NBSIR 75-724 (R)

Piezoelectric Polymer Films for

Fuze Applications

Philip E. Bloomfield

Electronic Technology Division

Institute for Applied Technology

National Bureau of Standards

Washington, D. C. 20234

August, 1975

Final Report Covering Period

September 1 , 1973 to December 31 , 1974

Prepared for U.S. Army Frankford Arsenal

Philadelphia, Pa. 19137

NBSIR 75-724 (R)

PIEZOELECTRIC POLYMER FILMS FOR

FUZE APPLICATIONS

Philip E. Bloomfield

Electronic Technology Division

Institute for Applied Technology

National Bureau of Standards

Washington, D. C. 20234

August, 1975

Final Report Covering Period

September 1, 1973 to December 31, 1974

This document has been prepared for the use of the U.S. Army

Frankford Arsenal. Responsibility for its further use rests with that agency. The measurement results reported are derived from a statistically small number of specimen devices,

and the conclusions that may be drawn are therefore tentative.

Prepared for U.S. Army Frankford Arsenal

Philadeiphia, Pa. 19137

/ V \

\.^/ •"•tAU 0«

U.S. DEPARTMENT OF COMMERCE, Rogers C.B. Morton, Secretary

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Acting Director
'^S'i -'CU 'JtiVf t'-l i

^'is fv-i.r'^v^;>^-"' JrtS'.'n^.i,

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ABSTRACT

Described is the development of polymer piezoelectric devices with sufficient

electrical output in response to impact to demonstrate the feasibility of

using polymer elements in place of ceramic elements in ordnance fuze systems.

To satisfy this requirement, piezoelectric polymer elements have been con- structed capable oi" providing at least 0.3 mJ into a 4-Q load, in response to

a force of 89 kN (10 tonsf). Total activities of up to 360 pC/N have been

achieved from elements built up from 30 sheets of 25-i;m poly(vinyl idene

fluoride). Test methods developed to characterize both single-sheet and

total -el ement activities are described, as are poling procedures and details

of element construction. Mathematical analyses of polymer piezoelectric

output into resistive-capacitive loads are presented for static and dynamic

cases. Experimental results and theoretical predictions are shovjn to be in

agreement to within the 10% estimate of measurement accuracy.

ii

SYMBOLS USED IN THIS REPORT

A, area

C, polymer capacitance [first defined in appendix A, (10)]

d, piezoelectric strain tensor [with elements d ] nm d33» longitudinal piezoelectric strain constant (an element of d)

D, electric displacement vector [with components D^]

E, electric field vector [with components E^]

E, energy delivered to L

E^i energy parameter [first defined in appendix B, (39)]

^, function relating E and Y [first defined in appendix B, (40), for

triangular force pulse]

F, force vector applied normal to polymer piezoid surface

Fq, applied force amplitude

g, piezoelectric stress tensor [with elements g ]

g33, longitudinal piezoelectric stress constant (an element of g)

g, function relating E and Y [first defined in appendix B, (59), for

haversine force Dulse]

i , current

ig, current generated by P [first defined in appendix A, (13)]

t, polymer thickness

L, electrical load; the subscript "L" denotes that the quantity so modified refers to the electric load; examples: Ci_, capacitance

of L; Rl, resistance of L; Ql, charge on C|_; ii_, current through

L; Vl, voltage across L.

P, poled active portion of polymer; the subscriot "P" denotes that the

quantity so modified refers to the oolymer P portion as distinguished

from the S portion; examples: Cp, capacitance of P; Rp, resi stance of

P; Qp, charge on Cn; ip, current through Rp.

i i i
ehargt

optn circuit charge [first daflntd 1n apptncHx B, (30)]

resistance

unpoled shunt portion of polymer; the subscript "S" denotes that the

quantity so modified refers to the polymer S portion as distinguished

from the P portion; examples: 0$, capacitance of S; R5, resistance

of S; Q5, charge on Cs; 15, current through R^.

time

pulse width of Vj^ [first defined in appendix B, (44)]

piezoelectric stress vector [with components Tp^]

voltage

"

open-circuit voltage [first defined in appendix B, (40)]

the product vt :

dielectric impermeability tensor [with elements 3^^,]

dielectric permittivity tensor [with elements enm]

capacitive parameter [first defined in appendix A, (23)]

phase shift in oscillatory component of Vl = VL(t), for haversine

force pulse.

the reciprocal of the circuit decay time [first defined in appendix A,

(23)]

pulse half-width of applied force

phase shift in oscillatory component of Vl = VL(t), for half-sine

force pulse.

iv
Piezoelectric Polymer

Phi 1 iD

Films for Fuze Applications

E. Bloomfield

1. INTRODUCTION

1.1 Objectives

The central objective of this project v/as to develop polymer devices v/hich

v/ould be as effective as the piezoelectric ceramic disks presently used in

fuze systems and which would provide low-cost construction, mechanical flexibility, and versatility of electrical arrangement. In order to achieve this

goal, it was necessary to increase the outputs of piezoid devices by developing more effective poling and stacking techniques. (The term "poling" refers

to the process by which as-received polymer is rendered piezoelectrically

active and is described in 2.2. A stack incorporates a sandwich of two or more individual sheets into a single structure, with leads, shields, and the

like. The active polymer part of the stack is referred to as a piezoid.)

The final goal was the production and delivery of polymer stacks having an electrical output comparable to that of the ceramic discs. This would enable

the sponsor to set an achievable standard for piezoid devices to be employed

in fuzing systems.

Objectives related to the development and delivery of polymer devices included

(1) the development of methods to increase piezoelectric activity, (2) the

development of methods to characterize piezoelectric activity under a number

of loading conditions, (3) the application of such methods to piezoids

developed in the work, and (4) the derivation and formulation of expressions

relating piezoid output to applied force.

1.2 Accomplishments

This project was characterized by (1) a growth of understanding of the nature

of piezoelectricity in polymers and of the practical difficulties in manipulating fairly large areas of polymer sheet, together with (2) a similar growth

of understanding of the differences and similarities between the electrical

properties of piezoelectric polymers and piezoelectric ceramics. This improvement in understanding was shared by the NBS staff and the project monitors of

Frankford Arsenal through frequency formal and informal contacts.

The initial work statement of July 9, 1973, called for NBS to "fabricate

samples and carry out joint experimental studies with the Pitman-Dunn

Laboratories, Frankford Arsenal on piezo-polymer films." Shortly after

experimental work was begun, it was found that the intended active areas of

30 cm^ could not be effectively electroded, poled, and stacked by the techniques previously developed and used on smaller areas. With continued

improvement in poling procedures, NBS was able to produce sheets with

increasingly higher activity. Thus the available energy output per stack

was continuously raised over the course of the project and the minimum

desirable device figure of merit was raised accordingly by the sponsor.

By September, 1974, the characteristics of piezo-polymer stacks delivered
2-

to the sponsor by NBS represented greater activity from a smaller piezoid

volume than had originally been considered attainable. Accordingly, the new values were taken by the sponsor as defining goals for any future polymeric

devices for the intended test purposes: piezoid stacks with an active area

of 30 cm^ and a maximum thickness of 0.7 mm having a static piezoelectric

activity of 280 pC/N and capable of delivering 2000 ergs (2.0 x lO"'* J) into

a 4-n load in response to a force of 89 kN (10 tonsf) which has a rise time

of 10 ys.

With respect to cost goals, at the time of project termination the estimated

direct labor cost of producing and testing a sixteen-sheet stack had been

brought below $500. This figure applies to laboratory conditions; presumably,

the cost would be considerably reduced in mass production.

