Phase diagram and commensurate-incommensurate transitions in the phase field crystal model with an external pinning potential.
ABSTRACT We study the phase diagram and the commensurate-incommensurate transitions in a phase field model of a two-dimensional crystal lattice in the presence of an external pinning potential. The model allows for both elastic and plastic deformations and provides a continuum description of lattice systems, such as for adsorbed atomic layers or two-dimensional vortex lattices. Analytically, a mode expansion analysis is used to determine the ground states and the commensurate-incommensurate transitions in the model as a function of the strength of the pinning potential and the lattice mismatch parameter. Numerical minimization of the corresponding free energy shows reasonable agreement with the analytical predictions and provides details on the topological defects in the transition region. We find that for small mismatch the transition is of first order, and it remains so for the largest values of mismatch studied here. Our results are consistent with results of simulations for atomistic models of adsorbed overlayers.
- SourceAvailable from: Tapio Ala-Nissila[Show abstract] [Hide abstract]
ABSTRACT: We study the nonlinear driven response and sliding friction behavior of the phase-field-crystal (PFC) model with pinning including both thermal fluctuations and inertial effects. The model provides a continuous description of adsorbed layers on a substrate under the action of an external driving force at finite temperatures, allowing for both elastic and plastic deformations. We derive general stochastic dynamical equations for the particle and momentum densities including both thermal fluctuations and inertial effects. The resulting coupled equations for the PFC model are studied numerically. At sufficiently low temperatures, we find that the velocity response of an initially pinned commensurate layer shows hysteresis with dynamical melting and freezing transitions for increasing and decreasing applied forces at different critical values. The main features of the nonlinear response in the PFC model are similar to the results obtained previously with molecular dynamics simulations of particle models for adsorbed layers.Physical Review E 01/2010; 81(1 Pt 1):011121. · 2.31 Impact Factor
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ABSTRACT: In this contribution our focus is on the phase-field crystal method, which can be viewed as the youngest methodology in the field of interface computation based on recent work by Elder et al. (Phys. Rev. Lett. 88, 245701 (2002)). It bridges the gap between the molecular simulation approaches and the phase-field approach by operating on diffusive time scales yet atomic length scales. Here we review the fundaments of the phase-field crystal method as well as different models established so far with the aim to capture the main features of the wide range of phase diagrams found in materials science more and more comprehensively.European Physical Journal Plus 126(10). · 1.30 Impact Factor
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ABSTRACT: We determine the phase diagram of the phase field crystal model in three dimensions by using numerical free energy minimization methods. Previously published results, based on single mode approximations, have indicated that in addition to the uniform (liquid) phase, there would be regions of stability of body-centered cubic, hexagonal and stripe phases. We find that in addition to these, there are also regions of stability of face-centered cubic and hexagonal close packed structures in this model.Journal of Physics Condensed Matter 05/2010; 22(20):205402. · 2.22 Impact Factor
arXiv:cond-mat/0608192v1 [cond-mat.other] 8 Aug 2006
Phase Diagram and Commensurate-Incommensurate Transitions in the Phase Field Crystal Model
with an External Pinning Potential
C. V. Achim
Laboratory of Physics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 TKK, Finland
Department of Applied Mathematics, The University of Western Ontario, London (ON), Canada
Department of Physics, Oakland University, Rochester, Michigan, 48309-4487, USA
Laborat´ orio Associado de Sensores e Materiais, Instituto Nacional de Pesquisas Espaciais, S˜ ao Jos´ e dos Campos, SP Brazil
Laboratory of Physics, Helsinki University of Technology,
P.O. Box 1100, FIN-02015 TKK, Finland and Department of Physics,
Brown University, Providence, R.I. 02912-1843, U.S.A.
Department of Physics, Brown University, Providence, R.I. 02912-1843, U.S.A.
(Dated: February 6, 2008)
We study the phase diagram and the commensurate-incommensurate transitions in a phase field model of
a two-dimensional crystal lattice in the presence of an external pinning potential. The model allows for both
elastic and plastic deformations and provides a continuum description of lattice systems, such as for adsorbed
atomic layers or two-dimensional vortex lattices. Analytically, a mode expansion analysis is used to determine
the ground states and the commensurate-incommensurate transitions in the model as a function of the strength
of the pinning potential and the lattice mismatch parameter. Numerical minimization of the corresponding free
energy shows good agreement with the analytical predictions and provides details on the topological defects
in the transition region. We find that for small mismatch the transition is of first-order, and it remains so for
the largest values of mismatch studied here. Our results are consistent with results of simulations for atomistic
models of adsorbed overlayers.
