AN EMPIRICAL MINIMUM DIELECTRIC LOSS RELATIONSHIP G . E. CROUCH, JR. Plaaties Laboratory, Princeton Cniversaty, Princeton, New Jersey Received June 2, 1960 I . INTRODUCTION
The electric behavior of a dielectric is intimately related to the mechanism of charge displacement, so that the dielectric properties of a material may be approached from a consideration of this mechanism. Such an approach to the dielectric properties of a hypothetical material is given here, rather than the usual and more rigorous consideration of the chemical composition. Indeed, the chemical composition and the mechanism of charge displacement in a material are interrelated to such an extent that a study of one leads to the other; however, no attempt is made to carry the analysis that far. An exact analysis from either standpoint would necessitate a knowledge of force constants and damping factors for displaced charges in various chemical combinations. Although measurement of mechanical and electrical properties affords some understanding, there remains much to learn about these constants. Nevertheless, by introducing approximations for the mechanisms of charge displacement, some progress can be made in predicting what dielectric properties one could expect for a given material (the idea considered here). From this approach an attempt is finally made to draw some useful conclusions with regard to developing a dielectric material with certain electrical properties. Only simple models for charge displacement will be employed. Such an analysis is of real physical importance and gives results in close agreement with those obtained from quantum theory, although the physical interpretation of the two is different. 11. CHARGE DISPLACEMENT AND DIELECTRIC DISPERSION
When an electric 'freld is applied to any material there is an effective displacement of charge. This charge displacement is against a restoring force, part of which is conservative and part of which is nonconservative. Considering an applied force varying with time, we introduce an inertial force varying with acceleration, a damping force varying with velocity, and a restoring force varying with displacement so that
M d2x dt2
dx + KX = eEej"' +Ddt
M is the mass associated with the displaced charge e , and D and K are appropriate parameters. E is the electric field intensity. Replacing D by Mg and K by w:M in equation 1 the vector displacement is given by:
x=
d - w2 + j w g 1279
1280
E. CROUCH, JR.
0.
If we assume N , charges of type x per unit volume the dipole moment per milliliter is given as: N.
e2 =
The summation is to include all types of oscillating charges. g is a constant. To obtain the dielectric contribution due to conductivity let W: + 0, so that from equation 1 the conductivity is u =
Ne2 Mg
+ jMu
-
For the materials considered here u ~ . ~ .2 X mhos per meter and therefore the D. c. conductivity contribution to e’’ will be assumed negligible. If the Lorens-Lorentz internal field relationship is used,
- jc” - 1 - jd’ + 2 =
N , e’ M€o(u: - W2 jug.) Separating real and imaginary parts we obtain the usual relations (2) : e’
€’
e t’
+
x 2
I
F
?
+
N,ea (W: - 0’) M €0 (uf
- a2+ u2gf
Here C is a constant and eo is the dielectric constant of free space. Slight approximations have been made in that (e’’)’ < < (e’)’, a reasonable assumption for the materials with which we are concerned. Since the electronic contributions to the dielectric constant show no systematic variation in different materials but remain somewhat constant, it seems to be a reasonable approximation to separate the electronic contributions by considering the charges to be immersed in a ,continuous medium of dielectric constant n2. Where n is the refractive index, equations 2 and 3 accordingly become:
Cl is a constant.
DIELECTRIC LOSS RELATIONSHIP
1281
If only one type of oscillator is considered, for the atomic polarization equation 4 becomes:
- 'n - CzN? --+ 2nz nzK t'
e'
Ct and K are constants. Here we consider w < < w z , which seems valid for data taken at frequencies of 1 m.c.p.s. and 1000 m.c.p.s. Equation 6 is identical with an equation given by Frohlich (1) for the static dielectric constant which holds in a very general way for any noncrystalline solid material. In obtaining equation 6 Frohlich follows a classical statistical mechanics treatment of the Lorenz spherical region with the Onsager modification, If one is dealing with a permanent dipole that can rotate, Onsager's formula for the dielectric behavior is obtained, while if the only displaced charge is bound elastically, equation 6 is obtained.
