2478
Langmuir 1993,9, 2478-2480
Formation of Lenticular Nuclei from an Insoluble Monolayer at the Air/Water Interface: A Model U. Retter'9t and D. Vollhardtj Bundesanstalt fiir Materialforschung und -priifung, Berlin, Germany, and Max-Planck-Institut fiir Kolloid- und Grenzflirchenforschung,Berlin, Germany Received March 16,1993. In Final Form: June 18,199P The formation of lenticular nuclei from insoluble monolayers at the airlwater interface has been theoreticallyinvestigated. On the basis of the classical nucleation theory it has been derived in which way the stationary nucleation rate depends on the surface pressure, the contact angles, the nucleus-tosurroundings interfacial tensions, the line tension of the three-phase contact airlwaterlnucleus, and the Zeldovich nonequilibrium factor. From that a method has been derived to obtain the critical size of the nucleus and the free energy of formation of the critical nucleus from the surface pressure dependence of the stationary nucleation rate. Introduction The surface pressure-area isotherms of monolayers of many long-chain compounds show a wide metastable region above the equilibrium spreading pressure. At present, it is evident that this metastable state corresponds to a certain supersaturation leading to a monolayer relaxation at constant surface pressure or constant molecular area. The relaxation involves a transformation of monolayer material to overgrown 3D structures which has been quantitatively described by a nucleation-growthcollisionmechanismexperimentallyverified.13 The model supposes a hemispherical and a cylindrical growth not considering the contact angle conditions. On the other hand, we have shown that the Prout and Tompkins law can be applied to the theoretical description of nucleation and growth of branching linear chains only when overlapping of the chains can be e ~ c l u d e d . ~ The formation of lenticular nuclei on the surface of an immiscible liquid has been considered in some Here, the nuclei are either assumed to be isolated from each othel.5s6 or to be formed from a bulk vapor phase.' Until now the nucleation of lenticular-shaped nuclei from an insoluble monolayer at the airlwater interface has only been theoretically investigated by ref 8. In this paper the theoretical dependence of the nucleation rate on the oversaturation pressure has been derived on the basis of the Defay modelgneglecting the line tension effect. The experimental data for long-chain aliphatic alcohols and fatty acids are consistent with the theory. The subject of the present paper is to develop a model on the basis of the classicalnucleation theory (macroscopic *Address correspondence to Utz Retter, Bundesanstalt fw Materialforschung und -priifung, Abteilung 10, Rudower Chaussee 5, 12489 Berlin, Germany. t Bundesanstalt f i u Materialforschung und -priifung. t Max-Planck-Institut ftir Kolloid- und Grenzflichenforschung. Abstract published in Advance ACS Abstracts, August 16,1993. (1) Vollhardt, D.; Retter, U. J . Phys. Chem. 1991,95,3723. ( 2 ) Vollhardt, D.; Retter, U.; Siegel, S. Thin Solid F i l m 1991, 199, 198.
(3) Vollhardt, D.; Retter, U. Langmuir 1992,8,309. (4) Retter, U.; Vollhardt, D. Langmuir 1992,8, 1693. ( 5 ) Scheludko, A,; Toshev, B.; Platikanov, D. Proceedings of 31st
International Congress of Pure and Applied Chemistry; Pergamon: Oxford and New York, 1987; p 180. (6) Toshev, B.; Platikanov, D.; Scheludko, A. Longmuir 1988,4,489. (7) Sheu, S. J.; Maa,R. J.; Katz, J. L. J. Statistical Phys. 1988, 52, 1143. (8) De Keyeer, P.; JOOS,P. J. Phys. Chem. 1984,88, 275. (9) Defay, R.; Prigogine, I.; Bellemane, A.; Everett, D. In Surface Tension and Adsorption; Longmans: London, 1966; Chapter 18.
