A MODIFIED GAMBLERS RUIN MODEL OF POLYETHYLENE CHAINS IN THE AMORPHOUS REGION

Polyethylene chains in the amorphous region between two crystalline la mellae M unit apart are modeled as random walks with one-step memory o n a cubic lattice between two absorbing boundaries. These walks avoid the two preceding steps, though they are not true self-avoiding walks. Systems of difference equations are introduced to calculate the stati stics of the restricted random walks, They yield that the fraction of loops is (2M - 2)/(2M + 1), the fraction of ties 3/(2M + 1), the avera ge length of loops 2M - 0.5, the average length of ties 2/3M(2) + 2/3M - 4/3, the average length of walks equals 3M - 3, the variance of the loop length 16/15M(3) + O(M(2)), the variance of the tie length 28/45 M(4) + O(M(3)), and the variance of the walk length 2M(3) + O(M(2)).