 C4]2S0.17H20 and CD31017H20 with varying composition x were measured. The maximum temperature range covered was 0 to -120 "C. RESULTS The clathrate components water (D20 was 99.9 atom %) and methyl iodide (99.5 atom % D) or tetrahydrofuran (99.0 atom % D) were separately freed from oxygen by several freeze-pump-thaw cycles and then distilled into an n.m.r.tube. A nitrogen I I I I I 3.7 L.0 4.5 5.0 5.5 6.0 I -10 I -30 I -50 1 -70 I -90 I -100 I -110 inner scale lo3 K/T outer scale T/"C FIG. 2.-Proton relaxation rates for the clathrate (CH31~;(CD31) -;17D20 top curve x = 1 ; middle curve x = 0.5; lower curve x = 0.25. M. JUNIO D. 0. LEICHT AND M. D. ZEIDLER 5.01 1.51 4.5 5.0 5.5 6.O 6.5 r -50 I -70 I -90 I -100 I -110 inner scale lo3 K/T outer scale T/"C FIG.3.-Proton relaxation rates for the clathrate (C4H80);(C4D80)l-,*17 D20 top curve x = 1; middle curve x = 0.6; lower curve x = 0.1. pressure of 0.4-0.5 bar was applied prior to sealing the tubes. Finally the clathrates were formed by keeping the samples between 0 and 4 "Cfor several hours occasionally shaking them to initiate crystallisation.To avoid inclusion of the liquids several freeze-thaw cycles were performed and it was ascertained that the relaxation-rate measurements were always reproducible after preparation of the clathrate. TI relaxation measurements were performed with a pulse spectrometer operating at 12 MHz using the 9O0-r-9O0 method. The probe temperature was controlled to A1 "C by a thermostatted nitrogen-gas stream flowing around the sample. 200-d ml-lvJ -_ 80-60-g LO-U 1 20 1 1 I I I I 3.7 4.0 L.5 5.0 5.5 6.0 I I I I 1 I I -10 -30 -50 -70 -90 -100 -110 inner scale lo3 K/T outer scale T/"C FIG.4.-Deuteron relaxation rates of the clathrates upper curve CD31*17H20; lower curve C4D80'17H20.N.M.R. INVESTIGATION OF CLATHRATES Fig. 2 and 3 show the proton relaxation rates of methyl iodide and tetrahydrofuran clathrates respectively as a function of temperature; the corresponding deuteron relaxation rates are summarized in fig. 4. DISCUSSION It is obvious from fig. 2 and 3 that dilution of the protonated guest molecule with deuterated species leads to a decrease of the proton relaxation rate. This behaviour is expected if an intermolecular relaxation contribution is present. Since the magnetic moment of the deuteron is considerably smaller than that of the proton the dipolar magnetic interaction is correspondingly reduced. In quantitative terms where l/Tl is the measured proton relaxation rate at the mole fraction x of protonated species.As required by eqn (l) plots of I/Tl against x proved to be linear and could TABLE 1.-PROTONRELAXATION CONTRIBUTIONSOF METHYL IODIDEAND TETRAHYDROFURAN CLATHRATES methyl iodide tetrahydrofur an lo3 KIT 3.8 6.44 4.96 4.0 7.36 5.74 4.2 8.37 6.73 4.4 9.76 8.03 - - 4.5 - - 1.75 0.46 4.6 11.6 9.4 - - 4.7 - - 1.82 0.56 4.8 14.4 11.4 - - 4.9 - - 1.91 0.64 5.O 17.7 15.1 - - 5.1 - - 2.04 0.70 5.2 21.6 20.4 - - 5.3 - - 2.19 0.74 5.4 29.1 25.4 - - 5.5 - - 2.37 0.77 5.6 40.6 32.6 - - 5.7 - - 2.57 0.79 5.8 54.0 47.5 - - 5.9 - - 2.77 0.82 6.1 2.98 0.87 6.3 3.21 0.93 6.5 3.53 0.94 be evaluated for the intra- and inter-molecular contributions.The results are sum- marized in table 1. The large intermolecular rates ca. 45% for methyl iodide and 23% for tetra- hydrofuran are surprising. We have not been able to account for this large rate in quantitative terms by any current theory. M. JUNIO D. 0. LEICHT AND M. D. ZEIDLER 29 The intramolecular contribution is evaluated using the extreme-narrowing formula i# i which relates the rate to a rotational correlation time zD the term " extreme narrow- ing " expressing the condition COT, < 1 where cr) = 27cv stands for the n.m.r. fre-quency. This condition was experimentally verified by a few test measurements at v = 60 MHz and the results were identical with those at v = 12 MHz. The geo- metrical factor containing the interatomic distances rijaveraged over all n protons of the molecule is calculated from the known molecular structure y is the magnetogyric ratio of the proton.The total factor in front of zDamounts to 4.49 x 1o'O and 3.81 X 1O'O s-~ for methyl iodide and tetrahydrofuran respectively the error in these figures being 10%. The correlation times therefore lie in the range 1-12 ps for methyl iodide and 0.5-1 ps for tetrahydrofuran depending on temperature (see table 2). For TABLE2.-ROTATIONAL CORRELATION TIMES OF METHYL IODIDE AND TETRAHYDROFURAN CLATHRATES FROM PROTON (ZD)AND DEUTERON (ZQ)RELAXATION RATES methyl iodide tetrahydrofuran 103~/~ TD/10-l2 s ZQ/lO-s ZD/lO-l2 s ZQ/10-l2s 3.8 1.43 0.68 4.0 1.64 0.79 4.2 1.86 0.92 4.4 2.17 1.09 - 4.5 -0.46 0.52 4.6 2.58 1.31 4.7 -0.48 0.52 4.8 3.21 1.60 - 4.9 -0.50 0.52 5.0 3.94 2.00 - 5.1 -0.54 0.54 5.2 4.81 2.54 5.3 -0.58 0.56 5.4 6.49 3.28 5.5 -0.62 -5.6 9.04 4.39 5.7 -0.67 5.8 12.03 6.06 - 5.9 -0.73 6.1 0.78 6.3 0.84 6.5 0.93 comparison the rotational correlation times in the pure liquids at 25 "Care 1.2 ps for methyl iodide and 0.6 ps for tetrahydrofuran.* The motion of the guest molecules is therefore as fast as in the liquid.From the deuteron relaxation measurements in fig. 4 another rotational correlation time zQ can be calculated through the relation N.M.R. INVESTIGATION OF CLATHRATES where the quantity in brackets is termed the quadrupole coupling constant.This quantity is taken as 153 kHz for methyl iodide determined in the solid ~tate,~ and 198 kHz for tetrahydrofuran a value derived from the liquid-state relaxation measure-ments ' assuming isotropic rotation. The correlation times calculated from eqn (3) are included in table 2. If the correlation times derived from proton relaxation Z and from deuteron relaxation zQ in table 2 are compared it is seen that about the same figures for tetra- hydrofuran but by a factor of 2 differing values for methyl iodide exist. Different correlation times can arise only if the rotational motion is anisotropic because they are associated with either proton-proton vectors (z,) or carbon-hydrogen bonds (zQ but only if the electric field gradient has cylindrical symmetry with its main axis directed along this bond) and both must have different directions within the molecule.Thus it seems that tetrahydrofuran reorients isotropically but this statement is valid only if the same is true for the pure liquid because the quadrupole coupling constant was derived under this assumption. On the other hand methyl iodide re- orients anisotropically behaviour which is also observed in the pure 1iquid;'O however in the clathrate it is even more pronounced. These conclusions can be drawn without going into detailed model calculations which seem not be justified in the clathrate case. However if one does it turns out that motion around the molecular symmetry axis is faster than perpendicular to it the latter motion being slower in comparison to the pure liquid by a factor of 2.We gratefully acknowledge continuing generous financial support by the Fonds der Chemischen Industrie. D. W. Davidson in Water-A Comprehensive Treatise ed. F. Franks (Plenum Press New York 1973) vol. 2 chap. 3. ' C. A. McDowell and P. Ragunathan J. Mol. Struct. 1968 2 359. C. A. McDowell and P. Ragunathan J. Mol. Struct. 1970,5 433. M. B. Dunn and C. A. McDowell Chern. Phys. Lett. 1972,15 508. S. K. Garg D. W. Davidson and J. A. Ripmeester J. Magn. Reson. 1974 15 295. M. D. Zeidler Ber. Bunsenges. Phjs. Chem. 1965 69 659. ' A. Abragam The Principles of Nuclear Magnetism (Oxford University Press London 1970). E. v. Goldammer and M. D. Zeidler Ber. Bunsenges. Phys. Chern. 1969 73 4. M. Rinn6 and J. Depireux Ado. Nucl. Quadrupole Reson. 1974,7 357. lo K. T. Gillen M. Schwartz and J. H. Noggle Mol. Phys. 1971 20 899. H. Versmold 2.Naturforsch. Teil A 1970,25 367.

