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Effect of temperature on the rupture of newton black foam films
Colloids and Surfaces, 36 (1989) 339-351 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
339
E f f e c t of T e m p e r a t u r e on the R u p t u r e of N e w t o n Black Foam Films A. NIKOLOVA, D. KASHCHIEV and D. EXEROWA
Institute of Physical Chemistry, Bulgarian Academy of Sciences, Sofia 1040 (Bulgaria) (Received 5 July 1988; accepted 9 December 1988)
ABSTRACT
The effect of temperature on the rupture of foam bilayers is studied experimentally. The foam bilayers are microscopic horizontal Newton black foam films of radius 0.25 mm in contact with an aqueous solution of surfactant. The dependence of the mean time for rupture of the bilayers on the concentration of surfactant (sodium dodecyl sulfate, NaDoS ) dissolved in the solution, in the presence of 0.5 mol dm -:~ NaC1, is obtained at 10, 22 and 30 ° C. The following thermodynamic parameters: size of the nucleus hole, nucleation work and equilibrium surfactant concentration are determined for the above temperatures according to the theory of rupture of foam bilayers by hole nucleation. Values of 3.1.10- '1 j m ~for the specific edge free energy of bilayer circular hole and 6.7" 10 -~° J for the binding energy of NaDoS molecule in the foam bilayer are calculated.
INTRODUCTION ' I n a series of w o r k s [ 1 - 8 ] t h e N e w t o n b l a c k f o a m film ( t h e f o a m b i l a y e r ) is c o n s i d e r e d as a t w o - d i m e n s i o n a l s y s t e m in w h i c h t h e s h o r t - r a n g e m o l e c u l a r i n t e r a c t i o n s are d o m i n a n t . A f t e r t h e a n a l y s i s of De Vries [9] of r u p t u r e of t h i c k e r f o a m films b y t h e m e c h a n i s m of f l u c t u a t i o n f o r m a t i o n of holes in t h e m , t h e r u p t u r e of t h e f o a m b i l a y e r s was t r e a t e d b y D e r j a g u i n et al. [ 1 0 - 1 2 ] . C o n s i d e r i n g t h e b i l a y e r as a t w o - d i m e n s i o n a l , viscous, n o n v o l a t i l e liquid, t h e y d e t e r m i n e d t h e w o r k of form a t i o n for a n u c l e u s hole in t h e f o a m b i i a y e r a n d t h e p r o b a b i l i t y of b i l a y e r r u p t u r e . L a t e r , a t h e o r y [1,6-8] was developed, a c c o r d i n g to w h i c h b i l a y e r r u p t u r e results f r o m f l u c t u a t i o n a p p e a r a n c e of holes in t h e b i l a y e r b y aggreg a t i o n of e x i s t i n g v a c a n c i e s of s u r f a c t a n t molecules. T h i s t h e o r y e n v i s a g e s t h e p o s s i b i l i t y of f o r m a t i o n of infinitely long living f o a m a n d m e m b r a n e b i l a y e r s in c o n t a c t w i t h a solution, w i t h a high e n o u g h s u r f a c t a n t c o n c e n t r a t i o n . A n u m b e r of e x p e r i m e n t a l r e s u l t s were o b t a i n e d in s u p p o r t of t h i s theory: t h e d e p e n d e n c e (i) of t h e b i l a y e r m e a n lifetime in free [4] a n d a - p a r t i c l e irradiat i o n - i n d u c e d r u p t u r e [13], (ii) of t h e p r o b a b i l i t y of o b s e r v a t i o n of a f o a m bil a y e r in a f o a m film [5] a n d (iii) of t h e p e r m e a b i l i t y of t h e f o a m b i l a y e r to air
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© 1989 Elsevier Science Publishers B.V.
340
[ 14 ] on the concentration of surfactant in the solution. Similar investigations were carried out in the case of emulsion bilayers [ 15 ]. As a result, some important thermodynamic characteristics of the foam bilayers were obtained: the hole specific edge free energy (which allowed evaluation of the energy of lateral nearest neighbor interaction of the surfactant molecules in the bilayer), the work of formation of the nucleus hole, the number of vacancies of surfactant molecules in the nucleus hole, the equilibrium concentration of surfactant in the solution, etc. Some theoretical and experimental results on the rupture and permeability of foam bilayers are reviewed by Exerowa and Kashchiev [7 ]. The aim of the present work is to study the effect of the temperature on the rupture (or stability) of foam bilayers by investigating the dependence of the bilayer mean lifetime on the concentration of surfactant in the solution at different temperatures. This makes it possible, by using the nucleation theory of rupture of bilayers, to determine some unknown characteristics of the foam bilayers investigated. THEORY The theory of nucleation-mediated rupture of bilayers by aggregation of vacancies of amphiphile molecules into holes [ 1,6-8 ] is based on the following physical model. The two surfaces of the foam bilayer are in contact with a gas phase, and its periphery contacts a bulk solution (meniscus) with a concentration C of monomer surfactant. The foam bilayer, itself, is constituted of two m u t u a l l y adsorbed surfactant monolayers which are extensions of the two monolayers of surfactant adsorbed at the gas/meniscus interface. There are vacancies of surfactant molecules in the bilayer which assemble by fluctuations and form holes (or vacancy clusters) of different size. After reaching the socalled nucleus size, the holes begin an irreversible lateral growth which causes the rupture of the bilayer. A measure of the stability of the foam bilayer is its mean lifetime r, i.e. the mean time from the formation of a bilayer with a fixed radius till its rupture. This time is defined as r = i tdP(t)
(1)
o
The probability P for bilayer rupture (assumed to be identical with the probability for the appearance of at least one nucleus hole in the bilayer) until time t can be written down as [16] P ( t ) = l - e -N(t)
(2)
where the number N of nuclei formed until time t is expressed by the nucleation rate J:
341 t
N ( t ) = f J(t')dt'
(3)
0
The density p of the probability for rupture of the bilayer film is defined as
p(t) =dP/dt from which, keeping in mind Eqns (2) and (3), it follows that
p(t) =J(t)e -N(t)
(4)
As seen from Eqns (2) and (4), the time variation of the nucleation rate can be determined by the experimentally obtainable probabilities p (t) and P (t) with the help of the relation
J(t) = p ( t ) / [ 1 - P ( t ) ]
(5)
In the case of steady-state nucleation J (t) = J~ -- constant. T h e n
P(t) = l - e -J~t
(6)
and from Eqn (1) it follows that in this case z is connected with the steadystate nucleation rate J~ by the simple formula
z=l/J~
(7)
From the classical theory of nucleation it is known [ 17 ] that J~ depends on the nucleation work W according to
J~ =A'e -W/kT
(8)
Here k is the Boltzmann constant, T is the absolute temperature, and the preexponential factor A' is controlled by the kinetics of vacancy a t t a c h m e n t to the nucleus hole. From Eqns (7) and (8), for the mean lifetime of the foam bilayer in steady-state nucleation one finds that [ 1 ]
•=Ae W/kT
(9)
where A-- 1/A'. T h e r m o d y n a m i c considerations [ 1,6-8 ] lead to the following expression for the work W for formation of a large enough circular nucleus hole
W= 7~Aefy2/A~
(10)
In this formula Aefis the effective area of a surfactant vacancy in the hole, y is the specific edge free energy of the hole, and d p is the supersaturation defined by [ 1,6-8 ]
Att=kT ln ( Ce/C)
(11)
Here Ce is the equilibrium bulk concentration of surfactant in the solution (i.e.