PACS numbers: 64.60.Cn Order-disorder transformations; statistical mechanics of model systems, 64.70.Rh Commensurate-
incommensurate transitions, 68.43.De Statistical mechanics of adsorbates, 05.40.-a Fluctuation phenomena, random pro-
cesses, noise, and Brownian motion
In nature there exist many modulated structures which posses two or more competing length scales. Such systems often
exhibit commensurate-incommensurate (CI) phase transitions [1, 2], which are characterized by structural changes induced by
competition between these different scales. Systems in this class that have received intense attention for many years include
spin-density-waves [3, 4] in Cr and charge-density-wave systems  e.g. in ”blue bronze”. They are characterized by an order
parameter (e.g., charge or spin density) that is modulated in space with a given wave vector q. In these systems the transition
from a commensurate (C) state to an incommensurate (I) state is controlled by temperature and interaction with defects of the
lattice . Other systems of interest are materials which exhibit magnetic phase with spiral-like structures [7, 8]. Finally, vortex
lattices in superconductingfilms with pinningcenters  andweakly adsorbedmonolayers[10, 11] on a substrate comprisea 2D
realization of systems exhibiting CI transitions. In these systems the interparticle interactions are minimized by a configuration
witha lattice constanta, whilethesubstrate-adsorbedmonolayerinteractionis minimizedbya configurationwith latticeconstant
b usually incommensurate with a.
The simplest theoretical model for a CI transition is the 1D Frenkel-Kontorova (FK) model [12, 13]. It consists of a chain of
particles interconnected by springs, representing an adsorbate layer, and placed in a periodic potential describing the effects of a
substrate. The potential energy of a such system is given by
2(xn+1− xn− a)2+ V (xn)
where a is the equilibrium lattice spacing of the chain in the absence of a potential and λ is the stiffness. The potential function
V (xn) has periodicity b and can be approximated by a cosine function
V (xn) = V0
1 − cos
When V0is sufficiently small, the adsorbate lattice will be independent of the potential. This structure is called a ”floating”
phase and the lattice spacing ˜ a = limx→∞(xn− x0)/n of the adsorbate lattice can be an arbitrary multiple of the substrate
periodicity. In general, the floating phase is incommensurate for almost all values of the ratio ˜ a/b.
In the opposite limit, when the potential is very strong, one can expect the lattice to relax into a commensuratestructure where
the average lattice spacing of the adsorbed atoms is a simple rational fraction of the period b. In the I phase, close to the CI
transition, it is energetically more favorable for the system to form C regions separated by domain walls in which the springs
are stretched or compressed and the commensurate registry with the potential is lost. These domain walls are usually called
discommensurations and the corresponding region in the phase diagram within the I phase will be referred to as a modulated
(M) phase. A positive (negative) discommensuration leads to a reduction (increase) in the density of adsorbate atoms and these
regions are referred to as light (heavy) walls.
The CI transition can be determined by examining the behavior of the winding number˜Ω = ˜ a/b as a function of Ω = a/b
for fixed V0. In the continuum version of the FK model the CI transition is second order with a correlation length, identified as
the domain wall separation, which diverges logarithmical near the transition. If V0is larger then a certain critical value, Vc, the
system will be commensuratefor all the values of Ω. The function˜Ω(Ω) has a staircase type of appearanceand thus for V0< Vc
it is called an incomplete devil’s staircase and for V0> Vca complete devil’s staircase [2, 14]. If there are discontinuities or first
order jumps between commensurate states it is called a harmless staircase .
The FK modelcan be extendedto two dimensionsto describe, for example,adsorbedlayers on crystal surfaces. In its simplest
version the 2D FK model  describes the adsorbate interactions by a pure harmonic potential in a periodic pinning potential.
Although the FK model takes into account topological defects in the form of domain walls it leaves out plastic deformations of
the layer due to topological defects such as dislocations. These defects are particularly important when the CI transition occurs
between two different crystal structures or in presence of temperature fluctuations or quenched disorder.
