摘要:

Nuclear Magnetic Resonance in the Study of LiquidsBY J. A. POPLECarnegie Institute of Technology and Mellon InstituteReceived 7th February, 1967Mechanisms of nuclear spin relaxation in liquids are listed and discussed. It is pointed out thatmeasurements of spin relaxation times (and also dielectric relaxation times) give information aboutdifferent aspects of the process of molecular rotation and re-orientation in liquids. Comparativestudies of different mechanisms should lead to a more detailed understanding of these movements.1. INTRODUCTIOKNuclear magnetic resonance experiments are most directly affected by the structureand behaviour of the surrounding medium through the process of nuclear spin relaxa-tion. This is the process whereby a set of nuclear spins tends to re-orient towards arandom distribution competing with the orienting forces due to the externally appliedmagnetic fields.It can be studied experimentally 1 by measurements of the spin-lattice relaxation time TI which is the time constant for approach of nuclear spins tothermal equilibrium in the presence of a strong external magnetic field H. Thesubject of this review is the relationship between 7'1 and the properties of the liquidmedium, particularly the rotation and re-orientation of molecules.One important way in which nuclear spin relaxation differs from other molecularrelaxation processes is that the time scale is much longer. Proton spins in an organicliquid may take many seconds to achieve equilibrium compared with perhaps 10-10sec for dielectric relaxation.This is because the nuclear spins are oiily very weaklycoupled to the other motions of the system (translations, rotations and vibrations,usually inaccurately called lattice motions), so that any particular molecule undergoesmany rotations and other movements in the liquid before the direction of its nuclearspins are significantly changed. However, nuclear spin relaxation only occursbecause of these weak interactions and the interpretation of experimental data has tobe in terms of them. As with the theory of any relaxation process, two main featuresare involved : (i) the nature and magnitude of the interaction energy ; (ii) the way inwhich the interaction changes with time due to the random motions of the moleculesin the liquid.If there is sufficient understanding of the first item, nuclear spin relaxa-tion experiments can provide useful information about the second and hence about themolecular dynamics.There is a number of mechanisms by which nuclear spins may be coupled to thelattice motions. It is useful to begin by listing them before proceeding to moredetailed considerations.(1) The interaction between the electric quadrupole moment of the nucleus and theelectric field gradient produced by the surrounding electrons and nuclei. Nuclearquadrupole moments only exist for nuclei with spins greater than 1/2 (N14, 017, etc.),but, if present, this mechanism is usually dominant.(2) The interaction between the magnetic moment of the nucleus and magneticfields due to the dipoles of other magnetic nuclei.As the molecules move about19J . A . POPLE 193in the liquid, the local magnetic field at one particular nucleus will fluctuate andrelaxation may occur. This process may be subdivided into intramolecular and inter-molecular contributions. The intramolecular parts involve molecular re-orientationand the intermolecular parts relative diffusion of neighbours.(3) The interaction between a nuclear magnetic moment and the magnetic fieldsgenerated by the orbital motion of the nuclei and electrons as charged particles. Asthe molecule moves in the liquid, it generates a fluctuating magnetic field which will beexperienced by each nuclear spin. This mechanism is frequently referred to as aspin-rotation interaction, but similar effects can arise from fluctuations of relativetranslational motion.(4) Another magnetic mechanism arises from fluctuating magnetic fields producedby anisotropic chemical shifts.As the molecules rotate, the screening field, producedby the diamagnetic electronic currents induced by the primary external field willfluctuate and may cause relaxation. This mechanism can only be significant if theexternal field is large.(5) Finally, relaxation may occur by interaction of nuclear spins with the unpairedelectron spins of paramagnetic species present in the system. This may be by meansof long-range dipolar interactions or by a contact mechanism if the unpaired electrondensity becomes significant at the positions of the nucleus under investigation.2. RELAXATION I N FLUCTUATING FIELDSThe general theory of the relaxation of nuclear spins in liquids involves time-dependent perturbation theory where the perturbation is the Hamiltonian Yf’(t)representing the interaction between the nuclear spin and the other (lattice) degrees offreedom.As noted in the introduction, this perturbation may arise by severalmechanisms, but the quantum mechanical treatment of the relaxation process is similarin all cases. The perturbation, which will fluctuate with the time because of molecularmotion, will have a matrix element between initial and final spin states 1 and 2 whichmay be writtenwhere the root-mean-square value should be calculable from equilibrium statisticalmechanics and u(t) is a fluctuating function with mean-square modulus unity.In time-dependent perturbation theory 2 the transition probability per unit time isgiven by(2 I Jw) I 1) = Kp 1 2r ’I 1) I 2 1 a v ) + m (2.1)where hco is the energy difference between states 1 and 2 and p(z) is the autocorrelationfunction of (2 1 A?‘’ I 1) defined asIf p(z) is supposed to decay exponentially,where rc is a correlation time, (2.2) becomesThese results are of general validity and may be applied to any type of relaxation byusing appropriate expressions for the coupling Hamiltonian 2’.P(7) = Cu(t>u(t + z)lav.(2.3)= exp (-z/%>, (2.4)(2.5) PI,, = 2 f q (2 I %’I 1) j2]l,~cl(l +W2TC).3. RELAXATION BY ELECTRON FIELD GRADIENTSFor nuclei with a spin of 1 or higher, relaxation occurs principally by means of theinteraction between the electric field gradient and the nuclear position and the nuclear194 NUCLEAR MAGNETIC RESONANCEelectric quadrupole moment eQ.If the electric field gradient is axially symmetricand has the value eq, the full expression for the spin-lattice relaxation time is~ l l = e4q2QZR2[&/ exp (iw,~) . exp (2ic0,z) . pz(z)dz4OPwhere 00 is the (angular) Larmor frequency and pz(z) is the autocorrelation functionfor P~(COS O), that is the average value of Pz(cos 0), 8 being the angle between the axisof the electric field gradient at times zero and 7. With the exponential form (2.4) forp(z), this gives4. RELAXATION BY NUCLEAR DIPOLE-DIPOLE INTERACTIONThis mechanism arises because the magnetic field at one nuclear position fluctuatesbecause of the re-orientation of the line joining the two particles.If the secondnucleus is of a different species (or has a large chemical shift with respect to the nucleusundergoing transitions), TI is given bywhere pZ(z) is again the autocorrelation function of Pz(cos O), 8 now being the anglebetween new and old positions of the line joining the two nuclei. y and y' are themagnetogyric ratios of the two nuclei and b is the distance separating them. Clearly,the re-orientation function pz(z) in (4 1) is very similar to that used in the theory ofelectric quadrupole relaxation (eqn. (3.1)). Eqn (4.1) is explicitly for intramolecularrelaxation where the intervening distance b is constant.Relaxation can also occur byinteractions with spins in other neighbouring molecules.The theory of nuclear dipole-dipole relaxation has the considerable advantage thatthe mean square of fluctuating Hamiltonian is known (if 6, y and y' are known) sodirect information can be obtained about the correlation function p2(z) if this mecha-nism is dominant.TF1 = +h2y2y'2b-6p2(z), (4.1)5. RELAXATION BY MOLECULAR MOTIONThe theory of spin relaxation by the magnetic fields produced by the relative motionof other electrons and nuclei in the same and neighbouring molecules differs from theprevious treatment because the strength of the fluctuating fields depends on thevdocities (linear and angular) of the molecules rather than their positions. If weconsider only intramolecular contributions to the local magnetic field, the couplingHamiltonian will be proportional to the angular momentum J and may be written(In this equation a summation convention is used for tensor suffixes.) C is usuallyreferred to as the spin-rotation interaction tensor.It may, in principle, be calculatedquantum-mechanically from a knowledge of molecular wave functions, but suchcalculations have only been carried through so far for small molecules. It can alsobe measured directly by microwave methods for small molecules. For example, therehave been recent studies of formaldehyde.3The general discussion is simplified if the full expression (5.1) is replaced by a scalarinteraction,&" = CI . J. ( 5 J . A . POPLE 195Clearly, the interaction (5.2) gives a magnetic field at the nucleus which is propor-tional to the angular momentum.The general equation (2.2) then involves theangular momentum autocorrelation function p&), which describes the average rateat which a molecule changes its rotational velocity.6 . RE-ORIENTATION OF MOLECULES I N LIQUIDSFrom the preceding discussion, all intramolecular nuclear spin relaxation processesare related to the way in which molecules re-orient in the liquid phase. Dielectricrelaxation times of polar molecules are also determined by re-orientation rates, so itshould be possible to probe the details of such movements by careful comparison of thevarious physical measurements.In the theory of dielectric relaxation, the loss (or imaginary part of the complexdielectric constant) at angular frequency o is proportional toexp ( i o z ) .p,(z)dz,where p1 is the autocorrelation function for the dipole moment,Pl(7) = "3 - P ~ ~ ~ l a v / r P 2 ( ~ ~ l a v * (6.2)This is clearly the autocorrelation function of cos 8, where 8 is the angle betweenorientations of the dipolar axis at times 0 and t.This means that experimental information should be available on the Fouriercomponent (at angular frequency a) of the autocorrelation functions of both p1(cos 8) and ~ ~ ( C O S 8) and also of the angular momentum J. These three quantitiesare strongly interrelated since the angle change 8 occurs only through the rotationassociated with the angular momentum. Comparison of these quantities shouldprovide a powerful test of any detailed model for re-orientation.The most widely used model is that of Brownian rotation motion which has beenapplied in theories of dielectric and nuclear spin relaxation.This leads to exponentialdecay functions of the type (2.4) for bothpl andpz. Using a model of a rough sphere ofradius a rotating in a viscous fluid, Bloembergen, Purcell and Pound 1 foundz, = 4nqa3/3kT (6.3)where q is the viscosity. This is the same model as was previously used by Debye 4for dielectric dispersion.Although the agreement with experimental nuclear spin relaxation times wasquite good using (6.3), the underlying assumptions of the Brownian motion theoryare unrealistic for the rotational motion. In brief, they are that the molecules movesas if in a viscous liquid except for short intervals of time during which strong randomtorques act to maintain the equilibrium thermal distribution of angular momentum.In a real liquid, the motion is more continuous and it should be possible to set up othermodels connecting the decay rates of angles and angular moments. Such a theoryshould give more direct information on molecular re-orientation by avoidingappeal to a semi-macroscopic description as in (6.3).1 N. Bloembergen, E. M. Purcell and R. V. Pound, Phys. Rev., 1948, 73, 679.2 J. A. Pople, W. G. Schneider and H. J. Bernstein, High Resolution Nuclear Magnetic Resonance3 W. H. Flygare, V. W. Weiss, J. Chem. Physics, 1966, 45, 2785.4 P. Debye, PoZar MoZecuZes (Dover Publications, New York, 1945).(McGraw-Hill, New York, 1959)

