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Energy transfer in hydrogen-bonded chains with asymmetric double-well potentials of hydrogen bonds
Solid State Communications, Vol. 72, No. 2, pp. 223-225, 1989. Printed in Great Britain.
0038-1098/89 $3.00 + .00 Pergamon Press plc
ENERGY T R A N S F E R IN H Y D R O G E N - B O N D E D CHAINS WITH ASYMMETRIC DOUBLE-WELL POTENTIALS OF H Y D R O G E N BONDS Alexander Gordon Department of Physics, Oranim-School of Education of the Kibbutz Movement, Haifa University, Tivon 36910, Israel
(Received 1 March 1989 by S. Amelinckx) It is shown that the two-component soliton of a new type travels in hydrogen-bonded chains if we take into account the asymmetry of the double-minimum potential of hydrogen bonds and the interaction between the proton and heavy ion sublattices. The calculated energy of the soliton for a weak proton-ion interaction is close to the activation energy measured in experiments for the proton conductivity in ice crystals. ON THE basis of both experimental and theoretical investigations it is generally known that the potentialenergy relief of many hydrogen bonds has two minima in the region available for protonic motion. In most cases the minima are not equivalent [1, 2]. Recently the soliton dynamics in solids containing hydrogenbonded solids has been considered using the asymmetric double-minimum potential relief of the hydrogen bond [3]. In this note we show that the energy transfer in hydrogen-bonded chains may be realized by the motion of the bell-shape soliton if we take into account the energy asymmetry between the two wells of the double-minimum potential of hydrogen bonds. According to the model [3, 4] the hydrogenbonded chain consists of two interacting sublattices of harmonically coupled protons (mass m) and heavy ions (mass M). Each proton is in an asymmetric double-well potential created by the pair of nearest neighbour heavy ions. In the continuum limit the model Hamiltonian is of the form 1
H = 7 ~H(x' t) dx,
m[(Ou] 2 (Ou~2] -~ Lkat/ + c] \ax/ J + V(u),
(
1
4dC~'/21
B2 ,] ] .
(5)
The Hamiltonian density for the heavy ion sublattice
Hi°n
M F(ae~2 (ae'~21 T l\&Y + ft°2e2 + v°2 \ ~ x J ] '
=
(6)
yields the dispersion relation of small-amplitude oscillations of heavy ions n 2 = D~ + v2qz, (7) where the frequency D.0 and v0 are the characteristic parameters of the heavy ion sublattice and Q(x, t) is the relative displacement field of heavy ions. The proton-ion interaction is given by the following Hamiltonian density [4]
(2)
where k is the constant of the proton-ion interaction. The equations of motion corresponding to this Hamiltonian are
kQ(u 2 -
=
aZu m ~-~
-- meg ~a2u
u2),
(8)
+ Art - - B u 2 + C u 3 - -
2k~u = O, (9)
(3)
(02¢
\~7-
where u is the protonic displacement and V(u) represents an asymmetric double-minimum potential A u2 - ~B u3 + ~Cu 4, V(u) = ~-
~-~
Hin t
where l is the lattice spacing. Here Hp is the Hamiltonian density for the proton sublattice given by
np=
B [1+
u0 =
(1)
with
H(x, t) = np(x, t) + nion(X , t) + Hint(x , t),
where constants A, B and C are positive; Co is the velocity of protonic low-amplitude sound waves. The upper minimum of V(u) is for u = 0 and the lower one is for u = u0, where
(4)
v02 02¢
~
+n~e
)
+ ~ ( ~ - U o ~) = o. (10)
Partial differential equations (9) and (lO) in independent variables x and t can be reduced to ordinary
223
224
ENERGY TRANSFER IN H Y D R O G E N - B O N D E D CHAINS
differential equations in the variable s by the substitutions = x - vt m(c~ -- v 2)
d2u
= Au - 2 k o u
- Bu 2 + Cu 3,
boring heavy atoms (l - Q) connected by the hydrogen bond. In case (2) the two-component soliton travels in the system. We calculate the soliton energy for a weak proton-ion interaction: case (1). The kinetic energy of the soliton (17) is given by
(11) M(v 2 -
d2~°
v0~) h-Ts~
(12)
M f ~ O - k(u 2 - u~).
