Abstract and Keywords
The physical properties of ferroelectric polymers, like polyvinylidene fluoride and its copolymers are briefly discussed. It is shown that polar nanoclusters are induced either by compositional disorder, as in the case of the terpolymer P(VDF‐TrFE‐CFE) or by electron irradiation as in P(VDF‐TrFE) giving rise to relaxor behaviour. A spherical random‐field–random‐bond model of relaxor polymers is presented and the electrostrictive coefficients are calculated as well as the linear and non‐linear dielectric properties. It is shown that the main difference between inorganic and polymeric relaxors is the physical character of polar nanoregions, the nature of the interaction between them and the origin of the random fields.
Electroactive polymers with a high strain response to an applied electric field are attractive for a broad range of applications, such as ultrasonic transducers for medical diagnostics, sonars, robots, and artificial muscles. Typical representatives are the electron‐irradiated ferroelectric copolymer poly(vinylidene fluoride‐trifluoroethylene) or P(VDF‐TrFE) and its terpolymer with chlorofluoroethylene P(VDF‐TrFE‐CFE). High‐energy electron irradiation is needed in order to transform the normal ferroelectric structure into an amorphous one, thus strongly enhancing their electromechanical properties. Recently, it has been demonstrated that these polymers exhibit physical properties, which are analogous to inorganic relaxor ferroelectrics such as PMN, PLZT, etc. [7.1]. Similar to their inorganic counterparts, relaxor‐like polymer systems are characterized by slow relaxation and a strong frequency dispersion of the dielectric permittivity. In the low‐frequency limit, a broad temperature maximum of the dielectric response is observed, and the longest relaxation time diverges at the freezing temperature T f according to the Vogel–Fulcher law: .
Following the physical analogy between inorganic relaxor ferroelectrics and relaxor polymers, a theoretical model has been proposed based on the assumption that electron irradiation creates a random set of polar nanoregions (PNRs) embedded in an amorphous matrix [7.2]. Meanwhile, in terpolymers disorder is introduced by the fluctuations of chemical composition alone, and PNRs appear without the need for irradiation.
Ferroelectricity is a property of well‐ordered solids that belong to one of the ten pyroelectric crystal classes. It is therefore surprising that ferroelectric polymers (Fig. 7.1) exist too [7.3–7.5]. The best known ferroelectric (FE) polymer is polyvinylidene fluoride (abbreviated as PVDF) [7.3–7.6].
The characteristic properties of this system are:
i) PVDF is a semicrystalline polymer with ≈ 50% crystallinity. The monomer is strongly polar [‐CH2‐CF2‐] and contains in addition to the linear carbon backbone two dipolar units, one associated with the ‐CH2 and the other with the ‐CF2 unit (Fig. 7.2).
ii) There are many modifications of PVDF. The most important are the anti‐polar α‐phase (C2h) and the polar β‐hase (C2v) both dispersed in the amorphous phase (Fig. 7.2). By stretching, the non‐polar α‐films (C2h) can be transformed into the polar β‐phase (orthorhombic C2v) with two parallel chains per unit cell.
for a ferroelectric to paraelectric (FE to PE) phase transition. In many cases the polymer melts before this condition is reached. This situation is changed in copolymers VDF/TrFE that exhibit the ferroelectric transition below the melting point T M. Here, TrFE [‐CF2‐CFH‐] stands for trifluoroethylene and TeFE for tetrafluoroethylene [‐CF2‐CFH‐]. The TeFE and TrFE molecules are randomly distributed in the polymer and can be viewed also as defects. These copolymers(p.189) are important technologically, e.g., as active piezoelectric films. Films can be produced directly from the melt and no stretching is necessary. The degree of crystallinity is higher than in pure PVDF and so are the piezo‐ and pyroelectric coefficients.
iv) The polymers are highly compressible and hydrostatic pressure is expected to produce large changes in their properties [7.7]. As can be seen from Fig. 7.3, pressure shifts the dielectric peaks to higher temperatures. This can be understood in terms of a closer packing of the polymer chains that hinders the motion necessary for the FE to PE transition. Therefore, higher temperatures are necessary to induce this motion. At still higher pressures there is a large decrease in the area under the ε″ peak, demonstrating a gradual disappearance of the PE to FE motion.