ISSN:0366-9033    

DOI:10.1039/DF9674300192

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

Study of Molecular Motion in Liquids by Measurement ofNuclear RelaxationBY R. A. DWEK AND R. E. RICHARDSPhysical Chemistry Laboratory, OxfordReceived 23rd January, 1967Nuclear magnetic relaxation times depend on the strengths of the magnetic components of randommolecular motion at nuclear or electron resonance frequencies. The methods by which these spectraldensities can be obtained are outlined. Measurements of correlation times for different types ofmolecular motion and their dependence on temperature are reviewed.Magnetic resonance methods may be used to study molecular motion in liquidsand in suitable cases can provide detailed information about specific interactionsbetween molecules in the liquid phase. In this paper the principles involved in suchstudies are set out in an attempt to illustrate the scope and limitations of the method.Molecular motion in liquids directly affects the nuclear relaxation times TI andT2.The spin lattice relaxation time TI is a measure of the time taken for the nucleito achieve an equilibrium population among the allowed energy levels in the appliedmagnetic field HO ; it is sometimes referred to as the longitudinal relaxation time as it isconcerned with the relaxation of the nuclear magnetization along the direction of theapplied field Ho, usually called the z direction. TI is a measure of the time requiredfor the nuclei to exchange energy with their surroundings. The spin-spin relaxationtime T2 measures the time taken for nuclei to exchange energy among themselves byan adiabatic process ; it is sometimes referred to as the transverse relaxation timebecause it is concerned with the relaxation of the nuclear magnetization in thetransverse (xu) plane at right angles to Ho.In liquids, TI and T2 are usually nearlyequal, but in some special cases their differences can give valuable information.It is convenient to consider first the mechanism of spin lattice relaxation for asystem of nuclei of spin 3. In an applied magnetic field HO these nuclei are distributedbetween two energy levels, which by convention we label + (lower) and - (higher),and which correspond to the two allowed orientations of the nuclear magnets in Ho.When radiation of angular frequency mn is applied to the system in the appropriateway, transitions are induced among the energy levels ; the probability of a quantumof radiation inducing an upward transition is the same as for a downward transition,so a net exchange of energy between the nuclear spins and the radiation can onlyoccur if the populations of the two nuclear energy levels are unequal.If the lowerenergy level has an excess population (as it would have at thermal equilibrium) thenenergy is absorbed from the radiation, but if the upper energy level were more highlypopulated for some reason, then an emission spectrum would be induced.The angular frequency 0% is equal to YnHO, where yn is the nuclear magnetogyricratio, and HO is the magnetic field actually experienced by the nucleus. For magneticfields of the order of 104 gauss, 0% is near lo7 c/sec.The prcbability that a nucleuswill make a spontaneous jump from one energy level to another is vanishingly small,19R. A . DWEK AND R . E. RICHARDS 197and for nuclei of spin +, there is no mechanism other than the oscillating field at confor the particular applied field Ho, which can cause transitions.When a sample of nuclei of spin + is placed suddenly in Ho, the initial populationof the two nuclear energy levels must be equal because the nuclei have randomorientations in zero field. The nuclei then relax to the populations correspondingto thermal equilibrium with the characteristic time TI. The oscillating magneticfields at con, needed to induce the transitions, are derived from the random motion ofthe molecules in the liquid.The magnetic moments of the nuclei in the liquid set up local fields in theirenvironment which are proportional to ,u/d3, where p is the nuclear magnetic momentand d is the distance from it to the point at which the field is measured. The pro-portionality constant depends on the angle between HO and the vector joining thenucleus to the point considered.These local magnetic fields fluctuate according to the molecular motion.Rotationof a molecule will cause the local field at one nucleus due to another in the same mole-cule to fluctuate at the rotational frequency. Relative diffusion of molecules causesthe local fields at the nuclei of one molecule, due to those of another, to change. Thefrequency of the molecular motion is usually characterized by a correlation time, 7,.for rotation and zd for diffusion.For any random motion there is the whole spectrumof frequencies and the variation of the intensity of the fluctuations with frequencymust depend on the type of motion concerned. However in all cases, the variationof the “spectral density” J(co) with CL) has a similar form which is roughly representedin fig. 1. J ( o ) is independent of co until wz approaches unity, when the intensity fallsto zero as (o increases. W-hen T is small, J(co) is low and extends to high values of01 ; when z is large, J(w) is greater but falls to zero at a lower frequency.The relaxation times are therefore dependent on the values of J(m) at the particularvalue of con. J ( u ) may be writteng(z) exp (- icoz) dz, (1)where g(z) is the correlation function.The form of this function must be known ifcorrelation times are to be calculated from relaxation data. It is common to use anexponential function, which is physically reasonable for some types of motion and forwhich the Fourier transform is particularly simple. If we writej(o) = J(co)/J(O) thenand in several cases, this has been shown to be a good approximation.However, more precise correlation functions can only be calculated on the assump-tion of a suitable molecular model, and Abragam 1 has dealt with several cases insome detail. Two further examples, however, are given to illustrate the type ofprogress that is being made towards our understanding of molecular motion. Thefirst is that of Hubbard 2 who has considered the dipolar interactions between spinspositioned off the centre of spheres.He assumed that both relative translation androtation of the spheres could be described by a diffusion equation and has shown thatto a first approximation, it is valid to neglect the contribution of the rotational motionto the relaxation rates, and to treat the spins as being at the centre of the spheres.jd(co) is then given bywhereandj(0) = 1/(1 +02z2) (2).id@) = (WWW,u = ~oz/*, z = d2/D,I(U) = ~ - ~ ( ~ ~ - 2 + e - ~ [ ( ~ ~ - - 2 ) sin u+(u2+4u+2) cos u ] ] 198 MEASUREMENT OF NUCLEAR RELAXATIONd is the distance of closest approach of the two spins, and D is the diffusion coefficientin the equation describing the translational diffusion of the spheres.Although theform ofj(w) appears to be quite different from eqn. (2), it still has a shape similar tothose of fig. 1.The second example is simpler, and invokes the concept of a distribution ofcorrelation times. It is often found that discrepancies between theory and experimentcan be removed by such a postulate. A recent paper concerned with the hydrationof keratin 3 shows this well. It is assumed that the water molecules can exist in arange of different environments, each characterized by its own correlation time, andproton exchange occurs between these sites in a time short compared with T2. Alog-normal distribution of correlation times was used in this case, implying that forrotation of the water molecules, there is a Gaussian distribution of free energies ofactivation, and the experimental measurements could be reproduced well.On theother hand, there may be a quite different correlation function with a single value ofzc. A recent paper by Waugh4 is concerned with the nature of the correlationI II 1I II III1IA ! In 3%vIIIIII! I \FIG. 1. 0A, large T ; B, small T.function for rotation in liquids and gases, and points out how in certain special casesadditional information about the form of the correlation function may be obtained.For most liquids, zc is near 10-10 sec, and con usually lies near 107 sec-1. wnzc istherefore much less than 1 (fig. 1) and we are well back on the flat part of the correla-tion spectrum, in the so-called “white spectrum”.The relaxation time is thereforeindependent ofmn and hence of Ho. In special cases, such as rotationally relaxednuclei in molecules at infinite dilution in a non-magnetic solvent, the theory ofrelaxation allows T~ to be obtained from measurements of T1.5 Usually, however,we are not able to do this, but the effect of changing conditions on z can often befollowed directly by measuring the change of TI. For example, when the temperatureis changed, z changes and this is reflected by a corresponding change of T I . Fromsuch measurements, activation energies for molecular motions can be obtaineddirectly. For example, in benzene, the nuclear relaxation rate is determined bymodulation of the dipole-dipole interactions. These may be either inter- or intra-molecular, and we may writef/Ti intrs depends only on the rotation of the molecules while l/T1 inter dependsTI = 1/Tl inter+ l/T1 intraR .A . DWEK AND R . E. RICHARDS 199primarily on translation. By measuring the proton TI for mixtures of benzene witha non-magnetic diluent such as carbon disulphide, or better, with fully deuteratedbenzene, and extrapolation to zero proton concentration, these two contributionscan be separated.6 The activation energies obtained from relaxation measurementsfor these processes have been compared with those corresponding to various physicalproperties 6 and are listed in table 1.TABLE AP APPARENT ACTIVATION ENERGIES FOR LIQUID BENZENE IN kcal mole-1 AT ROOMTEMPERATUREself diffusion 3.1 kcal mole-1viscosity/absolute temperature 3.2TI inter 3.0Rayleigh light scattering 1.35deuteron TI 1-86TI intra 1.2TI intra(4 1.86One might expect agreement between the activation energies of TI inter and thoseof self-diffusion and viscosity, as all three depend mainly on translational motion.Similarly, those of Rayleigh light-scattering, deuteron T I , and TI intra agree reasonablywell, since all depend principally on rotation.The rotation of a group or molecule may give rise to a group or molecular magneticmoment, and interaction of the nuclear magnetic moment with this may provide anadditional relaxation mechanism.This interaction is termed the spin-rotationinteraction and contributes to l/Tl intra. By considering the two contributions toTI intra the dipolar one, TI intra (d) and the spin rotation TlSR, we can writel/Tl intra = l/Tl intra ( d ) + l/TISR,The apparent activation energy for TI intra ( d ) is then found to be 1-86 kcal/mole-1in excellent agreement with that found from the deuteron T1.6 It may be that aredetermination from Rayleigh light-scattering is desirable.The nuclear relaxation times in a diamagnetic liquid are often reduced remarkedlywhen very small concentrations of paramagnetic solutes are added.The presence ofdissolved oxygen, for example, in benzene, reduces the proton relaxation time by afactor of 5. The reason for this is that the magnetic moments of unpaired electronsare about three orders of magnitude greater than nuclear moments, so that the localfields generated by them are correspondingly greater ; the increased intensity of thefluctuating fields induce more efficient nuclear spin relaxation. The situation is morecomplicated than this, however, and can lead to more important information aboutmolecular motion in the liquid.In a dilute solution of a paramagnetic solute, the nuclear relaxation is often entirelydominated by pairwise interactions between an unpaired electron, S, and nucleus, I ;the rapid diffusion of the solute through the solvent ensures that all the nuclei areequntry affected so that the relaxation is shared among them all.The strong localfields ?rodwed by the electron can be coupled to the nuclei by simple dipole-dipoleinterat :tion, or sometimes by a scalar coupling transmitted through a chemical bond,which may be transient, by the same mechanism which is responsible for the hyperfinestructure of e.s.r.spectra or the spin-spin multiplets in n.m.r. spectra.For purely dipole-dipole coupling, the pairwise interaction of a nucleus and anelectron can induce three types of transition involving the nuclei, which are illustratedin fig. 2. There are four energy levels, corresponding to the combinations of th200 MEASUREMENT OF NUCLEAR RELAXATIONS and I spin quantum numbers. The lowest nuclear energy level is by conventionlabelled+, and because the magnetic moment of the electron has the opposite signto that of the proton, we label the lower electronic energy levels negative. Therandom magnetic field fluctuations can induce nuclear transitions which may bedenoted conveniently by the shorthand I- and I+, which represent nuclear transitionsfrom the-and from the + levels respectively, As well as these transitions, thedipolar coupling can introduce the coupled transitions S+Z+(S-Z-) and S+I-(S-Z+)(fig.2) in which the electron and nucleus make simultaneous flips, and these termshave larger coefficients than the I+ or I- terms. The frequency involved in the I+, I-transitions is con, but the coupled transitions involve frequencies (638 -0,) and (me +con)which, bccause me is so much larger than@%, can both be taken to be me, the electronicLarmor frequency. The expression for the effect of the paramagnetic solute on theS Ii-+ +1nuclear relaxation time therefore contains two terms, one invoving an and the otherWe.If there is also a scalar interaction between the nuclei and electrons, this will adda further term, and since scalar coupling can induce only the coupled transitionsS+L(S-Z+), it also will involve me. The resulting equation may be written :where Ne and Nn are the molar concentrations of unpaired electrons and of nucleirespectively, p is the number of nuclei affected at each encounter, s is the electron <pinquantum number, g is the electronic g factor, B the Bohr magneton, A the scalarcoupling constant and r e is the correlation time for the modulation of the scalarcoupling.Whereas co,2z,2 in liquids is usually much smaller than 1, this is not necessarily sofor co:z,2.For example, if zc is 10-10 sec, then at 104 gauss, oe is about 1.8 x 1010sec-1 and co,2~,2 is 3-24. It therefore follows that an important term in eqn. (4) willbe magnetic-field-dependent, and a study of the variation of TI with magnetic fieldcan allow zc to be measured if the scalar contribution to TI is unimportant. Thisvalue of zc will be the correlation time for relative motion of nuclei and electrons;it may be a measure of the rate of rotation of a weak complex of solvent and solutemolecules or it may be the diffusional correlation time of solute molecules with respectto solvent molecules. An example of the former would be the relaxation of nucleiin a large ligand bound to a paramagnetic metal atom ; the latter situation arises whenthere is no significant binding between solute and solvent as in the case of a free radicaldissolved in an inert solvent.Several measurements of molecular correlation times in liquids by this methodhave been performed.Hausser, Kruger and Noack 7 have measured the frequencyand temperature dependence of the relaxation times of several protonic systemR. A . DWEK AND R. E. RICHARDS 201containing various free radicals. The experimental results agree with the theoreticalpredictions based on the translational diffusion of spheres with only dipole-dipoleinteractions between radical and solvent molecules. The correlation times werefound to depend on the diffusion constant of the solvent and on the temperature.However, for solutions of free radicals in toluene, discrepancies between this theoryand experiment were apparent at low temperatures.7 Reinvestigation of toluenesolutions over an extended temperature and frequericy range* showed that theexperimental results could be interpreted in terms of purely dipolar interactions, thetime dependence of which arises from the relative translational motions and also therotational tumbling of an associated complex.The activation energies for theseprocesses were found to be 3.1 and 7.8 kcal mole-1 respectively. The activationenergy calculated in previous work 7 assuming only translational motion to be contri-buting to TI, was 4.4 kcal mole-1. Thus it is essential to analyze relaxation data insome detail, if meaningful activation energies for molecular motions are to be calculated.Kramer, Muller-Warmuth and Schindler 9 have also investigated the temperaturedependence of the correlation times of several protonic systems containing free radicals.Since translational diffusion had been shown to be the main mechanism of relaxation,the activation energies for viscosity should be similar to those of the correlation times,and indeed this was the case.On the other hand, the activation energies fromdielectric relaxation measurements would be expected to be different, since the molecu-lar motion in this case is rotation. This has been illustrated for diethyl ether, for whichthe activation energies for the viscosity and those of the translational correlation timesare about 1.8 kcal mole-1. This value can be contrasted with that of 1.3 kcal mole-1obtained from dielectric relaxation.If scalar interaction makes an important contribution to the nuclear relaxation,there are two correlation times to contend with and it is also of interest to be able tomeasure the value of the scalar coupling constant. For this purpose, additionalinformation can be obtained from the field and temperature dependence of the nuclearT2.Any mechanism of interaction between nucleus and electron which can affect TImust also affect T2, but scalar coupling can also shorten T2 by a mechanism whichcannot affect T I . T2 measures the time taken for the nuclear magnetization in the xyplane to decay by a gradual loss of phase of the precessing nuclei. This phase losscan be induced by any static component of a local field which can cause some nuclei toprecess at different frequencies from others ; scalar interaction produces such acomponent which does not vary as the complex rotates.The longer the componentis applied the more the nuclei lose phase, so the effect will be proportional to ze, thelifetime of the scalar coupling. The expression for T2 is therefore similar to that forTI but contains an additional term in Te in the scalar part :Note that Tl/T2 = 1 except when scalar coupling becomes important, when Tl>T2.The importance of the scalar term for relaxation depends not only on A but also onze and we and hence on temperature and applied field strength. The detailed studyof systems to evaluate all these parameters has been carried out only in a few cases ;a recent paper by Pfeifer et al.10 describes some interesting examples and gives manyreferences.Further information about molecular correlation times for diffusion may beobtained from nuclear electron double resonance experiments.If the electronresonance of the free radical dissolved at low concentration in a diamagnetic liqui202 MEASUREMENT OF NUCLEAR RELAXATIONis saturated, remarkable changes in the nuclear resonance intensities often occur. Thisphenomenon depends on the spin-lattice relaxation processes already described.When the electron resonance of the radical is strongly irradiated, the populationsof the electron energy levels (+ - and - +) may be equalized ; the resonance issaturated.The spin-lattice relaxation processes attempt to restore the populationsof the energy levels to the thermal equilibrium values in which more spins are in thelower two levels than in the upper two (fig. 2). For dipolar coupling, the processesare S+I+ and &I-, but the first has the greater importance. The electronic transitionis from the (- +) to the (+ +) level, and from the (- -) to the (+ -) levels (fig. 2).The dominant relaxation is from (+ +) to (- -) by the S+I+ process, so that nucleiinitially in the lowest (- +) energy level are transferred to the (- -) upper nuclearlevel. Under optimum conditions the populations of the two nuclear levels can bechanged by a fraction +ye/yn, which for protons is - 330 ; the population of the uppernuclear level becomes 330 times greater than that of the lower level is greater in theupper at thermal equilibrium.The nuclear resonance becomes an emission insteadof an absorption spectrum and is greatly increased in intensity.The theory of the effect is well understood2 and depends entirely on the spin-lattice relaxation of the nuclei and of the electrons. Since this mechanism dependson the spectral density of the thermal motion in the liquid at we as described above, sothe enhancement of the nuclear resonance must also depend on this spectral density ;when coe is small enough to lie on the flat part of the correlation spectrum (fig. l), themaximum enhancement (of - 330 for protons) may be obtained, but at higher valuesof we the spectral density decreases and the nuclear resonance enhancement becomessmaller.Measurements of the enhancement of the nuclear resonance at differentmagnetic field strengths (and hence of we) can often be satisfactorily interpreted by amodel of diffusing spheres with a single translation correlation time 99 1 1 s 12If there is also scalar coupling between the electrons and the nuclei, there is anadditional relaxation mechanism of the S+L type (see above). If this relaxationmechanism is stronger than the dipolar mechanism? saturation of the electron reson-ance equalizes the population of the two electronic energy levels, and the S+I-relaxation returns nuclei to the lower (- +) nuclear energy level. Under optimumconditions the population of the lower level is increased by a fraction -ye/yn,whichis about + 660 for protons ; the nuclear resonance absorption is increased in intensity.There is thus a striking and qualititative difference in the results for dipolar and scalarcoupling.If the scalar coupling is weak and comparable with the dipolar coupling,the nuclear resonance intensity lies somewhere between - 330 and + 660.Various attempts have been made to interpret quantitative measurements of thesenuclear electron double resonance experiments in terms of simple models for thescalar coupling.ll9 l3 These studies give values for correlation times and couplingconstants which are in approximate agreement with other measurements, but un-fortunately none of them have so far proved entirely satisfactory.13~ 14 When scalarcoupling is present, therefore? double resonance experiments provide an extremelysensitive indication of its importance, although the correlation times derived may notbe accurate. On the other hand, when scalar coupling is absent, the double resonancemeasurements provide a useful addition to direct measurements of TI, and the valuesof translational correlation times and their field dependence, as obtained from boththese methods are often in excellent agreement.9For nuclei with spin quantum numbers greater than 3, the distribution of positivecharge over the nucleus may be aspherical, and this situation may be described interms of a nuclear electric-quadrupole moment.If the electron distribution aboutthe nucleus has less than cubic symmetry, the resulting electric field gradient caR. A .DWEK AND R . E. RICHARDS 203couple with the nuclear electric quadrupole moment.1 This nuclear quadrupolecoupling provides a further mechanism in addition to the nuclear magnetic momentby which the nucleus can relate its orientation to that of other particles around it.Thus fluctuation of the electric field gradient coupled to the quadrupole moment canprovide an additional mechanism for nuclear relaxation; the relaxation rate l/Tl =1/T2 is proportional to the square of the electric quadrupole coupling constant.1Because the relaxation is governed by fluctuations of the quadrupole interaction atthe nuclear resonance frequency con, the relaxation rate is also proportional to z inthe “white spectrum” approximation, where z is the correlation time for the motionwhich modulates the quadrupole interaction.The electric field gradients produced by chemical bonds are often very strong andthe electric quadrupole interaction often completely dominates the relaxation timesof the nuclei.The quadrupole relaxation is often so strong that nuclear resonancesbecome very broad and difficult to detect.In amolecule such as carbon tetrachloride, the quadrupole coupling arises from theelectric field produced by the C-C1 bond, so that relaxation is produced by rotationof the molecule. On the other hand, the electron distribution about the brominenucleus in the bromide ion in solution is symmetrical, but the symmetry is momentarilyreduced by collisions with solvent molecules or with other ions.In this case thetransient quadrupole coupling is associated mainly with a diffusional correlationtime.15The relevant correlation time may be rotational or diffusional in a liquid.TABLE 2A E expt. AE viscosity(kcal mole-1) (kcal mole-1) moleculeCC14 1-3 k0.1 2.3HCC13 1.4 f0.1 1.8ClC6H5 1.6 f0.2 2.3Tic14 1.0 f0.1 2-1ClO; (in H20) 1 a 5 f0.4 3.7If the quadrupole coupling constant is known from other measurements, theexperimental values of TI or T2 can be used to obtain a value for z. Herbison-Evansand Richards made such measurements on some nitrogen compounds 16 and foundthat their results supported the theory of Wirtz which relates the correlation time tothe microviscosity.The temperature dependence of correlation times of some chlorine compoundshave also been determined from measurements of T2 of the 35Cl resonances.17 Thistemperature dependence is quite different from that of the bulk viscosity as is evidentfrom the activation energies in table 2.In this case the motion involved is molecularrotation so the activation energies are different from those for viscosity which dependon translation. The authors suggest that the correct temperature dependence of thecorrelation time would be obtained from considerations of the microviscosity tempera-ture dependence. This would also seem to be the case for the results of Herbison-Evans and Richards. It will be interesting to compare the activation energies for thequadrupole relaxation of halide ions in solutions of electrolytes, which depend ontranslational motion, with the appropriate energies for bulk viscosity.Moniz and Gutowsky 18 have studied the relaxation times of 14N in differenttypes of groups and again using known values of 14N quadrupole coupling constants,correlation times for the molecular reorientation were calculated.The temperatur204 MEASUREMENT OF NUCLEAR RELAXATIONdependence of TI was also measured for several compounds, and the range of activa-tion energies was found to lie between 1.4 and 3.2 kcal mole-1, the differences generallyreflecting the relative sizes and shapes of the molecules. The temperature dependenceof the proton and 14N spin lattice relaxation times in methyl cyanide were found tobe 1.4 kcal mole-1 and 1.9 kcal mole-1 respectively.This reflects the differenttypes of motion contributing to the 7'1 in each case ; rotation for the 14N, rotation andtranslation for protons. The relaxation time of the nitrogen nucleus is determinedmainly by rotation about the axes perpendicular to the C-CN bond, while rotationabout all axes could contribute to the relaxation rate of the protons. Zeidler 19 hasfound the rotational correlation time for the protons to be in good agreement withthat obtained by the nitrogen TI. This could be understood if rotation about theC-CN axis makes little contribution to the proton 7'1 because of very fast reorienta-tion about the C3 axis of the molecule, as would seem reasonable. Thus rotationabout this bond would have an extremely short correlation time and would not makemuch contribution to TI.In conclusion, it must be emphasized that from measurements of nuclear relaxationtimes or of the enhancement of the nuclear resonance in a nuclear electron doubleresonance experiment, we are measuring the spectral density of the molecular inter-action at particular frequencies, con or we.We can measure how these functions varywith co by making measurements at different magnetic field strengths ; their tempera-ture dependence at each frequency can also be measured. It is therefore possiblein principle to plot out the correlation spectrum at any temperature from purelyexperimental results. If a model for the molecular motion is assumed, the correlationfunction can be obtained and the experimental results expressed in terms of one ormore correlation times. These can then be compared with values measured in otherways.When there are specific intermolecular interactions during a collision betweentwo molecules in the liquid, these can often be detected in the nuclear resonancespectrum. If one of the molecules is paramagnetic, measurements of T1 and 7'2 andespecially of the enhancement of the nuclear resonance in a nuclear electron doubleresonance experiment, can give sensitive indications of scalar coupling formed byweak transient chemical binding.1 A.Abragam, The Princigles of Nuclear Magnetism, chap. VII, (Clarendon Press, Oxford, 1961)2 P. S. Hubbard, Proc. Roy. SOC. A , 1966,291, 537.3 B. §heard and J. Clifford, Biopolymers, 1966,4,1057.4 J. S. Waugh, Molecular Relaxation Processes, (Chem. SOC. Spec. Publ. no. 20), p. 113. (Academic5 see e.g., A. M. Pritchard and R. E. Richards, Trans. Faraday Soc., 1966, 62, 1388.6 J. G. Powles and R. Figgins, Mol. Physics, 1966, 10, 155.7 K. H. Hausser, G. J. Kruger and F. Noack, Z. Naturforsch, 1965, 20a, 91.8 G. J. Kruger, W. Muller-Warmuth and R. Van Steenwinkel, 2. Naturforsch. 1966,21a, 1224.9 K. D. Kramer, W. Muller-Warmuth and J. Schindler, J. Chem. Physics, 1965, 43, 31.10 H. Pfeifer, D. Michel, D. Sames and H. Spring, Mol. Physics, 1966, 11, 591.11 K. D. Kramer, W. Muller-Warmuth, and N. Roth, 2. Naturforsch., 1965, 20a, 1391.12 R. A. Dwek, J. G. Kenworthy, D. F. S. Natusch, D. J. Shields and R. E. Richards, Proc. Roy.13 W. Muller-Warmuth, 2. Naturforsch., 1966, 21, 153.14 R. A. Dwek, J. G. Kenworthy, J. A. Ladd and R. E. Richards, Mol. Physics, 1966,11, 287.15 C. Deverell, D. J. Frost and R. E. Richards, Mol. Physics, 1965,9, 565.16 D. Herbison-Evans and R. E. Richards, MoZ. Physics, 1964, 1, 515.l7 D. E. O'Reilly and G. E. Schacher, J. Chem. Physics, 1963,39, 1768.18 W. B. Moniz and H. S. Gutowsky, J. Chem. Physics, 1963,38, 1155.19 M. D. Zeidler, Ber. Bunsenges. Physik. Chem., 1965, 69, 659.Press, 1966).SOC. A , 1966,291,487

