When dealing with 3D nucleation, it is evident that a substrate (which is not repulsive ) lowers the interface energy and then homogeneous nucleation is less probable than the heterogeneous one. But, when working with 2D nucleation, the problem is reversed. 2D-crystal nuclei can easily form on a given face of the same crystal (since only the edge energy of the nuclei should be spent), while the heterogeneous nucleation is more difficult to be obtained because also the new interface energy (2D nuclei/substrate) has to be taken into account. We can discuss about that, if you like, since there is a rich historical literature on this topics.

This depends of the number of heterogeneous nucleation sites. As these sites are saturated the rate will decrease, and the homogenous nucleation rate could at some point be higher. For example, this has been studied by magnetic levitation of a sample that is heated to liquid (either by laser or induction heating). On cooling since there is no interface with a solid, which is the primary source of nucleation sites for nucleation from the melt, homogeneous nucleation will be dominant.

When dealing with 3D nucleation, it is evident that a substrate (which is not repulsive ) lowers the interface energy and then homogeneous nucleation is less probable than the heterogeneous one. But, when working with 2D nucleation, the problem is reversed. 2D-crystal nuclei can easily form on a given face of the same crystal (since only the edge energy of the nuclei should be spent), while the heterogeneous nucleation is more difficult to be obtained because also the new interface energy (2D nuclei/substrate) has to be taken into account. We can discuss about that, if you like, since there is a rich historical literature on this topics.

The number of  the articles can proved that especially in CBD. High rate of homogeneous nucleation make the solution turbidity. and high rate of   heterogeneous nucleation display clear.

In classical nucleation theory, nucleation rate is proportional to N0*exp[-DeltaG/(kT)], where N0 is the density of nucleation site and DeltaG is the nucleation barrier. So, it is possible that heterogeneous nucleation rate is lower, although having a lower nucleation barrier because there it has a lower nucleation site density. The answer of your question is then "No, your statement is impossible to prove simply because it is wrong".

I agree with Michel Perez when taking a strict point of view. Absolute questions deserve absolute answers. In this case, the answer is apparently no. However, in homogeneous nucleation, the amount of nucleation sites is only related to the deviation from the thermodynamical equilibrium conditions and the value of the nucleation barrier is only determined by the composition of the compound or agent concerned. In heterogeneous nucleation, the agents operating as nucleation sites may influence both the number of nucleation sites as well as the value of the nucleation barrier, thereby in many cases increasing the nucleation rate or decreasing the deviation from thermodynamical equilibrium conditions necessary for nucleation. Hence the overall rate of nucleation and growth is in many cases larger for heterogeneous nucleation than for homogeneous nucleation, as is commonly observed. But this is not a law of nature.

In order of magnitude, the number of nucleation sites typically are Nhom-10 29,   N grain boundary -10 23 N graincorner-10 11. In addition, we may have distribution of sites of varying potency, the greater the potency, the lower is corresponding delta G star. Delta G star het  < delta G star hom  dominates the result through the exponential factor, even though the pre-exponential factor may be much smaller for heterogeneous nucleation as compared to the homogeneous case. A measurable nucleation rate is about 10 6 per m3per sec. then for measurable rate of the exponent  must be about 85. The nucleation rate is very sensitive to the value of this exponent, if this is decreased to   about 70 say by increasing the driving force (-delta G) through a change in reaction temperature, Nucleation rate ( I ) can be increased by a factor of 10 6  i.e, to 10 12 .

Dear S.N. Ojha, 

As such the data provided by your latest remarks correspond to the reasoning of my original answer. However, these data are not sufficient for an affirmative answer to your original question. For questions containing the word 'always', one negative exception is sufficient for a negative answer. So, in my opinion, the answer remains 'no'.

Even though it seems to be widely accepted the claim you posted above, I agree with the previous answers saying that the answer remains no. From a theoretical point of view, there are weaknesses in assuming "the lower the free energy the higher the nucleation rate". Indeed, this is only true when the nucleation mechanism remains the same, e.g. 3D homogeneous nucleation, otherwise all the ingredients playing an important role within the nucleation rate expression may vary in such a way that, even with a lower free energy barrier, the kinetic pre-exponential factor affects the nucleation rate compensating the exponential disadvantage of homogeneous nucleation. Thus, there is no general way to demonstrate what you want, since your question is ill-posed in my humble opinion. If you want to read a more extensive discussion about these competing processes (within the context of the classical paradigm) you may be interested in the book on "Nucleation" of Kashchiev.