Lecture 7-1 Transduction Based on Changes in the Energy Stored in an Electrical Field - Electrostriction
The electrostrictive effect is a quadratic dependence of strain or stress on the polarization P or electric field. This is an universally existing phenomenon in any Dielectric Materials (ceramics and polymers) whether it has center of inversion or not, S = Q P 2 where Q is the charge-related electrostrictive coefficient. More explicitly, S ij = Q ijkl P k P l For an isotropic polymer, S 3 = Q 33 P 2 and S 1 = Q 13 P 2 S 3 and S 1 are the strains along and perpendicular to the polarization direction, known as the longitudinal and transverse strains, respectively. For an isotropic polymer, experimental evidence and theoretical consideration indicate that Q 33 <0 and Q 13 >0. Hence, for a polymer, an increase in polarization will result in an contraction along the polarization direction. Generalized Displacement vs. voltage behavior of Piezoelectric PZT (dash line) and Electrostrictive PMN (solid line) actuators
For a linear dielectric polymer, the polarization is related to the dielectric permittivity as P = (ε - ε 0 ) E ε 0 is the vacuum dielectric permittivity (=8.85x10-12 F/m). The equation can be converted into S = Q (ε - ε 0 ) 2 E 2 = M E 2 For an isotropic solid, M is knownas the electric field-related electrostrictive coefficient. The longitudinal strain and the transverse strain S 3 =M 33 E 2, and S 1 =M 13 E 2. For an isotropic polymer, M 33 <0 and M 13 >0. i.e., it will contract along the thickness direction and expand along the film direction when an electric field is applied across the thickness.
For an electrostrictive polymer, one can apply a DC electric bias field and super-impose on it a smaller AC field. When the DC field is much larger than the AC field, the AC strain response term is S= 2 Q P D P P D is the DC bias field induced polarization P is the polarization change induced by the small AC field Under weak AC fields, the polarization response is dominantly linear, P=(ε ε 0 ) E, which leads to S= 2 Q P D (ε ε 0 ) E This Equation is valid for both linear and non-linear dielectric materials as long as E is small.
The linear relationship between the strain and field in the above equation resembles piezoelectric effect, Therefore, one can introduce an effective piezoelectric coefficient here for DC field biased electrostrictive polymers d = 2 Q P D (ε ε 0 ) Using the phenomenological theory, it can be shown that in DC field biased state, d coefficient in describes both the converse and direct piezoelectric effects. For an isotropic polymer, we have d 33 = 2 Q 33 P D (ε ε 0 ) d 31 = 2 Q 13 P D (ε ε 0 ) where d 33 and d 31 are the piezoelectric coefficients along and perpendicular to the induced polarization direction (or the DC bias field direction). For an isotropic polymer, there is only one independent dielectric permittivity ε.
For electromechanical applications, electromechanical coupling factor k, which measures the efficiency of the material in converting energy between the electric and mechanical forms, is one of the most important parameters 2 k = converted mechanical energy input electrical energy 2 k = converted electrica energy input mechanicall energy In electromechanical applications, an electric field is applied along certain direction and the electromechanical actuation along the same direction or other direction is used. The coupling factor depends on the direction of electric field and the mechanical strain (or stress) direction. Hence, there are many electromechanical coupling factors, corresponding to different combination of the directions of electric and mechanical variables.
For example, an actuator can be made with the electric field along the 3-direction and that the actuation along the same direction is used. In this case, the coupling factor is the longitudinal electromechanical coupling factor k 33 k 332 = d 332 /(ε 33T s 33E ) On the other hand, if the mechanical actuation is along the direction (the 1-direction, for instance) perpendicular to the applied electric field (the 3-direction, for instance), the coupling factor now is k 31 k 312 = d 312 /(ε 33T s 11E ).
