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Adsorption Kinetics of Oxyethylated Polyglycol Ethers at the Water-Nonane Interface V. B. Fainerman,†,‡ S. A. Zholob,† and R. Miller*,‡ Institute of Technical Ecology, 25 Shevchenko Boulevard, 340017 Donetsk, Ukraine, and Max-Planck-Institut fu¨ r Kolloid und Grenzfla¨ chenforschung, Rudower Chaussee 5, D-12489 Berlin-Adlershof, Germany Received January 22, 1996. In Final Form: September 30, 1996X A special measuring cell modifies the maximum bubble pressure tensiometer MPT1 to a drop volume apparatus which allows us to analyze the stability of the formation of single drops and to detect instabilities as observed recently in the form of drop volume bifurcations. This modified apparatus was used to study the adsorption kinetics of different Tritons (X-45, X-100, X-165, and X-405) at the water-nonane interface. The results of the measured dynamic interfacial tensions for long adsorption times are fully in line with a diffusion-controlled adsorption model. The short adsorption time data show adsorption rates faster than expected from diffusion, which may be caused by reorientation processes of the Triton molecules at low interfacial coverage. The experiments performed under different conditions (Triton dissolved in water, in nonane, or in both phases, respectively) allow us to estimate the distribution coefficient of the Tritons between water and oil, which increases with a decrease in the number of EO groups.
Introduction The oxyethylene chain of nonionic surfactants is able to adsorb at the water-air interface. This peculiarity of a nonionic surfactant with a long polyethylene glycol chain allows a self-regulation of the partial molar area of the surfactant molecule at the interface. The mechanism of self-regulation is based on the of Braun-Le Chatelier principle. This principle was discussed with respect to surfactant mixtures by Joos and Serrien1 and with respect to surfactants with changing molar area by Fainerman et al.2 At low surface pressure the nonionic molecule (including the hydrophobic and hydrophilic parts) lays flat in the surface covering a maximum area, in agreement with the model. The molar area decreases with a decrease in the surface tension at higher bulk concentration (under equilibrium conditions) or at longer adsorption times (under dynamic adsorption conditions). First, the oxyethylene chain is removed from the surface into the bulk, and second, the aliphatic chains change from a horizontal to a more or less normal orientation. One special feature of this effect is an unexpected fast decrease of surface tension γ of nonionic solutions. At a short adsorption time the rate of decrease of the surface tension is about 10 times faster than that calculated by a simple diffusion model which does not consider a reorientation.2 Recently, the dynamic surface tension of aqueous solutions of six polyglycol oxyethylene ethers with different numbers of EO groups (Triton X surfactants with 4.540.5 EO groups) was studied systematically.3 For all samples, the adsorption rate was faster than was expected from a classical diffusion model at surface pressure:
Π ) γo - γ(t) > 2 mN/m This effect was stronger at larger EO numbers and higher temperatures. * To whom all correspondence should be addressed. † Institute of Technical Ecology. ‡ Max-Planck-Institute. X Abstract published in Advance ACS Abstracts, December 15, 1996. (1) Joos, P.: Serrien, A. J. Colloid Interface Sci. 1991, 145, 291. (2) Fainerman, V. B.; Makievski, A. V.; Joos, P. Colloids Surf. A 1994, 90, 213. (3) Fainerman, V. B.; Miller, R.; Makievski, A. V. Langmuir 1995, 11, 3054.
