Dilational Viscoelasticity of Adsorption Layers Measured by Drop and

Article pubs.acs.org/Langmuir

Dilational Viscoelasticity of Adsorption Layers Measured by Drop and Bubble Profile Analysis: Reason for Different Results V. B. Fainerman,† V. I. Kovalchuk,‡ E. V. Aksenenko,§ and R. Miller*,∥ †

Medical University Donetsk, Donetsk, Ukraine Institute of Biocolloid Chemistry, Kyiv (Kiev), Ukraine § Institute of Colloid Chemistry and Chemistry of Water, Kyiv (Kiev), Ukraine ∥ MPI Colloids and Interfaces, Potsdam, Germany ‡

ABSTRACT: The dilational viscoelasticity of adsorption layer was measured at different frequencies of drop and bubble surface area oscillations for aqueous C12EO5 solutions. The modulus values obtained by the two experimental protocols are the same for Π < 15 mN/m, while for higher surface pressures the values from drop experiments exceed those from bubble profile analysis. The nature of this phenomenon was studied using stress deformation experiments. At high surfactant concentrations the magnitude of surface tension variations is essentially higher for drops as compared with bubbles, leading to an increased viscoelasticity modulus for oscillating drops. The observed effects are analyzed quantitatively using a diffusion controlled exchange of matter model. The viscoelasticity moduli for a number of surfactants (different CnEOm and Tritons, C13DMPO, and SDS) are reported, and it is shown that the discrepancies between the data obtained by the two methods for many surfactants agree well with the predictions made here.

1. INTRODUCTION Many technical and natural systems with fluid interfaces are to a great extent determined by surface rheological properties. Examples of such systems are foams and emulsions, food colloids, coating flows, biological liquids, and others. Studies of the dilational viscoelastic behavior of liquid/fluid interfaces are also of fundamental importance for understanding adsorption and relaxation mechanisms. The history and the actual state of the art of dilational and shear rheology has been published in many papers, chapters, and books.1−22 The dilational viscoelasticity modulus can be obtained from measurements of the interfacial tension response to respective area perturbations. There is a variety of methods suitable for these studies, such as the oscillating barrier, drop and bubble profile, capillary waves, and oscillating bubble methods. The surface dilational modulus can be presented as a complex quantity where the real and imaginary parts reflect the elastic and viscous response of the interfacial layer. The surface dilational modulus appears to be the most suitable characteristic parameter of an interfacial layer to compare results from different dynamic experiments. The experimental techniques based on drop or bubble profile analysis tensiometry can be employed to obtain information about the adsorption behavior of surfactants. From comparison of surface tension isotherms and employing the surfactant mass balance in the drop and around the bubble, the surfactant’s adsorbed amount can be calculated.23,24 The theoretical problems regarding the harmonic oscillations used in the drop and bubble profile methods were analyzed in refs 11, 12, © XXXX American Chemical Society

25, and 26. The pendant drop and bubble methods were employed in ref 9 to study the surface tension, the amount of C14EO8 adsorbed from the solution at the interface and the rheological characteristics of the adsorbed layers subjected to harmonic oscillations of the surface area. In27,28 the differences between the rheological data obtained using the drop and bubble profile methods were reported; however, the effect of solution concentration on these differences was not systematically analyzed. In the present study, the pendant drop and buoyant bubble methods were used; the surface rheological characteristics were measured at various frequencies of drop and bubble surface area oscillations for C12EO5 solutions of several concentrations, and the viscoelasticity modules for C12EO5 adsorption layers were calculated. The module values obtained by the two protocols were found to be virtually the same at surface pressure values below 15 mN/m; however, at higher surface pressure values the drop profile method yields higher modulus values as compared with those obtained by the bubble profile method. Therefore, also surface stress deformation experiments were performed with drops and bubbles and the factors leading to differences between the results obtained by the two methods were analyzed. The values of the viscoelasticity modulus for a number of surfactants mainly from the literature (C10EO5, C10EO8, C14EO8, C13DMPO, Triton X-45, Triton X-100, Received: March 23, 2016 Revised: May 8, 2016

A

DOI: 10.1021/acs.langmuir.6b01134 Langmuir XXXX, XXX, XXX−XXX

Article

Langmuir

dimensional compressibility coefficient, and a is the intermolecular interaction constant. The adsorption isotherms for states 1 and 2 are, respectively,

Triton X-165, Triton X-405 and SDS) are summarized, and it is shown that the discrepancy between the data obtained by the two methods agree well with the predictions made based on the theory developed here.

2. EXPERIMENTAL SECTION The experimental procedures using bubble/drop profile analysis tensiometers (PAT-1, SINTERFACE Technologies, Germany) were described previously.23,29 The buoyant bubble and pendant drop techniques are suitable to study the dynamic surface tension of surfactant solutions until complete equilibration, even when it takes many hours depending on the surfactant bulk concentration. In the present studies, the temperature of the measuring cell with a volume of V = 20 mL was kept constant at 25 °C. For the buoyant (oblate) bubble configuration, the bubble was formed at the bottom tip of a vertical Teflon capillary with an external diameter of 3 mm (the bubble was spread over the entire section of the capillary). In the pendant drop method, a steel capillary, thin-walled in its bottom part, with a conical internal profile and an internal diameter at its end of 2.8 mm was used. The surface areas of the drop or bubble, respectively, were chosen to be 38 mm2, and were automatically kept equal to this value during the experiments. The volumes of drop and bubble were slightly different: the initial volume of the bubble (at surface tension about 70 mN/m) was ca. 29 mm3, and decreased to 28 mm3 at a surface tension value of about 40 mN/m, while the drop volume was 27 and 25 mm3, respectively, at these surface tension values. The profiles of the drop or bubble were nonspherical: the bubble was slightly oblate, while the drop was prolate. To study the dilational elasticity, after having reached the adsorption equilibrium the bubble or drop was subjected to harmonic oscillations with frequencies f between 0.005 and 0.2 Hz and surface area oscillation amplitudes of 5−6%. The results of oscillation experiments were analyzed using the Fourier transformation. The oxyethylated alcohol C12EO5 was purchased from Sigma Chemical and used without further purification. The solutions were prepared in Milli-Q water with a surface tension of 72.0 ± 0.2 mN/m at 25 °C, which was constant over a time interval of up to 105 s (about 28 h).

