2879
Magnetic Properties of Metal-Ammonia Solutions T. TUTTLE. How does the mobility of the solvated electron in ethers compare with that in ammonia? G. R. FREEMAN.The electron mobility in low temperature ethers is more ion-like than that in ammonia. The ratio of mobilities (electron/anion) is about five in ammonia a t 240 K and only two in low temperature ethers.
M. H. COHEN.The fact that the activation energy for electron mobility a t lower temperatures is the same as that of the heavy ion mobility suggests that the mobility mechanism is electron diffusion rather than small polaron hopping from trap to trap. The
problem of the larger value of the mobility for the electron occurs also for NH3 and may arise from the fact that the electron does not possess a rigid structure, as an ion does, and so more rearrangements of the fluid can lead to electron diffusion.
G . R. FREEMAN.The low temperature mechanism of electron migration might simply be diffusion as you say, and as I speculated in the written manuscript. However, the low temperature activation energy is one half of the high temperature activation energy. If the high temperature mechanism involves a conduction band, then that at low temperatures might involve thermal activated hopping in the small polaron fashion.
Magnetic Properties and the Metal-Nonmetal Transition in Metal-Ammonia Solutions J. P. Lelieur,’ P. Damay, and G. Lepoutre Laboratoire de Chirnie Physique, ERA 126 du CNRS, 13, rue de Toul, 59000 Lille, France (Received July 23, 1975)
The magnetic properties of metal-ammonia solutions are examined in the dilute, intermediate, and concentrated ranges in the light of the present understanding of these solutions. More attention is given to magnetic data in the intermediate concentration range and to the metal-nonmetal transition problem. A recent model of this transition is analyzed. The analysis of experimental thermodynamic data (activity), in conjunction with a fluctuation correlation length (from neutron scattering), gives the distribution of the concentration fluctuations. This concentration fluctuation distribution around the mean concentration can be used to determine physical properties.
Introduction In his review given at Colloque Weyl 11, Catteralll analyzed the magnetic properties of dilute metal-ammonia solutions. Since that time, relatively few papers have been published in this field; however, there has been a number of experimental works dealing with the intermediate concentration range. Therefore, in this paper more attention will be given to the intermediate concentration range and to the problem of metal-nonmetal (MNM) transition. The only solvent considered in this paper will be ammonia. 1. Dilute Solutions
1.1 What about Models of Dilute Metal-Ammonia Solutions? Many measurements of magnetic properties have been performed in the past in the electrolytic dilute metalammonia solutions. They have been reviewed in an excellent paper by Cattera1l;l the “dilemma” raised by these dilute solutions have been analyzed by Dye2 and Lagowski3: Are solvated electrons associated (magnetic susceptibilities) or nonassociated (optical spectra)? Very dilute solutions contain two species: paramagnetic solvated electrons and solvated cations. An increase in metal concentration leads to the formation of electrically neutral ion pairs which, however, remain paramagnetic. It is often assumed that a t concentrations larger than about M , ion pairs associate into quadrupoles where the solvated electrons are spin-paired. The equilibrium constant for the pairing of ions has the order of magnitude predicted by Bjerrum’s theory. Demor-
tier4 has shown that the constant of association of ion pairs in quadrupoles is much larger than expected. He has therefore suggested that the attractive interactions between solvated electrons are much stronger than between conventional ions. Justice5 has shown that Bjerrum’s constant for ion pairing, K(+-), can be derived from a statistical treatment of binary interactions between ions, according to Mayer’s and Friedman’s6 views. Two interesting results are that the expression of IC(+-) is exactly as predicted by Bjerrum, and that two other constants of ion pairing appear as a consequence of the statistical approach K(--)and K(++). In most cases, these latter constants of association for ions of same sign are very small. Around a given ion, the probability of finding another ion of same sign a t a distance shorter than the Bjerrum’s distance is usually small and can be neglected. It is clear that such a statistical treatment of binary interactions cannot lead to quadrupoles. However, it seems to be adequate for most other dilute solutions, and it can be applied a t fairly high concentrations for nontransport properties. Finally, as Schettler7 shows a t the present Colloque, there are unusual attractive interactions between solvated electrons related to their high polarizability and their spinpairing. There may be pairs of solvated electrons at distances shorter than the Bjerrum’s distance. (This is quite different from two electrons in one cavity. In the present picture, The Journal of Phydcal Cbmistty, Vol. 79, No. 26, 1975
