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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