Situation seems very like to corrosion processes. Try to seek solution of diffusion problem for reaction diffusion. As I remember thickness of layer of product of corrosion increase as SQRT of time.

Dear Dr. Titov, here there is no chemical reaction  taking place. (See: W.JOST, CH II) There is diffusion of copper into the matrix of  TiN  intermetallic compound, where a new second  phase  Cua Ti1-a N  forms as a surface layer when the concentration of copper reaches to a certain level denoted by  'a'  and the rest of  the matrix contains copper as a diffuse solute species, and its concentration may be   described by the complementary error function. As you have guest properly  the time dependence of  the surface layer thickness follows universal behavior namely h2= k (t).  The problem is here to answer the question of 'how the coefficient  k  is connected to the diffusivity associated with the  stochiometric compound, which has a very small deviation due to existence of vacancies in the range  nV=10-4 .   

The method that we employed to assess the activation energy of the diffusion of indium through a very thin layer of alumina (towards a TiON layer) are described in J. Appl. Phys. 112, 053527 (2012). It is assumed that all of the indium that arrives to the TiON layer is trapped. Only the total amount of diffused material is employed in the calculations.

Dear Alberto, I have read your article as you cited above. It was a excellent experimental work but it came to the diffusion phenomena you guys all flipped! Whole your analysis relies on the assumption that there is a constant source of indium, which keeps the flux of this species was constant of time, which of course implies you were dealing with steady state diffusion. This was a false presumption. Secondly, you were not aware of the mechanism of diffusion of indium in this system, whether it was interstitial type of atomic hopping motion or vacancy exchange was operating. No matter what, the temperature dependence of the solubility of indium doesn't enter into the scenario. Energetically the plausible mechanism is the vacancy exchange which involves much less enthalpy of formation then the enthapy of formation of indium self-interstitials because of size consideration. Thirdly; you were using only two temperatures, namely; 500C and-700C to evaluate the activation energy(enthalpy) of diffusion, which doesn't assure you by looking at the semi-logarithmic plot that the Arrhenius relationship is operating for this system. You needed at least three experimental points lying along the straight line with a reasonable scattering.

What you were supposed to do,  was  to plot the square  of the  measured  thickness  h2   of In-O  layer as a function of time   t , which definitely results straight line as suggested by Dr. Titov, and get the slope, k.   This slope involves   k  involves bi-product of interdiffusion coefficient  and  the gradient of indium  Delta over the whole In-O layer.

where   k= 8/3 (D Delta)o exp(- Q/RT).   This is obtained theoretically from the asymptotic solution of the time dependent Diffusion Equation .taking D infinite and  Delta zero but  keep  the bi-product finite!!!  Delta is the stochiometric deviation (i.e., due to vacancies) over the layer, which may be in the range of 10-4

 See: Kinetics of Diffusion in the Nb-Al System, Ogurtani , Metallurgical Trans. V.3, (19729 pp 421-425. See also:  Borisov et al. and Wagner.  for the different approach to the problem.

Dear Tarik, thank you for taking the time for reading our paper. The enthalpy for indium hoping between interstitial sites is actually zero. Even more, the enthalpy for vacancy exchange could be negative (depending on the atoms involved). However, the important quantity for predicting the temperature dependence is the activation energy. As you can see from Table IV, we did considered vacancy exchange but the activation energy was too high (please take a look at the text below Figure 9 where it specifically compares these two mechanisms). And yes, we neglected the transient times. The method of h2 vs t doesn't apply to our case since we are not forming a thicker and thicker layer through which the indium has to travel. Since the amount of diffused material is very small, it varies linearly with time (again, neglecting transients). We considered a constant source of indium for the same reason. 

Dear Alberto, unfortunately you haven't answer my question, namely, what is the connection  between h versus t. which  means that the thickness of the newly developed phase changing with time.  Dr. Titov was  immediately anticipated my question since surface oxidation and corrosion problems results parabolic reactions at the surface. 

Since you didn't mention  clearly in your paper that there was no new phase appearing but just the vacant sides in TiO2  were  filling by Indium.  Where the main driving force is the chemical potential difference of Indium between the host phase: InGaAS and newly formed phase TiO2.   Where  Al2O3  doesn't play active role but just the easy passage through by Indium atoms!!