ISSN:0366-9033    

DOI:10.1039/DF9674300196

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

Angular Correlation in LiquidsBY A. D. BUCKINGHAMSchool of Chemistry, The University of BristolReceived 16th January 1967Distribution functions describing the correlation between the positions and orientations of groupsof one, two, three, . . ., molecules are defined and discussed. The one-particle distribution functionof an anisotropic fluid-such as a liquid in a uniform field-is expanded in orthogonal functions Sthat are simply related to the mean values of the Legendre polynomials describing its orientation. Avariety of observables, including dielectric polarization, optical birefringence, Rayleigh and harmonicscattering, and magnetic resonance spectra, are described in terms of the appropriate S-coefficientsand the orientation distribution functions.The microscopic structure and the bulk equilibrium properties of fluids areexpressible in terms of distribution functions, which give the probability of findingmolecules in various configurations.19 2 The pair distribution function is by far themost important, and is adequate for the description of the equilibrium propertiesof imperfect gases at low pressure, and of dense fluids, if the configurational energyis a sum of pair potentials.The distribution function dh)(r1 ;r2; .. . ;rh) is defined so thatd")(r1;r2; . . . ;rh)drldr2 . . . dr,is the probability of finding one molecule in each of the h volume elements drl, . . .,drh at rl, . . ., rh. Thus, the singlet distribution function n(')(rl) is the number den-sity; in a fluid it is independent of the position r1 and equal to n = N/V, where Nmolecules are confined to the volume V.The pair distribution function d2)(rl ; r2)in a fluid is a function only of the difference of rl and r2 and may be written d2)(r).The distribution function dh) for a system at equilibrium obeying the laws ofclassical statistical mechanics may be expressed in terms of the configurational energyU(r1 ;r2 ; . . . ;rN) = U :N ! 1 . . . J exp (-U/kT)dr,+, . . . dr,exp (- U/kT)dr,dr, . . . dr,n(")(r1;r2; . . . ;r,) = - (1)( N - h ) ! 1 . . .The factor N!/(N-h)! is the number of ways h objects may be chosen in order fromN objects. The denominator in eqn. (1) may be expressed in terms of the configura-tional partition function Q, where1 Q = -1. . . N ! exp (- U/kT)dr,dr, .. . dr,determines the bulk properties of a system at equilibrium.may also be related to the distribution functions.molecule 2 at a distance r from the molecule 1, and is defined by the equationHence these propertiesThe radial distribution function g(r) is proportional to the probability of findingd2)(r) = [N(N - 1)/V2]g(r). (3)20206 ANGULAR CORRELATION IN LIQUIDSFor large N, g(r) = d2)(r)/n2. This function is equal to zero for small r because ofthe repulsion between overlapping molecules, and in a fluid is unity for large r ; itrises to a maximum at the most favoured separation where the configurational energyis a minimum. For a dilute gas of spherical moleculesg ( r ) = exp (-U12lkT) (4)where ulz is the interaction energy of molecules 1 and 2.ANGLE-DEPENDENT DISTRIBUTION FUNCTIONSThe concept of radial correlation can be extended to angular correlation.Thus,the distribution function n(h)(rl,ol ;r2,02 ; . . . ;rh,oh) is defined so thatdh)(rl,o1;rz,o2; . . . ;r,,o,)dr,do,dr,do, . . . dr,do,is the probability of finding a molecule in each of the h volume elements drl, dr2,. . ., drh at r1, r2, . . ., rh and in the orientational elements dol, dm2, . . ., doh at 01, o2,. . ., oh. The minimum number of coordinates needed to specify oh depends on thesymmetry of the molecules comprising the medium-it is two if the molecules arelinear, and three if they are non-linear and rigid. Also1 . . J @)(r1,o,;r2,o2; . . . ;rh,o,Jdoido2 . . . do, = #)(rl;r2; . . . ;r,). ( 5 )In an isotropic fluid, the distribution functions ared1)(rI,ol) = n / ~ ,d2)(r1,ol ;r2,02) = d2)(r,o),where r,o are coordinates relative to rl, a1 and 0 1 = Idol.The mean value of some functionf(o1) of the orientation of a molecule is given by -f(oi> = J f ( ~ i > ~ ( ~ i ) d ~ i (8)where P(o1)dwl = n-ldl)(rl,ol)dol is the probability a inolecule has an orientationbetween o1 and 01 +dw, and may be called the singlet orientation distributionfunction.If one direction is favoured over others in a fluid (as when a uniform fieldis applied), 9(o) gives the mean orientation of the molecule and may convenientlybe described in terms of the direction cosines ai = cos& (i = 1,2,3) of the favoureddirection in a Cartesian frame, 1,2,3 fixed in the molecule.Thus~ ( o ) = n-ln(')(r,o) = R - ~ [ I + 3 C ~ , a ~ + S C a , a ~ ~ , ~ + 7 C aiajakSijk+1 i, j9CaiajakalSijkl+ . . . 1, (9)i j k lwhere the S-coefficients describe the ordering and, if 6ij = 1 if i = j and = 0 if i+j,-ai = .P,(cos ei) = a i P ( o ) d o = Si, (10)$aiaj-*Sij = s,,, (11)-5-a.a 2 J .a k-3(ai6jk+aj6ki+akSij) = S i j k , (12)-- - - -- - ___ - - - -yaiajakal -$(aiaj6kl + aiak8jl + aialSjk -+ ajak6, + ajalSik -+ akaLhij) -+In general there are three independent coefficients Si (viz., S1, S2, S3), fiveindependentcoefficients Sij (S11, S22, S12, S23, S31) (for S11+S22+S33 = 0), seven independent$(6ijdk1+ 6ikSjl-k Silajk) = S i j k l . (13A . D. BUCKINGHAM 207coefficients S i ~ k (S111, S112, S113, S122, S123, S222, &3), nine independent coefficientsSijkl, etc.However, if the molecule possesses elements of symmetry, these numbersare reduced. Thus, just one coefficient of any order (&; & I ; S111; S1111; . . .) issufficient to describe the mean orientation of a linear molecule in space. For a mole-cule with C3v symmetry (e.g., NH3, CH&l), the following are independent : & ; S11;S111, S222; s1111, s2221 ; for D6h (e.g., C6H6) there are Sl1 ; sllll ; and for c 2 v (e-g.H20, CH20, CH2C12) Si ; Sii, 5'22; Sill, s 2 2 1 ; ~ i i i i , s1122, s2222.The five P2 coefficients S'j were introduced by Saupe 3 to describe the averagealignment of molecules dissolved in liquid crystal solvents in a uniform magnetic field.In isotropic fluids every S is zero, and in a completely oriented arrangement of linearmolecules, S1 = S11 = S111 = S1111 = 1.The coefficients have the following limits :d S(14)so that (Vi2)-lg(r,co)drdo is the probability molecule 2 is centred at a point betweenr and r + dr from molecule 1 and at a relative orientation between o and w + dw. Fora dilute gasg(r,a) = exp (-U12IkT), (15)where 2412 is now dependent on orientation as well as separation.In a mixture of species a and b, there are two singlet distribution functions di),such that n(:)(rl,wl) = na/n1, and dB)(rl,col> = nb/R1, where nu = N ~ / v , nb = Nb/v.However, there are now three pair distribution functions, nf?(r,w), n(zd(r,o) andtzy&r,w> ; these functions give rise to the pair orientation distribution functions :goa(r,a) = Q2ni2n$(r,w),g,,(r,cu) = R2n, In; '&)(r,w), (16)2 -2 (2) gbb(r,co) = nb nbb (r,w),defined so that (~~)-lg,b(r,w)drdw, for example, is the probability a particular mole-cule of species b is at a point between r and r+dr from molecule 1 of species a,and at a relative orientation between w and w+do.MOLECULES I N A N ELECTRIC FIELDUsing the method of Kirkwood,4 it is possible to derive a rigorous expression forSl, the mean value of the cosine of the angle between p1, the permanent dipole momentof molecule 1, and the electric field E;3~ KE2&+1 3kTs1 = -- -where E is the static dielectric constant and where 6 is the mean dipole moment, inthe direction of pl, of molecule 1 and its neighbours in a small macroscopic spherecentred on molecule 1.If the molecules are not polarizable, = gp, whereg = 1 + ni2-l J J cos yg(r,co)drdco (18208 ANGULAR CORRELATION I N LIQUIDSis the angular correlation function,s y being the angle between the dipole moments ofmolecules 1 and 2; the integration over r is limited to this sphere inside the fluid.Hence cos 81 is determined by (cos y).In polar liquids in electric fields -100e.s.u., & N 10-2.Distortions of molecules through interactions with neighbours lead to difficultiesin the general theory of dielectric polarization. However, if the actual dipole mlis related to the permanent moment p by the Onsager relation 6( 2 E + l ) ( E + 2)3 ( 2 ~ + E,)m = - Pwhere E , is the high-frequency dielectric constant, then 7 9 88-1 ~ , - l - 4nnp2g 3 ~ ( ~ , + 2 )~ + 2 e,+2 9kT ( E + ~ ) ( ~ E + E , ) ’---_-Table 1 shows correlation parameters g derived from eqn.(20) for a variety ofpolar liquids. It is clear that g> 1 for molecules whose dipole axis is the short axis(chloroform, paraldehyde, trimethylamine), and gc 1 if the dipole axis is the longaxis (acetonitrile, chlorobenzene, nitrobenzene). In water, the strong hydrogenbonds linking a molecule to its four neighbours produce an abnormally large valueof g which decreases at T increases?TABLE ~.-DIELECTRTC DATA AND MOLAR VOLUMES Vm AND DERIVED ANGULAR CORRELATIONFUNCTIONS, gsubstance TCC) Vm(cm3) & Em AD)CH3Cl 25 57.47 9.68 1.93 1.86 0.90CH2C12 25 64.47 8.93 2-35 1.57 1.01CHC13 25 80.72 4.72 2.37 1.01 1.19CH3CN 25 52-82 36.7 1.8 3-96 0.82(CH3)2CO 25 73.86 19.11 2.0 2-89 0.97paraldehyde 25 133.5 12.93 2.5 1-44 3.56No3313 25 94.3 2.44 1.8 0.61 1.5c6H~C1 25 102.2 5.612 2.55 1.73 0.61C6H5N02 25 102.7 34-89 2.63 4.24 0.87p-CH3.c6H4. NO2 58 120.4 22-2 2.4 4.5 0.70H20 0 18.02 88.2 1 -797 1 -84 2.89H20 25 18.07 78-5 1 -795 1 -84 2.8 1H20 83 18-58 60.4 1 -767 1.84 2.69N H 3 - 33 25-4 22.4 1 -47 1 -47 1-66NH3 25 28.3 16.9 1.53 1 -47 1.65C6H5N02 200 121 15.95 2.29 4-24 0.83OPTICAL BIREFRINGENCE I N FLUIDSIn optical birefringence experiments, the observable is generally the differencebetween the index of refraction in the x and y directions, n,-n,. In a fluid thisdifference may be induced by a strong electric field (the Kerr effect), a strong magneticfield (the Cotton-Mouton effect), an electric field gradient, or the field of an intenselight beam.1 N - -n, - n, = -(n2 + 2)227r-(a,, - a,,,)9n Vwhere a is the polarizability in the presence of the external field, and the bars denotestatistical averages. If the molecules are anisotropically polarizable, the effect of thA . D .BUCKINGHAM 209fields on the polarizability can generally be neglected. If the uniform electric andmagnetic fields, or the optical field, are in the x-direction,1 N9n = -(n2 + 2)22nvaap~ap,where Sap is one of the components of n(l)(r,co) in eqn. (9). Actually, the mean valueof the component A,, of any second-rank tensor is A,, = &4,a+$4apS,p. Thedifferent Sap coefficients in eqn.(22) could be measured by observing the dispersionof the anisotropy nx - nu in the vicinity of absorption bands of known polarization.Birefringence studies are particularly valuable when the molecules possess Cnv (n 233)symmetry when a22 = a33 = al and only S1 is independent. Then eqn. (22) becomes1 N9n V (23) n, - n, = - (n2 + 2)22n-(a 1, - aJS1 1,makingH,, (nz-nIzy) is related to the Kerr and Cotton-Mouton constants B and C:lOwhere K is the wavelength of the radiation in air at 1 atm. For nitrobenzene,B = 4 x 10-5 e.s.u. and C = 2.4 x 10-12 e.m.u. for yellow light at 25"C, so S11 - 10-8E,2 or 5 x 10-16H:, where Ex and H, are in e.s.u. and e.m.u.In a strong magnetic field, diamagnetic molecules are oriented through theanisotropy in their susceptibility tensors x, anda directly observable quantity.In electric and magnetic fields Ez andn,- ny = B I E ~ , , or CAH:, (24)where (22)) is the mean susceptibility of molecule i in the direction of the ap axesof molecule 1 and f i ) = $xC,i: is the mean susceptibility of molecule i.If the orienting field is that of an intense beam of light polarized in the x-direction,NSap = ( a 3 - a(% )(n2+2)22nIX/5ckT,i = 1 9nwhere 1% is the intensity (energy crossing unit area in unit time) of the light beam, anddi) is the polarizability for the frequency of the orienting light beam.In strong static electric fields, the difficult question of the local field arises, but fornon-polar media the Lorentz field +(&+2)E, is reliable andN s aB = c (C@(i)-- + 2)2E$/lOkT, (27)i = 1where do) is the static polarizability tensor. In polar liquids 11where M is the total dipole moment of the macroscopic sphere in vacuo in the fieldEo = Q(&+2)Ez.The short and long-range contributions to the mean value of(MaMp-4M') must be evaluated in a similar way to that used by Kirkwood 4 fo210 ANGULAR CORRELATION I N LIQUIDSanalyzing (W). For highly polar axially symmetric molecules satisfying eqn.(1 91, - .90k2T2S11 27kT E(E-&,) 6e2(&, + 2)2( E + 2)2E,2--+ p2(g - Q(& + 2)2(2& + &,)2(M:-*M2) = -27111 (8 +2)2(2e + 6,)n'2Yr2902)dr2d02+ J J J J (+ cos 712 cos 713- 3 ~ 0 s 723)n( )(r2 ,a2 ;r3 ,o,)dr,do,dr,do , (29) 1 so the Kerr constant of a dipolar liquid depends on the two- and the three-particledistribution functions.12Eqn.(25), (26) and (27) reduce to expressions of the following form for symmetrictop molecules :Sl = +(XI, - xl)[1+ nR-l J J P2(c0s y)g(r,o)drdo] ~ , 2 / 1 0 k ~ . (30)Birefringence experiments therefore yield valuable information about angular correla-tion.The field dependence of the dielectric constant of a liquid is proportional to3( M4) - 5(M2)2, and this could similarly be related to the two, three and four particledistribution functions.RAYLEIGH LIGHT SCATTERING BY DENSE FLUIDSIf there is correlation between the positions of scattering centres, the wavescoherently scattered interfere and affect the intensity of the radiation emitted. If theincident beam travelling in the x-direction is plane polarized in the z-direction, thenthe intensities scattered by dipole radiation in the y direction with x and z-polarizationhave the ratio 13where $tj = (2n/A)(xt9-~+,) is the difference of phase at the detector between thewaves scattered from molecules i and j .If correlation extends over distances thatare short compared to the wavelength R (this condition does not apply near thecritical point), eqn. (3 1) becomeswhereH: = l+nSz-lJ 1 (g(r,o)-1)drdo (33)is the isothermal compressibility - V-l(d V / d p ) ~ provided the configurational energyU is the sum of pair contributions.When R is comparable to, or shorter than, the correlation distances, it is no longerpossible to use the dipole approximation. The scattering of X-rays or neutrons ofwavelength - 1 .$, gives valuable information about structure in a fluid-the Fouriertransform of the intensity of scattering at various angles gives the radial distributioA .D. BUCKINGHAM 21 1of the scattering centres. The observable is therefore the number of atoms at adistance Y from a given atom, and this can be interpreted in terms of distributionfunctions. The technique is most valuable when applied to very simple molecules.14HARMONIC SCATTERINGIn strong electric fields, the dipole moment of a molecule can be written 15where p and y are hyperpolarizabilities describing the non-linear polarization. If Ehas the frequency a, then p and y produce dipoles oscillating, and therefore radiating,at the second and third harmonic frequencies.If the molecules are centro-symmetric,p and p are zero. The second harmomc analogue of eqn. (31) ism a = pa + UafiED + +PabyEpEy + iyaByaEfiEy Ea + * 9 (34)If +A% the extent of correlation, the numerator and denominator in this equation canbe reduced to expressions similar to the numerator in eqn. (32).16 However, theresult is complicated and is limited by the uncertain effects of the environment on thehyperpolarizability p.Unlike 1,(2co), the third harmonic component 1,(3w) is reduced similarly to 1, inthe liquid. This is because cczz and yzzzz, unlike Pzzz, have non-vanishing mean values.MAGNETIC RESONANCE OF PARTIALLY ORIENTED MOLECULESThe magnetic resonance spectra of molecules in anisotropic fluids show interestingeffects, which can be interpreted in terms of the orientation coefficients Sap ofSaupe.3, 17 All five independent coefficients could, in principle, be obtained for amolecule of low symmetry possessing sufficient magnetic nuclei in known positions.Successful studies have been made in strong electric fields 18 and particularly inliquid crystal solvents. In nematic liquid crystals, the magnetic field aIigns thesolvent molecules and the solute derives its alignment through the angular dependenceof the distribution function gab(r,co). Large degrees of alignment, I Sap I -10-l)and as high as 0-7, have been achieved.1 J. de Boer, Rep. Progr. Physics., 1949, 12, 305.2 J. 0. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular Theory of Gases and Liquids, (Wiley,3 A. Saupe, 2. Naturforsch., 1964, 19a, 161.4 J. G. Kirkwood, J. Chem. Physics. 1939, 7, 91 1.5 G. Oster and J. G. Kirkwood, J. Chem. Physics, 1943, 11, 175.6 L. Onsager, J. Amer. Chem. Soc., 1936,58, 1486.7 H. Frohlich, Theory of Dielectrics, (Oxford University Press, Oxford, 2nd ed., 1958).8 A. D. Buckingham, Proc. Roy. SOC. A , 1956,238,235.9 J. A. Pople, Proc. Roy. Soc. A , 1951, 205, 163.10 J. W. Beams, Rev. Mod. Physics, 1932, 4, 133.11 A. D. Buckingham and R. E. Raab, J. Chem. Soc., 1957, 2341.12 J. M. Deutch and J. S. Waugh, J. Chem. Physics, 1965,43,2568 ; 1966,44,4366.13 A. D. Buckingham and M. J. Stephen, Trans. Faraday Soc., 1957, 53, 884.14E. E. Bray and N. S. Gingrich, J. Chem. Physics, 1943, 11, 351.15 A. D. Buckingham and J. A. Pople, Proc. Physic. Soc. A , 1955, 68,905.16 R. Bersohn, Y.-H. Pao and H. L. Frisch, J. Chem. Physics, 1966, 45, 3184.17 A. D. Buckingham and K. A. McLauchlan, Progress in N.M.R. Spectroscopy, Vol. 11, (ed.18 A. D. Buckingham and K. A. McLauchlan, Proc. Chem. Soc., 1963, 144 ; Chem. in Britain,New York, 1954).J. W. Emsley, J. Feeney and L. H. Sutcliffe), (Pergamon Press, Oxford, 1967), p. 63.1965, 1, 54