They are automatically included in a full microscopic model involving interacting atoms in the presence of a substrate po-
tential. However, the full complexities of the microscopic model limits the actual numerical computation to systems of relative
small sizes. The size effect could be very strong for CI transition involving extended domains or topological defects. Recently a
phase field crystal model was introduced [16, 17] that allows for both elastic and plastic deformations in the solid phase. In this
formulation a free energy functional is introduced which depends on a density averaged over microscopic times scales, Φ(? r,t).
The free energy is minimized when Φ is spatially periodic (i.e., crystalline) in the solid phase and constant in the liquid phase.
By incorporating phenomena on atomic length scales the approach naturally includes elastic and plastic deformations, multiple
crystal orientations and anisotropic structures in a manner similar to other microscopic approaches such as molecular dynamics.
However, the PFC method describes the density on a diffusive and not the real microscopic times scales. It is therefore com-
putationally much more efficient. Thus this model should provide a suitable description of the CI transition when topological
defects, domain walls and dislocations are present.
Inthis workwe extendthe2D phasefield crystal(PFC)model[16, 17]toincludethe influenceofan externalpinningpotential.
Such a model should provide a suitable continuum description of lattice systems such as weakly adsorbed atomic overlayers or
2D vortex lattices with pinning. The pinning potential is chosen such that it induces CI transitions between ground states of
different symmetries in the model. The outline of the paper is as follows. We first define the model and carry out an analytic
mode expansion to determine the crystal structure and the location of the CI transition lines. We analyze these transitions as
a function of the lattice mismatch and strength of the potential. Following this we carry out a numerical minimization of the
full free energy functional which gives good agreement with the theoretical predictions and provides details on the nature of
the topological defects near the transition. Finally we discuss the nature of the CI transitions and the relation of our results to
atomistic simulations for adsorbed overlayers. .
II.THE PHASE FIELD CRYSTAL MODEL
In the phase field crystal model [16, 17], the free energy functional is written as
where G(∇2) = λ?q2
0+ ∇2?2, and its eigenvalues can be related to the experimental structure factor of e.g. Ar . Eq. (3)
describes a crystal with a lattice constant of 2π/q0, while the elastic properties can be adjusted by λ, u and q0. For numerical
calculation it is convenient to rewrite the free energy in dimensionless units [16, 17] as
? x = ? rq0,ψ = Φ
,τ = Γλq4
In these units the free energy becomes
where ω?∇2?= r +?1 + ∇2?2. Since ψ is a conserved field it satisfies the following equation of motion,
δψ+ ζ = ∇2?ω?∇2?ψ + ψ3?+ ζ,
where ζ has zero mean, ?ζ (? x1,τ1)ζ (? x2,τ2)? = D∇2δ(? x1− ? x2)δ (τ1− τ2) and D is a constant. Equations (5) and (6)
have been used to study a variety of phenomena involving elastic and plastic deformation including grain boundary energies
between misaligned grains, buckling and dislocation nucleation in liquid phase epitaxial growth, the reverse Hall-Petch effect in
nano-crystalline materials, grain growth and ductile fracture .
In 2D the free energyin Eq. (5) in the absence of external potential is minimized by three distinct solutions for the dimension-
less field ψ; constant, stripes and dots. For the purposes of this work only the latter solution is relevant. This solution consists of
a triangular distribution of density maxima corresponding to a crystalline phase. In general this solution can be written as
ψ(¯ r) =
where?Gnm≡ n?b1+ m?b2,?b1and?b2are the reciprocal lattice vectors and¯ψ is the average value of ψ. For a triangular lattice?b1
and?b2can be written as
?√3/2ˆ x+ ˆ y/2
where atis the distance between nearest-neighbor (local) maxima of ψ (corresponding to the ”atomic” positions). The ampli-
tudes am,nand lattice spacing atare determined by minimizing the free energy functional. For simplicity it is useful to first
consider a simple one-mode approximation (OMA) in which only pairs (n,m) that correspond to |?Gn,m| = 2π/(at
retained. In this limit ψ can be written,
ψt= At[cos(qtx)cos(qty/√3) − cos(2qty/√3)/2] + ψ,
where Atis an unknown constant and qt = 2π/at. Substituting Eq. (9) in Eq. (5) and minimizing with respect to Atand qt
gives, At= 4(ψ + (−15r − 36ψ
2)1/2/3)/5, qt=√3/2 and
min/S = −1
−15r − 36ψ
As shown in Ref. 17 this approximation is valid in the limit of small r. In the next section an external pinning potential with
square symmetry and incommensurate with will be introduced in the PFC model. As will be shown , depending on the strength
of the external potential and the lattice mismatch, a commensurate incommensurate transition would occur.