The oxyethylene group should also adsorb at the wateroil interface. For unsaturated oils with a low interfacial tension compared to that of water, the penetration of the EO group should be very weak. The aim of this work is to study the dynamic interfacial tensions of Triton X solutions (X-45, X-100, X-165, and X-405) at the waternonane interface. In a forthcoming paper we will discuss the dynamic adsorption behavior of these surfactants at other water-oil interfaces. The study of dynamic interfacial tensions is a comparatively difficult problem, experimentally as well as theoretically.4-8 The theoretical analysis is complicated by the additional process of the transfer of surfactant molecules across the interface. Experimental difficulties arising especially at short adsorption times are described in detail in refs 6-8. Only recently it was shown that the application of the drop volume technique as one of the most frequently used methods to measure dynamic interfacial tensions yields undefined drop volumes at short drop times. Stable bifurcations in the drop volume-flow rate dependencies were observed.9 Thus, much care is necessary to obtain physically reasonable experimental data. Theory There are three principal cases for the adsorption process at a liquid-liquid interface: case a, the surfactant is present in the water phase only; case b, the surfactant is present in the oil phase only; and case c, the surfactant is present in both phases with an equilibrium surfactant concentration distribution. The theoretical model for the cases a and b (eqs 1) is a generalization of the theory of Ward and Tordai10 and was first proposed by Hansen5 where Γ(t) is the adsorption as a function of time t, ci and Di are the bulk concentration and diffusion coefficient in phase i (i ) 1, water; i ) 2, oil), (4) Van Hunsel, J.; Joos, P. Langmuir 1987, 3, 1069. (5) Hansen, R. S. J. Phys. Chem. 1960, 69, 637. (6) Van Hunsel, J.; Bleys, G.; Joos, P. J. Colloid Interface Sci. 1986, 114, 432. (7) MacLeod, C. A.; Radke, C. J. J. Colloid Interface Sci., 1993, 160, 435. (8) Miller, R.; Schano, K.-H.; Hofmann, A. Colloids Surf., A 1994, 92, 189. (9) Fainerman, V. B.; Miller, R. Colloids Surf., A 1995, 97, 255. (10) Ward, A. F. H.; Tordai, L. J. Chem. Phys. 1946, 14, 453.
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Γ(t) )
2 [c1oxD1t xπ (xD1 + KxD2)
Γ(t) )
Fainerman et al.
∫0xtc1(0,t - τ) dxτ]
(1a)
2 [Kc1oxD2t xπ (xD1 + KxD2)∫0xtc1(0,t - τ) dxτ] (1b)
and K ) c2o/c1o is the equilibrium distribution coefficient of the surfactant between the two liquid phases. The subscript “o” indicates the equilibrium bulk concentration. If the surfactant distribution between the two liquids is in equilibrium (case c), we obtain
(xD1 + KxD2) [c1oxt xπ
Γ(t) ) 2
∫0xtc1(0,t - τ) dxτ] (1c)
The eqs 1 assume a constant interfacial area and no liquid flow. For the drop volume experiment, a radial liquid flow can be used to describe the adsorption process at the surface of a growing drop. The equation corresponding to case c then reads11,12
Γ(t) )
[ x
(xD1 + KxD2)
xπ
3t 7
2c1o
∫0(3/7)t
t-2/3
7/3
( )
]
c1(0,(3/7)t7/3 - τ) dxτ (2c)
x)0
( )
- D2
∂c2 ∂x
x)0
(3a)
where x is the coordinate normal to the interface and t is the effective time (for a growing drop with an assumed spherical geometry, we have t ) (3/7)texp where texp is the experimental time). Following Van Hunsel and Joos,4 the thickness of the diffusion layer is (πDt)1/2 and from eq 3a we obtain
dΓ(t) ) dt
x
D1 (c - c1(0,t)) πt 1o
x
D2 Kc (0,t) (4a) πt 1
In analogous way the other two cases yield
dΓ(t) ) dt
x
x (x x )
D2 K(c1o - c1(0,t)) πt
dΓ(t) ) (c1o - c1(0,t)) dt
D1 + πt
D1 c (0,t) (4b) πt 1 D2 K πt
(
c1(0,t) ) aL
(4c)
(11) Davies, J. T.; Smith, J. A. C.; Humphreys, Proc. Int. Conf. Surf. Act. Subst. 1957, 2, 281. (12) Miller, R. Colloid Polym. Sci. 1980, 258, 179. (13) Miller, R. Colloid Polym. Sci. 1981, 259, 375. (14) Ziller, M.; Miller, R. Colloid Polym. Sci. 1986, 264, 611.
