A dislocation-dipole in one dimensional lattice model
AA dislocation-dipole in one dimensional latticemodel
Basant Lal Sharma ∗ Department of Mechanical Engineering , Indian Institute of Technology Kanpur , Kanpur, U.P. 208016, India
Abstract
A family of equilibria corresponding to dislocation-dipole, with variableseparation between the two dislocations of opposite sign, is constructed in aone dimensional lattice model. A suitable path connecting certain membersof this family is found which exhibits the familiar Peierls relief. A landscapefor the variation of energy has been presented to highlight certain sequen-tial transition between these equilibria that allows an interpretation in termsof quasi-statically separating pair of dislocations of opposite sign from theviewpoint of closely related Frenkel-Kontorova model. Closed form expres-sions are provided for the case of a piecewise-quadratic potential wherein ananalysis of the effect of an intermediate spinodal region is included.
Introduction
The subject of defect nucleation and quasi-static propagation of defects forms thecore of the subject of plasticity as well as the wider topic of irreversibility in nature.The emergence of a dislocation-dipole , referring to a configuration of two disloca-tions of opposite sign, is fundamental to the study of dislocation nucleation. In thiscontext, it is relevant to recall two achievements that occurred several decades ago.First is the mechanism that Frank and Read [7] suggested for the nucleation of adislocation loop from an existing dislocation. Second, arguably less known [28], isa classical model corresponding to a dislocation-dipole that has been presented byNabarro [19], in the framework of Peierls’ model of dislocation [20, 18]. Within onedimensional models, such as the Frenkel-Kontorova model [8], an equivalent entityis a kink-antikink pair, sometimes the same is referred as dislocation-dipole too. ∗ Email: [email protected] a r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p islocation-dipoleSome numerical experiments, for example see [3, 4], have also demonstrated thepossibility of the creation of a kink-antikink pair in the Frenkel-Kontorova modeldue to the interaction of two breathers, but in the absence of a driving force. Fora two dimensional Frenkel-Kontorova model at non-zero temperature, nucleationand propagation of kink-antikink pair has been studied through numerical simula-tions as well, for example see [9]. It is generally argued by physical considerationsthat the presence of thermal fluctuations can trigger changes in the lattice con-figurations that lets the particles explore the energy landscape. Naturally, thespace of lattice configurations also contains the metastable equilibria which areinterlaced with unstable (saddle point) equilibria leading to certain barriers forchanges between ‘neighboring’ lattice equilibria and consequent ‘lattice trapping’.As described by Seeger and Schiller [23], the calculation of the kink-antikink pairgeneration rate at a finite temperature may be related to energy barriers in theenergy landscape. These conceptual foundations prepare the background and amotivation for this paper where several simple and analytically tractable resultshave been presented; these are anticipated to build an understanding of the effectof lattice trapping and energy barriers in this context by further augmenting theresults, from a discrete viewpoint, in the corresponding continuum models [10].Going back to the model of Frenkel and Kontorova [8], recall that it involves anonsite potential which is periodic, in particular it assumes more than one energywell. Keeping the objective underlined above, and with an explanation in nextfew sentences, it is suffices for the purpose of this paper to consider the situationwhen just two energy wells are permitted for each particle in lattice. To utilizea workable vocabulary in the paper, when two particles lie in ‘two different en-ergy wells’ of onsite potential, it is referred as that the corresponding particlesare in ‘different phases’. It is immediately clear then that at least two types ofequilibrium configurations can be studied in such a one dimensional lattice withonsite potential. In one case the particles at ±∞ are in different phases so thatthis equilibrium configuration may be interpreted as dislocation or kink; in fact,it has been extensively studied during last five decades in the context of phasetransformation and plasticity [22]. In another case, however, with an exception ofa finite set of particles, all particles till ±∞ are in the same phase so that thisconfiguration can be associated with a dislocation-dipole or kink-antikink pair.From the point of view of a continuum limit of Frenkel-Kontorova model, tradi-tionally known as Sine-Gordon equation, the former type of configurations appearas heteroclinic orbits. In the same limit, the latter type of configurations appearas homoclinic orbits; in this sense, the present paper deals with equilibria of thistype. As it is assumed that all particles, except a few localised within a neighbour-hood of second phase, are in one phase, it is sufficient that the onsite potentialhas at least two local minima that are also global minimum. The conventional2islocation-dipoleFrenkel-Kontorova model with a periodic onsite potential can be thus replaced bya model with two-well potential . Sanders [30] and Atkinson and Cabrera [1] pre-sented the Frenkel-Kontorova lattice model for a special choice of onsite potentialthat allowed representation of kink-like equilibria in closed form. The choice ofonsite potentials in this paper is same as that of Atkinson and Cabrera [1] andWeiner and Sanders [30].The simplicity of the chosen framework of Frenkel-Kontorova lattice modelallows the construction of a family of dislocation-dipole-like equilibria, with sep-aration between the two dislocations of opposite sign taken as a variable. Theseequilibria are connected by a suitable path that is found to exhibit a familiarPeierls relief [13, 5] (see also Fig. 15-9 on page 544 of [10]). The foundationsof such energy landscape based approach for one dimensional lattice model areformed by the researches of Hobart [11] (see also [29] and [27]). This formulationeffectively relates to a projection of the energy landscape which is helpful in an-swering questions dealing with transient as well as steady state motion. In thepresence of a constant external force applied on each particle of the dislocation-dipole configuration, for a given number of particles in the second phase, a pathconnecting several possible equilibria is found using the concept of order parameter[11]. The highlight of the paper is a statement of sequential transition betweenthese equilibria that allows an interpretation in terms of the growth of separationbetween two dislocations of opposite sign. In this context, an exact solution isprovided for the case of a piecewise-quadratic potential with two wells as well asthe one with an intermediate spinodal region.The paper is organised as follows. In first section, the one-dimensional latticemodel, motivated by Frenkel-Kontorova model, is formulated and the equations ofequilibrium are presented. Second section contains the exact expression of equi-librium configurations for two-quadratic wells. Subsequent section deals with achange in energy for a path connecting two equilibrium configurations, which dif-fer by one or more particles in second phase in the presence of constant force.The energy landscape for a special case when equilibrium configurations differ byone particle in second phase is presented along with the the energy changes fortransition between configurations which differ by two particles in second phase. Infourth section, the effect of spinodal region is studied within the quadratic well ap-proximation. Fifth section provides discussion of a cascade of transitions betweenequilibria such that the number of particles in second phase changes sequentially,i.e., one at a time. Some remarks related to the Peierls refief and the effect of finitetemperature are given in the final section on discussion. Four appendices appear-ing at the end of the paper provide some additional expressions, few derivations,and accessory details of calculations. 3islocation-dipole Let the set of integers Z be identified with the particles constituting a one-dimensional lattice with lattice constant ε . Let ˜ u n denote the displacement of n th particle, which is located at position nε in the lattice, for each n ∈ Z . Supposethat the lattice is attached to a rigid foundation with on-site energy density ˜ w such that ˜ w (0) = ˜ w ( a ) = 0 , ˜ w (cid:48) (0) = ˜ w (cid:48) ( a ) = 0 , and ˜ w (cid:48)(cid:48) (0) = ˜ w (cid:48)(cid:48) ( a ) = c > , forsome a > . Throughout the paper, the notation f (cid:48) stands for the derivative ofthe function f with respect to its argument. It is assumed that each particle in-teracts with only its nearest neighbour particles through harmonic forces capturedby elastic modulus E so that the discrepancy between ˜ u n and the displacementsof nearest neighbours; thus, ˜ u n − and ˜ u n +1 contribute to this interaction. Due tothe on-site potential, the n th particle also experiences a force due to the potentialenergy ε ˜ w (˜ u n ) . As a representative of external bias in this lattice model, an exter-nal force per unit length is also considered and it is assumed to be independent of n ; suppose that this is denoted by ˜ σ. The total potential energy as a function ofthe displacement field { ˜ u i } i ∈ Z of all particles in the lattice is˜ E ( { ˜ u i } i ∈ Z ):= (cid:88) n ∈ Z { Eε ((˜ u n +1 − ˜ u n ) /ε ) + ε ˜ w (˜ u n ) − ε ˜ σ ˜ u n } . (1.1)Above leads to the equation of equilibrium: Eε (˜ u n +1 − u n + ˜ u n − ) − ε [ ˜ w (cid:48) (˜ u n ) − ˜ σ ] =0 , ∀ n ∈ Z . In this paper, the definitions are emphasized by := symbol in place ofequal sign. In order to reduce the number of physical parameters, let u n :=(2˜ u n /a − , σ :=˜ σ/ ( ac ) and w ( u n ):=2 / ( ca ) ˜ w (˜ u n ) . (1.2)Let u n for n ∈ Z represent the displacement at lattice site n . Then the equationof equilibrium, in above dimensionless formulation of the lattice model, can berewritten as ( u n +1 − u n + u n − ) − κ [ w (cid:48) ( u n ) − σ ] = 0 , ∀ n ∈ Z , (1.3)where κ := ε (cid:112) c/E. (1.4)In view of several applications of its expression in the sequel, the potential energyfunction as a counterpart to (1.1) is stated as E ( { u i } i ∈ Z ):= (cid:88) n ∈ Z {
12 ( u n +1 − u n ) + κ [ w ( u n ) − σu n ] } . (1.5)4islocation-dipoleFigure 1: Double-well onsite potential.
