J. Phys. Chem. 1991,95, 3723-3727 zation and the chemistries at the monolayer interface and subphase will provide a much better control of semiconductor particulate film formations and permit the colloid chemical generation of periodically and spatially modulated superlattices. Most importantly, substrate-supported, size-quantized semiconductor films
3723
can be fully characterized by solid-state methodologies and can be exploited as novel electronic devices. Acknowledgment. Support of this work by a grant from the Department of Energy is gratefully acknowledged.
Nucleation In Insoluble Monolayers. 1. Nucleation and Growth Model for Relaxation of Metastable Monolayers D. Vollhardt*lt and U. Retter* Central Institute of Organic Chemistry and Central Institute of Physical Chemistry, Academy of Sciences, Berlin, 1 1 99, Germany (Received: July 20, 1990)
The nucleation theory introduced for apparent area relaxation of insoluble monolayers at constant surface pressure is characterized by two main features: (i) the overall rate of the process is described by convolution of the nucleation rate and growth rate and (ii) the overlap of the growing centers is taken into consideration. The theoretical model allows us to distinguish between diffmnt nucleation mechanisms. Two experimental examples of octadecanoic acid monolayers have been given for the Occurrence of different nucleation mechanisms. Excellent correspondenceof calculated and experimental relaxation kinetics has been obtained.
Introduction Definite mechanisms of monolayer instability are a subject of growing attention. A better understanding of the stability properties of monolayers seems possible by overcoming the difficulties in the determination of the equilibrium spreading pressure, I&,' and investigating the metastable monolayer states above In the past, literature data on the instability phenomena of insoluble monolayers were often conflicting? At present, however, it is evident that, when compressing insoluble monolayers in conventional pressure-area measurements beyond the stability limit, the highly irreversible collapse observed depends on various experimental parameters such as nature of the substances, compression rate, temperature, and prehistory of the monolayer i t ~ e l f . ~ ? ~ Thus, the metastable, region of a compressed monolayer is confined by the equilibrium spreading pressure and the "collapse pressure" where the irreversible monolayer fracture starts. Relaxation measurements at constant surface pressures7 or constant area* have been proposed to reveal monolayer instability in the metastable surface pressure region of an appropriate material. An important step toward a quantitative analysis of the observed monolayer relaxations is the possibility of distinguishing the transformation of monolayer material to any 3D structures (mostly denoted as monolayer collapse) from other loss mechanisms, e.g., desorption, evaporation. An accelerating decrease of monolayer area at constant surface pressure has been accepted as criterion for the collapse p r o c e ~ s . ~ Over the past decade, there have been several attempts to explain such defined collapse processes as nucleation mechanisms. Smith and Berglo were the first to propose the idea of modeling the monolayer collapse by homogeneous nucleation with subsequent growth of "bulk surfactant particles at the surface". However, the interpretation of these experiments is based on theories which are still imperfectly developed. For example, the assumption of additivity for the loss rates due to the formation of the critical nuclei themselves and the loss attributable to their subsequent growth is incorrect. A similar concept has been pursued to describe the effect of pH on the monolayer stability 'Address correspondence to this author at Central Institute of Organic Chemistry, Academy of Sciences, Rudower Chaussce 5, Berlin 1199, Germany. Central Institute of Organic Chemistry. $Central Institute of Physical Chemistry.
with nucleation." Furthermore, both theoretical models can only describe the initial phase of the collapse process because of the neglect of the overlap of the growing centers. The primary objective of the present work is to supply a new model free of these defiencies and to obtain a rigorous problem solution of experimental relevance. The model is based on the assumption of homogeneous nucleation and growth of the centers, describing the total collapse rate by convolution of nucleation rate and growth rate and taking into consideration the overlap of the growing centers. Nucleation-Growth Theory The present nucleation-growth theory is focused on a theoretical description of the transformation kinetics of monolayer material to overgrown 3D phase. Therefore, the extended field of all the other dynamic processes of monolayers, such as adsorption, desorption, compression-expansion cycles, collapse, and chemical or structural changes, is excluded. Based on relaxation experiments giving molecular areas as a function of time at constant surface pressure, the facets of our nucleation model are as follows. At first, a quantitative analysis is made for the material loss of an insoluble monolayer by the above transformation process. In the second subsection, the growth rate of a single center is calculated assuming definite geometrical forms and the area where the molecules are transferred. In the third subsection the nucleation law is considered and then the overall rate of nucleation and growth is calculated for two limiting cases. In the concluding theoretical section it can be seen how the model is extended by considering (1) Jalal, I. M.; Zografi, G. J . Colloid Inreflace Sci. 1979, 68, 196. (2) Vollhardt, D.; Zastrow, L.; Wiistneck, R. Colloid Polym. Sci. 1978, 256, 973. (3) Gain-, G. L., Jr. Insoluble Monolayers at Liquid-Gas Interfaces; Interscience: New York, 1966. (4) Rabinowitch, W.; Robertson, F. R.; Mason, S. C. Can. J . Chem. 1981. 33, 1881. ( 5 ) Heikkila, R. E.; Deamer, D. W.; Cornwell, D. G. J . Lipid Res. 1970, 11, 195. (6) Archer, R. J.; La Mer, V. K. J . Phys. Chem. 1955, 59, 200. (7) Sims, B.; Zigrafi, G. J. Colloid Interface Sci. 1972, 41, 35. (8) Ncuman, R. D. J . Colloid Interface Sci. 1976, 56, 505. (9) Brooks, J. H.; Alexander, A. E. In Refardotion of Evaporation by Monolayers; La Mer, V. K., Ed.;Academic Press: New York, 1962; p 245. (10) Smith, R. D.; Berg, J. C. J . Colloid Interface Sci. 1980, 74, 273. (1 1) Xu, S.;Miyano, K.; Abraham, B. M. J . Colloid Interface Sci. 1982, 89, 581.
