Network morphology

An important class of bicontinuous structures studied extensively in the mathematics community are periodic minimal (surface-area minimizing) surfaces (e.g., Schwartz and Shoen’s G surfaces).^{Reference Brakke19} Given the large number of periodic surfaces, it is important to determine if the dealloying or subsequent coarsening process yields a bicontinuous two-phase structure similar to these other types of bicontinuous structures.

The morphology of a surface is linked to the principal radii of curvature of a small patch of interface, as in Figure 2a. The inverse of the principal radii of curvatures are the curvatures κ₁ = 1/R ₁, κ₂ = 1/R ₂. The shape of the patch can be described by the principal radii of curvature or combinations thereof. A particularly advantageous combination is the Gaussian curvature, K, and mean curvature, H:

(1)$$H = {1 \over 2}\left( {{\kappa _1} + {\kappa _2}} \right),$$

(2)$$K = {\kappa _1}{\kappa _2}.$$

Figure 2.

(a) A patch of interface with normal n, and radii of curvature R ₁ and R ₂ at a point p. In this case, both curvatures are positive, and the normal points from the gold into the pore; if both curvatures are negative, the normal points from the pore into the gold. (b) Interfacial shape distribution of nanoporous gold; five rectangular regions correspond to particular curvatures shown in the same colors in (c–e). (c) Convex (red) and concave (blue) regions; the circle indicates fission or pinch-off of a ligament into two knobs. (d) Regions with cylindrical patches (yellow and green). (e) Regions with zero mean curvature (purple).^{Reference Chen, Chu, Yi, McNulty, Shen, Voorhees and Dunand12}

The mean and Gaussian curvatures are invariants of the curvature tensor. The mean curvature can be directly connected to the chemical potential of gold atoms of the interface through the Gibbs–Thomson–Freundlich equation:^{Reference DeHoff20}

(3)$${\rm{\mu }} = {{\rm{\mu }}_{\rm{o}}} + 2{V_m}{\rm{\gamma }}H,$$

where μ is the chemical potential, μ_o is the chemical potential at a flat interface, V _m is the molar volume, and γ is the surface energy. The importance of the Gaussian curvature is now clear, since H = 0 for both a flat interface and a hyperbolic or saddle-shaped surface, where κ₁= −κ₂. For the flat surface, K = 0, and for the saddle-shaped surface, $K = - \kappa _1^2$. A surface with one of the principal curvatures equal to 0 is cylindrical, and if κ₁ = κ₂ > 0 the patch is spherical. If, unlike in Figure 2a, both curvatures are negative, the normal points from the pore into the gold. An alternate combination of principal curvatures is helpful if one is interested in just the morphology of the surface and not the connection to thermodynamics.^{Reference Koenderink and Vandoorn21–Reference Shahani, Xiao, Skinner, Peters and Voorhees24}

Using this description of a patch, or small region of interface, it is possible to construct the interface shape distribution (ISD), which is the probability of finding a patch of the interface with a given (κ₁,κ₂) or (H, K).^{Reference Mendoza, Savin, Thornton and Voorhees25} Specifically, I(κ₁,κ₂) is the fraction of interfacial area in the structure with the pair of curvatures κ₁,κ₂. This is directly related to the particle-size-distribution function for an array of spherical particles. In this case, only one parameter R = H is needed, since K is fixed, see Equations 1 and 2. The ISD for nanoporous gold has been measured by a number of groups.^{Reference Chen, Chu, Yi, McNulty, Shen, Voorhees and Dunand12,Reference Chen-Wiegart, Wang, Chu, Liu, McNulty, Voorhees and Dunand13,Reference Mangipudi, Epler and Volkert26–Reference Ziehmer, Hu, Wang and Lilleodden28} Shown in Figure 2b–e is an example for a nanoporous gold sample with a 50% volume fraction of gold. The peak of the ISD is along the dotted line H = 0 (Figure 2b), and the majority of the surface area is composed of patches that are saddle shaped. Larger magnitudes in H are parallel to this line, displaced to the right or left.

