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)

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?

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

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

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

What is available? Simple models for polarization in PVDF “bond-dipole” picture “structural-unit dipole” picture 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): 0.131 Mopsik 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: 0.086 Al-Jishi and Taylor, JAP 1985: 0.127 Carbeck, Lacks and Rutledge, J Chem Phys, 1995: 0.182 Nobody knows what these “structural-unit” dipoles are and how they change

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 = 0.178 C/m2

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

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

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

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

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

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

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

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

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

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

Now we know why simple models disagree! Empirical models (100% crystalline) Polarization (C/m2) Rigid dipoles (no dipole-dipole interaction): 0.131 Mopsik 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: 0.086 Al-Jishi and Taylor, JAP 1985: 0.127 Carbeck, Lacks and Rutledge, J Chem Phys, 1995: 0.182 β-PVDF crystal noninteracting chains Most models fit to this point and then use this value in calculations 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 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!

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 Notice that structural units become staggered dihedral 1 2 1 3

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

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?

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?

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

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

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

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