Due to the choice of scaling, the on-site potential w has (global) minima at ± w isconsidered, namely, w ( u ):= 12 ( u + 1) Θ( − u ) + 12 ( u − Θ( u ) , (1.6)where Θ is the Heaviside function defined by Θ( x ):=1 for x > , Θ( x ):=0 , x < . (1.7)The advantage of such choice of quadratic well potential function w lies in thefact that w (cid:48) is a piecewise linear function. The particular w is also shown inFig. 1 (as blue curve) alongwith its modified form incorporating an intermediatequadratic region as spinodal region (green curve, with the intermediate curvature µ <
0) as well as a quartic well potential function w ( u ) = ( u − ( u + 1) (purplecurve) and a sinusoidal potential function w ( u ) = π (1 + cos πx ) (brown curve)with the restriction that all these functions have the same curvature at u = ± . In the context of dislocation, from historical viewpoint, an approximation of anonlinear function by piecewise linear function has been used by Maradudin [17],Sanders [21], Celli and Flytzanis [2] and Ishioka [12] for a screw dislocation, andby Kratochvil and Indenbom [14], Weiner and Sanders [30], and Atkinson andCabrera [1] for Frenkel-Kontorova model. In this paper, the value 0 . κ is often used. Θ(0) does not affect the analysis presented in this paper so it can be left undefined, however,it can be assumed to be 0 for simplicity.
In the case of the quadratic well potential (1.6), the difference equation (1.3)becomes a piecewise-linear difference equation that describes the equilibrium con-figurations of assumed lattice model, namely,( u n +1 − u n + u n − ) − κ [ u n − σ + 1 − u n )] = 0 , ∀ n ∈ Z (2.1)with the boundary conditionslim n → + ∞ u n = lim n →−∞ u n = const . Figure 2:
Equilibria (local minima) for κ = 0 . σ = 0 .
4) and 6 ( σ = 0 .
05) particles inthe second phase (gray portion).
Assuming n l < n r , with n l , n r ∈ Z such that u n > , ∀ n ∈ ( n l , n r ) ∩ Z , and u n < , ∀ n / ∈ ( n l , n r ) ∩ Z , (2.2)the equation (2.1) becomes a system of coupled linear equations and these can besolved using well known methods for linear difference equations [16]. Eventually, afamily of dislocation-dipole-like stable equilibrium configurations can be expressedas u n = − σ + A n l − n r η n − n l , n ≤ n l B ( η n − n r + η n l − n ) , n l < n < n r A n l − n r η n r − n , n r ≤ n (2.3)with η := η ( κ ) = 1 + κ κ √ κ + 4 (2.4)6islocation-dipoleand A m := A m ( η ) = 2(1 − η m +1 ) / ( η + 1) , B := B ( η ) = − η/ ( η + 1) . (2.5)An example of the equilibrium configurations given by (2.3) is shown in Fig.2. Due to the form of the chosen on-site potential (1.6) and the assumptions(2.2), the equilibria (2.3) exist for a specific range of values of σ . In fact, in (2.3) u n r ≤ ⇒ σ ≤ σ upper and u n r − ≥ ⇒ σ ≥ σ lower with σ upper = ( η − η − n r + n l +1 ) / ( η + 1) , σ lower = (1 − η + 2 η − n r + n l +2 ) / ( η + 1) . In other words, theformal solution prescribed by (2.3), with N = n r − n l − admissible when σ ∈ [ σ lower , σ upper ] (2.6)where σ upper :=( η − η − N ) / ( η + 1) and σ lower :=(1 − η + 2 η − N +1 ) / ( η + 1) . (2.7)Here, the phrase ‘ n th particle is in the first (resp. second) phase’, it is meant that u n < > n ∈ ( n r , n l ) ∩ Z are in second phaseand remaining are in first phase according to (2.3) provided (2.6) holds. Remark 2.1
In the case of single particle in the second phase, say at n = 0 with n l = − , n r = 1 , the expression corresponding to (2.3) is given by u n = σ − η −| n | ( η − / ( η + 1) , ∀ n ∈ Z . Remark 2.2
There are also unstable (saddle point) equilibria as described by (A.1) , (A.2) , and (A.3) in the appendix A where the particles with n ∈ { n l } , n ∈{ n r } , and n ∈ { n l , n r } are in degenerate spinodal region, respectively; in this paperthis means that the displacement is zero for the particles at n ∈ { n l , n r } . All suchequilibria (2.3) , (A.1) , (A.2) , and (A.3) , coincide when σ = σ upper or σ = σ lower for given n l , n r . Within the simple framework of the case of the quadratic well potential (1.6),when σ varies but remains within the upper and lower bounds (2.7) for givennumber of particles in the second phase, the solution profile essentially shifts alongthe u axis (see Fig. 2 for the axes referred) in view of (2.3). As described in theappendix A, the above mentioned stable equilibria (2.3) are local minima andunstable equilibria (A.1), (A.2), and (A.3) are saddle points of the energy (1.5).When σ equals either σ lower or σ upper then two particles located at left and rightside of the second phase region are in the degenerate spinodal region and the latticeconfiguration becomes unstable. As soon as σ increases beyond these limits theequilibrium configuration transforms into another equilibrium with an increase ordecrease in the number of particles in the second phase and the transition continuesuntil a configuration is reached for which upper and lower bounds contain the givenvalue of external force σ, if such configuration exists. Indeed there may be manylocal minima for the same external force and a path connecting such configurationsby using the concept of order parameter is studied in the remainder of this paper.7islocation-dipole By its nature, near a saddle point equilibrium there are certain directions in theconfigurational space of lattice where energy decreases and along all other direc-tions energy increases. Thus a reduction of the entire configurational space ispossible in the overdamped limit of dynamics so that only a few directions are rel-evant to describe transition between metastable equilibria. The concept of orderparameter describes such reduction. Any two equilibrium configurations in theconfigurational space of the lattice which can be distinguished, based on this no-tion of order, can be connected by a path defined by the order parameter varyingbetween certain limits. In the following, the change in energy as a function of anorder parameter is studied.The equilibrium configuration of lattice is obtained by minimisation of a con-strained energy, i.e., the energy (1.5) minus a term accounting for order parameterbased constraints: E C ( { u i } i ∈ Z ):= E ( { u i } i ∈ Z ) − (cid:88) i ∈ Λ λ i u i , (3.1)with u i = u ( α ) i , ∀ i ∈ Λ , (3.2)in terms of a parameter α described below. Each Lagrange multiplier λ i is aperturbing force that scouts for the displacement constraint at i th particle foreach i ∈ Λ . Let Λ:= { p i } N i =1 be a subset of integers containing N number of positions in thelattice where the displacement is constrained. The equations of equilibrium foreach n ∈ Z obtained from minimisation of E C (3.1) are − ( u n +1 − u n + u n − ) + κ [ u n − σ + 1 − u n )] − (cid:88) i ∈ Λ λ i δ i,n = 0 , ∀ n ∈ Z (3.3)along with the constraints (3.2). In (3.3), δ i,n = 1 if i = n and 0 otherwise is theKronecker delta. Here, it is convenient to denote a general list of order parametersby α = ( α i ) i ∈ Λ with the number of components equal to N . A transition from oneconfiguration { u (0) n } n ∈ Z (with α i = 0 for all i ) to another configuration { u (1) } n ∈ Z (with α i = 1 for all i ) is considered so that n th particle in the two equilibriumconfigurations does not change its phase except on the set Λ where it is different.The transition is carried out by varying u ( α ) i continuously between u (0) i and u (1) i for each i ∈ Λ. By using α , the constrained value of displacement (3.2) can beexpressed as u ( α ) i :=(1 − α i ) u (0) i + α i u (1) i , ∀ i ∈ Λ . (3.4)8islocation-dipoleEach component of α lies in the interval [0 ,
1] and, therefore, α can be identifiedwith a vector in the unit cube [0 , N ⊂ R N . Moreover each vertex of this unit cubelies at, or in proximity of, a stable equilibrium configuration (local minimum ofenergy) of the lattice which are given by the condition that all λ i s are zero (thisincludes clearly, [0 , . . . ,
