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Journal Club Forum for April 15th: Fracture of Ferroelectrics

Ferroelectric materials have seen applications as actuators (the fuel injection system in the latest BMWs), sensors (naval sonar systems), and ferroelectric nonvolatile random access memories (FRAMs). For actuators and sensors it is the piezoelectric behavior of these materials that is exploited, while for FRAMs the ability of the material to “switch” polarization states is the essential feature for the application. 

For those who are not intimately familiar with these materials it is perhaps useful to describe these features at the crystal lattice level. The most widely used and studied ferroelectric materials have a perovskite crystal structure with an ABO3 chemistry.  This structure is shown schematically in the illustrations.  The A ions lie at the corners of the pseudo-cubic cell, B ions in the body center, and the oxygen ions are on the faces making up the “oxygen octahedron”.  Perhaps the most widely studied materials are barium titanate, BaTiO3, and lead titanate, PbTiO3.  Above the Curie temperature these materials are cubic, centro-symmetric, and non-piezoelectric.  However, below the Curie temperature these materials undergo a displacive phase transition to a low symmetry phase.  For the sake of this discussion let's assume that this phase is tetragonal, but in some cases this phase can be rhombohedral or orthorhombic.  Below the Curie temperature the free energy has local minima associated with the six possible tetragonal states (simplistically, the central B ion shifts towards one of the pseudo-cubic cell faces creating an electrical dipole and stretching the unit cell in this direction).  A schematic 1-D energy landscape is illustrated in Figure 1.

Figure1

Figure 1.  Free energy as a function of polarization.  Note the direction of the central B ion. 

The constitutive response can now be derived from this free energy functional.  The electric field is the derivative of the free energy with respect to polarization, and due to considerations of symmetry the strain is proportional to the square of the polarization (at least to leading order).  Hence, the polarization and strain versus the applied electric field take on the shapes shown in Figure 2.  Note that the red portions of the curves are unstable, and if the material could respond in a homogeneous manner the dashed blue lines would be followed creating the standard shapes for the dielectric hysteresis and strain butterfly loops.

 

Figure2.  The polarization-electric field hysteresis and the strain-electric field butterfly loops.

Again, a sensor or actuator utilizes the small signal response, i.e. the behavior near one of the energy minima that leads to linear elastic, dielectric and piezoelectric effects.  An electrical field applied in the direction of the polarization will cause the lattice to stretch, and this is used for actuating applications.  On the other hand, the sensing applications use the feature that a mechanical stress, e.g. a sound wave, will cause a change in polarization which can be detected with an electrical circuit.  Finally, the FRAM application requires the large signal response, treating “up” and “down” polarization states as 1’s and 0’s.

My apologies to those who already know all of this, but perhaps it is useful for the uninitiated.  There is a large literature dealing with continuum modeling of the constitutive behavior of ferroelectrics from both the physics community and the mechanics community.  The physics community tends to approach the problem from the microscopic scale by modeling the explicit evolution of the domain walls and domain structures in these materials.  For the “physics” approach to this scale see for example,

Chodhury,S., Li, Y.L., Krill, C.E., and Chen, L.Q. 2005. Phase-field simulation ofpolarization switching and domain evolution in ferroelectric polycrystals. Acta Materialia 53, 5313-5321.

But, the mechanic-ers are also looking at these phase-field theories of domain structure evolution, and a nice example is,

Zhang, W. and Bhattacharya, K. 2005a. A computational model of ferroelectric domains. Part I: model formulation and domain switching. Acta Materialia 53, 185-198.

Zhang, W. and Bhattacharya, K. 2005b. A computational model of ferroelectric domains. Part II: grain boundaries and defect pinning. Acta Materialia 53, 199-209.

For problems requiring a constitutive description of a more “macroscopic” material point, say the response of several grains and hence many domains, a more traditional mechanics approach has been taken that parallels J2 flow theory for metal polycrystals, and continuum slip plasticity for large single crystals. For reviews of these modeling approaches I would refer you to,

Kamlah M. 2001. Ferroelectric and ferroelastic piezoceramics – modeling and electromechanical hysteresis phenomena. Continuum Mechanics and Thermodynamics 13, 219-268.

Huber, J.E. 2005. Micromechanics modelling of ferroelectrics. Current Opinion in Solid State and Materials Science 9, 100-106.

One of the big drawbacks of ferroelectrics is that they are very brittle and susceptible to fracture.  In my opinion, a detailed understanding of fracture in ferroelectrics and a predictive framework for the failure of these materials is still an open problem.  For an excellent review of both the experimental and theoretical investigations on the fracture of ferroelectrics I refer you to,

Schneider, G.A. 2007. Influence of Electric Field and Mechanical Stress on the Fracture of Ferroelectrics.  Annual Review of Materials Research 37,491-538.

