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The non-linear dielectric effect in xenon. The density dependence
Volume 115, number 2
-l-DE NON-LINEAR J. VAN
DER
ELSKEN
DIELECTRIC and J.C.F.
Laboralorrwn voor Fysrsche Cheme, Recelvecl
29 March 1985
CHEMICALPHYSICSLE-ITERS
EFFECT
IN XENON.
THE
DENSITY
DEPENDENCE
MICHIELSEN
Ntemve Achrergrachr 127, 1Oi8 IVS Anlsrerdanr, 77le Nerherfands
2 January 1985; In final form 19 January
1985
The densely dependence of Ihe non-hnear dielectric effect m xenon IS measured Expenments were perCormed along an ~sothcrm \nth a reduced temperature T’ = 1 01 m a reduced densrty range 0.1 IO 1.8. There IS a large drsagreement wrrh theory, wiuch m first order leads IO elecuostncnon Arguments are grven why expenmenlal condrtrons and the consuucuon ot the
measunng cell m our case exclude elecuostncuon The results are explarned m terms of first- and second-order eflects Use 1s made of the close relation wrth the theory of hght scattenng rn first and second order it IS possrble IO wnte the non-linear &electnc the value
effect as a sum of IWO terms. one hnear m the density and the other mcreasmg WI& Pan. Quanutallve esumates of the coetficrent of p*’ cannot be grven because of lack of knowledge of four-poml correlation funcuons
1. Introduction In a prevrous paper [I] we reported measurements of the non-linear dielectric effect in a number of nonpolar molecular liquids and in liquid xenon. We compared the results with values that are predicted by a theory that estabhshes the connection between the polanzation fluctuations and the non-linear dielectric constant [?I. The agreement between experiment and theory turned out to be very poor. The experimental values of the non-lmear dielectric effect k/E2 are more than an order of magmtude huger than predicted for a collection of non-polar simple organic substances. For fluid xenon the discrepancy was less but for xenon at about the critrcal density where the theory predicts a very large value because of the compressibility that appears in the theoretical expression, the experimental values stayed very far behind. This made us doubt the applicability of the theory to the results of the experiments. Smce the dependence on density can be used to descrimrnate the influence of terms of different order m the theory, we undertook to measure the non-linear dielectric effect of xenon along an isotherm over as large a density range as feasible.
for
2. Experimental A detatled description given in our earlier
paper
of the apparatus has been [ 11. A measuring capacitor,
consisting of two electrodes at a separation of 135 W, forms part of au oscillatirlg LCcircuit. A schematic drawing of the cell is given in fig. 1. The lowvoltage measuring field has a frequency of about 28 MHz. The oscillating frequency of this circuit is changed if high-voltage field is apphed across the capacitor.
The field changes the permittivity of the fluid in the capacitor and therefore the frequency of the circuit. This frequency is determined by making tune-interval measurements, that is, measuring the time LIT necessary for 2500 oscillations of the measuring field. To increase the signal-to-noise ratio, a high-voltage pulse 1 I ’I I ----_-~_I-___7 ! 1
r _-------I
I
L-______-----
STAINLESS
STEEL ---_
I
-----__-I
--TEFLON --GOLD Rg.
230
0 009-26
l_ Schematic drawmg of the meas-g
14fSS/$O3.30 0 Eisevier Science
(North-Holland
Physics
Publrshing
Drvision)
cell.
Publishers
B.V.
Volume 115, number 2
with a duration
CHEMICAL
PHYSICSLETTERS
of IO ms is used, and 15 measurements
are taken just before
and 15 after
this HV pulse.
Dur-
ing the pulse 18 points are measured, all with the time interval AT. This measurmg sequence is repeated 2000 times with a frequency of 0.5 Hz. The NLDE is calculated according to the formula Ae/E*
= E -*
[(AT - A&)/AT,]
(E + CJC,)
.
(1)
In this formula E 1s the HV field, ranging up to 2.2 X lo6 V/m. ATO is the average time Interval measured in the absence of the HV field, this value also gives the relative dielectric constant of the fluid between the condenser plates. AT - ATE is the averaged tune-interval increase If the l-IV field 1s apphed. C, is the stray capacitance m the leads to the cell, Co is the capacitance of the measurmg cell under vacuum. C, and Co are determineci by iow-frequency measurements with a Wayne-Kerr B221 adrmttance bridge on hexane and cyclohexane which have a well-determined pemuttivlty and with HV experiments on vacuum. We found C,-,= 15.96pF,
Cs=4.58pF.