Πω0 = ln(1 − Γω) + Γ(ω − ω0) + a(Γω)2 RT

ω1/ ω0

(1 − Γω)

b2 c =

⎛ ω ⎞ exp⎜ −2aΓω 1 ⎟ ω0 ⎠ ⎝

Γ2ω0 α

ω2 / ω0

(ω2 /ω1) (1 − Γω)

(2)

⎛ ω ⎞ exp⎜ −2aΓω 2 ⎟ ω0 ⎠ ⎝

(3)

where c is the concentration of the surfactant, α is the power law exponent, and bi are the adsorption equilibrium constants. The ratio of adsorptions in the two possible states of the adsorbed molecules is expressed by a relationship which follows from eqs 2 and (3) when assuming b1 = b2: ⎛ Γ1 (ω1/ω2)α (ω − ω1) ⎞ = exp⎜ −2aΓω 2 ⎟ ( ω − ω )/ ω Γ2 ω0 ⎝ ⎠ (1 − Γω) 2 1 0

(4)

Note, the assumed two orientation states refer only to the most probable states and a whole spectrum of orientations around these states exists. A consideration of more than two states for surfactants of the type CnEOm would not lead to a better description of the adsorption properties, as it was shown in ref 31. In compression/expansion perturbations, the surface dilational modulus E is defined as the increase in surface tension γ for a small relative increase of the surface area A at constant temperature:

E=

dγ d ln A

(5)

T

The viscoelastic modulus can be presented as a complex number E = Er + iEi, where the real part is called the storage modulus equal to the dilational elasticity, and the imaginary part is called the loss modulus, representing the dilational viscosity contribution. In the case of a surfactant adsorption layer with a pure diffusional relaxation mechanism the real and imaginary parts of the surface dilational modulus are described by the equations1,2

3. THEORY Surfactant adsorption layers can be described by different adsorption models. The most simple ones are those of Langmuir and Frumkin. It was shown, however, that many oxyethylated surfactants, such as the C10EO5 studied here, are best described by the so-called reorientation model.23 The derivations of this model used to describe the equilibrium surface tension and adsorption of surfactants including a possible internal compressibility of the adsorption layer were given in detail in refs 5, 23, and 30. This theory assumes that the adsorbed molecule can exist in one of two possible states, referred to below by the subscripts 1 and 2. The main equations for the reorientation model applied in this study are given here. The surface layer equation of state reads −

Γ1ω0

b1c =

E r (ν ) = E 0

1+ζ , 1 + 2ζ + 2ζ 2

Ei(ν) = E0

ζ 1 + 2ζ + 2ζ 2 (6)

where ζ = νD/2ν , E0(c) = −dγ/d ln Γ is the limiting elasticity, and νD(c) = D·(dΓ/dc)−2 is the characteristic frequency of diffusion relaxation; Γ, c, D, and ν are the surfactant adsorption, bulk concentration, bulk diffusion coefficient, and the angular frequency of the generated oscillations, respectively. Suitable expressions for the viscoelasticity modulus |E| and phase angle ϕ between stress (dγ) and strain (dA) follow from eqs 6:

(1)

where Π = γ0 − γ is the surface pressure, γ and γ0 are the surface tension of the solution and pure solvent (water), respectively, R is the gas law constant, T is the temperature, ω = (ω1Γ1 + ω2Γ2)/Γ is the average molar area, θ = ωΓ= ω1Γ1 + ω2Γ2 is the surface coverage, Γ = Γ1 + Γ2 is the total adsorption, ω1 and ω2 are the molar areas of the two orientations of the molecules adsorbed at the interface, ω2 > ω1 and ω1 = ω0(1 − εΠθ), ω0 is the molar area of the surfactant at Π = 0 or the molar area of the solvent, the factor ε is the relative two-

|E| = E0(1 + 2ζ + 2ζ 2)−1/2 ,

ϕ = arctg[ζ /(1 + ζ )] (7)

Equations 6 and 7 describe the dilation viscoelasticity of the surfactant layer adsorbed from solutions onto a flat surface. However, for low oscillation frequencies and small oscillating bubbles or drops (in our study, the radii were 1.5−2 mm) the actual geometry of the oscillating object can play a significant role. The respective equations to be used instead of eqs 6 and 7 B


Article

Langmuir were derived by Joos,26 and for the complex elasticity of spherical objects the following are obtained: For the adsorption from the solution bulk onto a bubble surface: ⎧ ⎫−1 D dc E(ν) = E0⎨1 − i (1 + nr0)⎬ νr0 dΓ ⎩ ⎭

(8)

For the adsorption from the drop bulk onto its surface: ⎧ ⎫−1 D dc E(ν) = E0⎨1 − i [nR coth(nr0) − 1]⎬ νr0 dΓ ⎩ ⎭

(9)

where n = iν/D, ν = 2πf, r0 is the drop/bubble radius, and f is the oscillation frequency. For high values of r0 and ν, eqs 8 and 9 become equal to eqs 6. However, for low oscillation frequencies (below 0.1 Hz) and values for r0 as those used in this study, eq 8 yields a decrease of the modulus and increase of the phase angle for bubbles (as compared with a flat surface), while, for drops, eq 9, on the contrary, predicts an increase in the modulus and decrease in the phase angle. If the adsorption mechanism of surfactant is based on the diffusional transport of molecules to and from the interface, then diffusion is governed by Fick’s law, which in spherical coordinates reads: 2

⎛ ∂ 2c ∂c 2 ∂c ⎞ = D⎜ 2 + ⎟ ∂t r ∂r ⎠ ⎝ ∂r

Figure 1. Dynamic surface tension of aqueous C12EO5 solutions (black curves) at various concentrations as measured by the drop (D) and bubble (B) profile analysis method. Shown are the pairs of curves for which the equilibrium surface tension is almost the same: 1, 0.05 (B) and 1.0 (D) μmol/L; 2, 0.5 (B) and 1.7 (D) μmol/L; 3, 1.0 (B) and 3.0 (D) μmol/L; 4, 3.5 (B) and 5.0 (D) μmol/L; 5, 15.0 (B) and 15.0 (D) μmol/L. Red dashed curves, theoretical values calculated using the diffusion governed adsorption kinetics model.

