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Interferometric measurement of the apparent longitudinal electrostrictive effect in isotactic polypropylene.

INTRODUCTION

A Michelson interferometer is used to measure the strain induced in samples of isotactic polypropylene by an applied electric field. Polypropylene was chosen as a model substance with a defined structure and a weak dielectric relaxation. Classical methods of polymer analysis like isothermal dynamic mechanical analysis (1) mainly consist of the measurement of the mechanical response to a periodic mechanical input. Here, an AC electric field is the sinusoidal excitation signal. The strain is detected at the double frequency. This corresponds to the measurement of the proportionality factor [Gamma][prime] between strain and the square of the electric field strength.

There are two different mechanisms of second-order electromechanical coupling, which contribute to the observed quantity [Gamma][prime] in the purely elastic case: The Coulomb attraction between the electrodes on the sample and the forces present on the boundary between dielectric and air cause stresses that are linked to the observed strain by Hooke's law. These Maxwell stresses have been treated (2), so their part, [[Gamma].sup.MS] can be predicted for a nonviscoelastic material. Quadratic electrostriction, which is related to the microscopic structure of the dielectric, is another effect of electromechanical coupling. To obtain the material constant electrostriction [Gamma] from measurements of the apparent electrostriction coefficient, a correction in the following form should be applied

[Gamma] = [Gamma][prime] - [[Gamma].sup.MS] (1)

Another measurement of electrostriction in polypropylene that has been reported (3) combined the use of a capacitive method and a strain gage.

EXPERIMENTAL PROCEDURE

Measurement System

Figure 1 gives a diagrammatic overview of the interferometric dilatometer that has been built by Van Sterkenburg et al. and has been presented in (4). The main features of the instrument are as follows:

* Electronic stabilization is used to keep the optical path length difference at the point of highest sensitivity and to compensate low-frequency distortions.

* Detection with two photodiodes from one optical path serves to increase the signal-to-noise ratio. The polarization optics creates a phase shift of [Pi].

* Double-face detection minimizes the influence of vibrations of the sample holder due to Coulomb forces. To this end, the part of the system measuring the strain of the upper side of the sample is repeated to detect the strain of the bottom, which rests on the sample holder.

It might be useful to note that an alternative solution to the same technical problem has been developed by Zhang et al. (5, 6). They used a Mach-Zehnder interferometer with a sophisticated polarization-optical scheme to measure two opposite faces of the specimen. This way, they were able to compensate for spurious vibrations using only one light source.

Samples

Eight homopolymers of isotactic polypropylene (Solvay ELTEX P) with different melt flow indices were measured. The melt flow index is a measure for the amount of polymer that can be extruded in 10 min under specified conditions (7).

The samples and the data on melt flow index and flexural modulus presented in Table 1 were provided by Solvay Benelux. To characterize the viscoelastic properties of the polymers under test at room temperature, the glass transition temperature [T.sub.g] was measured by dynamic mechanical thermal analysis (DMTA) (8) in tensile mode with an estimated error of [+ or -]2K.

The polymer samples were produced by injection molding. They were measured as received. The samples contain antioxidants and counteracidic agents but no filler materials or UV stabilizer. Samples HF 405 X6530, HW 255, and HZ 206 X4565 contain an antistatic agent.
Table 1. Thermoelastic Properties of Eight Isotactic Polypropylenes

The Data on Melt Flow Index and Flexural Modulus Were Provided by
Solvay, Brussels

The Glass Transition Temperature Was Measured by Dynamic Mechanical
Thermal Analysis with an Estimated Error of [+ or -]2 Degree
Celsius.

 Glass Transition
 Melt Flow Flexural Temperature
 Index 0.09 Modulus Degree
 g/min GPa Celsius

HF 405 x 6530 0.9 1.35 8
HL 200 1.8 1.50 10
HS 250 5 1.40 12
HV 252 10 1.40 13
HW 255 20 1.60 13
HY 202 x 6725 50 1.59 8
HZ 212 x 96VS 100 1.80 10
HZ 206 x 4565 200 2.00 8


RESULTS AND DISCUSSION

Measurements

Dilatations were measured in the thickness direction of samples with typical dimensions of 10 x 6 x 2.1 [([10.sup.-3] m).sup.3]. The strain was induced by application of a sinusoidally varying voltage of 900 V rms. The longitudinal effect was observed, meaning that the electric field vector was parallel to the normal of the vibrating surface of the polymeric sample. Values of the proportionality factor [Gamma][prime] between observed strain and the square of the applied electric field are given in Fig. 2.