Figure 1. Fluid lenticular cluster formed from an unsoluble monolayer at the aidwater interface: u , ~ u, ~and , a, interfacial tensions; 01 and 02,contact angles;R1 and R2,radii of the contact perimeters; a,impingement flux;r,surface concentration of the monomers of the monolayer. approach) which involvesthe dependenceof the stationary nucleation rate of lenticular nuclei, formed from a monolayer, on the surface pressure, the contact angles, the nucleus-to-surroundings interfacial tensions, the line tension of the three-phase contact airlwaterlcluster, and the nonequilibriumZeldovichfactor for the airfwaterinterface. The organization of the paper is as follows: in the first part of the paper the volume, the circumference, and the nucleus-to-surroundings interfacial areas of a lenticular cluster will be calculated. In the second part of the paper it will be derived in which way the free energy of the formationof the nucleus-to-surrroundingssurface,the size of a criticalnucleus,the free energyof formationof a critical nucleus, and the stationary nucleation rate depend on the surface pressure, the contact angles, the nucleue-tosurroundings interfacial tensions, and the line tension of the three-phase contact airlwaterlcluster. Volume, Circumference, and Interfacial Areas of a Lenticular Cluster as Function of the Contact Angles and the Interfacial Tensions Let us s t a r t with Figure 1which shows schematically a lenticular cluster at the airlwater interface. Here, a,,, a, and uaware the air-cluster, the cluster-water, and the airwater interfacial tensions. 01 is the contact angle between the air-cluster interface and the air-water interface and
0~43-7463/93/2409-2478$04.oo/o0 1993 American Chemical Society
Formation of Lenticular Nuclei
Langmuir, Vol. 9, No. 9, 1993 2479
R1 the radius of the corresponding contact perimeter. Let
U = 2rR1 sin 8,
(17)
A, = 2rR12(1- COS e,)
(18)
A, = 2~R,~(u,~/u~:)(l -COS 0,)
(19)
A, = rR12sin28,
(20)
be B2 the contact angle between the cluster-water interface and the air-water interface. We denote with K the line tension of the three-phase contact air/water/cluster. Neglecting the influence of gravity, the generalized Neumann-Young equilibrium conditions are6 Q,
- uaCcos 8, - Q, cos 62 - K/R,sin 8, = 0
(1)
sin 0, - Q, sin 8, = 0
(2)
a,,
From that it follows with
2
e; = Qac
COS
The free energy of formation of a lenticular cluster is
=0
K
2
+ Qaw
=
-~
2
~ c w+ Qaw
-
2
c w
2QacQaw 2
cos
(3)
- Qac2
(4) 2Qcwuaw Let us now consider the cap which is formed by the interface cluster-air. The volume of this cap is = ( 1 / 3 ) ~ h(3R1t h,) (5) where R1 and hl are the radius of the contact perimeter and the height of the cap, respectively. Inserting
h, = R,(I
Stationary Nucleation Rate of Lenticular Nuclei
- cos e,)
(6)
d a d a w+ KU (21) AG,(i) = i AG, + aaJac + Q&, , Here AG, is the free energy change for the transformation of the monolayer parent phase to the bulk nucleating phase per monomer. -AG, is equal to the supersaturation S defined analogous to that for nucleation from vapor
S = -AG, = kT In II/n, (22) Here, 11and 11, are the surface pressure and the equilibrium surface pressure, respectively. From eqs 15 and 17-22 one obtains AGl(i) = -is + @2i2/3 + @,ill3 with
a2= T 1 / 3 (3v,/4@,)2/3(2uac(i - COS e,) +
leads to
2(um3/ua:)(l-
v, = ( 1 / 3 ) ~ ~(2, 3- 3 COS e, + cos3el)
(7) Defining the volume of a sphere as VIS it follows from eq 7 (2 - 3 COS e, + cos34)/4 (8) Analogous to VI the volume of the cap formed by the cluster-water interface is
v, = v,,
v, = v,(2
+ cos38,114
- 3 COS e,
(9)
From Figure 1 it can be concluded R, sin 0, = R, sin 0,
(10)
Combining this equation with eq 2 it follows that RiIR2 = uac/u,
(11)
and 3
v1dV2, = uac lucw Inserting this relation in eq 9 gives
3
(12)
v2= ~ , , ( ~ , ~ / ~ ~ , -3 3) (COS 2 e, + cos38,114
COS
+ V2 = Vmi
d2) - uaWsin20,) (24)
and
a3= 2TK sin e , ( 3 ~ , / 4 @ , ~ ) ~ / ~
(25)
The condition for a maximum in AGl(i) is 6 AGl(i) --0
6i AGl(i) passes through a maximum, the free energy of formation of the critical nucleus AGl*, at a value i* known as the critical size of the nucleus and at the critical radii R1* and R2*. The critical size, the critical radii, and the free energy of formation of the critical nucleus are found from eqs 11,15,16,23,24,25, and 26 under the generalized Neumann-Young equilibrium conditions, eqs 1and 2, and using the relation
+
02* (1/2)@3*(i*)-1/3= ~ ~ / ~ 4 ~ / Q~~(@,*)'/~ ~ ( 3 V ~ (27) ) ~ / ~
Then it results
(13)
Let us defiie V, as molecular volume. Then the following relation is valid with i the number of molecules involved in the cluster V,
(23)
(29)
(14)
(30)
From eqs 8,13, and 14 one obtains7
R, = (3v,i/4~0,)'/~
(15)
with
a, = (i/4)(2 - 3 COS e, + cos3e, + (u,3/ga,3)(2 - 3 COS e,
+ cos38,))
(16)
Now the following quantities of the lenticular cluster can be derived: circumference U,air-cluster interfacial area A,,, cluster-water interfacial area A, and the air-water interfacial area Aaw covered by the cluster
16?rua~01*Vm22?r sin el*uacV,K (31) S 3s2 Here, @I* is the value of @I at the critical contact angles 81* and 82* which follow from eqs 1and 2 for R1= R1*.R1* AG1* =
+
(10) Smith, R.;Berg, J. J. Colloid Interface Sci. 1980, 74, 273. (11) Sipbee, R. In Vapor to Condensed-Phase Heterogeneoua Nucleation. In Nucleation; Zettlemoyer, Ed.; Marcel Dekker Inc.: New York, 1969; p 151; Chapter 4. (12) Lothe, J.; Pound, G. Statistical Mechanics of Nucleation. In Nucleation; Zettlemoyer, Ed.; Marcel Dekker, Inc.: New York, 1969; Chapter 3, p 109.
