Ind. Eng. Chem. Res. 1997, 36, 2915-2923
2915
Etching of Mass, Surface, and Porous Fractal Solids C. Gavrilov and M. Sheintuch* Department of Chemical Engineering, Technion, Haifa, Israel 32000
To study fractal aspects of reactive etching, we consider three types of solids with surface, mass, and pore fractality. An analytical solution for etching of the solid boundary in the form of a Koch curve demonstrates that in the absence of diffusion resistance bifractal structures are formed due to screening effects. Over a certain domain the surface area scales like t1-Df. Stochastic simulation of etching of solid clusters, obtained by diffusion-limited aggregation (DLA), shows that the dynamics strongly depend on the value of fractal dimension. Analytical approximation for the dynamics of etching based on the DLAs density distribution was obtained, showing that the mass scales like (1 - Kt)Df, where K is a certain constant, and was in excellent agreement with the simulation. The simulation of etching of a Sierpinski-gasket fractal and the corresponding uniform-pore object of the same size, porosity, and reactive area demonstrates the combined influence of diffusion resistance optimization and fractal geometry on the character of etching. The etching of fractal structures is significantly faster in the regions of strong and moderate diffusion resistance. In the region of strong diffusion resistance the rates of etching declined monotonically with time. The reaction occurred in a narrow penetration layer, and fractality did not break. No simple scaling is evident in this case. 1. Introduction The analysis of many chemical reaction engineering problems is based on the comparison of two characteristic length or time scales. Fractal structures are interesting objects since they cannot be characterized by one length scale. The interest in fractal porous structures, as models for real catalysts, stems from the novelty of the structure as well as from the large amount of experimental evidence showing that many porous solids exhibit scaling behavior that indicate fractality (Avnir et al., 1985; Rothschild, 1991). Also, models of solid aggregation that produce fractal structures like DLA (diffusion-limited aggregation) and CCA (clustercluster aggregation) resemble preparation methods of certain porous materials (Elias-Kochav et al., 1991). Recent studies of fractal structures focused on problems of reaction and diffusion in a porous catalyst (e.g., Sheintuch and Brandon, 1989; Aris, 1991; Giona et al., 1996; Mougin et al., 1996); the works of Coppens and Froment (1994, 1995, 1996) bring these ideas one step closer to implementation for catalyst design purposes. Problems of liquid-solid or gas-liquid noncatalytic reactions, in which the solid changes its geometry and (or) pore structure due to reaction, are central to chemical reaction engineering. Coal gasification and coal combustion are just a few commercially important processes of this large class. Many samples of coal have been found to admit pore- or surface-fractality properties, evident from SAXS measurements or from adsorption of a homologic series of components (e.g., Friesen and Mikula, 1987; Avnir and Farin, 1984). The study of reactive etching processes of fractal objects is the subject of the present investigation. Unlike problems of reaction and diffusion in a porous catalyst, in which the pore properties can be lumped into an effective parameter (the Thiele modulus), etching problems require one to define the pore structure. We are interested in characterizing, therefore, the kinetics and dynamics of etching and the changes in pore structure with time. * Author to whom all correspondence should be addressed (phone, 972-4-8292820; fax, 972-4-8230476; e-mail, cermsll@ tx.technion.ac.il). S0888-5885(96)00605-7 CCC: $14.00
Several types of etching problems were considered in the literature. The grain model, introduced by Szekely and co-workers (Szekely and Evans, 1970, 1971; Sohn and Szekely, 1972), and its generalization for various grain size distributions (Szekely and Propster, 1975) can be used to describe the problem of the solid clusters etching. The solid phase consists of small spherical particles or grains. The space between the grains constitutes the porous network. These models account for two levels of grains, and the fractal model can be viewed as an extension to many such levels. Different capillary models (e.g., see Peterson, 1957; Hashimoto and Silveston, 1973; Simons and Finson, 1979; Simons, 1979a,b) were used to describe the diffusion-reaction problem in a porous solid where material is supplied into the solid by diffusion. The porous network is modeled as a set of randomly oriented cylindrical pores (capillaries) of different sizes. Different continuum models were used to describe the problem of evolution of external surface area exposed to etching conditions. The solid multiphase continuous system can be described by subdividing the space into a set of different polyhedra (e.g., see Meijering, 1953; Winerfeld, 1981). Modeling of the pore network by random structures and the effect of critical phenomena like percolation was the subject of a recent investigation (Sahimi et al., 1990). In studying etching in a fractal catalyst, we are interested in looking at new qualitative behaviors (scalings) and in comparing fractal and uniform catalysts. Catalytic fractal objects may be classified as surface, mass, or pore fractals, and each class is likely to exhibit different scaling. The objects may be ordered, for which analytical results can be obtained, or stochastic. Furthermore, the limiting step may be kinetics or diffusion through the fluid or the pore space. Thus, there exists a wide variety of problems to be posed, and here we show just three cases: Etching of a surface fractal in the absence of diffusion resistance, diffusionlimited etching of mass fractal, and diffusion-influenced etching of a pore fractal. We choose a Koch curve (Figure 1) to study etching of solids with fractal surfaces in the absence of diffusion resistance since analytical results can be obtained then, showing the influence of geometry aspects on kinetics and dynamics of etching. © 1997 American Chemical Society
2916 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997
Figure 1. A body limited by a Koch curve.