J. P. Lelieur, P.
2880
two solvated electrons are associated and spin-paired but still apart from each other.) It may therefore be possible to describe dilute metalammonia solutions with two ion-pairing constants, IC(+-) and K ( - - ) , for the two simultaneous equilibria
Ma,+
+ earn2eam-
Mam+-eam-
* (eam-)z
(1)
(2)
There would possibly be no need for introducing interactions between more than two particles. However, the use of pair interactions is not sufficient when the potential energy between two particles is perturbed by a third particle. Such a perturbation occurs when the particles are polarizable. This is the case for solvated electrons, so that triple ions and quadrupole as well as pairs of solvated electrons may occur. A good theory of interactions between solvated electrons would allow a comparison between these various associations. In any case the pairs of solvated electrons, be it in simple pairs or in triple ions or in quadrupoles, should not be viewed as contact pairs but as a statistical distribution within a critical distance. The solvated electron remains, therefore, an individual entity in the paramagnetic pair (Mam+-eam-),the diamagnetic pair (eem-)2, in the triple ion, or in the quadrupde. This should help to resolve Dye's dilemma, and reduce the problems of dilute metal-ammonia solutions to the general theories of electrolytic solutions, given a good understanding of interactions between solvated electrons. The range of application to static properties such as magnetic susceptibilities may hopefully be extended to fairly high concentrations, up to the onset of metallic properties (0.4 M ) . Preliminary results of Tehan, Lok, and Dye8 show that the temperature dependence of the spin-pairing equilibrium in dilute solutions is relatively insensitive to the nature of the cation. This suggests that the diamagnetic species is relatively independent of the cation. 1.2 The Solvated Electron Species from Nuclear Relaxation. All magnetic measurements related to dilute metalammonia solutions will not be analyzed here. Only experiments dealing with the correlation time of the interaction between electron and nitrogen, and with the solvation number of the solvated electron, will be considered here. The electron spin relaxation mechanism, which gives rise to the extremely narrow ESR line width, is the modulated hyperfine interaction of the solvated electron with nitrogen atoms of the solvation shell. The electron relaxation rate is then given by9 7 1 2 = -Z(Z + 1)nA2 Ti 3 1+ ( W N - WS)2T2
(3)
where Z is the nitrogen spin, n is the number of nitrogen atoms taking part in the relaxation mechanism, T is the correlation time of the interaction, WN and ws are the nitrogen and electron-spin Larmor frequency, respectively, and A is such that A h 1 8 is the energy of the interaction. The latter quantity A can be directly expressed in terms of the spin density PN on nitrogen atom in the solvated electron: A = (8r/3)YNYShPN/n
(4)
where Y N and 7s are the gyromagnetic ratios of nitrogen and electron, respectively. PN is directly given by the nitrogen Knight shift The Journal of Physical Chemistry, Vol. 79, No. 26, 1975
Darnay, and G.Lepoutse
K(N) =-I 8a Xm PN 3 No R where xm is the molar paramagnetic susceptibility a i d R the ammonia-to-metal ratio. I t should be noted that this expression defines the average spin density on nitrogen. In the motional narrowing situation, ( ~ J Nis small with respect tQ (l),and is neglected. From relations 3, 4, and 5 , rln can be written as:
with
Electron relaxation time 2'1 is deduced from ESR linewidth measurements. Published data are in agreement with the values T I -3.1 psec a t room temperature, and T I -1.1 psec a t 240 K (Hutchison and Pastor,lo O'Reilly,ll B l ~ m e , 'Pollak13). ~ Spin density on nitrogen, P N , is deduced from nitrogen Knight shift measurements (McConne1 and Holm,14 Acrivos and Pitzer,15 O'Reillyl6) and from molar paramagnetic susceptibility. For infinitely dilute solutions, it is expected that PNis independent of the cation in the solution, and PN can be taken as equal i o the values ded.uced16 from Na-NHS solutions, i.e., 0.77u0-~at room temperature and 1 . 4 2 ~ a0t ~240 ~ K. With these values of P N and T I ,T / n is equal to 0.85 X sec a t room temperature and 0.75 X sec a t 240 K. The value at room temperature is close to the value given by Catteralll in his review (0.62 X sec). In literature, authors usually give values of T or n. It should be realized that values of T or n require hypotheses on n or 7 , respectively, because eq 6, for sln prevents the independent determination of T and n. If hypotheses made on r or n in various papers are removed, it can be shown that r / n deduced from experiments (and not from theoretical estimation of P N ) h a w quite close values with different authors. In fact, such an analysis is only a first approximation. lt is well known that, when metal concentration increases, the spin-orbit coupling of the electron with solvent molecules and metal ions makes the electronic relaxation time decrease. With the realistic hypothesis that both relaxaiion mechanisms (hyperfine interaction and spin-orbit cou. pling) intervene simultaneously, and with the hypothesis that n = 24, Q'Reillyl7 was able to obtain the variations of T with metal concentration. However, if the hypothesis that n = 24 is dropped, and if the previously mentioned values of PN are used, values of T/n are obtained. The values of r / n which take account of the spin-orbit coupling are 0.46 X sec a t 300 K (compared to 0.85 X sec without spin-orbit coupling) and 0.58 X sec (compared to 0.75 X sec). Therefore T/n values are smaller even at infinite dilution if spin-orbit coupling is accounted for, and the temperature variation of s / n i s inversed. It should be concluded from this work of O'Rei1lyl7 that the spin -orbit coupling has to be taken into account for the determination of r / n values. Swift et al>lsJ9determined T and n through the analysis of the high-resolution proton magnetic resonance line shape. This proton spectrum in pure liquid ammonia i s well known to be a triplet arising from 14N--H coupling. The features of this triplet are determined by the I4N electric quadrupolar relaxation rate. These authors showed that, in dilute K-NHs solutions, a large contribution to the
Magnetic Properties of Metal-Ammonia Solutions
280 I
.. . ’
..
0
-
3n.
