OFFICE OF NAVAL RESEARCH

Gov't Order NAONR 26-74

Task. No. 74-622

TECHNICAL REPORT NO. 2

MOLECULAR DIPOLE ELECTRETS

by

Frederick I. Mopsik and Martin G. Broadhurst

Institute for Materials Research, National Bureau of Standards

Washington, D. C. 20234

November 27, 1974

Reproduction in whole or in part is permitted for

any purpose of the United States Government

Approved for Public Release; Distribution Unlimited

MOLECULAR DIPOLE ELECTRETS

Frederick I. Mopsik and Martin G. Broadhurst

Institute for Materials Research, National Bureau of Standards

Washington, D. C. 20234

ABSTRACT

The total polarization due to molecular dipoles in a glassy electret is

computed using an Onsager cavity approach. From this result, all the possible

contributions to the piezo- and pyroelectric coefficients are considered. It

is shown that there are major contributions from the variation in dielectric

constant and, for pyroelectricity , from thermal motion. These results account

well for experimental data for polyvinyl chloride.

INTRODUCTION

Solid organic materials which are cooled from the liquid state in the presence

of an electric field form electrets. This behavior was first postulated by

1 2

Heavisi'3e and confirmed by Eguchi . These electrets have a permanent or slowly

decaying electric polarization (dipole moment per unit volume) and, in general,

exhibit piezoelectric and pyroelectric properties. That is, they respond

electrically to mechanical and thermal stress, as was originally demonstrated

3 4 experimentally for waxy solids ' and first explained with a simple dipole

model,

In recent years a variety of synthetic organic polymers have been shown to

5 6 be piezoelectric and pyroelectric after suitable polarizing treatment. ' Even

though the technology of making polymer film transducers has developed sufficiently

7 8

to yield commercial instrumentation and devices there is still no consensus

regarding the underlying mechanism on a molecular level nor is there a generally

accepted equation relating transducer response to other basic structural or physical

properties. Such an. equation would aid in systematically adjusting parameters to

optimize the effect and help predict limiting sensitivities.

Previous work on ceramic piezoelectric and pyroelectric materials like barium

titanate and triglycine sulfate has provided a well established body of data and

theory.^' Fukada has applied the classical group theory approach to polymers."'""''

This formulation seems inappropriate for totally amorphous polymers, many of which

form piezo- and pyroelectric electrets, and has not led to an understanding of

fundamental behavior. A phenomenological model for polymers was published by

12 Gubkin and again the results did not adequately connect piezoelectricity to basic

molecular properties. Aslaksen provided a theory for pyroelectricity in polyvinylidene

13 fluoride (PVF2). This theory described an important effect of molecular librations

but did not include the volume effects originally described by Adams in his early

- 2 -

paper. ^ Other effects such as molecular reorientation"''^ and charge injection"'"^ have

also been proposed as contributing to piezo- and pyroelectricity in polymers.

In earlier papers^' '^^ we provided an equation connecting piezo- and pyroelectricity

in amorphous polymers to dielectric, mechanical and thermal properties. The model was

3 like that of Adams and in addition included librational effects. The results for this

model were deduced in an ad hoc manner however and missed an important contributing

factor from the variation in dielectric constant. In this paper we treat the model

rigorously and calculate the piezoelectric and pyroelectric responses from first

principles.

Specifically the model is for an amorphous solid containing molecular dipoles

which have been partially aligned in an electric field and then immobilized by

cooling below the glass transition temperature. An expression for the permanent

polarization is derived by means of a cavity reaction field calculation like that

used by Onsager.'''^ The temperature and pressure derivatives of this polarization

yield the piezo- and pyroelectric coefficients in terms of other independently known

physical properties. The pressure and temperature derivatives are shown to be

3 closely related to each other as originally proposed. The piezoelectric result

is verified for an amorphous polymer. Verification of the pyroelectric result

depends on knowing a value for a molecular libration amplitude, and a reasonable

assumption about its value provides results that support the model.

PERMANENT POLARIZATION

Consider the model as illustrated in figure 1. A dipole vzith permanent moment

j£ and polarizability a is at the center of a cavity of radius a. The cavity is

embedded in a slab of medium with dielectric constant e and also has an electrical

polarization P^ in the z direction which is perpendicular to the plane of the slab.