342
the lowest surfactant concentration at which the bilayer can live infinitely long). For Ce, an Arrhenius-type temperature dependence is predicted [1,6]: C e : C o e - Q/2kT
( 12 )
where Q is the binding energy of a surfactant molecule in the foam bilayer, and Co is a reference surfactant concentration. Theoretically [1,6], in terms of nearest-neighbor interactions only, Q = z e + z o e o , where ¢ and ¢o are, respectively, the lateral and normal bond energies of a surfactant molecule in the bilayer, and z and Zo are the corresponding coordination numbers of the molecule. For the number n of single vacancies constituting a circular nucleus hole the following equation holds [ 1,6-8 ] : n = 7CAef ~22//A~/2 = B / l n 2 ( C e / C )
( 13 )
where B = 7~Aefy2/(kT) 2
(14)
Substituting Eqn (10) in Eqn (9) and using Eqns (11) and (14), for the mean lifetime of the bilayer in a steady-state nucleation process we obtain [1,6,7] = Aexp [ 7~Aef7 2 / k T A ~ t ] = Aexp [B/ln (C~ /C ) ]
( 15 )
As shown by Kashchiev [ 18 ], and as can be easily checked in our case with W a n d n from Eqns (10) and (13), the quantities W, n and A/~ are related by dW/dA/~=
- n
Keeping in mind this formula and neglecting the relatively minor dependence of the preexponential factor A on the concentration of surfactant in the solution, with the help of Eqns (9), (11) and (15) we arrive at the expression n = dln z/dln C
(16)
This equation demonstrates a possibility for directly determining the number of surfactant vacancies in the nucleus hole from the slope of an experimentally obtained In r(ln C) dependence. EXPERIMENTAL Materials
The foam bilayers were formed from aqueous solutions of sodium dodecyl sulfate (NaDoS), containing NaC1 electrolyte at a concentration of 0.5 mol dm -3. The electrolyte concentration was chosen to ensure close packing of surfactant molecules at the air/solution interface (necessary for the formation
343 of Newton black foam films [19-21] ) at the three investigated temperatures (10, 22 and 30°C) over the whole N a D o S concentration range studied. The experiments were carried out at N a D o S concentrations lower than the critical micelle concentration [22]. The ionic N a D o S surfactant used was specially synthesized by Henkel KGaA, Diisseldorf, F.R.G. and did not display a minim u m in the surface tension isotherm. "Suprapur" NaC1 (Merck) was used as the electrolyte, heated to 500 ° C to remove any surface-active contaminations. The solutions were prepared with doubly distilled water.
Methods The experiments were carried out by the microinterferometric method for investigation of single foam films [7,23-25]. The microscopic horizontal foam bilayer was formed in the middle of a biconcave drop in a specially constructed measuring cell, described elsewhere [24 ]. The measuring cell was put in a thermostatting device combined with a microscope with a special system for observation and registration of the light reflected and interfered from the film. For more precise control around the lower part of the cell where the foam bilayer was formed, an additional thermostatting jacket was mounted, so that the temperature was constant to + 0.05 ° C. The microscopic Newton black foam films investigated were formed by growth of black spots and had a fixed radius of 0.25 mm. The measurement of the lifetimes of the successively formed foam bilayers started after about 2 h which were necessary for equilibration of the investigated system [ 20 ]. In the case of N a D o S bilayers the rate of lateral growth of the black spots is relatively high [ 19 ] and the bilayer formation time is commensurate with the observer's eye reaction time (about 0.5 s) [5]. Thus, it was possible to measure the individual lifetime of each foam bilayer practically from the moment of its formation up to a lifetime of 300 s. The experiments for determining the mean lifetime of the foam bilayers were carried out at 10, 22 and 30 °C in different N a D o S concentration ranges. Given the temperature, at each of the experimentally studied N a D o S concentrations the mean lifetime ~ of the foam bilayer, in the case of the steady-state nucleation process, was calculated statistically from the results for the lifetimes of about 240 successively formed foam bilayers, measured on average in 6 different experiments. According to their duration, the lifetimes obtained at given C and T values were divided into 15 groups including 10-20 measurements each. This statistical data gave the experimental values of p and P at the respective values of t directly. The results f o r p (t) and P(t) were then used in Eqn (5) for the calculation of J ( t ) and, thereby, of J~ which is connected to according to Eqn (7). The comparison of the various theoretical dependences with the experimental ones was done by nonlinear regression.
344 RESULTS AND DISCUSSION
Under the conditions of the experiment, deviations from the P(t) dependence, Eqn (6), corresponding to a steady-state process, are observed in the initial period after the formation of the foam bilayers [4]. However, straightforward application of the theory [ 1 ] is possible only when nucleation is stationary. For that reason, the determination of the mean lifetime z of the foam bilayers in the steady-state nucleation case was done by fitting the experimental time dependences of the probability P for rupture and of the nucleation rate J [calculated from Eqn (5) ] to empirical P(t) and J(t) dependences. The points in Fig. 1 show the experimental dependence of the hole nucleation rate J on time t at C = 1 . 9 - 1 0 -4 mol dm -3 N a D o S and T--22°C. The increased nucleation rate at shorter times is probably due to transition processes related to the formation of the foam bilayer. Depending on the experimental conditions, after a certain transient time (about 50 s in this case) a constant (steady-state ) hole nucleation rate is reached. The value of the steadystate nucleation rate Js, determined from the plateau of the J(t) dependence after arithmetical averaging, is (11 _+3)" 10 -3 s-1. Unfortunately, there were cases for which the transient time for reaching constant nucleation rate was relatively long so that the determination of Js from the J(t) plateau was impossible within the 300 s time scale of the experiment. Because of that, in order to determine J~, we used an empirical dependence for the nucleation rate as a function of time. This dependence was used in the form
J(t) =Js + (Jo --Js) e-t/a
(17)
where Jo is the initial nucleation rate (at t--0 ), and a is the time constant for
012' 0,09
0_06 0.03 I
I
100
•
,.--T--'7-~-, 200
300t [s] Fig. 1. Dependence of the hole nucleation rate J on time t for foam bilayers of radius 0.25 mm, formed at a N a D o S concentration of 1.9.10 -4 mol d m -3 in the presence of 0.5 mol d m -3 NaC1 at 22°C: points, experimental results from calculations according to E q n (5); solid line, theoretical curve according to E q n {17); dashed line, value of the steady-state nucleation rate calculated according to E q n (18).
