Polyurethanes have been extensively studied for their strong electromechanical response. Previous studies have mainly investigated the impact of electrical charges on these polymers' DC conductivity, which was measured on the order of 10−10 S/m. The movement of electric charges is responsible for the macroscopic deformation of polymer films under an electric field. However, this study focused primarily on electric fields below 107 V/m, where the electric current complies with Ohm's law. In this paper, we examine the electric current above this field value and observe a deviation in the current from linearity with the applied field, especially for a high electric field of 106–107 V/m. This change is known in polymers under high electric fields but has never been observed in polyurethane. This suggests the injection of electrode charges into the polymer material. This article provides the threshold at which the transition occurs from linear (Ohm's law) to nonlinear conduction as a result of injected electric charges.

Polyurethane (PU) is a copolymer that exhibits significant electrostrictive behavior. This phenomenon involves the deformation S of the material under an electric field E0, where E0 = V0/tf, with V0 representing the applied voltage and tf the polymer thickness. Classical electrostriction is induced by electric polarization P, and the electrostrictive deformation is expressed as S = QpP2(E) with the electrostrictive coefficient Qp.1 At low electric field strength, the polarization is linear according to P(E) = ɛ0χE, where χ is the electric susceptibility and ɛ0 is the void permittivity, leading to the usual expression S = ME02. This can be reinterpreted as the compression of the polymer sandwiched between the positive and negative electrodes as they are attracted to each other. This force per unit surface area is the Maxwell stress, and the resulting strain can be expressed as
S m = M m E 0 2 = ε r ε 0 2 Y E 0 2 ,
(1)
where the coefficient Mm equals −ɛ/2Y, ɛ is the permittivity (defined as D = ɛE), and Y is the Young modulus. In the case of PU, the measured coefficient M is larger than the coefficient Mm.2–5 Amorphous polymers have a larger coefficient Qp than other solid materials,1 and that of polyurethane (PU) is one of the largest, bringing more interest to this class of materials. Some experimental modifications of the polymer chemistry or insertion of dielectrics and/or conductive particles have succeeded in further increasing M (or Qp).6–10 Apart from the experiment, theoretical work on increasing electrostriction can achieve understanding of why the measured deformation is typically larger than the Maxwell deformation. The electrostrictive coefficient 1/ɛY is used in Refs. 1 and 10–12. However, its formula is an empirical expression, and it approximates opposite cation and anion displacements for an ionic crystal1,13 under an electric field, so it may not be suitable for amorphous materials.

An important consideration involves dynamic behavior. When measured over time, the deformation increases gradually before reaching a stable amplitude. This effect takes 1–102 s to fully develop.2,4,14–19 The theoretical expressions of Mm have an implicit time dependency through the permittivity ɛ and Young modulus Y. Dynamic characterization of PU reveals relaxations of the dielectric constant and the mechanical shear modulus.17 These relaxations, recorded for several PUs, appear at a much shorter time (by more than four decades) than the measured characteristic time of electrostriction (τ ∼ 102 s).17,18 This demonstrates that the mechanisms are more complex than expected in a pure dielectric material with elastic characteristics. To explain the time dependence of electrostriction, another hypothesis involves the internal movement of electric charges. A polymer chemically doped with NaCl has been shown to exhibit a slow drift of Na+ and Cl ions along its thickness due to the electric field.19 Two charge layers were created close to the electrodes. A numerical calculation estimated that each charge layer would expand laterally, thus reducing the polymer thickness. Furthermore, because the two charge layers do not possess the same characteristics (time, size, and charge), the expansions of their two interfaces do not necessarily compensate each other, and bending is observed.

