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Ordering of guest-molecule dipoles in the structure I clathrate hydrate of trimethylene oxides
CHEMICAL PHYSICS 3 (1974) 239-247.0
NORTWHOLLAND
PUBLISHING COMPANY
ORDERING OF GUEST-MOLECULE DIPOLES IN THE STRUCTURE I CLATHRATE HYDRATE OF TRIMETH YLENE OXIDE* S.R. COUGH,S.K. CARG and D.W. DAVIDSON Division of Chemistry.
Na!ional Resemch
Council of Cam&.
Ottawa,
Gmada KlA
OR9
Received 22 October 1973 The rotationl mobility of encaged trimethylene oxide (TMO) molecules was studied down to 1.8”K by subMHz dielectric measurements of the structure I Hz0 chthrate and by proton magnetic resonance meaarrements of the corresponding D20 clathrate. The results indicate that below a transitional temperature range about 10S°K most TM0 dipoles assume parallel alignment along the 5 axes of the cages. Below the transition the proton second moment
suggests the presence of hindered rotation of TM0 about its polar axis until the rigid-lattice condition is reached below 5°K. Some residual very broad dielectric absorption (activation energy 2.1 kcatfmole) persists to very low temperatures. Guest-guest and guest-host interaction energies are wlculated for simple models.
1. Introduction
Trimethyleneoxide (TMO) forms two distinct hydrates [l-3] which melt incongruently at -9 and -21°C. The fast has been identified [2] as a clathrate of von Stackelberg’s structure II [4] with a composition close to TM0 - 17 H20. Previous dielectric studies [l] of the water dispersion region of the second hydrate gave a composition within 1 to 2% of TM0 - 7: Hz0 which suggested that this hydrate is a structure 1 [4] clathrate in which only the larger of the two kinds of cage is occupied by TM0 molecules. This structure is now confirmed. We present results of dielectric and continuous-wave NMR studies to low temperatures of the rotational mobility of TM0 molecules encaged in the structure I hydrate. These show, for the first time in a clathrate hydrate, the presence of an ordering transition in which nearest neighbor guest-molecule dipoles acquire a predominantly parallel alignment.
2. Experimental
methods and results
2.1. Structure and composition of the TMO-rich hydrate
This hydrate is confiied
as structure I by the X-ray
measurements of L.D. Caivert, to whom we are indebted for the following results. A solution ofcomposition TM0 - 725H,O, annealed for 140 hours at -30 + S’C, gave, by methods previously outlined (21, a powder diffraction pattern in which no ice Ih or structure II hydrate lines were detected. Structure I was identified, with the cubic cell parameter u = 12.02 f 0.05 A at -13O’C and 12.15 5 0.02 A at -30°C. The limits shown are the spreads of measurements of eight lines. Lattice parameters near O’C of 18 other structure 1 hydrates (mainly from von Stackelberg and Jahns [S]) range from 11.97 to 12.14 A, the average being 12.05 A. Thus TM0 appears 10 be large enough to expand slightly the usual unit cell dimension. The large size of the TM0 molecule (largest van der Waals diameter 5.5 a) makes appreciable occupancy of the smaIl cages (mean free diameter 5.1 A) unlikely. The composition TM0 .61_Hz0 recently reported [3] from thermalanalysis, on the other hand, requires more than 50% occupancy of the small cages. Formation of structure I from structure II and TMQrich liquid is slow and erroneous analysis may result from lack
of equilibration. In the discussion which follows we assumethat only the large cages are occupied.
’ Issued as NRCC No. 13897.
-:“O -
1..
-
. .
3.2. Dkhiic
.
S.R. Cough et a& ckdering of gue.usin the smutwe I ti0
hydme
.
meosutetinrs
l4O .’ Measurements were made at frequencies between
I JO HZ and 1 MHz and at temperatures between I .8 and 200°K as previously described [G]. Hydrate sampleswere formed in a cylindricallowtemperaturedielectric cell [7] by conditioning solu_ tions of composition close to the ideal TM0 - 75 H20 at -3O’C for about a week, after which the dielectric properties were changing at an inappreciable rate. Probably as a result of the volatihty of TMO, none of five samples prepared was free of structure II-after the conditioning period, as shown both by the behavior in the water relaxation region [ 11 at high temperatures and by the presence of absorption characteristic of TM0 encaged @ structure II [61 at low temperatures. Permittivity results shown in fig. 1 refer to a sample which contained a volume fraction x = 0.08 of structure II, the complex permittivity e+(I) of “pure” structure I being estimated from e’(exp) = (l-z+*(l)
+X*(H)
,
I2
IO
1
&’ a
(1)
where B*(U) at the relevantfrequencies and temperatures was obtained from the previous study of struthue II (61. Results for higher structure 11content were generally consistent with eq. (1). No appreciable dependence of reIaxation times on sample composition Fig. 1. (A) Temperahue dependence of to1 = epl (heavy line) was observed (cf. fig. 3). and dispersion curves of structure 1 TM0 hydrate. (6) Abworp The dielectric behavior in the_water relaxation retion curves at a munbct of frequencies. Dashed curves for the gion at relatively high temperatures agreed closely structure II hydrate are from ref. [6]. with that found before [ 11. _ The main results of the permittivity measurements are shown in fq. 1A. The relatively large perrnittivity e,. 1 measured on the high-frequency side of the water dispersion region reachesa maximumvalue of 13 at ,about 135°K and falls fairly sharply at temperatures around 1CtS’Kto an almost temperature-independent value of about P.8 below 7S’K. Below 100°K the per. mittivity again becomes dependent on frequency at sub-megahertz frequencies. Here E,,~ is determined as eo2, the low frequency permittivity limit of this +cond dispersionregion, which is assignedto orientatinalielaxation of entiged TM0 mole+les. is ‘shdwk ‘- - Th& conesponding dielectric.ab,sorp~jon fpr ,somi frequencks in 0; lB,which also shdws the .E’ .kell:tip+qd absbrption at 10 kHz by TM0 in strut, [email protected]. conipkx pcnuititvituplot8 of the TM0 d@rsion m_e ?I [i],@!e that the additional p&k at jIMoK gion in the structure I hydrate. Frc&a~&r arc &own in ,_ ‘. ,_ __:k&. : .’ _- L .;ti t+.lM~ loss curve re<J from the rapid incre~ : _._
SAMPLE
slow for ~02 to be determined. Below about 40°K. it is possible to define e, 2, the high-frequency limit of the permittivity, at the temperature-insensitive vaiue of 3.45 * 0.15. This value, as in other clathrate hydrates [6,8,9], probably includes an important contribution fromrotationaloscillationsof the polar guest molecules at far infrared frequencies. The amplitudes of the dispersion and loss measured at a number of fmed temperatures between 100 and
Y. STmJCTUFE 1
+ 0 0
55 70 70
l
92
10 0 f/kHZ
4O’K were found to de&ease slowly with time, a behavior not observedin other clathrate hydrates studied. For example, the time dependence of the 1 IWz loss
a 9
I-
\
0.1 -
10=/T am
I2
I
I
14
16
,
I 18
20
+ 0
of a sample which was rapidly cooled from 140°K by immersion in liquid nitrogen could be represented during the course of 290 hours by
+
\ 22
Fig. 3. Temperature dependence of rhe frequency of maximum absorption by TM0 in the struchue I hydrate for Samples of different composition.