345
reaching the regime of steady-state nucleation. We found that the above formula gave a good fit to the experimental results in all cases in which, statistically, enough measurements were done. The theoretical J (t) dependence [Eqn (17) ], obtained from the best fit to the experimental J(t) data in Fig. 1 with Js = (11 4- 2 ). 10-3 s - 1, J0 = (85 4- 6 )" 10-3 s - 1 and a = 10 + 2 s, is shown in the same figure by the solid line. There is a very good agreement between the values of Js calculated from the plateau in Fig. 1 and from the fit of Eqn (17). The reliability of Eqn ( 17 ) allows the use of it in Eqns (2) and (3) to obtain, explicitly, the time dependence of the probability P of bilayer rupture in the form
P(t) = 1 - e x p [ - J s t + a ( J o -,.Is) (e - t / a - 1) ]
(18)
Rather than fitting J from Eqn (17) to the experimentally obtained J from Eqn (5), it is more advantageous to fit P from Eqn (18) directly to the experimental P(t) data, because then the error introduced b y p when using Eqn (5) is avoided. The experimental values for the probability P of rupture until time t at C = 1.9-10 -4 mol dm -3 and T = 2 2 ° C [4] are shown in Fig. 2. The solid curve is the best fit theoretical dependence from Eqn (18) with J,= (10 4-1). 10 -3 s- 1, Jo-- (79 + 5)" 10 -3 s - 1 and a= 10 ___1 s. Comparison of the values of the steady-state nucleation rate J , obtained by the different methods of calculation, shows that they coincide within the error limits. This value of J~ coincides also (within the error limits) with the one obtained by another method [26]. All of this allowed us to calculate Js by using Eqn (18) directly with the P (t) data. The values of J , obtained, were then used in Eqn (7) for determination of r corresponding to steady-state hole nucleation. The experimental results for the mean lifetime • of foam bilayers under conditions of steady-state nucleation at different NaDoS concentrations at the P 10 025 0.5 0.25 I
100
i
L
2o0
i
I
•
300
t is] Fig. 2. Dependence of the p r o b a b i l i t y P for bilayer rupture on time t for foam bilayers of radius 0.25 mm, formed at a N a D o S concentration of 1.9-10-4 mo| dm -3 in the presence of 0.5 mo] dm -3 NaC| at 22 ° C: points, experimental data; solid line, theoretical curve according to Eqn (18).
346
three investigated temperatures are presented in Fig. 3. The standard deviation of the ~ values is not greater than about 25%. Some of the results for the individual lifetimes of the foam bilayers at 22 °C are taken from previous work by Exerowa et al. [4]. The experimentally obtained decrease in the stability of the foam bilayers with increasing temperature is clearly demonstrated in Fig. 3. The experimental results for the initial nucleation rate Jo and for the time constant a are given in Table 1, the standard deviation being within 20%. A new step in the treatment of the experimental data for the mean lifetime of the foam bilayers at different N a D o S concentrations is the direct determination of the nucleus size n, i.e. the number of surfactant vacancies in the nucleus hole. Indeed, according to Eqn (16), the slope of the experimental r(C) dependence in In r versus In C coordinates yields directly the size of the nucleus hole at the respective values of C. It turned out that a third-order polynomial was sufficient for the numerical differentiation we used to calculate n. The n values obtained (accuracy _+0.5) at the various temperatures and N a D o S concentrations are depicted in Fig. 4. Table 2 shows the range of n values under the experimental conditions. The experimental n (C) dependences at the different temperatures were used for determination of the parameter B and the equilibrium N a D o S concentration Ce at the corresponding temperature. To this end, the best fit of Eqn (13) to the experimental n (C) data was made and the resulting theoretical dependences are illustrated in Fig. 4. The values of Ce and B from the best fit, as well as of W and A/2 (calculated according to Eqns (10) and (11), respectively), are given in Table 2. The results for 22 °C are in good agreement with those found earlier [4] by another fitting procedure. 500
1
/..00 300
I
3
200 100 0 1.5
2 2.5 Cx~O~ [tool dm'3]
Fig. 3. Dependence of the m e a n lifetime v on the concentration C of surfactant in the solution for NaDoS foam bilayers of radius 0.25 mm, obtained in the presence of 0.5 mol d m -3 NaCl: (()), ( O ) a n d ( • ), experimental results at 10, 22 and 30 ° C, respectively; curves 1, 2 a n d 3, theoretical dependences according to E q n (15).
347 TABLE1 I n i t i a l n u c l e a t i o n rate Jo a n d t i m e c o n s t a n t a for hole n u c l e a t i o n in N e w t o n b l a c k f o a m f i l m s of r a d i u s 0.25 m m , f o r m e d a t d i f f e r e n t t e m p e r a t u r e s f r o m N a D o S s o l u t i o n of c o n c e n t r a t i o n C in t h e p r e s e n c e of 0.5 m o l d m -3 N a C l C,10 4 (moldm -3)
10°C
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.4
22oc
30°C
Jo
a
Jo
a
Jo
a
(s-')
(s)
(s -~)
(s)
(s-')
(s)
0.665 0.442 0.190 0.099 0.102
1.50 3.01 8.83 10.31 8.78
3.33 4.15 9.96 9.18 6.79 10.40 13.95 21.00 -
0.579 0.283 0.075 0.119 0.059 0.016
-
0.579 0.292 0.162 0.184 0.136 0.079 0.043 0.026
1.35 3.22 10.38 7.88 17.13 77.97
r~
1
15
40 30 /f
20
/ //
IO
/
1/
."
11 OL
8.2
i
8!.
i
8.6
i
8.~ -t,-,c
A
s.o
[~o,~31
O
2
6
10 14 106/T2 [K"2]
Fig. 4. (left) D e p e n d e n c e o f t h e n u m b e r n of s u r f a c t a n t v a c a n c i e s in t h e n u c l e u s hole o n t h e c o n c e n t r a t i o n C of s u r f a c t a n t in t h e s o l u t i o n for N a D o S f o a m bilayers f o r m e d in t h e p r e s e n c e of 0.5 m o l d m -3 N a C h ( ¢ ) , ( O ) a n d ( • ) , e x p e r i m e n t a l r e s u l t s at 10, 22 a n d 30 ° C, respectively; c u r v e s 1, 2 a n d 3, t h e o r e t i c a l d e p e n d e n c e s a c c o r d i n g to E q n ( 13 ). Fig. 5. ( r i g h t ) D e p e n d e n c e of B o n 1 / T 2 for N a D o S f o a m bilayers f o r m e d in t h e p r e s e n c e of 0.5 m o l d m - 3 N a C h p o i n t s , e x p e r i m e n t a l d a t a ( T a b l e 2); s t r a i g h t line, t h e o r e t i c a l d e p e n d e n c e acc o r d i n g to E q n (14).