ISSN:0301-5696    

DOI:10.1039/FS9821700025

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC.,1982 17,31-40 'Infrared and Nuclear Magnetic Resonance Studies Pertaining to the Cage Model for Solutions of Acetone in Water* BY MARTYN C. R. SYMONS R. EATON AND GRAHAM Department of Chemistry The University Leicester LEI 7RH Received 3rd September 1982 Solutions of acetone in water show a single C=O stretching mode in the infrared (1697 cm-I), which is assigned to acetone hydrogen bonded to two water molecules. On cooling below 0 "Cnew features appear which are characteristic of concentrated solutions of acetone formed by growth of ice crystals. These bands are assigned to acetone forming single hydrogen bonds (1708 cm-') and no hydrogen bonds (1715 cm-l). The latter band is characteristic of bulk acetone. On standing at cu. -30 "C,all these features decay and ultimately only a single narrow band is detectable at 1722 cm-I which is characteristic of isolated acetone molecules in completely inert solvents.This change is clearly a result of clathrate formation. These results demonstrate unequivocally that acetone in liquid water is solvated in a manner that differs completely from cage "solvation". This remarkable difference between solid and fluid systems is interpreted in terms of the presence of large concentrations of free OH groups in liquid water. The concept of hydrophobic bonding is probably unique to aqueous systems. It is manifested in a gross manner in such systems as micelles and membranes and reflects the fact that every water molecule wants to form four hydrogen bonds water thus tending to reject molecules with which it cannot form bonds.For molecules that have regions that can form bonds and others that cannot there are a number of compromises micelles being a typical and satisfying example. Another related concept that is often invoked to explain the behaviour of water- rich aqueous systems is that of incipient clathrate cage formation. Water has a remarkable propensity to form fully hydrogen-bonded cages around inert solutes in the solid phase and although such cages are often nearly spherical they are a remark- able variety of more complex shapes in water's repertoire. An as yet unproven hypothesis invoked to explain a multitude of phenomena is that such cages are also important in the liquid phase at least at low temperatures.In our own studies such incipient cage formation was invoked to explain the appearance of intense ultrasonic relaxation effects in water + t-butyl alcohol systems,' proton resonance shifts for aqueous tetra-alkylammonium salts,2 asymmetry in the 6.s.r. spectrum for m-dinitro- benzene anions in aqueous sol~tions,~ and linewidths in the e.s.r. spectra for di-t- butylnitroxide in aqueous system~.~ Assuming for the moment that incipient cage effects may sometimes be significant in liquid-phase systems we need to enquire about possible links between these pheno- mena and hydrophobic effects. These are not obvious since in the solid clathrate compounds there is generally a filigree of water molecules separating each solute molecule thus even in these concentrated systems there is no tendency towards dimerisation or aggregation.This is not always true. For example two (C,H,),S+ * Taken as Solvation Spectra Part 73. CAGE MODEL FOR ACETONE SOLUTIONS cations share a single cage in the clathrate (C4H9)3S+F-8H20.5Thus cage-sharing may only be important when there are not enough water molecules to form mono- cages. Indeed the absence of ultrasonic relaxation behaviour in the 0-0.03 mole fraction range for t-butyl alcohol in water was taken as evidence that cage-sharing only set in at mole fractions > 0.03.' If this is correct we can argue that in systems favouring clathrate cage formation direct " hydrophobic bonding " will be avoided when the ratio H20 solute molecules is greater than that for the clathrate but increasingly favoured for smaller ratios.However another factor needs to be considered.'j In solid clathrates not only are the cages formed so as to give nearly perfect hydrogen bonding for the water molecules defining each cage but all cages are perfectly linked together. (By "per-fect '' we mean four strong nearly linear hydrogen bonds per water molecule.) Hence we can visualise a single cage as acting as a template for adjacent cages in the sense that a significant proportion of a second cage has already been constructed. This is even more the case for the construction of subsequent cages on a double-cage unit. These considerations lead to the tentative postulate that "cages breed cages ". If this were true then it would also lead to a kind of dimerisation or even oligomeris- ation of the solute.The two cases can be compared with the concept of contact and solvent-shared ion pairse7 If these two types of structure can be linked under the genetic term "hydrophobic bonding " perhaps they should be similarly differentiated as contact and solvent-shared units. The aim of the present work was to use infrared spectroscopy to probe similarities or differences between the state of a solute in liquid water and in its solid clathrate compound. We selected acetone as one example,' because of our extensive studies on the use of acetone as a spectroscopic probe of pure and mixed solvent ~ystems.~ In one of our previous spectroscopic excursions into systems that might be in- fluenced by water cages we suggested that the proton resonance for water molecules at low temperatures was shifted to low fields by R4N+ions the low-field contribution increasing as the size of R increased.2 This seemed to correlate with the well known ability for R,N+ salts to form clathrate hydrates.This trend was opposite to that previously detected 'OJ' at higher temperatures and we confirmed that the trend was indeed reversed on heating. This marked temperature effect accords with the fact that cage formation requires considerable organisation of many water molecules but the reversal of the trend seems to imply more than a loss of specific encagement at higher temperatures. It is relevant that shifts caused by R4N+ions in methanol are negligibly sma11.12 There are serious problems associated with this work,2 since we neglected to allow for changes induced in water by the halide ions.13 We postulate that there is a relatively high concentration of water molecules forming three hydrogen bonds in liquid water and we describe these broken bonds as (OH)f,ee and (LP)f,ee,where LP represents the " lone pair " of electrons not involved in hydrogen bonding.Despite the fact that this postulate is not generally accepted we use it herein without further ju~tification.'~ Although (LP)freegroups have never been detected spectroscopically it is usually considered that groups make a considerable contribution to the 0-H stretch overtone band^.'^-'^ When tetra-alkylammonium salts are added to water there is a sharp decrease in the overtone band at 7120 cm-' assigned to groups.This fall has quite reasonably been assigned to the effect of the R4N+ions on water since alkali halide salts induce much smaller changes in this The marked loss of (OH)freegroups was therefore taken as evidence for the " structure-making " ability of R4N+ions. This seemed unlikely to us since there is no major temperature effect on this behaviour and hence no correlation with the proton reson- M. C. R. SYMONS AND G. R. EATON ance results. In our view l8 the major part of this decrease in (OH)free groups is due to the reaction hal-+ n(OH)f,ee-+ hal(H0); (1) when n is the primary solvation number of the halide ion hal-. Given that R4N+ ions do not interact significantly with (LP)r,eegroups as is the case for methanolic solutions reaction (1) must cause a rapid loss of (OH)freegroups.For alkali-metal salts the metal ions generally interact with about the same number (n) of (LP)r,ee groups so that the net effect on equilibrium (1) is small. We expect that there is a small positive or negative contribution from R,N+ ions depending on the temperature but the experimental errors are unfortunately large and in our view such changes are not yet established. If these ideas are correct then the interpretation of the n.m.r. shifts previously proposed must be revised. This is in hand. In our view the e.s.r. results for rn-dinitrobenzene anions in water are more com- pelling and may constitute one of the best pieces of evidence in favour of significant structure around non-bonded groups in liquid water at low temperature^.^ Theory and e.s.r.studies in aprotic solvents require that m-dinitrobenzene anions be sym- metrical with two equivalent nitro group^.'^ However on forming ion-pairs with alkali-metal cations there is a dramatic loss of symmetry the spin-density on one NO2 group increasing by a factor of ca. 2 with that on the other falling to ca. 0.20 This unique modification can be understood in terms of the diagram shown in fig. 1.21*22 FIG.1.-(u) Combination of the ground-state SOMO (wScl,) for m-dinitrobenzene anions and a low-lying excited state ('yscz,) to give two asymmetric levels (yAs(l)+ I+Y*~(~)). These can be thought of as being primarily localised on one or other nitro group the asymmetric wavefunction being stabilised by asymmetric hydration as indicated 'in (6).The important result is that the e.s.r. spectrum in water is that of the asymmetric ion. From the linewidths governed by the transfer of charge and spin-density from one NO2group to the other a residence time z of 4.5 x s was calculated at 0 "C. This fell to ca. 0.8 x s at 25 "Cand rapidly decreased on heating. The lifetime CAGE MODEL FOR ACETONE SOLUTIONS increased by a factor of ca. 2 on adding 0.02 mole fraction of t-butyl alcohol but ul- timately fell to ca. s in the pure alcohol. Times in this region were also obtained for solutions in methanol. Although heating decreased z for aqueous solutions the 14N hyperfine coupling was unchanged showing that the extent of hydrogen bonding at the (NO2)- unit is independent of temperature.The only explanation that we can discover which can accommodate these results is that in protic solvents hydrogen bonding at one NO group pulls the negative charge onto this and away from the other NO2 group. This descent in symmetry tips the wavefunction into one of the two asymmetric forms (fig. 1). This perturbation will switch when some hydrogen bonding builds up fortuitously on the neutral NO2group and is concomitantly shed at the (NO2)- group so that they become effectively equivalent. Why should this be ca. times slower in water than in methanol? Our suggestion is that there is a cage around the neutral portion of the anion (fig. 1) and this needs to be destroyed as solvation builds up at neutral NO2.This very extensive reorganisation is not required for alcoholic solutions. The reinforcing effect of t-butyl alcohol in the 0-0.02 mole-fraction range may be evidence in favour of solvent-shared hydrophobic interactions. Another observation that led us to postulate incipient cage formation in aqueous solution is the marked downfield shift observed in the OH proton resonance for aqueous t-butyl alcohol. This shift which is much greater than that caused by methanol was initially interpreted by ourselves and by Hertz and coworkers in terms of incipient cage f~rmation.~~-~~ However Covington and Newman 26 concluded that the average shift observed was caused by a large downfield shift of the Me,COH protons and the water proton shift was unexceptional.Since then we have been able to obtain conditions of slow exchange and hence have observed both OH protons ~eparately.~' The results confirm our original interpretation it is indeed the water protons that display an anomalously large downfield shift in the usual 0-0.03 mole-fraction range. The only reasonable interpretation of this excess shift is that the cage- forming water molecules experience slightly stronger bonding than those in bulk water. The effect may also shift equilibrium (1) slightly to the left. Finally our studies of aqueous di-t-butyl nitroxide radicals in aqueous solutions In fact this claim was also made very give strong support to the cage the~ry.~.~~ strongly by Jolicoeur and Friedman 29 for similar systems but it seems to us that unfortunately their interpretation is in error.They interpreted their results in terms of a marked increase in zJ the spin-rotation correlation constant on cooling. This would mean that the R2N0molecules were gaining in rotational freedom on cooling which is most unexpected. They claim that this implies a gain of encaged R2N0 molecules which are not anchored to the cages and therefore are free to rotate. We showed that the increase in linewidth on cooling is due to an increase in zc the rotational correlation rather than in zJ. Furthermore the large magnitude of the hyperfine splitting A(I4N) which is indicative of strong hydrogen bonding does not alter in the significant temperature range (0-25 "C),and hence there cannot be any major trend towards non-hydrogen-bonded encaged molecules.Nevertheless we agree that the results imply cage formation although these must be anchored guests. Thus the cages confer unusual lack of freedom. On the addition of t-butyl alcohol zc increases still further in the 0-0.03 mole fraction range. We postulate that both additives are solvated in an elaborate manner at ca. 0 "C with hydrogen bonds linking both guests to partial clathrate cages and with cages reinforcing cages. On further addition of the alcohol zJ does begin to increase showing that the R2N0 molecules are gaining their freedom. Furthermore there is then a dramatic fall in A(14N) confirming that the N-0 HO bonds are lost. Evidently at this stage if cages M. C. R. SYMONS AND G.R. EATON are important the alcohol has tipped the equilibrium to the " free " stage envisaged by Jolicoeur and F~iedman.~~ EXPERIMENTAL Acetone was purified by standard procedures directly prior to use. D20 (99.8 atom %) was used instead of H20 because of the strong underlying absorption in the region of interest of H20. Infrared spectra were recorded on a Perkin-Elmer 580 spectrometer using demountable cells with Irtran windows and 0.025 mm path-length. Temperature was maintained using a Specac controller and thermostatted cell holder. ''0 n.m.r. spectra were recorded on a Bruker WM 400 Fourier-transform spectrometer and n.m.r. spectra on a Jeol FX 60 Fourier-transform spectrometer. Two methods of forming the clathrate were found to be successful (a) A suitable sample of acetone in D20(0.1 mole fraction) was cooled to -30 "C and held at this temperature for several hours the clathrate band was slowly seen to "grow" the effect being complete after cu.3 h. (6) The sample [as in (a) above] was cooled to below -100 "C and then allowed to warm up slowly at 2 "C min-I. The clathrate band was first observed at ca. -40 "C being fully formed at -38 "C. It disappeared rapidly at -19 "C. Method (6) was utilised for the work where alcohol was added to the system. RESULTS AND DISCUSSION INFRARED SPECTROSCOPY The key result of our infrared studies of the C=O stretching band for acetone in fluid water and in its clathrate is that there is a major shift together with a marked change in band shape (fig.2). We interpret this in terms of acetone forming two hydrogen bonds to OH groups in fluid water but no hydrogen bonds in the clathrate but this statement requires justification. wavenumber/cm-FIG.2.-Infrared spectra (C=O stretch) for acetone in water (0.05 mole fraction acetone) (a) at 0 "C (b)at -30 "Con storage for 7 h. Band maxima for acetone in (i) clathrate cage (ii) hexane (iii) bulk acetone (iv) concentrated and (v) dilute aqueous solutions are given as vertical bars. CAGE MODEL FOR ACETONE SOLUTIONS The clathrate band (1722 cm-l) is close to that for acetone in inert solvents such as cyclohexane (table l) so clearly there can be no significant hydrogen bonding in the cage. This result accords with conclusions from X-ray measurement^,^^ dielectric relaxation data 31 and proton n.m.r.When the concentration of ace-tone in liquid water is increased the absorbance of the band at 1697 cm-l falls and a new band at ca. 1708 cm-' grows in. This band is close to that observed for TABLE 1.-INFRARED DATA FOR ACETONE CARBONYL STRETCH IN VARIOUS SOLVENTS solvent band maximum/cm-' hexane 1721 carbon tetrachloride 1718 tetra hydro furan 1717 cyanomethane 1714 dimethyl sulphoxide 1709 methanol 1708 water (25 "C) 1697 water clathrate 1722 perfluoro-t-butyl alcohol 1688 acetone in methanol. It is easier experimentally to study the behaviour of acetone in low concentration as different aprotic cosolvents are added. This is the technique used in our work on R2N0 solutions and we refer to the acetone as a " probe " of the medium.When this is done with cosolvents such as cyanomethane we can again follow the loss of the 1697 cm-I band and growth of the 1708 cm" band. In solutions rich in MeCN this band loses intensity and a third band characteristic of acetone in pure MeCN grows in. Thus there are three species which we identify as the dihydrate the monohydrate and non-hydrated acetone. We conclude firmly that acetone sheds two hydrogen bonds on moving from liquid water into the clathrate. N.M.R. STUDIES As a check on the infrared results we have examined the 13C(CO)and 170n.m.r. spectra for acetone water systems. In our studies on triethylphosphineoxide (Et,PO) as a probe of mixed solvent systems we found that there was a good correlation between the 31Pchemical shifts and the P-0 stretch infrared shifts for a range of pure solvents which could be used to reconstruct the 31P shifts in mixed solvents.33 This was useful because the 31P spectra in mixed solvents are.rapid averages of the shifts for the different solvates involved whereas the infrared spectral features for all the sol- vates are in principle resolved. The 13C studies were very noisy despite long accumulation times but we were able to detect features for solutions in water at room temperature (the dihydrate) in partially frozen systems corresponding to the monohydrate and of course for pure acetone. Unfortunately the band for acetone in the solid clathrate was too broad to detect with our spectrometer. However all these bands were detected in the I7O spectrum including that for the clathrate and these correlate well with the infrared results (fig.3). The fact that the I7O resonance for the solid clathrate was no broader than the liquid-phase bands con- firms the remarkable rotational freedom of the encaged acetone molecules. M. C. R. SYMONS AND G. R. EATON r A0 -20 h Ob" lk E a W 3 x 0 -2 MeOHo -LO .3 0 /o frozen-30"~; 2 \:exone -60 hrate \ J I I I I 6 12 18 24 30 Av (carbonyl)/cm-' FIG.3.-Relationship between n.m.r. and infrared data for the carbonyl group of acetone in various states of solvation. ANCHORED US FREE CLATHRATES We need to ask why acetone changes its state so drastically on becoming enclath- rated.We stress that anchored guests are quite common and these include basic aprotic molecules such as amine~.~~ In our view the major reason for this change lies in the fact that in order that all the water molecules be tetrahedrally hydrogen- bonded in the clathrate cages the excess of (LP)freegroups must react with the (OH) groups bound to acetone Me,CO(HO) + 2(LP)fr,e +-Me2C0 + 2(0 * * HO) (2) where (0 * * HO) represents cage hydrogen bonds. Reaction (2) is favoured for acetone because it forms weak hydrogen bonds but evidently such a process is dis- favoured for most amines which form much stronger bonds. We conclude that acetone forms two weak hydrogen bonds in liquid water because of the availability of (OH)freegroups which are willing to form hydrogen bonds whenever possible.These are not available in normal clathrate cages where each water is tetrahedrally co- ordinated to four neighbours. INFRARED LINESHAPES The change in lineshape is in accord with these conclusions. The C=O stretch band in fluid aqueous solutions is Gaussian (fig. 2). This means that the width stems primarily from a range of subtly differing structures having long short and bent hydrogen bonds just as occurs for water-water interactions. In marked contrast the band for the clathrate is Lorentzian. This means that the width is kinetically controlled indicating considerable rotational freedom within these cage^.^^'^^ CAGE MODEL FOR ACETONE SOLUTIONS THE CASE FOR LIQUID-PHASE CAGES Our results show that if cages are important for liquid-phase systems they do not resemble those in the solid state and hence theories based on liquid-phase cages should not be justified on the basis that acetone forms a clathrate h~drate.~~~~~ What is important is that full allowance must be made for the fact that acetone forms two hydrogen bonds.In our view this reaction is likely to dominate most of the properties of aqueous acetone rather than cage formation. In the following we briefly consider alternative explanations for some observations that have been explained in terms of a tendency to form liquid-phase cages. EVIDENCE FROM RELAXATION STUDIES Both n.m.r.34*36 and dielectric 37 relaxation studies show that the correlation time (7,) for water molecules lengthens as basic aprotic cosolvents are added.This is interpreted in terms of increased water-water interactions and hence of cage formation. We wonder if the postulate of this structure-forming effect in the liquid phase is really necessary. We tentatively suggest that it is possible to understand the decrease in rotational freedom for water molecules in terms of eqn (2). We postulate that a major contribu- tion to rotational motion in normal water comes from the (OH)free water molecules. Their transformation into water bonded to acetone will inhibit such rotation. We stress that loss of (0H)free is balanced by a gain in (LP)f,,e units and hence we need to postulate that their rotational behaviour is less "free " than that of (OH)freeunits.This is reasonable since both protons remain hydrogen-bonded to bulk water in (LP)rreeunits. EVIDENCE FROM SPECTROSCOPIC STUDIES It has been suggested that the slight downfield shift 38 or plateau 25 that is initially observed in the water proton resonance as acetone is added is indicative of structure formation involving cages. However we have pointed out that this behaviour is a necessary consequence of the scavenging of groups.39 These groups must have their OH resonance close to that for monomeric water and hence they will con- tribute a considerable upfield shift to the time-averaged resonance. Their loss gives a downfield component that is almost balanced by other upfield shifts as we have dem~nstrated.~~ Similarly the loss of the band in the overtone infrared spectrum when basic aprotic solvents are added arises directly from the scavenging of groups by the base.It cannot be taken as proof of the formation of extra water structure. Again the ultraviolet shifts in the n+n* band for acetone in the water-rich region 40 can be fully explained in terms of the change from di-hydrogen-bonded carbonyl to mono-hydrogen-bonded and non-hydrogen-bonded units and in no sense requires the formation of cages. ADDITION OF ALCOHOLS Here we refer to some studies of the effect of added t-butyl alcohol and of methanol on the acetone clathrate. This was undertaken in the hope that with Me,COH unusual effects due to cage-cage affects might be detected. In fact nothing unusual was noticed.Methanol lowered the temperature at which the clathrates formed in the warming cycle [see experimental method (b)],whereas t-butyl alcohol raised it. M. C. R. SYMONS AND G. R. EATON 39 This presumably simply reflects the gain or loss of fluidity at low temperatures. The apparent decomposition temperature range was lowered for both systems and we were unable to form the clathrate at mole fractions (of alcohol) >0.016. BONDING IN T-BUTYL ALCOHOL CLATHRATES Finally we should mention a very interesting study of t-butyl alcohol clathrates by Gorbunov et aL41which in some ways resembles our own work. An infrared study in the fundamental 0-H stretch region for the mixed alcohol-H,S clathrate showed a narrow component in the 3608 cm-l region well removed from the absorption for the bound water molecules assigned to (OH)freeoscillators.It was reasonably concluded that the Me,COH molecules are not bonded (anchored) to the cage. This is not a necessary requirement since ROH molecules by forming two hydrogen bonds to water can knit into the cage structure with minimum disturbance. Indeed it is probable that this is the way ethanol molecules are en~lathrated.~~ On the other hand the structurally similar molecule Me,CNH, is freely enclathrated with no guest- host hydrogen These results which seem convincing suggest that Me,COH molecules like acetone undergo a dramatic loss of solvation on enclathration. We stress that in this case there is as discussed above extensive evidence for incipient cage formation in the liquid phase.M. C. R. Symons and M. J. Blandamer in Hydrogen-bonded Solvent Systems ed. A. K. Coving- ton and P. Jones (Taylor & Francis London 1968) p. 211. J. Davies S. Ormondroyd and M. C. R. Symons Trans. Faraday SOC. 1972,68 686. D. Jones and M. C. R. Symons Trans. Faraday SOC. 1971,67,961. Y. Y.Lim E. A. Smith and M. C. R. Symons J. Chem. SOC. Faraday Trans. 1,1976,72 2876. P. T. Beurskens and G. A. Jeffrey J. Chem. Phys. 1964,40,2800. M. C. R. Symons Nature 1972,239,257; Philos. Trans. R. SOC. London Ser. B 1975,272 13. ’T. R. Griffiths and M. C. R. Symons Mol. Phys. 1960,3 174. M. C. R. Symons and G. R. Eaton Chem. Phys. Lett. 1981,83,292. M. C. R. Symons G. R. Eaton T. A. Shippey and J. M. Harvey Chem. Phys. Lett.1980,69,344. lo H. G. Hertz and W. Spalthoff 2.Electrochem. Ber. Bunsenges. Phys. Chem. 1959 63 1096. E. Wicke Angew. Chem. Int. Ed. Engl. 1966 5 106. l2 R. N. Butler and M. C. R. Symons Trans. Faraday SOC. 1969 65 2559. l3 M. C. R. Symons Acc. Chem. Res. 1981 14 179. l4 J. D. Worley and I. M. Klotz J. Chem. Phys. 1966 45 2868. W. A. P. Luck and W. Ditter 2.Naturforsch. Teil B 1969 24 482. l6 0. D. Bonner and C. F. Jumper Infrared Phys. 1973 13 233. l7 P. R. Philip and C. Jolicoeur J. Phys. Chem. 1973,77 3071. S. E. Jackson and M. C. R. Symons Chem. Phys. Lett. 1976,37 551. l9 G. K. Fraenkel J. Phys. Chem. 1967 71 139. ’O T. A. Claxton W. M. Fox and M. C. R. Symons Trans. Faraday SOC. 1967 63 2570. C. J. W. Gutch W. A. Waters and M. C. R. Symons J.Chem. SOC. B 1970 1261. 22 M. C. R. Symons S. P. Maj D. E. Pratt and L. Portwood J. Chem. SOC. Perkin Trans. 2 1982 191. 23 R. G. Anderson and M. C. R. Symons Trans. Faraday SOC. 1969,65 2550. 24 B. Kingston and M. C. R. Symons J. Chem. SOC. Faraday Trans. I 1973,69 978. 25 W. Y. Wen and H. G. Hertz J. Solution Chem. 1972 1 17. 26 A. K. Covington and K. E. Newman in Modern Aspects of Electrochemistry ed. J. O’M. Bockris and B. E. Conway (Plenum Press New York 1973) no. 12 p. 41. ”J. M. Harvey S. E. Jackson and M. C. R. Symons Chem. Phys. Lett. 1977,47? 440. ” S. E. Jackson E. A. Smith and M. C. R. Symons Faraday Discuss. Chem. SOC.,1977 64 173. 29 C. Jolicoeur and H. L. Friedman Ber. Bunsenges. Phys. Chem. 1971,76,248; J. Solution Chem. 1974,3 15.30 A. S. Quist and H. S. Frank J. Phys. Chem. 1961 65 560 31 B. Morris and D. W. Davidson Can.J. Chem. 1971,49 1243; G. J. Wilson and D. W. David- son Can. J. Chem. 1963 41 264. CAGE MODEL FOR ACETONE SOLUTIONS 32 D. W. Davidson in Water-A Comprehensive Treatise ed. F. Franks (Plenum Press New York 1973) vol. 2 p. 218. 33 M. C. R. Symons and G. R. Eaton J. Chem. SOC.,Faraday Trans. I 1982 78 3033. 34 E. V. Goldammer and H. G. Hertz J. Phys. Chem. 1970 74 3734. 35 M. J. Blandamer in Water-A Comprehensive Treatise ed. F. Franks (Plenum Press New York,1973) vol. 2 p. 488. 36 M. D. Zeidler in Wafer-A Comprehensive Treatise ed. F. Franks (Plenum Press New York 1973) vol. 2 p. 540. 37 J. B. Hasted in Water-A Comprehensive Treatise ed.F. Franks (Plenum Press New York 1973) vol. 2 p. 441. 38 D. N. Glew D. H. Mak and N. S. Rath J. Chem. SOC.,Chern. Commun. 1968 264. 39 M. C. R. Symons J. M. Harvey and S. E. Jackson J. Chem. SOC., Faraday Trans. I 1980,76 256. 40 M. F. Fox J. Chem. SOC., Faraday Trans. I 1972 68 1294. 41 B. Z. Gorbunov and Yu. I. Naberukhin Chem. Phys. Lett. 1973 19 2. 42 A. D. Potts and D. W. Davidson J. Phys. Chem. 1965 69 996. 43 R. K. McMullan G. A. Jeffrey and T. H. Jordan J. Chem. Phys. 1967,47 1229.