To assess the anisotropy of the samples, the Laue X-ray diffraction method was used. Comparison with typical diffraction patterns presented by Young (9) leads to the conclusion that HS 250, HV 252, HW 255, HY 202, HZ 206, and HZ 212 can be regarded as isotropic, while HF 405 and HL 200 show slight deviations from the ideal isotropic pattern. A typical X-ray diffraction pattern is presented in Fig. 3.

To characterize the crystallinity of the samples, differential scanning calorimetry was employed. The results obtained using a Perkin Elmer DSC 6 are given in Table 2.

Corrections

Electrostriction is directly related to the polarization of the dielectric. The measured dilatation is caused by the combined effect of thermal stresses, internal mechanical stresses, Maxwell stresses, and electrostrictive stresses (10). In the static case, the measured values can be corrected for caloric and electrostatic cross-effects: If the measurement is performed under adiabatic conditions and the material is assumed to be isotropic, the following correction formula applies:

[Mathematical Expression Omitted] (2)

where [C.sup.[Sigma]]/[T.sub.0] is the heat capacity at constant stress divided by the reference temperature (in units of J/([m.sup.3][K.sup.2])) and [[Alpha].sub.11] is the coefficient of thermal strain. To determine the magnitude of [[Gamma].sup.TS], a value of [C.sup.[Sigma]] = 1, 75 x [10.sup.6] J/([m.sup.3]K) is calculated from [c.sup.[Sigma]] = 1860 J/(kgK) and a reported density of 940 kg/[m.sup.3] (11). A value for [[Alpha].sub.11] given in (11) is 10.5 x [10.sup.-5] 1/K. To estimate the value of the derivative of the dielectric constant with respect to temperature, the expression ([Delta][Epsilon]/[Delta]T) = 2[[Epsilon].sub.0]n([Delta]n/[Delta]T) and values for the refractive index and its derivative for PMMA below the [T.sub.g] (12) are used: ([Delta][Epsilon]/[Delta]T) = 29 x [10.sup.-16] As/VmK. The value of the correction term obtained this way is [Mathematical Expression Omitted]. Comparison with the measured values shows that the influence of thermal stresses can be neglected.
Table 2. Heat of Fusion [Delta]H of Seven Isotactic Polypropylenes
Measured by DSC

The Degree of Crystallinity is Calculated Assuming a Value of
[Delta]H(100%) = (165 [+ or -] 18) J/g (24) for a 100% Crystalline
Sample of Polypropylene.

 Degree of
 [Delta]H J/g Crystallinity %

HF 405 x 6530 89.6 54
HL 200 not measured -
HS 250 93.1 56
HV 252 91.6 56
HW 255 97.8 59
HY 202 x 6725 98.5 60
HZ 212 x 96VS 108.1 66
HZ 206 x 4565 113.6 69


Correction for Maxwell Stresses

The correction formula for electrostatic forces in the case of the longitudinal effect measured on an isotropic polymer reads:

[Mathematical Expression Omitted] (3)

The derivation of formula (4) has appeared elsewhere (13). From this formula the influence of Maxwell stresses can be estimated using [[Epsilon].sub.r] = 2.28 (14), and typical values for the elastic compliance and the Poisson number [Mu] = -[s.sub.1122]/[s.sub.1111]:[s.sub.1111] [approximately equal to] [10.sup.-9] 1/Pa (11) and [Mu] = 0.35 (15). It follows that [Mathematical Expression Omitted]. Under the approximation of isotropy, i.e., with-out taking the full anisotropy into account, the strains originating from Maxwell stresses have the same order of magnitude as the measured strains.