The development of polymer devices for ordnance fuze applications has resulted

in the following specific accomplishments:

1. Development and improvement of poling techniques to achieve piezoelectric

activities of 12 picocolombs per newton (pC/N) for a single sheet of poly-

(vinylidene fluoride) [PVF2] 25-micrometers (25-ym) thick, with 14 pC/N

realisable on the basis of a 50% success ratio,

2. Development of stacking methodology to achieve a piezoelectric activity

of 360 pC/N from a piezoid of 30 sheets of 25-ym PVF2,

3. Delivery of stacks representative of the current level of activity, (see

table 5)

,

4. Development of a quasi-static, or d-c, method of test to characterize

piezoelectric activity of both single sheets and stacks, using a press,

5. Development of a dynamic method of test to characterize piezoelectric

activity of both single sheets and stacks, using a drop-test machine,

6. Derivation of an expression relating piezoelectric and pyroelectric

activity, so that measurements of either type of activity may be used to

characterize the other [1]*,

7. Development of the theoretical basis for a method of test to measure

the compliance of bonding layers in a stack [1], and

8. Development of theoretical analyses describing the quantitative behavior

of piezoelectric polymer in response to applied force, including derivation

of the relationship between electrical and mechanical parameters, and of

expressions for optimizing piezoid output into a given electrical load.

^Figures in brackets indicate the literature references in section 6.
-3-

In developing piezoids with higher levels of energy output, it was necessary

to electrode, pole, and test large numbers of single sheets and to assemble

and test large numbers of stacks. In the course of this work test methods

were developed, discarded, or modified as needed; test equipment was designed,

tried, modified, and improved, and many sheets and some stacks were tested to

destruction. (As one example, a stack was tested in an air gun at Harry

Diamond Laboratories to investigate the ability of a new type of electrical

connection to survive violent impact; as another example, several small stacks were destroyed while various bonding agents and techniques were tried in the

development of a method to attach leads and glue the poled sheets together.)

The understanding and improved techniques derived from this testing program

made it possible to achieve the more important of the specific accomplishments

listed above.

2. EXPERIMENTAL DEVELOPMENT

2.1 Background

Piezoelectricity in stretched and polarized sheets of PVF2 was first reported and

studied by workers at several Japanese research institutes [2, 3, 4]. Recently,

very thin, uniform films of this polymer have become available. Among the

advantages of this material are that it has low density, that it is available

in sheets (hence devices may be fabricated with surface areas larae compared

to those of ceramic piezoelectrics) , and that it is sufficiently flexible to

permit fabrication into a range of shapes. Through the cnoice of sheet thickness, number of sheets, and electrical connection, a stack may be designed to

have specified capacitance and to deliver a maximized electrical output into

a given load, in response to an applied force.

At the inception of this project, limited experience was available with active

areas as large as 30 cm^; further, single-sheet piezoelectric elements had been

made on what was for the most part a one-at-a-time basis, and there was no developed stacking technology. Thus, new methods and apparatus were required

to: (1) extend the vacuum deposition technique to routinely apply metal-film

electrodes of 30-cm2 area, to apply electrodes to one side of a series of poly- mer elements in quick succession (i.e., without requiring separate pump-down

and cycling of the vacuum chamber for each element), and to support the polymer

sheet while the electrodes were being applied in such a manner as to minimize

the effects from heating (especially wrinkling of the polymer sheet), (2) improve

and extend poling procedures to enable the poling of several large samples at

once, to utilize higher electric fields and temperatures without introducing

wrinkles in the polymer, and to minimize the number of short circuits occurring

through the sample during poling, (3) produce a bonded stack from a number of

single-sheet elements, and (4) attach leads for parallel electrical connection of

the single-sheet elements in a stack.

In addition, methods were required that were appropriate for the tasks of

characterizing the piezoelectric activity of single-sheet elements, of sandwiches of elements being built into a stack, and of completed stacks.
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2.2 Sampli Preparation

2,B,1 DeposiHon of EUatrodea TM Aluminum electrodes approximately O.lO-ym

thick were deposited on each side of the 25-ym polymer sheets using vacuum evaporation apparatus. Short lengths of the pure aluminum wire used were

coiled around a resistively heated tungsten helix; the heating current was

not applied until the chamber reached a pressure Of 2.7 x 10-^ pascal (Pa)

during pump down.

Polymer surfaces to receive electrodes were rinsed in ethyl or i sol propyl

alcohol before being placed in the vacuum chamber. A fixture was designed

with two rollers. The polymer sheet to receive electrodes was in the form

of a long strip and was initially wound on the feed roller oriented so that

when unwound the intended side would face the evaporation apparatus. The

take-up roller was driven by means of a mechanical link through a vacuum

seal to a crank outside the vacuum chamber. The number of elements that

could be produced in one run was limited in principle by the diameter of

the rollers and associated clearances; however, in practice the limit was

the number of evaporations that would be carried out before more aluminum

wire had to be affixed to the tungsten helix.

The electrodes were deposited through a mask which consisted of four circular

holes 30 cm^ in area (diameter = 6.2 cm) in a 24-cm-square sheet of stainless

steel. The holes were symmetrically oriented about the center of the mask

with a center- to-center spacing of 7.5 cm in one direction (across the roll

width) and 8.7 cm in the other (along roll length). Each of these holes

had an associated slot, 3 by 12 mm, cut in the mask from the edge of

the hole outward along an extension of the hole center line parallel to the

intended long axis of the polymer film, with all slots on the same side of

the holes. See figure 1. After one side of the polymer sheet strip had

electrodes applied, the strip was rewound on the feed roller so as to have

been effectively rotated through an angle of 180° (the original "beginning"

end wound on the feed roller first) and turned over with former bottom side

now top side. The result of the orientations so achieved was successively

to evaporate electrode "tails" on opposite sides of the strip pointing in

opposite directions. During poling and in subsequent use, electrical

connections were made to these projections.

When the electrode thickness is less than 0.07 ym, the resistance across the

30-cm2 circular face of the electrode, measured from near edge to near edge

along a diameter, exceeds 2 J2. This measurement was made in air using probe

tips with a radius of curvature of 1.25 mm. The resulting activity of the

poled films is several orders of magnitude less for electrode thicknesses

corresponding to resistances greater than 6 f2 than for those corresponding

to 2 fi; the activity was found to be independent of electrode thickness for

thicknesses greater than 0.10 ym. (At this thickness, the resistance was

near 1 fi.) If the electrode thickness were permitted to exceed 0.15 ym, the

aluminum tended to flake off. An aluminum electrode thickness of 0.1 ym

(1000 A) has proven satisfactory and is recommended. Aluminum was chosen for

the electrode material because it is satisfactorily conductive at reasonable

thickness, because at such thicknesses it adheres well to alcohol-rinsed PVF2,

and because it is an easy material to evaporate, having a sufficiently low

melting temperature not to require the use of a crucible.
-5-

2.2.2 PoHng — In preparation for poling, a sandwich v;as made by placing a 125-ym sheet of PVF2 on a bottom 0.32-cm sheet of polytetraf 1 uoroethyl ene

(PTFE), followed by the electroded polymer element to be poled, another

125-ym sheet of PVF2, and a top 0.32-cm sheet of PTFE. Electrical connection to the top and bottom electrodes was made by using two thin strips of

12-ym-thick tin foil. This assembly was then placed between the platens

of an hydraulic press. A load of approximately 30 kN was aoplied to press

the sample and to prevent it from wrinkling during poling. The desired

electrical field was applied across the sample by connecting the tin-foil

leads to a high-voltage d-c power supply. The sample was heated to the

desired temperature by means of heaters embedded in the platens of the press,

and then cooled to room temperature by running water through cooling passages

also in the platens. The field was applied to the polymer during the entire

heating and cooling cycle. The PTFE served to provide electrical isolation

of the sample from the hydraulic press structure. The 125-ym sheets of PVF2 were used as liners because the surface of the insulating sheets was not

smooth. The temperature was monitored by means of thermocouples inserted

into the sandwich.

During the assembly of the sandwich, care was taken that the polymer element

and the inner sides of the 125-ym PVF2 were clean and free of dust in order

to prevent dirt particles from being pressed into the polymer surface during

poling. The thickness of the polymer would be reduced at the embedded particle,

and therefore the effective electrical field would be locally high. Embedded

particles are to be avoided as potential failure sites.

