Jun 20, 2006 - 2Idaho National Laboratory, P.O. Box 1625, Mail Stop 2211, Idaho Falls, Idaho 83415-2211, USA. (Received 3 January 2006; published 20 ...
PHYSICAL REVIEW LETTERS
PRL 96, 245501 (2006)
week ending 23 JUNE 2006
Fracture Patterns Generated by Diffusion Controlled Volume Changing Reactions A. Malthe-Sørenssen,1 B. Jamtveit,1 and P. Meakin2 1
Physics of Geological Processes, University of Oslo, Box 1048 Blindern, N-0316 Oslo, Norway Idaho National Laboratory, P.O. Box 1625, Mail Stop 2211, Idaho Falls, Idaho 83415-2211, USA (Received 3 January 2006; published 20 June 2006)
2
A simple two-dimensional model was developed for the growth of fractures in a chemically decomposing solid. Simulations were carried out under rapid chemical decomposition conditions for which the kinetics of fracture growth is controlled by diffusion of the volatile reaction product or the kinetics of evaporation. The growth of the fracture pattern is self-sustaining due to the volume reduction associated with the decomposition process. Consistent with the theoretical analysis of Yakobson [Phys. Rev. Lett. 67, 1590 (1991)], the fracture front propagates with a constant velocity v ’ k2=3 D‘0 1=3 under evaporation controlled conditions and v ’ D=‘0 under diffusion controlled conditions, where k is the evaporation rate constant, D is the diffusion constant for the volatile reaction product in the solid, and ‘0 is the critical stable crack length. Under diffusion controlled conditions, the front width w scales as w ’ k‘0 =D. DOI: 10.1103/PhysRevLett.96.245501
PACS numbers: 62.20.Mk, 05.45.Df, 46.50.+a
In many far from equilibrium processes, pattern formation is governed by an interplay between the growing structure and the physical fields that control growth [1– 3]. Familiar examples include disordered dendritic patterns formed by diffusion limited aggregation or growth [4] and fracture networks [5]. An important class of growth processes, in which longrange interactions between different parts of the growing surface play a critical role, is the formation of crack arrays [6] driven by volume reducing processes. The volume contraction and concomitant fracturing may enhance the penetration of the fluid toward the reaction front. Volume reduction can be a consequence of devolatilization [6], phase transformation, chemical diffusion and reaction [7], or thermally induced volume change due to invasion of a hot or cold fluid [8,9]. In this Letter we describe direct numerical simulation of fracture patterns generated by volume reducing chemical decomposition. An important example of pattern formation driven by fracture growth coupled with volume reducing reactions is high temperature decomposition described by the chemical process ABsolid ! Asolid Bvol , where B is a volatile reaction product. We assume that the volume reduction is controlled by the diffusion of B through unfractured solid, followed by evaporation at the interface between the solid and the fracture network and rapid release to the exterior via the growing fracture network. For the sake of simplicity, we assume that the diffusion coefficient and other material properties are constant. The growth of a single crack in a moving thermal gradient has been addressed experimentally [10], and explained theoretically [11,12]. The behavior of the decomposition front depends on the interplay between the cracking process and thermal diffusion. While processes of this type are responsible for striking patterns, such as the formation of polygonal cooling joints [8], and the process has been studied experimentally in analogue systems [13], 0031-9007=06=96(24)=245501(4)
theoretical progress has been confined primarily to stability analysis. Yakobson [6] introduced a one-dimensional model, which predicts front velocities and typical crack lengths, but does not provide a detailed picture of the emerging fracture pattern. Boeck et al. [9] developed a two-dimensional model of thermal cracking for a regular array of linear fractures propagating with a constant velocity into an unfractured solid. However, there have been no studies of the geometry of the growing fracture patterns. Yakobson [6] found two different regimes with constant fracture front propagation velocities. If the fracture growth process is controlled by transport across the fracture surface, then the front propagation velocity, v, is given by v c1 k2=3 D=‘0 1=3 , and if the fracture growth is controlled by diffusion through the solid to the fracture surface v c2 D=‘0 , where k is the evaporation rate constant, D is the diffusion coefficient for the volatile reaction product in the solid and ‘0 is the critical stable crack length. The coefficients c1 and c2 depended on the elastic properties of the solid. In this Letter we describe a computational model, which confirms the predictions of Yakobson [6]. For a process in which a solid, AB, decomposes instantaneously and completely into a volatile reaction product, B, dissolved in the solid, A, the local volume reduction depends on the local concentration, cB , of B in A. The mobile decomposition product, B, diffuses to the boundaries of the solid, where it escapes rapidly. A flux boundary condition given by Drn cB kcsB ceq B ;
