Surface Dilatational Elasticity of Poly(oxy ethylene)-Based Surfactants

Nov 29, 2007 - Daniela Georgieva , Véronique Schmitt , Fernando Leal-Calderon and Dominique Langevin. Langmuir 2009 25 (10), 5565-5573. Abstract | Fu...
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Langmuir 2008, 24, 189-197

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Surface Dilatational Elasticity of Poly(oxy ethylene)-Based Surfactants by Oscillation and Relaxation Measurements of Sessile Bubbles Finn Knut Hansen* Department of Chemistry, UniVersity of Oslo, P. O. Box 1033 Blindern, 0315 Oslo, Norway ReceiVed August 9, 2007. In Final Form: October 2, 2007 Surface dilatational elasticities and viscosities have been measured by means of the axisymmetric bubble shape method. Two different techniques using sinusoidal oscillations and step relaxations have been used, and the results are treated by means of the bulk/surface diffusional exchange model. Three different nonionic surfactants based on poly(oxy ethylene) as the hydrophilic group have been used: one simple surfactant, Brij 35, and two block copolymers, Pluronic F68 and P9400. Step relaxation and oscillation give mostly the same limiting surface dilatational elasticities, but step relaxation is a more model-dependent method. In the cases where the bulk/surface diffusion model is correct, the two methods give the same results, but otherwise step relaxation gives average values of the limiting elasticity E0 and the typical relaxation frequency ω0. Limiting elasticities up to ca. 25 mN m-1 have been found for these substances. The surface/bulk diffusion model describes quite well the two relatively hydrophilic substances Brij 35 and F68, especially at low concentrations, but less so for the more hydrophobic P9400. The surface dilatational elasticity as a function of the surface pressure of the surface-active polymers goes through at least one maximum as a result of surface conformational changes.

Introduction In many practical applications, the effect of surface-active substances cannot be explained only by the static/equilibrium values of adsorption and surface tension. For instance, the stability of emulsions and foams is often more dependent on dynamic properties of the surface pressure than on the static value. When the surface is expanded or contracted, the resulting change in the interfacial tension will be important for the behavior of the surface film. This surface tension response is traditionally denoted by the surface elasticity, E0, as defined by Gibbs

E0 )

dγ d ln A

(1)

where γ is the surface tension and A the surface area. This simple term does not, however, include the possibility of time-dependent responses, in which case the so-called Marangoni elasticity has been named and is connected to the surface viscosity. The more rigorous mathematical description of these phenomena is now well-established,1-6 and in these treatments, the Gibbs surface elasticity has been replaced by the complex surface elasticity

E* ) E′ + iE′′

(2)

where E′ is the storage modulus and E′′ the loss modulus. The latter will then describe relaxation processes. Only in the cases where E′′) 0 will E′ be equal to E0. There are two main types of processes that are thought to be responsible for surface elasticity * Tel. (+47) 22855554. Fax: (+47) 22855542. E-mail: f.k.hansen@ kjemi.uio.no. (1) Lucassen-Reynders, E. H. In Surfactant Science Series; Marcel Dekker: New York, 1981; Vol. 11, pp 173-213. (2) Loglio, G.; Tesei, U.; Degli Innocenti, N.; Miller, R.; Cini, R. Colloids Surf. 1991, 57 (3-4), 335-42. (3) Loglio, G.; Tesei, U.; Miller, R.; Cini, R. Colloids Surf. 1991, 61, 219-26. (4) Edwards, D. A.; Brenner, H.; Wasan, D. T. Interfacial Transport Processes and Rheology; Butterworths-Heinemann Publishers: Boston, 1991. (5) Dhukin, S. S.; Kretzschmar, G.; Miller, R. Dynamics of Adsorption at Liquid Interfaces; Elsevier: Amsterdam, 1995. (6) Joos, P. Dynamic surface phenomena; VSP: Utrecht, 1999; p XIII, 360 s.

variations. One is in-surface processes such as longitudinal diffusion and molecular relaxation processes; the other is outof-surface processes, i.e., transport between the bulk and the surface. Which of these processes dominates will depend on the type of surfactant, surface pressure, bulk concentration, and frequency. A clean liquid surface has no measurable elasticity, and ordinary low molar surfactants (e.g., SDS) are usually quite soluble and have a high diffusion coefficient. Therefore, these surfactants do not give measurable relaxation effects, except when measured at very high frequencies or very low concentrations. In order to measure the bulk-surface transport rate, rather high rates of deformation are necessary. The measurement of dynamic properties is usually carried out by analyzing the surface tension response from either a single transient surface area change or by a periodic oscillatory deformation. The transient changes can also have different shapesslinear, square, trapezoidal, and so forth. Several instruments exist that can be used to measure such dynamic properties of surfaces, e.g., the Langmuir surface balance, the elastic ring, the dynamic drop or bubble methods, the oscillating jet, surface waves, and so forth. The different instruments often have different typical relaxation time constants, from slow to very fast, but also overlap to a certain degree. Among these, the Langmuir surface balance and the oscillating drop/bubble are the most common and are also readily commercially available. The oscillating bubble was originally developed by Lunkenheimer7 and is based on the measurement of the resulting bubble pressure in the gas phase when oscillation the gas volume. The method has been developed further by others, especially at the MPRI in Berlin/Potsdam and University of Florenze.8-13 The bubble size used in the original (7) Lunkenheimer, K.; Kretzschmar, G. Z. Phys. Chem. 1975, 256, 593. (8) Fruhner, H.; Wantke, K.-D. Colloids Surf. A 1996, 114, 53-59. (9) Kovalchuk, V. I.; Kragel, J.; Aksenenko, E. V.; Loglio, G.; Liggieri, L. Stud. Interface Sci. 2001, 11 (Novel Methods to Study Interfacial Layers), 485516. (10) Loglio, G.; Pandolfini, P.; Miller, R.; Makievski, A. V.; Ravera, F.; Ferrari, M.; Liggieri, L. Stud. Interface Sci. 2001, 11 (Novel Methods to Study Interfacial Layers), 439-483. (11) Kovalchuk, V. I.; Zholkovskij, E. K.; Kragel, J.; Miller, R.; Fainerman, V. B.; Wustneck, R.; Loglio, G.; Dukhin, S. S. J. Colloid Interface Sci. 2000, 224 (2), 245-254.

