Quantum Theory of Polymers

Jean­Marie André

EC ­ Socrates ­ Erasmusprogramme

FUNDP, NamurUniversity of Warsaw

Quantum Theory of Polymers

1° Role of a quantum theory of polymers and its relations with photoelectron spectroscopy

2° LCAO Bloch theory for electronic structure analysis of polymers

3° Electronic structure of conducting and semi­conducting polymers

4° Examples of applications of LCAO Bloch theory:a) graphite and Boron nitrideb) planar, cyclically belt and Möbius polyacenes

5° Electron transfer in polymers: Marcus classical theory

6° Electron Transfer in polymers: Marcus semi­classical theory

7° Energy transfers in polymers: Förster and Dexter mechanisms

8° Non­linear optical (NLO) effects in polymers

Methodology

­ Concepts

­Theory at the simple orbital (Hückel) level

­ Examplified by timely subjects­(semi) conducting polymers­Nonlinear Optics (NLO)­Electron and Energy transfer

­ Combined with personal « trajectory » and « souvenirs »

PS. More examples in the .ppt files

than in the oral lecture

1991

2003

p. 426 et sq.

Quantum Theory of Polymers1° Role of a quantum theory of polymers and its relations with photoelectron spectroscopy

Jean­Marie André

FUNDP, NamurUniversita’ degli Studi dell’Insubria, Como

Methane CH4 - 161 °C -182 °CEthane C2H6 H-(CH2-CH2)-H -89 °C - 183 °CPropane C3H8 - 42 °C - 187 °CButane C4H10 H-(CH2-CH2)-(CH2-CH2)-H + 0,5 °C - 138 °CPentane C5H12 + 36 °C - 130 °CHexane C6H14 H-(CH2-CH2)-(CH2-CH2)-(CH2-CH2)-H + 69 °C - 95°CHeptane C7H16 + 98,4 °C - 91 °COctane C8H18 H-(CH2-CH2)4-H + 126,6 °C - 57 °C

PE HDPE 2,000 – 10,000 CH2-CH2 unitsUHMWPE > 100,000 CH2-CH2 units

Criticism against Staudinger’s ideas ( ≅ 1920)

Dear Colleague, Leave the concept of large molecules well alone; Organic molecules with a molecular weight above 5000 do not exist. Purify your products, such as rubber, then they will crystallize and prove to be lower molecular substances. Organic molecules with more than 40 carbon atoms do not exist. Molecules cannot be larger than the crystallographic unit cell, so there can be no such things as a macromolecule

Quoted in: R. Olby, J. Chem. Educ., 47, 168 (1970).

Polypropylene -CH2-CH(CH3)- CH2-CH(CH3)- CH2-CH(CH3)- CH2-CH(CH3)- -CH3

PVC -CH2-CH(Cl)- CH2-CH(Cl)- CH2-CH(Cl)- CH2-CH(Cl)- -Cl

1999 Polyethylene Production Data from C&EN, June 26, 2000

1,30189013,906HDPE

1,8561,59215,681LDPE

Japan**Canada**US*

* millions of pounds** milliers de tons

1999 European polyethylene production

France 2,770 millions of poundsGermany 2,097Italy 2,585U.K. 734

Ethylene C2H4 H-(CH=CH)-H -103,7 °C - 169 °

C

Butadiene C4H6 H-(CH=CH)-(CH=CH)-H - 4,4 °C - 108,9

°C

Hexatriene C6H8 H-(CH=CH)-(CH=CH)-(CH=CH)-H 78 °C - 12 °C

CH4

C2H6

C3H8

C4H10

C5H12

polyethylene

Basis of Bloch theory

ρ r ja =ρ r ∣ f n r ja ∣2

=∣ f n r ∣2

f n r ja =eikja f n r f n¿ r ja =e−ikja f n

¿ r

Consequencesf n r = f n , k r e k =en k

en k =en klg

en k =en −k

First Brillouin Zone

Half first Brillouin zone

Polymer quantum chemistry ≠1D solid state physics

1D periodicity but 3D orbitals

Band structures, density of states and XPS spectra

Band structures, density of states and XPS spectra

Ab initio calculationAndré, Leroy (1968)

ESCA spectrumAndré, Delhalle, Caudano, … (1972)

Angle-resolved ultraviolet photoelectron spectroscopyARUPS

UenoSeki

FujimotoKuramochi

SigitaInokuchi

Physical Review B 41, 1176 (1990)

