Influences of Structure Disorder and Temperature on Properties of Proton Conductivity in Hydrogen-Bond Molecular Systems

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1 Commun. Theor. Phys. Beijing, China) ) pp c International Academic Publishers Vol. 47, No. 2, February 15, 2007 Influences of Structure Disorder and Temperature on Properties of Proton Conductivity in Hydrogen-Bond Molecular Systems PANG Xiao-Feng 1,2, and YU Jia-Feng 1 1 Institute of Life Science and Technology, University of Electronic Science and Technology, Chengdu , China 2 International Centre for Materials Physics, the Chinese Academy of Sciences, Shenyang , China Received December 5, 2005) Abstract The dynamic properties of proton conductivity along hydrogen-bonded molecular systems, for example, ice crystal, with structure disorder or damping and finite temperatures exposed in an externally applied electric-field have been numerically studied by Runge Kutta way in our soliton model. The results obtained show that the proton-soliton is very robust against the structure disorder including the fluctuation of the force constant and disorder in the sequence of masses and thermal perturbation and damping of medium, the velocity of its conductivity increases with increasing of the externally applied electric-field and decreasing of the damping coefficient of medium, but the proton-soliton disperses for quite great fluctuation of the force constant and damping coefficient. In the numerical simulation we find that the proton-soliton in our model is thermally stable in a large region of temperature of T 273 K under influences of damping and externally applied electric-field in ice crystal. This shows that our model is available and appropriate to ice. PACS numbers: w, Kf, Dn, Jn Key words: proton conductivity, structure disorder, temperature, hydrogen bonded system 1 Introduction It is well known that the hydrogen-bonded systems occur extensively in a lot of condensed matters, polymers, and biological systems, for instance, ice, imidazole, hydrogen halides, carbohydrates, solid alcohol, bacteriorhodopsin, and protein. Experiments discovered that there is a considerable electrical conductivity, [1 6] even though electron transport through these systems is hardly present. This is quite an intriguing feature that has always stimulated wide interest in recently three decades. [1 18] Studies show that this phenomenon is related to the proton transfer along the hydrogen bonded chains, which is associated with motions of ionic and bonded defects appearing in the systems. [1 18] From present studies we know that solution of this problem is helpful to understanding formation of ionic channel on the biomembrane and mechanism of photosynthesis of plant in life science. [3,5,6] The hydrogen-bonded system is characterized by the formation of long chains of hydrogen-bonded molecules X H X X H, where X is a heavy ion or atomic group or oxygen atom in ice. Each hydrogen ion or proton H + ) can be transferred inside the X H X bridge interchanging the role of the covalent ) and the hydrogen ) bonds with the heavy ion group X ). [3 18] When such a transfer of proton in the intrabond occurs the chain is locally disturbing its neutral charge distribution and generating an ionic defect. An additional degree of freedom allows the group X H to rotate in such a way that a proton transfer in interbond is possible, which generates a bonding or Bjerrum or orientational) defect. Any of the two defects could be the majority charge carriers through the chains. Many scientists, [1 4] trying to explain the protonic conductivity in ice, formulated theories based on the hopping migration of the ionic and bonding defects along the hydrogen-bonded networks in ice crystals. However, these theories fail to explain the dynamics of the protons in the systems. New ideas from the