The free-growth model was introduced by Greer et al. in 2000. This model bases on hypothetical size distribution of nucleation substrates. The aim of this work is to develop a log-normal distribution of heterogeneous nucleation substrates for the Al-Cu alloy. The computational algorithm allowing to restore the nucleation substrates distribution was created. The input data for algorithm, that is grains density of aluminum primary phase and supercooling, were taken from literature. These data are important for inverse modelling which is based on the numerical optimization methods that allows to identify parameters of substrate distribution. The distribution described in this way may be then applied in simulation based on the free-growth model. Numerical simulations based on the free-growth model can predict the grains density of the examined Al-Cu alloy. These predictions are compared with the experimental data.
The authors acknowledge financial support from the statcment contract no. 11.11 .170 .318 task 1 (Faculty of Foundry Enginecring. AGH).
 A. I_ Greer, A. M. Rumn, A. Trmachr, P V Fvans, D. J. Bristow, Acta Materialia 48 (2000) $-2823-2835$
 T. E Quested, A. L. Greer, Acta Materialia $52(2004) 3859-3868$
I31 T. E. Quested, A. T. Dinsdale, A. L. Grecr, Acta Materialia 53 (2005) $1323-1334$.
 M. Gómy, G. Sikora, Joumal of Materials Engincering and Performance 24 3 (2015) 1150. 1156
 J. Lelito, P. L. Zak, A. A. Shirzadi, A. L. Greer, W. K. Krajewski, J. S. Suchy, K. Haberl, P.- Schumacher, Acta Materialia 60(2012) 2950-2958.
 J. Iwanciw, D. Podorska, I. Wypartowicz, Archives of Metallurgy and Materials 56 issue 3 $-(2011) 635-644$
 J. Iwanciw, D. Fodorska, J. Wypuntowice, Auchives uf Metalluryy and Materials 56 issue 4 (2011) 999-1005.
 J. S. Suchy, J. Lelito, B. Gracz, P. Leseck $Z$ ak, China Foundry 9 issue 2(2012) 184-188.
 J. Lelito, P. L. Zak, A. L. Grecr, J. S. Suchy, W. K. Krajewski, B. Gracz, M. Szucki, Composites Part B - Engineering 43 issue $8(2012) 3306-3309$
 J. Lellio, P. L. Zak, B. Gracz, M. Szucki, U. Kalisz, P. Malinowski, J. S. Suchy, w. K. A Krajewski, Mctalurgija 54 issue $1(2015) 204-206$
 W. Wolczyiski, W. Krajewski, R. Ebner, J. Kloch, Calphad- Computer Coupling and Phase Diaprams and Thermochemistry 25 issue $3(2001) 401-408$.
 P. K. Krajewski, G. Piwowarski, P. L. Zak, W. K. Krajewski, Archives of Metallurgy and Materials 59 issue $4(2014) 1405-1408$.
 R. Gonther, Ch. Hartig. R. Bormann, Acta Meterialia 54 (2006) 5591-5597.
 L.H. Cupido, P.L. Zak, N. Mahomed, J. Lelito, G. Piwowarski, P.K. Krajewski, Archives of Metallurgy and Materials $60(3)(2015) 2397-2402$.
 P.K. Krajewski, G. Piwowarski, P.L. Zak, W.K. Kraicuski. Archives of Metallurgy and Materials $59(4)(2014) 1405-1408$
 M. Gundaz and J.D. Hunt, Acta Mctall. 33 (9) ( 1985 ) 1651-1672.