Ion clusters in molten K2MgCl4

Volume

15, number

2

1 August 1972

CHEMICAL PHYSICS LETTERS

ION CLUSTERS

IN IMOLTEN KZiMgC14

Benson Department

of Chemistry.

R. SUNDHEIM

AJew York University, New York. Ne\v York IOOO3, USA

and Leslie

V. WOODCOCK*

Deportment of Chemistry. Uuiversity of So~~thar~lpton, Southarnptott. Uk’ Received 24 April 1372

A molecular dynamics comput;ltion on 3. model of KzhIgC1~ wherein the ionic interactions involve only central forces is reported. The predominant structure is B very well defined trigowl M&l; ion cluster. The Raman spectrum is discussed in terms of the potential of average force.

The Raman spectrum of a fused salt solution conof one mole of magnesium chIoride and two moles of potassium chloride (K2MgC14) displays four more or less resolved Raman bands, one of which is partially polarized [ 1,2] . The experimenters concluded that this spectrum apparently arose from the tetrahedral species M&l:-. Support for this conclusion was drawn from comparison with the Raman spectra of other clivalent species, namely, CdCli-, PbCl;-, and HgCl~-, as well as from heat of mixing results. The tetrahedral structure proposed for K2MgC14 was also proposed for the K,LJgBr, and K,MgI, and appeared to be the principal species in solutions where the halide to magnesium ratio varied from 3 to 5. In halide complexes of transition metal ions, the geometry and coordination number are presumably the result, in part, of angle-dependent interactions with the central ion reflecting its electronic structure. Group IIA ions, however, would not be expected to how angle-dependent interactions with Iigands. Consequently, the specific structure of a complex species in a fused salt presumably comes about through the interplay of coulombic (plus hard core) forces, the sisting

* Ramay

hfemorial

Fellow.

in the liquid state and the vagaries of thermal motion. It is not immediately clear Sow any such species finds iiself in an environment which is of definite geometry for long enough to lead to fairly sharp packing

Ramnn

lines.

One would

like to know

whether

non-

directional pair-potentials when combined with the exigencies of packing at !iquid densities can, in principle, produce a model system which will give Raman lines or whether the occurrence of Raman lines implies “chemical”, i.e., directional, bonds. The details of the average structure and fluctuations about the average for a plausible, though simplified, model of the fused salt, are accessible to molecular dynamics modeling on a computer. We report here on such calculations and on the properties of the liquid found in this way with particular emphasis cn their implications for the interpretation of the Raman spectrum. The potential was taken to be the sum of pairwise additive terms each composed of a coulombic and an exponential part. That is, l&&r) = zi zj ,2,-l

+ B exp [.4 (ai+ uj- r)] ,

where e is the electronic charge, ui and Oj are distance parameters characteristic of the “ionic radii”, Zi is the ionic valence, B = 0.338 X 10-l” erg, and .4 = 3 A-1. The values of A and P are based on the analysis of 191

Volume

CHEMICAL PHYSICS LETTERS

15, number 2 Table

ionic pnrxnetels

I

use6 in the calculations

t

= (‘4)

In 00-=4gI

bfg’+

+t

cl-

+1 -1

0.714 1.463 I.585

40.35 64.93 58.92

.&f -

1 August 1972

crystz! data by TOG and Fumi [3f. IIIverse rixth and eiehtll power terms representing dispcrsion forces f&e been omitted as well as any other pro+ion for ckctronic polarization. Values of OK+ and +t- are given by TOSS ;!nd Fumi f3] and CI~~~Z+ is tic!orminrJ by the corlditicln a);+/~~~ 2+ = K. Tables of i;xGc radii (Pauling or C oudsmit) E41 lead to K = .Zf?5. The i~div~duai ionic parameters arc given in tabtc 1. The RICK!:! is composed of 27 Mg2+ ions, 54 K” io::s, and 108 Cl- ions confined withiii a cube of fizzed miumc and subjected to the usual periodic boundary conditions. Tfre mearl kinetic energy corresponds to a tcmprattirc of82O’K and the density is 1.341 g cmm3. The method of computation is rssentitiiy the sz-ne as described previously [S 1. The preliminary results I,eported here are based ilpl>n two separate computations. For the first, the iitkafi hs!idc

