Nonlinear Localized Excitations in Helix Macromolecules
1 NONLINEAR LOCALIZED EXCITATIONS IN HELIX MACROMOLECULES
Natalya Kovaleva, Leonid Manevitch
Semenov Institute of Chemical Physics, Russian Academy of Science, Moscow 119991, Russia Abstract:
Physico-mechanical properties of polymers in solid state, in particular conditions of their structural transformations, are substantially defined by existence and mobility of elementary nonlinear excitations. The localized oscillatory excitations (breathers) which have been revealed in crystalline PE are important for thermodynamic studies. In the given work the breathers in more complex crystalline of helix macromolecules are analytically investigated for the first time. We show that introduction of adequate variables allows to construct the continuum model describing spatially localized excitations with internal degree of freedom. Characteristic parameters of breathers and conditions of their mobility are defined.
1. Introduction
It was shown recently numerically [1] and analytically [2] that vibrational localized excitations can exist in realistic model of zig-zag polymer chain. Such excitations existing in the attenuation bands of linear spectrum, introduce a noticeable contribution into thermal capacity. In this paper we consider essentially more complex case of helix polymer chain (in application to polytetrafluoethylene (PTFE)). The long wavelength localized excitations (acoustic solitons) in helix chains have been studied numerically in the paper [3]. We deal with oscillatory localized excitations which have several deformation components: radial, angular and longitudinal.
2. The Model
The ground state of macromolecule PTFE is a helix 1*13/6. In the approximation of “united atoms” we consider the CF -group as a particle with mass М =50 а .m.u. Then its equilibrium state can be characterized by radius vector: ),),sin(),cos(( ϕϕϕ ∆∆∆= nnRnRR где R =0,41Å – radius of helix. The angular and longitudinal steps of the helix are πϕ=∆ , =∆ z Å correspondingly. Let us introduce transversal n r , angular n ϕ and longitudinal n h displacements of n -th node of the chain with respect to its equilibrium state. Then the Hamilton function of the chain has a view: } )()()()([21 nnnnnnnn WUVhrRrMH δθρϕ ++++++ = ∑ &&& (1) Here dot denotes derivative by time t , n ρ – length of n -th valence bond, n θ – n -th valence angle and n δ – n -th rotation (conformational) angle. The potential of valence bond is ))](exp(1[)( ρραρ −−−= nn DV . In accordance with the data [4] = D kJ/mol, parameter =α Å. The potential of valence angle )]cos()[cos(2)( θθθ θ −= nn KU , where θ K =529kJ/mol [5]. The potential of internal rotation )( n W δ characterizes the rotation stiffness of n -th valence bond. The PTFE-molecule has 4 isomers. Two of them [trans(+) and trans(–)] have identical energies [ ==−== δδδπδδδ WW ], two others [gauche(+) and gauche(-)] possess more high energy [ >=≈≈ δδπδπδ WW ]. Potential of internal rotation is characterized by four values: the height of the potential barriers between both trans conformations )0( W =ε , and between trans and gauche conformations )3/( πε W = , the level of the gauche conformation energy )3/2( πε W = , and the height of the barrier between gauche conformations )( πε W = . According to [6] =ε kJ/mol, =ε kJ/mol, =ε kJ/mol and =ε kJ/mol. We approximate the potential of rotation by the expression [3] ])()([)( CZCZCW −+= δδδ γβ , where )( δ β Z and )( δ γ Z are one parametric