)
Γ(t) Γ∞ - Γ(t)
(5)
The parameter aL is the concentration at one-half of the interfacial coverage, and Γ∞ is the maximum interfacial excess concentration. Together with eq 5, the eqs 4 transform into
dΓ(t) ) dt
xπt1 (xD c
dΓ(t) ) dt
aLΓ(t)
1 1o
-
(x
x
1 πt
dΓ(t) ) dt
Equations 1a and 1b are transformed in an analogous way. The resulting equations, as well as eq 1, are very complex and cannot be solved analytically but numerically by following known algorithms.13,14 Another description of the adsorption process at the liquid-liquid interface is based on the diffusion-penetration theory proposed by Van Hunsel and Joos.4 The equation corresponding to an aqueous solution drop growing into an oil phase (case a) reads
∂c1 dΓ(t) ) D1 dt ∂x
In all derived equations we used the subsurface concentration in the aqueous phase c1 (0,t), which transforms into relationships with the subsurface concentration in the oil phase c2 (0,t) through the distribution law K ) c2 (0,t)/c1 (0,t), which holds at any time. For a further use of these equations, the subsurface concentration has to be replaced by the interfacial excess concentration (adsorption) Γ(t). A possible relationship is the Langmuir isotherm which can be used when we assume a pure diffusion-controlled adsorption mechanism,
(xD1 + KxD2)
Γ∞ - Γ(t)
D2Kc1o -
aL Γ(t)
)
(xD1 + KxD2)
Γ∞ - Γ(t)
xπt1 (xD + KxD )(c 1
2
(6a)
aL Γ(t) 1o
-
)
Γ∞ - Γ(t)
)
(6b) (6c)
The eqs 6 can easily be numerically integrated. To obtain a relationship for the dynamic interfacial tension, the function Γ(t) has to be replaced by γ(t). The equivalent relationship for a Langmuir isotherm is the Langmuirvon Szyszkowski equation
Π(t) ) γo - γ(t) ) -RTΓ∞ ln(1-Γ(t)/Γ∞)
(7)
where γo is the interfacial tension in the absence of any surfactant, R and T are the gas constant and absolute temperature, and Π(t) is the dynamic surface pressure. For the interpretation of our data some approximate equations are very useful. For short adsorption time t, eqs 6a and 7 yield
Π(t)|tf0 ) 2RTc1o
x
D1t π
(8)
and after differentiation with respect to t1/2, we have
( ) dγ dxt
) -2RTc1o tf0
x
D1 π
(9a)
For the other two cases (eqs 6b and 6c) we obtain
( ) dγ dxt
( ) dγ dxt
tf0
) -2RTc1o tf0
x
) -2RTc1oK
(
D2 π
(9b)
)
xD1 + KxD2 xπ
(9c)
From eq 9c, we see that at a short time the slope dγ/d(t1/2) for the adsorption from both phases is equal to the sum of the slopes for the adsorption from each of the two phases (cases a and b). At dΓ/dt ) 0 we obtain the equilibrium of adsorption Γ0 in case c, while for the other two cases, a and b, only a
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stationary state of adsorption Γ* is established,
c1o Γ* ) Γ∞ a1 + c1o
(10a)
c1o Γ* ) Γ∞ a2 + c1o
(10b)
c1o Γo ) Γ∞ aL + c1o
(10c)
or Figure 1. Drop formation time of water drops in toluene at two different values of water flow per unit time; rcap ) 0.135 cm.