0022-3654/91/2095-3723$02.50/00 1991 American Chemical Society
3124 The Journal of Physical Chemistry. Vol. 95, No. 9, 1991
the overlap of the growing centers. Dynamics of the Transformation of Monolayer Material to Overgrown 30 Phase. Let us consider the case where the constant surface pressure relaxation of the molecular area is based only on the transformation kinetics of monolayer material to overgrown 3D structures. Under these conditions the surface pressure (n) can be described as a function of the surface concentration (r(t))
n = n(r(t)) Thus, the total time derivative of r is obtained as
(1)
To describe the process of mass transformation into an overgrown three-dimensional (3D) phase, the time dependence of r, that is, dI'/dt, has to be known. Equation 2 provides the basis for correlating r(r)with n(t),which is accessible to direct measurements. Constant surface pressure means dn/dt = 0
Vollhardt and Retter 1. hemispherical
edge growth 2. cylindrical
L------/
h emisph erica I gr o wt h from the basal a r e a
Figure 1. Overgrowth of centers on an insoluble monolayer.
and inserting eq 10 into eq 9 the growth rate for a single cylindrical center is obtained
(3)
With eq 2 one can infer dF/dt = 0
(4) Since the surface concentration r(t)at any instant is defined as the quotient of the number of molecules in the monolayer, n(t), and the total monolayer area, A ( t ) , the eq 4 yields the expression
Now the case is considered where a hemispherical center is growing at constant rate from that peripheral edge of the molecular height d with the monolayer. Then the growth rate for one hemisphere can be derived analogously to the above12
that means Here k2 is the rate constant for hemispherical growth. Finally, the growth of one hemispherical center from the basal area can be obtained rather trivially:
dn - -- r-dA dt dt In the absence of other loss processes
dn,/dt = (M2/4p2)k33t2 (7)
where n, is the number of molecules forming 3D centers on the monolayer. Integrating eq 6 and considering eq 7 leads to the result n, = r ( A , - A ) (8) It is important to note that eq 8 represents the transformation process of monolayer material to overgrown 3D phase expressed by area changing. Growth of a Single Center. In the model proposed it is reasonable to assume that discrete 3D centers formed on the insoluble monolayer grow in a shape-preserving way. First, let us consider the nonconfined free growth of a single center on the monolayer. The laws of growth for a single center then result from assumptions about the geometry of the individual center and the faces where the molecules are deposited. To simplify matters, the growth of one single center with regular geometrical shape on an insoluble monolayer is treated. The growth of such a center will be quantified for two types of material transfer (Figure 1): (i) edge growth and (ii) growth from the basal area. Thus, three cases represented in Figure 1 will be regarded. First, a cylindrical center growing at constant rate from the circular contact edge with the monolayer is considered. Hence, for the growth rate, dn,/dt, the following balance equation holds (9)
where h is the constant height, r the radius, kl the rate constant for cylindrical growth, p the density of the center, and M the molecular weight of the molecules forming the center. Accordingly, the deposition of molecules at the periphery of the cylinder leads to volume change with time, dV/dt. Integrating eq 9 r = (Mk,/P)t (10)
(13)
k3 is the rate constant for the growth from the basal area. Equations 11-13 form the basis for a comparison of three modes of growth. Law of Nucleation. Now let us consider the nucleation regarding the formation of 3D nuclei from the monolayer material. In this system there is a uniform probability to form nuclei over time. Thus, the nucleation law is of first orderl3-lS N = N"(1
- exp(-k,t))
(14)
where N is the number of nuclei at any time, N, the total number of nuclei, and k, the nucleation rate constant. Two limiting cases of special interest can be obtained instantaneous nucleation for large k, progressive nucleation for small k, As symbolized in Figure 2, both limiting cases differ in their dispersity. In case of instantaneous nucleation there exist monodisperse centers whereas progressive nucleation leads to polydispersity of the centers. Both cases are important for deriving the overall rate of nucleation and growth. Overall Rate of Nucleation and Growth. Now the overall rates of nucleation and growth for the two typical nucleation modes will be considered. (12) Vollhardt, D.; Retter, U. In VI. Inrernurionule Tugung fiber GrenzflBchenukrive Srofle; Abh. Akad. Wiss. D D R Akademic-Verlag: Berlin, 1987; pp 172-183. (13) Avrami, M. J . Chem. Phys. 1939, 7 , 1103. (14) Avrami, M. J . Chem. Phys. 1940, 8, 212. (15) Avrami, M. J . Chem. Phys. 1941, 9, 177.