(1)
is used at the fracture surfaces. Here, rn cB is the concentration gradient normal to the surface, cB is the concentration of B in the solid at the boundary, and ceq is the B concentration of volatile reaction product in the solid at equilibrium with the volatile reaction product in the fracture network. The rate constant k, with units of lt1 ,
245501-1
© 2006 The American Physical Society
PHYSICAL REVIEW LETTERS
PRL 96, 245501 (2006)
determines the rate of evaporation. The volatile material is assumed to escape from the system as soon as it enters the growing fracture network, and under these conditions s s ceq B ! 0 and Drn cB kcB , where cB is the concentration in the solid at the fracture surface. The initial condition is an unstrained two-dimensional isotropic, linear elastic solid body with a constant concen~ c0B ]. While the behavior of twotration of B [cB x dimensional and three-dimensional systems will be quantitatively different, two-dimensional simulations provide valuable insight into three-dimensional behavior. The mechanical behavior of the solid is simulated using a discrete element model [14], in which the elastic material is represented by a network of nodes connected by elastic beams [15] as illustrated in Fig. 1. The force and moment acting on a node i at a position x~ i from a beam connected to node j are F~ i;j n di;j d0i;j n~ i;j s 12i;j j;i ~si;j ; ~ i;j s d M
(2)
1 2 1 i;j j;i i;j j;i : (3) 12 2 3 3
At equilibrium the net forces and moments are zero for each node. Here, di;j jx~ i x~ j j, and d0i;j ri rj , where ri is the radius of node i. The unit vectors n~ i;j and s~i;j point parallel and normal to the center line, respectively, and i;j is the angle between n~ i;j and the tangent t~i;j , as illustrated in Fig. 1. The beams connecting the particles are assumed to have a length of d0i;j with a rectangular cross sections of width hy in the x; y plane and depth hz in the out of plane, z, direction. The parameters used in the discrete element component of the model are related to the Young’s modulus E0 , shear modulus G0 , and Poisson’s ratio 0 by n E0 hz hy =d, s 12E0 I=d2 1 , and 12E0 I=G0 Ad2 , where I is the geometric part of (a)
(b) s i,j t i,j n i,j
φ i,j i
n i,j
j
φ j,i t j,i
FIG. 1. (a) Coupling between the elastic and diffusion components of the model. The local concentration is indicated by the gray scale with a darker shade representing a lower concentration, cB , of the diffusing volatile reaction product. Only unbroken bonds are shown. (b) A bending beam from node i to node j used for the discrete element component of the model. The unit vectors n~ i;j and s~i;j are parallel and normal to the straight line connecting nodes i and j. The deflection angles of the beam are i;j and j;i .
week ending 23 JUNE 2006
the moment of inertia. For hy d=2 and 0 1=3, the p effective elastic constants are E ’ E0 3, and ’ 0 . The volume reduction due to a reduction of the concentration of the volatile material dissolved in the solid can be included by changing the equilibrium bond lengths, ri . If the volume associated with the ith node is reduced from vi 0 to vi t at time t, then the particle radius ri t is reduced from ri 0 to ri ri 0vi t=vi 01=d , where d is the dimensionality of the system. We assumed that vi t vi 0 1 cB t=cB 0:
(4)
A volume reduction factor of vi 1=vi 0 0:9 was used in all of the simulations. The diffusion equation, with the boundary condition given in Eq. (1), is solved using a finite difference method. The location of the boundaries between the solid in which the volatile substance B is diffusing and the fracture aperture network is determined from the positions of and connection between the nodes in the elastic network. We assume that the mechanical relaxation is rapid compared with molecular diffusion and the associated volume reduc~ and the boundary tion. Given the concentration field cB x conditions (a free surface at y 0, periodic boundary conditions in the x direction and zero displacement at y ymax ), the equilibrium configuration of the elastic network was calculated using an over-relaxation method [16]. Quenched disorder was represented by a Gaussian distribution of elastic constants E0 . Essentially all real materials are disordered because of defects, and this disorder has an important influence on fracture growth. Fracturing is simulated by the irreversible removal of a beam if Fi;j =Fc 2 maxjMi;j =Mc j; jMj;i =Mc j > 1;
(5)