The charge movement, representing an electric current, is then correlated to the electrostriction/electrobending.20–22 Electrostriction is typically observed at a range of relatively high applied electric fields, and it is possible to inject electric charges (electrons and holes) from the electrodes into the polymer under such fields. Such injections are well documented for high-voltage cable polymeric insulation.23,24 According to Watanabe,25–28 these additional charges significantly contribute to PU electrobending. A typical fingerprint of charge injection can be seen in the deviation of PU conductivity from Ohm's law under high electric fields. Ohm's law is the drift of the internal charges expressed by J = σE with the electrical conductivity σ = n0, where n0 is the density of thermally generated free charges and/or impurities, q = ±e (e = 1.6 × 10−19 C), and μ is the charge mobility. At metal–semiconductor interfaces, charges (electrons/holes) can be injected easily with strong fields through Schottky and Poole–Frenkel processes, adding another charge density n. Ohm's law occurs at a low applied electric field where the injection is low. Above a threshold electric field ET, the density of injected charges, n, is then larger than n0, and a deviation from linear conductivity is observed.29,30

This article examines the high-electric-field conductivity of PU and probes this electric field threshold. The investigation measures PU's response to DC and AC fields to identify such deviation. The results are reported and analyzed in detail. This is important for the physics of electrostriction in polymers and for optimizing electrostrictive polymers for actuation applications.

The samples consisted of PU, specifically co-block polymers with a hard segment composed of 4.4′ methylene-bis-(phenyl isocyanate) (MDI) and 1,4-butanediol (BDO) and soft segments consisting of poly(tetramethylene oxide) (PTMO). The HS weight fraction was 45% with a molecular weight of 1000 g mol−1. Estane 58888 NAT021 PU was procured from Lubrizol. The PU granules underwent heating at 350 K for 3 h and were then dissolved in DMF at a concentration of 25 wt. % and stirred for 2 h at 350 K. The homogeneous solution that was obtained was left for 24 h to remove air. It was then deposited using an Elcometer 3700 film applicator. The resulting film was dried at 335 K for a day to evaporate the DMF and, subsequently, a second heating treatment was applied for 3 h at 400 K. For electric measurements, a time-varying step voltage was applied, and the film thickness was measured with a digital caliper to be approximately tf= 100 μm. For pulsed electro-acoustic (PEA) testing, the thickness was tf = 50 μm. An SC-701 quick coater from Sanyu Denshi was used to sputter two Au electrodes (work function ∼5.1–5.3 eV) as disks each with a diameter of 2 cm onto the top and bottom surfaces of the PU film under an Ar pressure of 0.05 MPa. The Au thickness was estimated as 10–20 nm from the sputtering time and the ion current (10 mA).

The electric measurements were conducted with a custom-built apparatus comprising two solid electrodes: a copper piece situated on top of the sample and a transparent disk with a thin conductive Ti layer placed below the sample. The top electrode was grounded, and the bottom electrode was used to apply the voltage. An amplification system (up to 10 kV) was used to generate the high voltage according to the command signal. labview program was used to monitor the applied voltage wave function and measure the current simultaneously.

The PEA technique was used to characterize space charge.31,32 An applied electric pulse causes the electric charges inside the polymer to move, which creates a pressure wave propagating through the polymer. A piezoelectric material is used to measure the amplitude of this pressure wave and the signal time delay, and software is used to reconstruct the charge density position and amplitude.

Figure 1 shows the measured current when step voltages with different amplitudes were applied to the PU. The current under a low applied voltage remains stable. For an applied voltage of 2.0 kV, the current first increases, reaches a peak, and then slowly decays until the voltage is switched off. The current at low voltage exhibits the same time dependency as the voltage, which implies a linear relationship between the current and applied voltage. At higher voltage, the current exhibits the same time dependency as the voltage along with another contribution. This will be discussed below. A change in the behavior of the step response of the current is observed by comparing the peak current (Ipeak) with the middle (Imiddle) and last (Ilast) currents.

FIG. 1.

Time response to step voltage with different amplitudes. Red dotted lines are linear fits of the step response.

FIG. 1.

Time response to step voltage with different amplitudes. Red dotted lines are linear fits of the step response.

Close modal

These current values were then converted into current densities using J = I/S, where S is the estimated surface area of each electrode. The voltage was used to estimate the electric field to be E0 ∼ 106–107 V/m, depending on the voltage.

