SOLITONS IN HYDROGEN-BONDED CHAINS

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SOLITONS IN HYDROGEN-BONDED CHAINS

H. Desfontaines, M. Peyrard, D. Hochstrasser, H. Büttner

To cite this version:

H. Desfontaines, M. Peyrard, D. Hochstrasser, H. Büttner. SOLITONS IN HYDROGEN-BONDED CHAINS. Journal de Physique Colloques, 1989, 50 (C3), pp.C3-11-C3-20. �10.1051/jphyscol:1989302�.

�jpa-00229441�

J O U R N A L D E P H Y S I Q U E

Colloque C3, supplkment au n03, T o m e 50, mars 1989

SOLITONS IN HYDROGEN-BONDED CHAINS

H. D E S F O N T A I N E S , M. P E Y R A R D , D. H O C H S T R A S S E R * and H. BUTTNER*

Laboratoire ORC, Universite de Dijon, F-21100 Dijon, France

"physikalisches Institut, Universitat Bayreuth, D-8580 Bayreuth, F.R.G.

Abstract : A one dimensional model for hydrogen bonded chains is investigated by including the dynamical degrees of freedom of the oxygens and thus generalizing the original Antonchenko-Bavydov-Zolotaryuk model. Analytical solutions are derived within a continuum approximation up to second order so that rather narrow solutions could be derived accurately enough. Numerical simulations with a set of parameters derived from infrared spectra show stable low and high speed solutions separated by a forbidden gap. There exists also an intermediate velocity range where the initial solution is unstable in the sense that it is transformed to a high speed solution with the same energy.

Resume : Un modele unidirnensionnel de chaines a liaisons hydrogenes est etudie en incluant les degres de liberte dynamiques des Oxygenes afin de generaliser le rnodele originel de Davydov-Antonchenko-Zolotariuk.

Des solutions analytiques sont obtenues dans le cadre de ['approximation des milieux continus conduites a I'ordre deux de sorte que les solutions assez etroites peuvent etre calculess avec une precision suffisante.Des simulations numeriques effectuees avec un jeu de paramaires convenablement choisi montrent la stabilite des solutions dans des dornaines de hautes et basses vitesses separes par une zone interdite . II existe aussi un domaine de vitesse intermediaire ou la solution initiale est instable dans le sens ou elle evolue vers une solution de vitesse plus elevee ayant la meme energie.

I . Introduction

Since the basic work of Antonchenko et all. and Zobtaryuk et a12. A great interest has grown considerably in the soliton dynamics in one-dimensional hydrogen-bonded systems. These models seem to be an effective description of the proton mobility in hydrogen-bonded chains, and therefore m?y also pl?y a role in irferpreting certain biological processes in such systems3. Since the work of Bernal and Fowler the hyciroger; bonds are described by some kind of double-minimum potential. The high proton mobility is given by a hopping contribution for the protons from one of the minima into the other. The pallisular mechanism, however, is not yet understood in detail. On the other hand the soliton picture for transport in biological macromolecules has been discussed intensively in the literature and seems to be one possible mechanism in some of these processes5-7.

In this model the hydrogens alternate within the chain with oxygen ions. This simple ice-structure ,where, an harmonic oxygen-oxygen and hydrogen-hydrogen coupling, a nonlinear hydrogen onsite-potential and a nonlinear hydrogen-oxygen interaction leads to two coupled nonlinear differencs equations. The origin of the nonlinear onsite potential for the hydrogen is not further described here, but one assumes that the detailed interaction mechanism may result in a such potential. In the same sense the oxygen-hydrogen interaction is regarded as an effective term. Since one assumes an onsite-potential one expects as excitations a topological solitary wave for the hydrogens and a'non- topological excitation for the oxygens.

The original model of Antonchenko et all. did not include all degrees of freedom of the oxygens because a single variable describing their relative motion was used for a pair of oxygen atoms. The model that we treat in this paper includes all degrees of freedom for the oxygens.. We start here from a discrete model because we found that, when reasonable parameter values are considered, the solutions are rather narrow. Although we have to use a continuum approximation to derive an analytical solution, we include higher order terms which are necessary to provide a correct description of the lattice effects. Part 2 presents the model. In chapter 3 we discuss analytic solutions for the continuum model in the whole range of velocities. While in Chapter 4 the numerical investigations are discussed in some detail.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989302

C3-12

2. The model Hamiltonian

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DE

PHYSIQUE

Fig. 1 : The hydrogen bond. The proton in the

o4

potential is drawn. The large circles are corresponding to the oxygens.

