On the difference between dielectric and piezoelectric relaxation

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On the difference between dielectric and piezoelectric relaxation

K. Nishinari, S. Koide

To cite this version:

K. Nishinari, S. Koide. On the difference between dielectric and piezoelectric relaxation. Journal de

Physique, 1978, 39 (7), pp.771-775. �10.1051/jphys:01978003907077100�. �jpa-00208812�

ON THE DIFFERENCE BETWEEN DIELECTRIC

AND PIEZOELECTRIC RELAXATION

K. NISHINARI

National Food Research

Institute, Ministry

^of

Agriculture

^and

Forestry, Shiohama, Koto-ku, Tokyo, Japan 135

and S. KOIDE

Institute of

Physics, College

^{of General}

^Education, University

^of

Tokyo, Komaba, Meguro-ku, Tokyo, Japan

¹⁵³

(Reçu

^{le 19 décembre}

1977,

^{révisé le 21}

février 1978, accepté

^{le 9 mars}

1978 )

Resume. ²⁰¹⁴ Les relaxations dielectrique ^et

piezoelectrique

^{sont traitées sur} la base du modèle à deux états séparés ^{par une} barrière de potentiel ^pour clarifier leur différence. On suppose que le mécanisme moléculaire responsable ^du phénomène ^est la réorientation des molécules ayant ^des dipôles électriques permanents. ^On suggère que la réponse instantanée au stimulus est essentiellement

importante ^{dans le cas}

piézoélectrique.

^La dépendance ^en fréquence ^{et en} température ^{est discutée}

pour ^un film de polymère. ^On peut caractériser les relaxations au moyen de ce modèle simple, ^en supposant ^un changement approprié ^de

potentiel

induit par l’application (ou ^la coupure) ^du champ électrique ^{ou de la} déformation externe.

Abstract. ²⁰¹⁴ Dielectric and

piezoelectric

relaxations are treated in the same way by the two-site model in order to clarify the difference. The molecular mechanism for these phenomena ^{is assumed}

to be the reorientation of molecules

having

permanent dipole ^{moments. It is} suggested ^{that the}

instantaneous response

(polarization)

^to the stimulus is essentially

important

^{in the}

piezoelectric

case. The

frequency

^and temperature

dependence

is discussed for a polymer film. It is shown that many characteristic features of the relaxation

phenomena

^{can be well} explained

qualitatively

by ^this simple ^{model if} one assumes an appropriate potential change ^{for the}

dipole

^caused by ^the

application

(or removal) of the external electric field or strain.

Classification Physics ^Abstracts 36.20 ^- 77.40 ^- 77.60

1. Introduction. ^- In recent years there has been ^a

growing

^{interest in}

piezoelectricity

^of

polymers, especially

^{in its} relaxational behaviour

[l, 2]. Although phenomenological

theories such as a mechanical

and

electrical model treatment

[3]

^{and a}

two-phase

^treat-

ment

[4]

^{have been}

proposed,

^{no molecular}

theory

^for

the relaxation

phenomenon

^{has been}

proposed

^so

far. As concerns dielectric relaxation, ^on the other

hand,

many

investigations

^{have been made on a}

molecular basis

[5-8]

and the methods are

applicable

to

piezoelectric

relaxation with minor alterations.

In

solids,

^the

following

^{two cases are} considered to be

important

^as mechanisms in which the

displacement

of electric

charges

contributes to the mean electric moment

[9] : (i)

^{When the electric}

charge

^is constrained to the

equilibrium position by

^{an elastic}

restoring

force as in ionic

crystals,

its behaviour is

governed mainly by

^the

restoring

^{force and} electric field. Born

[10]

calculated the

piezoelectric

^{constant in this case.}

This case

gives

^a resonance-type

absorption. (ii)

^In

many

polymeric

^solids,

on , the

contrary, ^{there are} several

equilibrium positions

^{for electric}

charges

^and

each

equilibrium position

^is

occupied

^{with a certain}

statistical

probability

^which

depends

^{on the local}

field

acting

^{on the}

charges.