initisf con~~uratjon and momenta w-e taken from an aged system composed of 108 K* and 108 Ci- ions nt lQ4S”K, ix., the melting temperature of KCl. 27 of the #* ions were removed s.nd 27 more were transformed into Mg2’ ions on ihe same sites. The volume and temperature were adjusted to the required values, and the dynamic computations resumed. Stabilization of the first order properties indicated that equilibrium

had been reached in less than 1O-12 set, and the fiidi configuration was used to initialize the production run. These computations were pfxformed at intervals of 5 X lo-t5 ac for a totat of 2.5 X 1O-12 sec. The second (production) run computation was made at steps of 1.0 X lo-l4 see for a total time of 5.0 X lo-12 sec. The mean internal energy was -1 I59 W mole- 1 (per mob: of Cl ions) and the mean pressure was 5.25 kbar. The molecular dynamics data obtained at each step after equilibrium was established were used to compute the v&ious correlation functions. After describing briefly the salient features of the resufts, we wiii turn to the overall discussion,

Fig. I. Rndiai distribution

function

for MpCIc.

Fig. 2. Radial distribution

function

for Cl-Cl”-.

The radial distribution

function

of an CY@ pair is

taken to be g,,W

= [dTip(~)/d~l/477~2~p 5

where ii&-) is the mean number of ions of type 0 around an ion of type 01in a sphere of radius I-, and pp is the number density of 8. The six different radial distribution functions describing the system are shown in figs. 14. The graph of gkrgCl(r)l fig. 1, clearly indicates a single coordination sphere centered at abcut 1.5 A. The area under this peak up to the first minimum in r2g(r) corresponds to 3.0 ions. The distance of closest approach is I.15 A and there is no indication of further ordering beyond the tist coordination sphere. The radia! flucturition function is defmed as

Voiume 15, number 2

2.

9h-1 I*

I

2

f.

3

CHEMKAL

PHYSICS LETTERS

L-. 4

5

6

7

8

YA

l-

Fi,r 3.

Radial distribution function for K-CL

ft is a measure of the broadness of the (discrete) distribution. This function $,&(~) (not shown) has the value 0.1 ion at the minimum ing&f,.&) at 2.55 A. At the maximum in &+a@) at 1S A it”is 2.25 ion. Thus the distribution is sharply peaked. The very small mean squared fluctuation at the minimum in gCr>means that not only is the average coordination number 3 but also that this is by far the predominant value (i.e., 2 or 4 are very improbable), The radial distribution function for gclcl(r), fig. 2, has a sharp peak at about 2.85 x and has an area corresponding to 2.0 nearest neighbors. There is also some interpenetration within about 4 a. Outside of this sphere the distribution is almost completely random. The curve forgKa(r), fig. 3, resembles the distribution function found in pure RCI [S, 61 except for the peaks being somewhat smaller and broader, The long range osc5lkttior.s damp out quickly. The Mg--Mg and K-K distribution functions, fig. 3, are gas-like in character, showing very little order after the “collision” peak. On the other hand,ghl, (r), fig. 3, has a tall, broad peak centered at 4.5 PFand another smearedout peak at nearly twice that distance. The radial distribution functions show unmistakab& that the magnesium ion is surrounded by three symmetrically placed chloride ions (i.e., C,, or i&h symmetry) and furthermore that the Iigand distribution is quite sharply peaked. The Cl-Cl distribution again reflects this structure in that there are just two