ISSN:0366-9033    

DOI:10.1039/DF9674300205

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

Frequency-Dependent Direct Correlation FunctionBY J. STEWInstitute for Physical Chemistry, Polish Academy of Sciences,Warszawa 42, PolandReceived 23rd January, 1967Generalization of the Ornstein-Zernike direct correlation function to a frequency-dependentfunction appropriate for a description of time-dependent fluctuations in an equilibrium fluid, isdiscussed.The standard distribution functions of equilibrium statistical mechanics can berepresented as single-time averages of microscopic density functions taken at differentpoints of the fluid. Thus ifNwhere Rg(t) is the position of particle i at time t,and so on. Here g(r) is the radial distribution of the fluid and the angular bracketdenotes the equilibrium average. In terms of Fourier-transformed quantitiesPk = & exp(ik .Rt),The direct correlation function is introduced via an integral equation which in termsof Fourier coefficients is 1P - l < P k ( f ) p - k ( t ) ) = 1+Pak= I(kO)* (4)(1 + p O k ) ( l + p p V k ) = 1, p = (kT)-? (5)This may be taken as a definition of Vk although (5) was derived as an approximaterelation and vn: were interpreted as parameters of the effective interaction. Relation(5) is an important element of the original derivation of the Percus-Yevick equation 1for g(r) in terms of the true intermolecular potential V(r).The generalization of expressions like (3) or (4) to the time-dependent descriptionis generally achieved by taking the microscopic densities not only at two differentspatial points but also at two different times.Thus, the 1.h.s. of (3) is generalizedto the average of p(rt)p(r’t’) and the van Hove function G(r,t) with its double Fouriertransform S(k,o), the “scattering law” of neutron or light scattering, make theirappearance. These quantities, of fundamental importance for the moleculardynamics of equilibrium systems, have been widely discussed and their propertieshave been listed, e.g., by Martin.2 However, as for g(r) some time ago, formalrelations and limiting properties are known and explicit calculations are possible forsystems such as the Boltzmann gas, slightly anharmonic solid, etc. Several detailedmodels have been proposed which will not be discussed here. One can also write21 J . STECKI 213down formal (linearized) kinetic equations 3 applying the procedures of Prigogine-RCsibois4 or of Zwanzig.5 What one lacks, are approximations that would formthe counterpart of the known integro-differential equations for g(r) such as Born-Green, Kirkwood-Salsburg, or Percus-Yevick equation.One also lacks a virial expansion in powers of density and recent developments ofkinetic equations seem to prove convincingly6 that such a virial expansion is notpossible for the autocorrelation functions. Moreover, the three-particle contributionin a moderately dilute gas is composed of two parts.One corresponds to the Enskogcorrection and is well-behaved; the other part contains re-collisions of the sameparticles, was always neglected in earlier work as a minor correction, and yet isdivergent for the hard-sphere potential This, to me, casts serious doubts on theattempts to use Enskog-type kinetic equations for systems of hard spheres even if onedid not have such doubts before.This can also be viewed as a one reason more fora search for a collective description of the fluctuations in a fluid. The direct correla-tion function may enter in such a description.Thus, we would like to see eqn. (5) as connecting two single-time quantities and asa particular case to which a relation between two double-time (or frequency-dependent) quantities would reduce. Also, eqn. (5) becomes an exact relation for adilute equilibrium (Debye) plasma if the true (Coulomb) potential is substituted forthe effective potential vk. Thus, conversely, vk is the effective potential to use if theelectron-gas relations are forced to fit any other system. This effective potentialwill guarantee that at least the static pair-distribution function is given correctly.We also note that the dilute plasma is one more example of the few physical systemsfor which S(kco) can be calculated. One obtains 7(6)I +cQS(km) = d o exp (imt)I(k,t) = -(&+(kO)- l)-'lm 277: 7cmwithI(k,t) = P-l<Pk(oP-k(0)).(7)(8)Here E+ is the frequency-dependent dielectric function of the plasma. Explicitly,1 +cQc+(k,o) = - f l p v k J dp(ku)&)~ukv' 1mm+O+.-00The one-dimensional Maxwellian is &I) and p = mv. This quantity E+ is of funda-mental importance ; it enters the short-time reversible Vlassov equation as well as theirreversible long-time Balescu-Lenard equation.It is directly related to the density(potential) response to an external perturbing charge density (potential) and can begenerally defined by1&+(km) 1 = pp v k [ imQ'( k i m ) + I( k,O)], (9) ~-withQ+(k,iw) = dz exp (ioz)I(k,z), S = n-'ReQ+, Imw+O+ (10) so Despite the simplicity of (6) and (8), these results 7 are far from trivial.Just as S(ko) and E-1 are, so is I(k,O) directly related to a density response to a(static now) external potential U(r) and the r.h.s. of (3) can be represented 8 as afunctional derivative Sp(r,)/SP U. Wertheim's relations become ordinary algebraic interms of Fourier coefficients. The direct correlation function or rather (1 +&v&)214 FREQUENCY-DEPENDENT DIRECT CORRELATION FUNCTIONwas expressed by Wertheim as a functional derivative inverse to that just quoted.Thus, if S(ko) is associated with 8-1, a proper frequency dependent v(kco) should beassociated with the inverse response function, i.e., with E.In fact, for an electron gas,relation ( 5 ) is simply rewritten as~&+(k,O)]-l[E+(k,O)] = 1, (1 1)I(k,O) = 1 +pak = doS(k,o) = [~+(k,O)l-' (12)&+(k,O) = 1 +ppr/k. (13)[e+(kw)]-l[e+(kw)] = 1, (14)because r: by Kramers-Kronig relations, andAn immediate generalization of eqn. (1 1) and thus of ( 5 ) is then obvious :withe+(kw) = 1 +ppv(k,w), (15)where e(ko) is a suitably defined screening function that should generalize the dielectricfunction for a system with Coulomb intermolecular potential.The function v(kco)generalizes the direct correlation function vk = v(k,w = 0). Thus, at zero frequencyA general set of definitions involving &+(kw) independently of the nature of theinterparticle potential and of the external perturbing potential, has been described byGlick.9 His definitions lead directly to (8), Vk being now the intermolecular potential.However, (13) is not preserved and the quantity PpYk is thus not eliminated in asimple way. It is, however, possible to define e(ko) so as to preserve the appearanceof eqn. (6) ; thus settinge(k0) = 1 +ppv,. (1 6)S( k,w) = (nw)- [ e ' (kO) - 11 - 'Im [ e+(ko)- '3.[e+(kw)]-l - 1 = Bpvk[ioQ'(k,iw)+I(k,O)].(17)(18)HenceEqn. (15) then defines v(kw).The bracketed expression in (18) fulfils Kramers-Kronig relations. The static structure factor is thus always given correctly asOne can also presume that in a stable system, e(ko)- 1 fulfils Kramers-Kronigrelations.Rewriting now eqn. (14) in the form [ (f - 1) + l][l + ppv(ko)] = 1,one can invert this relation to the kt and to the r,t space, thus obtaining integralequations relating the inverse transform of v to G(r,t) and its time derivative, or toI(k,t) or I'(k,t). Moment relations can be investigated, the hydrodynamic limitexamined, and so on. These calculations will not be recorded here; the usefulnessof C(r,t) or v(k,o) and of the particular choice of the definitions, are to be judged bythe feasibility of proposing simple approximations to v.One such approximation iJ . STECKI 215suggested by Zwanzig 10 and is also straightforward in the present context. We areinvestigating now a particular formv(ko) = -v,(kV(w-kv)-'), Imo+O+, (21)in which thecu-dependence is the same as in true plasma and we hope to report soonon the results to which this form may lead. In any case, a relatively simple expressionfor e(kw) ought to give already sufficiently interesting results for S. Also, otherpossibilities of simple approximations to v seem to be present in the subtracteddispersion relations, of Kadanoff and Martin and of Puff,11 which involve 3-1.1 J. K. Percus and G. J. Yevick, Physic. Rev., 1958, 110, 1.2 P. C. Martin, Proc. Int. Symp. (Aachen 1964).3 J. Stecki, unpublished.41. Prigogine and P. Rbibois, Physica, 1961,27, 621.5 R. Zwanzig, 1959 Boulder (Colorado) Lectures in Theoretical Physics, 1960, vol. 111.6 J. M. J. van Leeuwen and A. Weijland, reported at I.U.P.A.P. Inf. Symp. (Copenhagen, 1966).J. Sengers,L. K. Haines, J. R. Dorfman and M. H. Ernst, Technical Note BN-6197 D. C. Montgomery and D. A. Tidman, Plasma Kinetic Theory (1964) where further references8 M. Wertheim, Lecture Notes on Percus- Yevick Equation, (Los Alamos).9 A. J. Glick in Lectures on the Many-Body Problem, ed. E. R. Caianello (1962).10 R. Zwanzig, Physic. Rev., 1966, 144, 170.11 R. D. Puff, Physic Rev., 1965, 137, 407.K. Kawasaki and I. Oppenheim, ibid. E. G. D. Cohen, ibid. J. Stecki, ibid.preprints, 1965.(University of Maryland), Oct. 1965, where further references can be found.can be found