^The

simplest

^{model of this}

picture

^was introduced first

by Debye [11]

for dielectric relaxation. He treated the transition between these

equilibrium positions,

which leads to a

macroscopic change

ⁱⁿ dielectric

polarization by

^a

purely

^stochastic

process.

^{In the} present paper, ^we will

adopt

^{the latter}

point

^{of view to} handle the

piezoelectric

^{and dielectric}

relaxation on a common basis.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01978003907077100

772

Treating

^a distorted helical

polymer

chain with effective

charges

^or

dipole

^{moments on the main chain,}

Kasai

[121

^{has shown that the}

piezoelectric

^effects

occur

only

^{in an}

optical

^{shear mode of} lattice vibra- tion

along

^the helical axis. But the relaxational

phenomenon

^{is not in} the scope of his work.

We confine our attention to the

piezoelectric

^direct

effect

only

^and exclude the inverse effect, ^{i.e. the}

production

of a mechanical _response due to an electric stimulus

[13]. Among

^{the four}

piezoelectric

^constants,

the

piezoelectric

stress constant e

(i.e.

^{the constant}

for electric _response to an external

strain)

^{will be}

treated in the

following together

with the dielectric constant a, because motion of the

dipoles

^will

depend

on strain

directly

^{rather than on stress. We assume}

that the

microscopic

^entities

responsible

^{for the}

relaxation in

question

^are permanent

dipoles

^asso-

ciated with molecular chains and that

they

^are

making strongly

hindered rotations or rather vibrations around several favourable directions

accompanied by

occasional transitions from one to the other. Such a

situation will be most

suitably

^described

by

^{the so-}

called site model.

According

^{to linear} response

theory, the frequency dependent f,*(w)

^{= F’ - iF" and}

e*(w)

^{= e’ - ie" can both be derived from the}

stochastic behaviour of the

dipoles (reorientation)

after

switching

^{off the constant external electric}

field or mechanical strain. We can, therefore,

develop

our molecular

theory

of F* and e* in a unified scheme,

and, by doing

^so,

clarify

the difference between these two

quantities.

2. Site model. ^- Dielectric and

piezoelectric

^dis-

persions

^{are both the}

frequency dependence

^{of the}

polarization

^{M as a} response ^{to an} external force F, i.e. electric field E, and, mechanical strain S

respecti- vely.

Since the response ^M is common to both

phe-

nomena, the difference must be ascribed to the nature of the stimulus. It must be remembered that we are

considering

^the

piezoelectric

^{stress constant}

e*,

^{i.e. a}

response ^to strain rather than to stress. The

dipole

is forced to follow the

change

^{in strain}

promptly

because it causes some

change

in the favourable directions of

dipoles.

^In the dielectric case, this effect may be

negligible

and the instantaneous _response is

essentially

^{due to the}

change

in the electronic motions, which are not considered in the

present

paper. If the response is

completely instantaneous,

^{there is no}

possibility

of relaxation. The

dipole

reorientation with finite

velocity

^{in a solid is}

appropriately

^described

by

the site model. We

will, therefore,

^{treat these}

dispersions by

^the

simplest

^site

model,

^{i.e, two-site}

or bistable

model,

in order to see the difference between the dielectric and

piezoelectric dispersions.

We will first consider the case where there is no instan- taneous response.

We will assume that 0

symbolically

stands for the

angular

^variables

describing

^{the motion of the}

dipole.

In the two-site

model,

^{there are two}

potential

energy

minima

V(Ol)

^and

V(02)

^{for the}

dipole

^{rotation as}

shown in

figure

^{1. For}

simplicity,

^{we will confine}

ourselves to the

symmetric

two-site model in which U = 0 under no external

field,

since the extension to the

asymmetric

^{case is} easy. The site model is

appli-

cable to the case where the

potential

barrier between the minima is much

higher

^{than kT and most of the}

dipoles

possess insufficient _energy to rotate

by getting

over the barriers.

FIG. 1. ^- Potential _energy curve for the dipole orientation. The

angular variable 6 represents symbolically the direction of the dipole

orientation in the three dimensional _space.