7
Y
Cs-NH, 0 No-Nb
mole fractlon
sodium
%
Figure 1. Paramagnetic susceptibility of Na-NH3 and Cs-NH3 sohtions, per mole of metal in solution, taken from ref 20, vs. metal concentration. Data on the dilute side (-0-0-) are from ref 46.
proton line shape comes from 14N spin-lattice relaxation through magnetic dipolar interaction with unpaired electrons. The fit between theoretical and experimental line shapes gives the value of r / n . The important point is that the r / n value is field dependent if w s z r z (where us is the electron Larmor frequency) is not taken as negligible with respect to (1).Therefore proton line shapes at two different frequencies give the values of r and n . However, the accuracy of these determinations, particularly for n , was limited for the following reasons. The determination of PN is based upon the conflicting nitrogen Knight shift values reported in the literature.14-16 The proton spectra have been taken a t two frequencies (60 and 100 MHz) not largely different. The values of r were found to be of the order of 1 to 2 X 1O”l2 sec, and the solvation number ranged from approximately 20 to 40. These values are not surprising. However, this technique used with largely different frequencies, combined with simultaneous nitrogen Knight shift measurements to obtain a better accuracy, could prove to be a very powerful tool for the study of metal-ammonia solutions. 2. Intermediate Concentrations
2.1 Magnetic Susceptibility Measurements. In the MNM transition range, Le., between approximately 1 and 10 MPM, some experiments have been performed since the review made by Catterall for Colloque Weyl 11. The magnetic susceptibility20*z1has been measured for Na-NH3 and Cs-NH3 by the classical Gouy technique (Figure 1). For Na-NHs solutions, the paramagnetic susceptibility has also been measured20 directly by the Schumacher-Slichter resonance technique. Both techniques give the same trend of results. The paramagnetic suweptibility increases with the metal concentration; the temperature coefficient of the paramagnetic susceptibility decreases when metal concentration increases. It should be noted that the magnetic susceptibility is not known between approximately 0.5 and 2 MPM. It will be useful to know the concentration of the minimum of the magnetic susceptibility and the concentration of the maximum of its temperature coefficient. The magnetic susceptibility data show that between about 0.5 and 1 MPM most of the solvated electrons are paired. When the metal concentration increases, these solvated electrons become progressively delocalized and behave somewhat as free electrons.
0
10
20
30
A0
CONCENTRATION [MPMI
Figure 2. Nitrogen Knight shift, K(jAN) (ppm), vs. metal concentration, in Na-NH3 and Cs-”3. The data (V)are taken from ref 25. I t may be pointed out that the paramagnetic susceptibility is obtained from the Gouy technique, by subtracting the diamagnetic contributions of the ammonia molecules, of the metal cation, and of the Landau diamagnetism of free electrons. This last contribution, taken as equal to -y3xp as for a free electron gas, is questionable. However, no other estimation can be made, and it does not seem that the paramagnetic susceptibility trend vs. metal concentration would change significantly, even on assuming that the exact diamagnetic contributions could be known and taken into account. It should certainly be useful, as for dilute solutions, to know the magnetic susceptibility deduced from the Gouy technique, and the paramagnetic susceptibility from the resonance technique. However, it is unlikely that the accuracies of these results could make their difference meaningful. With the measurements of the paramagnetic susceptibility, the ESR line width has been measuredz0at 15 MHz for Na-NH3 solutions. It has also been measured by Chan, Austin, and Paezzz in the X-band frequency range for NaNHs solutions. Damayzs measured the ESR line width of Li-NH3 solutions in the X-band frequency range. Those results show the change from an ESR line width, typical of the dilute concentration range, i.e., of the order of 50 mG (electronic relaxation time of the order of 1 bsec), to an ESR line width of about 5 G for 10 MPM Na-NH3 solutions, and of about 4 G for 20 MPM Li-NHs solutions. Therefore the ESR line width displays a rather important change in the intermediate concentration range. On the other hand, it should be noted that there is always a single electron resonance signal, at 15 MHz as a t 9000 MHz. As a consequence, if a model of the metal-nonmetal transition assumes the existence of two different electronic states in the solution, for instance, solvated electrons in dilute clusters and free electrons in metallic clusters, the exchange between these states has to be fast. 2.2 Nuclear Resonance Measurements. Nuclear magnetic resonance measurements have been made for sodiumand cesium-ammonia ~ o l u t i o n s In . ~ ~the ~ ~intermediate ~ concentration range, the 14N Knight shift has the same values and variations in Na-NHa and CS-NH:~solutions up to 15 MPM (Figure 2). This confirms previous observations The Journal of Physical Chemistry, Vol. 79, No. 26, 1975
2882
J. P. Lelieur, P. Damay, and G. Lepoutre
1:I
0.3
\
I \
- 0
I
/cs
5
10
CONCENTRATION
15 (MPM)
Figure 3. &(k) = ( I \ k p J 2 ) ~(aoh3) , for Na-NH3 at -3OOC (righthand scale) and P&s) for Cs-NH3 at -4OOC (left-hand scale) vs. metal: concentration expressed in mole per cent (MPM) of metal,