Our model similar to the one by Antonchenko et a l l . consists of a diatomic chain of protons and oxygens (Fig.1). The total Harniltonian consists of three parts. The proton part is given by

with the onsite potential of strength :

The proton coordinate Un is the displacement of the hydrogen from the middle of the bond as it is shown in Fig. 1.

The oxygen Hamiltonian is written as

where qn describes the displacement from the equilibrium position. In the original model of Antonchenko et al. only optical vibrations were considered, whereas in our model acoustic vibrations are included.

The interaction is as in Antonchenko's model :

The physical content of this interaction is the lowering of the double potential barrier due to the oxygen displacements

The parameters in our model are the harmonic lattice frequencies ol, Q I the coupling constant

X ,

the barrier energy Eo, and the width uo of the potential. The actual values of these parameters will be discussed later. The classical equations of motion, that derive from the Hamiltonian, are :

We note here that we know of no solution for the discrete lattice equations, and therefore we try to find consistent equations in the continuum approximation. This approximation has to be done up to a certain order in the lattice parameter a. We will approximate the Hamiltonian up to second order in this parameter.

3. Solitary solutions i n the continuum approximation

For solutions which change slowly compared to the lattice spacing one can use the continuum approximation.

The resulting equations of motion are as follows :

where co = a 1 is the sound speed in the proton sublattice and vo = a R I is the sound speed in the oxygen sublattice.

From the special form of the interaction Hamiltonian it follows that the interaction between the two sublattices is quadratic in these expressions. Since we are only interested in running wave solutions with velocity v the partial differential equations can be transformed to ordinary differential equations for a single variable s = x-vt. From the equations of motion one then can do one integration in Eq. (3.2) and one finds :

Here we have used the following boundary condition for solitary solutions, which are appropriate for a solitary wave :

l i m

l u(s)l

=

uo I SI ->

m

and all derivatives are vanishing at large s. Now the two coupled ordinary differential equations are separated and the remaining equation in u(s) can be integrated at once with the result :

where we have used the following parameterfunctions

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2 2 4

& where & = E0[ l -

X

a

uo

/ 2M

C0 (vf-

v 2 )

1

^{( 3 6 ,}

& , =

7

mUo

is the effective potential barrier of the model which depends especially on the coupling constant

X .

Depending on these parameters B and C two different kink-type solutions are found :

For B c 0 and C r 1 one finds

( c - l ) l R ¹ /2

s

⁼

.

^[-

2

+ ('-I)

I

⁺ a r c sin

~UIJEI

^{( 3 .}

7)

1 /2 ( c - u 2 ) I l 2 - (C-1) U

For B > 0 and C c 0 we have instead :

where we used the following abbreviation : q =

4

_& 1/4uo.These somewhat complicated looking functions result in kink- like structures for u(s).

The velocity ranges of the two kink-like solution (Eqs. (3.7), (3.8) ) follow from the definitions of the parameters A, B and C in Eq. (3.5) and & ^l in Eq. (3.6), one presents the different cases in the next section.

4. Numerical investigations of the dynamics of the nonlinear excitations.

We have investigated numerically the dynamics of the solutions presented. in the previous sections in order to determine their stability. Moreover the simulations have been performed on the discrete lattice and not in the continuum limit so that discreteness effects, which may be important in these systems, have been studied. In order to make the comparison with real systems easier we have worked with physical units, i.e. we have used the equations of motions (2.5) and (2.6) that derive directly from the Hamiltonian defined by equations (2.1) - (2.4) .

These equations are solved with a fourth order Runge Kutta scheme and the time step is chosen small enough to preserve total energy to an accuracy better than 10-3 for the full duration of a simulation. The initial positions and velocities of the proton and oxygen atoms are obtained from the analytical solutions presented in the previous sections.

In terms of the physical parameters, the solutions are defined in two velocity domains :

(i) v > vo . This domain corresponds to the case B > 0, C c 0 defined previously.