Imagine

^{now that the external force, which is}

applied

^{at a constant value}

(unity)

^{for a}

long

^{time, is}

suddenly

^{reduced to zero t = 0. The}

behaviour

^{of the} system ^{is described}

by

^{the static}

susceptibility

^for

t 0 and

approximately by

^a relaxation time for t > 0. As is well known, ^{the static}

susceptibility

a, is

proportional

^to

1 IkT

for the site model. The relaxa- tion time i is

equal

^to

(k12

⁺

k21)-I.,

^where

kl2

^and

k21

^{are the} transition

probabilities

^{for the rotation}

of a

dipole

^{from site 1 to site 2 and vice versa. The}

probabilities w;(t) (i

^{= 1,}

2)

^{that the}

dipole

stays ⁱⁿ site i at time t > 0 are

expressed

^as

For a small force,

w;(0) - n,,(oo.)

^is

proportional

^to

the static

susceptibility

a,. Let the

dipole

^{moment of a}

dipole

^{at site} i be denoted

by

pi. The whole

polari-

zation

is

given by

^{use of}

equation (1).

^{Then, as the} response for a sinusoidal force

F(t)

⁼

Fo exp(iwt),

^the

complex

electric

susceptibility x*( w)

^or

complex piezoelectric

constant

e*(m)

^is

given by

The

complex

dielectric constant F* is’related to

x*

^as

F* = 1 + 4

Jtx*.

^As all the factors except

(t

⁺

icor)

in

equation (2)

^are

w-independent,

^thé

frequency

dependence

^{of such} f,* and e* becomes the well-known

Debye

type

single

relaxation as shown for the dielectric

dispersion

in many textbooks

[5,

^6,

9].

Let us now consider the instantaneous response in the

piezoelectric

^case. The sudden removal of strain will be

expressed by

^{a sudden}

change

^{in the}

potential V(O) itself,

and hence there can be no retardation as

might

^be

expected

in the relation between

V(O)

^{and an}

external mechanical stress. This

change

ⁱⁿ

V(0)

^{will be}

classified into two

categories.

One is the shift of the

potential minima 01

^and

02,

which will be illustrated

by

^some horizontal movement of the

potential

^curve

in

figure

^{2. The molecular}

dipole

^{will then be forced}

to follow this shift

promptly

because the shift

changes

the form of the container in which it is confined. This prompt rotation of

dipoles

will result in an instan- taneous

change

in the bulk

polarization.

At the same

time,

the removal of strain will

generally give

^{rise to some vertical}

change

^{in the}

potential

^curve,

i.e.

change

in the values of its minima. This

yields changes

in the Boltzmann factors associated with these

minima,

^and

begins

^the statistical redistribution

(i.e. relaxation)

^{to the}

equilibrium corresponding

^to

the minimum values for F ⁼ 0. This relaxation will be described

by

^a finite relaxation time i in the two-

site model.

In the dielectric case, the latter effect is

predominant

because the

potential change

^is

given by p.E

^{and is}

expressed by

^{a vertical}

change. Asi mentioned

^above,

electronic motions are not considered in the present paper. What we want to

emphasize

^{in the} present work is the

equal importance

^{of the above two effects}

on the rotational motion of the permanent

dipoles

for

piezoelectric

relaxation.

Since there is no

general

relation between the direc- tions of the

polarization

^and of the strain in the

piezoelectric

^{case, the cases}

(b), (c)

^and

(d)

^{shown in}

FIG. 2. ^- a) 1) ^{Potential curve under the constant} external force.

2) ^{Potential curve} immediately after the removal of the external force. 3) ^{Potential curve in} equilibrium in the absence of the external force. Dots represent ^the population ^{of the} dipoles. b) ^The change

in the dipole ^{moment due to} the removal of the éxternal force.

Dipole ^moments correspond ^{to the case} 1), 2) ^and 3) respectively.

Sgure 3

may take

place depending

^{on the direction}

of the shift of the

potential

^minima. The upper

figures

show the _response to the step stimulus :

F(t)

^{= 1 for}

t 0 and

F(t)

^{= 0 for t} > 0. The value _al represents the static _response, while _a2 represents ^{the instanta-}

neous

change following

the sudden removal of the external force F.