Correction for Constraints Caused by Electroding

Zhenyi et al. (16), proposed a formula to quantify possible constraints caused by the conducting layers used as electrodes:

[Mathematical Expression Omitted]

where the index 1 stands for the sample and the index 2 for the electrodes. [t.sub.i] are the corresponding thicknesses, [E.sub.i] are the elastic stiffnesses, and [[Mu].sub.i] are Poisson's numbers. The thickness of the samples is [t.sub.1] = 2.1 x [10.sup.-3] m. For [E.sub.1], [10.sup.9] Pa is taken as typical value. The thickness of the silver electrode is estimated as [t.sub.2] = 5 x [10.sup.-6] m. The values [E.sub.2] = 123 x [10.sup.9] Pa and [[Mu].sub.2] = 0.43 are taken from Ref. 17. Poisson's ratio [[Mu].sub.1] is varied from 0.35 to 0.55, which corresponds to the cases of undrawn and drawn polypropylene, respectively (18). For the isotropic case, one finds [K.sub.constraint] = 0.81, i.e., 81% of the strain signal of an unstrained sample. For [[Mu].sub.1] = 0.55 the predicted constraint is even more pronounced: 57%.

It seems as if this formula was derived without taking the position dependence of the strain tensor into account. An example of how this can be done to study the influence of electromechanical resonances on an electrostriction measurement can be found in Kloos (19). It should be mentioned that Ref. 19 contains an inconsistency because the dispersion relation is not completely satisfied.

Frequency Dependence

Relaxation Curve

In interpreting the measurements, a central question is how viscoelasticity influences the strain signal. To find an equation for the strain response of the viscoelastic material, Nowick and Berry's thermodynamic description of anelasticity (20) is extended to include second-order electromechanical coupling.

In order to adapt to the experimental situation in which the electric field strength E can be controlled and the strain signal is observed, it is appropriate to use the Gibbs energy function that depends on these variables. On the assumption of isothermal conditions the expansion of this thermodynamic potential reads:

G([T.sub.0], [Sigma], E) = [G.sub.0] - 1/2 s[[Sigma].sup.2] - 1/2 [Epsilon][E.sup.2] (4)

where s is the elastic compliance at constant electric field, [Epsilon] = [[Epsilon].sub.0][[Epsilon].sub.r] is the dielectric constant at constant stress. To describe electro-elastic coupling, i.e., the combined effect of electrostriction and Maxwell stresses, a term linear in mechanical stress and quadratic in the electric field strength is added in an ad-hoc manner, introducing [[Gamma].sup.td] as a coupling constant:

G([T.sub.0], [Sigma], E) = [G.sub.0] - 1/2 s[[Sigma].sup.2] - 1/2 [Epsilon][E.sup.2] - [[Gamma].sup.td][Sigma][E.sup.2] (5)

Now, an internal variable [Xi] that couples with elastic stress (-[Chi][Sigma][Xi]) and the second power of the electric field strength (-[Eta][E.sup.2][Xi]) is introduced, to take account of the time dependence of the observed strain because of relaxation of the polymer chains:

G([T.sub.0], [Sigma], E, [Xi]) = G([T.sub.0], [Sigma], E) + 1/2 v[[Xi].sup.2] - [Chi][Sigma][Xi] - [Eta][E.sup.2][Xi] (6)

v, [Chi], and [Eta] are coupling constants.

Following (20), the assumption is made that the value of the internal variable decays exponentially to an equilibrium value [[Xi].sub.s]:

d[Xi]/dt = -1/[Tau] ([Xi] - [[Xi].sub.s]) (7)

If we consider

A = [([Delta]G/[Delta][Xi]).sub.[Sigma],E] = v[Xi] - [Chi][Sigma] - [Eta][E.sup.2] (8)

as a thermodynamic driving force to equilibrium, we find the following for the value of [Xi] under equilibrium conditions:

[([Delta]G/[Delta][Xi]).sub.[Sigma],E] = 0 [if amd only if] [[Xi].sub.s] = 1/v ([Chi][Sigma] + [Eta][E.sup.2]) (9)

A combination of this result with the differential equation for [Xi] and the derivative with respect to time of the mechanical equation of state that reads

[Epsilon] = -[([Delta]G/[Delta][Sigma]).sub.E,[Xi]] = s[Sigma] + [[Gamma].sup.td][E.sup.2] + [Chi][Xi] (10)

allows one to eliminate the internal variable

[Tau] d[Epsilon]/dt + [Epsilon] = s[Tau] d[Sigma]/dt + [s + [[Chi].sup.2]/v][Sigma] + 2[[Gamma].sup.td][Tau] E dE/dt + [[[Gamma].sup.td] + [Chi][Eta]/v][E.sup.2] (11)

Now it is possible to identify [[Gamma].sup.td] = [[Gamma][prime].sub.[infinity]] with the coupling constant measured at high frequencies and

[[Gamma][prime].sub.[infinity]] + [Chi][Eta]/v = [[Gamma][prime].sub.s] (12)

with the low-frequency value.

