Studies of Ion-ion and Ion-molecule Interactions using Far-infrared Interferometry BY COLIN BARKER AND JACK YARWOOD* Department of Chemistry University of Durham Durham City DHl 3LE Received 1I th August 1976 The far-infrared spectra of tetra-n-butylammonium halides in benzene chloroform and carbon tetrachloride are interpreted with the aid of a dynamic model based on the stochastically modulated oscillator theory of Kubo. Although the model is not necessarily unique for this particular (and com- plicated) system it does enable a reasonable interpretation of the observed band frequencies widths and intensities and their variation (or otherwise) with changes of salt solvent and temperature. All the data are consistent with situations in which the ion-pair (or aggregate) vibrations are stochastically but relatively slowly modulated by interaction with the surrounding solvent molecules The perturbation of the solvent molecules mainly by dipole-induced dipole interaction is shown to be severe.There is also evidence of strong coupling of the solvent "collision mode " and the ion-pair vibration and that the latter is strongly overdamped by a high Langevin friction constant. The implications of this work for the interpretation of data obtained at lower frequencies are considered to centre on the large solvent-solute interaction and this large microscopic viscosity coefficient. The far-infrared (submillimetre) absorption by electrolyte solutions has been examined both directly 1-3 and indirectly 43 (using high frequency microwave measure- ments).Recent work in our laboratory has been aimed at a comprehensive quanti- tative study of a range of tetra-alkylammonium salts in (supposedly) poorly solvating media in an effort to throw further light on the nature and origin of these absorptions. In this paper we examine the possibility of constructing a dynamic model for the ion- pairs (or aggregates) which will allow us to interpret the experimental data in a meaningful way and which will (hopefully) provide further insight into the vibrational and relaxational behaviour of the system. We also consider here the ways in which our model may be developed and tested and the similarities and differences between this and other models-particularly ones used to interpret ''dielectric " (microwave) data.49 5.7-12 EXPERIMENTAL The measurements were made using a Beckman-R.I.I.C.Ltd. FS720 interferometer and fixed pathlength cells of stainless steel with high density polyethylene windows. A Beckman-R.I.I.C. Ltd. variable temperature cell holder (VLT-2) and temperature controller were used for measurements between 285 and 350 K. The tetra-n-butylammonium salts were pur- chased from Eastman-Kodak Ltd. They were dried (when necessary) and the water content of the solutions was monitored by the Karl-Fischer method. The solvents were either " AnalaR " or " Spectro " grades and were dried over molecular sieves immediately prior to use. The spectra shown here are ratioed absorbance spectra (corrected for gain differences) of the appropriate solution against pure solvent background.The spectral resolution is -2.5 cm-'. Our methods of estimating the precision of the data of constructing a meaning- COLIN BARKER AND JACK YARWOOD ful " base line " (for intensity measurements) and of checking on the effects of small amounts of water in the solutions have been published previ~usly.~ RESULTS Typical spectra of Bu,"N+Cl' in the three solvents used and of Bu,"N+Br- in benzene are shown in fig. 1-4 together with the decomposition (for Bu,"N+CI- solutions) into their separate component^.^ The resulting band frequencies half- widths and intensities are given in table 1. Our initial interpretations of these spectra have already been published3 but since any model which is constructed must at least be consistent with the experimental data it is appropriate here to summarise the principal features of the observed spectra.(a) The band system (for Bu,"N+Cl- solutions at least) can be decomposed into three separate bands (called A B and C for convenience) each of which appears to be consistent (both statistically and phenomenologically) with the (assumed) Gaussian pr0fi1e.l~ (b) The three bands are of distinctly different widths and intensities and these parameters are to some degree (see discussion below) dependent on variables such as salt concentration and temperature. However the main features of the spectra remain essentially unchanged for a given anion (but change when the anion is changed). In particular the spectra are to a large extent solvent independent.