A regulated 10 kV d-c power supply was used to provide the poling field. The

poling voltage typically ranged between 2000 and 2400 V across the 25-ym sheets,

with some of the early sheets poled at 1200 V. The 2400 V potential produced a higher activity than did 2000 V, but also resulted in a higher percentage of

the sheets failing. For example, the percent success-failure ratio experienced

in the laboratory for poling at 2400 V and 120°C was 50-50; for sheets poled at

2000 V and 110°C the percentage of failures dropped to 10. The sheets were

heated to approximately 120°C and then cooled. The temperature cycle was approximately 15 min for heating and 10 min for cooling.

2.2.3 Stack Construation — Each poled sheet had a "high side," this being the

side with electrodes connected to the positive, or high, side of the d-c poling

supply. As the first step in making a multi-sheet stack, two sheets at a time were bonded together, high side to high side, with a tin-foil lead providing

electrical contact to the internal high electrodes. Tin-foil leads were attached

to the electrode tails with silver-filled rubber paint. Bi laminate pairs were

then bonded together with tin-foil leads providing contact to the internal low

electrodes and extending from the opposite edge (i.e., 180° away from the edge

with high-side leads). Successive high-side leads were aligned one over the

other, as were low-side leads. The bonding agent was a solution of one part

contact cement to five parts toluene*. Experience with stack construction

showed that it was desirable during bonding to prevent entrapment of air bubbles

and avoid wrinkling sheets, as both conditions degraded performance.

*Considerabl e experimentation was conducted with various bonding agents.

Contact cement meeting Federal Specification MMii-A-130a has been

satisfactory.
In the manner described the stack was built up of 2n iheitl. The partially

completed stack had n superposed high-side leads extending from one edge

and n - 1 superposed low-side leads extending from the other edge. Two more tin-foil leads were then attached to the outside low-side electrodes.

The bundle of high-side leads was bonded to form a single lead with silverfilled rubber paint and reinforced with either copper or aluminum tape. An

exterior coat of silver-filled rubber was applied to the reinforced lead.

Similar treatment was given to the low-side leads. Figure 2 shows a laminated stack at the bottom (A), before the outer coatings of heavy aluminum

tape (providing electrical shielding and mechanical protection) were

applied. B shows a single sheet with electrodes deposited on both sides;

C shows a similar sheet with one foil electrode attached (which was destined

to be an outer low-side lead for the stack); and D shows two sheets bonded

together with the high-side lead at the left. E shows another sandwich of

two bonded sheets with high-side lead. The laminated stack was made up by

alternating D-and-E sandwiches.

2.3 Static Testing Procedure

Each poled sheet separately, and the various small stacks which were an intermediate stage in the fabrication of completed stacks, were tested quasi

statically on an hydraulic press.

When the piezoid sheet or stack is compressed, negative charge collects at

the high-side electrode; after discharge, and upon release of the mechanical

force, negative charge collects at the low-side electrode. The test procedure

was as follows: With an open-circuit configuration, (see figure 3), either

the piezoid by itself (with capacitance C) or in parallel electrical connection with a standard capacitor (capacitance C|_) was charged by means of com- pression or release of pressure (static force loading Fq) by means of the

single-pole switch show in the figure. Discharge of the charged piezoid into

the preamplifier of a storage oscilloscope permitted measurement on the stored

trace of the voltage developed across the capacitor(s)

.

The magnitude of the charge developed by the polymer is Qq. The voltage Vq

measured when the piezoid is not connected to C|_ is Qq/C.

With C|_ in the piezoid circuit, the charge developed Qq is shared between Ci and C. Under the open-circuit condition there is a common voltage across trie

two capacitors. In order to obtain a measure of charge output and the quasi

static capacitance of a given piezoid, two voltages, Vq and Vl were measured:

\ - Qq/C and

' Vl = \l\ = (Qq - Ql)/C, (1)

where Ql and Qq - Ql are the equilibrium values of the charge on capacitors C, and C , respectively. From (1),

C = (Vq/Vl - l)-i C^, and = CV^, (2)
-7-

From these two formulas and from measurements of Vq and Vi_, the static capacitance Cthe charge released Qq, and the activity Qq/Fq may be calculated.

2.3.1 S-ummary of Single-Sheet and Small-Staak Activities — Presented are data taken from August 28, 1974, through September 14, 1974, while stack

No. 7 (30 sheets; see table 5) was being constructed. Four samples in each

15 X 18-cm sheet (as obtained from the vacuum aluminizing apparatus) were

poled together in parallel; then each sample in the batch of four was assigned

a common batch number and a sample code letter a_ through d_. The active areas were cut apart from each other into four separate single sheets.

Poling voltage (V) and maximum temperature reached during the poling cycle (°C) are given in table 1, along with capacitance C (nF) and piezoelectric activity

Qq/Fq (pC/N) for eleven batches. C and Qq/Fq are the averages of the measured

values for the four samples in each numbered batch. Both quantities were

obtained by measurement and calculation using (2) for applied forces up to

45 kN, The variance in the capacitance of a single sample depends on the

reproducibility of the force imposed under the measurements implied by (1).

Analysis of repeated measurements resulted in an error estimation 5Q/Q ~ 6C/C =

0. 092 for single sheets. The data for the individual sheets are contained in

table 2 (parts A and B) in the rows labeled n = 1. The variation among samples

poled under identical conditions is exemplified by results from measurements on samples 34a-d, 36a-d, and 38a-d. The average values and the variation as determined by the standard deviations for these batches are:

Batch 34 C = 14.1 ±1.1 nF, Q/F = 9.7 ±0.9 pC/N;

Batch 36 C = 13.6 ±0.1 nF, Q/F = 9.5 ±0.7 pC/N;

Batch 38 C = 13.7 ±0.0 nF, Q/F = 9.8 ±0.5 pC/N.

In the two parts of table 2 are shown measured values for the capacitance and

piezoelectric activity for stacks having n = 1, 2, 4, 6, 8, 9, 10, 14, and 15

sheets. The entries are arranged in chart form to show from which particular

sheets each stack or intermediate stage was built. For n > 2 there are three

sets of entries for each stack or stage: the first set gives the measured

values [capacitance (nF) on top, piezoelectric activity (pC/N) on bottom] for

the particular stack of n sheets; the second set gives average sheet values,

1. e., the first set values divided by n\ and the third set gives weighted

average values based on the expected contribution of the constituents. That

is, the values at the previous stage of each of the constituents were averaged

with weighting factor equal to the number of sheets in the constituent stack.

The two intermediate stacks of 15 sheets were assembled into stack 7. Onethirtieth of the measured value of 360 pC/N yields C = 12,0 nF, and Q/F = 12.0 pC/N. The average values for all 30 of the single sheets are C = 13.6

±0.5 nF and Q/F = 9.8 ±0.7 pC/N. Also, the average of the measured values

per layer for the two stacks of 15 sheets is C = 12.9 nF and Q/F = 11.5 pC/N.

In this example, the measured value for the piezoelectric activity per layer

increases as the stack thickness (proportional to n) increases. An explanation may be that sheets located near the center of the stack undergo greater

distortion than sheets near the surface when a given force is applied to the

stack.
-8-

2.4 Analyses of P1ezo-Polymer Electrical Outputs

B.4,1 Surmary of Mathmatioal D§v§lopmmt = Appendices A through E are mathematical analyses of the output from a polymer plezold under various conditions.

Appendix A Is a circuit analysis of an active polymer element under dynamic

loading conditions. This analysis permits calculation of energy and voltage

outputs as seen by an R-C load as the plezold experiences Impact. Appendix B contains an evaluation of the formulae for the output due to triangular, halfsine, and haverslne force pulses (the haverslne pulse has the shape of that of

the half sine squared). Appendix C constitutes an analysis of the open-circuit,

static-loading voltage and charge levels developed across a piezo-polymer and

the subsequent electrical output upon application of an electrical load.

Appendix D is a table based on the results developed in appendices B and C and

constitutes a summary of polymer piezoid outputs (voltage, energy, charge)

under given conditions.

