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NOVEL MATHEMATICAL METHODS IN SOLITON THEORY AND THEIR APPLICATION TO
DISLOCATION MOVEMENT
W. Lay, A. Seeger, J. Esslinger
To cite this version:
W. Lay, A. Seeger, J. Esslinger. NOVEL MATHEMATICAL METHODS IN SOLITON THEORY
AND THEIR APPLICATION TO DISLOCATION MOVEMENT. Journal de Physique Colloques,
1989, 50 (C3), pp.C3-107-C3-112. �10.1051/jphyscol:1989316�. �jpa-00229457�
JOURNAL DE PHYSIQUE
Colloque C3, supplément au n°3, Tome 50, Mars 1989 C3-107
NOVEL MATHEMATICAL METHODS IN S0LIT0N THEORY AND THEIR APPLICATION TO DISLOCATION MOVEMENT
W. LAY, A. SEEGER and J. ESSLINGER
Unlversitat Stuttgart, Instltut fiir Theoretlsche und Rngewandte Physxk and Max-Planck-Institut fiir Metallforschung, Instxtut fiir Physik, Postfach 80 06 65, D-7000 Stuttgart 80, F.R.G.
R é s u m é : Presqu'en même temps, que la découverte des propriétés solitordques de ce que l'on appelle l'équation de sine-Gordon (A. Seeger et collaborateurs 1950/53), en relation avec des applications à la théorie des dislocations, on s'intéressait à la généralisation suivante:
d2f/dx2 - d2$/dt2 = F ( # ) - s, où F($) est une fonction périodique et s est une constante.
Cette équation joue un rôle important dans la théorie du mouvement des dislocations cristallines sous une tension appliquée. En particulier, on s'intéresse à la vitesse de formation de paires de décrochement qui dépend de s et de la température. Pour le calcul de cette vitesse, on se voit con- fronté à des problèmes relatifs à la thermodynamique statistique, à la théorie des réactions chimi- ques, au comportement solitonique des solutions particulières de l'équation pré-citée, à la théorie des équations différentielles ordinaires, etc. L'exposé mettra les développements récents dans une perspective historique et discutera en particulier des nouveaux développements mathématiques pour la résolution des problèmes aux valeurs propres découlant de la linéarisation de l'équation pré-citée autour des solutions statiques particulières.
A b s t r a c t : Almost simultaneously with the discovery of the soliton properties of the so-called sine-Gordon equation (A.Seeger and collaborators 1950/53), in connection with applications to dislocation theory interest arose in the generalization d2$/dx2 — d2$/dt2 = F($) — s, where F( £ ) is a periodic function and s a constant representing an external stress. This equation plays an important role in the theory of the behaviour of dislocations in crystals under applied stress.
A key problem is the calculation of the rate of formation of so-called kink pairs as a function of 3 and temperature. This involves statistical thermodynamics, rate theory, soliton behaviour of special solutions of the above mentioned equation, and the theory of ordinary differential equations. The paper puts recent developments in this field into historical perspective and discusses novel mathematical techniques for treating eigenvalue problems emerging from the linearization of the aforementioned equation in the neighbourhood of special static solutions.
1 D i s l o c a t i o n M o v e m e n t , S o l i t o n s , a n d K i n k P a i r s
Th.v.Karman's review of 1913 [1] of the attempts to understand the nature of the elastic hysteresis and the elastic after-effect of solids appears t o be the oldest written record of L.Prandtl's model of a special type of imperfection in crystals. Prandtl's model was published in detail only in 1928 [2] and had by that time been conceived independently by U.Dehlinger [3]. It was subsequently developed further by J.Frenkel and T.Kontorova [4] as well as U.Dehlinger and A.Kochendorfer [5], and is now known as the 'Prandtl-Dehlinger-Prenkel-Kontorova model' of dislocations ([6]).
The model is sketched in Fig. 1 (a) ; for details see (6). Starting from the PDFK model and performing a continuum approximation, A.Seeger [7] introduced, as far as we know for the first time in a physical context, the 'sine-Gordon equation'
(1) By re-interpretating $(x) as the transverse displacement of a line-element of a dislocation running on the average parallel to a crystallographic direction x, equation (1) may be used as an approximate description of an entire dislocation line in a crystal. The two terms on the left-hand side of (1) represent the line tension and the inertia of the dislocation line, respectively. The term sin # derives from a potential 1 — cos $, representing the spacially periodic variation of the line energy of a dislocation in a crystal. This form of the potential may be regarded as the first harmonic of a more general periodic potential V{ # ) , the so-called Peierls potential [8]. For our purposes we have to add to (1) a constant force term s (normalized to 0 < s < 1), representing an external applied shear stress. It causes a force acting perpendicular to a dislocation line element and tends to move the dislocation on its glide plane.