ISSN:0301-5696    

DOI:10.1039/FS9821700031

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC., 1982 17 41-53 Isothermal Transport Properties in Solutions of Symmetrical Tetra-alkyl ammonium Bromides BY LAWRENCE A. WOOLF Diffusion Research Unit School of Physical Sciences The Australian National University Canberra ACT 2600 Australia AND HERMANN WEINGARTNER Institut fur Physikalische Chemie und Elektrochemie der Universitat Karlsruhe Kaiserstr. 12 D 7500 Karlsruhe West Germany Received 1st September 1982 Density molar conductance transference number tracer diffusion (cation anion water) and mutual diffusion data are reported for aqueous solutions of the following symmetrical tetra-alkyl- ammonium halides at 25 "C up to 2 mol kg-' concentration Me4NBr Et4NBr n-Pr4NBr and n- Bu4NBr. These data were used to obtain information on the interactions between pairs of species c a w (cation anion water) using Hertz's velocity-correlation formalism.Within this formalism the transport coefficients are related to time integrals over velocity cross-correlation functions between the various interaction partners. Results obtained for solutions of tetra-alkylammonium salts differ characteristically from those reported for alkali halides and reflect structural features such as hydro- phobic hydration cation-cation and anion-anion interactions. As pointed out by Frank and Evans,' aqueous solutions of symmetrical tetra- alkylammonium halides show a remarkably high viscosity attributed to particular water structures around the hydrocarbon groups of the cations. Clearly the problem of the cationic hydration in such systems is closely related to the respective problem of the hydration of neutral molecules containing non-polar groups.2 Thus solutions of symmetrical tetra-alkylammonium salts have been used as model systems for studies on hydrophobic interaction^.^ Extensive studies made on the various proper- ties of these systems have included thermodynamic spectroscopic and relaxation techniques3 It is however curious that apart from viscosity data almost no data on transport properties exist.The only systematic investigations have been conducted by Kay and Evan~,~.~ who made extensive studies on the conductance of quaternary ammonium salts in the dilute concentration range. Above concentrations of ca. 0.01 mol kg-l only a little work on transport properties has been reported including data on self-diffusion of cations and water,6 electrical conductance ' and mutual diffusion.8 The present study provides systematic data on isothermal transport properties in aqueous solutions of tetramethyl- tetraethyl- tetrapropyl- and tetrabutyl-ammonium bromides (Me,NBr Et4NBr n-Pr4NBr and n-Bu,NBr) at 25 "C over the concentration range 0.05-2 mol kg-'.The data include self-diffusion coefficients of water cations and anions mutual diffusion coefficients electrical conductance and transference numbers. As a by-product we have obtained some density data. As suggested by various author^,^-^' a comprehensive set of experimental trans- TRANSPORT PROPERTIES port data can be used to obtain information on the interaction between pairs of species in concentrated electrolyte solutions.We will use a formalism first applied by Hertz," who has given a representation of transport coefficients in terms of velocity cross- correlation coefficients (v.c.c.) derived from linear-response theory. As pointed out by Friedman and Mills l2 and other^,'^-'^ these coefficients are more related to the basic physical processes than are other sets of cofficients (e.g. Miller's generalized Zij coefficients which have their basis in the thermodynamics of irreversible processes 9*16). If present specific interactions such as hydrophobic hydration or ion association should reveal themselves in the respective V.C.C. EXPERIMENTAL MATERIALS The tetra-alkylammonium salts were obtained from Eastman Organic Chemicals.They were purified as described below. Me4NBr was repeatedly recrystallized from a 1 1 methanol water mixture centrifugally drained and dried at 80 "C under vacuum for 3 days. A similar procedure was used for the purification of Pr4NBr and Bu4NBr in the latter case however with the use of an acetone diethylether solvent system. Et4NBr was precipitated from chloroform by adding diethylether ; the crystals were centrifugally drained and dried under vacuum for 5 days. Water used in the experiments was distilled and passed through an ion-exchange column. It was then heated and partially degassed at a water pump. The specific conductance was 1 x R-l cm-'. Stock solutions and all dilutions were prepared by weight.Some suitably diluted portions were analysed by potentiometric titration against AgN03 solutions. The uncertainty in composition is estimated as 0.1%. 14C-labelled materials were obtained from the Nuclear England Corporation. The tracers were mixed with purified inactive materials and dried after recrystallization as above. 82Br-labelled bromide was obtained as a 0.1mol dm-3 aqueous solution of HBr from the Australian Atomic Energy Commission Lucas Heights Australia. It was diluted in normal water and used without further purification. HTO was obtained from the Radiochemical Centre Amersham. DENSITY MEASUREMENTS Highly accurate density data were required as we intended to analyse concentrations in mutual diffusion experiments via density measurements.Therefore we performed some density measurements with an Anton Paar type densimeter. In general we found close agreement with results reported by Wen and Saito." Therefore values obtained by these authors and our values have been fitted by least squares to the equation n d = do +.Z Alrn'l2. 1=2 The resulting coefficients At and the standard deviations are shown in table 1. do = 0.997 05 g cm+. CONDUCTANCE MEASUREMENTS Conductance measurements were performed with a Leeds and Northrup Jones bridge in conjunction with an oil thermostat. The temperature measured with a certified platinum resistance thermometer was 25 rrt 0.003 "C. Jones and Bradshaw demal KCI standards were used for calibration.'8 16 data points between 0.05 and 2.25 mol kg-' for each system were fitted to polynomials in order to obtain values at rounded concentrations shown later in table 3.L. A. WOOLF AND H. WEINGARTNER TABLE 1.xOEFFICIENTS OF EQN (1) maximum system A2 A3 A4 A5 deviation standard /mol kg-' concentration Me4NBr 0.036710 0.006384 -0.010504 0.002515 1.3 X 3 Et,NBr 0.036 632 2.266 8 -0.004 883 9.789 1 1.3 x 10-4 3 x 10-4 x 10-4 Pr4NBr 0.026043 0.004 022 -0.003 912 1.1 x 10-4 3 Bu4NBr 0.023 673 0.001 008 -0.003 81 1 0.9 x 10-4 3.3 TRANSFERENCE NUMBERS E.m.f. measurements of concentration cells with transference were performed to obtain transference numbers. Standard techniques as described by Spiro l9 for example were applied. Only some additional remarks are given here.The cell was similar to that described by Stokes and Levien." Pairs of Ag AgBr electrodes were prepared by the thermal- electrolytic method.21 Bias potentials were <0.01 mV. The e.m.f. was determined by a certified digital voltmeter with a resolution of 0.001 mV. E.m.f. values of the corresponding cells without transference were calculated from the activity data reported by Lindenbaum and Boyd,22 but for Me4NBr Levien's data were preferred because Wen's comment indicated that the latter data may be more accurate than those of Lindenbaum and Boyd. MUTUAL DIFFUSION COEFFICIENTS Mutual diffusion coefficients were determined by the diaphragm technique as described by Stokes l8 and modified by Mills and W00lf.*~ After some trials we used potentiornetric titration of Br- against AgN03 to analyse the solutions in the top and bottom compartments of the cell; additional density measurements have been performed in some cases.We noted slight differences (1% at maximum) between values obtained from experiments with a glass sinter and those obtained with a platinum sinter. Comparison with the optical data of Pepela et aL8 for solutions of Pr4NBr indicated that use of the platinum sinter gave the correct values.* Differential diffusion coefficients were calculated from the measured integral ones by the procedure outlined in ref. (23). The final data are listed later in table 3. The estimated accuracies of the values at rounded concentrations are as follows better than 1 % (Pr4NBr and Bu4NBr) 1yo(Et4NBr) 2% (Me4NBr).TRACER DIFFUSION COEFFICIENTS Tracer diffusion coefficients of cations anions and HTO were measured with diaphragm cells by standard techniques outlined in ref. (23). The radiotracers were 14C,82Br and HTO. Measured values of DHTo by a correction factor 1.03,result-were converted to those for DH~O ing from the mass extrapolation procedure applied by Mills.24 Original data for the tracer diffusion measurements are given in table 2 and values at rounded concentrations are sum- marized in table 3. Limiting ionic diffusion coefficients were calculated from the following limiting ionic conductance^:^ 44.42 (Me4N+),32.22 (Et4N+),23.22 (Pr4N+),19.31 (Bu4N+) 78.22 (Br-) where all values are given in cm2 R-' equiv.-'. * This result is supported by some further diffusion measurements using the Schlieren method.These measurements and some additional conductance measurements have been performed at the Institut fur Physikalische Chemie der Universitat Aachen. Prof. H. Schonert is thanked for his hospitality. TRANSPORT PROPERTIES TABLE 2.-sELF-DIFFUSION COEFFICIENTS OF CATIONS ANIONS AND WATER IN SOLUTIONS OF TETRA-ALKYLAMMONIUM BROMIDES AT 25 "C Me4NBr 0 1.182 0 2.081 0 2.30 0.049 98 1.154 0.045 12 1.9664 0.1012 2.370 0.1Ooo 1.136 0.1110 1.903 0.2722 2.421 0.4998 1.072 0.2516 1.841 0.5532 2.334 0.9997 1.01 3 0.5056 1.759 1.0103 2.231 1.9989 0.885 1.015 1.607 1.9978 1.872 1.997 1.402 2.005 1.875 2.917 1.218 3.882 1.017 Et,NBr 0 0.858 0 2.08 1 0 2.30 0.049 95 0.823 0.099 91 1.867 0.1003 2.289 0.099 97 0.816 0.2453 1.699 0.2641 2.154 0.4998 0.734 0.5192 1.516 0.5003 1.985 0.5193 0.728 1.065 1.219 1.023 1.693 1.003 0.622 2.01 6 0.865 1.992 1.260 1.033 0.618 3.032 0.653 2.989 0.993 1.999 0.512 3.821 0.51 1 Pr4NBr 0 0.61 8 0 2.08 1 0 2.30 0.062 75 0.099 97 (0.620) 0.608 0.1006 0.2523 1.700 1.468 0.450 0.868 1.57 1.12 0.1532 0.582 0.6653 1.111 1.088 0.912 0.251 3 0.551 1.043 0.878 1.289 0.765 0.4996 0.466 1.376 0.788 0.9999 0.354 1.998 0.548 1.999 0.21 8 3.756 0.282 Bu4NBr 0 0.514 0 2.08 1 0 2.30 0.065 16 0.499 0.1003 1.652 0.480 1.57 0.098 02 0,486 0.2293 1.415 1.002 1.16 0.2501 0.419 0.5101 1.121 1.362 0.934 0.9991 0.232 1.1032 0.768 2.553 0.790 1.9955 0.137 2.045 0.4020 3.775 0.1892 "min mol kg-' all diffusion coefficients in m2s-'; value not used for further evaluation of data; measured by 'H n.m.r.in Pr4NBr + D20 and corrected by the ratio of self-diffusion co-efficients of the pure liquids (1.26). L. A. WOOLF AND H. WEINGARTNER TABLE3.-ISOTHERMAL TRANSPORT COEFFICIENTS IN AQUEOUS SOLUTIONS OF TETRA-ALKYL-AMMONIUM BROMIDES AT 25 "CAT ROUNDED CONCENTRATIONS m c d D,w D a Dw A 1 1 + m(dlny/drn) Me4NBr 0 0 0.997 05 1SO8 1.182 2.081 2.30 122.67 0.362 1 0.05 0.04956 0.998 93 1.350 1.153 1.964 2.34 104.14 0.349 0.8945 0.1 0.098 56 1.0o0 82 1.308 1.137 1.921 2.37 98.200 0.346 0.863 1 0.25 0.2422 1.006 43 1.266 1.109 1.838 2.42 88.547 0.344 0.8168 0.5 0.4714 1.015 50 1.230 1.072 1.748 2.35 80.096 0.342 0.7861 0.75 0.6884 1.024 01 1.229 1.042 1.677 2.31 74.717 0.341 0.7761 1 0.8943 1.032 11 1.250 1.01 1 1.61 6 2.22 70.634 0.341 0.7766 1.5 1.2758 1.047 08 1.258 0.949 1 SO6 2.05 64.209 0.34 1 0.7980 2 1.6216 1.060 66 1.269 0.885 1.403 1.88 59.165 0.341 0.8397 Et4NBr 0 0 0.997 05 1.214 0.858 2.081 2.30 110.44 0.291 1 0.05 0.049 42 0.998 88 1.070 0.824 1.930 2.29 91.019 0.271 0.8796 0.1 0.098 01 1.O00 67 1.024 0.814 1.858 2.28 84.179 0.269 0,8392 0.25 0.2389 1.005 96 0.949 0.786 1.706 2.15 72.587 0.266 0.7790 0.5 0.4589 1.014 39 0.855 0.733 1.520 1.98 61.968 0.264 0.7417 0.75 0.6624 0.022 44 0.805 0.680 1.373 1.84 55.089 0.262 0.7335 1 0.851 1 1.030 00 0.772 0.633 1.248 1.68 49.908 0.260 0.7397 1.5 1.1907 1.044 12 0.738 0.575 1.043 1.44 42.107 0.260 0.7794 2 1.4883 1.056 96 0.725 0.512 0.881 1.23 36.145 0.260 0.8462 Pr4NBr 0 0 0.997 05 0.953 0.618 2.081 2.30 101.40 0.229 1 0.05 0.049 26 0.998 38 0.816 0.620 1.805 2.19 81.623 0.210 0.8637 0.1 0.097 38 0.999 74 0.764 0.608 1.700 2.12 74.132 0.205 0.8166 0.25 0.2353 1.003 81 0.668 0.551 1.471 1.86 60.719 0.201 0.7540 0.5 0.4459 1.010 51 0.580 0.475 1.224 1.51 47.870 0.195 0.7345 0.75 0.6358 1.01699 0.531 0.415 1.052 1.23 39.576 0.193 0.7517 1 0.8080 1.023 20 0.502 0.354 0.907 0.98 33.586 0.192 0.7845 1.5 1.1091 1.034 70 0.469 0.277 0.702 0.66 25.437 0.191 0.8725 2 0.3636 1.044 86 0.444 0.210 0.550 0.51" 20.218 0.190 0.9776 Bu4NBr 0 0 0.997 05 0.821 0.514 2.081 2.30 97.50 0.198 1 0.05 0.049 12 0.998 23 0.687 0.500 1.785 2.21 75.984 0.177 0.8711 0.1 0.096 81 0.999 41 0.633 0.480 1.658 2.15 66.725 0.172 0.8251 0.25 0.2320 1.002 85 0.534 0.422 1.403 1.86 50.350 0.166 0.7532 0.5 0.4341 1.008 29 0.441 0.340 1.139 1.65 36.824 0.162 0.7002 0.75 0.61 19 1.013 31 0.381 0.275 0.958 1.43 30.1 18 0.159 0.6741 1 0.7697 1.017 92 0.336 0.225 0.808 1.17 26.422 0.158 0.6583 1.5 2 1.0371 1.2548 1.025 83 1.03200 0,258 0.197 0.159 0.138 0.593 0.430 0.82 0.65' 22.415 19.209 0.157 0.156 0.6381 0.6227 rn in mol kg-' C in mol dm-3 d in g ~m-~ A in cm' f2-l equiv.-' all diffusion coefficients in m2 s-l; * maximum of D,within range of experimental error; extrapolated.TRANSPORT PROPERTIES ACTIVITY COEFFICIENTS Accurate data on activity coefficients are needed since a term of the form dlny l+m-=v,+m* dm dm (where y is the activity coefficient and v is the osmotic coefficient) occurs in the evaluation of the mutual diffusion data.After some trials we have used Pitzer's equations 25 and the respective parameters obtained by Pitzer and Mayorga 26 as a representation of the activity data of Lindenbaum and Boyd 22 and Levien.' The expression for the derivative 1 + m(dlny/drn) in terms of Pitzer's equations is discussed in the Appendix; values at rounded concentrations are given in table 3. DISCUSSION GENERAL The question of how to approach a description of transport properties in con- centrated electrolyte solutions has been discussed from different points of vie~.~-I~ We will use Hertz's velocity correlation formalism which utilizes the fact that the phenomenological coefficients of irreversible thermodynamics can be related to time integrals over velocity cross-correlation functions." Velocity cross-correlation coefficientshj for pairs of constituents i,j = c;a,w are defined by fa = La = fcc = fww = where Va* denotes the velocity of particle a of constituent i with respect to the local centre of mass vector notation is not used in eqn (2)-(5).The pointed brackets denote the ensemble average. c is the salt concentration cw the water concentration both expressed in mol ~m-~; N is Avogadro's number and Y is the volume of the so-lution. Eqn (2)-(5) correspond to the definition of the self-diffusion coefficient D in terms of the velocity autocorrelation function of constituent i v,'(t)) dt.(6) Di= T/$v2(0) 1" As shown by Hertz," V.C.C. can be calculated from diffusion conductance and trans- ference data. We have calculated V.C.C. following Hertz's method from the data summarized in table 3 where the following denotations have been used Dt (self-diffusion coefficients) D, (mutual diffusion coefficient) A (equivalent conductance) and t (transference number of the cation). Concentrations are given in molality units rnlmol kg-I and molarity units C/mol dm-3. For explicit expressions the reader is referred to ref. (1 1). Note that the relation of transport coefficients to V.C.C. has been discussed from a different point of view by one of the present authors and by Miller. l6 L. A. WOOLF AND H. WEINGARTNER CATION-ANION CORRELATIONS The concentration dependence offac for the four tetra-alkylammonium bromides is shown in fig.1. It is not the purpose of this paper to give a quantitative account of the shape of these curves. Rather we will compare them with results recently ob- tained for alkali halides 15*27 and with predictions for V.C.C. in certain reference 0. 