摘要:

GENERAL DISCUSSIONMr. G. Mason (University of Bristol) said: I should like to report some workon random sphere packings using a computer to generate the co-ordinates of thesphere centres. This work is directed more towards a model of porous materialsthan to the problem of liquid structure, but the results are of interest in both fields.The method of generating the packing was first developed in two dimensions, packingcircles on a plane, and a short film has been made to illustrate this.* Three-dimensional packings have also been obtained, although these are as yet only of 100and 200 spheres.I20I008 8 0 g 1 8 60ru09c8 5 4 02 0 rr1o ! -+--=I I I I I0.3 0 - 4 0 . 5 0- 6 0*7 0- 8packing densityFIG. 1 .-Tetrahedron densities.bulk packing density = 0-63(4) ; number of spheres in packing = 200 ; total number of tetra-hedra = 727Initially a number of points are generated at random in a " box " in space.Thesepoints are then considered to be small spheres of a chosen radius. If two spheresoverlap, they are moved apart along the line of centres until they are just touching.The computer continues moving spheres apart until none overlap. The spheres arethen increased in size by a chosen increment and the process repeated. Spheres arefree to move out of the " box " during the process. The method may be regardedas a means of creating a random distribution of spheres in space, and bringing themtogether toward a point in the centre of the packing as if under the influence of a radialgravitational field.Packings of even these small numbers of spheres reach a limitingpacking density of 0.63-0.64 which is close to the density of experimentally produced* This film was shown during the discussion.75G. D. Scott, Nature, 1960, 188,90876 GENERAL DISCUSSIONpackings. The value cannot yet be determined precisely because of the small numberof spheres in the packings.The packings have been analyzed in terms of tetrahedral sub-units rather thanVoronoi polyhedra. There are several reasons for choosing tetrahedral sub-units.The analysis is simpler; there are more tetrahedra than Voronoi polyhedra in apacking of given size, so that the results are statistically more significant ; and it iseasier to correct for edge effects. Radial distribution functions have been calculatedover a wide range of packing density and detailed analyses of the tetrahedra in thedenser packings obtained.The packing density of each tetrahedron in one particularpacking was determined and fig. 1 is a histogram of the results. As would be expected,the histogram is spread over a wider range of packing density than the polyhedraof Bernal and Finney. It is hoped with more machine time to pack as many as 1000spheres and to vary their " hardness ".The present method has advantages over that of Bernal and his co-workers. Thepacking density may be varied in a more realistic manner than by the random creationof discrete holes, while the use in effect of a radial confining force avoids the asymmetrywhich may result from packing real macroscopic spheres under the influence of gravity.Mr.R. H. Beresford (University of Technology, Loughborough) said : Some of ourwork may contribute to the development of a statistical geometry of liquids and tothe study of Voronoi polyhedra. The work is concerned with the structure of randomheaps of random-sized hard spheres. Consequently it has the Bernal liquid modelas a limiting case ; it may allow a generalization by allowing an appropriate variationin size of molecules, with a resulting spread in closest approach distance. The basicapproach is to consider the Delauney graph (lines joining the centres of touching ornearly touching spheres) which includes the shortest total length of lines betweennon-touching spheres, and which still divides space into tetrahedral pieces.Lines__---(I XFIG. 1.corresponding to contact points are " edges ", and the others are "diagonals".Each edge represents a constraint-that the sphere centre separation equals the sumof the radii-and so the number of edges can be related to the number of degrees offreedom of the system. An attempt has been made to overcome boundary effects,and to tie up with crystal structures, by considering a heap as the limiting case ofa repeating primitive lattice. As the number of random spheres in this latticeapproaches a completely random heap.A 3-dimensional repeating lattice (fig. 1) can be divided into 6 tetrahedra per cellby 7 lines per cell (12 cell edges, each shared by 4 cells ; 6 face diagonals, each sharedby 2 cells; one body diagonal.The other two body diagonals are redundant).There is one lattice point per cell, and if a sphere is centred on this point, the numberof spheres per cell (= m) is 1. This system has 6 degrees of freedom, for if the sphereradius is taken as unity, the X vector has a 1 degree of freedom, the Y vector 2 andthe Z vector 3. Any additional spheres in this lattice are either " random "-witGENERAL DISCUSSION 77centres falling within an existing tetrahedron, or " ordered ", falling on a tetrahedronboundary. Ignoring the ordered case, an additional random sphere can be joinedby four lines to the four vertices of its surrounding tetrahedron (fig. 2). This addsFIG. 2.3 tetrahedra to the system by creating 4 new ones whilst destroying the original one.Each sphere will have (3 +a) degrees of freedom, where the 3 corresponds to the threeco-ordinates, and a is the number of degrees of freedom associated with the particleshape.If monosized spheres are employed a = 0. If any size of sphere is permiss-ible, a = 1. Then for a total of m spheres :no. of tetrahedra = T = 3m+3no. of lines = L = 4m+3no. of degrees of freedom = Ndf = 6+(m+ 1)(3+a)(The constants are obtained by reference to the case where m = 1.)This analysis gives the minimum number of lines needed to divide the cell intotetrahedra. It appears likely that the shortest set of lines can be obtained by anexchange system which replaces one long line by one or more shorter ones. Forinstance, consider two spheres which have n equatorial neighbours, and are joined bya " polar " line (fig.3). The resulting n tetrahedra can be replaced by (2n-4)~ _...... i I I ,_.,,,.......!FIG 3 . 4 1 2 = 5), . . . equatorial ; - . - . - . polar.tetrahedra if (n-3) equatorial lines replace the polar line. This increases both thenumber of tetrahedra and the number of lines by (n - 4). The original constructionensures that n >47 and a later analysis shows n+4+ as m-, 00. The net effect thereforeappears to be an increase in the number of lines. Let the shortest set of lines beL'>L = 4m-1-3; as EGN,,,D = L'-E>4m+3-6-(m-l)(3+a).and a large random assembly of equal spheres must have an average of at least onediagonal per sphere. For m = 1, this reduces to E<6, D 2 1 ; and the equalitieshold for crystalline close packing, which cannot have more than 12(= 2E) contactsper sphere.:.D>m-a(m- 1 ) = ( 1 -a)m+a78 GENERAL DISCUSSIONConsideration of a bounded cluster of spheres gives the approximation :L' 4m-6(m*- ; D (1 --ct)m--6(m*-- l)2.As both approaches giveas rn--+co,limit (D/m)>l-alimit (L'lm) 4FIG. 4.-bLATIVE FREQUENCIES OF TETRAHEDRAL TYPESstructures. . + ..face types .Dfrequency 2764-number012a20303a3c4a40562764-face nameoooo001 101 1211111122022211131223222222333333t,2 >964-frequency(L - 0)6L66(L-0)50L612(~-0)40*L63(L- 0 ) 4 PL612(~-00)303L64(L-0)303L64(L- 0)303L612(~-0)204L63(L- 0 ) 2 0 4L66(L- 0)D50 6L6-p.3 ; >*Y164-frequency ifL=4, D = 1 .7294096-145840969724096--24340963244096--108409610840961084096---27409618409614096--_IGENERAL DISCUSSION 79it would appear to be a useful hypothesis that a large random close-packed assemblyof equal spheres has 3 contacts and one near contact per sphere. This only appliesto spheres with an infinite range of sizes, Probably for equal spheres, L’/m--+7,0+4, the chances are 4/7 and 3/7, and there are tetrahedra round each line.Asecond hypothesis is that the elements of the Delaunay graph of such an assemblyhave a chance 4 of being a diagonal and $ of being an edge.This allows estimatesof the frequency of occurrence of closed tetrahedra in such an assembly, togetherwith the 10 other types of tetrahedra which can have two types of edge. These aregiven in fig. 4. Further statistical results should arise from the consideration of thesurface patterns on each sphere (fig. 4). One such result is that there are 4+ tetrahedraround each edge. This arises because the average number of points on each sphere( = 2L) is P = 8 and so the average number of triangles is T,. = 2P-4 = 12. Thusthere are 36 apices round 8 points. These correspond physically to tetrahedraround lines, and so the average must be 36/8 = 43 tetrahedra round each line.Edge.- _ _ _ - -DiagonalFIG. 5.Although there is a large number of possible surface patterns, the frequency ofoccurrence of those with a high density of diagonals must be low.Also the numberof possible patterns with a high density of edges is reduced by the geometrical require-ments for monosize spheres, and so it is hoped that enumeration of the likely patternswill be of value. The relationship of these patterns to Voronoi polyhedra has yetto be investigated. The number of redundant diagonals adjacent to an includeddiagonal can be evaluated for each section of a surface pattern ; the total number ofdiagonals and edges will be related to the number of faces on the Voronoi polyhedron.Adjacent Voronoi polyhedra will result in mirror image patterns round the line joiningtwo spheres, and will also set conditions for the surrounding polyhedra.This maybe relevant to the study of packing density. Finally, it is hoped that the cases whereD > m will shed light on less dense assemblies, and that the total length of diagonalswill be related to the separation of spheres, as discussed by Kohler and Springer(this Discussion).Prof. J. Walkley (Simon Fraser University) and Dr. I. H. Hillier (Universityof Manchester) said : Many of the problems associated with the prediction of thethermodynamic properties of a system at fluid densities arises from the conceptualdifficulty of describing a fluid. Bernal’s work lays particular emphasis on thestructural identity in a liquid. The concept of a “ coherent structure ” existing inthe high density fluid region agrees well with the molecular dynamics studies of hardsphere systems.2 We have made computer studies at the other end of the liquiddensity range where it might be expected that the coherent structure gives way to acompletely random structure.J.D. Bernal, Nature, 1960, 185, 68.B. J. Alder and T. E. Wainwright, J. Chem. Physics, 1960, 33, 143980 GENERAL DISCUSSIONThe generation of random configurations of hard-sphere molecules at a liquiddensity presents its own problems. By adding spheres to a “ box ” in a consecutivemanner in random positions (as simulated by the choice of random coordinates on acomputer) leads to maximum densities that are unexpectedly low. Any processinvolving the “ swelling ” of spheres already in the box at random positions has thedisadvantage that high densities can only be achieved at the expense of having only afew sample spheres in the box.A method of generating random configurations atthe required density is as follows.TABLE DISTRIBUTION OF FINAL PACKING DENSITY ; Np = 32, Ni = 80 ; PERIODICBOUNDARY CONDITIONnumber of boxes218507044142number of remaining Nj13141516171819The box is packed with Ni spheres by the assignment of random, x, y, z coordinatesto each molecule, irrespective of any overlap of any two or more spheres. The initialpacking number ATi may be greater or smaller than N,, where Np is the maximumnumber of spheres able to be placed in the unit cube ‘‘ box ” on a primitive cubiclattice. This defines the hard-sphere diameter.The coordinates of the centres ofeach sphere allows the scalar distance between centres to be found and by comparison26 c20 6 0 100 140 I00NiFIG. 1 .-Peak value of theNj distribution for Np = 32 ;curveA, without periodicboundaryconditions;curve B, with periodic boundary conditions.to the molecular diameter the number of spheres Nk which overlap the M,th spheremay be determined. Spheres are now removed from the box such that the sphereMq which incurs the greatest number of overlaps is removed. If spheres M,, M,and Mt each have the same (maximum) number of overlaps then a random choicebetween these is made to determine the one to be rejected. A re-evaluation of Nkfor each of the remaining (Ni- 1) spheres is now made and again the one incurring thegreatest overlap is removed. This procedure is repeated until there is no overlapbetween any two of the remaining N, spheres in the box.This process was carried out for Ni as small as 5 to values as large as 300 for GENERAL DISCUSSION 81typical box where Np = 64.In general, any ‘‘ run ”, i.e., the generation of a seriesof randomly packed boxes from a given Ni value, was continued until 200 sampleboxes were obtained. The distribution of the final N j population for any initial Nivalues was always narrow. A typical result for a ‘‘ box ” for which Np = 32 andfor Ni = 80 is given in table 1.In fig. 1 the peak value of the N j distribution for any given Ni value is plottedagainst Ni. Curves are given for a “ box ” with a torroidal (periodic) boundarycondition and for a ‘‘ box ” without such a condition.In each case the randompacking density approaches a maximum value, independent, of ATi. For a box size,Np = 32, this limiting random density is N j = 18 if periodic boundary conditionsare imposed and N j = 22 is they are not.The tentative result is that the random distribution of spheres can only occur to alimiting density. Bernal in an earlier paper comments upon the liquid-gas transitionand the implication of a A-point in the constant-pressure heat-capacity curve. Thedensity limit observed in the present study agrees well with the limit suggested byBernal for the break up of the liquid “ coherent structure ” and the inherent implica-tion in the heat-capacity curve.Mr. R. Collins (University of Salford) said: Bernal and King (I) and Bernal andFinney (11) use the term “polyhedron” in two different contexts, which perhapsought to be distinguished.Polyhedron in I1 means Voronoi polyhedron (VP),but polyhedron in I does not. The lattice of polyhedra in I is a subset of the latticeof tetrahedra topologically inverse to the lattice of VP’s. The definition of neighbourin I1 is much more precise than the J2 criterion in I. If the I1 definition of neighbouris adopted, then the average number of neighbours of a given atom in a real liquidincreases with decreasing density since it is 14 for (slightly perturbed) close-packing,14.28 at the intermediate density described in I1 and for vanishingly small densitiesapproaches the random distribution (perfect gas) figure of 2+48n2/35 = 15.54.lThe basic problems in using a realistic ‘‘ soft ” potential #(Z) appropriate to aninteratomic distance I are to calculate the energy and the entropy of any irregularconfiguration.Both are easier to express in terms of the lattice of tetrahedra inverseto that of the Voronoi polyhedra rather than the VP lattice itself. In this connectionthe work described by Beresford also breaks the structure down into tetrahedra only.In addition to the entropy contribution arising from the local atomic co-ordinationnumbers 8, discussed by Everett, there is also the contribution arising from theprobability function $(I) of the neighbour distances 1. We have estimated thisusing information theory arguments, and even if the entropy contribution from the8, is completely neglected, there results an equation of state for a general #(I) whichgives reasonable qualitative agreement with experiment.2 Contrary to a widely heldview, the geometric theory is capable of yielding thermodynamic results for a realistic4(0.The histogram of what are effectively atomic co-ordination numbers 6, in fig.6of I1 is valuable since it provides a starting point for evaluating the correspondingentropy contribution, which has so far proved an intractable analytic problem.Real progress would become possible in this direction if fig. 6 could be repeated forvarious densities, and extended to cover configuration generated by a soft (e.g.,Lennard-Jones) potential. I would expect that the other statistics (such as illustratedin fig.7) will prove of much less thermodynamic importance, although of sometheoretical interest.J. L. Meijering, Philips Res. Rep., 1953, 8, 270.R. Collins, Proc. Physic. Soc., 1965, 86, 199. D. C. S. Allison and R. Collins, to be published82 GENERAL DISCUSSIONIn an attempt to develop a realistic physical formalism, I would suggest that inthe first instance it is easier to consider initially a two-dimensional liquid. Here thetopology is much easier ; the Voronoi polygons have an inverse lattice which consistsof triangles only, and the mean co-ordination number lV is a strict topological constantequal to 6 at all densities. Also, one can see directly how the long-range orderappears in the crystalline state on freezing.If an adequate mathematical formalismcould be developed on this basis, it would not have the defect, inherent to the lattice-gas model of a liquid, that a long-range ordered lattice of sites is used to describe anessentially disordered physical state. If the problem cannot be solved in two-dimensions, there seems little hope of a three-dimensional solution.Prof. D. H. Everett (University of Bristol) said : Although work on the randompacking of spheres gives a valuable insight into the structural properties of liquids,it is not immediately apparent how the results of such studies can be applied to thecalculation of thermodynamic properties. In introducing these papers, Finneyhas mentioned that the results can be used successfully to calculate the energy of aliquid and hence the heat of fusion: but it is also necessary to be able to use themodel to calculate the entropy. An approximate approach to this problem seemspossible along the following lines.Simple cell theories of liquids predict too low an entropy for the liquid state.One feature of these theories which makes a major contribution to this discrepancyis the assumption that each molecule can be regarded as moving in the mean field ofits neighbours (the average potential model): the energy of a captive molecule at thecentre of its cell is the same for all cells. A more realistic model-which might becalled the lzeterogevleotis cell model-would taken account of the fact that at anyinstant a molecule finds itself in a field determined by a particular arrangement ofneighbours.We might therefore regard the liquid as divided into cells, each of whichdetermines a characteristic potential energy controlling the motion of the captivemolecule. The depth of the potential energy minimum in a cell will differ from cellto cell, and through time a given molecule will sample cells of all kinds with a pro-bability proportional to the frequency with which each type of cell occurs in the liquid.We could use this consideration to justify an intuitive evaluation of the configurationalentropy. A rigorous calculation of the configurational entropy arising from thedifferent arrangements of these heterogeneous cells would, however, presumably needa detailed knowledge of the number of ways of dividing space into cells.Studies ofthe statistical geometry of sphere packing, such as those described by Bernal and hisco-workers, are, however, based on an analysis of only one (or a very few) specificrandom packings. Thus, only one particular division of space is examined, fromwhich, when some arbitrary method of distinguishing cells of different kinds hasbeen chosen, enables us to evaluate the number of cells of different kinds in therandom packing. If a large enough number of spheres is considered, then the dis-tribution function derived for one particular random packing will approximate tothe mean distribution function derived from a large number of different packings.It seems reasonable to suppose that an approximation to the configurational entropycan be derived by equating the number of ways of dividing space to the number ofways of arranging the cells of a given packing among themselves.It may be arguedthat not all arrangements will completely fill space. The error so introduced willdepend on the precision with which we define a " kind " of cell. If we were to specifythe precise shape of each cell, then the error would be larger than if we specified, forR. Collins, Proc. Battelle Coll. Phase Changes in Metals (Geneva-Villars, March 1966), to bepublished by McGraw-HillGENERAL DISCUSSION 83example, the potential energy of a molecule at the centre of the cell : minor changesin cell shape needed to fit a cell into its surroundings can be made without seriouslyaffecting the energy.For the purposes of a preliminary estimate of the configurationalentropy corresponding to a random packing of spheres, we may classify cells accordingto the number of nearest neighbours which define the cell without specifying theprecise geometry of the cell. If a fraction ei of the cells are formed from i nearestneighbours, then the configurational entropy is simply - Rx6, In Bi. The intuitiveargument mentioned above would lead to the same result. The histogram given byBernal and King for close-packed spheres leads to a value of 2.77 cal deg.-l mole-l.This is certainly of the correct order of magnitude for the configurational entropy ofmelting of a substance with freely rotating spherical molecules (e.g., CH4, 2.5;CC14, 2.4).Other contributions to the entropy of fusion will arise from the looseningof the vibrational degrees of freedom in cells with smaller numbers of nearest neigh-bours; and from the fact that Bernal’s analysis refers to spheres in closest randompacking, to which liquids will tend only at the lowest temperatures. At the meltingpoint of most substances the liquid will probably have a significantly less densestructure so that we shall expect the figure of 2.77 cal deg.-l mole-1 to represent aminimum value for the entropy of fusion : in fact, only metals have substantiallylower values, while the noble gases (e.g., Ar, 3.4) have rather higher values.It will be of considerable interest to see how the configurational entropy of randomsphere packs varies with the density.However, we should not expect the methodof reducing the density by random removal of single spheres used by Bernal and Kingto be realistic : random sphere packs containing discrete holes of “ molecular ”size are highly improbable distributions and calculations based on them are unlikelyto be reliable. The method employed by Mason in which a continuous series ofrandom packings of varying density is generated should (provided that surface effectscan be eliminated) lead to much more realistic values.To carry the theory further it would be necessary to show how the density variedwith temperature : this is clearly a more complex problem and is not easily solublewithout a much more detailed analysis requiring the introduction of an assumedintermolecular force law between pairs of spheres.The real value of Bernal’s modelis that it brings out clearly what is implied by the term randomness as applied to thestructure of liquids, and emphasizes the fundamental geometrical distinction betweensolids and liquids.Dr. J. Finney, Birkbeck College, London, (communicated). The advantages of atetrahedral description of a random packing have been stressed by Beresford, Collinsand Mason. Beresford and Mason are particularly interested in porosity and hencethis approach is of greater use than polyhedra, while Collins’ approach via neighbourdistances implies the convenience of a tetrahedral lattice. We are interested in thegeometry of the array and the packing problem itself and therefore the Voronoipolyhedron is the obvious unit as it defines completely unambiguously a region ofspace associated with one point only.The polyhedron shapes may be complex,but they do give us information which could be used to elucidate the thermodynamics.For instance, Everett’s use of the co-ordination polyhedra to estimate the con-figurational entropy shows an approach which could yield fruitful results, althoughuse of the Voronoi geometrical co-ordination might be more realistic. To justifysuch an approach to configurational entropy, we must establish a relationship betweenthe particular histogram (here the coordination number) and the “ site energy ” of acentre. Further work on this has shown the site energy to be effectively unrelatedto geometrical coordination, but suggests a remarkably high correlation of 0.83-0.9084 GENERAL DISCUSSIONbetween the Voronoi volume and the centre energy.The physical reason for this isobscure-it may be connected to the packing restrictions-but the existence of sucha high correlation shows promise for evaluating an improved approximation to theentropy and facilitates the use of a large number of small samples to this end, whererealistic energy calculations could not normally be made.The general problem of calculating the entropy of a random array is fraught withuncertainties. The approach of Collins and Allison via information theory entailsthe difficulty of sorting out what information is relevant, and how much is needed togive an unambiguous minimum description of the system.In this connection the useof either coordination number or geometric neighbour histograms is an insufficientdescription, for the information contained about each site is incomplete in energyterms.Regarding the use of soft potentials suggested by Collins, we have in fact carriedout analyses on some high density Monte Carlo runs kindly made available to us byDr. Singer and Dr. Mcdonald of Royal Holloway College, in an attempt to comparea random packing with the structure of a “ liquid like ” arrangement generated in thecomputer. Initial results show a remarkable similarity between fig. 6 for the hardsphere model and the total histogram for several configurations at different timesin the same Monte Carlo run.Moreover, there are interesting variations betweenthe different configurations, suggesting complete inadequacy of single samples ofabout 100 centres for approaching the problem via geometrical neighbours.As for the two-dimensional liquid, we think it is essentially different in kind fromthe three-dimensional system. For example, = 6 exactly in two dimensions, whileiV is a function of density in three; moreover, the generation programme of Masoncrystallizes in two dimensions but not in three, suggesting differences in the basicnatures of the two problems.Beresford’s attempts to develop a statistical geometry in terms of a tetrahedraldescription are interesting, but entail an ambiguity in the choice of “ diagonals ”,or near contacts.This ambiguity could be removed simply by defining tetrahedradirectly from the Voronoi set-i.e. by choosing the inter-centre links to be theVoronoi face normals. With each Voronoi vertex are associated four centres, thusdefining uniquely a set of tetrahedra completely filling space. After removing thisambiguity, it would be interesting to see what this approach to the statistical geometryyielded.Mason’s random model generation programme could be a step forward inproducing data for analysis on a large scale as similar programmes constructedpreviously have not converged to a limiting density. We cannot see a theoreticalargument to show that Mason’s programme would converge. Moreover, it is notcertain what the characteristics of the packing are, how they depend upon the exactmechanisms written into the programme, and how they compare with physicallybuilt models.For example, the final result may depend on exactly how we removean overlap, and what system we use for picking out centres during the search foroverlaps. The construction of a random model under a central gravitational fieldis an improvement over the physical models, but out present inability to describethe essential characteristics of a random model make it difficult to compare explicitlythe results of the two modes of construction.One of the features of such a generating programme is that it gives random packingwith a large density variation up to the maximum of a random close packing andcould give the data necessary for investigating thermodynamic properties with tempera-ture variation.There are serious difficulties, however, in the long machine timesnecessary even for 200 spheres, and the rate of increase of time with sample size seemGENERAL DISCUSSION 85to prohibit the generation of much larger ones. Thus physical models still have animportant place. However, if the thermodynamic significance of the Voronoivolume can be firmly established, a large number of smaller packings will be invaluable.Prof. C. Domb (King’s Cottege, London) said: I wish to speak about the behaviourof the specific heat of a fluid in the critical region, and more particularly about com-paring experimental results with the predictions of the lattice gas model. Followingthe accurate experimental work of Voronel and his collaborators, various analyseswere attempted to establish the experimental values of the critical indices for thisspecific heat.There were differences of opinion as to whether the data could best befitted by a logarithm or a power law.2 Since the data cover a limited range withappreciable experimental errors, a formula of the typeinvolving several disposable parameters was used, and it is not surprising that the datacould be fitted by a range of values of a. (The in the formula refers to the regionjust above and just below the critical point, and the value a = 0 corresponds to alogarithm.) In fact, for a theoretical model there are no disposable parameters,and to test the validity of the model, its predictions should be compared directly withexperimental results.This has not been possible previously because of the lack ofsufficiently precise theoretical information on the three-dimensional lattice gas model.A. V. Voronel, V. G. Gorbunova, Yu. R. Chashkin and V. V. Schekochikhina, Soviet PhysicsJETP, 1966,23, 597. A. V. Voronel, Yu. R. Chashkin, V. A. Popov and V. G. Simkin, SovietPhysics JETP, 1964, 18, 568. M. I. Bagatskii, A. V. Voronel and V. G. Gusak, Soviet PhysicsJETP, 1963,16, 517.M. E. Fisher, Physic. Rev. 1964, 126, 159986 GENERAL DISCUSSIONRecent calculations by members of the theoretical research group at King’s Collegehowever, greatly improved the accuracy of the theoretical calculations.and his collaborators havesucceeded in adding four new terms to the high temperature series expansion for thespecific heat.This means, that series are available as far as l/TI3. (The terml/TI4 should be available shortly.) The result of plotting the ratio of successivecoefficients as a function of 1/N is shown in fig. 1 ; the upper points refer to thesusceptibility and the lower points to the specific heat. The critical index is deter-mined by the limiting slope of this ratio for large N and the evidence is convincingthat the specific heat index is &. It is then possible with so many exact terms andBy a remarkable co-ordination of techniques, Sykes654e --- 832Io_ - 5 - 4 - 3 - 2 - I 0log,, (1 -Tc/T)FIG. 2.a limiting asymptotic form to calculate the specific heat accurately, and the result ofsuch calculations for a number of three-dimensional lattices is presented in fig.2.It will be seen that the specific heat depends little on crystal structure. At lowtemperatures, the position is much less satisfactory since the series behave less regularly.However, Gaunt has examined other critical indices and has thereby producedindirect evidence that the critical index has the same value of $. For the diamondlattice, the low-temperature series are all positive, and Gaunt has made an estimateof the specific heat.When we compare with experiment, we follow the approach of F i ~ h e r , ~ remember-ing at the same time that a lattice gas is a crude model and we should not expect toomuch. In fact, we wish to determine how adequate it is in accounting for criticalM.F. Sykes, J. L. Martin and D. L. Hunter, Proc. Physic. SOC., in press.C. Domb and M. F. Sykes, J. Math. Physics, 1961, 2, 63.D. S. Gaunt, Proc. Physic. Soc., in press.M. E. Fisher, Physic. Rev. 1964, 126, 1599GENERAL DISCUSSION 87behaviour. Fig. 3 (prepared by D. L. Hunter) shows such a comparison; it willbe seen that, when the critical point is closely approached, the experimental resultsshow small but definite deviations from the theoretical calculations. The low-temperature comparison is shown in Fig. 4. Gaunt points out that in the range shownv-5 - 4 - 3 - 2 - Ilog,o (1 - TdT)FIG. 3.Points : Date for N2 (Voronel et al.)Curves : king model (a : F.C.C. ; b : S.C.)the theoretical curve could easily be mistaken for a logarithm since it is almost linearon the logarithmic scale.The disagreement between experiment and theory is moremarked than on the high-temperature side; this was already noted by Fisher whoregarded it as a serious defect of the model. We may conclude that, although theM. E. Fisher, Physic. Rev. 1964, 126, 169988 GENERAL DISCUSSIONlattice gas model has considerable success in accounting for critical behaviour, thereis room for improvement which could perhaps be obtained by relaxing the rigidityof the model and allowing holes of different shapes and sizes.Dr. B. L. Smith (University of Sussex) (communicated): In connection with thecritical point properties of simple fluids, I report the results of some preliminarymeasurements of the refractive index, surface tension and density of xenon in thisregion.The surface tension was measured in the temperature range 189 to 286°Kby a capillary rise method. The results may be represented by y = yo(l -T/289.74)*with yo = 54.6f0.1 dyne cm-I and p = 1.287+0.017. According to the law ofcorresponding states, p should be a universal constant ( N 1.22). Previous resultsfor simple molecules 2 p indicate that for argon p : 1.28, and for nitrogen, p = 1.24.It seems unlikely that the lack of sphericity of the nitrogen molecule (acentric factorl ~ ) = 0.04) is sufficient to account for the difference between the value ofp for nitrogenand those for argon and xenon. A more likely explanation would appear to be thatthe (pz - p,) data used to compute the surface tension are unreliable, and that moreaccurate values would result in better agreement.It is noted thatp N 1.28 is in excellentagreement with a calculation by Widom for a lattice gas model, based on a reformul-ated van der Waals, Cahn-Hilliard theory of surface tension.Refractive index measurements were carried out on xenon liquid and vapour incoexistence and also on fluid xenon at temperatures above the critical point. Studies,e.g., by Abbiss et aZ.,6 have suggested that an anomalously large deviation from theLorentz-Lorenz function might occur in this region, much greater than that predictedby * The results of over 500 measurements lead us to the conclusion thatthe Lorentz-Lorenz function (n2 - l)/(n +2)p remains constant to within f 3 %over the whole range studied (0~002-0~024mole~m-~) and that at the critical pointn, = 1.1383fO-0008, L.L. = 10.5f0.1 cm3 mole-l. The variation may be muchsmaller, since most of the possible error in the Lorentz-Lorenz function arises fromuncertainty in density.The lack of accurate density data, in particular, reliable values for (pz - p,), haslead us to develop a direct experimental method for obtaining this information.According to Guggenheim,l the difference in density between a liquid in coexistencewith its vapour is given by pz - p , = A( 1 - T/T,)P, where fl N 0.33. The results of ourpreliminary observations suggest that /3 = 0.343+0*010. This agrees well with thevalue = 0.345f0.015 obtained by Fisher lo from an analysis of the results ofWeinberger and Schneider,ll and also with /3 = 0.341 obtained from our refractiveindex measurements by assuming that the Lorentz-Lorenz function is independent ofdensity, i.e.,[(n2 - l)/(n2 + 2)p]z - [(n2 - l)/(n2 + 2)pJ, = K( 1 - T/Tc)tE. A. Guggenheim, J. Chem. Physics, 1945,13,253.F. B. Sprow and J. M. Prausitz, Trans. Faraday Soc., 1966, 62, 1097.D. Stansfield, Proc. Physic. Soc. A, 1958,72,854.G. N. Lewis, M. Randall, K. S. Pitzer and L. Brewer, Thermodynamics (McGraw-Hill, NewYork, 1961).B. Widom, J. Chem. Physics, 1965,43, 3892.C. P. Abiss, C. M. Knobler, R. K. Teague and C. J. Pings, J. Chem. Physics, 1965,42,4145.L. S . Taylor, J. Math. Physics, 1963, 4, 824.S. Y. Larsen, R. D. Mountain and R. Zwanzig, J. Chem. Physics, 1965, 42, 2187.B. L. Smith, J. Sci. Instr., 1966, 43, 958.lo M. E. Fisher, J. Math. Physics, 1964,5,944.l1 M. A. Weinberger and W. G. Schneider, Can. J. Chem., 1952,30,422