It is of interest to know whether the interface has an anisotropic morphology. This is straightforward to assess once the 3D shape of an object has been measured experimentally, since the orientation of the interface follows from the normals of the polygons that make up the mesh used to visualize the object. The interfacial normal distribution (IND) is the probability of finding a normal with a given orientation.^{Reference Kammer and Voorhees29,Reference Rohrer, Saylor, El Dasher, Adams, Rollett and Wynblatt30} While this provides the orientation distribution in the laboratory frame, it is possible to relate the normals measured in the laboratory frame to the crystallographic frame, if we assume that the NPG is a single crystal.^{Reference Chen-Wiegart, Wang, Chu, Liu, McNulty, Voorhees and Dunand13,Reference Rohrer, Saylor, El Dasher, Adams, Rollett and Wynblatt30} Since the temperatures at which the NPG structure are created are below the roughening temperature of gold,^{Reference Barnard31} the NPG interface can display facets.^{Reference Chen-Wiegart, Wang, Chu, Liu, McNulty, Voorhees and Dunand13}

Topology

Topology is a morphology-independent measure of the shape of an object. In three dimensions, the surface average of the mean curvature has units of length, and thus can be dependent on the size scale of the object. However, the surface integral of the Gaussian curvature is length-independent, and thus is a simple number. This number is directly related to the genus, g, of the object by the Gauss–Bonnet theorem:^{Reference Pressley32}

(4)$$g = 1 - {1 \over {4{\rm{\pi }}}}\mathop \smallint \limits_S^{} Kda,$$

where S is the surface. For a sphere K =1/R ², and thus g = 0. For a sphere with an internal pore, however, g = −1.

It is possible to generalize our description beyond spheres to show that the genus is defined as the number of cuts that can be made upon a closed surface without separating it into two disconnected bodies. Thus, a cube, a sphere, and a pyramid are all topologically equivalent (g = 0) while a cube with a handle and a doughnut are equivalent (g = 1). Given the difficulty in identifying internal pores in nanoporous gold,^{Reference Erlebacher33} we ignore them and thus have equivalent terms, genus, and connectivity.

In a bicontinuous structure, topological singularities can occur, for example, interfacial energy driven pinching of ligaments, which decrease the genus. These singularities typically occur on time scales that are much shorter than the overall evolution of the structure itself, as a result of the divergence of the mean curvature and hence chemical potential at a pinching event.^{Reference Wong, Miksis, Voorhees and Davis34,Reference Aagesen, Johnson, Fife, Voorhees, Miksis, Poulsen, Lauridsen, Marone and Stampanoni35} At the very smallest scales just before pinch-off, other issues may be important, such as plastic flow of the ligament.^{Reference Kolluri and Demkowicz36} The genus of 3D bicontinuous structures with no internal pores corresponds to the first Betti number (β₁) used in homology. The genus can be measured using the Gauss–Bonnet theorem, or through a polygonized mesh that is used to visualize the structure, since the genus is related to the Euler characteristic χ = 2(1 – g) = n – e + f, where n is the number of nodes, e is the number of edges, and f is the number of faces. This implies that the genus or connectivity can be calculated for faceted surfaces as well.

Internal microstructure

Having covered descriptions of network morphology and topology, both of which can be described as structural geometry characteristics, we now discuss the internal crystallographic structure of the NPG ligaments. While the microstructure, considered here to describe the internal structure of a material such as crystal orientation and defect distribution, is often the key structural characteristic of interest for bulk crystalline solids, it is often ignored in describing the structure of NPG. This is largely due to the assumption, supported in part by orientation imaging analyses,^{Reference Volkert, Lilleodden, Kramer and Wiessmüller8–Reference Jin, Kurmanaeva, Schmauch, Rösner, Ivanisenko and Weissmüller10} that the NPG structure preserves the underlying grain structure of its parent bulk alloy.

Given that the length scales of the ligament structure vary from nanometers up to a micron, whereas the parent alloys typically display a grain size of tens of microns, the RVEs of NPG are often considered single crystalline. Several groups have investigated the local misorientations induced in NPG via dealloying,^{Reference Petegem, Brandstetter, Maass, Hodge, El-Dasher, Biener and van Swygenhoven9} annealing,^{Reference Sun, Burger and Balk37} or plastic deformation.^{Reference Jin, Kurmanaeva, Schmauch, Rösner, Ivanisenko and Weissmüller10} While measurements of misorientations due to residual dislocations and elastic strains may be difficult to achieve with EBSD due to the lack of sufficient angular resolution, high-resolution XRD measurements have revealed small misorientations during dealloying,^{Reference Petegem, Brandstetter, Maass, Hodge, El-Dasher, Biener and van Swygenhoven9} as shown in Figure 3c. At significant plastic strains mosaic-like structures result while preserving the initial grain structure,^{Reference Jin, Kurmanaeva, Schmauch, Rösner, Ivanisenko and Weissmüller10} shown in Figure 3d–f, but annealing of thin-film NPG structures has led to significant reorientations, resulting in new grain boundaries.^{Reference Sun, Burger and Balk37} Clearly, consensus on the evolution of local crystallographic orientations in nanoporous metals is yet to be achieved.

