  We study the conduction of heat across a narrow solid strip trapped by an
external potential and in contact with its own liquid. Structural changes,
consisting of addition and deletion of crystal layers in the trapped solid, are
produced by altering the depth of the confining potential. Nonequilibrium
molecular dynamics simulations and, wherever possible, simple analytical
calculations are used to obtain the thermal resistance in the liquid, solid and
interfacial regions (Kapitza or contact resistance). We show that these
layering transitions are accompanied by sharp jumps in the contact thermal
resistance. Dislocations, if present, are shown to increase the thermal
resistance of the strip drastically.
