text: The excess volume is another important property in the characterization of grain boundaries. Excess volume was first proposed by Bishop in a private communication to Aaron and Bolling in 1972. It describes how much expansion is induced by the presence of a GB and is thought that the degree and susceptibility of segregation is directly proportional to this. Despite the name the excess volume is actually a change in length, this is because of the 2D nature of GBs the length of interest is the expansion normal to the GB plane. The excess volume () is defined in the following way,
  at constant temperature , pressure  and number of atoms . Although a rough linear relationship between GB energy and excess volume exists the orientations where this relationship is violated can behave significantly differently affecting mechanical and electrical properties.
  Experimental techniques have been developed which directly probe the excess volume and have been used to explore the properties of nanocrystalline copper and nickel. Theoretical methods have also been developed  and are in good agreement. A key observation is that there is an inverse relationship with the bulk modulus meaning that the larger the bulk modulus (the ability to compress a material) the smaller the excess volume will be, there is also direct relationship with the lattice constant, this provides methodology to find materials with a desirable excess volume for a specific application.