nonlinear dynamics and soliton motion provide a possibility to find a solution of this issue. The soliton model of the proton transport was first proposed by Antonchenko et al. [7] ADZ model) in ice, which is a two-component model. The main characteristic for this model is concentrated in the structure of the double-well potential, i.e., the proton transport in this system is determined by a double-well potential, the coupling interaction between the protons and heavy ions provides only one mechanism to reduce the height of the barrier that the proton has to overcome to pass from one molecule to other. Therefore, this model can only explain the ionic defect, but not the bonded defect. Also, since two coupled degrees of freedom at each lattice site are included in the ADZ model, it complicates also investigation of this problem. Even when a continuum limit is taken, the equations of motion coupling two fields are difficult to solve. So, no exact analytical solution is known for this model. Another difficulty appears, when one considers realistic parameters for the hydrogen-bonded systems, i.e., ranges of validity of the solutions are narrowing with respect to the lattice spacing, and the continuum approximation, then, fails. This is the case in ice in which the H 3 O + or OH is almost point defects. [8 15] Therefore, this model needs to develop further. The project supported by National Natural Science Foundation of China under Grant No pangxf@mail.sc.cninfo.net

2 236 PANG Xiao-Feng and YU Jia-Feng Vol. 47 We recently worked on a new model to study the dynamic properties of the proton transfer in this system. In this model we have included a double-well potential for the proton represented by UR n ) = U 0 [1 R n /R 0 ) 2 ] 2, the elastic interaction caused by the covalent interaction, the coupled interaction between the protons and heavy ions and the resonant or dipole-dipole interaction between neighboring the protons and the changes of relative positions of neighboring heavy ions resulting from this interaction were included. If assuming again the harmonic model with acoustic vibrations for the heavy ionic sublattice, the Hamiltonian of the systems is expressed by [15 20] H = H p + H ion + H int = { 1 2m p2 n mω2 0Rn 2 1 [ Rn ) 2 ] 2 } 2 mω2 1R n R n+1 + U [ 1 R n 0 2M P n W u n u n 1 ) 2] n + [ 1 ] 2 χ 1mu n+1 u n 1 )Rn 2 + mχ 2 u n+1 u n )R n R n+1, 1) n where the proton displacements and momentum are R n and p n = mṙn, respectively, the first being the displacement of the hydrogen atom from the middle of the bond between the n-th and the n+1)-th heavy ions in the static case. R 0 is the distance between the central maximum and one of the minima of the double-well, U 0 is the height of the barrier of the double-well potential. Similarly, u n and P n = M u n are the displacement of the heavy ion from its equilibrium position and its conjugate momentum, respectively. χ 1 = ω0/ u 2 n and χ 2 = ω1/ u 2 n are coupling constants between the proton and heavy ion sublattices which represent the changes of the energy of vibration of the protons and of the coupled energy between neighboring protons due to a unit extension of the heavy ionic sublattice, respectively. mω1r 2 n R n+1 /2 shows the correlation interaction between neighboring protons caused by the dipole-dipole interactions. ω 0 and ω 1 are diagonal and nondiagonal elements of the dynamic matrix of the proton, respectively. ω 0 is also the Einstein resonant frequency of the protonic sublattice. W is the elastic constant of the heavy ionic sublattice. m and M are the masses of the proton and heavy ion, respectively. C 0 = u 0 β/m) 1/2 is the velocity of sound in the heavy ionic sublattice, and u 0 is the lattice constant. The part H P of H is the Hamiltonian of the protonic sublattice with an on-site double-well potential UR n ), H ion is the Hamiltonian of