-
Vol. 72, No. 2
f
E~
d x m (du'] 2
J 5- ~ \dt/"
(20)
The integration in equation (20) gives The system of equations (11) and (12) can be solved in two cases: (1) k -o 0; (2) v = %. For the case (2) equation (12) gives k f~02M (u 2 - Uo2),
0 -
(13)
3A2mv 2 ( 5 7 ~_~2 ) 5B2l A F ,2,-2, '
Ek --
where F is the hypergeometric function. According to Ek = ½Mso~V2 and consequently we obtain the equation for the soliton mass Mso~
so that equation (11) takes the following form d2u m(c] - v 2) ds 2 -
A'u
-- B u 2 +
C ' u 3,
(14)
where A' and C' are constants A and C renormalized by the proton-ion interaction A'
= A
C' =
(16)
The solution of equation (14) has been obtained in [3] (2A'/C') 1/2 [//+(//2_
,
(17)
1),/2 cosh A]
where B (
2 "]'/2
// = -X \ A--rU )
'
and A is the width of the bell-shape soliton (17) given by A -
m(cg -- v2) m A,~/2
(18)
Substituting equations (5) and (17) into equation (13) we obtain 2..
6A2m (~ 7 ~ z) 5B2l~ F ,2,~, .
(22)
Then the total energy of the slowly moving (v 2 < c~) soliton (17) E0 = M~oiCg is given by 6A2mc2 ( ~ 7 ~-~2 ) 5B2lA F ,2,~, .
(23)
(15)
2k 2 C + D.~M'
u =
M~o, -
E0 -
2k2u~ D.~M'
(21)
. ~-~) [l q'-(I
_[//+ (//2_ i),/2cosh~]-:}.
In the following calculations we take v 2 < Co 2 (slowing solitons). We use the experimental data for ice. We take u = 0.375 A [5] and suppose that the asymmetry energy (the energy difference between the two minima of the potential of the hydrogen bond) A V = V(uo) hVH, where v. is the frequency the O-H stretching vibration in ice (v. = 3250 cm-t [6]). We take A V = 650cm -1 [2]. Then we estimate approximately A according to (2gVH) 2 = 2A/m: A = 4.92 x 105erg cm 2 [8]. Using the condition f12 > l we take f12 = ). For d = 2.76/~ [6] and Co = 1.1 x 106cms l [7] we obtain E0 = 0.32 eV. This value is close to the experimental ones of the activation energy measured by the proton conductivity in ice at T = -10°C: E0 = 0.30eV [8], E0 = (0.34 _ 0.02)eV [9], E0 = 0.37eV [ 10]. The obtained agreement with experiments shows that the bell-shape soliton propagation in hydrogenbonded chains may be a mechanism of the protonic conduction.
8 "~I/2l (19,
Consequently the propagation of the heavy ion displacement is described by the bell-shape antisoliton (inverted bell) (19). Therefore the propagation of the protonic soliton is accompanied with a localized extension of the relative distance between two neigh-
REFERENCES 1. 2. 3. 4. 5. 6.
R. Blinc & D. Had2i, Mol. Phys. 1, 391 (1958). R.L. Somorjai & D.F. Hornig, J. Chem. Phys. 36, 1980 (1962). A. Gordon, Solid State Commun. 68, 885 (1988). V.Ya. Antonchenko, A.S. Davydov & A.V. Zolotariuk, Phys. Status Solidi (b) 115, 631 (1983). R.J. Nelmes, Ferroelectrics 24, 237 (1980). G. Pimentel & A. McClellan, The Hydrogen Bond (Freeman, San Francisco) (1960).
Vol. 72, No. 2 7. 8. 9.
ENERGY TRANSFER IN HYDROGEN-BONDED CHAINS
A. Gordon, Physica 146B, 373 (1987). R. Ruepp & M. Kass, in Physics oflce, (Edited by N. Riehl, B. Bullemer & H. Engelhardt), p. 555, Plenum, New York (1969). B. Bullemer, H. Engelhardt & N. Riehl, in Physics
225
of Ice, (Edited by N. Riehl, B. Bullemer & 10.