0 1 2 rnlmol kg-l FIG. 1.-Velocity correlation coefficients f, for aqueous solutions of Me4NBr (l) Et4NBr (2) Pr4NBr (3) and Bu4NBr (4). For details see text. systems 11v2* in a more qualitative way. Nevertheless such a procedure should be adequate to detect specific effects if present. In the dilute concentration range fac increases rapidly with increasing salt con- centration. At very high dilution fa must go to zero as a consequence of the in- dependent motions of cations and anions.Miller l6 has considered limiting-law expressions according to which.fac should obey a m1/2 law with a positive initial slope. Roughly an m1/2 dependence seems to be fulfilled but because of the accumulation of errors in the calculation of small V.C.C. this can only be a tentative conclusion. In particular the magnitude of the initial slopes cannot be determined with the desired accuracy. In any case the observed positive correlations are a consequence of the coupling of cations and anions through an electrostatic potential. Effects in more concentrated solutions (>0.25 mol kg-l) indicate a decrease of correlations asfa goes through a maximum. In going from Me,NBr to Bu,NBr this maximum is systematically shifted to lower concentrations and its height decreases.Similar maxima were observed in solutions of alkali halides and the shape of the curves is roughly the same. Therefore we may conclude that there are no dramatic differences between solutions of alkali halides and tetra-alkylammonium bromides with respect to cation-anion correlations. The observed decrease offac with increasing salt concentration is attributed to the mutual averaging of the electrostatic forces by the presence of other ions close to a -0.11 TRANSPORT PROPERTIES given pair. This averaging diminishes the tendency to form ion pairs with a suficiently deep potential well to cause extended correlated motions between cations and anions.Obviously such effects set in at concentrations ca. 0.25 mol kg-'. Intuitively one may relate positive values offac (or more precisely values which are more positive than values predicted for ideal reference systems) to the degree of ion association. It is therefore important to emphasize the differences between the conventional concepts of ion association and our findings. Usually interaction partners are classified as being associated if their distance is smaller than a given reference distance and the degree of ion association is not expected to decrease with increasing concentration.18 At very high concentrations structural definitions may require the presence of solvent- separated or even contact ion pairs due to simple stoichiometric arguments.On the other hand cation-anion encounters are not necessarily related to the presence of correlated motions. In contrast the experimental findings show that extended correlations are absent. Indeed at high concentrations even negative values have been observed for solutions of the structure-breaking salts KI,27CSCI,~~ NaI l5 and RbC1.lS In contrast for LiCl and NaClf, remains positive over the whole con- centration range.27 From a rough extrapolation to high concentrations we conclude that for Me,NBr Et,NBr and Pr,NBr fac may remain positive whereas for Bu,NBr faC becomes negative at ca. 2 mol kg-'. The origin of negative V.C.C. and in particular off, at high concentrations has been discussed by Hertz:27 If specific interactions are averaged out momentum conservation will become the dominating effect which determines the magnitude offac.Indeed relations based on momentum conservation predict negative values off,,.11*28 0 1 2 rnlmol kg-FIG.2.-Velocity correlation coefficients f,= for aqueous solutions of Me4NBr (l) Et4NBr (2), Pr4NBr(3) and Bu4NBr (4). For details see text. L. A. WOOLF AND H. WEINGARTNER 0 i -0. -0. 0 1 2 rnlmol kg-’ FIG.3.-Velocity correlation coefficients ,La for aqueous solutions of Me4NBr (l) Et4NBr (2) Pr4NBr (3) and Bu4NBr (4). For details see text. In summary we believe that there are no indications for specific interactions between cations and anions which exceed those present in solutions of alkali halides. CATION-CATION AND ANION-ANION CORRELATIONS As noted by Hertz It and Friedman and Mills,” time integrals over velocity cross- correlation functions may take negative values in contrast to the behaviour of the respective time integrals over velocity autocorrelation functions (i.e.the self-diffusion coefficients) which must be positive. Hertz has attributed negative values off, to the requirement of momentum conservation. On this basis Hertz has derived ex- pressions for so-called “ standard coefficients” 28 fi, which predict the actual values f;:,in near-ideal systems rather well.27929 Applying Hertz’s approach we expect negative values offaa andf,. over the whole concentration range. Indeed for solutions of alkali halides these coefficients are negative and decrease monotonically with in- creasing salt concentration their values being more negative for structure-forming than for structure-breaking The curves shown in fig.2 and 3 for Me,NBr show the expected behaviour ofha andf, in solutions of KBr as interpolated from the experimental results for KC1 and KI.27 Turning to results for fc in solutions of tetra-alkylammonium bromides rather specific features are observed (see fig. 2). The limiting slopes are negative as expected. However at higher concentrations onlyf, values for Me,NBr and Et,NBr are in accordance with the expected monotonic decrease with increasing salt concentration. Curves obtained for Pr4NBr and Bu,NBr show minima at ca. 0.7 and 0.3 mol kg-’ respectively. For Et4NBr a minimum slightly above 2 mol kg-I is indicated from its TRANSPORT PROPERTIES behaviour at ca.2 mol kg-l. Obviously the initial decrease offcc is superimposed by a positive contribution the inset of which is shifted to lower concentrations in going from Et,NBr to Bu,NBr. One scarcely needs detailed consideration of the behaviour of reference systems to recognize the peculiarities occurring in these systems. Even if fCc values do not become positive on an absolute scale the marked positive deviations from the behaviour of alkali halides are obvious. We believe that these findings are a definite manifestation of particular cation-cation (hydrophobic) interactions in such systems which have often been proposed in the literat~re.~ Whereas cation-cation interactions have been considered to the authors' know- ledge no similar statements on the existence of anion-anion interactions are reported.It is therefore interesting to note that our method yields correlated motions between anions of a similar magnitude as observed for cations. The predicted negative slope and the montonic decrease ofha is only observed for Me,NBr. For the other tetra- alkylammonium bromides a minimum ofha is observed which is located at ca. 0.5 mol kg-'. The anomalous behaviour offaa seems to be more pronounced than the corresponding behaviour offcc for BudNBrf, extrapolates to positive values at con- centrations slightly above 2 mol kg-'. Thus we arrive at the conclusion that apart from cation-cation interactions discussed in the literature there is strong evidence for specific anion-anion interactions.It is however not easy to see whether the occur- rence of correlated motions between anions is a necessary consequence of the presence of correlated motions between cations and vice versa in particular since fa does not show anomalous behaviour. WATER-WATER CORRELATIONS fww coefficients in solutions of tetra-alkylammonium bromides are shown in fig. 4. In pure water i.e. at m = 0 according to the definitionf, is equal to the self-diffusion coefficient of water multiplied by -1. For m # 0 fww increases with increasing salt concentration. Hertz has noted 27 that the behaviour of the structure-forming alkali halides is very close to the predicted standard values f;,. The respective experimental curves and the standard values are of the same order of magnitude as observed for Me,NBr in the present work.Thus the curve for Me,NBr in fig. 4 at the same time gives a qualitative representation of the behaviour of the structure- forming alkali halides. In contrast for the structure-breaking CsCl positive values of fww have been observed at high concentrations. Positive values off, have therefore been interpreted as an important feature of the structure-breaking effect.27 Turning to tetra-alkylammonium bromidesf, is also less negative than predicted with the exception of Me,NBr. Moreover the curves seem to extrapolate to positive values at high concentrations. In fact in aqueous solutions of non-electrolytes such as acetone and simple alcohols 29~30 positive water-water correlations have been observed also.Thus we conclude that positive water-water correlations are an obvious manifestation of the hydrophobic hydration effect. INTERIONIC CONTRIBUTIONS TO THE EQUIVALENT CONDUCTANCE The anomalous behaviour of the like-ion coefficientsf, andf, has some important consequences for the concentration dependence of the equivalent conductance as will be shown for Bu,NBr. In terms of V.C.C.the equivalent conductance is given by L. A. WOOLF AND H. WEINGARTNER If all cross-terms were to vanish A would be given by the dashed line in fig. 5 which represents the sum of the contributions of the cationic and anionic self-diffusion co- efficientsA’. At concentrations below 0.3 mol kg-l,faa,fcc andf, are still sniall com- pared with the self-diffusion coefficients but they act in the same direction which results in a marked overall reduction of the conductance.Its magnitude can be seen by comparing the experimental curve with the dashed line. At concentrations above 1-4 Iv) “E h -31 0 1 2 rnlmol kg-I FIG.4.-Velocity correlation coefficients fww for aqueous solutions of Me4NBr (l) Et4NBr (2) Pr4NBr(3) and Bu4NBr (4). For details see text. 0.3 mol kg-l the anomalous behaviour offa andf,. sets in. The related effects on the conductance may be estimated as follows the initial curves fork andf, are extra- polated to higher concentrations assuming a similar shape of these curves as ob-served for Me,NBr. The result for the equivalent conductance in a hypothetical solution where specific cation-cation and anion-anion interactions are absent is also shown in fig.5. It is immediately obvious that the observed correlated motions between like ions shift the equivalent conductance to higher values. At 2 mol kg-l ca. 15% of its actual value may be attributed to these effects. At ca. 2 mol kg-l the contributions due tofaoh andf, nearly cancel one another and A is close to the value given by the sum of the ionic self-diffusion coefficients A’. More important at high concentrations A may even become larger than A’ behaviour which has not yet been observed in other systems. TRANSPORT PROPERTIES 100 75 50 25 I 1 1 0.5 1 1.5 (rnlmol kg -I)* FIG.5.-Equivalent conductance in solutions of BuoNBr as a function of the square root of molality.The solid line represents the experimental curve. The dashed line gives the contributions from the ionic self-diffusion coefficients neglecting the cross-terms. The dotted line gives the equivalent conductance estimated for a hypothetical solution where hydrophobic cation-cation association and anion-anion is absent. For details see text. CONCLUSIONS We have characterized the dynamical properties of aqueous solutions of tetra- alkylammonium ions via the correlated translational motions between the various interaction partners. Intuitively one is attempting to relate these effects to the intermolecular structures present in these solutions. However we believe that the translation of this dynamical information into a structural picture will be a difficult task.There is evidence that the spatial extension of the observed correlations is rather long range.27 Thus the detection of" association " by the aid of the present method is certainly different from the usual concept of association as for example is inherent in Bjerrum's treatment of ion association.18 Nevertheless the main results of the present study are supported by similar conclusions derived from various other experimental results and any structural models should account for the facts presented here. H. W. thanks the authorities of the Australian National University for a visiting fellowship during the tenure of which most of the experimental results have been obtained. Wealso thank the Deutsche Forschungsgemeinschaft for financial support.Helpful discussions with Dr R. Mills and Prof. W-Y. Wen and H. G. Hertz are grate- fully acknowledged. L. A. WOOLF AND H. WEINGARTNER APPENDIX Equations derived by Pitzer 25 have been found to yield accurate representations of experimental data on activity and osmotic coefficients in electrolyte solutions.26 For 1-1 electrolytes at 25 “C these equations are given in ref. (25) and (26). The expression for the derivative (1 + rn dlnyldrn) in terms of Pitzer’s parameters is do = 1 + m-dlny = 1 -0.3920rn3 (1.5 + bm+) p + dm dm (1 + brn+)2 where a = 2.0 and b = 1.2 are characteristic for the class of electrolytes and B(O) 8“)and C9 are characteristic for each electrolyte. Pitzer and Mayorga 26 have fitted their equations to the activity data of Lindenbaum and Boyd,” but the set of parameters for tetra-alkylammon- ium bromides has been corrected in an additional note.26 H.S. Frank and M. W. Evans J. Chem. Phys. 1945,13 507. F. Franks in Water-A Comprehensiue Treatise ed. F. Franks (Plenum Press New York 1975) vol. 4 chap. 1 pp. 1-94. ’W-Y. Wen in Water and Aqueous Solutions ed. R.A. Horne (Wiley New York 1972) chap. 15 pp. 613-661. R. L. Kay and D. F. Evans 1.Phys. Chem. 1965,69,4216. ’R. L. Kay and D. F. Evans J. Phys. Chem. 1966,70,366. H. G. Hertz B. Lindman and V. Siepe Ber. Bunsenges. Phys. Chem. 1969 73 542. B. J. Levien Aust. J. Chem. 1965 18 1161. C. N. Pepela B. J. Steel and P. J. Dunlop J. Am. Chem. SOC. 1970 92 6743. D. G. Miller J.Phys. Chem. 1966 70 2639. lo R. Haase and J. Richter Z. Naturforsch. Teil A 1967 22 1761. H. G. Hertz Ber. Bunsenges. Phys. Chem. 1977 81 656. H. L. Friedman and R. Mills J. Solution Chem. 1981 10 395. l3 K. R. Harris and H. V. J. Tyrell J. Chem. SOC. Faraday Trans. 1 1982 78 957. l4 H. Latrous P. Turq and M. Chemla J. Chim. Phys. Phys. Chim. Biol. 1972,69 1650. ’’ L. A. Woolf and K. R. Harris J. Chem. SOC. Faraday Trans. 1 1978,74,933. l6 D. G. Miller J. Phys. Chem. 1981 85 1137. l7 W-Y. Wen and S. Saito J. Phys. Chem. 1965 69 3659. R. A. Robinson and R. H. Stokes in Electrolyte Solutions (Butterworths London 2nd edn 1970). l9 M. Spiro in Techniques ofchemistry ed. A. Weissberger and B. W. Rossiter (Wiley New York 1971) vol. 1 chap. 4 pp.206-295. ” R. H. Stokes and B. L. Levien J. Am. Chem. SOC. 1946,68 333. 21 D. J. Ives and G. J. Janz in Reference Electrodes (Academic Press New York 1961). 22 S. Lindenbaum and G. E. Boyd J. Phys. Chem. 1964 68 91 1. 23 R. Mills and L. A. Woolf in The Diaphragm Cell (A.N.U. Press Canberra Australia 1968). 24 R. Mills J. Phys. Chem. 1973 77 685. ” K. S. Pitzer J. Phjs. Chem. 1973 77 268. 26 K. S. Pitzerand G. Mayorga J.Phys. Chem. 1973,77,2300; erratum J.Phys. Chem. 1974,78 2698. 27 H. G. Hertz K. R. Harris R. Mills and L. A. Woolf Ber. Bunsenges. Phys. Chem. 1977 81 664. H. G. Hertz 2.Phys. Chem. (Frankfurt am Main) in press. 29 R. Mills and H. G. Hertz J. Phys. Chem. 1980 84 220. 30 H. Leiter Thesis (University of Karlsruhe 1982).