aLΓo c1o ) Γ∞ - Γo with a1 ) aL[1 + K (D2/D1)1/2] and a2 ) aL[1 + (D1/D2)1/2/K]. Finally, we need a relationship to describe the adsorption process at a long adsorption time. Case c is interesting because, at a long adsorption time, we can assume a state not too far from the distribution equilibrium. From eq 1c we obtain
∆c ) c1o - c1(0,t) )
x
Γ(t) 2
π 1 (11c) t xD + KxD 1 2
Combined with the Gibbs fundamental equation
dγ ) -RTΓ d ln c
(12)
the differentiation of eq 11c with respect to 1/t1/2 results in a long adsorption time approximation,
(
dγ(t)
)
d(1/xt)
) tf∞
RTΓ(t)2 xπ 2c1o xD + KxD 1 2
(13c)
which is a generalization of the well-known approximation of Joos-Hansen15 for the case of simultaneous adsorption from two immiscible liquid phases. Materials and Method Poly(ethylene glycol) octylphenyl ethers C14H21O(C2H4O)nH (purchased from Serva and Aldrich) with different EO units (Triton X-45 (n ) 4.5), Triton X-100 (n ) 10), Triton X-165 (n ) 16.5), and Triton X-405 (n ) 40.5)) were used without further purification. Toluene, nonane, 1-propanol, and 1-hexanol were also used without further purification. Solutions were prepared with doubly distilled water. All measurements were performed at 25 °C. The drop volume technique used for the measurement of dynamic interfacial tensions is a preferential method for liquidliquid interfaces.16 An automatic drop volume tensiometer TVT1 from LAUDA, Germany, was described in detail elsewhere (17). Because of two reasons a specially developed equipment was chosen. Reason 1. The appearance of bifurcations in the drop volume in certain liquid flow rate intervals at drop times t < 1 s requires an instrument with which individual drops can be detected. In the present state, the TVT1 allows only two and more drops per measurement. The TVT1 then averages all the measurements so that no information about each possible drop volume bifurcation is available. The data given in Figure 1 show, as an example, the change of drop time for water drops formed in toluene. The stability of drop formation with respect to the flow was (15) Fainerman, V. B.; Makievski, A. V.; Miller, R. Colloids Surf., A 1994, 87, 61. (16) Miller, R.; Joos, P.; Fainerman, V. B. Adv. Colloid Interface Sci. 1994, 49, 249. (17) Miller, R.; Hofmann, A.; Hartmann, R.; Schano, K.-H.; Halbig, A. Adv. Mater. 1992, 4, 370.
Figure 2. Schematic of the experimental setup: 1, air flow from the MPT1; 2, vessel with water or aqueous solution; 3, flow capillary; 4, steel capillary; 5, vessel with nonane; 6, light barrier for drop registration; 7, interface to the MPT1; and 8, liquid overflow. demonstrated recently for water and aqueous solutions.9 Only small changes in the liquid flow rate can lead from a stable drop formation to dramatic bifurcations in the drop volume. Reason 2. The need of varying the water-oil volume ratio to realize the cases b and c to determine the distribution constant K requires a special design of the vessel, which was comparatively simple when using the MPT1 setup. The special design of the experimental setup (Figure 2) is based on the maximum bubble pressure tensiometer MPT1 (LAUDA, Germany), which was described in detail elsewhere.18 The air flow from the MPT1 is fed into vessel 2 filled with water or the aqueous solution. A modified version of the standard MPT1 software allows us to keep the gas pressure in vessel 2 constant. The amount of liquid detaching from the capillary 4 is equal to the amount of air L pumped by the MPT1 into vessel 2. The volume of a drop V can now be calculated from the liquid (or gas) flow rate and the time interval between two drops V ) Ltexp. The drop time is measured by a light barrier, 6, which feeds a signal, 7, into the MPT1 (instead of the signal for a bubble detachment in the standard use). The constancy of the flow rate is better than 1%. The comparison of experimental data for standard liquids shows an excellent agreement between the results from this setup and those from the TVT1. The present setup allows measurements from 0.2 up to 10 s of drop time, which corresponds to about 0.1-4 s effective adsorption time t. Each experimental run consists of up to 50 different flow rates with 10 drops per flow rate used to calculate average values. To obtain real interfacial tensions, correction factors have to be applied. In the present work the correction factors of Wilkinson have been used.19 Beforehand, the measured drop volumes (data in intervals with bifurcations excluded) have to be corrected with respect to a so-called hydrodynamic effect.6,8,20-23 The drop volume as a function of 1/t for different pure liquids and highconcentrated surfactant solutions (for which γ ) const for t > 0.1 s) are shown in Figure 3. The drop detachment times for the (18) Fainerman, V. B.; Miller, R.; Joos, P. Colloid Polym. Sci. 1994, 272, 731. (19) Wilkinson, M. C. J. Colloid Interface Sci. 1972, 40, 14. (20) Kloubek, J.; Friml, K.; Krejci, F. Collect. Czech. Chem. Commun. 1976, 41, 1845. (21) Jho, C.; Burke, R. J . Colloid Interface Sci. 1983, 95, 61. (22) Van Hunsel, J.; Joos, P. Colloids Surf., A 1987, 24, 139. (23) Van Hunsel, J. Dynamic Interfacial Tension at Oil-Water Interfaces. Ph.D. Thesis, University of Antwerp, Antwerp, Belgium, 1987.
