Surface dilational viscoelasticity of aqueous surfactant solutions by surface quasi-elastic light scattering

Abstract

We investigated the surface dilational viscoelasticities of aqueous solutions of dihexanoyl-phosphastidylcholine (DC₆PC) and dioctanoyl-phosphastidylcholine (DC₈PC), dodecyltrimethylammonium bromide (DTABr) and DTABr and sodium bromide (NaBr), dodecyltrimethylammonium tetrafluoroborate (DTABF₄) and bisdodecyltrimethylammonium sulfate (DTA₂SO₄), and nonionic surfactants with oxyethylene units (C₁₀E₅). We measured surface quasi-elastic light scattering (SQELS) spectra and analyzed the solutions by both surface rheological and phenomenological fitting approaches. For the DC₆PC-DC₈PC mixture, the electrocapillary wave (ECW) method was also applied. The molecular interactions between DC₈PC and DC₆PC in the adsorbed film clearly influenced the relaxation process on the order of several milliseconds (ECW). However, the effects were less influential on the order of several hundreds of microseconds (SQELS). The characteristic frequencies for relaxation owing to the staggered structure formation likely occurred in the range from 150 to 300 Hz. The dilational elasticity at 250–400 kHz of DTABr suggested that the main relaxation process influencing the dilational properties was neither diffusion of surfactants from the bulk phase to the subsurface nor an adsorption-desorption of surfactant ions between the adsorbed layer and subsurface. However, the diffusion of counterions between the Stern and diffuse layers occurred on a time scale of several microseconds. Studies of C₁₀E₅ suggested that the relaxation process captured by the SQELS studies was neither related to surfactant diffusion nor surfactant reorientation, but rather was likely to be a relaxation process where distortion of the hydration sphere and/or hydration-dehydration to/from the hydration sphere was caused by diffusion of water molecules.

Introduction

Surface viscoelasticity represents the rigidity and fluidity of surface layers and is defined as the restoring force, which works against deformation of the surface layer. This topic has been well reviewed [1,2,3,4,5,6,7] because of its theoretical and practical importance. For example, interfacial rheology often plays an important role in understanding the dynamics of surfactant layers that are related to emulsion stability [8], collapse of foam films [9], plate coating [10], and membrane systems [11]. The interfacial rheology of polymer films has been intensively researched in recent years [4, 12, 13], because of its practical importance for industrial applications; however, further studies of the rheology of films of adsorbed surfactants over a wide range of frequencies of surface perturbation are required to consolidate the basic concepts of interfacial rheology and the applicability of certain proposed models.

There are shear and dilational parameters for surface viscoelastic properties. However, the magnitude of the former is typically smaller than that of the latter by several orders [14]; thus, we focus our investigation on the dilational properties. For adsorbed films of soluble surfactants, the dilational viscoelastic properties are strongly dependent on the perturbation frequency, and thus, several experimental methods have been used to clarify surface dilational properties at different frequencies: the combination of oscillating drops and bubbles can be used to access data in a frequency range between 10⁻³ and 100 Hz. The capillary wave damping method can be used to study frequencies up to 1000 Hz and the surface quasi-elastic light scattering (SQELS) method can be used to assess to those up to approximately 1 MHz [6]. The oscillating drop method enabled the surface viscoelasticity of aqueous solutions of surfactants to be studied at low frequencies by Miller et al. [6, 15, 16], from which information was obtained on slower dynamic processes, such as bulk diffusion and adsorption barriers. The electrocapillary wave (ECW) method has been used for studies at intermediate frequency regions (~kHz) for cationic surfactants, and the viscoelastic coefficients have been determined by applying dispersion equations and models describing the adsorption state and processes [17, 18]. Furthermore, surface light scattering techniques at higher frequencies (between 5 and several hundred kHz) have been applied to different kinds of surfactants and revealed that negative surface dilational viscosities are often evaluated [17, 19,20,21].

In this study, we focus on the dilational viscoelasticity of adsorption layers in the high frequency region with the use of the surface quasi-elastic light scattering (SQELS) method. SQELS is based on ripplons generated by the thermal motion of constituent molecules at the surface and is thus advantageous as non-perturbative and non-contact method. The SQELS method features certain experimental peculiarities, in that the spectrum of scattered light features considerable instrumental broadening and becomes less applicable for extremely large surface viscoelasticities. However, these issues are respectively resolved by calibrating the SQELS equipment with pure water and choosing an appropriate accessible region of the surface viscoelastic parameters from the SQELS [12]. The other difficulty of the SQELS method is the theoretical complexity of air/aqueous solution surfaces. The dilational parameters should be derived from analysis of the experimentally detected transversal waves (capillary waves) through the dispersion equation, which often provides an unrealistic negative viscosity. Although there are some theoretical approaches to resolving the problem of negative viscosity, this issue has yet to be clarified. For example, Hennenberg et al. explained the physical reasons for improving the van den Temple-Lucassen theory by taking into account contributions from surface deformation [22], and Monroy et al. proposed a new dispersion equation that accounts for internal coupling between capillary and dilational modes, which leads to non-anomalous dilational viscosities [23].