v) Recent dielectric experiments have revealed a striking analogy between inorganic relaxor systems and disordered organic polymers. We show that a statistical ensemble of randomly oriented polar nanoregions embedded in an amorphous matrix can be well described by the SRBRF model, which is exactly soluble [7.2]. The main difference between the SRBRF model for inorganic relaxors and the model for polymer ferroelectrics is in the physical characteristics of polar nanoregions and the nature of interaction among them. The calculated third‐ order non‐linear susceptibility indeed predicts a cross‐over between paraelectric behaviour at high temperature and a relaxor‐like behaviour near the freezing temperatures.
These results are typical for other ferroelectric polymers too. This is particularly true for the odd [n = 7, 11, etc.] nylons that also exhibit ferroelectric transitions (Fig. 7.4).
Odd polyamides – or nylons – are semicrystalline polymers with a hydrogen‐bonded sheet structure. The dipole moments are perpendicular to the chains and point in the same direction in all the sheets. The T dependence of the D vs. E hysteresis loop of nylon 11 is shown in Fig. 7.5.
7.1 2D ferroelectricity
Another important development in the field of FE polymers has been the discovery of ferroelectricity in Langmuir–Blodgett (L–B) VDF‐TrFE polymer films [7.12]. These films are relatively easy to make and have a thickness of 2 to 500 monolayers. The films have excellent order and are much better suited than semicrystalline thick films to study the FE properties.
In a crystalline Langmuir‐Blodgett deposited random copolymer of vinylidene fluoride with trifluoroethylene P(VDF – TrFE 70:30) on graphite, switchable ferroelectric films can be made to a 0.9 nm thickness. The minimum thickness is just two monolayers. Finite size effects thus seem here to impose no practical limitations on thin‐film memory capacitors though for some designs tunneling currents may become too large.
In P[VDF – TrFE] 70:30 crystalline films the first‐order phase‐transition temperature T c at 108°C on heating and 77°C on cooling was found to be nearly equal to the bulk value, even in films as thin as 10 Å [7.12]. This has been attributed to two‐ dimensional ferroelectricity where the ferroelectric state is generated only within the plane of the film. The absence of finite‐size effects when the film thickness is decreased from 150 Å to 10 Å seems to support this conclusion.
The interphase coupling here should be rather weak and the system can be described by the anisotropic Ising model. Fluctuations in two dimensions should not destroy ferroelectricity – as in the case of an isotropic Heisenberg ferromagnet – because of the anisotropy of the coupling.
The low‐temperature second‐order phase transition at T c ≈ 20°C in P(VDF – TrFE) films of 30 monolayers or less can be described as a surface layer transition controlled by the interaction with the substrate or the top electrode.
The structure of a L–B P(VDF0.7TrFE0.3) film is shown in Fig. 7.6.
7.2 Spherical model of relaxor polymers
Let us now treat the SRBRF model of relaxor polymers in greater detail.
7.2.1 Polar nanoregions
We consider the case of a disordered ferroelectric polymer, characterized by a lamellar crystalline structure surrounded by an amorphous configuration of polymer chains [7.3]. The chains are assumed to be twisted and broken into nanosized segments of variable lengths by either irradiation as in P(VDF‐TrFE) or by compositional disorder (p.192)
In a simplified description, we will think of a polarized nanoregion as a compact lamellar array of polar nanosegments, with a net dipole moment perpendicular to the segments [7.3]. The magnitude of the dipole moment depends on the volume of the region and on the degree of ordering of the dipoles, and can thus vary over a wide range. Since the orientations of the dipole moments will also change across the sample, we are dealing with a statistical ensemble of randomly oriented dipoles embedded in a continuous amorphous medium. With the ith nanoregion, I = 1,2,…, N, we therefore associate a vector , where the length of the dipole moment is represented by the dimensionless scalar variable S i, and its orientation is described by an m‐dimensional unit vector , where m is the effective dimensionality of the embedding medium. For N 〉〉 1, we can assume a continuous probability distribution of S on the interval (0, ∞). For convenience, we choose the generalized gamma distribution having the form
The first two moments are ⟨S⟩ = 21/2 aΓ((m+1)/2)/Γ(m/2) and ⟨S 2⟩ = ma 2. Returning to the discrete case we find . We can now rescale the fields and obtain
This is equivalent to a generalized spherical condition for the m‐component order parameter field with the width of the corresponding distribution ρ(S) equal to unity. We will, therefore, describe the relaxor polymer by the m‐component SRBRF model in which the order‐parameter field satisfies the spherical condition (3).