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Figure 3. Drop volume as a function of t-1 for different liquid systems: water-nonane (2); water-air with different capillaries (0, (, )); and Triton X-405 solution (9), cf. data in Table 1.
Figure 4. Dynamic interfacial tension of Triton X-45 solutions as a function of t1/2, co ) 1.2 × 10-8 mol/cm3, five subsequent runs (for details, see text). different systems determined from fitting the relationship in eq 14 to the experimental data are summarized in Table 1. The
(
Vcor ) V/ 1 +
)
to texp
(14)
drop detachment times are in good agreement with those obtained from experiments in a time interval 1 < t < 30 s using the same correction formula.8 The whole measuring procedure with the special design of the MPT1 was automated by the modified MPT1 software. The measurement with the Triton X solutions was performed as follows. At the beginning, vessel 5 contains only pure nonane into which drops of the surfactant solution detach from the capillary. The amount of aqueous solutions was about 300 mL while the amount of nonane was 10-20 mL. Thus, even at distribution equilibrium of surfactant the concentration in the aqueous phase is changed by less than 1%, assuming a coefficient K ≈ 1. This experiment refers to case a described above. The interfacial tension experiments at the water-nonane interface were repeated until the results of two subsequent runs were identical within the experimental error. Figure 4 shows the results of five subsequent runs for a Triton X-45 solution.
Figure 5. Dynamic interfacial tension of a Triton X-45 solution as a function of t1/2, co ) 2.4 × 10-8 mol/cm3, for the three different cases (a) (9), (b) ((), and (c) (0) and in the absence of Triton X-45 ()).
Figure 6. Dynamic interfacial tension of a Triton X-405 solution as a function of t1/2, co ) 2.54 × 10-8 mol/cm3, for the three different cases: (a) (9), (b) (), (), and (c) (0). The last run refers to the case of adsorption from both adjacent phases (case c). Only experiments for this case allow quantitative interpretation because the experimental conditions completely agree with the theoretical model. For the other two cases, there is always a systematic but unknown deviation of the experimental conditions from the conditions of the theory so that these results were only used to estimate the distribution coefficient K.
Results and Discussion The experimental results for solutions of Triton X-45 and X-405, respectively, of almost the same bulk concentration are shown in Figures 5 and 6. From the three cases (a, b, and c) studied, the distribution coefficient can be estimated, which is significantly higher for Triton X-45 than for Triton X-405. This can be explained by the larger difference in the dynamic interfacial tensions between case, a and b for Triton X-45 compared to Triton X-405. Second, the data for Triton X-45 for case b are close to those for case c. In addition, the dynamic interfacial tensions for the Triton X-405 for case b are very high.
Table 1. Detachment Time to Calculated from Experimental Data of Different Liquid-Fluid Systems (cf. Figure 3) no
interface
surfactant
concn [× 10-6 mol/cm3]
rcap [cm]
γeq [mN/m]
to [s]
1 2 3 4 5 6 7 8 9
water-air water-air water-air water-air water-air water-air water-air water-air water-nonane
Triton X-100 1-hexanol 1-propanol Triton X-405 1-propanol
7.75 26 1000 5 500
0.152 0.152 0.152 0.152 0.152 0.135 0.152 0.203 0.135
32 40.8 43 46.5 49 72.4 72.4 72.4 50
0.028 0.028 0.026 0.047 0.043 0.040 0.046 0.040 0.080
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Figure 7. Dynamic interfacial tension of Triton X-45 solutions as a function of 1/t1/2 measured under the condition of case c: co ) 1.2 × 10-8 (9), 2.4 × 10-8 (0), 4.3 × 10-8 ((), 12.0 × 10-8 ()) mol/cm3.
Figure 10. Dynamic interfacial tension of Triton X-405 solutions as a function of 1/t1/2 measured under the condition of case c: co ) 0.254 × 10-8 (9), 0.51 × 10-8 (0), 2.54 × 10-8 ((), 5.1 × 10-8 ()), 10.0 × 10-8 (2) mol/cm3.