According to Eqn (14), if both y and Aef are T-independent, B must be a linear function of 1 / T 2 with zero intercept and a slope allowing the calculation of the hole specific edge free energy 7. The experimental values of B at the
348 TABLE 2 Calculated values of parameters of hole nucleation in Newton black foam films of radius 0.25 mm, formed at different temperatures from NaDoS solution in the presence of 0.5 mol din-3 NaC1 T (°C)
n
10 22 30
10-14 8-11 6-9
C~"104 (tool dm -3)
B
8.8+0.8 13.3__+0.1 15.3_+1.9
42___+4 39__+3 33_+4
Az/kT
W/kT
A (s)
1.7-2.0 1.8-2.2 1.8-2.3
21-25 17-21 14-18
(8.5__+3.2)-10 -9 (2.5_+0.2).10 -7 (1.0_+0.3)-10 -5
~0 6.8I
32
3.3
3,~
3.s
36
103/T[K"l] Fig. 6. Dependence of the equilibrium surfactant concentration Ce on l i T for NaDoS foam bilayers formed in the presence of 0.5 mol dm-3 NaCh points, experimental data (Table 2 ); straight line, theoretical dependence according to Eqn {12).
three investigated temperatures are shown in Fig. 5 and the straight line is drawn by the method of least squares under the condition of zero intercept. As can be seen from Fig. 5, Eqn (14) is in conformity with the experimental data. From the slope of the straight line, with the help of Eqn (14), for bilayer circular holes (then Ae~=Ao/2 [6,7], Ao--4.2"10 -19 m 2 [20] being the area of a NaDoS molecule in the foam bilayer) we find that y-- (3.07 _+0.06)" 10-11 j m -1. It was supposed in the calculations that Ao does not change considerably at the investigated temperatures, which is likely for the closely packed [20] surfactant molecules in the bilayer. If the hole is of monolayer depth (then Aef---A o [6,7] ), y= (2.20+0.04) "10 -11 J m -1. These ~ values are in very good agreement with those found earlier [4,7] and are also consistent with results obtained for bilayer lipid membranes by other authors [27,28]. According to Eqn (12), the temperature dependence of Ce offers a possibility for the determination of the binding energy Q ofa NaDoS molecule in the foam bilayer. The points in Fig. 6 show the experimental values of C~ in In Ce versus 1 / T coordinates. The straight line represents the best fit of the Arrhenius-type dependence, Eqn (12), to the experimental results. For the binding energy of NaDoS molecule we obtain Q= (6.7 + 1.0)-10 -20 J, and In Co-1.6 + 1.2. The-
349 oretically, the value of the binding energy (Q = 16 kT at 30 ° C ) is high enough [1,6] to ensure the occurrence of the two-dimensional phase transition of "condensation" of the "gas" of NaDoS vacancies in the bilayer into holes leading to the bilayer rupture. Knowing B and Ce and using Eqn (15), we can calculate the preexponential factor A from the experimental ~(C) data. The A values obtained for the three investigated temperatures do not show a considerable dependence on C and their arithmetic means are presented in Table 2. The experimental fact that A increases with increasing temperature cannot be interpreted without knowing the actual mechanism of hole formation in the bilayer. A possible explanation of this dependence may be given in the framework of the mechanism of nucleation of holes by aggregation of vacancies on preferred sites (e.g. along line defects) in the bilayer. For instance, accumulation of vacancies along structural defects is well known to occur in crystalline solids [29]. If the increase of the temperature can cause a decrease in the density of these preferred sites in the bilayer, an increase of A may be expected, since A is inversely proportional to the density of the nucleation sites [ 1,6 ]. The theoretical ~(C) dependences for the three investigated temperatures, calculated according to Eqn (15) with the values of A, B and Co from Table 2, are shown in Fig. 3. As can be seen, there is a good agreement between the theoretical and the experimental dependences for the mean lifetime r of the foam bilayers at the different temperatures. The observed decrease in the bilayer stability (i.e. the shortening of 3) with increasing temperature is understandable in the light of the nucleation theory for rupture of foam bilayers [ 1 ], upon recalling the considerable decrease in the dimensionless work W/kT for formation of a nucleus hole with an increase in temperature (Table 2). CONCLUSION The experimental investigation of the effect of temperature on the mean lifetime ~ of Newton black foam films shows a decrease of their stability with increasing temperature. This fact is explained by the thermal activation of the process of fluctuation nucleation of holes in the foam bilayers which causes their rupture [1,6,7]. The values of important thermodynamic quantities of NaDoS foam bilayers are calculated by comparison of the experimentally obtained dependences of r on the bulk concentration of surfactant in the solution at different temperatures with the theoretical ones known from the nucleation theory of bilayer rupture [1,6,7]. A good agreement between theory and experiment is established, indicating that the hole specific edge free energy ~ is virtually T-independent in the temperature range studied. The temperature dependence of the kinetic preexponential factor A remains a problem for further study. For the first time, the binding energy Q of a surfactant molecule in foam
350
bilayer is obtained. Its value for the investigated NaDoS bilayer foam films justifies the assumption for the occurrence of a two-dimensional phase transition of "condensation" of the "gas" of surfactant vacancies in the bilayer into holes, which is the basis of the nucleation theory for rupture of foam bilayers [ 1 ]. Finding the value of Q is essential for understanding the behavior of the surfactant molecules in the bilayer and the interactions between them. ACKNOWLEDGEMENTS
The authors are grateful to Dr M. Paunov for providing them with his "Best Fit" computer program. This project has been completed with the financial support of the Committee for Science at the Council of Ministers under Contract No 287/17.IV.1987.
REFERENCES 1 D. Kashchiev and D. Exerowa, J. Colloid Interface Sci., 77 (1980) 501. 2 D. Exerowa, D. Kashchiev and B. Balinov, in J.M. Georges (Ed.), Microscopic Aspects of Adhesion and Lubrication, Elsevier, Amsterdam, 1982, p. 107. 3 D. Exerowa, D. Kashchiev and B. Balinov, in B.V. Derjaguin (Ed.), Poverkhnostnye Sily i Granichnye Sloi Zhidkostei, Nauka, Moscow, 1983, p. 208. 4 D. Exerowa, B. Balinov and D. Kashchiev, J. Colloid Interface Sci., 94 (1983) 45. 5 D. Exerowa, B. Balinov, A. Nikolova and D. Kashchiev, J. Colloid Interface Sci., 95 (1983) 289. 6 D. Kashchiev and D. Exerowa, Biochim. Biophys. Acta, 732 (1983) 133. 7 D. Exerowa and D. Kashchiev, Contemp. Phys., 27 (1986) 429. 8 D. Kashchiev, Colloid Polym. Sci., 265 (1987) 436. 9 A.J. de Vries, Rec. Trav. Chim., 77 (1958) 383; 441. 10 B.V. Derjaguin and Yu.V. Gutop, Kolloid. Zh., 24 (1962) 431. 11 B.V. Derjaguin and A.V. Prokhorov, J. Colloid Interface Sci., 81 (1981) 108. 12 B.V. Derjaguin and A.V. Prokhorov, Kolloid. Zh., 49 (1987) 903. 13 I. Penev, D. Exerowa and D. Kashchiev, Colloids Surfaces, 25 (1987) 67. 14 M. Nedyalkov, R. Krustev, D. Kashchiev, D. Platikanov and D. Exerowa, Colloid Polym. Sci., 266 (1988) 291. 15 H.J. Mtiller, B. Balinov and D. Exerowa, Colloid Polym. Sci., 266 (1988) 921. 16 S. Toschev, A. Milchev and S. Stoyanov, J. Crystal Growth, 13/14 (1972) 123. 17 A.C. Zettlemoyer (Ed.), Nucleation, Marcel Dekker, New York, 1969. 18 D. Kashchiev, J. Chem. Phys., 76 (1982) 5098. 19 T. Kolarov, A. Sheludko and D. Exerowa, Trans. Faraday Soc., 64 (1968) 2864. 20 D. Exerowa, A. Nikolov and M. Zacharieva, J. Colloid Interface Sci., 81 (1981) 419. 21 M. Jones, J. Mysels and P. Scholten, Trans. Faraday Soc., 62 (1966) 1336. 22 P. Mukerjee and K. Mysels, Nat. Stand. Ref. Data Ser. Nat. Bur. Stand., 36 (1971) 70. 23 A. Sheludko, Adv. Colloid Interface Sci., 1 (1967) 391. 24 D. Exerowa, M. Zacharieva, R. Cohen and D. Platikanov, Colloid Polym. Sci., 257 (1979) 1089. 25 D. Exerowa, Comm. Dept. Chem. Bulg. Acad. Sci., 11 {1978) 739. 26 B. Balinov and D. Exerowa, Colloid Polym. Sci., (1989) in press.