the heavy ionic sublattice with low-frequency harmonic vibration and H int is the interaction Hamiltonian between the protonic and heavy ionic sublattices. Obviously, this model is significantly different from the ADZ model [7] and Pnevmatikos et al. s models [10,12,15,16] due to the following reasons. i) As far as the state and motion of the heavy ion in our model are concerned, it is only a harmonic oscillator with low-frequency acousticvibration due to large mass containing a great number of atoms or atomic groups. However, the heavy ion has both acoustic and optical vibrations in the ADZ model, whose physical idea, which is rather vague, we think is because the optical and acoustic vibrations are two different forms of vibration. ii) As far as the state and motion of the proton lying in the double-well potential are concerned, we here adopt the model of harmonic oscillator with optical vibration that includes a non-diagonal factor, arising from the interaction between neighboring protons; and the interaction of the proton with the heavy ions; namely, its vibrational frequencies are related to displacements of the heavy ions. Therefore there are high corresponding relations for these interactions in the above Hamiltonian. However the ADZ s Hamiltonian does not, the vibration of the proton is acoustic, which is contrary to the heavy ion. This is not reasonable for the protonic model because the vibration frequency of the proton is very high relative to the heavy ion due to its small mass and strong interaction. Moreover, the relation between the protonic and interactional Hamiltonians in the ADZ model does not have the correspondence with each other. These result in the above difficulties for this model. Inversely, the Hamiltonian in our model includes not only the optical vibration of the protons, but also the resonant interaction and the changes of the relative displacement of the neighboring heavy ions. Therefore, it can represent reasonably the dynamic features of the systems when compared with the ADZ and other models. Utilizing the new model we suggest [15 20] that the motion of the proton between a pair of heavy ions cross over the barrier in the intrabond, which results in a change of the relative position between the proton and neighboring heavy ions and the occurrence of the ionic defect, is mainly determined by the double-well potential. The coupled interaction between the proton and heavy ion, which is weaker due to larger spacing between them, can only reduce the height of the barrier that the proton has to overcome to pass from one well to another. However, when the proton approaches the neighboring heavy ions the above coupled interaction will be greatly enhanced and can be so much larger than the double-well potential that the proton can shift over the barriers in the interbonds at the heavy ions from one side to another by this nonlinear coupling interaction through the mechanism of deformation of the heavy ionic sublattice, arising from its stretching and compression This is a quasi-self-trapping mechanism). Thus, the direction of the covalent bond between the proton and heavy ion is changed and a rotation of the bond or Bjerrum defect appears. In such a case, the properties of motion

3 No. 2 Influences of Structure Disorder and Temperature on Properties of Proton Conductivity in 237 of the protons crossed over the barrier in interbond at the heavy ion, or the rotation of the bond Bjerrum defect), is mainly caused by the coupled interaction between the protons and heavy ions. Therefore, both kinds of defects, ionic and bonded, can occur through the competition of the two kinds of nonlinear interaction, double-well potential and nonlinear coupled interaction. Thus the properties of the proton transfer via the ionic and bonded defects were well described in our model, the mobility and