If we use a more general form of the Peierls potential and define F := dU/d$ ( = - f o r c e per unit dislocation lenght!), the resulting equation reads
(2) Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989316
JOURNAL DE PHYSIQUE
Figure 1: (a) The Prandtl-Dehlinger-Fkenkel-Kontorova (PDFK) model of a dislocation. A one-dimensional row of atoms is situated in a periodic potential. Each atom is assumed to interact with its two nearest neighbours through harmonic forces (symbolized by the springs). The displacements @* of the atoms, counted from the stable equilibrium positions of the atoms in the minima of the potential, increase from zero on the left-hand side t o the distance between neighbouring minima on the right-hand side. In the re-interpretation of the model referred to in the text, the full circles represent straight dislocation segments running along the z-direction perpendicular to the plane of drawing.
The 'continuization' of the springs leads to the line tension of the dislocation, counteracting the increase of the dislocation length when segments cross from one potential valley to the next. This leads to the 'line-tension model' of dislocations (b), which pictures a dislocation line as a massive elastic string lying on a sloping corrugated iron sheet, the inclination of which increases with increasing applied shear stress s. The dashed line shows a straight dislocation, the full line a dislocation containing a kink pair.
This so-called 'line-tension model' is illustrated in Fig. 1 (b). We again refer to [6] for further information.
Equation (1) had been considered for the f i s t time in 1870 by A.Enneper [9] in the context of the differential geometry of pseudospherical surfaces and should therefore more appropriately be referred to as 'Enneper equation'.
In a series of papers [10,11,12], A.Seeger et. al. examined the mathematical treatment of the non-linear partial differential equation (1) and discovered the physical significance of Backlund's transformation [13] and of Bianchi's theorem of permutability [14]. The whole family of now famous solutions, such as single and multiple kinks, breathers, heavy phonons etc. as well as their interactions were obtained in this way by algebraic operations or differentiation from a few basic solutions (for details see (15)). The soliton properties of a non-linear partial differential equation were then established for the f i s t time, more than a decade before the corresponding discoveries were made for the Korteweg-de Vries equation. Important results such as the relativistic relationship between the total energy and the rest mass mk of moving kinks and the relationship E(v) between the energy E and the frequency v of breathers were obtained.
In the theory of crystal plasticity the temperature and stress dependence of the rate of slipping of a dislocation line from one 'Peierls valley' to the next plays a key r61e. It is now generally accepted that the overcoming of the Peierls barriers is initiated by the thermally activated formation of kink pairs in the dislocation lines (see Fig.1 (b)).
Kink pairs are unstable solutions of (2) with an energy that exceeds that of a straight dislocation by the kink-pair formation enthalpy Hb(s). A suitable decomposition of the degrees of freedom of the entire system (cf. [6,16]) allows us to write the net rate of kink-pair formation per unit dislocation length,
rkp,
as ( k ~ = Boltzmann's constant, 2' = absolute temperature, h = Planck's constant divided by 2s)In (3) two degrees of freedom, namely the translations of the two kinks constituting a kink pair [cf. Fig. 1 (b)], have been incorporated in the preexponential factors. The f i s t factor comes from treating the translational motion of the kink pairs as a whole as that of non-interacting particles of mass 2mk (= 'soliton gas', [16]). The factor takes into account the decay of the kink pairs by either mutual annihilation or further separation of the individual kinks. It may be calculated from a one-dimensional model representing the escape of a particle from a position of metastable equilibrium in a potential well (Fig. 2). The entropy of kink-pair formation, S b , allows for the modification of the vibrational degrees of freedom of the dislocation by the presence of kink pairs. Some of the questions arising in the calculation of and Skp will be discussed in Sections 2 and 3.
2 The Escape Rate
r
Consider the situation sketched in Fig. 2. Particles, 'generated' inside a potential well and endowed with mass m and mobility p, escape over a barrier at rate
r.
In the present application the barrier height is identified with the formation enthalpy Hkp(8) of a kink pair, the mobility p with twice that of a single kink, and the mass of the particle with half the rest mass mk of a single kink (= reduced mass). Following the ideas of Kramers' basic paper 1171, thepotential energy
\
I kink
separation t\ Figure 2: Particle-escape model for the
determination of
r.