FIG.1 . Dielectric constant uersun specific gravity for a group of plastic materials at 1 m.c.p.rr. 111. APPLICATION TO A HYPOTHETICAL MATERIAL
Before attempting to draw any generalizations from equations ,4 and 5, we examine their applicability to a group of plastic materials. In figure 1 the specific gravity is plotted versus the dielectric constant as measured at 1 m.c.p.s. I n figure 2 the measured dielectric constant a t loo0 m.c.p.5. is plotted versus
1282
0 . E. CROUCH, JR.
specific gravity. If we choose the constants in equation 4 so that we obtain a dielectric constant of 3.0 at a specific gravity of 1.17 and assume a constant value of nz of 2.0, we find that if we assume that the number of oscillators increases linearly with the specific gravity, we obtain curve 1 in figures 1 and 2. However, if we amume that the restoring force, K , varies linearly with l / f l -an assumption which, it happens, is approximately equivalent to applying the Clausius-Mosotti relation with a constant K-we obtain curve 2 in figures 1 and2.
FIG.2. Dielectric constant versus specific gravity for a group of plastic materials at loo0 m.c.p.8.
Since curve 2 seems to fit the observations as well as one could expect, it appears that empirically we can treat these high-polymeric materials as if they were nonpolar, provided the restoring force decreases as e’ increases. Further considerations of equations 4 and 5 indicate that if other parameters remain constant an increase in g. increases e” and decreases e’. If the assumption is made that g. is small compared with w., it is seen that e’’ increases almost linearly with 8 . and e’ varies only slightly. It is also interesting to note that for a large value of w., that is, a large restoring force with displacement, e” is much less than it would be with a small w.. An increase in w alone to increase e”, while e’ decreases only slightly. I n certain instances when an ion can occupy one of several equilibrium positions in a crystalline lattice under the influence of an electric field, a satisfactory
DIELECTRIC LOSS RELATIONSHIP
1283
choice of the parameters in equation 1 is possible. In fact, for any mechanism the forces acting on a displaced charge must satisfy equation 1 so that equation 1 is applicable for all types of polarization, even though it affords no physical interpretation of dielectric phenomena as does the chemical rate process. We now examine certain generalizations drawn from equations 4 and 5. Although difficult to consider analytically, because of chemical composition and lack of homogeneity, these generalizations seem consistent with observations on most plastic materials, so that a practical empirical deduction of a minimum loss relationship, as described in Section IV for these materials, therefore seems justified. I t should be remarked that certain materials-polystyrene and hard rubber, for instance-show a remarkably constant energy loss per cycle over wide ranges of frequency. I t has been suggested that the losses in these materials arise from a mechanism behaving like ordinary friction or certain types of plastic flow where dissipations are essentially velocity independent. Such a mechanism would mean that the velocity-dependent term in equation 1 should be replaced by a constant; however, the slight observed increase in e” and decrease in e’ with increase in frequency seems to indicate that even these can be represented satisfactorily by the oscillator mechanism of equation 1. IV. DISCUSSION AND CONCLUSIONS
From curve 2 of figure 1 it is seen that the internal field relations of Lorenz and Lorentz adequately represent conditions in some high-polymeric materials if K varies as 1/47. This is not surprising, since the Clausius-hfosotti relation is known to give even approximately correct correlation betxeen the dielectric constant of a liquid and its vapor in some instances. The materials in figure 1 show a wide variation in composition and represent a completely random choice. Suppose that one is concerned with some material for which the dielectric constant is given by equations 4 and 5. This mould presumably be a material with no permanent dipole and therefore would appear to be different from most of the materials given in table 1. The physical explanation of the behavior of these materials is therefore not represented by equations 4 and 5 ; however, since for data we have taken a t a frequency of 1000 m.c.p.s. we find a kind of empirical agreement between equations 4 and 5 and observations, we take these equations as a basis for making certain predictions to be discussed below. While realizing that the physical behavior of a dielectric is not adequately. represented by equations 4 and 5 , it is nevertheless seen that they offer a correlation between tan S with e‘. Since one must believe that there is some correlation between an increase in e’ and an increase in tan 6 and not merely a “happen so” of nature, it is interesting to make some predictions for a hypothetical material which one might wish to develop. More specifically, suppose one considers the mechanism of polystyrene and could maintain all parameters in equation 4 and 5 constant while varying N,, the amount of displaced charge. This is admittedly not possible physically, but, based on this assumption, it is seen that while e’ changes from
1284
G. E. CROUCH, JR.
TABLE
1
Materials shown in jigurea 1 and 9
m....Cellulose acetate butyrate
l . . . . . Polyethylene 2 . . . . . Modified isomerized rubber com- 21.. . .
