DISCRETE BREATHERS IN POLYMER CRYSTALS
L.I.Manevitch* and A.V.Savin**
N. N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, ul.
Kosygina 4, 117977 Moscow, Russia
*
lmanev@center.chph.ras.ru
**
asavin@center.chph.ras.ru
We present a study of nonlinear vibrations corresponding to localized nonlinear
normal modes – breathers. In the localization region periodic contraction-extension of
interparticle bonds occurs which is accompanied by decrease-increase of angles
between the bonds. Using computer simulation it is shown that the breathers presence
in the thermalized chain and their energetic contribution to the heat capacity can be
significant.
I. Introduction
Short wavelength localized nonlinear excitations (breathers) is now a subject
of quite interest. However all studies in this field relate to straight oscillatory chains.
Real polymer chains have the specific structural features which make their dynamical
study more difficult. To overcome the difficulties, we use an asymptotic procedure
which allows to reduce the problem to well-known nonlinear Schrodinger equation.
After analytical study we have performed numerical investigation to confirm
theoretical results.
II. Short wavelength normal modes (breathers) in zigzag oscillatory chain
The structure of planar zigzag oscillatory chain is schematically plotted at
figure 1.
Let us introduce the local coordinate system presented on this figure. Then the
Hamiltonian function of the chain in the case of planar dynamics can be written as
follows.
1
H   M u n2  w n2  V   n   U  n 
(1)
2


where the first term describes the kinetic energy of the n -th particle ( M is the mass of
every particle), the second – the deformation energy of the n -th bond between n th
and (n  1) -th particles, and the last term – the deformation energy of the n -th angle
between (n  1) and n -th bonds.
The length of the n -th bond is
(2)
 n  (an2,1  an2,3 )1 / 2 ,




where a n ,1  u n  u n 1  l x , a n ,3  wn 1  wn  l z ( l x   0 cos  0 / 2 l z   0 sin  0 / 2 )
are the transversal and longitudinal steps of the zigzag chain respectively.
The cosine of the n -th angle is
cos  0  (a n 1,1 a n ,1  a n 1,3 a n ,3 ) /  n 1 n .
(3)
Potentials of the bonds and angles are accepted in the form
1
1
V ( n )  K1 ( n   0 ) 2 , U ( n )   (cos  n  cos  n ) 2
(4)
2
2
4 - 15
Small-amplitude vibrations of isolated transzigzag were considered in [4,6,7] and
vibrations with account of interaction with immobile surrounding chains – in [5].
Small-amplitude vibrations can be divided to planar (in the zigzag plane) and
transversal ones. In turn, the plane motions are divided to low-frequency acoustic and
high-frequency optic vibrations. Accordingly, one can separate three dispersion
curves corresponding to 1) plane acoustic phonons    a (q) , 2)transversal
phonons   t (q) , 3)plane optic phonons   o (q) . These curves with account of
interchain interaction are presented at Fig. 1 (a).
Let us consider the long wavelength modulation waves in weakly nonlinear
limiting case when one can find the following main nonlinear continuum
approximations for  n and  n with respect to small ratio of interparticle space and
characteristic wavelength of the process (the smallness of this ratio is supposed to be
similar to ratio of cubic and quadratic (as well as quarter and cubic terms) in
Hamiltonian
 


u 1 2  2 u 
w 2
   n   0   cos 0  2u  l z
 l z 2  sin 0 l z
 sin 2 0 u 2 ,
2 
z 2 z 
2 z  0
2
(5)


4
2
w 8 2
(6)
   n   0 
sin 2 0 u  cos 0 l z
 u .
0
2
0
2 z  02
Besides, it is taken into account that linear analysis reveals smallness of longitudinal
displacement with respect the transversal one for considered type of motion. In this
approximation the equations of motion after certain transformations can be reduced to
nonlinear partial equation:
2
 2U
2 2  U
 c
 U  4U 2  82U 3  0,
(7)
2
2


where


K
u
2
U 
;  
K1 cos 2 0  4 22 sin 2 0 t ;
0
2
2
m
0
 
z
lz
;