where Fc and Mc are critical force and moment corresponding to a von Mises failure criterion [14]. The simulation consists of interleaved diffusion, mechanical relaxation, and beam breaking steps. The equilibrium configuration of the beam network provides the boundary conditions during a single time step in the solution of the diffusion equation. This provides an updated concentration field from which new volume reduction factors and beam lengths are calculated. The beam network is then relaxed, and beams are removed if the failure criterion is satisfied. Before a new diffusion step is executed, the relaxation, testing, and beam removal is repeated until a mechanically stable beam network is obtained. The mechanical relaxation and bond breaking is assumed to be instantaneous, and the time is incremented by a fixed amount, t, after each relaxation step. The resulting fracture patterns did not change significantly when the diffusion time step was reduced or the beam breaking order randomized. The beam network component of the model is based on the concept of a critical strain, c , beyond which failure
245501-2
PRL 96, 245501 (2006)
week ending 23 JUNE 2006
PHYSICAL REVIEW LETTERS
occurs rapidly. Typical values used here range from c 103 to 101 , which are realistic for rocks and many polycrystalline materials. For a fracture of length l that is small compared to cB =rcB (the distance over which the concentration changes significantly) the maximum stress max acting on the fracture due to a change in concentration is on the order of E [OE] and the corresponding stress intensity p at the tips of the fracture is O l. If the fracture propagates when the stress intensity factor exceeds the critical stress intensity factor, the smallest fracture that will propagate due to this stress, has a length l0 with a magnitude OKc =E2 . Consequently, l0 is a natural length scale, which is used as the size of the lattice unit, a. A more detailed argument that provides the numerical prefactor for l0 can be found in [9]. Figure 2 shows fracture patterns from two simulations. After an initial transient, in which the fracture front is strongly influenced by the initial conditions, the growth of the fracture pattern can be described in terms of the propagation of an ‘‘active zone,’’ which propagates with a velocity v into undamaged solid. A ‘‘dead’’ zone in which the fracture pattern does not change is formed in the wake of the active zone. After the influence of the initial conditions has decayed, the active zone has a statistically time invariant structure, with a characteristic fracture spacing L. The front position can be determined from the fracture density profile, f y nf y=Lx , where nf y is the number of fractures that are intersected by a cut in the x direction at a ‘‘height’’ of y, and Lx is the width of the computational domain. Alternatively, the front position can be defined in terms of the concentration profile hcB iy, the average concentration of the mobile reaction product at a height of y. Here, the position (y coordinate of the front) is the value of y at which hcB iy cB 0=2. Similarly, the
front width can be defined in terms of either the fracture density or the concentration. Here, the front width is the length over which the concentration hcB iy; t falls from 0:75cB 0 to 0:25cB 0. The critical dimensionless ratio controlling the kinetics and geometry of pattern growth is 0 k‘D0 , where k, ‘0 , and D have units of lt1 , l, and l2 t1 . Dimensional analysis shows that the dimensionless ratios describing the behavior of the system are related to 0 by equations with the general form 1
vl0 f0 ; D
2
L g0 : l0
(6)
The functions fu and gu were measured by direct numerical simulation, and the results are shown in Fig. 3. The functions have the form: 2 1 u u u1 u1 ; gu ; (7) fu c1 c2 u 1 u 1 where c1 4:2 0:1, 1 0:66 0:03, c2 7:6 0:1, and 2 0:66 0:03. The measured scaling exponents are consistent with the predictions of Yakobson [6,9] that
1 2 2=3, based on stability analysis arguments. The fracture pattern does not change significantly with k‘0 =D, although the characteristic lengths and velocities do change. We may therefore address the details of the structure of the fracture pattern in the 0 ! 0 limit in which the volume contraction is controlled by slow evaporation at the solid-fracture interface, and cB ! 0 at the solid-fracture interface. Figure 4 shows that the fracture pattern resembles a forest of trees. The two-point densitydensity correlation function for the fractures shows that the pattern is linear on short length scales [there is only one more-or-less straight fracture segment in a region with a size ‘ (area ‘ ‘) that is smaller than the fracture spacing, L] and compact with a fractal dimensionality of D 2 on long length scales (when ‘ L, the total length of the fracture segments in a region of size ‘ is proportional to ‘2 ). 2.5 (b)
(a)
log10(w/l0)
log10(vl0/D)
0.5 0
2 1.5
-0.5
1
FIG. 2 (color). Fracture patterns and concentration fields obtained from two simulations with a system size of a Lx Lx 400 with periodic boundary conditions in the x direction. Simulation parameters are (a) D0 5, k0 0:5, c 0:01, and c c =2. (b) D0 5 102 , k0 0:5, c 0:01, and c c =2. The thick line in the lower right corner shows the measured width w of the front. The color scale indicates the local concentration cB .