Phenomenological descriptions x The fundamental equations governing electrostriction can be derived from the expansion of the thermodynamic full Gibbs free energy function with independent variables of stress and electric field G(X,E), neglecting higher order terms: 1 1 G( X, E) = s X X E E M E E X 2 2 η ij s ijkl η mn M mnij G = X ij ijkl ij kl mn m n mnij m n ij is the elastic compliance with tensor notation is the linear dielectric susceptibility is the electrostriction coefficient with tensor notation. E, T P m G = E m X, T
The constitutive equations for electrostrictive materials can be obtained as P = η E + 2M E X m mn n mnij n ij x ij = sijkl X kl + M mnij Em En At zero mechanical stress, the electrostrictive strain becomes, xij = Mmnij Em En
A set of similar constitutive equations can also be derived by using thermodynamic elastic Gibbs free energy function with independent variables of stress and polarization G 1 (X, P), neglecting higher order terms, 1 1 G1 ( X, P) = sijklxij Xkl + α mnpm Pn QmnijPm Pn Xij 2 2 α mn Q mnij is linear reciprocal dielectric susceptibility is electrostriction coefficient with tensor notation. x ij G = 1 X ij P, T E m G = 1 P m X, T
The constitutive equations for electrostrictive materials can then be obtained as x ij = sijkl X kl + Qmnij Pm Pn E = α P 2Q P X m mn n mnij n ij At zero stress, the electrostrictive strain become xij = Qmnij Pm Pn For linear material, the electrostriction coefficients M and Q have the relationship, M = Q η η where η is dielectric susceptibility. mnij opij om pn
There are two converse electrostriction effect. The first converse effect of electrostriction is described by Q mnij = 1 2 α X mn ij and M mnij = 1 2 η X The second converse effect of electrostriction is the electric field dependence of the piezoelectric coefficient d mij 1 d 1 bmij mij M = and Q mnij mnij = 2 E 2 n Pn The electric field dependence of the piezoelectric coefficient d mij dx de ij m = d = 2M E ijm ijmn n d = η g = 2η Q P ijk mk ijm mk ijmn n, dx dp ij m mn ij = g = 2Q P The change of piezoelectric coefficient b nij with polarization ijm ijmn n
If we compare the above electrostrictive constitutive equations with the linear piezoelectric constitutive equations, it can be seen that electrostriction is the origin of piezoelectricity in ferroelectric materials such as barium titanate BaTiO 3, lead zirconate titanate Pb(Zr x Ti 1-x )O 3 (PZT), relaxor ferroelectric PMN-PT, as well as polymeric ferroelectric PVDF and copolymers, etc. In a ferroelectric material that exhibit both spontaneous and induced polarizations, the total strains in the ferroelectric materials arising from spontaneous polarization, piezoelectricity, and electrostriction may be expressed as, x = Q Ps Ps + 2Q Ps P + Q P P ij ijkl k l ijkl k l ijkl k l Ps k and Ps l are spontaneous polarization P k and P l induced polarization In ferroelectric state, the piezoelectric coefficient can be written as, d ijk = 2Qijmnη mk Psn
The electrostriction tensor is a fourth rank tensor that may be cast in matrix form For crystals of PMN having the point group m3m x x x x x x 1 2 3 4 5 6 Q11 Q12 Q12 0 0 0 Q12 Q11 Q12 0 0 0 Q12 Q12 Q11 0 0 0 = 0 0 0 Q44 0 0 0 0 0 0 Q44 0 0 0 0 0 0 Q 44 P1 2 P2 2 P3 2 P2 P3 P3 P1 P P 1 2 This matrix contains three independent elements, while that for an isotropic material is of identical form, but with one less independent element, as in: ( ) Q = 4Q = 2 Q Q 44 1212 11 12
Electrostrictive Actuators Electrostrictive (such as PMN, PMN-PT, etc.) actuators are solid state actuators similar to Piezoelectric PZTs. Although sometimes advertised as a recent discovery, the material has been around for many years. PMN Electrostrictive actuators are made of a lead-magnesium-niobate (PMN) ceramic material. PMN is a non-poled ceramic with displacement proportional to the square of the applied voltage (more accurately to the square of the polarization). PMN unit cells are centro-symmetric at zero volts. An electrical field separates the positively and negatively charged ions, changing the Dimensionss of the cell and resulting in an expansion. Electrostrictive actuators are operated above the Curie temperature which is typically very low when compared to PZT materials.