Any free charge that is trapped in the medium will be considered to be confined to the

surface. Electrodes are applied to the surface and shorted together so that the applied

field E is equal to zero. There will be an internal field in the cavity acting

on the dipole so that the total moment m is given by

- 3 -

m = jj + a ^ f^2_)

For this case, since there are no free charges in the interior of the dielectric,

the potential can be written in the form

* = - + Br ^

cos 9 , (2)

where A and B are constants that will have different values in the external

medium than in the cavity. Linearity and hence superposition will be assumed since fields

commonly used to pole materials are still below those needed for saturation to be

observed , and hysteresis effects are excluded. Then, the field in the cavity will be

the superposition of the applied field, the permanent polarization and 'the dipole

18

field. Following Frohlich, we evaluate the potential assuming E^, and jj_ are

all present and parallel to each other. The boundary conditions are

E = E , r -V CO (3) — —o ,1

• - m cos 9 .. f,y. $ = —-— , r o, (4)

47Te 2 o r D = e eE + P , r > a, (5) o — —

D = e E, r < a, (6) — o ~

where D is the electric displacement 6^ is the permittivity of vacuum and

e is the angle measured from the z axis to the dioole axis. Then, one can \7rite

$ = - f-Ar + E r\ cos 9, r > a,

$ = - /—-^^^-y + BrVcos 9, r < a . VAtte r /

(7)

(8)

One can evaluate19$, A and B by imposing the conditions that the tangential component

8$ of E, E- = - — -TT , is continuous and the normal component of D, D = £ e — + P — or 00 — rodri

where is the normal component of P^ , must be continuous at a. For this last

condition it should be remembered that P^ = 0 inside the cavity and P^ is a constant

outside the cavity. Then one finds that

(c - 1) 1 Z

(2 e+ 1) ^o (2 e + 1)

m

(2 £+ 1) • 3

' Att e a

o

(9)

^ ~ (2 e + 1) o (2 £ + 1)

m

o 17

+

2(£ - 1)

(2 £ + 1) , 3

^ 4tt £ a (10)

With this result, the Onsager derivation can be suitably modified to allow

for a permanent polarization. For the electret between shorted electrodes, E^ = 0

and the cavity field is now given by

1 P

G =

(2 £ + 1)

There is also a reaction field, R, parallel to m given by

(11)

m

(2£+ 1)

This gives for the internal field F^,

4tt£ a" o

(12)

F = G + R = - ^ 2(£- 1)

m

^ ^+ ^ S (2 £+ 1) 4. £ o Using Eq. 1, we oDtam for the moment

m = y 'u + a — o —

I 2(£ - 1) m

2 £+ 1 +

(2 £+ l)a^ 4tt e o

(13)

(14)

where _u is a unit vector parallel to the permanent dipole and is the magnitude

of the permanent dipole moment. Solving for m and using the relation

e - 1 a =

£ + 2

00 we obtain for the total moment in the cavity

4 TT e a , o (15)

- 5 -

(2e + 1) (e + 2) —

m = _____ y u

3 (2 e + e ) o "

oo

3 e - 1

To obtain the total polarization it should be remembered that P and m are not parallel in general. However, P is uniform throughout the dielectric and only

the component of m parallel to P will not vanish on averaging. Then, since the

material is composed of the dipoles,

! = — (17)

where N is Avogardro's Number and V is the molar volume. We allow the volume of

the material to equal the sum of the volumes of the cavities so that

. - -f-

^ = 1 . (18)

If the average component of u parallel to P is given by < cos G > , then

combining Eqs. 16, 17 and 18, we obtain

(e^ + 2) C2e + 1) ^ - i (19)

P - 7^—T ^ T7~ M COS 6>+ p

3 (2e + ) V o +

where the vector notation has been suppressed since there is no question as to

orientation. We can now solve for P to give

+ 2)

P = — -~ y < cos 9 > . (20)

J V o

The result obtained for the permanent polarization using an Onsager type

model is quite interesting. In the derivation, a distinction between e and

was kept although they are equal where no dipolar reorientation is allowed. This

was done to show that all the terms involving the medium external to the cavity do

- 6 -

not enter the final expression. In fact the same result could have been derived

1 P

from the Lorentz field —

F = —E + t3e by letting°ooE =0. However, the case of

o frozen-in permanent dipolar polarization has not been recognized as leading to a

Lorentz type field.

