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Nucleation in Insoluble Monolayers. 3. Overlapping Effect of the Growing Centers D. Vollhardt*lt and U. Retted Central Institute of Organic Chemistry and Central Institute of Physical Chemistry, 0-1199 Berlin, Germany Received June 18,1990. In Final Form: July 15, 1991 In parts 1and 2, a nucleation-growth model has been introduced for experimental constant pressure monolayer relaxation described as transformation of monolayer material to overgrown 3D phase. In part 3, the effect of overlap between the growing centers expected for the succeeding stages is presented. The fits of the nucleation-growthmodel, with both freely growing and overlapping centers, to the experimental data of the constant pressure relaxations of octadecanoic acid monolayers are compared. As expected, both models coincide with the experimentalresults only in the initial stages of the monolayer relaxation. Excellent agreement of the adequate nucleation-growthmechanisms to the experimental data has been obtained for the whole process of constant pressure monolayer relaxation when the overlap of the growing centers has been considered. The free growth of the centers becomes limited in the succeeding stages, as has been evidenced by increasing deviations from the calculations with the model for freely growing centers.
Introduction The steadily growing interest in monolayer structure and function stems from their wide range of scientific and technological application.lI2 Over the last decade, there have been considerable advances in developing new experimental techniques to obtain information about the microstructure on the molecular level of monolayer^.^-^ So far, surface pressure measurements have been the main source of understanding of insoluble monolayers. Constant-pressure relaxation of metastable monolayers was found to be an interesting experimental model for defined transformation of monolayer material to 3D structures. The nucleation model introduced in part 1for apparent area relaxation of insoluble monolayers at constant surface pressure has general validity to theoretically describe the transformation of monolayer material to overgrown 3D phases6 This is underlined by the excellent accordance of calculated and experimental relaxation The theoretical model is based on homogeneous nucleation and succeeding growth of the centers, convolution of nucleation rate and growth rate, and overlapping of the growing centers. In a previous study,2s8we proposed a nucleation model for monolayer transformation to overgrown 3D structures which is based on a similar theoretical background but restricted to the initial stage of growth without overlap. In this paper, we present the effect of overlap between
* Address correspondence to Dieter Vollhardt, Central Institute of Organic Chemistry, Rudower Chaussee 5,O-1199Berlin, Germany. + Central Institute of Organic Chemistry. Central Institute of Physical Chemistry. (1) Mann, J. A., Jr. Langmuir 1985, I, 10. (2) Vollhardt,D. InSurjactantsinSolution;Mittal,K. L.,Ed.;Plenum Press: New York, 1989; Vol. 9, p 277 ff. (3) MBhwald, H. Thin Solid F i l m 1988, 159, 10. (4)MBbius, D. Z . Phys. Chem. (Munich) 1987, 154, 121. (5) MBhwald, H.J. Mol. Electron. 1988,4, 47. (6)Vollhardt, D.; Retter, U. J. Phys. Chem. 1991, 95, 3724. (7)Vollhardt, D.; Retter, U.; Siegel, S. Thin Solid Films. 1991, 199, 198. (8) Vollhardt, D.; Retter, U. VI. Internationale Tagung uber Grenz-
fllchenaktive Stoffe, Abhandlungen der Akademie der Wissenschaften der DDR, Abteilung Mathematik, Naturwissenschaften,Technik;Akademie-Verlag: Berlin, 1987;p 172.
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the growing centers in the succeeding stages. When the fits of the model for nucleation and growth, with both freely growing and overlapping centers, are compared to the experimental data of constant pressure relaxations of octadecanoic acid monolayers, it can be shown where both models deviate from each other. As expected, both models coincide with the experimental results only in the initial stages of the monolayer relaxation whereas, beyond a certain time, the model derived for freely growing centers deviates increasingly.
Theoretical Background The model for nucleation and growth of 3D centers overgrown on insoluble monolayers comprehends the following key features: (i) the centers with assumed geometry grow in a shape-preserving way, (ii) the total relaxation rate is formed as a convolution of nucleation rate and growth rate, and (iii) overlapping of growing centers is considered.6 A generalized expression for different mechanisms of nucleation and growth has been derived:
A,-A -- 1 - exp[-K,(t A, - A,
- ti)"]
where A , is the initial monolayer area, A the monolayer area at time t , A , the monolayer area for t = 0 3 , ti the induction time, and the exponent x of time t a characteristic quantity of the particular mechanisms specified by the overall transformation constant, K,. Information on the actual mechanism has been obtained by the best fit of the experimental data. Now let us consider the overlapping effect of the centers. In the succeeding stages, the centers of all geometrical shapes assumed cannot grow freely in all directions since they will impinge on each other. For the formation of a 3D phase, growth has to stop at the contact points. Finally, the size of the centers becomes limited in all directions. The extent of the overlap is expressed by Avrami's limiting l a ~ . As ~ Jan~example, the overlap for both limiting cases (9) Avrami, M. J. Chem. Phys. 1939, 7, 1103. (10) Avrami, M.J. Chem. Phys. 1940,8, 212.
0 1992 American Chemical Society
Vollhardt and Retter
310 Langmuir, Vol. 8, No. 1, 1992
The exponential function of eq 1can be expanded into a series. For I-K,(t - ti)"[