III.PHASE DIAGRAM WITH EXTERNAL POTENTIAL
In the PFC model , an external potential V is introduced by adding a term coupling V linear to ψ in the free energyfunctional
given in Eq. (5), i.e.,
+ ψ V
In this study, we consider an external potential of square symmetry which is distinct from the symmetry of the triangular
lattice in the absence of the external potential (e.g., Eq. 9).
V = V0cos(qsx)cos(qsy).
where qs= 2π/(as
√2). We define the relative mismatch δmbetween the external potential and adsorbed monolayer as
δm= 1 − 2π/as
With the aboveexternalpotential, the PFC modelcoulddescribeforexamplean adsorbedlayer onthe (100)face of an fcc crystal
To understand the influence of V on the minimum energy solution, we can again Fourier analyze the equilibrium density.
Taking into account both the intrinsic triangular symmetry and the external potential of square symmetry, the system is best de-
scribed by a combinationof hexagonal and square modes. In this case the correspondinghexagonal-squaremode approximation
(HSMA) can be written as
ψhs = A1cos(qsx)cos(qsy) + A2(cos(2qsx) + cos(2qsy))
+ At[cos(qtx)cos(qty/√3) − cos(2qty/√3)/2] + ψ.
This Fourier expansion includes the basic mode for the triangular lattice as done in Eq. 9 and the first two harmonics for the
commensurate modes with square symmetry. In general, they describe an incommensurate phase correspondingto an triangular
phase distorted by the square symmetry external potential. However, when the mismatch is sufficiently large and/or the strength
of the external potential is sufficiently strong, the solution that minimize the free energy will correspond to a vanishing value
forAtand a commensurate phase.
The free-energy density in the HSMA approximation is given by
The coefficients At, A1and A2are unknown and must be chosen to minimize the free-energydensity.
The analytic expressions for the free energies can now be used to obtain the phase diagram of the pinned PFC model as a
function of the pinning strength V0and the mismatch δm. To this end Eq. (15) can be used to determine the critical value
of V0,Vc, at which the CI transition occurs, identified by the point where the amplitude of the triangular phase vanishes, i.e.,
when At(V0,δm) = 0. The results are shown in Fig. 1(a). Within the HSMA the CI transitions is discontinuous and becomes
continuous only for infinite δm. In the next section we will compare the analytically obtained phase diagram in the HSMA
approximation with a full numerical minimization of the total free energy.
While the HSMA yields a good qualitative understanding for the phase diagram describing the CI transition, it is not quan-
titatively accrate, particularly near the transition, since the HSMA cannot describe the structure of the domain walls and other
possible topological defects. In this section, we desribe the full numerical investigation of the CI transition in the PFC model.
This is obtained by direct minimization of the free energy functional without any assumed form for the density field.
In the presence of the external potential, the equation of motion for ψ becomes,
∂τ= ∇2?ω?∇2?ψ + ψ3+ V?+ ζ.
The minimal energy numerical solutions for the equilibrium states of ψ were obtained by solving δF/δψ = 0 using a simple
relaxational method similar to the usual molecular dynamics annealing scheme. The noise term ζ in Eq. (16) is only used as an
−0.20 0.2 0.4 0.6 0.8
FIG. 1: (a) The phase diagram calculated analytically using the HSMA approximation for the free energy. The region denoted by C is the
commensurate phase and I denotes the incommensurate phase. (b) The corresponding phase diagram determined numerically as explained in
Sec. IV. The dashed line corresponds to a crossover regime between the fully incommensurate I and the modulated M phases. See text for
annealing tool to escape from any metastable states. It is applied for a limited period of time only and then set back to zero. For
the remainder of the time, Eq. (16) was solved using a simple Euler algorithm, i.e.,
ψn+1,i,j= ψn,i,j+ ∆τ∇2???
r +?1 + ∇2?2?