in Eo2
-
241
et al. Ckdwingof guests in the structureI ZUO hydrate
S R Cough
E-2 with increasing temperature
in this range
and that the loss is nor greatest at I MHz at this temperature.) The TM0 absorption is much broader than that characteristic of a single relaxation time and the complex permittivity loci (fig. 2) are flat and asymmetric. These features are invariably present in the low-temperature relaxation behavior of polar guest molecules in clatluate hydrates [6,8,9]. The temperature dependence of the frequency of maximum absorption is shown for samples of various composition in fB.3. The “activation energy” EA of 2.1 kca!/mole is easily the largest so far found for reorientation of a
guest molecule in a clathrate hydrate (table 1). Below about 55°K the retaxation rate becomes too
~“(1 MHz, 77.4OK) = 0.173 + 0.228 exp(-0.157t) + 0.175 exp(-O.O112t), where t is in hours. This shows, at least on the time scale of weeks, that the absorption ultimately decays to a still appreciable equilibrium value at this temperature. At higher temperatures equilibration is faster, so that samples cooled through the transition region over several hours exhibit losses which do not greatly exceed the equilibrium values. The equilibrium losses are
estimated to be, for example, 84% of those shown (fii. 1B) for the 1 MHz cooliog curve and 67% of those shown for the separate cooling run at 0.1 and 10 kHz. Similar considerations apply to e’ - e_, 2. . 2.3. NMR measurements
Samples of structure I clathrate were prepared
Table 1 Dkkctric parameters for mme enclathrated molecules HydratC
k.2
\1
ethyleneoxide
II
tetIahydrofuran trimethykne o@de ratone
trimcthykne oxide
cydobutulonc .
Temp. CK) of maximum 1 kHz absorption
&A
28.1
Ref.
cL;avmok)
4.1 3.45
51.5
1.4 21
PJ this wtxk
3.5 3.7 4.0 3.6
21.6 12.1 19.9 31.8
0.91 0.41 1.02 1.4..
161
.
5:. 191 .
.
242
S.R
Cough et al., Ordering ofguests
II
TH0.7.2D20
0
rrro.7.5aD20
----5TmJcTu4c
0
50
100
Fig. 4. Second moment of st~c~urc I debtcrate.
243-
K
150
II
200
the proton absorption of l?dO
in the structure
I lM0
hydrate
with 99.7 atom % D20 by extensive conditioning and the derivative of the proton absorption signal obtained as for the ethylene oxide deuterate [8]. Results are given for two samples which consisted of substantially pure structure I. One sample was measured with a crosszd-coil spectrometer at temperatures down to 77°K and the other down to 1.8’K with a single coil coupled to a marginal oscillator [8]. A very narrow line obsarved in the fist sample at temperatures above about 170°K corresponded to the presence of less than 3% of TM0 as liquid at any temperature below 240°K. Saturation was not encountered at the = 5 mG rf levels used. Second moments were corrected for relatively small modulation broadening. Second moments of TM0 in the structure I D20 lattice are plotted, along with those of structure II [lo], in fig. 4. The second moment of structure I is seen to vary appreciably with temperature over almost the who!e temperature range of the measurements. It rises quite sharply with decrease of temperature through the same region where e02 is rapidly falling, passes through a plateau-like region between 95 and 70’K,
A
K
Pig. 5.
Dcrivativcabsorption tine &apes of the proton rcY)nan(Pof structure 1 TM0 deuterate.
243
S.R. Cough et aL, edering olguests in the sfrucnwe I TM0 hydrate
and then rises gradually to the rigid lattice value which is attained below about 5°K. In the plateau region the second moment is close to half the rigid lattice value of 13.4 + 0.4 G2. The latter is not significantly different from the rigid lattice moment of 13.7 G2 found [ 111 for the structure II deuterate. The variation of the derivative line shape with temperature is illustrated in fig. 5. At temperatures above about 110’K the line is featureless with a shape which differs from gaussian in the presence of less absorption in the wings. In this region the experimental SACond moment approaches half that of a gausdan line of the same width (as measured by the separation of the extrema of the derivative curve). With decrease of temperature the line shape gradually evolves into a doublet which is best defined near SOOKand which becomes obscured at lower temperatures by increasing absorption near the band center. The rigid lattice pattern is similar to those found for TM0 and cyclobutanone in the structure II D20 lattices [ 111.
3. Discussion 3.1. Rationale
2
.e
0.06
I
0.04 x/a
0.02
0
0
30
60
90
120
150
180
+ (deal Fig. 6. Equipotential contours of Lmmud-Jones guest-lattice interaction energy (in kcallmole) of TM.0 aligned with the 4 axis of the 14-hedral cage. Coordinates are angle of rotation about the polar axis and displacement of center of mass (ii units of LT.the cell dimension) from the center of ?he age.
of the transition
The structure I hydrate of TM0 is distinguished
from the previously-studied clathrate hydrates by a fairly abrupt fall in eo2 at relatively high temperatures and by a corresponding rise to a plateau in second moment at much less than the rigid lattice value. Both features suggest the presence of a transition resulting from the gradual ordering of the orientations of TM0 molecules with cooling through the region of 105°K. The dielectric behavior is similar to that found, for example, during the transition near 178°K in the F_ quinol &&rate of HCN which, as confirmed by specific heat studies, marks the onset of parallel ordering of HCN dipoles in chains as a result of electric dipole coupling between the guest molecules [ 121. Such an ordering of guestdipoles (along the 3 cage axis) is potentially possible in structure I hydrates provided alignment with the cage axis is favoured, or at least permitted, by the interactions of the guest molecule with the water molecules of the cage. In the case of ethy lene oxide (EO) hydrate, a degree of preference for such alignment was invoked to account for the electron density distribution given by the X-ray study
[13] at 248”K, but no clear evidence for such preference was found [8] in the dielectric and NMR behaviour of EO hydrate. The shape of TM0 appears to be more favourable than that of EO to alignment of the symmetry axis with the cage axis. A rough model of the short range van der Waals interactions between TM0 and the water molecules of the cage (appendix A) suggests that the most stable configurations are coaxial ones in which the 0 atom of TM0 approaches the center of a hexagonal ring of water molecules untii it is as near to van der Waals contact with the water molecules as permitted by repulsion between the (LH atoms and adjacent water molecules. Six potential minima (fig. 6) correspond to rotation about the axis so as to bring the “plane” of TM0 nearly into the planes through the cage center which contain successive pairs of opposite water molecules of the hexagon. The symmetry of the above model of the van der Waals interactions is distorted by the electrostatic guest-host interactions. The resultant electrostatic fields of the dipole moments of the water molecules, which deperid on the details of the orientations of the
244
S R Gough et aL, Ordering of guests in the structure I TM0 hydmte
Table 2 Resultant dipole fiilds in kes V/cm at axial positions within
tile 14-h&al cages (averages of 60 cage confwrarions) Axial displacement from cageanter 0
0.03a 0.060 0.090 0.12a
Total
iqb)
-iE,,_.I =)
(x) field 18 27
46 70 100
+ 0.7 +9
I1 14
14 2t
+17 +25 +31
21 30 41
38 60 85
3 Arithmetic mean axial component. ~Mennaxial cornponenc without regard to sign. 3 Mean amplitude of Md perpendicular to axis.
