Langmuir 1993,9,3208-3211
3208
Generalized Nucleation-Growth-Collision Theory for the Formation of Lenticular Centers from Insoluble Monolayerst D. Vollhardt,*J M. Ziller, and U. Retterg Max-Planck-Institut fiir Kolloid- und Grenzflhchenforschung, 12 489 Berlin, Germany, and Bundesanstalt fiir Materialforschung und -priifung, 12 489 Berlin, Germany Received April 20, 199P A general nucleation-growth-collision theory for the formation of fluid lenticular centers from a supersaturated monolayer at the airlwater interface has been developed. For asymmetric lens growth, a general solution of the exponential law for nucleation includes the complete range from the totally progressive nucleation (k,t 0) to the totally instantaneous nucleation (k,t O D ) . Main features of the theory are that (i) the overall rate of the process is described by convolution of nucleation rate and growth rate and (ii) the overlap of the growing centers is taken into consideration. The effect of the interfacial tensions at the three-phase contact airlwaterlcenter on the both contact angles of the asymmetric lens and, thus, its geometry fador has been taken into account. Application of the present theory to experimental data of monolayer relaxation at constant surface pressure offers a possibility to calculate the respective nucleation rate constants.
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Introduction Recently, highly sensitive experimentaltechniques have been introduced to study the two-dimensional phase behavior and the morphology of amphiphilic monolayers.“ A large number of surface pressure-area isotherms of amphiphilic monolayers have a wide region above the equilibrium surface pressure which can be characterized by monolayer relaxation at constant surface pressures. This range is in the state of supersaturation, as postulated previously,6y7so that the monolayer relaxation is caused by a transition of monolayer material to the overgrown 3D phase. Nucleation is the primary stage for the formation of a new phase. The kinetics of the formation and growth of small nuclei of a new phase is one of the main problems in the understanding of phase transitions* of amphiphilic monolayers. In recent time, we have developed a nucleation-growth theory to quantitatively describe the 2D-3D transition in the supersaturated region of the isotherm^.^ This theory is based on two main features: (i) the overall rate of the kinetics is described by convolution of nucleation rate and growth rate and (ii) the overlap of the growing nuclei is taken into consideration.1° The growth of nuclei has been assumed to occur in a simple geometric shape, e.g. hemispheres and cylinders, and the material transfer has been quantified as edge growth or growth from the basal area. The two limiting cases of special interest, namely t Dedicated to the memory of J. M.
H. M. Scheutjens.
t Max-Planck-Institut fiir Kolloid- und Grenzfliichenforschung. 8 Bundesanstalt fiir Materialforschung und -priifung. Abstract published in Advance ACS Abstracts, September 1, 1993.
(1) Kenn, R. M.; Bdhm, C.; Bibo, A. M.; Peterson, I. R.; Mdhwald, H.
J. Phys. Chem. 1991,95, 2092.
(2) Mbhwald, H. Thin Solid F i l m 1988, 159, 1. (3) Hbnig, D.; Overbeck, G. A.; Mdbius, D. Adu. Mater. 1992,4,419. (4) Hdnig, D.; Siegel, S.; Vollhardt, D.; Mbbius, D. J . Phys. Chem. 1992,96, 8157. (5) Vollhardt, D.; Gehlert, U.; Siegel, S. Colloids Surf., in press. (6)Smith, R. D.; Berg, J. C. J. Colloid Interface Sci. 1980, 74, 273. (7) De Keyser, P.; Jooa, P. J . Phys. Chem. 1984, 88,274. (8) Ruckenstein, E.: Nowakowski, B. J. Colloid Interface Sci. 1990, 137,583. (9) Vollhardt, D.; Retter, U. J.Phys. Chem. 1991,95, 3723. (10) Vollhardt, D.; Retter, U. Langmuir 1992,8, 309.