10.1021/la7024582 CCC: $40.75 © 2008 American Chemical Society Published on Web 11/29/2007

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instrument is smaller than the hemispherical shape (subhemisphere regime), which gives better stability at higher rates, but excludes quantitative visual shape analysis. It is especially useful for the measurement of fast dynamic processes such as surfactant adsorption, but also has its practical limitations such as reproducible initial conditions (filling time) and long-term stability. Few systems are also suitable for the measurement of liquid/liquid interfaces. By using a pendant or sessile drop or bubble larger than a hemisphere (super-hemisphere regime), axisymmetric drop shape analysis can be used to measure the interfacial tension, and simultaneously the drop volume and the surface area.14-17 By the advent of modern fast real-time computers, rapid surface tension analysis can be based on video image analysis. This method has recently become quite popular and has been developed by several groups. A comparison between sub- and super-hemisphere methods has been carried out by Kovalchuk et al.18 In this laboratory, we have developed measurement methods based on axisymmetric shape analysis of drops or bubbles,19,20 and by oscillatory drop deformations, the dynamic properties can also been measured.21,22 This method is ideally suited for long time measurements of dynamic surface tensions, but for oscillations, it has an upper frequency limit of a few hertz, above which super-harmonic shape effects come into play, especially with super-hemispherical shapes. The analysis of the dynamic surface tension responses have traditionally been carried out in a theoretical framework based on a surface equation of state and surface/bulk diffusion in the linear surface elastic regime.5,6,23-26 In this framework, the central parameters describing the relaxation process are the limiting elasticity, E0, and the so-called molecular exchange parameter, ω0, which is the inverse of the relaxation time constant, τ (or τD if it is a diffusional process). The theoretical expressions for the elastic modules according to this theory

1+ζ 1 + 2ζ + 2ζ2 ζ E′′ ) E0 1 + 2ζ + 2ζ2 E′ ) E0

(3)

(12) Ortegren, J.; Wantke, K. D.; Motschmann, H. ReV. Sci. Instrum. 2003, 74 (12), 5167-5172. (13) Andersen, A.; Oertegren, J.; Koelsch, P.; Wantke, D.; Motschmann, H. J. Phys. Chem. B 2006, 110 (37), 18466-18472. (14) Benjamins, J.; Cagna, A.; Lucassen-Reynders, E. H. Colloids Surf., A: Physicochem. Eng. Aspects 1996, 114 (Collection of Papers presented at the Workshop\“Bubble and Drop 95\”, 1995), 245-254. (15) Miller, R.; Policova, Z.; Sedev, R.; Neumann, A. W. Colloids Surf. 1993, 76, 179. (16) Loglio, G.; Tesei, U.; Pandolfini, P.; Cini, R. Colloids Surf., A: Physicochem. Eng. Aspects 1996, 114 (Collection of Papers presented at the Workshop\“Bubble and Drop 95\”, 1995), 23-30. (17) Myrvold, R.; Hansen, F. K. J. Colloid Interface Sci. 1998, 207, 97-105. (18) Kovalchuk, V. I.; Kraegel, J.; Makievski, A. V.; Loglio, G.; Ravera, F.; Liggieri, L.; Miller, R. J. Colloid Interface Sci. 2002, 252 (2), 433-442. (19) Hansen, F. K.; Rødsrud, G. J. Colloid Interface Sci. 1991, 141 (1), 1-9. (20) Hansen, F. K. J. Colloid Interface Sci. 1993, 160, 209-217. (21) Myrvold, R.; Hansen, F. K.; Lindman, B. In AssociatiVe Polymers in Aqueous Media, Glass, E., Ed.; American Chemical Society: Washington, DC, 2000; Vol. 765, pp 303-316. (22) Frømyr, T.; Hansen, F. K.; Kotzev, A.; Laschewsky, A. Langmuir 2001, 17, 5256-5264. (23) Loglio, G.; Tesei, U.; Cini, R. J. Colloid Interface Sci. 1979, 71 (2), 316-20. (24) Loglio, G.; Rillaerts, E.; Joos, P. Colloid Polym. Sci. 1981, 259 (12), 1221-7. (25) Loglio, G.; Miller, R.; Stortini, A.; Tesei, U.; Degli Innocenti, N.; Cini, R. Colloids Surf., A: Physicochem. Eng. Aspects 1994, 90 (2/3), 251-9. (26) Loglio, G.; Miller, R.; Stortini, A.; Tesei, U.; Degli Innocenti, N.; Cini, R. Colloids Surf., A: Physicochem. Eng. Aspects 1995, 95, (1) 63-8.

