GAMM-Mitt. 38, No. 1, 115 – 131 (2015) / DOI 10.1002/gamm.201510006
Coordinate-invariant phase field modeling of ferroelectrics, part II: Application to composites and polycrystals 2 M.-A. Keip∗1 , D. Schrade2 , H. Thai3 , J. Schr¨oder3 , B. Svendsen4,5 , R. Muller ¨ , 6 and D. Gross 1
University of Stuttgart, Institute of Applied Mechanics (CE), Chair I, Pfaffenwaldring 7, 70569 Stuttgart, Germany
2
Technische Universit¨at Kaiserslautern, Institute of Applied Mechanics, Department of Mechanical and Process Engineering, Gottlieb-Daimler-Straße, 67653 Kaiserslautern, Germany
3
University of Duisburg-Essen, Institute of Mechanics, Department of Civil Engineering, Faculty of Engineering, Universit¨atsstraße 15, 45141 Essen, Germany
4
RWTH Aachen University, Material Mechanics, Schinkelstraße 2, 52062 Aachen, Germany
5
Max-Planck Institute for Iron Research, Microstructure Physics and Alloy Design, MaxPlanck-Str. 1, 40237 D¨usseldorf, Germany
6
Technische Universit¨at Darmstadt, FB 13, Institute of Mechanics, Franziska-BraunStraße 7, 64287 Darmstadt, Germany
Received 18 September 2014, revised 27 November 2014, accepted 2 December 2014 Published online 11 March 2015 Key words ferroelectrics, phase field method, homogenization, composites, polycrystals This paper deals with the application of the model presented in the first part S CHRADE ET AL . [1] to ferroelectric composites filled with electrically conducting inclusions as well as to ferroelectric polycrystals. Composites are analyzed through the use of a computational ¨ & M IEHE [2]. This will homogenization framework for phase field methods proposed in Z AH give insights into the coupled phenomena taking place on the microscale and on their relation to the overall behavior. Both will be of special interest for the development of advanced composite materials with tailored properties like, for example, particulate magneto-electric composites, which are composed of a ferroelectric matrix and magnetic rare-earth elements or metals. Furthermore, we analyze the behavior of ferroelectric polycrystals with a focus on size effects. This will enable us to reveal preferred microstructure configurations depending on the system and grain size. In addition to that, it will serve as basis for the extraction of the directional properties of polycrystals with respect to their switching behavior in the different grains of the polycrystal. Associated simulations could then be used to supply coarser models with the needed directional informations. c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
∗ Corresponding author: +49 711 685 66347
e-mail:
[email protected], Phone:
+49 711 685 69261, Fax:
c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
116 M.-A. Keip et al.: Coordinate-invariant phase field modeling of ferroelectrics, part II: Applications
1
Introduction
The theoretical prediction of the effective properties of smart materials plays an important role for the development of new functional devices with tailored, superior properties. There exist a variety of powerful methods which allow for the computation of effective parameters of composites based on, for example, analytical and semi-analytical homogenization schemes. In the area of electro-mechanical coupling one should mention the works by D UNN & TAYA [3, 4], B ENVENISTE [5, 6], amongst others, for a general treatise on homogenization see N EMAT-NASSER & H ORI [7]. When applied to non-linear problems or complicated microstructural morphologies, such methods experience certain limitations. In this case, computational schemes become attractive since they allow for the prediction of the effective properties of composites with any micro-morphology. Amongst the computational homogenization schemes for electro-mechanically coupled materials we would like to refer to P OIZAT & S ESTER [8] and B ERGER ET AL . [9, 10] as well as to the associated direct two-scale for¨ ¨ [11], S CHR ODER & K EIP [12], and K EIP, S TEINMANN & mulations given in S CHR ODER ¨ S CHR ODER [13]. Of particular relevance for the present contribution is the recent extension of computational homogenization to the incorporation of phase field models for ferroelectrics ¨ & M IEHE [2]. In their paper, the authors homogenized a microscopic ferproposed in Z AH roelectric phase field model (M IEHE ET AL . [14]) by a transition to a standard Boltzmann continuum on the macroscale. We will adopt this idea and apply it to the homogenization of ferroelectric/metallic composites. By doing so, we will investigate the evolution of ferroelectric microstructures under applied macroscopic loading as well as the resulting overall poling behavior. The poling behavior of composite materials made of ferroelectrics is of particular interest for a range of applications. Amongst those, we would like to point out composites with magnetoelectric (ME) coupling, and here specifically ME composites of particulate (0–3) type. These composites are characterized by magnetoactive particles embedded into a ferroelecric matrix. After manufacturing, the composites do not exhibit any ME coupling – the coupling needs to be activated through an electric pre-polarization process that is performed on the bulk composite, see S HVARTSMAN ET AL . [15]; a computational realization was presented in L ABUSCH ET AL . [16]. Since the magnetic materials used in ME composites are mostly electrical conductors, this pre-polarization has to be handled with care. The challenge is to avoid electric breakdown while still generating a sufficient (or optimal) state of polarization in the matrix material. The present paper tries to contribute to the understanding of the microscopic phenomena taking place in ferroelecric/metallic (however non-magnetic) composites exposed to electric (pre-polarization) fields. By homogenization we establish direct connections between these fundamental interactions on the microscale and the associated macroscopic loading and overall response. This constitutes one step towards the simulation-assisted experimental design of particulate ME composites with desired coupling properties. The other scope of the paper is to investigate size effects in ferroelecric polycrystals. In order to do so, we will first analyze the microstructure evolution in polycrystals of different size. Then, based on that, we simulate the effective hysteretic behavior of the considered specimens. Here, a focus will be on the assumption made on the (dis-)continuity of polarization across grain boundaries and its effect on the overall response. The simulations will give insights into the effective properties of polycrystalline ferroelectrics and reveal dependencies on modeling assumptions and system size.
