1. Self-polarization in ferroelectric polymers
Serge Nakhmanson
Self-polarization in ferroelectric polymers
I. Polyvinylidene fluoride (PVDF) and its relatives
[a brief reminder]
II. Polarization via maximally-localized Wannier functions
and why it is so good to study polymers [a brief reminder]
III. Projects:
a. Self-polarization in individual polymer
(and copolymer) chains
b. Self-polarization in PVDF: from a chain to a crystal
c. Self-polarization in PVDF/copolymer crystals
IV. Conclusions
Collaborators: Jerry Bernholc and Marco Buongiorno Nardelli (NC State and ORNL)

2. The nature of polarization in PVDF and its relatives
Representatives: polyvinylidene fluoride (PVDF), PVDF copolymers,
odd nylons, polyurea, etc.
PVDF copolymers
PVDF structural unit
with trifluoroethylene
P(VDF/TrFE)
with tetrafluoroethylene
P(VDF/TeFE)
Spontaneous polarization:
Piezoelectric const (stress): up to
Mechanical/Environmental properties: light, flexible, non-toxic, cheap to produce
Applications: sensors, transducers, hydrophone probes, sonar equipment
Weaker than in
perovskite ferroelectrics?

3. Growth and manufacturing
Pictures from A. J. Lovinger, Science 1983

4. Growth and manufacturing
β-PVDF
Pictures from A. J. Lovinger, Science 1983

5. Growth and manufacturing
β-PVDF
PVDF: grown approx. 50% crystalline,
which spoils its polar properties
PVDF copolymers (with TrFE and TeFE):
can be grown very (90-100%) crystalline
can be grown as thin films
stay ferroelectric in films only a few Å thick
People (experimentalists especially) are
very interested in learning more about
copolymer systems, but not much
theoretical data is available

6. What is available? Simple models for polarization in PVDF
“bond-dipole” picture
“structural-unit
dipole” picture
Experimental polarization for approx. 50% crystalline samples: C/m2
Empirical models (100% crystalline) Polarization (C/m2)
Rigid dipoles (no dipole-dipole interaction):
Mopsik and Broadhurst, JAP, 1975; Kakutani, J Polym Sci, 1970:
Tashiro et al. Macromolecules 1980:
Purvis and Taylor, PRB 1982, JAP 1983:
Al-Jishi and Taylor, JAP 1985:
Carbeck, Lacks and Rutledge, J Chem Phys, 1995:
Nobody knows what these “structural-unit” dipoles are and how they change

7. β-phase layout b = 4.91 Å a = 8.58 Å c = 2.56 Å
Orthorhombic cell for β-PVDF:
We will consider:
Chains: 4 x [unit] or 8 x [unit]
Crystalline systems:
4 x [chain with 4 units]
orthorhombic box ~ 10x10x10 Å
a = 8.58 Å
b = 4.91 Å
c = 2.56 Å
We will usually have a large
supercell with no symmetry
Berry phase method with DFT/GGA:
P3 = C/m2

8. Polarization in polymers with Wannier functions
Electronic polarization looks especially simple when using Wannier functions:
Ionic polarization is also a simple sum:
Unlike in a typical Berry-phase calculation, we can attach a dipole moment
to every structural unit:
Unlike in a typical Born-effective-charge calculation for perovskite-type
materials (e.g., “layer-by-layer” polarization), our analysis will be precise
We use the simultaneous diagonalization algorithm at Γ-point to compute
maximally-localized Wannier functions within our real-space multigrid method
(GGA with non-local, norm-conserving pseudopotentials)
See previous Serge’s talk for details
See also Gygi, Fattebert, Schwegler, Comp. Phys. Commun. 2003
See E. L. Briggs, D. J. Sullivan and J. Bernholc, PRB 1996 for the multigrid method description

9. Example: Wannier functions in a β-PVDF chain
centers (WFCs)
in a β-PVDF chain:

10. Example: Wannier functions in a β-PVDF chain
In a VDF monomer
Debye
(1 Debye ≈ 3.336×10-30 Cm)
Wannier function
centers (WFCs)
in a β-PVDF chain:
~ WFC

11. Structural-unit dipole moments in individual chains
A dipole moment of a structural unit in a chain gives us a good “natural” starting
value for a dipole moment of a particular monomer:
VDF
TrFE
TeFE
Debye
Debye
Debye

