Twist grain boundaries in three-dimensional lamellar Turing structures

Abstract

Steady spatial self-organization of three-dimensional chemical reaction-diffusion systems is discussed with the emphasis put on the possible defects that may alter the Turing patterns. It is shown that one of the stable defects of a three-dimensional lamellar Turing structure is a twist grain boundary embedding a Scherk minimal surface.

An ever increasing number of physico-chemical systems in or driven away from equilibrium are found to exhibit spatial symmetry breaking phase transitions or bifurcations leading to periodic patterns resulting from the spatial modulation of some property playing the role of order parameter (1–3). The large majority of nonequilibrium systems gives rise to two-dimensional (2D) patterns and, in absence of particular symmetries, hexagons, or stripes (rolls) are observed. When the aspect ratio of the system (its size measured in terms of the wavelength of the pattern) is large, or the boundaries not congruent, defects such as dislocations, disclinations, foci, grain boundaries, etc. (2, 4) are formed as a result of the underlying rotational symmetry and local biases present during the nucleation of the pattern.

On the contrary, because of their seldom occurrence, three-dimensional (3D) spatial symmetry breaking instabilities remain little studied. A rare example is that of the chemical dissipative structures (5) (so-called Turing patterns, ref. 6) obtained recently with the chlorite-iodode-malonic acid reaction under the influence of diffusive processes (7). Other diffusive instabilities to which the following discussion may be relevant arise in fields ranging from electron-hole plasmas in semiconductors (8), to gas discharges (9), semiconductor devices (p-n junctions, p-i-n diodes) (10), materials irradiated by energetic particles (11) or light (12). Because the wavelength of these chemical patterns is intrinsic, depending only on some reaction rates and diffusion coefficients, the dimensions of the reactor and the concentrations of the fed in reactants may be chosen in such a way that the experimentally emerging patterns are 3D (13). Taking into account the focal depth of the observation lens, the geometry of such patterns is however difficult to resolve precisely because the experimental pictures correspond to projections of the actual 3D structure. Although the “basic” structures that may appear are known from theory (14, 15), as we will recall shortly, they may again be distorted by the presence of defects. In fact the experimental situation is even more difficult as the observed structures arise under nonuniform conditions because of the feeding into the reactor, from two opposite boundary planes, of the reacting chemical species (16, 17). Therefore various structures may coexist side by side as these axial concentration profiles tend to unfold the bifurcation diagram in space (18). The presence of defects may furthermore help the structure adjusting to the spatially varying conditions but also blur the observation. Because of these difficulties the trend with the experiments has been toward using conditions such that the structures are strongly confined in a plane perpendicular to the axial feeding gradients. In this plane, where conditions are now uniform, quasi-2D structures develop with the standard symmetries of hexagons or stripes (16, 19). With the aid of beveled reactors it has recently been shown experimentally that, as one slightly modifies these strong axial constraints, bilayers or multilayers of 2D structures may arise (20, 21).

In this perspective, if one wishes to unravel the structure of a genuine experimental 3D organization, it is clear that all theoretical allowable “perfect” structures, and their defects, must be cataloged and placed on a bifurcation diagram. Help in this direction may be obtained by the direct numerical integration of reaction-diffusion models. Along this line we have obtained distorted lamellar structures in which a so-called minimal isoconcentration surface is contained. The mere existence of such an object opens up new vistas regarding the possible classes of 3D Turing structures. We will further discuss these issues after first recalling the results of the standard theoretical 3D weakly nonlinear analysis and the defects of the structures that may arise.

Pattern Selection

Pattern selection, i.e., the determination for given post bifurcation conditions of the dominating planform, or spatial tessellation, among the possible structures with different symmetries permitted by the rotational invariance, is usually understood in terms of the weakly nonlinear analysis carried out in the vicinity of the spatial symmetry breaking bifurcation point. For instance in 2D the bifurcation scenario often features the subcritical appearance of hexagons when the bifurcation parameter is varied followed supercritically by stripes (22, 23). This standard picture may nevertheless be altered by the presence of other nearby bifurcations (24–26) or by corrections to the weakly nonlinear theory (reentrance of hexagonal phases) (27).