conductivity of the proton transfer obtained from this theory is consistent with experimental result in ice crystal. [15 20] Therefore, the new theory is available and credible. However, the above results are obtained by the analytic way in which the hydrogen-bond molecular system is thought to be a periodic system, all physical parameters of the system are taken to be their average values, and some approximate ways, including long-wave approximation, continuum approximation, and so on, are used in the calculation. In practice, the hydrogen-bond system consists of different atomic groups with molecular weights, thus they are not periodic, but aperiodic and nonuniform systems, in which there is structure disorder. Thus all physical parameters and the corresponding states of the soliton in the new model will be affected due to the structure disorder. In such a case it is very necessary to study influences of the structure disorders on the solitons at different temperatures. In this paper we will study the states and features of the soliton in nonuniform and aperiodic hydrogen-bond system by numerical simulation and Runge Kutta way. [21] We will see that the soliton is still stable at the biological temperature 300 K and robust against these structure disorders of the system. In Sec. 2 we introduce the calculated method; the results and discussions are described in Sec. 3. In Sec. 4 we state the conclusions of this paper. 2 Calculation Method Utilizing Eq. 1) and from formulae t p n = H, u n t P n = H, R n we can get the equations of motion for the proton and heavy ion as follows: m R n = mω0r 2 n mω2 1 2 R n+1 + R n 1 ) [ + m χ 1 u n+1 u n 1 ) 4U ] 0 mr0 2 R n + mχ 2 [u n+1 u n )R n+1 + u n u n 1 )R n 1 ] + 4U 0 R0 4 R n 2 R n, 2) Mü n t) = W u n+1 + u n 1 2u n ) + mχ 1 /2 R n+1 2 R n 1 2 ) + mχ 2 R n+1 R n R n R n 1 ). 3) We can represent Eqs. 2) and 3) as the following forms: Ṙ n,t = y n,t m, 4) ẏ n,t = mω0r 2 n,t mω2 1R n+1,t + R n 1,t ) [ + 4U 0 R0 2 Rn,t ) 2 ] 1 R n,t R 0 mχ 1 u n+1,t u n 1,t )R n,t mχ 2 [u n+1,t u n,t )R n+1,t + u n,t u n 1,t )R n 1,t ], 5) u n,t = z n,t M, 6) ż n,t = W u n+1,t + u n 1,t 2u n,t ) + mχ 1 R 2 n+1,t R 2 n 1,t) + mχ 2 R n,t R n+1,t R n,t R n 1,t ). 7) The above equations can determine states and behaviors of the new soliton. There are four equations for one atomic group. Hence for the hydrogen bonded system constructed by N atomic groups there are 4N associated equations. When the fourth-order Runge Kutta way [21] is used to calculate numerically the solutions of the above equations we should discritize them. Thus n denotes the site of atomic group, time is denoted by t, and the step length of the space variable is denoted by h in the equation. Thus, we can get the solutions of the above equations as follows: R n,t+1 = R n,t + h 6 K1 n + K2 n + K3 n + K4 n ), y n,t+1 = y n,t + h 6 L1 n + L2 n + L3 n + L4 n ), u n,t+1 = u n,t + h 6 M1 n + M2 n + M3 n + M4 n ), z n,t+1 = z n,t + h 6 N1 n + N2 n + N3 n + N4 n ), where 8a) 8b) 8c) 8d) K1 n = y n,t m, L1 n = mω 2 0R n,t mω2 1R n+1,t + R n 1,t ) + 4U 0 R 2 0 [ Rn,t 1 mχ 2 [u n+1,t u n,t )R n+1,t + u n,t u n 1,t )R n 1,t ], ) 2 ] R n,t mχ 1 u n+1,t u n 1,t )R n,t R 0 M1 n = z n,t M, N1 n = W u n+1,t + u n 1,t 2u n,t ) + mχ 1 R 2 n+1,t R 2 n 1,t) + mχ 2 R n,t R n+1,t R n,t R n 1,t ),

