Why does thermal conductivity of an alloy becomes nearly flat from $0.2I am trying to understand the effect of alloying on the thermal conductivity of an crystalline alloy. I have found a great many papers where I see thermal conductivity sharply decreases from 0 to 0.2 X of the alloy composition, then becomes nearly flat, then they start increasing again after 0.8X composition. But not one paper properly explains what thermal conductivity becomes flat from 0.2<x<0.8. Can anyone please help me with this?
 A: Heat transport in solids occurs by conduction. Under the most commonly encountered circumstances (e.g. under ambient conditions) the dominant heat carriers are typically electrons for heat transport in metals and phonons for heat transport in insulators.
The thermal conductivity $\kappa$ can be calculated as
$\kappa = \frac{1}{3} C_v v \lambda$
where $C_v$ is the heat capacity at constant volume, $ν$ is the mean velocity of the particles transporting the heat, and λ is their mean free path. What this says is that the thermal conductivity is the product of the amount of energy that can be carried by a particle (electrons in a metal, phonons in an insulator), the speed at which the given particle moves, and the average distance the particle travels before scattering.
In impure metals or in disordered alloys, the electron mean free path can be reduced by impurities. Roughly speaking, a pure metal exhibits uniform crystal structure over small length scales. The introduction of impurity elements interrupts this regular crystal structure, and increases the likelihood the electron will scatter. For very pure metals, the amount of additional scattering is likely proportional to the impurity fraction/doping. This explains the sharp change in thermal conductivity observed near X=0 and X=1.
In practice neither $C_v$ nor $v$ are likely to vary too much as the doping fraction X is changed. So with the above explanation we expect best thermal conductivity of an alloy for values of X near either 0 or 1, and to exhibit substantially worse conductivity in between. As to the exact shape of the curve versus X, that is definitely beyond the scope of my knowledge. But I will hazard a guess. According to Kittel, if the electron scattering is bad enough, the thermal conductivity due to electrons in a metal may drop low enough to be comparable to that of the phonon contribution. That is one hypothesis for the flat region you observe for 0.2 < X < 0.8.
Interestingly, the overall effect of a U-shaped thermal conductivity vs. dopant fraction is observed with insulators as well. In insulators (diamond, sapphire), phonons effect thermal transport in the material. It is observed that the thermal conductivity of both pure 12C diamond and 13C diamond is higher than that of regular isotopic diamond (98.9% 12C and 1.1% 13C). See here: https://patents.google.com/patent/US5419276
References:

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*Kittel, Introduction to Solid State Physics, pg 156

*Simon R. Phillpot, Alan J.H. McGaughey, Introduction to thermal transport, Materials Today, Volume 8, Issue 6, 2005, Pages 18-20 (https://www.sciencedirect.com/science/article/pii/S1369702105709330)

A: bufferlab gives a good answer, but I'll dumb it down a bit. Since you tag phonons (and not electrons), I'm assuming that most of your thermal transport is due to phonons.
When $x$ is near 0 or 1, then you basically have a crystal with some impurities. E.g. if you're dealing with $Si_xGe_{1-x}$, then if $x$ is close to 0, then you more or less have Ge, and if $x$ is close to 1, then you more or less have Si.
However, if $x$ is in the middle, then you just have a disordered mess where your phonons are scattered all over the place. In fact, it's a little difficult to even define phonons at this point, because phonons as we normally think of them exist only in regular lattice --- not alloys. Arguably, $x=0.4$ is "more" disordered than $x=0.2$, but in either case, it's so disordered that it doesn't make a huge difference. Hence, thermal conductivity is basically the same for a fairly wide range of $x$.