ISSN:0366-9033    

DOI:10.1039/DF9674300075

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

Structure of LiquidsPart 7.-Determination of Intermolecular Potential Functions and CorrelationFunctions in Fluid Argon by X-Ray Diffraction Techniques *BY C. J. PINGSDivision of Chemistry and Chemical EngineeringCalifornia Institute of Technology, Pasadena, CaliforniaReceived 16th January, 1967Methods of obtaining the radial distribution function and the direct correlation function forsimple liquids from X-ray diffraction data are reviewed. A new theoretical analysis demonstratesthat certain of the cluster integrals appearing in the density expansion of the radial distributionfunction can be evaluated directly from Fourier transforms of appropriate powers of the X-rayscattering function. These results are used to develop expressions which permit determination ofthe intermolecular potential function from moderately dense fluids using essentially only X-raydiffraction data.Non-additivity effects may also be treated. Experimental data available areinadequate for conclusive determination of the potential function ; however, experimental resultsfrom one state of argon give plausible values for the lowest-order cluster integral.The direct determination of the radial distribution function of a fluid from X-raydiffraction data is generally well understood. This paper will briefly review thepertinent equations relating the structure of the fluid to the X-ray measurements,and will summarize some current results for argon from the author's laboratory. Wefurther call attention to the possibility of exploiting the diffraction data to obtainother quantities, viz., the direct correlation function and certain of the Mayer clusterintegrals.Used in proper combination, the Fourier transforms of several differentfunctions of the X-ray scattering data then lead to an expression in closed form for theintermolecular potential function.RADIAL DISTRIBUTION AND DIRECT CORRELATION FUNCTIONFor simple systems, the conventional radial distribution function is computedfrom the Fourier transform of the X-ray scattering functionG(r) = g(r)- 1 = (2n2rp)-' si(s) sin rds = Il(r). (1) 1: In this expression, g(r) is the radial distribution function, G(r) is the net radial distribu-tion function, p is the density, s = 47d-1 sin 8 is the scattering variable where 28 isthe angle between the incident and scattered beam, and A is the wavelength of themonochromatic incident radiation.The scattering function, i(s), for X-rays isdefined aswhere I(s) is the intensity of coherently scattered radiation, fully corrected for polariza-tion absorption, etc. ;f(s) is the atomic scattering factor. The derivations 1 of these*Work supported by the Directorate of Chemical Sciences, U.S. Air Force Office of ScientificResearch, under Contract AF 49(638)-1273.i(s) = I(s)/f '(s) - 1, (2)890 X-RAY DIFFRACTION TECHNIQUESformulae for monatomic substances and extensive discussion of problems of normaliza-tion, pretruncation of the integral, etc., are available.2, 3 Review articles by Gingrich,4Furukawa,s and Kruh 6 summarize results of measurements on many liquid systems.Although the number of substances studied has been large, the range of states hasusually been limited to ambient temperature and pressure, or in some cases to statesalong the vapour pressure curve.The classic study for a simple system was reportedin 1940 by Eisenstein and Gingrich.7 That work included X-ray scattering patte nsfor 26 states of confined argon specimens ; six of these were Fourier analyzed to prro-vide values of radial distribution function. Recently, the author's laboratory 8 hasreported diffraction experiments on liquid argon at 13 states on a systematic grid intemperature and density. A sample of the final radial distribution functions fromthis latter study is shown in fig. 1 and fig.2. The latter is a comparison with the21-1--; - r 1 1 ] 1 i v lt ARGON It.-125Oc 10I5 10 15FIG. 1.-Net radial distribution function 8 of liquid and gaseous argon along the isotherm t = - 125°C. Runs 31 and 32 correspond, respectively, to essentially saturated liquid and saturated gas.earlier data from the Eisenstein-Gingrich study. In spite of the more moderndetection systems and more sophisticated data-handling techniques, the recentexperiments still reveal considerable uncertainty due both to experimental error andalso to seemingly inevitable aberrations introduced in the numerical approximationsto the Fourier integral. These uncertainties are reflected in fig. 1 by the shaded areason each of the four radial distribution curves.The direct correlation function, C(r), as introduced by Ornstein and Zernike 9has played a central role in theory of optical scattering near the critical state and alsohas figured in the development of several integral equations describing the generalbehaviour of fluids.This direct correlation function is defined formally in terms ofthe net radial distribution function by the following equatioC . J . PINGS 91I I ! ; I I 1 ! I I in 2 4 6 84)Eisenstein and Gingrichr(&this laboratoryFIG. 2.-Comparison of radial distribution function determinations from two laboratories. Parti-cularly note that the subsidiary feature at about 5.0 8, in the top state is absent in the second work.ARGONt = - 125OC -1 -I --0--0 ---n 2 . -0 --t r-*0 ]I5r(&FIG.3.--Direct correlation function 14 for liquid and gaseous argon along the isotherm t = - 125°C.These curves might be comr>ared stateby-state with the net radial distribution function in fig. 192 X-RAY DIFFRACTION TECHNIQUESSolution of eqn. (1) and (3) results in the following expression for the directcorrelation function in terms of the Fourier transform of a particular function 10of the scattering intensity.This expression was applied by Goldstein 11 to scattering data for helium, but withinconclusive results. Reetz and Lund 12 applied the same expression to the earlierEisenstein-Gingrich data for argon, obtaining transforms for C(r) that were shortrange and generally plausible, except for erratic behaviour at small radii.Johnson,Hutchinson, and March 13 analyzed some diffraction data from liquid metals in orderto obtain C(r), with results suggestive of some long-range structure, i.e., second-orderpeaks, etc. Whether this reflects innate behaviour of liquid metals, or whether itmight be attributable to errors arising from omitted low-angle data (a particularlysensitive region for the highly incompressible liquid metal systems), is still not clear.Mikolaj and Pings 14 analyzed their X-ray diffraction data from argon at 13 states toobtain the direct correlation function. In all instances the computed direct correla-tion function exhibits a single maximum which decays rapidly to zero. The resultsfor 4 states are shown in fig. 3. These are the same 4 states for which the radialdistribution function is shown in fig.1 ; state-by-state comparison of G(r) and C(r)is informative.Direct subtraction of eqn. (1) and (4) results in the following expression for thedifference between the net radial distribution function and the direct correlation.function in terms of a Fourier transform of a certain function of the scatteringintensity.G(r)- C(r) = (2n2rp)- sin srds = 13(r).Although this is an obvious result, it seems to have only recently been pointed out.15As developed in a following section, this expression is useful since the difference ofthese two distribution functions in the low-density limit isolates certain of the low-order cluster integrals.DENSITY EXPANSIONSWe represent the expansion in powers of density of the radial distribution function,the net radial distribution function, and the direct correlation function as follows ;The expansion coefficients are the well-known cluster integrals,l% 17 some of whichare listed belowg o = 1 (9)91= 6\93 C .J . PINGSG1 = +g1(r)Eqn. (6) has been used 18 as a means of computing a theoretical g(r) as a functionof temperature and density using some assumed form of the intermolecular potentialfunction. The equation also offers the prospect of reverse usage, viz., computationof the intermolecular function from an experimentally determined radial distributionfunction. In the limit of zero density, the intermolecular potential function wouldbe exactly - kTln g(r). Such a direct determination would hold considerable attrac-tion since there is no model or parameterization involved.However, in this simpleform the computation is unrealizable since it is impossible to take meaningful diffrac-tion data at extremely low densities for the simple reason that the diffracted intensityabove background is proportional to the density of the sample. Mikolaj and Pings l9did demonstrate that plausible values of the intermolecular potential function couldbe obtained by applying an extended form of eqn. (6) to diffraction data taken onargon at about half the critical density. However, their procedure involved iterationtechniques in order to evaluate the gl(r) term, and some question of convergence didarise.We show here that the term p g l ( r ) and all but one of cluster integrals representedin thep2 term in the expansion of g(r) can be rigorously expressed as Fourier transformsof certain functions of the scattering data.Eqn. (7) and (8) are used to obtain thedensity series expansion of the difference between G(r) and C(r), and that differenceis then equated to the integral of eqn. (9, yielding the followingOne other of the cluster integrals in eqn. (6) can be related to an experimentallyaccessible integral.Before assembling the preceding expressions into a final expression, one furthereffect may be included, viz., possible non-additivity of the intermolecular forces.20-22We will concern ourselves here only with corrections to gl(r) which, in the presenceof non-additivity effects, can be represented as follows By combining these preceding results, the following final expression then is obtainedfrom eqn.(6)Unfortunately, there seems to be no simple representation of the five-bondedcluster integral in the p 2 term, and this is carried into the analysis in the final form.If the expression is being used as a means of determining the intermolecular potentialfunction, presumably an iteration would be possible since the cluster integral has a0 0 2 04 0.6 08 1.0density (g cm-3)FIG. 4.-Thermodynaniic plane for argon showing our estimates of region of applicability of equationsdeveloped in this paper for obtaining the intermolecular potential function from experimentaldiffraction data. The triangles represent states studied by X-ray by Eisenstein and Gingrich, thecircles the states of Mikolaj and Pings.The shaded area to the right is precIuded because of increasingcontribution from g3(r) and higher terms ; the shaded area to the left is essentially precluded becauseof too low bulk scattering from the low-density specimen ; temperatures below - 120°C are unfavour-able because of increasing contribution from the five-bonded double-rooted cluster integral.known dependence on 4 ( r ) . However, fortunately this five-bounded cluster integralis the smallest term in g2(r) ; in particular for temperatures greater than approximately1.2 Tc the contribution is almost negligible. Therefore, a perfectly acceptableprocedure is to use the above expression along with theoretical evaluations of thisone remaining cluster integral, which has already been evaluated by Henderson andOden 23 based on the Lennard-Jones 6-12 potential.We have indicated that termsof the order of p3 have been neglected ; however, we have also neglected non-additivityeffects of the order of p2. The immediately preceding equation has thus been putinto a form where it can be used in conjunction with experimental data to significantdensity-roughly up to a density for which PVT data could be adequately representedin terms up the fourth virial coefficient, but neglecting fifth and higher virial coefficientsC. J . PINGS 95Taking all these factors into consideration, we have shown in fig. 4 the domain oflikely applicability of eqn. (22) for determining the intermolecular potential functionand the non-additivity effect.Eqn.(23) can be put into a more practical form for comparison with experimentaldata. LetThen we have the following expression :Thus we have a simple functional relationship between the experimental quantities500400n+ 83 3005 nW G 2001000I 1 I 1 I I2 4 6 8 104)FIG. 5.-Comparison 15 of the three-body double-rooted cluster integral. The theoretical curvesare taken from the reduced g t ( r ) of Henderson and Oden. The experimental curve is the result ofapplying eqn. (18) to X-ray data at the state labelled 36 in fig. 4.-- (3-409)3 g?(r) at T* = 1- (3.409)3 gf(r) at T* = 1.4- - - - gl(r), expt. at T = 163~1°K= (1.37)X (119.5"K)A(Z1+ 1)-1 and p(l1+ 1)-1, with the slope determining the non-additivity effect andthe intercept the intermolecular potential function. Although the non-additivityterm may have dependence on temperature, the intermolecular potential function istemperature-independent ; therefore, the application of above expression to the samesubstance at two different temperatures must yield the same value of the intermolecularpotential function if the data are correct96 X-RAY DIFFRACTION TECHNIQUESA study of X-ray diffraction study of fluid argon by Mikolaj and Pings 8 includedseveral states in a region where the above expression should be applicable.However,the points were widely spaced in density, and too few in number to provide anystrong test of the applicability of the method suggested here.Nevertheless, thesedata were sufficient to lend some credibility to an important facet of the development,viz., computation of plausible values of the cluster integrals from eqn. (18). Fig. 5reproduces a comparison of theoretical values of gl(r) with values calculated fromeqn. (18), using data from a low-density state. Our laboratory is now conductingan experiment to accumulate detailed data for fluid argon with the best possibleaccuracy in a range of densities for a maximum utility of eqn. (23).1 R. W. James, The Optical Principles of the Difraction of X-Rays, (Bell, London, 1954), p. 477.2 J. Waser and V. Schomaker, Rev. Mod. Physics, 1953, 25, 671.3 H. H. Paalman and C. J. Pings, Rev. Mod. Physics, 1963, 35, no. 2, 389.4N. S. Gingrich, Rev. Mod. Physics, 1943, 15, 90.5 K. Furukawa, Progr. Physics, 1962, 25, 395.6R. F. Kruh, Chem. Rev., 1962,62, 319.7 A. Eisenstein and N. S. Gingrich, Physic. Rev., 1940, 58, 307.8 P. G. Mikolaj and C. J. Pings, J. Chem. Physics, in press.9 L. S. Ornstein and F. Zernike, Proc. Acad. Sci. Amst., 1914, 17, 793.10 L. Goldstein, Physic. Rev., 1951, 84,466.11 L. Goldstein, Physic. Rev., 1955, 100, 981.12 A. Reetz and L. H. Lund, J, Chem. Physics, 1957, 26, 518.13 M. D. Johnson, P. Hutchinson and N. H. March, Proc. Roy. SOC. A , 1964,282, 283.14 G. Mikolaj and C. J. Pings, J. Chem. Physics, in press.15 C. J. Pings, Mul. Physics, in press.16 G. S. Rushbrooke, in J. Meixner, ed., Statistical Mechanics of Equilibrium and Non-Equilibrium17 G. S. Rushbrooke and H. I. Scoins, Proc. Roy. SOC. A , 1953, 216,203.18 J. de Boer, Rept. Progr. Physics, 1949, 12, 305.19 P. G. Mikolaj and C. J. Pings, Physic. Rev. Letters, 1966, 16,4.20 H. W. Graben and R. D. Present, Physic. Rev. Letters, 1962, 9, 247.21 N. R. Kestner and 0. Sinanoglu, J. Chem. Physics, 1963, 38, 1730.22 A. F. Sherwood and J. M. Prausnitz, J. Chem. Physics, 1964, 41,413.23 D. Henderson and L. Oden, MoE. Physics, 1966, 10,405.(North Holland Publishing Co., Amsterdam, 1965), p. 222

ISSN:0366-9033    

DOI:10.1039/DF9674300089

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

X-Ray Diffraction Study of Liquid Water in the TemperatureRange 4-20Ooc*BY A. H. NARTEN, M. D. DANFORD AND H. A. LEVYChemistry Division, Oak Ridge National Laboratory, Oak Ridge, TennesseeReceived 9th December, 1966The scattering of X-rays from the free surface of liquid water in equilibrium with water vapourhas been analyzed at 4, 25, 50, 75, 100, 150 and 200°C. Deuterium oxide at 4°C was also studied.The diffractometer used was specially designed for the study of liquid structure. The radial distribu-tion functions derived from the experiments are in agreement with most of the previously publishedwork on water showing, however, much higher resolution.Intensity and radial distribution functions have been computed for a model structure and comparedto those derived from experiment.The model assumes an anisotropically expanded ice-I structure,surrounded by a continuous distribution of distances, as an adequate description of the short-rangeorder in water. Occupancy of the large dodecahedra1 cavities typical for this structure was permitted.The model intensity and radial distribution functions are in quantitative agreement with those derivedfrom experiment at all temperatures. Some thermodynamic properties estimated for the modelstructure are in essential agreement with those of liquid water.The first X-ray diffraction patterns from liquid water were obtained by Meyer,lStewart 2 and Amaldi.3 This work, together with the known properties of the isolatedwater molecule and the ice lattice, led Bernal and Fowler 11 to propose their model ofwater structure.Katzoff 4 was the first to apply to water the method of Fourieranalysis. Morgan and Warren (1938) 5 analyzed the X-ray scattering from liquidwater at five temperatures between the melting and boiling points. Their work ledto the abandonment of the Bernal and Fowler model of water structure.Since 1938, various X-ray diffraction studies of water have been published 6-10 ;these investigations have not added to Morgan and Warren’s results. It has been thepurpose of our work to obtain a set of diffraction data on liquid water which extendsand improves upon Morgan and Warren’s work both in resolution and temperaturerange.EXPERIMENTALThe diffraction measurements were made with a diffractometer specially designed for thestudy of liquids.The diffraction pattern from the horizontal surface of liquid water wasobtained with a divergent beam technique similar to the Bragg-Brentano system used forpowder samples. The instrument provides for simultaneous angular motion of the X-raytube and the detector about a horizontal axis lying in the liquid surface. This methodeliminates sample holder absorption and scattering. Monochromatic MoKa radiation isobtained through the use of a bent and ground crystal monochromator mounted in thediffracted beam. The diffractometer, the procedure for data collection, the various correc-tions applied to the raw data, and their final evaluation have been discussed elsewhere.12Scattered intensities were measured with various beam divergences, ranging from 0.5deg.at the lowest scattering angles to 4.0 deg. at the highest angles. The times for a fixed* Research sponsored by the U.S. Atomic Energy Commission under contract with the UnionCarbide Corporation.4 998 X-RAY DIFFRACTION STUDY OF WATERnumber of counts, ranging from 100,000 to 600,000, were measured at 0.25 to 1.0 deg.intervals in scattering angle. As the diffraction pattern showed interference throughout theobservable range of the instrument 12 (to smaX = (471/A) sin Om, = 16, 8 being half thescattering angle), the data are appreciably more extensive than the earlier ones.1-10We have studied the X-ray diffraction pattern of liquid water at 4,25, 50,75, 100, 150 and200°C; deuterium oxide at 4°C was also studied.The experiments at and below 100°Cwere done at atmospheric pressure, and those above 100°C at the vapour pressure of thesample. Both our raw data and the reduced intensity and radial distribution functionsderived from them are available in tabulated f0rm.13RESULTSThe reduced intensity i(s) represents the structurally sensitive part of the totalcoherent intensity I(s)lz9 13 in electron units. It is computed from the equationwhere Zf?(s) is the part of the scattering ascribable to independent atoms, and the sumis over the stoichiometric unit (one water molecule). A radial distribution functionD(r) is obtained by Fourier inversion of (1) according toiD(r) = 4nr2p, + ( 2 r / n ) l p s i(s)M(s) sin (sr)ds,where po is the average number density of molecules, and M(s) is a modification func-tion included to sharpen the features of the radial distribution function (RDF).Themodification function used wasM(s) = ccm1- 2, (3)iwhere f refers to the coherent scattering amplitudes of oxygen and hydrogen, andsummation is again over the stoichiometric unit. This modification function removesthe average breadth of the distribution of electron density in the atoms. It also changesthe scale so that if i(s) is in electron units, D(r) is in units characteristic of one molecule.The physical meaning of the function D(r) has been discussed elsewhere.121 14The reduced intensity and radial distribution functions for liquid water are shownin fig. 2 and 3 (circles). There are deviations from a uniform distribution of distancesout to 8 A at room temperature.The first prominent maximum in the RDF,corresponding to nearest neighbour interactions, shifts gradually from 2.82 A at4°C to 2.94A at 200°C. From the area under this peak, a coordination number of4.4-4-5 can be computed,* which remains constant over the whole temperature rangecovered by our experiments. Maxima around 4-5 and 7 A, corresponding to secondand third neighbour interactions, are distinct at room temperature, but disappeargradually with increasing temperature. There is no evidence for a sudden breakdownin the structure of water ; if changes in the average configuration around any watermolecule take place, they must occur gradually with increasing temperature.Thecurves for light and heavy water at 4°C are almost identical within experimental error.There is no significant difference in the arrangement of oxygen atoms at this tempera-ture.*The calculation of average coordination numbers from the area under a RDF peak is notunambiguous. In this case, the lower bound of the first maximum is fairly well defined, but the upperbound is not. The coordination number given here was computed from the parameters of our watermodel (for details see below and table 1). Its accuracy should be judged by the agreement of themodel radial distribution function (solid line, fig. 3) with those derived from experiment (circles, fig.3) in the region below 3.5 AA . H . NARTEN, M . D . DANFORD A N D H . A . LEVY 99All of the earlier diffraction studies of liquid water, with the exception of those ofthe Amsterdam gr0up,9~ 10 have confirmed the results of Morgan and Warren.5 It istherefore sufficient to discuss only these two studies.Our results are in essentialagreement with those of Morgan and Warren, whose data extended to a value ofsmax = 12 at room temperature. Slight differences arise from the higher resolution ofour work. The disagreement between Morgan and Warren’s work and that of theAmsterdam group (which yields a nearest neighbour distance of 3 A, and a coordina-tion number of 5 ) is thus confirmed. We have shown 13 that a value of Smax = 8, asused in the Amsterdam work, yields radial distribution functions which, for water,show erroneous features in the region of short radial distances.STRUCTURAL MODEL OF LIQUID WATERIt is not possible to compute radial distribution functions from a known or assumedpair potential for a system interacting with long-range, many-body, non-centralforces.However, the experimental radial distribution function for a liquid, togetherwith the known atomic arrangement in the solid and gaseous states, may suggest amodel for the liquid structure, which may then be tested against the diffraction data.For a model of liquid structure to be useful in the present context, it should have asufficiently detailed geometric basis to permit computation of intensity and radialdistribution functions. It must give a detailed description of the arrangement of allatoms around each atom in the stoichiometric unit taken as the origin, and should beconsistent with the fact that the distance spectra about each such stoichiometric unitare alike.A distribution of suitably chosen stoichiometric units over a latticeextending to the limits of the sample, would have the required properties. With adescription of this kind, it is possible to specify a set of mean radial distances rgjwhich is needed for the computation of reduced intensity functions according to theequation 12 :i,(s) = C C exp (- bijs2)fi(s)fj(s) sir (srij)/srij. ( 4 4i jHere, i is summed over the stoichiometric unit and j over all atoms in the structure.A “ temperature factor ” exp (- bijs3) is included in (4a) ; it takes into account thedistribution of instantaneous interatomic distances about their respective means rtjboth in time and space.The coefficient bij is one-half the mean square variation inthe distance rij between atom pairs.12If such a model is to be realistic, it must account for the fact that a liquid is not acrystal. We have, in a liquid, only a distribution of distances with maxima in posi-tions for which a latticc is a convenient frame of reference. These positions, which aresharp in a crystal, become increasingly diffuse with radial distance from any originatom. Thus, the coefficients bfj in (4a) must increase with the distance rij.For a model of liquid structure to be tractable, we can terminate the series ofterms (4a), which become smaller with increasing distance rij, at some arbitrarydistance, and assume a uniform distribution of distances (continuum) beyond. Thiscorresponds to making a “ hole ” in a uniform, structureless medium in which toplace the discrete interactions of the model structure (4a).The reduced intensityfrom the continuum of distances with a pair of atoms ik in the stoichiometric unitbeginning at riko, and extending to the limits of the sample, is given by the equation12 :i c ( S ) = eXp (- b,,oS2)>S,(S)f,(S)4.p~[S~iko COS ( S r i k O ) - Sin ( S r i , 0 ) ] / S 3 . (4b)i kHere, both i and k are summed over the stoichiometric unit. A ‘‘ temperature factor ”is included in (46) since the emergence of the continuum will not be sharp. Th100 X-RAY DIFFRACTION STUDY OF WATERreduced intensity curve of the model structure can now be computed as the sum of(4a) and (4b) :The model intensity function (4) can be systematically refined to seek a satisfactory fitof the experimental data over the whole angular range.i(s) = i D ( S ) + i&). (4)DESCRIPTION OF THE WATER MODELSeveral crystal structures were examined as a point of departure towards a struc-tural model for liquid water : the a-quartz structure, the cubic ice structure, the chlorinehydrate (clathrate) structure, and the ice-I structure.The clathrate type of model forliquid water proposed by Pauling 16 was investigated in detail. The RDF for theclathrate model was calculated for a nearest neighbour distance of 2-76 A. Expansionof this distance to that corresponding to the first peak in the observed RDF (2.88 Aat 25°C) leads to an increase of the fraction of molecules inside the clathrate cages inorder that the proper density be maintained.As a result, the area under the firstpeak becomes too small. In view of these difficulties, the Pauling model does not seemadequate to explain our X-ray data.17TRIAsD AXIS&7 MOLECULETRIAD AXISFIG 1.-The water model (ball/stick ratio not to scale).Among the crystal structures examined, the ice-I lattice was the most promisingone ; it is shown in fig. 1. Each oxygen atom is tetrahedrally surrounded by otheroxygen atoms, forming layers of six-membered puckered rings. Two adjacent layers,related by mirror symmetry, form dodecahedra1 cavities. In our water model, theice-I lattice is anisotropically expanded (PI + P2 in fig.l), and both vacant lattice sitesand occupancy of the cavities by water molecules are permitted, but constrained toconform to the experimental density. For simplicity, the model retains the hexagonalsymmetry of the ice-like network.Discreteinteractions were included for all distances out to approximately lOA, with a con-tinuum beyond. The initial agreement being promising, the model was subjected tosystematic refinement by iterative non-linear least squares,l5 in which the modelIntensity curves for this model structure were computed by eqn. (4)A . H. NARTEN, M. D . DANFORD AND H. A. LEVY 101intensity function (4) was fitted to values derived from experiment. At this stage, anumber of constraints were introduced in order to keep the model as simple as possibleand to reduce the number of variable parameters to a minimum.The position of thecavity molecule was restricted to the triad axis (fig. 1). Since exploratory calculationshad shown that the inclusion of discrete 0-H and H-H interactions did not changethe model intensity functions significantly (because of the low scattering power of theone electron in the hydrogen atom), continuous distributions were assumed for thesedistances. All " temperature factors " corresponding to longer range distances wererelated to those of the third coordination shell about any origin atom, under theassumption that bfjcc rij.18 Also, a number of " temperature factors '' correspondingL-L--'200G-'' A ' I ' 4 ' I ' ' ' ' 8 10 42 14 16s[A-l]FIG. 2.-Observed and model reduced intensities for liquid water.to first, second and third neighbour distances were arbitrarily set equal.Finally,only physically reasonable values of the model parameters were allowed in order tominimize the time required for convergence of the fit. Only three distances weretreated as independent variables : the two near neighbour 0-0 network distancesparallel and roughly perpendicular to the triad axis (PI and P2 in fig. I), and theshortest distance from the cavity oxygen atom to the network (P3 in fig. 1). Allother distances (out to 10 A) were properly related to these near neighbour distances.These three adjustable parameters, together with their associated " temperaturefactors ", were sufficient to yield a good fit to all but the low angle region of theintensity function.In order to fit this region also, additional " temperature factors "associated with second and third neighbour interactions, were varied and also thedistance characteristic of the start of the continuum, and a parameter specifying theoccupancy of lattice sites102 X-RAY DIFFRACTION STUDY O F WATERReduced intensity functions obtained from the least-squares refined model areshown in fig. 2 (solid lines), along with the observed data (circles). The correspondingradial distribution functions are shown in fig. 3. For the Fourier inversion of theobserved intensity functions, extrapolation from the lowest scattering angle accessibleby our technique to zero angle was made using the model intensity curves.PARAMETERS OF THE MODEL A N D THEIR TEMPERATURE DEPENDENCEThe refined parameters used for the calculation of the model intensity and radialdistribution functions are shown in fig.4 and 5. It should emphasized again, thatonly the three near neighbour distances PI, P2, and P3, the '' temperature factor "associated with the PI and Pz distances, and the parameterfi describing occupancy of1 I 7-I 11, ,Q LQ i 2 3 4 5 6 7 8 9r, AFIG. 3.-Observed and model radial distribu-tion functions of water.network sites are truly independent variables.All other parameters (especially the " tem-perature factor " associated with the P3distance, see below) are, due to the con-straints imposed on the model, the result ofsome arbitrary averaging process and there-fore of limited meaning.Of the features of the radial distributionfunctions (fig.3), only the first maximumremains pronounced at all temperaturescovered by the experiments. The c c tempera-ture factors " associated with second andhigher neighbour interactions show a con-tinuous increase with temperature, and thedistance characteristic of the start of thecontinuum changes from about lOA at 4°Cto about 6 A at 200°C. These variationsreflect the loss of long-range structure withincreasing temperature.The first peak in the radial distributionfunction of water is explained by the modelin terms of one network-network interactionat a distance P2 (fig. I), three network-net-work interactions at a distance PI, andthree network-cavity interactions at a dis-tance P3, each of them properly weightedaccording to the network and cavity occu-pancy factors.At 4"C, the distance P2, between network molecules related by mirror symmetry,is only slightly larger than the corresponding ice distance ; with increasing temperature,this distance (fig.4) contracts below the nearest neighbour distance in ice. Thedistance P1 between network molecules related by a centre of symmetry is, at 4"C,about 6 % larger than the corresponding ice distance, and with increasing tempera-ture, this distance goes through a maximum of 3-02A at 100°C. The root-mean-square (rms) displacements associated with these network distances (fig. 4) showroughly the same temperature dependence.At temperatures above lOO"C, the shortnetwork distance P2 expands, and the long PI distance contracts; at 200°C the twonetwork distances differ by an amount much smaller than their respective rms-displacements3.13.0-2-92.82.7A 0.6-0.70.5090.30.20.1OI I I l I I I I I-MODEL DISTANCES(Std. Error ~ 0 . 0 2 A ) -- --- -- +++ p,x x x P2- 000 P3-I - I;, f -- ’+,, 1 - ‘‘1 g _---- R MS- DJS PLACE ME NTS -OF MODEL DISTANCES ,,. I \ :I .t..rb’ ,x ___---- - f‘, I . I - -+: r-..,I ‘ +--- --.__ + ____. ,II-l:o -do -40 A 40 :o 40 ,Lo 260- - +--.I..- :\ /*-..t.--t***’-4 ::- X--- -0.90.80-70-60.50 40.30.20.4Oi I n c - ,. -l ” ” l ” I ” I I I-0--0-- -- XX X-X - XXX-- -- -i +- + + + + + -0 FRACTION OF NETWORK SITES OCCUPIED ( f , )- X FRACTION OF INTERSTITIAL SITES OCCUPIED ( f 2 ) -+ FRACTION OF MOLECULES I N INTERSTITIALPOSITIONS ( f - ~ )-40 A do e‘o 1:o d o 2d104 X-RAY DIFFRACTION STUDY OF WATERindicate that the position of the molecule is not at the cavity centre. At the positionfound, the angle subtended by two nearest network oxygen atoms has approximatelythe tetrahedral value.Thus each cavity molecules has three nearest network neigh-bours at a distance P3 (fig. 1) which is close to the longer network distance PI.In the refinement of the model parameters, the " temperature factors " associatedwith the distance P3 and the distance between a cavity molecule and its three secondnearest neighbours * were arbitrarily set equal.The rms-displacement shown in fig. 4as associated with distance P3 is thus an average of the amplitude characteristic of thisand the second neighbour distance, Pi. With increasing temperature, P3 expands, asdoes PI, and P; contracts, as does the short network distance P2 ; if the rms-displace-ments associated with P3 and Pi have the same temperature dependence, their averagecould show the observed minimum between 50 and 75°C.The occupancyfi of the network sites is 100 % up to 100°C ; at this temperature,empty network sites begin to appear. The value offi (fig. 5) drops to 87 % at 150°Cand 79 % at 200°C. The occupancyf:! of the cavity sites increases from 45 % at4°C to 57 % at 200°C.As a result, the fraction w of water molecules in networkpositions, decreases from 82 % at 4°C to 73 % at 200°C. The quantitiesfi,fi, and ware related bywhere z is the number of cavity sites per network site; for the ice lattice z = 3.PROPERTIES OF MODELThe structural model of liquid water was developed in order to explain the diffrac-tion data, and the agreement between the intensity and radial distribution functionscomputed for the model and those derived from experiment is gratifying. Thisagreement is necessary for the model to be tenable, but in itself not sufficient prooffor its reality. We therefore ask what other properties the model might have, andhow they compare to those of liquid water. Since many constraints were imposed onthe model parameters so as to keep the description as simple as possible, thesepredictions can be only of a crude nature; the numbers given in this section shouldtherefore be used with some caution.Since the experimental density of liquid water was used as a constraint on theoccupancy of network and cavity sites, the model is in quantitative agreement with theP-V--T behaviour over the temperature and pressure range covered by the diffractionexperiments. Occupancy of cavity sites overrides network expansion up to 4"C,leading to the familiar increase in density; above 4"C, lattice expansion at almostconstmt cavity occupancy results in a continuous decrease in density.The anisotropy of the network portion of our water model, amounting to a 10 %difference between the P1 and P2 near neighbour distances (at 75"C), remains pro-nounced up to 100°C and above.The distance between water molecules related bymirror symmetry (P2) is shorter, and thus the corresponding hydrogen bond appearsto be stronger than that between water molecules related by a centre of symmetry(PI). A similar small effect appears to be present in ice.19 Nothing can be saidconcerning 0--H interactions from this study ; these must await neutron diffractioninvestigations. Under these circumstances, it is difficult to make definite statementsas to the degree of hydrogen bonding in water that our model would predict. If in theice-like network a hydrogen atom were located between each pair of adjacent oxygens,* This distance, which would be equal to P3 if, as in ice, PI and P2 were equal, is, for the watermodel, about 10 % larger than P3A .H. NARTEN, M. D. DANFORD AND H . A. LEVY 105forming a hydrogen bond, there would be, per mole of water, a fraction wfi suchinteractions ; of these, 0.25 wfi would be of the mirror symmetric and 0-75 wfi of thecentrosyrnmetric type. Since the distance P2 is comparable to the near neighbourdistance in ice-I, the mirror symmetric bonds would presumably be of similar strengthas the hydrogen bonds in ice. Around an empty network site there would be afraction w(1 -fi) " broken " hydrogen bonds. Since, with our assumptions, neithera hydrogen atom nor an unshared electron pair belonging to a network water moleculewould point in the direction of a cavity molecule, the network-cavity moleculeinteractions, fraction (1 - w), would be of a different and probably weaker kind than thenetwork-network bonds.If we apply the term hydrogen bond to any 0-H .. . 0 interaction for which the0-0 distance is smaller than that computed from the van der Waals radii of oxygenand hydrogen, then the model would predict a distribution of hydrogen bonds; itwould change with temperature as does the near neighbour distance distribution, i.e.,as the first peak in the radial distribution function of water. This would be inagreement with spectroscopic investigations of the 0-H stretching frequency inwater3A crude estimate of the frequencies associated with the translational motion of thenetwork molecules can also be made. If the rms-displacements of the networkdistances were due to translational vibrations only, and if a network molecule movedindependently in the field of its nearest neighbours with a frequency v = (f/M)3/27r,where f is the force constant and M the reduced mass, f could be eliminated by thecorresponding expression for the energy: E = +kT = +fz; here 3 is the meansquare amplitude associated with the translational motion.Using the properconstants, the frequency (in cm-1) is related to the amplitude (in A) by the approximateexpression v E (T)f/X. At 4"C, X = 0.093 A for the mirror symmetric P2 bond, andthe frequency would be 200 cm-1. Similarly, with X = 0.203 for the motion along thecentrosymmetric PI bond the frequency computes as 90 cm-1.Bands around 160 and60 cm-1 have been observed in the Raman and inelastic neutron spectrum of water,and assignment of the 160 and 60 cm-1 frequency to the motion of water networkmolecules along the mirror and centrosymmetric hydrogen bonds has been proposedpreviously on a more intuitive basis.21 Our model is not in disagreement with thisview.Our model system in a state of statistical equilibrium can be described as a saturatedsolution of N' vacant network sites (holes) and N" cavity molecules in an ideal lattice,constituted by N-N" regularly arranged molecules and containing N-N" + N' sites,among which these molecules and the N' holes can be distributed at random. Thisdescription corresponds to the assumption that the holes and cavity molecules are notinteracting with each other ; this assumption is justified so long as N' and N" aresmall compared with N, which is not the case with our water model.The number ofnetwork holes and cavity molecules can be determined as a function of temperaturefrom the condition of the minimum of that part of the free energy which is due to theirpresence. The corresponding part of the entropy is proportional to the logarithm ofthe number of different ways they can be distributed over the respective sites. Employ-ing the customary approximations, the counterpart of the entropy of mixing of aternary solution isSM/R = - (Wlfl)C.fI In fi + (1 - fd In (1 - fdl - (1 - W l l f 2 L 7 . 2 In f 2 +The first term describes the entropy change associated with the formation of vacan106 X-RAY DIFFRACTION STUDY OF WATERnetwork sites (holes), and the second that corresponding to the formation of cavitymolecules.For the molal Gibbs free energy of the model we writeHere G&, Ggand GT are hypothetical standard molal free energies of network molecules,network holes, and cavity molecules, respectively, and S M is defined by eqn. (6).Introducing AG; = Gg-GS, and AG; = Gi;-G;, eqn. (7) rearranges toFrom the condition dG(fi,fJ = 0 we obtain for the standard free energy changeassociated with the formation of vacant network sites :G = w[flGz+(l- f1)Gi?J+(1-w)G;-TSM. (7)G = G,"+w[AG,"-(l- fJAGg]-TSM. (8)(9) AG; = (RTlf3Cln (1. - f l ) + In (1 - f2)lYand for the free energy change associated with the transformation of cavity moleculesinto the network :AGI" = RTT'Pn f2/(1-f2)-(4f1)1n ( ~ - f 2 ) l - ( ~ ~ / f l ) ~ f l ~ ~ f l + ( 1 - f l ) ~ n (l--fl)l+(1 -fdAG;;.(10)Eqn. (6-10) reduce to those derived by Frank and Quist 22 in their study of the thermo-dynamic properties of the Pauling model, for the binary case with fi = 1.The quantities AGhand AG,"can be calculated from eqn. (9) and (10) and the modelparametersfiJ2 and w ; they are shown in table 1. From the temperature dependenceof AG;i and AG; the average enthalpy changes associated with the formation ofnetwork holes and cavity molecules, AH& and AH,", can be obtained (table 1). Theenthalpy change associated with the formation of a network hole must be, at leastTABLE TH THERMODYNAMIC PROPERTIES OF THE WATER MODEL.ENERGY UNITS IN kcal/mole ;ENTROPIES IN kcal/mole deg. ; w IS THE FRACTION OF MOLECULES IN LATTICE POSITIONS ; fiTHE LATTICE OCCUPANCY, AND N THE AVERAGE NUMBER OF NEAREST NEIGHBOURSW AHv ASv T(OK) f l AGh AG: model water model water277.2 0.815 1 4.37 - 0.062 9.7 10.72 0.035 0.0387298.2 0.796 1 4-41 - 0.244 9.7 10.51 0.033 0.0352323.2 0.803 1 4.39 - 0.193 9.7 10-25 0.030 0.0317348.2 0.792 1 4-42 - 0.325 9.7 9.99 0.028 0.0287373.2 0.78 0.99 4.4 -4 0.45 9.7 9.71 0.026 0.0260473.2 0.73 0.79 4.5 -3.0 0.77 9.6 8.35 0.020 0.0177423.2 0.74 0.8, 4.5 -298 0.88 9.6 9-09 0.023 0-0215AH2 = -9.9f4.3 AH? = - 1.06 f0.23within the accuracy of this discussion, approximately equal to the evaporation energyof a network molecule, in our water model.For the total evaporation enthalpy of themodel, we may thus writeThe model evaporation enthalpies and the corresponding entropies A& = A&/Tare compared with the experimental values in table 1. From the temperaturedependence of the model evaporation enthalpies, the specific heat of the model canbe obtained as approximately 9 cal/mole at room temperature. The experimentalvalue for water is 18 cal/mole, but we do not consider this disagreement as significantin view of the approximate nature of this thermodynamic treatment. The assumptionAHy = wAH~ + (1 - w)(AH; -AH,"). (1 1A . H. NARTEN, M. D . DANFORD A N D H. A . LEVY 107of temperature dependent values for AH& and AH," would raise the model cp value intothe neighbourhood of the experimental value for liquid water.CONCLUSIONSThe X-ray diffraction data on liquid water presented here yield information on theaverage atomic arrangement around any oxygen atom taken as the origin.Theproposed structural model for water was developed to explain the diffraction data.Reduced intensity functions computed for the model are in quantitative agreementwith the X-ray data on liquid water over the whole angular range covered by ourexperiments. Radial distribution functions obtained by Fourier inversion of themodel intensity functions are in agreement with those derived from experiment todistances of lOA and beyond. The same model, with proper adjustment of itsparameters, explains the X-ray data over the whole temperature range covered by ourexperiments.Agreement of this model with the diffraction data is necessary but initself not sufficient proof of its reality. On the other hand, the model has propertieswhich are not in disagreement with the thermodynamic properties of water, and it maybe helpful in the interpretation of the many strange properties of water. In similarmanner, other proposed models of water23 which have a sufficiently detailed geomet-rical basis to permit computation of intensity and radial distribution functions canbe tested against the diffraction data, with agreement a necessary condition for themodel to be tenable.1 H. H. Meyer, Ann. Physik, 1930,5,701.2 G. W. Stewart, Physic Rev., 1931,37,9.3 E. Amaldi, Physik. Z., 1931, 32, 914.4 S. Katzoff, J. Chem. Physics, 1934,2, 841.5 J. Morgan and B. E. Warren, J. Chem. Physics, 1938, 6,666.6 L. Simons, SOC. Sci. Fennica, Commentationes Phys. Math., 1939, 10, no. 9.7 C. Finbak and H. Viervoll, Tidsskr. Kjemi, Bergvesen Met., 1943, 3, 36.8 G. W. Brady and W. J. Romanow, J. Chem. Physics, 1960,32,306.9 C. L. van Panthaleon van Eck, H. Mendel and W. Boog, Disc. Faruday SOC., 1957, 24,200 ;10 J. Heemskerk, Rec. T'av. Chim., 1962,81,904.11 J. D. Bernal and R. H. Fowler, J. Chem. Physics, 1933, 1, 515.12 H. A. Levy, M. D. Danford and A. H. Narten, Data Collection and Evaluation With an X-RayDiflractometer Designed for the Study of Liquid Structure (ORNL-3960, May 5, 1966).13 A. H. Narten, M. D. Danford and H. A. Levy, X-Ray Diflraction Data on Liquid Water in theTemperature Range 4 to 200°C (ORNL-3997, September, 1966).14 J. Waser and V. Schomaker, Rev. Mod. Physics, 1953, 25, 671.15 W. R. Busing and H. A. Levy, OR GLS, A General Fortran Least Squures Program (ORNL-16L. Pauling, in Hydrogen Bonding @. Hadzi and H. W. Thompson, ed.), (Pergamon Press,17 M. D. Danford and H. A. Levy, J. Amer. Chem. SOC., 1962, 84,3965.18 J. Frenkel, Kinetic Theory of Liquids (Dover Publications, New York, 1955).19 S. La Placa and B. Post, Acta Cryst., 1960, 13, 503.20 T. T. Wall and D. F. Hornig, J. Chem. Physics, 1965,43,2079.21 Y. V. Gurikov, Zhur. Struct. Khim., 1963, 4,824.22 H. S. Frank and A. S. Quist,.J. Chem. Physics, 1961, 34, 604.23 J. L. Kavanau, Water and Solute- Water Interactions (Holden-Day, San Francisco, 1964).Proc. Roy. SOC. A , 1958, 247,472.TM-271, August 1962).London, 1959).M. Falk and T. A. Ford, Can.J. Chem., 1966,44, 1699