Electrostrictive Actuators PMN electrostrictive actuators exhibit less hysteresis (on the order of 3 %) than PZT actuators, in a limited temperature range. Despite of the reduced hysteresis, they provide highly nonlinear motion because of the quadratic relationship between voltage and displacement. Also, they cannot be used in a bipolar mode with reduced electric field strength (see because reversing the electric field does not result in contraction. Furthermore, PMN actuators show an electrical capacitance four to five times as high as piezo actuators requiring significantly higher driving currents for dynamic applications.
Electrostrictive poly(vinylidene fluoride-trifluoroethylene) copolymers Polarization loop for P(VDF-TrFE) 65/35 mol% film before and after irradiation Z. -Y. Cheng et al. Sensors and Actuators A: Physical, Volume 90, Issues 1-2, 1 May 2001, Pages 138-147
Electrostrictive poly(vinylidene fluoride-trifluoroethylene) copolymers Amplitude of the longitudinal strain as a function of the amplitude of the electric field at 1 Hz: (a) Strain measured at room temperature for the 65/35 mol% copolymer films irradiated with 2.55 MeV electrons. Curves A and B correspond to the stretched film irradiated at 120 C with 80 Mrad dose and unstretched film irradiated at RT with 100 Mrad dose, respectively; (b) Strain measured at different temperatures, which are indicated in figure, for unstretched 68/32 mol% copolymer film irradiated at 105 C with 70 Mrad dose of 1.0 MeV electrons; and (c) The temperature dependence of the longitudinal strain amplitude for the film shown in (b) under a constant electric field of 150 MV/m.
Electrostrictive poly(vinylidene fluoride-trifluoroethylene) copolymers Amplitude of the transverse strain along stretching direction as a function of amplitude of the electric field at 1 Hz for stretched samples: (a) curves A, B, C, and D correspond to the strain measured at room temperature for copolymer films under different irradiation conditions, A: 68/32 mol% film irradiated at 100 C with 65 Mrad doses of 1.2 MeV electrons, B: 65/35 mol% film irradiated at 105 C with 70 Mrad dose of 1.0 MeV electrons, C: 65/35 mol% film irradiated at 95 C with 60 Mrad dose of 2.55 MeV, and D: 65/35 mol% film irradiated at 77 C with 80 Mrad dose of 2.55 MeV; (b) strain measured at different temperatures, which are indicated in figure, for stretched 68/32 mol% copolymer film irradiated at 100 C with 70 Mrad dose of 1.2 MeV electrons.
Electrostrictive poly(vinylidene fluoride-trifluoroethylene) copolymers (a) (b) Longitudinal strain measured at room temperature vs. P 2 for unstretched 68/32 mol% copolymer film irradiated at 105 C with 70 Mrad dose of 1.0 MeV electrons, where P is the polarization; Electrostrictive coefficient Q 13 vs. electrical field amplitude measured at 22, 25, and 30 C for stretched 65/35 film irradiated at 95 C with 60 Mrad dose of 2.5 MeV electrons, where Q 130 is the weak field Q 13 (measured at about 10 MV/m and 30 C).
Electrostrictive poly(vinylidene fluoride-trifluoroethylene) copolymers Electromechanical coupling factor as a function of the electric field: (a) k 33 for unstretched P(VDF-TrFE) 68/32 mol% film irradiated at 105 C with 70 Mrad dose of 1.0 MeV electrons; (b) k 31 for stretched P(VDF-TrFE) 68/32 mol% film irradiated at 100 C with 70 Mrad dose of 1.2 MeV electrons.
Electrostrictive poly(vinylidene fluoride-trifluoroethylene) copolymers Frequency dependence of longitudinal strain response for unstretched 68/32 mol% copolymer under 20 MV/m field. The film was irradiated at 105 C with 70 Mrad of 1.0 MeV electrons.