(c) The assignments5 of bands A and B to phenomena associated with solvent (v,) and ion-aggregate (vCA) (ion-pair in dilute solution) respectively is fairly straight- forward. Thus v corresponds to a (perturbed) collision model4 of the non-polar solvent while v, is taken as a vibration of anion against cation in an ionic aggregate- whose exact nature in concentrated solutions is somewhat obscure. The assignment of band C is more difficult but we shall see that our model points to a reasonable interpretation in terms of a combination band of vs + v, (see below). THEORETICAL CONSIDERATIONS Previous models5*7-9 relating to the relaxation of a system of solvated ions have been aimed at interpreting data c011ected~~~~~-~~ at much lower frequencies (typically between 100 Hz and 300 GHz i.e.up to 10 cm-l) on the dipolar relaxation of the perturbed solvent and of the ion-pairs (or aggregates of ion-pairs). As pointed out by Lestrade et aZ.,5 the relaxation (or other) phenomena which are observed depend on the frequency of the radiation used in the experiment and so in the far-infrared region (at frequencies corresponding to > 1000 GHz) we expect to " see '' phenomena which are characterised by relaxation times in the 0.1-1 .O ps region. Nevertheless it is valid to observe that microwave measurements lead to relaxation times for the ionic aggregate lifetime (7,in Lestrade's notation)7 and for the elapsed time between ionic collisions (z,) of the order of 100-300 ps.597-9 This means that as far as the far-infrared region is concerned the ion-pairs can be treated as though they are quasi- stationary.This is one of the assumptions inherent in the treatment given below. The other major assumption is that the vibrational mode considered is that of a solvent-surrounded ion-pair (see fig. 5). Except in very dilute solution this is unlikely to be realistic of course but we note that the spectra vary little over the concentration range studied down to 0.05 mol dme3. Further we have now studied (with the aid of a polarising optical system and a cryogenic detector)6*15 some of these systems down below 0.01 mol dm-3 without any major spectral changes. So there may be some empirical justification for considering an ion-pair in pseudo-isolation.138 STUDIES OF ION-ION AND ION-MOLECULE INTERACTIONS -D wavenumber/crn'l RG.1.-Far-infrared Spectrum of Bu4*N+C1- in benzene. X observed spectrum; A vs band at -65 cm-'; ByvcA band at -115 cm-'; C vS + v,A band at -185 cm-'; D total computed band envelope. wovenumber / cm" FIG. 2.F-r-infrared spectrum of Bu4"N+C1' in carbon tetrachloride. X observed spectrum; A v band at -70 cm-'; B vCAband at -120 cm-'; C v + vcA at -185 cm-'; D total computed band envelope. COLIN BARKER AND JACK YARWOOD w avenu mber/ cm-1 FIG.3.-Far-infrared spectrum of Bu4"N+C1- in chloroform. X,observed spectrum; A v band at -70 an-'; B v,A band at -120 cm-I; C,v + vcA band at -180 cm-*; D total computed band envelope. w avenumber/cm-l FIG.4.-Far-infrared spectrum of Bu4"N+Br-in benzene.Bands A and B are now very much closer together and absorption in 150 cm-l region is very small. The observed spectra and initial interpretation3 (see above) point to the need to consider a stable vibrating ion-pair (with a lifetime which is long compared with the vibrationalperiod) whose solvation shell of surrounding solvent molecules is consider- ably perturbed by interactions with the charged species. Consider (fig. 5) a representative solvent molecule (S) at radial distance R from the ion-pair CA (of finite length rCJ. A multipole expansion16 of the potential at S due to the ion-pair is TABLE1.