depletion of the solution within the drop bulk due to adsorption.23,24 For pairs 3 and 5, the shown theoretical curves are obtained assuming a diffusion-governed adsorption mechanism, eqs 10 and 11 for the adsorption model described by eqs 1−4 with the parameters listed in Table 1. In the

(10)

where c = c(r,t) is the surfactant concentration at time t and distance r from the center of the drop, and D is the diffusion coefficient. The main parameters of this adsorption model are the diffusion coefficient D and the initial surfactant bulk concentration c0. To derive the mathematical description of the adsorption, Fick’s law (eq 10) should be complemented by the boundary condition at the interface: dΓ(t ) ∂c(r , t ) = −D dt ∂r

r = r0

Table 1. Values for the Model Parameters for C12EO5 Solutions Used in the Model Calculations

(11)

which reflects the balance between the surfactant flux to or from the interface located at r = r0, expressed by the right-hand side of eq 11, and the temporal change of the adsorbed amount Γ(t). It is necessary to use the model of the adsorption layer, which relates the subsurface concentration cS(t) to the adsorption Γ(t). Let us assume that the model equations 1−4 are also applicable under dynamic conditions. For small-amplitude surface area variations, the solution of eqs 10 and 11 gives the expressions for the complex elasticity via eq 8 or 9 for the case of a bubble or drop geometry, respectively. In this case, the complex elasticity is expressed through E0(c) and dΓ/dc in a general form, which makes eqs 8 and 9 valid for any model of adsorption layer.

model parameter

interface W/A

dimension

a α ω10 ω2 ε b1

0.5 1.5 4.6 1.4 7.0 4.58

105 m2/mol 106 m2/mol 10−3 m/mN 103 m3/mol

calculations, the diffusion coefficients for the pair 3 were taken to be 4.0 × 10−10 and 2.5 × 10−10 m2/s, respectively, for the bubble and drop configuration. This difference is presumably caused by the geometrical difference between the objects (for the drop, the area-to-volume ratio is higher than that for the bubble), and by possible contributions of a convective diffusion for the bubble in a large volume of solution. However, for pair 5 with larger concentrations, the obtained diffusion coefficients are closer to each other: (3.0 ± 0.2) × 10−10 m2/s. Note that in the drop method, significant initial adsorption exists caused by the previous residual drop, while in the buoyant bubble method the surface renovation is much more significant due to the quick expansion of the bubble during its departure from the capillary. Therefore, the initial adsorption is lower at the bubble surface. Figure 2 illustrates the equilibrium surface tension of C12EO5 solutions measured by the drop and bubble methods; the experimental data obtained in ref 23 with the bubble method are also shown. The theoretical curve in Figure 2 for the bubble method was calculated using eqs 1−4 for the reorientation model with the parameters listed in Table 1; this model and

4. RESULTS AND DISCUSSION The dynamic surface pressure of C12EO5 solutions with various concentrations, measured using the drop (D) and bubble (B) protocols, respectively, is shown in Figure 1 for pairs of solutions. The equilibrium surface tensions measured by the two methods are approximately the same; however, to obtain these surface tensions, the concentrations for the drop experiments should be higher than those used in the bubble experiments. This phenomenon is obviously caused by the C


Article

Langmuir

pressure for the data pairs presented in Figures 3 and 4: data pair 1−4.5 mN/m, data pair 2−8.0 mN/m, data pair 3−13.2

Figure 2. Equilibrium surface tension of C12EO5 solutions as measured using the drop (D, ▲) and bubble (B, ◇) method; +, data measured in ref 23 using the bubble profile method. Solid curve B is calculated using eqs 1−4 with the parameters listed in Table 1; dashed curve D is calculated using Frumkin’s model.

Figure 4. Dependence of viscoelasticity modulus measured by the drop (◆,▲) and bubble (◇,Δ) methods for C12EO5 solutions with concentrations corresponding to pairs 4 (◆,◇) and 5 (▲,Δ) shown in Figure 1. Lines are only guides for eyes.

parameters were also used to calculate the values of dynamic surface tension shown in Figure 1 (theoretical curves for pairs 3 and 5). It is seen that the calculated surface tension isotherm provides a good fit for the experimental data, in line with the conclusions reported in refs 32−34 that the surface tension of various oxyethylated surfactants solutions obeys the reorientation model. The parameters summarized in Table 1 were also used to calculate the rheological characteristics of C12EO5 solutions discussed below. For all C12EO5 solution concentrations, after equilibration, the dilational rheological parameters were measured in the frequency range 0.005−0.5 Hz; in some experiments, the minimum frequency was 0.001 Hz. The values of the viscoelasticity modulus for the pairs of solutions denoted by numbers 1−3 in Figure 1 are shown in Figure 3. Note, from the legends to Figure 1 we can obtain the corresponding surface

mN/m, data pair 4−20.0 mN/m, and data pair 5−30.5 mN/m. It is seen from the comparison of the modulus for the drop (curve 1) and bubble method (curve 3) measured for the same concentration of 1.0 μmol/L that there are significant differences between the values obtained by the two methods; this fact can be attributed to the decrease of bulk concentration by depletion in the drop method. Note that for each pair of solutions, the equilibrium surface tension is the same, as shown in Figure 1. Therefore, the results for Π < 15 mN/m, if expressed in terms of the dependencies on surface pressure, are practically the same for both methods, because, in contrast to the dependencies on the concentration, the adsorption-related losses of bulk concentration in this case are irrelevant. It has to be noted here that the dependencies on surface pressure were used in refs 35 and 36 to analyze the rheological properties of protein solutions. The smooth red solid lines in Figure 3 correspond to theoretical calculations for bubble, eq 8, while dashed red line shows the theoretical calculations for a drop, eq 9; in all cases, the parameters listed in Table 1 were used. It is seen that for the drop at low frequencies, the calculated values are somewhat higher than for the bubble. However, at high surface pressures (Π > 15 mN/m), the modulus becomes dependent on the measurement method, as is seen from Figure 4, where the results obtained for the pairs 4 and 5 defined in Figure 1 are shown. For these systems the modulus values measured by the bubble method are lower than those obtained from the drop method, although the equilibrium surface tensions are equal as one can see in Figure 1. Note that the relative differences between the modules obtained in the drop and bubble experiments depend on the oscillation frequency: at 0.001 Hz for pair 5 (both concentrations 15 μmol/L) the difference is 33%, while at 0.5 Hz this difference is 17%. At the same time, the difference between the theoretical values calculated using eqs 8 and 9 is about 8% at 0.001 Hz, and less than 1% at 0.5 Hz. Figure 5 illustrates the dependence of the viscoelasticity modulus for the oscillation frequencies 0.1 and 0.01 Hz measured by both methods, on surface pressure. It is seen that,