Electrostrictive poly(vinylidene fluoride-trifluoroethylene) copolymers (a) (b) Transverse strain vs. tensile stress along the stretching direction for stretched 65/35 mol% copolymer film; Longitudinal strain vs. hydrostatic pressure for unstretched 65/35 mol% copolymer film. The strains were measured at room temperature. The films were irradiated at 95 C with 60 Mrad of 2.55 MeV electrons.
Electrostrictive poly(vinylidene fluoride-trifluoroethylene) copolymers (a) Frequency dependence of the impedance at room temperature for the stretched 68/32 mol% copolymer under different dc bias fields (from bottom to top, the dc bias is 0, 20, 40, 50, 60, 70 MV/m, respectively), the film was irradiated at 100 C with 70 Mrad of 1.2 MeV electrons; (b) and (c): The Young's modulus and coupling factor k 31 deduced from the data in (a).
Electrostrictive poly(vinylidene fluoride-trifluoroethylene) copolymers The performance of an unimorph with one electrostrictive P(VDF-TrFE) copolymer layer of 22 µm bonded to an inactive polymer of the same thickness: (a) The unimorph without electric field; (b) The actuation of the unimorph under an electric field of 65 MV/m.
Principle of operation of dielectric elastomers At small strains, this characteristic is mathematically described by assuming a Poisson ratio v of 0.5 Mechanical Constraint: incompressible volume A z=constant Z: the thickness of the polymer film A: area
Dielectric elastomers The stored electrostatic energy W of a film with opposite charges Q and Q placed on its surface can be written as: Where dw 2 2 Q Q z W = = 2C 2ε o ε r A ε r A C o ε = is the capacitance z 2 2 Q Q z = dz o r A o r A 2ε ε 2ε ε da A Since da A = dz z dw = 2 Q dz o r A ε ε
Dielectric elastomers The effective pressure p is So we have Considering that p 2 Q = ε ε E o r Q = ε o ε r A p = 1 A We can express the effective pressure as dw dz p = ε o ε E This effective pressure is exactly twice the pressure in a parallel plate capacitor If the electrostatic pressure is balanced by the elastic pressure of the 2 film, the thickness strain is V sz = p Y = ε oε r Y z r 2
Dielectric elastomers The in-plane strain s x =s y =s a ( 1 s )( 1+ s )( 1+ s ) = 1 + x y z 0. 5 ( 1+ ) 1 s a = s z Often, measuring s a is experimentally easier than measuring s z, z 2 2 ( s + s ) ( 1+ 2s s ) 2 a a a a s = + One of the more useful metrics for comparing actuator materials, independent of size, is the energy density of the material. The actuator energy density is the maximum mechanical energy output per cycle and per unit volume of material. The actuator energy density depends on the loading conditions. For small strains with free boundary conditions, the actuator energy density, e a, of the material can be written as (Conventionally, the elastic energy density e e =1/2 Ys z2 is often used) e a =ps z =Ys z2 =(ε o ε r ) 2 (V/z) 4 /Y.
For large strains with a linear stress strain relation the formula must be modified because as the thickness strain becomes increasingly negative, the film flattens out and the area over which the pressure must be applied increases. A more detailed derivation for large strains gives the formula for the elastic energy density of materials with a linear stress strain relation as e e =Y[(s z -Ln(1+s z )]. This equation agrees with the more common formula at small strains but is significantly higher for strains greater than 20%.
Dielectric elastomers Maximum response of representative elastomers Average engineering modulus at the maximum strain.
Comparison of dielectric elastomers with other actuator technologies
Zig-zag gold electrodes undergoing large strains. Uniform gold electrodes easily crack, and strains greater than about 4% are difficult to achieve. Recently, It has been demonstrated that patterned, rather than uniform gold electrodes can remain conductive at much greater film strains. Zig-zag patterns of gold traces deposited on silicone film, remain highly conductive at film strains up to 80%. The gold traces are sputtered uniformly on the film, then patterned via photolithography.