CHAPTER-!

NBN-LINEAR DYNAMICS OF PHASE TRANSITIONS AND PROTON MOTION IN MONOATOMIC AND DIATOMIC LATTICE CHAINS

1 .1 Introduction

1.2 Non-linear lattice Dynamics

1.3 Limitations of Linear Lattice Model

1.4 Solitary Waves in Non Linear Monoatomic and Diatomic Lattices

(i) Krumhansl - Schrieffer Lattice model for a monoatomic Lattice with Non Linear onsite potential

(ii) Dynamics of Diatomic chain containing Non Linear onsite potential: 'Henry-Oitmaa model

(iii) Ferro electric Phase Transition and the Dynamics of Diatomic chain with anion containing anharmonic Polarisable Shell-Core: Bilz model

(a) with SPA

(b) without SPA

1.5 Non Linear Excitations and Proton Motion in Hydrogen bonded system

(i) Hydrogen bonded system

(ii) ADZ model

(i i i) Pnevmat i kos model

(iv) Doubly periodic potential model

introduction

The main thrust of this work is to develop a non-linear lattice dynamical model to

explain the motion and transport of protons in hydrogen bonded ferroelectrics. The modcl is

essentially one-dimensional. During the past 15 years Bilz and his coworkers [I-51 have

developed a one di~ncnsional diatomic latticc non-linear dynamical modcl for explaining the

ferroelectric phase transitions in perovskites. Pnevmatikos and others [6-101 independently

developed a diatomic non-linear lattice dynamical model to explain the proton motion in

hydrogen bonded protonic conductors which are present in biological and chemical systems.

The proposed model in this work is a combination of the two above mentioned studies with

some modifications. Given bclow is Lhc rcvicw of non-linear lallicc dynamical studics on

monatomic and diatomic lattices with special emphasis on ferroelectric structural phase

transitions and prolon transport.

1.2 Non Linear Lattice Dynamics

Fermi - Pasta - Ulam problem [Ill together with the explanation by Zabusky and

Krushkal El21 can be considered as the origin of lattice solitons. After their work, the study of

nonlinear excitations in lallicc dynamics has bccome a very active Iicld of research . Till thcn

one dcalt with the problcm by lincar methods. But majority of the physical problcms arc

nonlinear in nature and the linear approach is unsuitable. Equipped with modern Icchniyucs for

the study of non linear systems through analytical (IST, Lax pair, Logistic map, ...) as well as

numerical methods (with the advent of fast modern computers), there are excellent prospects to

investigate the general non linear lattice dynamics, in which linear dynamics is the special

case. With thcsc tcchniqucs on hand, old physical problems can also be rcvicwcd.

2

120r example, if a syslcm undcrgocs slructural phase transition, the atoms arc displaced to

a largc extent Srom Lhcir equilibrium posilions near lhc critical poinl. The siluation is bcst

described by nonlinear dynamical equations. As there are no known analytical solutions,

perturbation and linearising the nonlinear terms are followed. But these schemes are

unsatisfactory. Hence solving such equations retaining nonlinear terms are necessary, as it will

give better insight into the dynamics.

1.3 Limitations of Linear Lattice model

CrysLallography generally concerns wilh the slalic propcrtics of crystais, describing

Scaturcs such as the average position of an atom and the symmetry of a crystal. Thc static

models ( Free electron model, Sommerfield model and Kronig Penny model ) which take into

consideration the average position of core atoms and neglect their motion ( but deal with the

dynamics of electron ) can explain a large number of material features such as chemical

propcrlics, malcrial hardness, shapes of cryslals, optical propcrtics, Bragg scattering of

X-ray, clcclron and neutron beams, classification of solids, electronic structure and electrical

propcllies etc..

However there are large number of solid properties that cannot be explained by a static

model. They are

Thermal properties, c.g. heat capacity

ESSect of tempcraturc on the lattice, c.g. Ihermal expansion

The existence of phase Lransilion, including mclling.

0 Transport properties. e.g. thermal conductivity, sound propagation

Certain electrical properties like superconductivity.

Interaction of radiation ( light, neutron,.) with matter.

3

To overcome these difficulties, linear lattice model which concerns the vibration of atoms about

their cquilibriuln was inlroduced. In this modcl cacla mass point is conncctcd with its neighbour

by harmonic springs. The springs provide the restoring force on the mass points when they are

displaccd li'om their cquilibriutn posilion. Using this modcl, bchaviour o f monoatomic and

diatomic chains were studied. Finding normal modes and characteristic frequencies of a crystal

is the ultimate aim of thcse studies. Quantum mechanical treatment of the modes lead to Ihe

phonon, the quasiparticle solution.

The harmonic phonon model has given us many reasonable results such as phonon

frequencies, normal mode amplitude, mean square atomic displacerncnts, elastic constanls ctc.

Howcvcr thcrc are a number of material properties that still cannot be cxpcctcd within the

Sramework of harmonic approximation. They are

Temperature dependence of equilibrium properties.

t thermal expansion

t temperature dependence of elastic constant

t temperature dependence of phonon frequencies

+ natural linewidth of phonon frequencies

Occurrence of phase transition.

9 Transport properties . e.g. thermal conductivity

This is because of two assumptions in a harmonic crystal.

(i) Small oscillation: Although the atoms are not rigidly contained to their equilibrium sites,

their displacement from those equilibrium sites arc small.

(ii) Narrno~zic un~roximatiorz: One retains only leading non vanishing term in thc cxpansion

of the inlcraction potential about its equilibrium value.