351 27 28 29
Yu.A. Chizmadzhev, L.V. Chernomordik, V.F. Patushenko and I.G. Abidor, in P.G. Kostyuk (Ed.), Biofizika Membran, Vol. 2, AN USSR, Moscow, 1982, p. 161. A. Petrov, M. Mitov and A. Derzhanski, in L. Bata (Ed.), Advances in Liquid Crystal Research and Applications, ¥ol. 2, Pergamon, Oxford, 1980, p. 695. H.G. Van Bueren, Imperfections in Crystals, North-Holland, Amsterdam, 1960.
339
E f f e c t of T e m p e r a t u r e on the R u p t u r e of N e w t o n Black Foam Films A. NIKOLOVA, D. KASHCHIEV and D. EXEROWA
Institute of Physical Chemistry, Bulgarian Academy of Sciences, Sofia 1040 (Bulgaria) (Received 5 July 1988; accepted 9 December 1988)
ABSTRACT
The effect of temperature on the rupture of foam bilayers is studied experimentally. The foam bilayers are microscopic horizontal Newton black foam films of radius 0.25 mm in contact with an aqueous solution of surfactant. The dependence of the mean time for rupture of the bilayers on the concentration of surfactant (sodium dodecyl sulfate, NaDoS ) dissolved in the solution, in the presence of 0.5 mol dm -:~ NaC1, is obtained at 10, 22 and 30 ° C. The following thermodynamic parameters: size of the nucleus hole, nucleation work and equilibrium surfactant concentration are determined for the above temperatures according to the theory of rupture of foam bilayers by hole nucleation. Values of 3.1.10- '1 j m ~for the specific edge free energy of bilayer circular hole and 6.7" 10 -~° J for the binding energy of NaDoS molecule in the foam bilayer are calculated.
INTRODUCTION ' I n a series of w o r k s [ 1 - 8 ] t h e N e w t o n b l a c k f o a m film ( t h e f o a m b i l a y e r ) is c o n s i d e r e d as a t w o - d i m e n s i o n a l s y s t e m in w h i c h t h e s h o r t - r a n g e m o l e c u l a r i n t e r a c t i o n s are d o m i n a n t . A f t e r t h e a n a l y s i s of De Vries [9] of r u p t u r e of t h i c k e r f o a m films b y t h e m e c h a n i s m of f l u c t u a t i o n f o r m a t i o n of holes in t h e m , t h e r u p t u r e of t h e f o a m b i l a y e r s was t r e a t e d b y D e r j a g u i n et al. [ 1 0 - 1 2 ] . C o n s i d e r i n g t h e b i l a y e r as a t w o - d i m e n s i o n a l , viscous, n o n v o l a t i l e liquid, t h e y d e t e r m i n e d t h e w o r k of form a t i o n for a n u c l e u s hole in t h e f o a m b i i a y e r a n d t h e p r o b a b i l i t y of b i l a y e r r u p t u r e . L a t e r , a t h e o r y [1,6-8] was developed, a c c o r d i n g to w h i c h b i l a y e r r u p t u r e results f r o m f l u c t u a t i o n a p p e a r a n c e of holes in t h e b i l a y e r b y aggreg a t i o n of e x i s t i n g v a c a n c i e s of s u r f a c t a n t molecules. T h i s t h e o r y e n v i s a g e s t h e p o s s i b i l i t y of f o r m a t i o n of infinitely long living f o a m a n d m e m b r a n e b i l a y e r s in c o n t a c t w i t h a solution, w i t h a high e n o u g h s u r f a c t a n t c o n c e n t r a t i o n . A n u m b e r of e x p e r i m e n t a l r e s u l t s were o b t a i n e d in s u p p o r t of t h i s theory: t h e d e p e n d e n c e (i) of t h e b i l a y e r m e a n lifetime in free [4] a n d a - p a r t i c l e irradiat i o n - i n d u c e d r u p t u r e [13], (ii) of t h e p r o b a b i l i t y of o b s e r v a t i o n of a f o a m bil a y e r in a f o a m film [5] a n d (iii) of t h e p e r m e a b i l i t y of t h e f o a m b i l a y e r to air
0166-6622/89/$03.50
© 1989 Elsevier Science Publishers B.V.
