Kinetics and fractal properties of the random sequential adsorption of line segments

The random sequential adsorption (RSA) of infinitely-thin line segments is studied by computer simulation. It is shown that the exponent for the time dependence of the surface coverage increases beyond the value 0.33, recently reported by Sherwood, to a value of 0.38, at later times. It is also argued that the fractal dimension of the adsorbed lines is about 1.8, making use of a previous result on the RSA of rectangular objects. The problem of random sequential adsorption, although an old one (e.g., Flory 1939, Renyi 1958, Mackenzie 1962, Widom 1966, Solomon 1967, Gonzales et a1 1974) has recently been attracting renewed interest (e.g., Rosen er a1 1986, Burgos and Bonadeo 1987, Meakin et a1 1987, Schaaf and Talbot 1989, Baram and Kutasov 1989). In two dimensions, where most results can be found only by extensive computer simulation, earlier work generally concerned simple discs (e.g., Feder 1980), aligned squares (Tory and Pickard 1979, Brosilow et a1 1990) and lattice models (Evans et al 1983, Nakamura 1986). The study of unsymmetrical objects has begun in earnest fairly recently. On lattices, various animals have been considered by Nord and Evans (1985) and Barker and Grimson (1988), while on a continuum surface, randomly-oriented ellipses have been investigated by Talbot er a1 (1989), and rectangles by Vigil and Ziff (1989, 1990). These last two systems are the subject of this comment. In their work, Talbot et a1 argued that the saturation coverage, OSAT, should follow the behaviour @ s A T @ ( f ) kr-P for t+w (1) where r is the time and k is a constant, with p = f. Here time is proportional to the number of trials T, and is made dimensionless by defining it by t = TAObj/Atot, where A,, is the area of the object being adsorbed and A,,, is the total area of the surface. Talbot et al’s result is in contrast to the behaviour for discs, where (1) is followed, with p = f (‘Feder’s law’) (Feder 1980, Pomeau 1980, Swendsen 1981). Before the work of Talbot er al, Feder’s law was believed to be rather universal, but now it appears that discs are a special case, as Talbot et a1 argue that p = f should apply to any non-circular object that is adsorbed in random orientation, on a continuum two dimensional surface. Talbot et a1 carried out simulations with ellipses which showed fair agreement with their prediction. It was also shown to be followed fairly well for randomly-oriented rectangles by Vigil and Ziff (1989), as described in the note at the end of that paper and in more detail in a forthcoming publication (Vigil and Ziff 1990). t Present address: Department of Physics, University of Texas, Austin 78712, USA 0305-4470/90/215103 +06%03.50 @J 1990 IOP Publishing Ltd 5103 5104 R M Zi$ and R D Vigil In a recent work, Sherwood (1990) provided further numerical evidence that p = f for ellipses. Sherwood also noted that the saturation coverage @SAT reaches a maximum at aspect ratio (or eccentricity) b / a = 0.5, a result that was also indicated in Talbot et al’s work. Here we wish to point out that this same behaviour was seen previously for rectangles by Vigil and Ziff (1989), where it was found that @SAT also reaches a maximum at an aspect ratio of just about 0.5. In figure 1, we plot Sherwood’s results for ellipses (taken off the figure), and the result for rectangles from Vigil and Ziff (1989), corrected for p = 4 as mentioned in the note at the end of that paper. The two curves can be seen to show similar qualitative behaviour, with the saturation coverage for ellipses somewhat higher than that for rectangles at the same aspect ratio.