The resulting current densities (peak, middle, and last) are plotted in Fig. 2 against the applied electric field E0. From the graph, it can be concluded that the current is linear for low E0. There is a threshold in the applied field estimated at E0 = ET = 7 × 106 V/m, beyond which the current no longer follows a linear trend. A difference in current behavior is clearly observed between the low field (E0 < ET) and the high field (E0 > ET). This result agrees well with our previous study on the long-term kinetics of PU, where a linear relationship was observed between current density and electric field below 8 MV/m.19 

FIG. 2.

Extracted measurements of current at the beginning (peak), middle (middle), and end (last) of the step response. The dotted line represents a linear fitted extrapolation.

FIG. 2.

Extracted measurements of current at the beginning (peak), middle (middle), and end (last) of the step response. The dotted line represents a linear fitted extrapolation.

Close modal

It is clear that Imiddle and Ilast were obtained arbitrarily by turning off the voltage, and that these values could be lower if a longer step voltage was applied. Nevertheless, this figure quantifies the observation that the current response was rectangular (similar to the applied step voltage) at low voltage, but was more trapezoidal at high voltage.

According to Sekiguchi,33 the current resulting from applying a step voltage is a combination of three components: the charging current, which is the current from the time at which V = VON to the peak, and after the peak to the time at which V = VOFF; the absorption current, which is a decaying component; and the leak (or conduction) current, which is a steady current.28,33 Over an extended period, the value Ilast is expected to be predominantly the conduction current. With a low field, E0 < ET, the current exhibits linear behavior, as in Fig. 2, and the polymer conductivity σ can be estimated from the fitted line to reveal an electrical conductivity of σ = J/E0 ≃ 2.7 × 10−9 S/m, a value similar to the results of a prior investigation.19 

To analyze the effect of time on a PU sample under a high field, the step voltage was applied on a longer time scale. The plot in Fig. 3(a) shows the current resulting from a step voltage with an amplitude of 2.0 kV applied to the polymer. This current also demonstrates strong decay over time, a typical asymptotic behavior seen in such curves.34 Such decay is usually modeled with a power law known as the Curie–Von Schweidler equation35,36
I ( t ) = A ( T ) t n ,
(2)
where A is a coefficient dependent on temperature and n is an exponent less than 1. The curve in Fig. 3(a) was fitted with Eq. (2) and yielded the experimental value n = 0.7.
FIG. 3.

Current vs time response from a 2.0-kV step function on (a) linear and (b) log–log scales.

FIG. 3.

Current vs time response from a 2.0-kV step function on (a) linear and (b) log–log scales.

Close modal

Extracting additional information from this power law curve is challenging. However, the log–log plot in Fig. 3(b) reveals certain characteristics. Inflections are noticeable in this plot. This suggests that the absorption current's time dependence varies at different time scales during observation.

To analyze and quantify these data, we used the following decaying function:36,37
i ( t ) = Y 0 + i = 1 N A i e t / τ i .
(3)

The current curve of PU exposed to the 2.0-kV step function was more accurately modeled with the sum of two decreasing exponentials (N = 2). The time scales were determined to be τ1 = 10 s and τ2 = 94 s.

This phenomenon can be interpreted as the trapping of charge carriers, which results in the formation of an electric charge Q(t) within the polymer. This relationship is mathematically written as38,39
Q ( t ) = i = 1 N B i ( 1 e t / τ i ) .
(4)
By the law of charge conservation, the increase in Q(t) results in a decrease in current contribution. Initially, this decrease is significant, and because t ≫ τi, Q(t) saturates and the resulting current becomes negligible. In reality, only a DC (or conduction) current persists.33 This is succinctly summarized in the equation
i ( t ) = i c + i Q = σ E + d Q ( t ) d t .
(5)
The time constants τi of the traps can be associated with trap energy levels. These energies can be estimated using the Lewis formula40–42,
t D , i = 1 ν e ( U i / k B T ) ,
(6)
where tD represents the time constant, U denotes the trap energy depth, kB refers to the Boltzmann constant (1.38 × 10−23 J K−1), T is the temperature, and υ is the attempt frequency υ = kBT/h with the Planck constant h = 6.62 × 10−34 J s. Although polymers can be characterized by a continuous distribution of trap energy levels, two energy levels, shallow and deep traps (defined by their energies), are commonly used to describe these materials.38,39 This hypothesis of two traps is also supported by the results presented in Fig. 3(b).

