PROTONIC CONDUCTIVITY : A NEW APPLICATION OF SOLITON THEORY

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PROTONIC CONDUCTIVITY : A NEW APPLICATION OF SOLITON THEORY

S. Pnevmatikos, G. Tsironis

To cite this version:

S. Pnevmatikos, G. Tsironis. PROTONIC CONDUCTIVITY : A NEW APPLICATION OF SOLITON THEORY. Journal de Physique Colloques, 1989, 50 (C3), pp.C3-3-C3-10.

�10.1051/jphyscol:1989301�. �jpa-00229440�

PROTONIC CONDUCTIVITY : A NEW APPLICATION OF SOLITON THEORY

S.N. PNEVMATIKOS and G.P. TSIRONIS*

R e s e a r c h C e n t e r of C r e t e , P O Box 1 5 2 7 , GR-711 1 0 H e r a k l i o , C r e t e , a n d U n i v e r s i t y o f t h e Aegean, GR-83200 K a r l o v a s s i , Samos, Greece

* ~ n s t i t u t f o r N o n l i n e a r S c i e n c e , R-002, a n d D e p a r t m e n t of C h e m i s t r y , B-040, U n i v e r s i t y o f C a l i f o r n i a a t S a n D i e g o , La J o l l a , CA 92093,

U.S.A.

Resume

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Nous proposons un model B deux sous-rbeaux avec un potentiel de substrat doublement periodique pour decrire le transport protonique dans un reseau quasi-unidimensionel a liaison hydrogene. Le systeme discret est reduit a une equation continue du type "double Sine-Gordon" pour les protons et une equation differentielle simple pour les ions lourds. Les deux types de solutions pour les solitons kink & deux composantes correspondent aux d6fauts ioniques et orientationels (Bjerrum) respectivement. La reponse realiste de ces defauts solitoniques aux champs exterieurs fnit de ce systeme un excellent modele pour decrire tant qualitativement que quantitativement la conductiviti protonique des reseaux a liaisons hydrogene.

Abstract

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We propose a two sublattice model with a doubly periodic on-site potential t o describe the proton transport in hydrogen-bonded quasi-one-dimensional networks. The discrete system is reduced t o a continuum double Sine-Gordon equation for the protonic part plus a simple differential equation for the heavy ion part. Its two-component kink solitons correspond t o the ionic and B j e m defects.

The correct response of these solitonic defects t o an externally applied dc electric field makes this system an excellent model for qualitative and quantitative description of the protonic conductivity in hydrogen-bonded networks.

1

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INTRODUCTION

Proton conductivity along quasi-one-dimensional hydrogen-bonded networks in some molecular systems can exceed the conductivity in the orthogonal direction by a factor of 1000. Ice-like structures of hydrogen-bonded systems show a proton mobility only an order of magnitude less than that in metals. For that reason, such a crystal is often considered as a "protonic semiconductor". Experimental evidence 11-41 show that high proton mobility in ice-like systems is due t o their transfer along the hydrogen bonds. In spite of the profound importance that the hydrogen bond plays in Condensed Matter Physics and Biology, very little is known about the detailed mechanisms for proton transport along hydrogen-bonded networks.

The hydrogen-bonded networks which we consider here are quasi one-dimensional clusters of molecular aggregates interacting with their first neighbors through hydrogen bonds. Considering the simplest case and focussing our attention on the main degrees of freedom, we obtain the following diatomic structure of hydrogen-bonded dimers :

where the full line segments indicate a covalent or an, ionic bond, the dotted line segments indicate a hydrogen bond, and X denotes a negative ion. The atoms that are usually involved with hydrogen bonds representing the negative proton-accepting side of the heavy molecule o r aggregate are 0, N, and F. In such a chain because of the symmetric one-dimensional environment of the hydrogen ion, protons can easily jump

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989301

C3-4 JOURNAL DE PHYSIQUE

from their left equilibrium position to the energetically equivalent position at the right near the next X ion. This physical situation is usually modeled by the association with each proton of a double-well substrate potential representing the two degenerate proton equilibrium positions per unit cell [S].