4

Botll assumptions are invalid at phase transition point, because at transition the amplitude of

oscillation is high cnough Lo canccl Lllc restoring forcc ( soil modc ) and thc atoms can occupy

ncw equilibrium silcs. Far horn i.ransilion poinl, assumplion oS small oscillation is acccplable.

One might think that anharmonic terms are cosrection to the harmonic approximation giving

rise to higher precision but this is incorrect. The reason is due to the neglect of higher order

terms in the expansion of the interaction potential about its equilibrium value.

1.4 Solitary waves in non linear monoatomic and diatomic lattices

(i) Kru~~xhansl - Schrieffer (KS) model for a monoatomic lattice with non-linear onsite

potential

Both FPU and Toda monoatomic lattice chain models assumed neighbouring intersite

interaction in a chain as anharmonic[ll, 131, They can explain thermal conduction in solids.

Krumhansl and Schxieffer proposed a monoatomic lattice chain model containing nonlinear

onsite potential[l4]. This model explains structural phase transition successfully.

Structural phase transition takes place due to the instability of the crystal with respect to

some lattice displacement pattern, which takes the system from a stable high temperature phase

to a different low temperature lattice configuration.

The dynamics of such system is characterised by a vibrational mode whose frequency

decreases rapidly near the critical temperature. The associated restoring forcc wcakcns or

soltens and hence the term "soft modc". Such a soft mode assisted phase transition is known as

displacive transition.

5

K_lumlransl and Schricl'lcr [14] priposcd a simplc ~nodcl capablc of explaining slruclural

phase Lransi~ion. The displacivc transition was carlicr lrcalcd by pcrturbalion theory using somc

set of self consistent high temperature phonons as a basis. But at the transition teinpcrature the

displacement relative to that lattice becomes large and no perturbation technique is expected to

be satisfactory. Even far from equilibrium at T< T, , this soft mode model approach could not

explain ferroelectric properties. Obviously higher order terms in the interaction potential should

The one dimensional model consists of a linear chain of atoms wilh one atom per basis.

The atoms interact harmonically only wilh their nearesl neighbours. Each alonl which is a1 a

lattice site lies in an anharmonic doublc well potential created by the rest of the lattice.

Tlis model is suitable for explaining structural phase transition because there is a

provision for two different equilibrium positions which is essential for this kind of a phase

transition. The model is shown in fig (1.1). Springs arc uscd LO show the ncarcsl ncighbour

hasmonic interaction and double well potential is used to indicate subslrate onsite potential for

each species.

m + Unil +

cell

Fig. l . la One Dimensional Lattice Chain: KS Model

Fig. l . l b Double Well Potential

The onsite potential is given by

where ui is the displacement of ifi atom.

The Hamiltonian 01 the system is

anharmonic harmonic K.E. onsite potenlial intersite potential

where

C, (+vc) - Force constant between neighbouring displaced atoms,

i, j - Neighbouring lattice sites,

Ui - Displacement of the ions with respect to reference lattice,

Ui - Velocity of the displaced atom,

m - Mass of the atom.

From equation (1.1)

or -A + U U: = 0 where Ui = uo at the minima

. u, = ~t ~ A I B

This model can be used to explain both order-disorder and displacive type transitions

(a) If the depth of the well is large (i.e, large barrier height) then the interacfion

energy is not large enough to lift the particle over the barrier.

barrier height interaction energy

where 'a' is lattice constant.

Thermal fluctuation at individual sites can lift the particle across the barrier. Effectively we

have a collection of weakly coupled anharmonic oscillators. Hence the atoms occupy one of the

wells and randomly oscillate about rt u, .

(b) In displacive transition the intersite potential is strong enough to lift the particle

over the barrier.

Fig. 1.2 Transition from one Minima to another Minima over several Unit cells.

The figure (1.2) shows transition from one minimum of the double well potential (at -u, ) to the

other minima (u,) as x -+ x . This appears as kink solution for displacive phase transition. In

the following we shall be concerned only with displacive transition. Real ferroelectrics are three

dimensional and have long range force. The methods available for soft mode treatment at finite

lcmpcralurc arc:

( I ) Harlcc approximalion ----+ <ui2>=constant i.c. pscudo Harmonic Harniltonian.

(2) Mean field approximation --, Depends on inter ion displacement.

(3)Continuum approximation --+ This approximation limits the displacement field which

does not change radically over a lattice spacing (also

called long wavclcnglh approximation).

Continuum approximation

Using Taylor's scrics cxpansio~l

ui ---+ u(x,t) ---+ u

+ u(x+a,t) + u + au' + a2u"/2

where d u d2u

u' = - ; u " = - dx dx2

a2ci, where c:= -

C, is the velocity of low amplitudc sound wavc (phonon) which would occur if A and 13 arc

negligible i.e. only interaction between displacing atoms are important.

Hamiltonian in continuum approximation is

where a is the lattice spacing and p is the momentum.

xj = x locates a species in continuum representation.

From the abovc Hamiltonian one can arrive at thc equation of motion in the continuum form.

Introducing the following dimensionless quantities

(c:-v2) u ( x- vt) m = 52 ; - = ? ? ; = S

I A l UO 5

and using the fact U: = NB (equation 1.4)

equation of motion becomes

The solutions of the equation are discussed below.

Small Amplitude Solutions :

(i) For small amplitude q3 c< q << 1, the equation of motion is q" + q = 0 and the solution

is

q = a sin (st-6) or u = a u,sin[ (x-vt) / E + 0) ]

where a is amplitude and 0 is phase.