ISSN:0366-9033    

DOI:10.1039/DF9674300097

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

Infra-Red Absorption of HDO in Water at High Pressuresand TemperaturesBY E. U. FRANCK AND K. ROTHInstitut fur Physikalische Chemie, Technische Hochschule,Karlsruhe, Englerstr. 1 l/GermanyReceived 16th Jamary, 1967By use of a specially designed cell with a sapphire window the absorption of the OD-stretchingvibration of HDO in H20 has been measured at frequencies from 2200 to 2900 cm-1 and at tempera-tures and pressures from 30 to 400°C and from 50 to 4000 bars respectively. At the supercriticaltemperature of 400°C and pressures below 200 bars (corresponding to a density of water of 0.1 g/cm3)the rotational structure of the vibration band of free water molecules is observed. At higher densityonly one intensive absorption maximum with simple shape is observed at all temperatures.This isconsidered as support for the continuum model for water in the liquid and in the dense supercriticalstate.New information on the structure of liquid water has recently been derived fromnear infrared and Raman spectra 1-12. The frequency and intensity of the absorptionof the oxygen-hydrogen vibrations indicate the existence and the extent of hydrogenbonding. The OD-vibration of dilute HDO in H2O is particularly suited for suchinvestigations because of the absence of interference from other frequencies and forother reasons.1, 2, 8 , 13 The temperature dependence of this vibration is of specialinterest. Wall and Hornig 1 and Falk and Ford 2 have obtained important resultswhich support the assumption of a continuum model for liquid water.The infra-red spectrum of liquid water in the overtone region above 5000 cm-1 atsaturation pressure up to the critical point at 374°C and 221 bars has been measuredby Luck.9 Other authors have also recently published spectra in this frequencyregion for liquid water at saturation pressure and elevated temperatures.10-12 TheOD-vibration of HDO, however, has only been determined up to 13OOC.2 No infor-mation about the pressure or density dependence of this vibration was available.Therefore, it was the purpose of this work, to investigate separately the influence oftemperature and density on the OD-stretching frequency of HDO.These investiga-tions had to be extended to supercritical temperatures in order to observe the variationof the absorption from very low to very high density of wster at constant temperature.It should be possible to determine the conditions under which OD-vibrationnot affected by hydrogen bonding is detectable.To obtain this information measure-ments were made to temperatures and pressures up to 400°C and 4000 bars respectively.EXPERIMENTALAn optical cell of the reflection type similar in some ways to that used by Welsh 19 wasdesigned. The cell has a single window, made of a colourless synthetic sapphire of 10 minthickness. Attached to the surface of the sapphite is a platinum-iridium mirror. Spacersof gold foil determine the distance between the mirror and sapphire. This space, fdled withthe fluid, provides a path length of twice the distance between the mirror and sapphire10E.U . FRANCK AND K. ROTH 109surfaces. Thus the path length is independent of the applied pressure. The aperture ofthe sapphire window is 8 mm. The body of the cell is made from a non-corrosive nickelalloy. It is covered by the coils of the heating wire and enclosed in a double-walled brasscase cooled with water. The cell body contains two thermocouple wells. A detaileddescription of the cell will be given elsewhere.The housing of the Perkin-Elmer model 521 grating spectrometer was flushed withdry air. It was furnished with a Micro Specular Reflectance Accessory which had beenslightly changed in order to mount the high-pressure cell outside the normal sample area.A 1 : 1 slit image could be produced inside the cell on the Pt-Ir mirror from which the beamwas reflected.The beam is refocused on the entrance aperture of the instrument by atoroidal mirror. A narrow stainless-steel capillary connects the cell to the pressure generatingunit. This unit consists of a hand pump, a pressure intensifier* and a separator vessel inwhich the pressure is transmitted from oil to water. The pressure is measured by calibratedBourdon gauges. The DzOt was 99-7 % pure ; the mixtures of D20 and H2O were preparedby weighing.0.HDO---------I I I I 8 1 I t2 8 0 0 2600 2400v [cm-11FIG. 1.-One set of original curves of the % transmittance as a function of frequency v in cm-1 at400OC and 3900 bars (0.9 g/cm3) ; path length, 35 p. 1, background absorption of H20 andsapphire; 2, absorption of a 8.5 mole % solution of HDO in H2O; 3, zero line, representinglight reflected by the water-sapphire surface and light emitted by the hot cell body ; --- region of C02absorption with reduced reliability.The accuracy of the temperature measurements, at the highest temperatures is f5"C.The pressure is correct within f 5 bars below 250 bars and within 340 bars beyond loo0bars.The water density has been calculated using the VDI-steam-tables 21 up to 1000 barsand the experimental PVT-data of Maier and Franck 14 at higher pressures. The spectro-meter was calibrated using the water vapour bands, the C02 doublet and polystyrene foilwith the micro-reflectance unit and the empty cell in position. The frequencies are correctto within f2 cm-1.The path length was determined using the interference fringes of theempty cell. For every transmittance curve of HDO in HzO the spectrum of pure H20 atequal temperatures and pressures had to be determined. Fig. 1 gives an example for 400°Cand 3900 bars. Curve 1 is the transmittance of pure H20 and sapphire ; curve 2 representsthe solution ofHDO in H20 with a path length as before. Curve 3 was determined withoutthe Pt-Ir mirror; it represents the black-body radiation of the heated cell filled with the*Hamood EngineeringTFarbwerke Hoechs110 INFRA-RED ABSORPTION OF HDOHDO+H20 mixture together with the light reflected by the inner surface of the sapphire.From such curves the extinction coefficient IC of HDO and the integrated band intensity Bhave been evaluated using the relationsv2 .=I Icdv.Vi -M is the average molecular weight of the mixtures, X ~ O the mole fraction of HDO usingan equilibrium constant of 3~96~3~ 1 5 ~ ~ 2 0 denotes the density of water in g/cm3 at temperatureT and pressure P. d is the path length in cm and JO and J in % are the transmittance valuesaccording to fig.1.RESULTSThe HDO absorption was determined in the range 2200-2900cm-1. At 30, 100,200 and 300"C, different pressures selected to produce water densities of 1.1, 1.0 and0.9 g/cm3 were applied. At 400°C nine different pressures between 48 and 3900 barswere used. Table 1 gives a compilation of experimental conditions together withthe main properties of the spectra observed.TABLE ABSORPTION OF THE OD-VIBRATION OF HDO (8.5 MOLE % IN H20). EXPERI-MENTAL CONDITIONS, BAND MAXIMA, HALF-BANDWIDTHS, MOLAR EXTINCTION COEFFICIENTSAND INTEGRATED INTENSITIESTOC4004004004004004004004004003002003002001003010030Pbar489614419324128010652055390021 8060050002800100010044003080.Pg/cm30-0 1 650.0360.060.0950.150.3 10.70.80.90.90.91.01.01.01.01.11.1dcm0.10-10.10.10.13.5 x 10-33.5 x 10-33.5 x 10-33.5 x 10-33.5 x 10-33.5 x 10-33.5 x 10-33.5 x 10-33.5 x 10-33-5 x 10-33-5 x 10-33.5 x 10-3v maxcm-12719271 526602655264226372619261 3260525952578258725682540250725352505A v tcm-1I-157148152150157167-153195195168-K X 10-3cm*/mole-4.5610152122232835293342554556BX 10-5cm/mole78.510121828384145-58708599Fig. 2 demonstrates the variation of the OD-absorption with temperature atconstant density.The decrease of frequency and maximum extinction was alreadyexpected from the isobaric infra-red 2, 5 and Raman 1, 6 observations. At densitiesof 0.9 g/cm3 and higher, no trace of a second absorption is detectable. At lowertemperatures the shape of the bands is almost Gaussian, a fact which has been empha-sized by Falk and Ford.2 Above 200°C, however, the shape becomes increasinglyasymmetric. The wings of the bands seem to disappear always in the same frequencyregion ; only the maxima are shifted to higher frequency with increasing temperatureE .U. FRANCK AND K . ROTH 111The isothermal behaviour of the OD-absorption at one supercritical temperatureis shown by fig. 3. At the lowest density, which is 28 times the density of water--50 X 10’I . I * l . l . l . , 2200 2800 2600 2400r. _ _ _ I iv25001 1 , , 1 , , , , , 1 1 ,2aoo 2 6 0 0 2400 2 200v [cm-11FIG. 2.-Molar extinction coefficient K in crnz/mole of the OD-stretching vibration as a function offrequency v in cm-1. Absorption curves at constant densities of water of 0.9, 1.0 and 1-1 g/crn3and at temperatures of 30, 100,200,300 and 400°C.2e00 2 6 0 0 2 4 0 0K2 8 0 0 2600 2400v [cm-11FIG. 3.-Molar extinction coefficient K in cm2/mole of the OD-stretching vibration as a function offrequency v in cm-1 at a constant temperature of 400°C and at different densities of water (glcm3).Note the different base lines of the individual curves.vapour at normal boiling conditions, the R-, Q- and P- branches of the OD-vibration band are clearly observable.The peak of the Q-branch is at 2719 cm-1.Benedict, Gailar and Plyler give 2720 & 5 cm-1 for dilute HDO gas at room tempera-ture. This Q-branch was no longer observable in the present work at densities highe112 INFRA-RED ABSORPTION OF HDOthan 0-1 g/cm3, e.g., at pressures beyond 200 bars. The structures in the P- andR-branches at 400°C and 0.0165 g/cm3 (48 bars) are in accord with the contours ofthe vibration-rotation band of the free HDO molecule observed in the gas at atmos-pheric pressure.16. 17DISCUSSIONThe temperature dependence of the frequency Vmax, of maximum absorption andof the integrated intensity B is shown in fig.4. The results for Vmax at a density of1.0 g/cm3 corroborate the frequency increase found by Falk and Ford 2 at isobaricconditions. The decrease of B with temperatures was expected from earlier observa-tions. Fig. 5 presents the density dependence of Vmax and of B at 400°C. The curvesare almost linear between 0.2 and 0-9 g/cm3. No effect of the critical density (0.32g/cm3) is observable.0 I 2 0 0 400 IvrE:XQI IT K IFIG. 4.-Frequency vmax (in cm-1) of the absorption maximum and integrated intensity B (in cm/mole) as a function of temperature ; --- results of Falk and Ford 2 at atmospheric pressure. Thecurves are for constant density of water of 0.9, 1.0 and 1-1 g/cm3 respectively.The curves in fig.4 and fig. 5 suggest that a simple relation between 7max andBmay exist. This assumption is verified by the plot of fig. 6. All the experimentalpoints, although extending over nearly 400°C and over a wide region of densities,are lying within the range of experimental uncertainty on one curve, which is onlyslightly curved. The extrapolated value of V m a = 2670 cm-1 for B = 0 is lowerthan the frequency of maximum absorbance of the observed Q-branch of free watermolecules at the lowest density investigated (see fig. 3). One might presume that theextrapolated value of Vmax reflects the combined influence of the surrounding polarmolecules on the OD-groups without hydrogen bonding.Other authors haveobtained curves of the type of fig. 6 by compiling frequency shifts and intensitychanges for several different hydrogen-bonded compounds in one diagrain.2E. U. FRANCK AND K. ROTH2 7 0 0 -.1 -113** - 0 -branch 4 0 0°C-2700u €2 ’ 2 6 0 0 -1 1 1FIG. 5.-Frequency Vmm (in cm-1) of the absorption maximum and integrated intensity B (in cmlmole) as a function of density of water (in glcm3) at 400°C. In the upper left corner the maxima ofthe Q-branch at densities of 0.0165 and 0-036 g/cm3 are indicated (see fig. 3).- 2 6 0 00 so I0033 x 1 O-5[cm/mole]FIG. 6.-Frequency Vma (in cm-1) of the absorption maximum plotted against the integrated intensityB (in cmlmole).I, points at 400°C and different densities of water : *, points at different tempera-tures and density 1*0g/cm3; +, maxima of the Q-branch at densities of 0.0165 and 0.036 g/cm3(see fig. 3) ; ---, extrapolation114 INFRA-RED ABSORPTION OF HDOThe frequency increase of the maximum absorption of hydrogen bonded OH-and OD-vibrations has been related to an increase of the 0-0 distance of thehydrogen bond. If such a relation does apply here, the curves of fig. 2 and fig. 5imply, that the 0-0 distance increases with increasing temperature although thedensity of the water remains constant.At all densities of water higher than 0.1 g/cm3 and in the whole region of tempera-tures up to 400"C, there is only one absorption maximum for the OD-vibration ofHDO in H20 having a simple shape and no shoulder.Its frequency graduallyincreases from 2505 cm-1 at 30°C and 1.1 g/cm3 to 2655 at 400°C and 0.095 g/cm3.If one considers this absorption as being characteristic for hydrogen-bonded OD-groups and assumes that free OD-groups should be indicated by a separate absorptionaround 2700cm-1, then almost all of the oxygen-hydrogen groups in this rangeshould be to some extent hydrogen bonded. Only at densities below 0.1 g/cm3 at400°C is the occurrence of free water molecules clearly demonstrated by the rotationalstructure of the water spectrum. It is reasonable to assume a wide distribution ofhydrogen bonds with different energies and 0-0 distances corresponding to thebroadness of the absorption.One might presume that the character of the spectrumat higher density is entirely a consequence of the increased dipole interaction becauseof closer intermolecular approach. This, however, would not account for theincrease of the integrated intensity of the band by a factor of 14 when proceedingfrom 0.0165 to 0.9 g/cm3 at 400°C. For HCl the density increase caused only atwofold or threefold rise of intensity.18Thus, the infra-red spectrum of the OD-vibration of HDO in H20 gives noindication of non-hydrogen-bonded OD groups or of defined small clusters of watermolecules at a density higher than 0.1 g/cm3. In accordance with the conclusion ofWall and Hornig 1 and of Falk and Ford 2 these spectra are considered as supportfor the continuum model of liquid and of dense supercritical water.1 T.T. Wall and D. F. Hornig. J . Chem. Physics, 1965, 43, 2079.2 M. Falk and T. A. Ford, Can. J. Chem., 1966,44, 1699.3 C. A. Swenson, Spectrochim. Acta, 1965, 21, 987.4 J. G. Bayly, V. B. Kartha and W. H. Stevens, Infra-red Physics, 1963, 3, 221.5 K. A. Hartmann, J. Physic. Chem. 1966,70,270.6 G. E. Walrafen, J. Chem. Physics, 1966,44, 1546; 1964,40, 3249.7D. P. Stevenson, J. Physic. Chem., 1965, 69, 2145.8 R. E. Weston, Spectrochim. Acta 1962, 18, 1257.9 W. A. P. Luck, Ber. Bunsenges. Physik. Chem., 1965,69,626.10 W. C. Waggener, A. J. Weinberger and R. W. Stoughton, 149th Nat. Meeting A.C.S., 1965.11 R. Goldstein and S. S. Penner, J. Quant. Spectr. Rad. Transfer, 1964, 4, 359,441.12 M. R. Thomas, H. A. Scheraga and E. E. Schrier, J. Physic. Chem., 1965,69,3722.13 R. D. Waldron, J. Chem. Physics, 1957, 26, SO9.14 S. Maier and E. U. Franck, Ber. Bunsenges. Physik. Chem., 1966,70,639.15 R. E. Weston, J. Chem. Physics, 1965, 42,2635.16 W. S. Benedict, N. Gailar and E. K. Plyler, J. Chem. Physics, 1956, 24, 1139.l7 E. F. Barker and W. W. Sleator, J. Chem. Physics, 1935,3, 660.18 W. West, J. Chem. Physics, 1939,7, 795.19 W. F. J. Hare, and H. L. Welsh, Can. J. Physics, 1958, 36, 88.20 G. C. Pimentel and A. L. McClellan, The Hydrogen Bond, (W. H. Freeman and Comp. San21 VDI-Steam Tables, Springer-Verlag and Verlag R. Oldenbourg, Berlin-Gott ingen-Heidelberg-Francisco and London, 1960), p. 96.Miinchen 6th ed., 1963