Moreover, Watanabe28 observed an interesting behavior: the measured electromechanical bending of PU was directed to the right during a positive step voltage. This was followed by a negative step voltage, under which the PU first bent again to the right and then to the left, still under the same negative step voltage. It is thus interesting to focus on the corresponding electrical behavior. It is known that the time-dependent behavior of the current is significantly affected by charge trapping. In Fig. 4, a long experiment was conducted by repeatedly applying a step voltage with a polarity change of ±1 kV. At first, three positive (+1 kV) step voltages (p1, p2, and p3) were applied with different durations and intervals, and then six negative (−1 kV) step voltages (n1, n2, n3, n4, n5, and n6) were applied, also with different durations and intervals. The reason was to observe the charging effect for different polarities and step durations.

FIG. 4.

Time dependence of the applied voltage (black dotted line) and the measured current (blue curve). The black dashed line serves as a guide to highlight the leakage current.

FIG. 4.

Time dependence of the applied voltage (black dotted line) and the measured current (blue curve). The black dashed line serves as a guide to highlight the leakage current.

Close modal

In Fig. 4, a 180-s step voltage (p1) of +1 kV is first applied from t = 60 to 240 s, followed by two shorter steps (p2 and p3) of the same voltage amplitude for 30 s. As in the previous experiment, the charging current decreases with each application of positive voltage. However, a decrease in the peak current amplitude is observed when comparing the resulting currents from p1, p2, and p3. This indicates a reduction in the charging effect.39 However, the DC current appears to remain constant during each iteration. These observations support the interpretation of trapped charges. Prior to the first step, it was assumed that traps were empty and as p1 was applied, electric charges were easily trapped. During p1, a greater number of charges became trapped within the polymer. After an initial decay, the current decreased to a stable level as fewer free traps were accessible at the level of the applied electric field. After p1, the voltage was removed for 60 s, during which it is assumed that some of the trapped charges were de-trapping. As the second positive step (p2) began, only the de-trapped portion of the trap sites was accessible, meaning only this part of the traps could be recharged. The amplitude of this peak corresponds to the number of accessible trap sites. This resulted in the current peak of p2, which was smaller than the initial peak. Before p1, all of the traps were empty, while just before p2, only a fraction of the accessible traps was empty. This trend is also visible in the third positive step (p3) voltage application.

Then, after these positive steps, a long step voltage (n1) of −1 kV was applied for 300 s. This was followed by shorter steps (n2 to n6) of 100 s, all with the same amplitude and sign. During the first negative step (n1), where the voltage polarity was reversed, a broad current peak was observed. The subsequent negative steps (n2 to n6) showed a more usual decaying shape. It is interesting to note that the amplitude of n3 was smaller than that of n2, and this was also observed for the n5 and n4 peaks. However, the amplitudes of the n4 and n6 peaks were larger than those of the negative n3 and n5 peaks, respectively. This is believed to be a result of the time delay between voltage applications. Specifically, there was a delay of Δtn2–n3 = 30 s between peaks n2 and n3 and of Δtn3–n4 = 90 s between n3 and n4. Therefore, fewer charges could be de-trapped between n2 and n3 than between n3 and n4. This pattern also occurred for peaks n4, n5, and n6, where the time delays were 30 s between n4 and n5 and 200 s between n5 and n6.

