Cellular Ceramics / p2

87

2.3

Three-Dimensional Periodic Structures

Jennifer A. Lewis and James E. Smay

2.3.1

Introduction

Three-dimensional (3D) periodic structures comprised of interconnected cylindrical rods may find widespread technological application as advanced ceramics [1, 2], sensors [3, 4], composites [5, 6], tissue engineering scaffolds [7], and photonic materials [8, 9]. Robotic deposition processes are direct-write techniques utilizing the extrusion of particle-filled inks and are capable of assembling such structures with submillimeter precision. Examples include fused deposition [4, 10] and Robocasting [11–13]. In both techniques, the extruded ink forms a continuous filament that is patterned in a layer-by-layer sequence to assemble complex 3D architectures. The primary distinction between these two approaches lies in ink design and postdeposition processing. Robocasting utilizes concentrated colloidal gels as inks [12, 13], whereas fused deposition uses particle-filled, polymeric inks [10]. Process control of and geometries attainable by either technique depend intimately on the dynamic viscoelastic properties of the ink during extrusion and patterning. The principal advantage of using colloidal gel-based inks is that the as-formed structures do not have high organic content and therefore are not subject to the lengthy binderremoval process typically associated with fused deposition. While fused deposition is a viable technique and other methods [14] (e.g., 3D printing [15], stereolithography [16]) may produce similar structures, this chapter focuses on the manufacturing of 3D periodic structures by extrusion-based robotic deposition using colloidal gelbased inks. First, a brief overview of this technique is provided. This is followed by detailed description of the design of colloidal gel-based inks, ink flow during deposition, and shape retention of the as-deposited features. Finally, examples of the types of 3D structures that can be produced by this assembly route are provided.

2.3.2

Direct-Write Assembly

Robotic deposition, illustrated schematically in Fig. 1, employs an ink delivery system mounted on a z-axis motion control stage for agile printing on a moving x–y platform. Coordinated three-axis motion is controlled by a computer program,

Cellular Ceramics: Structure, Manufacturing, Properties and Applications.

Michael Scheffler, Paolo Colombo (Eds.)

Copyright 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ISBN: 3-527-31320-6

88 Part 2 Manufacturing

z-stage

ink reservoirs

nozzle

x-y stage

Fig. 1 Schematic illustrations of a) robotic deposition apparatus and b) ink extrusion through a cylindrical nozzle onto a substrate.

which allows for the design and assembly of complex 3D structures in a layer-by- layer deposition scheme. The ink(s) are housed in multiple reservoirs affixed to the z-axis stage and extruded through a cylindrical nozzle oriented vertically above the x–y stage. The composition of the extruded filament can be varied by mixing various inks (shown schematically in Fig. 1a), or different ink compositions can be deposited discretely from different nozzles. Upon exiting the nozzle, the cylindrical rod (or filament) of ink bends by 90 to lie parallel to the x–y plane, as shown in Fig. 1b. The volumetric flow rate of the ink is proportional to the x–y table speed, which is typically 1–10 mm s–1. A multireservoir ink delivery system facilitates compositional grading along a given rod, between rods in a given layer, or between layers. Fused deposition uses a similar motion-control platform; however, the ink delivery system requires a heated nozzle to liquefy the particle-filled polymer ink during extrusion.

Direct-write assembly techniques allow any conceivable 2D pattern to be printed within a layer. Upon solidification, the layer then serves as a platform for the deposition of subsequent layers to give the desired 3D architecture. In the original conception of Robocasting, concentrated colloidal suspensions were utilized for the construction of space-filling solid components. These inks solidified after deposition by a drying-induced pseudoplastic-to-dilatent transition [11]. Space-filling components benefited from this initial ink fluidity, which allowed for the creation of a continuous structure. To fabricate self-supporting spanning structures, the ink design [12, 13] was changed to utilize the inherent viscoelastic properties of colloidal gels to enable the assembly of 3D periodic structures.

2.3 Three-Dimensional Periodic Structures 89

2.3.3

Colloidal Inks

Colloidal inks developed for direct-write assembly of 3D periodic structures with spanning features must satisfy two important criteria. First, they must exhibit a well-controlled viscoelastic response, that is, they must be able to flow through a deposition nozzle and then “set” immediately to facilitate shape retention of the deposited features even as they span gaps in the underlying layer(s). Second, they must contain high volume fractions of colloid to minimize drying-induced shrinkage after assembly, that is, the particle network must be able to resist compressive stresses arising from capillary tension [17]. These criteria require careful control of colloidal forces to first generate a highly concentrated, stable dispersion followed by inducing a system change (e.g., DpH, ionic strength, or solvent quality) that promotes a fluid-to-gel transition [13].