www.gamm-mitteilungen.org
c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
GAMM-Mitt. 38, No. 1 (2015)
117
The paper is structured as follows. In section 2 the theoretical background of the computational homogenization approach used in this paper is documented. Thereafter, in section 3, we apply the method to the homogenization of ferroelectric/metallic composites and analyze the evolution of microstructures and the overall hysteretic response. Furthermore, we analyze the behavior of polycrystals in consideration of different specimen sizes. Finally, we summarize the main findings of the paper in section 5.
2
Homogenization framework for ferroelectrics described with the phase field method
In order to compute the effective response of ferroelectric composites with metallic inclusions, we will use the concept of computational homogenization. In detail, we will follow an approach based on our previous works on the homogenization of electro-elastic materials ¨ ¨ ¨ [11], S CHR ODER & K EIP [12], K EIP, S TEINMANN & S CHR ODER [13] and the S CHR ODER ¨ & recent work on the homogenization of ferroelectrics based on phase field models by Z AH M IEHE [2]. The next few paragraphs will be devoted to a short summary of the above ideas that are relevant for the subsequent numerical computations. 2.1 Boundary conditions for the representative volume element ¨ In what follows we adopt ideas of the recently proposed homogenization approach by Z AH & M IEHE [2]. A basic assumption of their work is that the phase field formulation is applied on the microscale only. On the macroscale, they formulate a standard Boltzmann continuum without any order parameter. Under these assumptions, the general form of the Hill-Mandel condition, which states that the rate of the total macroscopic work W is equal to the rate of the averaged total microscopic work W, appears as1 [2] ˙ = W ˙ B W
⇒
˙ = H˙ + D , H B
(1)
in which H and H are the macroscopic and microscopic electric enthalpy function, respectively, and D is the microscopic dissipation. Note that on the macroscale we do not consider a phase field so that dissipation needs not to be taken into account explicitly. Dissipation enters the macroscopic constitutive response through the averaging process over the microscopic fields which contain dissipative contributions from the microscopic polarization. The microscopic and macroscopic electric enthalpy function as well as the dissipation are assumed to be of the form ˆ H := H(ε, E, P , ∇P ),
ˆ H := H(ε, E),
and D :=
∂Φ ˙ ·P ∂ P˙
(2)
in which Φ is a convex function of the rate of polarization P˙ in order to guarantee positive dissipation, see, for example, H ILL [17], B IOT [18], E DELEN [19], H ALPHEN & N GUYEN [20], 1 In order to simplify the notation and thus ease readability we will write surface and volume integrals as
•∂B :=
1 V
∂B
• da
and
•B :=
1 V
B
• dv,
respectively, where B denotes the domain of a representative volume element (RVE) with volume V .
www.gamm-mitteilungen.org
c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
118 M.-A. Keip et al.: Coordinate-invariant phase field modeling of ferroelectrics, part II: Applications
and Z IEGLER & W EHRLI [21], or the more recent works O RTIZ & R EPETTO [22], H OULSBY & P UZRIN [23], M IEHE , S CHOTTE & L AMBRECHT [24], M IEHE [25, 26], and I DIART [27]. ¨ & M IEHE [2] given by In the following we will adopt the assumption of Φ from Z AH Φ :=
1 βP˙ 2 2
(3)
where β > 0 is an inverse mobility parameter. Insertion of (2) together with (3) into (1) formally gives ∂H ∂H ˙ ∂H ˙ ∂H ∂H ˙ ∂H ˙ : ε+ : ε˙ + ·E+ ·P + : ∇P˙ + βP˙ 2 (4) ·E = ∂ε ∂ε ∂E ∂P ∂∇P ∂E B in which we identify the macroscopic and microscopic counterparts of the Cauchy stress, the macroscopic and microscopic counterparts of the electric displacement as well as the microforce and -stress σ :=
∂H ∂H ∂H ∂H ∂H ∂H , D := − , D := − , η := , Σ := . (5) , σ := ∂ε ∂ε ∂E ∂P ∂∇P ∂E
Thus, we may write in short form ˙ = σ : ε˙ − D · E˙ + η · P˙ + Σ : ∇P˙ + βP˙ 2 σ : ε˙ − D · E B
(6)
¨ & M IEHE [2] and which differs from the which is identical to the condition given in Z AH condition given in [11, 12] exactly in the terms associated with the order parameter P . Based on the above equation we are able to derive energetically consistent boundary conditions for the microscopic problem. In order to do so we reformulate eq. (6) to ˙ − D · E ˙ B + η · P˙ + Σ : ∇P˙ + βP˙ 2 B = 0, (7) ˙ B − σ : ε˙ + D · E P := σ : ε P1
P2
P3
and consider each of the three terms P1,2,3 independently by setting P1 = 0, P2 = 0, and P3 = 0, cf. [11]. Suitable boundary conditions on the RVE are then derived by transforming the above volume integrals into surface integrals, see [13] for a step-by-step reformulation of P2 which can analogously be applied to P1 . In order to reformulate P3 we use the AllenCahn-type Ginzburg-Landau evolution equation for P that was given in eq. (15) of the first part of this paper. For better clarity it is written down in the above notation as 1 ∂H ∂H 1 δH 1 =− − div (8) P˙ = − = − (η − divΣ) . β δP β ∂P ∂∇P β By inserting this into (7) we obtain the associated surface expressions ˙ · x] + π · P˙ , P = [t − σ · n] · [u˙ − ε˙ · x]∂B + [ρs + D · n][φ˙ + E ∂B ∂B 1 P
2 P
(9)
3 P
where the polarization flux has been introduced as π = Σ · n. We will now list some suitable
1 = 0, P
2 = 0, and P
3 = 0, see also Z AH ¨ boundary conditions that fulfill eq. (9) by setting P & M IEHE [2].