The volume of xenon in the pressurizable stamlesssteel box is about 2 cm3, that IS, many orders of magmtude larger than the volume of xenon in the measuring cell. There IS good contact between both volumes. The temperature was measured with a R-1 00 thermometer and kept at 2O.O”C with an accuracy of 0.1 K.
29 hlarch 1985
results of Michels et al. [3] and Streett et al. [4]. Because data on the T’ = 1.01 Isotherm cannot be found in the literature, an interpolation was made between the critlcal Isotherm and the isotherms with reduced densities of 1.03 and 1.05, for which data are avadable. Especially in the neighbourhood of the critical dennty, the change m density IS very large compared with the change in pressure. So we used an independent check to establish the density at a given pressure near the critical point. We made use of the value of the dielectric constant whch was also measured together with the non&near dielectric effect. The values of the dielectric constant as a function of the density are in fig. 2. Also given in this figure are the values of the dielectric constant as calculated from the Clausius-Mossottl relation. A correction of the density as gwen by de Boer et al. [5] IS apphed. As can be seen, only for very high values of the denslty are there discrepancies between experimental and calculated values of the relative dielectric constant. Vidal and LalIemand [6] measured the ClausiusMossotti function for xenon at 25OC. That is (E - l)/(e + 2)~ versus the number density. This function is a smooth function starting at p = 0 with the value ; ncuN, with ~1the polarizabdity and IV Avogadro’s number. Our high-density data are quite accurate, so we are able to construct a ClausrusMossotti function, using the experimental data of Vldal and Lallemand as a scaling function. We are able then to calculate from the dielectric constant the ap-
3. Results We performed NLJDE experrments along the isotherm for the reduced temperature T* = 1.01 in a reduced density range between p* = 0.1 and p * = 1.8. T* is the temperature divided by the crltical temperature. The choice of this isotherm was made on the consideration that the compressibihty shows a large variation along this isotherm and that hence the proposed dependence of the effect on the compresability can sensitively be tested. Furthermore such a supercritical isotherm allows a test of the density dependence over the largest density range possible within the experimental limitations. The experiments were done with various pressures ranging from 1 to 140 bar. The P, V,T data we used were compiled from
Fg. 2. Dzlectnc constant versus reduced dennty. Experlmen+al (*) and calculated values (A, ref. [S]). Reduced dennty is density cbnded by the cntical densty.
231
Volume
CHEWCAL
115, number 2
PHYSICS
Table 1 The experrmental pressures and &electric constants, the calculated densities, the mterpobted measured non&near dielectnc effect of xenon along the 293 K isotherm
Eothermal
compressibities
P’
139
342
238
1.82
15191
155+03
31.1 f 2.0
125.5
336
2.166
l.785
1.5084
I.63 5 0.3
31.7 f 2.0
110.5
325
1.892
1.73
1A973
2.37 f 03
26 8 2 2.0
96.5
314
1.625
1.67
1.4815
3.24 2 0.3
21.0 f 1.3
a3 5
297
1.430
1.58
1.4589
4.93 f 0.3
20.8 f 1.8
78
290
1.336
154
1.4556
5.79 f 0.4
20.2 f 1.8
75
284
1.284
1.51
1.4468
8.27 + 0.5
19.9 f 1.8
725
276
1.241
l-47
1.4273
10.4
* 2
17.1 f 1.8
645
241 = 10
1.104
1.28 + 0.06
1.3770
37.2
I- 6
16.9 2 1.5
625
2252;
1.070
1.20 * 0.2
1.3081
59.5
115 -20 +t5 loo* 5
1.019
0.61 f 0.09
1.1861
81.3
f 20
0.967
053
1.1580
46.0
i 10
2.7 + 1.0
44.8
f 5
4.0 f 1.5 1.5 f 1.5
56.5
Pa)
(amaeat)
E
P*
XT
(10
f 0.03
53
83
0.908
0.44
1.3336
Pa”)
m
10.1 + 1.0
47
66
0.805
0.35
1.1049
38.6
~4
44
58
0.753
0.31
1.0921
40.0
f 4
42
55
0.719
029
1.0859
34.8
24
34.5
39
0.519
0.21
I.0619
37.5
f 5
24
24
0.411
0.13
1.0399
42.8
= 8
21
22
0.360
0.115
1.0437
483
f 10
20
21
0.342
0.11
1.0317
50 6
2 10
9
0.162
0.05
1.002
9.3
Fig. 3. Non-bear tielectnc effect versus reduced dennty. The full tie IS the fitted sum of p* and pti contributions (eq. (9)). The dashed line is the calculated value for electrostnction.
