A mathematical derivation of an analytical expression is presented to evaluate the van der Waals interaction between a sphere and a cylindrical rod. This expression then is applied to study the growth of one-dimensional nanostructures, such as nanorods, using a common growth mechanism in colloidal chemistry, the oriented attachment growth mechanism. Parameters associated with the dimensions and the separation of nanoparticles and nanorods are varied in calculations to assess their influence on the magnitude of the van der Waals interaction.