Retter and Vollhardt
2480 Langmuir, Vol. 9, No. 9, 1993
depends on the supersaturation S (see eq 29) and therefore 81* and 192* as well as @I*depend implicitly on S. The exception is the case K = 0. Then el* and Bz*are constant (see eqs 3 and 4). Equations 29 and 31 correspond to eqs 13 and 14, respectively, of ref 13 where a heterogeneous nucleation theory of cap-shaped nuclei has been derived. Equations 29 and 30 show that the Gibbs-Thompson relationship for homogeneous nucleation is preserved. From eq 31 it results that AG1* is inversely proportional to the supersaturation for negligible K , but for nonnegligible K an additional term occurs which is inversely proportional to S. From AG1* the equilibrium concentration of critical nuclei can be determined using (ref 11, p 166) n(i*) = n(1) exp(-AG1*/kT) (32) In our case, n(1) is the surface concentration I' of the monomers of the monolayer per square centimeters. The nuclei are assumed to grow from these monomers with an impingement flux CY on the nucleus circumference preserving the lenticular shape of the nucleus. The stationary nucleation rate I of three-dimensional nuclei is defined11J2 as the product of the equilibrium concentration of the critical nuclei n(i*),the impingement flux of monomers on the critical nucleus surface, and the nonequilibrium Zeldovich factor 2. In the present case, the nuclei are formed by an impingement flux CY of monomers of surface concentration r on the critical circumference U*.Defining R1* as R1 at i = i* and using eq 32, it yields
I = Z2xR1* sin B1*crI' exp(-AG,*/kT) (33) I, is defined as preexponential factor or the limiting rate of nucleation for infinite supersaturation I, = Z2xR1*sin O1*d (34) which is assumed to be constant in comparison to the (13)Navascues, G.;Tarazona, P.J. Chem. Phys. 1981, 75,2441.
exponentialterm in eq 33. Then the followingapprosimate relation results using eqs 22, 33, and 34 1 6 ~ ~ , ~ @ ~ * 2a V ,sine , , ~ *u 3k3P ln2II/II,
-
v
k2pln II/II,
K
]
(35)
For negligible line tension of the three-phase contact air/ water/cluster, eq 35 predicts that In I depends linearly on the reciprocal of ln2II/lb.The slope gives uaC3@1* and from that one obtains the critical size of the nucleus i* and the free energy of formation of the critical nucleus AGi* using eqs 28 and 31 respectively. For non-negligible line tension K, 81* and @I*in eq 35 depend implicitly on the supersaturation S = kT ln n/& as mentioned above. Furthermore, In I depends on a s u m of two terms where one term is inversely proportional to ln2 lI/& and the other term is proportional to K and inversely proportional to ln II/&. The experimental I-ln II/& dependence can be modeled by meam of the nonlinear regression analysis starting from eq 35 with the generalized Neumann-Young equilibrium conditions eqs 1 and 2 for R1 = R1* (eq 29). Here, gat,,,a and K are treated as parameters to be determined (uawis known). This gives the dependence of &* and 82* as well as @I* on the supersaturation which can be inserted in eqs 28,29, 30, and 31 to obtain i*, R1*, R2*, and AG1*, respectively. Summing up the results, the paper shows in which way the stationary nucleation rate depends on the supersaturation, considering the formation of lenticularnuclei from an insoluble monolayer at the aidwater interface and including the line tension effect. The line tension of the three-phase contact air/water/nucleus leads to a dependence of the critical nucleation contact angles on the supersaturation and for the free energy of formation of the critical nucleus, to an additional dependence on the reciprocal supersaturation. By means of the surface pressure dependence of the stationary nucleation rate, the critical size of the nucleus and the free energy of the formation of the critical nucleus can be determined.