The solution is no thing but a phonon with wave vector q = 6-' , frequency is v/{ and phase

velocity is v. These small amplitude phonon oscillates about u=u,. The dispersive relation is

(ii) Another set of small amplitude oscillation can occur if all particles are displaced and

lowered in energy to the bottom of one of the wells. Let q = I+ y where y is small

dimensionless displacement. The equation of motion becomes y" - 2y + 0 (y) = 0 for

which the solutions is small oscillation about ku, and of the form

with the dispersion relation o+2 = c;q2 + 21 All m

Exact Solutions:

d2rl One can obtain solution for the general equation - + q - q 3 = 0

ds2

11

wilhout ally approximation by elliptical function method or by integration wilh suitable

houndiu'y condition. Thc solulion is

q = tanh ($42 )

u = u, tan11 [ (x-vt)/ 42 ]

The displacement is constant at -u, over (x-vt) < 0 and + u, over (x-vl) > O. The transition

takes place through a wall of approximate thickness 242 5 and the wall moves with velocity v.

From the definition of 5 , C, is the upper limit for the drift velocity.

Fig.l.4 Kink Solution.

This solution is a kink type soliton of the ( P ~ type.

(ii) Dynamics of Diatomic Chain containing non-linear onsite potential: Henry - Oitmaa

Model.

Lattice solitons have bccn extensively studied for a diatomic linear chain in both discrctc

and continuum limits with non-linear onsite potentials [I-101. Many of the solids which undergo

displacive phase transition have diatomic structure in the (100) direction, for example Barrio3 .

The diatomic model [15-181 for displacive phase transition is an obvious extension of the

monoatomic model examined in the previous section. This model [I61 is an extension of KS

12

model [Id]. It consists of a diatomic chain of harmonically coupled nearest neighbour atoms M1

and Mz wilh a non lincar potential lor spccics M1 as in fig (1.5).

B---------- Unit cell

Fig. 1.5 Orle Dimensional Diatomic Chain: f.10 Model.

The Hamiltonian for the discrete lattice is

where u, and v, are the displacements of MI and M2 atoms in the nth unit cell.

The corresponding equations of molion arc

For displacive transitions, y > I V(u,) I / u:

where k u, define the position 01 minima of the double well potential. To go over to thc

continuum we apply Taylor's scries expansion and neglect higher ordcr terms.

un L u(x,~) L U

v, - v ( ~ + ~ / ~ , t ) - v +l /2av7 ( '12 a ) 2 v " +...

v,.~- V(X - a l ~ , t) - v - av' + ('12 a )' v" - ...

a is the lattice constant. The continuum Hamiltonian is

H =I dn ['I2 MI b2 + 'h ~2 3 + y ( u - v12 + 'L y a2 u' v' + ~ ( u ) ] ...( 1.15)

Thc equations o l motioll are

(a) Linearised Periodic Solution (Small amplitude) :

V(u) = - '12 A U ~ + '14 B U ~ , (A, B > 0) wilh potential minima at i.e. u, =kl/ NB

These linearised solutions are low energy phonons and represent oscillation of MI atoms in one

of lhe double well (ku, ). The solutions are

u = If: u, + UL sin (kx - mt + Q) ) , UL << 1

v = k u , + v L s i n ( k x - @ t + ( D ) , v L < < l

with the dispersion relation

The lower branch of the above equation R2 (-) cannot be identified as the acoustic branch

as ~t~ (-) I k = ~ + O , unlikc in the case of harmonic diatomic chain. 13ul in liic lo~lg

wavelenglh limit ie. ka << 1, the lattice will bc highly displacivc and we can show that

(b) Solitasy wave solution (large amplitude solution) :

The lincariscd pcriodic solutions arc low amplitude solutions. The field cquations also

support largc amplitudc solutions. ']The simplcsl large amplitudc solutions dcscribc a slalic u

field and a static or oscillating v field.

U = + u o

v = u i: v, sin (kct + ) sin (kx t Qr2)

where k = (8/a2)'" and c =(ya2/4~2)112

The particular solution v = u = +u, is the lowest energy (ground state) solution. The cnergy

of this solution is derined as the reference level for the other solutions which can be regardcd as

excitations above lhis level. The energy level difference

i s the cxcitalion cnergy oS thc soliton.

15

A nlorc interesting class of large amplitude solutions is a solitary wave which is defincd as

localised travelling wave [19]. In order to investigate the field equations for solitary wave

solutions we make the following substitution.

with s = x-ct whcrc c is lhc spced of the travelling wavc in the (ficld) equation of motion. This

gives a set of equations.

For the particular characteristic velocity c = c, = (ya2/4~2)1". The equation (1.25b) tells us

that t l~e Iwo displacement fields will be equal. The first equation gives us the structure of these +

fields. i.e.

By including $"oublc well polenlid1 in equation (1.27), the following solilary wavc solutions

are obtained.

(i) For M1 < M2, c = 2 V (u,) yields the kink solution.

v (x,t) = u(x,t) = rt u, tanh (slk k )

(ii) For MI > M2, c = 0 yields pulse solution

v (x,t) = u(x,t) = f d2u, sech (sft )

A morc general solution than 11icsc equal displacement field, is a large amplitude solitary wave

solution which includes an oscillation ( acoustic phonon ) in the v field.