individual water molecules of the cage, are relatively small at positions near the cage center [ 141. With the dipole moment of the water molecule in the hydrate lattice taken as 2.6 D, a value calculated for ice [ 151, the mean field at the center of the 14hedral cage given by averaging 60 randomly-generated cage configgations was found to be 18 kes V/cm, less than half the field of a single Water dipole a cage radius awayAverage fields and field components at a number of positions (in units of u, the cell dimension) are given in table 2. The effective position of the dipole moment in ThlO is removed from the center of mass (c.m.) by perhaps 0.5 A in the direction of the 0 atom. The displacement of the c.m. by 0.060 from the cage center favored by the van der Waals forces (fig. 6) therefore corresponds to a dipole position about O.lOrr from the cage center. Here the resultant average field of 80 kes V/cm includes a mean component of some 27 kes V/cm in the direction of displacement which is opposite to the dipole direction. This results in an electrostatic energy which averages almost 1 kcal/mole higher than at the cage center. In this case, there is a statistical tendency for the dipolar fields to counteract the displacements favoured by the van der Waals interactions. It is unlikely that this conclusion is affected by inadequacy of the point dipole approximation, although it is emphasized that fields of quadrupole and higher moments of the water molecules have been ignored [ 141. Interactions between the guest-molecule dipoles favour parallel ordering of nearest-neighbour guest dipoles. For the ideal lattice consisting only of non-
polarizable dipoles located at the centers of the large cages the most stable configuration is that in which adjacent dipoles point in the same sense along chains extending in the x, y. and z directions, the sense alternating between adjacent chains. For this fully ordered lattice the field in the direction of an individual dipole is found in appendix B to be S = 38.47 p/a3. The dipole moment, 1.93 D for the isolated TM0 molecule, is effectively enhanced by the reaction field factor tosbout 2.3 D. On the other hand the contributions to S are reduced by the shielding effect produced by the polarizability of the intervening medium. For large separations in the rigid lattice the shielding factor is l/e-2. This factor is certainly too small, however, for the dominant contributions to S from the two nearest-neighbour dipoles.The field due to dipoles induced by a nearest-neighbour dipole in its sheath of 24 water molecules is tantamount to a shielding factor [ 141 of 1Il.6. We approximate the shielding effects by separating the lattice sum into 32 p/l .6n3, contributed by nearest neighbours, and 6.47 ~13.45 a3 contributed by all other dipoles. These assumptions lead to a field of some 29 kes V/cm in the direction of alignment of an ordered dipole, a field almost twice the mean field of the dipoles of the water molecules at the cage center. For the ordering transition to occur it is apparent that guest-guest dipole interactions must be more
important than the electrostatic interactions with the water molecules in most cages and therefore that the fields of the quadrupole moments of the water molecules do not seriously dominate those of the dipole moments. This in turn implies that the quadrupole moment tensor of the fully hydrogen-bonded water molecule is considerably more isotropic than in the isolated water molecule [ 141. The persistence of some disorder to temperatures much below the general ordering range is seen to be a likely consequence of exceptionally large fields of the water molecules in some cages. The time dependence observed is possibly the result of some increase in order associated with a slow irreversible readjustment of those water-molecule configurations which give large fields. The above field of 29 kes V/cm gives an energy of -0.72 kcallmole for the completely ordered state, which suggests that the ordering transition should occur at T S=720/R C=360°K. It is not surprising that
S.R. Gough et aL. Ordering o/guests in the structure I 7MO hydrate
the perturbing effects of the fields of the water molecules result in a lower transition temperature. 3.2. Roton magnetic resonance second moments The experimental second moment of the rigid-lattice proton absorption of structure I TM0 deuterate is about 1 G* smaller than the value of 14.45 G2 calculated with no vibrational corrections. The latter includes an intramolecular contribution of 13.7, G2 derived from the gas-phase microwave structure [ 161 of TMO. Inter-TM0 and TMO-D,O contributions of 0.37 and 0.38 C2, respectively, were evaluated as for ethylene oxide deuterate [8] except for the contribution from nearest-neighbour TM0 molecules. Here the assumption of parallel alignment gave a smaller contribution (0.13 G2) than the assumption [8] of random orientations. At the highest temperatures the second moment (e.g., 0.31 i 0.04 G2 at 243°K) approaches the value 0.24 G2 calculated for the inter-guest contribution when the reorientation is isotropic. It is possible to interpret the second moment in
the ‘plateau” temperature range in terms of reorientation about the polar axis of TM0 between the preferred positions determined by cage geometry. In conformity with fig. 6 it is assumed that there are six such positions and that these fall into a set of two and a set of four equivalent sites. The effect of rapid interchange between preferred positions on the rigid lattice second moment is related in appendix C to the difference of energy of the two sets for the case in which interproton vectors are perpendicular to the axis of rotation. This condition is satisfied for the interproton vectors of each CH2 group and for the vectors connecting the protons of different a-CH2 groups. In toto the corresponding interactions contribute about 10.9 G* to the rigid lattice second moment or 80% of the whole intramolecular moment. The same rotational jumps will reduce the remaining intra- and inter-molecular contributions to the second moment to a different extent. Jiowever, it is clear that the required second moment x 7 C2 in the plateau region around 80°K may be achieved by a reduction of the 10.9 G2 contribution by about afictoroftw. From eq. (9) of appendix C this requires an occupancy factor p = 0.36 for each position of the Z-fold set, which is then more stable by AE Z=250 Cal/mole than
24.5
each position of the 4-fold set. (A reduction factor of the right order (7116) resulting fromp = 0 and occupancy of the 4-fold sites only is a less likelyaltemative.) Although the results of this model are of Jimited quantitative significance, they serve to show that the second moment below the transition is consistent with the type of order proposed.
Appendices A. Shownznge guest-host interactions The crude Lennard-Jones model previously used [8] to estimate the position and orientation dependence of the energy of ethylene oxide in the 14.hedral cage has been employed to estimate the potential energy of short-range interaction between TM0 and the water molecules of the cage. The atom-atom Lennard-Jones parameters were the same as before and again the hydrogen atoms of the water molecules were neglected. Fig. 6 shows equipotential contours for encaged TM0 molecules aligned with the 4 cage axis as a function of displacement of center of mass along this axis from the cage center (in units of the cell dimension) and of rotation about the polar axis of the molecule. Positive displacements refer to approach of the oxygen atom to the nearest hexagon ring and + = 0 to coplanarity of the TM0 “ring” with the plane of symmetry of the cage which contains the two “c” water molecules [13] of this hexagon. One complete rotation is seen to eccompass six positions of potential minimum spaced at 60’ intervals in which the center of mass is some 0.7 A from the cage center. Witbin the limitations of the mode1 the energies of the 6 preferred orientations are not greatly different, although the two-fold set is slightly more stable than the four-fold set. The least energy required for Interchange between the sets is * 0.4 kcallmole, while that required for interchange within the &et is z 0.6 kcal/mole. The model is too crude for these energies to be of great significance. However, the conclusions that the 0 end of the molecule is appreciabiy displaced toward the center of a hexagon ring, that there are 6 preferred orientations of comparable energy, and that the barriers to rotation are less than 1 kcallmole are not sensitively dependent on the particular potential function and parameters employed.