340
[ 14 ] on the concentration of surfactant in the solution. Similar investigations were carried out in the case of emulsion bilayers [ 15 ]. As a result, some important thermodynamic characteristics of the foam bilayers were obtained: the hole specific edge free energy (which allowed evaluation of the energy of lateral nearest neighbor interaction of the surfactant molecules in the bilayer), the work of formation of the nucleus hole, the number of vacancies of surfactant molecules in the nucleus hole, the equilibrium concentration of surfactant in the solution, etc. Some theoretical and experimental results on the rupture and permeability of foam bilayers are reviewed by Exerowa and Kashchiev [7 ]. The aim of the present work is to study the effect of the temperature on the rupture (or stability) of foam bilayers by investigating the dependence of the bilayer mean lifetime on the concentration of surfactant in the solution at different temperatures. This makes it possible, by using the nucleation theory of rupture of bilayers, to determine some unknown characteristics of the foam bilayers investigated. THEORY The theory of nucleation-mediated rupture of bilayers by aggregation of vacancies of amphiphile molecules into holes [ 1,6-8 ] is based on the following physical model. The two surfaces of the foam bilayer are in contact with a gas phase, and its periphery contacts a bulk solution (meniscus) with a concentration C of monomer surfactant. The foam bilayer, itself, is constituted of two m u t u a l l y adsorbed surfactant monolayers which are extensions of the two monolayers of surfactant adsorbed at the gas/meniscus interface. There are vacancies of surfactant molecules in the bilayer which assemble by fluctuations and form holes (or vacancy clusters) of different size. After reaching the socalled nucleus size, the holes begin an irreversible lateral growth which causes the rupture of the bilayer. A measure of the stability of the foam bilayer is its mean lifetime r, i.e. the mean time from the formation of a bilayer with a fixed radius till its rupture. This time is defined as r = i tdP(t)
(1)
o
The probability P for bilayer rupture (assumed to be identical with the probability for the appearance of at least one nucleus hole in the bilayer) until time t can be written down as [16] P ( t ) = l - e -N(t)
(2)
where the number N of nuclei formed until time t is expressed by the nucleation rate J:
341 t
N ( t ) = f J(t')dt'
(3)
0
The density p of the probability for rupture of the bilayer film is defined as
p(t) =dP/dt from which, keeping in mind Eqns (2) and (3), it follows that
p(t) =J(t)e -N(t)
(4)
As seen from Eqns (2) and (4), the time variation of the nucleation rate can be determined by the experimentally obtainable probabilities p (t) and P (t) with the help of the relation
J(t) = p ( t ) / [ 1 - P ( t ) ]
(5)
In the case of steady-state nucleation J (t) = J~ -- constant. T h e n
P(t) = l - e -J~t
(6)
and from Eqn (1) it follows that in this case z is connected with the steadystate nucleation rate J~ by the simple formula
z=l/J~
(7)
From the classical theory of nucleation it is known [ 17 ] that J~ depends on the nucleation work W according to
J~ =A'e -W/kT
(8)
Here k is the Boltzmann constant, T is the absolute temperature, and the preexponential factor A' is controlled by the kinetics of vacancy a t t a c h m e n t to the nucleus hole. From Eqns (7) and (8), for the mean lifetime of the foam bilayer in steady-state nucleation one finds that [ 1 ]
•=Ae W/kT
(9)
where A-- 1/A'. T h e r m o d y n a m i c considerations [ 1,6-8 ] lead to the following expression for the work W for formation of a large enough circular nucleus hole
W= 7~Aefy2/A~
(10)
In this formula Aefis the effective area of a surfactant vacancy in the hole, y is the specific edge free energy of the hole, and d p is the supersaturation defined by [ 1,6-8 ]
Att=kT ln ( Ce/C)
(11)
Here Ce is the equilibrium bulk concentration of surfactant in the solution (i.e.
342
the lowest surfactant concentration at which the bilayer can live infinitely long). For Ce, an Arrhenius-type temperature dependence is predicted [1,6]: C e : C o e - Q/2kT
( 12 )
where Q is the binding energy of a surfactant molecule in the foam bilayer, and Co is a reference surfactant concentration. Theoretically [1,6], in terms of nearest-neighbor interactions only, Q = z e + z o e o , where ¢ and ¢o are, respectively, the lateral and normal bond energies of a surfactant molecule in the bilayer, and z and Zo are the corresponding coordination numbers of the molecule. For the number n of single vacancies constituting a circular nucleus hole the following equation holds [ 1,6-8 ] : n = 7CAef ~22//A~/2 = B / l n 2 ( C e / C )
( 13 )
where B = 7~Aefy2/(kT) 2
(14)
Substituting Eqn (10) in Eqn (9) and using Eqns (11) and (14), for the mean lifetime of the bilayer in a steady-state nucleation process we obtain [1,6,7] = Aexp [ 7~Aef7 2 / k T A ~ t ] = Aexp [B/ln (C~ /C ) ]
( 15 )
As shown by Kashchiev [ 18 ], and as can be easily checked in our case with W a n d n from Eqns (10) and (13), the quantities W, n and A/~ are related by dW/dA/~=
- n
Keeping in mind this formula and neglecting the relatively minor dependence of the preexponential factor A on the concentration of surfactant in the solution, with the help of Eqns (9), (11) and (15) we arrive at the expression n = dln z/dln C
(16)
This equation demonstrates a possibility for directly determining the number of surfactant vacancies in the nucleus hole from the slope of an experimentally obtained In r(ln C) dependence. EXPERIMENTAL Materials
The foam bilayers were formed from aqueous solutions of sodium dodecyl sulfate (NaDoS), containing NaC1 electrolyte at a concentration of 0.5 mol dm -3. The electrolyte concentration was chosen to ensure close packing of surfactant molecules at the air/solution interface (necessary for the formation
343 of Newton black foam films [19-21] ) at the three investigated temperatures (10, 22 and 30°C) over the whole N a D o S concentration range studied. The experiments were carried out at N a D o S concentrations lower than the critical micelle concentration [22]. The ionic N a D o S surfactant used was specially synthesized by Henkel KGaA, Diisseldorf, F.R.G. and did not display a minim u m in the surface tension isotherm. "Suprapur" NaC1 (Merck) was used as the electrolyte, heated to 500 ° C to remove any surface-active contaminations. The solutions were prepared with doubly distilled water.
Methods The experiments were carried out by the microinterferometric method for investigation of single foam films [7,23-25]. The microscopic horizontal foam bilayer was formed in the middle of a biconcave drop in a specially constructed measuring cell, described elsewhere [24 ]. The measuring cell was put in a thermostatting device combined with a microscope with a special system for observation and registration of the light reflected and interfered from the film. For more precise control around the lower part of the cell where the foam bilayer was formed, an additional thermostatting jacket was mounted, so that the temperature was constant to + 0.05 ° C. The microscopic Newton black foam films investigated were formed by growth of black spots and had a fixed radius of 0.25 mm. The measurement of the lifetimes of the successively formed foam bilayers started after about 2 h which were necessary for equilibration of the investigated system [ 20 ]. In the case of N a D o S bilayers the rate of lateral growth of the black spots is relatively high [ 19 ] and the bilayer formation time is commensurate with the observer's eye reaction time (about 0.5 s) [5]. Thus, it was possible to measure the individual lifetime of each foam bilayer practically from the moment of its formation up to a lifetime of 300 s. The experiments for determining the mean lifetime of the foam bilayers were carried out at 10, 22 and 30 °C in different N a D o S concentration ranges. Given the temperature, at each of the experimentally studied N a D o S concentrations the mean lifetime ~ of the foam bilayer, in the case of the steady-state nucleation process, was calculated statistically from the results for the lifetimes of about 240 successively formed foam bilayers, measured on average in 6 different experiments. According to their duration, the lifetimes obtained at given C and T values were divided into 15 groups including 10-20 measurements each. This statistical data gave the experimental values of p and P at the respective values of t directly. The results f o r p (t) and P(t) were then used in Eqn (5) for the calculation of J ( t ) and, thereby, of J~ which is connected to according to Eqn (7). The comparison of the various theoretical dependences with the experimental ones was done by nonlinear regression.