Appendix E is a discussion of the output from a piezoid sample which has non- uniform conductivity and dielectric constant. It was assumed in the analysis

given in appendix A that conductivity and dielectric constant are uniform

over the sample thickness. It is shown that under practical conditions no modification of the formulae developed in appendices A through C is necessary.

The theoretical conclusions show that to obtain the largest possible transfer

of energy stored in the stack, either a series- or parallel -connected stack

of n sheets may be constructed, and a switch arrangement should be used with

a value of RlC much smaller than the transfer time (the series stack value

of C decreases as n"^). For the dynamic loading arrangement, RiC should be

chosen to satisfy the relation R|^C z j then in time t an energy transfer

of approximately 40% of the energy stored in the stack, 40% of Q2/2C, will

take place. [As a numerical example, consider the values t ^ 10 ys, n = 20,

C = 20 X 12 nF, and Rl 20 fi. For Q/Fq 300 pC/N (2.7 yC/tonf), the maxi- mum output from Fq = 89 kN (10 tonsf) would be approximately 0.73 x lO"^ J

(7,300 ergs). For the same conditions, but using Rl = 100 M«, the energy

transferred would be reduced by a factor of 0.67 x 10-^; while if Rl = 4fi, the

energy transferred would be reduced to 0.5 of the maximum value.]

The theory also shows that to obtain the largest output voltage from a stack,

the stack should be series connected and the quantity vt should be << 1 (vt

is defined as the product of the reciprocal of the circuit decay time and the

pulse half-width of applied force).

2,4.2 Energy-Release Eocperimenta on Piezo-Polymer Stacks Conneoted Aoroes an

EleotHaat Load - Described are dynamic and static energy-release experiments

on three different multi layered stacks, each with individual component sheets

connected in parallel

2.4.2.1 The Voltage output of stack No. 7 was measured across various electrical loads under impact conditions. The stack was mounted on the impact table

(a steel cylinder) of a drop-test machine, and the electrical output into a given load was measured and recorded during the impact sequence. A commercially
-9-

available accelerometer was attached to the upper face of the dropping carriage

and had a specified sensitivity of 7.9 mV/g'n*over the frequency range 5 Hz to

15 kHz. The outputs from stack and accelerometer were stored in a two-channel

transient recorder and could be plotted as a function of time on an X-Y1-Y2

plotter and displayed on the screen of a monitor storage oscilloscope. A mass

of 2.59 kg (corresponding to a weight of 5.69 lb), consisting of an aluminum

impactor mounted on the carriage, was released from a height of 5 cm (2 in) to

fall freely and impact the polymer stack. The impactor surface was a portion

of a sphere with radius 23 cm (9 in). The voltage output from the calibrated

accelerometer showed that the maximum force (Fq) developed during impact was 23.8 kN (2.67 tonsf). The resistive load (Rl) was varied between 4 and

40 kf2. The voltage was measured both with no capacitive load and with a capacitive load (C|_) of 0.5 yF. Actually the transient recorder has an imped- ance (R^p = 1 Mfi, C^p = 15 pF) which is in parallel with the load; thus the

net resistive load was equal to the product of Rl and R^p divided by their sum,

and the net capacitive load was equal to the sum of Cl and Ctr- The measured

.data are given in table 3. The energy delivered to the load during the force

rise time t was calculated from the data and is also given in table 3.

Table 3 shows that the voltage maxima and minima for different resistive and

capacitive loads behave as predicted by the analysis of appendix B. The output

pulse width to also agrees with the calculated value. However, the output

voltage waveform for large vt (small Rl) was qualitatively different from that

predicted for triangular and half-sine force pulses. For large Rl a square wave would be the likely form expected, with the signal rising in a time much

MIN

less than t and then falling to V at t, with a fast decay time comparable

MAX

to the rise time. In fact a gradual rise to V^^ at the time entered in

table 3 was observed. This apparent discrepancy from predicted behavior is

the result of the fact that both the triangular and half-sine waveform force

pulses have a discontinuity in the time derivative of the force, dF(t)/dt,

at t = 0; thus the current iG(t) would be abruptly turned on at t = 0. Because

of the nature of the elastic properties of the polymer, the impacting force is

better characterized by a waveform which has both zero derivative and instantaneous value of impacting force at t = 0. After t = 0 both F(t) and iG(t) should

gradually rise from zero value. The voltage and energy integrals for the haversine waveform force pulse (which has these desired properties) have been evaluated in appendix B. There is agreement between the voltage pulse shapes predicted in appendix B, section 3, and those that have been observed during the

drop-test evaluations of stack output.

A noticeable difference between the static and dynamic capacity of the test

polymer stack was detected in the course of measurements. The capacitance of

stack No. 7 was measured statically on the hydraulic press as described in 2.3

(and as also follows from the open-circuit analysis of appendix C) and found

to be 0.36 yF. The stack capacitance was then measured at 1 kHz on an impedance bridge with the result of 0.25 yF. According to (31) or (51) of appendix

B, the capacitance may also be calculated from the peak voltage data at large

Rl given in table 3:

= 1 + C, /C = 22.5/7.5 = 3. (3)

* 2 9^ = 9.81 m/s
-10-

Since Cl = 0.5 yF, C = (3 - 1)-^ x 0.5 yF = 0.25 yF. This result agrees with

that of the bridge measurement.

The piezoid may be characterized by its frequency-independent static voltage

response to unit force Vq/Fq. For stack No. 7,

" (355 pC/N)/(0.36 yF) = 1 mV/N. (4)

O 0

A comparison value for a typical piezoelectric ceramic (lead zirconium-lead

titanate-4) is 0.2 mV/N.

The individual polymer sheets (average capacitance, 13.6 nF; average piezoelectric activity, 9.8 pC/N) also yield this characteristic voltage. Since

Vq/Fq " das/Cp, and Cp is proportional to A (30 cm^ for the piezoid sheets),

another quantity that may be used to characterize sheet performance is voltage

output in a unit pressure field:

VqA/Fq = 3 yV/Pa. (5)

The difference between the effective charge output observed at low and high

frequencies (between 0.35 yF x Vq and 0.25 yF x Vq) may be explained by assuming that C is frequency dependent while Cp is not (see appendix B). The dipole

orientational contribution to the dielectric constant is frequency dependent

according to Debye's theory of polar molecules [4]. The polarizability and

relaxation frequency should be much lower for the polarized portion of the

piezoid than for the unpoled portion according to the following explanation.

The unpoled portion can respond in synchronism to a driving field, while the

poled portion of the polymer is saturated. This situation causes the dielectric

constant and the resistivity of the poled portion to be less than that of the

unpoled portion [5]. The analyses in appendices A through C were carried out

under the assumption that the dielectric constant and resistivity were homogeneous throughout the piezoid. In appendix E it is shown that under practical

conditions even when the dielectric constant and resistivity vary through the

material, the formulas are still valid.

The data in tablie 3 may be interpreted in terms of the havers ine force pulse

analysis presented in appendix B, and comparisons made between measured and

theoretically predicted values. Consider as examples; { 1 } the time tM at

MAX which = .V|^ and the duration t^ of regarded as a pulse, { 2 } the

behavior of the maximum and minimum (positive and negative) voltage peaks of

Vl, and { 3 } the relationship between the energy E delivered to the load and

the load resistance Rj^.

In example { 1 }, for large values of Rl (corresponding to small values of

the product vr), the theory predicts that the time at which Vi reaches maximum value is equal to the rise time t of the force pulse and that to equals

2t. The measured values for Ri = 40 kn confirm this prediction; that is,

twice the time (measured from 0) at which Vl is maximum equals Iq (2 x 225 = 450). For small values of Ri (corresponding to large values of vt), the

theory predicts that to should be approximately equal to t with the Vl
-nmaximum occurring at t/2. Again, the measured values for = 4 confirm

this prediction, with to = x = 225, although the agreement is not as good

for the time at which Vj^^^'^ occurs (2 x 100 is only approximately equal to

225).