www.gamm-mitteilungen.org
c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
GAMM-Mitt. 38, No. 1 (2015)
119
Linear Dirichlet boundary conditions. The numerical problem to be solved will be driven by the set of primary variables u, φ, and P . Thus, we identify suitable Dirichlet-type boundary conditions that fulfill eq. (9) by u := ε · x + cu ,
φ := −E · x + cφ ,
and P := cP
(10)
where cu , cφ , and cP are some constants. Linear Neumann boundary conditions. Suitable Neumann-type boundary conditions that satisfy eq. (9) are given by t := σ · n,
Q := −D · n,
and π := 0.
(11)
Periodic boundary conditions. In order to obtain periodic boundary conditions that automatically fulfill eq. (9), we decompose the boundary of the RVE into ∂B + and ∂B − with coordinates x± ∈ ∂B ± and associated outward unit normals2 n± := n(x± ). For geometrical reasons, periodicity of the unit cell requires that n+ = −n− . In order to derive periodic boundary conditions for the primary fields, we first assume that the deformation u and the electric field E on the microscale can be decomposed into a constant macroscopic part and a fluctuation field as follows and φ := −E · x + φ u := ε · x + u
(12)
ε = ε + ε and E = E + E.
(13)
so that
Insertion of these relations into eq. (9)1,2 leads with t± = σ · n± and Q± = −D · n± and the periodicity requirements + = u − , u
φ + = φ − ,
and P + = P −
(14)
to the expressions
1 = (t+ +t− )·u ˙ + ∂B+ , P
˙+
2 = (Q+ +Q− )·φ P ∂B+ ,
P 3 = (π + +π − )·P˙ + ∂B+ . (15)
These are fulfilled if t+ = −t− ,
Q+ = −Q− ,
and π + = −π − ,
(16)
which constitute antiperiodic fluxes that have to be satisfied in addition to the constraint (14). 2 In the following the indices + and − will be used to identify variables that are located on either the positive or negative boundary of the RVE, i.e. •± := •(x± ). The associated coordinates x+ and x− are assumed to lie exactly opposite on the faces of the RVE.
www.gamm-mitteilungen.org
c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
120 M.-A. Keip et al.: Coordinate-invariant phase field modeling of ferroelectrics, part II: Applications
2.2 Micro-to-macro transition In the given homogenization framework, the macroscopic fields {ε, σ, E, D} are related to their microscopic counterparts {ε, σ, E, D} through surface integrals along the boundary ∂B of the microscopic representative volume element . For simplicity we assume continuity of the displacements and the electric potential across the RVE, so that we can define the macroscopic strains and electric field through volume integrals. Then we have ε := sym(u ⊗ n)∂B = εB
and E := −φ n∂B = EB ,
(17)
where n is a unit normal vector pointing outwards from the RVE. Next to the assumption of continuity of the primary fields, we assume that mechanical body forces and free electric charge carriers are negligible on the microscale3. Then we can define the macroscopic Cauchy stresses and electric displacements as σ := sym(t ⊗ x)∂B = σB
and D := −Q x∂B = DB .
(18)
¨ For a detailed derivation of (17) and (18) see [28]. Furthermore, we define according to Z AH & M IEHE [2] that P := P B
(19)
is the macroscopic polarization.
3
Homogenization of ferroelectric/metallic composites
In the present contribution, we will investigate the poling behavior of ferroelectrics with electrically conducting (e.g. metallic) inclusions. As mentioned earlier, we approximate the electric conductivity of the inclusions by setting their permittivity to a high value compared to the permittivity of the surrounding material. This will of course not perfectly resemble the behavior of real conductors (since, for example, the density of free charge carriers is still assumed to be zero on the microscale). It is, however, a reasonable approximation for the studies below. For modeling approaches that explicitely account for the presence of electric conductivity see, for example, X IAO & B HATTACHARYA [29], S URYANARAYANA & B HATTACHARYA [30], and S CHWAAB ET AL . [31]. The application of suitable boundary conditions for the above described coupled homogenization framework is a non-trivial task. This is due to the fact that the driving macroscopic fields – the strain ε and the electric field E – are assumed to be prescribed. Generally, it cannot be assumed that they are known a priori. The problem boils down to the fact that the effective properties of the composite are not known beforehand (i.e. before homogenization). If they were known, one could of course compute the desired relations between loading and effective response – but then homogenization would be unnecessary. A solution to this problem can be achieved through solving the computation iteratively. In this way it is possible to adjust 3 The present contributions aims at the computation of the effective behavior of ferroelectric/metallic composites. Of course, metals are conductors and are thus characterized by a non-zero amount of free electric charge carriers. We will, however, approximate the behavior of the inclusions through the assumption of a quasi-infinite electric permittivity . Practically this will be implemented by means of numerical values of that are much higher than those of the surrounding matrix material.
