The crystallization of a binary alloy from a cooled boundary is analys
ed and the structure of the resulting solid product examined. The anal
ysis includes effects due to differences in densities between solid an
d liquid phases and due to a non-zero segregation coefficient of the a
lloy. We determine the general conditions under which the solidificati
on takes place either at a planar interface or throughout a mushy laye
r of solid dendrites bathed in interstitial fluid. We are able to make
significant analytical progress by finding appropriate similarity sol
utions of the governing differential equations, and then simplifying t
hese solutions in the asymptotic limit of practical interest wherein t
he compositional diffusivity is very much less than the thermal diffus
ivity. Thereby we determine a simple criterion that distinguishes betw
een the solidification at a planar front being controlled mainly by th
e transport of the rejected component away from the phase boundary rat
her than by the associated thermal transports. We also calculate the s
olid fraction and the degree of micro-segregation in the mushy layer.
In addition, the density difference between the liquid and solid phase
s is shown to induce a flow of the liquid phase during solidification,
which causes macro-segregation of the resulting solid product.