In example { 2 }, the relative magnitudes of the positive and negative

MAX MIN

voltage peaks Vj^ and Vj^ agree with theoretical prediction: for large

values of R. , the measured values of V.'*^^^ tend toward zero [see (62)], and

MAX

depends on the value of as given in (63) [see also (23)]. For small

values of R|_, the theory (60) predicts that V^^'^^^ = -V^^^^^, and the measured

values agree for Ri as large as 40 f2. Further, theory predicts that the peak

values exhibit no dependence on Cl for small values of Ri ; comparison of the

MAX measured values of given in table 3 for two values of confirms this

independence (0.14 z 0.15; 0.40 = 0.40).

In example { 3 }, the theory predicts (65) that the value of vt for maximum

energy transfer into the load should be approximately equal to t/R[_C + 2.1.

225

R^ therefore would be predicted to be

(q 25) (2 1

)

^' ^'^''^'^ ''^ ''"'^ reasonable agreement with the measured value of 400 fi. The normalized value,

for maximum energy transferred into the load, E(t)/Fo^, was computed from

measurements to be 61.6 femtojoules per newton squared (fJ/N^), which agrees

well with the theoretically predicted value of 59 fJ/N^. This value and other

calculated values given in table 4 were calculated from equations derived in

the haversine force pulse analysis (and also summarized in appendix D, condition 3)

:

(V^/F^)TM = / = ••724 Rj_ (yV/N)

0 I VT >> 1 R, ttCV

0

(V^/Fq)^^ = C Vq/F^ = 1000 K (yV/N)

(6)

VT << 1

.

Comparison of the measured and calculated values of the various parameters

given in table 4 shows good agreement between the experimental results and

theoretical prediction.*

*From the general expression for V, , (56), v, was calculated to occur at

MAX

t = 0.78t = 175 ys, and /F^ was calculated to have the value (V /2F ) x

L 0 0 0

(0.831) X (1.156) = 480 yV/N.
2.4.2.2 By means of mechanically charging an initially open-circuited piezoid

which subsequently had its energy released when a preset stack voltage was reached, a pyrotechnic actuator (the electrical load) supplied by the sponsor

was fired.

In order to test the predictions of appendix C, the circuit shown in figure 4 was designed to establish an open-circuit condition across the polymer stack.

When the voltage resulting from a force applied to the piezoid rises to a predetermined level (the value used was 20 V), the Zener diode triggers the gate

of the SCR, which in turn permits the energy stored in the polymer device to

flow into the load. The load selected was a pyrotechnic actuator which on receiving 800 ergs (8 x 10"^ J) changed its dc resistance from 3 to 2 kn

and its capacitance from 1 yF to 1 nF. The values given are approximate.

Stack No. 5 was used for this test with successful results. It was found that

a force of 73 kN (8.2 tonsf) triggered the circuit and produced a signal of

23 V with the load being the input circuit of an oscilloscope (input impedance

1 Mn, shunted by 18 pF. Stack No. 5 (88 sheets; see table 5) had a measured

static output of 290 pC/N with a static capacitance of 1 yF. The static voltage level was calculated according to (4) to be Vq/Fq = 0.29 mV/N. The expected

voltage level for the force (applied with the hydraulic press) was Vq = 21 V.

For these parameters (c"^ = 2 and z i;-^RiC 2 6 x 10"^), the energy output

to the actuator was calculated using (56) and (58) to be 1100 ergs delivered at

approximately 10 V. .

.

Note that if the actuator had been cotinected directly across the piezoid, a force of 73 kN with a rise time of t = 10 ys* would have yielded Vl(t) z 6 V

and E(t) = 418 ergs. At 6 V, the threshold energy for the actuator is 1000

ergs; direct coupling therefore would not provide sufficient energy to the

load.

2.4.2.3 An impact test was carried out at the air-gun installation in the

Harry Diamond Laboratory in which a stack was directly impacted by a highvelocity projectile. A flat-nosed projectile with a mass of 1.38 kg was fired from a gun with a bore of 10 cm (4 in) at a polymer piezoid made from

20 sheets. The velocity at impact was determined to be 154 m/s (504 ft/s).

The force sustained by the polymer stack was estimated by the air-gun test

personnel on the basis of experience to be 450 kN (50 tonsf). A highresistance probe (Rl = 100 Mn, C|_ = 3 pF) was placed across the sample, and

voltage output was recorded photographically from calibrated oscilloscopes

which were triggered by means of a trip wire 40 ys before impact. Following

impact, the recorded voltage rose linearly to a peak of 200 V at 0.48 ys after

impact. According to calculation on the basis of the formulas given in

appendix B, the voltage produced should be 400 V."^ During the initial impact

*VT = 1.67; j^(1.67) = 0.379 = 0.381 , its maximum value; see appendix B. ^t/R|_C - 0.48 ys/(100 m) (0.26 yF) = 1.8 x 10"^, which is much less than one;

Vl(t) = Qq/C = (233 pC/N) (4.5 x 10^ N)/0.26 yF = 400 V.
-13-

duration of 0.48 ys, the energy delivered to the load was only 64 pJ because

of the extreme mismatch of the 100 Mfi load. Note that if Rl had been chosen

as 4 fi, then instead of 200 V, a voltage peak of 160 V and an energy output

of 1.15 mJ would have been expected; while if Rl had been chosen to optimize

the energy output (Rl = 1 n), a voltage peak of 90 V and E^^^ (x) = 1.98 mJ

would have resulted.

3. DELIVERIES

A tabulation of the PVF2 piezopolymer stacks which were delivered to Frankford

Arsenal is given in table 5. The entries in the column labelled "Total Activity"

were measured with an estimated sample single standard deviation of ±10% by the

static open-circuit test method described in 2.3. The final column gives the

average activity per sheet, the quotient of the values in the "Total Activity"

column and the number of sheets in the given stack.

In all cases, the individually poled PVF2 sheets had congruent SO-cm^ circular

aluminum electrodes vacuum deposited on both sides of the sheet. The thickness

of each stack is made up of contributions from n polymer sheets, 2n electrodes,

and n - 1 bonds. The nominal poling conditions for stack Nos. 1-6 were 95° C for

the maximum temperature and 1200 V for the poling potential; to achieve more effective poling, the nominal poling conditions for stack Nos. 7-10 were 120°C

for the maximum temperature and 2000 V for the poling potential.

4. SUMMARY AND RECOMMENDATIONS

4.1 Summary

A piezoelectric polymer transducer, with an active element of area 30 cm^ and

thickness less than 0.9 mm, has been developed that is capable of delivering

into a several -ohm load an energy of 0.32 mJ in response to a 89-kN impact

having a rise time of 10 ys.* In support of this development, specific poling methods and methods of stack construction were evolved.

Static and dynamic test methods have been developed to characterize the electrical output and piezoelectric activity of polymer elements, or piezoids.

The static method makes use of an hydraulic press to apply force to the piezoid;

in the dynamic method, the piezoid is impacted with a known force in a droptest machine.

Theoretical analyses, in particular applying to both test-method situations,

have been formulated to predict the electrical output of a stack under various

conditions. The theoretical predictions are in good agreement with the experimental measurements that have been made, especially for the haversine-force

pulse model of the drop-test situation.

*The values cited refer to Stack No. 7, which was slightly thicker than the

0.7-mm value specified. Stack No. 8 met all the requirements given in 1.2,

with a thickness of 0.68 mm and an output under the specified conditions

and into the specified load of 0 24 mJ.
-14-

4.2 Recommendations

Theoretical [6] and experimental studies [7] In our laboratory and elsewhere

indicate that most of the activity of a polymer film occurs in very thin

layers near each electrode and that the bulk of the material is relatively

inert. A conclusion that requires further experimental verification is that

higher activity should be achieved with the use of thinner sheets, since less

"inert" material would be present. The activity of two 9-ym PVFg samples

was measured in our laboratory and found to be approximately twice that of

two 30-ym samples said to be poled with the same field strength according to

the supplier. If more empirical evidence becomes available supporting the use of thin polymer sheets or films to achieve high piezoelectric activities,

techniques of depositing very thin polymer films on suitable substrate material

should be considered.