where the dimensionless parameter ζ is given by

ζ≡

x

ω0 2ω

(4)

and where ζ is connected to the oscillatory phase angle δ by

tan δ )

ζ 1+ζ

(5)

Such results from oscillating bubble measurements have been presented by several research groups.5,8,22,27-35 A common observation from many experiments is that, while pure diffusive description of the relaxation process is satisfactory for simple surfactants at lower deformation rates, and also often at lower surfactant concentrations, this model cannot explain all results for, e.g., surface-active polymers and for surfactants at higher deformation rates. Different nondiffusive models have been proposed, and research on more complicated systems is an ongoing effort. Results from transient surface area changes have also been presented, although not in a high number. The first transient surface tension relaxations were performed by Loglio et al.23 They also later gave experimental evidence for model (pure) surfactant2 and compared transient and harmonic perturbations.25,26 In these experiments, the instrument was the elastic ring in a Langmuir surface balance, and the model surfactant n-dodecyldimethylphosphine oxide was measured by a trapezoidal pulse perturbations and sinusoidal perturbations at different frequencies. Because of instrumental limitations, the frequencies used here were rather low (up to 0.1 Hz). They concluded that the different methods under these conditions give essentially the same results. Surfactants of the alkylpoly(oxy ethylene) type on the hexane/water interface have been studied by the dynamic drop volume method36 and serum albumin (HSA) at the air/ water interface by the drop shape method.15 In both these investigations, (almost) square-shaped surface area deformations were used. In the present paper, oscillation and square step surface tension relaxation are applied on the same sessile bubbles, and the results from these methods are compared. By such a comparison, one may evaluate to what degree these two methods can be used to extract the same information from polymeric surfactant systems and under which conditions the square step relaxation is a satisfactory method to characterize the system. This may also be an advantage in practical instrumentation, because a step function is usually easier to control and analyze than an oscillatory movement. Materials and Methods Materials. Three different types of poly(oxy ethylene)-based surfactants were investigated in this work, one simple nonionic surfactant and two surface-active block copolymers. The simple (27) Ravera, F.; Ferrari, M.; Miller, R.; Liggieri, L. J. Phys. Chem. B 2001, 105, 195-203. (28) Fruhner, H.; Wantke, K.-D.; Lunkenheimer, K. Colloids Surf., A: Physicochem. Eng. Aspects 1999, 162, 193-202. (29) Johnson, D.; Stebe, K. Colloids Surf., A: Physicochem. Eng. Aspects 1996, 114, 41-51. (30) Johnson, D.; Stebe, K. J. Colloid Interface Sci. 1996, 182 (2), 526-538. (31) Johnson, D. O.; Stebe, K. J. J. Colloid Interface Sci. 1994, 168 (1), 2131. (32) Loglio, G.; Pandolfini, P.; Tesei, U.; Noskov, B. Colloids Surf., A: Physicochem. Eng. Aspects 1998, 143 (2-3), 301-310. (33) Loglio, G.; Noskov, B.; Pandolfini, P.; Kragel, J.; Tesei, U. Colloids Surf., A: Physicochem. Eng. Aspects 1999, 156 (1-3), 449-453. (34) Loglio, G.; Pandolfini, P.; Miller, R.; Makievski, A.; Kraegel, J.; Ravera, F. Phys. Chem. Chem. Phys. 2004, 6 (7), 1375-1379. (35) Loglio, G.; Pandolfini, P.; Miller, R.; Makievski, A. V.; Kraegel, J.; Ravera, F.; Noskov, B. A. Colloids Surf., A: Physicochem. Eng. Aspects 2005, 261 (1-3), 57-63. (36) Van Hunsel, J.; Bleys, G.; Joos, P. J. Colloid Interface Sci. 1986, 114, 432.

