Physica D 38 (1989) 315-321 Norlh-Holland. Amsterdam GEOMETRICAL SCALING OF MICROSPHERE-DEPOSITED MONOLAYERS WITH HOLES A.T. SKJELTORP Insntute for Energp Technolo~, N-2007 Kjeller. Norway Uniformly sized microspheres dispersed in water are spread as a thin film on a flat surface. Dunng the drying process, random capillary forces between the spheres produce a random packing resulting in a broad distribution of holes in an otherwise compact. grainy structure. The patterns are shown to have a fractal geometry and the cumulative numbers of holes, n(s), with size larger than s scale as n(s) ~s-'. 1. Introduction The availability of uniformly sized polystyrene microspheres with diameter in the 1-100 lum range [ I ] has proven to be very useful as a means to con- struct experimental many-body model systems to study non-equilibrium processes like aggregalion [2,3], fracturing [4], ar ' deposition [5]. The re- sulting, often fractal [6] s,,-uctures have been stud- ied using optical microscopy and have been com- pared with computer s~mulations. They have provided a basis for obtaining a better understanding of growth processes far from equilibrium. Here, we will show how thin films ofsuch particles dispersed in water can be used to study deposition onto a substrate. The attractive capillary force be- tween the spheres during drying is much stronger than the bonding to the substrate. This non-equilibrium process produces a variety of structures with holes depending on the sphere concentration. The purpose ofthis work is to show that the two-dimensional "'po- rous" system has fracta! prope~ies and that the cu- mulative numbers of holes, n(s), with sizes larger than s scale as n(s),,xs -~ ~ ~ ) Here, the scaling exponent r characterizes the hole formation. So far, there is no analytical means to pre- dict the values of r based on the complicated dynam- Essays ,..i honour ofBenml B Mandelbrol Fractals in Physics - -~. ,~ha: :n~ and J. Feder (editors) ics in the formation of the present system. it may be noted that there have been earlier at- tempts to use eq. ( ! ) to express the cumulative num- bers of objects for other natural processes like: pieces ofdrift ice [7], lunar craters [8], and dust particles in atmospheres aerosols [ 9 ]. Compuler simulations have also been used to show that the cluster-size dis- tribution in diffusion-limited cluster-cluster aggre- gation follows this power-law dependence [ l0 ]. 2. Experimental results The most suitable spheres for the experiments were ofsize 2-5 pm. For the results reported here 4.8 ~tm spheres were used. The spheres are dispersed in water and spread to a thin film on a microscope slide, fig. la. During the drp.,iitg process, capillar~ and surface Spheres ~ater Glass Plat(, (a) .... . ,r~', - tl (hi Fig. I. Schemat,c. thm-f, lm depositton expenmems: (a) mono- laser of m~crosphcres dispersed in v,aien (b) partl~ dried monoia~er 0167-2789/~9/'t,03.50 ~, Else~ ,er Science Publishers B.V. ! North-Holland Physics Pub:ishing Division )