of O’Reilly. I t is realized that Knight shifts result from the product of the paramagnetic susceptibility and the electronic density on the nucleus. On the other hand, it has been foundz1 by independent measurements that the paramagnetic susceptibilities are similar for Na-NH3 and Cs“3. It is therefore concluded that the electron densities on the nucleus are similar in both types of solutions. This suggests that the interactions between the nitrogen nucleus and the metal valence electron are similar in Na-NHS and in Cs-”3. As a consequence it is believed that the solvations of Na and Cs in NH3 cannot be differentiated on the basis of the 14N Knight shifts. Here again, it should be noted, that a single 14N nuclear resonance signal has been observed. However, 14N nuclear resonance has not been investigated in Li-NH3 and divalent metal-ammonia solutions. Cesium Knight shifts have been measured as a function of concentration and t e m p e r a t ~ r e . ~ With * * ~ ~our data of paramagnetic susceptibility,21 the electronic probability density averaged on the Fermi electronic states, Le., PF(CS) = (I \ k ~ ( 0 ) 1 ~has ) , been obtained. The corresponding quantity PF(Na) has been obtained from the 23NaKnight shifts measurement^^^ and the paramagnetic susceptibility data (Figure 3). The electronic probability densities PF(Na) and PF(Cs) decrease up to about 3 MPM. When metal concentration increases above 3 MPM, PF(Na) is almost constant, while PF(CS) increases. This is a quantity where Na-NH3 and Cs-NH3 display different trends. These differences could be attributed to the different solvations of Na+ and Cs+ cations; but with this hypothesis the similar values of K(14N) for Na-NH3 and Cs-NH3 are surprising. It should be noted that a single metal nuclear resonance signal has been observed in alkali metal-ammonia solutions. Therefore, in the framework of an inhomogeneous model, this experimental fact implies a fast exchange between the components of the inhomogeneous model. Temperature coefficients have also been for the magnetic susceptibilities and cesium and nitrogen Knight shifts. All these temperature coefficients display the same trend: large positive values at the low concentration end of the intermediate region, and marked decrease when metal concentration increases. 2.3 The Metal-Nonmetal Transition. The main experimental magnetic properties of metal-ammonia solutions in The Journal of Physlcal Chemistry, Vol. 79, No. 26, 1975
the intermediate concentration range have been summarized. Let us see now which interpretations can be given of these properties. I t was recognized early that metal-ammonia solutions display a metal-nonmetal (MNM) transition when metal concentration decreases. However, this idea was discussed only on a qualitative basis for a long time, and often in connection with the liquid-liquid phase separation, the critical concentration of which is about 4 MPM. First, SienkoZ6showed that the criterion for a “Mott transition” predicted a MNM transition a t about 4 MPM. The first qualitative inhomogeneous model was proposed by Thompsonz7 and Cohen and Thompson.28 They suggested that between 0.1 and 3 MPM the solutions contain large metallic clusters. At Colloque Weyl 11, Lepoutre and Leli e ~ suggested r ~ ~ thqt, in the range of concentration where phase separation occurs, the solution is composed of a mixture of metallic clusters and the bulk of the solution, metallic clusters beginning to form a t about 1 MPM. These suggestions were only qualitative. Lelieur20 showed that magnetic susceptibility and metal Knight shift measurements could be interpreted in an inhomogeneous model where the solution is composed of dilute and concentrated clusters, a fast exchange between these clusters being assumed. Acrivos and Mott30,31suggested that physical properties of metal-ammonia solutions in the intermediate concentration range could be accounted for in terms of a diffusion type of electron transport. The basic parameter of this model is the Mott g factor (ratio of the density of states to the corresponding free electron density of states), which determines the depth of the pseudogap introduced in this model. It must be noted that this model is basically an homogeneous model. I t is difficult to obtain a coherent set of g values from different physical properties. Jortner and C ~ h e argue n ~ ~ that the log-log plot of the electrical conductivity vs. the paramagnetic susceptibility of Na-NHS solutions displays a slope of 2.3 instead of 2 as expected in the Mott model, More recently, Jortner and C ~ h e proposed n ~ ~ the first quantitative inhomogeneous model. They proposed that in the intermediate concentration range, metal-ammonia solutions are microscopically inhomogeneous with a volume fraction C of the material occupied by metallic clusters of a mean concentration 9 MPM, the remaining volume (1- C) consisting of small solvated electron-cation diamagnetic complexes of a mean concentration of about 1 MPM. It was previously showns4 that the electrical conductivity problem in such a system, i.e., percolation in a random potential, could be solved with a classical effective medium theory (EMT). Jortner and Cohen were able, with EMT equations, to compute the electrical conductivity of Li-NH3 sol u t i o n ~The . ~ ~EMT does not imply a size for metallic or dilute clusters. Jortner and Cohen modified the EMT to account for scattering from the boundaries of the metallic regions. As a consequence, the modified effective medium theory (EMTZ) requires a size for the metallic clusters. Jortner and Cohen found that a good fit with experimental data was obtained with a metallic cluster size of the order of 25 A. L e l i e ~ used r ~ ~ the Jortner-Cohen model to determine the pressure and temperature coefficients of the electrical conductivity, the pressure and temperature dependences of the metallic volume fraction being deduced from the isothermal compressibility and density, respectively. These calculations were able to reproduce the experimental trends of the pressure and temperature coefficients of the electrical conductivity. In the extended version of their
Magnetic Properties of Metal-Ammonia Solutions
0
CONCENTRATION
X
- "1
.02
.03
CONCENTRATION
"r * "2
Flgure 4. Concentration fluctuations distribution for an average metal concentration of about 5 MPM.
.01
N
.04 -
.05
2
nt+n2
Concentration fluctuatlons distribution for an average metal concentration of about 2 MPM. Figure 5.