4 238 PANG Xiao-Feng and YU Jia-Feng Vol. 47 K2 n = K1 n + hl1 n 2m, L2 n = mω0 2 R n,t + 1 ) 2 hk1 n + 1 [ 2 mω2 1 R n+1,t + R n 1,t + 1 ] 2 hk1 n+1 + K1 n 1 ) [ 1 + 4U 0 R0 2 1 R n,t + 1 )) 2 ] R 0 2 hk1 n R n,t + 1 ) 2 hk1 n mχ 1 [u n+1,t u n 1,t + 1 ] 2 hm1 n+1 M1 n 1 ) R n,t + 1 ) [ 2 hk1 n mχ 2 u n+1,t u n,t + 1 ] 2 hm1 n+1 M1 n ) R n+1,t + 1 ) 2 hk1 n+1 + u n,t u n 1,t + 1 ) 2 hm1 n M1 n 1 ) R n 1,t + 1 )] 2 hk1 n 1, M2 n = M1 n + hn1 n 2M, N2 n = W [u n+1,t + u n 1,t 2u n,t + 1 [ 2 hm1 n+1 + M1 n 1 2M1 n ) + mχ 1 R n+1,t hk1 n+1 R n 1,t + 1 ) 2 ] [ 2 hk1 n 1 + mχ 2 R n,t + 1 ) 2 hk1 n R n+1,t + 1 ) 2 hk1 n+1 R n,t + 1 ) 2 hk1 n R n 1,t + 1 )] 2 hk1 n 1, K3 n = K1 n + hl2 n 2m, L3 n = mω0 2 R n,t + 1 ) 2 hk2 n mω2 1 [R n+1,t + R n 1,t h K2 n )] 2 K2 n 1 [ 1 + 4U 0 R0 2 1 R n,t + 1 )) 2 ] R 0 2 hk2 n R n,t + 1 ) 2 hk2 n mχ 1 [u n+1,t u n 1,t + 1 ] 2 hm2 n+1 M2 n 1 ) R n,t + 1 ) 2 hk2 n mχ 2 {[u n+1,t u n,t + 1 ] 2 hm2 n+1 M2 n ) R n+1,t + 1 ) 2 hk2 n+1 [ + u n,t u n 1,t + 1 ] 2 hm2 n M2 n 1 ) R n 1,t + 1 )} 2 hk2 n 1, M3 n = M1 n + hn2 n 2M, [ N3 n = W u n+1,t + u n 1,t 2u n,t + 1 ] [ 2 hm2 n+1 + M2 n 1 2M2 n ) + mχ 1 R n+1,t hk2 n+1 R n 1,t + 1 ) 2 ] [ 2 hk2 n 1 + mχ 2 R n,t + 1 ) 2 hk2 n R n+1,t + 1 ) 2 hk2 n+1 R n,t + 1 ) 2 hk2 n R n 1,t + 1 )] 2 hk2 n 1, K4 n = K1 n + hl3 n m, L4 n = mω0r 2 n,t + hk3 n ) + 1 [ 2 mω2 1 Rn+1,t + R n 1,t + hk3 n+1 + K3 n 1 ) ] [ 1 ) 2 ] + 4U 0 R0 2 1 R n,t + hk3 n ) R n,t + hk3 n ) R 0 mχ 1 u n+1,t u n 1,t + hm3 n+1 M3 n 1 ))R n,t + hk3 n ) mχ 2 [u n+1,t u n,t + hm3 n+1 M3 n ))R n+1,t + hk3 n+1 ) + u n,t u n 1,t + hm3 n M3 n 1 ))R n 1,t + hk3 n 1 )], M4 n = M1 n + hn3 n M, N4 n = W u n+1,t + u n 1,t 2u n,t + hm3 n+1 + M3 n 1 2M3 n )) + mχ 1 R + n+1,t hk3 n+1) 2 R + n 1,t hk3 n 1) 2 ) + mχ 2 R n,t + hk3 n )R n+1,t + hk3 n+1 ) R n,t + hk3 n )R n 1,t + hk3 n 1 )). The system of units ev for energy, Å for length, and ps for time prove to be suitable for the numerical solutions of Eqs. 2) and 3). In the numerical simulation by the fourth-order Runge Kutta way, [21] we require that following conditions must be satisfied: the total energy E = constant up to %), a possible imaginary part of the energy which can occur due to numerical in accuracy is zero to an accuracy of fev; the norm was conserved up to ) 2 ) 2

5 No. 2 Influences of Structure Disorder and Temperature on Properties of Proton Conductivity in pp parts per million). An initial excitation is required in this calculation, it is chosen as, R n 0) = A tanh[n n 0 ) hχ 1 + χ 2 ) 2 /4ω 0 ω 2 1W ] where A is normalization constant) at the size n, for the lattice, u n 0) = 0 is applied. The molecular chain is fixed, N is chosen to be N = 100, a time step size of is used in the simulations. Total numerical simulation was performed by data parallel algorithm and MALAB language. 3 Calculation Results and Discussion 3.1 Numerical Result in Uniform and Periodic Hydrogen-Bond Chains If utilizing the above average values for the parameter, M, Ū0, ϖ o, ϖ 1, W, J, χ1, and χ 2 as shown in Table 1, here m = m P, we calculate numerically the solution of Eqs. 2) and 3) by the fourth-order. Runge Kutta way [21] in a uniform and periodic hydrogen bonded system. The result is shown in Fig. 1. This figure shows that the amplitude of the solution can retain constancy. In Fig. 2 we show the collision property of two new solitons, respectively. From the figures we see that the solution is very stable and can cross with each other in the collision of two solitons. Therefore, equations 2) and 3) have exactly soliton solution in the uniform and periodic systems. Table 1 The average values of physical parameters in ice crystal. W N/m) M kg) R m) U J ) ω s 1 ) ω s 1 ) χ /s 2 m) χ /s 2 m) m Fig. 1 The new soliton solution of Eqs. 2) and 3) for the average parameter values in ice crystal. Fig. 2 Collision behavior of the two solitons in Eqs. 2) and 3) in uniform system. 