Figure 3. Dependence of the viscoelasticity modulus as measured by the drop (▲) and bubble (◇) profile methods for C12EO5 solutions with concentrations corresponding to pairs 1, 2, and 3 shown in Figure 1. For detailed explanation see text. Dotted lines are only guides for the eye. D


Article

Langmuir

drop surface oscillations, and therefore, to an increase of the dilational modulus. The analysis of the mass balance within the drop is the simplest for very slow oscillations, when the surfactant remains almost homogeneously distributed throughout the drop bulk. This corresponds to the frequencies ν ≪ D/r02 (we assume, however, that the diffusion front does not penetrate deeply into the capillary yet). In this case eq A.3 in the Appendix gives the variation of the concentration in the drop: Δc = − dΓ dc

Γe +

r0 3

·

ΔA A0

(12)

where Γe and A0 are the values of adsorption and surface area at equilibrium, and ΔA is the surface area variation. It is seen from eq 12 that any finite variation of the drop surface area results in a finite variation of the concentration. This variation of the concentration is increasing with decreasing drop radius r0 and the derivative dΓ/dc. On the contrary, for a bubble or a plane boundary being in contact with a large solution reservoir, with a decreasing oscillation frequency, the concentration variation becomes infinitely small. The variation of the surface tension related to the concentration variation expressed by eq 12 obeys eq A.7 of the Appendix, i.e.:

Figure 5. Dependencies of the viscoelasticity modulus for C12EO5 solutions on surface pressure for the surface oscillation frequencies 0.1 Hz (lines 1, 2, points ▲,Δ) and 0.01 Hz (lines 3, 4, points ◆,◇), obtained by the drop (◆,▲) and bubble (◇,Δ) profile method. Smooth red curves were calculated using the diffusion model for the frequency 0.01 Hz for drop (dashed line) and bubble (solid line); open (○) and filled (●) red circles, corrected values for the drop at frequencies 0.1 and 0.01 Hz, respectively. For details, see text. Dotted and dash-dotted lines are guides for eyes.

Δγ =

below 15 mN/m, the results obtained by the two methods are almost the same, while for higher surface pressures, the values obtained by the drop method are higher than those measured in the experiments with bubbles. Similar differences between the results obtained by the two methods were also reported in refs 27 and 28. There are several possible reasons for the difference between the results obtained by the drop and bubble profile methods. Among them are a certain difference in the geometric shapes of these objects (for the drop method, the surface area to volume ratio at the same volume or area is higher than for the bubble method), and a convective-governed transfer of surfactant from the solution around the bubble, which could result in an increased apparent diffusion coefficient. In Figure 5, the theoretical curves calculated using the diffusion model eqs 8 and 9 for the frequency 0.01 Hz for the drop method with D = 4.0 × 10−10 m2/s (red dashed curve), and for the bubble method with D = 8.0 × 10−10 m2/s (red solid curve) are shown. While these theoretical curves exhibit rather good correspondence with the experimental values, this “convective-related” explanation seems rather irrelevant, because in the analysis of the dynamic surface tension, the diffusion coefficients at high C12EO5 concentrations for bubble and drop were found to be equal to one another (cf. results shown in Figure 1). Another reason for the differences between the values of viscoelasticity modulus measured in the drop and bubble configurations could be the difference in the surfactant mass balance, because the surface area oscillations lead to oscillations of the adsorption values, which in turn results in variations of both subsurface and bulk concentrations of the surfactant being rather different for these two methods. More specifically, the expansion of the drop surface leads to additional adsorption, and therefore to a decrease of the bulk concentration, while the compression has the opposite effect. This effect results in an increase of the surface tension variation amplitudes during the

dγ 1 dγ Δc = − dc c d ln c

Γe dΓ dc

+

r0 3

·

ΔA A0

(13)

Therefore, even for very slow variations of the drop surface area, the surface tension is subject to variations; on the contrary, very slow oscillations of the volume and surface area of a bubble do not result in surface tension changes. Hence, the surface elasticity of a drop is higher than that of a bubble due to, among other factors, the changes of concentration within the drop bulk. Note that it is seen from the analysis presented in the Appendix that eq 9 accounts for the variations of the concentration within the drop bulk that result from surface area variations. At higher frequencies (ν ≥ D/r02) the concentration variations which arise near the surface are too fast, and cannot propagate into the drop bulk or into the solution around the bubble. Therefore, the amplitude of concentration oscillations far off the bubble or drop surface becomes lower, while it increases in the vicinity of the surface. This results in an increase of the surface viscoelasticity modulus with the oscillation frequency; at the same time, the difference between the modules for drops and bubbles becomes lower with increasing frequency (cf. theoretical curves in Figure 3). It follows from eq 13, and also from eqs A.7 and A.8 derived in the Appendix that the extent to which the surface radius r0 influences the viscoelasticity modulus depends on the derivative dΓ/dc. At low bulk concentrations, this derivative is large enough, and therefore the curvature effect is negligible; this is clearly seen from Figures 3 and 5 in the surface pressure range below 15 mN/m. However, with increasing concentration, the adsorption tends to saturation, and the derivative dΓ/dc becomes smaller, which in turn results in an increased influence of r0 on the viscoelasticity modulus. This is also demonstrated by Figures 4 and 5 in the range Π > 15 mN/m. Therefore, there is at least a qualitative correspondence between the experimental data and the theoretical predictions that follow from eqs 8 and 9. However, the quantitative estimations appear E