0] and [1 , . . . , i th vertex of the unit cube as theset S i . The boundary of each set S i is such that those particles which are membersof a particular subset of Λ , depending on i, lie in degenerate spinodal region.The equations of constrained equilibrium (3.3) can be solved explictly and thedisplacement field at the sites in Λ, i.e., { u n ( { λ i , S i , } i ∈ Λ , α ) } n ∈ Z , can be found.Following this λ i = λ i ( { S i } i ∈ Λ , α ) can be obtained from the constraints (3.2) andfinally { u n ( { S i } i ∈ Λ , α ) } n ∈ Z can be determined. The boundaries of the sets S i aredetermined by the condition u m ( { S i } i ∈ Λ , α ) = 0 for each m ∈ Λ and thus, finally, { u n ( α ) } n ∈ Z . The piecewise-linear difference equations (3.3) allow closed form solution thatcan be written as u ( α ) n = u (0) n + (cid:88) m ∈ Λ (cid:36) m α m U n − m , ∀ n ∈ Z (3.5)where U n = 2 η −| n | ( η − / ( η + 1) ∀ n ∈ Z , (3.6)and the weights ( (cid:36) i ) i ∈ Λ come from the solution of the linear set of equations (cid:88) m ∈ Λ (cid:36) m U n − m = u (1) n − u (0) n , ∀ n ∈ Λ . (3.7)Recall that η is defined by (2.4). Remark 3.1
The expression σ − U n is same as the solution (2.3) for n ≤ n l = − , n ≥ n r = 1; recall Remark 2.1. It is easy to verify that U n is in fact theGreen’s function in the sense that it is the solution of the equation − ( u n +1 − u n + u n − ) + κ u n = κ δ i, . (3.8)The Lagrange multipliers, that are forces holding the constraints (3.4), are givenby λ m = 2 (cid:36) m ( α m − Θ( −| α − S m | ))(1 − η ) /η, ∀ m ∈ Λ , (using (2.4), it is easy tosee that (1 − η ) /η = (1 + η − η ) /η = κ + 2 − κ ) so that λ m = 2 κ (cid:36) m ( α m − Θ( −| α − S m | )) , ∀ m ∈ Λ . (3.9)Recall (1.7) for the definition of Θ. The boundary of S m is given by u (0) m + (cid:88) n ∈ Λ (cid:36) n α n U n − m = 0 , ∀ m ∈ Λ . (3.10)9islocation-dipoleThe expression for Lagrange multipliers can be also rewritten as λ m := λ m ( α ) = 2 κ (cid:36) m ( α m − Θ( u (0) m + (cid:88) n ∈ Λ (cid:36) n α n U n − m )) , ∀ m ∈ Λ . (3.11)The expression (3.5) describes a constrained path, in the configurational spaceof the lattice, connecting the two equilibrium configurations { u (0) n } n ∈ Z and { u (1) n } n ∈ Z with the constraining forces given by { λ i } i ∈ Λ . The energy E of each configurationcould be infinite since the lattice contains infinite number of particles. But thechange in energy is finite and it is possible to find it along a path connectingtwo equilibrium configurations. Define Ψ as the change in the energy E moduloits value at initial configuration and then an expression for Ψ can be derived asshown in the appendix B. In the next section the change in energy associated withthe transition from one local minimum to another that includes one more particlein the second phase is presented. n to n + 1 particles in second phase Consider a transition from one equilibrium configuration { u (0) n } n ∈ Z with n r − n l − >
0) particles in second phase to another configuration { u (1) n } n ∈ Z with n r − n l − { n r − } and let the order parameter be α. Therefore, according to (3.4), u ( α ) n r − = (1 − α ) u (0) n r − + αu (1) n r − . In fact, u (0) n r − = σ − A n l − n r +1 , u (1) n r − = σ − B ( η − + η n l − n r +1 ) . Recall that η is definedby (2.4). There exists α = α ∗ ∈ (0 ,
1) when the particle located at n r − α ∗ is givenby α ∗ := u (0) n r − ( u (0) n r − − u (1) n r − ) = (( σ − η + 1) + 2(1 − η − n r + n l +2 )) 12(1 − η ) . (3.12)The dependence of α ∗ on σ and the number of particles in the second phase leadsto certain special cases of σ and regions of solutions as shown in Fig. 3.The energy barrier for trapping regions is maximum when α ∗ = 0 . σ M given by σ M := 2 η − n r + n l +2 ( η + 1) . (3.13)Thus, (3.12) can be expressed as α ∗ = 12 + ( η + 1)2( η −
1) ( σ M − σ ) . (3.14) Here σ M is synonymous to the Maxwell stress in the terminology of phase transitions [6, 29]. Effect of κ over the limits on σ for transition between equilibria with a specifiednumber of particles in second phase: (a) κ = 0 .
5, (b) κ = 1 .
2. Star denotes σ M , upper andlower solid box denotes σ upper and σ lower respectively. The upper and lower line segments ofeach rectangle denote σ max and σ min . The energy barrier for forward transition region is maximum when α ∗ = 1 and thecorresponding stress is σ = σ L = − ( η − η +1) + σ M . The energy barrier for forward tran-sition is minimum when α ∗ = 0 and the corresponding stress is σ = σ U = ( η − η +1) + σ M . Using (2.7), (3.13), since the configuration u (0) contains n r − n l − σ (0) lower = (1 − η + 2 η − n r + n l +2+1 ) / ( η + 1) = − ( η − / ( η + 1) + ησ M > σ L (as η >
1) whereas σ (0) upper = σ U ; also σ (1) lower = σ L <σ (0) lower and σ (1) upper = ( η − η − n r + n l +1 ) / ( η + 1) = ( η − / ( η + 1) + η − σ M < σ U = σ (0) upper . The requirement that { u (0) n } n ∈ Z and { u (1) n } n ∈ Z are equilibria at the same external force σ implies that the transition may occur only for σ ∈ ( σ min , σ max ) = ( σ (0) lower , σ (1) upper ) , (3.15)with σ min = max { σ (0) lower , σ (1) lower } = σ (0) lower and σ max = min { σ (0) upper , σ (1) upper } = σ (1) upper . (3.16)Note that the interval [ σ L , σ U ] may not necessarily equal the range of σ for admis-sible transitions as σ (0) lower > σ L and σ (1) upper < σ U . Remark 3.2
For the special case of nucleation of two dislocations with oppositesign, (as shown in Fig. 4), there is one particle in the second phase (recall Remark2.1) in the final equilibrium configuration { u (1) n } n ∈ Z while the initial configuration { u (0) n } n ∈ Z is the single phase (perfect) lattice, also addressed as homogenous state,with u (0) n = − σ, ∀ n ∈ Z . The critical value of order parameter is α ∗ = (1 − Example of a Peierls Landscape with minimum and maximum values of σ for κ = 0 . σ max and σ min correspond to the square dots shown at n r − n l − σ )( η + 1) / ( η − . Using (3.12) also, it is found that same expression holds with n l = − , n r = 1 . Thus in the absence of thermal agitation, the external forcerequired to nucleate two dislocations with opposite sign is given by σ = 1 . Therefore, [ σ min , σ max ] = [ σ (1) lower , σ (1) upper ] . For α ∗ = 1 it is found that σ L = σ (1) lower = 1 − η − / ( η + 1) and for α ∗ = 0 . , it is easy to see that σ M = 2 / ( η + 1) , whichmay or may not lie inside the admissible range of σ . On the other hand, σ (1) upper =( η − η − ) / ( η + 1) < . In particular, σ (1) upper − σ (1) lower = 2 κ / ( η + 1) and there isa non-trivial barrier at all admissible σ . At finite temperature, the energy barrierfor may be overcome at any stress σ ∈ [ σ min , σ max ] and the dislocations may benucleated in the lattice. After this nucleation, depending on σ and persistenceof certain minimum temperature, the two fronts may begin to separate from eachother leading to plastic slip of the lattice. Remark 3.3
For the special case of one dislocation (as shown in Fig. 5(d)), n r is finite and n l → −∞ which gives ∞ α ∗ = ( − σ ( η +1)( η − + 1) . In order to initiate themotion of a dislocation the Peierls stress required is ∞ σ P = ( η − / ( η + 1) whichagrees with the results of Atkinson and Cabrera [1] (see also [15, 25, 26]). Inthis scenario, the maximal energy barrier for both forward and backward transitionexists with σ M = 0 (according to (3.13) ) and this is because the configuration withone more or less particle in the second phase is identical to the previous one. The change in energy, in one-dimensional case, is given by a very simple ex-pression Ψ( α ) = 2 κ U ( 12 (cid:36) α + (cid:36) ( α ∗ − α )Θ( α − α ∗ )) , (3.17)12islocation-dipoleFigure 5: Example of a Peierls Landscape with different values of σ for κ = 0 .