(ii) 0 c v c VM c vo which corresponds to the case B c 0, C > 1, where VM is a maximum speed in the low velocity range v c vo defined by the conditions :

which states that the effective barrier in the proton sublattice coupled to the oxygen has to be positive, and

2 2 4

2 2 2 2

v < v, with 2 v 2 = (c0 + v0) - [(co - v:)~ ⁺

4X u o a

M

I

^{' I 2 (4.} 2)

mM

which is a condition for the definiteness of the integral used in the calculation of u. For the parameter sets that we have used in the simulations v l c v2. For the protons the implicit solutions (3.7) (case (ii) ) or (3.8) (case (i) ) written in terms of the physical variables are inverted numerically and the displacements Un and velocities dun/dt are simply obtained from the values of the continuum solution for x = na.

For the oxygens the analytical solution gives only the derivative dq/ds according to Eq. (3.3) and a spatial integration is required to get initial conditions for the simulations. In order to get the best accuracy for the discrete lattice we have used instead a discrete summation along the finite chain comprising N cell that we simulate, according to

in which f(uk) is the expression of dqlds (Eq. 3.3). As the model is translationally invariant for the oxygens, the value of q1 is not defined. It is chosen in our calculations so that the solution in the oxygen sublattice is centered along the chain in the same cell as the proton kink.

We have defined a system of units which are convenient for the computations : energies are expressed in eV, masses in atomic mass units (a.t.u,), displacements in A. This defines a time unit (t.u) equal to 1.0217 10-I4s. The Slmulatrons have been performed with the two sets of model parameters listed in table 1. Most of the simulations have been performed with set ( I ) which is identical to the one we used in a previous work8 while set (2) has been used to check that the results are preserved in this case which is close to the parameters expected for protons in ice. All simulations have been performed with a chain of 500 cells and the initial kinks were always localized at cell number 20.

a) Stability of the analytical solutions

Figure 2 (a and b) compares the velocity of the two component excitation (Fig. 2a) and the amplitude of the kink in the oxygen sublattice (Fig. 26) in the analytical solution used as the initial condition, and in the final solution which is selected by the system as a function of the velocity v of the initial condition.

-

In the low velocity range (v c v l ) the initial condition is stable. The analytical solution obtained in the continuum limit provides a good approximation for the two component solitary wave which propagates as a steady state in the system shown in Fig. 3.

- As shown in Fig. 2 the behavior of the solitary wave in the high velocity range vo < v < co is more complicated.

There exists a velocity domain vo ^c v 5 VL in which the initial condition is instable. Any initial condition launched in this domain with an initial speed vi accelerates to the final speed VF larger than ^VL. Simultaneously the shape of the solitary wave changes and the excitation which emerges after some transient is the same as the solitary wave that is generated by an initial condition almost equal to VF. Fig. 4 and Fig. 5 show an example of this behavior. In Fig. 4 the proton soliton velocity versus time is drawn. The acceleration towards the final speed VF is clearly visible. As shown in Fig. 5 the variation in the amplitude of the oxygen kink between the initial condition at speed ^Vi and the solution at speed VF is very large. This causes a large distortion in the oxygen sublattice to be left behind the two component solitary wave while the kink in the proton sublattice is almost undisturbed by the change in speed. On the contrary, an initial excitation lauilched with a speed higher than VL is stable and the final excitation is very close to the initial condition as can be seen in Figs. 2a and 2b.

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(a)

v m

10

-

_=: ^8.

- .

d

'5

2-6.

.-

-

_U

D .

-

_a

>

4 .

Fig. 2 : Velocity (a) and amplitude of the soliton kink (b) versus initial velocity. The full lines show the values for the analytical solutions, the crosses show the values of the final velocity (a) or amplitude of the oxygen kink (b) when the steady state of the two-component solitaty wave has been reached. The dotted line in the range v > vo indicates the parameters of the high speed analytical solution which has the same energy as the initial condition (parameter set 1).

A

0.5-

0.4

'3 -

-0.3.

-

=

-

.-

P

0.2

0.1

0

I I I 1 I I

j

^VL

I

V

2

3 4 5

6 - 7

0

9

' 0 1

Initial Velocity ( A / t.u.)

