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Instabilities and Self-Organization in MaterialsVolume I: Fundamentals of Nanoscience, Volume II: Applications in Materials Design and Nanotechnology$

Nasr Ghoniem and Daniel Walgraef

Print publication date: 2008

Print ISBN-13: 9780199298686

Published to Oxford Scholarship Online: May 2008

DOI: 10.1093/acprof:oso/9780199298686.001.0001

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Instabilities and Self-Organization in Materials
Oxford University Press

We already discussed the fact that Landau free energy is a powerful tool for the analysis of symmetry-breaking phase transitions. It may be also be applied to the freezing transition of monatomic fluids of spherical particles and should thus mimic the liquid solid transition [1033]. Its minimum should thus determine the selected crystalline structure at given temperature and density.

Consider the Landau expansion of the free energy, near the liquid phase, in terms of density ρ:

F = Φ 0 + Φ 2 + Φ 3 + Φ 4
where Φ0 is the free energy of the liquid phase. The quadratic term is written as

Φ 2 = d 3 q A ( q ) ρ q ρ q
where ρ q are the Fourier components of the density. In isotropic systems A(q) only depends on wavenumber |q|. Furthermore, near the solid–liquid phase transition, A(q) is minimum for some well-defined wavevenumber q c and one may write

A ( q ) = A ( q ) T T c T c + d 0 ( q 2 q c 2 ) 2
The order parameter is thus associated with an irreducible representation of the rotational group described by the sphere of radius q c, and

Φ 2 = A ( q c ) d Ω Q ρ Q ρ Q
where Q are critical wavevectors of length q c. It is thus independent of explicit combinations of ρ Q. The cubic invariant on the sphere may be cast in the following form

Φ 3 = B ( q c , T , P ) d Ω 1 d Ω 2 d Ω 3 δ ( Q 1 + Q 2 + Q 3 ) ρ Q 1 ρ Q 2 ρ Q 3
where the wavevectors Q i have to form equilateral triangles to get nonvanishing contributions. Since ρ Q = ρ Q this requires at least three ρ Q i in the order parameter which may thus be written as


ρ q c ( r ) = 1 2 Σ i = 1 n [ ρ Q i ( r ) + ρ Q i ( r ) ] , n 3
As already discussed in the context of pattern selection in nonequilibrium systems, the simplest case is the triangle, n = 3, end the order parameter is

ρ 3 ( r ) = ± ( 2 3 ) 1 / 2 [ cos q c x + 2 cos q c x 2 cos 3 q c y 2 ]
and | Φ 3 h e x | = 2 B ρ q c 3 / 3 3 . In three-dimensional space, the corresponding structure is rod-like with two-dimensional triangular or honeycomb periodicities.

When n = 6, triangles can be arranged to form octahedrons, and

ρ 6 ( r ) = ± ( 2 3 ) 1 / 2 [ cos q c x 2 cos q c y 2 + cos q c y 2 cos q c z 2 + cos q c z 2 cos q c x 2 ]
and | Φ 3 b c c | = 4 B ρ q c 3 / 3 6 . The corresponding lattice structure in real space is BCC. Note that | Φ 3 b c c | > | Φ 3 h e x | , favoring BCC lattices.

More intricate wavevectors combinations also give nonvanishing contributions to cubic invariants, such as dodecahedrons, giving n = 15. The corresponding spatial structure is based on icosahedrons and has five-fold symmetry. It is thus not periodic but only quasi-periodic. Furthermore | Φ 3 b c c | > | Φ 3 i c o s | , and FCC crystal lattices are only favored by third-order terms.

Other structures, such as FCC crystallattices (n = 4), give no cubic contributions. Their study requires thus the analysis of the next higher-order invariant, Φ4, which may be written as

Φ 4 = Π i = 1 4 d Ω i C ( Q i ) δ ( Σ i Q i ) ρ Q 1 ρ Q 2 ρ Q 3 ρ Q 4
Since the wavevectore must form closed quadrangles to get nonvanishing contributions, C(Q i) may depend on two independent angles between wavevectors on the sphere. As a result, Φ4 may depend on specific materials properties, such as bond angles, packing properties, electronic structure, etc. At large densities, thus far away from the transition, these effects may dominate over universal features and favor other structures than FCC crystal lattices.

The previous discussion leads thus to the following conclusion. In the case of weak first-order transitions, which occur for low densities, and as long as specific forces are not too important, universal, model independent effects favor the formation of BCC crystal lattices. However, when specific forces are dominant, other structures, such as FCC lattices could emerge through a second-order mean field transition. Such a transition implies the existence of a solid–liquid critical point. This is in contradiction with Landau argument saying that such transition may not (p.1085) be continuous since liquid and solid phases have different symmetry groups. However, agreement with Landau theory is recovered when fluctuations are taken into account.

Effectively, for rotationally symmetric liquids, fluctuations dispersion relation, ω q T T c T c + d 0 ( q 2 q c 2 ) 2 , only depends on the length of the wavevector q and not on its orientation. For deviations from critical wavevectors, it has thus a one-dimensional behavior. Indeed, on writing q = q c1x + k (the orientation of the x-axis being arbitrary), one has ω k T T c T c + 4 d 0 q c 2 k x 2 + As shown by Brazovskii [1034], this implies diverging fluctuations which transform the transition from continuous to first-order.

Brazovskii’s argument may be sketched as follows in the simplest case of transitions to one-dimensional periodic structures in d-dimensional systems. The corresponding (Φ2, Φ4) Landau free energy, including noncritical fluctuations, may be written as

F = d d q [ T T c T c + d 0 ( q 2 q c 2 ) 2 ] ρ q ρ q + Π i = 1 4 d d q i δ ( Σ i q i ) ρ q 1 ρ q 2 ρ q 3 ρ q 4
and minimization equations for densities ρ q c , with qc = q c1x, including two-point correlation functions of fluctuations ρ q, with q = qc + k, are

T T c T c ρ q c + d d k < | ρ k | 2 > ρ q c + | ρ q c | 2 ρ q c = 0

< | ρ k | 2 > k T c T T c T c + 3 | ρ q c | 2 + 4 d 0 q c 2 k x 2

d d k < | ρ k | 2 > k T c 2 q c d 0 ( T T c T c + 3 | ρ q c | 2 )
These equations may be solved iteratively and, for T < T c, one has

( T T c T c + Γ T c T T c ) ρ q c + | ρ q c | 2 ρ q c = 0
Hence the transition is shifted downwards at T = T c(1 − Γ2/3) and becomes first order.