5. (a) transitionfrom a configuration with 1 particle in second phase to a configuration with 2 particles in secondphase, (b) transition from 2 particles in second phase to 3, (c) transition from 5 particles insecond phase to 6, (d) energy landscape for a single dislocation considered as a limiting case oftwo dislocations of opposite sign far away from each other. In (a), σ max and σ min correspond tothe square dots shown at n r − n l − where, in view of (3.7), (cid:36) is given by (cid:36) = ( u (1) n r − − u (0) n r − ) / U , and α ∗ is given by(3.12). Recall that the definition of Θ is given by (1.7). Remark 3.4
Using (2.3) (also recall the statements preceding (3.12) ), it is easyto simplify the above expression so that (cid:36) = 1 . The dependence of Ψ on the number of particles in the second phase is only throughdependence on α ∗ . In Fig. 5(a,b,c) the change in energy for the transition froma configuration involving 1 , , , , σ listed in the figure may be easilycomputed using the expressions presented in the previous paragraphs in equations(2.7) and (3.13). Remark 3.5
It is worth noting that from Fig. 5(a) that there is a very thin rangeof σ which is admissible; in view of Remark 3.2 this is significant as the transitionfrom to sites in the second phase appears as a bottleneck. n to n + 2 particles in second phase The transition from one local minimum to another that includes two more particlein the second phase is interesting from the perspective that there is no binding13islocation-dipoleon particles that prohibits simultaneous transition across the phases; in additionto this in some cases there is no intermediate equilibria permitted for a one-onesequential transition to be even possible. The order parameter is a vector ofdimension 2 . Figure 6:
Two-dimensional energy landscape for κ = 0 . n l = 1 , n r = 7 to n l = 0 , n r = 8 . (a) σ = σ min so that at this minimum value of σ the two particles in the secondphase will not prefer any more particles, (b) σ lies between σ min and σ M and in this situation,there is increase in energy or a positive energy barrier associated with particles on either onlyleft or only right side of second phase shift from first phase into second, (c) this is the situationwhen all energy wells are same except the one with two transitions and this occurs at σ = σ M ,(d) this is the case when the transition to either left or right side is feasible, moreover, when σ > σ max the energy barrier for the well located at (0,0) disappears. The lighter sections of thecontour plot indicate larger energy while darker sections imply lower. + sign marks the siteswith order parameter corresponding to (0 , , ,
1) and (1 , Consider a transition from { u (0) n } n ∈ Z involving n r − n l − { u (1) n } n ∈ Z involving n r − n l − { n l + 1 , n r − } andlet α n l +1 be denoted by α l , α n r − be denoted by α r . Then, according to (3.4), theconstraints are given by u ( α ) n l +1 = (1 − α l ) u (0) n l +1 + α l u (1) n l +1 , u ( α ) n r − = (1 − α l ) u (0) n r − + α r u (1) n r − . (3.18)The solution of the constrained problem is (3.5), i.e., u ( α ) n = u (0) n + (cid:36) l α l U n − ( n l +1) + (cid:36) r α r U n − ( n r − . 14islocation-dipole Remark 3.6
In this case, (cid:36) r = − − ( u (1) n r − − u (0) n r − ) U + ( u (1) n l +1 − u (0) n l +1 ) U n r − n l − U − U n r − n l − ,(cid:36) l = − ( u (1) n r − − u (0) n r − ) U n r − n l − + ( u (1) n l +1 − u (0) n l +1 ) U U − U n r − n l − . (3.19) In view of Remark 3.4, it follows that (cid:36) r = (cid:36) l = 1 . Figure 7:
Two-dimensional energy landscape for κ = 0 . n l = 1 , n r = 2to n l = 0 , n r = 3 , i.e., from a homogeneous state to a dislocation-dipole. + sign marks the siteswith order parameter corresponding to (0 , , ,
1) and (1 , In Fig. 6, the contours of the change in the energy for a transition fromequilibrium configuration with 5 particles in second phase to that with 7 particlesis shown and this is for various values of σ (relative the transition from 5 to 6particles in second phase) as listed above each contour plot Fig. 6(a, b, c, d).Due to reflection symmetry of the equilibria (2.3) there is a reflection symmetryabout the line α l = α r . In Fig. 6(c), it is also evident that the energy wells areidentical for order parameter (0 , , ,
1) due to σ = σ M . There is asaddle point corresponding to either of (A.1), (A.2), and (A.3) between any twoequilibria corresponding to the local minima of energy landscape. For example see15islocation-dipolein Fig. 6(b), there is a lowering of energy associated with simultaneous transitionson the left and right side of the second phase region as compared to the increasein energy associated with either only one particle at left side transforming or onlythe particle at right side changing its phase. This suggests that there is a trappingwhen the second phase region would not prefer any expansion. For dislocationsthis would mean that two dislocations with opposite sign may attract each otherrather than repel each other.
Remark 3.7
In this simple one dimensional model, the energy barrier for simul-taneous change is always greater than that for sequential transition, similar to thecase of a phase boundary studied by Sharma and Vainchtein [27]. Following thisresult, a sequential path connecting local minima of energy is described in AppendixC, such that a cascade of transitions occur for given constant force.
In the context of Remark 3.2 and 3.5, it is worth exploring the energy landspacefor 2 order parameters starting from the homogeneous state; this is shown in Fig.7. It can be noted in Fig. 7 from the location of + sign and whether there areany energy well contours around it, that there is an absence of the local minimumbesides the initial and final configuration (unlike for example the scenario of Fig.6) as a consequence of the Remark 3.5. This is alluded to in the opening sentencesof this section.
There are two types of symmetric configuration when the displacement across thepeak is equal. In one case however the particle is at the peak itself and to unravelsuch configurations it is useful to consider the potential with an intermediate regionso that w is a differentiable function. This brings the motivation to consider amodified form of the expression (1.6) so that the potential energy w is given by w ( u ):= 12 ( u + 1) , u < − χ (1 − χ ) + µu − µ χ , | u | ≤ χ ( u − , u > χ , (4.1)where χ = 11 − µ ∈ (0 , . (4.2)16islocation-dipoleAlternatively, above is re-written as w ( u ) = 12 ( u + 1) + ˆ w ( u ) , ˆ w ( u ):= 12 , u < − χ (1 − χ ) + µ ( u − χ ) − ( u + 1) , | u | ≤ χ − u, u > χ . (4.3)Notice that µ < µ → −∞ the spinodal region shrinks so that the modelreduces to the two-quadratic wells (1.6). It can be shown that those equilibria wheneven one particle is present in the spinodal region are not metastable equilibria. Inother words, the stable equilibrium configurations can be expressed by (2.3) withno particle in the spinodal region. On a careful reading of the conditions (2.6) on σ , it is also clear that the admissible range of σ also shrinks for finite µ .Next the transition of configurations similar to that studied in the previoussection is presented; in this portion of analysis too, the order parameter is restrictedto be first scalar then two dimensional case is considered.Consider a transition from homogeneous state u (0) to that u (1) with one atomin the second phase. Without loss of generality, it is assumed that the site of theleft dislocation and right dislocation is, respectively, n l = − g − , n r = 0 , (4.4)for the initial state u (0) . In this case, thus, there are g particles in the secondphase of a dislocation-dipole for g >
0. When g = 0, this corresponds to ahomogeneous state. The stable equilibrium configurations can be expressed as(2.3) with (4.4). In particular, when g = 1, according to (2.3), it can be consideredthat u (1) − = u − = σ + 1 − / ( η + 1) , which can be connected via a scalar orderparameter starting with the homogenous state u (0) n = − σ (with u (0) − having thesame constant value, for example). Recall that η is defined by (2.4). In general,consider the equilibria with { u (0) n } containing a site with particle at n = − g − u (1) n having it in the second phase; thus Λ = {− g − } . Theequilibrium configuration { u ( α ) n } of lattice, for such a transition from g − g in second phase, is obtained by minimisation of the energy(3.1). It satisfies − ( u ( α ) n +1 − u ( α ) n + u ( α ) n − ) + κ [ w (cid:48) ( u ( α ) n ) − σ ] − (cid:88) i ∈ Λ λ i δ i,n = 0 , Λ = {− g − } . (4.5)In this case, when α − g − = α c , the transition from first phase to spinodal happensand then when α − g − = α c the transition from spinodal to second phase. For17islocation-dipole α − g − ∈ [0 , α c ] ∪ [ α c , {− g − } , so that thesolution is given by (3.5), i.e., u ( α ) n = u (0) n + α − g − (cid:36) − g − U n + g +1 , (4.6)where (in view of (3.7)) (cid:36) − g − = ( u (1) − g − − u (0) − g − ) / U and U is given by (3.6), with λ − g − = 2 κ (cid:36) − g − (cid:26) α − g − , α − g − ∈ [0 , α c ] α − g − − , α − g − ∈ [ α c , . (4.7)The critical values of α − g − are given by u (0) − g − + α cr (cid:36) − g − U = − χ and u (0) − g − + α cr (cid:36) − g − U = + χ , i.e., α cr = − χ − u (0) − g − (cid:36) − g − U , α cr = + χ − u (0) − g − (cid:36) − g − U . (4.8)For α ∈ [ α c , α c ], − ( u ( α ) n +1 − u ( α ) n + u ( α ) n − ) + κ [ µu ( α ) n δ − g − ,n + (1 − δ − g − ,n )(1 + u ( α ) n ) − σ ] − λ − g − δ − g − ,n = 0 , (4.9)with (3.4). It is found that (4.6) holds again and that ( α = α − g − ) − α(cid:36) − g − ( U n + g +1+1 − U n + g +1 + U n + g +1 − )+ κ [(( µ − u (0) n + α(cid:36) − g − U n + g +1 ) − δ − g − ,n + α(cid:36) − g − U n + g +1 ] − λ − g − δ − g − ,n = 0 , (4.10)which yields λ − g − = κ (( µ − u (0) − g − − α(cid:36) − g − (( µ − U + 2)) . (4.11)As the work done by the constraint forces along a path in the space of orderparameter starting from 0, it is found thatΨ( α ) = (cid:90) α λ − g − ( t ) ddt u ( t ) − g − dt = 2 κ ( 12 α (cid:36) − g − U + 12 ˆ w ( u ( α ) − g − )) . (4.12) Remark 4.1 If Ψ (cid:48) ( α ) = 0 for α ∈ (0 , , suppose that such critical value of Ψ occurs at α = α ∗ , then α ∗ U + ˆ w (cid:48) ( u (0) − g − + α ∗ U ) U = 0 , i.e., α ∗ + 12 ( µu − u − | u =( u (0) − g − + α ∗ U ) = 0 , i.e., α ∗ (1 + ( µ − U ) + (( µ − u (0) − g − −
1) = 0 . In case of transition fromhomogeneous state (i.e., g = 0 ), α ∗ = − ( µ ( σ − − σ )1+ ( µ − U which equals for σ = σ l =18islocation-dipoleFigure 8: Two-dimensional energy landscape for σ = 0 . κ = 0 . µ = − / n l = 1 , n r = 7 to n l = 0 , n r = 8 (see also Fig. 6(c)). ( U − µ (1 −U )) / ( µ − and equals for σ = σ u = µ/ ( µ −
1) = 1 − χ (as expected);in fact these two limiting values are equal when µ = 1 − / U = − / ( η − . Thus,for µ < − / ( η − , there is a range of σ , precisely σ ∈ [ σ l , σ u ] , in which thetransition from u (0) to u (1) is possible with a barrier equal to Ψ( α ∗ ) . A similar analysis can be carried out for the transition from n to n + 2 particlesin second phase; the relevant details are provided in Appendix D. As a result ofthe expression (3.5), by (3.1), i.e., E C ( { u ( α ) n } ) = E ( { u ( α ) n } ) − (cid:80) i ∈ Λ λ i u ( α ) i , while d E C ( { u ( α ) n } ) /du ( α ) n = 0 , i.e., ddu ( α ) n E ( { u ( α ) n } ) = λ i δ i,n , the change in the energy isfound to beΨ( α ) = E ( { u ( α ) n } ) − E ( { u (0) n } ) = (cid:90) α ddt E ( { u ( t ) n } ) dt = (cid:90) α ddu ( α ) n E ( { u ( t ) n } ) ddt u ( α ) n dt = (cid:90) α (cid:88) n λ n ( α ) ddt u ( α ) n dt. (4.13)19islocation-dipoleFigure 9: Two-dimensional energy landscape for σ = 0 .