model, Jortner and C ~ h e nproposed ~ ~ that the inhomocan be computed from vapor pressure data. Ichikawa and geneous regime occurs between M I = 2.3 MPM and Mo = 9 Thompson39 have shown that emf data are consistent with MPM, mainly on the basis that a fit between experimental vapor pressure data. Damay40-41has shown that, if the two data and calculations is possible in this range. The main components are taken as pure ammonia and a solvated alphysical point of this microscopically inhomogeneous kali metal Na-(NH3),, the vapor pressure data can be repmodel is that the concentration fluctuations are bimodal, resented by an analytical function. The second and third varying locally around either of the two well-defined values derivatives of this analytical expression of the chemical POMo and M I , the concentration appearing to fluctuate tential reproduce the experimental critical temperature abruptly and randomly from Mo to M1 or vice versa, with a and concentration within 0.5%. fluctuation diameter of at least 30 %, for Li-"3. The number n of molecules involved in (8) (or the correThe most controversial idea of the Jortner-Cohen model sponding volume V) has to be chosen carefully. If fluctuais the bimodal concentration fluctuation. The physical oritions were studied on the overall solution, n would be of gin of such a bimodal fluctuation is unclear. More specifithe order of and the distribution function P ( x ) would cally, would it be possible for such clusters of concentrabe a 6 function. The fluctuations cancel each other, and tions Mo and M1 with a relatively large number of ions to there are no fluctuations on a macroscopic scale. exchange fast enough so that only one ESR and one NMR The local fluctuations do not cancel each other as long as signal would be o b ~ e r v e d As . ~ a consequence, the experithey are correlated. They are c ~ r r e l a t e dover ~ ~ the Debye mental activity data have been analyzed by D a m ~ y in , ~ ~ correlation length L. The smallest volume which has the connection with the fluctuation correlation length obtained same properties as the macroscopic properties of the soluby C h i e ~ from x ~ ~ neutron scattering, to obtain the concention is V = L3, and n is the number of molecules in V. tration distribution in a fluctuation model, and this distriThe correlation length has been recently determined by bution has been used to determine other physical properChieux3' on the Li-ND3 system by neutron scattering. His ties. Ornstein-Zernike length is proportional to L , with a nu2.4 Concentration Fluctuations Distribution of a Twomerical factor arising from dimensionality. His results Component Mixture near a Critical Point. The concentrashow that the Ornstein-Zernike-Debye variation of the tion fluctuations of a binary solution correspond to the correlation length with temperature is obeyed: density fluctuations of a single fluid. They can be repreL ( T ) = Lo[(T - T,)/T]"* (9) sented by38 where T , is the critical temperature and Lo is a constant P ( x ) = POexp(-nAp/RT] (8) which is characteristic of the actual range of interaction. with This constant Lo is assumed to be the same for all alkali metal-ammonia systems, which are much alike with reAW = ~ c i ( -~ Pi) i + xz(l.~z- F z ) spect to their critical phenomena. For Na-NH3 solutions at P ( x ) is the probability of finding a local concentration x 2 -35OC, the volume V corresponding to L ( T )contains a few within a volume V which has the average concentration RZ, thousand molecules, and thermodynamic properties remain n is the total number of molecules in V, and Ap is the local therefore meaningful. excess free energy. It is the difference between the local Figures 4 and 5 show the distribution curve which are free energy x1p1 + xzyz of the two components and their obtained for two concentrations. Such curves can be used free energy if they were a t the average chemical potentials, to compute the properties which depend on local fluctuaLe., x lit1 + x z j i z ; POis a normalization factor. tions. Relation 8 assumes that the pressure work associated 2.5 Determination of Physical Properties from the Conwith fluctuations is negligible, or that the excess volume of centration Fluctuation Distribution. Let us assume that a mixing is negligible. P ( x ) gives the distribution of concentheoretical model can predict the values f(x) of a given trations around an average concentration 2. It is a Gaussproperty as a function of concentrations for an hypothetiian-like function, more or less skewed. All parameters in Ap cal metal-ammonia solution without any critical fluctuaThe Journal of Physical Chemistry, Voi. 79,No. 26, 7975
J. P. Lelieur, P. Damay, and G. Lepoutre
0
04
02
CONCENTRATION
06 X
-
nor the resistivity adds up in a microscopically inhomogeneous system. He developed the effective medium theory (EMT) which gives good results for some nonhomogeneous alloys. For the application of EMT to metal-ammonia solutions, it may be assumed, as a first approximation, that the conductivity would be a step function if the system was free from fluctuations. The results thus obtained would not be essentially different from those obtained by Jortner and C ~ h e nThis . ~ ~comes from the fact that the ratio A2/A in (12) (Le., the percentage of the solution at a concentration larger than x , ) gives a concentration scale similar to the Jortner-Cohen scale. The ratio A2IA comes from the fluctuations and a step function, while the Jortner-Cohen scale comes from a bimodal model. However, it is possible to use EMT without the model of a step function. The distribution curve can be split into small intervals Ax. The area under the curve for an interval Ax is
.08
L
It
x,+Ax
"l'"2
Flgure 6. Experimental ESR line width of Na-NH3 solutions (taken from ref 20) and corresponding calculated values vs. metal concen-
tration (MPM).