111
and the
A EfE2 (lo*0 m21vq
P
P (10
232
29 March 1985
LElTERS
3.0 +_ 1.0
k 25
propriate densities. These calculated densities are UI good agreement with the interpolated values, and therefore it is possible to estimate the inaccuracy in the densities. In table 1 the experimental pressures and the dielectric constants, together with the cdculated densities, the isothermal compressibilities and the non-linear dielectric effects, are given. The NLDE is plotted in fig. 3 as a function of the density. For reduced densities less than 03 the NLDE is too small compared with the experimental inaccuracy to enable us to determine any signifkant figures. As can be seen from fig. 3, the NLDE as a function of the density does not show any tendency to follow the extreme behaviour of the compressibihty. If the NLDE is due to electrostriction, and the theory [2] claims explicitly that, in first order, electrostriction is the main contribution to the NIDE, the interpolated values for the isothermal compressibility make it possible to cakulate the NLDE as given by the formula
Volume 115, number 2 k/E2
= 27~ [;(e
CHEMICAL PHYSICS LETTERS
+ 2)14 a2p2xT.
(2)
These values are also plotted in fig. 3. It is obvious that electrostriction cannot explain the experimental results for the NLDE. Not only do the absolute values of the NLDE not fit, but even the trend in the experimental data is not in agreement with the calculated values. The experiments reported in our previous paper were made at densities where the NLDE results and the calculated electrostriction were almost equal. The agreement we claimed earlier was therefore purely fol-tultous.
4. Discussron The basic theory for the non-linear dielectnc effect has recently been expounded by Fulton [2] _ The full development of the theory was unfortunately limited to the first-order term in the expansion of the deviation of the average polarisability. We shall here confme ourselves to a somewhat schematic presentation of the theoretical expressions that allow us to find the dependence of the non-linear dielectric effect on the density and on the density correlations. The close relation with the theory of light scattering in first and second order can be used as aid. The starting point is eq. (84) and eq. (86) in ref. [2] : de/E2
= (CIkBT)
[f(e
+ 2)14 (3)
where the geometry of the problem determines the constant C and y = AD{1 + 47~LA/3)-~ III matrix notation, in which A/3 is the deviation from the average polarisability density, and L a propagator with a range limited by the core diameter of the atoms 173. By taking only the first-order term m the expansion in powers of AF one can substitute: Y(rl’2)
= 8 (r, - r2) 01[P(Q)
-PI
s
(44)
where p is the average densrty and Q the molecular polarisability. Transformation into k-space leads to (A&?)(l)=
(C/knT)[f
X JdkgI(k)S(k).
(E+ 2)14 cr2p (5)
29 March 1985
S(k) is the structure factor and gl (k) is a weight tiction that III the long-wavelength limit should be a delta function at k = 0, leadmg to AC/E* = C[$(E + 2)]4cr2p2xT.
05)
The first-order term, wrth XT the isothermal compressrbility, is the expression given by Fulton (eq. (87), ref. 123). It can also be derived by classical thermodynamics and the effect is called electrostriction 181. The crucial point is that the k = 0 limit should be taken, which physically means that flurd should flow from the bath mto the capacitor to balance the lowering of the chemical potential which is caused by the applied electric field. The result is an increase in density and consequently an increase in the dielectric constant. Under the experimental conditions it is, however, not likely that thermodynamic equilibrium is maintained. The cell that we used consists of two glass plates, gold-plated on one side. These two electrodes wrth dimensions 12 by 15 mm2 are kept at a separation of 135 m. One can estimate the pressure difference that would build up between the condenser and the reservoir as a consequence of the lowering of the chemical potential. Smce AP = (E - 1)E2/8n this pressure difference is about 12 Pa as a result of the 2.2 X lo6 V/m field during 10 ms. considering the construction of the cell and the small pressure difference it 1s reasonable to assume that no replenishment takes place. We find confirmation of this assumption by taking 18 time-interval samples in the tune the pulse is applied to the cell, that means that every 500 m the dielectric increment is measured. lf there were a stream of liquid going into the cell there would be an increase in the time-interval data in the time-sequence during the duration of the pulse. However no significant change in the measurements as a function of the time was observed. Smce we detect the difference between e for the system with and without the field a possible effect of the high field time-averaged over the repetition rime wti cancel. In the apparatus we used, and that has been used by many other experimentalists, the effect of electrostriction is completely suppressed. if electrostriction is not the cause of the measured effect, the question arises as to what contributions increase the dielectric constant under the influence of the high field. In fig. 3 we plotted the measured non233