Figure 8. Dynamic interfacial tension of Triton X-100 solutions as a function of 1/t1/2 measured under the condition of case c: co) 1.55 × 10-8 (9), 3.1 × 10-8 (0), 7.7 × 10-8 ((), 15.5 × 10-8 ()), 31.0 × 10-8 (2) mol/cm3.
Figure 11. Dynamic interfacial tension of Triton X-100 solutions as a function of 1/t1/2 measured under the condition of case b: co ) 1.55 × 10-8 (9), 3.1 × 10-8 (0), 7.7 × 10-8 ((), 15.5 × 10-8 ()), 31.0 × 10-8 (2) mol/cm3. Table 2. Interfacial Tension of Triton X Solutions at the Water-Nonane Interface at t f ∞ for the Three Cases a, b, and c As Described in the Text
surfactant Triton X-405
Triton X-165
Figure 9. Dynamic interfacial tension of Triton X-165 solutions as a function of 1/t1/2 measured under the condition of case c: co ) 0.54 × 10-8 (9), 2.14 × 10-8 (9), 5.4 × 10-8 ((), 10.7 × 10-8 ()) mol/cm3.
For a first estimation of the distribution coefficient K, the equilibrium or steady state interfacial tensions can be used. The equilibrium values are available as the extrapolated values σ(t-1/2)|tf∞ from the data (case c) given in Figures 7-10. The experiments of case b, for example, yield the same type of results, as shown in Figure 11 for solutions of Triton X-100 in nonane. The results of the extrapolations are summarized in Table 2. The presented data for Triton X-100 and X-405 for case c agree very well with those given by Van Hunsel and Joos.6,21-23 Also, the data for Triton X-100 of MacLeod and Radke7 are in good agreement with our results for case c.
Triton X-100
Triton X-45
a
γtf∞ [mN/m] c1o [× 10-8 mol/cm3] case a case b case c 0.254 0.51 1.00 2.54 5.10 10.00 0.54 1.10 2.14 5.40 10.70 1.555 3.10
34 29 25 22.5 20.5 18 36 34 29 20 17 24 18
36 33 30 28 26 24
7.70 15.50 31.00 1.20 2.40 4.70 12.00
12 7.5 4.5 28.5 18.5 16 11
19 17 15 26 18 14.5 10
38 34 32 26 20
(dγ/d[1/xt])tf∞ [s1/2 mN/m]
32 28 24 22 20 18 30 28 26 19 15 20 16.5 (17a) 11 6.5 3.5 24 17 13 9
10 10 11 9 6.2 3.4 7 6 5 5 4 7 6 3.5 2 1.2 5.5 4.5 3.5 3
Value from ref 7 at the water-dodecan interface.
Using extrapolated equilibrium interfacial tensions values and Gibbs equation (eq 12), the minimum area per molecule can be estimated. The values of Γ∞ obtained are almost the same for all studied Tritons, between 2.1 × 10-10 and 2.4 × 10-10 mol/cm2, again in good agreement
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Table 3. Average Distribution Coefficients of Triton X at the Water-Nonane Interface Expressed in Terms of K* surfactant
coeff K*
surfactant
coeff K*
Triton X-405 Triton X-165
0.30 ( 0.15 0.35 ( 0.15
Triton X-100 Triton X-45
0.45 ( 0.20 1.5 ( 0.8
with the literature.24 For the interface hexadecane-water solutions of the poly(ethylene glycol) dodecyl ethers C12H25O(C2H4O)xH, x ) 2, 3, 4, and 8, Rosen and Murphy25 also found Γ∞ ) constant. In contrast, the minimum area per Triton molecule at the water-air interface varies strongly with the number of ethylene oxide groups.26 This value increases from 1.4 × 10-10 mol/cm2 for Triton X-405 (40.5 EO groups) to 4.8 × 10-10 mol/cm2 for Triton X-45 (4.5 EO groups). Thus, while at the water-air interface the molecules can be partially located within the interface,1 this possibility seems to be unlikely at the water-oil interface. From eqs 7 and 10 we can obtain the so-called Szyszkowski isotherm (eq 15) where the parameter a takes
[
Π ) γo - γ∞ ) -RTΓ∞ ln 1 +
]
co a
(15)
values of a1, a2, or aL, depending on the experiment (cases a, b, or c). The interfacial tension of water-nonane was measured to be γo ) 50.0 mN/m. The values for γtf∞ given in Table 2 were used to calculate the values of the parameter a being between 1 × 10-10 and 4 × 10-10 mol/ cm3. The a values are not constant at different bulk concentration co so that the results from eq 15, as well as from eqs 7 and 10, are only estimates. Alternatively, Van Hunsel and Joos21-23 described their data by using a Temkin isotherm, but the fitting is of the same quality. To estimate the distribution coefficient K as a function of co, the values of a1, a2, and aL were used. The ratio D2/D1 can be estimated on the basis of the relationship given by Wilke and Chang:27