ISSN:0366-9033    

DOI:10.1039/DF9674300212

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

Influence of Molecular Rotation on Some PhysicalProperties of LiquidsBY D. B. DAVIES AND A. J. MATHESONDepartment of Chemistry, University of Essex, ColchesterReceived 2nd January 1967In those liquids which show an Arrhenius temperature dependence of viscosity, molecules are ableto rotate many times about at least two axes during the time between translational jumps. In thehigher temperature non-Arrhenius region molecules can rotate about only one axis during this time,while in the lower temperature non-Arrhenius region molecular rotation occurs primarily as a resultof translational motion. The distribution of shear relaxation times in liquids at low temperaturesarises from this restriction of molecular rotation, as does the structural contribution to the heatcapacity of non-associated liquids.TEMPERATURE DEPENDENCE OF VISCOSITY OF LIQUIDSThe temperature dependence of the viscosity v of liquids at temperatures wellabove their melting points is precisely described by the Arrhenius equationwhere A is a constant, R is the gas constant, T is the absolute temperature, and Eis the activation energy for viscous flow.This equation applies to some liquids overthe whole normal liquid range. The viscosity of most liquids is greater than thatpredicted by eqn. (1) at lower temperatures in the liquid range. In this region theviscosity is described by the free volume equation,or the equivalent TO equationHere uo is the limiting specific volume of the liquid at the second-order transitiontemperature TO, uf is the expansion free volume, and A', B', A" and 23'' are constants.The change from Arrhenius to non-Arrhenius viscosity behaviour occurs over 10 or20" at viscosities of approximately 1 cpoise.We define the Arrhenius temperatureTA as that temperature at which the viscosity deviates from eqn. (1). TO is the funda-mental reference temperature for molecular transport and relaxation processes inliquids, being that temperature at which there is no expansion free volume in theequilibrium glassy liquid and at which the viscosity becomes infinite.In order to fit the observed viscosity data in a few non-associated molecularliquids, eqn. (3) has to be applied to two regions with two sets of the constants A", B"and T0.1 This change in non-Arrhenius viscosity behaviour occurs over about 10"at or below the liquid melting point at a viscosity which is typically 1 poise.Wedefine the intersection temperature TK as that temperature at which this viscositychangeover occurs. TO of the higher temperature non-Arrhenius region is alwaysgreater than TO of the lower temperature region. Since the liquid density is linearwith temperature, this implies an increase in expansion free volume on cooling throughTK.216log v = A+E/RT, (1)log y = A'+B'VO/Uf, (2)log q = A"+B"/(T-To). (3D. B. DAVIES AND A . J . MATHESON 217MOLECULAR ROTATION AND VISCOSITY OF LIQUIDSThe viscosity of a liquid is a measure of the ease of translational movement of themolecules in the liquid. Eyring 2 suggests that this translational motion occurs as aseries of jumps by a molecule from one site in the liquid to another. The viscosity isapproximately related to the average time between jumps or shear relaxation timezs by Maxwell's relationwhere G , is the limiting rigidity modulus of the liquid.It is likely that molecular rotation has a considerable influence on the translationalmotion of molecules in a liquid.In considering liquid viscosities we say that amolecule exhibits free rotation about a given axis if it is able to rotate through 360" ormore during the time between translational jumps, while if a molecule makes severaltranslational jumps before rotation through 360" can occur we say that rotationabout that axis is restricted. A molecule whose rotation is restricted is able to trans-late less readily than one whose rotation is free : the restricted molecule is unable torotate to align itself into a favourable orientation to squeeze past its neighboursso that the observed viscosity is greater than it would have been if rotation were free.This suggests that liquids which are composed of spherical atoms or moleculesshould show an Arrhenius temperature dependence of viscosity throughout thenormal liquid range.The atoms or molecules are always in a suitable orientation tomake a translational jump, and the probability that they do so is determined solely bythe probability that they have sufficient energy to jump. For example, the viscositiesof the liquid inert gases and of liquid metals show Arrhenius viscosity behaviour.The same is true of those molten salts which contain only monatomic ions.3In molecular liquids the translational motion is severely restricted in comparisonwith the gas phase.In certain liquids such as Hz, 0 2 , N2 and CHI, however, Ramanspectroscopy has shown 4 that molecular rotation occurs as freely as in the gas phase.Hence many molecular rotations will occur per translational jump in the liquid, andso the viscosity of these liquids should show Arrhenius behaviour, as observed.3There is little direct evidence on the rotational behaviour of other quasi-sphericalliquid molecules, though it is reasonable that those molecules whose shapes areapproximately spherical should have relatively free molecular rotation. A usefulmeasure of molecular sphericity is the ratio Q of the longest to the shortest van derWaals' radii of the molecule.Smyth 5 found that when 0 is close to unity, as in thetetra-substituted methanes, molecules are able to rotate in the crystalline solid. Acomparison of the liquid viscosity behaviour with 0 is shown in table 1. The liquidviscosity shows Arrhenius behaviour 3 throughout the normal liquid range when themolecules are approximately spherical and Q is less than 1.5: on the other hand,when Q is greater than 1-5 and the molecules are non-spherical, non-Arrheniusviscosity is found at lower temperatures in the liquid range.3Further insight into the influence of molecular rotation on viscosity may beobtained from a calculation of the volumes required for rotation of a molecule aboutits three axes.These volumes have been calculated from molecular models 6 andtable 2 contains a comparison of these volumes, RA, RB and Rc with the averagevolume available to a molecule in a liquid at its melting-point, VM, at the Arrheniustemperature, VA, and at the boiling-point, VB. Although the approximations involvedin this comparison do not albw detailed, quantitative predictions to be made fromtable 2, some qualitative points emerge. Those substances which show Arrheniusviscosity behaviour throughout the normal liquid range have sufficient volume avail-able to a liquid molecule at the melting-point to permit molecular rotation about atr = M&cl, (4218 INFLUENCE OF MOLECULAR ROTATIONleast two molecular axes.In contrast to this, those substances which have non-Arrhenius viscosity behaviour at lower temperatures have free rotation about onlyone axis at temperatures in the region of the melting-point, while at higher tempera-tures rotation is possible about two or three axes. This suggests that the onset ofnon-Arrhenius viscosity behaviour occurs at that temperature at which rotation abouttwo molecular axes becomes restricted, while rotation about the third remains free.TABLE 1 .-VISCOSITY BEHAVIOUR AND MOLECULAR SHAPEArrhenius non-Arrheniussubstance U substance U1.21.41.21.01.01.01.01.11.01.01.31-51.1cyclohexyl chloridecyclohexanonecyclohexanolethanolglutaronitrilemethyl cyclopentanemethyl cyclohexanebenzene1-81.81.81.62.21.61.82.0In some liquids free rotation about all three axes is not possible even at the normalboiling-point, and the Arrhenius region coincides with the region in which the mole-cules have two axes with free rotation.It is probably not important that one axis ofrotation is restricted, since rotation occurring simultaneously and rapidly about theother two is equivalent to rotation about the third axis.By analogy with the change from Arrhenius to non-Arrhenius viscosity behaviour,it seems possible that the change in the non-Arrhenius viscosity behaviour at theTABLE 2.-MOLECULAR VOLUMES IN LIQUIDS (A3)liquid R.4 RB Rc VM v* VBethanepropanebutaneethylenepropylenebenzeneo-xylenep-xylenecyclohexane588287528414320218216772105158611 021372252651587213619668136127178203132761001267191145192203176- 91107 122142 153- 82100 141140 160200 227- 233- 194intersection temperature is also caused by the restriction of the rotation of the mole-cules in the liquid.To test this hypothesis, the minimum volumes required for rota-tion by molecules of some alkyl benzenes have been calculated : these substances havebeen chosen because the lower and higher members of the series do not show tworegions of non-Arrhenius viscosity behaviour, while intermediate members do havetwo such regions.1 For the purposes of this calculation it has been assumed that themolecules exist in a fully extended configuration.Table 3 contains a comparisonof the volumes required for rotation of a molecule about its three axes with the averagevolume available to a molecule in the liquid at its melting-point and at the intersectioD . B . DAVIES AND A . J . MATHESON 219temperature. The one rotational mode which remains free in the non-Arrhenius regionis still unrestricted at the melting-points of toluene, ethyl benzene and n-hexyl benzene.These liquids show a single type of non-Arrhenius viscosity behaviour throughout theexperimentally accessible liquid range, including the supercooled region. In contrastto this, the lowest rotational volumes of n- and i-propyl benzene and n- and sec-butylbenzene are all about the same as the volumes available to a liquid molecule at thetemperature at which the change in non-Arrhenius viscosity behaviour occurs.Although this correlation is less satisfactory for n-pentyl benzene, the figures for theother four liquids suggest that in the lower temperature non-Arrhenius region freerotation is not possible about any molecular axis, and that this restriction of rotationaround the intersection temperature leads to the change in the non-Arrhenius viscositybehaviour in liquids.TABLE 3liquidtolueneethyl benzenen-propyl benzenei-propyl benzenen-butyl benzenes-butyl benzenen-pentyl benzenen-hexyl benzene.-MOLECULAR VOLUMES IN AROMATIC HYDROCARBONS (A3)R A Rl3 Rc VK VIU149 210 167 - 158164 240 223 - 182210 33 1 273 208 20721 1 260 245 206 208227 409 316 235 235254 246 277 237 238252 -530 -420 264 264278 -680 -500 - 293The origin of the two TO values for certain liquids is now clear. In the non-Arrhenius region at temperatures above the intersection temperature, molecules arerotating freely on the definition given above.Hence the TO which governs viscosityin this region is that temperature at which translational motion would cease althoughmolecular rotation about one axis would still be possible. The lower temperatureTo is that temperature at which both molecular translation and rotation are impossible.It follows that the apparent increase in expansion free volume on cooling through theintersection temperature corresponds to the difference between the rotational envelopeof a molecule and its van der Waals volume.This argument requires that only one change in viscosity behaviour can occur inthe non-Arrhenius region.Hence, if we define the glass transition temperature Te ofa liquid as occurring when the viscosity attains 1013 poise, we would expect that ob-served T, values would correspond to values predicted for the lower temperatureregion of the liquid by eqn. (3). Experimentally this is so for the twelve liquids forwhich sufficient viscosity data for the two non-Arrhenius regions are available.7We conclude that at high temperatures in the liquid range where molecules areable to rotate several times about at least two axes during the time between transla-tional jumps, the viscosity of a liquid may be described by the Arrhenius eqn.(I).At lower temperatures molecular rotation is possible about only one axis on this timescale, and the viscosity is governed by the TO eqn. (3). If the liquid can be cooled tosufficiently low temperatures, the remaining rotational mode becomes restricted, anda change in the non-Arrhenius viscosity behaviour is observed.MOLECULAR ROTATION AND RELAXATION I N LIQUIDSIf the times required for a liquid molecule to translate and rotate are different,it would be expected that the dielectric (ZD) and shear (7s) relaxation times wouldalso be different. In the non-Arrhenius region rotation about the short molecula220 INFLUENCE OF MOLECULAR ROTATIONaxes occurs less frequently than does molecular translation. Since dielectric relaxa-tion generally requires molecular reorientation about a short axis, ZD should be longerthan zs in the non-Arrhenius region.This has been observed by Litovitzs in anumber of liquids. In the Arrhenius region, however, several molecular rotationsoccur per translational jump and hence ZD should be less than ZS. Experimentaldifficulties preclude the measurement of 7s in the low viscosity Arrhenius region, but itis significant that the ratio ZD/ZS in the non-Arrhenius region of n-propanol decreasesas the temperature is increased.9This concept of restricted molecular rotation in liquids also offers a possibleexplanation of the distribution of shear relaxation times observed in liquids.10 Inorder to separate the true relaxation region from the non-relaxing contribution causedby the dependence of G, on temperature, it is experimentally necessary to work atlow temperatures where molecular rotation about all three axes is usually restricted.The translational motion of a molecule is the vector sum of three mutually perpen-dicular translational motions, one along the long axis of the molecule which occursrelatively easily and two at right angles to this. One can envisage the translationalmotion perpendicular to the long axis of the molecule occurring either directly as atranslational jump or as a molecular rotation about a short axis followed by a transla-tion along the long axis of the molecule.Either type of motion has a relaxation timelonger than that for translation along the long axis of the molecule : in the formercase three distinct processes contribute to the distribution of shear relaxation times,and in the latter case five.The observed 10 distribution may be precisely described bysumming five single relaxation times. It may also be described to within the experi-mental error by a summation of three single relaxation times in the ratio 480 : 35 : 1.The ratio of the longest to the shortest relaxation time is comparable to the observedZD/ZS at low temperatures in n-propanol.9As the temperature increases and the various molecular rotations occur morerapidly than translation, the molecule is more readily able to align itself into a favour-able position for translation to occur. In the Arrhenius region a molecule rotatesmany times about at least two molecular axes during the time between jumps, and soit can translate equally readily in any direction. Experimentally, a single relaxationtime is adequate to describe the shear relaxation in the Arrhenius region.11MOLECULAR ROTATION AND THE HEAT CAPACITY OF LIQUIDSIn order to examine further this suggestion that molecular rotation is of importancein determining the properties of liquids, some studies have been made of the tempera-ture dependence of the heat capacity of liquids at constant volume, Cv.Cv of aliquid contains contributions from four sources : 12 (i) motion of the centre of gravityof the molecule, Ctr ; (ii) molecular rotation, Grot ; (iii) internal molecular vibrations,Cvib ; and (iv) changes in the structure of the liquid, C8t.Ctr can lie between the idealgas value of 3R/2 and 3R, the latter corresponding to the motion of a classical three-dimensional harmonic oscillator with potential and kinetic energy. Likewise Grotof a nonlinear molecule can be between 3R/2 corresponding to free rotation and 3Rcorresponding to fully restricted libration about the three molecular axes. Cvib isassumed to have approximately the same value in the liquid as in the ideal gas. Theorigin of Cst is obscure in liquids where molecular association (e.g., hydrogen bonding)is absent.The following discussion refers to Cv at atmospheric pressure since liquid vis-cosities are usually measured at this pressure. Few data are available for CV ofliquids over a large range of temperature and accordingly CV has been calculated 1D . B.DAVIES AND A . J . MATHESON 221from known values of the heat capacity at constant pressure, the expansion coefficient,and the velocity of sound. In order to investigate the effect of molecular interactionsin the liquid on CV, CV of the ideal gas has been subtracted from CV of the liquid togive the residual CV. Figs. 1 and 2 show the temperature dependence of the residualCV for some Arrhenius and non-Arrhenius substances, From fig. 1, those substances01 I 1 I 4 I 1 I0 50 Kx) ho v r j o rn 3wT"KFIG. 1 .-Variation with temperature of the residual heat capacity of liquids with Arrhenius viscositybehaviour: -- , maximum contribution from fully restricted rotation and translation : 1, 0 2 ;2, NZ ; 3, A ; 4, CH4 ; 5, CCl4 ; 6, neopentane ; 7, cyclohexane.01 f0 200 250 300 350T"KFIG.2.-Variation with temperature of the residual heat capacity of liquids with non-Arrheniusviscosity behaviour : - - , maximum contribution from fully restricted rotation and translation :-, Arrhenius temperature : 6, n-hexane ; 7, n-heptane ; 8, n-octane ; 9, n-nonane ; 10, n-decane ;12, n-dodecane.which have Arrhenius viscosities throughout the normal liquid range 3 have residualCV values which are less than those corresponding to fully restricted molecular rota-tion and translation.It was argued that, at the Arrhenius temperature, two molecular rotations becomefully restricted while one remains free on the time scale of the translational timebetween jumps ; and also that restricted rotation leads to restricted translation.Hence at the Arrhenius temperature the residual CV should have a value close to 3R222 INFLUENCE OF MOLECULAR ROTATIONFig.2 shows that this is approximately true for several n-alkanes to within the un-certainty of at least +lo" involved in estimating the Arrhenius temperature. Thesame effect is found in other non-associated liquids such as CS2, CHC13, diethyl ether,benzene, toluene, hept-1-ene and oct-1-ene.7~ 14At temperatures below TA there is a substantial structural contribution to CV.Thus we require that there exist in the liquid at low temperatures some type of struc-ture which progressively breaks up with increasing temperature and which disappearsabout the Arrhenius temperature.It has been suggested 15 that this structure arisesfrom the presence of small quasi-crystalline clusters of molecules in the liquid, butthere is little direct evidence about the nature of such clusters.The structural contribution to the heat capacity can result from the restrictionof molecular rotation in the liquid. The conventional melting point of a solidrepresents the onset of translational motion. Although in some substances un-restricted molecular rotation about certain molecular axes may also begin at themelting point, this is by no means universal. Spherical molecules such as CH451exhibit rotational melting in the solid at temperatures below the conventional meltingpoint. Other substances such as benzene and the n-alkanes 16 exhibit rotationalmelting about one molecular axis in the solid, while some quasi-spherical moleculessuch as cyclohexane have two rotational melting points in the solid.16It seems reasonable to state that complete rotational melting has occurred when amolecule is able to rotate about a given molecular axis independently of moleculartranslational motion so that several molecular rotations take place during the timebetween translational jumps.In the non-Arrhenius region complete rotationalmelting has not occurred because rotation about two molecular axes does not takeplace during the time between translational jumps. The energy required for normaltranslational melting or for rotational melting in the solid is usually several 100 cal/mole.In the non-Arrhenius region rotational melting is occurring in the liquid over aconsiderable temperature range, and hence the residual heat capacity of the liquidexceeds 3R. In the Arrhenius region, molecular rotation occurs independently oftranslation, the structural contribution to Cv disappears, and the residual CV fallsbelow 3R. Thus the restriction of molecular rotation in liquids is responsible bothfor non-Arrhenius viscosity behaviour and for the structural contribution to theheat capacity of non-associated liquids.1 A. J. Barlow, J. Lamb and A. J. Matheson, Proc. Roy. SOC. A , 1966,292, 322.2 S. Glasstone, H. Eyring and K. Laidler, Theory of Rate Processes, (McGraw Hill, New York,3 D. B. Davies and A. J. Matheson, Trans. Faraday Soc., 1967, 63, 596.4 W. J. Jones and N. Sheppard, 1962, Report Conf. Hydrocarbun Research Group (Institute of5 C. P. Smyth, J. Physics Chem. Solids, 1961,18,40.6 D. B. Davies and A. J. Matheson, J. Chem. Physics, 1966,45, 1000.7 D. B. Davies and A. J. Matheson, unpublished work.8 T. A. Litovitz and C. M. Davis, 1965, Physical Acoustics, Vol. IIA. ed. W. P. Mason, (Academic9 R. Kono, T. A. Litovitz and G. E. McDuffie, J. Chem. Physics, 1966,45, 1790.10 A. J. Barlow, J. Lamb, A. J. Matheson and J. Richter, 1966, Chem. SOC. Spec. Publ., no. 20,203.11 P. Macedo and T. A. Litovitz, Physics Chem. Glasses, 1965, 6, 69.12 D. Harrison and E. A. Moelwyn-Hughes, Proc. Roy. SOC. A , 1957, 239,230.13 J. S. Rowlinson, Liquids and Liquid Mixtures, (Buttenvorths, London, 1959).14 L. A. K. Staveley, K. R. Hart and W. I. Tupman, Disc. Faroday Suc., 1953,15,130.15 A. R. Ubbelohde, Melting and Crystal Structure (Clarendon, Oxford, 1965).16E. R . Andrew, J. Physics Chem. Solids, 1961, 18, 9.1941).Petroleum, London), 181.Press, New York)

ISSN:0366-9033    

DOI:10.1039/DF9674300216

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

Viscoelastic Relaxation in Supercooled LiquidsBY A. J. BARLOW AND J. LAMBDept. of Electrical Engineering, The University of Glasgow, Glasgow, W.2Received 2nd Jamary, 1967A study has been made of the viscoelastic behaviour of a range of supercooled liquids each ofwhich has a steady-flow viscosity -q fitting the equation,where TO is the reference temperature at which no free-volume is available for molecular translationalmotion. For all the liquids investigated, the limiting shear rigidity modulus G , varies with tempera-ture according to the relationship,lnq = A'+B'/(T-To),- = - + 5 (T-To),G , Go GIwhere c10 is the coefficient of thermal expansion and Go the shear modulus at temperature TO, and G1is a shear modulus associated with the weakening of the intermolecular force field due to an increasein free-volume with increasing temperature.The complete linear viscoelastic behaviour can becalculated from a knowledge of the dependence upon temperature of the density, steady-flow viscosityand limiting shear rigidity modulus. Details are given of the new liquid model on which thesepredictions are based, together with ample experimental confirmation.Viscoelastic relaxation in liquids can be observed by using alternating shear stressin which the frequency is an experimental variable. At sufficiently low frequencies ofthe applied shear, all liquids behave as Newtonian, the response being entirely viscouswith a phase difference of 90" between the sinusoidal oscillations of shear stress andthe resulting strain.At sufficiently high frequencies it is possible to achieve conditionssuch that the molecules of the liquid are sensibly unable to flow during the short timeperiod of the rapid stress alternations. Stress and strain are then in phase and theliquid behaves essentially as an elastic solid with a shear rigidity modulus G,, whichis of the order 1010 dyne/cm2. At intermediate frequencies the phase angle betweenstress and strain changes progressively with increasing frequency from 90" at lowfrequencies to zero at high frequencies and this constitutes the viscoelastic relaxationregion.In practice, by use of the principle of time-temperature superposition, it is possibleto employ temperature as an equivalent and supplementary variable to frequency,thereby enabling measurements to be made over the complete region of viscoelasticrelaxation. This implies that the variation with temperature of the limiting para-meters q, the steady-flow viscosity, and G,, the shear rigidity modulus, must be deter-mined over the temperature range employed.Although the frequency range whichcan be covered by the experimental systems developed in this laboratory extends fromapproximately 104 to 109 c/sec 1-5 better accuracy is obtained for the techniquesoperating above 106c/sec provided that the liquid has a steady-flow viscosity qexceeding 10 cpoise. For this reason it is generally necessary to work with liquidswhich can be appreciably supercooled.The Arrhenius equation fails to describe the relatively steep temperature depen-dence of the viscosity of most liquids having viscosities above about 0.1 poise and is22224 VISCOELASTIC RELAXATION I N SUPERCOOLED LIQUIDScompletely inapplicable to supercooled liquids.The viscosities of such liquids canbe described by the Doolittle free-volume equation 6 :where A and B are constants, Vf is the free-volume in the liquid and VO is the occupiedvolume. For the supercooled liquids considered here, the density is a linear functionof temperature and under this condition, eqn. (1) can be modified to givean equation identical with one proposed empirically by Tammann and Hesse 7 andsubsequently established on theoretical grounds.899The temperature TO, which can be calculated from viscosity data for the liquid overthe non-Arrhenius region, is considered as the fundamental reference temperature forall transport and relaxation processes and, at this temperature, no free-volume wouldbe available for molecular translational motion in the equilibrium liquid.However,in practice, the slowness of molecular motion below the glass transition temperatureT’,. precludes the attainment of an equilibrium liquid at TO( < T,) in an experiment offinite duration. For present purposes, T, is evaluated as that temperature at whichthe steady-flow viscosity reaches 1013 poise, this temperature being close to values ofTg recorded by differential thermal analysis.10Complete information on the details of the work reported here, is given in twoforthcoming papers.119 12 The purpose of the present paper is to review the presentstate of knowledge in this field and to outline the information which can be obtainedfrom measurements of viscoelastic relaxation.In q = A + B( V,/ V,), (1)In q = A’+ B’/(T-T& (2)VISCOELASTIC PARAMETERSIt is convenient to describe the Maxwell model for viscoelastic behaviour althoughsubsequently a new model is presented which supercedes previous analyses based onthe assumption of a summation of Maxwell elements.Consider a simple shear in theX- Y plane with transverse displacement u, caused by the shear stress T, and a shearstrain S = du/dz. In order to account for viscoelastic behaviour, Maxwell postulatedthe stress-strain relationship,where z is the viscolastic relaxation time.T+zaT/dt = qdS/dt, (3)For alternating shear stress of the form exp ( j u t ) we replace aJat byjw and obtainAt low frequencies where oz g 1 , T = juqS, and the response is entirely viscous. Athigh frequencies such that uz $1, T = (q/z)S, and the response is purely elastic.Theelastic modulus G , is therefore equal to (y/z), or q = z . G,.T(l + j o z ) = juqS. (4)Defining the ratio T/S as the complex shear modulus, G* = G’+jG”, we obtainG’ = G,co2z2/(1 +w2r2); G = G,uz/(l +w2z2), ( 5 )(6)and the dynamic viscosity,q’ = G/CO = q/(l +u2x2).In practice the quantities G’, G“ or q’ are not measured directly but are obtainedfrom the components RL and XL of the shear mechanical impedance ZJ, which areexperimentally determined. This impedance is defined by ZL = RL +~XL = - T’gA .J . BARLOW AND J . LAMB 225and solution of the wave equation, which is given by equating the net force on avolume element to the product of mass and acceleration, gives the general relationship :HenceFor a Newtonian liquid, RI, = XL = (nfqp)* ; G’ = 0 and q‘ = q, the steady flowviscosity.The predicted behaviour obtained from the Maxwell model for a single relaxationtime, z = q/G,, is shown in fig. 1 ; the viscosity ratio q’/q falls from 0.9 to 0.1 inZ t = pG*. (7)(8) pG’ = Rl-XL and pG” = 2R,X,.1.0-0 . 5 -10log ( q / G , )FIG. l.-Calculated behaviour of the viscoelastic relaxation parameters according to the Maxwellmodel with a single relaxation time, T = q/G,.approximately one decade of frequency.In order to represent experimental resultsin the normalized form shown for the Maxwell model in fig. 1 it is necessary to knowthe variation with temperature of the density p, the steady-flow viscosity q, and theshear rigidity modulus G,. Both p and q can be measured by conventional methodsbut the determination of the temperature dependence of G, requires shear-wavemeasurements to be made at temperatures above Ts in the region where (cq/G,) ismuch greater than unity.EXPERIMENTALThe liquids were obtained from various suppliers and were samples of high purity. Allsamples were dried before use in a desiccator containing P2O5. Densities were measuredto an accuracy of f0.1 % by the standard procedure of weighing a calibrated flask ofknown volume.Viscosities were measured with calibrated suspended-level viscometersaccording to B.S.S. no. 188. Values obtained are estimated to be accurate to f0-5 %226below 100 poise and to f l % above 100 poise. Measurements of the shear mechanicalimpedance were carried out by using techniques described previously.l-6$11 The estimatedaccuracy of measurement for RL and XL is &500 dyne sec/cm3, these being the c.g.s.units in which the shear mechanical impedance is expressed.VISCOELASTIC RELAXATION I N SUPERCOOLED LIQUIDSRESULTSDEPENDENCE O N TEMPERATURE OF SHEAR RIGIDITY MODULUS, G,The shear rigidity modulus G, can only be studied by making measurements inthe region where the liquid is exhibiting purely elastic behaviour, i.e., at relativelyhigh frequencies and in the temperature range between the glass transition temperatureand that at which viscoelastic relaxation becomes significant. In this region thereactive component XL of the shear mechanical impedance tends to zero and hencefrom eqn.(8), G, -Rz/p.T"CFIG. 2.-Measwed values of p/R: against temperature in the elastic region, showing the linearvariation of 1/G, with temperature ; f = 30 Mc/sec. A, sec-butyl benzene ; +, tetra-(2-ethylhexyl) silicate ; v , squalane ; [XI , 6,6,11 , 1 1-tetramethyl hexadecane : 0 , squalene ; 0, tri-chloroethylMeasurements made under these conditions on a wide range of different liquidshave all been found to give results which within experimental accuracy obey therelations hip,(9)Values of To are obtained from viscosity data, a0 is the thermal expansion coefficientand Go the modulus of the close-packed state at temperature TO and GI is a constantmodulus for a given liquid.(10)and with the valid approximation that Go 9 GI, eqn.(9) can be written as(1 1-1The modulus GI is therefore associated with the weakening of the intermolecularforce field due to an increase of free-volume. Typical plots of p/RZ against T areshown in fig. 2 : values of G1 are obtained from the slope of the straight line and Gois the intercept at T = To. Behaviour represented by eqn. (9) has been confirmed forover 20 liquids measured.phosphate ; 0, tri (m-tolyl) phosphate. f = 450 Mc/sec ; A , sec-butyl benzene.(1IGaJ = ( W O ) + (ao/WT- To).Since the density can be expressed asP = POL1 - ao(T- Toll,VIG, = (VOIG,) + (VflGl)A . J .BARLOW AND J . LAMB 221RELAXATIONAL BEHAVIOURHaving determined the dependence of G , upon temperature according to eqn. (9)-extrapolated values for G, can be obtained for somewhat higher temperatures cover,ing the relaxation region. It is then possible to normalize the measured values of RLlog10 (q/G,)FIG. 3.-Normalized plots of R&G,)) and of X&G,)) against loglo (oq/G,) for squalene (a), andfor 6,6,11,11 -tetramethyl hexadecane (m). The full curves are calculated from the liquid modelusing eqn. (14) and (15). The corresponding predictions from the Maxwell model are shown by thedashed curves.FIG. 4.-Normalized plots of R~l(pG,)t and XL/(~G,)+ against loglo (o-q/G,). The curves are cal-culated according to the liquid model from eqn.(14) and (15); v, squalane; 0, trichloroethylphosphate ; 0, tri(m-tolyl) phosphate ; x , tris(2-ethyl hexyl) phosphate ; + , tetra (2-ethyl hexyl)silicate ; A, bis (m-phenoxy phenoxy) phenyl ether ; 0, di(isobuty1) phthalate ; 0, di (n-butyl)phthalate ; m, iso-propyl benzene ; 0, n-propyl benzene ; A, sec-butyl benzene228 VISCOELASTIC RELAXATION I N SUPERCOOLED LIQUIDSand XL by dividing by (pG,)* and plotting the quantities R-&G,)+ and X~l(pG,)tagainst (wq/G,). Experimental results for squalene and for 6,6,11,ll-tetramethylhexadecane," represented in this manner, are shown in fig. 3 and compared with thecorresponding behaviour according to the Maxwell model. Clearly the observedrelaxation extends over a wider range of frequency than does that given by theMaxwell model and moreover, a single curve for RL/(~G,)) can be drawn through theexperimental results for the two liquids.The same curve has been found to fit theresults for a number of other liquids 11, 12 (fig. 4), the curves drawn on fig. 3 and 4being calculated from the liquid model, which will now be described.VISCOELASTIC RELAXATION MODEL POR LIQUIDSThe results given in fig. (3) and (4) show that the viscoelastic properties of theseliquids can be represented by two standard curves, one for RL/(~G,)* and one forXL/(~G,)+ against loglo(oq/G,). The curves drawn on these figures have beencalculated from a new model which describes the viscoelastic behaviour of liquidswhich have a viscosity-temperature variation according to eqn.(1).At sufficiently low frequencies where elastic effects are negligible, all liquids behaveas Newtonian fluids with a shear mechanical impedance,l3At sufficiently high frequencies, the liquid behaves as an elastic solid with a shearrigidity modulus G,, and the corresponding shear mechanical impedance for anassumed loss-free system is given byZN = R N + j X N = (1 +j)(n fqp)'. (12)Zs = Rs = (pG,)'+. (13)It is found that the behaviour of the liquids studied is represented by a parallelcombination of ZN and ZS, leading to the following expressions for the shear mechan-ical impedance of the liquid l/ZL = ( 1 / 2 ~ + I/&) from which are derived correspond-ing relationships for the components for the shear modulus G* and of the complexcompliance J*( = l/G*).Z, = RL + j X ,G,J' = 1 +(wq/2G,)-*,G, J" = (G,loq) + (oq/2G,)-*.* This compound was prepared by R.M. Schilsa, Monsanto Chemical Co., St. Louis, U.S.AA . J . BARLOW AND 3 . LAMB 229The curves of fig. 3 and 4 have been plotted by calculation from eqn. (14) and (15) ;there is no disposable parameter.DISCUSSIONThe work reviewed here establishes that the viscoelastic relaxation curves for awide range of pure liquids are identical when represented in terms of normalizedco-ordinates. The standard curves can be predicted from a simple model involvingonly density, viscosity and the limiting shear modulus of the liquid without anydisposable parameter.Many different types of liquid have been studied, includingsimple benzene derivatives, phosphate silicate and phthalate esters, polyphenylethers, and relatively long-chain hydrocarbons. Moreover, this general pattern ofviscoelastic relaxation is not confined to liquids which are normally regarded as super-cooled. The same behaviour has been found for a poly-l-butene liquid 14 of lowmolecular weight, this being monodisperse material with eight repeat units permolecule.It follows that viscoelastic relaxation is governed only by physical variables anddoes not depend directly on the type of molecule involved. All of the liquids whichhave been found to conform to the proposed model have a viscosity-temperaturedependence given by eqn. (1).It is therefore reasonable to suppose that this form offree-volume equation for viscosity must hold for the model to apply.At low frequencies (coq/G, 4 1) the variation with frequency of the componentsof the complex shear modulus according to the Maxwell model are given from eqn.(5) as G’/G, cc o2 and G”/G, cc o. However, according to the liquid model describedhere the corresponding variations given by eqn. (16) and (17) are G’/G,ccco~ andG”/G, cc m. Hence measurements of the frequency-dependence of G‘/G, in thisregion of the spectrum indicate which model is applicable.It has been suggested that liquids, which have a viscosity given by the Arrheniusequationshould exhibit viscoelastic properties which conform to the simple Maxwell model.Unfortunately most such liquids have values of viscosity which are too low for theirviscoelastic relaxation to be studied by existing experimental techniques.However,certain molten compounds are exceptional in that they have viscosities described byeqn. (20) with values exceeding 100 poise. Two such liquids are molten zinc chlorideand molten boron trioxide, and in each of these a single relaxation process has beenfound, the measured behaviour conforming to the Maxwell m0del.15~ 16Since the liquids reviewed here have viscosities which at higher temperatures aredescribed by the Arrhenius equation, it follows that if this approach is of generalvalidity then a single relaxation time would be expected if the viscoelastic relaxationcould be studied in this region.With decreasing temperature a gradual transitionfrom the Maxwell model to that described by the present liquid model would beexpected to occur as the degree of co-operative molecular motion increases. Evidencefor such a transition may be obtainable from measurements of the relaxation regionmade at high frequency atnd at temperatures close to the lower limit of Arrheniusbe haviou r .Recent measurements on mixtures of two liquids, each component of whichconforms to the model for supercooled liquids have shown that a small departure fromliquid purity, amounting to only a few percent, can give rise to significant deviationsof the values of &/(pG,)+ and XL/(~G,)+ from the standard curves over the visco-elastic relaxation region. However, large amounts of impurity do not give grossIn q = A+B/T, (20230 VISCOELASTIC RELAXATION IN SUPERCOOLED LIQUIDSdeviations, and preliminary evidence indicates that for certain critical mixture ratiosthe relaxational behaviour reverts to the standard curves.This suggests that forsuch compositions the mixture behaves as a homogeneous liquid as far as the pro-pagation of shear waves is concerned.This research has been supported in part by a contract with the National Engineer-ing Laboratory, Ministry of Technology. Assistance has also been given by ShellResearch Ltd. and by Imperial Chemical Industries Ltd. Many helpful discussionshave been held with Dr. G. Harrison and Dr. A. J. Matheson, and thanks are due toMiss A. Erginsav for her diligent experimental work.1 A. J. Barlow and J. Lamb, Proc. Roy. SOC. A, 1959,253,52.2 A. J. Barlow, G. Harrison, J. Richter, H. Seguin and J. Lamb, Lab. Pract., 1961, 10, 786.3 A. J. Barlow, G. Harrison and J. Lamb, Proc. Roy. SOC. A, 1964, 282, 228.4 J. Lamb and J. Richter, Proc. Roy. SOC. A, 1966, 293,479.5 J. Lamb and J. Richter, J. Acoust. SOC. Amer., 1967, to be published.6A. J. Barlow, J. Lamb and A. J. Matheson, Proc. Roy. SOC. A, 1966, 292, 322.7 G. Tammann and W. Hesse, 2. anorg. Chem., 1926,156,245.8 M. H. Cohen and D. Turnbull, J. Chem. Physics, 1959,31,1164.9 S . F. Kumar, Physics and Chem. of Glasses, 1963, 4, 106.10 M. R. Carpenter, D. B. Davies and A. J. Matheson, J. Chem. Physics, 1967, to be published.11 A. J. Barlow, J. Lamb, A. J. Matheson, P. R. K. L. Padmini and J. Richter, Proc. Roy. SOC. A,12 A. J. Barlow, A. Erginsav and J. Lamb, Proc. Roy. SOC. A , 1967,298, 481.13 W. P. Mason, Piezoelectric Crystals and their Applications to Ultrasonics, chap. 14 (Van Nostrand,14 A. J. Barlow, R. A. Dickie and J. Lamb, Viscoelastic Relaxation in Poly-I-Butene Liquids of Low15 G. J. Gruber and T. A. Litovitz, J. Chem. Physics, 1964, 40, 13.16 P. Macedo and T. A. Litovitz, Physics and Chem. of Classes, 1965, 6, 69.1967,298,467.Princeton, 1950).Molecular Weight, 1967, Proc. Roy. SOC. A, 1967, in press