functions )2/(sin1 )2/(sin)1()( δβ δβδ β −+= Z , )2/3sin(1 )2/3sin()1()( −+= δγ δγδ γ Z The magnitudes of the parameters = C (kJ/mol) , = C (kJ/mol) , = C (kJ/mol) , =β , − ×=γ . The potential under given parameters is presented in Fig.1. W ( kJ / m o l ) δ Fig.1. Potential of rotation around the valence bond C-C )( δ W for macromolecule PTFE. Dispersion relation for given system is algebraic equation of third order with respect to square of the frequency. Corresponding dispersion curves has three branches: two acoustic )(),( qq lt ωωωω == and one optic )).()()()(( qqqq olto ωωωωω ≤≤= These curves are plotted in Fig.2. The lower curve )( q t ωω= corresponds to dispersion law for torsional phonons, the middle curve )( q l ωω= – longitudinal phonons and upper one )( q o ωω= – to high frequency optic phonons [3]. The equations of motion corresponding to Hamilton function (1) can be presented as follows: ,0)( =∂∂++− nnnn rPrRMrM ϕ &&& ,0)(2)( =∂∂++++ nnnnnn PrrRMrRM ϕϕϕ &&&& ,0 =∂∂+ nn hPhM && where ∑ ++= n nnn WUVP )}()()({ δθρ
Let us perform analytical study of nonlinear oscillations in the vicinity of wave number = q and = ω cm -1 (left edge of optic dispersion curve). To extend the normal modes analysis on the nonlinear case let us Fig. 2. The frequency spectrum curves )( q t ωω = (1), )( q l ωω = (2), and )( q o ωω = (3), for the spiral isolated macromolecule PTFE. introduce the slow modulating functions. We use the denotations Rr n = , Φ= n ϕ , Hh n = , Rr n ~ = + , Φ= + ~ n ϕ , Hh n ~ = + and consider the power expansions in the vicinity of an arbitrary nth particle using modulating functions HHRR ~,,~,,~, ΦΦ [2]: ∂∂+∂∂+= + ξεξε RmRmRmkr mn ,~21~~)sin( L + ∂∂+∂∂++ ξεξε RmRmRmk ∂ Φ∂+∂Φ∂+Φ= + ξεξεϕ mmmk mn ,~21~~)sin( L + ∂ Φ∂+∂Φ∂+Φ+ ξεξε mmmk ∂∂+∂∂+= + ξεξε HmHmHmkh mn ,~21~~)sin( L + ∂∂+∂∂++ ξεξε HmHmHmk where small parameter ε characterizes the distance between the particles in the units z lz / εξ= , and z, z l - coordinate and helix step along the helix axis, respectively K ,2,1 = m . In the same way we consider the expansions in the vicinity of the )1( + n -th particle ∂∂+∂∂+−= ++ ξεξε RmRmRmkr mn ,~21~~)cos( L + ∂∂+∂∂++ ξεξε RmRmRmk ∂ Φ∂+∂Φ∂+Φ−= ++ ξεξεϕ mmmk mn ,~21~~)cos( L + ∂ Φ∂+∂Φ∂+Φ+ ξεξε mmmk ∂∂+∂∂+−= ++ ξεξε HmHmHmkh mn ,~21~~)cos( L + ∂∂+∂∂++ ξεξε HmHmHmk
The expansions should be considered in the vicinities of two particles because double degeneracy of linear normal modes. By substituting the modulating functions in linearized equations for ,,,,, +++ nnnnnn hhrr ϕϕ we obtain six partial differential equations taking into account the linear parts of the equations for modulating functions. These equations have to be in accordance with foregoing dispersion relations. In the particular case k=0 the equations are reduced to tree equations: ] [ ( =∂∂−∂Φ∂+−∂∂+−∂∂+ ∂Φ∂++∂∂ ξξξεξξε HaaRRaRAARaHaaRatR = ∂∂+∂∂+∂ Φ∂−∂∂−∂ Φ∂ ξξξεξε RrbHBBBBRbt = ∂∂−∂∂+∂ Φ∂−∂∂−∂∂ ξξξεξε RRcHCCCCRctH
Let us introduce the complex function iRRu +∂∂= τ , where ta =τ . Then the equations of motion are written as ( ) + −+∂∂+ ∂Φ∂++∂∂ uuAHAAiuu ξξετ ( = ∂∂−∂∂−− −−−∂ −∂ −+ ξξΦξε HAAuui uuAiuuAi **572222212 *422 = ∂−∂−−∂∂+∂∂− −∂−∂+∂∂ ξξξΦε ξετΦ uuuuBHBB uuBi **572222212*422 = ∂−∂−+∂∂+∂ Φ∂−∂−∂+∂∂ ξξξεξετ uuuuCHCCuuCiH We introduce, alongside with fast time ττ = , the slow times K ,, τετεττ == and expand the unknown functions into power series with respect to small parameter ε : )( K +++= ψεεψψ τ i eu ; )( K +Φ+Φ+Φ=Φ εε ; )( K +++= HHHH εε Selecting the terms of the same order by ε and equating them to zero one can find: 1) : ε ),,( ξτψψ K = , ),,( ξτ K Φ=Φ , ),,( ξτ K HH = : ε