37 with κ = 0 . µ = − / n l = 1 , n r = 2 to n l = 0 , n r = 3 , (similar to Fig. 7(b)). As a generalization of (4.12), it is found that above expression yieldsΨ( α ) = 2 κ { (cid:88) m ∈ Λ (cid:88) n ∈ Λ (cid:36) n α n U n − m (cid:36) m α m + 12 (cid:88) n ∈ Λ ˆ w ( u ( α n ) − ) } . (4.14) Remark 4.2
In the case considered earlier in § ˆ w ( u ) = (( u − − ( u +1) )Θ( u ) = − u Θ( u ) (with u = u (0) + α(cid:36) U = − α ∗ (cid:36) U + α(cid:36) U = ( α − α ∗ ) (cid:36) U ). An illustration of the two dimensional order parameter based energy landscape isprovided in Fig. 8 for a transition from configuration with 5 particles in secondphase to that with 7. The figure also reveals the nature of the Lagrange multipliersin this case and also the energy changes along four different kinds of paths inthe order parameter space. Similarly, a transition from homogeneous state to aconfiguration with 2 particles in second phase is illustrated in Fig. 9.20islocation-dipoleOverall, it is clear, from a mathematical viewpoint, that the incorporation ofthe spinodal region via (4.2) leads to smoothening of the Peierls landscape fortransition involving configurations that involves one particle crossing the (regularor degenerate) spinodal region in each of the two dislocations (of opposite sign)in the dislocation-dipole. However, within the confines of the assumed model,the equilibria (2.3) remain admissible for the model (4.1) only with small size ofspinodal region. As soon as the size (4.2) of spinodal region becomes larger theadmissible range of σ shrinks and the equilibria that exist are not stable any more. Let { u (0) n } n ∈ Z be the initial configuration with n particles in the second phaseand the final configuration be { u (1) n } n ∈ Z with n particles in second phase where n > n . Consider the quadratic well model without the spinodal region as discussedin previous section (before its last part). Then the change in energy for every singletransition can be calculated using the expression (3.17). Let α i denote the scalarorder parameter for the transition from a configuration with n + i − n + i particles and for this transition thechange in energy can be expressed asΨ( α i ) = 2 κ U ( 12 (cid:36) i α i + (cid:36) i ( α ∗ i − α i )Θ( α i − α ∗ i )) , (5.1)where (cid:36) i (for transition at site n i which lies at the left or right dislocation in thedipole) is (cid:36) i = u (1) n i − u (0) n i U , (5.2)with (in accordance with (3.12)) α ∗ i = { ( σ − η + 1) + 2(1 − η − n − i +1 ) } / (2(1 − η )) , i = 1 . . . N, N = n − n . Recall that η is defined by (2.4) and the definition ofΘ is given by (1.7). In view of Remark 3.4, it is noted that (cid:36) i = 1. Thus the totalchange ˆΨ in energy for last stage of the transition from the initial order parameterconfiguration { u (0) n } n ∈ Z to { u (1) n } n ∈ Z is given by (using (5.1))ˆΨ( α N ):=2 κ U (cid:88) N − i =1 ( 12 (cid:36) i α i + (cid:36) i ( α ∗ i − α i )Θ( α i − α ∗ i )) + Ψ( α N ) (5.3)If σ is such that α ∗ i ∈ [0 , , i = 1 . . . N (in other words, σ must belong to theintersection of admissible ranges for all such i ) then using this assumption ˆΨ( α N )simplifies toˆΨ( α N ) = 2 κ U { σ (1 − η )( N − − η − n ( η − N +1 − } / (2(1 − η ) ) + Ψ( α N )(5.4)21islocation-dipoleFigure 10: Example of a Peierls Landscape with various values of σ for κ = 0 .
5. The changein energy for a cascade of transitions is shown. In part (a), the change in energy is shown forthe cascade of transitions starting with 2 particle in second phase. In part (b), for 3 particlesas it is clear from the origin of the figures corresponding to Ψ = 0 and value of n for this, andsimilarly it can be seen in parts (c), (d), (e) and (f). Solid curve denotes the change in energy,using (5.4), for a transition from n particles in second phase to n + 1 particles where n is thehorizontal axis. Blue dashed curve (not grid lines) represents that the Peierls barrier has beencrossed indicating the motion of fronts with separation away from one another. Dotted curvedenotes the local maxima of the energy barrier between any two stable equilibria. In (b), (c), (d),(e), (f) the top solid curve represents the case σ = σ min , the second from top curve represents σ = 0, and the third curve is for σ = σ M , the fourth curve is drawn for σ = ( σ max + σ M ) /
2, andthe lowest curve is for σ = σ max , where these values of σ are evaluated for the transition frominitial configuration with n number of particles in the second phase. In part (a) the curve for σ = 0 is absent because 0 < σ M < σ min for one particle in second phase. Here, σ min etc refer tothe initial configuration corresponding to the lowest value of n in each plot. σ for κ = 0 .