ISSN:0366-9033    

DOI:10.1039/DF9674300108

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

Spectroscopic Studies Concerning the Structure and theThermodynamic Behaviour of H20, CH30H and C2H50HBY WERNER A. P. LUCKHauptlaboratorium der Badischen Anilin- & Soda-Fabrik AG67 Ludwigshafen/Rh., GermanyReceived 16th Jantrary 1967The i.-r. spectroscopic determination of non-H-bonded OH-groups in water and alcohols are inagreement with the thermodynamic behaviour of these liquids. It is possible to calculate the specificheats, the heats of vaporization, the surface energies and the density with these spectroscopic resultstogether with a simple model of the liquids. This method also gives the density maximum and theminimum of the specific heat of H20. The conformity with the thermodynamic properties showsthat these measurements can be employed to develop better theories of liquids and that the approxi-mation methods used for the interpretation of the i.r.spectra are satisfactory.Infra-red spectral-7 of H20, CH30H and C~HSOH in the temperature regionfrom - 50°C to + 400°C and comparison with solution spectra 8 have shown that inliquids these compounds have less non-H-bonded OH-groups than the most theoriesof liquids claim. An exact interpretation of the i.-r. spectra is not possible.9-1450 100 200 2I I300 350temperature, "CFIG. 1 .-Percentage P of non-H-bonded H20-molecules determined by 0.9488 p- and 1.140 p4-r.bands.5~ 6 ; middle line : mean values.The spectra of liquids and solutions are similar, but in liquids there are many morebands 14 of energetic uiifavourable H-bonds from Pople's point of view.15 Therefore,only approximation methods can be used to interpret the spectra.We think the bestapproximation method is to determine the percentage of non-H-bonded OH-groups(fig. 1). To take into account the cooperative effects 39 16 we need a model of the11116 1.R.-SPECTRA AND BEHAVIOUR OF H20, CH3OH AND CZH~OHwater structure (fig. 2) with fissure planes of free OH-groups (compare the Bjerrumdefects in the theory of the dielectric constantl7) and closed H-bonds. Assuming aflickering cluster model we have intermediate states of H-bonds with unfavourableangles or distances.9~ 11, 18 To prove the applicability of this method we attempt inwater modelH-bonds ; Q, non-H-bonded 0-H groups at the cluster surfaces.FIG. 2.-Simplified cluster model of the water structure at about 0°C.0-0, 0-atoms bound withthis paper to calculate the thermodynamic behaviour from the spectroscopic resultsassuming a two-species model of free OH-groups and ice-like H-bonded groups.Energetically unfavourable H-bonds are partially counted as free OH and lessperturbed H-bonds as linearly bonded.SPECIFIC HEATTo further our knowledge of the liquid structure we try first, to obtain a betterapproximation by refinement of the spectroscopic methods ; secondly, to find whetherthere is any thermodynamic behaviour in contradiction to this simple model.WATERThe specific heat Cv of H20 of our model isCv, vapour id is the specific heat of the intermolecular degrees of freedom. dP/aTis thechange of the number of H-bonds per degree and can be determined from fig.1.To obtain good results we made use of a mirror derivativemeter." AUH is the H-bondenergy per mole ; we take AUH = AUs - WH = 8 kcal/rnole. 2 is the coordinationnumber in the liquid state ; a is a constant, R = 1.986 cal/mole ; ZaR/2 is the contri-bution of the intermolecular forces of non-H-bond type to the specific heat, and is notexactly known. In addition, we assume that(2)and use the experimental temperature function (1 - P) of fig. 1. Fig. 3 shows thatZ(T)aR/2 = (4/2)( 1 -P)R,*Ott, Kempten, GermanyW. A . P. LUCK 117eqn. (1) gives good agreement between the experimental values of C and the valuesderived from our simple model. The disagreement in the temperature region525-625°K is of the similar magnitude to the approximations in eqn.(2). This isexpected of our model because the cluster size in this temperature region is smallerthan 6 molecules. We should correct the value 2 = 4, and take account of theT["KIFIG. 3.-Specific heats of water - obs.19; 0, calc. with 0.95 p band and eqn. (1); A, calc. with1-14 p band and eqn. (1).number of free molecules, which are neglected in our model. The spectroscopicanalysis has shown that the number of free molecules can be neglected (smaller than1 %) for t<200°C ; its increase is more than 1 % for t>200"C. Other disagreementsin fig. 3 are of the same magnitude as the errors in the spectroscopic method. Table 1TABLE 1 .-SPECIFIC HEAT OF H20!! AU 6T 2(1--P)R cv thwr.273 63 23 6373 6.1423 6.1473 6.15523 6.2573 6.3623 6-4638 6.40 .9 4 ~8.89.29.310.310.5512.219.531.276.71-lp7.758.59.912.114-517.320.329.651-20.94,~3.563.43.152.92.62.41.981.41.11-1p3.63.43-162.92.62.21.71.10.90 . 9 4 ~18-4618.618-5519.319.320.827-839.084.21-lp18.118.219.121.123.225-728-337.159.5shows the magnitude of the different factors in eqn. (1). aP/aT of the 1 . 1 , ~ band,which gives our best values, is nearly constant up to 35°C and then increases slowlyfor increasing temperatures. Thus, eqn. (1) and (2) together with fig. 1 give a minimumof the specific heat at about 35°C in agreement with experiments.This is caused bythe factor 2R(l-P)cal/mole deg. which decreases from 0°C to 35°C from 3.62118 1.R.-SPECTRA AND BEHAVIOUR OF H20, CH3OH AND C2HsOHto 3-481 cal/mole deg. The difference is 0.14 The experimental difference isCp(OoC)- Cp(35"C) = 18.1 1 - 17.95 = 0.16 cal/mole deg.We have shown that additions of salt change P in the sequence of the lyotropic ionseries.4 Small change of P and especially of @/dT should change the temperatureof the specific heat minimum. The specific heats of salt solutions are in keeping withthis conclusion.29CH30H A N D C2H-jOHExperimental values for Cp for alcohols are not so uniform and values of Cv andfor higher temperature are not available. Fig. 4 shows the spectroscopically deter-mined values of free OH for CH30H and C~HSOH.We now calculate in a similarL-100t ["CIFIG. 4.-Percentage P of non-H-bonded CH30H and C2H50H molecules in the liquid state, deter-mined by the 1.4 p band.14 *FIG. 5.-Simplified model for the liquid state of alcohols :chains of H-bonded molecules perpendicular to the paperplane, coordination number of these chains in bulk, 2 =6 ; at the surface, X = 4.way the C,-values. We take as a model CH30H chains of H-bonded molecules.Every chain has six neighbour-chains (fig. 5). It is possible to obtain good values ofCv with the formula,(3)ap 8cv = c ~ , vap. i d . + z A U H - f ( l -pbR, 2and a value A& = 4 kcal/mole.*The peaks near Tc are caused by the density gradient in the gravity field.6 For T>Tc theoptical density of the CH-band is nearly constant in the upper side of our cell.The OH-bandis constant only for T>T, (H,O). That means that for alcohols at Tc not all H-bonds are openW. A . P . LUCK 119The agreement between Cv, expt. and Cv determined by eqn. (3) is of similarmagnitude as the errors in our spectroscopic values P, of the differences of the experi-mental values and the uncertainty between C, and Cv in the liquid state (fig. 6).T["KIFIG. 6.-Specific heats of CH30H and C2H50H. 0, obs. value of different authors;20 A, calc.with fig. 4 and eqn. (3).Most doubtful is the factor ZaR/2. Fortunately it is the smallest one, and itsinfluence is not very large. (Our assumption has some similarity to that of Kincaidand Eyring,21 who have assumed that some rotational degrees of freedom becomelibrations in the liquids and increase the specific heat by about 1 cal/mole deg.).HEAT OF VAPORIZATIONWATERThe inner vaporization energy Lv can be determined fromL~ = (l - p ) A u S + p wH- (l - (Uvap.id. - Uvap. real). (4)(1 -P)AUs is the energy of the closed H-bonds. We take AUS = AUE + WH =11-64 kcal/mole, and AUs = L,("C)+LM is the sum of H-bond AUH and van derTABLE 2.-mAT OF VAPORIZATION H20 (KCAL/MOLE)(l-P)AUs T"K 0.95c( 1.lp (1-p) (1-p)Aus PWH +PW, (I--PPRT wnxil Lv d c . Lvexpt.300 309 o m io.ia 0.45 10.63 1-07 o 9.6 9-8353 363 0.815 9-49 0.67 10.16 1.16 0.03 9.0 9-07410 416 0.745 8.67 0.93 9.60 1-22 0.07 8.3 8-4507 4-87 0.625 7.27 1.36 8-43 1.22 0.32 7.1 7.2573 535 0.5 5.82 1.8 7.62 1.11 0-8 5.7 5.72617 594 0.375 4.36 2.27 6.63 0.91 1.47 4.2 4640 631 0.25 2.91 2.7 5.61 0.64 2.4 2-47 2.4643 640 0.2 2-33 2-91 5-24 0.51 3.54 1-2 1.6610 585 0.4 4.66 2.18 6-84 0.95 1.3 4-59 4.3628 608 0.333 3.88 2.4 6.28 0.82 1-76 3.7 3.5Waals interaction WH.We obtain this value as the sum of the heat of vaporizationat 0°C and the energy of melting LM. (LM = 1.43 kcallmole is the sum H-bond120 1.R.-SPECTRA AND BEHAVIOUR OF H2O, CH3OH AND C2H5OHof the energy 0-8 kcal/mole to open 10 % of the H-bonds, the energy 0.35 kcal/moleto change 20 % of the H-bonds in energetically unfavourable bonds and the changein the energy of the van der Waals interaction of about 0-25 kcal/mole). PWH isthe van der Waals interaction of the free H-bonds. We take WH = 3.64 kcal/mole.2016 -b -f [“CIFIG.7.-Heat function H and internal energy U above, saturated vapour state ;I9 below, liquidstate ;I9 straight lines, extrapolated ideal gas state.I0 0 a p tiTIT,FIG. 8.-Intemd heat of vaporization LO of H20 19 0, expt. ; -. calc. with 0.9 p and 1-1 p bandW . A . P . LUCK 121The factor (1 -P)2RT has the same meaning as in the calculation of C,. It givesthe heat content of the intermolecular degrees of freedom. The experimental valuesL,, expt. give the energy to transfer 1 mole from the liquid to the real vapour state.Our calculation gives the energy of transfer of liquid to the ideal vapour state. There-fore we require the value of Wreal in eqn. (4). We obtain it from the difference betweenthe real heat content of the vapour from thermodynamic tables19 and the linearextrapolated ideal vapour state (fig.7). Table 2 shows the different factors of eqn. (4).Table 2 and fig. 8 show that we obtain a good agreement between eqn. (4) combinedwith the experimental P-values, and the experimental L, values within the limits oferror of our spectroscopic method.CH30H AND C2H50HIn a similar way, we calculate L, of CH30H and C2HsOH from the equation,For alcohols we have only found L,, expt. for temperatures less than 130°C. Thereforewe do not need a correction for the real gas state, which is only important for highertemperatures with higher vapour pressures. We get Lv, expt. from Lp -PAY = L,, expt.Lv=(1-P)US+PW-8/2RT (5)We take AUs(CH30H) = 1C187 kcal/mole ; AUs(C2H50H) = 12 kcal/mole."0.4 05 06 0.7 08TIT,FIG. 9.Tnternal heat of vaporization of CH3OI-I and C2H50H x , expt.22; 0, calc. by eqn.(6)and fig. 4.These values are obtained by extrapolation of L,, expt. to the melting point andadding the heat of melting. Wis the van der Waals interaction of the free OH-groups.We take W = 3.6 kcal/mole, the same value taken for H20.Fig. 9 shows that eqn. (5) together with our experimental P-values are in goodagreement with the experiments. Especially the greater change of L, with tempera-tures above 50°C (T/Tc>O-63) agrees with the greater increase of P with T in thistemperature range.In the same way we can write eqn. (5) for lower P-values :L, = (1 - P)AUH + (612) WD + P W - (8/2)RT ( 6 )with AUE = (4 + 3.6) kcal/mole = 7-6 kcal/mole ; WD(CH~OH) = 1-03 kcal/mole ;WD(C~H~OH) = 1.47 kcal/mole.In eqn. (6) we separate the van der Waals inter-action WD of the rest of the molecules in the H-bonded chains. This interaction3-6 kcal/mole of the H-bonded OH groups is included in UH122 I.R.-SPECTRA AND BEHAVIOUR OF H20, CH30H AND C2H5OHSURFACE ENERGYThe surface energy Ua is defined by 23,249 14Uu = O M - TZCTMldT, (7)where OM is the molar surface tension. To calculate Ua we need a specific model.WATERWe take for water a model of tridymite-like clusters and calculate the energycontent of the (0001) basic plane for clusters of 324 and 96 molecules. The surfaceenergy is given byu u = C(1 -Wus +PWH)lc~uster- [(I - P O W U S +PBWH-Olsudace,u, = Ucluster - uo.(8)PO is the amount of non-H-bonded OH-groups at a surface.PB is the amount ofnon-H-bonded OH-groups at the border-line, which have van der Waals interactionwith neighbouring clusters. We obtain the values given in table 3.TABLE 3.-sURFACE ENERGY OF WATER : UCT (KCAL/MOLE)uu expt. VO uo t.,C cluster32 324 0.805 9.37 0.15 -0.30 9.22 1.41(1*59) 1.6183 96 0.75 8.73 0.22 -0.22 8.73 144(1*62) 1.66size (I--po) (1-po)AuS PBWH 0A comparison between table 2 and 3 shows Uu is similar in magnitude to the van derWaals interaction, which is considered in our model only in a crude manner. Secondlythe results of table 3 are differences of two values of similar magnitude therefore theaccuracy of table 2 cannot be good (see fig.10). In table 3 we made two assumptionsabout the van der Waals forces. The van der Waals interaction of the surfacemolecules is 36/2 kcal/mole, because the interaction with the vapour state is neglected.There are many molecules in the surface with 4-H-bonded neighbours. In ourcalculation we assumed the interaction of such molecules to be 11.64 kcal/mole.Therefore we made the correction 0, assuming that the van der Waals interaction ofsuch molecules is only 75 % of the interaction inside a cluster. 0 is the percentageof four-coordinated molecules multiplied by 3-64/4 kcal/mole. The van der Waalsinteraction of three-coordinated molecules at the surface is only included in ourcalculation as the interaction in the three H-bonds, i.e.corresponding to 75 %.The interaction of these 3-coordinated molecules should be smaller by the smallerinteraction with the second next neighbours. Assuming this effect to be 10 % of3.6 kcal/mole we get the results in parenthesis in table 3. In addition, the surfaceenergy of the other planes of the tridymite-like clusters would be larger. The (0001)planes in table 3 are the planes with the highest energy content and would be preferredin a crystal-like state. X-ray investigations have shown that the ice structure grownat the surface of undisturbed water is orientated with (0001) parallel to the watersurface.30 This may point to the preference of this plane at the surface of liquidwater. But in the flickering cluster model we would expect a fraction of energeticunfavourable faces too.It would be unwarranted to give exact values of all theseeffects in our rough model.However, we can state that our model is not in disagreement with the observedsurface energy. The increase of U, with temperature is only known for liquids withH-bonds. Our model gives the possibility to understand this anomalous effect bW. A . P . LUCK 123the cluster structure. It is only possible to make this rough approximation in the lowtemperature region where there are large clusters. For higher temperatures we haveto take into account that the partition function of the cluster size would be differentat the surface and in bulk. The percentage of unfavourable H-bonds would be'0.4 0.5 0.6 07 0.8 09 10TITCFIG.10.-Surface energy U, expt. values : 0, C6HsC1; 0, H2O ; -, C6H6 ; X , C2H50H ; 0, cc14 ;A, CH30H ; calc. with eqn. (8) A, H2O ; calc. with eqn. (lo), A, C2H50H ; A, CH30H.different between the surface and in bulk too. Only for large cluster sizes may weexpect that both these effects are not large. The increase of U, in the temperatureregion T/Tc>O-7 can only be understood by these effects. The decrease of U, inthe region T/T,> 0.9 for all liquids is due to the increase of the vapour pressure in thisregion, i.e., the difference of the coordination numbers at the surface and in bulkdecreases.ALCOHOLSWe calculate the surface energy of alcohols from eqn. (9),u, = [( 1 -P)AUH + (612) WD +P W] - [(I -P)oAUH + (4/2)w~ -k 2PoW/3] (9)(see our model, fig.5). For low temperatures, for low P-values, we put P = Po andFig. 10 shows that eqn. (10) together with the spectroscopic P-values (fig. 4) agreeswell with the experimental values in a way not expected from our rough model withP = Po.u, =wD+*Pw (10)4-COORDINATION NUMBERFor normal liquids, eqn. (1 1) is valid :I4u, = C(Z - X)/21(3/2)RTcwhere 2 is coordination number in bulk, and Xis coordination number at the surface124 1.R.-SPECTRA A N D BEHAVIOUR OF H20, CH3OH AND CzHsOHFor a face-centred cubic configuration one would expect 2- X = 3. For c c 4 ,C6H6, C ~ H S C ~ and C~HS-O-C~H~, 2- X is 2-88-2.76.14 This may mean that forthese ideal liquids the coordination number 12 is valid with defects of 5-10 %.(The percentage disagreement of the value 2- X = 3 is of similar magnitude as thepercentage volume change during melting.)For H2O and the alcohols, U,, is temperature dependent in contradiction toeqn.(9) and RTc is not a measure of the interaction energy between two molecules.For an interaction energy U between two H20 molecules of 4.9 kcal/mole we get2U,,/U-0.6. For ice, 2 - X = 0.5 The quotient 0.6 agrees with the assumption oflarge tridymite-like clusters and the spectroscopic result and the neglect of the presenceof free molecules. We would expect that free molecules would be concentrated atthe surface and would have higher values of 2UJU. X-ray scattering results25give the mean coordination numbers :Z(1*5"C) = 4 4 ; Z(83"C) = 4.8.Assuming 2 = 4-4-2 and R = 6-7, R is the coordination number at the cluster-surfaceinside the liquid andwhere p is the fraction of cluster surface molecules.For p - 2PpR+(l-p)Z = 2, (12)Pcalc. (83") N 0.12-0.20 ; Poba. (83")0*18.The observed P values would mean a mean cluster radius r in the tridymite-like clustermodel of F(0"C) N 14A ; F(30"C) - 10 A ; F(85"C) - 6 A. The X-ray experiments showordered zones around a standard molecule up to a distance of r(l-5"C)>8.5A;r(30"C) > 8.5 A ; r(83"C) N 7A.The radial partition curves of masses around a standard molecule given by theX-ray method agree with the assumption that the disorder mainly causes a decreaseof the-distance between the second-next neighbours. This could mean an effect ofthe H-bonds of unfavourable angles, which would not greatly change the distancebetween the first-next neighbours.In a similar way, an opening of H-bonds bymolecular rotation could decrease the distance to the second-next molecules.DENSITYThe density-temperature function of H20 was interpreted by a two-species model ;26more disordered groups were found than estimated from our experiments. Thisarises because the assumption that the density of the ordered volumes is the sameas in ice. This neglects the energetically unfavourable H-bonds, which the spectrashow are present. Therefore we believe that in a two-species model we should assumea higher mean density of the ordered zones than in ice. Our spectroscopic resultscannot give direct information on the density of the different H-bonded and non-H-bonded zones.For a model similar to that used to deduce the other eqn. of thispaper, we can evaluate the density curve (fig. 11) by the formula :l/p = (1 - P)v,+ PV,;volume of ice-like regions :&(crn3g-l) = 1.0350(1+ 1.741 10-4t(0C)),volume of disordered regions :VB(cm3g- ') = 0*64006(1+5-7249 10-3t(0C))W. A. P . LUCK 125The volume expansion of the H-bonded groups is taken to be similar to the expansionof ice. The expansion of the unbonded groups is adjusted from the experimentaldensity at 100°C. Eqn. (14) gives a rough approximation of the density (fig. 11).The deviation for higher temperatures means that we cannot neglect free moleculeswith a higher volume in this region and one cannot assume a linear volume expansiontill Tc (this has to be so for normal liquids, too).26 The known density maximum at1 I4°C may be understood by eqn.(14), which, however, is very sensitive to smallchanges of P. With P (0°C) = 0.088 and P(1O"C) = 0.101, in agreement with ourmeasurements, the density maximum is given by eqn. (14) too (see fig. 11). Thissensitivity of the density maximum onP agrees with the observation that the maximumis changed by salt additions.27 We have found salt additions change P t00.4 Wedid not search for the best constants in eqn. (14) ; it is possible that there are betterones.DISCUSSIONEvery equation of this paper by itself is somewhat hypothetical ; but the combina-tion of them with the same assumptions has more significance and are also consistentwith each other.In this way, fig. 1 and 4 would be interesting if they were theoreticalfigures; but we recall that they are experimental ones. We do not believe that ourequations and our model are correct for the liquid structure. New experiments anddiscussions of the relaxation times z should give new information to improve thisrough model126 I.R.-SPECTRA AND BEHAVIOUR OF H20, CH3OH AND C2HsOHThe i.-r. ice band-maximum changes little with decreased temperatures ; i.e.,the energy of the H-bonds of ice increases slightly at lower temperatures. But zincreases with lower temperatures. This could mean that z is coupled with a proton-rotation.The smaller proton mobility in liquid water in comparison with that in ice has toconform with the presence of large clusters.The mobility at the cluster surfacesmay determine the mobility in the liquid state.An interpretation of a sharp melting point in a model with small change of thedegree of order is given in ref. (28) ; and an explanation of the anomalous behaviourof € 3 2 0 in the critical region above Tc in ref. (6). Solubility experiments of ions ororganic molecules can also be discussed in terms of the cluster model. We mustexpect primary and secondary hydrates of ions and of lyophilic molecules 33 4 insuch solutions.The behaviour of H-bond systems are more complicated than the normal liquids.But the theory of the H-bonded liquids is simplified because the effect of the second-next neighbours is easier to neglect. We may hope that this new knowledges ofthe H-bonded liquids is also interesting for normal liquids too.The similarity of theordering effects determined by X-rays in both systems supports this statement.I thank my coworkers especially Mr W. Ditter for their assistance and for helpfuldiscussions.1 W. Luck, Z. Elektrochem., 1962, 66, 766.2 W. Luck, Ber. Bunsenges. physik. Chem., 1963, 67,186.3 W. Luck, Fortschr. chem. Forschung, 1964, 4, 653.4 W. Luck, Ber. Bunsenges. physik. Chem., 1965, 69, 69.5 W. A. P. Luck, Ber. Bunsenges. physik. Chem., 1965, 69, 626.6 W. A. P. Luck, Ber. Bunsenges. physik. Chem., 1966, 70, 1113.7 W. A. P. Luck, Plzysik. Blatter, 1966, 22, 347.8 W. A. P. Luck, Spectrochim. Acta, in press.9 T. T. Wall and D. F. Hornig, J.Chem. Physics, 1965,43,2079.10 M. Falk and T. A. Ford, Can. J. Chem., 1966,44, 1699.11 V. Vand and W. A. Senior, J. Chem. Physics., 1965,43, 1869, 1873, 1878.12 G. Bottger, H. Harders and W. A. P. Luck, J. Physic. Chem., 1967,71,459.13 W. A. P. Luck, Can. J. Chem., in press.14 W. A. P. Luck, Bey. Bunsenges. physik. Chem., 1967, in preparation.15 J. A. Pople, Proc. Roy. SOC. A , 1951, 205, 163.16H. S. Frank, Proc. Roy. Soc. A, 1958, 247,481. H. S. Frank and W. Y . Wen, Disc. Faradayl7 C. Jacard, Ann. N.Y. Acad. Sci., 1965, 125, 390. A. Steineman and H. Granicher, Helv.l8 W. Luck, Naturwiss., 1965, 52,25, 49. W. A. P. Luck, Naturwiss., 1967, 54, inpress.19 J. H. Keenan and F. G. Keyes, Thermodynamic Properties of Steam (John Wiley & Sons,New York, 1936).20 0.Maass and L. T. Waldbauer, J. Amer. Chem. Soc., 1925,47, I. G. S. Parks, J. Amer. Chem.SOC., 1925,47, 338. K. K.Kelley, J. Amer. Chem. Suc., 1929, 51, 180. E. F. Fiock, D. C. Ginnings and W. B. Holton,Bur. Stand. J. Res., 1931, 6, 886. W. Timofejew, Iswiestja d. Kiew. polyt. Inst., 1905, I. Diss.(Kiew 1905), 340 S. J. C. M. Li, K. S. Pitzer and E. V. Ivash, J. Chem. Physics, 1955,23,1814.G. S . Parks, J. Amer. Chem. Soc., 1925, 47, 338. K. K. Kelley, J. Amer. Chem. SOC., 1929,51,779. E. F. Fiock, D. C.Ginnings and W. B. Holton, Bur. Stand. J. Res., 1931,6,886. G. E. Gibson, G. S. Parks andW. M. Latimer, J. Amer. Chem. Suc., 1920, 42, 1537. Blacet, Leighton and Bartlett, J. Physic.Chenz., 1931, 35, 1935. A. Battelli, Rend. Lincei, 1907, 16, [l], 243, Cim., 1907, 13, 418.Regnault, Mem. I’Acad., 1562, 26, 262. Hirn,Ann. Chim. Physique, 1867,10,32. K. A. Kobeand R. E. Pennington, Petr. R e f , 1950, 29, nr. 9, 135.SOC., 1957, 24, 133.physic. Acta, 1957, 30, 554.C . Drucker and H. Weissbach, Z. physik. Chem., 1925,117,223.G. S. Parks and H. M. Huffmann, J. Physic. Chem. 1927, 31,1842.W. Sutherland, Phil. Mag., 1888, 26, 298.H. A. G. Cherrin, Petr. Ref., 1961, 40, 127W. A . P . LUCK 12721 J. F. Kincaid and H. Eyring, J. Chem. Physics, 1938, 6, 625.22 E. F .Fiock, D. C. Ginnings and W. B. Holton, Bur. Stand. J. Res., 1931, 6, 886.23 G. Kortum, Einfuhrung in die chem. Thermodynamik (Verlag Chemie, Weinheim, 1960), 3.24 K. L. Wolf, Physik und Chemie der Crenzfiichen (Springer-Verlag Berlin-Gottingen-Heidelberg,25 J. Morgan and B. E. Warren, J. Chem. Physics, 1938,6,666. M. D. Danford and H. A. Levi,0. J. Samoilow, Die Struktur von wassrigen Elektrolyt-Aufl., S. 420.1957).J, Amer. Chem. Soc., 1962, 84, 3965.fiisungen (Teubner, Leipzig, 1961), S. 38.26 K. Grjotheim and J. Krogh-Moe, Acta Chim. Scand., 1954, 8, 1193.27 R. Wright, J. Chem. SOC., 1919, 65, 119.28 S. E. Bresler, Acta Physicochim, 1939,10,491.29 M. Eigen and E. Wicke, 2. Elektrochem., 1951, 55, 354.30 U. Yoshida and T. Tsuboi, Mem. Sci. Kyoto Uniu. A, 1929, 12, 203.J. Frenkel, Kiizetische Theorie der FfissigkeitenE. Wicke, M. Eigen and Th. Acker-(VEB Verlag Wissenschaften, Berlin ; Clarendon Press Oxford, 1957).mann, 2. physik. Chem., 1954,1, 340. Th. Ackermann, 2. Elektrochem., 1958,62,411