Ice is a very common example of a hydrogen-bonded system. In the more stable Ih hexagonal ice structure 111, one can distinguish hydrogen-bonded networks either following a given crystallographic direction (for example, a zig-zag hydrogen-bonded linkage) or just following a Bemal-Fowler filarnental path [61. Before proceeding to better-defined but more complex quasi-one dimensional biological or polymeric hydrogen-bonded macromolecules, let us first study the zig-zag diatomic structure in Ih ice because of the extensive experimental data that are available for ice [I-31.

One well-known property of ice, as well as of other hydrogen-bonded sytems, is the spontaneous formation of ionic and Bjenum-type defects. In the zig-zag hydrogen-bonded network (Fig.0, one can realize a configuration where a number of successive protons jumping in their second equilibrium position produce one hydroxyl ion (OH-) with negative effective charge and one hydronium ion (H30+) with a positive effective charge [Fig.l(a)l. In an 'infinite' chain we can generate a large number of negative and positive ionic defects without changing the number of protons in the system. On the other hand, following Bjerrum's theory 171, we can rotate a number of successive water dipoles in such a way that an 0-0 bond with two protons and positive effective charge and one without proton and with negative effective charge is generated [Fig.l(b)l. The number of positive (D) and negative (L) Bjerrum defects can increase or decrease by injection or removal of protons, respectively.

IONIC -DEFECTS

BJERRUM DEFECTS

Fig. 1 : (a) Ionic and (b) orientational (Bjemun) defects ⁱⁿ icelike structure.

Onsager was the first to connect proton conductivity to a ho ing mechanism that allows proton transport along hydrogen-bonded atomic channels

67.

Weiner and Askar discussed the idea of a collective transition that leads to a formation of ionic defects using a double well atomic model and suggested the analogy between the motion of protons in hydrogen-bonded chains and the motion of dislocations in crystals [ 9 ] . Antonchenko et al have introduced a two-sublattice model (ADZ model) through which the ionic transport in hydrogen-bonded chains can be done collectively via the propagation of two-component q4 soliton with one characteristic velocity v, [lo].

Stability properties [I 11 and interesting dynamical features [I21 were systematically studied for the ionic solitons in the ADZ model. Also, some extensions to other hydrogen-bonded configurations have been introduced [13,141 as well as the thermal activation mechanism for ionic defects has been studied [IS]. SO, a remarquable effort have been devoted to connect energy transfer, carried by ionic-defects, to a collective dynamics of soliton type which represents a localized excitation of displaced hydrogens forming a smoothly relaxed in space ionic defect 19-16]. On the other hand, some models have been introduced to describe with a rotational soliton mechanism the Bjermm-defect formation and propagation through the system 117,18J. Although these models are able to describe (each one with its own mechanism) energy transport, dielectric polarization and proton storage in hydrogen-bonded systems, none of them is able to explain the protonic mass transport in such a network that is necessary to support the saturated (non-transient) protonic conductivity in ice and other hydrogen-bonded 'protonic semiconductors

.

The aim of this paper is to introduce a new atomic model that can describe

2

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THE MODEL

Let us consider the system with the diatomic zig-zag structure of Fig.1, and in a first approximation let us focus our attention on the longitudinal vibrations of both the heavy and light (protons) ions of the chain, neglecting any interaction with the rest of the 3D Lattice. The transition of a proton from one to the other equilibrium position of the hydrogen bond requires the overcoming of the potential banier that separates the two degenerate minima of the potential well. Using the same picture for the rotation of the X-H covalent bond, we can introduce a second barrier representing the energy that one proton needs in order t o rotate from the one X-X bond t o the next, thereby generating a Bjerrum defect. These potential barriers can be determined consistently from experimental data on the activation energy of the ionic and Bjerrum defects (for ice, see Refs.1-3). The relative size of the two barriers depends on the particular structure of the system. For ice, one expects t o have a large interbond barrier and a smaller intrabond barrier (0.2 eV). This may be not so for an HF crystal [19].