5. The details areavailable in the figure caption.
Remark 5.1
As mentioned before the changes in energy of the infinite one-dimensionallattice when the external force σ changes cannot be determined. The results pre-sented in Fig. 10 show the change in energy as superimposed curves only forconvenience. In the situations when σ changes, all particles in the lattice aredisplaced and so mathematically there is infinite increment to the energy (1.5) . A visual depiction of the cascade of transitions is also shown in Fig. 11 where asequence of equilibria (2.3) are plotted in (a) and (b) for two values of σ respectivelyas stated in captions. Upon ignoring the corners or hills in Fig. 10, due to energybarriers caused by lattice trapping in each successive transition, it can be notedthat there is an ‘approximate’ curve with a visibly negative curvature and there isone point along the n axis where the slope of this curve becomes zero and beyondthis point the slope continues to decrease. Call the point at which the maximumof this ‘approximate’ curve is located as (cid:96) a . The maximum of ˆΨ is located at (cid:96) c and it is near (cid:96) a . As shown below, (cid:96) c can be found analytically. The ‘approximate’curve connecting all local minima is given by∆ ˆΨ( (cid:96) ) = 2 κ U { σ (1 − η )( (cid:96) − − η − n ( η − (cid:96) +1 − } / (2(1 − η ) ) . (6.1)By this definition when (cid:96) = N, ˆΨ( α N ) = ∆ ˆΨ( N ) + Ψ( α N ) . Recall that η is definedby (2.4).The maximum of ∆ ˆΨ is (cid:96) a = − η ln( 12 ln η σ ( η − η n − ) , (6.2)and ∆ ˆΨ( (cid:96) a ) = κ U { σ (1 − η )( (cid:96) a − η ) + 2 η − n } / (1 − η ) . The energy barrierfor trapping of a dipole is given by∆ E H :=∆ ˆΨ( (cid:96) a ) + Ψ( α ∗(cid:98) (cid:96) a (cid:99) ) , (6.3)and clearly, (cid:96) c = (cid:96) a + α ∗(cid:98) (cid:96) a (cid:99) , where (cid:98) (cid:96) (cid:99) denotes the greatest integer less than (cid:96). ∆ E H is larger than the energy barrier due to first lattice trapping ∆ E L :=Ψ( α ∗ n ) , if σ < σ M (for example, see Fig. 12 (a)). Also, ∆ E H = ∆ E L if σ ≥ σ M . As σ increases towards σ max , (cid:96) c decreases towards the initial number of particles in23islocation-dipoleFigure 11: A family of equilibria (with dipole expanding in the + n r direction assuming n l = 0)beginning with 5 particles in the second phase for κ = 0 . σ = σ M , (b) σ = σ M , where σ M corresponds to the transition from 5 particles in second phase to 6 (the red curve in Fig.3(a) shows σ M dependence). The gray strip corresponds to the spinodal region for the modeldiscussed in § second configuration, n , and indeed (cid:96) c = α ∗ when σ = σ M (for example, see Fig.12 (b)).If σ ∈ ( σ min , σ M ) there is preference towards annihilation and if σ ∈ ( σ M , σ upper )(as previously stated, σ upper = σ max ) there is a preference towards separation offronts. This is also seen in Fig. 10. For σ ∈ ( σ min , σ M ) , due to thermal excitation,the two fronts can move towards each other, as ∆ E H > ∆ E L , and annihilatethe dipole. However as soon as σ > σ M , σ ∈ ( σ M , σ upper ) there is a preferencetowards a motion that leads to separation of the two fronts away from one anotherunless during motion there is a coherence between the two fronts as solitary wavesdescribed in [24]. This confirms the general principle regarding dislocations in alattice that two dislocations of opposite sign repel each other and they may attractone another if either the applied stress is too small or the separation between themis small.In Fig. 12, the solid blue curve refers to the energy profile for the cascade oftransitions in the presence of a spinodal region, using the results of § calcspin. Fromthis it is clear that an important role in terms of ∆ E L is played by the nature ofonsite potential model. With an increase in the size of spinodal region, accordingto (4.2) there is a decrease in the value of | µ | so that the energy barrier naturallyreduces. At this point, the last sentences of § Energy landspace corresponding to the cascade of transitions of Fig. 11(a) and (b).Light solid curve denotes the curve connecting local minima. Thick solid black curve denotes theclimb and brown curve denotes sliding. Solid blue curve represents the model having spinodalregion with µ = − / barrier per particle for a transition from n to n + 1 particles in second phase is˜Ψ( α ∗ ) = E ε a Ψ( α ∗ ) = E ε a κ U (cid:36) α ∗ (6.4)As n = n r − n l − → ∞ , ˜Ψ( α ∗ ) → εc { − σ − η +1 } . Using η = 1 + κ + κ / o ( κ ) , as κ → , U = κ − κ / o ( κ ) . Choosing the Young’s modulus (in threedimensions) E , the elastic modulus (in one dimension) is E ∼ E a , (cid:36) ∼ a ∼ ε, then T = ˜Ψ( α ∗ ) /k B = E a k B (cid:36) κ α ∗ + o ( κ ) α ∗ . (6.5)Let E ∼
100 GPa and a ∼ − m, so T = ˜Ψ( α ∗ ) /k B = k B κ α ∗ × − + o ( κ ) α ∗ J per particle. Using k B = 1 . × − J per deg K per particle, T =1 . × κ α ∗ + o ( κ ) α ∗ K. With this rough estimate, it can be stated thatthe energy barriers may be large compared compared to thermal fluctuations atlow temperatures. At small σ and high temperature, the energy barrier ∆ E L may be comparable with thermal fluctuations but ∆ E H may not be overcome bythermal fluctuations along and the external force may need to be increased so thatnucleation of dislocation-dipole and propagation of two dislocations is possible.The presence of a relatively small value of the onsite potential elastic constant κ (1.4), in the presence of spinodal region, may lead to a reduction in the energybarrier so it can be overcome even at low temperature. Acknowledgements : The partial support of SERB MATRICS grant MTR/2017/000013 isgratefully acknowledged.
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Physical ReviewA-General Physics (4A), 1007–1015 (1964) A Unstable equilibria u n = σ − A n l − n r η n − n l , n ≤ n l B n l − n r η n l − n + B n l − n r η n − n r , n l < n < n r − ( σ − η n r − n , n r ≤ n (A.1) u n = σ − − ( σ − η n − n l , n ≤ n l B n l − n r η n l − n + B n l − n r η n − n r , n l < n < n r A n l − n r η n r − n , n r ≤ n (A.2) u n = σ − − ( σ − η n − n l , n ≤ n l − ( σ − η n − n r + η n l − n ) / (1 + η n l − n r ) , n l < n < n r − ( σ − η n r − n , n r ≤ n (A.3)with η given by (2.4) and B n = − (2 η + η n ( η − − ( η + 1) σ )) / (( η + 1)(1 − η n ))and B n = − (1 + η − η n ) η n + ( η + 1) σ ) / (( η + 1)(1 − η n )) . { u si } i ∈ Z , described by(2.3). Let u n = u sn + εv n with ε small so that u n still satisfies the consistency withrespect to number of particles in each phase. Then using the fact that { u si } i ∈ Z isan extrema, it is found that E ( { u i } i ∈ Z ) − E ( { u si } i ∈ Z ) = ε (cid:88) n ∈ Z {
12 ( v n +1 − v n ) + κ v n } > , for all ε for which the consistency mentioned is not violated. From the constructionit can be seen that such ε exists for σ ∈ ( σ lower , σ upper ) for all directions { v i } i ∈ Z .So equilibria (2.3) are local minima. If σ = σ upper (or σ = σ lower ) so that u sN = 0for some N , then one can show that E ( { u i } i ∈ Z ) − E ( { u si } i ∈ Z ) = ε (cid:88) n ∈ Z {
12 ( v n +1 − v n ) + κ v n } + ε (cid:88) n ∈ Z { ( u sn +1 − u sn )( v n +1 − v n ) + κ [(1 ± u sn ) v n − σv n ] } which is positive for all sufficiently small ε if v N = 0 using that { u si } i ∈ Z is anextrema, but it may be negative for some ε if v N (cid:54) = 0 (the directional deriva-tive of E along { v i } i ∈ Z may not exist). For example of the latter, consider v n =0 , n (cid:54) = N, v N (cid:54) = 0 , then E ( { u i } i ∈ Z ) − E ( { u si } i ∈ Z ) < , for v N > > ε > − κ / ((1 + κ ) v N ( η + 1)) . Therefore the equilibria (A.1), (A.2), and (A.3) aresaddle points.