To obtain similar information in a system where 3D patterns may form, one follows an identical approach (23). The concentration vector C obeys the reaction-diffusion equations

where F(C) describes the local reaction dynamics and D is the species diffusion matrix. The concentration field C is expanded as a superposition of the m active modes defining the tessellations the stability of which one wishes to study:

The associated 2 m wavevectors are taken with modulus |k_i| = k_c given by the linear stability analysis of the uniform reference state C_o; A_i = R_ie^iφ_i is the complex amplitude of the active mode with wavevector k_i. The evolution equations of the amplitudes A_i are then obtained using the standard techniques of bifurcation analysis (2) and take the form (neglecting the slow spatial modulations for the time being):

where the couplings g_ND(ij) and w(ijkl) are functions of the angles between the wavevectors of the chosen set of m active modes. The w(ijkl) couplings arise from the subsets of active modes whose wavevectors form noncoplanar closed loops with k_i. The analysis of the relative stability of the different solutions to this set of equations allows the construction of the theoretical bifurcation diagram for the allowable tessellations of the 3D system.

For resonant structures, involving triplet interactions among modes forming equilateral triangles in k-space, the phases of the complex amplitudes of the active modes are not arbitrary: some of their combinations are locked and fixed by the nonlinear dynamics. In 2D the phases of the three active modes defining the resonant structures (m = 3) may sum up either to 0 or π. As a result one can distinguish two types of structures: the so-called h0 or hπ hexagons corresponding, respectively, to a triangular or honeycomb lattice for the maxima of the concentration of one component of the concentration fields (27). The other components are then either in phase or phase opposition.

This also solves the phase problem for the m = 3 3D structures that correspond to hexagonally packed cylinders (hpc). Indeed we may have either hpc0 or hpcπ as they are the respective 3D extensions of the 2D triangular and honeycomb lattices. Similarly, one can show that only two out of all structures characterized by six pairs of wavevectors (m = 6) are dynamically stable with respect to phase perturbations. There the wavevectors form the backbone of an octahedron with each wavevector partaking in two equilateral triangles in k-space. In one type the maxima of concentrations then order on a body-centered cubic (bcc) lattice while the minima organize along a filamental structure with cubic symmetry (bcc0); in the other type the role of the maxima and minima are interchanged (bccπ) (see Fig. 1). We have obtained both, and no other, in numerical integration of a chemical reaction-diffusion model such as the Brusselator.

Figure 1.

3D Turing bccπ pattern obtained by numerically integrating the reaction-diffusion Brusselator model: ∂X/∂t = A − (B + 1)X + X²Y + D_X▿²X, ∂Y/∂t = BX − X²Y + D_Y▿²Y in a cube of size l_x = l_y = l_z = 40 with periodic boundary conditions along the three axes. The Turing instability occurs when B = B_c^T = [1 + A(D_X/D_Y)^1/2]². The parameters are A = 4.5, D_X = 2, D_Y = 16, B = 6.9 (B_c^T = 6.71). Isoconcentration surfaces of the variable X are looked at along the [111] axis of the cube in which the equations are discretized. (A) The spheres of lower isoconcentrations (as an example here X = 2.737) are organized with the bcc symmetry. (B) The higher isoconcentrations (here X = 5.153) fill in the interspace between the lower isoconcentrations bcc.

On the theoretical bifurcation diagram one then has the successive appearance, subcritically, of the two resonant structures, i.e., bcc and hpc, followed supercritically by lamellae (lam). The latter is naturally also the 3D extension of the 2D stripes (m = 1). All other structures may be shown to be unstable in the weakly nonlinear approximation. The existence in succession of the theoretically predicted bcc, hpc, and lam patterns generated by a Turing instability was corroborated by the results of our simulations (15). Here we go further by exhibiting the first numerically obtained bifurcation diagram for the chosen reaction-diffusion model in 3D (Fig. 2). This succession of structures is analogous to that observed at equilibrium in amphiphilic systems (28) or diblock copolymers melts (29).

Figure 2.

Numerical bifurcation diagram for the amplitude of the concentration X of the Brusselator model with A = 4.5, D_X = 2, D_Y = 16 in the system with size l_x = l_y = l_z = 40. The amplitude is here defined as the difference between the maximum and minimum value of the variable X in the system.

Twist Grain Boundary (TGB)

Distortions from perfect patterns and defects are usually discussed in the framework of phase diffusion equations (2, 4). Such equations for the resonant 3D patterns have recently been derived and analyzed (30). There it was shown that the resonance condition causes confinement of dislocations. Whereas the nature of the defects of the bcc structures have been classified for a long time by solid state physicists, those of hpc patterns have only been studied more recently in the context of discotic liquid crystalline phases (31).