(c) Non linear periodic solution

l;or c + c,, the displaccrnent iicld is no1 equal and the travelling wavcs lnay bcco~xcs

extcndcd inslead of bcing localiscd. In this casc Lhc cquation of motion (1.25) may bc

decoupled in which casc a Sou1111 ordcr non linear differential equation for f is obtained.

with

...( 1.30b) V is given by the double well potential

Considering the solution for a particular velocity c = [ ya2 / ( M1 M2 ) ' I2 lU2, equation

(1.30a) now becomes second ordcr nonlinear diffcrcnlial equation

Equalion (11.31) can be solved by standard techniques [20] using exact intcgral arlci hq

clliptic integral Lcchniques. T i ~ c solutions arc

f = So sin (ks) g = go sin (ks) + gl sin (3ks) ...( 1.32)

the ainplitudcs of the non linear periodic solutions are

I

2 y - ya2 k2 B with go = f S I = fO3 ; c = -

2 y - M 2 c k 9 y a 2 k 2 - g y [$J t'M] I2

These solution oscillate with characteristic frequency o = ck = ( 2y / 9 M2-) 1 i2

The non linear phonon of equations (1-32) represent high energy phonons.

The solutions for non-linear oscillations in the discrete lattice arc given as

un = Fo sin ( w t - n q a) ...( 1.34)

v, = G , s i n [ w t - ( n + '12)qa] + Glsin [ 3 ( o , l - ( n + ' / 2 ) y a ) j

with amplitudes

2 y cos ('12 q a) B Go = " G1 = - I;:

2'y- W ' M ~ 8 y cos (3/2 qa)

Tllus the nonlinear phonon is tlic continuum analogue of discrete lattice pcriodon.

18

( i i i) Il'erroeleclric Pliase Transition; and the dynanlics of diatomic chair! wit11 arkion

containing anharmonic polarisable shell - core: Bilz model.

It has been observed that 90 % of the ferroelectric materials contain anions of the

chalcogenic group and oxygen. Migoni et a1 [21] proposed that the large anharmonic and

anisotropic polarisability of the oxygen ion is responsible for the lattice instability in

ferroclectrics.

(a) Non linear shell model with SPA approximalion

Bilz et. al. [I] considered a diatomic chain with non linear polarisability in one species

and harmonic coupling between nearest and second nearest neighbours as shown in 13g.1.6.

They obtained the dispersion relation using self consistent phonon approximation(SPA).

f--------- Unit cell

Fig. 1.6 One Dimensional Diatomic chain with Polarisable anion

The equation of motion for the abovc system arc

wlmc u l n , Urn and v, --- displacclncnt of nIh spccics of M1 , M2 and shell rcspcclivcly,

w,, = (v, - uln) --- rclalivc shcll core displacemcnt of the anion.

Applying the adiabatic condition Me ?, = 0, equations of motion become

In Sl'A tile cubic term is replaced by a lincarised term.

where <w:> is the self consistent thermal average over the square of shell displacement w,, at

ternpcraturc T. Assunling the solution

one gets the dispersion relation

- where

g f f = - and g = g, + g2 and q is the wave vector.

2f+g

At q=O, the frequency of the acoustic mode a- becomes zero and the frequency of

- 2 C %2 g

the optical mode is (q=O) = - = - P 2 f + g

2f The optical mode corresponds to the I'erroelectric mode (q ) , a2 = - is the rigid ion

P lrcqucncy and p is the rcduccd mass. On decreasing the tcrnperaturc, g decrcascs and hcnce w

decreases. This lowers the ferroelectric branch into the low frequency acoustic region.

At phase transition g = 0 and cq2 = 0, the transverse optical branch degeneraies into a pseudo

acoustic branch. The solutions are

For small g , the splitting of the pseudo acoustic branch can be neglected.

21

q2 (q) is tcrnperaturc dcpcndent. As the temperature varies g changes giving rise to a changc in

frcqucncy. Thus t11c inclusion oS the highly localised electron-ion coupling could explain the

temperature dependence of the ferroelectric softmode in SbSI and AB03 perovskites.

(b) Non linear shell model without SPA approximation

BuUner and Bilz [2] Scll Lhal a nlorc accurate solutioil can bc obtaincd by rc~aining Lhe

non linear tcnns and succcedcd in finding tllc cxact solulion for the modcl. Thc equations 01'

notion art:

g4 where 3

Zn = (vn - u~n) + - (vn - utn) S2

By applying the adiabatic condition ( Mi?= 0) and after some transformation one arrives at the

4' order differential equation containing w, and z, where w, = v, -uln

4('w,+ f c ) - ( u I 2 + @')D;,+ Y ( 4 0 $ 2 - ~ 2 ~ ) ' i ,

= m2 D [ -'/4 @2 D wn + Y ( + a 2 ) zn ] ...( 1-46]

wherc D X, = X,I f X,.I - 2X,, and ...( 1.47)

~01 '=2f lMi , ~ ' = 2 f l M ~ , W 1 2 = 4 f ' / ~ I , (U ,2=2f /p= 0:+ a2 , y = g 2 121,

y determines whether the transition is of displacive or order disorder type.

22

From Lhc 4"' ordcr dil'l'cscncc dil'lcscntial cquation Butlncr and Bilz obtaincd t l~c foliowing

solulions, viz

i) Static solution ( ground state and excited state solutions)

ii) Time dependent solution (running kinks and oscillatory solution viz phonon &

periodon)

l i ) Static solution

They are defined by kn =O

(1 .a) Ground state ( independent of n )

The solution with lowest energy is w: = (v, - uln?: = - g2 1 g4

and (~2n)o = (vn)o ; MI U l n + M2 ~ 2 n = 0

In the limit , M2 >> M1 , ul, >> u2, = v, + 0 and W: + ulo 2

In the limit , M2 << MI , ulo <c u~~ = V, and W: 3 vlo 2

The ferroelectric ground state is represented in this case by a non linear displacement of the ionic

core against electronic charge density at the anion cluster . The finite electron-core displacement

w, is analogous to the static ionic displacement in the double well problem. Howevcr the

double well results in this model from a non linear interaction between electrons and ions. Thc

vanishing of the displacernent ul , u2 in the limit M2 << MI does not mean that all parlicles arc

still sitting at the high temperature undistorted lattice sites, but displacing electronic shell

against their core to leave the cluster mass center at rest.