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instantaneous and progressive nucleation, have been concerned with. Different mechanisms of this model have been experimentally verified, in particular with hemispherical edge growth.” The objective of this work is to develop a more general nucleation-growth-collision theory. It can be assumed that the formation of 3D centers at insoluble monolayers is based on nucleation involving a single step so that the nucleation rate can be described by the “exponential law” which is now introduced into the overall rate of kinetics. The exponential law includes two limiting cases: instantaneous nucleation and progressive nucleation. The growing centers form a new 3D phase at the monolayercovered surface of the aqueous solution. Therefore the three-phase contact airlwaterlcenter has to be taken into consideration so as to postulate lenticular growth. In a very recent paper,lZwe have developed a model for the formation of lenticular nuclei from an insoluble monolayer at the airlwater interface on the basis of the classical nucleation theory providing the possibility to obtain the critical size of nuclei, the free energy of formation of the critical nucleus-to-surroundings surface, the free energy of the critical nucleus, and the Zeldovich nonequilibrium factor. Finally, in the subsequent section of this paper, a kinetic equation is established for a generalized nucleation-growth mechanism, thus obtaining nucleation rate constants from the experimental data of constant pressure monolayer relaxations.
Nucleation-Growth-Collision Theory The theoretical description of the transformation kinetics of monolayer material to overgrown three-dimensional phase is based on relaxation experimentsa t constant surface pressures. Thus, the problem studied comprises the formation and the growth of lenticular-shaped threedimensional centers from a supersaturated monolayer at the airfwater interface. In the following subsection, first the geometric conditions of an asymmetric lens will be studied. Then the lenticular shape expressed by a geometry factor is being (11) Vollhardt, D.; Retter, U.; Siegel, S. Thin Solid Film 1991,199, 198. (12) Retter, U.; Vollhardt, D. Langmuir 1993,9, 2478.
Q743-7463/93/24Q9-3208$04.oOlQ 0 1993 American Chemical Society
Langmuir, Vol. 9, No.11, 1993 3209
Formation of Fluid Lenticular Centers
AM2
the lens from the spherical shape. Accordingly, G is defied
G = f(41)+ f(&)
(6)
If the dependence of the geometry factor on the contact angle is neglected, e.g. for hemispherical growth, as considered in our previous paper, it would be G = l/2. Both contact angles are, however, not independent of each other; on the contrary, they are coupled as a result of the three interfacialtensions of the three-phase contact. Lenticular Geometry as a Function of the Contact Angles. The mechanical equilibrium conditions of a 3D phase center at the airlwater interface is determined by interfacial tensions at the nucleus/air/water three-phase contact. The center is formed as an asymmetric lens and the Neumann-Young equations can be applied when the effects of line lension and gravity are neglected
Figure 1. Fluid lenticular nucleus formed by collisions from an insoluble monolayer at the air/water interface for a,, = 42.3 mN m-l, um = 22.1 mN m-l, and um = 33.6 mN m-l: a, air; w, water; n, center (nucleus);m, monolayer. discussed as a function of the two contact angles and the respective interfacial tensions. In the subsequent subsections,the growth of a lenticularcenter, the law of nucleus formation, the total size of all centers growing freely, and the overlap of the growing centers are being considered. Finally, the expression of the generalized nucleationgrowth model will be presented and discussed. Geometric Study of the Asymmetric Lens. When a center is formed from an insoluble monolayer at the airlwater interface the geometric shape of the new phase is determined by the conditions of the nucleuslwaterlair three-phase contact and has the form of an asymmetric lens as discussed in the following subsection. The geometric conditions of an asymmetric lens are presented in Figure 1. As can be seen, the lens volume consists of two spherical segments and can be calculated additively. The volume of a spherical segment, Vi, is defined as
Vi= (r/6)(3r2hi+ h:)
(1) where r is the radius of the contact perimeter and hi the height of the spherical segment as presented in Figure 1. According to the geometric analysis of any triangle ABMi, the volume of a spherical segment can be expressed as a function of the angle 4i
Vi= (2/3)rr3 ((1+ cot24i)3/2- (312 + cot2
cot
(2) By defining the function of the angle f(&) in the following way cot 4i f(4i) = 1 2 (1 cot2 4)3/2 - 4 (3 + 2 cot2(bi) (3)
+
,u
= uancos 41 + uwncos 42
(7)
sin 4l= uwnsin 42
(8)
u,
where uaw, am, and uwnare the interfacial tensions of the three phases air (a), water (w), and center (n). 41 is the contact angle of the center with air and 4 2 the contact angle of the center with water. Hence it followsthat the contact angles can be calculated as a function of the three interfacial tensions uaw9ban,and UWll
(10) Hence, both contact angles of a 3D center can be calculated as function of the surface pressure, rB,for monolayer relaxation at constant surface pressures (rs= up 7 uaw, where the index p means pure aqueous surface and index aw represents the monolayer-covered surface). It is a prerequisite that the respective data of uan and a , are available. On insertion of eq 4 into eq 6 the geometry factor, G, of the asymmetric lens can be obtained for the calculated contact angles 41 and 42. Growth of a Single Center. Now the unconfined free growth of a single lenticular center formed from an insoluble monolayer is considered. The calculation is based on the following conditions: (i) growth in shapepreserving way; (ii) constant growth rate; (iii) material transfer from the monolayer to the nucleus according to edge growth, i.e. across the three-phase contact line of the center. Consequently, the growth rate, dn,ldt, is proportional, on the one hand, to the circumference of a circle with the radius r (Figure 1)and, on the other hand, to the volume change of the asymmetric lens with time.