Using these abbreviations, Eq 11 reads

[Tau] d[Epsilon]/dt + [Epsilon] = [s.sub.[infinity]][Tau] d[Sigma]/dt + [s.sub.s][Sigma] + 2[[Gamma][prime].sub.[infinity]][Tau]E dE/dt + [[Gamma][prime].sub.s][E.sup.2] (13)

Assuming an input periodic in time and a response of the system at the double frequency, one finds:

[Epsilon] = [[s.sub.[infinity]] + [s.sub.s] - [s.sub.[infinity]]/1 + [(2[Omega]).sup.2][[Tau].sup.2]][Sigma] + [[[Gamma][prime].sub.[infinity]] + [[Gamma][prime].sub.s] - [[Gamma][prime].sub.[infinity]]/1 + [(2[Omega]).sup.2][[Tau].sup.2]][E.sup.2] (14)

In the following, it will be assumed that the sample is mounted almost stress-free, so that [Sigma] can be treated as equal to zero. The observed proportionality factor [Gamma][prime] is a function of frequency due to viscoelastic relaxation. Two limiting cases can be distinguished:

The instantaneous value

[[Gamma][prime].sub.[infinity]] = 1/2 [([[Delta].sup.2][Epsilon]/[Delta][E.sup.2]).sub.[Sigma],[Xi]] (15)

that is observed when no time is allowed for a relaxation to take place and the static value

[[Gamma][prime].sub.s] = 1/2 [([[Delta].sup.2][Epsilon]/[Delta][E.sup.2]).sub.[Sigma],A = 0] (16)

that is measured under equilibrium conditions. It is the instantaneous value corrected for Maxwell stresses that is predicted by theory if no viscoelastic relaxation has been taken into account. Therefore, in viscoelastic material an additional effect becomes important, which causes a strain superimposed on the strain originating from quadratic electrostriction as given by Eq 15.

Relationship of the Relaxation Strength With Other Material Constants

The relaxation strength [Delta][Gamma][prime] is the difference between the equilibrium value and the instantaneous value of the proportionality factor:

[Delta][Gamma][prime] = [[Gamma][prime].sub.s] - [[Gamma][prime].sub.[infinity]] (17)

An expression of the relaxation strength in terms of other material constants can be derived using thermodynamics:

[Delta][Gamma][prime] = [Delta]s [([Delta][Epsilon]/[Delta][Epsilon]).sub.T,[Sigma],E] (18)

In this approximation, the relaxation strength is linear in the relaxation strength of the elastic compliance and proportional to the derivative of the dielectric function with respect to strain. From linear theory, it is known that the [Delta]s = [s.sub.s] - [s.sub.[infinity]] is always positive. The sign of the relaxation strength is therefore entirely determined by the sign of the strain derivative. A value of [Delta][Epsilon]/[Delta][Epsilon] measured transversally (3) is -0.78 x [[Epsilon].sub.0].

Comparison With Literature

A measurement of the transversal electrostrictive effect of polypropylene has been reported by Nakamura and Wada (3). They determined a component of the derivative of the dielectric tensor with respect to strain: [Delta][[Epsilon].sub.r]/[Delta][Epsilon] = -0.78. Even if the case of highest symmetry is assumed, i.e., a mechanically isotropic polymer, the knowledge of another component of this tensor is necessary (21) to transform to the electrostrictive tensor defined as

[[Gamma].sub.ijkl] = 1/2 ([[Delta].sup.2][[Epsilon].sub.ij]/[Delta][E.sub.k][Delta][E.sub.l]) (19)

It follows that Nakamura and Wada's value cannot be directly converted to [[Gamma].sub.1111], as necessary for comparison. A rough estimation leads to the same order of magnitude as observed in this experiment.

CONCLUSIONS

The analysis shows that relaxation of isotactic polypropylene can strongly influence the measurement at the double frequency. The expression given for the relaxation strength leads to the conclusion that its contribution should be taken into account if measurements are performed in the relaxed state. Without additional information no decision can be made if the relaxed or unrelaxed value of the coupling constant or a value in between has been determined.