246
SR
Gough et aI.. Ordering of guests in the smtcture I TM0 hydmte
B. Guest dipole lattice sums
The cubic unit cell (space group Pm3n) of the structure I hydrate lattice defined by the average positions of the 46 water molecules contains six 14hedral cages centered at (4, a, )), (5, *. $), (0, ), Oh (0. 2, 01, (), 0,;). and (i. 0.4). The cage symmetry is 32 m with the 2 axis coincident with the line joining the centers of nearest neighbor cages. Cage centers lie on lines parallel to the three orthogonal axes. We locate a guest dipole at the center of each cage and evaluate the field at the site of a dipole arising from other guest dipoles for the condition that all the dipoles in a Qne of nearest neighbors point in the same direction along that line. The symmetry is such that dipoles pointing along lines parallel to the x and y axes produce no net field at the site of a L dipole so that only dipoles lying in parallel chains need be considered. We consider first the “antiferroelectric” case in which the sense of dipoles in adjacent parallel chains alternates between t and -z. The problem is then the evaluation of the fields of two sub-lattices containing, respectively, dipoles which are parallel (p) and antiparallel (a) to the dip016 at which the field is evaluated: S=S,_S,J+
c ax
[ e,,)
I
(2)
=8-t.
The individual sums of eq. (2) are conditionally convergent, i.e., depend on the order of summation and therefore on the shape of the crystal. The difference of the sums is, however, absolutely convergent since the “antiferroelectric” crystal possesses no net pOlilliZ3tiOIl.
Because of the slowness of convergence
ti ki exp (2nIlj,o)-1
for the parallel lattice, and hjjCOS(i+J>ll
S(a,&#O)
= 8n21.c d a,3 i,i
exp (2nhija)
for direct
- 1
(4)
for the antiparallel lattice. The summations are now over square lattices which are the reciprocals of the 2-dimension4 lattices which form the x. y planes: hii = (i2 +j2)f where i, j take on integral values, 0, kl, k2, . . . , with i = j = 0 excluded. Both series rapidly converge to give S(p,h#O)
= 82.289p/aj
S(a,&#O)
= -19.043
,
p/a;
(5)
after 13 X 13 terms. For hz = 0, S(p, b = 0) = -9.0336
p/a:
(6)
[ 181. Moreover,
since reversal of the antiparallel lattice based on u. Thus S(a,%=O)=
where the summations are over tetragonal cells, each cell being identified by integral coordinates X,, X,,, h in terms of the basis set a,, a,, (=a,), a,. in the first sum, from which Xx = hy = 4 = 0 is excluded, pzp = Ai + A2 + a2 *; h the second pza = 9 + a2$ In both, as in cos 13, = (xr+$)2+&+;) c&IpA,4=az/ax
= 8* a,
S(p, 4 =0) + S(a, X, = 0) = -9.0336
XP
--c h da1 2P2kOS
Jlij
S(p, h#O)
has already been evaluated
2p2;;0AP) h
summation we resort to the method of de Wette [17] in which each sum of eq. (2) is separated into a sum over & = 0 and a sum over h, # 0. The latter may be replaced by
-16.5173IJa?
p/as,
dipoles gives a single
.
Addition of the contributions of eqs. (S)-(7) the total field for alternating chains of dipoles S = 108.82 p/a2 = 38.474 p/a3 .
(7) gives
09
For the “ferroelectric” case reversal of the sign of S, in eq. (2) givesS = 13327 p/a3, in which the numerical factor may be compared with the value of 13.32 gken by de Wette [ 191 for a single tetragonal lattice with u = i. From the method of summation the result for this polar lattice is for a slab&aped crystal with the polar axis perpendicular to the surface of the slab. A polarization contribution of 4nrJo3 must therefore be added.
S.R.
Gough
et aL,
Ordering
of guests
C. Second moment for rotational flips between six preferred orientations
structwe
I nf0
[ 11 R.E. Hawkins and D.W. Davidson, 1. Phys. Chem. 70 (1966) 1889. (21 D.F. Sargent and L.D. Calvert, J. Phys. Chem. 70 (196612689. [ 31 J.C. Rosso and L. Carbonnel, Compt. Rend. Acad. Sci
(Paris) 274C (1972) 1108. 141 M. von Stackelberg and H.R. Mulfer, 2. Elektrochem.
d(t)>),
58 (1954) 162. ]6] S.R. Cough, RE. Hawkins, B. Morris and D.W. Davidson, J. Phyr Chem. 77 (1974) 2969. [I] S.R Cough. Can. J. Chem. 50 (1972) 3046. [S] S.K. Carg, B. Mcrris and D.W. Davidxln. J. Chem. Sot.
Faraday Tranr II 68 (1972) 481. [9] B. Morrisand D.W. Davidson, Can. I. Chem. 49 (1971) 1243. [ 101 S.K. Garg and D.W. Davison, In: Physics and chemistry
1Li.
where pi specifies the degree of occupancy of site i and tii= fio+(i-l)7r/3
9
i=
25.
[ 51 hf. van Stackelberg and W. JamMS, Z. Elekirochem.
where 0 is the angle between the magnetic field and the axis of rotation and ti the angle of rotation of the H-H vector in the plane perpendicular to the rotation. The expectation value Cpjsin2
241
References
58 (1954)
(sin2 9(t))=
hj-dnate
2-fold site, the reduction factor may assume values between $ for AE = 0 and 1 for AE > RT.
We consider the effect of rapid uniati rotational jumps between six possible orientations on the contribution to the proton second moment associated with inter-proton vectors which are perpendicular to the axis of rotation. The reduction factor by which the polycrystalline rigid lattice second moment is to be multiplied is [20] : (I -3tin~ehrl*
in the
of ice, ed. E. Whalley (Roy. Sot. Canada, 0tIaw-a. 1973). [ 1 l] S.K. Carg and D.W. Davidson, Chem. Phys. Letters 13 (1972) 73. [ 121 T. Malsuo, H. Sup and S. Seki, I. Phys. Sot. Japan 30
1,6.
To conform to fig. 6, we take pt = p4 = p for the 2.fold sites and p2 = pg = ps = pg = (1 - 2p)/4, whence
(1971) 785.
[ 131 RK. McMullanand G.A. Jeffxy, [ 14) [ 151
(sin2$(t)>=i-qp+(3p-4)sin2JI0.
[16]
Pol crystalline averaging with the help of sin2 Go = f , &=$,ao=i h gives 9and==
[ 171
reduction factor = yp* - zp * & .
