edited by JAMES P. BIRK Arizona State University
computer series, 154 The Fractal Nature of Polymer Conformations C. vijayani Deptartment of Physics Pondicherry University Pondicherry, India, 605 014
M. Ravikumar Pondicherry University Pondicherry. India, 605 014 The concept of fractals has been applied successfully to understand a variety of phenomena in physical and chemical sciences during the last few years ( I ) . This has pmvided new insights .. into the shape-related properties of random structu~(!sthat exhibit statistical sklf-similarity and invariance on scaling. For example, characterization of polymer ronfnrmations by their fractal dimensionality pmvldes an efficient bas19fir exploring the correlation hetween the complex geometrical & d G e and the interesting physical properties of polymeric materials. Theoretical investigation based on computer modelling of polymer conformations has been used extensively to gain insight into the relation between the structure and physical properties of polymem (2). The basic concepts of random-walk modelling of ~ o l v m e rconformations and their characterization gy fr&l dimension can be presented to the students of chemistrv with the help of a simple computer-modelling experiment. ~~
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Random-Walk Modelling Polymers in dilute solution always assume some arbitrarv shape that is mostly neither perfectly straight nor com&eteG coiled. In other words, the monomer units are linked to one another a t random angles. This can be modelled by a restricted random walk with fxed step length on a square lattice. The restrictions arise due to the physical constraints, such as the excluded-volume consideration. This can be achieved by a self-avoiding walk (SAW), in which the walker is never allowed to visit the same site more than once. Differentt w e s of SAW'S have been ~ . r o. ~ o s toe dmodel polymcrs by imposing various types ot'constraints on the walker and bv usina different schemes of statist~calsam-
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Fractal Dimensionality Fractals are objects having self-similarity in either a geometrical sense or a statistical sense. The fractal dimension is a measure of their space4lling property. It is the index of how the mass of a n object scales with the linear dimension of the obiect. For example. the fractal dimension of a thin string "is taken as 1because the mass is proportional to the length. Similarly, the fractal dimensions of a thin disk and a sphere are 2 and 3, respectively Thus, for obiects with rewlar eeometrical shapes the fractal dimension coincides with the topological dimension. However,
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'NOW at the Department of Physics, Indian Institute of Technology, Madras. India 600 036.
830
Journal of Chemical Education
Tempe. AZ 85281
the scaling indices for geometrically or statistically selfsimilar objects tend to be fractional. The path resulting from random walks is self-similar in a statistical sense due to its random nature. Here, the "mass" is measured in terms of the number of monomer units and is equal to the number of steps taken by the walker. The characteristic linear dimension in a geometric sense is the mean end-bend length of the polymer, measured by the mean displacement of the random walker. The number of steps in the case of a random walk on a plane is known to scale with the mean displacement as N~~
where d is a fraction. Thus, we see that d can be taken as the fractal dimension of the conformation because i t measures the scaling of the "mass" of the conformation with its linear dimension. The value of d is known to be exactly 2 for uncorrelated random walks, and it is a fraction between 1 and 2 for restricted random walks (4). I t is quite simple to write-a computer program to model a SAW. Elementary can be used to draw the sim. maphics - . ulated polymer conformations on a wmputer screen. In the present work we used a basic promam to simulate a SAW on two-dimensional square iattices. The walker starts from the origin, chooses one of the four nearest neighbor sites randomly, and makes the first move to the chosen site. This will be one step in any of the four perpendicular directions. Step length is kept constant because the monomer units of a polymer have the same length. From the new position the walker chooses the next site randomly from the four nearest neighbors on the square lattice, avoiding the site from which it has just moved. The visited sites are remembered by assigning a tag, such as some number, to each. The walk continues until the newly selected site is one that has already been visited. This is verified by checking for the tag attached to the site. Now there is no further move, and the walk is terminated. After drawing each conformation the number of steps N and the net displacument R are measured. A large number of walks are Only those containing the desired value of N are considered for statistical analysis to estimate the mean displacement 61>for that particular N. One hundred conformations were averaged for each N. The fractal dimension is measured as the slope ofthe plot ofthe logarithm of N against the logarithm of a>. Results and Discussion Students were able to visualize the conformations on the monitor and compare qualitatively the patterns arising from uncorrelated random walks with those from walks with constraints. The concepts of nonreversibility and selfavoidance can be easily presented to the class with the help of many wnformations drawn according to the algorithm described. A number of such polymer ronformations generated in the present work are shown in F i p r e 1.The wrwiiponding