Allow me to outline some of the issues and then open the floor for a hopefully fruitful discussion.

1.  Fracture in ferroelectrics is inherently “mixed mode” involving both mechanical and electrical singularities at the crack tip.

2.  Unlike purely mechanical fracture, the crack-face boundary conditions are not well understood for ferroelectrics.  Furthermore, these boundary conditions have a considerable effect on the electromechanical fields around the crack tip.  One physical feature that is not well understood and may play an importantrole in fracture is the occurrence of electrical discharge between the crack faces.

3.  Generally, electric fields do not like cracks in the sense that they “go around” the crack tip in much the same way that mechanical stresses do.  However, in the electrical case, and within the framework of linear piezoelectric fracture mechanics (LPFM), this leads to a negative contribution to the energy release rate, suggesting that the application of electric fields should retard crack growth.  This feature of LPFM has not always been bourn out by observations.

4.  Ferroelectric switching must occur near crack tips due to the elevated stress and electric field states near such defects.  Such switching is akin to transformation toughening and leads to R-curve behavior.  Some, but not all, aspects of ferroelectric fracture have been explained using detailed constitutive models that account for this switching behavior.

5.  There is not a great deal of understanding on how the separation process occurs in ferroelectrics, even in a qualitative way.  For example, in mechanical fracture we have a good idea of what goes on in ductile and brittle materials, and in both cases we can describe this with a traction-separation law.  Bonds must be broken in ferroelectrics as well, but during this process what happens to the charge carriers? 

I am sure there are many other questions to explore and these are just a few to start with. 

 

Comments

Hi Dr Landis

  This is a pretty interesting topic...i was intrigued by the possibility of using electric fields to retard crack growth in ferroelectrics. I do have a couple of questions...

* Is my understanding correct that the "negative" energy release rate should be higher for high dielectric materials like certain perovskites? Also, i noticed in your article that experimental observations have not confirmed the idea of a negative energy release rate due to applied electric fields working against the mechanical positive energy release rate to retard crack growth...while this is a little disheartening, has any body put forward some theoretical explanation for why the effects of the electric field are not being observed?

* In calculations, what is the dielectric permittivity of the crack taken to be? Talking about FRAMs...how defect free should they be? 

* On a side note, the phase transition of a ferroelectric from its cubic centrosymmetric phase to a lower symmetry phase is associated with the soft optical phonon becoming unstable at the curie temperature. Is there a "mechanics " interpretation behind this? 

 Thanks! 

 

 

Ravi,

Let me be a little more careful.   Aside from a recent attempt by Gerold Schneider and his group, the energy release rate has never been unambiguously measured in a piezoelectric/ferroelectric fracture experiment.  So when I talk about negative energy release rates I am referring to predictions from linear piezoelectric fracture mechanics (LPFM) with impermeable or semi-permeable crack face boundary conditions.  So one point is that the negative energy release rate contributions predicted by LPFM may not actually exist in the experiments.  Some possible mechanisms for a reduction in the retarding effects of electric fields include electrical discharge in the crack gap, grain bridges in the crack gap, and large scale domain switching.  I have done some theoretical work on the effects of electrical discharge.

http://www.ae.utexas.edu/~landis/Landis/Research_files/IJSS2004b.pdf 

In calculations the dielectric permittivity of the crack is usually taken to be that of free space (or air).  This assumption is not a requirement in the models, but it is what is most commonly used.  Recently Scheinder et al have measured the opening displacement and electric potential drop across a crack and found that the permittivity is higher than that of free space.  Again, some possible mechanisms for this observation include crack bridges (although they did not observe any) and electrical discharge.

Schneider GA, Felten F, McMeeking RM. The electrical potential difference across cracks in PZT measured by Kelvin probe microscopy and the implications for fracture. Acta Materialia 51:2235-2241, 2003. 

I am not an expert on FRAMs, but my understanding is that defects, generally charge defects in the form of oxygen vacancies or free electrons, are bad.  Charge defects act to do two things, 1) they pin the motion of domain walls which leads to an increase in the coercive field and 2) they shield the polarization of domains leading to a reduced current signal during switching.  The migration of charge defects leads to the phenomenon of electrical fatigue (very different from mechanical fatigue) in these materials.

Scott JF, Dawber M. Oxygen-vacancy ordering as a fatigue mechanism in perovskite ferroelectrics. Applied Physics Letters 76:3801-3803, 2000.

I will get back to you on the optical phonon question.  I do believe that there is a "mechanics" interpretation, but I need to make sure I get my terminology right.

Chad 

OK, I'll have a go at this, and my understanding comes from some work by Nick Triantafyllidis, John Shaw  and Ryan Elliot.