ISSN:0301-5696    

DOI:10.1039/FS9821700041

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Symp. Chem. Soc. 1982 17,55-67 Thermodynamics of Cavity Formation in Water A Molecular Dynamics Study BY JOHAN P. M. POSTMA J. C. BERENDSEN HERMAN AND JAN R. HAAK Laboratory of Physical Chemistry The University of Groningen Nijenborgh 16 9747 AG Groningen The Netherlands Received 26th August 1982 Thermodynamic quantities related to the solvation of hydrophobic solutes in water can be approxi- mated by the application of scaled-particle theory. The crucial quantity is the Gibbs free energy of cavity formation. A series of six molecular-dynamics simulations of water including repulsive cavities of various sizes has been carried out. Using perturbation statistical mechanics the free energy has been derived as a function of cavity radius up to 0.32 nm (0.32 nm approach of water oxygens to the cavity centre).The free energy agrees well with predictions from scaled-particle theory and the experimental surface tension is predicted to within 5%. The radial distribution of water molecules with respect to the cavity has been determined for five cavity sizes; for one size (radius of approach of water oxygens of 0.3 nm) the orientational distribution and the residence-time distribution in the hydration shells has been determined. 1. INTRODUCTION The thermodynamics of hydrophobic hydration can be reasonably well understood by the application of scaled-particle theory (SPT). This theory originally devised for hard-sphere solutes and solvents,' has been applied to aqueous solutions by Pierotti 2*3 and refined for water by Stillinger.4 Several later applications and comments are a~ailable.~-~ SPT essentially computes the free energy of formation of a hard-sphere cavity of diameter a2 in a hard-sphere solvent of molecular diameter a and number density p by fitting a third-degree polynomial to the exact solution available for small cavity sizes.For application to real solutions a perturbation treatment is used to allow for realistic interactions between solute and solvent. The reasonable success of SPT for aqueous solutions of hydrophobic solutes suggests that the main factor determining the negative entropy of hydration is the packing of solvent molecules around the solute. The determining parameters are the hard-sphere size of the solvent set equal to the normal hydrogen-bonding approach of two water molecules and the experimental density of water.In section 2.1 the results of SPT are summarized. The applicability of SPT to water remains to be assessed because rather severe approximations are made. The most important of these is the hard-sphere assump- tion water may be expected to be capable of the formation of a variety of hydrogen-bonded structure (" icebergs '' in a rather dated terminology) that could well deviate in thermodynamic properties from hard-sphere solvent structures. In as far as the radial distribution function of liquid water gives a clue to such deviations corrections have been derived by Stilli~~ger.~ A second important approximation is that the introduction of solutes into hard-sphere cavities is treated as a perturbation of the hard- sphere cavity reference state.Since such a state does not contain water configurations protruding into the cavity soft potentials cannot be reliably introduced. Moreover SIMULATION OF CAVITY FORMATION solute perturbation is usually not small. In practice the entropic contribution to the perturbation is negle~ted.~ Molecular-dynamics (MD) simulations using a fast computer provide the means to study the process of cavity formation in detail. Both structural and dynamical properties of the hydration shell follow from the simulations. The treatment can incorporate any soft cavity potential. Most importantly if simulations are carried out over a range of solute sizes such that overlapping statistics result the full thermo- dynamic solvation quantities can be derived by integration over a scaling parameter starting at the pure liquid.Molecular dynamics is thus an extremely powerful way to answer any pertinent question on hydration and is limited in principle only by the accuracy of the potentials used. MD and Monte Carlo simulations on hydrophobic hydration have been reported.10-14 We have chosen the following approach. Using a simple but well tested point- charge model for water l5 we have carried out isothermal-isobaric simulations of 216 water molecules in a periodic box including repulsive cavities of five different sizes. The repulsive interaction has the form Vcw= L(B/r)'*,where L is a scaling parameter B is the same parameter as used in the repulsive term of the water-water potential and r is the distance from an oxygen to the cavity centre.Simulations extended over 30-35 ps. Details of the simulations are given in section 3. By choosing appropriate cavity sizes and by monitoring appropriate perturbation functions the free energy for the formation of a cavity of sizes up to 0.32 nm (radius of approach of water mole- cules) was obtained. The perturbation method by which these values were determined is described in section 2.3 while results are given in section 4.1. Structural properties of the cavity hydration are described in section 4.2 with emphasis on one simulation (with cavity approach radius of 0.3 nm). Of the various dynamic properties (trans- lational and rotational characteristics of the hydration layers) we analyse the residence- time distribution in the hydration shells of one cavity diameter in section 4.3.A full account of all structural and dynamical properties including a more accurate evalua- tion of thermodynamic properties over a more extended range of cavity sizes will be reported elsewhere. Finally in section 5 we discuss the results of these simulations in comparison with scaled-particle theory thereby assessing the limitations and accuracy of SPT as applied to water. 2. THEORETICAL 2.1. SCALED-PARTICLE THEORY lt3 The standard Gibbs free energy of transfer of one mole of a substance X from its ideal-gas phase at standard pressurep and temperature T to an infinitely dilute solution at the same temperature and pressure referred to a standard of unit mole fraction of X is given by3 AGO = AGc + AGi + RTln-RT POGl where AGc is the isobaric reversible work to create a cavity with given radius and at a fixed position of its centre AGi is the reversible work to charge the cavity with the real interaction between solute and solvent and G1 is the molar volume of the solvent.It is assumed that the internal partition function of the solute is the same in solution and in the gas phase. AGO relates to Henry's constant KH(the ratio between gas fugacity and mole fraction in a saturated dilute solution of the gas) AGO = RTln KH. J. P. M. POSTMA €3. J. C. BERENDSEN AND J. R. HAAK The reversible work to create a mole of hard-sphere cavities with radius of approach r is given by’ 4AG = -RT In (1 -y3p) (r < a/2) (3) and AG = KO+ K,r + K2r2+ K3r3 (r > a/2) (4) where 6 + 18(Y)I+npa2 1-Y 4 -np.(44 K3-3 Here a is the (hard-sphere) diameter of the solvent y = 7ca3p/6is the volume fraction of the solvent spheres p is the number density of the pure solvent and p the pressure. The terms involving p are negligible for cavities of molecular size at atmospheric pressure. The radius of approach r is the smallest distance between a solvent mole- cule and the cavity centre; it is related to the diameter a of a hard-sphere cavity by r = (a + a2)/2. Eqn (3) is exact (1 -4nr3p/3)is the probability that the cavity centre is not within a distance r of a solvent centre (r < a/2; r = a/2 corresponds to a point cavity with a2= 0) and this probability is equal to exp(-AGJRT) where AG is the reversible work to create such a cavity.The creation of a cavity also requires enthalpy because of the temperature dependence of p and y. Values of AG are given in fig. 1 for T = 305 K using a = 0.2875 nm and a density of 0.983 g CM-~(see section 4.1). 2.2. THERMODYNAMICS OF SOFT-CAVITY FORMATION In our simulations we have introduced a soft cavity with cavity-water interaction potential (see section 3) VCw= A(B/r)12 (5) where B = 0.3428 (kJ mol-1)1/12 nm. The scaling parameter ilcan also be regarded as a scaling factor for the size of the cavity. We define the thermalradius rth of the cavity as the radius at which Y, =kT rth= B(L/kT)1’12 (6) Just as in the SPT for hard spheres we can derive a rigorous expression for AG for small cavities.For this we use VCwas a perturbing potential energy term in the Hamil- tonian of pure water. As shown in subsection 2.3 G -Go = -RT In ( exp(-VcW/kT) (7) )o SIMULATION OF CAVITY FORMATION 't -10 t ti +* I 1 I 1 I 1 0.1 0.2 0.3 thermal radius/nm FIG.1 .-Free energy of cavity formation derived from simulations (A) compared with scaled-particle theory (B). Points C denote relative AG values obtained by perturbation of cavity diameter in each of the five simulations. Thermal radii of the simulations are indicated by arrows. where GIand Goare the free energies of cavity solution and pure solvent respectively and ( )o denotes an average over the isobaric-isothermal ensemble for pure water.Now for small cavities interaction is only appreciable at small distances r. As long as r < 0.5 rmin,where rminis the distance below which the radial distribution function of water shows no neighbours (rmin x 0.24 nm) interaction takes place with maximally one water molecule. If po(r) denotes the normalised probability distri- bution of the distance of the nearest water molecule to the (arbitrary) cavity centre we obtain W WP(-VCWlkT) ) 0 = I exP(- Vcw/kT)Po(r)dr. (8) 0 For small r,po(r)=4nr2p and for small R the exponential in the integrand changes from zero to one in this range. Taking into account that po(r) is normalized it follows that for small A W (exp(-Vc,/kT)) = 1 + p [exp(- Vc,/kT) -1]4nr2dr.(9) 0 The integral can be expressed in terms of a gamma function and is also known from computation of second virial coefficients for power-law repulsion.16 The result is J. P. M. POSTMA H. J. C. BERENDSEN AND J. R. HAAK where r(:) = 1.226 41. Except for a small numerical factor eqn (10) is equivalent to the hard-sphere expres- sion (3) if we rewrite eqn (10) with eqn (6) in terms of the thermal radius lim AGc = -RTln rth Expression (11) should be valid for thermal radii up to ca. 0.1 nm. For radii between 0.5 rminand (1/43) rmin(ca. 0.14 nm) extension is possible using the radial pair distribution curve g(r) of pure water;Ip4 above (1/43) rminmore than two water molecules can exist at the cavity radius and higher-order distributions are required.We derive values for AG by overlapping perturbations of MD simulations. 2.3. THERMODYNAMIC VALUES FROM PERTURBATIONS IN STATISTICAL REFERENCE ENSEMBLES Let us assume that we have a representative classical statistical ensemble (from MD or Monte Carlo simulations) available containing a sufficient number of states to yield accurate averages. We restrict ourselves to an isobaric-isothermal ensemble but the results are easily modified for other ensembles. Let the Hamiltonian be given by Xo(p 4). The free energy G is related to the isobaric partition function A G= -kTlnA (12) where Of course this integration cannot be carried out. Now consider a system with a perturbed hamiltonian Z1=Xo+ AV,where A V is a perturbation of the potential energy.The difference in free energy AG is AG = G1 -Go= -kT In (Al/Ao) (14) where A1 is similar to eqn (13) involving Z instead of Zo.It follows that or AG = -kT In ( exp(-V/kT) ) o. (16) If AV is the perturbation resulting from the introduction of or change in the proper- ties of one particle (cavity) AG applies to one particle; per mole of particles we have AG = -RTln ( exp(-AV/kT) ) o. (17) The averaging is over the unperturbed isobaric-isothermal reference ensemble. In practice it is possible to derive reliable values for AG if IAGl does not exceed 2kT; for larger deviations the reference ensemble contains an insufficient number of con- figurations that are representative for the perturbed ensemble. Values for the enthalpy H can be derived directly from each ensemble so that the SIMULATION OF CAVITY FORMATION need for perturbative treatment does not arise.One can easily derive however that The first term indicates the direct interaction and the second is a contribution from relaxation of the ensemble. 3. MODEL AND METHOD The model used for the water molecule is the SPC (simple point charge) model which is an effective pair potential derived from MD ~imulations.~~ It reproduces the radial distribution curve of liquid water with a second neighbour peak at the correct position but less pronounced than the X-ray results. Density is ca. 2% too low; energy is correct as well as specific heat and compressibility.18 The diffusion constant is ca. 50% too high.The model is similar to the TIPS mode1;19 it consists of a negative charge of -0.82e at the oxygen nucleus and two positive charges of 0.41e on tetrahedral positions at 0.1 nm from the oxygen. No dielectric constant is used. In addition the oxygens have a Lennard-Jones interaction Simulations were carried out on 216 molecules in a periodic cube with intermolecular interactions taken into account for all water molecules for which roo < 0.9 nm. Cartesian coordinates were used and the internal degrees of freedom were constrained by the SHAKE method.20*21 A time step of 0.001 ps was used which gave a (short time) fluctuation of the total energy better than 15% of the kinetic-energy fluctuation. An algorithm was used that implies a coupling of the system to a constant tem- perature and pressure bath.22 The algorithm is based on a leap-frog scheme in which at each step both velocity and coordinate scaling (and volume scaling) occurs.The scaling involves delayed coupling to constant temperature and pressure; the time constant of each coupling is adjustable. A long time constant means a weak coupling and slow restoration of temperature and pressure deviations but also a weak pertur- bation of the system. We used 0.4 ps for the coupling to constant temperature and 0.5ps for the coupling to constant pressure. Dynamic runs were initiated by gradually introducing a cavity starting from a pure water configuration in 1 or 2 ps and then equilibrating over 5 ps. Runs extended over 25 or 30 ps. Averages over the equili- brating 5 psdid not differ significantly from any 5 ps section of the run but were never- theless not used for obtaining final averages.The runs were subdivided into 5 or 6 sections of 5 ps each; averages Aiwere obtained for each subsection and final averages A' and their estimated errors AA were obtained from the N subaverages and their statistical spread IN A=-2 Ai (204 N i=l N The following potential between cavity centre and oxygen of water molecules J. P. M. POSTMA H. J. C. BERENDSEN AND J. R. HAAK was introduced for five values of A. The cut-off radius used for this interaction was also 0.9 nm. The cavity was kept stationary. 4. RESULTS 4.1. THERMODYNAMIC QUANTITIES Data for six MD runs including pure water are given in table 1.It is apparent that neither the volume nor the enthalpy shows significant dependence on cavity size. This is because the full fluctuations of the large system of 216 molecules are present in such overall quantities while the cavity concerns effects on a single-molecule scale. TABLERESULTS FROM MD RUNS A t-th length Ew-wPot Ec-wPot V H ~p0-5 Inm /ps /kJ mol-' /kJ mol-' /nm3 /kJ mol-' cm2s-I 0 0 25 -8983 (13) 0 6.57 (1) -7325 (15) 4.2 (6) 0.100 25 -8996 (10) 0.1 (1) 6.63 (2) -7345 (22) 5.9 (4) 10-3 0.178 30 -8997 (12) 1.3 (1) 6.62 (1) -7340 (13) 4.6 (6) 0.0325 0.238 30 -8992 (10) 2.9 (1) 6.61 (2) -7332 (11) 7.1 (12) 0.5 0.299 25 -8971 (13) 6.2 (9) 6.65 (5) -7306 (15) 6.8 (7) 1 0.317 25 -9013 (6) 6.9 (6) 6.65 (2) -7343 (9) 4.8 (3) a Temperature for all runs is 305 K (rt0.5 K).Pressures are between 30 and 50 bar not corrected for cut-off. Values in parentheses are r.m.s. errors in last digit determined from statistics of the averages over subsets of 5 ps each. Free energy of cavity formation was determined in the following way. For each value of rth the perturbation treatment of section 2.3 [eqn (17)] was carried out for several deviations of rth on both sides of the thermal radius. Thus overlapping extra- polations from adjacent simulations were obtained that enable construction of the AG curve over the full range of cavity sizes studied. For each perturbation the error in AG was determined according to eqn (20) by performing the averages separately over 5 ps sections. Fig. 1 shows the values of AG obtained in this way for the various simulations as well as the AG curve constructed from these.The error in AG accumulates when the number of successive overlap adjustments increase; it is indicated by error bars in fig. 1. The connection between the simulation at rth = 0.100 and pure water deserves extra attention since extrapolation from this simulation downward to water is not possible. This is due to the fact that a cavity does not contain any water molecules; hence statistics rapidly become poor when extrapolations to smaller cavity size are made configurations are then required that are not available in the reference simu- lation. This is true for all reference sizes and extrapolation upward is always more reliable then downward.Even the error gives an unreliable estimate when insufficient statistics exist. Thus for the connection between rth = 0.1 nm and water several more simulations would be required. Fortunately the theoretical value for AG of eqn (1 1) is reliable up to 0.1 nm and can be used to obtain the connection. This was in fact done; fig. 2 illustrates this point. Fig. 1 also gives values of AGc for perturbations of each reference state. The shifts in free energy obtained by comparison of adjacent simulations and their SIMULATION OF CAVITY FORMATION I 01 I I I I 0-0.06 0.08 0.10 0.12 0.14 FIG.2.-Theoretical free energy of cavity formation (drawn curve) and points derived by perturbation of the Ych = 0.1 nm simulation illustrating inadequacy of extension from simulation to lower values of the cavity radius.accumulated values are given in table 2. We see that connection between Yth = 0.238 and 0.299 involves a rather large shift that may require an intermediate simu- lation. From the present analysis we cannot construct the AH curves with acceptable accuracy. Also the volume allows no definite conclusions; compared to the volume of the cavity itself the indications are that a volume contraction occurs that com- pensates the cavity volume itself. TABLE 2.-ADJUSTMENT OF FREE-ENERGY DIFFERENCES BY PERTURBATIONS OF MD SIMULATIONS connection shifts accumulated between rlh/nm /kJ mol-' shift/kJ rno1-I ~~ ~ ~~ 0.100+0.178 3.5 f0.5 3.5 f0.5 0.178 +0.238 7.3 0.6 10.8 f0.7 0.238 -+0.299 0.299 -f 0.317 11.1 -+0.6 4.5 -+ 0.6 21.8 f0.9 26.3 f1.1 4.2.STRUCTURAL PROPERTIES The distribution of water molecules around a cavity is characterized by the cavity- water pair distribution gcw(r),and by the water-water pair distribution function on the cavity surface. Water orientations are characterized by orientational distribution functions of OH HH and dipole vectors. Fig. 3 gives the cavity-water pair distribution function for all five cavities studied. It is evident that the cavity of 0.178 nm thermal radius has the most pronounced radial shell structure with a high first-neighbour peak and a well developed second-neighbour peak. Also the 0.299 nm cavity has relatively better resolved structure. Considering the orientational distributions for one simulation (rth = 0.299 nm) we J.P. M. POSTMA H. J. C. BERENDSEN AND J. R. HAAK observe a higher occurrence of OH orientation of both 0" (radially outward) and 120° while the dipole is mostly oriented around 70-80" (fig. 4). Very roughly a combin- ation of configurations as given in fig. 5 would account qualitatively for the orien- t ational distribution curves. The pair distribution curve of molecules in the first shell can best be expressed in terms of the distribution of the angle 8 between two molecules of any pair in the shell.4 Fig. 6 shows the distribution of cos 8 in the first solvation shell (rth = 0.299 nm) n fi R I x Y .-.I > 3 c5 rlnm FIG.3.-Cavity-oxygen radial distribution curves for five values of the thermal radius of the cavity (A) 0.100 (B) 0.178 (C) 0.238 (D) 0.299 (E) 0.317 nm.Curves are obtained by spline smoothing with deviation from the data not exceeding 0.05. No smoothing was applied for distances below the first maximum where the shell has been divided into two subshells. The subshell directly on the surface of the cavity shows a pronounced first-neighbour peak and a resolved second- neighbour peak. The first peak corresponds to hydrogen-bonded distances of neighbours. Solvation shell structures are different for different cavity sizes; a more complete analysis of these structures is in progress. 4.3. DYNAMIC PROPERTIES As far as dynamic properties of cavity hydration are concerned we limit our analysis here to the residence-time distribution of molecules in the solvation shells and defer the analysis of reorientational motions to a later publication.The residence- time distribution is a measure of the stability of a solvation shell; it is determined both by the size of the shell and the diffusion constant essentially in radial direction. The residence-time distribution was determined by monitoring what fraction of the molecules present in a shell at t = 0 were present at time t. This was averaged over initial times and corrected for the fraction of molecules that are expected to occur in the shell on the average. This distribution is equivalent to the correlation function of a variable that is equal to 1 when a molecule is in the shell and zero when it is not in the SIMULATION OF CAVITY FORMATION 0.8 0.6 0.4 0.2 c -1 -0.5 0 0.5 1 P!' ! ' ! ' I 114 180150 120 90 60 30 0 inner scale cos 0 outer scale O/" FIG.4.-Probability density of orientation of OH dipole and HH direction with respect to radius from cavity centre to oxygen expressed as distribution over cos 8.Data apply to molecules in first shell (Y < 0.475 nm) from simulation with T(h = 0.299 nm. shell. The resulting curves for the first second and third shell of the 0.299 nm cavity are given in fig. 7. After an initial fast partial decay the distribution decays very slowly reaching half-lives of several picoseconds. The interest in these curves lies in the prediction of an effective diffusion constant FIG.5.-TWO possible orientations of water consistent with the orientational distributions of fig.4. J. P. M. POSTMA H. J. C. BERENDSEN AND J. R. HAAK for the molecules in a shell. The exact calculation of the residence time distribution in a spherical shell with a non-uniform distribution is prohibitive but a good approxi- mation is possible. For this we consider a planar slab with thickness a uniformly filled with particles in an infinite medium. The particles diffuse with a diffusion V!'! I ! I 1 1 I4 180150 120 90 60 30 0 inner scale cos 8 outer scale S/" FIG.6.-Pair correlation of water molecules on cavity surface for rth = 0.299 nm. G(cos 0)dcos 8 gives the fraction of the total number of neighbours in the indicated shell that are in an angular range (cos 8 cos 8 + dcos S) given the presence of one molecule at 8 = 0".(-) rcav-ox < 0.4 nm; (--) 0.4 < rlnm < 0.475.constant D through the slab and the medium. The fraction P(t) of the particles originally present in the slab that are still in the slab at time t is given by P(t) = 2adnDt fixExf exp (w2). In the case that one of the walls of the slab is reflecting the lower bound of the second integral should be -a. Solving the integrals we obtain P(z) = 1 -d(z/n)[I -J(z-+)] where J(x) is the normalised integral of the error function also indicated as which is available in tabulated form.23 The variable z is given by z = Dt/a2 for a reflecting slab and z = 4Dtla2 for an open slab. SIMULATION OF CAVITY FORMATION 0 1 2 3 4 5 t ime/ps FIG.7.-Residence-time distributions in (a)first (r< 0.475 nm) (b)second (0.475<r/nm<0.76) and (c)third (0.76<r/nm<0.9) shell of cavity with thermal radius of 0.299 nm.Drawn curves give the fractionP(t) of molecules present in the shell at t = 0 that are also present at time t. P(t)is corrected for the probability of findinga molecule in the shell at t =co due to the finite size of the system. Broken curves are theoretical residence time distributions based on a diffusion model. Applying eqn (23a) and (23c) for the first solvation shell and eqn (23b) and (23d) for the second and third solvation shells we can determine the diffusion constant. The thickness a was chosen as 0.175 0.285 and 0.140 nm for the three shells.Using values for D of 8 x 6 x lov5and 5 x cm2s-' for the first second and third shell respectively we obtain the broken curves given in fig. 7. The measured curve agrees very well with the theoretical one. The values of the diffusion constants in the three shells do not differ significantly from the bulk value of (6.8 0.7) x cm2 s-' of the simulation. The conclusion must be that at least the translational dynamics are not slower in the solvation shell than in the bulk liquid. 5. DISCUSSION We limit our discussion to the comparison with scaled-particle theory; a discus-sion of structural and dynamic properties requires a more complete analysis of the data. We have shown that it is quite possible to obtain values for the free energy of cavity formation by scaling a cavity up from zero to its size through a limited number of discrete simulations.It is much more difficult to obtain enthalpy and entropy values with comparable accuracy. It is of special interest to compare our " experimental " AGc values with those predicted from scaled-particle theory. There are two difficulties (a) what is the hard-sphere radius of a cavity with given thermal radius and (b)what is the hard- 67 J. P. M. POSTMA H. J. C. BERENDSEN AND J. R. HAAK sphere radius a of the water molecule? To the first question there is a unique answer the theoretical exact relation for small hard cavities in a hard-sphere solvent [eqn (3)] is equivalent to the exact relation for soft cavities [eqn (lo)] when the hard-sphere radius rhs is related to the thermal radius as rhs = 1.0704 rth.(24) The numerical factor is equal to [l-(3/4)]1’3. This relation we shall apply throughout. The hard-sphere radius a of a water molecule cannot be determined independently. When a = 0.2875 nm is taken using the density of our water model (0.983 g ~m-~) SPT gives the values of AG as shown in fig. 1. The agreement with “experiment ” is excellent. This value of a is slightly (but significantly) larger than the values of 0.275 or 0.277 nm used in the literature. The parameter y has a value of 0.4092. For the coefficients K in eqn (4) neglecting the term in r3 we find KO= 6.651 (254 Kl = -110.39 (25b) K2 = 507.82. (254 The excellent agreement supports the applicability of scaled-particle theory to water provided a reliable estimate for the effective hard-sphere radius can be made.The quadratic term in the expansion of AG [eqn (4)] relates to the surface tension y. Thus we can derive the surface tension from our simulations. We find y = K2/(4nNAv)kJ nm-2 (26) or y = 0.067 N m-’ which agrees closely with the experimental value at 305 K of 0.071 N m-l. We gratefully acknowledge the generous support of the Computer Centre of the University of Groningen. This research was supported by the Foundation for Chemical Research (S.O.N.) under the auspices of the Netherlands Organisation for the Advancement of Pure Research (Z.W.O.). See for a review H. Reiss Adv. Chem. Phys. 1965 lX,1 and references quoted therein. ’R. A. Pierotti J. Phys.Chem. 1965 69 281. R. A. Pierotti Chem. Rev. 1976 76 717. F. H. Stillinger J. Solution Chem. 1973 2 141. H. M. Neumann J. Solution Chem. 1977 6 33. H. DeVoe J. Am. Chem. SOC.,1976,98 1724. ’M. Lucas J. Phys. Chem. 1976,80 359. M. Lucas and R. Bury J. Phys. Chem. 1976 80 999. N. Morel-Desrosiers and J-P. Morel Can. J. Chem. 1981 59 1. lo V. G. Dashevsky and G. N. Sarkisov Mol. Phys. 1974,27 1271. A. Geiger A. Rahman and F. H. Stillinger J. Chem. Phys. 1979 70,263. l2 C. Pangali M. Rao and B. J. Berne J. Chem. Phys. 1979 71 2975; 2982. l3 G. Alagona and A. Tani J. Chern. Phys. 1980,72 580. l4 K. Nakanishi S. Okazaki K. Ikari and H. Touhara Chem. Phys. Lett. 1981 84 428. l5 H. J. C. Berendsen J. P. M. Postma W. F. van Gunsteren and J. Hermans in Intermolecular Forces ed.B. Pullman (D. Reidel Dordrecht 1981) pp. 331-342. l6 J. 0. Hirschfelder C. F. Curtiss and R. B. Bird Molecular Theory of Gases and Liquids (J. Wiley New York 1954) p. 157. D. A. McQuarrie Statistical Mechanics (Harper and Row New York 1976). J. P. M. Postma unpublished results. l9 W. L. Jorgensen J. Am. Chem. SOC. 1981 103 335. ’O J. P. Ryckaert G. Cicotti and H. J. C. Berendsen J. Comput. Phys. 1977,23 327. W. F. van Gunsteren and H. J. C. Berendsen Mol. Phys. 1977,34,1311. ”H. J. C. Berendsen J. P. M. Postma A. Di Nola W. F. van Gunsteren and J. R. Haak to be published. 23 M. Abramowitz and I. A. Stegun Handbook of MathematicalFunctions (Dover Publication Inc. New York 1965).