ISSN:0366-9033    

DOI:10.1039/DF9674300115

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

GENERAL DISCUSSIONDr. Manse1 Davies (Aberystwyth) said : The spherical-molecule liquids naturallyoccupy a special position in theoretical treatments and have great areas of interestin the monatomic liquids, metals and fused salts, but in the field of molecular liquidsthose of nearly spherical form (HCl, CC14, CH3CC13, camphor, etc.) are far fromtypical liquids. They are a major sub-set of the class I1 liquids of Hi1l.l Respectsin which they are “ abnormal ” liquids include : (i) the marked differences betweenthe activation energies for the viscosity and for the molecular reorientation (i.e.,rigid dipole relaxation) processes :AH*(z rotational) < AH*(q) ;(ii) the differences between the effective specific volume parameters b> b’ in (1,’~) =A(u - b) ; (1 /4) = A‘(v - b’), where u is the specific volume, A and A’ are (Batschinski)constants, and 4 = 2kTz ; (iii) the abnormally small enthalpy (or entropy) of crystal-lization of the spherical-molecule liquids ; (iv) other “ abnormalities ” are listed inTimmermans’ As Nora Hill has suggested, features (i) and (ii) can beused as criteria to predict that such spherical-molecule liquids will crystallize to solidshaving considerable rotational mobility in the solid (“ rotator phase ”) state, whichis the immediate cause of (iii).Although local anisotropy plays an important role in most molecular liquids thedetailed structural study of the spherical-molecule examples is greatly to be desiredand could be markedly advanced by first studying the corresponding rotator-phasecrystal feature^.^ The correlation of the changes in properties on melting, with theaddition in these cases of only translational molecular mobility, should help to unravelthe patterns in molecularly anisotropic liquids.Dr.J. N. Sherwood (Strathclyde University) said : Following their remarks uponthe similarity between the molecular properties of the rotator phase solids and thoseof the corresponding melt, Davies, and PowIes, have commented that these solidscould well serve as model systems for the study of the structure and properties of theliquid state. With this possibility in mind we have been examining the structuralproperties of rotator phase solids. Previously, comments upon the structure ofthese materials were confined to observations upon the lack of long-range order, asinstanced by X-ray diffraction ~tudies,~ and speculations upon the extreme plasticityof these solid^.^ Our initial studies of self-diffusion and plastic flow in several ofthese solids permit some further speculations on the nature of this unique phase.Self-diffusion measurements show that translational molecular mobility is high.The self-diffusion coefficient at the melting point Dm= cm2 sec-l compared toa much lower value for non-rotator phase solids, Dm* cm2 sec-l.If weconsider this phase as a solid, then the rate of self-diffusion will be proportional tothe defect concentration and hence these figures imply a considerably higher defectcontent in the rotator phase solid. The energies AHd, involved in the self-diffusionN.E. Hill, Proc. Roy. SOC. A , 1957, 240, 101 ; Trans. Faraday SOC., 1959, 55, 2000.J. Timmermans, Les Consfantes Physiques des Conzposks Organiques Cristallisks (Masson et Cie.,Paris, 1953).C. Clemett and M. Davies, Trans. Faraday Soc., 1962, 58, 1705, 1718. G. Corfield and M.Davies, ibid., 1964, 60, 10.W. J. Dunning, J. Physics Chern. Solids, 1961, 18, 21.12GENERAL DISCUSSION 129process can be rationalized on the basis of a vacancy diffusion process for bothtypes of solid.' The pre-exponential factors in the equationD = Do exp (-AHd/RT)are much higher, however, than would be expected for a normal vacancy process(Do = lo6 cm2 sec-', cyclohexane2). On the basis of this, we have suggested thatthe principal point defect in the rotator phase solids may be a highly relaxed vacancy.This would correspond to a small liquid-like region in the lattice and could result in0.1 % of the lattice being in a disordered state.Studies of diffusion in impurity-doped crystals confirm this interpretation.In such a highly disordered system, the considerable plasticity of the solid mightresult from a vacancy creep process in the single ~ r y s t a l . ~ (For metals such a mechan-ism only holds for micro-crystalline solids). Creep studies have confirmed thatthe rotator phase solids are highly plastic (100 times more so than metals at similarreduced temperatures) and that the activation energy AHc for the creep process isequal to that for self-diffusion (Pivalic acid, AHd = 21.8_+0.1 kcalmole-l, AHc =22.4+0.7 kcal mole-').The creep strain, however, is proportional to the 5th powerof the stress whereas an exponent of unity would be expected for vacancy creep.Exponents of this order are characteristic of creep taking place by a dislocation climbmechanism. Analysis of the experimental data indicates that the dislocation contentof the crystals studied is - lo5 per em2. This figure is in reasonable agreement withetch pit counts which we have made; these yield apparent dislocation contents of> lo4 per cm2. Thus, there is sufficient crystallinity in the solid for dislocations tobe present and to have an effect upon the properties of the crystals.We conclude from this evidence that the rotator phase solids are fundamentallycrystalline but that there is a high proportion of disorder in the solids.At themelting point the solid loses its remaining degree of crystallinity with the characteristic-ally low entropy of fusion, AS,<5 cal/mole deg. How close in structure thesematerials are to liquids can only be decided after further structural studies, but as anintermediate phase such studies may well shed further light upon the nature of theliquid state; the above comments refer to polyatomic rotator phase solids and notto rare gas solids. The latter appear to behave normally.'Dr. G. Caglioti, Dr. ME. Corchia and Dr. G. Rizzi (CNEN, Ispra, Italy) said:In a programme on the structure of liquid and resuming a previous in-vestigation on the crystal physics of zinc,7 Corchia, Rizzi and myself have recentlymeasured at the CNEN laboratory of Ispra the structure factor S(Q) of liquid zincat 470°C, by conventional diffraction of 1 A neutrons in a well collimated ( N 25 minarc) geometry.Our S(Q) does not exhibit any evidence for the existence of theunusual bump obtained in 1941 by X-ray diffraction in the region of wave vectortransfer Q N 1.5 A-1,8 while in the rest of the diffraction pattern the agreement betweenX-ray and neutron data is only qualitative. The radial distribution function 4nr2g(r)obtained by Fourier inversion of our neutron data is consistent with the short-rangeJ. N. Sherwood, Conference on Point Defects in Non-metallic Solids (Proc. British Ceram. SOC.,19671, to be published.G.M. Hood and J. N. Sherwood, Molecular Crystals, 1966, 1, 97.H. M. Hawthorne and J. N. Sherwood, to be published.W. J. Dunning, J. Physics Chem. Solids, 1961, 18, 21.P. Ascarelli, Physic. Rev., 1966, 143, 36.P. Ascarelli and G. Caglioti, Nuovo Cimento, B, 1966, 43, 375.C. Gamertsfelder, J. Chem. Physics, 1941, 9, 450.' G. Borgonovi, G. Caglioti and J. J. Antal, Physic. Rev., 1963, 132, 683.130 GENERAL DISCUSSIONstructure of the liquid to be expected on the basis of the structure and dynamicalproperties of the corresponding crystal. It shows a sharp peak representing a well-separated shell of ten first nearest neighbours, at a distance (2.60A) slightly smallerthan the distance (2.67 A) between the first six nearest neighbours in the correspondingcrystal.Further studies are under way to obtain the ion-ion potential from thediffraction data, as well as to ascertain the nature of a small anomaly seeminglyappearing at 2.72 A-l, just before the first diffraction peak (at Q = 2.9 A-') in theexperimental S(Q). The results reported here will appear soon in Nuovo CimentoB, 1967, 49, 222.Dr. D. I. Page (A.E.R.E., Harwell) said: ResuIts of some work at Harwell on thediffraction scattering of neutrons by heavy water, to compare with the X-ray datan nQ (A-1)FIG. 1 .-Neutron diffraction scattering from D20 at 295°K. (a) A,, = 0-835 A; (b) A,, = 1.365 A.r t 1 II"I0 . 5 -nr(0 , ;- 0 5 - \\1 ,~ 1- _-LL I ' L - I I 15 10Q @-'IFIG. 2.-Comparison of X-ray and neutron scattering data from water at room temperature.(a) X-ray.Narten, Danford & Levy. O.R.N.L. 3997. (b) Neutron.of Dr. Narten, are shown. The measurements were made on a conventional diffracto-meter using a neutron beam from the " hot source " in DIDO. This is a block ofmoderator at high temperature which enhances the flux of short wavelength neutronsallowing large momentum transfers to be measuredGENERAL DISCUSSION 131Fig. 1 shows the best fit to the raw data and the resolution function of the instru-ment at two incident wavelengths. The statistical error on a single point is abouttwice the line thickness. With the incident wavelength of 0.835 A, a value of Q = 14was reached. There is little structure apart from the main peaks.Fig. 2 shows the corrected data plotted as a function of Q(1- 1) to compare withthe X-ray data.There is a correlation between the first two maxima but little similarityafter that. This can be qualitatively accounted for by realizing the 2-rays arescattered predominantly by the oxygen atoms while the neutrons are mainly “ seeing ”the hydrogen atoms. Thus the oxygen atoms which are much more localized thanthe hydrogen atoms give rise to a sharp first peak in the X-ray radial distributionfunction; on transforming this, the almost sinusoidal variation of the X-ray datais obtained. The less localized hydrogen atoms give a more blurred R.D.F. andhence a much smoother diffraction pattern. The first peak in the diffraction patterns(corresponding to intermolecular scattering) appears as a doublet in both cases ;again, the difference in shape could be due to the radiations “ seeing ” different atomsand consequently the location of the molecular scattering centres would not be thesame for neutrons and X-rays.We are trying to find a suitable model to predictthe neutron scattering data-as Dr. Narten has done for the X-ray case. We concludethat X-ray and neutron scattering are complementary techniques rather than com-peting ones.Mr. J. W. Perram and Dr. S . Levine (Manchester University) said: We havebeen examining lattice models with the view to explaining some of the properties ofliquid water in terms of co-operative hydrogen bonding, as envisaged by Frank andWen and elaborated on by Nemethy and Scheraga (N.S.).There has been muchcriticism of N.S.’s “ significant structure ” theory of water, on spectroscopic groundsand on their use of adjustable parameters. There are also objections to their statisticalmechanical treatment, which should be discussed. We start with their partitionfunction for N moleculesA 4QNS = z ( N ! / n ni!) exp (- E(n)/kT) . n fli, (1)n i = O i = Owhere n, is the number of water molecules with i(= O,l, ..., 4) hydrogen bonds, E(n)the energy of a particular distribution n(= no, ..., n,) of hydrogen bonds andf, is thesingle molecule partition function which accounts for vibrational, rotational andtranslational states. The combinatorial factor N ! / n n i ! , which is supposed to bethe degeneracy of the energy level E(n), is, however, in error. For example, manyof the configurations included in this factor are geometrically impossible.ApparentlyN.S. partly avoid this difficulty by restricting the terms in the partition sum to thosecorresponding to the hydrogen-bonded clusters of Frank and Wen. However, theydo not show that clusters follow from a proper statistical mechanical theory.To illustrate the inadequacy of (l), we have considered the following simplemodel : (i) no a priuri restrictions, due to clusters, are assumed ; (ii) the energy E(n)of a configuration is proportional to the total number of bonds, i.e., E(n) = -rEH,where EH is the so-called energy of the hydrogen bond and r = ini is the numberof such bonds. This is equivalent to N.S.’s assumption of equal energy spacingsbetween the 5 hydrogen-bonded molecular species.(iii) All the fr( = f) are equal.4i = O4i = OH. S. Frank and W.-Y. Wen, Disc. Faraday SOC., 1957,24,133.* G. Nemethy and H. A. Scheraga, J. Chem. Physics, 1962, 36, 3382132 GENERAL DISCUSSIONThis may be a reasonable approximation for all species except i = 0, but experimentssuggest that the number no is sma1l.l If we imagine the 0 atoms as forming a tetra-hedral lattice, then the 2N mid-points of the lines joining adjacent 0 atoms also forma tetrahedral lattice and we can assign energy states 0 and -EH (one hydrogen bond)to each of these 2N points. The degeneracy corresponding to Y hydrogen bonds is (';">, the number of ways of distributing r objects among 2N sites. The partitionfunction is therefore2NQ = .f N x ( r ) e x p w 2N 2iv rEH = f N ( l + e x p g ) .r=OIn contrast, (1) is a multinomial expansion, viz.,E H EH 3EHQ N ~ = f 1 +exp - +exp - +exp __ +exp [ 2kT kT 2kT(3)We have compared the entropy S, due to the hydrogen bond distributions in (2) and(3), taking the value EH = 1.4 kcal/mole. In the temperature range 280-370°K,(2) gives an increase in SH from 1-1 to 1.6 (cal/deg. mole) whereas, according to (3),this increase is from 1.7 to 2.1. In our simplified model, Y ~ N , i.e., half the bondsremain unbroken, as T-+co. All thermodynamic functions are well behaved, ourwater neither boils nor freezes and it possesses no short-range order compatible withthe existence of clusters. Furthermore? both this model and that of N.S.have neglectedthe residual entropy (considered by Nagle 2, due to the two directions available tothe hydrogen bond.The basis of Frank and Wen's concept of clusters is that one hydrogen bondpromotes the formation of other hydrogen bonds. Thus, the energy spacing EHshould depend on the number i of hydrogen bonds per molecule. N.S. discuss thisbriefly and suggest that allowance is made for unequal spacing by restricting thesummation in (l), but that nevertheless equal spacing is an acceptable approximation.Our calculations show, however, that equal spacing is incompatible with the assump-tion of cluster formation? a point which has also been stressed by G ~ r i k o v . ~ Itseems necessary, therefore, to postulate unequal spacing from the start and work onthis problem is in progress.One technique which can be used is familiar in the Isingproblem, the absence or presence of hydrogen bonds corresponding to the two direc-tions of electron spin. In addition, the implications of the Samoilov model astreated by G ~ r i k o v , ~ who introduces interstitial molecules and vacant sites (similarlyas Narten, Danford and Levy) as well as hydrogen bonding, are being studied. Ourinvestigations indicate that the quantitative predictions of N.S.'s theory and of thesimilar theory by Vand and Senior should be viewed with some reserve.Dr. W. A. P. Luck (LudwigshafenlRh.) said: In answer to Perram and Levine Iwould mention that the N.-S. paper was a pioneering one but only a first approxima-tion.The " unsharp " isosbestic points and the differences between the spectra ofD. P. Stevenson, J. Physic. Chern., 1965, 69, 2145.Yu. V. Gurikov, Zhur. Strukt. Khim. 1965, 6,817.0. Ya. Samoilov, Structure of Aqueous Solution of Electrolytes afid Hydration of Ions (Englishtrans., Consultants Bureau, New York, 1965).V. Vand and W. A. Senior, J. Chem. Physics, 1965,43, 1869, 1874, 1878.* J. F. Nagle, J. Math. Physics, 1966, 7, 1484GENERAL DISCUSSION 133pure alcohol and the solution show that the assumption of an equal energy spacing isapproximate. In addition, the energy difference of the so-called free molecules inthe N.-S. paper compared with ice-like bonded molecules is only 2.64 kcal/mole.The sublimation energy of 1 1.6 kcal/mole gives the real difference (ice-vapour like)molecules. Therefore we do not agree with some spectroscopic tests of the N.-S.theory and the bands allied to the different N.-S.species. In our nomenclature, thefree species of the N.-S. paper are energetically unfavourable to H-bonding. The freeOH-groups of our nomenclature are neglected in the N.-S. theory. This is the reasonthat our percentage of free OH is much smaller that in the N.-S. theory. Scheragahas now devised a new theory which is much better as a first approximation.Dr. W. A. P. Luck (LudwigshafenlRh.) said: It is easy to show that Franckand Roth’s and our experiments agree. If we have an equilibrium of two specieswith two different bands, then it depends on the frequency distance Av of these bands,the half width Av+ and the quotient of theopticaldensity whether we observetwo bands,C ---Em 33006Ooo4vm4 v ----&D1600580020021.5En1VmAvAv+AI 0.8 A 4 0.2 62 0.6 A + 04 B3 0.4 A 4 06 84 9.2 A + 0 8 62 l 1300 6007000 6400- 6005 Q 8 C + 0.2 D6 0*6C+O& D7 0.4C + 0.8 D8 0*2C+ 08DD1.06000 5500one main band with peaks, one unsymmetrical band or only one band.For example,the overlapping of two bands with the same extinction-coefficients always gives onlyone band if AY < 0+85Av,. There are many experimental observations of one-bandsystems of an equilibrium of two species with two overlapping bands.l Fig. 1 showstwo theoretical examples of an equilibrium corresponding to two overlapping bands :W.Luck, J. SOC. Dyers Col., 1958, 74, 221 ; Angew. Chem., 1960, 72, 57134 GENERAL DISCUSSIONFig. l a corresponds to the experimental conditions of the HOD overtone band.Fig. lb corresponds to the experimental conditions of the HOD combination bandand should be similar in the fundamental region. Fig. 1 shows that it is not unusualto observe one band without peaks in the combination or fundamental region, andto observe separate bands in the overtone region. The experimental H 2 0 spectraare more complicated because the H-bond bands are composed of a system of bands.In OH H-bonding the Av between the free OH-band and the H-bonded band is smallerin the fundamental and combination bands than in the overtone regi0n.l In addition,the intensity of the H-bond-band is higher in the fundamental region than the freeOH-band.In the overtone region it is reversed : the intensity of the H-bond bandis smaller than the free OH-band. Therefore it is easier to observe free OH or NH-bands in the overtone region (cp. fig. 1 and 3 in ref. (2)). This outstanding advantageof the overtone bands can be observed especially with the HOD spectra 1-3-1-8 pFIG. 2-1-4 p HOD overtone and 1.6 p HOD combination-band ; 50 ml H20/1. in D20 (D20-spectrum is subtracted).(fig. 2). At 1.6 p appears a combination band of the fundamental vibrations with aquotient of the extinction-coefficients E of the H-bond band to the free OH-bandof 4 q 1 . At 1.4 p appears the second overtone of the OH-stretching vibration withan intensity quotient 0.6/1.The combination band 1-6 p has similar overlappingeffects as the 4 p band which Franck and Roth have observed. I agree this band isnot useful for finding different states of H,O.* But the 1.4 p overtone shows a bandmaximum belonging to more or less free OH. Taking the supercritical spectra-which are temperature independent for constant density-as the standard state fora fluid-like interaction of non-H-bonding we obtain from this isolated 1 . 4 ~ HODovertone-band the same results for free OH that we obtain with four different H20bands (fig. 3). This good agreement shows clearly that in the overtone-region theoverlapping of different bands does not affect our method. Even so, this experimentalW.A. P. Luck, Ber. Bunsenges, 1965,69, 626.W. Luck, 2. Elektrochem., 1961, 65, 355.* For an attempt to analyze different species with the 3 p fundamental band, see W. K. Thompson,W. A. Senior and B. A. Pethica, Nature, 1966, 211, 1086GENERAL DISCUSSION 135agreement (fig. 3) of different bands shows that a variation of E with temperature maybe unimportant too. The statement that the HOD fundamental stretching bandwould be the best one to study the water structure is not valid, if we include the over-tone-spect ca.I50 100 150 200 250 300 3 9 380- ~~~1 ("aFIG. 3.-The % free OH P determined by the two H20 bands 1.14 and 0-95 p agrees well with thedetermination of the 1.39 p HOD band.T ("C)HzO 1.143 pFIG. 4.-H20 extinction coefficient E at the wavelength 1.143 p (free OH-region) : e, liquid state pc ; +, vapour state pc ; 0, liquid state 2pc.Franck and Roth observe with decreasing T an increase of the high intensityH-bond band with the corresponding Av.The same change was observed in thesupercritical T-region with increasing p from pc to 3p,. That means an increasingpfrom 240 to 2000 atm. This could be an indication of induced H-bonding by p or p 136 GENERAL DISCUSSIONSince the H-bonding is an equilibrium this is expected. We have made measurementsof the 1.1 H20 band at 2p, (fig. 4). For T> 320"C, and this density 2p,, the opticaldensity in the free OH-region is more or less constant and lower than for the densityp,, as expected from Franck and Roth's results. Our band-system has a decreasingintensity with increasing H-bonds.Even so, we have observed at 2p, and T> 320°Ca constant frequency of the band maximum, which v,,, corresponds to the saturatedstate of 320°C. The increase of pressure from 120atm at 320°C to 1200atm at450°C has evidently the same effect on the H-bond bands as has AT from 320 to374°C under saturation conditions. A slow increase of E at 2p, above 430°C mayshow a further opening of H-bonds. But we stress that we have only made one4.03 53.02 50. W G M1 *5PO060115 KO 1.55 wo rc;5 1-701 (P-)FIG. 5.-Spectra of pure pyrrolidon show with higher temperatures more and more intensity anda maximum in the wavelength region of the spectrum of the free molecules in dilute CCl,-solutions.experiment with this high-density conditions because our cell exploded during themeasurements shown in fig.4.I think the H-bond overtone-studies in solutions provide the best equilibrium-knowledge which physical chemistry has. From these papers we cannot doubtthat we observe free OH or free NH in the overtone region. For example, for pyr-rolidon the equilibrium constant for dimerization is constant with change of weightconcentration of a factor of 103.1 We know in this case especially well the freeNH-band. Fig. 5 shows that heating pure pyrrolidon gives a change of spzctra inthe direction of this free NH-band. We obtained the same results with CM,QH,C2H50H, 1-3 propandiol, 1-4 butandiol, 1-5 pentandiol, 1-4 butindiol and N-ethyl-acetamide too.2 Comparing the CH30H solution-spectra with the CH30HW.A. P. Luck, Naturwiss., 1965,52,25.W. A. P. Luck, Spectrochim. Acta, 1967, in pressGENERAL DISCUSSION I37T-dependent-spectra in bulk we see that we not only have two species, free OH andlinear H-bonded, but also different types of energetically unfavourable H-bondbands.l I agree with Franck and Roth that in this region of different H-bondsthere may be, more or less, a continuous distribution of different H-bonds withenergy contents U<AUH. But all OH-groups with U>AUH are in similar statesfrom the point of view of H-bond bands. This is one reason why we prefer onlyto discuss quantitatively the percentage P of free OH. The second reason is that inthis frequency region all overlapping effects are minimal.The intensity of the free OH-bands may change with a change of molecularenvironment. But this change cannot be so large that it is impossible to makeapproximative statements.An exact interpretation of the H20 spectra is impossible,so we have to make approximative statements because a knowledge of water structureis so important.Many of own measurements at lower densities agree nevertheless with Franckand Roth in showing smaller intensities. In the small p-region there is an increaseof intensity of a rotation-structured spectra with increasing pressure.2 But we havelittle information what happens if the disturbed rotation changes to a libration.At a maximum a limit of 5-20 % free OH at 0°C is expected; this only meansthat we do not know how many hundreds of molecules are in the flickering clusters.But the exact number of molecules in a cluster is not so important for our understand-ing the water structure ; it was important to show that most theories have assumed asyet too many free OH-groups.Frof.H. S. Frank (University of Pittsburgh and iMellon Institute) said: I expressmy admiration of the work reported by Narten et al., by Franck et al. and by Luck.Each, by difficult and painstaking experiment, has established new conditions whichany acceptable model for water will have to satisfy, and each has, therefore, madean important contribution to the ultimate solution of the water problem. None ofthe authors has purported to say the last word about that problem, and each one haswisely emphasized that the interpretation of his data which he favours is not necessarilyunique.That their interpretations could not all be right is clear, for none of thethree is in agreement with either of the other two. Franck considers that only one" kind " of 0-H bonding in liquid water need be assumed at any temperature ordensity. Narten specifies two types of water molecules, which should thereforeshow two kinds of bonding, but makes the relative proportions of the two veryinsensitive to temperature changes, whereas Luck not only distinguishes between" bonded " and " unbonded " 0-H entities, but gives numbers which make theirproportions much more temperature-sensitive.One of the principal problems in discussion of the water structure is the difficultyin finding experimental evidence which requires either the acceptance or the rejectionof any given model.To find a model with which one set of new data are consistent,or which they suggest, is relatively easy, but uniqueness of interpretation is anothermatter. Thus, for all the attractiveness of the inferences Luck draws from the effectof temperature changes on his spectra, I worry about not really understanding thelaws which govern the addition of intensities in the overtone and combination bandswhich he has studied; and therefore consider his work likely to be more useful asillustrating and refining a model which will have been established on other grounds,than as providing the initial proof for such establishment.Similarly, while the dataof Narten, Danford and Levy seem to add in an important way to our experimentalW. Luck, Ber. Bunsenges, 1963, 67, 186.G. Kortum and W. Luck, 2. Natwfursch., 1951, 6a, 191, 305, 313138 GENERAL DISCUSSIONknowledge of liquid H 2 0 and D20, the model that they propose could be proved tobe unique only if it could be shown that no alternative starting structure, with anequal number of adjustable parameters, could be refined to give calculated radialdistribution curves which would fit the data as well as their present ones do. Andin the meantime, the fact that, on their present model, the fraction of water in theframework form changes so slowly with temperature has the thermodynamic conse-quence that there can be little difference in molal enthalpy between framework andinterstitial molecules, which, if true, would have remarkable implications for thenature of the bonding in the two states.I I i i0.9 -0 8 -0.7-7 0.6-h %W$2g oI & O S - Q a.04-nu-5Ip+ W 0.3-\\\'Tt ("C)FIG. 1.In this connection, it is worth looking at another set of data which suggestsstrongly that some form of water is indeed disappearing fairly rapidly as roomtemperature water is heated.These data, taken by G. E. Walrafen in the BellLaboratories, are intensities of Raman scattering at frequencies near 175 cm-l.This band, which presumably is the counterpart of the ice scattering near 220 cm-l,is thought to arise from the vibration against each other of oxygen atoms linked byhydrogen bonds.As shown in fig. 1, reproduced from Walrafen's paper, the correctedintensity of this band drops by almost half between 0 and 25°C and does so in theway that would be observed if the fraction f of the scattering material and that(1 -f), of the non-scattering material, were related by an equilibrium constantK = f/( 1 -f) which obeyed conventional thermodynamic laws.The question then arises how this result is compatible with the fact that Franckand Roth (along with others who have studied the 0-D stretch of HOD in H,O)get an infra-red band which superficially is simple. The answer is that bands which" superficially " are simple may, in fact, be resultants of 2 or more component bands.This is illustrated in fig. 2, which was constructed by Dr.W. A. Senior using theNational Heart Institute's curve analyzer located in Mellon Institute, which waskindly placed at our disposal by the Petroleum Fellowship. Here two Gaussiansare seen to add up to something which, on superficial inspection, looks like anotherGaussian. If there are two component bands which are farther separated in centralG. E. Walrafen, J. Chem. Physics, 1966,44, 1546.W. A. Senior, Ann. N. Y. Acad. Sci., 1964, 115, 644GENERAL DISCUSSION 139frequency a shoulder will appear, and if the component bands correspond to chemicalspecies which are in equilibrium, so that a change in conditions (temperature, forexample) causes more of one to be formed at the expense of the other, then, eitherwith or without overt shoulders in the individual plots, the curves corresponding todifferent conditions will often display the so-called isosbestic phenomenon (fig.3).A V ~ A = AV+B = 130 cm-I ; VA-VB = 100 cm-' ; EACA = 4 ; EBCB = 2. (See Curve 3).FIG. 2.A V ~ A = AV+B = 130 crn-l; VA-VB = 100 cm-' ; EA = ~ E B .FIG. 3.curveno. CAEA CBEB1 8 02 6 13 4 24 2 35 0 4CA+CB = const140 GENERAL DISCUSSIONThe appearance of such a pattern is often taken by chemical spectroscopists to betokenthe existence of an equilibrium, e.g., the mixture of acid and basic forms of a colouredindicator changes in the course of a titration and successive spectral curves showisosbestic behaviour. The existence of an equilibrium need not, however, producea " perfect " isosbestic. If, e.g., the 0-H stretch, say, of HOD in D20 consistedz 5 Pseudo - isos bes t ic"0-100 0 leo cm-'A V ~ A = 130-190 cm-I ; AV+B = 130 cm-I ; VA-VB = 100 cm-IFIG.4.frequency, cm-IFIG. 5.4mparison of the infra-red band YO-H of HDO molecules: A, HDO in in DCC13, 27";B, HDO in liquid DzO at various temperatures.of two bands, corresponding, respectively, to hydrogen-bonded and non-hydrogen-bonded 0-H vibrations, then the bonded band would be expected not only to decreasein intensity with rising temperature, but to become broader as well, and in this casethe plotting together of curves for a variety of temperatures would show a modified,or pseudo-isosbestic, behaviour. Fig. 4 shows how this would come about, andfig.5, a set of absorption curves taken by K. A. Hartman, Jr. of the du Pont CentraGENERAL DISCUSSION 141Experimental Station shows that some such thing really is observed. This isconsistent with, but does not prove, the idea that two distinguishable kinds of 0-Hstretching motions are in liquid water.The objection has been made that " unbonded" 0-H groups ought to absorbat much higher frequencies, where no absorption is observed. Here the new findingsof Franck and Roth assume great importance, for they seem to me to confirm thatan unbonded OH or OD in liquid water at a density of 1.0 g/cm3 need not be expectedto absorb at the frequency found in the dilute vapour. If the v,,, against Tcurve oftheir fig. 4, for p = 1-0 g/cm3 is extrapolated to higher temperatures, there seemslittle doubt that it will level off to an asymptotic v,,, well below 2700 cm-l, i.e., wellbelow the Q branch 0-D absorption in HOD vapour.But if the temperaturevaried from 400 to 500, 600 - - - 1000 - - - "C, there is also little doubt that it wouldbecome increasingly inappropriate to describe the HOD molecules as hydrogenbonded. That is, at high densities, Franck and Roth have shown-to me atleast-that there is a non-hydrogen-bonded interaction between HOD moleculeswhich lowers the frequency of the OD absorption in a way qualitatively similar to thatproduced by hydrogen bonding. This is consistent with the inference that there aretwo kinds of strong cohesive forces between water molecules that can be drawn fromthe fact that the vapour pressure curves of H 2 0 and D20 against temperature crosseach other.2 If this be granted, then the smoothness and apparent simplicity of the0-H and 0-D stretch curves found at any one temperature by Wall and Hornig,by Hartman, by Falk and Ford and by Franck and Roth, no longer has any cogencyas an argument for a one-species model, and the two-species inference from the pseudo-isosbestic curves of Hartman and from Walrafen's Raman data becomes credible.Dr.W. A. P . Luck (LudwigshafenlRh.) said: In answer to Frank's contribution,the different overtone bands of H20, D 2 0 and HOD that we have observed all haveisosbestic points that are not sharp. Therefore, as a first approximation we mayassume a two-species model.But all these " isosbestics " are not good ones5 Weobtained the same result from the argument that the sum of free OH and linear-bonded OH do not add up to 100 %,' i.e. the reason that the isosbestic points are notgood is because of the existence of more than two species.With free molecules, they have similar vibration frequencies as in the vapourstate; but they are disturbed in the fluid state. Therefore we have not obtained anyindication of rotation structure in the fluid state, but have observed disturbed rotationstructure of H20 in CC14 solutions and in the vapour state at smaller pressures as inthe saturated state.I agree that all the new experimental data from X-ray scattering, Raman- andi.-r.- spectra are not contradictory. Different authors have made different interpreta-tions with different approximations; we would need a special meeting of all thesepeople to make progress in our knowledge of the water structure.Dr.J . Padova (S.N.R.C., Yavne, Israel) said: With reference to the paper byLuck, I would like to state that measurements of the absorptivity of water at 960 mpin NaNO, solutions were carried out at 25°C in our laboratory by I. Abrahamer.The molar absorptivity E showed a linear increase with concentration of NaN03K. A. Hartman, Jr., J. Physic. Chem., 1966, 70, 270.I. Kirschenbaum, Physical Properties and AnaZysis of Heavy Water (McGraw-Hill, New York,1951), p. 25.F. T. Wall and D. F. Hornig, J. Cheni. Physics, 1965, 43, 2079.M. Falk and T. A. Ford, Can. J. Chem., 1966,44,1699.W.Luck, Ber. Bunsenges., 1963, 67, 186142 GENERAL DISCUSSIONup to saturated solutions. From the two-species model of free OH and ice-like Hbonded groups used by Luck, it may be shown that the percentage of free OH groupsis proportional to the absorptivity of water, pointing thereby to the similar effect oftemperature and salt concentration on free OH groups percentage (cf. fig. 1 of Luck’spaper) and confirming the structure breaking effect of NaNO,.Prof. M. L. Josien (Faculte‘ des Sciences de Paris) said : In the course of a systematicinfra-red study of intermolecular associations, we have obtained data concerning theassociation of water with proton acceptors and donors. (a) In ternary mixtures, when aproton acceptor is added to a carbon tetrachloride solution of water, the water moleculeyields two kinds of compIexes : HOH .. . B and B . . . HOH . . . B.l* Fig. 1shows, e.g., the evolution of absorption between 3800 and 3400 cm-l as a function ofv, cm-IFIG. 1 .-Stretching frequencies of water in carbon tetrachloride+ dioxane mixtures. f.m. : dioxanemolar fraction, water concentration = 0406-0-05 mole/l. ; pathlength = 30.3 mm. The product(water concentration x pathlength) has been kept constant.the molar fraction of dioxane added to a carbon tetrachloride solution. The freemolecules H20 absorb at 3613 (v,) and 3708 cm-I (v3), while the 1-1 complex absorbsat 3510 and 3683 cm-l, and the 1-2 complex at 3512 and 3583 cm-l. For all theproton acceptors studied, if a 1-1 complex is formed, there remains an absorptionaround 3700 cm-l ; on the contrary, in the 1-2 complexes (cf.table 1) the frequenciesof the v1 and v3 vibrations are lowered by a similar value, although the difference(v,-vl) decreases progressively with the strength of the proton acceptor and the in-tensity ratio Z(vl)/I(v3) increases to about 1.P. Saumagne and M. L. Josien, Bull. Soc. Chim., 1958, 813.P. Saumagne, ThBse (Bordeaux, 1961)GENERAL DISCUSSION 143Fig. 2 shows, for ethyloxide, a series of curves recorded in 1960 in order to analysethe influence of temperature on the equilibrium between the 1-1 and 1-2 complexes.The spectra were obtained with a single-beam spectrometer P.E. 112 ; the absorptionbands of atmospheric water vapour were not compensated for.At 40°C, waterwas found mainly in the state of a 1-2 complex, while the amount of 1-1 complex(vl 3700 cm-', v 3 3560 cm-l) was predominant around the temperature of 300°C.TABLE 1acetoni trile 3543 3636 93 0.88cyclohexanone 3530 3610 80 0.90diet hylether 351 8 3590 72 0.97solvent Y I (cm-1) v3 (cm-1) ( ~ 3 - ~ 1 ) , cm-1 I(Vi)/%)dimethylsulphoxide 3440 3505 65 1pyridine 341 1 3485 (sh.) -(I(vl)/(I(v3) in the gas is ca. 0.1).Vapour ofatmosphere water3518 I3400 3 5 0 0 3500 3700v, Cm-lFIG. 2.-Stretching frequencies of water in diethyl oxide ; effect of temperature ; saturated solution ;pathlength = 0-3 mm at high temperatures.(b) The stretching frequencies of water are not very sensitive to complexing byhydrogen bonding of the oxygen atom1 Under the influence of a very strong acid,such as trifluoroacetic acid, a decrease of only 22 cm-l is observed for the frequenciesv1 and v3.(c) The above data lead to the interpretation of the spectrum of liquid water interms of being a complex of the 1-2 type since, in the limits of sensitivity of infra-redspectroscopy, no absorption around 3700 cm-' can be observed.2J. de Villepin, A.Lautie and M. L. Josien, Aim. Chim., 1966, 1, 365.Congr. Slow Neutron Scattering (Vienna, 1960)144 GENERAL DISCUSSION(d) Fig. 3 shows the evolution of the water spectrum as a function of temperature.In view of the preceding remarks, we propose the following interpretation : the twomaxima which progressively occur around 3545 and 3650cm-l are related to thev1 and v3 vibrations of water associated mainly as 1-2 complexes; the frequencyv2, whose overtone appears as a shoulder around 3300 cm-l, is lowered as the tem-perature increases.Prof.J. B. Wyne (Calgary) said: While spectroscopic studies of water are of primeimportance in the elucidation of thestructure of this liquid conclusions drawn from suchevidence must be capable of accommodating the evidence obtained by other techniques.In particular, thermodynamic and kinetic evidence obtained from studies of aqueoussolutions and reactions in aqueous solutions (e.g., H. S. Frank, F. Frank and Ives,Arnett et al. etc.) leave little doubt that the initial effect of added cosolvents is to'' buttress " the existing pure water structure and not destroy it.Accordingly, anymolecular model of water structure must be capable of rationalizing these observedeffects. While the model of Norten, Danforth and Levy and that of Luck provideeither interstitial holes or interfaces between cluster surfaces which might be '' butt-ressed " by added non-aqueous species, the lack of" defined small clusters of water ''as suggested by Frank and Roth would render rationalization of the structure buttress-ing effect of additives difficult.While the conclusion of Luck that HzO, CH30H and C2H50H in the liquid statehave less non-H-bonded OH-groups than most theories claim is, in itself, an intui-tively acceptable one, the relative percentages of non-H-bonded molecules in watercompared with the two alcohols is surprising.In fig. 1 the percentage of" free " H,Omolecules is shown to be approximately 15 % at 50°C while in fig. 4 the correspondingamount of " free " alcohol is less than 5 %. This means that the alcohols, which forGENERAL DISCUSSION 145linear hydrogen bonded chains, are more hydrogen bonded than water which is capableof forming three-dimensional hydrogen bonded structures. It might also be noted thatat 50°C the alcohols are much closer to their boiling points than is water. AlthoughLuck does point out that the curves in fig. 1 and 4 are experimental and not theor-etical, would he care to comment on this question of the relative amounts of “ free ’’species in water compared with the alcohols.Dr. W. A. P . Luck (LudwigshafenlRh.) (communicated).In answer to PadovaI refer to an earlier paper,l in which is given a detailed discussion about the salteffects on free OH in aqueous solutions.In answer to Prof. Josien. The absorption at 3700 cm-l in the “ 1-1 complexes ”shows a residue of free OH, as is verified by the spectra given by Josien in fig. 1 and bythe temperature-effect given in fig. 2. We can recognize in fig. 1 with mediumconcentrations of acceptors a shoulder of free OH-band. With higher concentrationsthis shoulder increases to a single new peak. This means that this peak belongs to afree OH-group of a molecule whose second OH-group is H - b ~ n d e d . ~ - ~The second peak of the “ 2-1 complexes ” is also observed by Mohr, Wilk andB a r r ~ w . ~ But these authors refer to the two peaks belong to H-bond-bands of thev1 and the v3 vibrations which change the intensities with the change of symmetryduring the change from a 1-1 to a 1-2 complex.With different acceptors the differencebetween these two H-bond-bands is at about 9 0 ~ m - l . ~ The difference betweenv 1 and v3 is also 90 cm-l.The two peaks with Av- 100 cm-l observed in fig. 3 at higher temperatures belongto these two vibrations v1 and v3 of more or less unbonded OH-groups. Perhapsthese peaks may be disturbed by the density gradient near the critical point orby higher pressure in Saumagne’s cell. Josien’s remark shows the difficulty in theanalysis of the OH fundamental vibration. This is one more reason for preferringthe overtones or the HOD spectra.In answer to Hyne.I think the reason that the free OH of the alcohols is smallerthan the water values is the structure-difference between chain-like aggregation ofalcohols and network-building of water. Comparing the number of molecules peraggregate (or cluster) at corresponding temperatures we find there are not such largedifferences between alcohols and waterProf. M. Magat (Faculte‘ des Sciences de Paris) said: I have a few remarks to makeabout the problem of liquid water. The first remark concerns theoretical calculations.Rushbrooke has pointed out that no precise theoretical calculations were possible onany liquids except the monoatomic. I would appeal to the theoreticians to recognizethat water is of paramount importance-be it only for biology-and attempts shouldbe made to attempt a theoretical treatment of its structure and properties, evenusing oversimplified models for the charge distribution in the water molecule, as inthe old Bernal and Fowler model.Computers should allow one to go much furtherthan in the old calculation by these authors and by myself.I was gratified to hear Franck’s report on recent progress concerning the variationof the intensity of Raman bands attributed to intermolecular vibrations and librations(fig. 2.30).W. A. P. Luck, Ber. Bunsenges, 1965, 69, 69.W. A. P. Luck, Spectr. Chim. Acta in press.W. A. P. Luck, Ber. Bunsenges, 1965, 69,626.W. A. P. Luck, Fortschr. chem. Forschung, 1964, 4, 653.S. C. Mohr, W. D. Wilk and G. M. Barrow, J.Amer. Chem. Soc., 1965,87, 3048.W. A. P. Luck and W. Ditter, Ber. Bunsenges, 1966, 70, 1113.(Heinemann Educ. Books Ltd., London, 1967), p. 35.’ W. A. P. Luck in Physico-chemical Processes in Mixed Aqueous Solvents, ed. F. Franks146 GENERAL DISCUSSIONas a function of temperature. Indeed this careful and precise work confirms exactlymy findings of some 30 years ago based on much fewer and much less precise measure-ments : a particularly rapid decrease of these intensities is observed between 25 and50°C. The intensity at 70°C is much lower than at 20"C, which cannot be simplyinterpreted as a disappearance of hydrogen bonds.The third point concerns the problem of " free " water molecules. Mrs. Reinisch(in our laboratory) has recently made a careful survey of results of different methodson the question of the existence of" freely rotating " water molecules, which probablycan be also called " not hydrogen-bonded water molecules ".The results are un-fortunately contradictory. The situation as summarized by Mrs. Reinisch is asfollows.existence (+) orof H-bondsmethods used non-existence (-)- infra-redRaman spectra ? (a)neutron scattering + (6)n.m.r. - (4X-ray structure + (9 (4dielectric constant + (3 (4(a) Raman spectra give no definite evidence except perhaps the appearance above30°C of a band near 3600 cm-' of a frequency not very different from that of the gas.(b) Palevsky et aZ.l insists that the existence of some movements may be inter-preted as rotations of free molecules. This was not confirmed by Jacrot,2 Larssonis sceptical,2 and Palevsky did not mention it in his later papers.(c) Detailed information on n.m.r.results were given here by Powles.(d) The X-ray studies, e.g., that presented here by Narten, Danford and Levyindicate for water a co-ordination number of 4.4. Since a water molecule can formonly 4 hydrogen bonds, some molecules not involved in hydrogen bonds must bepresent even if the distances involved are close to H-bond distances.(e) This is the most difficult point. Ice I has at high frequencies a dielectricconstant cw = 3.2. This was established by Lamb a few years ago and never con-tested. Liquid water even close to 0°C has a dielectric constant cw = 5-0_+0.5 atfrequencies lying just below the intermolecular vibrational frequencies.The Onsagercorrelation g factor may be determined a priuri by statistical mechanical or electro-statical considerations. Thus, Kirkwood and Oster for water ; Onsager andD ~ p u i s , ~ Coulson and Ei~enberg,~ Hollins for ice (which g factor must not be verydifferent from that of water) found a g value 2.1 <g<2*9, in approximate agreementwith the experimental g that one obtains assuming E , for liquid water = n2 (which isimprobable), or cmwater = E , for ice = 3.2.However, if one takes E , = 5.0, the correlation factor becomes g = 1,7 i.e., thereis no correlation between water molecules, a highly improbable situation unless thereis some kind of compensation of effects. Hence, either the dielectric constant c , = 3-2of liquid water is due partly to molecules involved in H-bonds as in ice and partlyPhysic.Rev., 1960, 119, 872.Congr. Slow Neutron Scattering (Vienna, 1960).J. Chem. Physics, 1943, 11, 175.Rendiconti S. I. F., X Corso, 1960, p. 294.Proc. Roy. SOC. A , 1966, 291, 445.Proc. Physic. SOC., 1964, 84, 1001. ' Trans. Faraday SOC., 1963, 59, 344GENERAL DISCUSSION 147to free water molecules. Then the correlation factor is reasonable, but there is acomplete contradiction between infra-red and dielectric results. Or the differencebetween E , in ice and in water is to be explained by some unknown effects (deforma-tions?) and we have to explain why the correlation factor in water is g = 1.The fourth points concerns difficulties in interpreting the intermolecular frequencies.There is just one frequency, about 160-180cm-1, that was observed in Ramanspectra, in infra-red and in neutron scattering.Since it is not affected by isotopicsubstitution, it seems generally agreed that it is some kind of breathing frequency.Heinloth ' has observed by neutron scattering the same band in ice. A band notaffected by isotopic substitution and located at 60cm-l was observed by Bolla inRaman spectra and by neutron scattering. It was never observed in the infra-red,even where techniques as sensitive as those developed by Dr. Gebbie at the N.P.L.were used.The last point deals with the bands at 475 and 685 cm-l. Cartwright and Rubensobserved the 475cm-l band in the i.-r., but this was not confirmed by the laterauthorsY3 i.e., it exists only in the Raman spectrum.The 685 cm-1 is active in bothRaman and i.-r. spectra, but it was not observed by Palevsky in his neutron scatteringexperiments; this is surprising especially as it must be due to movements of thehydrogens. Indeed, these two bands are sensitive to substitution of D to H and musttherefore be attributed to the libration of the water molecule ; the first band corres-ponds then to the libration around the binary axis. But this is in contradiction withtheory : the calculations show that 500 cm-' corresponds to the frequency of librationaround an axis perpendicular to the molecular plane. Perhaps the i.-r. spectrashould be re-examined very carefully in this region.Finally, a suggestion concerning a possible exploitation of i.-r.results on the ODband presented by Franck, who traced the precise form of this band at differenttemperatures. From the accurate measurements of the radial distribution functionsgiven by Narten, Danford and Levy, we know the average distances at these varioustemperatures. We can hence correlate the average distance between neighbouringinolecules with the average frequency for the same compound. It may then be possibleto deduce from the intensity distribution of the i.-r.-band the distribution of distancesbetween first neighbours that could be compared with the radial distribution function.A significant disagreement could be an indication of orientational deviations of watermolecules of the ideal positions as the intermolecular distance increases.Prof.H. S . Frank (University of Pittsburgh and MeZZon Institute) said: It shouldbe made clear in connection with the discussion of water as a mixture that no onethat I know of considers that there are any free H20 molecules in the liquid, if bythis is meant free in the sense in which a molecule is free in the dilute vapour. Other-wise, there is not much definiteness, among the propments of a mixture model,regarding the " species " of which water is a mixture. Of the species that may bepresent, the only kind which seems relatively easy to characterize would consist ofclusters or chunks of 4-co-ordinated molecules connected by bonds which are in somesense space-filling, as in ice or in the clathrate frameworks.That such a structureshould support vibrational motions similar to the 220 cm-l " hindered translation "motion, or to the 400-1000 cm-1 '' hindered rotation ", or librational motions in iceseems reasonable, and that the intensities of the Raman or i.-r. bands which theselead to should decrease with rising temperature, also seems reasonable in a generalCongr. Slow Neutron Scattering (Vienna, 1960).Nature, 1966, 210, 790.Optics and Spectr., 1961, 10, 278 ; J. Opt. SOC. Amer., 1966, 56, 64148 GENERAL DISCUSSIONsense. What it is that disappears, however, and what is produced when this happens,is far from obvious. I have not seen, e.g., any clear description of the normal modesof the ice lattice with which the 220 cm-I frequency or the 400-1000 cm-I frequenciesare associated.In what sort of configuration space (e.g., a space of how manydimensions) must a representative point move to trace out these vibrations, and whatsort of trace does it describe? Walrafen considers that the vibrating unit has C,,symmetry and suggests that it be visualized as a tetrahedron of 5H20 molecules-a central one and the 4 that are bonded around it. This, according to him, is alsothe unit in liquid water, and when the intensity of the 175 cm-l Raman band falls offwith rising temperature (those in the librational range fall off at just the same rate)he thinks that the bonds break co-operatively, leaving “ unbonded ”, but denselypacked H20 molecules.It would appear, however, that another possibility exists, and that what disappearswith rising temperature is not in the first instance the bonding, but rather the sym-metry of the vibrating groups, and that this causes the disappearance of the scatteringpower at the frequencies under discussion.Even in that case, however, it wouldbe necessary to recognize two kinds of material-not, now, “ bonded ” and “ un-bonded ”, but the kind that can scatter at the intermolecular frequencies and the kindthat cannot-and this, again, would conflict with the “ uniform average ” models.Some people believe in the existence of some “ covalent ” contribution or “ mutualdistortion ” contribution to the electron configuration in a hydrogen bond, and thatthe magnitude of this contribution is expected to fall off sharply as the bond bends-i.e., as the 0-H line bends out of the 0 - - - 0 line of centres. When this bendingangle is small, the symmetry of the co-ordination pattern is favoured, as in ice.When the bending gets greater it might well do so co-operatively, and the “ electro-static plus covalent ” environment of the 0-H stretch might be replaced by an“ electrostatic only ” environment corresponding, perhaps, to the cosine-law energeticinteraction between water dipoles postulated by Pop1e.l In that case, the two speciesmight be “ covalent H-bond water ”, and “ Pople water ”, both characterized bylarge cohesive energy densities, but spectroscopically distinguishable. If they weresharply distinguishable in the overtone and combination absorptions this would alsoaccount for Luck’s observations.Dr. Brigitte Eckstein (Aachen) said : A poly-paracrystalline structure, analogousto the “ flickering cluster model ” postulated by Luck for water, was found to be anadmissible model for molten metals near their melting point. Thus assuming a strongconnection between the structure of the melt and the corresponding crystal, variousthermal properties of molten metals were calculated, based on a generalized theory oflattice defects in crystals., These calculated values of thermal expansion, com-pressibility, atomic heat, entropy of fusion and atomic radii agree within plausiblelimits with the experimental data. A theoretically derived linear relation betweenmelting temperature and activation energy of the vacancies in the crystal also is ingood agreement with the experimental findings. Furthermore, the model gives a goodunderstanding of the mechanisms of melting and of crystallization from the melt.3Notwithstanding their great chemical difference, water and molten metals exhibitcorresponding structural features. Probably the model will prove admissible forother simple melts, and it may be taken as a basis for a comprehensive theory ofcondensed matter.Pople, Proc. Roy. Sac. A , 1951, 205, 163.B. Eckstein, Phys. stat. SOL, 1967, 20, 83.B. Eckstein and H. Peibst, Rev. hautes temp., in press