Other means of improving per-sheet activity, for example, multiple poling cycles,

the use of high-frequency electrical poling pulses, and treatment such as stretching prior to electrical poling, require investigation.

The methods of test need to be further developed to improve the precision of

results. The use of a commercially available force transducer for calibration

is recommended.

Results of measurements on completed stacks show that the activity of the stack

as a whole tends to be greater than the sum of the activities of the individual

sheets. This phenomenon should be investigated, as it may hold clues to

improved stack design leading to higher per-stack activities.
-15-

APPENDIX A

Output from Piezoid Sample Under Dynamic Loading Conditions

Voltage and charge measurements have been conducted during static- and dynamicloading experiments on single polymer sheets and on stacks of active and inactive

polymer sheets electrically connected in both series and parallel.

Consideration of the results of these tests has led to the equivalent circuit

shown in figure 2. The polymer is modeled as consisting of two uniform parts:

An active part in series with an inactive region. The active part is understood

to coincide with the poled portion, is denoted by the letter "P", and has a thickness Ip. The inactive, unpoled portion may be regarded as a shunt with

respect to P, is denoted by the letter "S", and has a thickness 1$. Each part

is modeled separately as a parallel R-C circuit with a charge Q on C and a current i in R. Theoretical and experimental results for a poled sample show

that the surface regions are strongly poled, while the bulk of the interior is

less effectively poled. Figure 3 shows the polymer element with a series load

of resistance R|_ (with current i|_) and capacitance Ci_ (with charge Ql).

Consider a piezoelectric material. A pair of equivalent piezoelectric equations

relate electric field E, electric displacement D, and stress T.[8]

and

D = e"^ E + d T„, (8)

n nm m nm m' ^

'

expressed in Einstein summation convention.

In this convention, B is the dielectric impermeability tensor (elements B^^)

,

and g is the piezoelectric stress tensor (elements gnm)' which are related to

the dielectric permittivity e and the piezoelectric strain d, respectively by:

6 = and g = e'^d . (9)

If stress T is applied uniformly to the piezoid (in a direction perpendicular

to the plane of the polymer (i.e., in the negative-3 direction), a voltage V

develops across its thickness I and charge Q appears at the surface of area A.

It is assumed that in a polymer the polarization is induced predominantly

parallel or antiparallel to the poling field direction. After poling when a stress in any direction is applied to the poled material, the voltage along

directions other than the poling direction are negligible. TKis implies that

the dielectric tensor elements £3-1, ej3, Bsi, Bis are zero when i ^ 3, and

that the q^^^ and d^m tensors have non-zero elements only in the row n = the

poling direction.

Note that

Cp = epl^/lp C3 = ejA/^g, C"i = Cp"! + C^-^, and Q = CV. (10)
-16-

Integration of (7) undir the conditions D 0 Inside the plezold (the open

circuit condition), g ^ 0 only across the poled portion, and the applied

force F « AT, yields:

t

V = Edx3 = gF = ^

F and Q = ^ dssF.

' (11)

The quantity das is taken as the longitudinal piezoelectric constant of a poled polymer.

For short-circuit conditions, E is zero inside the piezoid, and it is convenient

to use equation (8). As a stress is imposed, a short-circuit current equal to

the displacement current is generated in the piezoid. This current is:

^ = * = "33 f

^ "saF. (12)

The sign of the force for compression is opposite to that for tension, and the

sign of the resulting charge is also opposite for compression to that for

tension.

During the application of a time-dependent force to the polymer, the active

portion of the piezoid (see figures 5 and 6) acts as a generator of current

ig. As a convention,

iG(t) = d33*F(t). (13)

The current, i = i(t), flowing in the external circuit is predominantly determined by the electrical load. Under dynamic loading [Cl ^ 0, Rl ^ <»], some of

the generated power is lost internally to Rp and Rs.

Application of Kirchoff's laws to the circuits in figures^ 5 and 6 yields:

1 = ig + ip + Qp = ig +

Qs

= V

+ Ql,

and (15)

Substituting in (14) and (15) for ip, i,., and i. in terms of Qp, Qg, and Q. as given in (16 yields:

Q3 + (RgCs)'^ " \ (RlC^)-! and (18)

Qp + (RpCp)-i

= \ ^^\^\)'^ \

'g- (1^)
-17-

The time derivative of (17) added to the product of (17) and the quantity

(RpCp)-^ give:

0 = + (RpCp)-^QL] + ^p''[Qp + (RpCp)-iQ3]. (20)

In solving (20), (18), and (19) it is convenient to assume that

RpCp = R3C3.* ^ (21)

This relation holds if the active and inactive portions have the same conductivity and dielectric constant, since C and R are inversely and directly proportional to thickness, respectively. Substituting first (19) into (20) and then

(18) with (R5C2)-^ replaced by (RpCp)-i into (20) gives the result:

"^L ^ = ^ C^^'g, (22)

where

1 1 C|

V = dpjx R^)' C = (1 + -^)-^ and C-i = Cp-i + C3-1 . (23)

The solution of (22) is given by

QL(t) = V dt'exp(vf - vt)ig(t'). (24)

From (16) and (13),

iL(t) = (RLCL)"^QL(t) = (RLCp)-icd33/^^dt'exp(vt' - vt)F(t'). (25)

The voltage across the load is

\l^{t) = R|_ijt), (26)

while the energy delivered thereto is given by

E(t) = /^^dt'RLii_'(t' ). (27)

For stacks consisting of n piezoid sheets electrically connected in series

or parallel {k indicates kth sheet]:

0 = i R -

Q, C -1 = i R3 - Q r -1 =1 R - Q Cp -\ (28)

For the parallel arrangement,

i, R, + in Rp + i<: Rq = 0, and (29)

^k^k ^k\

*See appendix E.
-18-

i = + Ql

= E 1^. (30)

where

For the series arrangement,

i,R, + S (ip Rp + ic Rc ) = 0, and (31)

^ k^] ^k ^k \

If the sJ>eets have identical electrical characteristics, the subscript k may

be dropped and the summations may be replaced by a multiplicative factor, n.

Then for the parallel case,

(1 QlM^j + r^) Ql

= S^1g' . (33)

For the series case,

nC| . 1 C, M Q, + (Ar +^) 0. = J^nU . . ^ (34)

Thus in (23) through (27) all symbols remain the same except that for the

parallel case

C > nC, (35)

while for the series case

C n-^C and ig ^ nig. (36)

Note that the polymer resistance term occurs only in the combination RpCp and

that the value of this product does not change whether the arrangement is

series or parallel

.
-19-

APPENDIX B

Evaluation of the Output for Triangular-, Half-Sine- and

Haversi ne-Waveform Force Pulses

1 . Triangular Force Pulse

The triangular force pulse is characterized by:

Then

Also

where

F(t) =

' (t/T)FQ, 0 < t < T

(2 - t/i) Fq, T < t < 2t

(1 - e"'^), 0 < t < T

VL(t) = ?V^(vt)-J (2e^" - 1 - €^^)e-^^ x < t < 2t.

(gVT

_ g-v.^

2t _< t

-vt

R, C

^(Y) = Y

- Y2 (3 - e-Y)(i - e-^), Y = vi, and = ^33^0

(37)

(38)

(39)

(40)

Since Rp is greater than lO^^Q, Ri_ << Rd; consequently, : Ri (C + C|_), and

E(=o) = (C + Cl)--Qo2 ^(Y) where

Y . Cd F^

3 3 0 - (41)

The voltage has its maximum at time t = t, where V|_(t) = Y"-(l - e~''')?Vo; this

expression is a monotonic decreasing function of Y. Then,

Vl(0 = i (42)

T 0

v., Y >>1
-20-

The minimum value of the voltage occurs at time t « 2t where Vl(2t) - "(1 - e"')V^^(T). This expression has Its least values for Y small and large

-YVl(t), Y «1

-Vl(t), Y »1

V, (2t) ' (43)

and its largest value at Y e, 1.2565, where

-0.407

Vl(2t) =

^ ^

= -0.407cVo = -0.715 y^{x).