In our electric current measurement in Fig. 4, the current amplitude response after the polarity inversion, n1, has changed its sign. However, it first decreased, then reached a minimum, and finally exhibited an asymptotic trend. As the field was applied, one electrode injected charges while the other electrode collected them after they passed through the polymer. However, some of the charges are trapped near the injector. If the field is cut, the layer can be removed with time as described. If the field is reapplied with the opposite polarity before full discharge of the previously injected charges, this layer continues to discharge and, at the same time, a new charge layer is being created on the other side of the polymer. There are then two layers: one that is discharging, and a second one that is charging. The minimum meant that the discharging layer was then fully discharged. In the bending28 during the field reversal, the film was bent first because of the discharging layer, before the new charging layer became dominant and reversed the bending direction.

After this two-step DC voltage experiment, an AC voltage experiment was carried out to separate the behaviors of both the conductivity and dielectric capacitance under a high electric field. A sinusoidal voltage was set with amplitude V0 = 0.5–2 kV, and the resulting applied field E0 = V0/tf then ranged from 5 to 20 MV/m. The frequency was set to 0.1 Hz. Data are presented in Fig. 5 for an applied voltage amplitude of 2 kV.

FIG. 5.

Example of the signal at V0 = 2 kV in the 100-μm-thick sample.

FIG. 5.

Example of the signal at V0 = 2 kV in the 100-μm-thick sample.

Close modal
The measured current consists of two components: one in phase and the other out of phase, corresponding to the conduction and capacitance, respectively. The equation for the current determined from capacitive and conduction effects is8,43
I = C V ˙ + V R ,
(7)
with the electric capacitance C given by
C = ε S t f ,
(8)
and the electric resistance R expressed as
R = 1 σ t f S ,
(9)
where tf denotes the sample thickness and S represents the area of each sputtered electrode. The ratio S/tf is then S/tf = π(10−2)2/10−4 = π. A geometrical factor must be calculated for both R and C.
Equation (7) can be reformulated using Eqs. (8) and (9) as
I = σ S t f V + ε S t f V ˙
(10)
for a harmonic applied voltage signal such as
V ( t ) = V 0 sin ( ω t + φ 0 ) .
(11)
After Eq. (11) is substituted into Eq. (10), the current becomes
I = σ S t f V + ε S t f V ˙ = [ A sin ( ω t + φ 0 ) + B cos ( ω t + φ 0 ) ] .
(12)
By identifying the contributions on the left and right sides of Eq. (12), we can express the in-phase part of the signal, A, and the out-of-phase part, B, as
( A B ) = V 0 S t f ( σ ω ε ) = E 0 S ( σ ω ε ) .
(13)

From Eq. (13), the constants A and B depend on the geometric parameters (S and tf) and the input voltage V0. These constants have a linear relationship with the material parameters σ (for A) and ɛ (for B). It is then possible to obtain the conductivity and relative permittivity.

The values of A and B extracted from Fig. 5 are plotted in Fig. 6(a) against the applied voltage. Parameter A exhibits nearly quadratic behavior with voltage, while B displays an almost linear trend. The behavior of A can be extracted by setting a power law relationship between the conduction current and the applied voltage. From Eq. (13), the conductive part of the current, A, and the voltage are now rewritten as
A = σ S t f V 0 c .
(14)
FIG. 6.

(a) Extracted contribution from the sinusoidal fit as a function of the applied electric field, and (b) conduction as a function of the applied electric field.

FIG. 6.

(a) Extracted contribution from the sinusoidal fit as a function of the applied electric field, and (b) conduction as a function of the applied electric field.

Close modal
To extract the exponent c at low or high voltage, this equation was transformed using the logarithmic function
ln ( A ) = ln ( σ S t f ) + c ln ( V 0 ) = ln ( A ) + c ln ( V 0 ) .
(15)

The slope of the corresponding curve was extracted at low and high voltage, as seen in Fig. 6(b). The exponent c was found to be 0.98 for low voltage and 1.80 for high voltage.

