# Electric-field-induced antiferroelectric to ferroelectric phase transition in a mechanically confined perovskite oxide

The electric-field-induced phase transition was investigated under mechanical confinements in bulk samples of an antiferroelectric perovskite oxide at room temperature. Profound impacts of mechanical confinements on the phase transition are observed due to the interplay of ferroelasticity and the volume expansion at the transition. The uniaxial compressive prestress delays while the radial compressive prestress suppresses it. The difference is rationalized with a phenomenological model of the phase transition accounting for the mechanical confinement.

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### Coupling free energy in terms of strain?

Dear Wei,

The free energy functions given in equations (1) and (3) are epxressed in terms of stress tensor and polariztion vectors. What if, alternatively, strain is used instead of stress? What would be the explicit expression of coupling free energy in terms of strain and polarization?

Regards,

Jinxiong

### Re: coupling energy in terms of strain

Dear Jinxiong,

The coupling free energy written in terms of stress is similar to the complimentary strain energy.  We can write an alternative total free energy in terms of strain through the Legendre transform:

W(P, Pbar, x) = W'(P, Pbar, X) + X:x

Then, the elastic contribution becomes the strain energy, and the coupling energy is simply

W_c(P, Pbar, X(P, Pbar, x))

where X(P, Pbar, x) is the constitutive relation of stress, Obtained from x = dW'/dX

Xkl = Sklij (xij - α Pi Pj - β Pbari Pbarj - γ P2 δij)

We use stress as the independent variable because in the experiment, we have the stress-free data but no 0 strain data.  It is mathematically simpler to extract the parameters directly from the stress-free data, if stress is used as an independent variable.

Wei

### The general principle of constructing coupling free energy?

Dear Wei,

Thanks a lot for your answer. Intuitively, the terms following xij in the last expression of your reply look like the eigen strain. And the constitutive relation is equivalently to state that stress is equal to the elastic coefficients multiply the difference of total strain and eigen strain.

You mentioned in your paper that the coupling energy should be a combination of invariants: Xii, P2, Pbar2, XijPiPj, XijPbariPbarj. I think the expression is similar if stress invariants is replaced by strain invariants.But can the coupling free energy contain a term with only Xii? Should the expression be constructed to fullfill the requirement of objectivity?

Finally, do you measure the coefficients alpha,beta, gama from experiments?

### Re: coupling free energy

Dear Jinxiong,

Yes, you are absolutely right.  The stress-free strain is the "eigen strain", and our assumption of the free-energy function is equivalent to say that the deformation from the eigen states is linear elastic.

The expression would be similar if we write it in terms of stain invariants, but with an additional term that consists of the eigen strain and polarization only.

A term with only stress should be included in the complimentary strain energy.  It represents the compressibility of the material.

The expression should be objective.  Here it is less of an issue because we are looking at small deformation.

Yes, the parameters α, β, γ are obtained directly from experimental data - Fig 4a - the eigen-strain measurements.

Wei