B Derivation of the change in energy
The change in the energy isΨ( α ) = E ( { u ( α ) i } ) − E ( { u (0) i } )= 12 (cid:88) n ( (cid:88) m ∈ Λ (cid:36) m α m ( U n +1 − m − U n − m )) + (cid:88) n ( u (0) n +1 − u (0) n )( (cid:88) m ∈ Λ (cid:36) m α m ( U n +1 − m − U n − m ))+ 12 κ (cid:88) n/ ∈ Λ [ ± ± u (0) n ) (cid:88) m ∈ Λ (cid:36) m α m U n − m + ( (cid:88) m ∈ Λ (cid:36) m α m U n − m ) ] − κ (cid:88) n ∈ Λ sgn[ u (0) n + (cid:88) m ∈ Λ (cid:36) m α m U n − m ][ u (0) n + (cid:88) m ∈ Λ (cid:36) m α m U n − m ]+ (cid:88) n ∈ Λ κ [2 u (0) n (cid:88) m ∈ Λ (cid:36) m α m U n − m + ( (cid:88) m ∈ Λ (cid:36) m α m U n − m ) − u (0) n ] − (cid:88) n κ σ ( (cid:88) m ∈ Λ (cid:36) m α m U n − m ) . (B.1)28islocation-dipoleNow observe that Ψ must have local extrema at any α for which all componentsare either zero or one. So using ∂ Ψ ∂α m | α =0 = 0 , obtainΨ( α ) = 12 (cid:88) n { ( (cid:88) m ∈ Λ (cid:36) m α m ( U n +1 − m − U n − m )) + κ ( (cid:88) m ∈ Λ (cid:36) m α m U n − m ) }− κ (cid:88) n ∈ Λ { u (0) n + (cid:88) m ∈ Λ (cid:36) m α m U n − m } Θ[ u (0) n + (cid:88) m ∈ Λ (cid:36) m α m U n − m ] , (B.2)and using ∂ Ψ ∂α m | α m =1 ,α n =0 ∀ n (cid:54) = m = 0 , Ψ( α ) = 2 κ { (cid:88) m ∈ Λ (cid:88) n ∈ Λ (cid:36) m α n U n − m (cid:36) m α m − (cid:88) n ∈ Λ ( u (0) n + (cid:88) m ∈ Λ (cid:36) m α m U n − m )Θ[ u (0) n + (cid:88) m ∈ Λ (cid:36) m α m U n − m ] } . (B.3)In one-dimensional case, by explicit calculation also, it can be shown that (cid:80) n { ( U n +1 −U n ) + κ U n } = 2 κ U . C Sequential vs simultaneous
The energy barrier for sequential transition, where transition occurs effectivelythrough scalar order parameter, isΨ sq ( α ∗ ) = max { κ U α ∗ l , κ U ( α ∗ r + 2 α ∗ l − } , (C.1)where α ∗ l , α ∗ r are given by u (0) n l +1 + α ∗ l U = 0 , u (0) n r − + U n l − n r +2 + α ∗ r U = 0 . It is easyto see that α ∗ r < α ∗ l as U n > , ∀ n and u (0) n l +1 = u (0) n r − < . The energy barrier forsimultaneous transition is Ψ sm ( β ∗ ) = κ ( U | β ∗ | + 2 β ∗ l U n l − n r +2 β ∗ r ) . Since β l = β r and U and u (0) have the reflection symmetry, Ψ sm ( β ∗ ) = 2 κ ( U + U n l − n r +2 ) β ∗ l . At β ∗ , u (0) n l +1 + β ∗ l ( U + U n l − n r +2 ) = 0 . Using u (0) n l +1 = u (0) n r − , Ψ sm ( β ∗ )Ψ sq ( α ∗ ) = min { U + U n l − n r +2 ) β ∗ l U α ∗ l , U + U n l − n r +2 ) β ∗ l U ( α ∗ r + 2 α ∗ l − } . (C.2)When 0 < α ∗ l < − α ∗ r < , it can shown thatΨ sm ( β ∗ )Ψ sq ( α ∗ ) = 2 U U + U n l − n r +2 > , (C.3)using U > U n l − n r +2 > . In the case 1 > α ∗ l > − α ∗ r > , Ψ sm ( β ∗ )Ψ sq ( α ∗ ) > U + U n l − n r +2 > − u (0) n l +1 (2 + 1 u (0) n l +1 ( U + U n l − n r +2 )) , (C.4)or β ∗ l + β ∗ l > , but this is an obvious inequality for β ∗ l ∈ (0 , . D Transition from n to n + 2 particles in secondphase for model with spinodal region D.1 Transition from homogeneous state
Let n l = − h, n r = − , (D.1)for the initial state u (0) . The stable equilibrium configurations can be expressed as(2.3) with (D.1). When h = 2, it can be considered that u (1) − = u (1) − = u − = u − = σ + 1 − η − , which can be connected with u (0) n = − σ with u (0) − , u (0) − havingthe same value; this corresponds to a transition from homogeneous state u (0) tothat u (1) with two atoms in the second phase. The equilibrium configuration oflattice, for such a transition from h − h in secondphase, is obtained by minimisation of the energy (3.1), i.e., solving (4.5) withΛ = {− , − } . In general, consider Λ = {− h, − } , and for now h = 2 , while thecase h > h = 2 is discussed here for Λ = {− , − } , . For α ∈ N cr = { ( α − , α − ) : u ( α ) − , u ( α ) − < − χ } , − ( u ( α ) n +1 − u ( α ) n + u ( α ) n − ) + κ [1 + u ( α ) n − σ ] − λ − δ − ,n − λ − δ − ,n = 0 . (D.2)For n / ∈ Λ , the solution can be written as u ( α ) n = − σ + (cid:26) A l η n +2+1 , n ≤ − − A r η − n , ≤ n, (D.3)such that (3.4) holds. Then, using (D.2) for n ∈ Λ, it is found that along with u ( α ) − = − σ + A r η and u ( α ) − = − σ + A l η , gives A l = 1 κ ( η − η + 1) η λ − + λ − ηη , A r = 1 κ ( η − η + 1) η λ − + λ − ηη . (D.4)It is found that (3.5) holds and λ − = 2 κ α − (cid:36) − , λ − = 2 κ α − (cid:36) − , (D.5)where (in view of (3.7)) (cid:36) − = − ( − ( u (1) − − u (0) − ) U + ( u (1) − − u (0) − ) U − ) / ( U − U − ) ,(cid:36) − = ( − ( u (1) − − u (0) − ) U − + ( u (1) − − u (0) − ) U ) / ( U − U − ) . The critical value ofthe components of α is given by u (0) − + α − (cid:36) − U − + α − (cid:36) − U − = − χ , or30islocation-dipole u (0) − + α − (cid:36) − U − + α − (cid:36) − U = − χ . In general, for α ∈ N cr , N cr , N cr , N cr , the following relations are obtained, α − ± ±
12 = 12 κ (cid:36) − λ − , α − ± ±
12 = 12 κ (cid:36) − λ − , (D.6)with critical value of α s given by appropriate conditions. Certain special points ofthe unit square are stable equilibria (local minima of energy) of the lattice if andonly all λ i s are zero (this includes clearly, [0 ,
0] and [1 , α / ∈ N cr ∪ N cr ∪ N cr ∪ N cr = { ( α − , α − ) : | u ( α ) − | , | u ( α ) − | > χ } , − ( u ( α ) n +1 − u ( α ) n + u ( α ) n − ) + κ [ µu ( α ) n δ Λ ,n + (1 − δ Λ ,n )(1 + u ( α ) n ) − σ ] − λ − δ − ,n − λ − δ − ,n = 0 . (D.7)Then − ( − σ + A r − u ( α ) − + u ( α ) − ) + κ [ µu ( α ) − − σ ] − λ − = 0 , (D.8) − ( u ( α ) − − u ( α ) − − σ + A l ) + κ [ µu ( α ) − − σ ] − λ − = 0 , (D.9)such that (3.4) holds. Then A l = ( η − − (1 + µ ( η − η − µ ( η − ) ( λ − η + λ − (2 η − µ ( η − ) + A ) , (D.10) A r = ( η − − (1 + µ ( η − η − µ ( η − ) ( λ − η + λ − (2 η − µ ( η − ) + A ) , (D.11)where A = κ (3 η − µ ( η − )( µ − ( µ − σ ) . (D.12)With R = 3 η − µ ( η − , it is found that A = κ R ( µ − ( µ − σ ) , while bysimplifying A l η n +3 and A r η − n , it is found that (3.5) holds, and λ − and λ − arefound as λ − = 2 κ η ( η + 1) ( (cid:36) − α − η (2 + µ ( η − (cid:36) − α − ( µ − η − − κ ( µ − ( µ − σ. (D.13) λ − = 2 κ η ( η + 1) ( (cid:36) − α − ( µ − η −
1) + (cid:36) − α − η (2 + µ ( η − − κ ( µ − ( µ − σ. (D.14)31islocation-dipoleIn a different regime, with α such that it lies in { ( α − , α − ) : | u ( α ) − | < χ , | u ( α ) − | > χ } , − ( − σ + A r − u ( α ) − + u ( α ) − ) + κ [1 + u ( α ) − − σ ] − λ − = 0 , (D.15) − ( u ( α ) − − u ( α ) − − σ + A l ) + κ [ µu ( α ) − − σ ] − λ − = 0 , (D.16)such that (3.4) holds. Then A l = ( η − − (2 + µ ( η − η ( λ − η + λ − η + κ η ( µ − σ ( µ − , (D.17) A r = ( η − − (2 + µ ( η − η ( λ − η + λ − (2 η − µ ( η − ) + κ η ( µ − σ ( µ − . (D.18)Simplifying A l η n +3 and A r η − n , it is found that (3.5) holds, and also (D.5) holds,while λ − = − κ (cid:36) − α − η − ( η − − µ ) η + 1 + (2 + µ ( η − η + 1 2 (cid:36) − α − κ − κ ( µ − σ ( µ − . (D.19)In another regime, with α such that it lies in { ( α − , α − ) : | u ( α ) − | < χ , | u ( α ) − | > χ } ,after simplifying A l η n +3 and A r η − n , it is found that (3.5) holds, and also it is foundthat λ − is given by (D.5) while λ − = − κ (cid:36) − α − η − ( η − − µ ) η + 1 + (2 + µ ( η − η + 1 2 (cid:36) − α − κ − ( µ − σ ( µ − . (D.20) D.2 Transition from existing dipole