Also, it is interesting to compare the total polarization given in Eq . 20 with

that of the Onsager liquid with free dipoles. The polarization ,of a fluid with

permanent dipoles e can be written

+ 2) (2. + 1 ) ^ 3cJ.„- 1) ^'^^

where < cos 0 > is carried explicitly and e is the liquid dielectric constant.

J-i

(e - 1) e E ^ ? (22)

Li O

Substituting Eq. 22 in Eq. 21 and solving for P, we obtain

p = (i^^) —T~ — % < 9 > • ^23)

When Eq . 23 is compared with Eq. 20, it is seen that at constant volume, the

(e^ - 1)

equations differ only by the factor —^—^—r- . Therefore, if < cos 0 > is frozen

Lt 00 in by going through a glass transition, at constant volume, one should freeze in a .

polarization equal to (e^-e^)e^E^ where E^ is the poling field. This is exactly

what one would predict from macroscopic arguments^ and shows the consistency of the

results for the electret with that of the Onsager fluid.

The last point that will be considered here is the question as to correlations

between the dipoles of the electret. In a fluid this leads to the replacement in the

2 2

Onsager formula of the factor by g where g is the average sum of projection

cosines of all dipoles upon any reference dipole in the fluid. The g factor comes

about from statistical mechanical averaging of squared total moment < PV.JPV > at

zero applied field to which the dielectric constant is related. In our notation,

- 7 -

this becomes N < m . E HLj^ > where the index numbers all dipoles in the sample.

It is the last factor in the angular brackets that leads to the g factor. By

comparison with the development here where < P^V > does not vanish for zero applied

field the g factor must be contained in the < cos 9 > term of Eq . 20. In any

event we expect the relation between Eq . 20 and Eq. 23 to still hold.

PIEZOELECTRIC AND PYROELECTRIC COEFFICIENTS

With the expression for the total polarization given by Eq . 20, it is possible

to evaluate the change in surface charge with a uniform change in volume due to

either a change in pressure or a change in temperature. That is, the change in

6 surface charge with temperature or pressure is given by the change in P_.

From Eq. 20, ...

e + 2 3P N «> ^ 3V

y < cos e > 3X ^2 3 "^o 3X

+ -ZTT- y < cos e > 3e

3V "^o 3X

N

^ 3< cos 6 > TT" ^ VI tt; , (24)

o 3X

where X refers to either temperature or pressure. The permanent dipole moment

y is assumed constant since the intra molecular force constants are much stronger

othan the inter molecular force constants. Also, unpublished dielectric measurements

- 8 -

on CCl^F in this laboratory were carefully analyzed for possible variation of

with pressure and none could be found. Of the three terms in Eq. 24 only the last

two need further evaluation and of these the second is the easiest to deal with.

19

It has been shown from pressure measurements on dipolar liquids over the range of

densities considered here that,

e - 1 D

e + 2

(25)

C and D are constants for the material, and one can carry out the differentiation

with respect to V to give

8e

8X

1_

V

U + 2)(e - 1) D + 2)

3X (26)

The term involving D in (25) corrects the usual Clausius-Mossotti relation. Its

value can be estimated for a particular polymer from data on structurally related

simple liquids, and is relatively small.

Using (26) we can write (24) as ,

•

_3P

3X

N 9V

e + 2 +

(e^ + 2)(e„ - 1) (^^ + 2)'

N

e + 2

J

3<cos e>

3X (27)

- 9 -

Except for the small term containing D, the terms inside the bracket are of

similar magnitude. The change in dielectric constant of the electret provides a

major contribution to its piezo- and pyroelectric response.

< cos 6 >, such as the Kirla^ood g factor will be ignored since they are liable

to be small and require a more specific model than we are using.

The variability of < cos 6 > at constant dipole orientation may seem surprising,

but fixed dipole orientation implies only . that <9> is constant. To evaluate

< cos 9 > we will assume that the dipole is a classical rotational harmonic oscillator

that oscillates about an angle 9 fixed with respect to the polarization direction.