Colloidal inks are produced by first preparing an aqueous dispersion of colloidal particles (e.g., ceramic, polymer, semiconducting, or metal colloid). Polyelectrolyte species such as poly(acrylic acid), PAA, are typically added as dispersants to provide suspension stability. PAA is a linear polymer that has one ionizable carboxylic acid group per monomer unit. Under appropriate conditions of pH and ionic strength, it is fully ionized (or electrostatically charged), and this allows it to provide the electrosteric stabilization necessary to create highly concentrated colloidal suspensions with a solids loading of about 50 vol %. By altering solution conditions such as decreasing pH or increasing ionic strength, the colloidal stability can be reduced to cause the desired fluid-to-gel transition and yield a concentrated, colloidal gel-based ink, as illustrated schematically in Fig. 2.

∆

fluid

gel

Fig. 2 Schematic illustration of the fluid-to-gel transition that occurs for a polyelectrolyte-stabilized colloidal dispersion

(fluid phase) upon changing pH or increasing ionic strength of the solution to induce attractive interactions between colloidal particles (gel phase).

Colloidal gels consist of a percolating network of attractive particles capable of transmitting stress above a critical volume fraction ugel, known as the gel point. If the colloid volume fraction u within the gelled ink is held constant, the elastic properties of the ink can be controlled by altering the strength of the interparticle attrac-

tions according to the scaling relationship [18] given by:

!x

90 Part 2 Manufacturing

where y is the elastic property of interest (shear yield stress sy or elastic modulus G¢) k is a constant, and x is a scaling exponent (ca. 2.5). The equilibrium mechanical properties of colloidal gels are governed by two parameters: u, which is representative of the interparticle bond density, and ugel, which scales inversely with bond strength. As the interparticle forces are made more attractive, colloidal gels (of constant u) experience a significant and controllable rise in their elastic properties [18–26] (see Fig. 3). In addition to the ability to control equilibrium elastic properties, colloidal gels have a dynamic ability to break down (when the applied shear stress s exceeds sy) and rebuild network structure (under quiescent conditions, s fi 0). This makes colloidal gel-based inks well suited to fulfill the requirements of flow through the deposition nozzle while maintaining the elasticity necessary to promote shape retention.

(a)

(b)

Fig. 3 a) Schematic illustration of attractive particle network that forms upon gelation and b) log–log plot of elastic modulus as a function of shear stress for both a weak and strong colloidal gel. (Adopted from Ref. [13].)

2.3 Three-Dimensional Periodic Structures 91

2.3.4

Ink Flow during Deposition

During direct-write assembly, the concentrated colloidal-gel-based ink must flow through a fine deposition nozzle (ca. 100 mm to 1 mm in diameter) at the volumetric flow rate required to maintain a constant deposition speed v of 1–10 mm s–1. The shear rate profile that the ink experiences during the deposition process depends on the nozzle diameter, the deposition speed, and the rheological properties of the ink [7, 12]. Recently, the flow behavior of viscous Newtonian fluids within a cylindrical deposition nozzle has been modeled by Baer et al. [27]. The computed shear rate profile across the nozzle cross section is shown in Fig. 4 for a 100 Pa s fluid flowing

at rest

(a)

at rest

(b)

Fig. 4 Contour plots illustrating the calculated shear rate profile within the deposition nozzle (arrow denotes end of nozzle) for a viscous Newtonian fluid (100 Pa s viscosity) deposited at a table speed of 5 mm s–1 through cylindrical nozzles of diameter of

a) 250 mm and b) 1.37 mm. (Adopted from Ref. [27].)

92 Part 2 Manufacturing

through nozzles of diameter 0.25 and 1.37 mm at a constant deposition speed of 5 mm s–1. As expected, the fluid experiences a maximum shear rate (or stress) along the nozzle walls. The maximum values observed were approximately 250 s–1 in the 250 mm nozzle and 50 s–1 in the 1.37 mm nozzle under these conditions. In both profiles, it is evident that the observed shear rate (or stress) experienced by the fluid decreases towards the center of the nozzle to a nearly stress-free state. Immediately upon exiting the nozzle, the ink experiences a 90 bend and all regions within the filament undergo some deformation. However, once the ink filament is deposited onto the underlying layer (or substrate), it returns to a quiescent state.