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surfactant was Brij 35, dodecanol poly(oxy ethylene)23, (C12EO23). It has a molecular weight of 1214 and contains 86% (w/w) OE. This surfactant is well-soluble in water, relatively stable, and the adsorption kinetics is sufficiently slow to be easily investigated with the techniques used here. The surfactant has also been investigated previously by step relaxation methods.37 It has a critical micelle concentration (CMC) of 0.092 mM38 (0.11 g/L) and is a commonly used detergent in HPLC applications. It was delivered by Fluka. The block copolymer surfactants were of the Pluronic type, poly(oxy ethylene)-poly(oxy propylene)-poly(oxy ethylene) (PEO-PPOPEO) block copolymers. The two polymers were Pluroinc F68 and PE9400. F68 has an average molecular weight of 8400 and contains 80% (w/w) poly(oxy ethylene), i.e., its formula is in shorthand EO76PO30-EO76. The CMC is >10% at 25 °C,39 which means that this surfactant does not form micelles in dilute solutions at room temperature. It is made by BASF but was delivered by Fluka. PE9400 has an average molecule weight of 4600 and contains 40% (w/w) poly(oxy ethylene), i.e., it may be written EO21-PO50-EO21. The CMC is ca. 2.5 g/L at room temperature.39 It was delivered by BASF. All surfactants were used as received. Water was MilliQ grade II cleaned by reverse osmosis and ultrafiltration. Instrument. The instrument for drop shape analysis has been described elsewhere.20,22 The instrument is a modified Rame´-Hart goniometer (rame´-hart instruments co., Netcong, NJ) running the Dropimage AdVanced software. The video camera has a frame rate of 30 fps (non-interlaced). It is equipped with a computer-controlled dispenser (rame´-hart) and a custom-built oscillation unit consisting of a syringe with an excenter-mounted piston that is driven by a stepper motor. The oscillation unit and dispenser are mounted in series with water-filled tubes. The water is thus acting as a hydraulic liquid that conveys the volume changes of the dispenser and oscillator to the drop or bubble. The drops and bubbles are extended from the tip of a small PTFE tube into a cuvette inside a thermostated environmental chamber with glass windows. The PTFE tube contains an air pocket toward the water in the hydraulic tube. All glassware and PTFE was cleaned by chromic sulfuric acid and rinsed in purified water. The instrument computer controls the dispenser, oscillator, and image acquisition, makes the measurement sequences, and performs all calculations. Procedure. The method for calculating the surface tension from sessile drops and bubbles used in the program has been described earlier.19,20 In an adsorption (dynamic surface tension) experiment, a new bubble of 25 µL is created by the dispenser in ca. 0.5 s. The surface tension is then followed as a function of time, keeping the surface area constant by means of a feedback procedure controlling the dispenser. The surface tension may thus be measured over time intervals from a few seconds to several days. During this time, the constant area measurement may also be interrupted by oscillation and/or step relaxation experiments. The main experiment may then be continued by the original conditions. For an oscillating bubble experiment, the surface area, A, is varied by changing the bubble volume in a sinusoidal manner, resulting in a corresponding sinusoidal variation in the bubble surface area, provided that the volume change is small (typically 2-3 µL) A ) A0 + Aa exp(iωt)

(6)

where Aa is the surface area amplitude and ω the frequency (angular velocity). The resulting surface tension is γ ) γ0 - γa exp[i(ωt + δ)]

(7)

where γa is the surface tension amplitude and δ the phase angle. The measured surface area and surface tension data as a function of time are analyzed by fitting sine functions by means of a nonlinear curve fitting (Marquard) algorithm. Prior to fitting, the equilibrium values (37) Balbaert, I.; Joos, P. Colloids Surf., A: Physicochem. Eng. Aspects 1987, 23, 259. (38) Andres, J.; Maller, J. J. Biol. Chem. 1989, 264, 151-156. (39) Alexandridis, P.; Holzwarth, J. F.; Hatton, T. A. Macromolecules 1994, 27, 2414-2425.

(A0 and γ0) are first determined from the mean values, and the amplitudes (Aa and γa) from the standard deviations. These values are then used as starting values in the full nonlinear curve fit. In the fitting procedures, the surface area, which has the best accuracy, is fitted first, and the frequency, ω, is determined. Then, the surface tension curve is fitted, keeping the frequency the same. From the phase shift between the area and surface tension curves, the phase angle, δ, is determined. E′ and E′′ may then be calculated from E* ) E′ + iE′′ ) |E| cos δ + i|E| sin δ

(8)

where |E| )

γa Aa/A0

(9)

For the step perturbations, the dispenser is programmed to increase the bubble volume by typically 4 µL in ca. 0.1 s, followed by surface tension measurements in 0.2 s time intervals. A total of 10-40 s is measured, before a reverse step of equal volume is made. This sequence is then repeated, usually once, but several repetitions have also been used. The step relaxation data are analyzed by fitting the data to the so-called Sutherland equation40 ∆γ ) ∆γ0 exp(ω0t) erfc(xω0t)

(10)

where ∆γ0 is the initial surface tension change. E0 is then calculated from γ0 in a similar way as in eq 9, i.e., E0 ) ∆γ0A0/∆A, where ∆A is the surface area step. Equation 10 can be used for the first step, but if the surface area step is reversed after a given time t1, the surface tension relaxation will also change its sign, so that5 ∆γ2 ) ∆γ1(t) - ∆γ1(t - t1)

(11)

This procedure can again be applied on further step functions at times t2, t3, and so forth,41 depending on the step sequence number. The curve fit is again performed by a nonlinear (Marquard) algorithm, using start values taken from the relaxation curve and the measured surface area curve. It should be noted that the product exp(y) erfc( xy) goes to zero when y increases, even as exp(y) goes to infinite. At long times, numerical inaccuracies in the calculation of the error function can therefore play a role in this calculation, and to avoid this, the product can be approximated by exp(y) erfc(xy) ≈

1 1 12y xπy

(

)

(12)

at y > 2519 (note: this equation as given by Miller et al.41 contains a printing error). A special tool has been developed in the program to make this data analysis task. A possible correction for a slope in the surface tension data has been included in order to make it possible to do relaxation measurements during an adsorption experiment (changing γ0), although this may make some assumptions in the theoretical derivations somewhat invalid.