CHEMICAL PHYSICS LETTERS
Volume 115, number 2
linear dielectric effect corrected for the factor 1; (E + 2)14 as a function of the density. For denslties 1 .I there is a linear dependence, which is uptop’= m accordance with eq_ (5) if one assumes that the integral over k IS a weak function of the density and hence does not contain the k = 0 contributions. For larger denntws the density dependence becomes much stronger and this might well be caused by higher-order terms III the expansion of A/3_ The second-order term adds a contribution (A&@)(2) X_/drld v
= (C&T)[f(e
+ 2)14
‘2 dr3 (AO(rl )LA&)
A&)LA/Gq))(71
This is the type of term that appears in the theory of depolarized Light scattermg. For a full discussron of the assumptions and approximations we refer to the hterature on that subJect [9- 12]_ We can rearrange this second-order term and obtain an expression of a form analogous to eq. (5). (Ae/E2)(2) X ldk’
= (C/x-&
[f(~ + 2)14 o!p3
s d/c” g?(k’k”)
T(k’k”),
(8)
where T(k’k”) is now a four-point density correlation function and the weight-factor gz(k’k”) depends on the actual induction mechanism represented by the propagators L_ We can combine the contributions to the non-linear dielectric effect, eq. (5) and eq. (8). to derive AtzIE2=ap*
+bp*3
_
(9)
By plotting p*-1 (AE/E~)[$ (E + 2)] -4 versus p* we fimdthevaluesa=3_5X lO-*~m2V-2andb=I.7 X 10a20 m2 Vm2. In fig. 3 we show the result of plotting eq. (9) wth a subaitution of these values ofn and b. The result shows rather convincingly that there must be a term lmear in p* and one that increases with Pan. In the absence of a full second-order theory we can only draw some qualitative conclusions from our experimental findings. The density dependence of the measured non-linear dielectric effect shows that there is no electrostriction but that there are contnbutions from first-order as well as from second-order in the developmeat in the polarisabrlity. The second-order contribution is surprisingly large with respect to the 234
29 March 1985
fust-order contribution. This is caused by suppression of the first-order term by the way the-experiments are carried out. The first-order effect obeys a rather strict k = 0 selection rule and the geometry of the capacitor that we used, and that was used in most experiments reported in the literature, keeps the amplitude of the k = 0 modes low. This is borne out by the total absence of the effect of the diverging compressibility, eq. (6). That a small fust-order contribution persists nonetheless is probably caused by relaxation of the k-selection rules, eq. (5). On the other hand, there is no restriction to small kvectors for the second-order term. Probably g2(k’k”) will have the form of an integral over space of a spherical Bessel function and WIIJ favour k-values around the fust peak of S(k) [ 121. Very little is known about the four-point density correlation functions [ 131 and no quantitative estimate of the expected value of the constant b in eq. (9) can be made. However, the conclusion that the non-linear dielectric effect of xenon at high densities is pre-dominantiy a second-order effect is of consequence for the interpretation of analogous measurements on compounds of comparable density. The origin of the discrepancies that were noted between fnst-order predictions and various experimental values [l] lies in the fact that one measures mainly a second-order effect.
Acknowledgement This work is part of the research program of the Foundation for Fundamental Research of Matter (FOM) and the Netherlands Foundation for Chemical Research (SON), with financial support from the Netherlands Organization for Advancement of Pure Research (ZWO).
References [ 1) J. van der Elsken, P. van Zoonen
and J.C.F. hfictuelsen, Chem. Phys. Letters 106 (1984) 252. [2] R.L. Fdton, J. Chem. Phys. 78 (1983) 6865,6877. [ 3 ] A. Mxhels and T. Wassenaar. Physiu 16 (1950) 253. [4] W.B. Streett, L.S. Sagan and L.A.K. Staveley, J. Chem. Thermodynam. 5 (1973) 533. [S] J. de Boer, F. van der Maesen and CA. ten Scldam. Physiea 19 (1953) 265.
Volume 115. number 2
CHEMICAL
16) D. Vidal and M. Lallemand, J. Chem. Phys. 64 (1976) 4293; 66 (1977) 4776. [7] H.MJ. Boots. D. Bedaux and P. Mazur, Physica A79 (1975) 379. [ 81 H.S. Frank, J. Chem. Phys. 23 (1955) 2023; L. Landau and E. LAhitz, Electrodynamique des milieux contmus (MIR. Moscow, 1969) p. 79.
PHYSICS
LETTERS
29 March lS85
[9]
D. Frenkel and J-P. McTague, J. Chem. Phys. 72 (19I:O) 2801. [lo] B.U. Felderhof, Physica 76 (1974) 486. [ 111 T. Keyes, J. Chem. Phys. 70 (1979) 5438. [12] P. Madden, Mol. Phys. 36 (1978) 365. [ 131 J. van der Elsken and D.M. Heyes, Can. J. Phys. 59 (1981) 1532.