In the nonviscoelastic case thermal stresses can be neglected. The contribution of Maxwell stresses to the observed value of the electromechanical coupling constant cannot be ignored. In the frequency domain of 1 to 9 kHz an order of magnitude of [10.sup.-21] [m.sup.2]/[V.sup.2] is representative for the apparent electrostriction coefficient of the samples of isotactic propylene that have been investigated. The measured values seem to be insensitive to the influence of substances like antistatic agents. This can be concluded from the fact that while, the concentration of these substances differs for the polymers in Table 1, the observed value remains the same, with the exception of HL 200. No explanation has been found for the different behavior of this polymer. It might be related to its differing anisotropy. The values measured are relatively small and in the same order of magnitude as values reported for non-ferroelectric ionic crystals.

The method presented is also applicable to polymers with other features, i.e., with a different melt index or flexural modulus. Theoretical considerations (22, 23) allow the conclusion that higher electrostrictive strains can be assumed for polymers with higher elastic compliances [s.sub.1111] and/or higher dielectric constants.

ACKNOWLEDGMENT

The samples were kindly provided by Solvay Benelux, Brussels. I would like to thank C. Ponet and R. van Asbroek for their advice and the detailed information they gave me. I thank L. Govaert and J. M. M. de Kok of the Department of Mechanical Engineering of our university for introducing me to DMTA and letting me use their measurement system. I am indebted to J. Couwenberg for his technical help. I thank H. Dalderop of the Department of Technical Physics for teaching me how to use his X-ray equipment. I would like to thank A. Franken of the Department of Chemistry for introducing me to his DSC system and U. Goschel from the same department for his kind help to evaluate and interpret the results.

REFERENCES

1. D. W. Hardley and I. M. Ward, in Encycl. Polym. Sci. Eng. Vol. 9, Wiley, New York (1987).

2. M. Abraham and R. Becker, The Classical Theory of Electricity and Magnetism, London, Blackie (1959).

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5. Q. M. Zhang, S. J. Jang, and L. E. Cross, J. Appl. Phys., 65, 2807 (1989).

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7. International Standard ISO 1133: 1991(E).

8. J. N. Hay, in Analysis of Polymer Systems, pp. 169-99, L. S. Bark and N. S. Allen, eds., Applied Science Publishers, London (1982).

9. R. J. Young, Introduction to Polymers, p. 150, Chapman and Hall, London (1981).

10. E. Durand, Electrostatique Vol. 3, Masson, Paris (1966).

11. R. P. Quick and M. A. A. Alsamarraie, in Polymer Handbook, p. V/27, J. Brandrup and E. H. Immergut, eds., (1989).

12. W. Wunderlich, in Polymer Handbook, p. V/77, J. Brandrup and E. H. Immergut, eds. (1989).

13. G. Kloos, J. Phys. D.: Appl. Phys., 28, 939 (1995).

14. E. W. Anderson, D. W. Call, J. Polym. Sci., 31, 241 (1958).

15. D. W. van Krevelen, Properties of Polymers: Correlations with Chemical Structure, p. 157, Elsevier, Amsterdam (1972).

16. M. Zhenyi, J. I. Scheinbeim, J. W. Lee and B. A. Newman, J. Polym. Sci.: Part B: Polym. Phys., 32, 2721 (1994).

17. Landolt-Bornstein, Numerical Data and Functional Relationships in Science and Technology, NS III/11, p. 11, Springer-Verlag, Berlin (1979).

18. Yu. K. Godovsky, Thermophysical Properties of Polymers, p. 143, Springer-Verlag, Berlin (1992).

19. G. Kloos, Cryst. Res. Technol., 30, 963 (1995).

20. A. S. Nowick and B. S. Berry, Anelastic Relaxation in Crystalline Solids, Academic Press, New York (1972).

21. P. Preu and S. Haussuhl, Solid State Commun., 45, 619 (1983).

22. R. A. Anderson, Phys. Rev. B, 33, 1302 (1986).

23. G. Kloos, J. Phys. D: Appl. Phys., 28, 1680 (1995).

24. B. Wunderlich, Macroscopic Physics, Crystal Melting, Vol. 3, Academic Press, New York (1980).
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Author:Kloos, Gerhard
Publication:Polymer Engineering and Science
Date:Mar 1, 1997
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