The rheological behavior of colloidal gels is complex, and can be described by the Hershel–Bulkley model [28]:

where n is the shear thinning exponent and K is the viscosity parameter. Ink flows through the cylindrical nozzle when a pressure gradient (DP) is applied in the axial direction producing a radially varying shear stress sr parallel to the nozzle axis given by:

where r is the radial position within the nozzle (i.e., r = 0 at the center axis and r = R at the nozzle wall). The flow rate can be calculated from the constitutive equation (Eq. 2) and the applied stress field given by Eq. (3).

A Hershel–Bulkley material flowing through a fine nozzle may develop a threezone velocity profile consisting of 1) an unyielded core of radius rc moving at constant velocity surrounded by 2) a yielded shell experiencing laminar flow, and, possibly, 3) a particle-depleted slip layer at the nozzle wall [29, 30]. The slip layer is a thin fluid layer with thickness d << R, between the nozzle wall and the bulk ink. The volumetric flow rate Q as a function of applied pressure is found by summing the contributions from slip at the nozzle wall and the integrated velocity profile in the

where ms is the slip velocity, and f(ss) the integrated velocity profile of the core and shell region. The slip velocity is determined by:

where sR is the shear stress at the nozzle wall, ss the shear stress at the slip layer/gel interface, and gc = 30 mPa s. The function f(ss) is given by

2.3 Three-Dimensional Periodic Structures 93

where m= 1/n, n is the Hershel–Bulkley shear thinning exponent, K the viscosity parameter a= (1–sy/ss), and sy is the gel yield stress. These material parameters can be used to calculate a pressure gradient required for a desired flow rate (or deposition speed) using Eqs. (3)–(6), while the shear rate profile within the nozzle is derived from Eqs. (2) and (3). Under the condition of no slip boundary (ms = 0), Eq. (6) reduces to the Buckingham–Reiner relationship for an ideal Bingham fluid or the Hagan–Poiseuille relationship for a Newtonian fluid [31].

The shear rate profiles for a representative colloidal ink were calculated at varying deposition speeds under slip and no-slip boundary conditions by using Eq. (2), where the slip layer thickness was taken as the mean particle diameter. These profiles are plotted in Fig. 5, where the abscissa is the reduced radial position within the nozzle r/R. The no-slip boundary condition (solid lines) led to a wall shear rate of 100–400 s–1 for deposition speeds of 2–8 mm s–1. No core region is apparent on this scale except for the 2 mm s–1 profile, where rc/R » 0.01. Introduction of the slip layer (dashed curves) reduced the wall shear rate to between 0.8 and 5 s–1, with the core region expanding to rc/R » 0.1–0.3 with decreasing deposition speed. These calculations indicate that the no-slip condition leads to high shear rates throughout the extruded filament and thereby alters the ink structure. In contrast, the ink structure experiences significantly less change during extrusion under the slip boundary condition.

Fig. 5 Calculated shear rate profiles for a representative colloidal ink (sy = 100 Pa) in a 200 mm diameter capillary geometry as a function of varying deposition speed. Solid lines assume noslip condition, and dashed lines assume slip boundary of

0.65 mm. (Adopted from Ref. [12].)

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2.3.5

Shape Evolution of Spanning Filaments

To create 3D periodic structures by direct-write assembly, filaments are deposited in the form of a linear array of rods aligned along the x- or y-axis such that their orientation is orthogonal to the underlying layer. During deposition, the extruded filament must span gaps in the underlying layer, as it first anchors to one of the supporting filaments, then traverses the unsupported gap region, and finally anchors to an adjacent supporting filament. Despite the dynamic nature of this process, the spanning filament can be viewed to be reasonably stress-free in the initial state and to deform to the equilibrium shape allowed by the elastic properties of the filamentary elements (or rods) when anchored to the second support [12].

A static, simply supported beam is an idealized model of the spanning phenomena that captures the essential body forces and ink properties, which influence the structural evolution of spanning filaments after deposition. In this model, the spanning filament is represented as a beam of circular cross section that is simply supported at its ends (see Fig. 6b). Such a beam deflects in proportion to the distributed load of its own weight as a function of distance from the supports given by [32]

where W is the distributed load (= 0.25[ eff ]gopD2), eff the effective ink density ( ink– fluid), and fluid the density of the deposition medium (e.g., air or liquid), go the gravitational constant, y the position along the rod, E the Young’s modulus of the gel (E=(1+2m)G¢), m= 0.5 is Poisson’s ratio for the gel [18], and I the area moment of inertia of the circular cross section (=pD4/64). Thus, for a specific span distance, the modulus and/or effective moment of inertia may be chosen as degrees of freedom to capture the elastic behavior of the beam and deflections predicted by Eq. (7).