The idea is that the phase transition occurs due to some instability in the lattice.  I think that the easiest type of transition for mechanic-ers to understand is one that happens as a result of long wavelength perturbations, or essentially homogeneous deformations of the lattice.  For this type of transition one of the eignevalues of the tangent elasticity matrix goes to zero and a transition can occur.  This is not the soft optical phonon.  However, this is not the only type of instability that can occur.  The lattice must also be stable against short wavelength perturbations.  My rough mechanistic understanding of this type of instability is that the perturbation of one atom, or a few atoms, from its equilibrium position causes a cascade of the remainder of the crystal to a new lower energy state.  I believe that this type of instability is the of the "soft optical phonon" type.

I am really stretching the limits of my knowledge on this, but I think this explanation has some elements of truth to it. 

 I welcome any additional responses that directly or indirectly point out my ignorance here.

 Chad 

ellio167's picture

Chad has the right idea.  Soft phonons are deformation modes in a
crystal that have a low stiffness associated with them.  Thus,
they are likely to lead to instabilities.  However, these modes
generally involve all atoms in the crystal.  They can be described
as plane wave deformations or motions of the atoms.  The
distinction between these and more classical continuum deformations,
that most mechanicers are used to, is that the wavelength of the motion
is of the same order of magnitude as the lattice spacing.  For
deformations of on this scale, the discrete nature of the crystal leads
to different behavior (wave speed, stiffness, etc.) associated with
different wavelength phonons (eg. dispersion).  This is in
contrast to the continuum case where no dispersion exists (there is no
length scale).

The distinction between optical and acoustic phonons is most clear in
the "long wavelength limit".  On one hand, long wavelength
acoustic phonons can be directly correlated with "rank-one" uniform
deformations which mechanicers are well acquainted with.  Long
wavelength optic phonons, on the other hand, are atomic motions that
correspond to localized excitation of the crystal where the atoms in
each unit cell move about in lock step (Cauchy-Born type
shifts/shuffles) but there is no macroscopic (uniform) deformation.

The phase transformation in Ferroelectrics is often driven by a soft
optic phonon.  That is the optic phonon deformation mode is the
"primary order-parameter" (the mode that first goes unstable). 
The transformation strains that result are "secondary order-parameters"
that are directly coupled to the primary order-parameter and occur
because of the optic phonon mode deformation.  In Ferroelectrics
the primary oder-parameter corresponds to the shifting of the center
atom off of its central location (this happens in each unit cell, thus
illustrating the optic phonon nature of the deformation).  As a
result of the atom moving from its central location the lattice spacing
changes which leads to the overall transformation strain.

This type of behavior is not limited to soft optic phonons.  In
most Shape Memory Alloys it is a soft "zone-boundary phonon mode"
(neither optic or acoustic, really) that is responsible for the phase
transformation.  This type of instability also involves a
"period-extension" where the size of the unit cell changes as a result
of the transformation.  So that the martensite crystal structure
has a primitive unit cell with twice as many atoms, say, as the
austenite's primitive unit cell.

You are being kind to say I had the right idea.  My "ideas" have been refined by your very informative post.

 

Chad 

Arash_Yavari's picture

Dear Chad:

Thanks for the very interesting discussion. Could you mention a few key
papers that discuss the theoretical issues in fracture mechanics of
ferroelectrics?

Thanks,
Arash

Arash,



In my opinion the best paper on linear piezoelectric fracture mechanics is by Suo and co-workers.



Suo Z, Kuo CM, Barnett DM, Willis JR. 1992. Fracture mechanics for piezoelectric
ceramics. J. Mech. Phys. Solids 40:739–65. 
 
 

Ortiz and co-workers have done some work on a cohesive zone model.  In my opinion this is a good direction to go in, but more physical understanding is required to determine the coupling between the crack opening displacement and the electric potential drop across the crack faces.  I would also be interested to see what such a model predicts for the simple static semi-infinite crack.

Arias I, Serebrinsky S,Ortiz M, 2006. A phenomenological cohesive model of ferroelectric fatigue. Acta Materialia 54:975-984. 
 
 
 

For a more selfish response to your comment, I would refer you to my work on the electromechanical boundary conditions and on small scale domain switching,

http://www.ae.utexas.edu/~landis/Landis/Research_files/IJSS2004b.pdf

http://www.ae.utexas.edu/~landis/Landis/Research_files/ActaMat2004.pdf

http://www.ae.utexas.edu/~landis/Landis/Research_files/JOMMS2006.pdf

If you have a specific topic you are interested in, linear, non-linear, boundary conditions, etc.,then let me know and I can dig up some more specific references.