7.2.2 Free energy
The free energy of a quenched disordered system described by the spherical model can be calculated exactly either by using the eigenvalue spectrum of the random interaction matrix or by applying the replica method [7.16].
The Hamiltonian of a relaxor polymer is now written as
As usual, J ij are quenched random interactions between the polar nanoregions i and j, which are infinitely ranged and have a Gaussian probability distribution such that
In the last term of Eq. (4), h iμ are components of local random fields with zero mean and second cumulant
E μ is the applied field, and g an effective dipole moment. When E = 0, the model is characterized by three parameters, namely, J 0, J, and Δ.
In inorganic relaxors, compositional fluctuations give rise to so‐called chemical clusters, i.e. chemically ordered regions with a net non‐zero electric charge, which act as sources of random electric fields. This is to be contrasted with relaxor polymers, where such charged regions are less likely to be created by the random disorder. Therefore, we expect that random electric fields will be weaker in polymers. On the other hand, the breaking up of the layered ferroelectric structure results in large strain fields, which have a different symmetry from the electric fields, but may also affect (p.194) the polar nanoregions through piezoelectric and electrostrictive coupling. Thus, it is not surprising that strong electrostriction is observed in some polymer composites.
For a general direction of the electric field , where we choose E 1 = E and E 2 = … = E m = 0, we obtain three types of thermodynamic averages, namely
The μ ≠ ν averages are zero by symmetry.
It is well known that for a random spherical model the replica symmetric solution is exact. Thus we may choose the longitudinal (L) and transverse (T) components of the order parameters (8)–(10) as follows [7.17, 7.18]:
7.2.3 Order parameters
The equilibrium values of the order parameters q l,t,r l,t, and P are determined by the saddle point conditions ∂f/∂q L,T = ∂f/∂r L,T = ∂f/∂P = 0.
The saddle‐point conditions now lead to the following relations:
and thus B l = B T ≡ B, with B to be determined below.
The field‐cooled polarization is given by the equation
The order parameters q L,T and r l,t are in general the diagonal elements of m × m matrices q and r, respectively. It is convenient to consider the corresponding invariants
Analogous results have been derived earlier using symmetry arguments for inorganic relaxors (m = 3) in a field  and for [7.19].
It should be noted that in analogy to spin and dipolar glasses the quantities q and r are not thermodynamic order parameters, and thus cannot be derived by minimizing the free energy, but rather follow from the saddle‐point conditions employed in evaluating the free energy.
7.3 Dielectric susceptibility
7.3.1 Longitudinal and transverse susceptibilities
The linear longitudinal static field‐cooled susceptibility χ 1l of a relaxor polymer is obtained by differentiating the physical polarization P = (g/υ 0) with respect to the field E, where we introduce υ 0 = V/N as the effective average volume of a polar nanoregion. Equation (23) yields
The corresponding transverse susceptibility χ 1,T follows from the free energy by differentiating with respect to an infinitesimal field along any of the perpendicular directions μ ≠ 1, say, E 2,
and from Eq. (24) we obtain the relation
In Fig. 7.7, the calculated temperature dependence of the static field‐cooled longitudinal susceptibility χ 1 ≡ χ 1, L and the corresponding order parameter q obtained from Eq. (22) are shown for J 0/J = 0.9 and three representative values of the random field parameter Δ/J 2. Figure 7.8 shows a comparison between the experimental and theoretical values of the static field‐cooled dielectric constant ε s = 1 + χ 1/ε 0 in electron‐irradiated copolymer P(VDF‐TrFE) [7.20] and in terpolymer P(VDF‐TrFE‐ CFE) [7.21]. The solid lines were calculated from Eq. (24) and fitted to the data using a set of fit parameters listed in the caption, and by adjusting the amplitude of χ 1. The agreement between the experimental and theoretical values is reasonably good. It should also be noted that the fit values J/k = 276 ± 5 K and 266 ± 16 K are close
In vector spin glasses, the transverse order parameter q T vanishes above the so‐ called Gabay–Toulouse (G–T) line in the (E, T)‐plane [7.22]. In the present case, one has q T ≠ 0 at all temperatures and fields due to the presence of random fields, and there can be no GT line. However, even without random fields (Δ = 0) it can be shown that there is no GT line in the spherical model.