12. Playing “lego” with structural units in a chain
TrFE 5 3 4 8 7 6 2

13. Playing “lego” with structural units in a chain
TeFE 5 3 4 8 7 6 2

14. Playing “lego” with structural units in a chain
HTTH
defect 5 3 4 8 7 6 2

15. Playing “lego” with structural units in a chain
CHF-CHF 5 3 4 8 7 6 2

16. Now we start packing chains into a crystal and see what happens
Some general observations for chains:
All kinds of interesting structural-unit dipole arrangements along
a chain are possible (experimentalists can not yet synthesize
polymers with such precision, though)
Structural-unit dipoles on a chain like to keep their identities,
i.e., they stay close to their “natural values” and self-polarization
effects are weak
Now we start packing chains into a crystal and see what happens

17. Packing β-PVDF chains into a crystal
noninteracting
chains
Strong self-polarization effect!
weakly interacting
chains
crystal

18. Now we know why simple models disagree!
Empirical models (100% crystalline) Polarization (C/m2)
Rigid dipoles (no dipole-dipole interaction):
Mopsik and Broadhurst, JAP, 1975; Kakutani, J Polym Sci, 1970:
Tashiro et al. Macromolecules 1980:
Purvis and Taylor, PRB 1982, JAP 1983:
Al-Jishi and Taylor, JAP 1985:
Carbeck, Lacks and Rutledge, J Chem Phys, 1995:
β-PVDF crystal
noninteracting chains
Most models fit to this point
and then use this value in
calculations for β-PVDF crystal

19. On to more complex PVDF/copolymer crystals
Now when we know what is going on with β-PVDF crystal, let’s transform it into
a PVDF/copolymer crystal by turning some VDF units into the copolymer ones:
We will “randomly” change some VDF units into TrFE or TeFE taking
into account that they don’t like to sit too close to each other
Volume relaxations will be important
Our grid-based method can not do volume relaxation, we use PWscf/USPPs
to get us to the volume that is about right
Polarization will not be too sensitive to small stress variations
We will monitor structure
Volume and lattice constants
Dihedral angles between units
and polarization
Dipole moment values in structural units: will they keep their identities?
Total polarization
in our models as we change PVDF/copolymer concentration
Not for the faint of heart!

20. This is how a relaxed model looks like:
Example: P(VDF/TrFE) 62.5/37.5 model (6 units out of 16 changed into TrFE)
Front view
Side view 1 2 3 2

21. This is how a relaxed model looks like:
Example: P(VDF/TrFE) 62.5/37.5 model (6 units out of 16 changed into TrFE)
Front view
Top view
Notice that structural
units become staggered
dihedral 1 2 1 3

22. Volume relaxation in PVDF/copolymer models
Elementary cell
with two units a b c

23. Volume relaxation in PVDF/copolymer models
Elementary cell
with two units a b c
Models expand mostly along “1” direction.
There is no change along the direction of the backbone.
Unit staggering is to blame?

24. Dihedral unit-unit angle change
Elementary cell
with two units a b c
Models expand mostly along “1” direction.
There is no change along the direction of the backbone.
Unit staggering is to blame?

25. Dipole-moment change in VDF structural units
β-PVDF crystal
β-PVDF chain
VDF unit dipole moments change a lot when substantially diluted with less polar units
Close to linear drop in unit dipole strength with changing concentration

26. Dipole-moment change in copolymer structural units
TrFE chain
TeFE chain (nonpolar)
Copolymer units become strongly polarized when surrounded by more polar VDF units
Copolymer unit polarization decreases with concentration but never goes back to its “natural” chain value

27. Total polarization in PVDF/copolymer models
β-PVDF crystal
Mapped out the whole “polarization vs concentration” curve!
Linear to weakly parabolic (?) polarization drop with concentration
Considering the “estimative” character of calculations, remarkable agreement with experimental data
Volume relaxation is important: no agreement with experiment at fixed volume
Tajitsu et al. Jpn. J. Appl. Phys. 1987
Tasaka and Miyata, JAP 1985

28. Conclusions
Better understanding of polar polymers in chains and crystals
The nature of dipole-dipole interaction in polar polymer crystals is complex (although, the curves are simple)
Information about the structure and polarization in PVDF/copolymer compounds is now available. It can be used as a guide to design materials with preprogrammed properties.
We have the models now, so that we can do other things with them