The maximum of0 0.25 0.5 0.75
1 . O s 1 ' 2 the potential corresponds to the saddle Figure 3: Numerically calculated eigenvalue spectrum of (10) for ridge in the multidimensional configura- a kink pair in the so-called Eshelby potential [ 2 4 ] . The hatchedtion space. area corresponds to the continuous spectrum.
calculation of
r
may be handled by different approaches depending on whether we are in the regime of very low, intermediate, or very high p [16,18].In the limit of very high mobilities (which means weak coupling to the environment, represented by the non- translational degrees of freedom of the system) a particle will perform many oscillations inside the potential well before escaping, altering its energy E only slowly. As was shown by Kramers [ 1 7 ] , the latter process may be described as diffusion in the one-dimensional space of the action variable I associated with the oscillations offrequency v = v ( E ) . According to classical mechanics these three physical quantities are related to each other by
With the help of (4), the escape rate may be obtained from Kramers' diffusion equation in the form
where 3 is a functional of E [ I S ] . The subscripts denote parametric dependences, which furnish the required depen- dence of 'I on s and T. In the limit of small applied stresses ( s
<<
1 ) we may choose F ( @ ) = sin @ and take E ( v ) from the breather solution of Enneper's equation ( [ 1 1 ] , [ 1 6 ] ) . Karpman [19] has treated the perturbation of the breather solution of Enneper's equation by a constant applied stress by means of the adiabatic approximation. In the case of vanishing damping his &st-order coupled differential equations admit a &st integral which we have used to derive the energy and the frequency of the perturbed breather up to s 2 terms. Insertion of the E , ( v ) relationship into ( 5 ) allows us to calculate r ( s , T) for small s and high kink mobility.3 T h e C a l c u l a t i o n o f t h e E i g e n v a l u e S p e c t r u m by a N o v e l M a t h e m a t i c a l M e t h o d
Ln the following, we outline a novel mathematical method to calculate the spectrum of the differential equation result- ing from linearizing ( 2 ) in the neighbourhood of the kink-pair solution Go(%). Further details may be found elsewhere [20,21]. The spectrum is required in order to determine the decay rate of the unstable saddle-ridge configuration in situation of intermediate or small p on the one hand and the entropy Skp of kink-pair formation on the other hand.
Writing the solution of ( 2 ) in the form
and assuming
1
<< 1, we getBy means of the separation ansatz
Gl(x, t ) = q ( x ) exp(iwt) equation ( 7 ) may be reduced to the ordinary differential equation
C3-110 JOURNAL DE PHYSIQUE
We search for solutions of (9) which for 1x1 + w either vanish or which behave periodically in this limit. The former solutions correspo~ld to discrete eigenvalues w: and the latter to the continuous spectrum of a two-parameter eigenvalue problem.
Introduction of @o as independent variable transforms (9) into
If F = F(Go) is a polynomial of degree m - 1 with simple zeros only, (10) is a Fuchsia differential equation with m singularities. According to the general theory of these equations [22], their solutions may be written as generalized power series of the form
OJ
'P =
C
A,(Go - C)"+",
n=O
(11) where C denotes one of the singularities of (10) and where for each C the exponent u may assume two values. In the complex Go plane, these series converge uniformly within the circle around C going through the neighbouring singularity. This means that the exponents u determine the solution of (10) completely up to m - 3 so-called accessory parameters. We may express this by means of a generalisation of Riemann's P-symbol [cf. (12)] which shows in the rows of the left-hand block the singularities and the parameters u associated with them, on the right-hand side the independent variable and the accessory parameters ([22, pp. 3701).
The essential features of the present problem may be obtained by choosing U(dmo) as a polynomial of degree four, the so-called Eshelby potential [23]. Such a choice allows us to represent the transition of a dislocation line from one Peierls valley to the next. By a suitable homographic transformation of the independent variable (henceforth denoted by z) we may position the four singularities at O,a, 1, and w with 0 5 a 5 1
,
a = a($), in such a way that the P-symbol readsO a l w
p 0 0 3 t ; 3 ( l + a )
I . (12)
- p 112 112 -2
Here 3(1+ a) is the accessory parameter. For the discrete spectrum we have p E R 2 0, whereas for the continuous spectrum p = ip*
,
p* E R 2 0 holds.The two-parameter eigenvalue problem of (9) for the discrete spectrum requires that the series
represent a function which at z = a is either holomorphic or behaves as
The two cases may be treated by the same method, to be explained presently, if in the second one we introduce as independent variable.
The condition of holomorphy a t z = a is satisfied if and only if the radius of convergence exceeds a. Since the solutions of (10) are holomorphic in the entire z-plane save for the singularities, this radius of convergence must be 1.
We thus have to seek those pairs (w, s) for which the coefficients a, in (13) behave as
These pairs give us the discrete eigenvalues wi = w;(s) (i = 0,1, ...).
