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) chainsb. Self-polarization in PVDF: from a chain to a crystalc. Self-polarization in PVDF/copolymer crystals

IV. Conclusions

Self-polarization in ferroelectric polymersSerge Nakhmanson

Collaborators: Jerry Bernholc and Marco Buongiorno Nardelli (NC State and ORNL)

The nature of polarization in PVDF and its relatives

Spontaneous polarization:Piezoelectric const (stress): up to

Mechanical/Environmental properties: light, flexible, non-toxic, cheap to produceApplications: sensors, transducers, hydrophone probes, sonar equipment

2C/m 2.01.0 2C/m 2.0 Weaker than in

perovskite ferroelectrics?

Representatives: polyvinylidene fluoride (PVDF), PVDF copolymers, odd nylons, polyurea, etc.

PVDF copolymerswith trifluoroethylene

P(VDF/TrFE)

PVDF structural unit with tetrafluoroethyleneP(VDF/TeFE)

Growth and manufacturing

Pictures from A. J. Lovinger, Science 1983

Growth and manufacturing

Pictures from A. J. Lovinger, Science 1983β-PVDF

Growth and manufacturing

β-PVDF

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

PVDF: grown approx. 50% crystalline,which spoils its polar properties

People (experimentalists especially) arevery interested in learning more about

copolymer systems, but not muchtheoretical data is available

What is available? Simple models for polarization in PVDF

Experimental polarization for approx. 50% crystalline samples: 0.05-0.076 C/m2

Empirical models (100% crystalline) Polarization (C/m2)

Rigid dipoles (no dipole-dipole interaction): 0.131Mopsik and Broadhurst, JAP, 1975; Kakutani, J Polym Sci, 1970: 0.22 Tashiro et al. Macromolecules 1980: 0.140 Purvis and Taylor, PRB 1982, JAP 1983: 0.086Al-Jishi and Taylor, JAP 1985: 0.127Carbeck, Lacks and Rutledge, J Chem Phys, 1995: 0.182

“bond-dipole” picture “structural-unit dipole” picture

Nobody knows what these “structural-unit” dipoles are and how they change

β-phase layout

Berry phase method with DFT/GGA:P3 = 0.178 C/m2

a = 8.58 Å

b =

4.9

1 Å

c =

2.5

6 Å

We will consider:

Chains: 4 x [unit] or 8 x [unit]

Crystalline systems: 4 x [chain with 4 units] orthorhombic box ~ 10x10x10 Å

Orthorhombic cell for β-PVDF:

We will usually have a largesupercell with no symmetry

Polarization in polymers with Wannier functions

occ occ

22

nn

cellnnn

cell

el rV

eWrW

V

eP

l

llcell

ion beZV

P 1

unitn

nunitl

llunit rebZed

2

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

Example: Wannier functions in a β-PVDF chain

Wannier function centers (WFCs)

in a β-PVDF chain:

Example: Wannier functions in a β-PVDF chain

unitn

nunitl

llunit rebZed

2Wannier function centers (WFCs)

in a β-PVDF chain:

~ WFC

In a VDF monomer2unitd

Debye

(1 Debye ≈ 3.336×10-30 Cm)

Structural-unit dipole moments in individual chains

A dipole moment of a structural unit in a chain gives us a good “natural” startingvalue for a dipole moment of a particular monomer:

VDF

0unitd

Debye77.0unitd

Debye2unitd

Debye

TrFE TeFE

Playing “lego” with structural units in a chain

5

3

4

8

7

6

2

TrFE

Playing “lego” with structural units in a chain

5

3

4

8

7

6

2

TeFE

Playing “lego” with structural units in a chain

5

3

4

8

7

6

2

HTTHdefect

Playing “lego” with structural units in a chain

5

3

4

8

7

6

2

CHF-CHF

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

Packing β-PVDF chains into a crystalno

nint

erac

ting

chai

ns

wea

kly

inte

ract

ing

chai

ns

crys

tal

Strong self-polarization effect!

unitn

nunitl

llunit rebZed

2

Now we know why simple models disagree!

Empirical models (100% crystalline) Polarization (C/m2)

Rigid dipoles (no dipole-dipole interaction): 0.131Mopsik and Broadhurst, JAP, 1975; Kakutani, J Polym Sci, 1970: 0.22 Tashiro et al. Macromolecules 1980: 0.140 Purvis and Taylor, PRB 1982, JAP 1983: 0.086Al-Jishi and Taylor, JAP 1985: 0.127Carbeck, Lacks and Rutledge, J Chem Phys, 1995: 0.182

β-PVDF crystal

noninteracting chains

Most models fit to this pointand then use this value incalculations for β-PVDF crystal

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 importantOur 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 structureVolume and lattice constants

Dihedral angles between units

and polarizationDipole 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!

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

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

1

2

1

3

Notice that structuralunits become staggered

dihedral

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?

Elementary cell with two units

a

b

c

Volume relaxation in PVDF/copolymer models

Models expand mostly along “1” direction.

There is no change along the direction of the backbone.

Unit staggering is to blame?

Elementary cell with two units

a

b

c

Dihedral unit-unit angle change

Dipole-moment change in VDF structural units

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

β-PVDF crystal

β-PVDF chain

Dipole-moment change in copolymer structural units

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

TeFE chain (nonpolar)

TrFE chain

Total polarization in PVDF/copolymer models

β-PVDF crystal

Tajitsu et al. Jpn. J. Appl. Phys. 1987

Tasaka and Miyata, JAP 1985

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

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