ISSN:0301-5696    

DOI:10.1039/FS9821700055

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC.,1982 17 69-78 Molecular Librations and Solvent Orientational Correlations in Hydrophobic Phenomena BY PETERJ. ROSSKY A. ZICHI * AND DOMINIC Department of Chemistry University of Texas at Austin Austin Texas 78712 U.S.A. Received 5th August 1982 A detailed analysis of the energetic characteristics associated with the solvent in a simulation of a solution containing a pair of apolar spheres immersed in ST2 water solvent is presented. The spheres are separated by a distance corresponding to the so-called solvent-separated hydrophobic interaction. The analysis of local structure is facilitated by the consideration of only near-neighbour solvent-solvent interactions and the use of a coarse-grained time-averaging procedure which removes the high-frequency librational excitation from the underlying hydrogen-bond network.The struc- ture produced is analogous to that termed " V-structure " by Eisenberg and Kauzmann. The results manifest the expected enhancement in intermolecular interactions within the solvation shell characteristic of higher orientational correlation and this enhancement is markedly greater in the region which includes solvent that is shared by solute solvation shells. The V-structure obtained is consistent with an underlying network with further enhancement of intermolecular orientational correlations upon which is superimposed thermal librational excitation. The intermolecular bonding is consistent with local energetic stabilization of the solvent-separated configuration reinforcing the clathrate-like picture of solvation.1. INTRODUCTION The detail with which the structure of liquid water and aqueous solutions can be described at a molecular level has been rapidly refined by recent theoretical work primarily via the now standard techniques of computer simulation.' In particular numerous studies 2-10 have appeared in which the system examined includes one 3~7-10or more 2*4*6 apolar solutes or solutes containing substantial apolar groups.s The goal of these studies has been to elucidate the important details of both the geometric and energetic measures of structure associated with the hydrophobic effect. By the last term we mean both what has been come to be called hydrophobic hydration i.e. the solvation by water of isolated apolar solutes and hydrophobic interaction the mean interaction associated with the proximity of more than one apolar group in an aqueous environment.These theoretical studies have been in rather good accord in their qualitative conclusions regarding these phenomena. For water molecules adjacent to apolar solutes there is a clear clathrate-like orientational preference induced by the ten- dency of the hydrogen-bonding groups of the solvent to avoid being directed toward the " inert'' solute. Such an orientation permits the number of strong solvent- solvent hydrogen bonds in this solvation region to be close to that in bulk water. In fact the strength of these bonds appears to be typically greater than those in the bulk in accord with experiment." Hence the total binding energy of a molecule in the solvation layer of an apolar solute is typically at least as strong as one in the bulk despite the smaller number of solvent molecules in its own solvation shell.When two (or more) such solvated species come into close proximity the changes in * Alfred P. Sloan Foundation Fellow 1982-1984. 70 MOLECULAR LIBRATIONS AND ORIENTATIONAL CORRELATIONS the (already perturbed) solvation regions lead to the hydrophobic interaction. Both experimental results 12913 and more recent theoretical results 2*4314 have indicated that not only is a solute-solute “ contact ” separation important but at least for isolated pairs of relatively small solutes there exists a so-called solvent-separated hydrophobic bond at larger separation.For the former one expects that the driving forces with respect to infinitely separated solutes are dominated by the partial exclusion of non- polar solute-water contact. For the latter evidence indicates that the solvent “ structure ” is enhanced beyond that concommitant with the hydration pheno- and that the geometric characterization of the structure is most analogous to crystalline clathrate compounds with solvent participating in a shared clathrate face or edge.15 Such a structure would owe its relative stability at least in part to the particularly low potential energy which can be obtained by the solvent an aspect of the solvent separated configuration which would be absent in a simple fluid. The purpose of the present work is to characterize the non-polar solvation struc- ture associated with the solvent-separated configuration in more detail with particular emphasis on energetic measures.In particular we wish to elucidate the “under-lying ” hydrogen-bonding structure present in the solvation region as compared with bulk water. By the term “ underlying ” structure we refer to the relatively slowly changing network of intermolecular hydrogen bonds which is present in the solution. It is well established 16-18that it is both physically meaningful and productive to view the aqueous structure as consisting of a set of such quasistatic networks upon which is superimposed rapid molecular librations (hindered rotations) and intermolecular vibrations of centre-of-mass positions (hindered translations) which occur on a time-scale that is shorter than and well separated from that for network rearrangement.The underlying network we characterize here is closely analogous to that described by Eisenberg and Kauzmann l6 as “V-structure ”. The solution structure can then be characterized on a local level by the degree of orientational correlation between nearest neighbours in the underlying hydrogen-bond network (hydrogen-bond distortions) which will be manifest by the distribution of interaction energies present in the solvent. Superimposed on this are librational motions characterized by force constants which are a property of this network. Recently we have developed and implemented a coarse-grained time-averaging technique which leads to a vibrationally averaged “ V-structure ” from a standard molecular-dynamics simulation of the instantaneous “I-structure ”;l8 the technique is described briefly below.In application to pure water it has been demonstrated that the thermal excitation present as librational motion in the I-structure obscures the degree of intermolecular hydrogen bonding which exists in the liquid. In the present context then the potential advantage of this analytical approach lies in the ability more clearly to resolve the changes in intermolecular hydrogen bonding which are associated with hydrophobic phenomena and in addition obtain a more accurate view of the degree of distortion (or order) present in this underlying network. In particular one aspect which we wish to address is the development of a clearer description of the degree of librational freedom of molecules in the apolar solvation region.It is clear from solvent isotope-effect ~tudies,l~*~~ for example that there is an increase in the effective “ average ” librational force constant for solvent in solutions including “ structure-making ” solutes. It is of interest to make a direct connection between this change and the local hydrogen-bonding environment experienced by the solvent. In section 2 we briefly outline the computational techniques and describe the systems considered here. The polar-solute system consists of a pair of Lennard-Jones spheres in aqueous solution with an intersolute separation consistent with a locally P. J. ROSSKY AND D.A. ZICHI stable solvent-separated hydrophobic bond. In section 3 for completeness we first present representative results for pure water which were obtained previously." We then present corresponding results obtained for apolar solvation and discuss the implications for the molecular picture of hydrophobic phenomena. The conclusions are presented in section 4. 2. COMPUTATIONAL METHODS The results presented below were obtained from conventional molecular-dynamics simulations of aqueous systems.' The water-molecule interactions were modelled using the ST2 model of Stillinger and Rahman 21 and the equations of motion were integrated using a time-step of s and the SHAKE algorithm22 to maintain intramolecular constraints. In all cases the interactions were computed with periodic boundary conditions using a spherical cutoff of 8.46 A.In each simulation the molecular number density is 0.033 46 A-3. The pure-water simulation was carried out with a system of 216 molecules for 4 ps. The solution system consists of two Lennard-Jones spheres and 214 water molecules. The spheres interact with each water molecule via a potential which is identical to that employed by Pangali et aL4 I.e. the interaction consists solely of a Lennard-Jones potential UAW(r)which depends only on the separation r between the water oxygen atom and the centre of an apolar sphere and characterized by oAW= 3.43 8,and EAW/kB=77.82 K where kBis Boltzmann's constant corresponding very roughly to krypton atoms. For the present purposes we desired a simulation which generated configurations that represented only the solvent-separated hydrophobic interaction.Therefore the pair was rigidly constrained at a centre-to-centre distance of 7.13 A corresponding to the position of the local minimum found in the potential of mean force for the identical The pure-water trajectory has a mean temperature of 281 K. For the apolar sphere solution the initial configuration was obtained from an equilibrium simulation of pure water with two molecules replaced by the apolar spheres. The solution system was carefully re-equilibrated in order to produce a comparable temperature and of most importance to assure that the temperature characterizing the solvation shell was in fact in equilibrium with the bulk.It is unclear whether this is true in earlier molecular-dynamics studies of similar systems.2 Via periodic reassignment of velocities by those chosen from a Boltzmann distribution with a characteristic tem- perature T = 283 K over a period of 10 ps we obtain our initial configuration from which a simulation of 11 ps was generated. The overall mean system temperature from the simulation was 281.8 K and in very close agreement that of all water molecules within 5.5 A of either apolar sphere (a list which was updated every 0.2 ps) was 283.6 K. We note that the present simulation is substantially longer than those used in previous studies of the corresponding intersolute separation regi~n.~.~ The V-structure was generated from these simulations via a technique which has been described in detail elsewhere.'* The basic procedure involves dividing the dynamical history into segments each of a length zA which is of the order of an inter- molecular vibrational period.The desired information consists of the sequence of mean molecular positions each mean obtained from an average over the time zA of each segment. The averaging of coordinates was carried out in the six-dimensional space consisting of the centre-of-mass position and the Euler angles. From the mean centre-of-mass positions and Euler angles obtained we then recover a set of Cartesian coordinates for the sample of molecules. From the sequence of such sets the analysis of the liquid structure proceeds in precisely the same manner as would that of the initial sequence of positions.72 MOLECULAR LIB RAT1 ONS AND OR1 ENTAT1 ONA L CORRELATIONS We have shown previously l8 that over a reasonable range the results obtained are not sensitive to the choice of zA,and here we use zA = 0.2 ps. As is clear from the power spectrum for pure water shown in fig. 1 this value is sufficient to permit averaging over at least one period of even the low-frequency librations as well as of the higher-frequency regime of the hindered translations. The latter are presumably associated with relative motion of nearest-neighbour molecules. 0.05 t*=0.2 ps 0*06 t 0 40 80 120 160 w/ps-1 FIG.1.-Single-molecule velocity power spectra for centre-of-mass translational velocity (left-most curve) and principal axes rotational velocity (x,y,z).From the present simulation of apolar spheres in water we obtain 55 V-structure configurations. For the purposes of comparison the corresponding analysis of I-structure is carried out with 55 configurations which span the run these involve no averaging but are simply separated sequentially in time by 0.2 ps. For the purpose of analysis it is necessary to define separate regions of the solvent which we here denote as bulk (b) hydration shell (s) and interior shell (i). The water molecules in the " bulk " class are defined by an oxygen-apolar-sphere centre-to-centre separation > 4.5A. The shell is the complementary class to the bulk. The interior-shell region is defined as that subregion of the shell lying between a pair of parallel planes which are perpendicular to a line joining the centres of the apolar spheres and each of which intersects one apolar-sphere nucleus.This definition of the interior subregion corresponds directly to that employed earlier by other authors [see ref. (2) and (4)]. The choice of shell radius is not crucial here. The present choice is motivated in part by indications 'p7 of a possible multilayer substructure within the first broad peak of the radial distribution function for water-molecule oxygen atoms around the apolar spheres which here has its first minimum at 5.4 A. It is the innermost sphere which is apparently most strongly influenced by the apolar The present choice of 4.5A is smaller than the largest reasonable choice (5.4A) but is sufficiently large to incorporate all of the molecules that would be found in the present system if they contributed to the idealized clathrate-like structure proposed by Geiger [see ref.(21 fig. 71. In the present analysis we wish to focus on the local environment of water mole- cules. Hence in the following analysis we compute all quantities by considering only nearest-neighbour interactions operationally defined by including only water molecules with oxygen-atom separations within 3.5 A. This value corresponds to the position P. J. ROSSKY AND D. A. ZICHI of the first minimum in the pure solvent pair distribution function.21 For solvent energetic quantities evaluated here we have the pair interaction energy between sohent molecules denoted E for neighbouring pairs and a corresponding " local " binding energy which we denote by El.For each solvent molecule El is the sum of the values of E for each neighbouring solvent molecule and differs significantly in magnitude from the more usual total binding energy which includes also all more distant interactions. El is more directly interpretable in terms of the values of E observed than is the total binding energy since it is constituted from only a small number of increments from the pair energy distribution. The simulation of the apolar-sphere solution is analysed according to the division of solvent described in section 2. This division leads to an average number (N) in each region of ca. 179,24 and 11 for the bulk shell and interior shell respectively (see table 1).TABLEME MEAN ENERGETIC CHARACTERISTICS OF SOLVENT FOR BULK (b) SHELL (s) AND INTERIOR (i) REGIONS (IN kcal mol-') difference I-structure V-structure v-I -2.83 -3.32 -0.49 -2.82 -3.31 -0.49 -3.72 -4.31 -0.59 -3.68 -4.28 -0.60 -15.55 -18.19 -2.64 -16.10 -18.77 -2.67 -16.30 -19.14 -2.84 <Ns> 24.24 24.45 <N,> 11.15 11.09 Regions and energy quantities are defined in the text; all quantities in kcal mol-'; average numbers of molecules in regions. 3. RESULTS AND DISCUSSION In this section we describe the results obtained by the methods outlined above. We first present a representative V-structure result for pure liquid water to demonstrate the expected behaviour. In fig. 2 we show the normalized distribution of pair interaction energies i.e.the probability of finding a pair of molecules interacting with a potential energy E where here and nowhere else all intermolecular separations are included in the distribution. As is clear from the figure in the V-structure the pair energies are shifted to more negative values (the peak by ca. 0.7 kcal mol-') and the distribution is sharper. The observed shift is qualitatively very similar to that found to result from a decrease in temperature in bulk water. Also included are the results obtained with Z~ = 0.1 ps and as stated above the energetic structure is not sensitive to the choice of averaging time. Corresponding behaviour has been found in the distribution of molecular binding energies." The results obtained from the present analysis are presented in fig.3-6 and in table 1. The means obtained from all probability distributions are given in the table. For each quantity we present both I-structure and V-structure results. For pair 74 MOLECULAR LIBRATIONS AND ORIENTATIONAL CORRELATIONS -6 -4 -2 0 E/kcal mol-' FIG.2.-Probability distribution for intermolecular pair interaction energy E for pure water. I-structure (-); V-structure T~ = 0.2 ps (--); V-structure T~ = 0.1 ps (0). In this case only all intermolecular spatial separations are included. /\ !/ ' ' 0.00 I I' I I I I I I -8.00 -4.00 0.00 4 .OO 8.00 E/kcal mol-' FIG.3.-Probability distribution for near-neighbour interaction energy E in I-structure. b-b (solid) b-s (long dot-dash) s-s (short dot-dash).The i-i distribution is within statistical limits the same as s-s. The absolute minimum of E occurs at -6.839 kcal mol-'. P. J. ROSSKY AND D. A. ZICHI 0.00 I I I I 1 I I 1 -8.00 -L.OO 0.00 4.00 8.00 &/kcal mol-' FIG.4.-Probability distribution for near-neighbour interaction energy E in V-structure; otherwise as in fig. 3. interactions (E) the solution region in which each member of the pair is located must be specified and this is indicated in the figures and table. We consider first the hydration shell (s) as a whole and then return to a consider- ation of the interior region (i). 0.121 EJkcal mol-' FIG.5.-Probability distribution for local bonding energy El in I-structure bulk (short dot-dash) shell (solid) interior shell (long dot-dash).The absolute minimum of E in a perfectly bonded lattice occurs at -27.356 kcal mol-'. 76 MOLECULAR LIBRATIONS AND ORIENTATIONAL CORRELATIONS Considering the pair interaction energies for nearest neighbours the results shown in fig. 3 for I-structure are generally consistent with those obtained by others.2-10 The interaction between neighbours in the hydration shell is significantly more nega- tive than in the bulk the mean by ca. 0.9 kcal mo1-' (see table 1). In contrast the interaction between bulk and shell is indistinguishable in the present result from that within the bulk. Of particular interest is the reduction in the hydration shell of the proportion of higher-energy presumably distorted bonds compared with the bulk.Elkcal mol-' FIG.6.-Probability distribution for local bonding energy El in V-structure; otherwise as in fig. 5. In fig. 4 the V-structure corresponding to the results in fig. 3 are shown. The results manifest a further shift of ca. k,T for each distribution compared with fig. 3. The relatively narrow distribution of pair interactions within the hydrogen-bond net- work constituting the shell is quite striking; the great majority of nearest neighbours fall in the range > -4 kcal mol-I. The local binding energy results are shown in fig. 5 and 6. These results reflect the shift in interactions manifest in the pair energies although the total shift in El for an average shell molecule as compared with the bulk is only ca.0.5 kcal mol-' as shown in the table. This results from the fact that the total number of water-water nearest-neighbour interactions is reduced in the shell (by ca. ~ne),~~~ although those among the shell molecules are significantly stronger. The V-structure (fig. 6) reflects the relatively narrow distribution of environments compared with the bulk. The shell includes a significant number of molecules where after librational excitation is removed the local binding energy approaches that of a perfect lattice (-27.36 kcal mol -') . Of special relevance to the hydrophobic interaction is the defined interior shell region. We have found that the distribution of pair energies between molecules both of which are within the interior region is within the noise level apparent in the figures superimposable on that shown for the shell as a whole.Nevertheless the local bind- ing energy curve is shifted to even more negative values. This is particularly clear in the result shown from V-structure in fig. 6 where the peak occurs only ca. -5 kcal mol-' above the absolute minimum. The obvious interpretation of this fact is that P. J. ROSSKY AND D. A. ZICHI for a significant number of interior-shell molecules there are fewer (or no) shell-bulk interactions present. This view is consistent with the formation of shared clathrate- like cages with a portion of the shell molecules occupying shared vertices.” It is however not the individual bond strengths which are modified in the interior region but rather their distribution among molecules.Using the mean number of shell and interior-shell molecules present and the energetic mean values in table 1 the results above imply that the exterior solvation region (the subregion of the shell complementary to the interior) is characterized approximately by a local binding energy which is only ca. -0.3 kcal mol-’ more negative than in the bulk (-0.38 in I-structure -0.27 in V-structure). Thus the interior is further stabilized energetically locally by ca. -0.5 kcal mol-’ (-0.37 in I-structure -0.68 in V-structure). Returning to the general question of librational motion and the underlying hydro- gen-bonding structure we see that the observed comparison of V-and I-structures is completely consistent with the removal of an equal amount of librational energy from every molecule.The sharpening of distributions can be attributed to the fact that all energetic quantities have a lower bound (the region around which is sampled in the I-structure as well). An equal energetic shift for each molecule independent of local librational force constant is sensible; the same behaviour would be observed for a thermally excited harmonic oscillator. It is also clear that a consistent interpretation of experimentally observed libra- tional frequencies l9v2O follows from the picture above. We postulate reasonably only that molecules in a local (V-structure) environment with stronger (less distorted) hydrogen bonds are subject to greater librational force constants and that a different librational frequency is associated with each local environment.Then the shift in librational bands associated with hydrophobic solutes 2o is best described as a shift in the proportion of molecules in strongly bonded local environments as compared with bulk water as is clear from fig. 4 and 6. This need not imply the existence of a new local structure absent in the bulk. Finally we note that some time ago Franks l2 inferred that one should observe a “ sharpening ” of intermolecular solvent correlations induced by hydrophobic hydr- ation and that further enhancement was induced by hydrophobic interaction. The present results are in accord with this view. The increased proportion of molecules which exist in a local environment that includes improved near-neighbour interactions is consistent with a narrower distribution of intermolecular orientational correlations in the V-structure.The enhanced librational force constants inferred here should also produce narrower librational widths in the hydrophobic solvation region when observed via geometric measures although not energetically as we have seen here. 4. CONCLUSIONS We have presented a detailed analysis of the energetic character of the solvation region for a pair of apolar atomic-like solutes with an intersolute distance correspond- ing to the so-called solvent-separated hydrophobic bond. In general accord with earlier studies the analysis has shown that the solvation region is characterized by stronger intermolecular hydrogen bonds than are typical in the bulk.The relatively low probability of highly distorted near-neighbour interactions corresponding to a higher degree of intermolecular orientational correlation is emphasized in the V-structure which manifests a rather sharp distribution of local bonding environments. A consideration of the local bonding is consistent with the view that the solvation region is characterized by a higher average librational force constant than bulk water due to an increased proportion of molecules which are in a local environment charac- MOLECULAR LIBRATIONS AND ORIENTATIONAL CORRELATIONS terized by strong intermolecular bonding. The use of the local (near-neighbour) binding energy rather than the total has facilitated this analysis. The fact that in the interior-shell region the intermolecular bonds between shell molecules are comparable to those on the exterior but that in the interior there are fewer of the more distorted shell to bulk bonds is in accord with the picture described by Stillinger l5 in which the solvent-separated configuration is characterized by a re- duced strain associated with reduction of the shell-bulk interface.In this sense the present results reinforce the clathrate-like picture of hydrophobic phenomena. This picture is further reinforced by the V-structure characterization after librational excitation is removed. A more complete analysis of the present simulation is currently underway. Included in this is a study of the geometrical character of the solvation region and the corresponding comparison of I-and V-structures.In contrast to energetic character for geometric properties we anticipate that the difference induced by averaging out librational motions will vary in different regions of the solution. Support of the research reported here by grants from the Robert A. Welch Found- ation and the National Institute of General Medical Sciences (USPHS) is gratefully acknowledged. See for example D. W. Wood in Water-A Comprehensive Treatise ed. F. Franks (Plenum Press New York 1979) vol. 6 chap. 6. A. Geiger A. Rahman and F. H. Stillinger J. Chem. Phys. 1979 70 263. S. Swaminathan S. W. Harrison and D. L. Beveridge J. Am. Chem. SOC.,1978 100 5705. C. Pangali M. Rao and B. J. Berne J. Chem. Phys. 1979 71 2982. P. J. Rossky and M.Karplus J. Am. Chem. SOC.,1979 101 1913. D. C. Rapaport and H. A. Scheraga J. Phys. Chem. 1982 86 873. 'J. C. Owicki and H. A. Scheraga J. Am. Chem. SOC.,1977,99 7413. S. Okazaki K. Nakanishi H. Touhara and Y. Adachi J. Chem. Phys. 1979 71 2421. G. Alagona and A. Tani J. Chem. Phys. 1980,72 580. lo E. Bolis and E. Clementi Chem. Phys. Lett. 1981 82 147. l1 See for example (a) F. Franks in Water-A Comprehensive Trearise ed. F. Franks (Plenum Press New York 1973) vol. 2 chap 5; (6)E. Wilhelm R. Baltino and R. J. Wilcock Chem. Rev. 1977 77 219. F. Franks in Water-A Comprehensive Treatise ed. F. Franks (Plenum Press New York 1974) vol. 4 chap. 1 ; see in particular pp. 77-79. l3 W-Y. Wen and S. Saito J. Phys. Chem. 1964 68 2639. l4 L. R. Pratt and D.Chandler J. Chem. Phys. 1977 67 3683. l5 F. H. Stillinger Science 1980 209 451. l6 D. Eisenberg and W. Kauzmann The Structure and Properties of Wafer (Oxford University Press New York 1969). S. A. Rice Top. Curr. Chem. 1975 60 109; A. C. Belch S. A. Rice and M. G. Sceats Chem. Phys. Lett. 1981 77 455. F. Hirata and P. J. Rossky J. Chem. Phys. 1981 74 6867. l9 G. Jancso and W. A. Van Hook Chem. Rev. 1974 74 689 'O C. G. Swain and R. F. W. Bader Tetrahedron 1960 10 182. F. H. Stillinger and A. Rahman J. Chem. Phys. 1974 60 1545. 22 J. P. Ryckaert G. Ciccotti and H. J. C. Berendsen J. Comput.Phys. 1977 23 327. 23 C. Pangali M. Rao and B. J. Berne J. Chem. Phys. 1979,71 2975.