www.gamm-mitteilungen.org
c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
GAMM-Mitt. 38, No. 1 (2015)
121
the prescribed loading step-wise in order to finally obtain the desired macroscopic state. The complexity of such an approach however grows with the non-linearities on the microscale, ¨ see, amongst others, V OLKER [32]. Another method is to embed the unit cell into a discretized macro-continuum, so that the macroscopic quantities can be computed through the solution of an associated macroscopic boundary value problem (as, for example, in case of direct two-scale approaches like the FE2 -method, see S MIT ET AL . [33], M IEHE ET AL . [34,35], ¨ M ICHEL ET AL . [36], and S CHR ODER [37] amongst others). 3.1 Microscopic boundary value problem In what follows we will homogenize ferroelectric composites by using a pure unit-cell framework. We will investigate the poling behavior of periodic RVEs comprised of ferroelectric matrix and different volume ratios of conducting inclusions, see Fig. 1.
inclusion (dielectric)
matrix (ferroelectric) a
transversely isotropic phase-field model and material data from [1]
incl
Y = 211 · incl ν = 0.29
a
C Vm 9 N 10 m2
incl = 4.96 · 10−7
Fig. 1 (online colour at: www.gamm-mitteilungen.org) RVE of a ferroelectric solid with regularly distributed, square-shaped high-electric-permittivity inclusions that approximates a ferroelectric/metallic composite. The outer dimensions of the RVE are 12 × 12 nm2 and the volume fraction of the inclusions are adjusted through the length a. The ferroelectric matrix material is given by barium titanate (BaTiO3 ; BTO) and described with the phase field model presented in S CHRADE ET AL . [1, 38]. The metallic inclusion is modeled as an isotropic, perfectly dielectric material with high electric permittivity.
The volume fractions of the inclusions are chosen as 25%, 17.36%, 11.11%, 6.25%, and 0%. These correspond to simple rectangular inclusions for the mesh discretizations of the composite depicted in Fig. 2.
(25%)
(17.36%)
(11.11%)
(6.25%)
Fig. 2 (online colour at: www.gamm-mitteilungen.org) Each RVE is discretized with a regular mesh of 48 × 48 linear quadrilateral finite elements. The volume fractions of the inclusions are 25%, 17.36%, 11.11%, and 6.25%, which correspond to 24 × 24, 20 × 20, 16 × 16, and 12 × 12 finite elements, respectively.
www.gamm-mitteilungen.org
c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
122 M.-A. Keip et al.: Coordinate-invariant phase field modeling of ferroelectrics, part II: Applications
In order to analyze the microscopic behavior and the associated overall macrosopic response of the composites, we load the different RVEs with an alternating macroscopic electric and , φ, field E. The RVEs are discretized by employing periodic boundary conditions for u P . Before we apply the macroscopic load, we relax the system by computing an equilibriated state of spontaneous polarization. This is done under E = 0 and ε = 0. Once this state is obtained, we apply the macroscopic electric field in form of a sinusoidal function while keeping the macroscopic strains zero, i.e.
0 and ε = 0 E := (20) E max · sin(ωt) where the angular frequency ω = 2πf was optimized through a convergence study in such a way that it guarantees rate-independent results at short computation times. In the present calculations we use a frequency of 107 Hz. The reason for this comparably high rate-independent loading frequency may partly be found in the extremely small size of the simulated specimen. In experiments one measures the response on crystals that have a size in the range of several millimeters (lexp ∼ 10−3 m). For a full macroscopic switching process a domain wall should have the time to cross some rather large fraction of the specimen length. In our simulations we consider microstructures with dimensions of several nanometers (lsim ∼ 10−9 m), which is approximately six orders of magnitude smaller than in the experiment. Taking into account that experiments are often performed at frequencies of fexp ≈ 1 Hz, we could roughly deduce that in our simulations this corresponds to a loading frequency of fnum ≈ lexp /lsim · fexp ≈ 106 Hz. There are some additional sources for the deviation as for example the estimate of the mobility parameter β −1 which is always exposed to some uncertainties. Another important source may be found in the idealization of the model. 3.2 Microstructure evolution The evolution of spontaneous polarization on the microscale as a result of the macroscopic loading is shown in Fig. 3 for the case of 11.11% volume fraction. The initial state of the microstructure depicted in Fig. 3 (a) shows the equilibrium configuration after relaxation of the spontaneous polarization. We see clearly that the distribution of the polarization is not homogeneous. In some simulations, we also observed microstructures with a nearly homogeneous distribution of polarization, so that the the obatained structure is of course not the only possible metastable state. The presence of this inhomogeneous herringbone structure is, however, typical for mechanically highly constrained microstructures (recall, that initially we set E = 0 and ε = 0). As soon as the macroscopic electric field is applied, the spontaneous polarizations tend to align with it. However, since in the initial configuration no polarization is parallel to the applied field, the vertical polarizations are first absorbed into the domains with horizontal orientations (Fig. 3 b). Then, local reorientation in the direction of the applied field occurs (Figs. 3 c and d). This process goes on, so that now the horizontal domains are absorbed into the vertical domains (Figs. 3 e and f), until finally, all polarizations are pointing upwards (Fig. 3 g). After removal of the applied field, a completely aligned configuration of polarization remains so that an overall remanently polarized material is obtained (Fig. 3 h). When the electric field is reversed, it drives the domain walls outwards to the lateral sides (Fig. 3 i) until a microstructure with polarization in the downward direction is obtained. This state is