Ai(Ax) =
P ( x ) dx
(13)
and the fraction of the solution which has a concentration x i f (Ax/2) is
tion. It is then possible to calculate the values F(x) of the same property for the real fluctuating system, by use of the distribution curves. The measured property is given by
J ' W x ) dx
A JO
x l P ( x ) f ( x )dx F(x) =
--
(10)
In the simplest case f(x) is a step function corresponding to a sharp transition at a critical concentration x,, with f(x) = f l for x < x c and f(x) = f z for x > x,, Then F(x) is given by
If a theoretical conductivity can be ascribed to the nonfluctuating solutions of concentration x r , it is possible to extend the EMT as follows. For two states (step function), the conductivity calculated by EMT follows the condition: y l ( abll+- 25 5) + Y Z
+ 25- ) = o
(b2UZ
(15)
where y2 is the fraction of the solution where x 2 > x,. The extension to n states is straighfforward and the equation F(x)
fi
+ ( A 2 / A ) ( f z- f i )
(12)
where A2 is the area under the distribution curve for concentrations going from x c to 1 and A is the total area under the curve. The line width of the ESR signal is likely to behave as a step function. A reasonable choice for x c is the experimental critical concentration for phase separation. Constant values for the line widths may be chosen as 30 mG for the dilute solutions and 4.85 G for the concentrated Na-NH3 solutions, The values calculated through (12) for ESR line widths are compared with experimental data in Figure 6. The discrepancies a t x 2 > 0.05 ( 5 MPM) are explained by the choice of constant value for the line width in concentrated solutions, The difference in the slopes is sensitive to the choice of the correlation length. An almost perfect fit would be obtained if the size of the characteristic volume was decreased by 10 to 15%. Other properties like the paramagnetic susceptibility can be interpreted in the same way. The theoretical curve is more questionable than for the ESR line width. A step function gives nevertheless a good result. It is less straightforward to interpret the transport properties. Landauer4" showed that neither the conductivity The Journal of Physical Chemistry, Vol. 79, No. 26, 1975
gives 5 = f(xz). Thus even for the transport properties, the above treatment of the fluctuations can help to check a theoretical model against the experimental data, with or without the simplification offered by a step function. It is clear that the present ideas are profoundly different from the J ~ r t n e r - C o h e n model. ~~ Present ideas36 deal with thermodynamic functions. The concentration fluctuations distribution is calculated in a volume V = L3. This volume is large because of the closeness of the critical point. The concentration fluctuation distribution is around the average concentration. It is an inhomogeneous model because concentrations different from the average concentration have to be taken into account. The application of present ideas to the calculation of physical properties does not imply limits of applicability; Le., it is not necessary to give the lower and the upper concentration where these ideas are applied. Consequentlp, these ideas can be used in the whole range of concentrations where MNM transition is known to occur. More calculations are, of course, needed, especially a t various temperatures, to check this model.
Magnetic Properties of Metal-Ammonia Solutions However, it should be noted that the concentration fluctuations distribution does not depend on model specifically designed for this purpose. The concentration fluctuations distribution is based on experimental values of activity and correlation length, and on the analytical expression of excess thermodynamic functions. This analytical expression is itself based on the assumption that the solution is a mixture of pure ammonia and of solvated alkali metal, M(NH3),. The analytical expression of excess thermodynamic functions has been proved successful for the calculation of critical temperature and concentration. 3. Concentrated Solutions
Since the last review about magnetic properties of MAS, some new data have been published in the range of concentrated solutions. The magnetic susceptibilities and nuclear magnetic resonance will be successively considered. 3.1 Magnetic Susceptibility. The magnetic susceptibilities of Na-NH3 and Cs-NH3 have been measured20p21for concentrated solutions by the Gouy technique. With the Wiedeman additivity law, the paramagnetic susceptibilities have been obtained after the removal of the ammonia, metal cation, and Landau diamagnetic contributions. This last contribution, taken equal to the free electron Landau diamagnetism, is of course more realistic for concentrated than for dilute solutions. I t was found that the paramagnetic susceptibilities of solutions more concentrated than, say, 10 M P M have a free electronlike order of magnitude. It was found that, a t about 15 MPM, the experimental paramagnetic susceptibility is about twice as large as the noninteracting free electron gas value. This result is not surprising because the free electron formula neglects exchange and correlations effects. Theoretical and experimental works have shown that, for alkali metals, for instance, the ratio of the true spin susceptibility to the value given by the Pauli formula is between 1.5 and 2. From the Pauli formula, the paramagnetic susceptibility per gram atom, xpat, is expected to vary as n-2/3, where n is the electron density per unit volume. Therefore xpat is expected to increase when the metal concentration decreases from eaturation in Na-"3, or from, say, 15 MPM in Cs-NH3. This trend is, in fact, observed in Cs-"3, but the opposite trend is found in Na-NH3. I t is difficult to speculate about the origin of such a difference. The temperature coefficient of the paramagnetic susceptibility has been found to be always positive and decreasing when the metal concentration increases. For concentrations of the order of 15 MPM, the temperature coefficient of the paramagnetic susceptibility has been found to have the value expected from the thermal expansion. Therefore, as far as magnetic susceptibility is concerned, solutions of concentration close to saturation, or close to, say, 15 MPM, appear to have a free electron behavior. However deviations occur for lower metal concentrations. 3.2 Magnetic Resonance. Nitrogen (14N) Knight shifts K(N) have been m e a ~ u r e d ~ Ofor 3 ~ Na-NHS ~ and Cs-NN3 solutions. For concentrations up to about 15 MPM, K(N) is the same for both types of solutions, a t the same concentration. It was found also that for concentrations larger than about 20 MPM, K(14N) has constant value. I t was not possible to measure K(N) for Concentrations larger than about 40 MPM. The general expression of the 14N Knight shift