(p.198) To illustrate the above point further, let us consider the simpler case Δ = 0 and J 0 = 0, but E≠0. Let us assume that q T ≠ 0. According to Eq. (18), in which now q ̃l,t = q l,t, we have or
Since r l − q l = r T −q ?, we see that r L − q L = 1/(βJ), too, and Eq. (18a) cannot be fulfilled unless p = E = 0. Thus, for E ≠ 0 we cannot have q T = 0 at any temperature, and there can be no transition from q T = 0 to q T ≠ 0, i.e. no GT line.
It should finally be noted that a general feature of classical continuous models – and hence of the SRBRF model – is a negative value of the entropy in the limit T → 0. Thus, the model predictions should be treated with some caution on approaching the zero‐temperature limit.
7.3.2 Spontaneous polarization
A second illustrative example is the case E = 0 with both Δ ≠ 0 and J 0 ≠ 0. For there is no spontaneous polarization at any temperature. By symmetry we have r T = r l = 1 and q T = q L = q, where q is the solution of
We can show that for long‐range order (LRO) exists and the spontaneous polarization is non‐zero at T 〈 T c.
The E–A order parameter at T 〈 T c is obtained from Eq. (19) as
and the spontaneous polarization follows from Eq. (22) as
The critical temperature T c is, therefore, given by
The spontaneous polarization at T = 0 is
This shows that LRO exists only for
We introduce the third‐order non‐linear longitudinal susceptibility χ 3 by the usual relation P = χ 1 E χ 3 E 3, where the indices L have been omitted. By performing the derivatives of q and P with respect to E in Eqs. (22) and (23), respectively we obtain in the limit E → 0
This expression corresponds to the static field‐cooled third‐order non‐linear susceptibility of a relaxor. It is useful to introduce the scaled third‐order non‐linear susceptibility , [7.23], i.e.
By determining a 3 experimentally, we can simply discriminate between the para‐ electric, ferroelectric, and relaxor behaviour of a given system. For example, in a mean‐field‐type ferroelectric χ 3 diverges, but a 3 is finite at T c, whereas in a relaxor we expect the same type of behaviour to occur in both χ 3 and a 3.
It should be emphasized that the above model of a relaxor must include random bonds – i.e. a non‐zero parameter J– in addition to random fields in order to describe the anomaly in the scaled non‐linear response.
The temperature dependence of the static third‐order non‐linear response is shown in Fig. 7.9 for various values of the random field strength Δ/J 2, and for J 0 = 0. As is known from the theory of dipolar glasses and inorganic relaxors, for small values of Δ/J 2 the non‐linear susceptibility has a nearly divergent behaviour at the freezing temperature T f ≅ J/k, whereas in spin glasses (Δ = 0)χ 3 actually diverges as ≈ ǀT − T cǀ−1 at the glass‐transition temperature T g = J/k. According to the Landau theory for homogeneous ferroelectric systems, χ 3 should diverge at the critical temperature T c as χ 3 ≈ ǀT − T cǀ−4. This type of behaviour is also found in the present model for . For a trivial paraelectric system with , we see that χ 3 ≈ T. On the other hand, in the general case of a relaxor polymer with all parameters J, J 0, and Δ non‐zero and , we find for T 〉〉 J/k that q 〈〈 1, and Eq. (36) predicts a linear temperature dependence at high temperatures, similar to the paraelectric case.