www.gamm-mitteilungen.org
c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
GAMM-Mitt. 38, No. 1 (2015)
123
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
V E2 [ m ]
Fig. 3 (online colour at: www.gamm-mitteilungen.org) Exemplary snapshots of microstructure evolution with 11.11% volume fraction during the macroscopic loading process. State (a) represents the initial configuration in which the spontaneous polarization has reached an equilibrium state. Then, through the application of a macroscopic electric field E in positive vertical direction, the domain walls are driven out of the structure. This process is rather complex since initially no preferred direction of spontaneous polarization is parallel to the applied field. Thus, domains with polarizations pointing opposite to E are partly absorbed into the horizontal domains (b). Then, switching in the direction of the applied field is initiated locally (c and d). This switching is completed at the cost of horizontal domains (e and f), until finally all polarization directions are fully aligned with the outer field (g). When the electric field is removed, a completely poled microstructure remains (h). Thereafter, when the electric field is reversed, the polarization reorients first above and below the inclusion and the domain walls are driven laterally outwards (i). When the field is further increased and then removed, symmetric microstructures to (g) and (h) with spontaneous polarization vectors pointing downwards are obtained (not depicted here). All deformations are scaled by a factor of 50. The legend does not resemble the maximum and minimum obtained fields on the microstructure (local field concentrations appear which are not resolved for convenience).
www.gamm-mitteilungen.org
c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
124 M.-A. Keip et al.: Coordinate-invariant phase field modeling of ferroelectrics, part II: Applications
symmetric to the state (g) and not depicted in the figure (symmetry axis is the horizontal axis). After removal of the applied field the microstructure will be symmetric to state (h). 3.3 Macroscopic behavior We now analyze the overall macroscopic poling behavior of the individual composites with 25%, 17.36%, 11.11%, and 6.25% volume fraction of the inclusion. For comparison we also consider the behavior of a pure single-crystal (in the following associated with an RVE with 0% volume fraction of the inclusion). The corresponding effective hysteresis results are shown in Fig. 4. P 2 [ mC2 ]
εr22 [%]
0.3
0.6
0.2
0.5
0.1 0 -0.1
0.4
(1) (2) (3) (4) (5)
0.3 0.2
-0.2 -0.3 -50 -40 -30 -20 -10
0.1
0
10
20
30
40
kV E 2 [ mm ]0
50
(1) (2) (3) (4) (5)
-50 -40 -30 -20 -10
0
10
20
30
40
kV E 2 [ mm ]
50
Fig. 4 (online colour at: www.gamm-mitteilungen.org) Dielectric and electro-mechanical hysteresis for the different composites with (1) 6.25%, (2) 11.11%, (3) 17.36%, (4) 25%, and (5) 0% volume fraction of the inclusions. The initial poling curves are not shown for better clarity.
As expected, the volume fraction of the inclusions has a significant impact on the effective behavior. Two phenomena may be accentuated. First, we see that the effective coercive field depends strongly on the ratio of conducting inclusion and matrix. This effect can already be observed in the plots of the microstructure evolution shown in the previous paragraph, see Fig. 3. There we have seen that the reorientation of spontaneous polarization was initiated above and below the inclusion. This is due to the fact that the inclusion is approximated as a conductor and is – at least ideally – characterized by an electric equipotential (i.e. by a constant electric potential and thus by a zero electric field). Thus, the larger the size of the quadratic equipotential surface (the area of the inclusion), the higher the electric field above and below the inclusion. As a result, switching of the polarization is intiated earlier for the composites with high volume ratio of inclusions. The second effect, i.e. the decrease of overall remanent polarization with increasing volume fraction, is due to the fact that the inclusions are modeled as an ideal dielectric and thus cannot be polarized spontaneously. We would like to point out explicitly that the computed effective coercive fields are much higher than the coercive fields usually measured in experiments. The discrepancy observed here is roughly one order of magnitude, which can be partly explained by the chosen boundary conditions. This particular phenomenon has, however, also been observed in previous phase ¨ & K AMLAH [40], S U ET field studies, see, for example, C HOUDHURY ET AL . [39], V OLKER AL . [41], H ONG ET AL . [42], WANG & Z HANG [43], and S ANG ET AL . [44]. In these publications it was shown that the magnitude of the coercive field depends on several parameters
www.gamm-mitteilungen.org
c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
GAMM-Mitt. 38, No. 1 (2015)
125
like, for example, the size of the simulated specimen, the presence of stresses, the orientation ¨ of crystal axes, and inhomogeneities. More specifically, in the recent work by V OLKER & K AMLAH [40], the authors computed a coercive field of a single crystal that is about two orders of magnitude higher that the value measured in experiments. They showed that the effective coercive field could be reduced by accouting for point charges or polycrystals (for the latter see also section 4 of the present paper). We would, however, like to emphasize that the phase field predictions made in the present paper are of valuable qualitative nature. Although quantitatively some of the effective quantities might be slightly inaccurate, the simulations offer important insights into the driving phenomena taking place on the microscale. From such observations one can derive important ideas for the development of new composite materials with tailored properties. The interested reader might also consult the critical review on phase field methods by Q IN & B HADESHIA [45], in which the authors comment on the capabilities and limitations of phase field methods for quantitative predictions.