where the Laplacian operator ∇2represents the lattice second order derivatives. A so-called spherical Laplacian approximation
 was used for ∇2. For a thorough discussion on solving solving differential equations numerically using stencils, see
Eq. (17) was solved on a 512 × 512 grid with the spatial discretization dx = 1 and time step ∆τ = 0.02 using periodic
boundary conditions. The parameters r and¯ψ were chosen to correspond to a crystalline region of phase space, i.e., r = −1/4
and¯ψ = −1/4. However, a hexagonal lattice cannot be fitted in a square geometry and in order to satisfy the periodic boundary
conditions, the lattice will be distorted and it may exhibit domains of different orientations separated by walls, thereby giving
a free-energy density higher than that of the ground state. The value of dx also influences the free-energy density when lattice
constant is small. We havecheckedthe finite-size effects for ourlattice in the absenceof pinningpotential, the relative difference
between free-energy density Fn/S and the OMA is only about 0.56%.
B.Phase Diagram and Ground State Configurations
The free energy density and the structure factor S(|?k|) = S(k) = |˜ψ(|?k|)|2were calculated numerically for several different
values of 2π/asas allowed by the periodic boundary conditions. In Fig. 2 a set of configurations for the model with increasing
amplitude of the pinning potential V0are shown. As expected for Vp= 0 the groundstate has perfect triangular symmetry. With
increasing amplitude of the square pinning potential the configurations become spatially distorted and eventually the system
undergoes a transition to a square lattice.
We define the position of the CI transition by analyzing the structure factor S(k). The transition occurs when the peaks of the
structure factor that correspond to the square lattice begin to split as shown in Figs 3 and 4.
Next the approximate analytic solutions for the free energy are compared to the numerical ones as a function of pinning
potential for fixed mismatch. As seen in Fig. 5 the free-energydensity decreases as a function of the pinning strength. For small
values of δmthe transition to square symmetry occurs at relatively weak pinning strengths and is first-order. For larger values
of δmthe free energy decreases slowly and the CI transition appears continuous (see Fig. 5(a)). However, we have checked
the finite-size dependence up to systems of linear size 1024 and find that for up to at least δm = 0.52 the transition remains
first-order. We note that a first-order CI transition is in fact consistent with theoretical predictions based on models of interacting
domain walls on a pinning potential with square symmetry . In Fig. 5 we can also see that the analytical HSMA gives
reasonable agreement with the numerical results, especially for large misfit where domain structures are not as predominant.
(a) V0= 0.00
(b) V0= 0.02
(c) V0= 0.04
(d) V0= 0.05
FIG.2: Snapshots of ground statesand the corresponding structure factors (lower figures) showing thetransition of the system froma triangular
lattice for zero pinning to a square lattice for δm = 0.14 (only a small part of the lattice is shown here).
(a) V0= 0.04440
(b) V0= 0.04455
FIG. 3: Splitting of the peaks of the structure factor with decreasing pinning potential in the vicinity of the CI transition for δm = 0.14. The
splitting is discontinuous, suggesting a first-order like transition.
Finally, we determined the onset of the modulated (M) phase within the I phase by the value of V0where the peaks in the
structure factor correspondingto the triangular symmetry split in multiple peaks. In the present model the M phase corresponds
to regions commensurate with the pinning potential, but separated by heavy domains walls of excess of local density (see also
Fig. 7). The dashed lines indicates the threshold value where domain walls appear in the system, shown in Figs. 1 and 5(a)-(c)
(in the insets) by dashed lines.
(a) V0= 0.4275
(b) V0= 0.4455
FIG. 4: Splitting of the peaks of the structure factor with decreasing pinning potential in the vicinity of the CI transition for δm = 0.52. The
splitting is much smaller than in Fig. 3 but remains discontinuous. See text for details.
(a) δm= 0.52
(b) δm= 0.14
(c) δm= −0.01
FIG. 5: Comparison of the free-energy densities between analytical (HSMA) and numerical results for different values of δm. The inset
represents intensity of the peaks of the structure factor corresponding to the reciprocal vectors of magnitude k = 2π/as. The continuous
vertical lines represent the position of the CI transition, while the dashed lines marks the cross-over regime in the I phase.
