---
abstract: 'The profile of vapor around the droplet growing in the diffusion regime with known moving boundary have been calculated and expressed in terms of special functions.'
author:
- Victor Kurasov
date: 'Victor.Kurasov@pobox.spbu.ru'
title: About the depletion profile in nucleation kinetics 
---

The problem of determination of depletion zone around the growing droplet is widely investigated in modern publications, meanwhile the problem of nucleation kinetics construction has been already solved in [@kurasov]. Now this fact in widely recognized [@korsakov] but still several attempts [@zuv] were made to present some other (less sophisticated) models of metastable phase depletion. Since the approach of stationary solution [@zuv] was strongly criticized in [@lanl02] the new version [@schekin] of the theory mainly by the same authors of [@zuv] takes into account non-stationary effects. Namely the mentioned publication initiates this consideration.

One has to stress that there is no need to find the rate of droplet growth, this value has been determined with a great accuracy. Results on growing interface are widely published nowadays (see for example Physica A) and can be found elsewhere. So, it gave the opportunity to present in [@kurasov] solution based on given intensity of growth. This solution is absolutely balanced and was also discussed and refined in [@lanl97], [@lanl98], [@lanl02], [@vestnik_profile].

Nevertheless the authors of [@schekin] continued to construct the solution of self consistent problem of the growing droplet. They failed to solve the precise problem and they formulated some iteration procedure and wrote the solution as an integral in the first approximation. One has to note that earlier a more general solution was written in [@lanl02] in application to nucleation, but in principle it is well known [@eger]. In [@lanl02] the general solution, which closes the question of obtaining the profile on the base of the known moving boundary, was given. Certainly, one can fulfill the next step of this recurrent procedure but it is absolutely useless to concrete numerical results.

The profile of droplet was given in [@schekin] as a value proportional to the integral $$I = \int_z^{\infty} \exp(-x^2) (1 -
\frac{z^2}{x^2}) dx$$ Having corrected the evident misprint (in the next paper [@last] the authors of [@schekin] used the results of [@schekin] with corrections, i.e. the right variant of formulas) we can reduce it to $$I = \int_z^{\infty} \exp(-x^2) (1 -
\frac{z^2}{x^2})^{1/2} dx$$ One can easily calculate this integral which is equivalent to the integral calculated in [@lanl98].

Really $$\int_z^{\infty} \exp(-x^2) (1 - \frac{z^2}{x^2})^{1/2} dx
=
2 z^2  \Gamma(3/2)
\Psi(\frac{3}{2}, \frac{3}{2} ; z^2)$$ Here $\Gamma$ is the Gamma function, $\Psi $ is the Confluent Hypergeometric function.

We have to note that in [@kurasov], [@lanl98] the variable $z$ was shifted on a radius of droplet, which gives equivalent asymptotic expressions far from the embryo, i.e. in the regions important for nucleation. But

In [@schekin] none of publications [@lanl97], [@lanl98], [@lanl02], [@vestnik_profile] have been mentioned or cited in any way.

It is more than symbolic that in [@schekin] the authors stopped at uncalculated integral and present only numerical pictures. Why was the integral left uncalculated? To give the answer one has to keep in mind the following arguments:

-   Really, in [@kurasov], which was also unmentioned by authors of [@schekin] the integral was also left uncalculated.

-   Only in the little paper [@lanl98] lying in lanl-archive the representation via standard special functions was given. This representation is the most natural and the absence of this representation in [@kurasov] was the certain defect of narration.

-   The paper [@lanl98] was the only paper which has no analogous variants in Russian.

-   One has also to take into account that all authors of [@schekin] except H.Reiss work at the same department of Physics in St.Petersburg University, where the author of [@kurasov], [@lanl97], [@lanl98], [@lanl02], [@vestnik_profile] works. Contrary to [@lanl98] the papers [@lanl97], [@kurasov] have been published in the St.Petersburg University Scientific Journal or have been reflected there as Summaries.

As the result one can make further conclusions which are so trivial that can be omitted here.

[99]{}

V. Kurasov, Physica A , vol. 226 (1996) p.117 - 136

V.B. Kurasov, Deponirovano v VINITI (Deponed in VINITI) 5147-B from 1.06.1989, 50p. (in russian)

V.B. Kurasov, Deponirovano v VINITI (Deponed in VINITI) 2590-B95 from 19.09.1995, 47p. (in russian)

Kurasov V., Explicit role of density profile in nucleation kinetics Second international workshop NPT-2002. Book of abstracts, SPb, 2002, p. 39-40

A.P.Grinin, A.K.Schekin, F.M.Kuni, E.A.Grinina and H.Reiss, [*Non Steady profile of vapor concentration around the Growing Droplet: Balance of the condensing Vapor and the Moving Boundary of the Droplet*]{}, In: Nucleation and Atmospheric aerosols, 16-th International Conference, Kyoto, Japan 2004, Kyoto University Press, Editors Mikio Kusahara and Markku Kulmala, p.375-379

A.P.Grinin, I.A. Zhuvikina, F.M.Kuni and H.Reiss, [*Nearest-Neighbor Drop in Homogeneous Nucleation in a Supersaturated Vapor* ]{}, In: Nucleation and Atmospheric aerosols, 16-th International Conference, Kyoto, Japan 2004, Kyoto University Press, Editors Mikio Kusahara and Markku Kulmala, p.15-19

V. B. Korsakov , R. A. Suris Nucleation kinetics under low vapor supersaturation, Second International workshop “Nucleation and non-linear problems in first order phase transitions” (NPT’2002), Book of abstracts, 1 July - 5 July 2002 , St. Petersburg, Russia p. 68-69

Zuvikina I.A., Grinin A.P., Kuni F.M., Stochastic regularities of new phase nuclei ebullition in first order phase transitions, Second International workshop “Nucleation and non-linear problems in first order phase transitions” (NPT’2002), Book of abstracts, 1 July - 5 July 2002 , St. Petersburg, Russia, p.54-57

V.F.Zaitsev, A.D.Polianin, A handbook on differential equations in partial derivatives, Precise solutions, Moscow, 1996, 496 p.

Karsloy G., Eger D, Teploprovodnost’ tverduh tel (Heat conductivity of solid bodies), Moscow, Nauka, 1964, 488 p.

V.Kurasov Preprint cond-mat@lanl.xxx.gov get 9806342

Kurasov V.B., Vestnik (Herald)St.Petersburg University, Seria 4, 1999, Issue 2 (N 11) p. 80 - 83

V.Kurasov Preprint cond-mat@lanl.xxx.gov get 9810022

V.Kurasov Preprint cond-mat@lanl.xxx.gov get 0207772

Kurasov V.B., Vestnik (Herald)St.Petersburg University, Issue 3 (20), 2002, p. 19-31