In Fig. 7, the conductivity σ and the relative permittivity ɛr are plotted vs E0. The conductivity is constant at low field but linearly increases at high field, as expected from Fig. 6(b). The two behavior domains are separated by a field threshold of approximately ET ∼ 107 V/m. The experimental conductivity was fitted below this threshold (Fig. 7), and a real conductivity contribution of σ ∼ 1.65 × 10−10 S/m was obtained. This is much lower than the DC value. We should note that for the AC measurement, we had to process the electrical signal to extract the real and imaginary parts despite a rather low signal-to-noise ratio. We did not focus on the absolute value of the conductivity, but on how it was affected by the value of the applied electric field. Although this conductivity value is lower than previously reported for DC, the ET threshold remains very close (107 vs 7 × 106 V/m) to the previously reported value for DC.

FIG. 7.

Extracted relative permittivity ɛr and conductivity σ as functions of the applied electric field E0.

FIG. 7.

Extracted relative permittivity ɛr and conductivity σ as functions of the applied electric field E0.

Close modal

The obtained PU conductivity change with the applied field is consistent with other measurements. A decrease in the resistive part with an increase in the applied electric field has been noted44 for a field ranging from 5 to 20 MV/m, which is consistent with the field-dependent increase in conductivity observed here. The J–E plot in Ref. 19 was restricted to 10 MV/m, and linearity was assumed, although the points above 8 MV/m were a little higher than the fitted curve and might indicate non-linearity. More generally, Fig. 8 presents the field effect on conductivity for the PU.

FIG. 8.

Conductivities (AC and DC) and relative change in permittivity as functions of the applied electric field for polyurethane (PU) and polyethylene (PE).

FIG. 8.

Conductivities (AC and DC) and relative change in permittivity as functions of the applied electric field for polyurethane (PU) and polyethylene (PE).

Close modal

The extracted relative permittivity is also plotted in Fig. 7, and this remains almost constant at ɛr ∼ 4 in the entire experimental range of the applied electric field.

The PEA technique allows for objective observation of space charges in polymers.31–33,38,39,45 The amplitude of the space charge density (ρ) can be probed as a function of time under various parameters such as field, temperature, and electrode type. In this experiment, the applied electric field E0 = 107 V/m was similar to the threshold value determined in the previous section to enable visualizing the type of injected charge in the polymer. Figure 9 shows the charge density profile along the thickness just after the electric field is applied, where x = 0 μm and x = 50 μm represent the cathode and the anode, respectively.

FIG. 9.

Space charge density after application of an electric field of 107 V/m.

FIG. 9.

Space charge density after application of an electric field of 107 V/m.

Close modal

It is evident that the space charge exists at the two metal–polymer interfaces. This refers to the charges at the electrodes. Furthermore, there are two peaks inside the sample: a minor positive peak (Q+) at approximately x = 10 μm and a more substantial negative peak (Q−) at approximately x = 35 μm (or 15 μm from the anode). Thus, the polymer contains two layers of charges: a layer of positive charges near the cathode and a layer of negative charges near the anode.

The time evolution can be observed, for instance, in Fig. 10(a), where the peak Q− is plotted at various experimental times. This peak amplitude increases with time, and the minimum peak shifts further inside the polymer.

FIG. 10.

(a) Time evolution of negative space charge (Q−) and (b) resulting peak load vs time integration.

FIG. 10.

(a) Time evolution of negative space charge (Q−) and (b) resulting peak load vs time integration.

Close modal
Because the time evaluation of the charge is suspected to play a role in polymer electrostriction, the surface charge density Qs was calculated by integrating the space charge density across the entire sample thickness38 
Q S = ρ ( x ) d x .
(16)

This equation was used to estimate the surface charge Q− (or Q+) by integrating only this peak (or peak Q+).46 The resulting values are plotted in Fig. 10(b) along with the total charge Qtot = Q− + Q + . These are observed to increase (in the absolute value) over time. These curves were fitted with Eq. (6), and the same charging behavior described by the law (also with N = 2) was observed. The primary contribution to Qtot is attributed to Q−. The two fitted time constants were τ1 = 4 s and τ2 = 143 s. These time constants are consistent with those of the electric current.

The time dependences of both the electric current and space charge were similar, implying that both measurements were probing the same phenomenon.