The energy of the ground may bc obtained as

:vhere N is the no. oC unit cells in the chain.

(1.b) Excited static solution

It is controlled by the right hand side of equation (1-46) i.e.

Allalytical solutions cxist only in contilluum approximation, whcrc thc finite: diff'crcncc is

n:placed by derivatives of Taylor's series. This solution is given as

w, =w, tanh(px) where (2pa2) = 2 ly I l+ - [ I f ] (iii) Time dependent solution (oscillatory solution)

(2.a) Phonon

Small anlplitudc phonons abovc thc ground slalc arc lound by taking lfic

ansatz w, = w, + a, and linearising equation (1-46) with respect to a,. The solutions dcscribe

renormalised phonon which is identical to the harmonic case with gz replaced by 2 lgzl

(2. b) Periodon

From equation (1.46), periodic non linear waves(periodon) are obtained. The simplest

form of the stationary wavc is

w, = A sin (At - 211pa)

wikh thc Sscqucncy h

r

In the limit of 03 << o), (rigid ion phonon model) , equation (1.51) has the simple form

The corresponding dispersion of the rigid ion phonon model is obviously

u21,2 (pa) = 3 h212 ( p a 4 = (pa)

( h: (pa) is ihc rigid ion Sscqucncy of thc harmonic modcl )

This means that the frequencies of the non linear waves are subharmonics of the con-esponding

phonon frequencies with commensurate scaling by a factor of 3. At the same time, the phases

are multiplied by a factor 3 which leads to a reduction of the Brillouin zone to one third of its

size in the case of the phonon. In contrast to the linear problem the amplitudes arc also

determined by the equations of motion. The corresponding solutions for the individual ions arc

w, = A sin (kt -2npa)

ul, = B sin (kt - 2npa) + C sin [ 3(kt - 2npa) ]

- D sin [ ht - (2n+l)pa ] + E sin 3[ h t - (2n+l)pa ] U2n - ...( 1.53)

Ssom wllich a~nplitudcs can be found.

8 S [ h2j ,2 - w2 sin 2pa ] [ h2, ,2 - W22 sin ] A , 2 = - [ i -

3g.1 1 h21,z - h2+(pa) I 1 h21,2 - h2-(pa) I 1

In gencral these soluiions exist only as long as thc frequency is not equal to one of Lhc rigid ion

Non Adiabatic solution

If M,;;, t 0, one can obtain an equation which is of the sixth order instead of equation

(1.46) which is fourth order in time. Even though onc could not find exact solution of the

equation, one may obtain plasrnon likc cxcitalions of thc clcctrons which arc cohcrcntly moving

along with thc vcry low Srcqucncy cxcitalions oC thc latticc.

It can be shown that plasmons are not involved in the ferroelectric phasc transition since

they are stable even in harmonic case. So, it is the effective electron-lattice interaction ie.

negative g2 which. determines the ferroelectric properties.

26

1.5. Non linear Excitation and Proton Motion in Hydregen Bonded System.

( i ) l i ydrogcrl boildcd sysle~ai

Onsagar associated the conductivity in hydrogen bonded system , which is not electronic

but protonic in nature, to a hopping mechanism that allows the protons to move along hydrogen

atomic channels [22]. It has been recognised since the work of Bernal and Fowler that the

anamolously high mobility of prolons in hydrogcn bonded network cannot he accoun~cd lor by

slandaid thcories [23] . This old problem of clcctrical (proton) conductivity in hydrogen bondcd

crystals has been reviewed with the introduction of new techniques and ideas rrom non linear

physics [ 6-10, 24-34 1.

As hydrogen bonds are the building blocks of many biological materials llkc DNA,

protein etc., and condensed matter molecular systems like ice, imidazole, lithium hydrozonium

sulphate, ferroelectrics like KDP etc., understanding electrical properties of such systenls will

providc a wcallh of inrormation Ih r physical and biological systems and psoccs~cs like proion

transport across ccllular mcn~bsancs, thc proton pump, muscular contraction, change of

inl'ormation across neural networks clc.

Hydrogen bond and double well potential:

Hydrogen bond is formed due to dipole interaction. It occurs between molecules in which

one end is hydrogen atom. When a hydrogen atom is covalently bonded to relatively large

atoins such as oxygcn, nitrogen, flourinc, ctc. a largc pcrmancnt dipole momcnt is scl up. This

is bccause the cleclron clouds tcnds to bccomc conccntratcd around thc part ol' thc molcc~ilc

containing oxygen, thus leaving positively chargcd hydrogcn unprotected ic. hare proton. So, a

strong dipole is created that can bond to other similar dipole with a force called hydrogen i.sond.

Hydrogen bond

x - - - - - - - - ----- Covalent hond

Fig.P.7 Hydrogen Bonded System

Hydrogen bonds are formed always in addition to other bonds. For example, in the case

of icc, hydrogen atom forms covalent bond with oxygcn atom and the bare proton forms a

hydrogen bond with another oxygcn atom. In a network of alternate covalent and hydrogcn

bonds, thcse arc two cncrgclically cquivalcnl positions for each proton bctwccn two oxygcn

atoms. Hence the covalent and hydrogen bonds interchange with the proton between two

oxygen atoms. The covalent and hydrogen bond are interchanged due to the proton tunneling

t l ~ o u g h the potential barrier. Therefore each proton can be associated with a potential in the

form of a double well with two minima corresponding to the two equilibrium states [35]. The

doublc wcll potential is rcsponsihlc for the non linear nature of hydrogcn bonded network.