Figure 3.

(a–b) Electron backscatter diffraction (EBSD) inverse pole figure maps (1100 × 900 µm², step size 2.5 µm) showing the grain structure and orientation (a) before and (b) after dealloying. (c) (311) diffraction peak as a function of dealloying time. The maximum intensity is normalized to 1. (d–f) EBSD inverse pole figure maps of a magnified region within a grain of the parent alloy of a similar NPG sample (d) prior to deformation, (e) at 15% compressive strain (ε), and (f) at 65% compressive strain, showing the development of a deformation-induced mosaic structure. (a–c) Reprinted with permission from Reference Petegem, Brandstetter, Maass, Hodge, El-Dasher, Biener and van SwygenhovenReference 9. © 2009 American Chemical Society. (d–e) Reprinted with permission from Reference Jin, Kurmanaeva, Schmauch, Rösner, Ivanisenko and WeissmüllerReference 10. © 2009 Elsevier.

Self-similarity? Limitations and consensus

An interesting aspect of nanoporous metals is their ability to coarsen while maintaining their bicontinuous structure. The consensus is that coarsening occurs via surface diffusion. A more contentious debate, however, is whether the coarsening is self-similar, the implications of which will be discussed later. A structure is self-similar when the evolution of the morphology and topology can be made time-independent under the scaling of a time-dependent quantity, such as the inverse of the surface area per volume, $S_v^{ - 1}$, or the average ligament diameter, 〈D〉_lig. During coarsening, both quantities are monotonically increasing functions of time. It is possible to show that for a system of coarsening spherical particles in the limit, t → ∞, where t is time, any reasonable initial particle-size distribution will assume a unique time-independent form when the particle sizes are scaled by the time-dependent average particle size.^{Reference Lifshitz and Slyozov38} In this self-similar limit, the average particle size is given by a temporal power law.

Experiments have shown that the time required to reach the self-similar regime can be considerable, in some cases, requiring more than a factor of five change in the length scale of the system.^{Reference Alkemper, Snyder, Akaiwa and Voorhees39,Reference Snyder, Alkemper and Voorhees40} Simulations indicate that the average particle size can actually decrease in the transient regime and require greater than a factor of ten change in the average particle size to reach self-similarity when starting with a distribution that is quite different from the self-similar form.^{Reference Chen and Voorhees41} Nanoporous metals can be created via dealloying using a variety of temperatures, compositions, and electrochemical conditions. This leads to a variety of possible structures, most of which are unlikely to be immediately formed in the self-similar regime. This makes it challenging to compare results from different experiments simply by scaling results by a time-dependent length, since it is first necessary to determine that the self-similar regime has been attained.

In terms of the ISD, the self-similar regime is reached when the probability of finding a patch of interface of a given value of κ₁,κ₂, can be written as $P\left( {{\kappa _1},{\kappa _2},t} \right) = A(t)P({\tilde{\kappa }_1},{\tilde{\kappa }_2})$, where A(t) = kt ⁿ, k is a constant, ${\tilde{\kappa }_i} = {\kappa _i}B(t)$, with B(t) being a time-dependent length such as $S_v^{ - 1}$ or 〈D〉_lig; these length scales increase as a temporal power law in the self-similar regime. However, as we discussed for identifying the RVE of a structure, a rigorous classification of self-similarity is difficult; it is possible for the system to not be self-similar, but still display well-defined temporal power laws for the characteristic length scale of the system.^{Reference Kammer and Voorhees29,Reference Marsh and Glicksman42,Reference Mendoza, Alkemper and Voorhees43}