344 RESULTS AND DISCUSSION
Under the conditions of the experiment, deviations from the P(t) dependence, Eqn (6), corresponding to a steady-state process, are observed in the initial period after the formation of the foam bilayers [4]. However, straightforward application of the theory [ 1 ] is possible only when nucleation is stationary. For that reason, the determination of the mean lifetime z of the foam bilayers in the steady-state nucleation case was done by fitting the experimental time dependences of the probability P for rupture and of the nucleation rate J [calculated from Eqn (5) ] to empirical P(t) and J(t) dependences. The points in Fig. 1 show the experimental dependence of the hole nucleation rate J on time t at C = 1 . 9 - 1 0 -4 mol dm -3 N a D o S and T--22°C. The increased nucleation rate at shorter times is probably due to transition processes related to the formation of the foam bilayer. Depending on the experimental conditions, after a certain transient time (about 50 s in this case) a constant (steady-state ) hole nucleation rate is reached. The value of the steadystate nucleation rate Js, determined from the plateau of the J(t) dependence after arithmetical averaging, is (11 _+3)" 10 -3 s-1. Unfortunately, there were cases for which the transient time for reaching constant nucleation rate was relatively long so that the determination of Js from the J(t) plateau was impossible within the 300 s time scale of the experiment. Because of that, in order to determine J~, we used an empirical dependence for the nucleation rate as a function of time. This dependence was used in the form
J(t) =Js + (Jo --Js) e-t/a
(17)
where Jo is the initial nucleation rate (at t--0 ), and a is the time constant for
012' 0,09
0_06 0.03 I
I
100
•
,.--T--'7-~-, 200
300t [s] Fig. 1. Dependence of the hole nucleation rate J on time t for foam bilayers of radius 0.25 mm, formed at a N a D o S concentration of 1.9.10 -4 mol d m -3 in the presence of 0.5 mol d m -3 NaC1 at 22°C: points, experimental results from calculations according to E q n (5); solid line, theoretical curve according to E q n {17); dashed line, value of the steady-state nucleation rate calculated according to E q n (18).
345
reaching the regime of steady-state nucleation. We found that the above formula gave a good fit to the experimental results in all cases in which, statistically, enough measurements were done. The theoretical J (t) dependence [Eqn (17) ], obtained from the best fit to the experimental J(t) data in Fig. 1 with Js = (11 4- 2 ). 10-3 s - 1, J0 = (85 4- 6 )" 10-3 s - 1 and a = 10 + 2 s, is shown in the same figure by the solid line. There is a very good agreement between the values of Js calculated from the plateau in Fig. 1 and from the fit of Eqn (17). The reliability of Eqn ( 17 ) allows the use of it in Eqns (2) and (3) to obtain, explicitly, the time dependence of the probability P of bilayer rupture in the form
P(t) = 1 - e x p [ - J s t + a ( J o -,.Is) (e - t / a - 1) ]
(18)
Rather than fitting J from Eqn (17) to the experimentally obtained J from Eqn (5), it is more advantageous to fit P from Eqn (18) directly to the experimental P(t) data, because then the error introduced b y p when using Eqn (5) is avoided. The experimental values for the probability P of rupture until time t at C = 1.9-10 -4 mol dm -3 and T = 2 2 ° C [4] are shown in Fig. 2. The solid curve is the best fit theoretical dependence from Eqn (18) with J,= (10 4-1). 10 -3 s- 1, Jo-- (79 + 5)" 10 -3 s - 1 and a= 10 ___1 s. Comparison of the values of the steady-state nucleation rate J , obtained by the different methods of calculation, shows that they coincide within the error limits. This value of J~ coincides also (within the error limits) with the one obtained by another method [26]. All of this allowed us to calculate Js by using Eqn (18) directly with the P (t) data. The values of J , obtained, were then used in Eqn (7) for determination of r corresponding to steady-state hole nucleation. The experimental results for the mean lifetime • of foam bilayers under conditions of steady-state nucleation at different NaDoS concentrations at the P 10 025 0.5 0.25 I
100
i
L
2o0
i
I
•
300
t is] Fig. 2. Dependence of the p r o b a b i l i t y P for bilayer rupture on time t for foam bilayers of radius 0.25 mm, formed at a N a D o S concentration of 1.9-10-4 mo| dm -3 in the presence of 0.5 mo] dm -3 NaC| at 22 ° C: points, experimental data; solid line, theoretical curve according to Eqn (18).
346
three investigated temperatures are presented in Fig. 3. The standard deviation of the ~ values is not greater than about 25%. Some of the results for the individual lifetimes of the foam bilayers at 22 °C are taken from previous work by Exerowa et al. [4]. The experimentally obtained decrease in the stability of the foam bilayers with increasing temperature is clearly demonstrated in Fig. 3. The experimental results for the initial nucleation rate Jo and for the time constant a are given in Table 1, the standard deviation being within 20%. A new step in the treatment of the experimental data for the mean lifetime of the foam bilayers at different N a D o S concentrations is the direct determination of the nucleus size n, i.e. the number of surfactant vacancies in the nucleus hole. Indeed, according to Eqn (16), the slope of the experimental r(C) dependence in In r versus In C coordinates yields directly the size of the nucleus hole at the respective values of C. It turned out that a third-order polynomial was sufficient for the numerical differentiation we used to calculate n. The n values obtained (accuracy _+0.5) at the various temperatures and N a D o S concentrations are depicted in Fig. 4. Table 2 shows the range of n values under the experimental conditions. The experimental n (C) dependences at the different temperatures were used for determination of the parameter B and the equilibrium N a D o S concentration Ce at the corresponding temperature. To this end, the best fit of Eqn (13) to the experimental n (C) data was made and the resulting theoretical dependences are illustrated in Fig. 4. The values of Ce and B from the best fit, as well as of W and A/2 (calculated according to Eqns (10) and (11), respectively), are given in Table 2. The results for 22 °C are in good agreement with those found earlier [4] by another fitting procedure. 500
1
/..00 300
I
3
200 100 0 1.5
2 2.5 Cx~O~ [tool dm'3]
Fig. 3. Dependence of the m e a n lifetime v on the concentration C of surfactant in the solution for NaDoS foam bilayers of radius 0.25 mm, obtained in the presence of 0.5 mol d m -3 NaCl: (()), ( O ) a n d ( • ), experimental results at 10, 22 and 30 ° C, respectively; curves 1, 2 a n d 3, theoretical dependences according to E q n (15).
347 TABLE1 I n i t i a l n u c l e a t i o n rate Jo a n d t i m e c o n s t a n t a for hole n u c l e a t i o n in N e w t o n b l a c k f o a m f i l m s of r a d i u s 0.25 m m , f o r m e d a t d i f f e r e n t t e m p e r a t u r e s f r o m N a D o S s o l u t i o n of c o n c e n t r a t i o n C in t h e p r e s e n c e of 0.5 m o l d m -3 N a C l C,10 4 (moldm -3)
10°C
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.4
22oc
30°C
Jo
a
Jo
a
Jo
a
(s-')
(s)
(s -~)
(s)
(s-')
(s)
0.665 0.442 0.190 0.099 0.102
1.50 3.01 8.83 10.31 8.78
3.33 4.15 9.96 9.18 6.79 10.40 13.95 21.00 -
0.579 0.283 0.075 0.119 0.059 0.016
-
0.579 0.292 0.162 0.184 0.136 0.079 0.043 0.026
1.35 3.22 10.38 7.88 17.13 77.97
r~
1
15
40 30 /f
20
/ //
IO
/
1/
."