H. Engelhardt), p. 416, Plenum, New York (1969). B. Bullemer & N. Riehl, Solid State Commun. 4, 447 (1966).
0038-1098/89 $3.00 + .00 Pergamon Press plc
ENERGY T R A N S F E R IN H Y D R O G E N - B O N D E D CHAINS WITH ASYMMETRIC DOUBLE-WELL POTENTIALS OF H Y D R O G E N BONDS Alexander Gordon Department of Physics, Oranim-School of Education of the Kibbutz Movement, Haifa University, Tivon 36910, Israel
(Received 1 March 1989 by S. Amelinckx) It is shown that the two-component soliton of a new type travels in hydrogen-bonded chains if we take into account the asymmetry of the double-minimum potential of hydrogen bonds and the interaction between the proton and heavy ion sublattices. The calculated energy of the soliton for a weak proton-ion interaction is close to the activation energy measured in experiments for the proton conductivity in ice crystals. ON THE basis of both experimental and theoretical investigations it is generally known that the potentialenergy relief of many hydrogen bonds has two minima in the region available for protonic motion. In most cases the minima are not equivalent [1, 2]. Recently the soliton dynamics in solids containing hydrogenbonded solids has been considered using the asymmetric double-minimum potential relief of the hydrogen bond [3]. In this note we show that the energy transfer in hydrogen-bonded chains may be realized by the motion of the bell-shape soliton if we take into account the energy asymmetry between the two wells of the double-minimum potential of hydrogen bonds. According to the model [3, 4] the hydrogenbonded chain consists of two interacting sublattices of harmonically coupled protons (mass m) and heavy ions (mass M). Each proton is in an asymmetric double-well potential created by the pair of nearest neighbour heavy ions. In the continuum limit the model Hamiltonian is of the form 1
H = 7 ~H(x' t) dx,
m[(Ou] 2 (Ou~2] -~ Lkat/ + c] \ax/ J + V(u),
(
1
4dC~'/21
B2 ,] ] .
(5)
The Hamiltonian density for the heavy ion sublattice
Hi°n
M F(ae~2 (ae'~21 T l\&Y + ft°2e2 + v°2 \ ~ x J ] '
=
(6)
yields the dispersion relation of small-amplitude oscillations of heavy ions n 2 = D~ + v2qz, (7) where the frequency D.0 and v0 are the characteristic parameters of the heavy ion sublattice and Q(x, t) is the relative displacement field of heavy ions. The proton-ion interaction is given by the following Hamiltonian density [4]
(2)
where k is the constant of the proton-ion interaction. The equations of motion corresponding to this Hamiltonian are
kQ(u 2 -
=
aZu m ~-~
-- meg ~a2u
u2),
(8)
+ Art - - B u 2 + C u 3 - -
2k~u = O, (9)
(3)
(02¢
\~7-
where u is the protonic displacement and V(u) represents an asymmetric double-minimum potential A u2 - ~B u3 + ~Cu 4, V(u) = ~-
~-~
Hin t
where l is the lattice spacing. Here Hp is the Hamiltonian density for the proton sublattice given by
np=
B [1+
u0 =
(1)
with
H(x, t) = np(x, t) + nion(X , t) + Hint(x , t),
where constants A, B and C are positive; Co is the velocity of protonic low-amplitude sound waves. The upper minimum of V(u) is for u = 0 and the lower one is for u = u0, where
(4)
v02 02¢
~
+n~e
)
+ ~ ( ~ - U o ~) = o. (10)
Partial differential equations (9) and (lO) in independent variables x and t can be reduced to ordinary
223
224
ENERGY TRANSFER IN H Y D R O G E N - B O N D E D CHAINS
differential equations in the variable s by the substitutions = x - vt m(c~ -- v 2)
d2u
= Au - 2 k o u
- Bu 2 + Cu 3,
boring heavy atoms (l - Q) connected by the hydrogen bond. In case (2) the two-component soliton travels in the system. We calculate the soliton energy for a weak proton-ion interaction: case (1). The kinetic energy of the soliton (17) is given by
(11) M(v 2 -
d2~°
v0~) h-Ts~
(12)
M f ~ O - k(u 2 - u~).