ISSN:0301-5696    

DOI:10.1039/FS9821700069

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Symp.Chem. SOC., 1982 17 79-9 1 Monte Carlo Computer Simulation Study of the Hydrophobic Effect Potential of Mean Force for [(CH4>JaSat 25 and 50 "C GANESAN MIHALY MEZEI AND DAVIDL. BEVERIDGE RAVISHANKER Department of Chemistry Hunter College of the City University of New York 695 Park Avenue New York New York 10021 U.S.A. Received 1st September 1982 A series of liquid-state computer simulations has been performed on the system [(CH&laq at 25 "C as a function of the separation R between the dissolved CH molecules. The potential func- tions for HzO-H20 CH4-HzO and CH4-CH4 are based on quantum-mechanical calculations of the corresponding intermolecular interactions. The simulation was carried out by the Monte Carlo method augmented with convergence acceleration techniques involving force-bias and the method of preferential sampling.The potential of mean force wss(R)for the interaction of the apolar species in water was determined using umbrella sampling procedures on four windows on the intersolute co- ordinate. Convergence acceleration was found to be necessary for obtaining satisfactory results. Contact and solvent-separated contributions to the statistical state of the system are discussed in terms of the radial distribution functions quasi-component distribution functions for coordination number binding energy and pair interaction energies. Stereographic views of significant structures are presented and contributions of clathrate structures in the solvent are discussed. The results are discussed in the context of the experimentally observed concentration dependence of hydrophobic effects and related theoretical work on wss(R)by Marcelja Mitchell Ninham and Scully Pratt and Chandler and Pangali Rao and Berne.Prospects for quantitative study of the temperature depen- dence of the hydrophobic interaction and other quantitative extensions of the work are discussed. 1. INTRODUCTION The tendency of apolar molecular species to associate spontaneously in aqueous solution is presently recognized as one of the primary organizing principles in struc- tural biochemistry and biology and is widely known as the hydrophobic effect. Prevalent ideas about the hydrophobic effect at the molecular level involve contact interaction of apolar groups stabilized by entropic effects originating in the solvent water.The entropic nature of the process leads to the expectation of an inverse temperature dependence in the hydrophobic association of apolar solutes. Recent experimental data from diverse sources indicate that the nature of hydro- phobic association is possibly more complex than expected and involves competing processes. Direct experimental data on the nature of this problem at the molecular level are difficult if not impossible to obtain. This problem has currently become accessible to theoretical study at a high level of computational rigour by liquid-state computer simulation using both the method of molecular dynamics and the Monte Carlo method. Here additional perspectives on the nature of the hydrophobic effect at the molecular level may be gained.We present herein Monte Carlo simulation results on a prototype system for study of the hydrophobic interaction [(CH4)Jas and discuss our results in the context of other recent theoretical and experimental work on this topic. MC SIMULATION OF THE HYDROPHOBIC EFFECT 2. BACKGROUND The initial physicochemical account of the hydrophobic effect and the implications thereof in structural biochemistry were advanced by Kauzmann.' Experimental evidence for hydrophobic effects are collected together with numerous applications in the monograph by Tanford.' For the brief description that follows we employ the conventional terminology as much as possible and distinguish "hydrophobic hydra- tion" the solvation of apolar species in water from " hydrophobic interaction" the solute-solute association process.The latter effect is yet referred to as " hydrophobic bonding" a term which has inspired a controversy which has only been increased by current research. There is consistent theoretical and experimental evidence that in the hydrophobic hydration of an apolar species the local solution environment of the solute is best described in terms of contributions from clathrate-like cages of water molecules not found in the pure liquid.' Typical experimental indices of this phenomena are the thermodynamic data of Glew and the n.m.r. relaxation studies of Hertz and Ra~3e.~ Computer simulation studies of [CH& at 25 "C by Owicki and Scheragas and Swaminathan et aL6 directly implicate pentagonal dodecahedra1 clathrate structures in the aqueous solution of methane as shown in fig.10 of ref. (6). Hydrogen-bond analysis showed increased structuration in the water surrounding the solvated methane? The ordered water structure is expected to be entropically unfavourable and thus consistent with the sparing solubility of apolar solutes in water. Apolar groups incorporated as functional groups in polar molecules and macromolecules must of course contend with hydrophobic hydration and diverse clathrate structures can be formed to accommodate various sizes and shapes of ~olute.~' The hydrophobic interaction or association of apolar groups in water has been described as a partial reversal of hydrophobic hydration,8 entropically favoured when the solutes approach proximity and intervening clathrate water is expelled into bulk solvent.The currently prevalent textbook-level view of the hydrophobic effect has the apolar species in contact with one another henceforth referred to as contact hydrophobic interaction (CHI). In an extensive review article Franks collected evidence from diverse experimen- tal points of view for an anomalous concentration dependence of observable proper- ties of solutions of effectively hydrophobic solutes very near the limit of infinite dilution. Experimental data on partial molar volumes partial molar enthalpies apparent molar heat capacities and compressibilities and light scattering were among those cited. Clear physicochemical evidence that competing processes were operative in the association of hydrophobic solutes was presented.Although the nature of these processes could not be confirmed Franks discussed the possibilities in terms of contact hydrophobic interaction and a longer-range effect which is stabilized in spite of intervening solvent This latter situation is henceforth referred to as a solvent- separated hydrophobic interaction (SSHI). The formal statistical mechanics of the interaction of apolar species in aqueous solution is described by Ben-Naim lo in terms of a free energy &IHI@) related to the potential of mean force wss(R)or equivalently the radial distribution function gss(R) for solute-solute interactions in aqueous solution. A knowledge of wss(R) as a function of an intersolute coordinate R carries a full description of the association process in the hydrophobic interactions for apolar solutes in water.In an extensive study of the hydrophobic effect based on thermodynamic perturbation theory for liquids Pratt and Chandler studied prototype hydrophobic hydration and hydro- phobic interaction of a hard sphere in water with the available diffraction data incor- G. RAVISHANKER M. MEZEI AND D. L. BEVERIDGE porated for the description of the water-water interactions. They obtained a w,,(R) exhibiting oscillatory behaviour with one minimum corresponding to contact inter- action and another corresponding to a solvent-separated interaction. The well depth of the solvent-separated structure was considerably less than that of the contact minimum but since more configuration space is associated with the solvent-separated form the relative equilibrium constant K = [S W S]aq/[S S]aq was found to be 1.8-4.0.Thus a solvent-separated structure could have a significant statistical weight in the description of an aqueous solution of hydrophobic solutes at infinite dilution. Experimental data on the structure of liquid water at the molecular level have been obtained from diffraction experiments,12 but procedures for aqueous solutions are still under development. Molecular detail in the association process at low concen- trations is not readily accessible to experimental study. Thus computational chem- istry and liquid-state theory have a unique vantage point on this problem.The structure of liquid water was studied by liquid-state computer simulation beginning with work by Rahman and Stillinger l3 using molecular dynamics and by Barker and Watts l4 using the Monte Carlo method. Studies on aqueous solutions of hydro- phobic solutes were later reported by Owicki and Scheraga and Swaminathan et aL6 Early work on the interaction of apolar species in water was reported by Dash-evsky and Sarkiso~'~ and by Marcelja et all6 The initial large-scale computer simulation study of the hydrophobic interaction was the molecular-dynamics calcu- lation by Geiger et aZ.I7on two neon-like Lennard-Jones solutes in ST2 water. The simulation was initiated with the solute molecules in contact. In the course of a 4.5 ps simulation the solutes oscillated for 2.5 ps about the contact structure and moved apart for the remaining period to a solvent-separated structure.Quite recently a large molecular-dynamics study of apolar species in water by Rapaport and Scheraga l8 surprisingly revealed no tendencies towards aggregation. Monte Carlo calculations of hydrophobic interactions in prototype systems were reported first in terms of solvent averaged potential of mean force by Pangali et aZ.19 and by Swaminathan and Beveridge.20 Evidence for solvent-separated structures appeared in both these studies but with larger statistical errors due to limitations in the methodology. Pangali et aL2' extended their studies to the direct calculation of the potential of mean force for Lennard-Jones solutes in ST2water and confirmed the oscillatory behaviour first noted for this problem by Pratt and Chand1er.l' In the present study we have extended simulations based on quantum-mechanical potential functions for intermolecular interaction energies to study the prototype molecular system for hydrophobic interactions [(CHJ2laq.Improved methodologies for achieving convergence in the Monte Carlo simulation are employed. We focus on providing knowledge on the sensitivity of results to choice of potential function the molecular nature of the contact and solvent-separated structure and the temper- ature dependence of the hydrophobic interaction. 3. METHODOLOGY AND CALCULATIONS The theoretical basis for our study is the treatment of the interaction of molecules in condensed phases set forth by Ben-Naim lo for the constant-volume ensemble.Here the work of bringing two solutes from fixed positions at infinite separation to a distance R from one another is given by the Helmholtz free-energy term MC SIMULATION OF THE HYDROPHOBIC EFFECT where Ess(R) is the solute-solute potential energy of interaction and wss(R) is the potential of mean force due to solvent-induced interactions between the two particles. Liquid-state theory relates w,,(R) to the solute-solute radial distribution function gss(R) where k is the Boltzmann constant and Tis the temperature. The quantity w,,(R) is identical to Ben-Naim’s lo index of hydrophobic interaction 8AHI(R). The calculation of w,,(R) for the interaction of apolar solutes in water as described herein proceeds via the calculation of g,,(R) using Monte Carlo computer simulation along the methodological lines established by Patey and Valleau 22 and enhanced by Pangali et a1.” Patey and Valleau recognized that special methodology was required in order fully to sample the intersolute coordinate in a Monte Carlo simulation with only two solutes and devised an umbrella sampling procedure for this purpose in their study of the ion-ion interactions in a dipolar fluid.Here thegss(R) were obtained for small overlapping segments or windows on the R coordinate and are subsequently combined to produce g,,(R) over the range of interest. Pangali et al. adopted this approach in their study of Lennard-Jones solutes in ST2 water refined the methodology and introduced the idea of defining the sampling window by constraining the solute- solute separation using a harmonic restoring potential in the configurational energy expression to be determined by trial and error.The bias introduced by this harmonic restoring function is removed by standard procedures in umbrella sampling. The calculations described herein are intended as a direct extension of the earlier studies from this laboratory on hydrophobic hydration to the problem of hydrophobic interactions. The potential functions used for the evaluation of the water-water contributions to the configurational energy was that of Matsuoka et al.23based on quantum-mechanical calculations. The characteristics of this function used in Monte Carlo simulations for the description of liquid water have been treated in earlier papers from this laboratory and elsewhere; in summary this function produces a gss(R) in agreement with that observed from diffraction experiments and gives most thermo- dynamic variables in reasonable agreement with the experiment considering that the cooperative effects are neglected.The most serious problem with this potential in computer simulations is the inflated value computed for the pressure of the liquid indicating problems in describing the curvature of the potential in the equilibrium region. Methane-water interactions are treated by means of a quantum-mechanical potential function developed for this study by refitting the data base used in earlier work to a functional form with a more suitable long-range behaviour.The details of the determination of this function and related functions are described by Marchese et al.24in a current paper. The equilibrium binding energy in the methane-water potential function is -1.32 kcal mol-I attributed mainly to dipole-induced-dipole interactions. The parameters for the methane-methane interaction are taken from the transferable methane-water p~tential.’~ Monte Carlo computer simulations in these studies were carried out using the Metropolis method augmented with convergence acceleration procedures. Monte Carlo simulations applied to aqueous solutions and by inference for interactions in aqueous solutions are subject to convergence problems above and beyond those encountered in pure liquids due primarily to the loss of a significant statistical factor in the determination of configurationally averaged quantities referenced to the solute.A recent study from this laboratory by Mehrotra et aL2’ demonstrated that this difficulty can be overcome by the joint application of force-bias sampling and G. RAVISHANKER M. MEZEI AND D. L. BEVERIDGE preferential sampling technique in addition to the Metropolis method in computer simulations on aqueous solutions. In the present study all simulations reported involve Metropolis force-bias and preferential sampling procedures. The full details of the calculations follow the description given by Mehrotra et aZ.25 The calculations described herein are aimed at producing gss(R)and wss(R)for the intersolute coordinate in the system [(CH4)JaQ.The system for study consists of two methane molecules and 2 14 water molecules configured under face-centred cubic periodic boundary conditions with a spherical cutoff treatment of the potential. Four points of origin on the intersolute coordinate were chosen defined as 3.9 5.3 6.07 and 6.8 A. Harmonic constraining functions with a force constant of 1.5 kcal 8,-' were used to define the four windows used in the umbrella sampling procedure. The density of the system was computed from the experimental measurements of partial molar volumes of methane and water. Calculations for all four windows were carried out at T = 25 "C and for reasons described below at T = 50 "C for the win- dow centred at 5.3 A.For each window that was studied a total of 3 x lo6 con-figurations were sampled of which the first 1.5 x lo6 configurations were treated as equilibration. All the properties reported are the ensemble averages from the second 1.5 x lo6 configurations. 