Consider the coordinate system shown in Fig. 2 where z is the direction of the

permanent polarization and z' is the direction of the average position of the

dipole. Then for a unit vector representing the dipole, < cos 6 > is equal to

< z > , the projection of the unit vector along the z axis. If the x' axis is

chosen so that the dipole oscillates with amplitude (f)^ about the y' axis, then

the dipole remains in the x' z' plane. Then from the definition of the Eulerian

angles,

EVALUATION OF < cos 9 >

The only term that remains to be evaluated in Eq . 24 is that involving

°y ^— . In the discussion that follows any correlation contributions to

z = sin 9 sin x' + cos 9 z'.

where z' = cos 4) and x' = sin (J) . If now 4 = cos cot, then one obtains for

o the average projection

1

TT

O

(28)

+

o

10 -

Eq. 28 is easily evaluated by noting that the first integral vanishes by

symmetry and the second integral yields a Bessel Function of the first kind of

zero order, J ((t> ) . Therefore, one obtains

o o

< cos 6 >= cos 6 (4> ) . (29)

Using the properties of Bessel Functions, the derivative is then evaluated as

9< cos 9 > , . o , . = - cos 6 (c^^) — , (30)

where J, (4> ) is the Bessel Function of the first kind and first order.

1 o For a classical harmonic oscillator, the mean squared amplitude of vibration

is given by

.2 ' 2kT

^ *o ^ = ~Y~ (31)

lO)

where I is the moment of inertia. Replacing X with T we find

(32)

Assuming the quasiharmonic approximation that the frequency depends only on volume

by a Gruneisen constant y = - (V/w) (dco/dV) , then Eq. 32 becomes

/!M !o_ . !o_ /3v_\

I 3T I 2T V ^ I 3T 1 (33)

- 11 -

For X equal to p, the first term will not occur and we obtain

The first term in Eq. 33 is large compared to the second, and hence the response

to temperature will be greater than the response to pressure.

The. final expression for the derivative of the polarization can now be written

with the use of Eqs. 20, 27, 29 and 30 as

8P

3X

= - P

_9V

8X

1 + - (e + 2)

D

V

+

J, ) 1 o

o o

94)^

"ax" (35

where ^ is given by either Eq. 33 or Eq. 34.

RELATION TO MEASURED QUANTITIES

While the derivation until now has been in terms of the polarization, the

polarization is not a measured quantity. One customarily measures current, area

and potential. If the polarization is uniform in the sample, then the total charge

present on the surface due to the polarization is given by

Q = PA

- 12 -

where A is the area. JJif f erentiating and using the relation

P = (e -e ) e E = Aee E

^ L CO o p op

where E^ is the poling field used to form the electret, one obtains

1 3^

e E A 8X

The assumption of isotropy has been used to convert the area derivative to a volume

derivative. .

This result can be combined with Eq. 35 to yield

Ae _3P

3X

+

Ae

V

3V

9X (36)

e E A

o p

^ - 3X

Ae

SX

- + 2)

V

J ((f, ) ^ o

(37)

1 9V

If —rr- — is defined by V , Eq. 37 can be rewritten as V oX A

e E AV^

op X

M - 3X

= - A e - 3 (e +2) D

X W dx

(38)

- 13 -

where V /3 is the linear coefficient of expansion for the variable X. Equation

38 gives the desired result, the charge flow in an external circuit due to a change

in either temperature or pressure in terms of the relevant material properties.

Of thes§j (J)^ is the most difficult to measure. The term containing the constant D

can;- be determined directly from dielectric and dilatometric measurements and, judging

from materials where it has been measured, is probably small enough to be neglected.

In a constant-temperature, variable-pressure experiment, the term in (j) is also

o small since y for vibrations leading to (})^ will be small. For pressure variationonly the first term in the bracket will apply. For the constant-pressure, variabletemperature experiment there is an additional contribution from the term in <^ . o For PVC"'" the results^^ are shown in Table I. It is seen that the theory adequately

accounts for the piezoelectric data without adjustable parameters and that the angle

4)^ needed to account for the difference between the piezoelectric and pyroelectric

21 results is quite reasonable. (Recent measurements on polyethylene gave = 10°. )

Thus, the model employed in this paper accounts in a reasonable manner

for the electret properties due to frozen-in dipolar orientation in a glass of polar

molecules.

REFERENCES

1

0. Heaviside, Electrical Papers, Macmillan, London, !^, 488 (1892).

M. Eguchi, Phil. Mag. 4^, 178 (1925).

^ E. P. Adams, J. Franklin Inst. 204, 469 (1927).