ISSN:0366-9033    

DOI:10.1039/DF9674300128

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

Radiation Scattering Studies of the Structure and TransportProperties of Liquids*BY P. A. EGELSTAFFSolid State Physics Division, A.E.R.E., HarwellReceived 27th January, 1967The use of radiation scattering techniques to study problems in liquid state physics has severaladvantages. These include the detailed nature of the information which can be obtained onstructural and dynamical properties, the definiteness of the interpretation which may be given to thedata and the exceedingly wide range in types of sample and experimental circumstances which canbe covered. To deploy these advantages to the full requires powerful sources and special experi-mental methods which are still under development. A review is given of the techniques and theoryof radiation scattering woFk emphasizing the relation between neutron and X-ray methods.Thepossibility of studying electron shell movements by measuring the ratio of X-ray to neutron intensitiesis discussed.Broadly, the kinds of experiment fall into two classes : (a) those in which the scattered intensityis measured as a function of the momentum transferred (fie) by the radiation to the specimen,giving the function S(Q), and (b) those in which the intensity is measured as a function of both themomentum transferred and the energy transferred (h) by the radiation to the specimen, giving thefunction S(Q,m). The former are used to obtain information on atomic positions while the latterexperiments (coupled with the former) are used to derive dynamical information. Both kinds arediscussed and some methods of interpreting S(Q) and S(Q,o) are described.Co-operative modes of motion in the system, particularly for wavelengths of the order of thespacing between atoms, may be observed as peaks in S(Q,a). The relationship between this kindof data and general transport coefficients is discussed.In addition, the spectral density of the velocitycorrelation function for atomic motions may be obtained from S(Q,o), and this relationship is out-lined. This paper covers only the theoretical background to the field.1. INTRODUCTIONThe mathematical discussion of the properties of condensed matter is normallycarried out by considering either the system as a continuum or the details of theatomic structure and the atomic motions.It is the aim of radiation scattering workto derive information concerning the latter properties. Then it is possible (inprinciple) to construct a theory which will predict the macroscopic properties em-ployed in the continuum discussion. Considerable progress has been made forcrystalline solids, the primary reason being that the structure and the thermal motioncan be separated and that the structure is simplified through the symmetry relationsof the crystal lattice. Unfortunately this is not the case for glasses and liquids, wherethe various pictures tend to get inter-mingled. In these cases it is important thata detailed understanding of the microscopic behaviour is attempted to enhancetheoretical progress. With molecular structures the positions and the vibrational* This paper is a condensation of the following papers by the author :Microscopic Transport Phenomena in Liquids, Reports Progr.Physics, 1966,29, 333.Radiation Scattering Data on Liquid Metals (Conference on Properties of Liquid Metals,Use of Pulsed Neutron Sources in Solid and Liquid State Physics, (I.A.E.A. Panel on ResearchBrookhaven National Lab., Sept. 1966).Applications of Pulsed Reactors, Dubna, July 1966).14150 RADIATION SCATTERINGmodes of the different atoms can be studied and discussed theoretically with someconfidence, but the interaction of large molecules is a difficult problem and theexperimental material provided by radiation scattering work is of great value.It is important to study the structure and thermal motion of the different classesof condensed matter throughout the entire diagram of state, which represents aformidable task.There are many different types of radiation available and eachmay be used in a number of ways. Owing to the wide variety of samples, tempera-tures, pressures and techniques which one would like to cover, one must consider thedetailed properties of each technique to see where it may be best used. While thegeneral use of radiation scattering methods to study condensed matter cannot bequestioned, it is possible to discuss the detailed use of these methods for any givenproblem. Part 5 of this paper is therefore devoted to comments on the differentavailable techniques, their features and the experimental ranges which may be covered.Other sections of the paper are concerned with the results which may be obtainedwith these methods.It will be emphasized that the data obtained by neutron andX-ray methods should be combined. into one larger interpretation covering thenuclear and electronic positions (and motions) taken separately. In an ideal caseit would be possible to separate out these two kinds of effects from the completedata. Such a development, however, requires significant improvements in thequality of the experimental data, and it is therefore necessary to examine each sourceof error and attempt to minimize it. For this reason some of the major sourcesof error are listed in 93 and brief comments made; this list is illustrative ratherthan exhaustive.Only the theoretical background will be discussed here, for ex-perimental data reference should be made to the papers cited above.2. NUCLEAR SCATTERING OF NEUTRONSThe cross-section for the scattering of radiation by matter has already beendescribed (e.g., Van Hove,l Sjolander.2) The method followed is (essentially) thesame for each kind of radiation, differing only in detail, and consequently we give theprincipal steps which are followed in one case only, viz., the scattering of thermalneutrons. The steps in the cross-section calculation are as follows. (a) For lowenergy (thermal) neutrons only s-wave nuclear scattering is important (because theneutron wavelength 9 nuclear diameter). (b) +Wave scattering cross-sections (foran isolated stationary nucleus) are spherically symmetric and independent of neutronenergy.(c) The nuclear size is much smaller (-10-5) than the neutron wavelengthand the interatomic distances hence for the ith nuclear potential V(r) (after Fermi),V&r) = constant 6(r-ri)where ri is the position of the ith nucleus. (d) The overall perturbation is small(i.e., although the nuclear potential is strong within its range, the intensity scatteredby a single nucleus is small) so that the Born approximation is valid. (e) The constantin ( c ) is determined by fitting the Born cross-section for epi-thermal neutrons to themeasured " free atom '' cross-section. This constant is written as 2nZi2b/m, where mis the neutron mass and b is a length (positive for hard-sphere scattering).[Thefree-atom cross-section is equal to 4n(A/A+ 1)2b2, where A is the ratio of nuclearto neutron mass.] (f) Therefore, the scattering amplitude for a system of N atoms isa($) 1 bn dr' exp (iQ . r')6(r'- rJ.n = l ' JThe essential point is that the scattering amplitude in the Born approximation isSince the scattering expressed as the Fourier transform of the scattering potentialP. A . EGELSTAFF 151potential consists of a &function point for each atom we get a Fourier transform ofthe atomic positions. (g) If the atoms are moving about on a timescale comparablewith the value of h/dE in table 1 belbw, then the time dependence of the atomicpositions given in the formula in (f) above must be considered. Essentially theradiation suffers a Doppler shift, which is taken into account by considering theenergy spectrum of the scattered radiation, The condition for the conservation ofenergy may be represented as a Fourier transformation, so that the scattering amplitudeinvolves a second Fourier transformation between time and energy.In this case itdepends upon both the momentum and the energy transferred in the scattering process.One final point is that an observable cross-section is given by the square of thescattering amplitude and hence it involves a product of two terms of the kind givenunder cf). The two 6-functions give a “ pair correlation function ”, so that the cross-section finally becomes the Fourier transformation of the pair correlation given ateqn. (1) [the form of the correlation function shown here applies to a classical system,eqn. (1) and (2) are taken from Van Hove 1 to whom this formalism is due],1G(r,z) = - < 6(r + R,(O) - rm(z)) >N n,mwhere m ( z ) is the position of the nth atom at time z.The cross-section is(2) -- exp i(Q . r - 07) {G(r,z) - p}drdz,where b is the scattering length for a single bound atom and p is the mean density.It is possible to consider two types of correlation function. Referring to thefirst is the case for which the two atoms n and m are the same in eqn. (1) ; this self-correlation function describes the average motion of a single atom. The secondtype is that for which rz and rn may have any values, so that all atoms in the systemare considered. This function is the coherent correlation function.The techniqueswhich are used in neutron scattering work to separate the G function into these twoparts depend upon the fact that different isotopes of the same element may havedifferent scattering lengths and also that different spin states of the same isotopemay have different scattering lengths. Thus the separation depends upon makingmeasurements with the system in the same chemical state but in different nuclearstates.3. SCATTERING OF X-RADIATION BY ELECTRONSFor electromagnetic radiation the scattering potential is no longer a d-function asgiven under (c) above. Instead, it is a distributed potential having a size equal tothat of the atom, so that we take a convolution of the atomic positions with thepotential for a single atom.Since the Fourier transform of a convolution is aproduct of transforms, it is necessary to calculate the Fourier transform of thescattering potential for a single atom (i.e., the “ form factor ”) which appears as afactor in the observed scattering amplitude. This point may be understood byconsidering the general formula for the scattering of X-rays by a liquid written interms of an electronic space time correlation function.2 This function is defined byeqn. (3) :I fG,(r,z) = - dr’<p(r’,O) . p(r‘+r,z)>, 4 (3)where p(r,z) = ;pi(r- Rl(z),z) is the electron density distribution and p;(t,z) is th152 RADIATION SCATTERINGeIectron density around the Zth nucleus.cross-section may be written down in a form,+eqn. (4) analogous to eqn.(2) :In terms of this correlation function the(e) = g k p [i(Q . r-WT)] . (G,(r,z)-p,}drdz, (4)dQdw X-rayswhere B is a constant and pe is the mean electron density.Two assumptions are now made in the standard theory: (a) the internal statesof the atoms are not excited by the X-ray scattering process, and (b) the electronsfollow the nuclei rigidly. These two assumptions allow the cross-section to be re-written as(5) = " 4 e ) j ' f e . p [i(Q. r-coz)] . (G(r,t)-p}drdz, (5)dQdw X-rays 2~whereF(Q) = [exp (iQ . r) . <p"(r)>dr.This form is similar to eqn. (2) for neutrons and if the form factor F(Q) is known thenit is possible to derive from X-ray scattering the nuclear space time correlationfunction G(r,z).In X-ray scattering work the energy resolution is insufficient to allow the spectrumof the scattered X-rays to be determined.For this reason the X-ray scattering cross-section consists of an integral over all possible values of co. Provided Q is constantit is then possible to write down the expression obtained by integrating over co ineither eqn. (2) or (5). This result isS(Q) = S(Q,w)do = dwJ exp [i(Q . r-wz)] . (G(r,t)-p}drdt= J exp (iQ . r) . {G(r,O)-p)dr, (6)where S(Q) is defined by this equation. The usual pair distribution function g ( r ) isgiven by g(r) = [G(r,O)/p] --6(r).A physical description of these equations may be given for a classical system.In eqn. (1) the space time correlation function G is the probability of finding an atomat the position (r,z) given that there was an atom at the origin at time z == 0.Thusthe co-ordinates r and z are the relative positions of two atoms. The functionG(r,z) may be divided into two parts : (i) a " self " term which describes the motionof the atom which was at the origin at time t = 0, and hence the " self" diffusionof this atom, and (ii) a " distinct " term describes the average motion of all otheratoms relative to this origin. Eqn. (3) represents the same function for electrons.It describes the probability of finding electron density at the position ( r , ~ ) given thatthere was (effectively) an electron at the origin at time z = 0. In eqn. (5) this pro-bability was factored assuming that the electrons are associated with particularnuclei, and consequently the major effects are to be expressed through the positionsand motions of the nuclei. Assumptions (a) and (b) given before eqn.( 5 ) ensurethat the only time dependence appearing in the G function (eqn. (3)) arises from themotion of the nuclei. In this case the correlation function for z = 0 is a convolutionof the nuclear pair distribution function with a function representing the averageelectron cloud around a nucleus. The Fourier transformation of this convolutionyields the expression ( 5 ) for the X-ray cross-section.Previously, the function F(Q) was known by calculation to much greater accuracythan S(Q), so that it was natural to use the X-ray data to derive S(Q). But S(QP. A . EGELSTAFF I53may be now obtained by neutron scattering to an accuracy of -2 %, which is signifi-cantly smaller than the error of the calculated IF(Q)12. As a result it is desirable toreverse the normal procedure and to divide S(Q) out of the X-ray data leaving lF(Q)12as the experimentally determined quantity.The theoretical uncertainty in F( Q)arises from three causes : (i) the free atom values of F( Q) are uncertain to - 5 % ;(ii) the free atom value is modified in the liquid environment by several % ; (iii) thethermal motion modifies F( Q), and in particular introduces a small temperaturedependence.Someevidence for this occurrence may be obtained by considering crystalline solids,particularly materials containing easily polarizable atoms. Cochran 3 has shownthat in order to describe the thermal motion of the atoms in these crystals it is necessaryto assume that a shell of outer electrons moves with respect to the nucleus and thecore electrons.This view has been substantiated by a number of experiments onphonon dispersion curves.For liquids (both metallic and non-metallic) this effect should occur, although itsmagnitude is hard to calculate. A several % effect is likely at Q values correspondingto the size of an atom, i.e., Q-2Qo. Unfortunately, such an effect renders the usualX-ray theory invalid, but since the effect is small an approximation may be madeto givewhere P( Q) is the Fourier transform of an average electron distribution includingperturbations due to effects (i)-(iii) above.The function S(Q) is measured by recording the intensity of scattered X-rays orneutrons as a function of the momentum transfer, i.e.,Item (iii) arises when the electron cloud moves relative to the nucleus.(dddQ)-P(Q) .S(QL (7)By dividing the neutron result into the X-ray result it is possible to measure thefunction P( Q). This requires measurements of high accuracy ( - 1 %) which is justpossible using modern techniques. In this way the positions of both the nuclei andthe electrons can be studied.4. GENERAL TRANSPORT COEFFICIENTSThe phenomena occurring in a liquid are both frequency (co) and wave number (Q)dependent. In this section we define a transport coefficient which depends uponthese parameters, and then discuss its relation to a measurable function (e.g., thescattering law) of the same parameters.Chester 4 uses phenomenological equations to define the general transport co-efficient as follows.We imagine a system in equilibrium which is disturbed by anumber u of different driving forces X,, and then suppose this causes a number ,LL ofcurrents J, to flow in the system ; in general all currents are assumed to be producedby each driving force. Now we assume a linear relation between the forces and thecurrents, and also, that the driving forces and the coefficients in this relation areposition and time dependent. The general linear relationship isr t PJ,(r,t) = CJ dtfJ dr'Zp,,(r- r', t - tf)X,(r',tf). (9)0 -c154 RADIATION SCATTERINGFourier transformation of this equation gives the general transport coefficientZ,,(Q,d, i.e.,In order to measure this quantity it is necessary to relate it to S(Q,o).For thispurpose, matter transport modes u are defined byu = A(Q,co) exp [i(Q . r-wz)] for r>O ] ( a >A(Q,co) = y J o m d r S d r Imag G(r,z) exp [ i ( Q . r-wz)] (b) i (11)It may be shown that for a classical liquid,S(Qyw) = +n 1 exp [i(Q . r-oz)]G(r,z)drdzso that the amplitude of these modes may be calculated from measurements ofA further simplification is achieved if it is assumed that the matter transport modesatisfies a simple differential equation.4 For the self term of G(r,z) it is assumed thatthe diffusion equation is appropriate, i.e.,S( Q4.(MG ,o)at f - c V 2 ) u s M = Q 2 Z exp [i(Q . r-or)],where D(Q,o) is a general diffusion coefficient.that the longitudinal wave equation is appropriate, orFor the complete G(r,z) it is assumedwhere V(Q,o) is a general viscosity coefficient.Because of these definitions themacroscopic diffusion D and viscosity coefficients are given in terms of D(Q,o) andv(Q,4 by2D = -([real D(Q,~)IQ+O)CO-~O, (15) nwherestituted into (13) and (14) the relationshipsand [ are the shear and bulk viscosities respectively. If eqn. (110) is sub-kTQ2/M= kTQ2/MS(Q)- ioQ2V(Q,w)<”are found. These relationships when combined with (1 1) enable some experimentaldata on D(Q,cu) and V(Q,o) to be obtained. In particular, the spectral density of thevelocity correlation function is equal to D(0,co) and the spectral density of the (z,z)stress correlation function is equal to V(0,o)P .A . EGELSTAFF 1555. COMPARISON OF DIFFERENT RADIATION SCATTERING METHODSIn the discussion leading to the formula in item 2(f) it has been implicitly assumedthat the atomic positions are all determined at the same instant of time. If this werethe case the cross-section would correspond to the intensity integrated over allpossible values of the energy transfer (i.e., S(Q)). On the other hand, if elasticscattering only were measured the correlation function would give the time-averagedpositions of the atoms. Thus there are three kinds of measurement which may bemade. These are (a) the intensity as a function of the momentum and the energytransferred in the scattering process : this gives the most complete information onthe positions and motions of the atoms in the scattering system.(b) The intensityTABLE 1 .-FUNCTIONS ACCESSIBLE BY RADIATION SCATTERING METHODSself G-functionincoherent neutron scatteringneutron absorptionpray absorption/emissioncoherent G-functioncoherent neutron scatteringelectron scatteringX-ray or 7-ray scatteringRaman or Brillouin scatteringas a function of the momentum transfer but integrated over all possible energytransfers : this gives the Fourier transform of a " snapshot " of the atomic positions.(c) The intensity as a function of the momentum transfer considering only elasticscattering : this gives the Fourier transformation of the time averaged atomic positionsand is used to study crystal structures.For liquids measurement, (c) is not strictly possible as there is no elastic scattering,but where a quasi-elastic peak is well defined it has some meaning.Only measure-ments (a) and (6) are considered here, and they may be subdivided into measurementsof the self and coherent G-functions. Table 1 shows the functions accessible to thedifferent techniques.TABLE 2.-wAVELENGTH AND ENERGY OF VARIOUS TYPES OF RADIATIONwavelength approximate approximatemomentum energy d E (expt.),A = 27KIAA-l € \radiationneutrons 1- 10 2 100-1 meV 10-3- 10-1y-rays 1-2 1 -10 keV - 10-9light (Raman) 4,000 10-3 -10 eV - 10-1light (Brillouin) 4,000 10-3 -10 eV - 10-5electrons 0.1 0.1 -50 keV wideX-rays 0*5+2 1 -10 keV wide6E (expt) indicates the energy transfer range which may be covered.These different types of radiation may be compared through their wavelength andenergy.Table 2 shows the typical values of wavelength and energy for neutrons,electrons and several kinds of electromagnetic radiation.In order to measure the correlation function in detail it is necessary for the wavelengthsto be of the order of magnitude of interatomic distances in condensed matter, andfor the energy resolution to be of the order of the widths and spacings of the energylevels in condensed matter. From table 2, neutrons satisfy these requirements,while the other kinds of radiation satisfy either one or the other of the two require-ments. This is further illustrated in fig. 1, which is a momentum-energy space diagram156 RADIATION SCATTERINGThe range of the variables (0- Q) covered in typical measurements by each techniquehas been marked, and through the use of a logarithmic scale the axes have beenplaced at positions equal to the reciprocal spacing,and energy levels of atoms in atypical liquid.To conclude this section some of the difficulties which are met in measuringda/dfi for X-rays and neutrons are discussed.First, there is the question of whetherQ is constant in a measurement at constant angle. For neutrons suffering an energytransfer of the difference AQ in Q compared to elastic scattering isThis ratio may be as high as 3, (e.g., for scattering by molecular liquids) while forAQ/Q = -ho/Eo. (17)X-RAY i-t? 8ELECT N INTEGRATE)-9FIG. 1.-Momentum energy space diagram showing the regions covered by different types ofradiation : the axes are drawn at values of the momentum corresponding to the reciprocal atomicspacing and the energy corresponding to room temperature.Broadly, the optical scattering experi-ments follow the energy axis. Scattering of X-rays and electrons involve an integration over theenergy transfer and provides information as a function of one variable (momentum) only. Neutronabsorption experiments occupy a very small region and involve relatively poor resolution. Theonly technique which covers the central portion of momentum and energy space is that of neutronscattering.other cases it may be as low as 1/30, (e.g., liquid metals). In the former case a cor-rection is necessary if the proper value for S(Q) is to be derived by eqn.(6). Cor-rections of this kind have been considered by Placzek 6 and discussed in detail byEnderby,’ while Ascarelli and Caglioti * suggest using a model in order to derive thecorrect value. Placzek’s method consists of writing an expansion for the cross-section in terms of the energy moments of S(Q,m). One difficulty is that hisexpansion converges rather slowly but if a model is used, as proposed by Ascarelliand Caglioti,8 the correction obtained may depend upon the details of the model.Indeed, this method seems to lead to corrections which are too large. There is noreal alternative to using a higher incident energy thereby making AQ small, andprobably the incident energy normally employed (0-07 eV) is too low in someinstancesP .A. EGELSTAFF I57The second item to consider is the form factor for X-rays. This has been calculatedfor the elements, and tables are normally used giving its value (e.g., MacGillavry andRieck 9 and Croiner.10) In addition these values may be checked by measurementsof the scattering by crystals. The normal practice is to apply these IF(Q)12 valuesto each case without considering individual corrections. The error on this procedureis difficult to estimate (see 93) but may be of the order of 5 %. It is difficult todetermine the X-ray scattered intensity at a large value of Q where the form factor issmall and the corresponding intensity low. Several background effects occur whichhave to be calculated and subtracted (e.g., Compton scattering etc.).This introducesan intensity threshold and hence a limiting value of Q beyond which the data areunreliable.Another problem is that of multiple scattering; this is more severe in neutronwork than in X-rays because of the higher absorption cross-sections for X-rays.Vineyard 11 and by Cocking and Heard,l2 have shown that for successful neutronexperiments it is necessary to use samples which are both thin and divided into seg-ments. The correction for multipie scattering may then be readily calculated andaccurate corrections made to the data. Absorption corrections, on the other hand,are large for X-rays but usually small for neutrons. Paalman and Pings 13 havediscussed this correction for X-rays.A problem in both kinds of experiments is that of low angle scattering, which isnecessary in order to make observations at low values of Q.For X-rays this isparticularly difficult, because the high absorption cross-sections means that a reflectiongeometry has to be used at glancing angles. Moreover, since the X-rays do notpenetrate a liquid metal metal significantly the condition of the surface is importantin that case. In contrast, low-angle scattering of neutrons is more straightforwardbecause the neutrons can be transmitted through the body of the sample due to the lowabsorption cross-section, On the other hand, the corrections for multiple scatteringare large because the probability of two large angle scatterings contributing tointensity in the forward direction is high.The normalization of scattering data to give an absolute scale is important.Reference to eqn.(8) shows that it is the constants A and B which are required.One procedure with X-rays has been to use the fact that S(Q) = 1 at large values ofQ. However, this method is difficult to carry out accurately because the X-rayintensity is small at high values of Q. Thus, where the assumption is most accuratethe measurements are most inaccurate. Another method is to normalize to thedensity, but the integration cannot be done accurately. For these reasons, trial anderror techniques are frequently used in X-ray work.With neutrons the normalization procedure is more straightforward, because thescattering by a single nucleus is isotropic and, for incoherent neutron scattering, theintensity is uniform as a function of angle.Fortunately, vanadium is an almostpurely incoherent scattering material and therefore forms an ideal calibration sub-stance. By measuring the ratio of scattered intensities from the sample and fromvanadium it is possible to obtain an accurate absolute scale for each measured point.This technique has been employed recently (e.g., Egelstaff et a2.14) and the accuracyobtained was about 5 %. However, there is no reason, in principle, why accuraciesof about 1 % should not be achieved using this method.The above features are the major experimental problems met in X-ray and neutronscattering work. There are usually other problems associated with the particularequipment used for the experiment or with the particular sample and the temperaturerange employed.The overall accuracy which is now obtainable is several percent ineach case, and it is just becoming feasible to make comparisons between the tw158 RADIATION SCATTERINGkinds of data. The above discussion applies to measurements of S(Q), but many ofthe points apply to the neutron measurements of S(Q,co) also.6. DISCUSSIONThe radiation scattering data as a function of both momentum and energy transferhave been discussed, with the aim to bring out those features which are relevant torecent experiments and to indicate some of the directions which future research mighttake. By combining the neutron and X-ray data, it may be possible to obtaininformation on the electron distribution in the liquid.Such work may well forma valuable goal for high precision measurements. The limited data that are availableat the present time suggest that a real effect may be observed.In addition, the direct analysis of the function S(Q) to test certain hypotheses,such as that of the uniform expansion of liquids, should be emphasized. Withcrystalline solids it is sometimes advantageous to discuss the behaviour of the Fouriercomponents of the density (i.e., of S(Q)). Amongst the data which may be obtainedin this way are the range of crystalline order, the spacing and number of atoms in agiven direction, the expansion coefficient for the lattice itself and the size of the thermalcloud. A similar approach may be of value for liquids. For example, the tempera-ture variation of the position of the first two peaks in S(Q) may be compared to thechange in the cube root of the density.For a perfect anharmonic lattice the expansionof g(r) would be uniform and thus the contraction of S(Q) would be proportional top*. Other models of interest include the harmonic lattice, where the positions arefixed but the peaks fall and the valleys rise to give a constant density, and the hard-sphere gas where the main peak position is fixed and the amplitude scales as the densityto a first approximation. The experimental data show that the movements of thepeak positions are neither proportional to p* nor independent of p, thereby indicatingthat the liquid differs from these extremes in that some spacings change at a differentrate than others.More data on S(Q) are needed over wider ranges of the parameterQ and over wider temperature ranges, and that these data should be measured to anaccuracy of approximately 1 %.An understanding of microscopic transport phenomena in liquids requires atheoretical and experimental examination of modes of motion over wide ranges offrequency and wave number. The background for discussing such effects has beendeveloped in terms of correlation functions. With the aid of new radiation scatteringtechniques it is now possible to measure some parts of the correlation functionsused in the theory. For temperatures near to the melting point temperature, suchexperiments have shown that the high-frequency behaviour of the liquid state issimilar to the solid state in accordance with the visco-elastic theory of Frenkel andothers.Of the various functions now accessible experimentally the velocity correlationfunction has been studied in most detail, and in particular the shape of the spectraldensity has been determined.The time for a diffusive step has been measured fora number of systems and found to be fairly independent of material and temperature(- 1-2 x 10-12 sec). It seems to give one basic relaxation time in liquids. It isnot the shortest time involved since the damping of high-frequency modes is necessarilyfaster and gives another basic relaxation time for energy reorganization of - 3 of theformer time. Thus, both these relaxation times (and the ordering time mentionedbelow) are ca.10-12 sec.Experiments aimed at the study of the correlation functions used to discussviscosity and heat conduction are limited so far. The spectral density of the correla-tion function for viscosity has the same general shape as that for diffusion, but thP . A . EGLESTAFF 159magnitude and detailed shape are different. High-frequency co-operative modes ofmotion have been observed in several liquids, the maximum frequency being similarto that for the corresponding solid. With present techniques it is difficult to identifythe polarization of the mode, but significant evidence can be advanced for the existenceof high-frequency transverse modes in simple liquids. A detailed theoretical dis-cussion of all these points is awaited, since existing treatments are mainly phenomeno-logical.A simple picture of a dense liquid may be built up from recent structural anddynamic studies.It is postulated that a liquid is trying to be a structurally orderedmaterial but that at any instant a significant number of atoms is in flight betweenordered sites. The time in flight (i.e., the time to reach the ordered site) is relativelylong so that by the time the “ in-flight ” atoms have completed their move some of theatoms previously ordered have started to diffuse to new sites. Atoms within therange of order in the liquid will contribute to the high-frequency vibrational modes,but atoms outside the range of order appear to be (relative to the local atoms)stationary.On this model, these are the atoms which are in flight, so that the flighttime has to be longer than the time for a wave to propagate over the range of order ;the role of the “on-site ” and ‘‘ in-flight ” atoms can be interchanged. An equivalentargument is to say that the atoms outside the range of order appear stationarybecause many Fourier (frequency) components become important which from thecentral limit theorem leads to non-propagating motion. The lifetime for suchmotion can be measured and is found to be N 10-12 sec ; on this model this is thetime it takes for an atom to move to an ordered site.In the future it will be necessary to extend experimental measurements to wideranges of temperature and pressure, essentially covering the whole of the liquidregion of the condensed state. The complementary nature of the different radiationscattering techniques should be exploited and the results combined with measurementsusing the low-frequency techniques (ultrasonics, etc.). In this way, more detailedinformation on the correlation functions can be obtained and a more detailed under-standing of the liquid state achieved.1 L. Van Hove, Physic. Rev., 1954, 95, 249.2 A. Sjolander, Phonons and Phonon Interactions, ed. T. A. Bak (W. A. Benjamin, New York,3 W. Cochran, Proc. Roy. SOC. A , 1959,253, 260.4 G. Chester, Reports Progr. Physics 1963, 26, 41 1.5 P. A. Egelstaff, Reports Progr. Physics, 1966, 29, 333.6 G. Placzek, Physic. Rev., 1952, 86, 377.7 J. E. Enderby, The Physics of Simple Liquids, ed. J. S . Rowlinson, G. S. Rushbrooke and8 P. Ascarelli and G. Caglioti, Nuovo. Cim., 1966, X43, 375.9 C. H. MacGillavry and G. D. Rieck, International Tables for X-ray Crystallography, III.10D. T. Cromer, Acta Cryst., 1965, 19, 224.11 G. Vineyard, Physic. Rev., 1954, 96, 93.12 S. J. Cocking and C. R. T. Heard, AERE Report R 5016, 1965 (H.M.S.O., London).13 H. H. Paalman and C. J. Pings, J. Appl. Physics, 1962, 33, 2635.14 P. A. Egelstaff, C. Duffill, V. S . Rainey, J. E. Enderby and D. M. North, Physics Letts., 1966,1964), p. 76.H. N. V. Temperley (North Holland Publishing Co., 1966).Physical and Chemical Tables (The Kynoch Press, Birmingham, 1962).21, 286