Following these ideas one can model the longitudinal dynamics of the zig-zag structure of Fig.1 with the dynamics of protons in the deformable doubly periodic substrate potential of Fig.2.

Fig2 : The dynamics o f protons in the zig-zag hydrogen-bonded diatomic chain is eqmvalently described by the motion o f protons in a doubly periodic deformable substrate potential.

The total Hamiltonian of the system consists of three parts :

H = H, + Ho + Hi (2.1)

The proton part is given by :

where the first term is the kinetic energy of the proton masses (m=mp), the second term is the interaction energy between first proton neighbors, and the last term represents the substrate potential function that defines the protons' equilibrium positions.

K1 and Sp are two potential constants, and yn are the proton displacements from the rmddle of the X-X bond. The heavy-ion part of the Hamiltonian is given by

where the first term is the kinetic energy of the heavy masses M (for ice M=17 m ), the second term is the interaction energy between first heavy ion neighbors and the list

C3-6 JOURNAL DE PHYSIQUE

term represents an on-site potential that acts on the heavy ions. This potential is created from the interaction of the quasi-one-dimensional chain under study with the rest of the crystal. K and S are two potential constants and Yn is the heavy ion displacement from equi?ibrium. (\The two sublattices interact between each other following the third part of the Harniltonian :

where

x

is the coupling constant, @(4ny /lo) is a function of yn related t o the substrate potential V1(4xyn/lO), and 1, is the equilbnum distance between neighbor unit cells. It is appropriate t o introduce the following dimensionless quantities :

un = 4nyn/lo -

,

wn = Yn/l, (2.3)

One convenient function modeling the on-site potential V1(%) of Fig.2 is

Vl(%) = [2/(1-a2)1 [cos(un/2)-aI2 (2.4a)

where O<a<l is a free parameter that defines both minima and maxima of the potential. This is a disadvantage for the potential (2.4a), but with the proper choise of the parameter a, we can simulate the respective values of the ionic and Bjermm defect activation energies t o reasonable accuracy for several hydrogen-bonded systems. The potential V2(wn) is given for simplicity by

assuming that heavy ions oscillations amplitudes are small compared t o 1,.

For the given substrate potential V 1 ( ~ ) , let us introduce now the following interaction function for the third part (2.2~) of the Hamiitonian

@(u,) = COS(U,/~)

-

COS(U,/~) (2.4~)

where uo=2arccos(a) defies the proton equilibrium positions from the middle of the X-X bond. With this function, the part (2.2~) becomes physically meaningful because when one of the two sublattices is at rest (Y,

-

^{Yn-l =} 0 or u, = +uo,Vn) then Hi vanishes. Otherwise, for H.>O, the comblned motion of the two sublattices is energetically expensive, while, lor Hi<O, this combined motion is energetically favored.

The equations of motion for the Hamiltonian (2.2) can now be written, in dimensionless form, as follows :

In equations (2.5), as well as in what follows, we use the following units : Energy : E, = 1 . 9 8 6 ~ 1 0 - ~ ~ ~ , Time : to = w2-l = ( M / K ~ ) ~ / ~ and length : 1,.

With these choises, energy is measured in cm-l and for mass and forces we have the following derived units :

mass : m, = ~,t,~/l,, force : f, = ~,/1, and potential const. : KO = eO/lo2.

This system of units is introduced in order t o facilitate the numerical computations while with a simple transformation we can compare our results with experiments. Using the above fundamental and derived units, we determine the values of the constants and

x1

⁼ (4nI2 (x/fo)/(m/m0) = (4~)~t:x/(ml,)

x2

⁼ (x/fo)/(M/m0) = t:x/(~1,) Q, = 4 n ( ~ ~ / m ) ~ / ~ / 1 , R2 = (s,/M)'/~/~,

With these definitions, all coefficients are dimensionless. For a detailed presentation of the ground states of this system and its linearized (small amplitude) limit see ref. 1191.