4
Domain evolution and poling behavior of BTO polycrystals
In what follows, the simulations of the first part of this paper S CHRADE ET AL . [1], section 3.2, are repeated for a BTO polycrystal consisting of five grains with randomly oriented crystal axes. The equilibrated domain structures are shown in Fig. 5; the boundaries of the crystallites are indicated by black lines. The results are similar to those obtained for the single crystals. Due to the strain mismatch and the electrical fields at the grain boundaries, the domain structures are more complex compared to what is observed for the single crystals.
12 × 12 [nm2 ]
48 × 48 [nm2 ]
24 × 24 [nm2 ]
36 × 36 [nm2 ]
72 × 72 [nm2 ]
96 × 96 [nm2 ]
Fig. 5 (online colour at: www.gamm-mitteilungen.org) Equilibriated domain configurations for differently sized BTO polycrystals. The surfaces are charge free, and the boundary is mechanically clamped.
www.gamm-mitteilungen.org
c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
126 M.-A. Keip et al.: Coordinate-invariant phase field modeling of ferroelectrics, part II: Applications
In another set of simulations, we investigate the overall switching process of the polycrystals by applying a large-signal electric field. Phase field models have been used to study the poling behavior of ferroelectric single-crystals, see, for example, WANG ET AL . [46], Z HANG & B HATTACHARYA [47], S OH ET AL . [48], and C HOUDHURY ET AL . [39] as well as poly¨ crystals, see, for example, C HOUDHURY ET AL . [49], V OLKER & K AMLAH [40], and L IU & S U [50]. The focus of the present simulations is on the transition conditions at the grain boundaries and on size effects with regard to the ferroelectric hysteresis. While the mechanical displacement and the electric potential are commonly assumed to be continuous, the order parameter can be allowed to experience a jump across the grain boundary. In the following, the two transition conditions at the grain boundaries are labeled “coupled” (continuous order parameter) and “uncoupled” (discontinuous order parameter). The two conditions are realized by using separate meshes for each grain. The order parameter at coinciding nodes along the grain boundaries can then be coupled or left uncoupled. The mechanical displacement and the electric potential are coupled along all grain boundaries to guarantee continuity. The boundary conditions for the simulations are defined as follows: A time-dependent electric potential difference is applied between the top and bottom boundaries, and the lateral boundaries are assumed to be charge free. We fix the horizontal displacement at the top and the bottom surface and inhibit rigid body rotations. The time interval for the alternating triangular electric potential difference ϕ∗ (t) is sufficiently long to ensure a full poling of the polycrystals. A nominal external electric field E ∗ may be defined by E ∗ (t) = −ϕ∗ (t)/l, where l is the edge length of the polycrystal. The initial conditions for the order parameter are as before chosen to be small random perturbations around the cubic state of the crystallites. Figure 6 shows the resulting dielectric hysteresis curves for differently sized simulation boxes. The two transition conditions “coupled” and “uncoupled” lead to very different hysteresis behavior for smaller polycrystals: While the remanent polarizations are almost equal, the coercive field is considerably higher for the “uncoupled” condition. Furthermore, the hysteresis curves for the “uncoupled” case show that the corresponding polycrystals switch in two stages, which is not the case for the “coupled” polycrystals. These two differences become less relevant as the size of the crystal increases; for large crystals the hysteresis results are almost identical (apart from the initial poling). The electrical and strain hysteresis curves are plotted in Figs. 7 and 8. Since the grain structure is not geometrically symmetric, the hysteresis curves are not symmetric, i.e. there are two different coercive fields in each dielectric hysteresis. This asymmetry is also reflected in the shape of the strain hysteresis curves. To assess the size effects on the poling behavior, the averaged coercive field is plotted versus the crystal size in Fig. 9. The coupling condition for the order parameter only has a significant effect for the smaller crystals. The coercive field seems to be saturating at least for the “coupled” condition; additional simulations could provide clarity as to whether a size effect is still occurring for even larger crystals. Further studies are also required to assess the influence of a greater number of grains on the coercive field.
5
Conclusion
In this contribution we applied the phase field model presented in the first part of this paper to the homogenization of ferroelectric solids with metallic inclusions and to the simulation
www.gamm-mitteilungen.org
c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
GAMM-Mitt. 38, No. 1 (2015)
D3 /P0
127
12 × 12 nm2
D3 /P0
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
coupled uncoupled
-1 -3
24 × 24 nm2
-2
-1
0
1
2
coupled uncoupled
-1 3
-3
-2
-1
0
E ∗ [107 V/m] D3 /P0
36 × 36 nm2
D3 /P0 1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
48 × 48 nm2
-0.8
coupled uncoupled
-1 -2
-1
0
1
2
coupled uncoupled
-1 3
-3
-2
-1
0
E ∗ [107 V/m] D3 /P0
72 × 72 nm2
D3 /P0 1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
1
2
3
E ∗ [107 V/m]
1
96 × 96 nm2
-0.6
-0.8
-0.8
coupled uncoupled
-1 -3
3
-0.6
-0.8 -3
2
E ∗ [107 V/m]
1
-0.6
1
-2
-1
0
1
2
coupled uncoupled
-1 3
E ∗ [107 V/m]
-3
-2
-1
0
1
2
3
E ∗ [107 V/m]
Fig. 6 (online colour at: www.gamm-mitteilungen.org) Dielectric hysteresis results for the “coupled” and “uncoupled” transition conditions. D3 denotes the averaged electric displacement in the vertical direction.