235
-l-DE NON-LINEAR J. VAN
DER
ELSKEN
DIELECTRIC and J.C.F.
Laboralorrwn voor Fysrsche Cheme, Recelvecl
29 March 1985
CHEMICALPHYSICSLE-ITERS
EFFECT
IN XENON.
THE
DENSITY
DEPENDENCE
MICHIELSEN
Ntemve Achrergrachr 127, 1Oi8 IVS Anlsrerdanr, 77le Nerherfands
2 January 1985; In final form 19 January
1985
The densely dependence of Ihe non-hnear dielectric effect m xenon IS measured Expenments were perCormed along an ~sothcrm \nth a reduced temperature T’ = 1 01 m a reduced densrty range 0.1 IO 1.8. There IS a large drsagreement wrrh theory, wiuch m first order leads IO elecuostncnon Arguments are grven why expenmenlal condrtrons and the consuucuon ot the
measunng cell m our case exclude elecuostncuon The results are explarned m terms of first- and second-order eflects Use 1s made of the close relation wrth the theory of hght scattenng rn first and second order it IS possrble IO wnte the non-linear &electnc the value
effect as a sum of IWO terms. one hnear m the density and the other mcreasmg WI& Pan. Quanutallve esumates of the coetficrent of p*’ cannot be grven because of lack of knowledge of four-poml correlation funcuons
1. Introduction In a prevrous paper [I] we reported measurements of the non-linear dielectric effect in a number of nonpolar molecular liquids and in liquid xenon. We compared the results with values that are predicted by a theory that estabhshes the connection between the polanzation fluctuations and the non-linear dielectric constant [?I. The agreement between experiment and theory turned out to be very poor. The experimental values of the non-lmear dielectric effect k/E2 are more than an order of magmtude huger than predicted for a collection of non-polar simple organic substances. For fluid xenon the discrepancy was less but for xenon at about the critrcal density where the theory predicts a very large value because of the compressibility that appears in the theoretical expression, the experimental values stayed very far behind. This made us doubt the applicability of the theory to the results of the experiments. Smce the dependence on density can be used to descrimrnate the influence of terms of different order m the theory, we undertook to measure the non-linear dielectric effect of xenon along an isotherm over as large a density range as feasible.
for
2. Experimental A detatled description given in our earlier
paper
of the apparatus has been [ 11. A measuring capacitor,
consisting of two electrodes at a separation of 135 W, forms part of au oscillatirlg LCcircuit. A schematic drawing of the cell is given in fig. 1. The lowvoltage measuring field has a frequency of about 28 MHz. The oscillating frequency of this circuit is changed if high-voltage field is apphed across the capacitor.
The field changes the permittivity of the fluid in the capacitor and therefore the frequency of the circuit. This frequency is determined by making tune-interval measurements, that is, measuring the time LIT necessary for 2500 oscillations of the measuring field. To increase the signal-to-noise ratio, a high-voltage pulse 1 I ’I I ----_-~_I-___7 ! 1
r _-------I
I
L-______-----
STAINLESS
STEEL ---_
I
-----__-I
--TEFLON --GOLD Rg.
230
0 009-26
l_ Schematic drawmg of the meas-g
14fSS/$O3.30 0 Eisevier Science
(North-Holland
Physics
Publrshing
Drvision)
cell.
Publishers
B.V.
Volume 115, number 2
with a duration
CHEMICAL
PHYSICSLETTERS
of IO ms is used, and 15 measurements
are taken just before
and 15 after
this HV pulse.
Dur-
ing the pulse 18 points are measured, all with the time interval AT. This measurmg sequence is repeated 2000 times with a frequency of 0.5 Hz. The NLDE is calculated according to the formula Ae/E*
= E -*
[(AT - A&)/AT,]
(E + CJC,)
.
(1)
In this formula E 1s the HV field, ranging up to 2.2 X lo6 V/m. ATO is the average time Interval measured in the absence of the HV field, this value also gives the relative dielectric constant of the fluid between the condenser plates. AT - ATE is the averaged tune-interval increase If the l-IV field 1s apphed. C, is the stray capacitance m the leads to the cell, Co is the capacitance of the measurmg cell under vacuum. C, and Co are determineci by iow-frequency measurements with a Wayne-Kerr B221 adrmttance bridge on hexane and cyclohexane which have a well-determined pemuttivlty and with HV experiments on vacuum. We found C,-,= 15.96pF,
Cs=4.58pF.