To demonstrate the influence of ink rheology on shape evolution, V-shaped test structures were deposited by direct-write assembly, as shown in Fig. 6, and their deflected shape measured by noncontact laser profilometry. The height profiles observed for test structures produced from representative colloidal inks of varying elastic properties are shown in Fig. 7. The color scale representing height data was limited to a range from 0 to 200 mm corresponding to the diameter D of a single filament. The maximum rod deflection occurred midway between the inner supports and increased with span length (L= 0.53 mm near the apex of the structure to L = 2.60 mm at the base of the triangular support). Test structures assembled from the weakest colloidal gel experienced severe deformation even at modest span lengths (see Fig. 7a), whereas those assembled from the strongest ink exhibited maximum deflections of less than 0.25D for spanning distances up to 2 mm (10D; see Fig. 7b). Such observations provide important ink-design guidelines. Spanning-filament deflection is directly related to the elasticity of the colloidal gel in a form similar to Eq. (7). By tailoring ink rheology, 3D periodic structures comprised of spanning filaments can be created by direct-write assembly.

96Part 2 Manufacturing

2.3.6

Direct-Write Assembly of 3D Periodic Structures

Robotic deposition is a facile approach for fabricating 3D periodic structures from a wide variety of materials, including structural, functional, and bioactive ceramics. 3D periodic lattices of varying geometry, filament (rod) diameter, lattice spacing, and composition have been produced. Recent work in the area of piezoelectric ceramic/polymer composites will be highlighted to illustrate the structural features accessible by this technique.

3D periodic lattices consisting of alternating layers of parallel filaments with a 90

rotation between layers were assembled from a concentrated colloidal ink at a deposition speed of 6 mm s–1. The filament diameter D was held constant within

any single structure while the spacing between filaments within a layer was varied.

As an example, the rod spacing L varied from 300 mm to 1.2 mm when using a 200 mm nozzle and was fixed at 1.21 mm when using a 400 mm nozzle. To illustrate that the extruded filament need not be deposited in a straight line, radial arrays were assembled (D = 200 mm) by sequential deposition of layers with alternating patterns of concentric rings and a circular array of radially oriented rods. The first layer was an array of five equally spaced concentric rings between an inner and outer diameters of 3.8 and 9.7 mm. In the second layer, an array of radially oriented rods was deposited between the inner and outer radii. The angular spacing between the rods was 4.68 such that the arc length between adjacent rods varied from 0.38 to 0.19 mm on going from the outer to inner radius. The pore architecture can be easily designed by varying the deposition pattern and the spacing between filaments.

Representative 3D periodic lattices and a radial array assembled from a colloidal gel-based ink are shown in Fig. 8. The 3D lattice structures in Fig. 8a and b result if filaments aligned parallel to the x-axis are stacked in simple columns, and the filaments aligned parallel to the y-axis are stacked in a similar pattern. The 3D lattice in Fig. 8c results from an alternative pattern in which x-axis-aligned filaments are stacked with a 1/2 rod spacing offset in alternating layers. High-quality interlayer bonding is observed at the junction points between filaments in adjacent layers. These junctions strengthen the cellular ceramics, facilitating both green-body handling and post-densification processing. In fact, the cross-sectional images in Fig. 8 were obtained by carefully slicing the structure with a diamond saw, but no embedding in a support material was required. The representative radial array shown in Fig. 8c has an overall cylindrical symmetry. The visible top layer is an array of radial lines, and the underlying layer is a series of concentric rings. The rings maintained their circular shape during deposition despite the changing arc length between supports provided by the radial lines in the previous layer. In addition, the deposited rods maintained a circular cross section and spanned the gaps in underlying layers with minimal deflection.

The 3D periodic structures can be represented by unit cells, as highlighted in Fig. 8a and c. Schematic illustrations of unit cells for both simple and “face-centered” stacking patterns are shown in Fig. 9, along with accompanying illustrations of their pore space. For the simple stacking pattern, the unit cell has a height c defined as twice