Christopher J. Kimmer, Ph.D.

IU Southeast Informatics

Minimal Surfaces

The relationship between quasiperiodic structures and tilings arises from the idea that local structure in a tiling reflects the local interatomic interactions. This same statement can be made for a periodic tiling or lattice, too, but in periodic systems, the infinity of points possessing identical environments is essentially reflected in the fact that there is one unit cell that is tiling space in only one way in the crystal. For periodic systems, the constraints imposed by translational and point group symmetries restrict the structure to be completely expressed by its content in one fundamental domain or irreducible region of the symmetry group. These ideas carry over verbatim and are highly effective in the higher-dimensional periodic space used to model quasicrystals, but in physical space the fundamental domain is infinite and every point sits in a unique environment. The quasiperiodic tiling picture is blind to these distinctions between environments, but a quasiperiodic surface could represent the symmetry of a quasiperiodic structure and still maintain this fundamental distinction.

The use of surfaces as models in physics is ubiquitous; surfaces naturally represent membranes, films, or more generally interfaces where the thickness is small compared to the pore size. In the study of fluid membranes[ 16 ], self-assembled amphiphilic systems[ 17 ], and water-hydrocarbon interfaces[ 18 ] the interfaces themselves may be treated as a surface or as a curtain of atoms draped over both sides of the surface[ 19 ]. More abstractly, surfaces can be considered for their role as space partitioners and used to model phase separations, surfaces of constant energy[ 20 , 21 ], or bonding networks[ 22 ], to name but a few applications. In all of these areas, the study of the surfaces ultimately becomes the study of the system's geometry as a function of its state.

Physically, the study of the geometry of a surface or interface begins by relating the surface's energetics to its geometry. To first order, the energetics accompanying the creation of an interface involve overcoming an energy penalty proportional to area, the constant of proportionality being the surface tension, γ ; equilibrium interfaces minimize the energy penalty by minimizing their area. A small contour around a point on such an interface would delimit a surface patch that minimizes area for that boundary, and surfaces that locally minimize area in this way are known as minimal surfaces[ 23 ].

Not all mimimal surfaces are stable equilibrium interfaces, for the area of some minimal surfaces is a maximum with respect to certain deformation modes. To define a surface patch bounded by a contour as having minimal area, the surface patch must be a critical point of the area functional with respect to local deformations of the surface. These deformations can be parameterized by considering normal displacements of the surface, where the surface patch is displaced parallel to its normal vector. After deformation, the variation in area of a surface patch of area dA depends on the curvature of the surface. The leading correction to dA is proportional to the mean curvature H , defined at a point on the surface as the average curvature of all curves passing through the point and contained in a plane containing the normal vector. If H = 0 everywhere, the variation of the area functional is zero.

Alternatively, H is defined as the average of the principal curvatures, k ₁ and k ₂ , which are the maximum and minimum curvatures, respectively, of all curves through a point as defined above. Zero mean curvature then implies the principal curvatures are equal and opposite, and the local form for a minimal surface can now be seen. For a point on a minimal surface, the coordinates can always be chosen such that the x and y axes are the directions of the principal curvatures and z lies along the surface normal. Consequently, z can be written as a function of the x and y coordinates: ( x , y , z ) = f ( x , y ) . Since the mean curvature is zero, f ( x , y ) is expanded about a point on the surface ( x ₀ , y ₀ , z ₀ ) = f ( x ₀ , y ₀ ) as f ( x , y ) = z ₀ + k ₁ ( x ² - y ² ) . This is simply the equation of a saddle surface with ( x ₀ , y ₀ , z ₀ ) the saddle point; minimal surfaces can be characterized as the class of surfaces where every point's environment is locally a ``balanced'' saddle.

The equal and opposite principal curvatures reflect an essential symmetry of a minimal surface: locally it looks the same from either side. However, not all interfaces share this property. If there is a pressure difference between the two sides of an interface, as in a soap bubble, this symmetry is not present, and the interface is not minimal; the net force on a surface patch has led to the surface's curving. The relationship between the pressure difference across an interface and its bending is given by the Laplace-Young equation

This equation relates changes in volume (conjugate to the pressure) on each side of an interface, to its change in area (with leading correction proportional to H ) and states that the effect of these changes is seen in the curvature of the surface. Consequently, equilibrium interfaces are surfaces of constant mean curvature, and the additional constraint that the separated phases have the same pressure leads to interfaces with zero mean curvature.

The physical relevance of minimal surfaces is then that they separate homogeneous phases at the same pressure. For instance, minimal surfaces model lipid bilayers in solution[ 24 ]. The rich phase diagrams of these materials contain lamellar phases where regions of bulk water are separated by planes of bilayers. Of more striking visual appeal are the bicontinuous cubic phases which have lattice translational symmetry where two congruent regions of water are separated by the membrane. Together, the two labryinths fill space, and each region of water forms a labyrinth that may be traversed in its entirety without crossing the interface. The congruence of the two labyrinths implies that each occupies half of space, and minimal surfaces that are periodic and contain two congruent labyrinths that together fill space are known as balance periodic minimal surfaces (BPMSs).

The first periodic minimal surfaces were found over a century ago, and in the last few decades, they have been the focus of intense research efforts by both physicists and mathematicians[ 22 ]. Periodic minimal surfaces serve as powerful representations of the partitioning of space subject to symmetry constraints, and in the same manner, quasiperiodic minimal surfaces are splendid candidates for the description of quasiperiodic systems.

Studies of this possibility have been carried out by Sheng[ 25 ] and Elser[ 26 ] using a Landau-Ginzburg free energy functional to model an interface between two fluid phases in 6D. Fixing the symmetry so each phase occupies half of space ensures that the equilibrium interface will be a 5D BPMS in the ambient 6D space, and the physical space structure of the surface can be found by cutting 6-space with a 3D plane. A generic surface formed in 3D by a cut from a higher dimensional minimal surface will not be minimal in the 3D subspace, so a limiting procedure is necessary to create an interface that is minimal in physical space and ``stretched out'' in the unphysical perp space dimensions. Sheng studied the limiting process numerically and found the dominant low-order Fourier modes that described the topology of what appeared to be the limiting structure, although the limit is not proven to exist.

The quasiperiodic surface has degrees of freedom associated with moving the cut plane in perp space, and as the cut plane is moved, the surface in physical space undergoes abrupt topology changes analagous to phason jumps in an atomic quasicrystal or tile rearrangements in a quasiperiodic tiling. Associated with these topology changes is a two-dimensional ``critical surface'' in 6D which is the locus of points at which these topology jumps occur. In the limit of infinite stretch along the perp space directions, the critical surface should correspond to the acceptance domain of a tiling associated with the minimal surface. Sheng studied the critical surface and conjectured that it had the outward appearance of a triacontahedron when projected into perp space. Higher-resolution numerical studies allow the limit of infinite stretch in perp space to be approached more closely and also provide more insight into the structure of the critical surface.

Chris Kimmer 2011-06-01

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