Article

Langmuir to underestimate the values observed in the experiment, which could possibly be attributed to the approximate nature of eqs 9 and A.1. To study this problem in more detail, stress deformation experiments with bubbles and drops were performed. For C12EO5 solutions with concentrations corresponding to pairs 3, 4, and 5 defined in Figure 1, the dynamic surface tensions were measured at stress expansion/compression deformations with bubble and drop surfaces. The initial surface area was 40 mm2 for systems 3 and 4, and 38 mm2 for system 5. The C12EO5 solutions studied by the drop and bubble method were aged until equilibration (for 8000, 4000, and 3000 s for pairs 3, 4, and 5, respectively), and subsequently subjected to compressions by 2 mm2 during 1 s. Then, after equilibration (for 2000, 1500, and 1000 s for pairs 3, 4, and 5, respectively) the compression was repeated. A total of 5 compression steps were performed, and the results are illustrated in Figure 6a−c. With regard to the results of these experiments, two comments should be made. First, after equilibration, the surface tension of the drop remains by 0.3−0.5 mN/m lower than before each new compression. The total surface tension decreases caused by five compression steps were 2.0, 2.7, and 1.5 mN/m for pairs 3, 4, and 5, respectively. On the contrary, for the bubble at all concentrations, the total surface tension variation caused by five compression steps did not exceed 0.15 mN/m. Similar results were obtained in experiments with a stress expansion of the drop and bubble surface. The second important fact is the relationship between the magnitudes of surface tension variations caused by the repeated compressions of the drop and bubble surfaces. These steps are roughly the same in Figure 6a, while in Figure 6b the steps for the drop are approximately by a factor of 1.5 larger than those for the bubble, while at larger concentration in Figure 6c this factor is as large as 2.5−3. To be certain of the fact that the surface tension variations caused by the changes in the size of spherical objects described above cannot be attributed to some errors inherent to the drop or bubble profile analysis method, experiments with stepwise stress compression of pure toluene (surface tension ca. 30 mN/ m) were performed similarly to those with C12EO5 solutions discussed above. The results obtained in these experiments are illustrated in Figure 7. It is seen that the experimental error is insignificant: the decrease of the bubble surface area from 32 to 22 mm2 does not lead to a surface tension increase more than 0.1 mN/m, while in the corresponding drop experiments an increase of the surface tension does not exceed 0.2 mN/m. The difference in the absolute values (average surface tension values of 30.0 and 30.5 mN/m for the bubble and drop experiments, respectively) are obviously caused by small errors in the calibration of the instrument for the two experimental geometries, and are in the range of experimental accuracy of ±0.2 mN/m. Therefore, the difference between the surface tension jumps for the two methods shown in Figure 6 explains why the viscoelastic modules for the two methods are different. The magnitude of the drop/bubble surface area harmonic oscillations is approximately ±2 mm2, which amounts to about 6% of the total area 38 mm2. At a frequency of 0.1 Hz, the time during which this change of the surface area occurs is 2.5 s, while at frequency 0.01 Hz this time is 25 s. It follows from Figure 6a that for these time values after the compression, the surface tension variations for the drop and bubble are both equal to about 1 mN/m. In Figure 6b the surface tension

Figure 6. Time dependence of surface tension for C12EO5 solutions as measured by the drop (black curves, D) and bubble (red curves, B) profile methods for the concentrations corresponding to pairs 3 (a), 4 (b), and 5 (c) as defined in Figure 1 for five consecutive stress compressions of the surface by 2 mm2 each. The area values (numbers, in mm2) and time moments when the stress deformation was applied (arrows) are shown at the top of the figures.

variations for the bubble is by 0.3−0.4 mN/m lower, while for the solutions with high concentration (Figure 6c) the variations for the drop and the bubble are 0.7 and 0.3 mN/m, respectively. Even for the frequency of 0.001 Hz, the surface F


Article

Langmuir

Figure 7. Time dependence of toluene as measured by the drop and bubble profile methods for five consecutive stress compressions of the surface by 2 mm2 each. The notation is the same as in Figure 6.

Figure 8. Dependence of the derivative dΠ/dΓ on surface pressure for aqueous solutions of C10EO8, C12EO5, C10EO5, C14EO8, and C13DMPO.

area variation by 6% for 250 s also results in a significant difference between the amplitudes. Thus, the variation of the surface area and volume of the surfactant solution drop caused by its compression or expansion results in additional changes of the adsorption and bulk concentration, which lead to the increase of the viscoelasticity modulus. For the bubble that is surrounded by a large solution reservoir, these additional changes in bulk concentration and adsorption do not occur, and the viscoelasticity modulus measured in bubble oscillation experiments is lower than that for the drop. If a correction is introduced by the decrease of the amplitude of the oscillationgoverned surface tension variations for the drop by the differences shown in Figure 6b,c, the resulting values of viscoelasticity moduli become close to corresponding values for the bubble, as shown in Figure 5 by red circles. This phenomenon, the increase of the viscoelasticity modulus as measured by the drop method, is observed only for highly surface active surfactants (for which the adsorption essentially affects the bulk concentration), and is governed by the value of the derivative dΠ/dΓ. More precisely: the higher the surface pressure variation caused by the change of adsorption, the larger could be the difference between the viscoelasticity modules obtained from drop and bubble profile experiments. Let us consider the values of the derivative dΠ/dΓ. These values were determined from the surface tension and adsorption isotherms calculated using eqs 1−4 for nonionic surfactants and the Frumkin model for SDS (see refs 5, 9, 23, 37, and 38). The dependencies of dΠ/dΓ on surface pressure for different surfactants are shown in Figures 8 and 9, whereas Figures 10−13 illustrate the viscoelasticity moduli for these surfactants measured using the drop and bubble profile methods at different surface pressures. It could be supposed that the surfactants with short hydrocarbon chains exhibit highest derivative values (C10EO8, C10EO5, and C12EO5 in Figure 8); therefore for these surfactant solutions, the drop method should yield higher values of the viscoelasticity modulus compared to the bubble method, as is in fact seen from Figure 5 for C12EO5 and Figures 10 and 11 for C10EO8 and C10EO5. For C14EO8, the derivative is smaller and the values of the viscoelasticity modulus obtained by the two methods are quite close to each other (cf. Figure 8 and Figure 11).