[ 181
(9)
Since p = [2+4 exp (-M/RT)] -1 where AE = E(4)-E(2). the excess energy of a Cfold site over a
J. Cbem. Phyr 42 (1965) 2725. D.W. Davidson, Can. 1. Chem. 49 (1971) 1224. C.A. Coukon and D. Eisenberg. Proc. Roy. Sot. 291A (1966) 445. S.l. Ghan, J. Zinn and W.D. Cwinn, J. Chem. Phys 34 (1961) 1319. F.W. de Welle and G.E. Schacher. Pbyr Rev. 137A (1965) 78; MR. Philpott. J. Chem. Phys 58 (1973) 588. B.hl.E. van der Hoff and CC. Benron.Can. J. Phys. 31
(1953) 1087. 1191 F.W. de Weltc. Phys Rev. 123 (1961) 103. [20] D.C. Look and I J. Lowe, J. Chem. Phys. 44 (1966) 3437.
NORTWHOLLAND
PUBLISHING COMPANY
ORDERING OF GUEST-MOLECULE DIPOLES IN THE STRUCTURE I CLATHRATE HYDRATE OF TRIMETH YLENE OXIDE* S.R. COUGH,S.K. CARG and D.W. DAVIDSON Division of Chemistry.
Na!ional Resemch
Council of Cam&.
Ottawa,
Gmada KlA
OR9
Received 22 October 1973 The rotationl mobility of encaged trimethylene oxide (TMO) molecules was studied down to 1.8”K by subMHz dielectric measurements of the structure I Hz0 chthrate and by proton magnetic resonance meaarrements of the corresponding D20 clathrate. The results indicate that below a transitional temperature range about 10S°K most TM0 dipoles assume parallel alignment along the 5 axes of the cages. Below the transition the proton second moment
suggests the presence of hindered rotation of TM0 about its polar axis until the rigid-lattice condition is reached below 5°K. Some residual very broad dielectric absorption (activation energy 2.1 kcatfmole) persists to very low temperatures. Guest-guest and guest-host interaction energies are wlculated for simple models.
1. Introduction
Trimethyleneoxide (TMO) forms two distinct hydrates [l-3] which melt incongruently at -9 and -21°C. The fast has been identified [2] as a clathrate of von Stackelberg’s structure II [4] with a composition close to TM0 - 17 H20. Previous dielectric studies [l] of the water dispersion region of the second hydrate gave a composition within 1 to 2% of TM0 - 7: Hz0 which suggested that this hydrate is a structure 1 [4] clathrate in which only the larger of the two kinds of cage is occupied by TM0 molecules. This structure is now confirmed. We present results of dielectric and continuous-wave NMR studies to low temperatures of the rotational mobility of TM0 molecules encaged in the structure I hydrate. These show, for the first time in a clathrate hydrate, the presence of an ordering transition in which nearest neighbor guest-molecule dipoles acquire a predominantly parallel alignment.
2. Experimental
methods and results
2.1. Structure and composition of the TMO-rich hydrate
This hydrate is confiied
as structure I by the X-ray
measurements of L.D. Caivert, to whom we are indebted for the following results. A solution ofcomposition TM0 - 725H,O, annealed for 140 hours at -30 + S’C, gave, by methods previously outlined (21, a powder diffraction pattern in which no ice Ih or structure II hydrate lines were detected. Structure I was identified, with the cubic cell parameter u = 12.02 f 0.05 A at -13O’C and 12.15 5 0.02 A at -30°C. The limits shown are the spreads of measurements of eight lines. Lattice parameters near O’C of 18 other structure 1 hydrates (mainly from von Stackelberg and Jahns [S]) range from 11.97 to 12.14 A, the average being 12.05 A. Thus TM0 appears 10 be large enough to expand slightly the usual unit cell dimension. The large size of the TM0 molecule (largest van der Waals diameter 5.5 a) makes appreciable occupancy of the smaIl cages (mean free diameter 5.1 A) unlikely. The composition TM0 .61_Hz0 recently reported [3] from thermalanalysis, on the other hand, requires more than 50% occupancy of the small cages. Formation of structure I from structure II and TMQrich liquid is slow and erroneous analysis may result from lack
of equilibration. In the discussion which follows we assumethat only the large cages are occupied.
’ Issued as NRCC No. 13897.
-:“O -
1..
-
. .
3.2. Dkhiic
.
S.R. Cough et a& ckdering of gue.usin the smutwe I ti0
hydme
.
meosutetinrs
l4O .’ Measurements were made at frequencies between
I JO HZ and 1 MHz and at temperatures between I .8 and 200°K as previously described [G]. Hydrate sampleswere formed in a cylindricallowtemperaturedielectric cell [7] by conditioning solu_ tions of composition close to the ideal TM0 - 75 H20 at -3O’C for about a week, after which the dielectric properties were changing at an inappreciable rate. Probably as a result of the volatihty of TMO, none of five samples prepared was free of structure II-after the conditioning period, as shown both by the behavior in the water relaxation region [ 11 at high temperatures and by the presence of absorption characteristic of TM0 encaged @ structure II [61 at low temperatures. Permittivity results shown in fig. 1 refer to a sample which contained a volume fraction x = 0.08 of structure II, the complex permittivity e+(I) of “pure” structure I being estimated from e’(exp) = (l-z+*(l)
+X*(H)
,
I2
IO
1
&’ a
(1)
where B*(U) at the relevantfrequencies and temperatures was obtained from the previous study of struthue II (61. Results for higher structure 11content were generally consistent with eq. (1). No appreciable dependence of reIaxation times on sample composition Fig. 1. (A) Temperahue dependence of to1 = epl (heavy line) was observed (cf. fig. 3). and dispersion curves of structure 1 TM0 hydrate. (6) Abworp The dielectric behavior in the_water relaxation retion curves at a munbct of frequencies. Dashed curves for the gion at relatively high temperatures agreed closely structure II hydrate are from ref. [6]. with that found before [ 11. _ The main results of the permittivity measurements are shown in fq. 1A. The relatively large perrnittivity e,. 1 measured on the high-frequency side of the water dispersion region reachesa maximumvalue of 13 at ,about 135°K and falls fairly sharply at temperatures around 1CtS’Kto an almost temperature-independent value of about P.8 below 7S’K. Below 100°K the per. mittivity again becomes dependent on frequency at sub-megahertz frequencies. Here E,,~ is determined as eo2, the low frequency permittivity limit of this +cond dispersionregion, which is assignedto orientatinalielaxation of entiged TM0 mole+les. is ‘shdwk ‘- - Th& conesponding dielectric.ab,sorp~jon fpr ,somi frequencks in 0; lB,which also shdws the .E’ .kell:tip+qd absbrption at 10 kHz by TM0 in strut, [email protected]. conipkx pcnuititvituplot8 of the TM0 d@rsion m_e ?I [i],@!e that the additional p&k at jIMoK gion in the structure I hydrate. Frc&a~&r arc &own in ,_ ‘. ,_ __:k&. : .’ _- L .;ti t+.lM~ loss curve re<J from the rapid incre~ : _._
SAMPLE
slow for ~02 to be determined. Below about 40°K. it is possible to define e, 2, the high-frequency limit of the permittivity, at the temperature-insensitive vaiue of 3.45 * 0.15. This value, as in other clathrate hydrates [6,8,9], probably includes an important contribution fromrotationaloscillationsof the polar guest molecules at far infrared frequencies. The amplitudes of the dispersion and loss measured at a number of fmed temperatures between 100 and
Y. STmJCTUFE 1
+ 0 0
55 70 70
l
92
10 0 f/kHZ
4O’K were found to de&ease slowly with time, a behavior not observedin other clathrate hydrates studied. For example, the time dependence of the 1 IWz loss
a 9
I-
\
0.1 -
10=/T am
I2
I
I
14
16
,
I 18
20
+ 0
of a sample which was rapidly cooled from 140°K by immersion in liquid nitrogen could be represented during the course of 290 hours by
+
\ 22
Fig. 3. Temperature dependence of rhe frequency of maximum absorption by TM0 in the struchue I hydrate for Samples of different composition.