In a similar etching of a flat surface the boundary will move at a constant speed while the surface area and the etch rate are constant. A surface fractal, on the other hand, offers initially a very large area that will decline with time. We are interested in the scaling properties of such a surface. Stochastic simulations of etching of DLA solid clusters, using etchant molecules that are released from a fixed boundary, are used to study etching of a mass fractal. Etching dynamics of DLA clusters strongly depend on the value of fractal dimension. We obtained an analytical approximation for the dynamics of etching of a generalized DLA cluster showing excellent agreement with the simulation. The etching rate of a two-dimensional DLA, as well as that of a dense cylinder, decline with time due to diffusion limitation. The DLA, however, offers a very large area, and consequently we find different scaling characteristics. In the third example we simulate the etching of a Sierpinski gasket, a pore fractal, and the corresponding nonfractal uniform-pore object of the same size, porosity, and reactive area to demonstrate the combined influence of diffusion resistance and fractal geometry on the character of etching. The etching of fractal structures is significantly faster in the regions of strong and moderate diffusion ressitance. This is because a branching porous catalyst is optimal for reaction and diffusion purposes in a way similar to that of the lungs or of the blood vessels: the narrow pores, where effective diffusivity is small due to a small cross section or due to Knudsen diffusivity, are also the shorter pores, and the corresponding Thiele modulus is moderate. The narrow pores are supplied by longer but larger pores where the effective diffusivity is large. 2. Diffusion-Free Etching of a Surface Fractal (Koch Curve) We use a Koch curve as a model for surface fractals and study the influence of fractal geometry on the
Figure 2. Schematics and notation of a base line building block.
kinetics of surface etching. Assume that a solid body, the boundary of which is a Koch curve of finite number of generations (Figure 1), is exposed to etching conditions due to surface reaction. We are interested in portraying the dynamics of etching in the absence of diffusion resistance. The solid boundary propagates then at a constant speed in the direction normal to the surface, and the overall reaction rate is proportional to the total length of the boundary. As time progresses the surface area diminishes as the fine features of the object disappear. The problem, therefore, is translated into finding the length of the boundary (LΣ) as a function of normal direction (d) which is proportional to the elapsed time. A new boundary is formed by moving all points of the original boundary in the normal direction; all points of this moving boundary have the same distance (d) from the original boundary of the Koch curve. A true fractal of infinite number of generations does not have, of course, a normal direction. Such a fractal, however, will exhibit an infinite rate initially, and as the smaller triangles disappear the rate will decline and become finite. We start here with an object of finite number of generations, which represents a real solid surface. As fine features disappear we need to consider the critical distances (or times) for the elimination of every generation. To do that we will consider two types (base line and secondary) of building elements of the Koch curve. Base line building blocks are triangles generated on the main base of the curve, while the secondary building blocks are triangles on the sides of other triangles. Obviously, for certain self-similar parts the secondary elements play the rate of base line blocks. Since eventually the whole curve will be etched into a flat line, it will take an infinite time to eliminate base line building blocks. Secondary building blocks, on the other hand, will collapse in pairs and will be eliminated at finite times. The methodology we apply is the following: Below we derive the length-distance relationship for each type of building element. The derivation is based on geometric considerations, and the results are summarized in eqs 2 and 10. We will then use these results in the analysis section to find LΣ(d). A. Building Blocks. Base Line Building Block. Consider a base line triangular building block of the Koch curve (Figure 2). We want to find out how the geometry of this unit changes when moving along the normal direction of the boundary: Regular points are displaced by d, while corner points develop into arcs. If BB′′ and DD′′ are normal to BC and DC, then straight segments BC and DC disappear when the two corner points meet at a certain dcr ) L tan R. Prior to that distance (d < dcr; see Figure 2, part a), the resulting structure will consist of two lines of length l ) L - d
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 2917
one arc and its length is given by
LΣ(dcr2 < d < dcr3) ) dγ2
(6)
where
γ2 )
2l π d - β2, ) 2 sin R sin(π - R - β2)
(7)
When d > dcr3 the secondary block has disappeared and does not contribute to total length:
LΣ(d > dcr3) ) 0
(8)
Figure 3. Schematics and notation of a secondary building block.