The voltage has value zero for tg given by

t^ = v-i ln{2e^ - 1), T < t^ < 2t;

to ^ T, Y » 1; t^ <2t, Y « 1. (44)

The energy has a maximum when the load resistance is chosen to be an appropriate

value for a given rise time and polymer capacitance. That is, ^(Y) reaches its

maximum 0.381 at Y = 1.893. Then

MAY 0-381

- C^c' =0.38UCV^2. (45)

Note that iJ(Y = 0.25) = 0.139; i^(Y < 0.25) : | Y - i Y2; jJ(Y = 4) = 0.317; and

ii(Y > 4) z Y

- ji' When Y » 1 (very small R|_).

2R

EH z Qjj2. (46)

Calculation of the energy delivered to the load during time t = t and t = 2t

yields

E(t) = iE(<») and E (2t) = EJ^{vt). . (47)

where

1$2(Y) = m - (1 - e-Y)\

For the values x = lO'^s, Ri = 4fi, and C = 0.3 yF, Y = 8 and the approximation of (46) is valid. With the expression of (41) or (70) for the effective

charge Qo developed on the capacitor, (46) and (47) yield E(t) = RlQo^/j«
-21-

Under optimum conditions, as given in (45), E increases linearly with the

number of sheets in a stack, n, whether the sheets are connected in series

or parallel. For Y >> 1 , the energy is independent of n for series connection;

for parallel connection, the energy varies as n^. However for Y << 1 (i.e.,

very large R[_),

Qo^

^^"^ = 3R[ (C + C|_)^ = 31^ ^^^o^'»

and the remarks applying to series and parallel stacks are reversed.

2. Half-Sine Force Pulse

For the half-sine pulse given by

(48)

F(t) = Fq sin 1^, 0 < t < 2t

TTt

•2t

sin(5^ + (j)) - e"^^ sincj) , 0 < t< 2t

-(1 + e^'^'^) e'""^ sine}, , 2t < t

(49)

(50)

where

2

tar\(\> = - VT. (51

)

For continuous sinusoidal force the first equation of (50) applies with

0 < t < °°.

Note that if vx « 1, the voltage reaches its maximum at t = x. Then

V|_(t) . (52)

and the minimum occurs at t = 2t, where

Vl(2x) = - V (53)

(As a numerical example, consider the following: a frequency of 100 Hz

corresponds to x = 2.5 ms; a stack of n = 4 sheets in series corresponds to

a value of C = 0.0025 yF; for Rl = 10 Mfi, vx z x/R|_C ~ 0.10.)

If in addition Cl « C and if the stack is made up of sheets connected electrically in parallel, Vl(x) is unaffected, while Vl (2x) depends on n-^; but for

the series connection, V, (x) varies as n, and V[_ (2x) depends on n^.
-22-

On the other hand, if vt » 1, the voltage amplitude is reduced considerably.

The voltage minimum occurs at t = 2t, and the voltage maximum occurs at t on the order of v"^ R, C « t, since (j)

TI H

V

V = 2 2vT

•

^MAX

, !^Vq , .Vl(2t). ; , (54)

(As a numerical example, consider the following: a frequency of 1 kHz

corresponds to t = 0,25 ms; a stack of n = 4 sheets in parallel corresponds

to a value of C ? 0.40 yFj for Rl ^ 100 fi, vt « t/RlC =57.)

If a stack consists of n sheets electrically in parallel, Vl'^^ is increased

n-fold times; for the series connection, V^^^^^ is unaffected.

3. Evaluation of the Output for Havers ine Waveform Force Pulse

For the haversine pulse given by

F(t) = IFq (1 - cos^) = Fq sin2^, 0< t< 2t, (55)

VL(t) = 1 sine

fsin (^ - e) + e-''''^ sine, 0 < t < 2t

t

.(e2^" - 1) e-^* sine, 2t < t

(56)

where [see (51 )]

cote = ^ = ^

tan*. ^ (57)

The energy is calculated as

E(~) = 2E(t) = i B^givx, vt/tt)

E(2t) = 1 E^giZvx, vt/tt) ' (58)

where

g(a,b) = (1 + b2)-i [a + (1 + b2)-i(l - e'^^U. (59)

Here for vt >> 1 the voltage rises slowly to a maximum at t = falls

3t

gradually to zero at t = t, and then reaches a minimum at t = -y. Then since

Rl/rp « 1,

v,MAX

, Vo « -Vl^I^ . (60)
-23-

This behavior contrasts with that of the two force pulses discussed previously. For example, compare this temporal behavior to that described by

(38) and (50) for which the voltage rises very quickly in times on the order

of v-^ « T [see also (43)].

Returning to the haversine pulse, when vt >> 1 (very small Rl),

E(2t) = EH z ^ (TrQ^)2. (61)

On the other hand, for vx << 1, the voltage behaves as in the previous cases,

the maximum occurring at t = t where

. (62)

and the zero occurs almost simultaneously with the minimum at t = 2t where

Also if VT << 1 (very large Rj^),

^(2^) = . ^

^jALy, = ^ {,V^)^ (64)

Now the energy, E(<») is largest when vx is chosen so as to maximize the function

g(vx, vx/tt). This occurs at vx = 2.106 when g = 1.922. Then (with R|_/Rp z 0)

E^^(oo) = 0.481 ^^Tcj^ = 0.48UCVq2^ (65)

Because of the close identification between (42) through (48), (52) through

(54), and (60) through (65), the remarks in sections 2 and 3 herein about

series- and parallel-electrically connected stacks apply to all three pulse

waveforms. Note that the inequalities (44) satisfied by t^, the instant when

the voltage null occurs for the triangular pulse, apply also for the half-sine

and haversine pulses. Also, it can be proved in general that if the force

pulse is symmetric about x [i.e., if F(t) = F(2x - t), O < t < 2x and F(t) = O for all other t], then the identity E(co) = 2E(x) holds.
-24-

APPENDIX C

Piezoid Output Produced by Open-Circuit Charging

Followed by Application of an Electrical Load

The polymer stack may be electrically charged under open-circuit conditions

by applying pressure with an hydraulic press or by impacting the stack with

a moving mass. The developed charge and energy may then be dumped into a load by means of a switch. This procedure results in an optimal charge and

energy transfer from the polymer piezoid to the load. The process of charging under open-switch conditions followed by switching in a load is equivalent to turning on a force infinitely rapidly under closed-circuit conditions.

This process may be described by the relation

F(t) = Lim (1 - e-^^)FQ, t > 0. ^ - : - - (66)

If at t = O"*", i.e., at a time instantaneously greater than t = 0, the switch

across the load is closed (after application of pressure or after impact), then

the voltage level across the load is

VL(t) = cVQe"^\ ; . j;v (67)

the charge which flows off the polymer is given from (15), (22), and (25) of

appendix B by .

Q(t) « fl dt'i(t') - Q^(l +^)"'C1 - ?(1

and the energy delivered to the load is

E(t) = 1(1 - e-^^^)E^ w: (69)

For times long compared to ~ R|_C, and using the usual approximations

(Cl « C, Rl « Rp),

Q.Qo.e4cV„2= J. , (70)

This Q is the charge built up on the polymer stack. Q is independent of n for

series electrical connection and proportional to n for parallel connection.

The quantity E is the energy stored in the stack and is proportional to n for

both series and parallel connections. For times of the order of tens of microseconds and capacitances of the order of tenths of microfarads, Rl should be

somewhat less than 100 to have full energy transfer to the load.

(68)
-25-

However, if is large (certainly if Rl > 10 kn, and t< 1.0 ms),

= 5^1%*

Note that if C|_/C < vt < 1 , the second term in the expression for Q(t) in (71)

dominates; however if Cj_/C > vt, the first term is larger than the second.

