Exact differential equations for diffusion limited aggregation

PHYSICA ELSEVIER

Physica A 227 (1996) 269-276

Exact differential equations for diffusion limited aggregation S.F. E d w a r d s a'*' 1, M o s h e S c h w a r t z b ~Mortimer and Raymond Sackler Institute of Advanced Studies, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel bRaymond and Beverly Sackler Faculty of Exact Sciences, School of Physics and Astronomy, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel

Received 11 December 1995

Abstract A central difficulty in many problems involving statistical dynamics of surfaces or lines is the preservation in model equations of the correct topological integrity. This paper continues a series by the authors which study this problem, and contains equations for the evolution of surfaces by the deposition of particles out of an ensemble of diffusing particles onto the surface. It is shown that closed equations can be given to describe the process and that these equations honour the topological integrity of non-penetrating surfaces. Keywords: DLA; Deposition

1. Introduction Two areas of theoretical physics have had much attention in recent years. The first concerns the movement of systems which possess topological invariants or temporary invariants. F o r example, a particle in an emulsion is a liquid bounded by a membrane. It can move through fluid flow or Brownian motion but the membrane cannot enter itself Fig. 1. A polymer molecule cannot pass through itself, and although the molecule can find a way to any configuration, on a short time scale the motion is severely restricted Fig. 2. One can attempt to derive analytic expressions for topological invariants, e.g., Gaussian invariants, but these are difficult and incomplete. The alternative is to write equations of motion for the surface (or polymer) which preclude entering a regime which is mathematically possible but physically impossible. It is always possible to do this by using infinite repulsion at contact, but this is difficult because any point is in contact with its neighbouring points. An alternative is to define the surface by an

*Corresponding author. 1Permanent address, Cavendish Lab., Madingley Rd. Cambridge, CB30HE, UK. 0378-4371/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSD1 0 3 7 8 - 4 3 7 1 ( 9 5 ) 0 0 4 6 5 - 3

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S.F. Edwards, M. Schwartz/Physica A 227 (1996) 269-276

+ (o)

(b)

(c)

Fig. 1. The transition from shape (a) to shape (b) is allowed. The transition to (c) is forbidden.

except viQ

Fig. 2. The transition from the left configuration to the right one is possible but only through a middle stage like the lower right configuration.

equation

4,(x,

y, z; t) = 0 .

(1)

It has been shown that one can write equations of motion for ~, which are indeed the equations of motion of the globule bounded by the surface of equation (1) [1]. Moreover, the topological integrity is maintained when the exact equations are approximated in a useful way. It has been shown that standard results in the literature, i.e., Taylor's modification [2] of the Einstein suspension viscosity formula for emulsions [3], can be derived. In fact, it also led to useful results beyond the regime of applicability of Taylor's formula [4]. The second problem arises in the growth of surfaces by deposition. Such surfaces, provided they are smoothed over by the local arrangement of the particles, clearly have a topological integrity, i.e., one can deposit to find Fig. l(a) and (b), but not Fig. l(c). The complication associated with keeping the topological integrity is avoided in the case of ballistic deposition. It is assumed that the growth of the surface can be described as in Fig. 3, Fig. 3(c) being a smoothed version of Fig. 3(b). A parametrization of the surface in terms of height function, h(x, y, z, t), is thus possible and obviously, the topological integrity is trivially preserved. The full problem, however, is not so simple. A surface 0 = 0 grows by deposition of diffusing particles. Fig. 3 no longer describes the surface which can take up configurations like Fig. 4. It can be described by 0 = 0 but not by a parametrization analogous to h = h(x, y, z, t), not even in spherical or polar coordinates.

S.F. Edwards, M. Schwartz/Physica A 227 (1996) 269-276

E]

271

[]

(e)

(c)

Fig. 3. The deposition (a) leads to a surface (b); (c) is a smoothed version of (b).

Fig. 4. A shape that does not allow a simple parametrization.

In the case of ballistic deposition we were able to obtain the power laws describing the statistical properties of the growing surface, by going over from the stochastic K P Z equation to a differential functional equation for the distribution function of the heights at all point r in space [-5]. In this paper we derive the equations for the D L A in terms of 0 field and show that they have the topological integrity required. We arrive for the ~bfield at an analogue to the K P Z equation, i.e., an explicit stochastic evolution equation for the ~ field. The stage of the evolution equation for the distribution of ~b at all r's is not considered here and will require much more work.

2. The use of a guage field to describe the dynamics of surfaces

The dynamics of deformable bodies with a well defined surface can be well represented by a gauge field ~b such that ~bis non-positive inside each of the bodies and positive outside. The surface is the boundary ~b = 0. The key to the topological

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S.F. Edwards, M. Schwartz / Physica A 227 (1996) 269-276

integrity will be the equation of motion of ~ where a velocity field v (r, t) will be found such that

~0 0~- + (v. V)~, = O.