Corresponding

^{to these} types ^of response

(a) - (d),

^we get, ^as the Fourier transforms of their

negative

time derivative

(i.e.

response ^{to a} unit

FIG. 3. ^- The _upper figures ^{show the} response ^to the step stimulus : F(t) ^{= 1 for t 0 and} F(t) ^{= 0 for} t > 0. The value _al stands for the static _response, while _a2 expresses the instantaneous part. ^The sign of the strain is chosen so that _al > 0. a) a2 ⁼ 0, b) a2 > 0, c) a2 0, 1 (X2 [ al, d) a2 0, I a2 1 > oc,. The lower figures ^{show the} response to ^the sinusoidally varying stimulus. Dotted and solid curves

represent the real and imaginary part ^of x* ^{and e*} respectively.

774

pulse),

^the

frequency dependent

response

(a’) "" (d’)

to the

sinusoidally varying

^stimulus.

Now let us consider the temperature

dependence.

From viscoelastic and dielectric

investigations [14],

it is well known that the

time-temperature

^relation

in

polymers

^{is described}

by

the W.L.F.

equation

rather than

by

the Arrhenius

equation.

Since, however, the use of the W.L.F.

equation neglecting

^{the distri-}

bution of relaxation times does not

improve

^the

situation,

^{we will assume} the Arrhenius

equation

^as

the

simplest expression.

^{In this case,} the relaxation time i is written as

where the

frequency

^{factor A} may ^or may ^not

depend

on temperature

[6].

^{Some authors}

(e.g.

^Kauzmann

(1 5]) apply

^{the rate}

theory

^of

Eyring [16],

^which

yields

A r

kT/h

^{in the}

simplest

^{case, and assumes a linear}

dependence

^{on the} temperature. Hoffman and Pfeif- fer

[17]

^{assume it to be} constant, without any dis-

cussion,

^{while Lauritzen}

[18] simply

^{states that it is}

only slightly dependent

upon temperature. ^After

a rather detailed consideration of the motion of a

dipole

rotator, Bauer

[19]

obtained A oc T1/2. All that we can say ^at present is, therefore, that it is some

slowly increasing

function of temperature ^T. Then, in the case of

figure

^{3a’, the} temperature

dependence

of the real and the

imaginary

parts ^of

x*

^{and e* are}

described

by

These relations _{may be}

expressed graphically.

^In

drawing

^{such curves, however, the} temperature

depen-

dence of A does not

yield

^an

appreciable

^effects,

because the

exponential

^factor

exp(W/kT)

^dominates

the temperature

dependence overwhelmingly

^{as in the}

case of Richardson’s formula for the thermionic emission. The effect of the instantaneous _response

simply

^{adds a constant} a2 ^to the real parts

x’

^{and e’.}

3.

Application

^{to the}

piezoelectric

relaxation in

polymer

^{films. - When a} sinusoidal shear strain with

an

angular frequency

^is

applied

^to

uniaxially

^drawn

films as shown in

figure

^{4, the}

polarization

^{of the same}

frequency

^{is observed on the film surface. In- such a}

dynamic

^{case, the}

piezoelectric

stress constant is

usually

^{denoted as}

e*(o»

⁼ e’ - ie". In the case

shown in

figure

^{4, the}

polymer

^{film has the} symmetry

D oo( 00 22)

^{and the}

only non-vanishing

component of the

piezoelectric

^{tensor is}

e* - - e* -

^e

p].

Recently Hayakawa

^{et al.}

[20]

and Furukawa and Fukada

[211 ]

^{have made} measurements of the

piezo- electricity

^of

poly-y-methyl-L-glutamate (PMLG)

ⁱⁿ

detail.

Hayakawa

^{and Wada}

[4, 22]

suggest ^{that the}

FIG. 4. ^- A film sample ^and rectangular coordinates. The film is drawn along ^the z-axis and the strain is applied along the Z-axis.