-FAR-INFRARED SPECTRAL PARAMETERS OF Bu4"N+X-SOLUTIONS IN ORGANIC SOLVENTS band "A " (v,) band "B " (vcA) band "C " (vs + vca) salt solvent temp./K iimax/cm-' Av'i/ern-' in tensity" ijmax/Cm-' Ai+/cm-l intensi t yo iim,x/cm-l AiiJcm-l intensitya Bu,"N+Cl-C6& 285-305 75 f3 56 f 2 2400 118 4 2 52f 3 3800 182f 2 60f 10 -1400 305-328 73 f2 58 f2 2600 117 f2 54f 2 3900 181 f2 54f 5 -1200 328-350 75 f5 62rt 3 3300 118 & 2 52f 5 4200 180f 3 55% 5 -1500 293 73 f4' 724.8' 6300' 118 & 3' 53 f 4' 58Wb 181 f6b 724 19' ~3500~ ~ ~~~~~~~~ + Bu4"N C1-CC4 293 75 f 6 64f 6 3700 114f 2 50f 2 7200 184f 3 80f 15 2500 Bu"~N+ C1-CHC13 293 70f 7 50f 11 1800 116f 2 52f4 7500 175f 7 84f 16 1500 & 0 r C C Bu4"N+Br-C6H6 293 65 f 4 75 4 3300 78 f2 56 f3 1900 --C m 0 C C C Bu4"N+Br-CHCl 293 65 & 5 -3900d 80 f5 -3900d --C r m Intensities have -&1O-15% error in all cases.They are based on total solvent (band " A ") or total solute (bands "B " and "C ") concentrations. The units are dm3 mol-I cm-2. Data at 293 K are averages over whole range of concentration from 0.05 to 0.8 mol dm-3. Variable temperature data at 0.25 mol dm-3. Band not intense enough to measure. Total band intensity of usand oCA. COLIN BARKER AND JACK YARWOOD 141 where Q is the electronic charge. If R & rcAthen the 1st term (dipole term) in (1) is dominant and it will be a reasonable approximation to consider the solute-solvent interaction to be controlled mainly by dipole-induced dipole interactions. For a polarisable ion the dipole moment is,14 and for the tetrabutylammonium halides pcA is expe~ted~~pl~ to be about 30 -36 x 10-30Cm.Thus the dipole induced in a typical solvent molecule pg = a,E (3) [where the field strength is given'* by E = -grad (V(R,O)}] gives rise to a potential energy of interaction Udp-indp = -*O!,E2 (4) FIG,5.-The ion-pair and surrounding solvent molecules. Parameters needed for the calculation of the dipole induced in a representative solvent molecule by a point dipole pcA. where CC,is the polarisability of the solvent molecule. It should be noted that for benzene the polarisability is distinctly anisotr~pic,~~*~~ and this may result in orienta- tional effects around an ion-pair. The results of applying eqn (3) and (4) will of course be to calculate significant dipole induced interactions between the ion-pair and solvent.We now have to note that due to the motions of the solvent molecules and also due to the vibration of the ion-pair the dipole-induced dipole interaction of eqn (4) is time dependent. In particular it will depend on rcA(t)and R(t) and since in the simplest case (2) reduces to pcA= Qr,, it then follows from (l) (3) and (4) that Udpindpis proportional to the square of the total ion-pair-solvent distance [see ref. (18) p. 1261. However if R(t)is taken to be a slowly varying function of t (for the simplest case) then we can write Udp-indp = k(l)rz.4-(5) (However see discussion section for details of possible deviation from this simplified expression for U.) k(t) then represents the time dependence of the interaction potential and is a complicated function which depends on (i) the relative orientations and separation of the solvent and ion-pair and (ii) the (possibly anisotropic) l9polaris-ability of the solvent molecules.Thus as k(t) fluctuates the ion-pair vibration suffers a stochastic modulation which will of course in principle affect the spectrum. The ion-pair vibration may now be treated using the general formalism developed by Kubo20 for calculating the spectral line shape (or associated relaxation function) of a randomly modulated oscillator. (This treatment has recently been successfully applied2lVz2to the case of a weak hydrogen-bond vibration in a " bath " of surround-ing solvent molecules.) 142 STUDIES OF ION-ION AND ION-MOLECULE INTERACTIONS The Hamiltonian for the vibration of the ion-pair is written as =$&A/m + 3m rZA ($A + kr:A =$&$m 3-$m rgA (WgA + 2kfm) where m is the reduced mass givenB for a contact ion-pair as the subscripts c A and s referring to cation anion and solvent respectively.We may now regard the operator (6)as a stochastic operator written as x(t) = *p:$m + 4m rgA m2(t) where the effective time dependent oscillator frequency m(t) is obviously given by Thus o(t) = mo + ml(t) where wo= oCA and ml(t) =k(t)/mw, in our case. The transition dipole relaxation function p(t) of the:oscillator is given 2o by p(t) = (exp (iP(t')dtf)) (9) 0 and is easily calculated from the spectral distribution I(m) since the transition dipole moment autocorrelation function is and J -ca Although the calculation of q(t) is easy (see