4. RESULTS AND DISCUSSION The gss(R)for the four windows studied at 25 "Care shown in fig. 1. For window 1 the point of origin was chosen at 3.9 8,to coincide with the solute-solute contact 0.05 0.04 0.03 0.02 0.0 1 3 4 5 6 7 8 RIA FIG.1 ,-Calculated radial distribution function g&) plotted against intersolute separation R for each of the four windows of umbrella sampling for [(CH&laq at 25 ,OC. ,Window 1;0, window 2; ., window 3 ; A,window 4.interaction based on R = 1.92 A estimated from the position of the first peak in gss(R). In this realization the intersolute region from 3.2 to 5.2 8 is sampled. The maximum falls at 3.9 A with a bias in the distribution in favour of longer distances. Window 2 has the point of origin at 5.3 A and was observed to sample a quite broad range of the intersolute coordinate from R = 4.0 A to almost 7 A. The maxi- MC SIMULATION OF THE HYDROPHOBIC EFFECT mum in the probability distribution for this window was found at 6.4 A. The remaining two windows R = 6.07 and 6.8 A,both sample ca. 3.5 A of the intersolute coordinate and find their maxima in the region of 6.2 A. The probability distributions for the four windows were matched on the basis that the points in the overlapping region should coincide and the solute-solute radial distribution function gss(R) was generated.The corresponding potential of mean force w,,(R) was generated using eqn (2) and is plotted in fig. 2. The oscillatory I 01 2345678 RIA FIG.2.-Calculated potential of mean force wss(R)as a function of R,after matching for [(CH&las at 25 "C. behaviour in w,,(R) noted previously by Pratt and Chandler l1 and Pangali et aL2' is clearly evident. Two distinct minima one at R = 3.9 A and the other at R = 6.0 A are found. The former corresponds to the contact hydrophobic interaction and the latter to a solvent-separated structure. The results of Pratt and Chandler (P.C.) and Pangali et al. (P.R.B.) are included in fig.3 for comparison and contrast. Two essential points of difference are noted (a) the positions of the minima in our study occur at slightly shorter intersolute separations and (b)the solvent-separated minima in our study are deeper than the minima for the contact interaction whereas in the previous studies the contact minima were deeper. The difference in position of the minima in wss(R)is due primarily to a difference in the solute-water potential. The P.C. and P.R.B. studies both assume a Lennard-Jones form for the solute-water potential which lacks the granularity expected when the molecules interact with one another. The quantum-mechanical potential used herein contains a more detailed description of this interaction and this is reflected in the positions of the calculated minima in wss(R).The discrepancies in relative depth of the contact and solvent-separated minima in w,,(R) can also be traced back to differences in the potential function. The methane-water binding energy is negligible in the P.C. and P.R.B. studies whereas our function exhibits an equilibrium binding energy of -1.32 kcal mol-'. The description of the water-water interactions varies among the studies as well but since there is such close agreement between P.C. [using the experimental goo(R)] and P.R.B. G. RAVISHANKER M. MEZEI AND D. L. BEVERIDGE n a v 012345678 RIA FIG.3.-Comparison of the calculated wss(R)in this work (-0-) with that computed by Pangali et al. (0)and Pratt and Chandler (-). (using ST2 water which overemphasizes considerably the tetrahedral nature of water interactions) we feel that the origin of this discrepancy probably resides in the different solute-water binding energy.This effect was anticipated by Pratt and Chandler.26 The interpretation of the results is significantly affected by this point. The P.C. and P.R.B. results produce an equilibrium constant of the order of unity indicating sig- nificant contributions from both contact and solvent-separated structures to the statistical state of the system at 25 "C. Our results predict a strong preference for the solvent-separated form. There is at present no means of deciding which result is correct but the sensitivity of this crucial result to the choice of potential functions is 4 5 6 7 RIA FIG.4.-Calculated gss(R)as a function of R for window 2 for [(CH&], at 0, 25 and 0 50 "C.MC SIMULATION OF THE HYDROPHOBIC EFFECT notable. All studies concur in indicating that the solvent-separated structure should be considered seriously in the structural chemistry of the hydrophobic effect. A preliminary study of the temperature dependence of the hydrophobic interaction was carried out by means of a Monte Carlo calculation on [(CH4)Jaqat T = 50 "C. This study was carried out for window 2 only which was observed to span the contact and solvent-separated structures on the intersolute coordinate. The calculated g,,(R) 0 -2 -4 3.8 4.8 5.0 6.8 RIA FIG.5.-Calculated wss(R)plotted against R for window 2 for [(CH,),], at 0,25 and 0 50 "C.and w,,(R) for 50 "C are shown in fig. 4 and 5 respectively. The minimum in w,,(R) corresponding to the contact interaction is no longer present and the solvent-separated structure is preferentially stabilized by the increase in temperature. Thus the inverse temperature dependence of the hydrophobic effect is accounted for by computer simulation and linked to the preferential stabilization of the solvent-separated hydro- phobic interaction. 5. ANALYSIS OF RESULTS The results of the preceding section strongly implicate the solvent-separated structures in the hydrophobic interaction. In this section we inquire into the molecular nature of these structures. Two procedures are employed for this analysis (a) examination of the distribution functions goo(&) and g.H(R) describing the organiz- ation of solvent water with respect to the centre of mass (0)of the SS complex and (b)consideration of stereographic pictures of supermolecular structures from configur- ations that contribute significantly to the statistical state of the system.The g.,(R) for window 1 fig. 6 shows successive peaks at 2.8,4.1 and 5 A. Assignment of these peaks to waters in the first hydration shell follows straightforwardly from a consider-ation of the excluded volume of methane and water molecules as shown in the inset of fig. 6. The distribution function g.H(R) fig. 7 is more complex because of the greater orientational possibilities but the onset of probability at 1 8 is clearly con- sistent with a contact interaction.Structures representative of this type of configur- ation were extracted from the simulation and a typical example is shown in fig. 8. G. RAVISHANKER M. MEZEI AND D. L. BEVERIDGE I FIG.6.-Calculated goo(R) for [(CH,)~laqfor window 1. For window 2 the distribution functiongoo(R) fig. 9 shows a well developed peak at 1 A. Examination of g.H(R) fig. 10 shows dominant contribution from structures in which a single hydrogen atom of a water molecule intervenes between solutes and the peak in goo(R) is clearly consistent with significant contributions from this struc- ture. A structure corresponding to this configuration is shown in fig. 11. Windows 3 and 4 are centred on solvent-separated values of the intersolute co- ordinate. Thego,(&) and g.,(&) for these two windows are very similar in appear- ance and we discuss here only the results from window 4.The goo@) fig. 12 shows a strong peak at R = 0 corresponding to solvent-separated structures in which the 45 36- 27 - - 18 9- I 0 1 2 3 4 5 6 7 8 9 10 RIA FIG.7.-Calculated g0JR) for [(CH4)z]a,for window 1. MC SIMULATION OF THE HYDROPHOBIC EFFECT FIG.8.-Typical structure corresponding to the most probable configuration in window 1. water oxygen atom intervenes and a smaller peak at R= 1 A corresponding to struc- tures in which a water hydrogen intervenes. The g.,(R) for window 4 fig. 13 falls off more gradually than the corresponding plot for window 2 as a consequence of the significant contributions from the structures in which the water oxygen intervenes.A solvent-separated structure extracted from the configuration is shown in fig. 14. In conclusion some comments on the molecular structures in fig. 8 11 and 14 are in order. In viewing these structures keep in mind that no one structure in a simula-tion is necessarily representative of the statistical state of the system and structures arbitrarily chosen may be misleading. The urge to look at structures is nevertheless irresistible and we present them here along with a strong caveat emptor. The structures presented here are given as computer-drawn Dreiding models with all oxygens closer than 3.2 A shown as bonded. Hydrogen atoms are included only for the methanes and the solvent-separated water molecule.Only those water molecules within 6.5 8 of the methane molecules are included which amounts to ca. 70. Extensive networking is found in all three cases displayed. Three- four- and five- a -63 54 -k 45 -3 36- 0 3 I2 27- H H 18- 4 9-I I I I I 1 I I I I 0 12 3 4 5 6 7 8 9 10 RIA FIG.9.-Calculated goo(@ for [(CH2)4]aqfor window 2. G. RAVISHANKER M. MEZEI AND D. L. BEVERIDGE - 150 125 - h5 100- z 0 m?J' 2 75- - 50 - 25 I I I I I I I I I I I. 0 1 2 3 4 5 6 7 8 9 10 RIA FIG.10.-Calculated go@) for [(CH&las for window 2. FIG.11.-Typical a 175 150 125 3 100 0 3 I 75 50 25 0 structure corresponding to most probable configuration in window 2.I I I I I I I I I 1 12 3 4 5 6 7 8 9 10 RIA FIG.12.-Calculated goo(R) for [(CH4)2laq for window 4. 90 MC SIMULATION OF THE HYDROPHOBIC EFFECT 175 150 ,251 n 5 100- 0 l.9' 2 75--50 25 -0 1 2 3 4 5 6 7 8 9 10 RIA FIG.13.-Calculated g.,(R) for [(CH4)2]aq for window 4. coordinate waters are readily seen in all structures as expected. While the statistical indices of the calculations are consistent with the existence of clathrate-like contribu- tions the individual structures do not feature much which can be interpreted as regular clathrate polyhedra although quite good individual pentagons can be found in all structures. The intervening water molecule in the solvent-separated structures in fig.11 and 14 and the interaction of this molecule with the other waters is clearly evident. FIG 14.-Typical structure corresponding to most probable configuration in window 4. This research was supported by the General Medical Sciences Division of the U.S. National Institute of Health (grant GM24149) and a CUNY Faculty Research Award. W. Kauzrnann Adu. Protein Chem. 1959 14 1. C. Tanford The Hydrophobic Effect (Wiley New York 1973). D. N. Glew J. Phys. Chem. 1962 66 605. 'H. G. Hertz and C. Radle Ber. Bunsenges. Phys. Chem. 1973 77,521. J. C. Owicki and H. A. Scheraga J. Am. Chem. Soc. 1977,99 7413. S. Swaminathan S. W. Harrison and D. L. Beveridge J. Am. Chem. Soc. 1978 100 5705. (a) M. Mezei and D. L. Beveridge J. Chem. Phys. 1981 74 622; (6) G.A. Jeffries Ace. Chem Res. 1970 2,344. G. RAVISHANKER M. MEZEI AND D. L. BEVERIDGE G. Nemethy and H. A. Scheraga J. Chem. Phys. 1962 36 3401; G. Nemethy and H. A. Scheraga J. Phys. Chem. 1962,66 1773. F. Franks in Water-A Comprehensive Treatise ed. F. Franks (Plenum Press New York 1975) vol IV p. 1. lo A. Ben-Naim Water and Aqueous Solutions (Plenum Press New York 1974); A. Ben-Naim Hydrophobic Interactions (Plenum Press New York 1980). L. Pratt and D. Chandler J. Chem. Phys. 1977 67 3683. l2 A. H. Narten and H. A. Levy J. Chem. Phys. 1971,52,2263. l3 A. Rahman and F. H. Stillinger,J. Chem. Phys. 1971 55 336. l4 J. A. Barker and R. 0.Watts Chem. Phys. Lett. 1969 3 144. V. G. Dashevsky and G. N. Sarkisov Mol. Phys. 1974,27 1271.l6 S. Marcelja D. J. Mitchell B. W. Ninham and M. J. Scully J. Chem. SOC.,Faraday Trans. 2 1977,73 630. A. Geiger A. Rahman and F. H. Stillinger,J. Chem. Phys. 1979,70,263. l8 D. C. Rapaport and H. A. Scheraga J. Phys. Chem. 1982 86 873. l9 C. S. Pangali M. Rao and B. J. Berne in Computer Modeling ofMatter ed. P. Lykos (American Chemical Society Washington D.C. 1978). 2o S. Swaminathan and D. L. Beveridge J. Am. Chem. Soc. 1979,101 5832. 21 C. S. Pangali M. Rao and B. J. Berne J. Chem. Phys. 1979 71 2975. 22 G. N. Patey and J. P. Valleau J. Chem. Phys. 1975 63 2334. 23 0.Matsuoka E. Clementi and M. Yoshimine J. Chem. Phys. 1976,64 1351. 24 F. T. Marchese P. K. Mehrotra and D. L. Beveridge,J. Phys. Chem. 1982,72 2592. 25 P. K. Mehrotra M. Mezei and D. L. Beveridge J. Chem. Phys. in press. 26 L. Pratt and D. Chandler J. Chem. Phys. 1980,73 3434.

ISSN:0301-5696    

DOI:10.1039/FS9821700079

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

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. The authors also acknowledge gratefully discussions with Professors Fennel1 Evans and Bruce Benson. This is publication LBC 212 from the Laboratory for Biophysical Chemistry Univer- sity of Minnesota Minneapolis Minnesota 55455 U.S.A. D. D. Eley Trans. Faraday SOC. 1939 35,1281. J. A. V. Butler Trans. Faraday Soc. 1937 33 229. H. S. Frank and M. W. Evans J. Chem. Phys. 1945 13 507.J. D. Bernal and R. H. Fowler J. Chem. Phys. 1933 1 515. R. Lumry C. Jolicoeur E. Battistel L. Lemelin and A. Anusiem J. Solution Chem. in press. J. F. Alary M. A. Simard J. Dumont and C. Jolicoeur to be published. ’J. Semishin Gen. Chem. (U.S.S.R.) 1938 8 654. W. C. Schumb C. N. Satterfield and R. L. Wentworth in Hydrogen Peroxide (Reinhold New York 1955). G. Kell J. Chem. Eng. Data 1975 20 97. lo H. S. Frank in Water-a Comprehensive Treatise ed. F. Franks (Plenum New York 1972) vol. 1 chap. 14. H. B. Callen in Thermodynamics (J. Wiley New York London 1960). l2 G. Stey Dissertation (University of Pittsburgh 1967); G. Stey and H. S. Frank Abstracts 167th Meeting ofthe Am. Chem. Soc. (Los Angeles 1974) Phys. 52; C-H. Chen M. J. Wootten and H.Frank Phys. 51. 108 GEOMETRIC RELAXATION IN WATER l3 M. Oguni and C. A. Angell J. Chem. Phys. 1980,73 1948. l4 T. H. Benzinger Nature (London) 1971,229 100; T. H. Benzinger in Adolescent Nutrition and Growth ed. F. P. Heald (Meredith New York 1969) chap. 14. R. Lumry and H. S. Frank Proc. 6th Znt. Biophys. Congr. (1978) vol. VII p. 554. l6 R. Lumry in Bioenergetics and Thermodynamics; Model Systems ed. A. Braibanti (D. Reidel Dordrecht 1980) p. 405. l7 K. Pitzer and L. Brewer in Thermodynamics (McGraw-Hill London ,1961) table A7-1 p. 671. T. H. Benzinger and C. Hammer in Curr. Top. Cellular Regulation 1981 18 475. l9 A. Ben-Nairn J. Phys. Chem, 1968,72 2998. E. Wilhelm and R. Battino Chem. Rev. 1973 73 1 ; E. Wilhelm R. Battino and R.Wilcock, ’O Chem.Rev. 1977 77 219. E. Chang N. Gocken and T. Poston J. Phys. 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Danninger and G. Zundel J. Chem. Phys. 1981 74 2769. 45 H. S. Frank and F. Franks J. Chem. Phys. 1968 48 4746. 46 A. Ben-Naim J. Phys. Chem. 1965,69,3240; Water and Aqueous Solutions (Plenum Press New York 1974) p.309. 47 For recent references see L. R. Pratt and D. Chandler J. Chem. Phys. 1977 66 147; F. H. Stillinger and A. Rahman J. Chem. Phys. 1974 60 1545; H. E. Stanley and T. Teixeira J. Chem. Phys. 1980,73,3404; F. Hirata and P. Rossky J. Chem. Phys. 1981,74,6867. 48 A. Geiger A. Rahman and F. H. Stillinger J. Chem. Phys. 1979,70 263; L. R. Pratt and D. Chandler J. Chem. Phys. 1977,78 3863; S. Goldman J. Chem. Phys. 1981 75,4064; D. C. Rapaport and H. A. Scheraga J. Phys. Chem. 1982 86 873. 49 H. S. Frank Proc. 1st Int. Symp. Water Desalination (U.S. Department of the Interior. Office of the Saline Water Washington D.C. 1967) vol. 1 p. 292.

ISSN:0301-5696    

DOI:10.1039/FS9821700093

出版商:RSC    

年代:1982

数据来源: RSC