-

/+-+'++@

0 1

2

3 4

5 6 - 7

8 9 1 0 '

Initial Velocity ( A 1 t.u.) A

(b)

-

:di

I I

,

I I

,

1

+ +,'

=:

^{+ I}

4

^,

^y'i

^{4I I : V L}

^{-+ +'}

^{-+-+ I"}

Fig. 3 : Proton and oxygen displacements showing the steady state two cornpoent solitary wave which is generated by an initial condition in the low velocity range v < V i < vo (v = 3.0 ht.u. in this case with parameter set 1) where the kink is situated at cell n=20.

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P r o t o n soliton velocity (a)

o . o \

. . ^, , . . . , . . . . . , . , . .

0 100 200 300 400

T i m e I t . u ?

Fig. 4 : Velocity versus time of the two component solitary wave with an initial condition corresponding to a speed vo c v < v l (v = 5.3 h u . with parameter set 1).

The origin of the particular velocity VL which separates a domain in which the initial condition is instable from a domain in which it is stable can be understood from Fig. 6 in which the energy of the analytical solution is plotted versus its speed v.

vo VL

Velocity (A/t.u.)

Fig. 6 : Numerical Energy (derived directly from the hamiltonian) versus its speed

Fig. 5 : Proton and oxygen displacements showing the solitary wave and the large distorsion in the oxygen sublattice that is left behind (same initial condition as in Fig. 4).

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The speed VL corresponds to a minimum in the energy and the domain vo c v c VL in which the initial condition is instable is the only domain in which the energy decreases as the speed of the solitary wave increases. Due to the U shape of the energy curve in the velocity domain v > vo, a given energy corresponds to two possible solutions in this domain. Initial states vo c Vi c VL lie on the left (unstable) branch of the energy curve and tend to evolve toward the solution with the same energy on the right (stable) branch with VF > VL. Although some energy stays in the local deformation of the oxygen sublattice which is left behind the two components solitary wave which emerges, this is only a small part of the initial energy so that energy conservation gives a rather good estimate of the final state as shown in Fig. 2 where the dotted lines indicate for each vi the velocity VF and amplitude of the analytical solution lying on the right branch of the energy curve.

5. Summary

We have proposed a model for solitons in hydrogen bonded chains where the solutions can be given

analytically in a wide ranoe of velocities. In addition a.forbidden zone was found. where no solitary excitations exist. This seems to be a aeneral inirinsic feature of such models not deoendina on the details of the interaciion9. Such a forbidden region should perhaps be e$erimentally detectable

in

an &maly

in

the proton mobility. In order to do the calculations within a realistic model, we have succeeded in establishing a set of parameters which results in physical reasonable values for various energies and frequencies. The stability of the solutions was checked numerically by computer simulations of the discrete model. So one shows mainly that in a small intermediate velocity region (where for a given energy two solutions were possible) a transition to the high velocity solution was found.

Table 1 :

m M a Uo EO Oi vo co Eo v i

(a.m.u.) (a.m.u.) (A)

(A)

(ev) (t.u-1) (t.u-1) (e~/A3) (ht.u.) (h.u.) (ev) ( h . u . )

Set 1 1 1 6 5 1 2 6 1 0.15 5 30 15.939 4.999

REFERENCES

V.Ya. Antonchenko, A.S. Davydov and A.V. Zolotaryuk, Phys. Stat. sol. b

B,

631 (1983).

A.V. Zolotaryuk, K.H. Spatschek and E.W. Laedke, Phys. Lett. BLQL 517 (1984).

J.F. Nagle, M. Mille and H.J. Morovitz, J. Chem. Phys.

z,

3959 (1980).

J.D. Bemal and R.H. Fowler, J. Chem. Phys. 1,515 (1933).

A.S. Davydov, Physica Scripta

a,

387 (1979).

A.R. Bishop, J.A. Krumhansl and S.E. Trullinger, Physica

u,

^{1 (1980).}

A.S. Davydov, Solitons in Molecular Systems, D. Reidel Publishing Compacy, Dordrecht, Holland (1985).

M. Peyrard, St. Pnevmatikos and N. Flytzanis, Phys. Rev. A 3.

903 (1 987).

A.V. Zolotaryuk, TMPHAH 68,916 (1986).

( ) Submitted to Physical Review for Publications.