fig.6) in order to proceed further with the interpretation in terms of the molecular properties of the system it is necessary to know the form and distribution of the stochastic process k(t),ofeqn (8). Although expressions for k(t) are complicated (and we do not believe that with the present quality of spectral data band fitting is justified) we do have some experimental evidence as to the probable source and properties of this process. We first note that the observed spectral distribution of rcA and its associated p(t) (fig. 6) are well-approximated by Gaussian functions with relaxation times of about 0.4 ps. In the slow modulation limit20*22 [i.e.,for a k(t) process which is slow com- pared with the rate of decay of the dipole moment relaxation function p(t)] the spectral distribution follows the statistical distribution of k(t)-which would in that case be Gaussian.Further we note that the band profile is expected22 to be Gaussian (by the central limit if the number of near neighbour solvent molecules is large. Since it is unlikely that benzene molecules are rigidly "bound "to the ion-pair in this case it is doubtful if the solvation number means very much for these solutions. However the tetralkylammonium ions are known 26 to have effective ionic sizes (without solvation shell) of 4-5 A and the number of benzene near neigh- bours is estimated to be about 20-30. However k(t) is clearly not a stationary COLIN BARKER AND JACK YARWOOD timeips FIG.6.-Relaxation functions p(t) of the v6 and vcA vibrations in the Gaussian approximation.process. In the liquid phase we know that the solvent molecules collide with the ion- pair (or aggregates) and with each other. Indeed we see the effects of these collisions in the "A " band of the observed spectrum. So one component of k(t)will fluctuate at the solvent collision rate and another component will fluctuate at the solvent-ion collision rate. Since the motion of the ions is rather slow it is principally these motions of the solvent molecules which lead to the variation of k with time. Thus the collision mode of the solvent may be treated as a stochastic process. If the change in R caused by collisions is AR then R(t) = Ro + AR(t) (12) and since the effect on mcAis through dipole-induced dipole 16*18interactions k(t) = C'R-6 = C'(Ro + AR)-6.(13) Expansion of (13) gives terms in AR AR2 etc. which when used to modify the Hamiltonian will lead to terms in &*AR rEA AR2 etc. These will lead in principle to coupling of the collisional (v,) and ion-pair (vCA)modes. The resulting effect which is expected to be large because v and vcA are very similar in frequency is that a band at zFs + GcA is predicted. The presence of a band very close to 9 + tcAin our spectra-a band without any other obvious explanation-provides strong support for the general validity of our model and the preceding theoretical considerations. 144 STUDIES OF ION-ION AND ION-MOLECULE INTERACTIONS DISCUSSION Clearly the principal observed spectral features are adequately explained with the aid of the model of the electrolyte/solvent system which is outlined above.This model is based on a stochastically modulated ion-pair vibration the modulation being provided by the surrounding solvent molecules which are themselves severely per- turbed by the effects of dipole-induced dipole interactions. Nevertheless it is useful to consider the data in rather more detail in order to investigate more closely the nature of the molecular processes involved in giving rise to these spectra. Consider first the "A " band (vJ. The results in benzene solution show that both the width of this band and its intensity increase with increasing temperature while the band centre remains fairly constant (with a slight tendency to show a high frequency shift but the scatter of data is too high for this tendency to be confirmed).These results are of course what is expected if the band is due to collisions of the solvent molecules. For example Pardoe 27 found that collision bands of non-polar solvents increase slowly with an increase in temperature (in contrast to the behaviour of the "libra-tional" band27-29 of a polar solvent which decreases rapidly in frequency as the temperature is increased). Thus our data strongly support the idea that band "A " is caused by solvent molecules translating against one another (and against the ions in solution) the large perturbation being due to the greatly enhanced fluctuating electrical fields caused by changes in the solvent-ion distances.Although any solvent might be expected to show this effect a correlation is expected between the size of these fluctuating fields (and hence spectral intensity) and the polarisability. It is clear from table 1 that indeed the largest "A " band intensity occurs for benzene which is known to have high p~larisability.