First, we used SQELS to study the ripplon frequency at approximately 250 kHz of a short-chain phospholipid mixture of dihexanoyl-phosphatidylcholine (DC₆PC) and dioctanoyl-phosphatidylcholine (DC₈PC). This mixture was chosen because both storage and loss modulus values at approximately 0.2 kHz by the ECW show a maximum near the surface composition of the minimum of the excess Gibbs energy of adsorption, which can be attributed to a staggered arrangement of the mixed adsorbed film [24]. Thus, the effects of the perturbation frequency on the surface viscoelasticity will be extracted by comparing the SQELS results with those from ECW. For this purpose, we also performed some ECW experiments on the DC₆PC and DC₈PC mixtures at different frequencies from 0.2 kHz.

For a better understanding of the surface viscoelasticity from SQELS, we used aqueous solutions of different types of surfactant: (A) dodecyltrimethylammonium bromide (DTABr), (B) DTABr in the presence of sodium bromide (NaBr), (C) dodecyltrimethylammonium tetrafluoroborate (DTABF₄) and bis-dodecyltrimethylammonium sulfate (DTA₂SO₄), and (D) pentaethyleneglycol monodecyl ether (C₁₀E₅). DTABr was chosen because its equilibrium surface properties [25, 26] and dilational viscoelastic properties [17, 18, 27, 28] have been relatively well investigated. Comparing (A) with (B), we might gain information on the effects of the electrical double layer on the dilational viscoelasticity. Comparing (A) with (C), we might gain insight into the influence of the degree of counter ion binding to the surfactant ion on dilational viscoelasticity because the (A) and (C) systems contain the same surfactant ions but different counterions. Previous studies by surface tension [29,30,31,32] and total reflection X-ray absorption fine structure (TRXAFS) [33, 34] measurements have clarified that differences in the dehydration energy of counterions and their ionic atmosphere affect the surface density of adsorbed films. Thus, the SQELS measurement is expected to enable examination of how these factors influence the surface viscoelasticity of adsorbed films. Furthermore, (D) was chosen because it is nonionic, and its hydrophilic chain is considerably hydrated; thus, we might better understand the influence of water molecules on the dilational viscoelasticity.

This paper is constructed as follows. In the “Materials and methods” section, the materials and the thermodynamic equations for the surface tension analysis [35] are described and the SQELS apparatus is explained. Next, the theoretical background for evaluating the surface viscoelasticity parameters, that is, the dispersion relation [36], is presented and the two analytical methods of analyzing the SQELS spectrum, i.e., the surface rheological and phenomenological fitting analysis, are described. In the “Results and discussion” section, first, SQELS and ECW studies of short-chain phospholipid mixtures are presented and then the surface dilational elasticities from the SQELS method are explained and examined, where the Lucassen and van den Temple model [17, 37,38,39] is introduced to understand the adsorption dynamics from the viewpoint of viscoelasticity. Finally, we briefly mention the dependence of the damping coefficients on the coupling of dilational and capillary waves.

Materials and methods

Materials

1,2-Dihexanoyl-sn-glycero-3-phosphocoline (DC₆PC) and 1,2-dioctanoyl-sn-glycero-3-phosphocoline (DC₈PC) were purchased from Avanti Polar Lipids (USA) and used without further purification. DTABr (99%) purchased from Wako Pure Chemical Industries (Japan) was purified by recrystallizing it five times from the acetone/ethanol mixture (volume ratio = 5/1). NaBr (99.9%) purchased from Aldrich Chemical Co. Inc. (USA) was used without further purification. DTABF₄ was synthesized by exchanging Br⁻ ions for BF₄⁻ ions by adding an equimolar amount of sodium tetrafluoroborate (NaBF₄) into an aqueous solution of DTABr, followed by recrystallization twice from water. The NaBF₄ (98%) purchased from Kanto Kagaku Co., Ltd. (Japan) and was recrystallized once from water and then baked at 170 °C for 7 h under reduced pressure. DTA₂SO₄ was synthesized by exchanging Br⁻ ions for SO₄²⁻ ions by adding a half molar amount of Ag₂SO₄ into an ethanol solution of DTABr. The Ag₂SO₄ (99.5%) was purchased from Wako Pure Chemical Industries and used without further purification. The ethanol solution was ultra-sonicated for 9 h and then AgBr precipitates were removed with 0.2-μm pore filters. A white solid salt was yielded from the filtrates, which was then recrystallized twice from methanol/acetone mixtures (volume ratio = 1/4) and dried at 90 °C under reduced pressure. The final product contained a small amount of water, which was estimated to be about 1 wt% with a Kahl–Fischer moisture meter and taken into account in the sample preparation. C₁₀E₅ was purchased from Bachem AG (Switzerland) and purified by the three-phase extraction technique [40]; the weight ratio of water/hexane/surfactant was 42.5/42.5/15. Ultrapure MilliQ water was used to prepare all surfactant solutions.