(2)

Immediately one can see that the surface ~ = 0 cannot evolve into a shape which cuts itself because this would imply different velocities for elements of the surface which are closing in on each other, which cannot be so from Eq. (2). The value of v for a moving membrane has been given in a previous paper and exploited in later papers [-1,4,6,7]. In this paper we consider matter landing on and being incorporated into a swelling surface. The equations can easily be modified for loss as well as gain, but there is no difference in principle so we stay with the DLA case. The physical picture we have in mind is the following. The system is composed of two phases: a gas phase of diffusing particles and a solid phase of constant microscopic density P0- All the amount of material that hits the boundary of the solid phase is absorbed onto it, leaving thus the gas phase and becoming part of the solid phase. In the solid phase the density of solid is constant and the density of gas, po, is zero. We describe the solid phase by having a field ~ that is non-positive. In the gas phase p~ > 0 and ~ > 0 (Fig. 5). We view the physical process described above as being made up of two separate processes: (a) Gas particles diffuse and are being absorbed at the boundary of some evolving surface. The surface absorbs all the gas that hits it. (b) The material that disappears from the gas phase (absorbed) becomes part of the solid and changes the surface. The process (a) is described usually by the following diffusion equation: ~& = D V2pG + sources vp_y

in the region ~ > 0 ,

(3)

with the boundary condition PG = 0 on the boundary of that region. It will serve our purpose, however, to consider instead, a totally equivalent equation that describes Po throughout space, assuming initial conditions with po = 0 in the non-positive 0 region. The equation describing the diffusion for a given rate of absorption per unit time ~(r, t) at a given point of the boundary r and t, is •PG _ D V z p G q_ s o u r c e s

~t

lim ~(r, t)6(O -- e) [ V~9l. e--+0 +

(4)

For given qS(r, t) the solution of (4) is continuous at the boundary, while the derivative normal to the surface is discontinuous. If we require that ~b(r, t) be such, that the normal derivative is zero at the boundary on the non-positive 0 side, we find that because of the initial condition PG = 0 on that side, that pG remains zero in that

S.F. Edwards, M. Schwartz/Physica A 227 (1996) 269-276

273

~b >O ,o > o

Fig. 5. The region with ¢ > 0 PG > 0 is the gas region, the ¢ <~0 p~ = 0 is the solid region.

region for all times. This is because the flux emanating in the direction of non-positive from its boundary is the only source for that region, Since pa is continuous it is also zero on the boundary. This explains why Eqs. (3) and (4) (with the initial condition on pa and the requirement of total absorption on ~b(r, t) are equivalent. We find also that 0(r, t) (on the boundary) is qS(r, t) = D(Vpa)+-n,

(5)

where + denotes approaching the boundary from the region of positive ~ and n a unit vector normal to the surface and pointing into the region of positive ¢. The combined meaning of Eqs. (4) and (5) is just that the current density hitting the surface from the gas side is exactly that absorbed by the sink on the boundary. The amount of material absorbed from the gas phase at a point r of the boundary per unit time per unit area is given by Eq. (5). Therefore the normal velocity, V,(r), of the growing surface is just 1

V.(r) = - - D (VpG)+ "n.

Po

(6)

We define now a velocity field V(r) t h r o u g h o u t s p a c e , 1

V(r) = ~

D Vp~(r).

(7)

Po (This is identically zero in the region of non-positive ~.) We now write down an equation for the evolution of the ¢ field throughout space: a¢ ~ t + v ( r ) . v o = 0.

(8)

It is obvious that Eq. (8) implies that each point R on the boundary of the positive ¢ region can be viewed as evolving in time according to the equation [~ = V ( R ) ,

(9)

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274

so that indeed V.(r) is given by Eq. (6). Eq. (4)-(7) can now be reduced to a single equation for the ~ field. First we express Eq. (4) in the form 3p~ V2 po+sources+ & -D

lim

8

a~

(10)

_~x--Po~76(~--e).u+cr

The solution for po is

p~ = pS + mli0+e~ po f at' dr' GEr - r', t - t'] ~-i 0~ (r', t ' ) 6 [ ~ ( r ' , t')

c],

(11)

where pS is the density that would have existed just in the presence of the sources and G is the Green function for the free diffusion equation. The equation for the ~/field reads now 0~ 0t

lira D f dr' dr' V G ( r - r', t - t')" V~k(r, t) e --* 0+ - -

x ~ ; (r', t') 6[0(r', t') -- e] = -- - - D VpS" VO. Po

(12)

It is interesting to note the following points: (a) The equation is gauge covariant, i.e., the transformation ~--*f(O) where sign If(x)] = sign(x), leads to the same equation. This must be expected, since only the boundary has any physical significance. (b) When P0 = oo or in the absence of external sources, 0¢/49t = 0 as expected. Eq. (12) can be modified slightly if we assume that Po is very large compared to pa. In such a case the adjustment of the gas density to the growing boundary may be considered instantaneous (on the time scale of the growing boundary), so that G(r - r', t - t') can be replaced in three dimensions by 1

1

4reD Ir -- r'[

~(t - c ) .