cot R(C′B′′ and C′D′′) and two arcs of length larc ) βd (B′B′′ and D′D′′). Hence the initial 2L(BCD) segment translates into LΣ(d) given by
Solving eqs 5 and 7 and substituting R ) π/6, we find
γ1 ) 2 arcsin
π π L L - , γ2 ) arcsin 2d 3 2d 3
(9)
LΣ(d < dcr) ) 2l + 2larc ) 2(L - d cot R + βd) ) 2(L + d(β - tan β)) < 2L (1)
and obtain for the secondary blocks the following relations:
where the inequality holds since (β - tan β) < 0 for β > 0. When d g dcr (Figure 2, part b), the resulting structure will consist of two arcs of length larc ) d‚arcsin(L/d sin R) which becomes flat at infinite times. Since for Koch curve β ) π/3 and R ) π/6, we find for base line building blocks that
LΣ(d) )
LΣ(d) )
2L + 2d(π/3 - x3); d < d {2d‚arcsin(L/2d); dgd
cr
) L tan R ) L/x3
(2)
{
for 0 < d < dcr1 2l + 2d(π/3 - x3) d(3 arcsin L/2d + π/3) for dcr1 < d < dcr2 (10) d(arcsin L/d - π/3) for d > dcr3
Equations 2 and 10 allow us to build the LΣ(d) function for the Koch curve boundary. B. Analysis. We consider a Koch curve of n generations. The minimal element of the fractal is
cr
Secondary Building Block. Figure 3 presents a secondary block. We define now three critical lengths that describe the etching processes in a secondary block: dcr1 and dcr3 are intersections of normals to the base of the block, drawn from the corners of the base with their symmetric counterpart, and dcr2 is the normal to the side of the block drawn from the furthest corner of the base; equivalently, dcri can be found from the intersection of these normals with the symmetry axis. The lengths are given by
dcr1 ) L tan R )
x
L 5 , dcr2 ) L tan R ) 2 x3 2L 5 (3) L, dcr3 ) 2L tan R ) 6 x3
x
In the range d < dcr1 the secondary triangles change like base line blocks and the resulting structure consists of two lines and two arcs whose total length is given by eq 2. In the range dcr1 < d < dcr2 the secondary blocks begin to be etched and the structure consists of two arcs (one of which is partly screened); its total length is given by
(2dL ) + dγ
LΣ(dcr1 < d < dcr2) ) d‚arcsin
1
(4)
where
γ1 ) π - R1 - β1, cos R1 )
d L , ) 2d sin R L (5) sin(π - R - β1)
In the range dcr2 < d < dcr3 the structure consists of
rmin )
(31)
n
(11)
and the number of these elements is 4n. Hence, the initial length of the curve is given by
L0 )
(34)
n
1-Df ) (3n)Df-1 ) rmin
(12)
where Df ) ln 4/ln 3 is the fractal dimension of the Koch curve. We consider several cases of d values. For very short times, when 0 e d e dmin ) rmin/ cr 1/2 3 , none of the triangles has been eliminated but they all have changed according to eq 2. The change in length will be proportional to the total number of triangular blocks (N∆) of the Koch curve (see eq 2):
(π3 - x3) )
N∆(d) ) L0 - 2N∆d
(
L0 1 -
(
))
(
)
2 d π 2 π - x3 + d - x3 (13) 3 rmin 3 3 3
