## Modeling of 3D RD

The key feature of the etching process is the sharpness of the RD front. To understand its origin, let us first, as usual, write the equations governing the RD process. Denoting the metal by the subscript M, and the etchant by E, we have dCE/dt = DV2Ce - akCECM no n

where CM is the concentration of metal colloids/NPs (in terms of atoms) immobilized in the cube, CE is the concentration of etchant (limiting species), D is the diffusion coefficient of the etchant, k is the apparent reaction constant and a and b are the stoichiometric coefficients for the etching reaction: aE + bM ! soluble salt. These equations assume that metal particles do not diffuse through the agarose/PDMS matrix, and that the dissolved metal does not influence the reaction kinetics or the transport of the fresh etchant. The initial and boundary conditions are such that: (i) the concentration of metal is initially uniform throughout the cube, CM, and (ii) the concentration of etchant is kept constant, CE, at the surface of the cube (since the solution is well stirred). The RD equations can be further simplified by rescaling the variables: CE = CE/CE, Cm = CM/CM, X = x/L and t = Dt/L2,

Figure 10.3 Evolution of cores inside of cubical particles. (a) The top row shows experimental, side-view images of 1 mm agarose/copper cubes etched for 6,12, 18,24and 27min. The bottom row shows the corresponding 2D projections of modeled 3D structures. (b) Analogous experimental and simulated structures in 400 mm PDMS/AuNP cubes at 15, 30, 45, 60 and 75min. (c) Diameters of the copper and AuNP cores as a function of time (measured along the dashed line Z shown in (a)). (d) Corresponding plots of the sphericity index F obtained from 2D core projections by (i) fitting a circle to an actual core shape and (ii) calculating the difference, A, in area between the two; and (iii) calculating the value of F = (A& — A)/A&, where A& is the area of a perfect square circumscribed about the core. In this way, F = 1 for a perfectly circular core and F = 0 for a square one. In (c) and (d), the markers correspond to experimental data and the blue lines to the simulations. Standard deviations are based on the averages over at least ten cubes for each time point.

Figure 10.3 Evolution of cores inside of cubical particles. (a) The top row shows experimental, side-view images of 1 mm agarose/copper cubes etched for 6,12, 18,24and 27min. The bottom row shows the corresponding 2D projections of modeled 3D structures. (b) Analogous experimental and simulated structures in 400 mm PDMS/AuNP cubes at 15, 30, 45, 60 and 75min. (c) Diameters of the copper and AuNP cores as a function of time (measured along the dashed line Z shown in (a)). (d) Corresponding plots of the sphericity index F obtained from 2D core projections by (i) fitting a circle to an actual core shape and (ii) calculating the difference, A, in area between the two; and (iii) calculating the value of F = (A& — A)/A&, where A& is the area of a perfect square circumscribed about the core. In this way, F = 1 for a perfectly circular core and F = 0 for a square one. In (c) and (d), the markers correspond to experimental data and the blue lines to the simulations. Standard deviations are based on the averages over at least ten cubes for each time point.

where L is a characteristic length of the gel/polymer particle (e.g., the side of a cube). This procedure yields the nondimensional RD equations dCEl'dt = V2CE - aDaCECM 9cm/9ï = - bgDaCecm

with initial conditions in the cube CM(x, 0) = 1,CE(x, 0) = 0, boundary condition for the etchant concentration at the cube's surface CE (xS, t) = 1, and g = CE/CM-The most important parameter in these equations is the dimensionless Damkohler number, Da = kL2CM/D, which here can be interpreted as a ratio of the characteristic width of the reaction zone,9 LRZ = (D/kCM)1/2, to the dimensions of the cube, L, as Da = (L/LRZ)2. Thus, the 'sharpness' of the reaction zone, defined as Lrz/L, may be expressed in terms of the Damkohler number as Da -1/2. We note, however, that this relation between sharpness and Da holds only asymptotically as the front moves far from the initial boundaries of the cube; thus, for intermediate times, the other dimensionless parameter, g, may also influence the sharpness.

For the case of agarose/copper cubes, a = 1/2, b = 1 and CM = 31 mM. Since the etching process requires both HCl and O2 (without oxygen, etching rates are much smaller), the effective etchant concentration is determined by the concentration of limiting reagent, O2, dissolved in the etching solution (~8.4 mg L-1 at 25 °C) such that CE = 0.26 mM and g ~ 0.01. The diffusion coefficient of oxygen through the gel matrix is taken from the literature, D = 2.1 x 10-9 m2 s-1, asistherateconstantof thesurfacereaction,kS = 5.4 x 10-4m s-1,determinedpreviouslyforaplanarcopper surface etched with 100 mM HCl/O2. For our system, however, this surface reaction rate has to be translated into a bulk etching rate accounting for the finite sizes of the colloidal particles (radii, R ~ 5-10 nm). The procedure for doing so is detailed in Example 10.1 and gives a Damkohler number for the process of Da ~ 104.