The purity of these materials was confirmed by observing that the surface tension of aqueous solutions reached a constant value that agreed with literature values and did not change over time. Furthermore, for all surfactants, it was confirmed that the surface tension vs. molality curves did not show a minimum around the critical micelle concentrations.

Thermodynamic equations for surface tension analysis

The surface tension was analyzed as follows. Let us consider a system composed of air and aqueous solution of DTABr and NaBr. The total differential of the surface tension γ is expressed as a function of temperature T, pressure p, and the the electrochemical potential of the ions α, ${\overset{\sim }{\mu}}_{\alpha }\ \left(\alpha ={\mathrm{DTA}}^{+},{\mathrm{Na}}^{+},{\mathrm{Br}}^{-}\right)$ by

$$ \mathrm{d}\gamma =-{s}^{\mathrm{H}}\mathrm{d}T+{v}^{\mathrm{H}}\mathrm{d}p-{\varGamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}\mathrm{d}{\overset{\sim }{\mu}}_{{\mathrm{DTA}}^{+}}-{\varGamma}_{{\mathrm{Na}}^{+}}^{\mathrm{H}}\mathrm{d}{\overset{\sim }{\mu}}_{{\mathrm{Na}}^{+}}-{\varGamma}_{{\mathrm{Br}}^{-}}^{\mathrm{H}}\mathrm{d}{\overset{\sim }{\mu}}_{{\mathrm{Br}}^{-}}, $$

(1)

where s^H, v^H, and ${\varGamma}_{\alpha}^{\mathrm{H}}$ are respectively the surface excess entropy, volume, and surface density of ion α, which are defined with respect to the two dividing planes making the excess numbers of moles of air and water zero simultaneously [35, 41]. Introducing the electrochemical potential gives

$$ {\overset{\sim }{\mu}}_{\alpha }={\mu}_{\alpha }+{z}_{\alpha } F\phi, $$

(2)

where μ _α is the chemical potential of ion α, F is the Faraday constant, ϕ is the electrical potential, and z _α is the valence of ion α. The electroneutrality condition in the adsorbed film is

$$ {\varGamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}+{\varGamma}_{{\mathrm{Na}}^{+}}^{\mathrm{H}}={\varGamma}_{{\mathrm{Br}}^{-}}^{\mathrm{H}}, $$

(3)

which when inserted into Eq. 1 gives

$$ \mathrm{d}\gamma =-{\varGamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}\mathrm{d}{\mu}_{{\mathrm{DTA}}^{+}}-{\varGamma}_{{\mathrm{Na}}^{+}}^{\mathrm{H}}\mathrm{d}{\mu}_{{\mathrm{Na}}^{+}}-{\varGamma}_{{\mathrm{Br}}^{-}}^{\mathrm{H}}\mathrm{d}{\mu}_{{\mathrm{Br}}^{-}}. $$

(4)

at a given T and p. Assuming that the bulk solution is ideally dilute as in the present case, Eq. 4 can be rewritten as a function of the molality of DTABr m₁ and NaBr m₂ as

$$ \mathrm{d}\gamma =- RT{\left[{\varGamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}\left(\frac{1}{m_1}+\frac{1}{m_1+{m}_2}\right)+{\varGamma}_{{\mathrm{Na}}^{+}}^{\mathrm{H}}\left(\frac{1}{m_1+{m}_2}\right)\right]}_{T,p,{m}_2}\mathrm{d}{m}_1- RT{\left[{\varGamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}\left(\frac{1}{m_1+{m}_2}\right)+{\varGamma}_{{\mathrm{Na}}^{+}}^{\mathrm{H}}\left(\frac{1}{m_2}+\frac{1}{m_1+{m}_2}\right)\right]}_{T,p,{m}_1}\mathrm{d}{m}_2. $$

(5)