3

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DOUBLE SINE-GORDON LIMJT

Limiting our attention t o the case where the coupling between first neighbors of the same sublattice is strong enough t o ensure that variations of un and wn from cell to cell are smooth (at least at low temperature), we can solve this model in the

"displacive" limit for the proton sublattice taking into account the smooth influence of the heavy ion vibrations. In this continuum limit the N coupled difference-differential equations of motion reduce to the following two partial differential equations :

where ~ = t / t , and x=nl, are dimensionless.

In the special case when B -0, using the functions (2.4) and looking for travelling-wave solutions moving with the Lensionless velocity v, we introduce the new variable E=x-vz and obtain an easily integrable form for the second equation, which gives :

with zero integration constant. Substituting now this expression into the first equation in (3.1), we obtain the following double Sine-Gordon equation :

with

E = CZ12/(1-a2) + X1X2/4(w2

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vo2)

where c, and vo are the dimensionless speeds of sound in the light and heavy sublattice respectively (co=wl and vo= 1).

The double Sine-Gordon equation (3.2a) has been studied extensively both from dynamical and thermodynamical points of view 120-221. Following standard techniques [201, one can obtain two types of kink solutions for eq.(3.2a) : a small kink (kink I) corresponding t o the transition form the minimum at -u, to the minimum at +u,, mod(4x), and a large kink (kink II), corresponding t o the transition from the minimum at +uo to the minimum at 4~-u,, mod(4n).

uI(x,z) = 4nn ? 4arctan {R tanh CK,(x-x0)

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S2,~l) (3.3a)

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where

and wx=w5 is given by (3.2b). Integrating (3.2b) with respect x, using expressions (3.3), we obtain two equivalent kink type solutions for the heavy ion displacement w(x,t). For

x1,x2 -

0, we have h

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1 and kinks (3.3) take the standard form of the one-component double Sine-Gordon kinks [201. In Fig. 3, we represent the two components of the two soliton solutions of our system for Q2=0. The small kink (kink I) is associated t o a compressive pulse for the heavy sublattice, while the large kink (kink 11) is associated t o a rarefactive pulse. For the protonic chain, we represent the proton displacement u(x,t) given by (3.3), while for the heavy chain, we represent the derivative w, given by (3.2b). In Fig. 3, we considered v=O. For v>uo the heavy component change signs.

The total energy of these solutions is obtained by integrating the Harniltonian :

and

with

%

⁼ (rn/m,)/(4~)~, Mo = M/m, and the energy is given in cm-l.

WILS[E P : dw/Bx PULSE # : dw/@x

2Fi

%

20 40 0 20 40

lattlcm cello lettlcs cells

Fig. 3 : The two component of the two soliton solutions for %=0.

[I91 demonstrated that excitations in the heavy sublattice must have zero asymptotic values in order to correspond to finite energy travelling wave solutions. This is due to the presence of the one minimum on-site potential for the heavy ion subsystem.

However, due to the small coupling between the two sublattices, the protonic part of the solution is not appreciably affected.

4

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DISCUSSION AND CONCLUSIONS

In the context of this simpIe model, we have the following physical interpretation for our protonic kink soliton solutions :

kink I

-

I- ionic defect, kink I1

-

^L Bjerrum defect, antikink I

- ^I+

ionic defect, antikink I1

-

^D Bjenum defect

These protonic kinks are accompanied by compressions and rarefactions of the heavy sublattice around the protonic defect. Looking at the parameter E in ( 3 . 2 ~ ) we note that the soliton velocity v modulates the height of the potential barriers as a direct implication of the two-sublattice coupling.

In general, because of the small size

pf

the coupling constant X , the influence of the heavy sublattice vibrations on the proton dynamics is small. However, when u-u,, this coupling may have dramatic effects on the defect dynamics. Such effects will produce irregularities in the mobility properties of the soliton excitations and, insofar as soliton propagation is related to the dynamics of charged defects, should also produce striking effects on the conductivity properties of hydrogen-bonded systems. Numerical sirnufations show our solutions to be stable during the free propagation on the discrete lattice [19], while their annihilation and creation interaction properties are consistent with the expected dynamics of the ionic and Bjemvm defects in real hydrogen-bonded systems.

