A note on the Aboav-Weaire law

Abstract

Abstract In this paper we present an alternative derivation of the Aboav-Weaire law. By first making the assumption that the mean of the number of sides surrounding a cell is a function of the time, leads to Mn, the mean number of sides surrounding a cell of sides n, as a linear function of the second moment μ, and is independent of n. This indicates that the local mean increases with time. When written in the form of the general Aboav-Weaire relation we find in their notation that a = 1 and b = 6 7 . The analysis also leads to the relation that the deviation from the ensemble average, 6, is proportional to μ, and b is the coefficient of proportionality. The initial assumption is removed and the assumption that Mn is now a function of time and the number of sides is made. This assumption leads to Mn as a linear function of the ratio μ/(n+1). This applies in the limit of small μ and the assumption that Mn can be expanded as a Maclaurin series. When written in the form of the general Aboav-Weaire law a = 1 and b = 0. This implies that the deviation from the mean 6 is then proportional to μ/(n+1). A similar analysis is applied to three dimensions and when the faces of the cells are considered we find that the average number of faces of the cells surrounding a cell of face ⨍ is a linear function of μ/(⨍+;1), where μ is the second moment of the faces with mean 14. This analysis of the mean number of sides surrounding a cell is extended to finding the mean area surrounding a cell of sides n and area A, MAn. It is found that the mean area of the cells MAn is given by MAn = Aa + (Aa − A)/n, where Aa is the local cell area average which in the first approximation can be treated as the ensemble average area. Similarly the mean volume of cells surrounding a cell of faces ⨍ and volume v is given by M v ⨍ = v a + (v a − v)/⨍ , where va is the average local volume. In this case cells of small areas (volumes) are surrounded by cells of areas (volumes) greater than the average area (volume). We can reverse the argument to show that the large area (volume) cells are surrounded by small area (volume) cells. The results of this analysis indicates that the cells are not randomly distributed.