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Substitutional solid solutions that exist in both ordered and disordered states will exhibit markedly different physical properties depending on their exact crystallographic configuration. Many random substitutional solid solutions (alloys) will display a tendency to order given the appropriate kinetic and thermodynamic conditions. Such order-disorder transitions will result in major crystallographic reconfigurations, where the atomic basis, symmetry, and periodicity of the alloy change dramatically. Consequently, the dominant scattering mechanism in ordered alloys will be different than that in disordered alloys. In this study, we present a hypothesis that ordered alloys can exhibit lower thermal conductivities than their disordered counterparts at elevated temperatures. To validate this hypothesis, we investigate the phononic transport properties of disordered and ordered AB Lennard-Jones alloys via non-equilibrium molecular dynamics and harmonic lattice dynamics calculations. It is shown that the thermal conductivity of an ordered alloy is the same as the thermal conductivity of the disordered alloy at ≈0.6T(melt) and lower than that of the disordered alloy above 0.8T(melt).