The volume of xenon in the pressurizable stamlesssteel box is about 2 cm3, that IS, many orders of magmtude larger than the volume of xenon in the measuring cell. There IS good contact between both volumes. The temperature was measured with a R-1 00 thermometer and kept at 2O.O”C with an accuracy of 0.1 K.
29 hlarch 1985
results of Michels et al. [3] and Streett et al. [4]. Because data on the T’ = 1.01 Isotherm cannot be found in the literature, an interpolation was made between the critlcal Isotherm and the isotherms with reduced densities of 1.03 and 1.05, for which data are avadable. Especially in the neighbourhood of the critical dennty, the change m density IS very large compared with the change in pressure. So we used an independent check to establish the density at a given pressure near the critical point. We made use of the value of the dielectric constant whch was also measured together with the non&near dielectric effect. The values of the dielectric constant as a function of the density are in fig. 2. Also given in this figure are the values of the dielectric constant as calculated from the Clausius-Mossottl relation. A correction of the density as gwen by de Boer et al. [5] IS apphed. As can be seen, only for very high values of the denslty are there discrepancies between experimental and calculated values of the relative dielectric constant. Vidal and LalIemand [6] measured the ClausiusMossotti function for xenon at 25OC. That is (E - l)/(e + 2)~ versus the number density. This function is a smooth function starting at p = 0 with the value ; ncuN, with ~1the polarizabdity and IV Avogadro’s number. Our high-density data are quite accurate, so we are able to construct a ClausrusMossotti function, using the experimental data of Vldal and Lallemand as a scaling function. We are able then to calculate from the dielectric constant the ap-
3. Results We performed NLJDE experrments along the isotherm for the reduced temperature T* = 1.01 in a reduced density range between p* = 0.1 and p * = 1.8. T* is the temperature divided by the crltical temperature. The choice of this isotherm was made on the consideration that the compressibihty shows a large variation along this isotherm and that hence the proposed dependence of the effect on the compresability can sensitively be tested. Furthermore such a supercritical isotherm allows a test of the density dependence over the largest density range possible within the experimental limitations. The experiments were done with various pressures ranging from 1 to 140 bar. The P, V,T data we used were compiled from
Fg. 2. Dzlectnc constant versus reduced dennty. Experlmen+al (*) and calculated values (A, ref. [S]). Reduced dennty is density cbnded by the cntical densty.
231
Volume
CHEWCAL
115, number 2
PHYSICS
Table 1 The experrmental pressures and &electric constants, the calculated densities, the mterpobted measured non&near dielectnc effect of xenon along the 293 K isotherm
Eothermal
compressibities
P’
139
342
238
1.82
15191
155+03
31.1 f 2.0
125.5
336
2.166
l.785
1.5084
I.63 5 0.3
31.7 f 2.0
110.5
325
1.892
1.73
1A973
2.37 f 03
26 8 2 2.0
96.5
314
1.625
1.67
1.4815
3.24 2 0.3
21.0 f 1.3
a3 5
297
1.430
1.58
1.4589
4.93 f 0.3
20.8 f 1.8
78
290
1.336
154
1.4556
5.79 f 0.4
20.2 f 1.8
75
284
1.284
1.51
1.4468
8.27 + 0.5
19.9 f 1.8
725
276
1.241
l-47
1.4273
10.4
* 2
17.1 f 1.8
645
241 = 10
1.104
1.28 + 0.06
1.3770
37.2
I- 6
16.9 2 1.5
625
2252;
1.070
1.20 * 0.2
1.3081
59.5
115 -20 +t5 loo* 5
1.019
0.61 f 0.09
1.1861
81.3
f 20
0.967
053
1.1580
46.0
i 10
2.7 + 1.0
44.8
f 5
4.0 f 1.5 1.5 f 1.5
56.5
Pa)
(amaeat)
E
P*
XT
(10
f 0.03
53
83
0.908
0.44
1.3336
Pa”)
m
10.1 + 1.0
47
66
0.805
0.35
1.1049
38.6
~4
44
58
0.753
0.31
1.0921
40.0
f 4
42
55
0.719
029
1.0859
34.8
24
34.5
39
0.519
0.21
I.0619
37.5
f 5
24
24
0.411
0.13
1.0399
42.8
= 8
21
22
0.360
0.115
1.0437
483
f 10
20
21
0.342
0.11
1.0317
50 6
2 10
9
0.162
0.05
1.002
9.3
Fig. 3. Non-bear tielectnc effect versus reduced dennty. The full tie IS the fitted sum of p* and pti contributions (eq. (9)). The dashed line is the calculated value for electrostnction.