3.2 Effects of Fluctuation of Force Constant and Disorder in Sequence of Mass on New Soliton We now study numerically the influence of the changes of force constant W and variation of mass of the atomic group, M, arising from the structure distortion of the hydrogen bonded systems or impurity importing, on the stability of the new soliton. When the effects of the structure disorder are taken into account the states and features of the new soliton will be changed. In such a case we should use a random number generator to produce or represent the random sequences of the different parameters in the systems. To test the stability of the new soliton against disorder in the sequence of masses we have to introduce random number generator α k to create random sequences of the masses of amino acids, thus there is a random series of masses for the whole chain, M k = α k M, where αk are determined using a random-number generators with equal probability within a prescribed interval. The fluctuation of W arising from the structure disorder is designated by W = ±β W, β < 1. The results of simulations are shown in Fig. 3 for the four cases of fluctuation of W at 1.1 α k +1.1). From this figure we see that up to a random variation of ±30% W. We find no change in the dynamics for the new soliton. For ±40% W its velocity only is somewhat diminished as compared with the case of W. Finally, for ±50% W the new soliton disperses slowly and the propagation is irregular. In the case of W < ±40% W at 1.1 α k +1.1) virtually no change in the new soliton stability can be obtained. Therefore the ability of the new soliton against the fluctuation of the force constant and disorder in the sequence of masses is robust. Also, in this numerical simulation we find that the influences of the fluctuations of other physical parameters, for example, Ū 0, ϖ 0, ϖ 1, J, χ1, and χ 2, on the properties of the new soliton are small, thus we here do not list these

6 240 PANG Xiao-Feng and YU Jia-Feng Vol. 47 results. Fig. 3 The states of the new soliton resulting from the fluctuations of force constant W for W = ±20% W a), ±30% W b), ±40% W c), ±50% W d) at 1.1 α k +1.1). 3.3 Influences of Temperature and Damping of Medium on New Soliton Exposed in an Externally Applied Electric-Field Since the hydrogen-bonded system is always in an environment or heat bath) with finite temperatures, then it is necessary that the state and properties of the soliton are affected by the temperature and damping of the medium. One can safely assume that the heat bath primarily affects the soliton motion via only the lattice of heavy ion. [22,23] In accordance with thermo-dynamic theory, a decay term MΓ q n and random thermal-noise term, F n t), resulting from the interaction between the heat bath with temperature T and the hydrogen-bonded systems, should be added in displacement equation of the heavy ions. [22,23] Utilizing this idea and considering further the influence of an externally applied electric-field its strength is E), on the soliton, then equations 2) and 3) become m R n = mω0r 2 n + 1 [ 2 mω2 1R n+1 + R n 1 ) + 4U 0 R0 2 Rn ) 2 ] 1 R n mχ 1 u n+1 u n 1 )R n R 0 mχ 2 [u n+1 u n )R n+1 + u n u n 1 )R n 1 ] + qe, 9) Mü n = W u n+1 + u n 1 2u n ) + mχ 1 R 2 n+1 R 2 n 1) + mχ 2 R n R n+1 R n R n 1 ) MΓ u n + F n t), 10) where Γ is the damping coefficient of the medium, which is about 10 9 s 1 for the ice, and q is the charge of the proton. In such a case we must give the explicit representation of F n t). In accordance with statistic physics, F n t) is related with the temperature of the systems, the average value of its correlation function can be represented by [22,23] F x, t)f 0, 0) = 2MΓK B T δx)δt) 1 τ, where τ is damping constant. We assume that the deviation of the random noise satisfies the normal distribution with criterion deviation and has zero expectation value, thus it can be expressed by NF n ) = 1/ 2πσ) exp Fn/2σ), 2 where σ = 2MK B T Γ/τ, τ is time constant, Γ is measured by an inverse number of the time constant of the heat bath. Thus F n can be denoted by F n t) = σ L r=1 [X nrt) 1/2)], here the random number X nr t) is in the region of 0 X nr 1). We here assume L = 12, then the deviation of [X ur t) 1/2)] is 1/12, the domain of the random noise force is just F n t) 6 σ. Thus, F n t) can be represented by σ. Therefore the temperature of the systems is considered in this calculation.