www.gamm-mitteilungen.org
c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
128 M.-A. Keip et al.: Coordinate-invariant phase field modeling of ferroelectrics, part II: Applications
D3 /P0
D3 /P0
coupled OP
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
12x12 24x24 36x36 48x48
-0.8 -1 -3
coupled OP
-2
-1
0
1
2
48x48 72x72 96x96
-0.8 -1 3
-3
-2
-1
0
E ∗ [107 V/m] ε33 [%] 0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
2
3
E ∗ [107 V/m] ε33 [%]
coupled OP
1
coupled OP
0 12x12 24x24 36x36
-0.1 -3
-2
-1
0
1
2
48x48 72x72 96x96
-0.1 3
E ∗ [107 V/m]
-3
-2
-1
0
1
2
3
E ∗ [107 V/m]
Fig. 7 (online colour at: www.gamm-mitteilungen.org) Dielectric and butterfly hysteresis results for the “coupled” transition condition.
of polycrystals. The homogenization of ferroelectric composites showed a significant dependence between the effective response and the volume fraction of the inclusions. The simulations offered insights into driving microscopic mechanisms and their connections to the overall behavior. The polycrystal simulations revealed size effects associated with domain evolution and effective hysteretic response. Furthermore, it could be shown that the influence of a continuous/discontinuous transition of polarization across grain boundaries decreases with increasing grain size.
Acknowledgements The financial support of the German Research Foundation (DFG) in the framework of the research group FOR 1509 “Ferroische Funktionsmaterialien – Mehrskalige Modellierung und experimentelle Charakterisierung” (projects SCHR 570/12-1, MU 1370/8-1, SV 8/14-2) are gratefully acknowledged.
www.gamm-mitteilungen.org
c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
GAMM-Mitt. 38, No. 1 (2015)
D3 /P0
129
D3 /P0
uncoupled OP
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
12x12 24x24 36x36 48x48
-0.8 -1 -3
uncoupled OP
-2
-1
0
1
2
48x48 72x72 96x96
-0.8 -1 3
-3
-2
-1
0
E ∗ [107 V/m] ε33 [%] 0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1 12x12 24x24 36x36
0 -3
-2
-1
0
1
2
3
3
uncoupled OP
48x48 72x72 96x96
0
E ∗ [107 V/m]
2
E ∗ [107 V/m] ε33 [%]
uncoupled OP
1
-3
-2
-1
0
1
2
3
E ∗ [107 V/m]
Fig. 8 (online colour at: www.gamm-mitteilungen.org) Dielectric and butterfly hysteresis results for the “uncoupled” transition condition. The ragged graphs for ε33 are attributed to domain switching; this can be seen from the plots for D3 .
References [1] D. Schrade, M. A. Keip, H. N. M. Thai, J. Schr¨oder, B. Svendsen, R. M¨uller, and D. Gross, GAMM-Mitteilungen 38(1), 102–114 (2015). [2] D. Z¨ah and C. Miehe, Computer Methods in Applied Mechanics and Engineering 267, 487–510 (2013). [3] M. L. Dunn and M. Taya, International Journal of Solids and Structures 30, 161–175 (1993). [4] M. L. Dunn and M. Taya, Proceedings of the Royal Society London A 443(1918), 265–287 (1993). [5] Y. Benveniste, Journal of Applied Mechanics 60(2), 265–269 (1993). [6] Y. Benveniste, Journal of Applied Mechanics 60(2), 270–275 (1993). [7] S. Nemat-Nasser and M. Hori, Micromechanics: Overall properties of Heterogeneous Materials, second edition (North-Holland, 1999). [8] C. Poizat and M. Sester, Computational Materials Science 16(1–4), 89–97 (1999). [9] H. Berger, S. Kari, U. Gabbert, R. Rodr´ıguez-Ramos, J. Bravo-Castillero, and R. Guinovart-D´ıaz, Materials Science and Engineering: A 412(1–2), 53–60 (2005).
www.gamm-mitteilungen.org
c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
130 M.-A. Keip et al.: Coordinate-invariant phase field modeling of ferroelectrics, part II: Applications
Ec [107 V/m] 1.9
coupled uncoupled
1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 10
20
30
40
50
60
70
80
90
100
crystal size [ nm] Fig. 9 (online colour at: www.gamm-mitteilungen.org) Comparison of the (averaged) coercive fields for the two transition conditions. Both curves reveal a significant size effect in the poling behavior.