Here however we wish to concentrate on the nature of a particular defect appearing in frustrated lamellar patterns. The analogy then lies with the defects appearing in smectic A liquid crystals (31) as, in common with the stripes (2D) or lam (3D) generated in driven systems, the breaking of translational symmetry is only in one direction of space (32). Building on this analogy, the relaxation and the distortion of a convective roll pattern (2D) around an edge dislocation have for instance been analyzed in the case of the convective instability of a nematic subjected to an elliptic shear (33).

Consider our numerical work to draw the bifurcation diagram in 3D (Fig. 2) in the region where lam patterns are found: when starting from random initial conditions we often obtain structures exhibiting various perfect lamellar regions oriented orthogonally to one another and linked by boundaries whose thickness is of the order of the wavelength of the lam. This is reminiscent of the kind of TGB found in block copolymers (34) and amphiphilic systems or smectic liquid crystals (35).

In general lam in neighboring domains may meet at any given angle θ. Considering the TGB to have an orientation parallel to that of the x-y plane and, at large distances above and below the TGB, the normals to the two sets of lam, respectively, to be directed along the directions n₊ = (cos α, sin α, 0) and n₋ = (sin α, cos α, 0), then α = π/4 − θ/2. The normal thereby rotates through an angle θ from one side of the TGB to the other. If, as in our case, θ = π/2 and hence α = 0,n₊ is aligned along the x direction while n₋ points along the y direction.

In the polymer context this defect of the lam pattern has been characterized by the existence in the pattern of an embedded surface that approximates the so-called Scherk First surface (36) (Fig. 3). The region of space where the two sets of lam come into contact then consists in a doubly periodic array of saddle-surface regions. Such a Scherk surface belongs to the class of minimal surfaces whose local mean curvature is everywhere identically zero (thereby it also has a negative Gaussian curvature).

Figure 3.

Scherk surface obtained by numerically integrating the reaction-diffusion Brusselator model with A = 4.5, D_X = 2, D_Y = 16, and B = 7.2 in a box of sizes l_x = l_y = 60, l_z = 30. Periodic boundary conditions are applied along x and y while no-flux conditions are imposed in the z direction. To have a better visualization of the saddles region, typical of the Scherk surface, we zoom in on half the system. The representation is then l_x = l_y = l_z = 30 with l_z being on the vertical axis. The drawn isoconcentration surface corresponds to X = 4.5, the concentration of the uniform unstable reference state. The Scherk surface joins two orthogonal sets of three planes, one set being the x = constant planes while the other set is the y = constant planes.

To check the possible existence of such a minimal surface in our reaction-diffusion patterns we have seeded simulations with a small amplitude Scherk surface, using its known analytical expression cos x = e^z cos y, superimposed on the initial random noise. No-flux boundary conditions are chosen along z while periodic boundary conditions apply along the x and y directions to which the TGB will be parallel. One observes that during the numerical integration the amplitude of the Scherk surface grows until it becomes of the order of magnitude of that of the perfect lam and we recover a pattern morphology of the same type as that we had obtained when starting solely from random conditions.

As shown on the bifurcation diagram (Fig. 2) such structures containing one, or even various TGB, are stable in the same range of values of the control parameter as the perfect lam. The reason for this can be apprehended if one recalls that the long scale distortions of lamellar structures can be described by the rotationally covariant phase diffusion equation (37) that derives from a potential V:

In the harmonic approximation and neglecting the compression of the layers, this potential can be written in the form of the Helfrich curvature energy (38), that may also be used to discuss the relative stability of the bcc, hpc, and lam structures,

where H = (c₁ + c₂)/2 is the mean local curvature with c₁ and c₂ the local principal curvatures, whereas G = c₁ c₂ is the local Gaussian curvature. The deformation from the perfect lam can be described in terms of two coefficients, the bending modulus K and the saddle-splay modulus that measures the energy cost of saddlelike deformations.