The case h > {− h, − } , . For α ∈ N cr = { ( α − , α − h ) : u ( α ) − , u ( α ) − h < − χ } , the particles on the left and right sides of the bump are in thefirst phase while others are in the second phase. For example, the equation ofequilibrium is − ( u ( α ) n +1 − u ( α ) n + u ( α ) n − ) + κ [1 + u ( α ) n − σ ] − λ − δ − ,n − λ − h δ − h,n = 0 , (D.21)for n / ∈ {− h + 1 , . . . , − } . For n / ∈ Λ , the solution can be written as u ( α ) n = − σ + A l η n + h +1 , n ≤ − h −
12 + ( B l η n + h +1 + B r η − n ) , − h < n < − A r η − n , ≤ n, (D.22)32islocation-dipolesuch that (3.4) holds. Then, using (D.21) for n ∈ Λ, it is found that A l = 2 η − − η − h η + 1 + 1 η − λ − h + λ − η − h ) ,A r = 2 η − − η − h η + 1 + 1 η − λ − + λ − h η − h ) , (D.23)and B l = 2 1 − ηη − η − h + λ − η − η − h , B r = 2 1 − ηη − η − h + λ − h η − η − h . (D.24)After some simplifications, it is found that (D.5) holds, where (cid:36) − , (cid:36) − h are giventhe same expressions as in the context of (D.5), except that (cid:36) − is replaced by (cid:36) − h , and (3.5) holds. The critical value of α is given by either u (0) − + α − (cid:36) − U + α − h (cid:36) − h U − h = − χ , or u (0) − h + α − (cid:36) − U − h +1 + α − h (cid:36) − h U − h + h = − χ . In general,for α ∈ N cr , N cr , N cr , N cr , such that it lies in { ( α − , α − h ) : | u ( α ) − | , | u ( α ) − h | > χ } , itis easy to see that (D.6) holds. This completes the discussion for four regimes ofthe order parameter space, α ∈ N cr ∪ N cr ∪ N cr ∪ N cr . Certain special points of the unit square are stable equilibria (local minima ofenergy) of the lattice if and only all λ i s are zero (this includes clearly, [0 ,
0] and[1 , α / ∈ N cr ∪ N cr ∪ N cr ∪ N cr , such that it lies in { ( α − , α − h ) : | u ( α ) − | , | u ( α ) − h | < χ } , − ( u ( α ) n +1 − u ( α ) n + u ( α ) n − ) + κ [ µu ( α ) n δ Λ ,n + (1 − δ Λ ,n )(1 + u ( α ) n ) − σ ] − λ − δ − ,n − λ − h δ − h,n = 0 , (D.25)for n / ∈ {− h + 1 , . . . , − } . For n / ∈ Λ , the solution can be written as (same asbefore) u ( α ) n = − σ + A l η n + h +1 , n ≤ − h −
12 + ( B l η n + h +1 + B r η − n ) , − h < n < − A r η − n , ≤ n, (D.26)such that (3.4) holds. Also u ( α ) − = − σ + A r η = 1 + σ + B l η h + B r η and u ( α ) − h = − σ + A l η = 1 + σ + B l η + B r η h . Here, α n ∈ [0 , , n ∈ Λ . Then, using33islocation-dipole(D.25) for n ∈ Λ, such that (3.4) holds, it is found that A l = λ − η h +1 ( η + 1) − λ − h (( µ − η − η − (2 + µ ( η − η h ) R + A ,A r = − λ − (( µ − η − η − (2 + µ ( η − η h ) + λ − h η h +1 ( η + 1) R + A ,B l = λ − η h +1 (2 + µ ( η − − λ − h ( µ − η − η R + B ,B r = − λ − ( µ − η − η + λ − h η h +1 (2 + µ ( η − R + B , (D.27)where R = − ( µ − ( η − η + (2 + µ ( η − ( η − η h ,A = η ( η ( − σ ) − σ ) − µ ( η − η + η h )( σ −
1) + η h (2 + ( η − σ ) η (( µ − η − η + (2 + µ ( η − η h ) ,B = − η − σ − µ ( η − σ )(( µ − η − η + (2 + µ ( η − η h ) . (D.28)In fact, the solution can be written as u ( α ) n = u ( α ∗ ) n + ( A l − A ) η n + h +1 , n ≤ − h − B l − B ) η n + h +1 + ( B r − B ) η − n ) , − h < n < − A r − A ) η − n , ≤ n, , (D.29)where α ∗ corresponds to the equilibrium with 2 particles in the unstable spinodalregion; moreover, u ( α ∗ ) n = − σ + A η n + h +1 , n ≤ − h −
12 + B ( η n + h +1 + η − n ) , − h < n < − A η − n , ≤ n, = u (0) n + α − (cid:36) − U η −| n +1 | + α − h (cid:36) − h U η −| n + h | (D.30)so that (it can be easily checked that the equations with terms involving A arealso satisfied) α − (cid:36) − U = α − h (cid:36) − h U = ( η −
1) 2( µ − η + η h (2 + ( η + 1) σ − µ ( σ + 1 + η ( σ − µ ( η − η h ( η + 1) + ( − µ ) η ( η − , (D.31)34islocation-dipoleand let such α s be denoted by α ∗ = α − = α − h . (D.32)Recall (3.6), that is, U n +1 = U η −| n +1 | , U n + h = U η −| n + h | . By simplifying ( A l − A ) η n + h +1 , ( A r − A ) η − n , and (( B l − B ) η n + h +1 + ( B r − B ) η − n ), it is found that(3.5) can be written as u ( α ) n = u (0) n + (cid:80) m ∈ Λ α m (cid:36) m U n − m = u ( α ∗ ) n + (cid:80) m ∈ Λ ( α m − α ∗ ) (cid:36) m U n − m . These relations lead to( α − − α ∗ ) (cid:36) − U η − h R = λ − η h +1 (2 + µ ( η − − λ − h ( µ − η − η , ( α − h − α ∗ ) (cid:36) − h U η − h R = − λ − ( µ − η − η + λ − h η h +1 (2 + µ ( η − λ − and λ − h in terms of α − , α − h .In a different regime, such that it lies in { ( α − , α − h ) : | u ( α ) − | > χ , | u ( α ) − h | < χ } , − ( − σ + A r − u ( α ) − + u ( α ) − ) + κ [1 + u ( α ) − − σ ] − λ − = 0 , − ( u ( α ) − h +1 − u ( α ) − h − σ + A l ) + κ [ µu ( α ) − h − σ ] − λ − h = 0 , (D.34)such that (3.4) holds. Then A l = λ − η − h +1 + λ − h R ( η + 1) + A l ,A r = λ − η − − h ((1 − µ )( η − η + (2 + µ ( η − η h ) + λ − h η − h (1 + η ) R + A r , − η h ( η − − ( η − σ + µ ( η − σ ))) /R ,B l = λ − η − h η − B l ,B r = − λ − ( µ − η − η − h + λ − h η − h +1 ( η + 1) R + B r , (D.35)where R = (2 + µ ( η − η − ,A l = ( η − η − − h ( − η + η h (2 + ( η − σ + µ ( η − − σ )))( η + 1) /R ,A r = η − − h (2( µ − η − η + 2(2 + µ ( η − η − η h B l = − η − r − h η − ,B r = η − h (2( µ − η − η − η h ( η − − ( η − σ + µ ( η − σ ))) /R . (D.36)35islocation-dipoleIn fact, the solution can be written as u ( α ) n = u ( α ∗ ) n + ( A l − A l ) η n + h +1 , n ≤ − h − B l − B l ) η n + h +1 + ( B r − B r ) η − n ) , − h < n < − A r − A r ) η − n , ≤ n, , (D.37)where α ∗ corresponds to the equilibrium with 1 particle in the unstable spinodalregion. In fact, the solution can be written as u ( α ) n = u ( α ∗ ) n + ( A l − A l ) η n + h +1 , n ≤ − h − B l − B l ) η n + h +1 + ( B r − B r ) η − n ) , − h < n < − A r − A r ) η − n , ≤ n, , (D.38)where α ∗ corresponds to the equilibrium with 2 particles in the unstable spinodalregion. Moreover, u ( α ∗ ) n = − σ + A l η n + h +1 , n ≤ − h −
12 + ( B l η n + h +1 + B r η − n ) , − h < n < − A r η − n , ≤ n, = u (0) n + α − (cid:36) − U η −| n +1 | + α − h (cid:36) − h U η −| n + h | (D.39)so that α − (cid:36) − U = 0, and α − h (cid:36) − h U = ( η − η − h (2( µ − η + η h (2 + ( η + 1) σ − µ (1 + η ( σ −
1) + σ )))(2 + µ ( η − η + 1) ;(D.40)let such α s be denoted by α − = 0 , α ∗ = α − h . It can be easily checked thatthe equations with terms involving A l and A r are also satisfied; indeed, A l − η − − η − h +1 η +1 = α − h (cid:36) − h U η − , and A r − η − − η − h +1 η +1 = α − h (cid:36) − h U η − h . By simplifying( A l − A l ) η n + h +1 , ( A r − A r ) η − n , and ( B l − B l ) η n + h +1 + ( B r − B r ) η − n , it isfound that (3.5) can be written as u ( α ) n = u (0) n + (cid:80) m ∈ Λ α m (cid:36) m U n − m = u ( α ∗ ) n + (cid:80) m ∈ Λ ( α m − α ∗ δ m, − h ) (cid:36) m U n − m . Indeed, α − (cid:36) − U R = λ − η (2 + µ ( η − , ( α − h − α ∗ ) (cid:36) − h U R = − λ − ( µ − η − η − h + λ − h η ( η + 1) . Above relations can beinverted to obtain λ − and λ − h in terms of α − , α − hh