Figure 9. Dependence of the derivative dΠ/dΓ on surface pressure for aqueous solutions of various Triton X-n.

Figure 10. Dependencies of the viscoelasticity modulus on surface pressure of C10EO5 solutions for the surface oscillation frequencies 0.1 Hz (▲,Δ) and 0.01 Hz (◆,◇), obtained by drop (◆,▲) and bubble (◇,Δ) profile methods. Lines are guides for eyes. G


Article

Langmuir

Figure 11. Dependencies of the viscoelasticity modulus on surface pressure of C14EO8 (▲,Δ,◆,◇) and C10EO8 (●,○,■,□) solutions for the surface oscillation frequencies 0.1 Hz (▲,Δ,●,○) and 0.01 Hz (◆,◇,■,□), obtained by drop (◆,▲,●,■) and bubble (◇,Δ,○,□) profile methods. Lines are guides for eyes.

Figure 13. Dependence of the viscoelasticity modulus on frequency as measured by the drop (◆,▲,■,●) and bubble (◇,Δ,□,○) profile methods for C13DMPO solutions with Π = 15 mN/m (Δ,▲) and Π = 27 mN/m (◇,◆), Triton X-405 solutions with Π = 10 mN/m (■,□), and SDS in 0.01 M NaCl (●,○) solutions with Π = 25 mN/m. Lines are guides for eyes.

which evens the influence of the surface tension changes caused by oscillations on the surfactant concentration in the drop bulk. For the alkyl dimethyl phosphine oxides, this derivative is small, and virtually the same for C12DMPO, C13DMPO (shown in Figure 8), and C14DMPO. Also for ionic surfactants (SDS, CTAB, DoTAB), the derivative dΠ/dΓ does not exceed 1.5 × 104 N·m/mol. Therefore, it could be supposed that for these surfactants the modulus values obtained by the two methods will be almost equal to each other. The dependencies of the viscoelasticity modulus on frequency for the solutions which, according to these estimates, should not exhibit differences between the values obtained by the two methods at the same Π, are shown in Figure 13. It is seen that the results are indeed almost the same. These results confirm that the differences between the viscoelasticity moduli determined by the two methods are observed only for those surfactants that exhibit high values of the dΠ/dΓ derivative. Probably, more rigorous conditions would be provided by the product (dΠ/dΓ)·(dΠ/ dc): the higher this value is, the more probable are differences between the viscoelasticity moduli. For all the surfactants studied, this latter statement agrees with the experimental data.

Figure 12. Dependencies of the viscoelasticity modulus on surface pressure of Triton X-100 (●,○) and Triton X-165 (▲,Δ,◆,◇) solutions for the surface oscillation frequencies 0.1 Hz (●,○,▲,Δ) and 0.01 Hz (◆,◇), obtained by drop (●,◆,▲) and bubble (○,◇,Δ) profile methods. Lines are guides for eyes.

5. CONCLUSIONS In the present dilational rheology study of C12EO5 adsorption layers, measurements at various frequencies of drop or bubble surface area oscillations, respectively, were performed. Particular attention was paid to the influence of the surfactant solution concentration on the surface viscoelastic properties in these two experimental configurations. These results were used to calculate the dilational viscoelasticity modulus for C12EO5 adsorption layers. The values obtained by the two methods were found to be the same in the range of equilibrium surface pressures below 15 mN/m, while for higher surface pressures the values obtained from drop profile experiments exceed those obtained from bubble profile analysis. The nature of this phenomenon was studied using stress deformations of bubbles and drops. It was shown that at sufficiently high surfactant concentrations, the surface tension variations caused by the surface area stress are essentially higher for the drop as

Very interesting are the dependencies for the Tritons (polyethylene glycol octylphenyl ethers, Figure 9). The Tritons with long oxyethyl chains (Triton X-405, Triton X-165, and Triton X-100) exhibit a clearly pronounced maximum at a surface pressure of 7−9 mN/m. This fact can be attributed to the transfer from the state with maximum molar area (adsorption of oxyethyl and hydrocarbon groups) to the state with minimum area (adsorption of only hydrocarbon groups),30 which is accompanied by a significant increase of adsorption. Therefore, for the Triton X-165 and Triton X-100, one can expect increased viscoelasticity modulus values for the drop method at low surface pressure. Indeed for Triton X-165, the difference is observed only at surface pressure values below 15 mN/m (Figure 12). However, for Triton X-100, the difference is small at all surface pressures. For Triton X-405, the high dΠ/ dΓ derivative value is insufficient to produce this effect, because the CMC for this surfactant is quite high (about 900 μmol/L), H


Article

Langmuir compared with the bubble, due to the influence of adsorption at the drop surface on the concentration of solution enclosed in the limited drop volume. Therefore, the variations of the bulk concentration caused by surface expansions and compressions of the surfactant solution drop result in an increased viscoelasticity modulus. The influence of the drop surface deformations on the surfactant bulk concentrations at low surface oscillation frequencies is analyzed analytically. The values of viscoelasticity modulus for a number of surfactants from the literature (C10EO5, C10EO8, C14EO8, C13DMPO, Triton X-45, Triton X-100, Triton X-165, Triton X-405, and SDS) are discussed. It is shown that the discrepancies between the data obtained by the two experimental methods for many surfactants can be explained well with the theory developed here.