Results and Discussion The dynamic surface tension curves for the simple surfactant Brij 35 is shown in Figure 1. For the concentration 1000 ppm, which is far above the CMC (110 ppm), an equilibrium value is reached very rapidly and does not change with time. Below the CMC, however, the semilog plot shows that to obtain the true equilibrium value for these concentrations usually takes ca. 24 h. These rates are far too low if only diffusion through the aqueous phase is considered, so some additional surface mechanism must be involved. A popular explanation is that the diffusion to the so-called subsurface is fast, but there is additional steric hindrance (40) Sutherland, K. J. Phys. Chem. 1948, 52, 394. (41) Miller, R.; Loglio, G.; Tesei, U.; Schano, K.-H. AdV. Colloid Interface Sci. 1991, 37, 73.

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Figure 1. Dynamic surface tension curves for Brij 35 at four different concentrations.

Figure 3. Indicatrix from the oscillation measurements of Brij 35 surfactant. The lines are theoretical curves as described by eq 15 for different values of E0 indicated by the right-hand interception on the horizontal axis. Table 1. Step Relaxation Results for Brij 35 C (ppm)

γ0 (mN m-1)

E0 (mN m-1)

ω0 (s-1)

mean error

2 2 10 100

54.95 53.16 48.74 40.07

16.2 16.7 22.9 36.3

0.20 0.007 0.68 8.3

0.055 0.022 0.077 0.054

also a Cole-Cole plot in ordinary (3D) rheology and electromagnetic relaxation theory. It has the shape of a semicircle. In the surface relaxation context, E′ and E′′ are given by eq 3. If ζ is eliminated, we obtain the equation6

E′2 + E′′2 ) E0(E′ - E′′)

(14)

Figure 2. Results from oscillation measurements of the Brij 35 surfactant.

(E′ - E0/2)2 + (E′′ + E0/2)2 ) E02/2

(15)

upon adsorption. This can involve surface reorientations and possibly surface phase equilibria, and may again lead to nondiffusional relaxation effects. This will be further discussed below, but a detailed description of the adsorption mechanism as such is not the objective of this work. Some artifacts may also be due to competitive adsorption of contaminations and effects of molecular weight distributions, but this is not believed to be a major contribution here. The results from oscillation measurements for the Brij surfactant at the 2, 10, and 100 ppm concentrations are shown in Figure 2. For the 2 ppm concentration, one measurement taken during the adsorption process is also shown. At such low concentrations, this can be achieved because the adsorption process is very slow. The difference between the two curves shows that the elasticity is not only dependent on bulk concentration, but also on the surface pressure. To investigate whether the measured elasticity is due to pure diffusional exchange, we previously22 calculated the limiting elasticity, E0, from the theoretical equation

The equation describes a circle with origin (E0/2, -E0/2) and radius x2E0/2. By plotting E′′ as a function of E′, we can, by comparing with this circle, determine if the relaxation is only controlled by surface/bulk diffusion. This has been plotted in Figure 3 for the same results as in Figure 2. In the figure is also plotted E′′ against E′ as described by eq 15 for some values of E0 (the value of E0 can be seen in the right-hand value of E′ when E′′ ) 0). It is observed that the results do not fit exactly to the circular lines, even if many are quite close. The trend is also that the points are below the lines for lower values of E′, i.e., that E0 is increasing with frequency. This is the same trend that we observed earlier for other systems22 and means that relaxation in most systems is dominated by surface/ bulk diffusion, but there probably are additional relaxation mechanisms. It may also be that some of the assumptions in deriving the model equations are not correct, e.g., that the effective diffusion coefficient is not constant and/or that the linearization of the adsorption isotherm is invalid. In view of the slow adsorption rates at lower concentrations that indicate some surface reconfiguration process, this is not surprising. Step relaxation results for the three concentrations are shown in Table 1, and the relaxation curve for the concentration 100 ppm is shown in Figure 4. In addition is shown a curve fitted by eq 10, and the value for γ0 is indicated by the horizontal line. Apparently, the theory fits quite well to the experiments, and the value of E0 also is approximately equal to the value measured by oscillation, ca. 32 mN m-1 (Figure 3). The value of ω0 is

or

E0 ) |E| x1 + 2ζ + 2ζ2

(13)

by using the measured value of |E| and ζ given by eq 5. By plotting E0 as a function of ω, we may observe whether it is constant or varies with ω, with the latter as an indication of nondiffusional mechanism(s). Another method of analyzing this is to plot E′′ as a function of E′, which is called an indicatrix and

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τD )

1 1 dΓ 2 ) ω0 D dC

( )

(16)

where D is the diffusion coefficient of surfactant between the surface and the bulk, and dΓ/dC is the slope of the adsorption isotherm. It may be rewritten as

ω0 ) D/

dΓ (dC )

2

(17)

Often, a Langmuir model is used for the adsorption isotherm, and we may write

dΓ C d Γ ) ) dC dC m C + a

(

Figure 4. Step relaxation results for 100 ppm concentration of Brij 35. The curves are calculated from eq 10 with E0 ) 36.3 mN m-1 and ω0 ) 8.3 s-1.