7 No. 2 Influences of Structure Disorder and Temperature on Properties of Proton Conductivity in 241 Utilizing Eqs. 9) and 10) we can study the influences of the temperature and random thermalnoise forces of medium exposed in an externally applied electric-field E on the states of the new soliton in the ice by above fourthorder Runge Kutta way. We first study the effect of the electric-field on the soliton at T = Γ = 0. The states of the soliton for E = 100 kv/cm and 200 kv/cm are shown in Fig. 4. From this figure we see that the soliton is stable, and its velocity is changed. This electric-field dependence of velocity of the soliton is shown in Fig. 5, namely its velocity increases linearly with increasing of the electric-field. We further calculate the influence of the damping of the medium on the soliton at T = E = 0. The properties of the soliton for three different damping coefficients are shown Fig. 6. We see that no change in the new soliton stability is found for small damping, but at great damping the soliton disperses, its velocity decreases also with the increase of the damping coefficient, which is shown in Fig. 7. when Γ 0, T 0, and E 0, the states of the soliton differ from above results. In Fig. 8 we give the features of the soliton for four different temperatures, T = 190 K, 200 K, 210 K, and 273 K at E = 200 kv/cm and Γ = s 1 in the ice. From this result we find no variation in the dynamics for the new soliton under influences of these temperatures, i.e., the new soliton is thermally stable. This shows that our theoretical model of the proton transfer in hydrogen-bonded systems is available and successful. Fig. 4 The states of the soliton for E = 100 kv/cm a) and 200 kv/cm b) at T = Γ = 0. Fig. 5 This electric-field dependence of velocity of the soliton at T = Γ = 0, here the velocity is denoted by the lattice numbers in longitudinal axis. Fig. 6 The properties of the new soliton for the damping coefficients Γ = s 1 a), s 1 b), and s 1 c) at T = E = 0.

8 242 PANG Xiao-Feng and YU Jia-Feng Vol. 47 Fig. 7 The velocity of the soliton on the damping coefficient at T = Γ = 0, here the velocity is denoted by the lattice numbers in longitudinal axis. Fig. 8 The features of the soliton for T = 190 K a), 200 K b), 210 K c), and 273 K d) at E = 200 kv/cm and Γ = s 1 in the ice. 4 Conclusion We here study the dynamic properties of proton conductivity along hydrogen-bonded molecular systems, for example, ice crystal, with structure disorder or damping and finite temperatures exposed in an externally applied electric-field by numerical simulation and fourth-order Runge Kutta way in our soliton model. The results obtained show that the proton-soliton is very robust against the structure disorder including the different fluctuation of the force constant and disorder in the sequence of masses and thermal perturbation at different temperatures and damping coefficients of medium, the velocity of its conductivity increases with increasing of the externally applied electric-field and decreasing of the damping coefficient of medium, but the proton-soliton disperses for quite great fluctuation of the force constant and damping coefficient. In the numerical simulation we find that the proton-soliton in our model is thermally stable in large region of temperature of T 273 K under influences of damping and externally applied electric-field in ice crystal. This shows that our model is available and appropriate to ice. References [1] P. Schuster, G. Zundel, and C. Sandorfy, The Hydrogen Bond, Recent Developments in Theory and Experiments, North Holland, Amsterdam 1976). [2] M. Peyrard, Nonlinear Excitation in Biomolecules, Springer, Berlin 1995).

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