in Eo2
-
241
et al. Ckdwingof guests in the structureI ZUO hydrate
S R Cough
E-2 with increasing temperature
in this range
and that the loss is nor greatest at I MHz at this temperature.) The TM0 absorption is much broader than that characteristic of a single relaxation time and the complex permittivity loci (fig. 2) are flat and asymmetric. These features are invariably present in the low-temperature relaxation behavior of polar guest molecules in clatluate hydrates [6,8,9]. The temperature dependence of the frequency of maximum absorption is shown for samples of various composition in fB.3. The “activation energy” EA of 2.1 kca!/mole is easily the largest so far found for reorientation of a
guest molecule in a clathrate hydrate (table 1). Below about 55°K the retaxation rate becomes too
~“(1 MHz, 77.4OK) = 0.173 + 0.228 exp(-0.157t) + 0.175 exp(-O.O112t), where t is in hours. This shows, at least on the time scale of weeks, that the absorption ultimately decays to a still appreciable equilibrium value at this temperature. At higher temperatures equilibration is faster, so that samples cooled through the transition region over several hours exhibit losses which do not greatly exceed the equilibrium values. The equilibrium losses are
estimated to be, for example, 84% of those shown (fii. 1B) for the 1 MHz cooliog curve and 67% of those shown for the separate cooling run at 0.1 and 10 kHz. Similar considerations apply to e’ - e_, 2. . 2.3. NMR measurements
Samples of structure I clathrate were prepared
Table 1 Dkkctric parameters for mme enclathrated molecules HydratC
k.2
\1
ethyleneoxide
II
tetIahydrofuran trimethykne o@de ratone
trimcthykne oxide
cydobutulonc .
Temp. CK) of maximum 1 kHz absorption
&A
28.1
Ref.
cL;avmok)
4.1 3.45
51.5
1.4 21
PJ this wtxk
3.5 3.7 4.0 3.6
21.6 12.1 19.9 31.8
0.91 0.41 1.02 1.4..
161
.
5:. 191 .
.
242
S.R
Cough et al., Ordering ofguests
II
TH0.7.2D20
0
rrro.7.5aD20
----5TmJcTu4c
0
50
100
Fig. 4. Second moment of st~c~urc I debtcrate.
243-
K
150
II
200
the proton absorption of l?dO
in the structure
I lM0
hydrate
with 99.7 atom % D20 by extensive conditioning and the derivative of the proton absorption signal obtained as for the ethylene oxide deuterate [8]. Results are given for two samples which consisted of substantially pure structure I. One sample was measured with a crosszd-coil spectrometer at temperatures down to 77°K and the other down to 1.8’K with a single coil coupled to a marginal oscillator [8]. A very narrow line obsarved in the fist sample at temperatures above about 170°K corresponded to the presence of less than 3% of TM0 as liquid at any temperature below 240°K. Saturation was not encountered at the = 5 mG rf levels used. Second moments were corrected for relatively small modulation broadening. Second moments of TM0 in the structure I D20 lattice are plotted, along with those of structure II [lo], in fig. 4. The second moment of structure I is seen to vary appreciably with temperature over almost the who!e temperature range of the measurements. It rises quite sharply with decrease of temperature through the same region where e02 is rapidly falling, passes through a plateau-like region between 95 and 70’K,
A
K
Pig. 5.
Dcrivativcabsorption tine &apes of the proton rcY)nan(Pof structure 1 TM0 deuterate.
243
S.R. Cough et aL, edering olguests in the sfrucnwe I TM0 hydrate
and then rises gradually to the rigid lattice value which is attained below about 5°K. In the plateau region the second moment is close to half the rigid lattice value of 13.4 + 0.4 G2. The latter is not significantly different from the rigid lattice moment of 13.7 G2 found [ 111 for the structure II deuterate. The variation of the derivative line shape with temperature is illustrated in fig. 5. At temperatures above about 110’K the line is featureless with a shape which differs from gaussian in the presence of less absorption in the wings. In this region the experimental SACond moment approaches half that of a gausdan line of the same width (as measured by the separation of the extrema of the derivative curve). With decrease of temperature the line shape gradually evolves into a doublet which is best defined near SOOKand which becomes obscured at lower temperatures by increasing absorption near the band center. The rigid lattice pattern is similar to those found for TM0 and cyclobutanone in the structure II D20 lattices [ 111.
3. Discussion 3.1. Rationale
2
.e
0.06
I
0.04 x/a
0.02
0
0
30
60
90
120
150
180
+ (deal Fig. 6. Equipotential contours of Lmmud-Jones guest-lattice interaction energy (in kcallmole) of TM.0 aligned with the 4 axis of the 14-hedral cage. Coordinates are angle of rotation about the polar axis and displacement of center of mass (ii units of LT.the cell dimension) from the center of ?he age.
of the transition
The structure I hydrate of TM0 is distinguished
from the previously-studied clathrate hydrates by a fairly abrupt fall in eo2 at relatively high temperatures and by a corresponding rise to a plateau in second moment at much less than the rigid lattice value. Both features suggest the presence of a transition resulting from the gradual ordering of the orientations of TM0 molecules with cooling through the region of 105°K. The dielectric behavior is similar to that found, for example, during the transition near 178°K in the F_ quinol &&rate of HCN which, as confirmed by specific heat studies, marks the onset of parallel ordering of HCN dipoles in chains as a result of electric dipole coupling between the guest molecules [ 121. Such an ordering of guestdipoles (along the 3 cage axis) is potentially possible in structure I hydrates provided alignment with the cage axis is favoured, or at least permitted, by the interactions of the guest molecule with the water molecules of the cage. In the case of ethy lene oxide (EO) hydrate, a degree of preference for such alignment was invoked to account for the electron density distribution given by the X-ray study
[13] at 248”K, but no clear evidence for such preference was found [8] in the dielectric and NMR behaviour of EO hydrate. The shape of TM0 appears to be more favourable than that of EO to alignment of the symmetry axis with the cage axis. A rough model of the short range van der Waals interactions between TM0 and the water molecules of the cage (appendix A) suggests that the most stable configurations are coaxial ones in which the 0 atom of TM0 approaches the center of a hexagonal ring of water molecules untii it is as near to van der Waals contact with the water molecules as permitted by repulsion between the (LH atoms and adjacent water molecules. Six potential minima (fig. 6) correspond to rotation about the axis so as to bring the “plane” of TM0 nearly into the planes through the cage center which contain successive pairs of opposite water molecules of the hexagon. The symmetry of the above model of the van der Waals interactions is distorted by the electrostatic guest-host interactions. The resultant electrostatic fields of the dipole moments of the water molecules, which deperid on the details of the orientations of the
244
S R Gough et aL, Ordering of guests in the structure I TM0 hydmte
Table 2 Resultant dipole fiilds in kes V/cm at axial positions within
tile 14-h&al cages (averages of 60 cage confwrarions) Axial displacement from cageanter 0
0.03a 0.060 0.090 0.12a
Total
iqb)
-iE,,_.I =)
(x) field 18 27
46 70 100
+ 0.7 +9
I1 14
14 2t
+17 +25 +31
21 30 41
38 60 85
3 Arithmetic mean axial component. ~Mennaxial cornponenc without regard to sign. 3 Mean amplitude of Md perpendicular to axis.