ISSN:0366-9033    

DOI:10.1039/DF9674300223

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

Euclidean Geometry and the Flow of Generalized LiquidsBY F. W. SMITHDivision of Mechanical Engineering, National Research Council, OttawaReceived 14th November, 1966This paper is a review of recent research on liquid viscosity, derived from structural studies basedon aggregates of spheres in space, and from non-linear continuum mechanics. The use of Voronoipolyhedra and the Delaunay graph to characterize irregular aggregates is briefly described, and thetheory due to Bernal of a liquid as an aggregate of spheres is reviewed. The Bernal polyhedra, andan elementary rate theory where the activation volume is essentially a second rank tensor, giveexpressions for viscosity and for stress in lineal flow that are compatible with the non-linear theoryof stress of Coleman and Noll.1.AGGREGATES-VORONOI AND DELAUNAY STRUCTURESIn this section are reviewed some concepts in the geometry of irregular aggregatesin euclidean space. These concepts underlie, or are derived from, the work of Bernaldescribed in 5 2.The instantaneous structure of a simple liquid or other irregular aggregate can berepresented as an array of points in space. The most familiar way of conciselycharacterizing such an array is by means of its radial distribution function of density ;we deal here with a different characterization, that based on the Voronoi construc-tion.192 If lines are constructed between each point and all other points in its generalvicinity, the planes which perpendicularly bisect these lines cause each point to beenclosed in a polyhedron, the Voronoi polyhedron of the point.The dimensions andtopology of these polyhedra give a characterization of the aggregate.Each face of a polyhedron serves to relate two points as “geometric neighbours” ofeach other. The set of all lines between neighbouring points forms a network inspace. For irregular aggregates (or of regular aggregates whose points have beenslightly translated in a suitable fashion) this network of lines is the edges of tetrahedrawhich pack to fill space, and may be termed the Delaunay simplices, Delaunay graph,or simplicial graph of the aggregate.3-5 The average number of lines meeting eachpoint may be termed the “primitive coordination number” of the aggregate. Thisgraph is a purely geometric structure, whose lines in a physical case do not necessarilyimply a physical interaction between the particles connected by the line; for thediamond lattice, for example,s the primitive coordination number is 20.It is con-venient, therefore, to define a “structural graph’’ of a physical aggregate as a subsetof the simplicial graph such that its lines are the vectors of a significant interaction.The structural graph of the diamond crystal is thus the familiar diagram which connectseach atom to four neighbours.2. BERNAL LIQUIDBernal has described and reviewed experiments and calculations on irregularaggregate having varying degrees of geometric resemblance to monatomic liquids.6-13This work elucidates many structural features of liquids, such as radial distribution,23232 FLOW OF GENERALIZED LIQUIDScoordination, nucleation, and crystallization ; we mention here only certain aspectswith a bearing on the theory of viscosity.These experiments have included a study of the polyhedra obtained when anaggregate of plastic spheres is compressed into a compact mass.’ These shapes givean impression of some features of the Voronoi polyhedra of a monatomic liquid, suchas a prevalence of 5-sided faces, and indicate a primitive coordination number ofabout 14.Further observations on masses of steel balls and models of connectedrods led essentially to a description of the structural graph of a prototype liquid ofspherical molecules, the Bernal liquid, whose principal features are that its arrange-ment is irregular and that no cavities occur that are large enough to contain anadditional molecule.This structural graph connects all “near neighbours”-pairs of molecules whichare separated by distances ranging up to about 15 % above the minimum separation.The lines of this graph were shown to form the edges of a space-filling by polyhedraof only five types.These Vernal polyhedra” comprise the tetrahedron, the octa-hedron, and 10, 14 and 16-faced figures, all faces of which are triangles. This seriesof polyhedra encloses cavities of progressively greater size, surrounded by 4, 6, 8, 9and 10 molecules respectively. The dynamic behaviour of these cavities will bediscussed in the following section in connection with the theory of liquid viscosity.3.FLOW OF A BERNAL LIQUIDThe structural graph of a Bernal liquid which is not flowing but whose moleculeshave kinetic motion is a dynamic network of polyhedra which are continuouslyinterchanging their identities, but whose overall population statistics remain constant.For example, an octahedron may transform into four tetrahedra by the approach oftwo molecules along a diagonal, while compensating changes occur elsewhere in theliquid.This concept has been used to develop a theory of viscosity, along the followingroute.4 First, it is assumed that each step in which a polyhedron changes to thenext polyhedron in order of size can be treated as an activation process by the tech-niques of chemical rate theory,l4 and that an “activation volume” is involved whichis the difference in volume of the cavities of the two polyhedra.For the Bernalliquid this activation volume is about 10 % of the molar volume. Secondly, theselocal changes in free volume are related to the overall shear of the liquid by using theconcept of “volume production” described in the next section. Thirdly an expressionis derived for viscosity by relating the rates of local events in the liquid to a macro-scopic stress tensor. This stress tensor can be either that given by the classical linearNavier-Stokes theory,4or that given by the non-linear Coleman-No11 theory, mentionedbelow.These procedures yield expressions for a liquid viscosity which decreases withincreasing temperature and with high shear stress, and which increases with pressureand which exhibits the normal stresses discussed in $5.Because of the many assump-tions and simplifications in these theories, their possible field of application seems atpresent to be in inter-relating macroscopic parameters-eg., high stress viscosities interms of low-stress viscosities-rather than in the ab initio calculation of flow pro-perties from molecular properties.4. ACTIVATION VOLUME; THE KINEMATICS OF AGGREGATESThe treatment of liquid viscosity requires an analysis of the way local changes inintermolecular cavities combine into the macroscopic flow of a liquid which in principlF. W. SMITH 233need possess no cavity large enough to contain an additional molecule. The basicconcept used is that the creation or change in shape of a cavity in an aggregate is aprocess that involves direction as well as magnitude.The process has been termed“volume production” or “volume destruction”,4 and “tensor displacement”.l6 Forexample, in the Bernal liquid, an octahedron may collapse into tetrahedra along anyof three directions, in a process of volume destruction.In a liquid at rest but undergoing thermal kinetic motion, such processes areuniformly distributed in space ; the total volume production and destruction areisotropic. In a flowing liquid, the macroscopic shearing motion causes a perturbationof this distribution, so that the volume production is anisotropic; more cavities arecreated in one direction than another, and more destroyed in the second direction thanthe first.The local flow process must be represented mathematically as a second rank tensorin order to be integrated into the tensor representing the macroscopic flow of theliquid.In current applications of the theory a simplification has been necessary inwhich it is assumed that the local tensor has a single eigenvalue-the activationvolume.4~ 15 For example, the three-directional motion by which an octahedroncollapses is represented by a one-directional volume change aligned in a particulardirection in space.Towards generalization of the Bernal liquid, some discussion has been given of theflow of a “general aggregate”, whose members are arbitrary closed surfaces ineuclidean space.16 Such a structure would be a useful basis for discussing the flowof more complicated molecules such as those of lubricating oils, and would seemnecessary as a basis for more detailed treatments of the intermolecular cavities in theBernal liquid.One likely result in the latter case is that more than one “activationvolume” may be needed to describe a local flow event in any liquid, or to express themacroscopic response of the liquid to deformation.5. MODERN CONTINUUM MECHANICS-THEOREM OF COLEMAN AND NOLLModern non-linear continuum mechanics is a subject that has developed fromextensive analyses of the nature of deformation.17 As the “modern natural philo-sophy” of Truesdell, this subject has implications for wider problems in physicalchemistry.18 The description of a liquid in continuum mechanics is different fromthat given by the space-filling ideas underlying the previous sections, but the twomethods perhaps have in common that they each seek to incorporate completely therelevant properties of the euclidean space in which the liquid exists: generalizedsymmetry properties in continuum mechanics, metric properties in space-filling.For present purposes, a single result from modern continuum mechanics will beused.The classical, linear, or Navier-Stokes theory of the stress accompanying asteady lineal flow (simple shear) of a liquid is that a shear stress z acts across the planesof shear, in combination with an isotropic pressure p . The modern non-linear theoryof Coleman and No11 regards this pressure as essentially non-isotropic, or as anisotropic pressure p combined with two additional components of pressure-the“normal stresses”.l7~ 19 The latter stresses decrease with the rate of shear in such away that the Navier-Stokes theory is obeyed in sufficiently slow motions. Thesenormal stresses can be readily manifested as the Weissenberg effect in certain poly-meric liquids ; in liquids of simple molecular structure it may be difficult or impossibleto reach rates of shear at which normal stresses can be observed, but they retaintheoretical significance9The rate theory of local activation-dependent volume production processesdescribed in 3 3 and 4 gives rise to a non-linear stress system whose normal stresse234 FLOW OF GENERALIZED LIQUIDSdecrease with decreasing rate of shear in a manner functionally consistent with thetheory of Coleman and Noll.15 Essentially, the flow of a liquid is treated as a pertur-bation of its kinetic motion.If this perturbation is small, i.e., a " sufficiently slow "motion, it produces a linear response in the form of a Navier-Stokes stress field with aNewtonian viscosity ; a larger perturbation produces a non-linear response in the formof a Coleman-No11 stress field with a non-Newtonian viscosity.1 G. F. Voronoi, J. Reine Angew. Math., 1908, 134, 198.2 H. S. M. Coxeter, An Introduction to Geometry, (Wiley, New York, 1961).3 B. N. Delaunay (€3. N. Delone). Bull Acad. des Sci. URSS, Classe Sci. Math. Naturelles, 1933,4 F. W. Smith, Can. J. Physics, 1964, 42, 304.5 F. W. Smith, Can. J. Physics, 1965, 43, 2052.6 J.D. Bernal, Trans. F'raday Soc., 1937,33, 27.7 J. D. Bernal, Nature, 1959, 183, 141.8 J. D. Bernal, Nature, 1960, 185, 68.9 J. D. Bernal and J. Mason, Nature, 1960, 188, 910.10 J. D. Bernal, Sci. Amer., 1960, 203, 125.11 J. D. Bernal, J. Mason, and K. R. Knight, Nature, 1962, 194, 957.12 J. D. Bernal, Proc. Roy. Soc. A , 1964, 280, 299.13 J. D. Bernal in Liquids, Structure, Properties, SolidInteractions (Ed. T . J. Hughel), (Elsevier,14 S. Glasstone, K. J. Laidler and H. Eyring, The Theory of Rate Processes, (McGraw-Hill, New15 F. W. Smith, Cun. Congr. Appl. Mechanics (Quebec, 1967).16 F. W. Smith, in Modern Developments in Mechanics of Continuzrrn. (ed. S . Eskinazi), (Academic17 C. A. Truesdell and W. Noll, in Encyclopedia of Physics (ed. S. Fliigge), (Springer, Berlin-18 C. A. Truesdell, Six Lectures on Modern Natural Philosophy. (Springer, Berlin-Heidelberg, 1966).19B. D. Coleman, H. Markovitz, and W. Noll, Viscometric FZows of non-Newtonian FZuids.20 C. A. Truesdell, in Second Order Efects in Elasticity, Plasticity, and Fluid Dynamics. (ed. M.6, 793.Amsterdam, 1965.)York, 1941).Press, New York, 1967).Heidelberg, 1965, 3, part 3).(Springer, Berlin-Heidelberg, 1966).Reiner and D. Abir). (MacMillan, New York, 1964)