Thc double wcll potential function U(u) in terms of proton displacement can be written

where E, - Height of the potential barrier

u - Ilisplaccmcn~ of proton from the barrier lop

UO - Minimum position of the potential well.

28

Tiic study oS pmton conductivity in watcr molcculc chain is bascd oo tllc lollowing

concept. A proton niay bc iransfesscd along the chain in the form of

(a) Ionic defects and

(b) Bjerrum defects.

Thc ioiiic dclcct is (brnicd by ~ h c dissacialion uS walcr ~nolcculc 1120 inlo Ilydroxonium 0 1 3 0 ' )

and hydroxyl (OH- ) due lo the transfer 01 a proton from one water molcculc to ncighhooring

molecule in accordance with

2H20 - H3Q' + OH-

The Bjerrum defect is formed due to the rotation of water molecule and with respect to the sense

of rotation we delime positive (D) and negative (L) defects.

I H H

Fig.l.8 Ionic and Bjerrum Defects in a Hydrogen Bonded Chain

29

The ionic (01-1- and 1-130t) and the oricnta~ional (L and D) dcfccts arc f'ormcd only in Lhc iilicidk

of the chain in pairs. In such a chain each watcr molcculc supplics onc proton lo providc Ihc

hydrogen bonding formation. A second proton does not participate in the hydrogen bonding and

is linked with an oxygen atom and is considered as one unit ie. as the hydroxyl ion 013-

(Fig 1-8). So, thc chain of water molecule is divided into two subsystems -- the basic lallice

forrncd by the hydroxyl group and the proton subsystem.

(i i) ADZ nlodcl

The basic idea of the modcl proposed by Antonchenkov, Davydov and Zolotariuk [ 24 ]

is that the coupling between oxygen atom and proton can provide a mechanism which changes

the polentiai barricr that protons havc to overcome to jump from one molecule to anotl~cr and

thereby making their motion easier. They introduced a two sublattice modcl for ;?iI one

dimensional lauice. The Hamilronian of the system is

Here

is the I-famillonian of thc proton subsyslem.

where u, - Displacement of nth proton from the top of the potential barrier placcd in thl:

middle between the n~ and (n+l) 'h OH- ions

m - Mass of the proton

n1oI2 - Elasticity coefficient of the proton subsystem

is the f-Iamiltonian o l the basic lattice OH- ions with mass M, and R, & Ql arc the

charactcristic frequencies, p, is the relative displacement between OH- ions.

Finally,

2 Hint = x C pn (un2- uo 1 ...( 1.59)

is 111e Hamiltonian of the interaction between proton displacement and hydroxyl ion

displacement.

In the coiltinuurn approach

..-(1-60)

where a - Lattice constant.

c, = a o - Velocity in the proton subsystem (proton sound)

v, = a Ct1 . Characteristic velocity which takes into account the dispcrsion of sound

The equations of motion are

U r t -C;U,,- Q , ~ ( ~ - u ~ / u ~ ) u + ~ I H - ' X P U = O

p t i + ~ : p - v ? p x , + X M - ' ( n2-u,2) = O

Q,~ = 4~~ /mu? is tile quantity which characterizes the potential barrier.

Introducing d irncnsionlcss paramctcrs

2 { = ( x - x o - v t ) l a , s = v / c o , o o = v o / a Q , , h= c? - oo , @ = u I u o , \ I I = P / F ) ~

2 2 a = 2 ~ , / ( m o ) ~ u,), p = x ~ u ~ ~ / ( ~ w ~ ~ M ~ ~ ? ) and p, = X U , ~ / M Q ?

...( 1.62)

equations of molion become

(a) Proton motion without interaction between sublattices (X = 0)

Now the coupled equation becomes uncoupled and as we are interested only in proton

dynamics we take the f i s t equation

+ 2 p 2 ( 1 - a 2 ) @ = 0 ...( 1.64)

where p2 = a I (1- s2 ) and s2 < 1

Its solution is

= L t a n h ( p t ) 1=+1

or u, (5 > = 1 uo tanh (P 5)

32

The function will1 a positive sign ie I = 1 corresponds to the soliton(kink) and with a negative

sign ie 1 = -1 corresponds to the anti -soliton (anti-kink)

Fig. 1.9 Solution for ADZ Model

The antikink characteriscs an extended ionic defcct moving with a velocity v less than lliat of

"proton sound". Therefore the transition of a proton from the left potential well to the right one

occurs, not by a jump but step by step over several cells as denoted by fig (1.9).

(b) Proton motion with interaction between sublattices (X ;t 0)

If x f 0 , thcn the proton motion is dcscribcd by the coupled set of equatio~ls (1.63)

(bl) Wc Sirs1 consider tllc motion of a proton with llxcd vclocity v = v, (ie. no cxtcrnal forcc)

and h = 0. Then thc equation(l.63b) reduces to

substiluting this into equation (1.63a)

a{{ 4- 2 p 0 2 ( 1 - 0 2 ) @ = 0

whcrc pz = (a - 8) ( 1 - 502 ) -' ; so2 = (vo / c0 )2 < 1

33

comparing equation (1.67) with (1.641, it is evident that the effect of x ;t 0 is taken into account

by replacing a by (a - P) . According to equation (1.621, the coefficient a is proportional to

the potential barrier height. Thus the coupling between the protons and the basic sub systc~n

(OH- ) leads to an effective lowering of the height of the potential barrier by a factor

Thc lowcring of Ihc potential barricr height does not depend on the sign oP X. 'l'hcscforc lor

~$0, and for a fixed proton velocity v = v,, the solution of the set of equations (1.63a) &

(1.63b) takes the form

So, the inclusion of the interaction betwcen the displacement of ions OH- and prololls increase

t l ~ c distance over which a proton transfers from onc polcntial wcll to Ihe othcr. This is evident

from tllc fact that po is decreased comparcd to the case without coupling between sublatticcs.