Δc = −

Δγ =

■

■

where n = iν /D , r is the radial coordinate (r ≤ r0), c0 is the initial concentration in the drop, and H is a constant. In the case of very slow oscillations, |nr0| ≪ 1, the expression for the concentration distribution (eq A.1) becomes simpler:

c ≈ c0 + Hne

dc

+

r0 3

·

ΔA A0

■ ■

REFERENCES

(1) Lucassen, J.; van den Tempel, M. Dynamic Measurements of Dilational Properties of a Liquid Interface. Chem. Eng. Sci. 1972, 27, 1283−1291. (2) Lucassen, J.; Hansen, R. S. Damping of Waves on MonolayerCovered Surfaces: II. Influence of Bulk-to-Surface Diffusional Interchange on Ripple Characteristics. J. Colloid Interface Sci. 1967, 23, 319−328. (3) Benjamins, J.; Cagna, A.; Lucassen-Reynders, E. H. Studies for Determination of Elasticity of Adsorption Films of Soluble Surface Active Substances. Colloids Surf., A 1996, 114, 245−254. (4) Noskov, B. A. Dilational Surface Rheology of Polymer and Polymer/surfactant Solutions. Curr. Opin. Colloid Interface Sci. 2010, 15, 229−236. (5) Fainerman, V. B.; Lylyk, S. V.; Aksenenko, E. V.; Makievski, A. V.; Ravera, F.; Petkov, J. T.; Yorke, F.; Miller, R. Adsorption Layer Characteristics of Triton Surfactants. 3. Dilational visco-elasticity. Colloids Surf., A 2009, 334, 16−21. (6) Wantke, K. D.; Fruhner, H.; Fang, J.; Lunkenheimer, K. Measurements of the Surface Elasticity in Medium Frequency Range Using the Oscillating Bubble Method. J. Colloid Interface Sci. 1998, 208, 34−48. (7) Wu, D.; Feng, Y.; Xu, G.; Chen, Y.; Cao, X.; Li, Y. Dilational Rheological Properties of Gemini Surfactant 1,2-ethane bis(dimethyl Dodecyl Ammonium Bromide) at Air/water Interface. Colloids Surf., A 2007, 299, 117−123. (8) Ravera, F.; Ferrari, M.; Santini, E.; Liggieri, L. Influence of Surface Processes on the Dilational Visco-elasticity of Surfactant Solutions. Adv. Colloid Interface Sci. 2005, 117, 75−100.

0

(A.5)

AUTHOR INFORMATION

ACKNOWLEDGMENTS The authors are grateful to ESA for the support in the framework of the MAP “Soft Matter Dynamics”.

spherical drop), eq A.3 becomes the equation which describes the surfactant balance on the surface and in the drop bulk during the oscillations:

Δ(ΓA) + V0Δc = 0

(A.8)

The authors declare no competing financial interest.

where Γe and A0 are the values of adsorption and surface area at equilibrium, and ΔA is the surface area amplitude. After simple r V dΓ transformations, noting that ΔΓ = dc Δc and 30 = A0 (for a

(A.4)

dc r0 dΓ 3

Notes

(A.3)

A 0ΔΓ + V0Δc = −ΓeΔA

1+

*corresponding author: [email protected].

(A.2)

Γe

E0

Corresponding Author

This expression is independent of r; therefore the concentration oscillates synchronously with the same magnitude Hn throughout the entire drop (increases and decreases with respect to the initial value c0). Accounting for the surfactant balance at the surface, eq 11, one obtains the amplitude of concentration oscillations: Δc = Hn = − dΓ

(A.7)

The same equation follows also from eq 9 in the limiting case ν → 0. Therefore, eq 9 obtained in ref 26 based on the diffusional exchange of matter model accounts for the concentration variation in the drop caused by the variations of the drop surface area. This concentration variation results in a larger viscoelasticity modulus for drops compared to the bubbles.

(A.1)

iνt

E0 dγ 1 dγ dΓ ΔA Δc = Δc = · dc r dc Γe d ln Γ dc 1 + dΓ 30 A 0

where E0(c) = −dγ/d ln Γ is the limiting elasticity. Then the surface elasticity can be expressed as

APPENDIX To analyze the concentration variations within the drop bulk, we consider the case of slow oscillations when the surfactant has sufficient time to redistribute within the drop after a surface area change. For small harmonic oscillations of frequency ν, the solution of eq 10 yields the distribution of the surfactant concentration at any time moment t (see eq (8.66) in ref 26): H sinh(nr )eiνt r

(A.6)

This concentration variation is determined by the variation of the amount of surfactant adsorbed on the drop surface, and by the drop volume V0. The surface tension jump Δγ caused by the concentration jump Δc in the equilibrium conditions is expressed as

E=

c = c0 +

1 Δ(ΓA) V0

or

Note that this equation does not contain the term c0ΔV, i.e., the amount transferred with the solution to and from the capillary during the oscillations. This quantity remains constant and does not participate in the overall surfactant balance (we assume approximately that the diffusion front does not penetrate into the capillary). Therefore, the concentration variation within the drop bulk during slow oscillations is I


Article

Langmuir

(30) Kovalchuk, V. I.; Miller, R.; Fainerman, V. B.; Loglio, G. Dilational Rheology of Adsorbed Surfactant Layers  Role of the Intrinsic Two-Dimensional Compressibility. Adv. Colloid Interface Sci. 2005, 114−115, 303−313. (31) Fainerman, V. B.; Zholob, S. A.; Lucassen-Reynders, E. H.; Miller, R. Comparison of Various Models Describing the Adsorption of Surfactant Molecules Capable of Interfacial Reorientation. J. Colloid Interface Sci. 2003, 261, 180−183. (32) Valenzuela, M. Á .; Gárate, M. P.; Olea, A. F. Surface Activity of Alcohols Ethoxylates at the n-Heptane/Water Interface. Colloids Surf., A 2007, 307, 28−34. (33) Ritacco, H. A.; Busch, J. Dynamic Surface Tension of Polyelectrolyte/Surfactant Systems with Opposite Charges: Two States for the Surfactant at the Interface. Langmuir 2004, 20, 3648− 3656. (34) Lee, Y. C.; Liu, H. S.; Lin, S. Y. Adsorption Kinetics of C10E4 at the Air−Water Interface: Consider Molecular Interaction or Reorientation. Colloids Surf., A 2003, 212, 123−134. (35) Benjamins, J. Static and Dynamic Properties of Proteins Adsorbed at Liquid Interfaces. Ph.D. Thesis, Wageningen University, Wageningen, The Netherlands, 2000. (36) Lucassen-Reynders, E. H.; Benjamins, J.; Fainerman, V. B. Dilational Rheology of Protein Films Adsorbed at Fluid Interfaces. Curr. Opin. Colloid Interface Sci. 2010, 15, 264−270. (37) Fainerman, V. B.; Lylyk, S. V.; Aksenenko, E. V.; Petkov, J. T.; Yorke, J.; Miller, R. Surface Tension Isotherms, Adsorption Dynamics and Dilational Visco-elasticity of Sodium Dodecyl Sulphate. Colloids Surf., A 2010, 354, 8−15. (38) Fainerman, V. B.; Lylyk, S. V.; Aksenenko, E. V.; Makievski, A. V.; Petkov, J. T.; Yorke, J.; Miller, R. Adsorption Layer Characteristics of Triton Surfactants. 1. Surface Tension Isotherms. Colloids Surf., A 2009, 334, 1−7.