~~*~~ (Since carbon tetrachloride also has a high p~larisability~l it appears that the anisotropy of the polarisability of benzene does play an important role.) The data for carbon tetrachloride and chloroform show some interesting features in that the v band frequency is virtually solvent independent. Carbon tetrachloride a collisional band at 44 cm-l in the pure liquid while chloroform an absorption maximum at -35 cm-l.Of course the situation is complicated in chloroform by the presence of permanent solvent dipoles and the p~ssibility~~ of specific solvation of the anion but it appears that interaction with the ion-pairs (of aggregates) in solution is so strong that this produces virtually the same "effective " collisional frequency of the perturbed solvent molecules. This may be related to the change in vis~osity~~*~~ of the medium on going from solvent to solution (some of the solutions are highly viscous34 especially at the higher concentrations) but in that case it is surprising that benzene does not show the same effect. The results in benzene show that this band is strongly anion dependent the frequency and intensity of the band decreasing somewhat on going from CI-to Br- (table 1).This implies that the effects of ion-solvent interaction are smaller for the bromide salt. Such effects depend of course on the " effective " dipole moment of the ion- pair [pCAof eqn (2)] and some such values have been calculated from dielectric data.12J7*35 Since the effective interionic distance is expected to be greater for the bromide salt the effective dipole moment should be greater. Comparison of the observed dipole moment data33 for the corresponding Bu,"NH+X- (X = C1 Br I) salts shows that this is indeed the case. However the exact nature of the species present (and their electrical properties) depends on the extent of ionic aggregation so it is difficult at this stage to be sure of the effects to be expected.Measurements of the far-infrared spectra at much lower concentrations l5 should throw further light on this problem (and remove possible effects due to gross viscosity changes). Considering now the " B " band (designated vcA) we see from table 1 that the COLIN BARKER AND JACK YARWOOD most obvious feature of the data is the almost total invariability of the spectral para- meters for a given salt. The band centre and width remain constant (within the experimental error) over a concentration range of 0.8-0.05 mol dm- and a temperature range of nearly 70 K for three different solvents. (There is a suspected tendency to show some increase in intensity as a function of increasing temperature but this effect if real is small.) On the other hand there is a considerable shift of vCAon changing the anion and this is roughly in proportion to the increase in effective reduced ma^^.^*^*^^ The data serve to confirm that this band is due to an internal mode (or modes) of the ion-pair or aggregate.They also enable us to confirm that the principal effect of the presence of these ions is as expected a large dipole-induced dipole effect on the surrounding solvent molecules. Within the framework of the stochastic model outlined above we might expect that this vibration would be sensitive to both temperature and solvent since we have treated the collisions of the solvent as a stochas- tic process-giving rise to k(t)of eqn (5)-and these processes are obviously dependent on temperature and microscopic solvent properties (the solvent properties are usu- characterised by a "friction constant " p obtained by using the Langevin ally20*21 equation of motion for the displacement coordinate rcA [i.e.fcA + picA+ &ArCA = F(t)]. Such a friction constant (or damping factor) will