D≈
TxΦM η(Vm)0.6
(16)
where M, η, and Φ are the molecular mass, the viscosity, and the association parameter of the solvent, respectively, and Vm is the molecular volume of the solute. For the two solvents 1 and 2, the ratio D2/D1 is obtained from eq 16 and is
D2 xΦ2M2 η1 ) D1 xΦ M η2 1
(17)
1
For the water-nonane system, D2/D1 ) 2.3. Note that the values of D calculated directly from eq 16 are systematically higher than the experimentally determined values. For example, for Triton X-45 the “theoretical” D value is three times higher, and for Triton X-405, two times higher. An explanation of this effect could be the association and hydration of Triton molecules in water. As there are no experimental data on the diffusion coefficients of Tritons in nonane available, it seems reasonable to determine the coefficient K* ) K(D2/D1)1/2 instead of K. This is exactly the parameter which in (24) Abramson, A. A. Surfactants; Khimiya: Leningrad, 1979; p 91. (25) Rosen, M. J.; Murphy, D. S. Langmuir 1991, 7, 2630. (26) Fainerman, V. B.; Miller, R. Colloids Surf., A. 1995, 97, 65. (27) Wilke, C. R.; Chang, P. AlChE J. 1955, 1, 264.
Figure 12. Dynamic interfacial tension of Triton X-405 solutions as a function of t1/2 measured under the condition of case c: co ) 0.254 × 10-8 (9), 0.51 × 10-8 (0), 2.54 × 10-8 ((), 5.1 × 10-8 ()), 10.0 × 10-8 (2) mol/cm3. Solid lines represent the diffusion-controlled model for co ) 0.254 × 10-8 mol/cm3 and co ) 10-8 × mol/cm3.
addition to the known diffusion coefficient of the surfactant in water D1 controls the adsorption kinetics model, eqs 6c or 9c. As average values calculated for the entire concentration range, we obtained the results summarized in Table 3. The K* value increases with decreasing number of EO groups, which is in line with the characteristics of the dynamics interfacial tensions shown in Figures 5 and 6. We can conclude that the presaturation of the oil phase with surfactant is one important prerequisite for suitable studies of Tritons at the water-alkane interface. In Table 2, the values of (dγ/dt-1/2)tf∞ are summarised for all experiments of case c, which can be used to check the adsorption mechanism. The diffusion coefficients calculated on the basis of a Langmuir isotherm eqs 10c and 13c, are varying, which can be explained by the comparatively bad fitting of the experimental data by this isotherm. The results unambiguously show that the Tritons adsorb in a diffusion-controlled manner, which confirms the results obtained for all Tritons at the waterair interface3,4,6,21,26 and for Triton X-100 at the waterheptane interface.4,21 This conclusion agrees very well with the general ideas of adsorption kinetics models discussed elsewhere.15,28 The most interesting results from the present experiments can be obtained in the short-time adsorption interval. As examples, the dynamic surface tensions of Triton X-405 solutions (case c) are shown as γ ) γ(t1/2) plots in Figure 12, and for Triton X-100 for all three cases a, b, and c in Figure 13. In addition, the theoretical dependencies calculated from eq 9 are shown as solid lines. The experimental results are significantly lower than the data calculated from the theory. Similar results are obtained for all studied Tritons. The values of dγ/dt1/2 determined from the experiments are summarized in Table 4. As one can see, the values obtained for case c is about the sum of the values obtained for the cases a and b, while the contribution of case b increases with decreasing number of EO groups. The values of dγ/dt1/2 calculated from eq 9 are 10-30 times smaller than those given in Table 4. Unfortunately, there are very few literature data available on adsorption kinetics at the water-oil interface, and therefore, a direct comparison of the present results (28) Dukhin, S. S.; Kretzschmar, G.; Miller, R. In Studies of Interface Science; Mo¨bius, D., Miller, R., Eds.; Elsevier: Amsterdam, 1995; (Dynamics of Adsorption at Liquid Interfaces. Theory, Experiment, Application) Vol. 1. (29) Liggieri, L.; Ravera, F.; Passerone, A. J. Colloid Interface Sci. 1995, 169, 226.