Insertion of (13) into (10) yields the three-term recurrence relation where !F0,1,2(5) are polynomials of degree two. With
where r is determined uniquely by. the coefficients of (10) (in the present case T = 1/2), we may obtain another three-term recurrence relation by expanding (18) with respect to the last row:
From (19) it can be proved that
1.) lim D, n- = D(w,s)
,
2.) D(w,s) = 0 +-- w = w,(s).
m (20)
The numerical results are shown in Fig. 3. Since for kink pairs G0(z) is even, the eigenfunctions of (9) form pairs of even and odd functions. The eigenvalues coincide pairwise at s = 0 and s = 1 (cf. Fig. 3).
In Fig. 3
w i 5
0 describes the decay of the kink-pair configuration, wt 0 the translation of the kink pair along the dislocation line. w: and wi represent the frequencies of even and odd vibrations localized inside the kink pair. The behavior of these curves for small s is: wo(s) = -1.156 s+
O(s2) ; WZ(S) = 0.75-
0.73 s1I2+
0(s3I2) ; w3(s) = 0.75+
0.55 9'12+
0(s3/').
We now turn to the continuous spectrum. The corresponding generalized eigensolutions behave periodically as
121 -+ m and may therefore be characterized by wave-numbers k. The main information on these solutions, which again are either even (g) or odd (u), is contained in phase shifts a , , at z = f m between the solutions associated with the kink-pair configuration and straight dislocations. These phase shifts are connected to the change of the spectral density of phonons caused by the creation of a kink pair on a straight dislocation line.
This change has been obtained from special Wronskians containing the series
- 2
bnZn+P +C
c n Z n - ~ ;s2
= dn(t - a)"-1 =
n=O n = O n=O
in the even and 43 m m
, s3
= en(% - a)"+'I2 El = b,~"'~+
C . , Z ~ - ~.
n=O n=O n=O
(22) in the odd case and their first derivatives. Now p = ip* is imaginary. As in the discrete case, the coefficients b,, en, dn and en of (21) and (22) may be deduced from three-term recurrence relationships obtainable by inserting the series into (10). The functional form of El is determined by the condition that this function has to be real for real z . 2'2 and 2'3 have to represent even or odd functions in the z-frame. The zeros of these special Wronskians fix the continuous spectrum and one gets the phase shift a mentioned above explicitly. For details and curves a = a(pV; s) see [20].
The kid-pair entropy Skp is defined as the difference between the vibrational entropy of a dislocation containing a kink pair in unstable equilibrium under the external stress s and the vibrational entropy of a straight dislocation under the same stress. If we treat the various vibrational modes as harmonic oscillators, we obtain (see [16], [24],
with
In (23) the summation extends over the localized vibrational modes (in the case of the Eshelby potential i = 2,3).
Eq. (24) has to be evaluated for even (g) and odd (u) eigenfunctions separately. The dispersion reiation
is that of an unkinked dislocation [lo], where c denotes the group velocity of short-wavelenght perturbations. ,,k denotes a short-wavelenght cut-off determined by the discrete structure of the crystal along the dislocation line. In the high-temperature approximation tw/kBT
<<
1 this cut-off may be replaced by m. For any choice of U(%) we have a,,".(oo) = 0. The Eshelby potential gives in addition ag(0) = 3x12 (except for s = 0) and a,(O) = 2 r (except for s = 1). The information required for the evaluation of (23, 24) has been derived by the mathematical techniques described above. Approximate evaluations will be presented elsewhere.JOURNAL DE PHYSIQUE
4 Conclusions
The investigation of the rate of formation of kink pairs in dislocations as a function of temperature and applied stress has led us to a number of problems related to the soliton properties of the Enneper ('sine-Gordon') equation. These have been treated in terms of two related equations, namely the generalization of the Enneper equation to U t e external forces s, and that following from the replacement of the cosine potential by the so-called Eshelby potential.
In the first case we have succeeded in deriving up to s2 terms the relationship between energy and action variable for breathers, which is important for the calculation of the kink-pair formation rate under the condition of high kink mobilities. In the second case a novel mathematical method for solving an eigenvalue problem arising in the calculation of the kink-pair entropy and other quantities related to kink-pair formation has been developed.
Acknowledgement
The authors wish to express their appreciation to Dr. E. Mann for many stimulating discussions on the theory of kinks in dislocations and for his readiness to check several calculations.
References
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[4] J.Frenke1, T.Kontorova : J.Physics USSR 1, 137 (1939)
[5] U.Dehlinger, A.Kochendorfer : Zeitsckrift f. Physik 116, 576 (1940)
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[7] A.Seeger : Diploma thesis, TH Stuttgart 1948149 [8] R.Peierls : Proc. Phys. Soc. (London) 52,34 (1940)
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[14] L.Bianchi, 'Sulla trasformazione di Backlund per le superficie pseudosferiche' in : Rend. Acc. Naz. Lincei [5], 1, (2' sem. 1892) p. 3
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