pound 3 . . . . . Polystyrene 1..... Hard rubber (no filler) 5 . . . . . Ethyl cellulose 6 . . . . . Polyvinyl carbazole 7 . . . . . Methyl methacrylate resin molded 8 . . . . . Vinyl chloride-acetate resin 9..... Cellulose propionate 10.. . . . Vinyl formal resins 11:. . . . Vinyl butyral resins (rigid) 12... . . Aniline-formaldehyde resin (no filler) 13. . . . . Allyl resins (cast) 14. . . . . Glyceryl phthalate resin (cast) 15.. , . . Nylon resin (injection molded) 16.. . . . Vinyl butyral resin (flexible filled) chloride resins 17... . . Vinylidene (molded) 18.. . . . Cellulose acetate (sheet) 19... . . Cellulose acetate (molded)
Cellulose acetate propionate Phenol-formaldehyde resin (no filler) 2 3 . .. . Phenol-formaldehyde compound (wood flour filled) Phenol-formaldehyde compound 24... . (mica filled) Phenol-furfural compound (wood 25 . . . . flour filled) Phenol-furfural compound (fabric 26 . . . . filled) Silicone rubber (mineral filled) 27.. , . 28.... Melamine-formaldehyde compound (cellulose filled) 29.. . . Cold-molded nonrefractory (organic) 30.. . . Celluloae nitrate (pyroxylin) 31.. . . Inorganic plastic glass-bonded mica Melamine-formaldehyde com32.. . . pound (asbestos filled) 22 ....
C
FIQ.3. Expected tan 6 uemw dielectric constant for a plastic material
DIELECTRIC LOSS RELATIONSHIP
1285
2.75 to 7.00, tan 6 increases by a factor of 4.This is shomp in curve 1 of figure 3. Furthermore, it is seen that if one could retain the very efficient mechanism of polystyrene one could not hope, for instance, to obtain an e’ of 10 with a tan 6 < 0,0011, whereas for polystyrene tan 6 0.0003. If tan 6 = e”/e’ is considered an index of the efficiency of any mechanism, the efficiency of a barium titanate and a rotating dipole mechanism in solids is in many cases greater than that even of polystyrene. Curve 2 in figure3 is plotted for a vinyl formal resin system, density 1.25, e’ = 3.0, tan 6 = 0.02. One would conclude then that in a plastic material there is a minimum tan 6 for a given value of e‘ and that if the Lorenz-Lorentz internal field is an adequate representation it is inherent in charge displacement that the efficiency decreases with the charge that is displaced in any given system. The impurities in polystyrene as well as in all the other actual polymeric materials considered appreciably affect the dielectric properties. The dielectric contribution from the various polarizations is quite different as one goes from material to material, so that a more rigorous examination of this relation would have to include these considerations.
-
V. SUMMARY
An attempt is made to place on a common analytical basis the consideration of the dielectric behavior of a group of heterogeneous polymeric materials for the purpose of predicting certain dielectric properties of a hypothetical material, The possibility of incorporating these properties in an actual polymeric material is of practical interest. An expression for the complex dielectric constant of a solid is obtained. From this expression predictions of minimum dielectric loss for a hypothetical material are made and correlations with actual materials drawn. The research reported here was sponsored by the U. S. Army, the U. S. Navy, and the U. S. Air Force under Signal Corps Contract No. W-36-039-sc-32011. Reproduction in whole or in part is permitted for any purpose of the United States Government. REFERENCES (1) FROHLICH, H.: Trans. Faraday SOC.44,238 (1948). (2) SLAPER,J. C., AND FRANK, A.: Electroma&tism. McGraw-Hill Book Company, Inc., New York (1947).