This problem still diffusion and or mass transport problem since Indium moving from one phase to another phase by going through at least  three phases successively. 

I still believe that this problem is unique,  and it deserves  better analyticall treatment/ I will attempt to do.  Best regards.

As for me your problem is similar to the so-called Stefan problem. It was resolved many times in literature; originally for the temperature distribution in water – ice system, or so.

In your case this is diffusion with the diffusion coefficient depending on the concentration of diffusing element: Dcu= f(Ccu)

But, in any case: did you measure Cu distribution precisely? By what technique(s)?

Dear Dr Kudriavtsev; As a graduate student at Stanford, I was first introduced to the problem of diffusion in multiphase binary systems in the course entitled as Advanced Phase Transformation (1960-1961), where the book written by W. Jost (in English edition:1952) used as a reference [German Ed. ' Diffusion und chemische Reaction in festen stoffen' (Leipzig-Dresden, 1937)]. We had also the first edition of J. Crank (1956). The Stefan problem without spelling his name was extensively treated by C. Wagner related to the phase transformations, and his unpublished work appeared in unabridged form in Jost (1952). Unfortunately, J. Crack didn’t cite Wagner work even in his recent book on ‘Free and moving BC problems’. (1982). I think through mathematical treatment of multi-phase moving BC given by V. J. Borrisov et. al., Phys. Metals. Metallog. (USSR) 1965 vol. 20 p. 63. I had the English translation of this article accidentally. But the asymptotic solution where the concentration gradient goes to zero while diffusivity is approaching to infinite keeping the product is finite value was treated by me, and its application appeared in my paper on the diffusion in Al-Nb See: RG). C. Wagner mean while published a paper ‘Acta Met., 1969, v. 17, p.99 on tarnishing, and he also deal with the similar problem but he could get the desired answer on k dependence on D GradC product. No one still has spelt the name of Borrisov in the literature!!! Who applied his theory on Fe-Cr system successfully. Best Regards. Happy New Year.

Dear Yuriy, I was working on Niobium-Aluminum system. At the niobium surface (800-1300C) Al2Nb layer was formed, which is a stochiometric intermetallic compound according to Wells. Later I was informed by Bob Huggins of Stanford that this compound had the highest Superconducting transformation temperature known up to 1972. I obtained activation enthalpy of D Grad C product out of  measured k as well as its preexponential factor. Met. Trans. V. 3 1972 p.423.( See please for a  copy in RG), Regards

Dear Yuriy, I didn't study Cux Ti1-x N but I made extensive experimental and theoretical work on the Al3-Nb intermetallic compound form at the surface of the niobium as a new phase during the diffusion of Al atoms in the aluminum melt into the Nb bulk system (900-1300C). In this stochiometric system one has to rely on the atomic hopping motion of aluminum atoms in Al3-Nb via available aluminum vacant sites. The obtained activation energy 36.5 Kcal/mole is very similar to the one in pure aluminum (30.5 Kcal/mole).

Dear Tarik,

As I see, my knowledge in this area is not so extended to help you. I am SIMS analyst and sometimes deal with the diffusion process (or inter diffusion) in some experimental samples. This is a reason why I asked you about experimental profiles of Cu in TiN.

Last time, when I had some problems with quantification of an experimental diffusion profile I used the book of A.V. Luikov "Theory of heat transfer", Moscow 1967, (Vusshaya Shkola, in Russian). There are several solutions of the “Stephan problem” and several solutions for diffusion with phase transition and with variable coefficient of diffusion (heat transfer) in this book.I have no idea if it was translated, but in Russia this book is a classic in mass and heat tranfers area. I have an electronic copy of the book.

Below there are some references for this author.

1.Luikov A.V. Heat and mass transfer in capillary - porous bodies, Advanced in Heat Transfer, V.I. Academic Press, 1964

2. Luikov A.V. and Mikhailov Y.A. Theory of nergy and mass transfer, Prentice - Hall Inc, New York, 1961, Pergamon Press, London, 1965

3. A.V. Luikov, Analytical Heat Diffusion Theory, 1968

I think the last is translation of the book above mentioned.

With best wishes and happy New Year!