since for the fractal of n generations N∆ ) 1 + 4 + ... + 1-Df 4n ) (4n - 1)/3. (Recall that L0 ) rmin ) (4/3)n.) In the limit of a large number of iterations the second term of eq 13 is negligible. In this range of d the reactive area changes linearly with time. For intermediate values of dmin e d e d(2) cr cr3, we find that secondary triangles are eliminated at critical times (or distances). (d(n) cri values are critical distances for the triangular blocks of generation n.) We choose a set of d values that correspond to the elimination of secondary triangular blocks of generations larger than k:
d ) Lk tan R, 1 e k e n
(14)
2918 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997
The contribution of all other blocks will be analogous to that in the previous case. Using eq 13 we can write:
LΣ(d) )
( ) (1 - 32 x13(x3 - π3)) + 32 x13 31 (x3 - π3) + 4 3
k
k
Xn-k (15) where Xn-k is the amendment for contribution of base line blocks of iterations k + 1, k + 2, ..., n. For Xm we can write the following recursive equation
Xm+1 ) Xm ) Xm +
(
)
2k+m 1 1 + 2‚2k+m d arcsin k+m+1 k+m+1 2d 3 3
( )( 2 3
k
12m + 33
()
-
(
1 x3 2m arcsin k+m+1 2 2 3 x3
(32) [2x13 arcsin(2x1 3) - 31]
))
(16)
k
X1 )
d)
which follows from the self-similarity: at step (m + 1) we build base line blocks of iteration (k + m + 1) by cutting out 2k+m lines of size 3-(k+m+1) and exchanging each of them for two arcs of length arcsin(3-(k+m+1)/2d) as was derived in eq 2. We can find then that
Xm+1 )
() [ 2
k
m
2 ∑ i)0
3
2i
x3
( ) ( )] x3
arcsin
1
12
i
(17)
-
2 3i+1
33
At x , 1, arcsin(x) can be well approximated by arcsin(x) ≈ x + (x3/6). The largest argument of arcsin in the sum (17) is x3/6, so this approximation can be used in (16). We find then that
2
measures of size (31/2d). With growth of d this member becomes less important and (31/2d)1-Df/2 will become a leading member. It corresponds to probing of the reactive area of the external building blocks with measures of size (31/2d). The resulting structure will be bifractal with dimensions of Df and Df/2. The other two members can be neglected in most of the domain. Other critical and intermediate values of d in the range can be considered by applying formula 15 and the same algorithm of computation. It will lead, however, to the same basic resultssbifractal dynamics with leading exponents of (1 - Df) and (1 - Df/2) and a bifractal structure formed as a result of etching with values of dimension Df and Df/2. 2.5 the secondary At large values of d > d(2) cr3 ) 2/3 building blocks are totally wiped out. The base line blocks, however, remain and can be computed similarly to eqs and 15 and 16. To use the results obtained above, we will use again convenient values of d:
(
) () ( )
i x3 1 12 1 2 2i arcsin ≈ + i+1 2 3 33 72 27 x3
( )∑ ( ) 2
3
k m
1 2
i)072
()
m+1
3 1 - (2/27)
i
)
27
8
25
2
LΣ(d) ) 1 + Xn
(23)
where Xn is the correction for contribution of external blocks of iterations 1, 2, ..., n. The expression for Xm becomes now:
1 Xm+1 ) Xm - 2m m+1 + 3
(
2‚2md arcsin
1
3
m+1
)
1 1 ; X1 ) 9k (24) 2d 72
Substituting the values of d and using approximation arcsin(x) ≈ x + (x3/6), we get:
( )
1 2 m Xm+1 ) Xm + 9k 72 27
(18)
(25)
which translates to m
k
3
(22)
Then LΣ(d) can be computed as
i
Equation 17 can be approximated therefore by
Xm+1 )
1 1 , k