ISSN:0366-9033    

DOI:10.1039/DF9674300231

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

GENERAL DISCUSSIONDr. Manse1 Davies (Aberystwyth) said: Prof. Pople has reviewed the variousof nuclear spin relaxations in n.m.r. spectra and mentioned also molecular dipolerelaxations. It seems appropriate to offer a reminder of the large body of evidence onmolecular behaviour in liquids provided by dielectric relaxation methods.Three factors have to be distinguished: (i) an inertial effect which persists forthe very short time ( x 10-1 sec) needed for a molecule to achieve constant angularmomentum under the joint influence of an applied torque and viscous damping.It was neglected in Debye’s treatment (1913) and has the effect of pulling the dielectricloss factor E” down to zero at a frequency depending on [1/2kTz2].l As this occurswhere the loss is already very low andthe frequency high ( V ~ 3 0 cm-l) it hasonly recently been directly observed.(ii)The angular reorientation of the (rigid)dipole. For this, Debye introduced thesimple exponential factor exp (- t/z) withan unique time constant z. The singlevalued z-function accurately representsthe behaviour of a considerable numberof polar liquids and of polar solutes inmany different solvents. In the complexcompliance (dielectric) plane the simpleDebye relation leads to a semicircular arc(fig. la). Increasing the inhomogeneityof the local field around a molecule (i.e.,at lower temperatures ; at higher viscosi-ties ; with increased molecular anisotropy ;etc.) an apparent range of relaxationtimes whose incidence is weighed by adistribution function G(z) appears.Thisleads to a “ depressed circular arc ” andthe corresponding distribution functionscan be deduced in alternative forms+ &“FIG. 1.(Cole-Cole ; Fuoss-Kirkwood ; etc.). Further departures are found in some liquidsoften (not always) of complex structural character (di- and trihydroxylic molecules)or in a supercooled (i.e., glassy) state. These lead to “ skewed-arc” plots in thecomplex plane (c). Their analytical representation and the corresponding G(z) hasbeen given by Davidson and Cole, and alternative functions considered by Higasi.Important models for the co-operative molecular relaxation in liquids have beengiven by Kirkwood, Frohlich and others. Most significant is the development byCole and Glarum of a model where re-orientation is partly a localized molecularrotation promoted by the near presence of a “ defect ” that also diffuses through theliquid.Cole has shown how the treatment based on Kubo’s statistical formalismsuffices to represent many of the types of behaviour encountered in liquids.(iii) Some non-rigid polar molecules show distinctly separate dipole relaxations :usually from the essentially well-separated rates of component (orthogonal) dipoleJ. G. Powles, Trans. Farahy Soc., 1948,44,802.23236 GENERAL DISCUSSIONelements (d). These can lead to the direct evaluation of, e.g., energy barriers restrict-ing rotations of hydroxyl groups within a molecular framework. By transferring suchmolecules to a very " viscous '' medium (e.g., a polystyrene matrix) it becomes possiblegreatly to increase the separation of component motions, i.e., an intermolecularrotation can be fully resolved from the whole-molecule re-orientation.The individualactivation energies of these motions can be evaluated.Dipolar relaxations have been characterized from 10f3 to 10-1 sec. Whilstmost have dispersions broader than that for the Debye single relaxation-time, absorp-tions intimately related to the structure in the liquid state have been found in the10-60 cm-1 region of the infra-red which are of a degenerate resonance form : as such,they are appreciably sharper than the Debye type. The approximate positions ofthese absorptions which are characteristic of the liquid state were predicted by Hillin 1963.2 It is suggested that dielectric absorption results offer a major contributionto the appreciation of molecular behaviour in the liquid ~ t a t e .~Prof. A. Bellemans (University of Brussels, Belgium) said : In relation to Pople'spaper, I would mention that the autocorrelation function of the electric dipole momenthas recently been studied by the method of molecular dynamics by Gancberg, Kohlerand myself. We considered a two-dimensional circular array of about 400 rotatorslocated on the sites of a quadratic lattice and provided with a permanent electricdipole. The equations of motion of this system were integrated numerically on aIBM 7040 computer and therefrom the autocorrelation function was obtained.When the interactions between the rotators are negligible, but the angular momentadistribution nevertheless is Maxwellian, the autocorrelation function is of the formexp (--;'/2), where 7 is a reduced time (with the mean period of rotation as unity).When the rotators interact through strictly dipolar forces only, we observed that theautocorrelation function remains nearly the same as for free rotators, even when theabsolute mean potential energy is about equal to the kinetic one.This shows thatdipolar forces play little part in dielectric relaxation and do not lead to the Debyepicture.On the contrary, simple angular potential other than the dipolar one may bemuch more efficient in this respect and transform the free rotator autocorrelationfunction into a function much more like the exponential decaying function of Debye.This already occurs for rather weak coupling, e.g., an absolute ratio of about 20 %between mean potential and kinetic energies.Frof.H. G. Hertz (Technical University, Karlsruhe) said : Dr. Dwek introducedthe concept of a distribution of correlation times. In addition to the reference toa paper by Waugh given by Dr. Dwek, I would point out that the formulation of adistribution of correlation times should not be used too easily in future n.m.r. work.First, we almost never have a precise knowledge of the time correlation functions ofthe spherical harmonics of second order-or other functions-entering in the theoryof nuclear relaxation. Then, usually one starts from a simple model and the devia-tion of the experimental results from those predicted by this model are ascribed toa distribution of correlation times.However, the intramolecular and intermolecular relaxation rates are generally ofequal magnitude ; they clearly have different dispersion behaviour.The intra-molecular relaxation rate may be caused by anisotropic rotational motion, the latterbeing determined in a compIicated way by two or three microdynamic time constantsG. W. Chantry and H. A. Gebbie, Nature, 1965, 208, 398 ; and later publications.N. E. Hill, Proc. Physic. Soc., 1963, 82, 723.for further references, see C. P. Smyth, Ann. Rev. Physic. Chem., 1966, 17,433GENERAL DISCUSSION 237even if one uses the most simple rotational diffusion model. The intermolecularrelaxationrate depends as well on the special form of translational motion.Theremay be a finite jump-or infinitesimal jump-mechanism, there may be a superpositionof molecular rotation on the translation or partial rotational contribution to theintermolecular rate caused by association effects. These more detailed models arecharacterized by a small number of well-defined microdynamic time constants-withdifferent temperature dependence. All these facts produce a dispersion behaviourwhich generally is not explicitly known. Thus, if one uses the concept of adistributionof correlation times too easily one may veil the information buried in the dispersionbehaviour of the relaxation time in the form of a small number of microdynamictime constants.Moreover, even if one considers the name '' distribution of correlation times "to be another word for the more precise formulation of non-exponential decay of thecorrelation functions, the distribution of correlation times thus determined is notnecessarily the distribution of correlation times in the various environments in thesystem under consideration.The latter correlation times are those one would obtainif the particle is trapped for a sufficiently long time in a given environment. The twodistributions are no longer equal if the residence times of a molecule in the variousenvironments are shorter than the corresponding correlation times. Thus, in pureliquid, a distribution of correlation times should not be ascribed to different environ-ments.Finally, even for a purely translational intermolecular relaxation rate in a highlyfluid liquid we have a " distribution of correlation times " because the correlation timefor a spin-spin vector depends on the length of this vector.Writing z = d2/D in thecase of the most simple diffusion-determined intermolecular rate is only a formal ab-breviation; what determines the relaxation mechanism is the macroscopic self-diffusioncoefficient D and the distance of closest approach d. The relation z = d2/D onlytranscribes the diffusion coefficient into a quantity of dimension time and " micro-scopic '' magnitude.Another comment-which is related to the first one-concerns the separationof the motion of toluene in solutions of free radicals into translational and rotationalmotion of a solvation complex formed by the free radical and the toluene as reportedby Kruger, Muller-Wahrmut and van Steenwinkel. One should be sceptical aboutsuch a procedure if it yields an activation energy of the rotational motion 2+ timesas great as the one for translational motion. The correlation time z, of the rotationalmotion of the complex is roughly given by z,-4nqa3/3kT, a being the radius of thecomplex, thus we would expect to find the activation energy for the rotation to beabout equal to that of the viscosity q which is not very different from the one fortranslational motion.Furthermore, if for acetonitrile we find the correlation time for the rotationalmotion of the CN-bond to be equal to the correlation time of the rotational motionof the proton-proton vectors, then we must conclude that the rotation of the wholemolecule is isotropic.This is so because for an isotropic rotational motion of a rigidbody the correlation time for the rotation of one selected vector is identical with thecorrelation time of any other vector within the rigid body (the molecule). If we hadvery rapid motion about the C-CN bond, then we would find a smaller correlationtime for the proton relaxation. With acetonitrile one other possibility is that theproton relaxation is partly due to spin-rotational interaction. This would meanthat the correlation time as determined for the proton-proton vector considering onlymagnetic dipole-dipole interaction is not correct. This question seems not to besettled experimentally as yet238 GENERAL DISCUSSIONProf.A. D. Buckingham (University of Bristol) (partly communicated): In reply toMagat and Davies, it is true that if a larger value for 8, were used in the approximateequation for the dielectric polarization of a polar liquid (see eqn. (20) of my paper),the corresponding value of g would be smaller. The equation is based on a separationof the " distortion " and " orientation '' polarizations, and for water there may besome ambiguity in this separation. The " distortion " polarization is the meanpolarization induced in the system for fixed molecular orientations, and the conceptdepends on the molecule retaining its integrity when the field is applied; if protonexchange were induced by the field, the resulting polarization could not properly beincluded as distortion polarization. The librational motion of a molecule in inter-action with its neighbours does not contribute to the distortion polarization whicharises from electronic and nuclear motion at optical frequencies.This is the reasonthat, in my table 1, E , for water is only about 1.8. The fact that the resulting valuesof g are in agreement with the Pople model for water lends support to this interpreta-tion of the dielectric polarization of the liquid.The difficulty in assigning a correct value to E , for water has been discussed byHi1l.l She concluded that the infra-red dispersion in water should be attributed todistortion polarization and that the hydrogen bonding raises the dielectric constantby enhancing the polarizability rather than by modifying the dipole distribution.The ambiguity stems from simplifications inherent in the Onsager model in which themolecular dipole in the liquid is related to the permanent moment p and polarizabilitydo) of the molecule :where R is the " reaction " field.The high-frequency dielectric constant enters thetheory through the equationrn = p+a'O'R,Normally a(') is approximately independent of state, but in water it may not be.However, if the interaction is so strong that do) is increased by a factor of about 2.5,it is difficult to believe that the gas value of p is relevant.In reply to Stecki, it is true that the Kerr constant of a dipolar fluid is dependenton the three-particle distribution function.Studies of the relaxation of the Kerrconstant would give information about the time dependence of both P,(cos y12) and($ cos y12 cos 713 - cos 723), and might be a useful source of information about cor-relation times in liquids. It would be interesting to compare the results with studiesof dielectric relaxation (which depends on Pl(cos y1 J) and nuclear magnetic relaxationwhich is also related to P,(cos y12).The three-particle distribution function is also involved in dielectric relaxation inmonatomic fluids, for in this case, an interacting pair of molecules has no dipolemoment, whereas a trio does. However, we have little knowledge of the dependenceof the dipole moment on the positions of the three atoms, and the effect may be largelydetermined by short-range interactions.Dr.G. H. Findenegg (Bristol University) said: I wish to ask Dr. Matheson whatassumptions are made as to hindered rotation around the C-C bonds of the n-alkanemolecules in the calculation of the residual heat capacities given in fig. 2 of theirpaper. In the liquid state these internal librational modes are affected not only bythe intramolecular potential barriers but also by surrounding molecules ; this mayN. E. Hill, Trans. Faraday SOC., 1963, 59, 344GENERAL DISCUSSION 239alternatively be described in terms of a great number of conformational isomers ofdifferent potential energy. By raising the temperature, conformations of low energywill be converted into others of higher energy.The corresponding contribution tothe heat capacity cannot be calculated at present, and this forms a major problem foran interpretation of C, of these compounds. However, it is improbable that chainmolecules can rotate as one unit in the liquid state.Dr. A. J. Matheson (University of Essex) said: In “ simple ” liquids such aschloroform or toluene it is unlikely that the vibrational heat capacity will be signi-ficantly different from that of the substance in the ideal gas phase at the same tempera-ture. In such liquids a structural contribution to the residual heat capacity existsonly below the Arrhenius temperature. We have assumed that in the n-alkanes also,the vibrational heat capacity is little affected by the transition from the gas to theliquid : it is then found that, as in the “simple ” liquids, a structural heat capacityexists only below the Arrhenius temperature.Dr.R. A. Dwek (Oxford University) said: Hertz comments on the use by some auth-ors of a distribution of correlation times to interpret results which cannot be explainedon the Kubo and Tomita theory with a single value of 7,. It is true that there mustbe other models which could equally well be made to fit the experimental results. Allthat can be said is, that in many of the systems concerned, the postulate of a randomdistribution about a mean correlation time is a plausible one on which to base aninterpretation. Some examples are to be found under ref. (3) of our paper.Dr. I. Henderson (University of Southampton and Defence Research Board, Ottawa)(partly communicated). Smith has noted the importance of determining the effectof volume changes on transport properties.Recently developed techniques for high-pressure studies have already given rise to many data relating to isothermal pressurecoefficients and isochoric temperature coefficients, the interpretation of which islikely to lead to further insight into the structural features of liquids. The transportprocess most amenable to precise measurement is that of conductance, and over thelast few years, Hills and his co-workers have studied the variation over a wide rangeof pressure and temperature of the conductance of a number of aqueous and non-aqueous systems.Presented here are some preliminary results of a numerical analysis in progress atSouthampton of these and other data relating to transport processes.The influenceof isothermal volume changes, and in particular of changes of “ free volume ”, onfluidity, conductance and diffusion, has been the subject of several semi-empiricaltheories, one of the most recent being that of Macedo and Litowitz who relatedthe jump probability to the product of a free volume term and a Boltzmannfactor. Following Cohen and Turnbull,2 they expressed the free volume term asexp ([Y- Vg]/Yg), where V is the molar volume and Yg a corresponding limitingvolume characteristic of a glassy state. This relationship has not been found to beappropriate for ionic transport in a wide range of systems in which the free volumedefined as (Y- Yo) (where Yo is the low temperature volume of the solid) representeda large fraction (8-20 %) of the total volume.With solutions for which Walden’srule is a good approximation, namely those for which the solute ions are large inP. B. Macedo and T. A. Litowitz, J. Chem. Physics, 1965,42.245.M. H. Cohen and D. Turnbull, J . Chem. Physics, 1959,31, 1164240 GENERAL DISCUSSIONcomparison with the probable size of solvent voids, a simple relation similar to that ofBatchinskishould be valid. This model assumes that concurrently with ionic displacement thereoccurs a co-operative motion of solvent molecules and that the hydrodynamic pro-perties of the system are therefore involved.1/11 = A(V-VO)/V (1)Og2O t/ 0.001 1 I I I0 5 0 I00&IFIG.1 .-Limiting equivalent conductance of potassium picrate in dimethyl formamide againstIn the present analysis, values of a parameter V' were determined to yield the best(V- V')/V. *, 25°C; 0, 35°C; 0,45"C ; x , 55°C ; 0, 65°C.fit to the isothermal relations.where A is the equivalent conductance and K(T) a density-independent proportionalityfactor. The resultant values of V' were substantially temperature-independent, sothat K(T) could be further expressed aswhere E,, is an isochoric energy of activation. Fig. 1 shows the isothermal data ofBrummer for potassium picrate in dimethyl formamide plotted as a function of(Y-67*34)/V (corresponding to a pressure range of 1000 atm). Fig. 2 shows thesame data computer-plotted against values calculated from the equationSimilar analyses for tetra-methyl and tetra-butyl ammonium picrates in dimethylformamide and for four tetra-alkyl ammonium picrates in nitrobenzene (using thedata of Barreira 3), showed that all of the data for each system could be fitted by asimilar single relationship.When the solute ionic species are small in comparison with the dimensions ofsolvent vacancies, Walden's rule is a poor approximation.Ionic motion can thenA = K(T)(V- V')/V, (2)K(T) = K' exp (- EJRT), (3)A(calc) = 3.33 x lo3[( Y- 67-34)/V] exp (- 1100/RT). (4)A. J. Batchinski, 2. physik. Chem., 1913, 84, 643.* S. B. Brummer, J. Chem. Physics, 1965,42, 1636.F. C. Barreira, D.Z.C. Thesis (Imperial College, London, 1964)GENERAL DISCUSSION 241occur without concurrent rearrangement of solvent molecules, although such solventmotion must take place subsequently.In such a situation, the ionic jump probabilityshould be proportional to the probability that a vacancy exists adjacent to an ionin the plane normal to the direction of the applied field. Thus the overall conductanceequation should contain a factor ([ V - Y'] /V)', correspondingto the pv of MacedoandLit0witz.l have shown thatthis analysis is applicable to a number of ionic solutes in methanol.Preliminary calculations using the data of Howard5 0 I00A0FIG. 2.-Conductance calculated from eqn. (4) against limiting equivalent conductance ofDr. F. W. Smith (N.R.C., Ottawa) said: I would like to raise a general questionof the physical interpretation of volume parameters in liquids, notably activationvolume.These parameters are introduced into equations of thermodynamic type asempirical scalars with the dimensions of volume, e.g., 20 cm3/mole. A molecularliquid is an aggregate of closed surfaces moving in euclidean space, and the questionis " what is the geometric property or function of this aggregate that can be said tohave this value of 20 cm3/mole under the conditions of the experiment?" Theanswer evidently involves integral or statistical geometry, but a crude analogy suggeststhat an activation volume function may resemble a deformation tensor, at least tothe extent of being characterized by more than one invariant or eigenvalue. Differentexperiments on the same liquid may therefore yield different scalar " activationvolumes " which may be different invariants, or different functions of the invariants,of the underlying geometrical function.Dr.B. Cleaver (Southampton University) (communicated): Dr. F. W. Smith hasraised the question of the physical meaning of the activation volume for transportprocesses in liquids. It is wrong to ascribe to experimental activation volumes thesimple meaning which the term imp lie^.^ However, the empirical meaning of thisparameter is unambiguous. The relative importance of the isochoric activationenergy and the activation volume in determining the temperature dependence of atransport process at atmospheric pressure may be distinguished in the following way.The variation of the transport parameter with temperature at atmospheric pressuremay be represented by an Arrhenius equation.Using the self-diffusion coefficientpotassium picrate in dimethyl formamide. *, 25°C ; 0, 35°C ; 0,45"C ; x , 55°C ; 0, 65°C.l P. B. Macedo and T. A. Litowitz, J. Chem. Physics, 1965,42,245.B. Howard, Ph.D. Thesis (University of London, 1963).see for e.g., D. Lazarus and N. H. Nachtrieb, Solids under Pressure (McGraw Hill, 1963), p. 47242 GENERAL DISCUSSIONas an example,D1 atm = Do ~ X P (-EpIRT)*Defining the activation volume asand the isochoric activation energy asthenAV* = - RT[a In D/dpIT,E, = -R[d In D/d(l/T)lV,Ep = E,,+(aT/P)AV*.(This is strictly true in the limit of zero pressure, but is a sufficiently good approxima-tion for pressures up to about 1 kb.)Substituting (2) into ( l ) ,D1 atm = Do exp (- E,/RT) exp (- nAV’ jRT), (3)where n = aT/P, the “ internal pressure ” of the liquid.Similar equations applyfor the fluidity 1 / q , and for the equivalent conductance A for ionic liquids.The table shows data for different types of liquid. On substituting these valuesof E, and nAV* into (3), the relative importance of the two exponential terms changesdramatically as we pass down the table, in a way which one could not have predictedin the absence of experimental data. This reflects real differences in the microscopictransport EP E V nAV *process (kcal/moie) (kcal/mole) (kcal/mole) ref* liquidCC14 fluidity 3.02 1 -22 1.80 2benzene fluidity 3.24 1.12 2.12 2chlorobenzene fluidity 2.83 0.95 1-88 2ethanol fluidity 3.83 2.10 1 -73 2butanol fluidity 6.40 4-25 2-15 2mercury diffusion 1-00 0.96 0.04 3gallium diffusion 1.12 0.98 0.24 3fused LiN03 conductance 3-42 3.31 0.11 ownworkfused CsN03 conductance 3-69 1 -09 2.60 (unpublished)mechanisms of transport, but the exact nature of these is obscure until an adequatetheory of the liquid state is available.In the meantime, I suggest that the transportprocesses in these liquids be described as “ energy restrained ” or “ volume restrained ”according as E, is greater, or less, than zAV*. This distinction is based directly onempirical data, and can be applied to any type of liquid.The terms ‘‘ energy limited ” and ‘‘ volume limited ” have been used by Macedoand Litovitz in a slightly different sense to that proposed here. Their usage isbased on a model for the transport process and on an expression for the size distribu-tion of holes in the liquid, so it breaks down if these are incorrect. The purelyempirical terms suggested above are more appropriate at this stage, when satisfactorytheories are lacking for some types of liquid and experimental data are relativelysparse. There seems little chance that successful theories of transport will be developeduntil experimental investigations have been extended to cover a wider range of liquids.Transport measurements made at atmospheric pressure give no indication that thedistinction drawn above even exists, and there is need for more work on pressuredependence of transport processes.P. B. Macedo and T. A. Litovitz, J. Chem. Physics, 1965,42,245.M. K. Nagarajan and J. O’M. Bockris, J. Physic. Chern., 1966,70, 1854