ISSN:0366-9033    

DOI:10.1039/DF9674300149

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

Brillouin Scattering of Neutrons from LiquidsBY B. DORNER, TH. PLESSER AND H. STILLERKernforschungsanlage, Institut fur Neutronenphysik, 5 17 Julich, W.-GermanyReceived 16th January, 1967Inelastic coherent neutron scattering is called Brillouin scattering, if the momentum transferredto the scattering system is entirely taken up by collective particle motions. With this scattering,single excited states of such motions can be observed in polycrystals and in liquids in a similar wayas in single crystals, provided the measurements are not too much contaminated by multiple scatteringprocesses combining Bragg and inelastic scattering. Liquid and solid lead are investigated in thisway. The comparison of the measured dispersion laws shows that at 340°C, for frequencies between6-4 x 1011 and 2.3 x 1012 sec-1 and wavelengths between 7 and 21 A, sound waves propagate in theliquid by the same mechanism as in the solid. The contaminating double scattering is predominantlydetermined by maxima in the distribution of frequencies of collective motions.The observedpositions of such maxima indicate that transverse modes exist also in the liquid.As in liquids the kinetic energy of the particles is comparable to the interactions,the liquid phase cannot be extrapolated to an ideal state, for which the N-particle-problem may be reduced to a number of one-particle-problems. As a consequencethe liquid state can be described rigorously for two limiting regions of space and timeonly ; for very small times and distances, where all systems behave like a gas, and forvery large times and distances, where the equations of hydrodynamics are valid.For the intermediate region the most successful descriptions are based not on firstprinciples but on models, which require experimental tests.As the region of interestextends from approximately 1 to, say, 1008, and from approximately 10-14 to 10-10sec, inelastic scattering of slow neutrons appears to be the most promising tool forsuch tests ; neutron radiation taken from a reactor contains wavelengths between0.5 and approximately 20A and frequencies between 1014 and 5 x 1010 sec-1.* Ashas been shown by Van Hove 1 the most detailed information is obtained with coherentinelastic scattering. The measurable distribution, Scoh (K,w), of momentum transfersnK and energy transfers hco is related to a time-dependent correlation function FG(rJ) = Gs(r,t) + Gd(r,t) byScoh(K,m) = - G(r,t) exp [i(Kr - cot)]drdt.27G "sGs(r,t) is the self-correlation, G&,t) is the pair-correlation,where ko is the wave vector of the incoming, kl the wave vector of the scatteredneutrons, q5 the scattering angle, m the neutron mass.*In contrast, electromagnetic radiation of 10 8, wavelength has a frequency of 3 x 1017 sec-1.16B . DORNER, TH. PLESSER AND H. STILLER 161In general, a description of the space-time behaviour of a many-particle systemcan be tested in a direct way, if the relation, a(@, between the frequencies u) and thewave vectors q of collective particle motions can be predicted from the description or-the one hand and measured on the other hand.For instance, if the propagation ofsound in a liquid is phenomenologically described by a wave equation for the imaginarypart of G(r,t),* one obtains 3 :NA2K2 fiw 1Sgk(K,w)a--( 4x2m 1 + coth -){ 2kBT (a - w , ) ~ + ( 2 ~ 3 - ~ + (u, +withThe index s specifies the modes,? z is a relaxation time, c the sound velocity. Underthe assumption, that the dynamical behaviour of the liquid can be described by somesingle 2, the dependence of this quantity on q may be calculated from models.$ Themodels then can be tested by experimental determinations of a&).For the evaluation of corresponding experiments, the most difficult problem liesin the relation between q and K. With single crystals such a relation is given by theinterference condition K = 2xg -q, where 2xg is a reciprocal lattice vector and hg isthe momentum, which in the scattering process is taken up by the crystal withoutenergy change.For liquid systems the question, which part of the momentumtransfer EK may be taken up by the system without energy change, is extremelycomplicated. To answer it, one needs a detailed knowledge of the liquid structureand also of the distribution of frequencies in the system.6 These difficulties areavoided if the measurement is carried out within a region of reciprocal space wherewith certainty no momentum is transferred without energy change, i.e., in a regioncorresponding to the innermost Brillouin zone of a solid; we call this scatteringBrillouin scattering.§ The interference condition is theni.e., we have five equations, eqn. (3), (5) and (6), for five experimental variables,namely, a, q and one quantity varied in the experiment, for instance ikll; we obtainwithin &oh (K,w) peaks at those values of the variables which satisfy those equations.By repeating the measurement with changed values for quantities kept constantwithin the single experiment, for instance with different 4, we can determine certainbranches of co,(q), viz., those branches, where according to eqn.(6) K is parallel to q(for instance, the longitudinal mode for sound waves).In contrast to experiments, in which the system takes up some average momentumwithout energy change, with Brillouin scattering each peak in &oh (K,o) represents asingle excited state of collective particle motions.Hence, with this method the onlyaveraging, which enters the experiment, concerns a possible directional dependenceK = -q, (6)*ImG(r,f) represents the density disturbance 2 ; it is related to the real part of G(r,f) by thefluctuation-dissipation theorem.?The three values of s for acoustic motions are usually referred to as “longitudinal” and“transverse”, respectiveIy. A more precise meaning is discussed in ref. (4).$From hydrodynamics one has 5 : 7-1 = A@, where A is a constant given by the transport co-efficients. For a strict validity of hydrodynamics the transport coefficients should not depend onq or a.$Because with inelastic interference scattering of light 7 one always measures within the innermostBrillouin-zone, since, in contrast to neutron scattering? the total momentum transfer is nearly zerocompare the first footnote ref.(1)).162 BRILLOUIN SCATTERING OF NEUTRONSof w,(q), i.e., some possible short-range elastic anisotropy of the system. Even ifsuch an anisotropy exists, this averaging is not very serious 4 from an experimentalpoint of view, e.g. for the accuracy of frequency determinations or for the determina-Na, q = 3(2x/a)w [lo13 sec-11FIG. l.-Directional distribution of sound wave frequencies for fixed wave number 141 , gq(o), forpolycrystalline Al (small elastic anisotropy) and polycrystalline Na (large anisotropy). The secondand the third peak fa11 into symmetry directions.The first peak contains a mixture of modes;from ref. (4).tion of lifetimes, as long as the short range order is of sufficiently high symmetry,because then the directional distribution of frequencies is sharply peaked. This isshown in fig. 1.EXPERIMENTALMETHODUnfortunately, the simplicity of the experimental results obtained by measuring closeto the origin of reciprocal space is paid for by small scattered intensities. The reasonsare shown in fig. 2. The curves represent eqn. (2) and (3) after elimination of kl for severalEO = @k$/2m and for scattering angles 4 of 0 and 180". The region, for which for liquidlead one may be certain that no momentum is taken up without energy change, is shaded inthe figure. In order to stay within this region and to cross e.g., the lineco = cK, one needssmall k and large EO and, consequently, small 4.For small IKI the probability for excitationor de-excitation of collective motions is small (see e.g. ref. (8)). Small values of 4 and largevalues of EO (with correspondingly small resolutions AEo/Eo) restrict the accessible solidanglesB . DORNER, TH. PLESSER AND H . STILLER 163K (A-1) .-/’10 5 0 -5 - 10fia (meV)FIG. 2.--0, K plane : the branches of the broken curves limit the regions accessible with fixed EOThe shaded area is the region, where 2xg = 0 in liquid lead. The solid curves represent dispersionbranches for certain directions in crystalline lead. Values of t$ are given on each curve.shutterMonochromatorMonitor counter‘ B F ~ countorFIG.3.-Schematic representation of the 3-axes-spectrometer at the FRJ-2 reactor. OM = Braggangle of monochromizing single crystal, OA = Bragg angle analyzing crystal, 4 = scattering angle.The three angles can be varied simultaneously and automatically164 BRILLOUIN SCATTERING OF NEUTRONSMeasurements on liquid and powdered solid Pb were carried out at the FRJ-2 reactorwith a 3-axes-spectrometer (fig. 3), partly at constant 4 and partly with K kept constantby varying simultaneously kl and 4. Fig. 4 shows typical results for liquid and polycrystal-line lead. The data are given without any corrections.i :* .. .....'..... . ,, . _ _ . . .: ,. . - . . , .53' 54' 55' 56O 57' sao 59' 60' 61' 62' 63'20AFIG. 4.-Typical results of a measurement withK kept constant.kl = ~(dsin Od-1, with d =distance of lattice planes employed for Bragg re-flection from the analyzing crystal. For 20A =63" the scattering is elastic. The backgrounddata belong to the data for polycrystalline Pb.The background for the measurements on theliquid was correspondingly higher but of the sameshape. The background was determined withthe empty container and an absorber foil, thethickness of which was chosen such that thetransmission of the empty container was 85 %,as for scattering from the sample.- 0 -polycrystalline- x - liquidbackgroundq = 0-6 A-1 EO = 24.55 meVRESULTS AND DISCUSSIONFig. 5 shows the results of three measurements on liquid lead performed with K keptconstant. Fig.6 shows the positions of all inelastic scattering peaks observed,* i.e., thedispersion law co,(q). For comparison the results obtained with polycrystalline lead arealso shown. The open circles represent points found with liquid lead by Cocking andEgelstaff 10 in a time-of-flight experiment with much smaller incident energy; the dottedcurve indicates the region which could be covered in ref. (10). The solid curves representdispersion laws to be expected (a) from hydrodynamics and (b) from the kinetic descriptionof the liquid state by Nelkin, Van Leeuwen and Yip.11 Neither (a) nor (b) can describethe observations for ~ 0 . 4 A-1.The most striking festure of the measured curves is the appearance of two dispersionbranches in both the solid and the liquid.Two dispersion branches for sound waves inliquids are not expected by any theory. The reality of this observation was questioned byconsidering one special scattering process which may have contaminated the experiments,double scattering combining Bragg and coherent inelastic scattering. The possibility isillustrated in fig. 7 for a polycrystalline sample; neutrons, reaching the sample with wavevector ko, are first Bragg-scattered from planes 27cg (respresented in reciprocal space by asphere of radius 12ng1) and then scattered inelastically (with excitation or de-excitation ef*Measurements not included in fig. 5 were done with different ko and with 6 kept constant.Preliminary results, obtained with the same instrument at the BR-2 reactor in Mol, Belgium, havebeen reported already earlier.9 They are included in fig.(6)B . DORNER, TH. PLESSER AND H. STILLER 165c 10Aw [meV]FIG. 5-Results of three constant K measurements on liquid lead at 340( f4)"C. The backgroundis subtracted. The intensity, I(lio), has been calculated from data, n(BA), as shown in fig. 4. Thedata are corrected for the reflectivity, R(OA), of the analyzing crystal.0FIG. 6.-o,(q): 0, for liquid lead at 360°C (measured by Cocking and Egelstaff 10; 9, for liquidlead at 340°C (present measurements) ; w, for solid powdered lead at 20°C (present measurements) ;----, measured in certain symmetry directions of a Pb single crystal 12 ; -, (a) from hydrodynamics :eqn. (5) with c = 2100 msec-1 and l/z = (2y1/3p)@; 71 = 2.58 centipoise; the terms involving 1 2and the heat conductivity are neglected, (b) from the gas-like model of Bhatnagar et aZ.13 comparealso ref.(11); . . . . limits the region, which could be covered with the measurements.1166 BRILLOUIN SCATTERING OF NEUTRONSsound waves) into the direction of observation, i.e., into the direction of kl. After theBragg scattering we have neutrons with wave vectors kb, which fall on to a cone around ko.For the succeeding inelastic scattering the end-point of each vector kb can be considered asthe origin of a secondary reciprocal space, i.e., the coherence condition K’ = kh- kl = 2ng’-q now refers to spheres of radius 2719’ (lg’l = lgl) around the end-points of the vectorsk& ; two such spheres are drawn into fig.7. There will be interference scattering, wheneverfor some kl, i.e. for some u), the corresponding lql can be drawn to any one or several of such\I--- \I \ \\ /\FIG. 7.-Bragg scattering followed by de-excitation of sound waves, represented in the scatteringplane of reciprocal space for a polycrystal.spheres. An estimate on the contribution of this double scattering is given in the appendix.It shows that the contribution is not larger than about 10 % for the high frequency branch,but considerable for the lower branch. It also shows that the co-dependence of this doublescattering is predominantly determined by the distribution of sound frequencies; hence,even with this contamination the low branch frequencies are real. The same argumentshold if the sound wave de-excitation precedes the Bragg scattering, and they qualitativelyalso hold for the liquid, the only difference being that here the surfaces of the spheres inreciprocal space have a finite thickness with a structure corresponding to diffractionpatterns.*We now ask, do the two dispersion branches, observed in the liquid, both representlongitudinal modes or does the lower branch, as a consequence of the contamination dis-cussed above, also contain transverse modes? The first possibility cannot be excluded,because it has been shown,4 that, if the system has some elastic anisotropy, the modes aremixed within the frequency band (fig.l).?*The concept of a reciprocal lattice vector is not strictly valid for liquids. For the q-o-rangehere considered its introduction, however, does not invalidate the arguments given above.Themeasurements reported in ref. (lo), indicated in fig. 6 , were done with a ko smaller than the limitingvalue for Bragg scattering in solid Pb. This may be the reason for the points of w(q) being observedfor liquid Pb only.?Another explanation for a low-frequency branch has been suggested by Egelstaff 14 : stronginteractions among sound waves might give rise to secondary interference maxima called “pseudo-phonons”. For the present case, however, we believe this explanation to be unlikely, because thelower branch covers too high frequency values, and because it does not exhibit strong temperaturedependenceB . DORNER, TH. PLESSER AND H. STILLER 167The intensity of scattering with sound wave excitation or de-excitation is proportionalto c-2, if c is the phase velocity of the sound wave (see e.g.ref. (8)). If L1 is the relativeamount of longitudinal modes on the upper branch, L2 the relative amount on the lowerbranch (L1+L2 = l), D1 the relative contribution of double scattering on the upper, D2 onthe lower branch, then the ratio of scattered intensity for the lower branch to scatteredintensity for the upper branch should be(a) if only longitudinal modes exist k).=(zY(6) if both modes existAs L24 1,* we expect (I~/11)~<1, and (IZ/rl)b2:(cl/c2)2D2"-2'5, with 0 2 = 0-65 asestimated in the appendix. Since experimentally I2/11 was always found to be greater than1 (compare fig. 3, we conclude that transverse modes are c0ntributing.tThe coniparison of the dispersion of sound waves in liquid and in solid powdered leadthus show, that at 340°C for frequencies between 6 .4 ~ 1011 and 2 . 3 ~ 1012 sec-1 and wave-lengths between 7 and 21 A, liquid lead behaves much like a solid. The relaxation timesfor all modes are considerably longer than 5x 10-11 sec. q~O.4A-1 (Air 16& is certainlyan upper limit for the validity of hydrodynamics. Unfortunately the experimental resultscannot yet be compared to the most detailed solid-like model for the liquid state suggestedso far, the model of Singwi,l6 because the dependence ofw,(q) or of z on the parameter R,which in this model characterizes the mean extension of regions of solid-like dynamics, hasnot yet been worked out.The authors thank Dr T.Springer and Mr K. Mika for helpful discussions, Mr F.Fredel for his continuous assistance with the experiments and Mr K. W. Kutzbachfor his help with the computer calculations.APPENDIXThe relative amount of double scattering, combining Bragg scattering from a set ofplanes 27cg and de-excitation of sound waves of frequencyw, iswhere PB(g) is the probability for Bragg reflection from planes 2ng, Pl(K,w) is the probabilityfor de-excitation of sound waves with g = 0, &(g',q,w) the probability for sound wavede-excitation following Bragg scattering. For a liquid, PB(Q) is to be replaced by the relativeintensity of a diffraction peak. We have (see ref. (8) and (17))k k,Tko 2Mc2P,(K,w) = Nda2' - exp ( -u2K2), W.2)and (see ref.(6)),k zkJ Fffp f(wJ (e, . K:)2k, 4MV g 4i,s w,"P,(g',q,w) = N d a 2 L - Ic- - exp (- u2Ki2). (A.3)KfN is the density of scattering atoms, d is the sample thickness, a the coherent scatteringamplitude, c the phase velocity of sound waves, M the mass of the scattering atoms, Y thevolume of the unit cell, the multiplicity of planes 2nq', f(cos) the spectrum of sound wave*For q<04 A-1, &%as been estimated to be approximately 0-16.TAn indication for transverse modes also has been found 15 in liquid Al168 BRILLOUIN SCATTERING OF NEUTRONSfrequencies, e, a unit vector in the direction of the wave polarization, u2 the mean squaredisplacement of the atoms. The summation goes over all spheres, for which 2ng'-q = Kfis f a l e d (see fig.7). As PZ must go to zero for q+O (elastic scattering), Kf = 2ng' mustbe excluded from the summation.Since the origins of the Kf are uniformly distributed on the base circle of the cone aroundko, the sum over i in (A.3) can be written as an integral. We writeK f 2 = kg+kf-2k,kl cos 40(cos 4+sin 4 tan q),where #o is the angle between ko and kb, 4 the scattering angle, y the angle between ko andthe projection of kh on the plane of fig. 7 (the scattering plane). Thusx ~ X P [ - ~~K'~(q)]dq, (A.4)where K&n = 27qf-9, as long as Kf(4o)>2.ng'-q, and ICkin = K ' ( ~ o ) , if KAin\<2ng1-q,and correspondingly KAax = 2ng'+q, as long as K'(--0)<27rg'+q, and KA,, = K'(-&,)otherwise. The expression (A.4) has been evaluated numerically, with ex = q/q for the highfrequency branch, which is entirely longitudinal, and with (e, .K')2 = K'2( 1 - (q . K')2/qzK'2)for the lower branch, i.e., neglecting the small contribution of longitudinal modes to thelower branches (compare ref. (4)), and with f(wJ and q(coJ taken from an approximateisotropic dispersion tq = cx(q+bsq2). As a function of ci), (A.4) is predominantly deter-mined by f(coJ.P&) was determined experimentally as the elastic scattering density per sterradian onthe base circle of the cone around ko, - 2ng' - ' - where A~ is the vertical collimatordivergence. For 27rg = 2.18A-1 it was found PB = 1 . 6 ~ 10-3. After summing withappropriate weights over all sets of planes, from which reflectioas are possible with theincident wave length, we obtain for K = 0.6 A-1 : for the high frequency branch (bco =7 mew, D150.1, and for the low frequency branch (%o = 3-2 mev) D250.65.ko I0 A$1 L. Van Hove, Physic. Rev., 1954,95,249.2 L. Van Hove, Physica, 1958,24,404.3 T. Ruijgrok, Physica, 1963,29,617.4 K. Mika, to be published.5 L. Kadanoff and P. Martin, Ann. Physics., 1963,24,419.6P. Egelstaff, A.E.R.E. N/R 1164 (1953).7 see e.g., J. Frenkel, Kinetische Theorie der Fliissigkeiten (VEB 1957), chap. VI, 9.8 A. Sjoelander, Arkiv Fysik, 1958, 14, 315.9 B. Dorner, Th. Plesser and H. Stiller, Physica, 1965, 31, 1537.10 S. Cocking and P. Egelstaff, Physic. Letters, 1965, 16, 130.11 M. Nelkin, J. Van Leeuwen and S. Yip, Inel. Scatt. of Neutrons, 1965, vol. 11, p. 35.12 B. Brockhouse, T. Arase, G. Caglioti, K. Rao and A. Woods, Physic. Rev. 1962,128,1099.13 P. Bhatnagar, E. Gross and M. Krook, Physic Rev., 1954,94, 511.14 P. Egelstaff, Lattice Dynamics (Pergamon Press 1964), p. 699.15 K. Larsson, U. Dahlborg and D. Jovic, Inel. Scatt. of Neutrons, 1965, vol. IT, p. 117.16K. Singwi, Physic. Rev., A , 1964, 136, 969; Physica, 1965, 31, 1257.17 B. Dorner, Report Jiil.-412-NP, 1966

ISSN:0366-9033    

DOI:10.1039/DF9674300160

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

Neutron Scattering Spectroscopy of LiquidsBY B. K. ALDRED, R. C. EDEN AND J. W. WHITEPhysical Chemistry Laboratory, South Parks Road, OxfordReceived 3rd March 1967This paper reports the use of atomic substitution techiques in some preliminary studies by neutronscattering spectroscopy of the microscopic transport properties of acetic acid and methanol. Bymaking selective substitution of deuterium and fluorine for the hydrogens in the CH3 and OH groupsof these molecules, the separate contributions to the quantized and diffusive motions of each havebeen analyzed. This is possible because the substitution of fluorine or deuterium for hydrogen in aparticular group reduces the scattering cross-section of that group by at least an order of magnitude.The contributions of the group motions to the neutron scattering spectrum are then correspondinglyreduced. In the inelastic scattering region of the spectrum we have been able to make vibrationalassignments as a result.In the quasi-elastic region the intensity and angular dependence of thescattering show a marked dependency on the nature of the protons substituted. In this region theresults have been analyzed by determining an effective molecular diffusion coefficient and by comparingthis with values from bulk-phase studies. Particular attention has been devoted to the question ofthe extent to which intra-molecular hindered rotation contributes to the quasi-elastic scattering.In neutron scattering spectroscopy a beam of thermal neutrons, which has beenmade almost monoenergetic by velocity selection, is allowed to fall on the sample.Neutrons in the beam are scattered, often through large angles, in collisions withmoving atomic nuclei in the sample.The scattering collisions may be either elasticor inelastic. Elastic scattering alone would occur for a perfectly rigid scatteringcentre ; inelastic processes follow when energy can be exchanged between the neutronand the scattering nucleus which is in motion. For molecules, this inelastic exchangeof energy can occur with quantized motions, such as molecular vibrations androtations, or with non-quantized diffusional motion in the liquid and gaseous states.Experimentally, neutron scattering spectra are recorded by analyzing the energiesof all neutrons scattered over a particular angle by some method such as time of flight.In general, the spectra are recorded at several different angles of scattering (whichcorrespond to different momentum transfers for the scattered neutron) and they mayby divided up into a quasi-elastic and an inelastic region.The technique thus resembles Raman scattering except that a beam of mono-energetic neutrons rather than photons is scattered by the moving molecules.Thereare some important differences, however, which make neutron studies, especiallyon liquids, of unique interest. These stem from the fundamental difference inmechanism between neutron and photon excitation of molecular motion and from thegreatly different time scales involved in the neutron and in the photon ‘‘collision’’with the scattering centre.In molecules, photon excitation of molecular vibrationsoccurs via the electrons in the molecules which transmit the electric disturbance tothe charged nuclei and cause them to vibrate. Thus optical excitation invariablyinvolves dipole and induced dipole selection rules. Neutron excitation by contrastoccurs by the direct impact of the neutron and the scattering nucleus in the bondsuffering excitation. A consequence of this direct method of stimulation is that notonly can vibrational and rotational frequencies be determined as by optical means,161 70 NEUTRON SCATTERING SPECTROSCOPYbut the absolute intensities of the bands concerned can be measured. Furtherdeductions may follow this about the mean square amplitude of the motions con-cerned.Again, because the neutron velocity is much lower than that of light, itsinteraction time with the molecule during scattering is about 10-11 sec. This timeis long enough for some liquid molecules to move so that Doppler broadening ofthe energy spectrum, even for elastically scattered neutrons occurs. This quasi-elastic scattering often leads to lines in the elastic region which are considerablybroader than the energy spread of the incident neutron beam. The broadening isgreatest the faster the relative motion between the scattering molecule and theneutron during the time of the scattering event. It is thus greatest at highest anglesof scattering (largest momentum transfers) and for the most mobile liquids.Asdiscussed below, the broadening is directly related to microdiffusion of the moleculeand can sometimes be related to the bulk diffusion constant.In the inelastic region information about molecular vibration and rotationfrequencies (although not necessarily about their intensities) should be complementaryto that from optical spectra. In some cases difficulties have been encountered 1 inachieving a synthesis between these two different sets of results. This has beenpartly because of difficulty in making unambiguous assignments of bands in theneutron spectrum to particular vibrating groups as the resolution in neutron spectro-scopy is still lower than that in optical spectroscopy. The atomic substitutiontechnique, reviewed for two groups of molecules in this paper, suggests a methodof making more confident assignments.Secondly, there has been a considerablediscussion 2, 3 about whether the diffusion constants measured by neutron scatteringobservations in the quasi-elastic region are the same as those determined by tracerstudies in the bulk phase. In general, the diffusion constants measured by neutronshave been greater than those measured in the bulk, and it has been suggested thatsince the neutron method measures diffusion over a very small distance (typically1-2 A) microdiffusional processes, such as hindered internal rotation, contribute to thetransport of scattering nuclei during the ccevent’y. The atomic substitution techniquefor a molecule which contains two scattering centres, one susceptible to internalrotation and the other not, provides an experimental method of testing these views.The time development of the displacement of a representative incoherent scatteringcentre in a moving molecule may be described by the space-time correlation functionG(r,t).4 This function may be made sufficiently general to include both oscillatoryand diffusive motions of the scattering atom.The time Fourier transform of thisfunction gives the spectral density of the atomic motions as a function of frequency.This is the same spectral density function which is used to discuss the frequencydependence of nuclear and electron magnetic relaxation probabilities. The way inwhich the spectral density can be obtained from neutron measurements is illustratedelbow.For an incident neutron spectrum with a narrow energy spread, the observedintensity distribution of the scattered neutrons as a function of the energy transfer CL)and the solid angle of scatter $2 is identical to the differential scattering cross sectiona2alaaaW :-- exp (i(Q .r-cot)) . G(r,t)drdt.Here b is the scattering length of the nucleus; k, ko are the scattered and incidentwave vectors of the neutron respectively, Q = k - ko and is the momentum transferin the collision, w is the energy transferred in the collision, expressed in radianslsec,and G(r,t) is the space-time correlation function describing the motion of the atoB . K . ALDRED, R . C. EDEN AND J . W . WHITE 171It is a correlation function and is defined by containing the scattering nucleus.eqn.(2) :In this expression, rn(t) is the position of the atom rn at time t. The functionalrelationship between Y and t may be simply sinusoidal as for a vibrator, or it may havea complicated form as for diffusive motions in liquids. Suitable expressions forG(r,t) have been discussed by Larsson.5 For convenience, the concept of a scatteringlaw S(Q,o) is introduced to describe the distribution of the intensity of scatteringas a function of momentum transfer and energy transfer. This is defined from eqn.(1) in eqn. (3).The scattering law defined in eqn. (3) is not symmetrical because the anti-Stokesscattering is more probable than the Stokes scattering of cold neutrons from samplesat room temperature. The intensities of the Stokes and anti-Stokes regions areconnected by a Boltzmann factor and so S(Q,o) can be modified to a symmetricalscattering law, (Q,o) independent of populations, by extracting an exponential asin eqn. (4).d % / X i a o = b2kS( Q,o)/ko (3)Each of the parent molecules considered in this paper has two incoherent scatteringcentres.If these centres have scattering lengths bl and b2 then the differentialscattering cross-section can be written asThere are no cross-terms so long as the scattering from each centre is incoherent andso long as ortho and para states for the two centre system have sufficiently closeenergies. Since the neutron scattering spectrum is the simple sum of the effects dueto two independent scattering laws, whose relative importance is determined by thevalues of the bl and b2, it is possible to make clear assignments by isotopic and atomicsubstitution.In our case we have been able to change the relative values of bl andb2 (which are proportional to the square-root of the scattering cross-section) sothat the substituted spectrum is largely derived from one or other of the two scatteringlaws. The changes produced by this device not only occur in the inelastic scatteringregion of the spectrum but also in the quasi-elastic region since the scattering lawdescribes this as well.The spectral density of the atomic motional frequencies, as a function of frequencyis a most suitable function for comparisons of neutron scattering, magnetic resonanceand infra-red data since the neutron data overlap the regions of interest in bothtechniques. This function may be obtained from values of the reduced scatteringlaw defined in eqn.(4), by finding the limit defined in eqn. (6) :a20/8Qao = (k/ko)[b:S,(Q,w) + b f s , ( Q , ~ ) ] . ( 5 )f ( o ) = const. o2 iim (s(Q,co)/Q~) (6)Q-tOThe limit chosen in this expression is the optical limit of zero momentum transfer.For liquids a number of models have been proposed to describe the molecularmotions and hence to provide values of S(Q,o). Using these it is possible to analyzethe shape and angular dependence of the quasi-elastic region of the spectrum toobtain the diffusion coefficients associated with a particular scattering centre. Th172 NEUTRON SCATTERING SPECTROSCOPYvarious models have been discussed by Larsson 5.6 and Sjolander.7 For a moleculeundergoing simple diffusion the differential scattering cross section is given by eqn. (7) :a2a exp ( -x2Q2)DQ2(DQ2)2+m2 ' ~ = const. anamwhere Q 2 is the momentum transfer squared, cu2 is the energy transfer squared, D isthe diffusion constant defined by Fick's law and x2 is a Debye-Waller factor fromwhich it is possible to obtain the amplitude of the motion giving rise to diEusion.5The energy width of the quasi-elastic region for a liquid which obeys simple diffusion(eqn. (7)), should increase linearly with momentum transfer squared, as shown byeqn. (8) :where fi is Planck's constant divided by 27c.AE = 2ADQ2 (8)EXPERIMENTALThe experiments were carried out using the cold neutron time-of-flight spectrometer onthe DIDO reactor at A.E.R.E.Harwell. This has been described by Harris et aZ.8 and isshown diagramatically in fig. 1. Thermal neutrons from the moderator of the reactor arepassed into a liquid hydrogen moderator, 3 in. diani. and 2 in. long, which reduced theirD1 DOReactorCoreLiquid RefrigeratedHydroger. F i l t e i ASource?--LFIG. 1.-Schematic diagram of the cold neutron scattering spectrometer on the DIDO reactor atHamell. For the experiments described here the chopper was set to pass 5-3 A neutrons.energy to a Maxwell-Boltmann distribution about 30°K. Any fast neutrons unchangedby this process are removed from the beam by passing it through a filter of beryllium andbismuth cooled in liquid-nitrogen. The beam then passes to a rotating chopper which hasthe dual function of pulsing the beam and also transmitting only neutrons which have adefined small range of velocities.The velocities of the neutrons passed are determined bythe curvature of the slots in the rotor and by its spinning speed. The sample is inclined at45" to the neutron beam and the scattered neutrons are collected in boron trifluoride countersplaced at distances of 1-2 m from the sample and at 8 angles ranging from 5 to 90" of inclina-tion from the incident neutron beam. In these experiments the time of flight of the scatteredneutrons from the sample to each of the detectors is measured. The neutron scatteringspectrum built up from each counter is therefore a plot of the number of neutrons arrivingat the counter as a function of their time of flight from the sample.The chopper was set to deliver bursts of monokinetic neutrons every 2,760psec.Theenergy of these neutrons was such that they had a wavelength of 5.3~4. The incidentneutrons had a wavelength spread of 0.6 A, which corresponds to a resolution for the experi-ment of about 6 cm-1, All spectra were normalized to a run using vanadium. A typicaB . K . ALDRED, R . C. EDEN AND J . W. WHITE 113run consisted of two samples and the vanadium standard being cycled through the beamwith counting periods of 20min per sample. The total running time for good countingstatistics over the whole neutron scattering spectrum was generally 1-2 days. Care wastaken to use thin samples so that at no time did the scattering exceed 15 %.This procedureminimizes double scattering of the same neutron and so all peaks in the spectra may beattributed to single scattering processes.The acetic acids used in these experiments were carefully distilled several times to give thecorrect boiling point. The deuterated compounds were obtained from Merck, Sharp andDohme Limited, and their nuclear magnetic resonance spectra were run to make sure thatthe deuteration was complete. As no peaks due to mixed deuteration were visible in anyof the samples, it was coiicluded that deuteration was at least 99-5 % complete.RESULTSACETIC ACIDSThe neutron scattering spectra from carefully dried and purified acetic andtrifluoro-acetic acid were recorded using the spectrometer described above.Thespectra for the two materials at 20°C and at scattering angles of 20 and 90" to the2 0". . . . .* ..I .. . .- . . .. . - * :.*. *. .* '. . . . . ..*9 0"*.. L- .. . . .1 I I I I I I I . : . .time of flight (p-sec m-1)t .;- .I..400 I200 2000CH3COOH 20°CFIG. 2.-Neutron scattering spectrum from acetic acid, CHsCOOH, at 290°K and at scattering anglesof 20 and 90" to the incident beam.incident neutron beam, are shown in fig. 2 and 3. The substitution of fluorine(scattering cross-section 4 barns) for hydrogen (scattering cross-section approximately80 barns) produces marked changes in both the inelastic region of the spectrum atshort times of flight, and also in the elastic region174 NEUTRON SCATTERING SPECTROSCOPYINELASTIC REGIONSince the sample temperature is 20°C, the dominant inelastic scattering processis one whereby the neutrons gain energy in collisions with the molecules.The in-elastic scattering region then appears at shorter times of flight than the elastic region.In the spectra shown in fig. 2 and 3 little energy loss spectrum is observed because theincident neutrons have such low energies.Acetic acid shows a broad maximum at about 500 psec/m in the inelastic region.This has a long tail to lower energy transfers which extends right into the quasi-elastic region at about 1,350 psec/m. In this general region there are bands corres-ponding to quantized motions of both the CH3 group and OH group in the molecule.9 0"..I400 I 200 2000time of flight (psec m-1)CF3COOH 20°CFIG. 3.-Neutron scattering spectrum from trifluoroacetic acid, CFJCOOH, at 290°K and atscattering angles of 20 and 90" to the incident beam.The lowest frequency motions of the latter type will be associated with the inter-molecular hydrogen bond motions, which in this molecule should be reasonablysimple as dimer formation is a predominant mode of association. For simplicity,the spectrum may be thought of using the Raman spectrum analogy; the elasticpeak corresponds to the Rayleigh line and is associated with zero energy transfer inthe scattering process. The large inelastic feature at about 500 psec/m correspondsto an optical excitation frequency of about 155 cm-1.The present experimenttherefore measures the lowest energy motions of acetic acid. The spectrum cuts offat about 800 cm-1, where thermal population of excited vibrational states becomesinadequate to provide enough scattering intensity in these energy gain experiments.The angular dependence of the intensity in the inelastic region is characteristic ofsolid-like behaviour. Treating the vibrational motion associated with the 155 cm-B . K . ALDRED, R . C. EDEN A N D J . W . WHITE 175peak as harmonic in the one phonon approximation, the observed increase in intensitywith increased scattering angle can be described as the effect of a Q 2 term and asingle exponential containing a Debye-Waller factor. This procedure allows the rootmean square vibrational amplitude, JF) to be determined as 0*6A.Also as aresult, at all angles of scattering in our experiments the momentum transfers associatedwith this peak are sufficiently large that the “observation time” associated with theneutron molecule scattering is short compared to the time for gas like diffusive motionsto set in.QUASI-ELASTIC REGIONThe quasi-elastic region of scattering for acetic acid falls between about 1200 and1600 psec/m on the time-of-flight scale. The strong angular dependence shown bythe spectrum is the result of a fall-off in area with increasing momentum transfer andan increase in the peak breadth due to Doppler broadening during diffusive steps.The logarithm of the area of the quasi-elastic peak is shown plotted as a function ofthe square of the momentum transfer in fig.4. The dependence is almost exponentialQ2, A-2CH3COOH 20°CFIG. 4.-Debye-Waller plot of the log quasi-elastic area against momentum transfer squared, Q2,for acetic acid.and can be described by an effective Debye-Waller factor from which it is possible tocalculate 5 9 9 the mean amplitude of the thermal cloud developed by the diffusivemotions associated with this region to be 1.2&The other feature in the quasi-elastic region is the dependence of the peak widthon momentum transfer squared. Because it is difficult to subtract the inelasticcontribution at high angles of scattering, measurements of this width are uncertainat these angles. Nevertheless, since measurements at the lowest angles are the mostimportant for measuring molecular diffusion coefficients, the difficulties are noI76 NEUTRON SCATTERING SPECTROSCOPYinsuperable.Fig. 5 summarizes the results for both acetic acid and trifluoro-aceticacid. To determine the diffusion constant the peak broadening is measured relativeto the width of a vanadium peak at the same angle of scattering, and is calculated onQx 6XX0X 0I I0 I 2Q2, A-2FIG. 5.-Width of the quasi-elastic peaks for acetic and trifluoroacetic acids as a function of Q 2 at290°K.X , CH3COOH ; 0, CF3COOHthe basis that the observed peak shape is the effect of a Lorenzian-Gaussian convolu-tion. The incident neutron spectrum is assumed to have Gaussian form and this isbroadened by a Lorenzian expected from eqn.(7). The slope of the curve can beconverted to a diffusion constant for the molecule using eqn. (8). For acetic acid thetentative value is 2.1 x 10-5 cm2 sec-1.TRIFLUORO-ACETIC ACIDINELASTIC REGIONBy comparison with the spectrum of acetic acid, the most marked effect of sub-stituting 3 fluorines for hydrogens in the terminal carbon atom is to remove completelythe large inelastic feature peaking at about 155 cm-1. Some small inelastic featuresof low intensity remain, but these will not be discussed. Fluorine substitution there-fore allows us to assign the high intensity peak in the acetic acid spectrum to a CH3motion, most probably the torsional vibration. At 90" scattering angle, where theinelastic processes are more important compared to elastic ones, the small featuresjust visible in the spectrum of trifluoro-acetic acid at 20" angle of scattering, are moreapparent.The dominant feature in the 90" spectrum is the band between 500 and800 pseclm, which also extends on the low-energy side into the quasi-elastic region.We assign this band to the OH motion, most probably the intermolecular hydrogenbond stretching frequency. This band corresponds, in approximate position, to adiscontinuity in the acetic acid spectrum. We note that the OH band fuses with thequasi-elastic region at high angles of scatteringB. K . ALDRED, R. C. EDEN AND J. W. WHITE 177QUASI-ELASTIC REGIONThe quasi-elastic region for trifluoro-acetic acid has sharply contrasting featuresfrom that in acetic acid.At a scattering angle of 20” the scattering intensity betweenthe elastic peak and the inelastic region is almost down to background level, a featureonly commonly encountered with solids. Again the quasi-elastic area show a muchless marked dependence on momentum transfer than does the corresponding areafor the acetic acid. This is also a feature reminiscent of scattering from solids. Theeffective Debye-Waller factor is less easy to determine than for acetic acid because ofsome Bragg scattering particularly in the 60 and 75” counters, which arises from therelatively greater proportion of coherent scattering in this system. Despite this, anestimate has been made leading to a mean amplitude for the diffusion thermal cloudof 1.2 A.This is surprisingly close to the acetic acid value. The agreement may befortuitous, however, due to different mechanisms for diffusion of the OH protonsand so it cannot be discussed without more information on the angular dependence ofthe inelastic scattering in CF3COOH. The qualitative differences between theCH3COOH acd CF3COOH spectra remain, however, and suggest that the OHprotons are part of a relatively well-defined “lattice” which vibrates during the timescale of the neutron molecule scattering “event”. This situation contrasts with thealternative picture of a “gas-like” liquid of freely diffusing molecules with a continuumof state at low energies.The dependence of the width of the quasi-elastic peak, on momentum transfersquared, is given in fig.5.The dependence on momentum transfer shown by the width of the quasi-elasticpeak in trifluoroacetic acid’ is almost identical with that in acetic acid suggesting thattheir diffusion coefficients as measured by neutrons are also equal. An alternativesuggestion is that the motion of the OH hydrogen in both of these systems is thedominant contribution to the quasi-elastic broadening. In either case, a paradoxarises so that theories of broadening in similar systems where hindered internalrotation has been taken as an essential contribution to explain discrepancies betweenthe higher diffusion constants observed by neutron scattering and those observed bytracer elements must be reconsidered.5 A model system to explore these effectsfurther is the methanols which are discussed below.METHYL ALCOHOLSThe neutron scattering spectrum of methyl alcohol Hq has been measured over anextensive range of momentum transfers by Saunderson and Rainey.1 The spectrumand its angular dependence superficially resemble those of acetic acid shown in fig.2,although the details of band positions and broadenings, differ. This resemblancearises, however, because the scattering centres in the two molecules, the CH3 groupand the OH group, are the same. The spectra of the two deuterated analogues ofmethyl alcohol, CD30H and CH30D may therefore be compared with fig. 2 in arough quantitative way to see the large changes in intensity produced by deuteration,especially in the inelastic region of the spectrum.In these molecules the deuterationtechniques seems extremely powerful as the contributions from each of the scatteringcentres may be evaluated separately with a minimal change in the total mass of themolecule. Although deuterium is a coherent scattering nucleus, the present pre-liminary appraisal of the data shows that Bragg peaks and other coherent scatteringintrude only to a minor extent to mar the simplicity of the spectral changes. Thecoherent effects are noticed in the elastic peak and these only at high angles ofscattering. When they are present the area/Qz plot deviates from linearity178 NEUTRON SCATTERING SPECTROSCOPYMET HA N 0 L- d3The neutron scattering spectrum for liquid methanol d3 CD30H, at 290°K and atscattering angles of 20 and 90" to the incident neutron beam is shown in fig.6. Theeffect of deuterating the methyl group is to produce changes in both the inelastic andin the quasi-elastic region of the spectrum which are reminiscent of the changesproduced by fluorination of the methyl group in acetic acid.In the inelastic region the large peak in methanol at about 160cm-1 is almostabsent from the spectrum recorded at the 20" angle of scattering. At 90" this peakis slightly apparent. This is the most noticeable qualitative change. The spectralI II I I I I I I I I ...time of flight (pec m-1)"'.s... .....0 400 800 1200 1600 2 0 0 0CD30H 290°KFIG. &-Neutron scattering spectrum of CD30H at 290°K and at scattering angles of 2Q and 90"to the incident beam.distribution in the inelastic region is, however, quite different from that observed inthe acetic acids; there are, for instance, many more low frequency motions excitedbetween 800 and 1200psec/m time of flight.These probably correspond to quasi-acoustic modes of the liquid. There is also strong evidence of fine structure in thespectral distribution, from the number of spectra which have been recorded.Associated with the changes in the inelastic region of the spectrum there are markeddifferences between CD30H and CH30H for quasi-elastic scattering. Whereas inmethanol itself the quasi-elastic region has a strongly angular dependent intensityand width, this is much less noticeable for CD30H. In fig. 6, even at 90" of scattering,the quasi-elastic region is well-resolved and of greater intensity than the inelasticregion, at shorter times of flight than 1,000 psec/m.The intensity of the scatteringin the quasi-elastic for CD30H falls off almost exactly linearly as exp (-x2Q2)B . K . ALDRED, R . C. EDEN AND J. W. WHITE 179where x2 is a Debye-Waller factor and Q 2 is the momentum transfer squared. TheDebye-Waller factor is 0.14 which corresponds to a mean amplitude for the thermalcloud generated by the motion of 0.9 A.The width of the quasi-elastic peak, obtained after subtraction of a reasonableinelastic background, is also a simple function of momentum transfer squared.The simple diffusion model seems to hold over the momentum transfer range of ourmeasurements as is shown by fig. 7, where the width at half-height in pseclm isplotted as a function of Q2.After deconvoluting this width it is possible to determinethe value for the diffusion coefficient associated with the motion of the OH group.2 8 0n - 260 k 8 2 4 0v s 2203 18025 : 200Y.-3------160 xXx00x0Z00 I 2 3Q2, A-2CD3OH ; 0, 173°K ; X , 290°KFIG. 7.-Energy width of the quasi-elastic peaks in CD30H at 290 and 173°K as a function of Q2showing the jump diffusion behaviour at low temperatures.At 290°K this has the value of 1.9 x 10-5 cm2 sec-1. Measurements have also beenmade at 170°K near the freezing point of the liquid. At this temperature the freediffusion model seems to break down about 2 A-2 but below this a diffusion constantcan be found which is approximately 1.2 x 10-5 cm2 sec-1.Since great care was taken in preparing these samples to ensure that the materialswere extremely dry and free of any traces of acid or alkaline catalysts which mightpromote hydrogen exchange between molecules, we assume that the diffusion constantsmeasured here are characteristic not only of the OH but also of the whole moleculardiffusion since the residence time of the OH proton in this case is much longer thanthe time of observation in any of these neutron scattering experiments.METHANOL d lIn the inelastic region the reappearance of a strong band at an energy transfercorresponding to 160 em-1 provides confirmation of the assignment of this region ofthe spectrum to the methyl group motions.The peak height of this band shows astrong dependence on the momentum transfer associated with the scattering and thi180- 20° -- ---- * . --. . - --.... - .......... . 3El - . :.CIY 8 - :.& . .... .. '," .............. I I I I I t.... 90' -- . . - . '-- - ...... 0 . ..............I , I . . I ".;'" .....- - ..... . . . .:0 400 800 1200 I630 2000NEUTRON SCATTERING SPECTROSCOPY2407 220-E322000W 8J .- ' 180- 5AY6 160-QB .CII40--X* A -I 1B . K. ACDRED, R. C . EDEN AND 3. W . WHITE 181appears to obey the scattering law in the one phonon harmonic approximation.Fig. 8 shows the scattering spectrum of CH3QD at scattering angles of 20 and 90"for the sample temperature 290°K.At the highest momentum transfers used, ashoulder appeared on the peak at 800psec/m. This may also correspond to amethyl group motional frequency.Not only are there large changes in the inelastic region between CD30H andCH30D, but so also are there associated changes in the quasi-elastic region. Fig. 8shows the extremely sensitive variation of the quasi-elastic area with momentumtransfer. The peak height changes by almost a factor of 10 in going from the 20"angle of scattering to 90" angle of scattering. The intensity change obeys a Debye-Waller expression with a factor of 0.6 carresponding to a mean amplitude of motionThe quasi-elastic peak widths have a linear dependence on Q 2 and some resultsare shown in fig. 9. Because the strong inelastic scattering band makes widthmeasurements above 45" scattering angle difficult, only the results for lower anglesare shown, An inelastic background was subtracted before calculating the decon-voluted peak widths.Fig. 9 indicates that the molecular diffusion studied throughthe methyl group also fits the simple diffusion law and gives a diffusion constant of2.1 x 10-5 cm2 sec-1 which is approximately equal to that for CD30H at 290°K.The behaviour parallels the results for the acetic acids although again there are clearqualitative differences between the substituted and unsubstituted spectra.of 1.9 A.DISCUSSIONThese new experimental results are intended as a preliminary review of the powerof substitution methods in studying neutron scattering spectroscopy of liquids.Oneapplication is the assignment of particular parts of the inelastic scattering spectrumto particular atomic motions. The unique virtue in the neutron method is that aband associated with the vibration of any chosen atom can be almost completelyremoved from the spectrum rather than moved to lower frequencies where it couldincrease the complexity as in infra-red spectrometry. It has been possible to assignboth the CH3 and OH motions in acetic acid and in methanol by this technique. Thefrequencies of the band motions are listed in table 1, which also shows the availableinfra-red data.TABLE 1.-FREQUE"IES (Cm-l) OF PROMINENT BANDS IN THE INELASTIC NEUTRON SCATTERINGSPECTRACH3COOH 155CF3COOK 255 155 114 65(infra-red) 176 75CH3OD 160 50CD3OH (290°K) 550 150 (broad)CD30H (173°K) 550 1004440The second contribution that the atomic substitution studies have made is thatchanges produced in the inelastic region on substitution are accompanied by corres-ponding changes in the quasi-elastic scattering.Both the acetic acid and the methanolsystems show this. In particular, the spectra show that where there is strongscattering in the inelastic region from a methyl group so also there will be a strongand highly angular-dependent scattering in the quasi-elastic region. When only thescattering from the hydroxyl hydrogens in these hydrogen-bonded systems is recorded182 NEUTRON SCATTERING SPECTROSCOPYthe quasi-elastic region is sharp and well defined even at high angles of scattering.These liquids therefore show solid-like behaviour, the hydrogen-bonded protonseffectively remaining fixed in their positions during the observation time.Thisparticular view is corroborated for the methanols by the relatively high Debyetemperatures and small amplitudes associated with the motion in this region.The third interesting observation which has come from the atomic substitutionstudies (most particularly from the isotopic substitution experiments) is that theneutron scattering measurements of molecular diffusion constants do not seem todepend on which proton in the molecule is chosen as the scattering centre for themeasurement. In acetic acid and trifluoroacetic acid the diffusion constants arealmost equal.This may mean that the dominant motional process is that associatedwith some intermolecular exchange of hydrogens rather than with the motion of themolecule as a whole. Exchange is very unlikely in our systems since scrupulouscare was taken to avoid any catalysts for this process, and so the equality of the twodiffusion constants may mean that all protons in acetic acid are participating in thesame diffusive process as the single proton in trifluoroacetic acid. Under thesecircumstances the neutron measures, in acetic acid, the mean diffusion constant of allthe four protons and this is no different from the diffusion constant that would bemeasured from any one of them separately. Although this view is unexpectedbecause the molecular masses of trifluoroacetic acid and acetic acid differ considerably,it does fit with the observations on the substituted methanol.There again, especiallyfor CHsOD, we find diffusion constants associated with the CH3 motion and the OHmotion to be almost equal. Both values also agree closely with the value found forCH30H by Saunderson and Rainey (2.1 x 10-5 cm2 sec-1).If the methyl group in the molecule were rotating freely so that internal rotationwith respect to an oncoming neutron could give rise to preferential Doppler broadeningfor methyl protons, we should expect that the diffusion constant measured for CH3groups would be greater than that measured for either the molecule as a whole or forthe OH groups separately. The present analysis does not support this view and sofor the molecules studied and at the temperatures at which the experiments were done,the physical process of intramolecular rotation does not contribute appreciably tothe diffusive kinetic processes seen by a neutron.On the time scale of the neutronobservation range, the OH groups are relatively localized in the liquid. If freerotation is not important, then the predominant motion for the CH3 group is atorsional vibration around an equilibrium position.The problems posed by invoking free rotation in some systems to explain the highervalues for molecular diffusion constants obtained by neutron scattering measure-ments, compared to those obtained for bulk phase tracer studies, have been discussedby Egelstaff.3 Fig. 10 shows how the neutron scattering data at the lowest momentumtransfers must theoretically fit the values predicted by a simple diffusion theory.That they do not, e.g.for glycerol, at the normal momentum transfers used in neutronscattering experiments, must mean a simple Gaussian is no longer a suitable correlationfunction. Associated with the point of inflection needed to join the neutron data tothe bulk diffusion data there must come in the time Fourier transform of the correla-tion function (the spectral density of eqn. (6)) a maximum shown in fig. 10 as BCD.This maximum can only be associated with physical processes such as a quantizedinternal rotation. In our systems the only feature in the spectrum which could corres-pond to this is a vibration associated with the OH measurement, which under condi-tions of chemical exchange could easily merge into the quasi-elastic region. Asparticular care has been taken to avoid this exchange in our experiments, the OHpeak appears separately (energy transfer is approximately equal to 65 cm-1). AnB . K . ALDRED, R . C. EDEN AND J . W. WHITE 183so the spectral density as a function of frequency probably does not show the feature.It is possible that selective isotopic substitution experiments on glycerol and otherliquids, which do not show the simple diffusion behaviour, would reveal these effects.value of Q* -+value of wFIG. lO.--{i) Relationship between experimental neutron scattering results on the momentum transferdependence of the quasi-elastic peak width and the theoretical line predicted for some liquids usingbulk diffusion constant data. The experimental region is shown with points.(ii) Possible spectral density for a system showing the above width dependence.1 D. H. Saunderson and V. S. Rainey, Inelastic Scattering of Neutrons (I.A.E.A. Vienna), 1963,2 K. E. Larsson in Thermal Neutron Scattering, ed. P. A. Egelstd, 1965, p. 378.3 P. A. Egelstaff, Inelastic Scattering of Neutrons (I.A.E.A.), 1965, vol. 2, p. 553.4 L. Van Hove, Physic. Retr., 1954, 95, 249.5 K. E. Larsson Inelastic Scattering of Neutrons (I.A.E.A. Vienna) 1965, vol. 2. p. 3.6 K. E. Larsson in Thermal Neutron Scattering (ed. P. A. Egelstaff) (Academic Press, 1965), p. 347.7 A. Sjolander Thermal Neutron Scattering (ref. 6) p. 291.8 D. H. C. Harris, S. J. Cocking, P. A. Egelstaff and F. J. Webb, Inelastic Scattering ofNeutrons,9 J. S. Downes, P. A. Egelstaff and J. W. White, Chem. and Ind., 1967.vol. 1. p. 413.(I.A.E.A. Vienna), 1963, vol. 1, p. 107