K2
2 0


cos


2
sin
0
2
2
2


2
K1  0 sin  0
K1  0

;
c2  








K
K
K
2
2 
16 K1 cos 2 0  4 22 sin 2 0   1  2
 1  2
cos  0 
2
2


2
2  
0
K1  0 
K1  0 


sin  0 sin
4 
 0 
2 
3 K1 
8 K 2 
 02 
;




K
2 K1 cos 2 0  4 22 sin 2 0 

2
2 
0



K
8 
8  2  K1 sin 4 0  2 22 sin 2  0 .

2
 0 
0

4 - 16
Let us introduce the complex functions [2,3]:
 ,   V  iU ,  * ,   V  iU
where V  U /  . Partial equation is equivalent to the system of two equations
 
 
(8)
V
 2U
  2c 2
 U  4U 2  8 2U 3 ,
2


(9)
U
 V.

After substitution of (8) into (9) and certain transformation one can obtain

 2 (   * )
 i  i 2 c 2
 (   * ) 2  i 2 (   * ) 3

 2
The change of variables   e i (, ) leads to equation

 2 (  * e i )
 i 2 c 2
 (e i  * e i ) 2 e i  i2 (e i  * e i ) 3 e i

 2
(10)
Then we introduce, alongside with fast time    0 , the slow times
1  0 ,  2   2  0 
(11)
   0  1   2  2  
(12)
and power expansion
After substitution of the expansion (12) into (10) with taking into account (11) and
selecting the terms of similar order with respect to small parameter one can find
 0
 0,
 0
2
1  0

  02 e i  2  0 e i  ( 0 ) 2 e 3i  0,
 0 1
0
0
0
2
 2 1  0
2  0


 ic
 20 1e i  2*0 1* e 3i  20 1* e i  2*0 1e i 
2
 0 1  2

0
0
0
0
2
2


 i 30 e 2i  3  0  0  3  0 *0 e  2i  (*0 ) 3 e  4i   0.


0
0
0
The condition of absence of secular terms in first two equations of this system leads to
relations 0  0 (1 ,  2 ,) ,  0 1  0 , consequently  0   0 ( 2 ,  3 ,) . Then the
solution of the second equation can be written as follows
2
1
1  i02 e i  2i  0 e i  (*0 ) 2 e 3i  0.
3
The condition of absence of secular terms in third equation may be presented in the
form
0
0
4 - 17
0
2

 0
 2  0  16 2
 ic 2
i
  3   0  0  0.
(13)
2
3

 2



So we come to well known exactly integrable nonlinear Schrodinger equation (NSE).
If
 16

    2  3   0,
(14)
3



this equation possess localized soliton-like solution (envelope soliton)
2
 
 2S 
 


1/2  





(15)
 0 ( ,  2 ) 
exp i
 i 2 sec h S
  2 
  
 2c

c


 






where

2
 S,
4
(16)
To check the assumptions accepted for derivation of the equation (15) a numerical
study of the problem has been performed.
Fig.1. Dispersion curves    a (q) ,   t (q) ,   o (q) . (curves 1, 2, 3) for transzigzag interacting with immobile surrounding chains (a). Density of energy
distribution p on frequencies  for thermal vibrations with temperature T  1K (b),
T  100K (c), T  200K (d), and T  300K (e). Gray color corresponds to
frequencies region in which the discrete breathers occur
4 - 18
III. Numerical study of localized nonlinear vibrations of the chain
The equations of motion corresponding to Hamiltonian (1) have the form
H
H
H
n  
M un  
, M  n  
, Mw
, n  0,1,2, .
u n
 n
w n
The finite chain consisting of N  200 particles was considered. The viscous friction
providing absorption of phonons was introduced on the boundaries of the chain. The
system of equations (17) with n  1 , 2 , , N has been integrated numerically with
breather-like initial condition. If a discrete breather can exist in the chain it can
manifest itself in irradiation of superfluous (non-breather part of initial excitation)
phonons.
Numerical modelling has shown that localized periodic vibrations exist due to
tension-compression of bonds with cooperative change of angles in the plane of
transzigzag – see Fig. 2 (b) (c). The vibration proceeds in the plane of trans-zigzag
with nodes displacements transversal to main backbone axis [Fig. 2 (a)].
Fig2. Localized planar periodic vibrations of trans-zigzag . Vibrations are
schematically shown, the thickness of line corresponds to amplitude (a). The
magnitudes of valence bonds p n (b) and angles  n (c) are presented for ten different
instants. The frequency of breather   820.5 cm-1, energy En  26.4 kJ/mol, width
L  4.28
These vibrations are stable excitations, which are characterized by frequency  ,
energy E and dimensional width
N