(9) Fainerman, V. B.; Aksenenko, E. V.; Krägel, J.; Miller, R. Viscoelasticity Moduli of Aqueous C14EO8 Solutions as Studied by Drop and Bubble Shape Methods. Langmuir 2013, 29, 6964−6968. (10) Edwards, D. A.; Brenner, H.; Wasan, D. T. Interfacial Transport Process and Rheology; Butterworth-Heinemann: Boston, MA, 1991. (11) Ivanov, I. B.; Danov, K. D.; Ananthapadmanabhan, K. P.; Lips, A. Interfacial Rrheology of Adsorbed Layers with Surface Reaction: on the Origin of the Dilatational Surface Viscosity. Adv. Colloid Interface Sci. 2005, 114, 61−92. (12) Hoorfar, M.; Neumann, A. W. Recent Progress in Axisymmetric Drop Shape Analysis (ADSA). Adv. Colloid Interface Sci. 2006, 121, 25−49. (13) Liggieri, L.; Ferrari, M.; Mondelli, D.; Ravera, F. Surface Rheology as a Tool for the Investigation of Processes Internal to Surfactant Adsorption Layers. Faraday Discuss. 2005, 129, 125−140. (14) Hansen, F. K. Surface Dilatational Elasticity of poly(oxy ethylene)-based Surfactants by Oscillation and Relaxation Measurements of Sessile Bubbles. Langmuir 2008, 24, 189−197. (15) Loglio, G.; Tesei, U.; Cini, R. Spectral Data of Surface Viscoelastic Modulus Acquired via Digital Fourier Transformation. J. Colloid Interface Sci. 1979, 71, 316−320. (16) Georgieva, D.; Cagna, A.; Langevin, D. Link between surface elasticity and foam stability. Soft Matter 2009, 5, 2063−2071. (17) Huang, Y.-P.; Zhang, L.; Zhang, L.; Luo, L.; Zhao, S.; Yu, J.-Y. Dynamic Interfacial Dilational Properties of Hydroxy-substituted Alkyl Benzenesulfonates. J. Phys. Chem. B 2007, 111, 5640−5647. (18) Babak, V. G.; Desbrieres, J.; Tikhonov, V. E. Dynamic Surface Tension and Dilational Viscoelasticity of Adsorption Layers of a Hydrophobically Modified Chitosan. Colloids Surf., A 2005, 255, 119− 130. (19) Freer, E. M.; Wong, H.; Radke, C. J. Oscillating Drop/bubble Tensiometry: Effect of Viscous Forces on the Measurement of Interfacial Tension. J. Colloid Interface Sci. 2005, 282, 128−132. (20) Santini, E.; Ravera, F.; Ferrari, M.; Stubenrauch, C.; Makievski, A.; Krägel, J. A Surface Rheological Study of Non-ionic Surfactants at the Water−air Interface and the Stability of the Corresponding Thin Foam Films. Colloids Surf., A 2007, 298, 12−21. (21) Koelsch, P.; Motschmann, H. Relating Foam Lamella Stability and Surface Dilational Rheology. Langmuir 2005, 21, 6265−6269. (22) Ravera, F.; Ferrari, M.; Liggieri, L. Modelling of Dilational Viscoelasticity of Adsorbed Layers with Multiple Kinetic Processes. Colloids Surf., A 2006, 282−283, 210−216. (23) Fainerman, V. B.; Aksenenko, E. V.; Petkov, J. T.; Miller, R. Adsorption Layer Characteristics of Mixed Oxyethylated Surfactant Solutions. J. Phys. Chem. B 2010, 114, 4503−4508. (24) Fainerman, V. B.; Miller, R. Direct Determination of Protein and Surfactant Adsorption by Drop and Bubble Profile Tensiometry. In Bubble and Drop Interfaces; Progress in Colloid and Interface Science, Miller, R., Liggieri, L., Eds.; Brill Publishing: Leiden, The Netherlands, 2011; Vol. 2, pp 179−193. (25) Kotula, A. P.; Anna, Sh. L. Regular Perturbation Analysis of Small Amplitude Oscillatory Dilatation of an Interface in a Capillary Pressure Tensiometer. J. Rheol. 2015, 59, 85−117. (26) Joos, P. Dynamic Surface Phenomena; VSP: Dordrecht, The Netherlands, 1999. (27) Boos, J.; Preisig, N.; Stubenrauch, C. Dilational Surface Rheology Studies of n-dodecyl-β-D-maltoside, Hexaoxyethylene Dodecyl Ether, and their 1:1 Mixture. Adv. Colloid Interface Sci. 2013, 197−198, 108−117. (28) Reichert, M. D.; Alvarez, N. J.; Brooks, C. F.; Grillet, A. M.; Mondy, L. A.; Anna, Sh. L.; Walker, L. M. The Importance of Experimental Design on Measurement of Dynamic Interfacial Tension and Interfacial Rheology in Diffusion-limited Surfactant Systems. Colloids Surf., A 2015, 467, 135−142. (29) Zholob, S. A.; Makievski, A. V.; Miller, R.; Fainerman, V. B. Optimisation of Calculation Methods for Determination of Surface Tensions by Drop Profile Analysis Tensiometry. Adv. Colloid Interface Sci. 2007, 134−135, 322−329. J


Dilational Viscoelasticity of Adsorption Layers Measured by Drop and

Recommend Documents