Figure 11(a) illustrates the mechanism of the Maxwell strain by depicting a dielectric polymer with electrodes under a voltage. In this mechanism, the interfaces (i.e., electrodes) become charged and interact with the electric field (E0) to produce an effective pressure (p = ɛ0ɛrE02).47 This pressure results in strain, as expressed by Eq. (1).

FIG. 11.

Dielectric materials (a) without and (b) with space charges.

FIG. 11.

Dielectric materials (a) without and (b) with space charges.

Close modal

The behavior of polymers exposed to a high electric field differs significantly from that of a simple capacitor, as depicted in Fig. 11(a). Generally, PU polymers, like most other polymers, are amorphous, or at least have a low crystallinity ratio. Consequently, they exhibit a greater degree of disorder in their structure, and the electronic structure is different from that of crystal insulators.41,48 They have localized electronic levels within the insulator gap. This significantly affects electrical conduction in polymers subjected to high electric fields.

For instance, research is being conducted on polyethylene (PE) and its variations such as low-density polyethylene (LDPE), high-density polyethylene (HDPE), and cross-linked polyethylene (XLPE) to understand their deterioration and breakdown with time and high electric field because they are used as electrical cable insulators.24 The lifetime of these materials has been shown to be affected by the creation of a large space charge layer. This layer is determined by a combination of internal parameters (such as mobility) and the injection of charges (holes and electrons), which depend on both the electrode material and the dielectric polymer.23 

These space charges alter the dielectric and electrical characteristics of PE. High-field investigations of several polymeric insulating materials such as PE have demonstrated that the high-field threshold between linearity and non-linearity in the J–E plot can be attributed to the charging effect.49 The same threshold was indeed observed as a change in slope in the current density vs the voltage and as a sudden increase in charge vs voltage, measured with the PEA method. The presence of substantial space charges alters the local field, affecting the conduction.

The space charge also influences the dielectric properties: a slight increase in the electric capacitance has also been observed. For instance, the dielectric constant of PE was measured as ɛi = 2.23ɛ0 and ɛ = 2.25ɛ0 below and above the threshold, respectively.42,50 Therefore, the permittivity increase δɛ is relatively small with δɛ/ɛi = (2.25−2.23)/2.23 ∼ 1%. This is presented in Fig. 8. Measurements performed by Sakamoto provided the electric field threshold for both the permittivity and the conductivity, and it was shown that they occurred at the same field value. The injected space charges, therefore, increase the dielectric constant even for a small change of approximately 1%. The 1% can be explained by the fact that trapped charges are located in a very thin layer close to the electrodes, so the sample permittivity obtained through a volume average should yield a small change in this quantity. In our experiments, the noise was too significant to detect a comparable alteration of the dielectric constant in the PU, as shown in Fig. 7. However, previous investigations have reported the ɛ-vs-E curve in PU,14,51 and a relatively minor shift of approximately 1% was observed for two different PUs. This is shown in Fig. 8. The shapes of the ɛ-vs-E curves for both PU14,51 and PE42,50 are analogous, involving a constant value ɛ for E < ET and a larger constant value ɛ + δɛ for E > ET with δɛ > 0.

The alteration of the dielectric constant can be attributed to the formation of space charge layers resulting from charge injection at high electric fields. This has been observed in various polymers.49 Sakamoto50,52 estimated the increase in the capacitance of a PE sample with thickness d and area S, with a single space charge layer with thickness l, to be
C C 0 C 0 = ( S / ( d l / 2 ) ) ε ( S / d ) ε ( S / d ) ε = l 2 d l ,
(17)
where C0 and C represent the capacitances for E < ET and for E > ET, respectively, with C > C0. For a layer with small thickness l, the relative change is small. Reference 52 also mentions that the charge layer was thinner in the AC case than in the DC case. This might be a reason for the difference in threshold measured for DC (Fig. 2) and AC (Fig. 7) above.
Another model, presented in Ref. 53, involves the injection of electric charges through a corona discharge in a polypropylene (PP) with a thickness of l. This model consists of two charge layers with a thickness of t and charge densities of ±ρ (where ρ is negative). The calculated capacitance is
C = S ( l / ε ) + ( ρ / ε σ 0 ) t ( l t ) ,
(18)
so C(ρ ≠ 0)> C(ρ = 0). Interestingly, the PP charged by a corona discharge exhibited a larger electrostriction coefficient than the non-charged PP,54 which is another confirmation that the charge layers inside the polymer played a role in its electrostriction.28 In these cases, the presence of internal charges increased the polymer capacitance, whether with one [Eq. (17)] or two layers [Eq. (18)].