(b2) If the velocity v of the solitons does not coincide with v, (ie v ;t v,) , then the parameter h

is not equal to 0. For small h we may seek the solution y ( 5 ) of equation (1.63b) in thc form

of a series expansion in powers of h

34

wlsrc yf, = { 1 - 0' ) is liic soluiion of equation (1.63) wilh h =O, yi ( 5 ) is thc first ordcr

corrcclion. Subsliluting cquitlion(l.70) in (1.63b) and solvir~g for and (1> with approprialc:

boundary conditions

The kink and antikink may arise simultaneously in an arbitrary part of the chain. Thcy may arise

separately only at chain ends. The motion of protons from the lcfi to right occurs oriIy in cases

where all the protons are localised in the left wells. The transfer from the left well to the right

one lakes place under the motion of the antikink (H30') from the left to the right or of the kink

(OH-) in the opposite direction. The return of proton to their initial state (left usually) occurs

due to the tra.nsmission o l an oricntational negative Ujerrum defcct from tile right to tire lcft.

(iii) Y~levnlatikos Model

Pnevmatikos modified ADZ model to describe the dynamics in hydrogen bonded

macromolecular system [8] . In addition to ADZ model, he assumed

1. Harmonic substrate potential on the sublattice (OH).

2. Existence of next ncighbour harmonic interaction.

3. Applied force t damping term.

Tllc modilicd ADZ model is shown in i'ig. 1.10.

G2 Units cell

Ffg.1-10 Pnevmatikss Model for Hydrogen Bonded System.

The equations of lnotions are

3 ml 2, + Inl r1 ?, = G1 ( Z,+I + Z,.I - 2 ~ " ) + 6 ( wn - 22, + w,., ) - Ga Z, - Ba Z, -t 1'1

...( 1,72a) . .

m2 W, + m2 r2 w, = G2 ( wntl + w,.~ - 2w,) i- G ( z, - 2w, t- ) - Gb wn - I; ... (1.72b)

ml is in double well and mz is in harmonic well, T1 and T2 are damping coefficicnls.

Constant external force as well as the influence of dissipation mechanism are taken into

account. In the case of slrong enough coupling between protons and heavy ions one can usc tl~c

continu~~rn approximation to solve the system for simple localised travelling wavc solution

will1 s = x-vt

(v2 - u ~ ) z , , - r l v z , + [ ( G a + 2 G ) / m l ] z + ( ~ , / r n ~ ) z ~ = ( G 1 m l ) p +f l Iml

...( 1.73a)

(9 -v?)pSs - 1 - 2 ~ p s + [ ( G b + 2 G ) / m z ] p = ( ~ G / ~ ~ ) ( z + D ~ z , , ) + 2 f 2 / m 2

...( 1.73b)

36

where D is the M i c e spacing , p = p, = w, + w,-~ ,

u, = 2D (Gi 1 n?l )'" , v, = 2D ( 6 2 / m2 )'" are speeds of sound in anion sublatlice

and pro1011 sublaltice.

In the absence of an exterilal field and damping one can easily solve the system when v = v,. I11

this case we have

If v = v, the solution will be

T72 -T71 z(x,t) = k'\I(A2/B,)

1 + exp [ (x-vt)/(L+x, ) ] 1 where 71 , qz and ~ 7 3 are the roots of the polynomial

F(q) = q - q3 + Ro and R, = f B,"~ t22. 312

where v is the velocily of solit011

37

Pncvlnalikos showed that the prololl kink propagation is energetically favoured by " in pllase

motion" of the neighborn heavy ions which happen for 6=.0 while for hydrogen bonded chain

in ice crystals the " out of phase" oxygen motion is quite reasonable. For other macromolecules

with the same hydrogen bonded structure the "in phase motion" may be more probabie.

(iv) Doubly Periodic Potential Model.

Even though the previously discussed modcls Sos hydrogcn bonded systcln provide

qualitative information regarding the collective proton dynamics, they all suffer from thc same

defect viz they take into account only one or the other possible two types of defects in thc

hydrogen bonded networks. This is far from realistic since experimental evidence clearly

suggesls that both types of defects participate in the transfer of charge across the llydrogen

bonded network.

In the last section double well potential (q4) models on proton was rcvicwed. Such

modcls have bcen shown to had to kink solitons that reprcscnt the ionic delccts in the crystals.

The main disadvantage of such potential however is that it cannot take into account the

orientational defects that are known to be present in a hydrogen bonded system. Sincc

macroscopic charge transfer in hydrogen bonded systcm involves both kinds oS defects, all

modcls [6-81 that are based on such typc of potential can providc only parlial inSorn~ation on lhc

dynamics oC prolons. I11 particular, conduclion propcrlics oS Lhc proton cannol hc addrcsscd

with such potentials.

To overcome these difficulties, it is necessary to adopt model substrate potential for the

proton that on one hand retains the topology of the double well potential which is csscntial for

thc proper description of the hydrogen bond, and on the othcr hand, allow for an efibctivc

3 8

cilargc transkr bctwcc~l adjaccnt hydiogcn b o ~ ~ d s [hat colncs as a result of' ~ h c I3jciii1rn rolalion

[9,101. l'his can hc nccomplishcci will] t i ~ introduclion of a douhlv pcriodic subtra&2cl_c11tj:i!

that can accommodate both typcs of deScct formation that ase known expcrimcil~ally to play

important roles in the elccrrical properties.