[10] H. Berger, S. Kari, U. Gabbert, R. Rodr´ıguez-Ramos, R. Guinovart-D´ıaz, J. A. Otero, and J. Bravo-Castillero, International Journal of Solids and Structures 42(21–22), 5692–5714 (2005). [11] J. Schr¨oder, Computational Materials Science 46(3), 595–599 (2009). [12] J. Schr¨oder and M. A. Keip, Computational Mechanics 50, 229–244 (2012). [13] M. A. Keip, P. Steinmann, and J. Schr¨oder, Computer Methods in Applied Mechanics and Engineering 278, 62–79 (2014). [14] C. Miehe, D. Z¨ah, and D. Rosato, International Journal for Numerical Methods in Engineering 91, 115–141 (2012). [15] V. V. Shvartsman, F. Alawneh, P. Borisov, D. Kozodaev, and D. C. Lupascu, Smart Materials and Structures 20(7), 075006 (2011). [16] M. Labusch, M. Etier, D. C. Lupascu, J. Schr¨oder, and M. A. Keip, Computational Mechanics 54(1), 71–83 (2014). [17] R. Hill, Journal of the Mechanics and Physics of Solids 6, 236–249 (1958). [18] M. A. Biot, Mechanics of Incremental Deformations (John Wiley & Sons, New York, 1965). [19] D. G. B. Edelen, Archive of Rational Mechanics and Analysis 51, 218–227 (1973). [20] B. Halphen and Q. S. Nguyen, Journal de M´ecanique 40, 39–63 (1975). [21] H. Ziegler and C. Wehrli, The derivation of constitutive relations from the free energy and the dissipation function, in: Advances in Applied Mechanics, edited by T. Y. Wu and J. W. Hutchinson, Advances in Applied Mechanics Vol. 25 (Academic Press, San Diego, 1987), pp. 183–238. [22] M. Ortiz and E. Repetto, Journal of the Mechanics and Physics of Solids 47(2), 397–462 (1999). [23] G. Houlsby and A. Puzrin, International Journal of Plasticity 16(9), 1017–1047 (2000). [24] C. Miehe, J. Schotte, and M. Lambrecht, Journal of the Mechanics and Physics of Solids 50(10), 2123–2167 (2002). [25] C. Miehe, Journal of the Mechanics and Physics of Solids 59(4), 898–923 (2011). [26] C. Miehe, GAMM-Mitteilungen 35(1), 8–25 (2012). [27] M. I. Idiart, Modelling and Simulation in Materials Science and Engineering 22(2), 025010 (2014). [28] M. A. Keip, Modeling of electro-mechanically coupled materials on multiple scales, PhD thesis, Institute of Mechanics, Department of Civil Engineering, Faculty of Engineering, University of Duisburg-Essen, 2012. [29] Y. Xiao and K. Bhattacharya, Archive of Rational Mechanics and Analysis 189(1), 59–95 (2008). [30] P. Suryanarayana and K. Bhattacharya, Journal of Applied Physics 111, 034109 (2012).
www.gamm-mitteilungen.org
c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
GAMM-Mitt. 38, No. 1 (2015)
131
[31] H. Schwaab, M. Deluca, P. Supancic, and M. Kamlah, Journal of the Mechanics and Physics of Solids 61(2), 504–516 (2013). [32] B. V¨olker, Phase-field modeling for ferroelectrics in a multi-scale approach, PhD thesis, Karlsruhe Institute of Technology (KIT), 2010. [33] R. J. M. Smit, W. A. M. Brekelmans, and H. E. H. Meijer, Computer Methods in Applied Mechanics and Engineering 155, 181–192 (1998). [34] C. Miehe, J. Schotte, and J. Schr¨oder, Computational Materials Science 16(1–4), 372–382 (1999). [35] C. Miehe, J. Schr¨oder, and J. Schotte, Computer Methods in Applied Mechanics and Engineering 171, 387–418 (1999). [36] J. C. Michel, H. Moulinec, and P. Suquet, Computer Methods in Applied Mechanics and Engineering 172, 109–143 (1999). [37] J. Schr¨oder, A numerical two-scale homogenization scheme: the FE2 -method, in: Plasticity and Beyond, edited by J. Schr¨oder and K. Hackl, CISM International Centre for Mechanical Sciences Vol. 550 (Springer Vienna, 2014), pp. 1–64. [38] D. Schrade, R. M¨uller, D. Gross, M. A. Keip, H. Thai, and J. Schr¨oder, International Journal of Solids and Structures 51(11–12), 2144–2156 (2014). [39] S. Choudhury, Y. L. Li, C. Krill III, and L. Q. Chen, Acta Materialia 55(4), 1415–1426 (2007). [40] B. V¨olker and M. Kamlah, Smart Materials and Structures 21(5), 055013 (2012). [41] Y. Su, H. Chen, J. J. Li, A. K. Soh, and G. J. Weng, Journal of Applied Physics 110(084108), 1–6 (2011). [42] L. Hong, A. K. Soh, Y. C. Song, and L. C. Lim, Acta Materialia 56, 2966–2974 (2008). [43] J. Wang and T. Y. Zhang, Physical Review B 73, 144107 (2006). [44] Y. Sang, B. Liu, and D. Fang, Computational Materials Science 44(2), 404–410 (2008). [45] R. S. Qin and H. K. Bhadeshia, Materials Science and Technology 26(7), 803–811 (2010). [46] J. Wang, S. Q. Shi, L. Q. Chen, Y. Li, and T. Y. Zhang, Acta Materialia 52(3), 749–764 (2004). [47] W. Zhang and K. Bhattacharya, Acta Materialia 53(1), 185–198 (2005). [48] A. K. Soh, Y. C. Song, and Y. Ni, Journal of the American Ceramic Society 89, 652–661 (2006). [49] S. Choudhury, Y. L. Li, C. E. Krill III, and L. Q. Chen, Acta Materialia 53(20), 5313–5321 (2005). [50] N. Liu and Y. Su, Acta Mechanica 225(4–5), 1335–1345 (2014).
www.gamm-mitteilungen.org
c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