Example 10.1 Transforming Surface Rates into Apparent Bulk Rates

Values for reaction rates describing metal etching are usually expressed in terms of the velocity of the receding surface of a bulk metal, dd/dt, with units of length per time. If the etching rate is first order with respect to the etchant of concentration CE, the velocity of the receding surface can be written as (1/v)(dd/dt) = kSCE, where v is the molar volume of the bulk metal and ks is the surface rate constant of the etching reaction. Also, if the experimental values of this velocity are reported as (dd/dt)0 for a given etchant concentration, C^, the rate constant may be estimated as ks = (dd/dt)0/vCE.

The question relevant to 3D fabrication - and instructive practice in chemical kinetics - is how these experimentally measured surface rate constants can be translated into the apparent bulk rate constants describing the dissolution of finite-sized colloidal particles immersed in the gel matrix.

To answer this question, we first consider the etching of a single spherical particle of radius R and containing N metal atoms, related to R as N = 4pR3/ v. The etching rate may be defined as

4pR2 dt v dt

Integrating this expression, we find that for 0 < t < R0/kSnCE, the radius of the particle evolves as R = R0 — kSvCEt, where R0 is the initial radius of the nanoparticle. Substituting this expression back into the equation above, we can write dN/dt = —4pkSCE(R0 — kSvCEt)2. Because this rate varies with t, we will integrate it over the period R0/kSvCE required to etch the entire particle. This procedure yields the average rate of particle etching:

Now, for a collection of colloidal particles dispersed in some matrix, the rate at which metal atoms are consumed by the etching reaction is simply equal to the average rate for a single colloid times the total number of particles, NP, in solution:

Here, the number of colloidal particles, NP, is related to the number of metal atoms, NM, as Nm = (4pR3/3v)Np, where 4pR3/3 v is simply the number of atoms in a single particle of radius R0. Substituting this relation into the above equation and dividing by the total volume, V, we obtain dCM _ (ksv\n n

This rate equation, which describes how the total concentration of metal atoms, CM, evolves in time, is identical to the second-order rate equations introduced in Equation (10.1). Thus, by inspection, we can identify k = kSv/R0 as the apparent bulk rate constant for the dissolution of metal atoms into soluble ions. Also, the Damkohler number for the RD process of core formation, defined in the main text as Da = kL2CM/D, may now be related to the surface reaction rate as Da = kSvL2CM/DR0.

For the PDMS/AuNP cubes, I3_ is the reagent limiting gold etching. In this case, CM = 38 mM and CE = 8 mM such that g ~ 0.2; other parameters are a = 1/2, b = 1 and D = 5 x 10~12m2 s_1. Using the experimentally estimated rate constant for AuNP etching, kS = 1.4 x 10~6ms-1, the Damkohler number for L = 400 mm cubes is then estimated at Da ~ 6200.

Figure 10.4 shows that when the nondimensional RD equations (10.2) are solved numerically (using the Fluent® finite volume method, which is a 3D version of the finite element method discussed in Section 4.3.2; for the source code see dysa.northwestern.edu) with the estimated Damkoohler numbers, they give good quantitative agreement both in terms of the time evolution of the cores as well as their sphericities. Interestingly, the combination of core 'sharpness' (i.e., narrow reaction zone) and high sphericity is observed only for intermediate values of Da (^104; Figure 10.4(b)). When Da is smaller o •

Figure 10.4 Effects of Damkohler number on the sharpness and the sphericity of metal cores formed in cubical particles at two different times (t = 22.2 and 23.8 min) and for g = 0.01. (a) Da = 103; (b) Da = 104; (c) Da = 106. The intermediate values of Da give the best combination of core sharpness and sphericity. The numbers in the upper-right corners of the images give the sphericity indices, F

Figure 10.4 Effects of Damkohler number on the sharpness and the sphericity of metal cores formed in cubical particles at two different times (t = 22.2 and 23.8 min) and for g = 0.01. (a) Da = 103; (b) Da = 104; (c) Da = 106. The intermediate values of Da give the best combination of core sharpness and sphericity. The numbers in the upper-right corners of the images give the sphericity indices, F

(e.g., of the order of 103; Figure 10.4(a)), the width of the reaction zone is roughly Da~1/2 or 3% of the cube size, such that the contours of the cores - albeit more spherical - are significantly 'blurred'. In contrast, for high values of Da (e.g., 106; Figure 10.4(c)), etching becomes increasingly diffusion limited, and the etchant delivered to the reaction front is consumed immediately. As a result, the reaction front is sharp (~0.1% of L), but resembles more the initial, cubical contour.

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