If Cl/C « 1, Q and E are independent of n for the parallel electrical connection, while for the series electrical connection, Q is proportional to n, and

E is proportional to n^.
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-27-

APPENDIX E

Output from Piezold Sample Having Non-Uniform

Conductivity and Dielectric Constant

Both theoretical and experimental investigations show that the dielectric

constant and conductivity of the poled portion of a piezoelectric material

are different from the unpoled bulk portion.

In appendix A, the condition RsCs = RpCp, was assumed for convenience of

exposition. However, an equation for Ql can be found by substituting for

Q + (RpCp)"^Qp from (19) into (17). Then the resulting equation may.be

combined with (18) to determine an expression for Qs in terms of Ql>Ql» snd

Iq- Finally. this expression an_d its time derivative (the expression for Qs

in terms of Qi_, Q]_, and di^/dt) may be substituted in (18) and there results:

[c-i + c^-i]

Ql

+ [c-HR^c^]-^ + (c^-i + C3-I) (RpCp)-i +

(C^-i + Cp-i) (RsCj)'^] Ql

+ [(Rp^)"^ + + (RpRs)"^] X

^Wl^"' \ = Cp-i[d1g/dt + (RpCp)-i ig]. (72)

This equation is easily solved; however the solution is lengthly, and therefore

only the expression for Rl « Rp and R^, which holds in all practical situations,

is given here.

Il = (\Cl)-^ Ql

= dVi^iV) {(RlCp)-^ ^ exp[^(RLC)-^ (f - t)] +

+ (CpC)-^ [(RpCp)-^ - (Rf^)-'] exp[(Rp-^ + R3-M (Cp + C3)-^ (f - t)]}.

RlCl « RpCp. (73)

Only if T > RpCp z 10"s >> RlC, does the second term in (73) give a non-negligible contribution. Therefore retention of only the first term in (73), which

would result in equation (25), yields a useful formula even when the condition

RsCs = RpCp does not obtain.
-28-

6. REFERENCES

[1] Edelman, S., Piezoelectric Polymer Films for Fuze Applications -

Progress Report for the Period Ending December 31, 1973, Instrumentation

Applications Section Report 162 (unpublished).

[2] Kawai, H. , The Piezoelectricity of Poly(Vinylidene Fluoride), Japan,

J. Appl. Phys. 8, pp. 975-976 (1969).

[3] Fukada. E. , and Takashita, S., Piezoelectric Effect in Polarized

Poly(Vinylidene Fluoride), Japan, J. Appl. Phys. 8, p. 960 (1960).

[4] Nakamura, K. , and Wada, Y., Piezoelectricity, Pyroelectricity, and

the Electrostriction Constant of Poly(Vinylidene Fluoride), J. Polymer Sci.

A-2,9. pp. 161-173 (1971).

[5] Kittel, C, Introduction to Solid State Physics, 4th ed. (1971),

p. 463, J. Wiley & Sons, N.Y.

[6] Bloomfield, P.E., Lefkowitz, I., and Aronoff, A.D., Electric Field

Distributions in Dielectrics, With Special Emphasis on Near-Surface Regions

in Ferroelec tries, Phys. Rev. B4, pp. 974-987 (1971).

[7] Edelman, S., Unpublished Studies of the Distribution of Piezoactivity

Through a Stack of Poled Polymer Films (private communication, 1972).

[8] Mason, W.P., Piezoelectric Crystals and Their Application to Ultrasonics,

pp. 36-38 (D. Van Nostrand Company, Inc., N.Y., I960).

-30-

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-35-

FiGURE 1: Mask and resulting electrodes on PVF2 film. On the left

IS THE 24-cm-square stai nless-steel mask used in the

VACUUM deposition OF ALUMINUM ELECTRODES ONTO THE POLYMER

SHEET MATERIAL. On THE RIGHT ARE FOUR ELECTRODED AND POLED ACTIVE ELEMENTS. FoR EACH PIEZOID, THE TAIL SEEN

AT THE BOTTOM FORMS PART OF THE UPPER ELECTRODE AND THE TAIL SEEN AT THE TOP FORMS PART OF THE LOWER ELECTRODE

(hidden by the upper ELECTRODE IN THE PHOTOGRAPH).
Figure 2; Stack construction. At the bottom (A) is a completed

STACK, The other stages shown are (B) a single-sheet

PIEZOID ELEMENT/ (C) A SINGLE SHEET WITH ONE FOIL

electrode lead ATTACHED/ (D) TWO SHEETS BONDED TOGETHER

WITH THE HIGH-SIDE LEAD BETWEEN THE SHEETS (AT LEFT) AND TWO LOW-SIDE LEADS ATTACHED TO THE SHEET OUTER SURFACES (at right)/ AND (E) TWO SHEETS BONDED TOGETHER

WITH AN INTERIOR HIGH-SIDE LEAD,
Figure 3: Circuit used in static-testing procedure,

Storage oscilloscope vertical-axis inputs

are marked 0; P IS THE PIEZOID TO WHICH

FORCE Fq is APPLIED; AND C|_ IS THE STANDARD CAPACITOR/ AS DESCRIBED IN THE TEXT.
-38-

FiGURE ^: Circuit used to connect polymer device to load, 'low"

and "high" refer to polarities established at the time of

poling; "high" corresponds to positive.

For compression tests / the double-pole / double-throw switch

is in the position SHOWN/ AND THE LOW SIDE BECOMES CHARGED

positively WITH RESPECT TO THE HIGH SIDE. WhEN THE VOLTAGE

SENSED BY THE ZeNER DIODE REACHES THE ZeNER REFERENCE

VOLTAGE/ THE POLYMER DEVICE DISCHARGES THROUGH DIODE CR2

INTO THE LOAD. FoR TENSION TESTS/ THE SWITCH IS THROWN

TO POSITION 2^ AS THE DEVICE POLARITIES ARE REVERSED.
-39-

I I I P

Ig

Figure 5; Polymer element equivalent circuit,

I

1 poiYMi ;

I tLEKfr I

« 1 Ql Cl

II

Figure 6: Polymer element connected to electrical load.
NBS^IUA (RgV, 7-73)

U.S. DEPT. OF COMM. ]. PUBLICATION OR REPORT NO. 2. Gov't Aeeesiion

BIBLIOGRAPHIC DATA ,^ , , No. SHEET NBSIR 75-724 (R)

3i Recipient's Aecession No.

4. TITLE AND SUBTITLE

ri eZOe 1 cCir1 c roiynier r 1 iiiis Tor ruze nppn canons

S. Publication Dace August 1975

6. Performing Organization Code

7. AUTHOR(S)

Philip E. Bloomfield

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 4253557

11. Contract/Grant No. DAAA25 74 F0008

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

U.S. Army

Frankford Arsenal

Philadelphia, PA 19137

13. Type of Report St Period Covered p.jpaT

9/1/73 to 12/31/74

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-wortl or less factual summary oi moat aigpilicarit information. If document includes a significant

bibliography or literature survey, mention it here.)

Described is the development of polymer piezoelectric devices with sufficient

electrical output in response to impact to demonstrate the feasibility of

using polymer elements in place of ceramic elements in ordnance fuze systems.

To satisfy this requirement, piezoelectric polymer elements have been con- structed capable of providing at least 0.3 mJ into a A-Q load, in response to

a force of 89 kN (10 tonsf). Total activities of up to 360 pC/N have been

achieved from elements built up from 30 sheets of 25-ym poly(vinylidene

fluoride). Test methods developed to characterize both single-sheet and

total -element activities are described, as are poling procedures and details

of element construction. Mathematical analyses of polymer piezoelectric

output into resistive-capacitive loads are presented for static and dynamic

cases. Experimental results and theoretical predictions are shown to be in

agreement to within the 10% estimate of measurement accuracy.

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)

Calibration; electrical energy; electrical output; piezoelectric; piezoelectric

output; polymer; poly(vinylidene fluoride); stack construction; theory; voltage.

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