C.Voronoi Analysis of Domain Walls
It is of further interest to analyze the nature of the spatial configurations and domain walls in the I phase and in the transition
region. A convenient way to quantify the defects is to use the Voronoi analysis [21, 22]. By definition, the Voronoi cell of any
particle contains all the points that are closer to it than any other particle, i.e. it defines the Wigner-Seitz cell for each particle.
FIG. 6: Relative contributions of the Voronoi polygons with k NNs (pk, k = 4,5,6,7) as a function of the pinning strength, with δm = 0.14.
The continuous line represents the CI transition, while the the dashed line marks cross-over regime within the I phase.
The resulting cells are polygons with N sides that represents the number of nearest neighbors (NNs). In the present case this
is particularly useful since very close to the transition region there is some variation in the positions and sizes of the density
maxima, identified as effective ”particles” in our phase field crystal model. In Fig. 6 we show the results of Voronoi analysis of
our data as a function the pinning strength for a fixed mismatch δm= 0.14. For weak pinning, Voronoi cells with N=6 dominate
as expected. Closer to the CI transition line (shown with a vertical line) contributions from N=5 becomes important and in the
immediate vicinity of the CI transition virtually all Voronoi cells have N=4. The Voronoi cell analysis can be used to identify
the location and nature of the domain walls. Close to the CI transition, the system is composed of regions commensurate with
the pinning potential but separated by heavy walls. These configurations appear spontaneously and have been observed also
in Monte Carlo simulations of overlayers of atoms interacting via the Lennard-Jones potential adsorbed on the (100) face of
an fcc crystal . In Fig. 7 we show results for the present model in the corresponding region. The density of maxima is
smoothed with a gaussian function with width of approximately 3as. As can be seen in Fig. 7(a), the system indeed comprises
commensurate regions separated by walls. The regions of high density indicate ”heavy” domain walls with excess of ”particles”
comparedto the commensurateregions. In these regionsthe ”particles” are plottedwith blackcircles while in the commensurate
regions with white circles. In Fig. 7(b) we show the contribution of different Voronoi polygons in this state. The fraction of
polygons with N=5 (p5) is 0.53, while the fraction with N=4 (p4) and N=6 (p6) are 0.16 and 0.29, respectively, which agrees
well with results in Ref. 18.
FIG. 7: (a) Example of modulated phase obtained from system characterized by δm = 0.14,V0 = 0.043. The dark areas represent the I
regions, i.e. heavy walls, while the light areas are C regions. The positions of the ’particles’ are superimposed on the density maxima. (b) The
contribution of Voronoi polygons in this state. See text for details.
V.DISCUSSION AND CONCLUSIONS
In this work we have considered the recently developed crystal phase field model  in the presence of an external pinning
potential. As themodelnaturallyincorporatesbothelastic andplastic deformations,it providesa continuumdescriptionoflattice
systems such as adsorbedatomiclayers or2D vortexlattices, while still retainingthediscrete lattice symmetryof thesolid phase.
The main advantage of the model as compared to traditional approaches is that despite retaining spatial resolution on an atomic
scale its temporal evolution naturally follows diffusive time scales. Thus the numerical simulations studies of the dynamics
of the systems such as approach to equilibrium can be achieved over realistic time scales many order of magnitudes over the
microscopic atomic models. We have taken advantage of this and considered the full phase diagram of the model as a function
of the lattice mismatch and pinning strength, both analytically and numerically. A systematic mode expansion analysis has been
used to determine the ground states and the commensurate-incommensurate transitions in the model. Numerical minimization
of the corresponding free energy shows good agreement with the analytical predictions and provides details on the topological
defects in the transition region. In particular, we find that the transition remains discontinuous for all values of the mismatch
studied here. We have also performed a detailed Voronoi analysis of the domain walls throughout the transition region. Our
results are consistent with simulations for atomistic models of pinned overlayers on surfaces.
A particularly interesting application of the present model is to pinned systems which are driven by external force. Examples
of such systems include driven adsorbed monolayers , driven charge density waves  and driven flux lattices . Work
in these problems is already in progress.
This work has been supported in part by the Academy of Finland through its Center of Excellence grant for the COMP CoE.
M.K. has been supported by NSERC of Canada. E.G. has been supported by Fundac ¸˜ ao de Amparo ` a Pesquisa do Estado de S˜ ao
Paulo - FAPESP (grant no. 03/00541-0). K.R.E. acknowledges support from the National Science Foundation under Grant No.
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