Cellular Ceramics / p2
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2.3 Three-Dimensional Periodic Structures 97
(a) |
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(b) |
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Z |
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X |
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1mm |
Y |
100m |
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(c) |
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(d) |
1mm
5mm
Fig. 8 Optical and SEM images of 3D periodic lattices in a)–c) and a radial array in d) comprised of cylindrical rods (ca. 250 mm in diameter) interconnected in all three dimensions
[12]. (Reprinted with permission from Smay, J.E., Cesarano III, J., and Lewis, J.A., Langmuir 2002, 18, 5429–5437. Copyright 2002 American Chemical Society.)
the filament diameter, c = 2D. For the face-centered stacking pattern, the unit cell repeat height is c = 4D. In the lateral directions, the unit cell dimensions a, b are determined by the spacing of filaments within a layer. For the 3D lattices shown in Fig. 8, the unit cell lengths are identical (a = b); however, this is not required. Therefore, both simple and face-centered stacking patterns can be produced with tetragonal (a = b „ c) or orthorhombic (a „ b „ c) unit cells if alternating filament layers are perpendicular. Also note that the filament orientation between layers need not be restricted to 90 rotations.
Within any single layer the solid volume fraction Cp is defined by:
Cp ¼ |
p D |
(8) |
4 L |
with 1–Cp equal to the volume fraction of porosity. The number of pores per inch (ppi) or pores per meter (ppm) is simply the reciprocal of the filament spacing. Given the ease with which the filament spacing can be varied by robotic deposition, the ppi can be different in all three dimensions. The surface area (SA) to volume (V) ratio is another important feature of cellular ceramics, and is given by:
p |
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4CP |
(9) |
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SA=V ¼ |
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¼ |
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L |
D |
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where SA/V is given in units of inverse length (m–1). Note that Eq. (9) neglects the effect of the junction regions created due to rod–rod interfacial bonding. This ratio
98 Part 2 Manufacturing
Lattice |
Unit Cell |
Unit Cell |
Pore Space |
b
Simple
Stacking c
a
b
Face
Centered c
Stacking
a
(a) |
(b) |
Fig. 9 Schematic illustrations of the a) unit cells and b) corresponding pore space for simple (top row) and face-centered (bottom row) stacking patterns.
SA/V increases as the lateral dimension of the unit cell decreases; however, in the limit that L £ D, the pores become isolated and SA/V decreases dramatically.
The 3D periodic structures shown in Fig. 10 illustrate systematic variation of unit cell size. Each structure is comprised of a simple stacking pattern in which the c dimension of the unit cell is held constant. The lateral unit cell dimensions varied from 230 mm (sample a) to 937 mm (sample h). For these samples, the filament di-
ameter is D » 160 mm, such that in the lateral directions, Cp varies from 0.55 to 0.13, ppi varies from 110 in–1 to 27 in–1, and SA/V varies from 136 cm–1 to 33 cm–1. For
the radial array shown in Fig. 8d, the calculation of Cp depends on radial position. In this case, Cp in the radially oriented layer varied linearly from 0.85 at the inner radius to 0.4 at the outer radius.
2.3 Three-Dimensional Periodic Structures 99
a |
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b |
1 mm |
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1 mm |
Fig. 10 Optical images of 3D periodic structures with varying lattice constant, and corresponding higher magnification views of structures a) and h) with respective solids volume fractions of 0.17 and 0.70. (Adopted from Ref. [33].)
2.3.7
Summary
Direct-write assembly processes such as Robocasting and fused deposition provide remarkable control and flexibility in designing cellular ceramics with 3D periodic structures. As demonstrated above, use of colloidal gel-based inks allows one to create regular 3D patterns of varying lattice geometry, lattice constants, and rod diameter. Such 3D structures can be utilized directly or can serve as a constituent in 3D interpenetrating composites (e.g., ceramic–polymeric [33], ceramic–metal [34], or ceramic–ceramic composites) formed by infilling their intervening pore space with a second phase.
100Part 2 Manufacturing
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