The simplified equation reads 0~ •t -~

-

lim 1 (" 1 J dr' V - - e --* 0 + 47c Ir - r'l

- - - D V p ~ (r, 0 " V 0 ( r , t ) .

Po

V0(r, t) ~E4'(r', 0 - ~] ~-~0t(r', t)

(13)

S.F, Edwards, 34. Schwartz/Physica A 227 (1996) 269-276

275

3. High momentum cut-off The origin of the r a n d o m character in which the boundary evolves m a y be either in the fluctuations of the sources or in r a n d o m initial conditions. T h e first case would correspond to a contribution on the right-hand side of (12) or (13) that is r a n d o m in space and time. (This mimicks most of the numerical simulations in which single particles are generated at some boundary of the system and left to diffuse towards the growing cluster). In the second case a flat (or spherical) surface is considered in the presence of a uniform current due to the external sources. Without any disturbance the b o u n d a r y remains flat (or spherical) at any later time. Slight r a n d o m deformations of the surface at t = 0 will give rise to the typical violent D L A growth. M a n y investigators have realized that the continuous D L A problem (that we describe by Eqs. (12) and (13), is actually ill posed and needs a high m o m e n t u m cut-off [8]. The reason is that in the regime where the disturbances are still small, a disturbance with wave vector q will grow exponentially in time with an exponent that is proportional to q. That means that not only the growth is unstable (as expected) but that the surface that we have assumed to be smooth enough (to possess a well defined normal at each point) m a y cease to have this property in an infinitesimal time even if it were initially smooth. This problem has received attention in the literature. The numerical simulations have of course a natural ultraviolet cut-off related either to the lattice spacing or to the finite size of the particle if continuum diffusion is considered. We can introduce such a cut-off by hand into our Eq. (12) or (13) by smearing the velocity field (Eq. (7)),

Vsm(r ) = - -

1

Po

D | VpG(r') g (It -- r'l) dr',

3

(14)

where the smearing function, 9 has some finite range Ro. In such a case the growth exponent cq is replaced by cqO(q ), where O(q) is the Fourier transform of 9(0, so that for q < 1/Ro the fluctuations still grow as before but the surface cannot become infinitely spiky, because the relative weight in the large q disturbances decreases now with time (in the linear regime) due to the effective cut-off introduced by 0(q.) Another way of introducing an ultraviolet cut-off would be to consider the tendency of the surface to avoid large q fluctuations because that increases the energy. In such a case we have to consider some mechanism of redistribution of the mass of the solid phase. This can be accomplished by adding to v(r) defined by Eq. (7) another term

V(r) = __1 D VG(r) + Vs.T.(r) , P0

(15)

where VS.T.(r ) has to move the boundary in such a way as to decrease the surface tension. Since, our equations represent conservation of material we must require also v - v~.T,(r) = o.

S.F. Edwards, M. Schwartz / Physica A 227 (1996) 269 276

276

There are of course many possible choices. One example is Vs,a~,(r) = - ~:[(V" n)n]r,

(16)

(n is defined everywhere as normal to the surface ~ = const) T denotes the transverse part of a vector field.

Acknowledgements The work of one of the authors (Moshe Schwartz) was supported by in part by the (German-Israeli Foundation (GIF) and S.F.E. by the Sackler Institute.

References [1] [2] [3] [4] [5] [6] [7] [8]

M. Schwartz and S.F. Edwards, Physica A 153 (1986) 3551. G.I. Taylor, Proc. Roy. Soc. London A 138 (1932) 41. A. Einstein, Ann. Physik 19 (1906) 289, and correction to that papers Ann. Physik 34 (1991) 591. M. Schwartz and S.F. Edwards, Physica A 167 (1990) 589. M. Schwartz and S.F. Edwards, Europhys. Lett. 20 (1992), 301. S.F. Edwards and M. Schwartz, Physica A 167 (1990) 595. S.F. Edwards and M. Schwartz, Physica A 178 (1991) 236. B. Shraiman and C. Bensimon, Phys. Rev. A 30 (1984) 2840; D. Bensimon, L.P. Kadanoff, S. Liang, B. Shraiman and C. Tang, Rev. Mod. Phys. 58 (1986) 977; R. Blumenfled, Phys. Rev. E 50 (1994) 2952.