The low‐frequency third‐order non‐linear dielectric response χ 3 and the corresponding scaled response a 3 have been measured in the electron‐irradiated relaxor copolymer P(VDF‐TrFE) [7.20] and in the terpolymer P(VDF‐TrFE‐CFE) [7.21]. Specifically, in P(VDF‐TrFE) the non‐linear susceptibility χ 3 shows a peak near ~ 300 K, whereas the scaled non‐linear response a 3 shows a cross‐over between the paraelectric behaviour a 3 ~ T at high temperatures and a relaxor‐type behaviour a 3 ~ (T − J/k)−1 near ~ 300K [7.20]. Figure 7.10 shows the experimental data for a 3 in electron‐irradiated P(VDF‐TrFE) [7.20], which is in qualitative agreement with the theoretical predictions. (p.200)
It has recently been shown that relaxor polymers and their all‐organic composites exhibit a high value of electrostriction, namely large mechanical strains can be generated by application of an electric field. Typically the copper phtalocyanine (CuPc) oligomers have been dispersed in the electrostrictive copolymer P(VDF‐TrFE) matrix, leading to a high net dielectric constant, while retaining the elastic modulus of the copolymer as well as its flexibility [7.24]. It has been suggested that the ultrahigh strain response in relaxor polymers is due to the expansion of polar nanoregions under an electric field, coupled with a large difference in the lattice strain between the polar and unpolar phases [7.25]. Alternatively, an exchange coupling between CuPc and the copolymer matrix could lead to a dramatic enhancement of the electrostriction and dielectric constant in the CuPc‐P(VDF‐TrFE) composite [7.26, 7.27].
Here, we consider a three‐dimensional system with Cartesian indices μ, ν,…, = 1, 2, 3. The electrostrictive coefficients Qμνκλ are defined by the relation
where u μν is the strain tensor and P κ, etc., are components of the dielectric polarization induced by the applied field, i.e.
with χ μν representing the longitudinal field‐cooled dielectric susceptibility tensor.
The electrostrictive coefficients can be calculated from the thermodynamic Maxwell relation
where are components of the inverse susceptibility tensor and p κλ of the stress tensor. From Eq. (24) we have
At constant polarization P, the E–A order parameter q is independent of J 0 according to Eq. (22). The stress dependence of χ μν is expected to arise from the parameters J 0 and g 2 /υ 0. In the following we will limit ourselves to the case of hydrostatic pressure, where p μν = −pδ μν. At high temperatures, the static dielectric constant ε s behaves asymptotically as ε s ~ C/(T − T 0), where T 0 = J 0/k plays the role of an effective Curie–Weiss temperature and C = g 2/(kε 0 υ 0) is the corresponding Curie constant. Thus, by measuring dε s/dp at high temperatures an estimate for dT 0/dp and hence for dJ 0/dp can be obtained.
The first term is expected to be dominant in inorganic relaxors of PMN type [7.29] where dT 0/dp 〈 0 and Q h 〉 0 [7.30]. In electroactive polymers, such as P(VDF‐TrFE), however, we have Q h 〈 0 [7.28], and the pressure dependence of the parameters υ 0 and g must obviously be considered. We assume that dυ 0/dp 〈 0, i.e. , where B is the bulk modulus. Adopting the value [7.31] B ≅ 0.8 × 109 N/m2 as well as [7.32] ε s ≅ 25 for 60 Mrad irradiated conventional electrostrictive polymer P(VDF‐TrFE) at room temperature, we can estimate the contribution of the second term in Eq. (42) to be of the order Q h ~ −2.8m4/C2. A comparison with the experimental value [7.28] Q h = −6m4/C2 suggests that the contribution of the last term in Eq. (42) should also be included; however, there is no estimate of the dg/dp available at this time. The above value of Q h is roughly two orders of magnitude larger than in inorganic relaxors [7.30], and Q h is hence referred to as ‘giant electrostriction.’
The main differences between the present model and the SRBRF model of inorganic relaxors are in the physical character of polar nanoregions, the nature of the interactions between them, and the origin of the random fields. Formally, the two models are equivalent; however, the values of the physical parameters characterizing the model can vary to a significant degree. For example, relaxor polymers are usually softer than the inorganic systems and thus have a greater value of compressibility which strongly affects the hydrostatic electrostriction constant.