A significant feature of this model is the correct response of the four defects to an externally applied dc electric field. Following the qualitative picture of Fig.4, we observe that, when a dc field is applied to the positive direction, the negatively charged and L defects will propagate in the negative direction while the I+ and D defects will propagate in the positive direction. Simultaneously, in both cases, protons move in the positive direction. From Fig.4, we easily realize that the two types of defects propagate cooperatively, when a dc field is applied. Thus, with the introduction of suitable boundary conditions (periodic, or permanent injection of protons), our model appears able t o support a permanent flow of solitons which is necessary for a nontransient proton current in the hydrogen-bonded network. The two component solitons of both the ionic and Bjerrum type behave differently when the field value becomes strong enough. In this case, the heavy-ion part of the wave cannot follow the protonic part because of its high speed and the two components separate. This is reflected on the value of the mobility which now becomes larger.

The two barrier structures of the on-site potential are expected to produce very rich temperature dependence on the soliton properties. In a subsequent paper we examine analytically and numedcally the dynamical properties of our model in the presence of damping, dc electric field, and temperature, and we calculate several physical quantities, such as mobility. Finally we compare these results with experimental data that are available for relevant hydrogen-bonded systems.

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Fig. 4 : Collective ionic- and Bjemm-defect for- mation (qualitative pictu- re).

RIGHT

r n ~

Part o f this work was done at the Los Alamos National Laboratory under the auspices of the U.S. Department o f Energy. One of us (G.P.T.) acknowledges partial support of NSF grant No DMR 86-19650-Al.

REFERENCES

/1/ "Physics of Ice", edited by N. Riehl, B. Bullemer and H. Engelhardt (Plenum, New York 1969).

/ 2 / "Physics and Chemistry of Ice", edited by F. Whalley, S.J. Jones and L.W. Gold (Royal Society of Canada, Ottawa, 1973).

/3/ Hobbs P.B., 'Ice Physics", (Clarendon, Oxford 1974).

/4/ Glasser L., Chem. Reviews,

75

(1975) 21.

/ 5 / Pnevmatikos St., Flytzanis N. and Bishop A.R., J. Phys. (1987) 2829.

/ 6 / Bernal J.D. and Fowler R.H., J. Chem. Phys.

1

(1933) 515.

/ 7 / Bjerrum N., Science,

115

(1952) 385.

/ 8 / Onsager L., Science,

156.

(1967) 541;

166

(1969) 1359.

/ 9 / Weiner J.H. and Askar A., Nature, 226 (1970) 842.

/ l o / Antonchenko V.Ya., Davydov A.S. and Zolotaryuk A.V., phys.st.so1. (b),

115

(1983) 631; see also A.S. Davydov, "Solitons in Molecular Systems", (Reidel, Dordrecht 1985).

/11/ Laedke E.W., Spatschek K.H., Wilkens Jr M., and Zolotaryuk A.V., Phys. Rev. & (1985) 1161.

/12/ Peyrard M., Pnevmatikos St. and Flytzanis N., Phys. Rev.

A36

(1987) 903.

/13/ Zolotaryuk A.V., Spatschek K.H. and Laedke E.W., Phys. Let. &OJJ (1984) 517.

/14/ Pnevmatikos St., Phys. Let.

A122

(1987) 249.

/15/ Halding H. and Lomdahl P.S., Phys. Rev. A s (1988) 2608.

/16/ Yomosa S., J. Phys. Soc. Jpn.

52

(1983) 1866 and 51 (1982) 3318.

/17/ Gosar P., in ref.[ll, (1969) 401.

/ l 8 / Bagley R.J., Pnevmatikos St. and Campbell D.K., unpublished (1987).

/19/ Tsironis G.P. and Pnevmatikos St., Phys. Rev. B, submitted (1988).

/20/ DeLeonardis K.M. and Trullinger S.E., Phys. Rev. B27 (1983) 1867.

/21/ Condat C.A., Guyer R.A. and Miller M.D., Phys. Rev. B27 (1983) 274.

/22/ Campbell D.K., Peyrard M. and Sodano P., Physica (1986) 165.