Hi Ahmed,

It depends if the two components of the alloy are fully intermiscible (1) or if there is an intermiscibility gap in the phase diagram (2).

(1) good examples are Ag-Au, Pt-Pd. Here, you have a statistical mixture of atoms in the full phase diagram. The resistivity rho shows roughly a parabolic behaviour with a maximum somewhere near 50%. Here, not the resistivity, but the temperature gradient drho/dT follows (roughly) Vegard's law. For small amounts of impurities, the resistivity increases roughly linearly with amount. This is related to Matthiessen's rule. It does not matter if the single crystal of the impurity component has a lower resistivity - as atomic impurity it is a scattering center and will increase rho. If at some ratios of the constituents, ordered structures are preferred, the resistivity can drop nearly to the values of the single elements, Cu-Au is a good example with the ordered structures Cu3Au and CuAu with clear minima in rho. If these ordered structures on the other hand form more or less covalent bonds, their behaviour changes from metallic to semiconducting, and rho exhibits maxima at these ratios, examples are III-V systems. But this, similar to the superconductors previously mentioned, is a different story.

(2) Examples: Cu-Ag, Pb-Sn, Zn-Sn. Assuming no intermiscibility at all, one gets a mixture of crystallites of the components A and B, (n reality, both with a tiny amount of B and A, respectively (which for simplicity one can ignore)). Then the resistivity follows Kirchhoff's rule and depends on the microstructure. For lamellar microstructures perpendicluar to current the resistivities add up (resulting in Vegard's rule); parallel to current, the inverse add up. For a mixture or globular microstructures (or real samples for that matter), it is somehwere in between. A good fit is usually a logarithmic Vegard's rule ln(rho) = x_a  ln(rho_a) + (1- x_a) ln(rho_b). If these alloys are quenched from the molt and disorder is frozen in, case (1) is possible again.

(3) If there is just a partial miscibility gap, it is a mixture of (1)  at the edges of the phase diagram and (2) in the region of the miscibility gap.

The reason for this different behaviour is basically the size of the defects compared to the wavelength of the electrons. For large scattering probabilities, the size of the defect has to be of the order of the wave(length) to be scattered. That's the case for atomic disorder, but not for micron-sized precipitates (ignoring again impurities in the precipitates and the grain boundaries). A simple analogon would be: We can hear around the corner but not look around the corner.

Short answer: Both, Vergard's rule and larger rho, is possible.

Ideal mixing or Vegard's law can be applied to determine suitably weighted (usually mole fraction) properties in certain cases provided the components are mutually soluble in each other or form a solid solution (same phase).  However, in most cases one will encounter either positive or negative deviations from ideal mixing.  If the components are not mutually soluble in each other, then the deviations can be significant.  Single crystal alloys will behave differently compared to polycrystalline alloys since the additional component might segregate to grain boundaries at low concentrations.

It is not mandatory that all properties follow the Ideal mixing law. Certain deviations to exist when coming to conductivity properties.

The fully misible alloys are likely to show very similar properties (not always) but the intermetallic compounds if they form can have wdely different properties than the constituents.

Because of formation some phases which may be more resistant than the main components of the same alloy.

In metallic alloys it is very common that mixing of higher conductivity materail to other metals does not decrease the resistivity but increases resistivity of the parent metal. Is is due to increase in the scattering centres in the binary alloy as compared to singe elemental metals. In non - metallic alloys also this can happen as the mobility of charge carriers may be quite different in alloy as compared to their parent elements. Resistivity of the alloy may be increased due to mobility change even if the additive element is of low band gap as compared to parent element. But, there is no general rule for resistivity increase on alloying. In some cases, it may decrease also. It all depends on the mobility change and resultant band gap.

It doesn't have to be so, as Kumar explains. An extreme case would be superconducting alloys.

Hi Ahmed,

It depends if the two components of the alloy are fully intermiscible (1) or if there is an intermiscibility gap in the phase diagram (2).

(1) good examples are Ag-Au, Pt-Pd. Here, you have a statistical mixture of atoms in the full phase diagram. The resistivity rho shows roughly a parabolic behaviour with a maximum somewhere near 50%. Here, not the resistivity, but the temperature gradient drho/dT follows (roughly) Vegard's law. For small amounts of impurities, the resistivity increases roughly linearly with amount. This is related to Matthiessen's rule. It does not matter if the single crystal of the impurity component has a lower resistivity - as atomic impurity it is a scattering center and will increase rho. If at some ratios of the constituents, ordered structures are preferred, the resistivity can drop nearly to the values of the single elements, Cu-Au is a good example with the ordered structures Cu3Au and CuAu with clear minima in rho. If these ordered structures on the other hand form more or less covalent bonds, their behaviour changes from metallic to semiconducting, and rho exhibits maxima at these ratios, examples are III-V systems. But this, similar to the superconductors previously mentioned, is a different story.

(2) Examples: Cu-Ag, Pb-Sn, Zn-Sn. Assuming no intermiscibility at all, one gets a mixture of crystallites of the components A and B, (n reality, both with a tiny amount of B and A, respectively (which for simplicity one can ignore)). Then the resistivity follows Kirchhoff's rule and depends on the microstructure. For lamellar microstructures perpendicluar to current the resistivities add up (resulting in Vegard's rule); parallel to current, the inverse add up. For a mixture or globular microstructures (or real samples for that matter), it is somehwere in between. A good fit is usually a logarithmic Vegard's rule ln(rho) = x_a  ln(rho_a) + (1- x_a) ln(rho_b). If these alloys are quenched from the molt and disorder is frozen in, case (1) is possible again.

(3) If there is just a partial miscibility gap, it is a mixture of (1)  at the edges of the phase diagram and (2) in the region of the miscibility gap.

The reason for this different behaviour is basically the size of the defects compared to the wavelength of the electrons. For large scattering probabilities, the size of the defect has to be of the order of the wave(length) to be scattered. That's the case for atomic disorder, but not for micron-sized precipitates (ignoring again impurities in the precipitates and the grain boundaries). A simple analogon would be: We can hear around the corner but not look around the corner.

Short answer: Both, Vergard's rule and larger rho, is possible.

If one reads the comments made Jens it is very clear that the conductivity [that  is in general a tensorial property in the case of anisotropic, which  connects two vectorial quantities such as  current density and  the electric field. ] can' t obey any simple additive role in alloys and especially for the  intermetallic stochiometric compounds where one has to face the complete miscibility in chemistry language or the complete solid solubility of its constituents as we used to address in physical metallurgy.

Only if one have some  mechanical mixtures or composite material where the individual elements and/or  phases constituting overall micro/macro  structure while  keeping  their individual properties in tact  than the average values of those properties can be estimated from their volumetric and /or  areal fractions depending upon the micro/macro texture or distribution. I don't think here any mole or weight fraction enters directly into the scenario but only the space distribution matters.  

If the solvent element from group V the resistivity will higher than that of the solvent and solute. Also the behavior can be changed to semiconducting it depends on the type of the solute atom. Actually this can be explained in terms of the change in the band structure of the solvent.