individual water molecules of the cage, are relatively small at positions near the cage center [ 141. With the dipole moment of the water molecule in the hydrate lattice taken as 2.6 D, a value calculated for ice [ 151, the mean field at the center of the 14hedral cage given by averaging 60 randomly-generated cage configgations was found to be 18 kes V/cm, less than half the field of a single Water dipole a cage radius awayAverage fields and field components at a number of positions (in units of u, the cell dimension) are given in table 2. The effective position of the dipole moment in ThlO is removed from the center of mass (c.m.) by perhaps 0.5 A in the direction of the 0 atom. The displacement of the c.m. by 0.060 from the cage center favored by the van der Waals forces (fig. 6) therefore corresponds to a dipole position about O.lOrr from the cage center. Here the resultant average field of 80 kes V/cm includes a mean component of some 27 kes V/cm in the direction of displacement which is opposite to the dipole direction. This results in an electrostatic energy which averages almost 1 kcal/mole higher than at the cage center. In this case, there is a statistical tendency for the dipolar fields to counteract the displacements favoured by the van der Waals interactions. It is unlikely that this conclusion is affected by inadequacy of the point dipole approximation, although it is emphasized that fields of quadrupole and higher moments of the water molecules have been ignored [ 141. Interactions between the guest-molecule dipoles favour parallel ordering of nearest-neighbour guest dipoles. For the ideal lattice consisting only of non-
polarizable dipoles located at the centers of the large cages the most stable configuration is that in which adjacent dipoles point in the same sense along chains extending in the x, y. and z directions, the sense alternating between adjacent chains. For this fully ordered lattice the field in the direction of an individual dipole is found in appendix B to be S = 38.47 p/a3. The dipole moment, 1.93 D for the isolated TM0 molecule, is effectively enhanced by the reaction field factor tosbout 2.3 D. On the other hand the contributions to S are reduced by the shielding effect produced by the polarizability of the intervening medium. For large separations in the rigid lattice the shielding factor is l/e-2. This factor is certainly too small, however, for the dominant contributions to S from the two nearest-neighbour dipoles.The field due to dipoles induced by a nearest-neighbour dipole in its sheath of 24 water molecules is tantamount to a shielding factor [ 141 of 1Il.6. We approximate the shielding effects by separating the lattice sum into 32 p/l .6n3, contributed by nearest neighbours, and 6.47 ~13.45 a3 contributed by all other dipoles. These assumptions lead to a field of some 29 kes V/cm in the direction of alignment of an ordered dipole, a field almost twice the mean field of the dipoles of the water molecules at the cage center. For the ordering transition to occur it is apparent that guest-guest dipole interactions must be more
important than the electrostatic interactions with the water molecules in most cages and therefore that the fields of the quadrupole moments of the water molecules do not seriously dominate those of the dipole moments. This in turn implies that the quadrupole moment tensor of the fully hydrogen-bonded water molecule is considerably more isotropic than in the isolated water molecule [ 141. The persistence of some disorder to temperatures much below the general ordering range is seen to be a likely consequence of exceptionally large fields of the water molecules in some cages. The time dependence observed is possibly the result of some increase in order associated with a slow irreversible readjustment of those water-molecule configurations which give large fields. The above field of 29 kes V/cm gives an energy of -0.72 kcallmole for the completely ordered state, which suggests that the ordering transition should occur at T S=720/R C=360°K. It is not surprising that
S.R. Gough et aL. Ordering o/guests in the structure I 7MO hydrate
the perturbing effects of the fields of the water molecules result in a lower transition temperature. 3.2. Roton magnetic resonance second moments The experimental second moment of the rigid-lattice proton absorption of structure I TM0 deuterate is about 1 G* smaller than the value of 14.45 G2 calculated with no vibrational corrections. The latter includes an intramolecular contribution of 13.7, G2 derived from the gas-phase microwave structure [ 161 of TMO. Inter-TM0 and TMO-D,O contributions of 0.37 and 0.38 C2, respectively, were evaluated as for ethylene oxide deuterate [8] except for the contribution from nearest-neighbour TM0 molecules. Here the assumption of parallel alignment gave a smaller contribution (0.13 G2) than the assumption [8] of random orientations. At the highest temperatures the second moment (e.g., 0.31 i 0.04 G2 at 243°K) approaches the value 0.24 G2 calculated for the inter-guest contribution when the reorientation is isotropic. It is possible to interpret the second moment in
the ‘plateau” temperature range in terms of reorientation about the polar axis of TM0 between the preferred positions determined by cage geometry. In conformity with fig. 6 it is assumed that there are six such positions and that these fall into a set of two and a set of four equivalent sites. The effect of rapid interchange between preferred positions on the rigid lattice second moment is related in appendix C to the difference of energy of the two sets for the case in which interproton vectors are perpendicular to the axis of rotation. This condition is satisfied for the interproton vectors of each CH2 group and for the vectors connecting the protons of different a-CH2 groups. In toto the corresponding interactions contribute about 10.9 G* to the rigid lattice second moment or 80% of the whole intramolecular moment. The same rotational jumps will reduce the remaining intra- and inter-molecular contributions to the second moment to a different extent. Jiowever, it is clear that the required second moment x 7 C2 in the plateau region around 80°K may be achieved by a reduction of the 10.9 G2 contribution by about afictoroftw. From eq. (9) of appendix C this requires an occupancy factor p = 0.36 for each position of the Z-fold set, which is then more stable by AE Z=250 Cal/mole than
24.5
each position of the 4-fold set. (A reduction factor of the right order (7116) resulting fromp = 0 and occupancy of the 4-fold sites only is a less likelyaltemative.) Although the results of this model are of Jimited quantitative significance, they serve to show that the second moment below the transition is consistent with the type of order proposed.
Appendices A. Shownznge guest-host interactions The crude Lennard-Jones model previously used [8] to estimate the position and orientation dependence of the energy of ethylene oxide in the 14.hedral cage has been employed to estimate the potential energy of short-range interaction between TM0 and the water molecules of the cage. The atom-atom Lennard-Jones parameters were the same as before and again the hydrogen atoms of the water molecules were neglected. Fig. 6 shows equipotential contours for encaged TM0 molecules aligned with the 4 cage axis as a function of displacement of center of mass along this axis from the cage center (in units of the cell dimension) and of rotation about the polar axis of the molecule. Positive displacements refer to approach of the oxygen atom to the nearest hexagon ring and + = 0 to coplanarity of the TM0 “ring” with the plane of symmetry of the cage which contains the two “c” water molecules [13] of this hexagon. One complete rotation is seen to eccompass six positions of potential minimum spaced at 60’ intervals in which the center of mass is some 0.7 A from the cage center. Witbin the limitations of the mode1 the energies of the 6 preferred orientations are not greatly different, although the two-fold set is slightly more stable than the four-fold set. The least energy required for Interchange between the sets is * 0.4 kcallmole, while that required for interchange within the &et is z 0.6 kcal/mole. The model is too crude for these energies to be of great significance. However, the conclusions that the 0 end of the molecule is appreciabiy displaced toward the center of a hexagon ring, that there are 6 preferred orientations of comparable energy, and that the barriers to rotation are less than 1 kcallmole are not sensitively dependent on the particular potential function and parameters employed.