2005
where R is the mole ratio and xpat the paramagnetic susceptibility per gram atom. Therefore, the constancy of K ( N ) between 20 and 40 MPM for Cs-NH3 solutions means that x p a t P ~ ( N varies ) as R does. In fact xpat has been found to vary very slightly in this concentration range. Therefore, PF (N) varies approximately as R does, that is to say that the electron density a t the nitrogen nucleus decreases as R does when the cesium concentration increases. Cesium (133Cs) Knight shifts K(Cs) have been measured20J4 vs. concentration and temperature. The shifts increase with cesium concentration and with temperature. The values of K(Cs) increase rapidly up to about 25 MPM and show an inflection point between 30 and 35 MPM. With our magnetic susceptibility results, the electronic probability density Pfi- = ( ( \ k ~ ( 0 ) (was ~ ) obtained for the sodium and for the cesium nucleus and are shown in Figure 3. For concentrated Na-NH3 solutions, PF(Na) is almost constant, while for concentrated Cs-NH3 solutions PF (Cs) increases rather sharply. These differences in the trends of PF(Na) and PF(CS) are significant. They cannot be accounted for by the uncertainties of the paramagnetic susceptibilities values. These differences are probably a consequence of different solvation for Naf and Cs+ cations. However, no calculation has been made of the conduction electron wave function in concentrated solutions. Such calculations should lead to theoretical values of PF(CS) and PF(Na). Cesium Knight shifts measurements also gave evidence of the liquid-solid equilibrium in the phase diagram of Cs-NH3 solutions for metal concentrations larger than the eutectic concentration. Thermal variations of K(Cs) have been m e a ~ u r e d . ' ~ ,I' t~ was found that K(Cs) always increases with temperature (for concentrations below about 50 MPM), and for temperatures lower than -4OOC. K(Cs) should go through a maximum and its temperature coefficient should become negative, but Cs-NH3 solutions are not stable enough to permit useful measurements a t these temperatures. Experimental measurements showed that the positive temperature coefficient decreases when metal concentration increases. Therefore, even for concentrations larger than 10 MPM, increases of metal concentration or temperature make the solutions more metallic, which is a picture coherent with physical properties such as electrical conductivity. Other NMR measurements have been reported by Garroway and C ~ t t s . ~They * measured self-diffusion coefficients of %, 'L3Na,lH in Li- and Na-NH3 by spin-echo technique, in the intermediate and concentrated ranges up to saturation. The purpose of these experiments was to give direct evidence of the solvation of Li+ and Na+ by ammonia molecules. They observed that the measured ammonia self-diffusion coefficient is greater than the 7Li self-diffusion coefficient, except a t 20 MPM where they are essentially equal. This result suggests strongly that the Li+ cation is solvated by four NH3 molecules, which is an expected result. The similarity of the Na-NH3 and Li-NH3 diffusion data indicates that the solvation number is also 4 in the case of Na+ in Na-NH3 near saturation. I t has been suggested by Sienko that the diffusion coefficient should also be determined for I4N. I t would also be interesting to have diffusion measurements in Cs-NH3, since it is usually expected that the Cs+ solvation is different (much less tightly bound) from the solvation of other alkali cations. I t should also be noted that the relaxation times and relaxation mechanism for the metal nuclear spin for very The Journal of Physical Chemistry, Vol. 79, No. 26, 1975
J. P. Lelieur, P. Damay. and G. Lepoutre
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concentrated solutions are not reported. Let us note also that no other proton Knight shifts measurements have been reported for concentrated solutions since the Hughes measurement^^^ of proton Knight shifts in Na-NH3 solutions. Let us recall that the proton Knight shift K(") was found by Hughes to be negative a t all concentrations in Na-NH3 solutions. It should be noted that the absolute value of K(lH) increases with metal concentration. Therefore K('H) is negative even for concentrations close to saturation, i.e., for concentrations for which there are no more solvated electrons, and for which most of the ammonia molecules are in solvation shells of cations. Therefore, the origin of the negative value of K(lH) cannot be investigated only in the solvated electron. It should certainly be useful to have data of K(IH) for other metalammonia solutions. Conclusion It seems to us that at least the qualitative understanding of metal-ammonia solutions is improving in all the concentration ranges. However, important questions, such as the description of the MNM transition in the intermediate concentration range and the nature of the diamagnetic species in the dilute range, certainly need to receive more attention. About the experimental magnetic data, many have now been collected. However, the magnetic susceptibility is not known between about 0.5 and 2 MPM. Even in very dilute solutions it should be very useful to check the magnetic susceptibility by static and resonance techniques and for different cations. In the intermediate and concentrated ranges, valuable relaxation times studies must still be done. Acknowledgment. The authors express their thanks to Dr. A. Demortier for helpful discussions.
References and Notes R. Catterall, "Proceedings Colloque Weyl 11". J. J. Lagowski and M. J.
Sienko, Ed., Butterworths, London, 1970, pp 105-130. J. L. Dye, Ref 1, pp 1-17. J. J. Lagowski, "Electrons in Fluids", J. Jortner and N. R. Kestner, Ed., Springer-Verlag.Berlin, 1973, pp 29-36. A. Demortier, M. De Backer, and G. Lepoutre, J. Chimie Phys. 3, 380 (1972).