000 00 =+−+ ∂∂+∂∂+∂∂+∂∂ −− −− ψψψψ ξξΦτψτψ τττ ττ iii ii eAeAeA eHAeA (2) *0000401 0220 12 = ∂∂− ∂∂+∂∂ Φ∂+∂ Φ∂ − ξψξψτττ ττ ii eeBi , (3) *0000401 0220 12 = ∂∂− ∂∂+∂∂∂+∂∂ − ξψξψτττ ττ ii eeCiHH . (4) To avoid the appearance of growing (secular) terms, we suppose: =∂∂τψ , ),,( ξτψψ K = , ),,( ξτ K Φ=Φ , ),,( ξτ K HH = Integrating these equations (2-3) one can obtain −++ + ∂∂+∂∂−= −−− ττττ ψψψψ ξξΦψ iiii eeeAi HAAie ∂∂−∂∂=Φ − ττ ξψξψ ii eeBi ∂∂−∂∂= − ττ ξψξψ ii eeCiH Substituting the expressions for ,, H Φψ into equations of order ε selecting the secular terms and equating them to zero, we come to the main asymptotic approximation, that are the conditions of absence of growing(secular) terms in the solution. For : ε : =∂Φ∂−∂∂++∂∂−∂∂ ψξψξψψξψτψ iPHiPiPiP (5) =∂∂−∂ Φ∂−∂∂ ξψξξ PPHP (6) =∂∂+∂ Φ∂−∂∂ ξψξξ PPHP (7) where )(21
CABAAP −−= =0.124; )109(241
AAP += =244.7; )2(21 AAAP −= =14.345; )2(21 AAAP += =0.1036; BABP −= =0.0035; BABP −= =0.002; BBAP += =0.308; CACP −= =0.105; CACP −= =0.001; CACP −= =7.17. Then we express ξξ ∂Φ∂∂∂ , H via ψ using equations (6) and (7): , ψξ Q =∂Φ∂ (8) ψξ QH =∂∂ , (9) where PPPP PPPPQ −+= , PPPP PPPPQ −+=
Substitution of (8), (9) into (5) leads to final asymptotic equation: =−∂∂−∂∂ ψψξψτψ iQiP , (10) which is nonlinear Schrodinger equation – one of the fundamental equations of Nonlinear Dynamics. r n / R n (number of particle) n (number of particle) ϕ n , deg n (number of particle) h n / ∆ z Fig.3. Profiles for radial, angle and longitudinal variables. Thin line −=∆ω , thick line −=∆ω ; = U , =ε . The parameters are: )(21 CABAAP −−= =0.124;
PPPP PPPPPPPPPP PPPPPQ −+−−−+= =750 After change )v( τξψ ωτξ Ue iik r −= ∆− , if r kPU = we obtain that equation (10) is reduced to nonlinear ordinary equation: =+−′′ να , where Pk r ωα ∆−= , PQ = ν Its spatially localized solution has a view ))(sech()2(v τξανα U −= Fig.4. Time-coordinate maps for radial n r (a), angle n ϕ (b) and longitudinal n h (c) variables. if >> να . To satisfy these conditions it is necessary to provide PUPk r =<∆ ω Returning to initial variables, one can obtain: ))1(sin())(sech()2( ξεωεξαναε r ktataUr +∆−−= ))((th2 taUQ εξαν αεϕ −= ))(th(2 taUQh εξαν αε −= The plots describing the spatial profiles of radial angular and longitudinal coordinates corresponding to localized excitation are presented in Fig.3. It is interesting to note that breathers-like behavior of radial coordinate is accompanied by kink-like spatial dependence of angular and longitudinal coordinates. Corresponding time-coordinate maps are plotted in Fig.4.
3. Conclusion
The localized vibration excitation in helix polymer chain has been found analytically for the first time. For this the asymptotic approach with using the complex presentation of equations of motion and multiple scale expansions has been applied. At that, we deal with of nonlinear modulation of linear vibrations possessing optic branch of dispersion curve .It is shown that this excitation which is a bright breather has a rather complex structure corresponding to a combination of three types of deformation: radial, longitudinal and rotational. These results were confirmed by computer simulation with using of initial discrete equations of motion.
This research was supported by CRDF (grant RUB2-2920-MO-07).
References [1] Savin A.V., Manevitch L.I., Descrete breathers in a polyethylene chain,