ISSN:0366-9033    

DOI:10.1039/DF9674300235

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

SUMMARIZING REMARKSBY J. S . ROWLINSONDept. of Chemical Engineering and Chemical Technology,Imperial College, London, S.W.7This meeting has considered the experimental and theoretical methods now avail-able for determining the structure of fluids. In this attempt at summarizing thediscussions, it is perhaps appropriate to recall at each stage the formal notation inwhich our knowledge is codified. For a fluid of structureless spherical molecules,we start with the pair distribution function g2(r). At equilibrium this is a functionof the separation Y of two points in the fluid, and, of course, a function also of densityand temperature. It represents the probability that molecules are found at separationY, and is the minimum information that is of any value to us.This value is consider-able if the configurational energy of the fluid can be expressed as a sum of pairpotentials, for then all the thermodynamic properties can be expressed in terms ofg2(r). If uZ(r) is the pair potential, then we have the well-known results that theconfigurational energy U is the average of u2(r) over g2(r), and that the imperfectpart of the pressure is the average over g2(r) of the intermolecular virial functionu2(r) = du2(r)/d In r.UIN = +n2pm72(r)dry (1)p -nkT = -* n2 s u2(r)g2(r)dr, (2)where N is the number of molecules and n the number density. However, the entropyand free energy cannot generally be written in so simple a way but require a furtherintegration over density or, what is formally equivalent, an integration over a couplingparameter as proposed by Kirkwood.To determine g,(r) we turn usually to one of three theories, that of Kirkwood,Born and Green (KBG), that of Percus and Yevick (PY), or the hyper-netted chain(HNC) theory. The first has the disadvantage that it leads at once to the introductionof the triplet function g3, which has then to be eliminated.The traditional elimination-the superposition approximation of Kirkwood-is now known to be inadequate.Rice and Young have argued that it can be improved by the introduction of a fewterms of the expansion of g3 in powers of the density. Experience with attempts toimprove other approximations by this method suggests that we shall be fortunate ifthis approach is satisfactory at liquid densities.The PY and HNC approximations are more economical in the sense that theyboth require and yield information only about the pair functions.The former inparticular is closely related to the function cz(r), the direct correlation function ofOrnstein and Zernike. The importance of this function was brought out by Rush-brooke in his Spiers Memorial Lecture. I think that one can use it to answer aquestion raised in discussion by Levine, namely, what is the physical significanceof the PY approximation? The answer is that it is the only self-consistent theorywhich takes literally the original idea of Ornstein and Zernike that the direct correlation24244 SUMMARIZING REMARKSfunction c2(r) has a range no greater than that of the intermolecular forces, i.e., therange off2@) = exp [ - u2(r)/k7'l-- 1.This restriction on the range of cz(r) leads inevitably to the PY approximationif one requires also that the virial equation, (2), is to yield a unique value of the pressurefor discontinuous potentials as, e.g., for hard spheres or for a square-well potential.This result may be shown as follows.Define a function y ( r ) that is related to g(r) by the equationg w = [I +f(r)lY(r)* (3)(The subscripts 2 are omitted for simplicity.) Now if u(r) is discontinuous, thenv(r) in (2) is composed of one or more &functions, and if this equation is to yield aunique value of p then it follows that y(r) must be continuous even when u(r), f ( r )and hence g(r) are not.The Ornstein-Zernike equation istotal = direct +indirect correlation function.The indirect part of the total correlation function is clearly a continuous functionof r12, whatever the discontinuities in c(r) and g(r).Substitute for g(r) from (3),and obtainHence if c(r) is to be proportional tof(r), and if the solution of ( 5 ) is to lead alwaysto a smooth function for y(r), then there is only one admissible functional form forc(r), viz.,This is the PY approximation and insertion of (3) and (6) into (4) gives the PY integralequation.The nature of the HNC approximation is, as its name implies, expressible onlyin terms of its graphical expansion. PY is certainly the better approximation at hightemperatures. Henderson's results and Hutchinson's contribution to the Discussionshow that HNC may be the better at low temperatures and in the liquid state, butthe question is still open.Thus, these theories allow us to obtain g(r) from u(r) for fluids at moderate andhigh temperatures.They all yield qualitatively correct liquid-vapour equilibria ifu(r) has a negative portion, but do not apparently describe the solid state. The onlyindication these equations give of the onset of freezing at high densities are eitherfailure to yield a solution at all (KBG) or the existence of solutions in which g(r) hasbecome negative (PY).Machine calculations, such as those of McDonald and Singer and those reportedin the discussion by Verlet, are now giving us a reasonable " experimental " knowledgeof the behaviour of one model fluid of precisely known u(r), viz., that with a Lennard-Jones (12,6)-potential.We have U and p , from eqn. (1) and (2), for the homogeneousfluid at temperatures down to the triple point and for the solid. However, the moredifficult task of obtaining the free energy or chemical potential, which requires thefurther integration mentioned above, has not yet been attempted and is, perhaps,the most urgent task in this field. Until this is done, we have no knowledge of thevapour pressure curve.Pings has shown us how, in principle, we can obtain u(r) from the X-ray diffractionpattern of a moderately dense gas, and his work with Mikolaj adds to the recentevidence, discussed at the General Discussion of the Society on Intermolecular Forces[ 1 +f(r)]y(r) = 1 + c(r) + indirect term.( 5 )c(4 = fW(r). (63. S . ROWLINSON 245in 1965, that the true pair potential of the inert gases is deeper and narrower than thatof a (12,B)-potential. We have, therefore, the paradox that results such as those ofMcDonald and Singer show that a (12,6)-potential represents well the properties ofliquid argon, whilst studies of the gas phase show that the true pair potential is notof this form. The paradox can be resolved onIy by assuming that a triplet potentialu, is acting in the dense state and that the sum of the true u2 and this triplet term canbe treated approximately as an effective pair potential of (1 2,6) shape.We still know little about either u3 or about the triplet distribution function g,.Pople observed that g2 gives only the mean number of molecules in, say, the firstco-ordination shell around a given molecule, but nothing about the angular distribu-tion of molecules in that shell.This knowledge is formally represented in g3 and inhigher functions, and an increase in our understanding of these functions is an urgenttask in equilbrium statistical mechanics. We need to know more about g , in orderto improve KBG, PY and HNC theories in a systematic way, in order to discusscertain properties of non-spherical molecules (Pople, Buckingham), and in order tomake further advances in the theory of mixtures-a field which has scarcely movedfor 10 years.There has been much interest lately in the nature of the singularities in physicalproperties at the critical point, but we have only touched on this problem here.However, Domb and Sykes have shown that in at least one three-dimensional modelthe singularity in C, is not logarithmic as has been widely thought but is a little sharperand varies as (T-Tc)-*.Fixman has discussed the singularities in viscosity andthermal conductivity. Experimentally the latter singularity is found to be readilyobserved whilst the former, if present, is scarceIy detectable. He suggests the differ-ence is not one of functional form but merely of the size of the numerical coefficientsof the singular function (T- Tc)-*.With the work on water we reach a fluid that is so different from argon, etc.,that our methods of interpretation are much more crude. Narten, Danford and Levyhave made perhaps the most thorough study yet reported to the X-ray diffractionpattern and obtained from it what is essentially g(r) for the 0-0 separations.Page,in the discussion, has shown that neutron diffraction gives us primarily the H-Hdistribution function. A third technique, with a different balance between 0-0,0-H and H-H scattering is needed for a complete study.One striking difference between water and argon was pointed out by Egelstaff.In argon the first peak in g ( r ) moves to shorter r as the temperature rises along theorthobaric curve. This peak resembles that in a hard-sphere fluid in that it is a“packing” effect or a manifestation of the repulsive part of the potential. Theattractive forces serve merely to provide the “ internal pressure ” that holds the liquidtogether.Since the repulsive part of u(r) is not infinitely steep, then the effectivesize of the argon atoms decreases as the temperature rises.In water we have the opposite effect. The first peak in g(r) moves to larger ras the temperature rises. Here the maximum of the peak is fixed primarily by thelength of the 0--H . . . 0 hydrogen bond, and it is the weakening of this bond athigh temperatures that is reflected in the change of position of the maximum. Inshort, argon is a liquid in which the structure is determined primarily by the repulsiveforces and water a fluid in which the structure is determined primarily by the attractiveforces. The studies of random close-packing of Bernal and his colleagues tell ussomething of the geometrical nature of the first problem, at least, in the limit of steeppotentials and high pressure.It was evident from the discussion that the infra-red spectrum of water does notlead to an unambiguous description of the hydrogen-bonding as a function of densit246 SUMMARIZING REMARKSand temperature.The words “ weaken ”, “ deform ” and “ break ” were used todescribe the effect on these bonds of raising the temperature, whilst the words “ non-crystalline ”, “ quasi-crystalline ’’ and ‘‘ micro-crystalline ” were used to describe thestructure. However, it is clear also that studies such as those of Luck and of Franckand Roth provide a substantial body of experimental information with which anyproposed structural description of water must conform.Manse1 Davies drewattention to the fact that recent work on the far infra-red spectrum is here of greatpotential value.The last part of this meeting dealt with the time-dependent properties of fluids.Egelstaff described what is perhaps the most significant experimental advance of thelast decade-the use of scattering techniques, and in particular of neutron scattering,to study the evolution in time z of the pair distribution function. The symbolg(r,z) is generally used for this, rather than g(r,z), in order to emphasize that we areinterested in two experimentally separable phenomena. The motion of a pair ofmolecules initially at rl and r, can be described in terms of the diffusion of the firstmolecule from position r1 and in terms of the change with time of the probabilitythat the separation hasany particular value rl-r2.The first motionleads toincoherentscattering of neutrons and the second to coherent scattering. The experimentalknowledge of the structure of 9’(r,z) that we so obtain throws light on those morecomplex space and velocity correlation functions which are related to the transportproperties of a fluid. However, in this discussion, we confined ourselves to the studyof $(r,z) and did not explore this link with non-equilibrium statistical mechanics.The results of Dorner, Plesser and Stiller and those that Cocking reported indiscussion show that there is a surprisingly close resemblance between the dynamicproperties of solids and of fluids at low temperatures.This resemblance must besupposed to weaken as the temperature rises or as the time scale is extended. Themeasurements of Aldred, Eden and White and those quoted in discussion by Larssonare of particular interest to chemists, since they show how these scattering techniquescan now be used to tell us about the relative motions of different parts of complexmolecules.The earlier part of the discussion has shown how valuable a concept is the directcorrelation function at equilibrium c(r). It is, therefore, to be hoped that the effortsmade by Stecki to define a time-dependent direct function will be of equal value innon-equilibrium statistical mechanics. Clearly, such a concept need not be restrictedto a plasma which he discusses in his paper.The use of nuclear magnetic resonance to study molecular motions by means ofthe spin-lattice relaxation time Tl is probably more familiar to chemists than theneutron scattering work. Pople reviewed the different mechanisms that can leadto relaxation and Dwek and Richards discussed applications to particular molecules.It is often the temperature derivative of T,, an activation energy, which is of greaterinterest than its absolute value, since this can be compared with activation energiesfrom other types of translational and rotational relaxation processes. The thermo-dynamic aspects of molecular rotation were discussed by D.B. Davies and Matheson.Finally, from Barlow and Lamb we heard how some order can be brought to thecomplicated field of visco-elastic relaxation.R. 0. Davies pointed out in discussionthat their analysis, which is based on the assumption that the viscous and elasticcomponents of the mechanical impedance are in parallel (Le., their reciprocals are tobe added), is neither more nor less arbitrary than that of Maxwell. It is, however,equivalent in Maxwell’s language to a distribution of times of relaxation rather thanto a single time, and this inverse way of looking at the problem may be of value inother fields where more than one time of relaxation has to be invoked to explain thJ . S. ROWLINSON 247behaviour observed. Unfortunately, no molecular model of the process has yetbeen developed.A natural complement of work of this kind is that on Brillouin scattering of lightby liquids. This gives the speed and the coefficient of absorption of sound at fre-quencies above 1O1O sec-l, whereas the mechanically generated frequencies of Barlowand Lamb are restricted to lo9 sec-l and below. Brillouin scattering was not dis-cussed at this meeting, althou6h it was the subject of a paper that had to be withdrawnbecause of the illness of Prof. Litowitz. In spite of many difficulties of interpretation,it is clear that the last decade has seen the development of several new techniques whichshould soon allow us to describe the time-dependent microscopic properties of fluidswith the same confidence that we now have for the equilibrium properties. It willthen be for the new field of non-equilibrium statistical mechanics of dense systemsto relate this microscopic information on structure to the macroscopic transportproperties

ISSN:0366-9033    

DOI:10.1039/DF9674300243

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

AUTHOR INDEX*Aldred, B. K., 169Barker, J. A., 50Barlow, A. J., 218.Bellemans, A., 236.Beresford, R. H., 76.B e d , J. D., 60.Buckingham, A. D., 199,238.Cagloti, G., 129.Cleaver, B., 241.Cole, G. H. A., 56.Collins, R., 81.Corchia, M., 129.Danford, M. D., 97.Davies, D. B., 216.Davies, M., 128, 184, 235.Domb, C., 85.Dorner, B., 160.Dwek, R. A,, 191,239.Eckstein, B., 148.Eden, R. C., 169.Egelstaff, P. A., 149.Everett, D. H., 82.Findenegg, G. H., 238.Finney, J., 83.Fischer, J., 32.Fixman, M., 70.Franck, E. U., 109.Frank, H. S., 137, 147.Henderson, D., 26, 50.Henderson, I., 239.Hertz, H. G., 236.Hillier, I. H., 79.Hutchinson, P., 53.Hyne, J. B., 148.Josien, M. L., 142.Kim, S., 26.King, S. V., 60.Kohler, F., 32.Lamb, J., 218.Larsson, K.-E., 184.Levi, A., 190.Levine, S., 56, 131.Levy, H. A., 97.Luck, W. A. P., 115, 132, 133, 141, 144.Magat, M., 145.Mason, G., 75.Matheson, A. J., 216,239.McDonald, I. R., 40.Moreton, A., 56.Narten, A. H., 97.Ng, W. Y., 57.Oden, L., 26.Padova, J., 141.Page, D. I., 130.Perram, J. W., 131.Pings, C. J., 89.Plesser, Th., 160.Pople, J. A., 187.Rice, S. A., 16, 57.Richards, R. E., 191.Rizzi, G., 129.Roth, K., 109.Rowlinson, J. S., 55, 56,243.Rushbrooke, G. S., 7.Sherwood, J. N., 128.Singer, K., 40.Smith, F. W., 231, 241.Smith, B. L., 88.Stecki, J., 190,212.Stiller, H., 160.WaIkley, J., 57, 79.Weber, U. v., 59.White, J. W., 169.Young, D. A., 16, 57.* The references in heavy type indicate papers submitted for discussion.24

ISSN:0366-9033    

DOI:10.1039/DF9674300248

出版商:RSC    

年代:1967

数据来源: RSC