L  2   ( n  nc ) 2 p n 
n1

4 - 19
1/ 2
,
where the point nc  n n pn determines the position of the vibrations center and
sequence pn  n En / E – the density of energy distribution along the chain.
Essential nonlinearity of these vibrations is manifested in decrease of its frequency
with amplitude growth. So, the revealed excitation is actually discrete breather.
Dependence of energy E and width of the breather L on its frequency  is presented
at Fig. 3. The frequency spectrum of the breather is situated near lower boundary of
optic phonons.
Fig.3. Dependence of energy E (a), width L (b) of the breather upon frequency  in
isolated chain (curves 1, 3) and in the chain with substrate potential (curves 2, 4).
IV. Thermal vibrations of trans-zigzag as origin of discrete breathers
Let us consider the thermal vibrations of trans-zigzag. With this goal we
analyze the finite chain consisting of N segments. Their N0 segments near boundary
(from both sides) are situated into heat bath with temperature T. The dynamics of the
system is described by the systems of Langevin equations
H
M un  
  n  n M u n ,
u n
H
M n  
 n  n M n ,
(17)
 n
H
n  
Mw
  n  n M w n ,
wn
n  0,1,2,  .
 n where the Hamiltonian of the system H is given by Eq. (1),  n ,  n , and  n are
random normally distributed forces describing the interaction of n-th molecule with a
4 - 20
thermal bath, the coefficient of friction n  0 for N 0  n  N  N 0 and n   for
n  N 0 and N  N0  n  N . Coefficient of friction   1 / tr , where t r – the relaxation
time of the velocity of the molecule. The random forces  n ,  n , and  n have the
correlation functions
 n (t1 ) m (t 2 )   n (t1 ) m (t 2 )   n (t1 ) m (t 2 )  2 Mk B  nm (t1  t 2 ),
 n (t1 ) m (t 2 )   n (t1 ) m (t 2 )   n (t1 ) m (t 2 )  0,
1  n, m  N 0 , N  N 0  n , m  N ,
where k B is Boltzmann’s constant and T is the temperature of heat bath.
The system (17) was integrated numerically by the standard forth-order
Runge-Kutta method with a constant step of integration t . Numerically, the delta
function was represented as (t )  0 for t  t / 2 and (t )  1 / t for t  t / 2 , i.e.
the step of numerical integration corresponded to the correlation time of the random
force. In order to use the Langevin equation, it is necessary that t  t r . Therefore
we chose t =0.001 ps and the relaxation time t r =0.1 ps.
Let us consider a frequency distribution of kinetic energy of thermal
vibrations. For this goal the system (17) was integrated numerically for N=500,
N0.=N/2 While choosing the initial conditions as corresponding to ground state of the
chain the system was integrated during t  10 t r to bring it in the thermal equilibrium.
After that we calculated the density of molecules kinetic energy distribution on
frequencies p() . To increase an accuracy, the density of distribution was calculated
using 1000 independent realizations of the chain thermalization. The profile of the
density of distribution for different values of temperature is presented at Fig. 1 (it is
accepted that  p () d   3 ).
For temperature T =1 K the density of distribution almost coincides with
corresponding density for linearized system, so anharmonicity is not essential here.
All vibrations are linear and only phonons are thermalized. For T =100 K we see a
shift of density behind the low boundary of the spectrum of optic phonons which
becomes more pronounced with further increase of the temperature. High frequency
vibrations in this region can be identified as breathers because they exist in the
frequency region [b , o (0)] , corresponding to this type of excitations. The part of
energy corresponding to breathers may be found as
 ( 0)
pb   p() d  .
o
b