The increase in dielectric capacitance of the polymer is consistent with greater electrostriction, as seen in PP54 or PU.14,51 The presence of the charged layer amplifies the local electric field [E(x) = E0 + Eρ(ρ(x))], which affects the body force Fv = ρ(x) E(x). Measurements performed by Watanabe revealed a link between the charge layer and the bending effect.25 Indeed, a study on two PU films has shown that materials with a charged layer near one electrode can cause polymer bending towards the non-charged electrode. One PU was easily electrochemically oxidized and exhibited a significant positive space charge layer near the cathode. This charged layer stretched laterally in response to an external electric field,19,55 causing the entire film to bend towards the anode side. The second PU, which was reduced, produced a large negative charge layer near the anode and was then bent toward the cathode. The PU chemistry thus triggered the type of injection25 and the bending direction. Therefore, the illustration in Fig. 11(b) with one or more layers of internal charges is a better image for the high-field phenomenon than the conventional image in Fig. 11(a), because all charges, both at the interface (electrodes) and in the bulk, must be considered.

In this study, we examined the electric properties of PU at high electric fields. We conducted experiments using either a DC step voltage or a sinusoidal AC voltage. The results revealed that the conduction current behaves as I = V0c with c = 0.98 or 1.80 depending on the amplitude of this applied field. We defined a threshold ET for the applied electric field between two conduction mechanisms: the current was linear for E0 < ET and almost quadratic for E0 > ET. The threshold field was approximately 7 × 106 V/m for DC measurements and approximately 1 × 107 V/m for AC measurements. Such a transition has been reported for other polymers exposed to high electric fields but never measured for PU.

At this threshold ET, the injection of electric charges is reported to become gradually stronger compared with thermally generated and/or already existing free charges in the polymer. These charges are trapped by physical and/or chemical defects. These trapped charges create space charge layers. Here, two layers were observed through the PEA technique in PU at E0 ∼ ET close to the electrodes. For E0 < ET, the electrical conduction remained constant, whereas for E0 > ET, the electrical current was enhanced and decayed with time.

This threshold is significant for understanding electrostriction. The findings presented here suggest that for a field E0 ∼ ET, the trapped charges contribute to the polymer's deformation. Therefore, this result provides physical meaning to electrostriction as an effect of charge injection. Our previous simulation work on PU bending, which relied solely on electric carrier diffusion and did not consider injection, is still relevant because all values of E0 used for the simulations were in the domain j ∼ E0.19 

Future work will investigate controlling the amount of injected charges (holes/electrons). A simple way is to use different metals for the electrodes. Their work function is a key parameter for tuning the barrier energy of the metal/semiconductor interface. According to the Schottky process, this would change the amount of injected charges near the electrodes.

The PU threshold may be of interest to the community studying the behavior of polymers in a high electric field. The molecular formula of PU is quite different from that of the PE family, so having a different polymer threshold may help in understanding the general behavior of polymers for electroactive applications.

The authors acknowledge Professor Hiroaki Miyake for his support with PEA measurement at Tokyo City University. Mark R. Kurban from Edanz (https://www.edanz.com/ac) edited a draft of this paper.

The authors have no conflicts to disclose.

Gildas Diguet: Conceptualization (lead); Formal analysis (lead); Investigation (equal); Methodology (equal); Writing – original draft (equal). Jean-Yves Cavaillé: Methodology (equal); Validation (equal). Gildas Coativy: Methodology (equal); Validation (equal). Joel Courbon: Methodology (equal); Validation (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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