-3 -2 -1 0 1 2 log10(kl0/D)
3
4
-3 -2 -1 0 1 2 log10(kl0/D)
3
4
FIG. 3. (a) Dependence of the dimensionless crack front velocity, vl0 =D, on the dimensionless ratio 0 k‘D0 . The dotted line adjacent to the scaling function fk‘0 =D has a slope
2=3. (b) Dependence of the dimensionless characteristic front width w=‘0 on the dimensionless ratio 0 k‘D0 . A line with a slope of 2=3 is shown for comparison.
245501-3
PRL 96, 245501 (2006)
PHYSICAL REVIEW LETTERS
week ending 23 JUNE 2006
stants are usually very temperature sensitive the heat adsorbed by the reaction process and evaporation of the volatile decomposition product(s) at the solid-fracture interfaces might play an important role. However, thermal diffusion coefficients are usually much larger than molecular diffusion coefficients, and an isothermal approximation would be justified for many applications.
FIG. 4 (color). (a) Fracture pattern from a simulation on a 400 400 lattice for k‘0 =D 500. Each fracture tree starting from y 0 is shown in a separate color. (b) The probability distribution ‘0 Ph=‘0 of the heights, h, of fracture trees in the y direction. The dotted line has a slope of 1:5 corresponding to a distribution Ph / h3=2 . PXX is the number of trees with X x X X in the limit X ! 0.
Figure 4(b) shows that the tree height distribution has a power-law form, Ph / h1:5 0:05 . This can be understood in terms of a simple coalescing random-walker model. The sides of the regions occupied by each tree are similar to directed random walks, starting at y 0. If the directed random walks corresponding to two sides of a tree meet, then that tree can no longer grow. The contacting directed random walks then coalesce to form a single directed random walk, which is the boundary between the two taller trees on either side of tree that can no longer grow. Since the average lateral distance moved by a random walker after n steps is proportional to n1=2 , the number of trees Nh > h0 that reaches a height of h0 or greater is proportional to h01=2 . It follows directly from this R that the probability Ph0 is proportional to h03=2 [ Ph0 dh0 / Nh > h0 / h01=2 ] Figure 4(b) shows that the tree height distribution is consistent with this idea. An important direction for further investigation would be to include the kinetics of decomposition and reversibility of the decomposition process. Since reaction rate con-
[1] T. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore, 1992). [2] P. Meakin, Fractals, Scaling and Growth Far from Equilibrium (Cambridge University Press, Cambridge, England, 1998). [3] A.-L. Barabasi and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, England, 1995). [4] T. S. Witten and L. M. Sander, Phys. Rev. Lett. 47, 1400 (1981). [5] A. Skjeltorp and P. Meakin, Nature (London) 335, 424 (1988). [6] B. I. Yakobson, Phys. Rev. Lett. 67, 1590 (1991). [7] B. Jamtveit, H. Austrheim, and A. Malthe-Sørenssen, Nature (London) 408, 75 (2000). [8] A. Aydin and J. M. DeGraff, Science 239, 471 (1988). [9] T. Boeck, H.-A. Bahr, S. Lampenscherf, and U. Bahr, Phys. Rev. E 59, 1408 (1999). [10] A. Yuse and M. Sano, Nature (London) 362, 329 (1993). [11] M. Marder, Phys. Rev. E 49, R51 (1994). [12] E. Bouchbinder, H. Hentschel, and I. Procaccia, Phys. Rev. E 68, 036601 (2003). [13] G. Mu¨ller, J. Geophys. Res. 103, 15 239 (1998). [14] Statistical Models for the Fracture of Disordered Media, edited by H. J. Herrmann and S. Roux (North-Holland, Amsterdam, 1990). [15] S. Roux and E. Guyon, J. Phys. (Paris) 46, 999 (1985). [16] D. M. d. G. Allen, Relaxation Methods (McGraw-Hill, New York, 1954).
245501-4