111
and the
A EfE2 (lo*0 m21vq
P
P (10
232
29 March 1985
LElTERS
3.0 +_ 1.0
k 25
propriate densities. These calculated densities are UI good agreement with the interpolated values, and therefore it is possible to estimate the inaccuracy in the densities. In table 1 the experimental pressures and the dielectric constants, together with the cdculated densities, the isothermal compressibilities and the non-linear dielectric effects, are given. The NLDE is plotted in fig. 3 as a function of the density. For reduced densities less than 03 the NLDE is too small compared with the experimental inaccuracy to enable us to determine any signifkant figures. As can be seen from fig. 3, the NLDE as a function of the density does not show any tendency to follow the extreme behaviour of the compressibihty. If the NLDE is due to electrostriction, and the theory [2] claims explicitly that, in first order, electrostriction is the main contribution to the NIDE, the interpolated values for the isothermal compressibility make it possible to cakulate the NLDE as given by the formula
Volume 115, number 2 k/E2
= 27~ [;(e
CHEMICAL PHYSICS LETTERS
+ 2)14 a2p2xT.
(2)
These values are also plotted in fig. 3. It is obvious that electrostriction cannot explain the experimental results for the NLDE. Not only do the absolute values of the NLDE not fit, but even the trend in the experimental data is not in agreement with the calculated values. The experiments reported in our previous paper were made at densities where the NLDE results and the calculated electrostriction were almost equal. The agreement we claimed earlier was therefore purely fol-tultous.
4. Discussron The basic theory for the non-linear dielectnc effect has recently been expounded by Fulton [2] _ The full development of the theory was unfortunately limited to the first-order term in the expansion of the deviation of the average polarisability. We shall here confme ourselves to a somewhat schematic presentation of the theoretical expressions that allow us to find the dependence of the non-linear dielectric effect on the density and on the density correlations. The close relation with the theory of light scattering in first and second order can be used as aid. The starting point is eq. (84) and eq. (86) in ref. [2] : de/E2
= (CIkBT)
[f(e
+ 2)14 (3)
where the geometry of the problem determines the constant C and y = AD{1 + 47~LA/3)-~ III matrix notation, in which A/3 is the deviation from the average polarisability density, and L a propagator with a range limited by the core diameter of the atoms 173. By taking only the first-order term m the expansion in powers of AF one can substitute: Y(rl’2)
= 8 (r, - r2) 01[P(Q)
-PI
s
(44)
where p is the average densrty and Q the molecular polarisability. Transformation into k-space leads to (A&?)(l)=
(C/knT)[f
X JdkgI(k)S(k).
(E+ 2)14 cr2p (5)
29 March 1985
S(k) is the structure factor and gl (k) is a weight tiction that III the long-wavelength limit should be a delta function at k = 0, leadmg to AC/E* = C[$(E + 2)]4cr2p2xT.
05)
The first-order term, wrth XT the isothermal compressrbility, is the expression given by Fulton (eq. (87), ref. 123). It can also be derived by classical thermodynamics and the effect is called electrostriction 181. The crucial point is that the k = 0 limit should be taken, which physically means that flurd should flow from the bath mto the capacitor to balance the lowering of the chemical potential which is caused by the applied electric field. The result is an increase in density and consequently an increase in the dielectric constant. Under the experimental conditions it is, however, not likely that thermodynamic equilibrium is maintained. The cell that we used consists of two glass plates, gold-plated on one side. These two electrodes wrth dimensions 12 by 15 mm2 are kept at a separation of 135 m. One can estimate the pressure difference that would build up between the condenser and the reservoir as a consequence of the lowering of the chemical potential. Smce AP = (E - 1)E2/8n this pressure difference is about 12 Pa as a result of the 2.2 X lo6 V/m field during 10 ms. considering the construction of the cell and the small pressure difference it 1s reasonable to assume that no replenishment takes place. We find confirmation of this assumption by taking 18 time-interval samples in the tune the pulse is applied to the cell, that means that every 500 m the dielectric increment is measured. lf there were a stream of liquid going into the cell there would be an increase in the time-interval data in the time-sequence during the duration of the pulse. However no significant change in the measurements as a function of the time was observed. Smce we detect the difference between e for the system with and without the field a possible effect of the high field time-averaged over the repetition rime wti cancel. In the apparatus we used, and that has been used by many other experimentalists, the effect of electrostriction is completely suppressed. if electrostriction is not the cause of the measured effect, the question arises as to what contributions increase the dielectric constant under the influence of the high field. In fig. 3 we plotted the measured non233