Various groups have assessed the self-similarity of NPG with ISDs. Early work by Y. Chen using x-ray nanotomography found a lack of self-similarity,^{Reference Chen-Wiegart, Wang, Chu, Liu, McNulty, Voorhees and Dunand13} but the small volumes of material evaluated in this study limit the validity of such a conclusion. More recent work shows that NPG is nearly self-similar,^{Reference Hu, Ziehmer, Wang and Lilleodden16,Reference Ziehmer, Hu, Wang and Lilleodden28} as demonstrated by the 3D reconstructions for the as-dealloyed and coarsened structures shown in Figure 1, and the evolution of principal curvatures shown in Figure 4. The hallmark of these experiments is the large sample size and the significant change in length scale—greater than a factor of 10—achieved during coarsening. The sample used in the study had a volume fraction of gold of approximately 30%, as compared to the ISD shown in Figure 2 for 50% volume fraction gold. The location of the peak implies that most of the interfacial area of the sample has cylindrical shapes and that there is a considerable surface area with patches that have a nearly spherical shape. These patches are often the consequence of pinching events, where an hourglass-shaped gold ligament splits and yields two spherical caps. The ISDs shown in Figure 4 are qualitatively similar to those shown by others,^{Reference Mangipudi, Epler and Volkert26,Reference Erlebacher and McCue27} also using similar gold volume fraction samples.

Figure 4.

Interfacial shape distributions for a NPG sample undergoing coarsening at 300°C. The 2D probability density contour plots of the scaled (κ₁, κ₂) distribution core regions of five samples show some evolution from the very early to the later stages, but are otherwise almost symmetrical around κ₁ = 0. Here, the curvatures are scaled by the cube of the mean diameter. The 〈D〉_lig = 208 nm sample appears to be an outlier.^{Reference Ziehmer, Hu, Wang and Lilleodden28}

The morphology of NPG is different from that of other bicontinuous structures. By comparing the ISD formed during coarsening by bulk diffusion following spinodal decomposition,^{Reference Kwon, Thornton and Voorhees44} with approximately the same volume fraction of gold, the 30% gold NPG structure shows much more interfacial area in the region κ₁ > 0, κ₂ > 0 than the spinodal structure displaying many more saddle-shaped surfaces, where κ₁ > 0, κ₂ > 0. The reason for this discrepancy is an active area of investigation.^{Reference Andrews and Thornton45}

Nevertheless, a careful examination of the ISDs in Figure 4 show that they are not completely time-independent, and thus NPG is not rigorously self-similar. The shape of the ISDs shows broadening of κ₁ and sharpening of κ₂. A physical interpretation of these curvatures can be made, correlating their inverse values to structural length scales of the NPG network. The radius of curvature, R ₁ = 1/ κ₁ correlates extremely well to the average ligament radius, while the larger radius of curvature R ₂ = 1/ κ₂ might quantify the radii of interconnected tori serving as the functional structure of the network structure.^{Reference Ziehmer, Hu, Wang and Lilleodden28} Such a skeleton can be readily identified from skeletonizing the 3D tomograph.^{Reference Hu, Ziehmer, Wang and Lilleodden16}

Self-similarity of bicontinuous structures can alternatively be assessed via the genus (or connectivity). Since the magnitude of the genus is a function of the sample size, it is necessary to normalize its value by the volume of the region considered, g _v. During coarsening, the size scale of the system increases and thus the genus decays with time through topological singularities.^{Reference Erlebacher and McCue27,Reference Mendoza, Thornton, Savin and Voorhees46} In the self-similar regime, the decreasing genus can be made time independent by scaling with the cube of a characteristic length ${\tilde{g}_v} = {g_v}S_v^{ - 3}$ or in terms of connectivity and ligament diameter: ${\tilde{C}_v} = {C_v}D_{lig}^3$. The scaled genus of the bicontinuous structure found following volume diffusion or interfacial mobility-limited coarsening in the self-similar limit has nearly the same values of ${\tilde{g}_v} \approx 0.14$. Experimentally, ${\tilde{C}_v}$ was found to be approximately 0.11.^{Reference Hu, Ziehmer, Wang and Lilleodden16} The importance of the characteristic length scale for the normalization, and the associated challenges with experimentally determining them, is still under debate. Furthermore, the relationship between the shape of the ISD, the diffusion path responsible for the evolution of the structure, and the value of the scaled genus during coarsening remains an active area of research.