Coupled aggregation and sedimentation processes: The sticking probability effect G. Odriozola, 1 R. Leone, 1 A. Schmitt, 2 A. Moncho-Jorda ´ , 2 and R. Hidalgo-A ´ lvarez 2, * 1 Departamento de Quı ´mica Fı ´sica y Matema ´tica, Facultad de Quı ´mica, Universidad de la Repu ´blica, 11800 Montevideo, Uruguay 2 Departamento de Fı ´sica Aplicada, Universidad de Granada, Campus de Fuentenueva, E-18071 Granada, Spain ~Received 7 July 2002; revised manuscript received 20 December 2002; published 18 March 2003! The influence of the sticking probability P and the drift velocity on kinetics and structure formation arising in coupled aggregation and sedimentation processes was studied by means of simulations. For this purpose, a large prism with no periodical conditions for the sedimentation direction was considered allowing for sediment formation at the prism base. The time evolution of the cluster size distribution ~CSD! and weight-average cluster size ( n w ) were determined in three different regions of the prism. The cluster morphology and the sediment structure were also analyzed. We found that the coupled aggregation and sedimentation processes in the bulk are governed by P for short times, and controlled by the Pe ´clet number Pe for long times. In the lower part of the reaction volume, where the sediment grows, the local n w grows at sufficiently large times analyti- cally with an exponent of four. This behavior seems to be independent of Pe and P. The obtained results are in good agreement with the experimental data reported by C. Allain, M. Cloitre, and M. Wafra @Phys. Rev. Lett. 74, 1478 ~1995!# and support the idea of a possible internal cluster rearrangement for the experiments. Finally, we discuss how the scale dependent fractal character of the sediment is related to the different stages of the aggregation process. DOI: 10.1103/PhysRevE.67.031401 PACS number~s!: 82.70.Dd, 61.43.Hv, 02.50.2r I. INTRODUCTION Due to its presence in many natural and human-made pro- cesses, aggregation and sedimentation phenomena arising in mesoscopic systems have attracted a great deal of interest for pure science and industrial applications @2,3#. Although they are frequently found simultaneously and show a cooperative behavior, it was not until the last few years that scientists have started to study them as a whole. The reason for that may lie in the complexity of the equations that govern the overall process. Hence, it is not surprising that simulations have turned an important tool for understanding and predict- ing the behavior of such coupled phenomena. In the literature, pure irreversible aggregation processes, i.e., those where neither sedimentation effects nor cluster breakup take place, have been classified according to the cluster sticking probability @4–6#. Freely diffusing particles which always aggregate once they collide are doing so in the so-called diffusion-limited cluster aggregation ~DLCA! re- gime @7,8#. The regime in which a large number of collisions is needed before a new aggregate is formed, is known as reaction-limited cluster aggregation ~RLCA!@9,10#. The first regime is characterized by a fast evolving cluster-size distri- bution ~CSD! and a cluster fractal dimension d f close to 1.75 @11–14#. The latter regime develops much slower in time and is characterized by d f 52.1 @15,16#. To the best of our knowledge, all so far reported simula- tions of coupled aggregation and sedimentation processes have been performed imposing DLCA conditions @17–20#. Doing so, Gonzalez found a shift of d f to higher values, which was in good agreement with the experimental results reported by Allain, Cloitre, and Wafra @1,21,22#. He also pre- dicts that the critical mass concentration required for gelation increases for increasing settling effects. In a former paper, the authors show how the time evolution of the CSD depends on the position inside the reaction volume and determine how the sediment grows at the bottom @20#. In this paper, we lift the imposed DLCA restriction and study the influence of P on coupled aggregation and sedi- mentation processes by means of simulations. For this pur- pose, a large prism with no periodic boundary conditions for the sedimentation direction was considered, and P as well as Pe were introduced as free parameters. II. SIMULATIONS The simulations were performed off-lattice and on a square section prism of side L and height H. Inside the prism volume, N 0 identical particles of radius a were randomly placed avoiding particle overlap. Two contributions to the particle movement were considered: a random Brownian mo- tion and a vertical sedimentation velocity. The time step was defined by t 0 5l B 2 /6D 1 , where D 1 5 k B T /(6 p h a ) is the monomer diffusion coefficient, k B T is the thermal energy, h is the solvent viscosity, and l B is the Brownian step length. The monomeric particles are always moved a fixed distance l B in a random direction plus an additional vertical Stokes contribution due to sedimentation. The step length for the latter contribution is given by l S 5v S t 0 5l B 2 Pe/6a , where v S 52( r 2r 0 ) ga 2 /(9 h ) is the Stokes velocity for the mono- meric particles, r is the particle mass density, r 0 is the fluid density, g is the gravitational acceleration, and Pe54 p a 4 ( r 2r 0 ) g /3k B T is the Pe ´ clet number. For further details see Ref. @20#. When a collision takes place, a random number j uni- formly distributed in @0,1# is generated and compared with the given P. The cluster collision is considered effective only when j , P is verified. Periodic boundary conditions were imposed for the horizontal directions, x and y. In the settling *Email address: rhidalgo@ugr.es PHYSICAL REVIEW E 67, 031401 ~2003! 1063-651X/2003/67~3!/031401~5!/$20.00 ©2003 The American Physical Society 67 031401-1