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Adsorption Kinetics of Oxyethylated Polyglycol Ethers
Langmuir, Vol. 13, No. 2, 1997 289
improved agreement, we again suggest assuming a process of reorienting the molecules at low surface coverage in the short time interval, as was proposed for the waterair interface recently.3 Another additional effect of the all too fast interfacial tension decrease in the short adsorption time interval may be the initial loading of the residual drop after a drop detachment. A detailed analysis of the latter effect will be given in a forthcoming paper.
Figure 13. Dynamic interfacial tension of Triton X-100 solutions as a function of t1/2 measured under the condition of cases a ((), b (9), and c (0). Solid line represents the diffusioncontrolled model for case c. Table 4. Dynamic Interfacial Tension of Triton X Solutions at the Water-Nonane Interface at t f 0 for the Three Cases a, b, and c As Described in the Text (dγ/dxt)tf0 [mN/m s1/2] surfactant
c1o [× 10-8 mol/cm3]
case a
Triton X-405
0.254 0.51 1.00 2.54 0.54 1.10 2.14 1.555 3.10 1.20 2.40
2.5 8 12 16 5 13 15 20 30 8 30
Triton X-165 Triton X-100 Triton X-45
case b 8 9 9 12 15 16 22 20 40
case c 5 13.5 18 27 13 20 25 40 50 24 50
is not possible. We mention that there is a qualitative agreement of our results with those of Mac Leod and Radke7 obtained for Triton X-100 at the water-dodecane interface using a growing drop technique. Liggieri et al.28 studied the adsorption kinetics of Triton X-100 at the water-hexane interface and found that at short adsorption time the process is about eight times faster than expected from the classical diffusion model eq 9a. For a better agreement between theory and experiment, the authors chose a Freundlich instead of a Langmuir-type isotherm. Although this leads to an
Conclusions The adsorption kinetics of different Tritons at the water-nonane interface were studied experimentally in terms of the dynamic interfacial tensions using a special measuring cell for the maximum bubble pressure tensiometer MPT1. This version of a drop volume apparatus yields excellent interfacial tension data and allows us to analyze single drops with respect to drop formation instabilities as observed recently.9 These measurements enable us to discover the conditions for the fastest but most stable drop formation, which only allows an accurate interfacial tension calculation. The measured dynamic interfacial tensions for long adsorption times are fully in line with a diffusioncontrolled adsorption model, while the data at short adsorption times show adsorption rates faster than expected from diffusion. This effect may be caused by reorientation processes of the Triton molecules at low interfacial coverage or by a significant initial adsorption at the residual drop surface.30 The experiments were performed under different conditions: Triton dissolved in water (case a), in nonane (case b), or in both phases (case c). The distribution coefficient of the Tritons between water and oil can be estimated on the basis of the theory developed by Hansen.6 This coefficient K, defined as the equilibrium concentration ratio of the Tritons in water to that in oil, increases with a lower number of EO groups. Acknowledgment. The work was financially supported by a project of the European Community (INTAS 93-2463) and by the Fonds der Chemischen Industrie (Grant RM 400429). The authors also thank Prof. Paul Joos from the University of Antwerp for many helpful discussions. LA960068L (30) Zholob, S. A.; Fainerman, V. B.; Miller, R. J. Colloid Interface Sci. 1995, 185.