The Journal of Physical Chemistry, Vol. 95, No. 9, 1991 3725
Nucleation in Monolayers
INSTA N TANEOUS MONO DISPERSE
NUC L EA TlON CENTRES
zext = nng/nc-
000 00
The relationship between z and zat is given by the Avrami limiting law13-15
dz - = (1 dt
cylinders /hemispheres
PROGRESSIVE
POLYDISPERSE
CENTRES
z
Figure 2. Dispersity of both limited cases of nucleation.
In the case of instantaneous nucleation the overall growth rate is given by the product of the growth rate of one center with the total number of nuclei N,, dnng/dt = Nmx(dng/dt)
d%xt )
T
= 1 - exp(-zext)
(25) Inserting eq 22 and 23 into eq 25, a generalized relationship is obtained between the measured quantity A and the number of molecules forming freely growing centers by transformation of monolayer material
(17)
For progressive nucleation the overall rate of nucleation and growth can be obtained as follows. Consider the time t at any instant. The number of nuclei formed in the previous period is expressed by
!E( dt
-Z
which implies
OOGO
NUCLEATION
(23)
t-y
This expression has to be multiplied by the growth rate corresponding with the time ( t - y). Finally, it has to be integrated between 0 and t over all y. In other words, for progressive nucleation the overall rate dn,/dt of nucleation and growth is the convolution of nucleation rate and growth rate
Now the general eq 18 allows us to derive relationships for the special models with progressive nucleation under different growth conditions: for cylindrical centers with edge growth by inserting eq 1 1
This equation allows us to deduce expressions for the different kinds of nucleation and growth as well as the assumed geometrical forms of the centers: Instantaneous nucleation, cylindrical edge growth by inserting eqs 11 and 17 A0 - A -= 1 - exp(-Klt2) A0 - A , with
progressive nucleation, cylindrical edge growth by inserting eq 19 A0 - A -- 1 - ex~(-Kl't3) A0 - A , with
dn,
- = --MN,,xk,IIk12kt2
dt P for hemispherical centers with edge growth by inserting eq 12
for hemispherical centers with growth from the basal area by inserting eq 13
Uuerlap ofGrowing Cenrers. In the initial stages of the process the monolayer material is transformed to freely growing 3D centers. For this situation the overall rates of nucleation and growth are deduced in the above section. However, in the succeeding stages of the transformation process the centers cannot extend without overlapping. In the following the effect of overlapping centers is considered. It is useful to relate n, and (Ao- A) to the corresponding quantities for t = in eq 8 representing the fundamental relationship for the transformation process of monolayer material to 3D centers. Then the real normalized volume z can be expressed as nc Ao- A z=-=n,, A, - A, where A is the total monolayer area at time, t, A. the initial monolayer area, A, the monolayer area for t = m, and n,, the number of molecules transformed from the monolayer material to 3D centers. The appropriate normalized volume of freely growing centers without overlap is defined by
-
instantaneous nucleation, hemispherical edge growth by inserting eqs 12 and 17
with
progressive nucleation, hemispherical edge growth by inserting eq 20 A0 - A - I -ex~(-K~'t~/~) (33) A0 - A ,
--
with
instantaneous nucleation, hemispherical growth from the basal area by inserting eqs 13 and 17 A0
-= A0
-A
- A,
1 - exp(-K3t3)
(35)
with
progressive nucleation, hemispherical growth from the basal area by inserting eq 21
3726 The Journal of Physical Chemistry, Vol. 95, No. 9, 1991 A0
-= A0
-A - A,
1 - exp(-K