or
f(4Ji) =
+
2 - 3 COS 4i ~ 0 8 ~ 4 ~ 4 (1- cos24i)3’2
(4)
for an asymmetric lens consistingof two spherical segments with the volumes VIand V2 and the contact angles 41 and 42, we obtain
VL = (4/3)rr3 Wl) + f(42))
(5)
The consideration of eq 4 allows one to suggest that the two contact angle functions within the parentheses represent a geometrical factor, G, expressing the deviation of
where k is the growth rate constant, d the molecular height, the density, and M the molecular weight. Since V = (413)dG and, thus, dV/dr = 4rr2G, the differential equation (11)comprises the effect of the geometry factors of the asymmetric lens defined in eq 6. Accordingly, eq 11can be written in the form
p
2urdk =
dr & 4rr2Gdt
The solution of the differential equation (12) is
3210 Langmuir, Vol. 9, No. 11, 1993
Vollhardt et al. (13)
Finally, insertion of eq 12 into eq 11leads to the growth rate of a single lenticular nucleus
Law of Nucleus Formation. Now the formation of 3D nuclei from the monolayer material is considered. In this case, there is a uniform probability of converting sites into nuclei with time, that means the nucleation law is of first order.
0.2 o,L
li
N = Nmar (1- exp(-k,t))
(15) where N is the number of nuclei at any time, N m , the potential nucleus forming sites, and k, the nucleation rate constant. The corresponding rate of nucleus formation is
dN/dt = kniVm, exp(-k,t) (16) The fundamental equation (16) for the rate of nucleation has been designated as exponential law. In our recent paper, only two limiting cases have been analyzed in this nucleation-growth study (i) instantaneous nucleation for large k n with
N
-
"ax
0
20
6o
LO
k,t
Figure 2. Graphical illustration of the function F(k,t) of eq 28.
nuclei in all the directions. The extent of overlapping is expressed by Avrami's limiting law z = 1- exp(-ze,)
(23)
where z is the actual normalized volume and zext the extended normalized volume equivalent to the freely growing volume. According to eq 8 of our recent paper, the actual normalized volume z can be expressed as
(17)
(ii) progressive nucleation for small k n with W l d t = knNm, (18) The further considerations are based on the exponential law (eq 16) including the two boundary cases in a generalized model. Total Size of All Centers. The overall rate of nucleation and growth results from the convolution of nucleation rate and growth rate
By insertion eqs 14 and 16 into eq 19, the following expression is obtained
exp(-knt) sokty1/2exp y dy (20) The integral is not soluble analytically, its analysis rather leads to sokqy1/2exp y dy = (knt)1/2exp(knt) f(knt) (21) where f(k,t) is a function which can be calculated only approximately by series expansion (see Appendix eqs A5 and A6). After insertion eq 21 into eq 20 the expression for the overall rate takes the form
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where A is the total monolayer area at time, A0 the initial monolayer area, A, the monolayer area for t -, and n,, the total number of molecules transformed from the monolayer material to 3D centers. With the appropriate normalized volume of freely growing centers zext = nnglncm
(25)
the general expression based on monolayer relaxation experiments a t constant surface pressures is obtained
A , - A - 1- exp( - s) --
A, - A , nCGeneralized Nucleation-Growth-Collision Model. The application of eq 26 necessitates the integration of the overall rate of nucleation and growth given by eq 22. This is possible due to the relationship derived in the Appendix (A7 and A8).