In the case of lamellar Turing patterns arising in isotropic reaction-diffusion systems, one finds K = ξ_o²/4k_c² and = 0, where ξ_o is the coherence length related to the molecular diffusion coefficients of the model. The zero cost in the modification of the Gaussian curvature allows the formation of saddle-shaped minimal surfaces with zero mean curvature (H = 0) and nonpositive Gaussian curvature (G < 0). If were positive, the value of the potential associated with the saddle-shaped surface would be smaller than that of a perfect lam structure for the same value of all parameters and the twist would form spontaneously (39). However as = 0 the two types of structures have identical domain of stability.

It is known experimentally that besides the bcc, hpc, and lam structures, the equilibrium systems (e.g., block copolymers, amphiphiles, etc.) feature other organizations such as the bicontinuous double diamond, the lamellar catenoid or the gyroid structures involving various kinds of triperiodic minimal surfaces (40). The observation of Scherk surfaces in a reaction-diffusion system undergoing a Turing instability now favors the idea that these other structures may also be expected to arise in out-of-equilibrium chemical systems. This possibility of obtaining additional 3D Turing patterns besides the classical bcc, hpc, and lam was already alluded to in previous works (23, 41). However it results from theoretical work with block copolymers that the study of such patterns goes beyond the weakly nonlinear theory as for instance the bicontinuous double diamond structure, while characterized by the same basic wavevectors as the bcc structure, differs starting with the first harmonic modes (42). Nevertheless if one wishes to include the latter then it has been shown that other more conventional structures as the face centered cubic, simple cubic, square packed cylinders, or hexagonal closed packed structures may also be stabilized in some region of the bifurcation diagram (43). These ideas remain to be vindicated.

Discussion

In this work we report the existence of a novel class of organization in systems driven far from equilibrium, as exemplified by the existence of a Scherk isoconcentration surface in the chemical concentration fields. It may thus become useful to discuss some aspects of order in terms of the properties of characteristic surfaces rather than the position of particular centers on a lattice. Such surface description becomes particularly significant when discussing space symmetry breaking in systems described by continuous fields. This possibility arises because various patches of constant mean (in particular zero) curvature can be joined smoothly together to create doubly or triply periodic surfaces and give rise to “flexocrystals” (44) the structural elements of which are curved sheets rather than punctual elements. It is noteworthy that the full classification of such mathematical object is so far not complete. A remarkable aspect resides in the bicontinuous nature of these patterns. In the double diamond structure for instance, one phase resides in two intertwined but distinct labyrinthine networks, each exhibiting diamond cubic symmetry, while the other phase fills the continuous matrix between the two diamond channels. In such settings the matrix is bissected by a connected triply periodic minimal Schwarz D-surface.

When comparing dissipative structures with those arising in soft matter physics (e.g., amphiphiles and polymers) one expects that the characteristic wavelength will be larger in the former. For instance the wavelengths of the structures observed experimentally in the chlorite-iodide-malonic acid reaction-diffusion systems are of the order of 0.2 mm. As a further object of discussion let us mention that recently 2D-disordered labyrinthine chemical patterns have been obtained experimentally using the ferrocyanide-iodine-sulfite reaction using the same type of gel reactors discussed above (45). There the local chemical kinetics exhibits bistability between uniform steady states. When coupling by diffusion is allowed, fronts between the two states may be created. It has been shown that they may undergo a morphological instability leading to the observed labyrinthine structures. If the same experiment was carried out under conditions such that a 3D pattern results, one may wonder whether they would exhibit any similarity with the so-called sponge phases (46) that consist of randomly connected surfaces. Going further, far from equilibrium evolution is generally of a nonvariational type. The resulting effects may stabilize localized structures where a patch of one structure coexists with another. Another possibility resides in the competition between space symmetry and time symmetry breaking (24). In 3D new spatio-temporal organizations that have no counterpart in equilibrium systems are then likely to become possible.

In conclusion the present work provides another instance in the realm of chemistry where morphologies built around minimal surfaces come into play (47). Therefore, in physico-chemical systems driven out of equilibrium, our vision of 3D self-organization may be enriched by emphasizing isoconcentration surfaces, for which the salient variable is curvature, instead of the more traditional crystalline skeletal lattice of extrema of concentrations. These complementary and mutually supporting views open up a new path in the field of dissipative crystallography that should help in deciphering the experimental 3D textures.

ABBREVIATIONS

2D

two dimensional

3D

three dimensional

TGB

twist grain boundaries

hpc

hexagonally packed cylinders

bcc

body-centered cubic

lam

lamellae