Figure 5. Step relaxation results for 2 ppm concentration Brij 35. The curves are calculated from eq 10 with E0 ) 16.7 mN m-1 and ω0 ) 0.007 s-1.

equal to ω in oscillation measurements at the maximum in E′′, and this also fits quite well, as seen in Figure 2, although the curve for E′′ is rather flat, so the maximum is not easily observed. By scrutinizing Figure 4, we see there is a tendency of the theoretical curves to be somewhat less steep at short times, then crossing the experimental curves and going faster against equilibrium. This means that E0 should be higher at short times and lower at long times, which also agrees well with oscillation results, where short times correspond to high frequencies and vice versa. In other words, E0 increases with increasing frequency. At lower surfactant concentrations, the elasticity is lower because of lower surface pressure, but the relaxation time is longer (ω0 lower), because of lower surfactant bulk concentration. Figure 5 shows the relaxation curve for a 2 ppm concentration of Brij 35. At this concentration, the surface pressure is lower and therefore also the elasticity. The value of E0 from step relaxation is close to the average of E′ from oscillation (Figures 2 and 4) at this concentration, which is also as expected. The relaxation mechanism here seems to be well-described by the diffusional exchange model, although the diffusional rate here is quite low due to the low concentration. It is interesting to note that the value of ω0 varies strongly with concentration, and much more than E0. It is convenient to discuss this by considering the theoretical expression for ω0. In the diffusion theory, the relaxation time is given by

)

Γm

a a ≈ Γm 2 (C , a) (18) (C + a)2 C

where a is the Langmuir adsorption constant that is dependent on the free energy of adsorption, and so forth. This means that ω0 should be proportional to C4 at a given surface pressure (constant a) at high surface coverage, but more and more independent of C at low coverage (C < a). At very low bulk concentrations, ω0 is therefore always low, but at higher concentrations, the isotherm is close to saturation, dΓ/dC is much lower, and ω0 thus higher. From the results in Table 1, it is seen that even if ω0 increases strongly with increasing C, it is not proportional to C4 at low concentrations and the dependence of C further decreases with increasing C. The change in ω0 from C ) 2 to 10 ppm can be fitted to the Langmuir model with a ≈ 1.7 ppm, but the decreasing dependency at C ) 100 does not fit. The accuracy of the results are not sufficiently good to justify a more rigorous model discussion, but it may not be very surprising that adsorption of the Brij 35 surfactant does not fit a Langmuir model, as this may also be deduced from the surface tension data. If step relaxation is to be used to measure surface dilatational elasticity parameters, it is important that the model can be fitted fairly well to the experimental data. As shown above, this is possible with the diffusion model, but it is not necessarily a straightforward task. In the fitting process, it was discovered that some data sets do not necessarily give a unique combination of theoretical parameters, and especially that E0 and ω0 sometimes are cross-correlated, meaning that many combinations of these two parameters can give (almost) the same fitted curves. To illustrate this correlation, a contour plot of the mean fitting error (root-mean-square) as a function of ω0 and E0 or the 100 ppm concentration of Brij 35, as in Figure 4, is shown in Figure 6. From the figure, it is observed that the minimum error is obtained rather in a “valley” of ω0 and E0 combinations rather than at one unique combination. Even here it is, however, possible to find a unique optimum combination, as further illustrated in Figure 7. Here, the mean fitting error is plotted as a function of E0 at many values of ω0 (the step in ω0 is 0.5 s-1). It is seen that all error minima are below 0.06 (the lowest contour in Figure 6), but a unique solution is found at the combination E0 ) 36 mN m-1, ω0 ) 8 s-1. However, many other combinations give almost the same result, and just by visually comparing the experimental and theoretical curves, large errors in parameter estimation can be made. In other words, the standard errors in the estimated values are quite high (not computed here). The reason for this problem is of course that the model does not describe the physical process sufficiently well. If the description is better, a more unique fit to the model is also expected. This is the case for the Brij 35 concentration 2 ppm

194 Langmuir, Vol. 24, No. 1, 2008

Figure 6. Error contour plot of the curve fit to step relaxation of 100 ppm concentration of Brij 35. The error is the mean deviation (root-mean-square) between model and experimental surface tensions.

Figure 7. Error plots of the mean deviation (root-mean-square) against the limiting elasticity, E0, for different values of ω0. Results for 100 ppm concentration of Brij 35.

Figure 8. Error contour plot of the curve fit to step relaxation of 2 ppm concentration of Brij 35. The error is the mean deviation (root-mean-square) between model and experimental surface tensions.

which is shown in Figure 5. A corresponding contour map for this concentration is shown in Figure 8, and in this case, the error is smaller and ω0 is also small. It is an almost entirely elastic surface, and the diffusion model fits very well. The reason for this is probably that the surface layer at this lower surface pressure is less compact and thus the assumptions in the model derivation

Hansen

Figure 9. Dynamic surface tension curves for Pluronic F68 at five different concentrations.