Proton

@ Heavy ion

Periodicity of the potential is 4n

Fig 1.11 Doubly Periodic Substrate Onsite Potential

The total Hamiltonian for a quasi one dimensional system is

1-1 = Hp + I I o I ~ + Hi,,,

where H, - Hamiltonian for proton lattice,

1-1, - Hamiltonian Sor ion sub-lattice,

Hint - Interactio~l Hamiltonian.

where, m, M - Mass of the proton and ion respectively,

kl , k2 - Corrcspo~lding spring constants,

X - Coupling parameter,

J'n - Displacement of n~ proton , measured from the central unstable

equilibrium position in hydrogen bond i.e. from the middle of the

bond that links the ions,

y,, ' I>isplaccrncnL 01' ions , mcasurcd li*oni ils cquilibt-iu~n posiliot~,

lo - Lattice constant.

writing un = ( 4 n l l , ) yn ; w,= Y n l l o ;

The substrate potential is

For the interaction term, the Sunclion @(un) is defined as

The potential V1 (un ) is the onsite potential for the proton sublattice and it is chosen to satisfy

the physical requkemenls possessed by hydrogen bonded network. If we assume t i~a i thc rest

ion position is whcrc thc largc maximum occurs, then thc two local minima scp;u.alcd by lhc

slnallcr rnaximun~ represents the two prolon equilibrium positions withill thc hydrogen bond,

and the l uge barrier represents the energy nccessary for a Bjerrum rotation lo take place. If a

proton has enough energy, such rotation is possible and the proton can move to thc othcrsidc of

the large barrier.

The equations of motion in dimensionless form are

where mass m, = E t: 11: ; force io = E / 2, ; potential constant k, = E / 1;

ol = [ ( k l n b ) l ( k o m ) ] l " = t o (k l /m) ' 12

~ 0 1 = [ ( k 2 m o ) / ( k o m ) ] 1 1 2 =Lo(kz /m)112 = 1

= ( 4 7 ~ ) ~ ~ m , / f ~ m = (47~)~t :~ /n11,

~2 = X m o l f o M = t , 2 ~ 1 ~ l ~

521 = ( 4 n / l 0 ) ( S , / m ) ' " t , ; Q2=(1/10) ( S o / ~ ) 1 1 2 t o

Although the physical hydrogen bonded network is a system with discrete symmetry, analpica1

results can only be obtained in continuum limit, where the excitations are assumed to exlend

ovcr l uge dislances compared lo the ialtice spacing. The equations of motion in continuum l i~ni l

2 dV1 da, Uzz - C, U x x + q 2 - f X1 Wx - - - 0

du du

2 2 dV2 da, wrz - V, w x x + Q 2 -- X2 - = 0

dw dx

whcrc x, .t: are the dimcnsionlcss space and time variables.

c, = wl, v, = 1 represent the speed of sound in the protonic and ionic sublatticc

respectively.

, ~2 are propostional to and $21 , Q2 are proportional to sP1" , s,ln respectively.

In the special case when Q2 = 0, equation (1.85a) lead to a Double Sine-Gordon equation

for thc prolonic sublatticc.

while lor the heavy sublattice we have

X 2 Q? f

X1 X2 Ws = -

2 2 [ cos ( ' I2) - a ] with E = -

v -v, (1-a2) 4(v: - ?)

wl~crc s = x-vt and v is the travelling wave velocity. Thc parameter E dcl'incs ncw cCf'cclivc

barrier height for the Double Sinc Gordon potential(DSG).

42

When x = 0, we can find analytical result. For , x2 and x z 0 , this coefl'icient E

contains the inllucncc of Lhc hcavy sublatticc on thc ionic one. It was dcmonslraLcd Lti;il for

travelling velocities smaller than v,, the effective barrier decreases, whereas the opposite effect

occurs for v > v,. This holds independent of sign of coupling coefficient X. On th other hand ,

the sign of w, depends on the velocity v.

The non ~ravclling witvc solulions o l cqualio1l(l.86) arc obtaitlcd as

ill (x, 2 ) = 41-c~ i- 4 h c lan [ K Lanh { k, (x - x,) - Q, z ) ] ...( 1.88)

u1 I (x, z) = 4n (n+ '12) f 4 k c tan [ R-' tanh { k, (x - %) - QS z } ] ...( 1.W)

where R = [ (1-a)/(l+a) 1'" ks = y h / 2d ; Qs = kSv ; y = ( 1- v /c,)-'" ; d = cJQ1

and a = cos (uJ2)

2 xz R k, sech2 [ k, (x - x,) - Qs 2 1 u~(x,t) WI (x,z ) = - - sin -

Q~~ 1 + R~ tanh2 [ k, (X - x,) - fi, T I 2

2 x2 R-' k, sech2 [ k, (x - %) - 0, 2 1 u11(x,t) wll (x, z) = - - sin -

1 + 1 1 ' 2 t a n h 2 [ k , ( x - ~ ) - f i s ~ ] 2

The solutions given above provide two types of kinks for the protonic subIattice (small

and luge kinks and the corresponding antikinks), and tor each o l those a non-linear kink-type

excitatioil in the ionic sublattice. When a kink is present in the protonic sublattice a deformation

is created in the ionic sublattice that can travel with it as long as the velocity of the kinks do not

43

exceed the speed of sound in the ionic chain. When the velocity of the former is larger than that,

the ionic deformation lags behind.

Solution I in the proton lallice is a small kink soliton represcnling thc transport of tlic

protoll from one minimum to the next (ionic defect). Solution I1 is the large kink soliton and it

represents the Bjerrum defect.