M. C. Justice, TWse, doctorat d'etat. Paris, 1974.
H. L. Friedman, "Ionic Solution Theory", Wiley-lntersclence, New York, N.Y., 1962. P. D. Schettler, this colloque. F. J. Tehan, M. T. Lok, and J. L. Dye, 165th National Meeting of the American Chemical Society, Dallas, Tex., 1973. C. Lambert, ref 1, pp 301-308. C. A. Hutchison and R. C. Pastor, J. Chem. Phys., 21, 1959 (1953). 0. E. O'Reilly, J. Chem. Phys., 35, 1856 (1961). R. J. Blume, Phys. Rev., 109, 1867 (1958). V. L. Pollak, J. Chem. Phys., 34, 864 (1961). H. M. McConnel and C. H. Holm. J. Chem. Phvs.. 26. 1517 (1957). J. V. Acrivos and K. S. Pitzer, J: Phys. Chem, 66, 1693 (1962). D. E. O'Reilly, J. Chem. Phys., 41, 3729 (1964). D. E. O'Reilly, J. Chem. Phys., 50, 4743 (1969). T. J. Swift, S. B. Marks, and R. A. Pinkowitz, ref 1, pp 133-144. R. A. Pinkowitz and T. J. Swift, J. Chem. Phys.. 54, 2858 (1971). J. P. Lelieur, These, doctorat d'etat, Orsay, 1972. J. P. Lelieur and P. Rigny. J. Chem. Phys.. 59, 1142 (1973). S. I. Chan, J. A. Austin, and 0. A. Paez, ref 1, pp 425-438. P. Damay, this colloque. J. P. Lelieur and P. Rigny, J. Chem. Phys., 59, 1148 (1973). E. Duval, P. Rigny, and G. Lepoutre. Chem. Phys. Lett, 2, 237 (1968). M. J. Sienko, "Metal-Ammonia Solutions", G. Lepoutre and M. J. Sienko, W. A. Benjamin, New York, N.Y., 1964, pp 23-40. J. C. Thompson, Rev. Mod. Phys., 40, 704 (1968). M. H. Cohen and J. C. Thompson, Adv. Phys., 17,857 (1968). G. Lepoutre and J. P. Lelieur, ref 1. J. V. Acrivos and N. F. Mott, Philos. Mag., 24, 19 (1971). J. V. Acrivos, Philos. Mag., 25, 717 (1972). J. Jortner and M. H. Cohen, "The Metal-Nonmetal Transition in MetalAmmonia Solutions", unpublished. J. Jortner and M. H. Cohen, J. Chem. Phys., 58,5170 (1973). S. Kirkpatrick, Phys. Rev. Lett., 27, 1722 (1971). ~
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P. Lelieur, J. Chem. Phys., 59, 3510 (1973).
P. Damay and P. D. Schettler, J. Phys. Chem.. this Issue.
P. Chieux, Phys. Lett. A, 48, 493 (1974). T. L. Hill, "Statistical Mechanics", McGraw-Hill, New York. N.Y., 1956. K. lchlkawa and J. C. Thompson, J. Chem. Phys., 59, 1680 (1973). P. Damay, These, doctorat d'etat, Paris, 1972. P. Damay, ref 3, pp 195-202. P. Debye, J. Chem. Phys., 31, 680 (1959). R. Landauer. J. Appl. Phys., 23, 779 (1952). A. N. Garroway and R. M. Cotts, Phys. Rev. A, 7,635 (1973). T. R. Hughes, J. Chem. Phys., 38, 202 (1963). S. Freed and N. Sugarman, J. Chem. Phys., 11, 354, (1943).
Discussion M. H. COHEN AND J. JORTNER. This is an interesting and valuable paper. With regard to our own work, the basic physical idea advanced by us concerning the metal-nonmetal transition in metal-ammonia solutions (MAS) is that those materials fall within the general category of systems where microscopic inhomogeneities determine electronic structure and transport properties within the transition region. We welcome the authors' agreement on this point, but, in our opinion, the question of whether the fluctuations are unimodally or bimodally distributed remains open. Only unimodal fluctuations can occur within the conventional fluctuation theory used in this paper. Cases of bimodal distributions are, however, also known to occur. Short wavelength fluctuations have been shown to be bimodally distributed in Wilson's theory of phase transitions. The highly developed droplet model of condensation contains bimodally distributed density fluctuations near the liquid-gas critical point. Many examples of clustering are known both near and unrelated to critical points. As the classical thermodynamic analysis used by Lelieur, Damay, and Lepoutre is known not to give a good account of fluctuation phenomena near critical points, and as our proposed bimodal fluctuations are closely connected to the phase separation, the present paper does not clarify the question of unimodal vs. bimodal fluctuations. Our analysis of the electrical conductivity, the optical properties, and the thermoelectric power, however, suggest strongly that bimodally distributed concentration fluctuations occur. For the Cs solutions, where there is neither a phase separation nor anomalous concentration fluctuations, and for all the solutions a t high temperatures, the approach of Lelieur, Damay, and Lepoutre will be very useful. With regard to details, it is not clear to us why the authors have supposed the Debye correlation length to be proportional to the Ornstein-Zernike correlation length. A detailed discussion of unimodal inhomogeneous materials and reduction of the theory to that of an equivalent bimodal case has already been given in our paper on expanded liquid Hg [Phys. Rev., A10.978 (1974)]. The effective medium theory is inapplicable for low values of the metallic volume fraction, C < 0.4, when the ratio of, e.g., the conductivities, is smaller than 30, as is the case for Li and Na ammonia solutions. J. P. LELIEUR, P. DAMAY,AND G . LEPOUTRE.The bimodal fluctuation model is obviously successful for the determination of some physical properties. I t seems that the existence and origin of these bimodal fluctuations have not been clarified. The bimodal fluctuation model has essentially been introduced to match the electrical conductivity. The physical picture of metal-ammonia solutions in the intermediate range (unimodal or bimodal fluctuation model) has first to consider the structural experimental data (neutron or x ray) rather than the electrical conductivity. The results of Chieux were obtained for temperatures very near the critical point. In the temperature range of Chieux's results, the classical description of Ornstein-Zernike for critical fluctuations in fluids is known t o be correct. Chieux's data follow this description. In this framework, the Ornstein-Zernike fluctuation decay length is proportional to the Debye correlation length. Both lengths express the same quantity except for a dimensionality factor [B. Widom, J. Chem. Phys., 43, 3892 (1965); Sette, Essays in Physics (1973)]. I t should also be noted that the bimodal fluctuation model for the description of the nonmetal to metal transition is applicable, according to Jortner and Cohen, only for concentrations larger than 2.3 MPM. It is clear that the experimental onset of metallic characters appears rather for concentrations of the order of 1MPM. It would be difficult to apply the bimodal fluctuation model to concentrations down to 1 MPM.