hold quite well. Because the elasticity is lower and surface tension changes less with time during the relaxation, it is more probable that E0 is the same during the whole step relaxation process. It may thus also be expected that, if the relaxation measurements at higher concentrations (e.g., 100 ppm) are carried out over shorter time intervals, better curve fits will be obtained, as the assumption of a constant value of E0 and ω0 will be more correct. The main problem at high concentrations is that γ decreases very rapidly during the first few seconds and the bubble does not become sufficiently stable until ca. 0.2 s. The fast initial decrease is therefore difficult to observe with this methodology. It can be concluded that the surfactant Brij 35 behaves ideally according to the diffusional exchange theory at low concentrations, but less so at higher concentrations and especially during longer step time intervals. Dynamic surface tension curves for the polymer Pluronic F68 are shown in Figure 9. In comparison to the simple surfactant Brij 35 (Figure 1), this polymer adsorbs more slowly, which may be expected because of its higher molecular weight. The maximum surface pressure at high concentrations is, however, more or less the same, ca. 30 mN m-1. This may be a consequence of its content of poly(oxy ethylene), which is almost as high as that of Brij 35. The “bump” in the adsorption curves that is most clearly seen in the 10 ppm sample may be a result of trace impurities, but may also have other more interesting conformational reasons, as will be further discussed below. The adsorption curves hardly seem to reach an equilibrium value during the observed time period; to reach a final value would seem to take a very long time indeed. This feature is quite common for the adsorption of polymers and is probably due to very slow conformational changes that take place in a densely packed surface layer. However, at high concentrations (1%), a minimum value is still reached. It should be noted that this polymer does not form micelles at room temperature so that a minimum surface tension at the CMC is not expected. The results from oscillation measurements for Pluronic F68 are shown in Figure 10. In comparison to the Brij 35 surfactant, the elasticities are on the same order of magnitude, but generally slightly lower. For this polymer, the limiting elasticity, E0, is not very dependent on the bulk concentration. This is most easily seen from the indicatrix, the plot of E′ vs E′′ in Figure 11. This is so because E0 is mainly dependent on the surface pressure and not directly on the bulk concentration. However, the concentration affects more directly the frequency dependence of E′ and E′′ through the influence on ω0, and the points are shifted to higher

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Langmuir, Vol. 24, No. 1, 2008 195

Figure 10. Results from oscillation measurements of the Pluronic F68.

Figure 12. Error contour plot of the curve fit to step relaxation of 100 ppm concentration Pluronic F68. The error is the mean deviation (root-mean-square) between model and experimental surface tensions.

Figure 11. Indicatrix from the oscillation measurements Pluronic F68. The lines are theoretical curves as described by eq 15 for different values of E0 indicated by the right-hand interception on the horizontal axis.

Figure 13. Dynamic surface tension curves for Pluronic P9400.

Table 2. Step Relaxation Results for Pluronic F68 C (ppm)

γ0 (mN m-1)

E0 (mN m-1)

ω0 (s-1)

mean error

2 10 100 1000 10000

57.73 50.44 47.26 40.60 39.47

22.1 25.6 22.0 19.6 18.2

0.0015 0.055 0.401 0.883 1.83

0.067 0.057 0.044 0.061 0.023

E′ with increasing C. At higher surface pressures, E0 is lower; in addition to that, E′ at a given frequency is even lower because of the higher bulk concentration. This effect will be discussed further below. The step relaxation measurements of F68 give mostly similar results to the oscillation measurements, as shown in Table 2. For this polymer at most concentrations, E0 is not very frequency dependent, as is evident from Figure 11, meaning that the surface relaxation process is fairly well described by the bulk/surface diffusion model. This would also be expected to give more unique curve fits to eq 10 which is also the case. From the error contour plot for the 100 ppm F68 concentration in Figure 12, it is seen that a unique solution is found, even if there is also some degree of cross-correlation between E0 and ω0. At the 1000 ppm concentration, E0 is more dependent on frequency, as indicated by the higher value of the mean error (Table 2). But also here, step relaxation gives satisfactory results. The best fit to the model

is obtained for the 10000 ppm concentration, which is also logical because surface/bulk diffusion is expected to play a greater role at high concentrations as long as the surface pressure is not affected. The fact that E0 decreases with increasing surface pressure, rather than increases as is the case with Brij 35, may seem strange at first glance. This means that there must be a maximum in surface dilatational elasticity at some surface pressure below ca. 20 mN m-1. However, this observation is not unique; such maxima have also been observed previously for other surface-active polymers, e.g., proteins (unpublished results), and are most probably a surface configuration effect. This effect will be discussed further below. The value of the surface tension does not vary very much, considering the 3 orders of magnitude change in C. It is of course a consequence of the strong adsorption of the polymer at equilibrium and shows that the isotherm cannot be of the Langmuir type. The variation in ω0 with concentration is stronger, approximately a factor of 2-4 per order of magnitude, but this variation is even lower than for the Brij 35 surfactant and also refutes the Langmuir model. The polymer Pluronic P9400 is considerably more hydrophobic than F68 and would be expected to adsorb more strongly. This is shown in Figure 13, and even at the 2 ppm concentration, the adsorption reaches an equilibrium value at ca. 10 h adsorption. This polymer also forms micelles with the CMC at ca. 2500 ppm. The highest concentration in Figure 13, 10 000 ppm, is therefore above the CMC, while the others are below. This

196 Langmuir, Vol. 24, No. 1, 2008

Hansen Table 3. Step Relaxation Results for Pluronic P9400

Figure 14. Results from oscillation measurements of Pluronic P9400.