A

A. Meissner and R. Beckmann, Z. Tech. Phys. 9_, 174 and 430 (1928).

^ H. Kawai, Japan J. Appl. Phys. 8^, 975 (1969).

M. G. Broadhurst, W. P. Harris, F. I. Mopsik and C. G. Malmberg, Polymer Preprints,

ACS 14, No. 2, 820, August (1973).

^ Piezoelectric Analyzer, Toyo Seiki Ltd., Tokyo,

g Piezoelectric Headphones, Pioneer Electronic Corporation, Tokyo.

9

W. G. Cady, Piezoelectricity, McGraw-Hill Book Co., Inc., N. Y. (1946).

D. A. Berlincourt, D. R. Curran and H. Jaffe, Physical Acoustics , Vol. 1 Part A,

Academic Press, p 183 (1964).

E. Fukada, Progress in Polymer Science, Japan 2^, 329 (1972).

"'"^ A. N. Gubkin and W. S. Sorokin, Izv. Akad. Nauk SSSR, Ser. fiz. 24_, 246 (1960).

(Trans. Bull. Acad. Sciences 24^, 246 (I960).)

'^

E. W. Aslaksen, J. Chem. Phys. 57.» 2358 (1972).

14

R. Hazakawa, J. Kusuhara, K. Hattori and Y. Wada, Repts. Progr. Polymer Phys. Japan

16, 477 (1973).

"""^ G. Pfister, M. Abkowitz and R. G. Crystal, J. Appl. Phys. 44, 2064 (1973).

M. G. Broadhurst, C. G. Malmberg, F. I. Mopsik and W. P. Harris, Electrets, Charge

Storage and Transport in Dielectrics , M. M. Perlman, Ed., Electrochemical Society, Inc.

N. J. (1973) p. 492.

L. Onsager, J. Amer. Chem. Soc. 58^, 1486 (1936).

18

H. Frolich, Theory of Dielectrics , Clarendon Press, Oxford (1949).

"""^ F. I. Mopsik, J. Chem. Phys., Vol. 50, //6 , (1969).

20 These values have been corrected for fringing effects resulting from poled material

beyond the electrode edges.

21

K. lohura, K. Imada and M. Takayanagi, Polymer Journal 3^, 357 (1972).

Table I

Results for PVC

e = 13, e =3 — „ -^S^ Measured Predicted

eopE A V„ dX

X

X = p 27 30

X = T 44 44



Fig. 2 Coordinates and Eulerian Angles for Librating Dipole

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2a

tncJcsLjre

( /)

NBS-1 14A (REV. 7-73)

U .S. DEPT. OF COMM.

EiBLIOGRAPHIC DATA

SHfET

1. PLiHI.ICA I ION OK Ki:i'OR r NO.

NBSXJ^ 74-622

2. C lov 't Ae e e s s 1 on No.

3. Ki c ipienl's Aeeession No.

4. i rri.i-: an'd suii 1 1 1 i.i-: MOLECULAR DIPOLE ELECTRETS

5. I'lilij ic at ion Date

6. Performing (Organization C ode

^' ^reclerxck I. Mopsik and Martin G. Broadhurst

8. Performing Organ. Report No. NBSIR 74-622

9. FHRFORMING ORGANIZATION NA.MF AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Proieet/ Task/Work Unit No.

11. Contract/Grant No.

Gov't Order NAONR 26-74

12. Sponsoring Organization Name and Complete Address (Street, City, Stale, ZIP)

Office of Naval Research

Arlington, Va. 22217

13. Type of Report lit Period Covered

Technical Report No. 2

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. AI-iSTRACT (A 200-u-ord or less factual summary of most sifyiificant information. If document includes a significant

bibliography or literature survey, mention it here.)

The total polarization due to molecular dipoles in a glassy electret is

computed using an Onsager cavity approach. From this result, all the possible

contributions to the piezo- and pyroelectric coefficients are considered. It

is sho\"7n that there are major contributions from the variation in dielectric

constant and, for pyroelectricity , from thermal motion. These results account

well for experimental data for polyvinyl chloride.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons ) -i ^ . ^ • i -, electret; dipoles; glass; piezoelectric; polarization;

polyvinyl chloride; pyroelectric

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