Chad

Kaushik Dayal's picture

Dear Chad,

As you've worked with both the phase-field and the macroscopic models, can you provide your perspective on what inputs phase-field calculations can provide to macroscopic models?  I'm not aware of work that uses phase-field results in this manner directly, do you know of any references?

Thanks,
Kaushik

Kaushik,

The question to answer is "what does the macro constitutive law need".  If we take for example macro single crystal constitutive models (which are analogous to continuum slip plasticity models) we need to know the critical level of "driving force" required to cause switching on a given "transformation system", i.e. 180 or 90 degree switching for tetragonal ferroelectrics.  My paper with Yu Su studied how charge defects affect the coercive field for 180 and 90 degree switching.

http://www.ae.utexas.edu/~landis/Landis/Research_files/JMPS2007.pdf

So one element that the phase field models might be able to provide to the macro models is how the coercive field is dependent on the concentration of defects. These phase field calculations have also suggested that the critical driving force can also depend on the other non-driving stresses and electric fields.  For example, due to changes in piezoelectric properties across a 180 degree domain wall, the applied driving force for motion increases with combined electric field and tensile stress along the wall, suggesting that the observed critical electric field should decrease.  However, the phase field calculations indicate that the critical driving force increases faster due to the tensile stress than the applied driving force, and so the critical electric field increases.  This type of calculation suggests that the macro models should have a more complicated formulation than most (I think all) of them do.

Another part of the physics that is swept under the rug with the macro models is the distribution of residual stresses and electric fields associated with incompatible domain states.  For example we see pictures of needle-like domain tips all of the time, and there are complicated fields associated with these.  I have not thought about how to systematically study this (some type of lamination theory a la Jiangyu Li and Kaushik Bhattacharya is probably best suited for this problem), but it would be useful to have a better picture of how the residual energy evolves with the concentrations of the different variant types.  The problem here is that I do not think there is a unique answer to this question.  But then, I guess this is always what will happen when you represent a continuous set of internal variables (the positions of all domain walls) with a finite discrete set (volume concentrations of variant types).  My point is that there are certain components of the macro models that can be informed by this type of information.

The short answer to your question would have been that there haven't been many links made between the macroscopic models and phase field models. 

Chad 

Arash_Yavari's picture

Dear Chad:

Are you aware of any atomistic calculations of fracture in ferroelectric crystals?

Arash

Arash,

 No I am not.  I would be keenly interested in seeing something like this.  I think that some of the methods that you are working with might be able to look at this problem.  Perhaps you could post a reply detailing some of the issues that your methods would face while trying to address this.

Chad 

Arash_Yavari's picture

Dear Chad:

The best known and most common interatomic potentials for
ferroelectrics are shell potentials (here "shell" is very different
from what we usually understand as a shell in mechanics). These
potentials were introduced in the sixties by Dick and Overhauser:

Dick, B. G. and A. W., Overhauser [1964], Theory of the dielectric
constants of alkali halide crystals, Physical Review 112: 90-103.

In these potentials, each atom has a core (nucleus and inner electrons)
and a massless shell (valance electrons). The shell is assumed to be
spherical and uniformly charged. What enters into the energy expression
is the position of the center of the shell. Total energy has the
following three parts:

1) Short-range energy: Only shells contribute to this part and for the
most part this is a repulsive energy that prevents shells to get too
close to one another (Pauli repulsion).

2) Core-shell energy: In a given atoms core and shell interact usually
by a nonlinear spring (a fourth order polynomial in terms of the
relative distance). This prevents collapse of core and shell in the
same atom.

3) Electrostatic energy: This is the classical Coloumb potential. For a
given atoms shell and core do not interact electrostatically. This is
the troublesome part as long range interactions have to be very
carefully treated as the lattice sums representing energy and force are
conditionally convergent (Ewald summation method is usually used for
periodic systems).

In this model, spontaneous polarization (for example in the tetragonal
phase of PbTiO3) is mainly due to relative shifts of cores and shells
in the direction of polarization.

These potentials have been used by several groups in understanding the structure of domain walls, free surfaces, etc.

I have done some calculations for domain walls (surfaces on which
polarization is discontinuous, i.e. boundaries between different
energetically-equivalent variants in a single crystal). My experience
with these potentials is that they are very sensitive and the resulting
stiffness matrices could be close to ill-conditioned.

There have been analytic/semi-analytic lattice calculations for cracks
in the past mostly for idealized 2D lattices (an interesting observed
phenomenon was lattice trapping in early seventies). The main
contributers are Slepyan and Marder. However, as far as I can see it,
their methods cannot be used for analysis of something complicated like
PbTiO3 with its fairly complicated interatomic potential. My guess is
that the best approach would be MD.

Regards,
Arash

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