-
Vol. 72, No. 2
f
E~
d x m (du'] 2
J 5- ~ \dt/"
(20)
The integration in equation (20) gives The system of equations (11) and (12) can be solved in two cases: (1) k -o 0; (2) v = %. For the case (2) equation (12) gives k f~02M (u 2 - Uo2),
0 -
(13)
3A2mv 2 ( 5 7 ~_~2 ) 5B2l A F ,2,-2, '
Ek --
where F is the hypergeometric function. According to Ek = ½Mso~V2 and consequently we obtain the equation for the soliton mass Mso~
so that equation (11) takes the following form d2u m(c] - v 2) ds 2 -
A'u
-- B u 2 +
C ' u 3,
(14)
where A' and C' are constants A and C renormalized by the proton-ion interaction A'
= A
C' =
(16)
The solution of equation (14) has been obtained in [3] (2A'/C') 1/2 [//+(//2_
,
(17)
1),/2 cosh A]
where B (
2 "]'/2
// = -X \ A--rU )
'
and A is the width of the bell-shape soliton (17) given by A -
m(cg -- v2) m A,~/2
(18)
Substituting equations (5) and (17) into equation (13) we obtain 2..
6A2m (~ 7 ~ z) 5B2l~ F ,2,~, .
(22)
Then the total energy of the slowly moving (v 2 < c~) soliton (17) E0 = M~oiCg is given by 6A2mc2 ( ~ 7 ~-~2 ) 5B2lA F ,2,~, .
(23)
(15)
2k 2 C + D.~M'
u =
M~o, -
E0 -
2k2u~ D.~M'
(21)
. ~-~) [l q'-(I
_[//+ (//2_ i),/2cosh~]-:}.
In the following calculations we take v 2 < Co 2 (slowing solitons). We use the experimental data for ice. We take u = 0.375 A [5] and suppose that the asymmetry energy (the energy difference between the two minima of the potential of the hydrogen bond) A V = V(uo) hVH, where v. is the frequency the O-H stretching vibration in ice (v. = 3250 cm-t [6]). We take A V = 650cm -1 [2]. Then we estimate approximately A according to (2gVH) 2 = 2A/m: A = 4.92 x 105erg cm 2 [8]. Using the condition f12 > l we take f12 = ). For d = 2.76/~ [6] and Co = 1.1 x 106cms l [7] we obtain E0 = 0.32 eV. This value is close to the experimental ones of the activation energy measured by the proton conductivity in ice at T = -10°C: E0 = 0.30eV [8], E0 = (0.34 _ 0.02)eV [9], E0 = 0.37eV [ 10]. The obtained agreement with experiments shows that the bell-shape soliton propagation in hydrogenbonded chains may be a mechanism of the protonic conduction.
8 "~I/2l (19,
Consequently the propagation of the heavy ion displacement is described by the bell-shape antisoliton (inverted bell) (19). Therefore the propagation of the protonic soliton is accompanied with a localized extension of the relative distance between two neigh-
REFERENCES 1. 2. 3. 4. 5. 6.
R. Blinc & D. Had2i, Mol. Phys. 1, 391 (1958). R.L. Somorjai & D.F. Hornig, J. Chem. Phys. 36, 1980 (1962). A. Gordon, Solid State Commun. 68, 885 (1988). V.Ya. Antonchenko, A.S. Davydov & A.V. Zolotariuk, Phys. Status Solidi (b) 115, 631 (1983). R.J. Nelmes, Ferroelectrics 24, 237 (1980). G. Pimentel & A. McClellan, The Hydrogen Bond (Freeman, San Francisco) (1960).
Vol. 72, No. 2 7. 8. 9.
ENERGY TRANSFER IN HYDROGEN-BONDED CHAINS
A. Gordon, Physica 146B, 373 (1987). R. Ruepp & M. Kass, in Physics oflce, (Edited by N. Riehl, B. Bullemer & H. Engelhardt), p. 555, Plenum, New York (1969). B. Bullemer, H. Engelhardt & N. Riehl, in Physics
225
of Ice, (Edited by N. Riehl, B. Bullemer & 10.
H. Engelhardt), p. 416, Plenum, New York (1969). B. Bullemer & N. Riehl, Solid State Commun. 4, 447 (1966).