Figure 15. Indicatrix from the oscillation measurements of Pluronic P9400. The lines are theoretical curves as described by eq 15 for different values of E0 indicated by the right-hand interception on the horizontal axis.

stronger adsorption might also be expected to give differences in the elasticity, or in the elasticity-time relationship as shown in Figures 14 and 15. The values of the surface dilatational elasticities of P9400 are not very different from those of the other polymer, however, but E′ and E′′ generally vary less with frequency. The indicatrix in Figure 15 also shows that this polymer does not fit very well to the bulk/surface diffusion model, except at low concentrations (2 and 10 ppm) and higher frequencies (>ca 3 s-1). At lower frequencies, the loss modulus, E′′, and thus the surface dilatational viscosity, η ) E′′/ω, is less than should be expected from diffusional exchange. This result is also present for Brij 35 at the 2 ppm concentration (Figure 3) but is not seen for F68. It seems strange that diffusion should play a greater role at high than at low frequencies, as the opposite is expected, and some sort of artifact may be suspected. However, another way of looking at this is that E0 increases with frequency. For all the higher concentrations, this seems to be the case, while for the 2 and 10 ppm concentration, E0 increases up to ca. 3 s-1 (0.5 Hz) and then becomes essentially constant. Such an increase has been observed earlier22 and may be a result of in-surface relaxations, i.e., configurational changes. These changes may be sufficiently slow to be observable only at lower rates when the surface pressure is lower, while at higher pressures, they are detectable also at higher rates. The results from the step relaxation measurements of this polymer are given in Table 3. It is quite clear from the table that E0 here decreases strongly with increasing concentration and

C (ppm)

γ0 (mN m-1)

E0 (mN m-1)

ω0 (s-1)

mean error

2 2 10 100 1000 10000

45.58 43.05 42.15 37.89 34.88 32.69

18.4 16.1 12.9 7.26 7.25 4.4

0.011 0.027 0.108 0.023 0.010 0.017

0.049 0.089 0.071 0.035 0.032 0.0173

thus also with increasing surface pressure. The values of ω0 are always low, as is also apparent from the shape of the curves in Figure 14, but the accuracy in ω0 is not very good at such low values anyway, so we cannot read much more out of these. By comparing the values of E0 in Table 3 with the curves in Figure 14, we se that, even so, the step relaxation method results in more or less the same elasticities as oscillations. The diffusion model does not, however, fit too well for this polymer, even if some error values in Table 3 are quite low. They must in this connection be compared to the surface tension step response, which is quite low at high surface pressures of this polymer (∆γ ) 0.5 mN m-1) for 10 000 ppm), so the relative error is higher. The fact that the limiting elasticity E0 decreases with increasing surface pressure for both the Pluronic polymers seems to be quite common for these substances. This is opposite of what is observed for most ordinary surfactants, here represented by Brij 35, and also opposite that of some high polymers. However, Jiang and Chiew42 have shown that both the Pluronics F68 and F127 and also POE (Mw ) 31 000) have a maximum E0 of 11.5 mN m-1 at a surface pressure of ca. 6 mN m-1. The Pluronics also have a minimum of 5 mN m-1 at Π ) ca. 11 mN, whereafter the elasticity again increases, leading to a(n) (inverse) S-shaped curve. They did not investigate pressures above 15 mN m-1, and our results indicate that there is another maximum at higher pressures, around 20 mN m-1. We have previously43 shown that for some block copolymers the elasticity is proportional to the surface pressure up to ca. 20 mN m-1, whereafter it may increase more slowly. The proportionality constant may be calculated by scaling theory44 to be between 2.58 up to quite high values (ca. 50) depending on the solubility of the surfactant in the surface layer (the highest value at θ conditions). It is also well-know that the Π-A isotherm for insoluble monomolecular films may have several transition regions where the isotherm is more or less flat. This happens when different surface configurations are in equilibrium (e.g., gaseous and liquid monolayers), and the surface dilatational elasticity in such a region will be zero (if the time of observation is long enough for the equilibrium to be retained). The limiting elasticity below such a region must therefore go through a maximum, and the elasticity vs pressure curve may have several such maxima and minima if the surface configuration changes further. This is therefore probably the explanation for the elasticity variations of the Pluronics polymers and also for other surface-active polymers. A more detailed investigation of this phenomenon has previously been carried out for the protein HSA, and further work on the present polymers is also being done. One result from these measurements is that the second maximum in E0 for F68 is ca. 26 mN m-1 at a surface pressure of 20 mN m-1 and that E0 then decreases again.

Summary Step relaxation and oscillation give mostly the same limiting surface dilatational elasticities, but step relaxation is a more model(42) Jiang, Q.; Chiew, Y. C. Macromolecules 1994, 27, 32-34. (43) Kopperud, H. B. M.; Hansen, F. K. Macromolecules 2001, 34, 56355643. (44) Vilanove, R.; Poupinet, D.; Rondelez, F. Macromolecules 1988, 21, 28802887.

Surface Elasticity of PEO-Based Surfactants

dependent method. In the cases where the surface/bulk diffusion model is correct, the two methods give the same results, but otherwise step relaxation gives average values of the limiting elasticity E0 and the typical relaxation frequency ω0. In the case where E0 changes a lot with time, i.e., the model is not sufficiently accurate, it is often difficult to obtain unique curve fits to the model, and E0 and ω0 are intercorrelated. In these cases, better solutions may be obtained by measuring over shorter time intervals and/or using more complicated models. For the Brij 35 surfactant and the Pluronics polymers, that all are nonionic surfactants based on poly(oxy ethylene) as the hydrophilic group, limiting

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elasticities up to 20-25 mN m-1 have been found. The surface/ bulk diffusion model describes quite well the two more hydrophilic substances Brij 35 and F68, but less so the more hydrophobic P9400. It is also probable that the diffusing species here are not the whole molecule, but only (parts of) the hydrophilic chain. The surface dilatational elasticity as a function of the surface pressure of the surface-active polymers is a complex function of the surface pressure and goes through at least one maximum as a result of surface conformational changes. LA7024582