11 OL
8.2
i
8!.
i
8.6
i
8.~ -t,-,c
A
s.o
[~o,~31
O
2
6
10 14 106/T2 [K"2]
Fig. 4. (left) D e p e n d e n c e o f t h e n u m b e r n of s u r f a c t a n t v a c a n c i e s in t h e n u c l e u s hole o n t h e c o n c e n t r a t i o n C of s u r f a c t a n t in t h e s o l u t i o n for N a D o S f o a m bilayers f o r m e d in t h e p r e s e n c e of 0.5 m o l d m -3 N a C h ( ¢ ) , ( O ) a n d ( • ) , e x p e r i m e n t a l r e s u l t s at 10, 22 a n d 30 ° C, respectively; c u r v e s 1, 2 a n d 3, t h e o r e t i c a l d e p e n d e n c e s a c c o r d i n g to E q n ( 13 ). Fig. 5. ( r i g h t ) D e p e n d e n c e of B o n 1 / T 2 for N a D o S f o a m bilayers f o r m e d in t h e p r e s e n c e of 0.5 m o l d m - 3 N a C h p o i n t s , e x p e r i m e n t a l d a t a ( T a b l e 2); s t r a i g h t line, t h e o r e t i c a l d e p e n d e n c e acc o r d i n g to E q n (14).
According to Eqn (14), if both y and Aef are T-independent, B must be a linear function of 1 / T 2 with zero intercept and a slope allowing the calculation of the hole specific edge free energy 7. The experimental values of B at the
348 TABLE 2 Calculated values of parameters of hole nucleation in Newton black foam films of radius 0.25 mm, formed at different temperatures from NaDoS solution in the presence of 0.5 mol din-3 NaC1 T (°C)
n
10 22 30
10-14 8-11 6-9
C~"104 (tool dm -3)
B
8.8+0.8 13.3__+0.1 15.3_+1.9
42___+4 39__+3 33_+4
Az/kT
W/kT
A (s)
1.7-2.0 1.8-2.2 1.8-2.3
21-25 17-21 14-18
(8.5__+3.2)-10 -9 (2.5_+0.2).10 -7 (1.0_+0.3)-10 -5
~0 6.8I
32
3.3
3,~
3.s
36
103/T[K"l] Fig. 6. Dependence of the equilibrium surfactant concentration Ce on l i T for NaDoS foam bilayers formed in the presence of 0.5 mol dm-3 NaCh points, experimental data (Table 2 ); straight line, theoretical dependence according to Eqn {12).
three investigated temperatures are shown in Fig. 5 and the straight line is drawn by the method of least squares under the condition of zero intercept. As can be seen from Fig. 5, Eqn (14) is in conformity with the experimental data. From the slope of the straight line, with the help of Eqn (14), for bilayer circular holes (then Ae~=Ao/2 [6,7], Ao--4.2"10 -19 m 2 [20] being the area of a NaDoS molecule in the foam bilayer) we find that y-- (3.07 _+0.06)" 10-11 j m -1. It was supposed in the calculations that Ao does not change considerably at the investigated temperatures, which is likely for the closely packed [20] surfactant molecules in the bilayer. If the hole is of monolayer depth (then Aef---A o [6,7] ), y= (2.20+0.04) "10 -11 J m -1. These ~ values are in very good agreement with those found earlier [4,7] and are also consistent with results obtained for bilayer lipid membranes by other authors [27,28]. According to Eqn (12), the temperature dependence of Ce offers a possibility for the determination of the binding energy Q ofa NaDoS molecule in the foam bilayer. The points in Fig. 6 show the experimental values of C~ in In Ce versus 1 / T coordinates. The straight line represents the best fit of the Arrhenius-type dependence, Eqn (12), to the experimental results. For the binding energy of NaDoS molecule we obtain Q= (6.7 + 1.0)-10 -20 J, and In Co-1.6 + 1.2. The-
349 oretically, the value of the binding energy (Q = 16 kT at 30 ° C ) is high enough [1,6] to ensure the occurrence of the two-dimensional phase transition of "condensation" of the "gas" of NaDoS vacancies in the bilayer into holes leading to the bilayer rupture. Knowing B and Ce and using Eqn (15), we can calculate the preexponential factor A from the experimental ~(C) data. The A values obtained for the three investigated temperatures do not show a considerable dependence on C and their arithmetic means are presented in Table 2. The experimental fact that A increases with increasing temperature cannot be interpreted without knowing the actual mechanism of hole formation in the bilayer. A possible explanation of this dependence may be given in the framework of the mechanism of nucleation of holes by aggregation of vacancies on preferred sites (e.g. along line defects) in the bilayer. For instance, accumulation of vacancies along structural defects is well known to occur in crystalline solids [29]. If the increase of the temperature can cause a decrease in the density of these preferred sites in the bilayer, an increase of A may be expected, since A is inversely proportional to the density of the nucleation sites [ 1,6 ]. The theoretical ~(C) dependences for the three investigated temperatures, calculated according to Eqn (15) with the values of A, B and Co from Table 2, are shown in Fig. 3. As can be seen, there is a good agreement between the theoretical and the experimental dependences for the mean lifetime r of the foam bilayers at the different temperatures. The observed decrease in the bilayer stability (i.e. the shortening of 3) with increasing temperature is understandable in the light of the nucleation theory for rupture of foam bilayers [ 1 ], upon recalling the considerable decrease in the dimensionless work W/kT for formation of a nucleus hole with an increase in temperature (Table 2). CONCLUSION The experimental investigation of the effect of temperature on the mean lifetime ~ of Newton black foam films shows a decrease of their stability with increasing temperature. This fact is explained by the thermal activation of the process of fluctuation nucleation of holes in the foam bilayers which causes their rupture [1,6,7]. The values of important thermodynamic quantities of NaDoS foam bilayers are calculated by comparison of the experimentally obtained dependences of r on the bulk concentration of surfactant in the solution at different temperatures with the theoretical ones known from the nucleation theory of bilayer rupture [1,6,7]. A good agreement between theory and experiment is established, indicating that the hole specific edge free energy ~ is virtually T-independent in the temperature range studied. The temperature dependence of the kinetic preexponential factor A remains a problem for further study. For the first time, the binding energy Q of a surfactant molecule in foam
350
bilayer is obtained. Its value for the investigated NaDoS bilayer foam films justifies the assumption for the occurrence of a two-dimensional phase transition of "condensation" of the "gas" of surfactant vacancies in the bilayer into holes, which is the basis of the nucleation theory for rupture of foam bilayers [ 1 ]. Finding the value of Q is essential for understanding the behavior of the surfactant molecules in the bilayer and the interactions between them. ACKNOWLEDGEMENTS
The authors are grateful to Dr M. Paunov for providing them with his "Best Fit" computer program. This project has been completed with the financial support of the Committee for Science at the Council of Ministers under Contract No 287/17.IV.1987.
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