E : electrode, -P : the angle ^between the z-axis and the Z-axis.

piezoelectric

temperature

dispersion

^{near room} tempe-

rature is attributed to the

quasi-crystallite

^{which has a}

relaxing piezoelectricity.

^A

uniaxially

drawn film of PMLG is considered to be an oriented

one-phase

system. Further, ^a conformational

analysis [23]

^seems

to show that U ⁼ 0 in PMLG. We will,

therefore,

try

to

apply

^our

theory

^{to PMLG.}

The

frequency dependence

of F’ and F" observed

by

many authors

[21,

23,

24]

^{seems to be}

roughly

^{of the}

Debye

type.

Strictly speaking,

^{of course,} it should be différent from the _pure

Debye

type ^{because of the} distribution of the relaxation times. The

frequency dependence

^{of’ e’} and e" observed

by Hayakawa et

^al.

[20]

is shown in

figure

^{5. The}

imaginary part e"

^{shows a}

FIG. 5. ^- Frequency dependence ^{of the} piezoelectric constants e’

and e" of PMLG at 12 °C observed by Hayakawa ^{et al.} [20].

behaviour

quite

^{similar to that of f,"., while the curve}

of e’ seems to be obtained

by shifting

that of e,’ down- wards

along

the ordinate. This

corresponds

^{to case}

(d’)

in

figure

3. As stated in section 2, ^the

positions

^{of the}

potential

^{minima can} shift in the

piezoelectric

^case

because the

dipole

^{is forced to} follow the

change

^of

the strain

instantaneously.

^{Thus the}

frequency depen-

dence of the dielectric and

piezoelectric

^{constants in}

PMLG can be well

explained qualitatively by

^{the two-}

site model.

The temperature

dependence

^of

(e’, e")

^and

(B’., a")

observed

by

^Furukawa and Fukada

[2l],

^{and the}

curves calculated

by

^{means of}

equation (3) adopting

the value W ⁼ 23

kcal/mol [24, 25]

^and

assuming

A ⁼ 8.0 x 1019 are shown

together

ⁱⁿ

figure

^6.

FIG. 6. ^- Temperature dependence of dielectric and piezoelectric

constants of PMLG. Solid curve : observed by Furukawa and Fukada [21] ^{at 10 Hz. Dotted curve :} calculated by eq. (3). ^The calculated curve of e’ in this figure ^is shifted downwards by

1.8 x 101.

Again,

^{while the} temperature

dependence

of a" and e" shows almost the same

tendency

^{in the}

experiment,

the curve of e’ seems to be obtained

by shifting

^that

of e,’ downwards

along

the ordinate. Since the instan- taneous

change

^{in the}

position

^{of the}

potential

^minima

in the

piezoelectric

^{case is}

expected

^{to take}

place

^to

the same extent at any temperature, this result is

quite

reasonable. The calculated curves are too steep

by comparison

^{with the}

experimental

^{ones. This} may be attributed to our

simplified assumption

^{of a}

single

relaxation time.

Distribution of the relaxation times can be intro-

duced,

^for

example, by

the multi-site model treatment.

The two-site model used so far can be

easily

^extended

to the multi-site model as in the case of dielectric relaxation

[7,

^17,

18].

^In the multi-site model, the

decay

^of

polarization

^{after the removal at time t = 0}

of the external stimulus is

given by

where _in is the relaxation time of the n-th mode of inter-site transitions. The

coefficient cn

^{are determined}

by

the initial condition at t ⁼ 0, ^which

depends

^on

the nature of the stimulus under consideration. Thus the time

dependence

^of

M(t)

^{in the}

piezoelectric

^case

can be different from that in the dielectric case. Note

that,

apart ^{from the constant factor and} the additive constant in the real part, ^{there can be no} such diffe-

rence in the two-site model.

Acknowledgment.

^- The authors wish to thank Dr. E.

Fukada,

^{Dr. M.} Date, Dr. T. Furukawa at the Institute of

Physical

^{and Chemical Research, and}

Dr. R.

Hayakawa

^{at the}

University

^of

Tokyo

^{for their}

valuable advice.

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