1 August 1972

I

L 3

e

5

6

r

7

8

Fig. 4, Radial djst~bu(~on functions for cation pairs: (a) ~QKW> (b) &&-), Cc)sag&‘). nearest neighboring chloride ions for each chloride ion and these are at the same distance (one sharp pesk). There is no possibility of obtaining these distribution functions from a Ccoordinated species, IclgCl~-, in arly symmetrical arrangement. The K--C1 distribut~oR shows that the potassium ions occur in part as KCi ion pairs and the h4gK distribution t‘unctions suaest that there are potassium ions on the exterior of the MgCiS complexes. All in all, the distribution functions s&or&y suggest a sfructure composed ofMgCl$ dusters, KC1 ion pairs, and K+ counterions. There is no way to make the resutts compatible with an IV&&cluster in tetrahedral or other configuration. Some insight into the factors favoring trigonal over tetrahedral structures can be gained by calculating the potential ener,T of isolated ion clusters. The same ionic parameters used in the dynamics calculation were used to find the minimum potential. energy and other corresponding internuclear distances (table 2). We observe that MgCl; is more stable than M&l:by an energy difference which is many times X-Tat T< 103. Neglecting the effecis of the counterion and entropy considerations, the equilibrium concentration ~st~butio~ ratio of the two competing clusters MgC@ and MgCI~ can be roughly estimated from

= exp (3.8/O. 113) at 820% ee3s . 193

Volume

15. number

CHEMICAL PHYSICS LETTERS

2 ‘Table 2

Depth of potential energy well of ion clusters

K-cl

hfg-a+ hW2

Potential energy at lninimrlm (erg X lo-l21

InternucIear distance (a)

-7% -29.62 -39.70

2.65 1.07 1.29 hlg-Cl 2.58 Cl-Cl

(linear)

hlgcl~

-55.47

(trigdnal planarj hIgCq (tetrahedral)

-5 1.67

1.33 2.30 1.50 2.45

Mg-Cl Cl-Cl Mg-Cl Cl-Cl

Turning to the Raman data, we note first that the species MgCIF would be expected to show three Raman lines, one polarized. Comparing with the experimental observation of four Raman fines, one polarized, there seems to be a significant discrepancy. However, we note that the distribution functions all call for the ion pair K-CL This species would be expected to show one Raman line although molten KC1 itself shows only a pronounced Rayleigh wing. Thus, the calculated and observed spectra are quite similar. The dynamical calculations reveal that the pairwise additive potential energy of interactions together with the problems of packing at a given density and the entropic contributions to the free energy lead to a structure which has a surprisingly well-defined symmetry. That is, the pair distribution function for Mg-Cl, which shows a sharp peak, is equal to exp (-13higCl/kT), where Wrdscl is the potential of average force between the magnesium and chloride ions. That the pair distribution function has a sharp maximum implies that there is a sharp minimum in

194

1 August 1972

the potential of average force. When the equilibrium configuration of a species as determined by the minimum in the potentiaf energy curve is characterized by a given symmetry, one may expect that the motion of the system may be described as a superposition of dispkKements corresponding to the irreducible representations of the symmetry of the potential energy functions. It is clear that one must be very cautious in attributing any conventional interpretation to force constants which may be calculated in these circumstances. We may also note that it is possible in principle to compute the polarizabilities of the species found in the dynamical calculations and use them to predict the intensities of the Raman lines. A closer analysis (in progress) of the shape of the potential of average force and of the nature of the fluctuations around this quantity should make it possible to make quantitative predictions about the shape of Raman lines. L.V.W. is pleased to acknowledge financial assistance towards travel from NATO (grant no. 379). B.R.S. acknowledges with pleasure assistance from the National Science Foundation (grant no. GP 14615).

References r11 V.A. hlaroni, E. J. Hathaway and E.J. Cairns, J. Phys. Chem. 75 (1971) 155.

I21 V.A. hfaroni, J. Chem. Phys. 55 (1971j 4759. t31 h1.P. Tosi and F.G. Fumi, J. Phys. Chem. Solids 25 (1964) 31.

[41 A. Cottcn and G. Wilkinson, Advanced inorganic chemistry, 1st Ed. (Interscience, New York, 1964) p_ 43. ISI L.V. Woodccck, Chem. Phys. Letters 10 (1971) 257. [61 L.V. Woodcock and K. Sinser, Trans. Fanday Sot. 67 (1971) 12.