ISSN:0366-9033    

DOI:10.1039/DF9674300169

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

GENERAL DISCUSSIONDr. Manse1 Davies (Aberystwyth) said: Egelstaff’s sentence : “ In X-ray scatteringwork the energy resolution is insufficient to allow the spectrum of the scattered X-raysto be determined,” is a reminder that such an energy difference in the scattered X-rayswas precisely what Debye in the first place hoped to see in the work with Scherrer,initiated in 1915. Only currently is it becoming possible to achieve a sufficientresolution for the structure of the excited molecular state to be seen from the in-elastically scattered X-rays.In the discussion, it would appear that the time for a diffusive step (ca. 1-2 x 10-l2sec) “ seems to give one basic relaxation time in liquids ”. Taken as a relaxationtime, the factor quoted leads to a spectroscopic frequency corresponding to 2-5 cm-l.In polyatomic liquids, molecularly “ spherical ” or otherwise, collisional processesappear to be associated with spectroscopic features when plotted in terms of anabsorption coefficient a in nCper cm-l in the range 20-40 cm-l.Is this difference aresult of the different conditions in a polyatomic liquid or is the diffusive step not tobe seen spectroscopically ? A spectroscopic statement analogous to Egelstaff ’s-“ the high frequency behaviour of the liquid state is similar to the solid state ”-is also probably true, i.e., (low-frequency) lattice mode vibrations appear also inliquid-phase absorptions. Is there any correlation between Dr. Egelstaff’s frequenciesand such lattice-type modes ?Prof. Karl-Erik Larsson (Royal Institute of Technology, Stockholm) said : Theresults of a slow neutron scattering experiment on a hydrogenous liquid is stronglydependent upon the value of the viscosity or the self-diffusion constant for the liquidin question.Through a long series of experiments lm4 on various hydrogenousliquids that were performed at the Stockholm laboratory as well as from experimentsperformed at other laboratories 5 * it is found that one class of results is obtainedfrom liquids with viscosities q about 1 cp or less or D>10-5 cm2/sec, and anotherclass is obtained when the viscosity is of 0-1 p or more when D < cm2/sec.The main difference appears in the quasi-elastically scattered neutron intensity in such away that the width A o of this peak appears to vary in a way typical for simple diffusionwhen y is small (Arc is the momentum change in the scattering process) and in a waytypical for solid behaviourACO = 2 0 ~ ~ (1)Aco---+2/z (only approximately)K largewhen is large. Here z is some relaxation time.The approach to the almost constantor slowly rising vafue of A o occurs differently for different liquids. IntermediateK. E. Larsson and U. Dahlborg, Physica, 1964,30,1561.K. E. Larsson, Proc. ZAEA (Bombay, December, 1964) (Vienna, 1965), vol. 2, p. 3.K. E. Larsson and L. Bergstedt, Physic. Rev., 1966,151,117.K. E. Larsson, L. Queiroz do Amaral, N. Ivantchev, S. Ripeanu, L. Bergstedt and U. Dahlborg,Physic. Rev., 1966, 151, 126.Proc. Symp. Inelastic Scattering of Neutrons in Solids and Liquids (Vienna, 1960, Chalk River,1962 and Bombay, 1964).(The International Atomic Energy Agency, Vienna, 1961, 1963and 1965).Thermal Neutron Scattering, ed. P. A. Egelstaff (Academic Press, 1969, chap. 7 and 8.18GENERAL DISCUSSION 185behaviours of Am as a function of IC are also observed most often when 100 > q > 1 cp.The high viscosities are reached only in liquids with strong intermolecular bonds likethe hydrogen bonds. Examples are glycerol and n-propanol at temperatures belowroom temperature. Examples of low viscosity liquids are methane, pentane, methyl-and ethyl-alcohol, etc. The details of the experimentally observed line widths maybe understood if a neutron cross-section is formulated on the basis of the van Hoveformalism.The molecules considered consist of carbon chains with protons and hydroxylgroups hooked on to the chain at selected positions.The motion of this “line”molecule with respect to the centre of gravity is described (fig. 1) in the following way.At time t = 0 the molecule has the direction in space given by N1. A proton at r1vibrates round a quasi-equilibrium position, which in itself performs simple diffusivemotion. The diffusive motion which gradually changes the molecular direction inFIG. 1 .-The motion of the molecule relative to its centreof gravity : N1 is molecular direction at time t = 0 ; Nzis molecular direction at time t = T~. In between thesepositions the direction diffuses randomly. A proton atrl at time t = 0 diffuses and vibrates until it reachesposition r2.N3 is molecular direction after jump. Aproton starting at r1 is dragged with the molecule in itsdirectional change to the position r3. The jump length Iis covered in time q.space is described by a diffusion coefficient Dp and continues until the average timet = T~ when the molecular direction has changed from N1 to N,, resulting in a dis-placement of the equilibrium position of the regarded proton from rl to r2. Thephysical cause for these small displacements is small changes in the orientationsof neighbour molecules. After the elapse of zo the molecular direction in spacechanges abruptly from N2 to N3 in a short time zl. This large change of molecularorientation transfers the proton from r2 to r3, a distance I, the jump length, andresults from a larger change of the position or orientation of neighbour molecules.The equivalent to a sudden change of molecular orientation would be a partial rotationto form an isomeric molecular form.After the average time T , + Z ~ the wholeprocedure is repeated. The motion of the centre of gravity is described as eithera vibrational state when the molecule is bound to neighbours for the time T& whichshould be long in the solid state, or as a diffusive state for the time ~ i . The displace-ment of the centre of gravity is then described by Ax& = 202;. A mixture of thetwo cases might occur even in hydrogen bonded liquids of high temperatures. At theend of the average time T& which might be much longer than r0+z1, the molecularconfiguration round the regarded molecule has changed so much that the moleculeis free to diffuse for the time 7; to another position where it is again caught in a quasi-stable position.The same discussion can be carried through for each proton in thecomplex molecule.L. Van Hove, Physic. Reo., 1954,95,249186 GENERAL DISCUSSIONIn order to calculate the cross-section, five different partial self-correlationfunctions are needed describing respectively : the diffusion of the centre of gravityhe(r,t), the vibration of the centre of gravity ge(r,t), the vibration of the proton withrespect to the centre of gravity glvib(r,t) the diffusion of the vibrating proton g,,,,,(r,t)and the jump of the proton hi(r,t). It is first assumed that all five functions areGaussian functions of the form :centre of gravity motionthree dimensional motionrelative motionthree dimensional motionapproximated as motions in a planewith p e , l ( t ) = p e , i ( a ) = ( r z i ) / 6 for ge and givib, pe(t) = D t for he, pi(t) = D,t forgidiff and pi(t) = 31: for hi.(r&) is the mean square of the thermal cloud set up bythe vibrating centre of gravity or the vibrating proton relative to the centre of gravityrespectively. I,” is a mean square component of the jump length 1. It is assumed thatZ2 = 21;. The spherical surface on which the proton moves is approximated by aplane tangent to the sphere at r2. It is assumed that the relative proton motionduring zo is obtained through a folding gi(r,t) = J gldiff(r’,t)gtvib(r’ - r,t)dr’.If a technique developed by Singwi and Sjolander is used to calculate the cross-section one finds a Lorentzian form for the quasi-elastic peak.We shall consideronly the case when free rotations are hindered such that zo$zl. The full width athalf maximum of these cross-sections are :(a) high viscosity. 76 and zo $= z; and z,- I(44Here7;; =-T,l+tb-?If the molecular mass is much larger than the proton mass, We< W,. For largeIc-values the line width asymptotically approachesACO = 2(1/zoO + Dpx2). (4b)There is no horizontal tangent to the curve ACO =f(lc2) unless D, = 0 (solid state).As IC approaches zero one obtainsK. E. Larsson and L. Bergstedt, Physic. Rev., 1966, 151, 117.K.S. Singwi and A. Sjolander, Physic. Rev., 1960, 119, 863GENERAL DISCUSSION10 T1871 2 3 4 5 6 7 8 9 1 0x2, A-2FIG. 2.-Observed line width data on glycerol together with fitted theoretical line width forms givenat solid lines. The best fit to the asymptotic behaviour is shown for 266, 293, 334 and 369°K.The intercept on the AE axis is 2 f i / ~ ~ ~ . The slope of the asymptote is 2 f i D ~ .1 03 T, OK-'FIG. 3.Relaxation times zoo, -ro and T& for glycerol derived from neutron data. Measured dielectricrelaxation time TD ranges from lower temperatures up to 285°K. Thereafter an extrapolation ismade according to a formula given by McDuffie and Lit0witz.l Ultrasonic data from Piccirelliand Litowitz.2G. E. McDuffie, Jr. and T. A. Litowitz, J.Chern. Physics, 1962, 37, 1699.R. Piccirelli and T. A. Litowitz, J. Acuust. SUC. Amer., 1957, 29, 1009I88 GENERAL DISCUSSIONIn this case,ACO = (2jz0)[1 + (D - exp (- 2wi)@K21. (54Am = 2[ l/zo + ( D + +D,)K2].For large Ic-values the asymptotic form is(5b)When IC is small the asymptotic form is1c2 = 2(D,+ + Dre&c2 = 2'rD'r~2. (5c)In both cases (a) and (b) the slope of the line width curves at the origin is determinedby a combination of a relative and a centre of gravity diffusion coefficient.n-propanol;GIIIIla'2L2 3 4 5 6 7103 IT, OK- 1FIG. 4.-Relaxation times T ~ ~ , T~ and 76 for n-propanol derived from neutron data. The shadedareas show the possible range of the derived variables. The two dielectric relaxation times and~ f , are taken from Davidson and Cole.' According to Lyon and Litowitz the distribution ofultrasonic relaxation times may fall between the two limits shown.The cross-section result was used to fit to data on glycerol, n-propanol and pentanefor various temperatures.By carefully fitting the line width formulas to experimentalvalues were obtained for P , D,, zo and T& An example of the line width data andfit for glycerol is given in fig. 2. Similar results were obtained for n-propanol. Inboth cases Z2 is temperature-independent for each substance and Z2 = 2 . 6 A 2 forD. W. Davidson and R. H. Cole, J . Chem. Physics, 1951,19, 1484.R. H. Cole and D. W. Davidson, J. Chem. Physics, 1952, 20, 1389.T. Lyon and T. A. Litowitz, J. Appl. Physics, 1955, 27, 179GENERAL DISCUSSION 189glycerol and 8.0A2 for n-propanol.All the other quantities vary with temperature.The product 2 0 , ~ ~ describing the change in molecular direction in the time zo remainsabout constant for each substance and is about 0*7A2 for glycerol and 0-3A2 forn-propanol.Of particular interest are the temperature variation of the relaxation times T~ andzh. These are given for glycerol and n-propanol in fig. 3 and 4, where a comparisonis made with experimental data on dielectric and ultrasonic relaxation times. Inboth cases the time T(, and the dielectric-or some component thereof-relaxationtimes agree relatively well. The mechanical relaxation time is not in disagreementwith 76 but may be a mixture of ~6 and zo.Since 76 is identified with the mean life-time of a hydrogen bond, the agreement is gratifying. An example of how well theglycerol2.2 2-5 30 35103/T, OK-'Drel are shown as dashed lines. The full line is the sum of these components.FIG. 5.-Observed diffusion coefficient '' D ". The various theoretical components D, Dmix andcomponents of D when summed up describe the directly observed D is given in fig. 5.Both glycerol and n-propanol are non-Arrhenius, i.e., high-viscosity liquids.For pentane the viscosity is in the range 1 cp and no solid-like behaviour isobserved. Unfortunately the experimental accuracy is poor but it seems that anapplication of formula (5a) for the line width would fit the data. The observed slopeat the origin is about twice the value of the self-diffusion coefficient indicating con-tributions from the terms D, and T2/6~0. In general it is expected that for liquidsin or near the Arrhenius region simple diffusion will dominate and rotational effectsare difficult to observe in the width of the quasi-elastic peak.Using the present model the neutron data on several hydrogenous liquids areunderstood.The fact that the motion relative to the centre of gravity is dividedinto small and large fluctuations represented by two Gaussian functions indicatethat this motion-sum of two Gaussians-in non-Gaussian. The present analysi190 GENERAL DISCUSSIONalso permits a direct connection between the neutron data and dielectric and ultra-sonic data.Prof. J. Stecki (Warsaw) said: In Pings’ paper, the question arises of the physicalsignificance of the direct correlation function experimentally determined.Onesimple and straightforward interpretation is that - kTC(r) is an effective potential.The relation between g(r) and -kTC(r) is the same as between g(r) and the truepotential in a dilute (Debye) plasma. Thus C(r) is the effective potential to use ifrelations valid for a plasma are forced to fit any other system, i.e., if actual screeningis described by the Debye screening.Am I correct in supposing that the dispersion relation co(k) in Stiller and Dorner’spaper is determined by choosing a sharp peak in the S(k,co) against co plot and thenfollowing the shift of the peak maximum with k? What is the criterion of sharpness?Surely one would not attempt to construct the disperion relation by following astrongly damped peak or a hump that Egelstaff described.From this point of view,the concept of the dispersion curve appears ill-defined and the significance of theplots thus obtained not clear.At the end of my paper I drew attention to an approximation to v(k,cu) which isexactly equivalent to Zwanzig’s earliercalculationresulting fromavariational treatmentof the Liouville equation. We made some numerical calculations of S(k,cu) at fixed k.For low values of pvk, there is Gaussian behaviour about o = 0 and a damped peaknear the corresponding Landau pole. For large vk there is one damped peak not farfrom co = 0, so that the Gaussian behaviour is not discernible, and another peak withnegligible damping, from the Landau pole, corresponding to a large sound velocity.Thus the qualitative behaviour is not in agreement with hydrodynamical behavioursince this approximation is equivalent to a Vlassov equation which is a short-timeequation involving no true dissipation.It would be nevertheless interesting to see ifsome of the features of Zwanzig’s approximation are not found in what is called the‘‘ kinetic regime ”.In reply to Rowlinson about the reason for considering relations that are valid fora plasma. First, the interpretation of C(r) is I believe the “ right one ”, i.e., theeffective potential refers explicitly to the Debye screening. In particular, also,the Ornstein-Zernike approach is an example of plasma-like approximations found(or assumed!) to be valid for a particular dense system.Secondly, the relationbetween S(kc0) and the ‘‘ dielectric function ” E(kco) is generally valid for a slightlygeneralized definition of E and it is fortuitous that these relations were derived firstfor a quantum or classical electron gas.Dr. A, Levi (Rijksuniversiteit te Leiden) said : The velocity self-correlation functionF(t) = (i(O)2(t)) has acquired an increasing importance in the dynamical descriptionof liquids. It is useful to find a simple analytic form for it, containing a few para-meters which can be fitted from experiments or theories, and incorporating some of theinteresting properties of that function. Simple forms of this kind are afforded bystochastic processes, if one makes convenient assumptions about the atomic motions.It has often been assumed, by analogy with Brownian motion theory, that thevelocity vector contains a term behaving as a Gaussian, Markovian process.This,however, is in contrast to the short-time properties of the correlation function, whichcan be studied by Taylor expansion. A Markovian process is non-differentiable andwould correspond to infinite mean square force, while the mean square force is afinite quantity, well known from both theory and experiments. On the other hand,it is much more reasonable to assume Gaussian and Markovian properties for thestochastic pair formed by velocity and acceleration together, taking into accounGENERAL DISCUSSION 191explicitly the above information about the mean square force.In this case thevelocity self-correlation isexp (-zot)-z2 expwhere k is Boltzmann’s constant, T the temperature, m the molecular mass, w is givenby co2 = (V2u)/3m while the mean square force is kT(V2u), and z is related to theself-diffusion coefficient by D = (kT/mm)(z + z-l). Similarly, under more generalconditions, one can assume Gaussian and Markovian properties for the triplet formedby velocity, acceleration and time derivative of the acceleration; one gets a definiteanalytic form again, although slightly more complicated. One can even devise ageneral approximation scheme where in the nth approximation a stochastic vectorformed by the first n time-derivatives of displacement is assumed to perform aGaussian, Markovian process.This is described by the matrix equationR(t) = exp (QOW), (2)where R(t) is the correlation matrix and Q = R’(O)R-l(O). This equation becomesexact when the full set of derivatives is considered (n+m), but in general it is anapproximation. The situation is still simple, however, since the velocity self-correla-tion function is a sum of n exponentials, with coefficients and exponents connectedwith the initial correlations, except for one parameter which is left free in order to getdamping at long times. The n time factors are the eigenvalues of Q. The correlationfunctions obtained by this method have a clear mathematical meaning, so that theyare perhaps suited for an approximate description of the true correlation functions.These models were applied to liquid argon at the triple point. The parametero was taken from the isotopic separation factor and the experimental diffusioncoefficient D (from neutron scattering) was also used. The velocity correlationfunction was computed in the second and third approximations, and the results werecompared to Rahman’s molecular dynamics correlation functions. They agreebefore the first zero, but at later times the amplitude of the oscillations is too large,nor does the third approximation improve over the second.Presumably, a smoothing due to three-dimensional effects has been neglected.To be more precise, any coupling between motions in different directions has beenignored ; since the components of velocity are actually uncorrelated, any suchcoupling disappears in the Gaussian approximation. The latter should therefore bedropped in a more exact treatment, while the Markov property can still be expectedto hold to a reasonable extent. The correlation function (I) is the same as is obtainedfrom the equation of the kindF(t) = - a(z)F(t-z)dz s: (3)provided a(t) has an exponential form

ISSN:0366-9033    

DOI:10.1039/DF9674300184

出版商:RSC    

年代:1967

数据来源: RSC