246
SR
Gough et aI.. Ordering of guests in the smtcture I TM0 hydmte
B. Guest dipole lattice sums
The cubic unit cell (space group Pm3n) of the structure I hydrate lattice defined by the average positions of the 46 water molecules contains six 14hedral cages centered at (4, a, )), (5, *. $), (0, ), Oh (0. 2, 01, (), 0,;). and (i. 0.4). The cage symmetry is 32 m with the 2 axis coincident with the line joining the centers of nearest neighbor cages. Cage centers lie on lines parallel to the three orthogonal axes. We locate a guest dipole at the center of each cage and evaluate the field at the site of a dipole arising from other guest dipoles for the condition that all the dipoles in a Qne of nearest neighbors point in the same direction along that line. The symmetry is such that dipoles pointing along lines parallel to the x and y axes produce no net field at the site of a L dipole so that only dipoles lying in parallel chains need be considered. We consider first the “antiferroelectric” case in which the sense of dipoles in adjacent parallel chains alternates between t and -z. The problem is then the evaluation of the fields of two sub-lattices containing, respectively, dipoles which are parallel (p) and antiparallel (a) to the dip016 at which the field is evaluated: S=S,_S,J+
c ax
[ e,,)
I
(2)
=8-t.
The individual sums of eq. (2) are conditionally convergent, i.e., depend on the order of summation and therefore on the shape of the crystal. The difference of the sums is, however, absolutely convergent since the “antiferroelectric” crystal possesses no net pOlilliZ3tiOIl.
Because of the slowness of convergence
ti ki exp (2nIlj,o)-1
for the parallel lattice, and hjjCOS(i+J>ll
S(a,&#O)
= 8n21.c d a,3 i,i
exp (2nhija)
for direct
- 1
(4)
for the antiparallel lattice. The summations are now over square lattices which are the reciprocals of the 2-dimension4 lattices which form the x. y planes: hii = (i2 +j2)f where i, j take on integral values, 0, kl, k2, . . . , with i = j = 0 excluded. Both series rapidly converge to give S(p,h#O)
= 82.289p/aj
S(a,&#O)
= -19.043
,
p/a;
(5)
after 13 X 13 terms. For hz = 0, S(p, b = 0) = -9.0336
p/a:
(6)
[ 181. Moreover,
since reversal of the antiparallel lattice based on u. Thus S(a,%=O)=
where the summations are over tetragonal cells, each cell being identified by integral coordinates X,, X,,, h in terms of the basis set a,, a,, (=a,), a,. in the first sum, from which Xx = hy = 4 = 0 is excluded, pzp = Ai + A2 + a2 *; h the second pza = 9 + a2$ In both, as in cos 13, = (xr+$)2+&+;) c&IpA,4=az/ax
= 8* a,
S(p, 4 =0) + S(a, X, = 0) = -9.0336
XP
--c h da1 2P2kOS
Jlij
S(p, h#O)
has already been evaluated
2p2;;0AP) h
summation we resort to the method of de Wette [17] in which each sum of eq. (2) is separated into a sum over & = 0 and a sum over h, # 0. The latter may be replaced by
-16.5173IJa?
p/as,
dipoles gives a single
.
Addition of the contributions of eqs. (S)-(7) the total field for alternating chains of dipoles S = 108.82 p/a2 = 38.474 p/a3 .
(7) gives
09
For the “ferroelectric” case reversal of the sign of S, in eq. (2) givesS = 13327 p/a3, in which the numerical factor may be compared with the value of 13.32 gken by de Wette [ 191 for a single tetragonal lattice with u = i. From the method of summation the result for this polar lattice is for a slab&aped crystal with the polar axis perpendicular to the surface of the slab. A polarization contribution of 4nrJo3 must therefore be added.
S.R.
Gough
et aL,
Ordering
of guests
C. Second moment for rotational flips between six preferred orientations
structwe
I nf0
[ 11 R.E. Hawkins and D.W. Davidson, 1. Phys. Chem. 70 (1966) 1889. (21 D.F. Sargent and L.D. Calvert, J. Phys. Chem. 70 (196612689. [ 31 J.C. Rosso and L. Carbonnel, Compt. Rend. Acad. Sci
(Paris) 274C (1972) 1108. 141 M. von Stackelberg and H.R. Mulfer, 2. Elektrochem.
d(t)>),
58 (1954) 162. ]6] S.R. Cough, RE. Hawkins, B. Morris and D.W. Davidson, J. Phyr Chem. 77 (1974) 2969. [I] S.R Cough. Can. J. Chem. 50 (1972) 3046. [S] S.K. Carg, B. Mcrris and D.W. Davidxln. J. Chem. Sot.
Faraday Tranr II 68 (1972) 481. [9] B. Morrisand D.W. Davidson, Can. I. Chem. 49 (1971) 1243. [ 101 S.K. Garg and D.W. Davison, In: Physics and chemistry
1Li.
where pi specifies the degree of occupancy of site i and tii= fio+(i-l)7r/3
9
i=
25.
[ 51 hf. van Stackelberg and W. JamMS, Z. Elekirochem.
where 0 is the angle between the magnetic field and the axis of rotation and ti the angle of rotation of the H-H vector in the plane perpendicular to the rotation. The expectation value Cpjsin2
241
References
58 (1954)
(sin2 9(t))=
hj-dnate
2-fold site, the reduction factor may assume values between $ for AE = 0 and 1 for AE > RT.
We consider the effect of rapid uniati rotational jumps between six possible orientations on the contribution to the proton second moment associated with inter-proton vectors which are perpendicular to the axis of rotation. The reduction factor by which the polycrystalline rigid lattice second moment is to be multiplied is [20] : (I -3tin~ehrl*
in the
of ice, ed. E. Whalley (Roy. Sot. Canada, 0tIaw-a. 1973). [ 1 l] S.K. Carg and D.W. Davidson, Chem. Phys. Letters 13 (1972) 73. [ 121 T. Malsuo, H. Sup and S. Seki, I. Phys. Sot. Japan 30
1,6.
To conform to fig. 6, we take pt = p4 = p for the 2.fold sites and p2 = pg = ps = pg = (1 - 2p)/4, whence
(1971) 785.
[ 131 RK. McMullanand G.A. Jeffxy, [ 14) [ 151
(sin2$(t)>=i-qp+(3p-4)sin2JI0.
[16]
Pol crystalline averaging with the help of sin2 Go = f , &=$,ao=i h gives 9and==
[ 171
reduction factor = yp* - zp * & .
[ 181
(9)
Since p = [2+4 exp (-M/RT)] -1 where AE = E(4)-E(2). the excess energy of a Cfold site over a
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