be temperature- and solvent-dependent). However it is easily shown that in the classical (high tempera- ture) approximation the root-mean-square amplitude of the oscillator A (equivalent to (k2)lI2 in the slow modulation limit and measured by the band width in this approximation) is proportional to T1/2so over a range of 70 K the effect is unlikely to be large. The damping factor p being a viscosity coefficient is exponentially dependent on temperature and should also change from one solvent to another.The fact that little or no change occurs in the band profile with temperature or solvent strongly supports the assertion that one is dealing with the slow modulation limit this is further rationalized as follows. It seems that bis so large34 for the concentra- tions studied so far that the vibrational mode vCAis heavily overdamped20*21 (p/2c0,A > 1). Since the amplitude of the modulation <k2)1/2 is also large we see that the slow modulation limit (A p/2co2,A % 1 in this notation)20*21 arises as a natural conse- quence of the combined high viscosity and rapid rate of vibrational relaxation.22 We note here that the mean vibrational lifetime measured by the rate of decay of q(t)-see fig. 6-has a time constant of -0.4 ps. This is comparable with the rate of collision of the solvent molecules of course and means that the rate of modulation of the environment of the ion-pair oscillator is a considerably slower process than the rate of solvent-solvent collisions.It is known from dielectric studies5p7 that the time between ion-ion collisions is of the order of 200 ps and it is possible that it is this quasi-static situation of the solute particles which leads to the slow modulation limit. What is clear is that the variation of k(t) [eqn (5)] is slow compared with the decay of ~(t) and that this is caused (as far as the model is concerned) principally by a very high viscosity coefficient p. Our current studies are aimed at collecting more accurate data at reduced concentrations and extending the solvent and temperature range so that some of these ideas may be further tested.As we have already pointed out the band " C " in the spectrum is explained within the framework of our model as a combination band of v + vcA which arises through strong coupling of the solvent collisional and ion-pair vibrational motions. Interestingly the band is very weak in (or absent from) the spectra of Bu4"N+Br- solutions (fig. 4) but one can only speculate at the present time as to why this is. Only an investigation of salts with a range of anions will indicate whether or not there is a strong anion dependence of the intensity of this band. Finally it is interesting to consider what aspects of the model described and the 146 STUDIES OF ION-ION AND ION-MOLECULE INTERACTIONS conclusions drawn here have a bearing on models used to interpret microwave data on similar solutions.Two points emerge which need to be carefully considered in relation to the data obtained at lower frequencies. These are (i) the very severe perturbation of the solvent molecules and their large induced dipole moments; (ii) the very large viscosity coefficient-arising very probably from the large macroscopic viscosity of the solutions. Both these effects are likely to be important in the micro- wave region of the spectrum and indeed have been at least implicitly taken into acco~nt.~*'-~~ The real value of this work may well be that we have shown a com- bination of studies in different frequency regions leads to a better understanding of the fundamental resonant and relaxation processes involved in the behaviour of similar or identical systems.The help of Dr G. N. Robertson (University of Cape Town) with the theoretical aspects of this paper is gratefully acknowledged. Thanks are also due to Beckman- R.I.I.C. Ltd. for their continued support (through the C.A.S.E. scheme) and to the S.R.C. for the C.A.S.E. award (to C. B.) and equipment grants. We are grateful to Dr. R. N. Jones for copies of the N.R.C. band fitting programmes. J. C. Evans and G. Y. S-Lo J. Phys. Chem. 1965,69 3223. 'M. J. French and J. L. Wood J. Chem. Phys. 1968,49,2358. 'C. Barker and J. Yanvood J.C.S. Faraday II 1975,71 1322. 'H. Cachet F. F. 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