CHEMICAL PHYSICS LETTERS
Volume 115, number 2
linear dielectric effect corrected for the factor 1; (E + 2)14 as a function of the density. For denslties 1 .I there is a linear dependence, which is uptop’= m accordance with eq_ (5) if one assumes that the integral over k IS a weak function of the density and hence does not contain the k = 0 contributions. For larger denntws the density dependence becomes much stronger and this might well be caused by higher-order terms III the expansion of A/3_ The second-order term adds a contribution (A&@)(2) X_/drld v
= (C&T)[f(e
+ 2)14
‘2 dr3 (AO(rl )LA&)
A&)LA/Gq))(71
This is the type of term that appears in the theory of depolarized Light scattermg. For a full discussron of the assumptions and approximations we refer to the hterature on that subJect [9- 12]_ We can rearrange this second-order term and obtain an expression of a form analogous to eq. (5). (Ae/E2)(2) X ldk’
= (C/x-&
[f(~ + 2)14 o!p3
s d/c” g?(k’k”)
T(k’k”),
(8)
where T(k’k”) is now a four-point density correlation function and the weight-factor gz(k’k”) depends on the actual induction mechanism represented by the propagators L_ We can combine the contributions to the non-linear dielectric effect, eq. (5) and eq. (8). to derive AtzIE2=ap*
+bp*3
_
(9)
By plotting p*-1 (AE/E~)[$ (E + 2)] -4 versus p* we fimdthevaluesa=3_5X lO-*~m2V-2andb=I.7 X 10a20 m2 Vm2. In fig. 3 we show the result of plotting eq. (9) wth a subaitution of these values ofn and b. The result shows rather convincingly that there must be a term lmear in p* and one that increases with Pan. In the absence of a full second-order theory we can only draw some qualitative conclusions from our experimental findings. The density dependence of the measured non-linear dielectric effect shows that there is no electrostriction but that there are contnbutions from first-order as well as from second-order in the developmeat in the polarisabrlity. The second-order contribution is surprisingly large with respect to the 234
29 March 1985
fust-order contribution. This is caused by suppression of the first-order term by the way the-experiments are carried out. The first-order effect obeys a rather strict k = 0 selection rule and the geometry of the capacitor that we used, and that was used in most experiments reported in the literature, keeps the amplitude of the k = 0 modes low. This is borne out by the total absence of the effect of the diverging compressibility, eq. (6). That a small fust-order contribution persists nonetheless is probably caused by relaxation of the k-selection rules, eq. (5). On the other hand, there is no restriction to small kvectors for the second-order term. Probably g2(k’k”) will have the form of an integral over space of a spherical Bessel function and WIIJ favour k-values around the fust peak of S(k) [ 121. Very little is known about the four-point density correlation functions [ 131 and no quantitative estimate of the expected value of the constant b in eq. (9) can be made. However, the conclusion that the non-linear dielectric effect of xenon at high densities is pre-dominantiy a second-order effect is of consequence for the interpretation of analogous measurements on compounds of comparable density. The origin of the discrepancies that were noted between fnst-order predictions and various experimental values [l] lies in the fact that one measures mainly a second-order effect.
Acknowledgement This work is part of the research program of the Foundation for Fundamental Research of Matter (FOM) and the Netherlands Foundation for Chemical Research (SON), with financial support from the Netherlands Organization for Advancement of Pure Research (ZWO).
References [ 1) J. van der Elsken, P. van Zoonen
and J.C.F. hfictuelsen, Chem. Phys. Letters 106 (1984) 252. [2] R.L. Fdton, J. Chem. Phys. 78 (1983) 6865,6877. [ 3 ] A. Mxhels and T. Wassenaar. Physiu 16 (1950) 253. [4] W.B. Streett, L.S. Sagan and L.A.K. Staveley, J. Chem. Thermodynam. 5 (1973) 533. [S] J. de Boer, F. van der Maesen and CA. ten Scldam. Physiea 19 (1953) 265.
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CHEMICAL
16) D. Vidal and M. Lallemand, J. Chem. Phys. 64 (1976) 4293; 66 (1977) 4776. [7] H.MJ. Boots. D. Bedaux and P. Mazur, Physica A79 (1975) 379. [ 81 H.S. Frank, J. Chem. Phys. 23 (1955) 2023; L. Landau and E. LAhitz, Electrodynamique des milieux contmus (MIR. Moscow, 1969) p. 79.
PHYSICS
LETTERS
29 March lS85
[9]
D. Frenkel and J-P. McTague, J. Chem. Phys. 72 (19I:O) 2801. [lo] B.U. Felderhof, Physica 76 (1974) 486. [ 111 T. Keyes, J. Chem. Phys. 70 (1979) 5438. [12] P. Madden, Mol. Phys. 36 (1978) 365. [ 131 J. van der Elsken and D.M. Heyes, Can. J. Phys. 59 (1981) 1532.
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