JMA, Delhalle, …1975

Study of conformationaleffects on XPS valencespectra:

Polyethylene

Study of conformationaleffects on XPS valencespectra:

Polypropylene

JMA, Delhalle, …1979

CH bonds

CC bonds

-CH2 - CH2 - CH2 - CH2 - CH2 - CH2 - CH2 -

Polyethylene

- CH2 - CH(CH3) - CH2 - CH(CH3) - CH2 - CH(CH3) -

Polypropylene

MetalZero Gap

SemiconductorGap < 2 eV

InsulatorGap > 2 eV

Electrical conductivity

In order to have a net electrical current, electrons must jump from filled levels to empty levels across the bandgap (Eg).

If Eg is large, ≥≈ 2 eV → INSULATOR

σRT ≈ 10-10 Ω−1cm-1

≈ 10-10 S.cm-1

Electrical conductivity

For 0 < Eg ≤≈ 2 eV → SEMICONDUCTOR

10-10 ≤ σRT 10≤ 2 S.cm-1

polyacetylene Eg ≈ 1.5 eV σRT ≈ 103 - 105 S.cm-1

Si Eg ≈ 1.1 eV

For Eg = 0 → METAL

Upon applying an external electrical field, few e- at room temperature (RT) have the necessary energy to jump from valence band to conduction band.

Thermal energy : kT (per particle) or RT (per mole)at 300 K kT ≈ 0.025 eV

RT ≈ 0.6 kcal.mol-1

RT ≈ 2.5 kJ.mol-1

σRT 10≤ 2 S.cm-1

Cu σRT ≈ 6 105 S.cm-1

Ag, Au σRT ≈ 106 S.cm-1

Electrical conductivity

Mobility ≈ average speed of diffusion of the charge carriers (cm/s) as a function of applied electric field (V/cm):

m= cm

s V

cm

Dimension analysis:s=

Scm

=1 . cm

=q . m . n

q . m . n=Cb⋅cm2

V . s⋅1

cm3

¿CbV . s

⋅1cm

=1 .cm

R=

Vi

V . sCb

=

s=n⋅m⋅q N = density of charge carriers cm-3

µ = mobility of charge carrier cm2/V.sq = charge Cb

Electrical conductivityBand model ⇔ Hopping model

Band model: . perfectly ordered material. delocalized wave-functions of holes (in VB)and of electrons (in CB) over whole chain and various chains

Hopping model: . charges localized by vibrations on a single chain. charges “hop” from one chain to another

Time of residency of a carrier on a given polymeric unit =

must be less than characteristic time of a vibration ≈ 10-13 s

W(VB or CB) ≤≈ 0.1 eV → band model

t=hW

=23

10−1 5

W eV

W< kTIncoherent motion of localized chargesGeometry relaxation (polarons)Hopping regime controlled by MarcusActivation energy

W>kTExtended coherent electronic statesNo geometry relaxation (vibration=10-13s)Band-like regime

Electronic Structure of Polymers and Molecular Crystals, J. M. André and J. Ladik, Eds., Plenum, New York (1975).

Quantum Theory of Polymers, J. M. André, J. Delhalle, and J. Ladik, Eds., Reidel, Dordrecht (1978).

Recent Advances in the Quantum Theory of Polymers, J. M. André, J. L. Brédas, J. Delhalle, J. Ladik, G. Leroy, and C. Moser, Eds., Springer-Verlag, Berlin (1980).

Quantum Chemistry of Polymers: Solid State Aspects, J. Ladik and J. M. André, Eds., Reidel, Dordrecht (1984).

C. Pisani, R. Dovesi, and C. Roetti, Hartree-Fock Treatment of Crystalline Systems, Springer-Verlag, Berlin (1988).

R. Hoffmann, Solids and Surfaces: A Chemist’s View of Bonding in Extended Structures, VCH, New York (1988).

J. Ladik, Quantum Theory of Polymers as Solids, Plenum Press, New York (1988)

J. M. André, J. Delhalle, and J. L. Brédas, Quantum Chemistry Aided Design of Organic Polymers for Molecular Electronics, World Scientific, Singapore (1991).

J. M. André, D. H. Mosley, M. C. André, B. Champagne, E. Clementi, J. G. Fripiat, L. Leherte, L. Pisani, D. P. Vercauteren, M. Vracko, Exploring Aspects of Computational Chemistry, I. Concepts, II. Exercises, Presses Universitaires de Namur (1997).

Bibliography