The contribution of the breathers to thermal energy increases with growth of
temperature (for T=1 K it is pb=0.002, for T=100 K — pb=0.106), has a maximal
value pb=0.115 for T=200 K and then decreases (for T=300 K the breathers
contribution corresponds to pb=0.083).
Let us isolate the breathers from thermal vibrations. With this goal we consider the
chain consisting of N=500 segments with boundary cites (N0=50) connected to the heat
bath with temperature T. After thermalization, we put the temperature of thermal bath
T =0 and consider the irradiation of heat energy from internal region
( N 0  n  N  N 0 ) . The relaxation process for T=200 K is shown at Fig. 4. We see a
formation of several mobile localized excitations. Their detail analysis leads to
4 - 21
conclusion that they are discrete breathers with frequencies   b . So, one can
observe the presence of breathers in thermal vibrations.
Fig4. Formation of discrete breathers from thermal vibrations of zigzag chain (N=500,
T=200 K). The absorbing ends are considered (N0=50). Temporal dependence of
energy distribution En in the chain is presented.
Let us consider the interaction of discrete breather with thermal phonons. The
stationary discrete breather with frequency   820.5 cm-1 was situated in the center
of finite chain (N=200), their edges (N0=10) being situated in the thermal bath (T=10
K). As we can see from Fig. 5 the breaking of breather is observed just as the center
of the chain is thermalized (the energy loss is 50% for 20 ps).
Probability of thermally activated formation of discrete breathers in the chain
increases with growth of the temperature. Therefore their concentration has to
increase when temperature grows. However in a thermalized chain the breather has a
finite time of life, decreasing with growth of the temperature. It is a reason for nonmonotonous dependence of the concentration of breathers pb on temperature T — it
increases when T  200 K and decreases for T >200 K with maximal magnitude for
T=200 K. Numerical study shows that the breathers may be better distinguished from
thermal vibration namely when T=200 K.
One can see from Fig. 1 that revealed breathers is unique type of stable localized
periodic excitations in thermalized chain for given parameters of the crystal. Besides
the breathers only vibrations with frequencies corresponding to the spectrum of linear
oscillations can be thermalized. It confirms the conclusion that only one stable type of
discrete breathers corresponding to localized oscillations of valence C—C bonds
exists. The breathers are present in thermalized chain even for sufficiently
smalltemperatures.
4 - 22
Fig. 5. Breaking of the breather in thermalized chain (N=200, N0=10 ,T=10 K,
frequency of the breather   820.5 cm-1). Temporal dependence of current local
magnitudes of temperature (kinetic energy of chain segments) Tn is presented.
V. Conclusion
Stable localized nonlinear vibrations which are discrete breathers can exist in
polymer crystal. In PE macromolecules they are planar vibration of transzigzag with
periodic deformation of valence bonds C—C and valence angles CCC. The breathers
present in thermalized chain and their contribution in heat capacity may be essential.
The work was supported by RFBR (grant 04-03-32119)
REFERENCES
[1] Aubry S Physica D 103 201 (1997)
[2] Manevitch L I Nonlinear Dynamics 25 95-109 (2001)
[3] Manevitch L.I Polymer Science C 4(2) 117-181 (2001)
[4] Manevitch L I, Savin A V Phys. Rev. E 55 4713 (1997)
[5] Savin A V, Manevitch L I Phys. Rev. B 58 (17) 11386 (1998)
[6] Kirkwood J G J Chem. Phys. 7 (7) 506 (1939)
[7] Pitzer K J Chem. Phys. 8 (8) 711 (1940)
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