whereF(k,t) is, likewise,approximately calculable by series expansion. Finally the general nucleation model with asymmetric lens growth is described as
summarizing all the constants by This expression describes the overall rate of nucleation and growth for the condition of unconfined free growth in the form of asymmetric lenses. Overlap of the Growing Centers. As discussed in a recent paper, the centers cannot grow freely in all directions since they will impinge on each other in the succeeding stages of the transformation process. The growth has to stop at the contact point, so as to limit the size of the
A numerical description of the function F(knt) is shown in Figure 2. The two limiting cases of the general model discussed above follow directly from the properties of the function, F(y) (see Appendix A9 and A10):
Langmuir, Vol. 9, No. 11, 1993 3211
Formation of Fluid Lenticular Centers instantaneous nucleation lim k,t-m
F(knt) = 1
(30)
,
A,-A -- 1- e x p ( - ~ t ~ / ~ )
resulting in the series expansions
A, - A' progressive nucleation
Sy'I2 exp(y) dy = yl/' exp(y)
(32) A,-A
-
n # 1 (A2)
(33)
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Both limiting cases represent the tangent of the function and knt 0 as shown in Figure 2. F(k,t) for knt For asymmetric lens growth, the general expression (eq 28) describes the complete transition range from totally progressive nucleation (knt 0) to totally instantaneous nucleation (knt m). The equation allows calculation of the nucleation rate constant, k,, from the data of monolayer relaxation at constant surface pressure. For a further analysis of experimental results, it is noteworthy that besides the growth rate constant, k, the constant C also includes the geometry factor, G, and the mean number of molecules contained in a nucleus (iV& nd.
2
2y -
(2n + 1) (A3) This series has excellent convergence properties for y < 20 (10 ... 100 steps). To avoid bad convergence properties for large y the back-propagation of the partial integration was used in the numerical calculations resulting in the series expansion i=l
n-1
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Conclusions A theory for the formation and growth of fluid lenticular three-dimensional centers from a supersaturated monolayer a t the airlwater interface has been developed. For asymmetric lens growth, a general solution of the nucleation law which is based on the exponential law describes the complete range from totally progressive nucleation (knt 0) to totally instantaneous nucleation (knt In contrast to our recent nucleation-growth-collision model, an arbitrary geometry has not been assumed for the growth of the centers, but instead the conditions of a fluid 3D-phasecenter at the airlwater interface determined by the interfacial tensions at the centerlwatertair threephase contact have been taken into account. Anapplication of the present theory to the experimental data of monolayer relaxation at constant surface pressure provides a possibility for obtaining nucleation rate constants. Referring to the results of our recent paper,12the normalized free energy of formation of the cluster-tosurroundings surface can be calculated from the surfacepressure dependence of the stationary nucleation rate. Knowing this quantity, the critical size of the nucleus, the free energy of its formation, and the free energy of the formation of the critical nucleus-to-surroundings surface can be obtained.
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00).
Appendix For the integration of thefunctiony1/2(exp(y)arepeated application of partial integration can be used in form of Jy" exp(y) dy = Y" expW - n
J Y"-'
expW dy
(AI)
(A41 Sufficient results were obtained in the range of J = 10 20. Consequently, the function f ( u )which is calculable numerically can be defined for the whole definition range
...
2Y i=l
"=I
fW =
(2n
+ 1)
for y I15 (A51
(2n-3)
for y > 15
Thus the integral to be calculated can be written as hkncy'/' exp(y) dy = (k,t)'/' exp(knt) f(knt) (A6) which is identical to eq 21. Considering the special convergenceproperties, the elementwise integration of the above series (A51 leads to
where F(knt) is 2y "=I
(2n + 3)
for y I15
F(y) =
048) (2n-5)
for y > 15
The asymptotic behavior of the function F(u) can be concluded from the above series expansions as lim
F(y) = 1
F'
Acknowledgment. Financial assistance from the Deutsche Forschungsgemeinschaft and Fonds der chemischen Industrie is gratefully acknowledged.