The surface density of DTA⁺ is evaluated from the dependence of γ on m₁ and m₂ with the use of

$$ {\varGamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}=-\left[\left(1+{X}_2\right){\left(\frac{\partial \gamma }{\partial \ln {m}_1}\right)}_{T,p,{m}_2}-{X}_1{\left(\frac{\partial \gamma }{\partial \ln {m}_2}\right)}_{T,p,{m}_1}\right]/2\mathrm{R}T, $$

(6)

where X₂ = m₂/(m₁ + m₂). In the absence of added salt, Eq. 6 is reduced to

$$ {\varGamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}=-\frac{m_1}{2 RT}{\left(\frac{\partial \gamma }{\partial {m}_1}\right)}_{T,p}. $$

(7)

Surface quasi-elastic light scattering method

The SQELS apparatus was constructed based on that reported by Sakai et al. [42,43,44] as schematically illustrated in Fig. 1. The surface wave “ripplon” that was naturally excited by thermal motion of constituent molecules was detected and analyzed. The principles and details of the SQELS method have been described elsewhere [36, 42].

Fig. 1

Optical arrangement of the surface quasi-elastic light scattering (SQELS) device. Solid and dashed lines represent incident and reference lights, respectively

The incident ray of YAG (Y₃Al₅O₁₂) laser light (532 nm, 200 mW) was diffracted by the periodic surface displacement owing to the ripplon. The wavenumber k of the ripplon to be detected was determined by the scattering angle through the Bragg’s condition (see Fig. 2):

$$ k=K\sin \theta, $$

(8)

where K denotes the wavenumber of incident light and θ is the scattering angle. To gain sufficient signal intensity, we used the heterodyne method [42, 43]. A cell of the sample solutions was placed at the measurement point shown in Fig. 1 and covered with a thermostatic stage fitted with Peltier devices for temperature control. The difference of the frequencies between the incident and the scattered light was monitored in the form of a power spectrum (frequency domain) with the use of a spectrum analyzer. Note that the frequency of the reference light was shifted by 85 kHz by an acoustic optic (AO) modulator before it met the incident light, as shown in Fig. 1. This shift was introduced to remove the signal noise within the low frequency region. The apparatus set up here resulted in three peaks in the power spectrum; the central peak was related to the instruments and the two side peaks originated from the ripplon with k satisfying Eq. 8.

Fig. 2

Image of the light scattering from ripplon at the solution surface. Solid and dashed lines represent incident and reference lights, respectively

The main parts of the SQELS apparatus were as follows: YAG laser (Verdi-V5, Coherent), Acoustic Optic Modulator (TEM-85-2-532, Brimrose), Photomultiplier (C5331-11, Hamamatsu), and Spectrum Analyzer (MS2661C, Anritsu). The SQELS measurements were performed under atmospheric pressure for the respective systems as follows: (A), (C), and (D); at 298.15 K as a function of m₁, (B); at 298.15 and m₂/mmol kg⁻¹ = 5 and 10 as a function of m₁.

Dispersion relation for air/water interface

A liquid surface is always fluctuating owing to thermal motion of its constituent molecules. The vertical displacement of a given point of the surface r at a time t can be written as [36]:

$$ u\left(\mathbf{r},t\right)=\sum \limits_{\mathbf{k}}u\left(\mathbf{k}\right)\exp \left[i\left(\mathbf{k}\bullet \mathbf{r}-\omega (k)t\right)\right], $$

(9)

which is a sum of Fourier components at all wave vectors k. Taking account of that the surface tension γ is considered to be the main restoring force for surface transversal displacement and surface dilational viscoelasticity ε^∗(ω) works against the surface longitudinal displacement, where ω is the real angular frequency of the surface wave. We obtain the following dispersion relation for an air/water interface in which both the density and viscosity of air are negligible [36, 45];

$$ D\left(k,\omega \right)=L\left({\varepsilon}^{\ast };k,\omega \right)\times T\left(\gamma; k,\omega \right)+C\left(\eta, \rho \right)=0, $$

(10)

where

$$ L\left({\varepsilon}^{\ast };k,\omega \right)={\varepsilon}^{\ast }{k}^2+ i\omega \eta \left(k+\mu \right), $$

(11)

$$ T\left(\gamma; k,\omega \right)=\gamma {k}^2+ i\omega \eta \left(k+\mu \right)-\frac{\rho {\omega}^2}{k}, $$

(12)

and

$$ C\left(\eta, \rho \right)={\left[ i\omega \eta \left(k-\mu \right)\right]}^2, $$

(13)

respectively. Here, C(η, ρ) depends on density ρ and viscosity η of the aqueous phase and couples with the longitudinal quantity L(ε^∗; k, ω) given by Eq. 11 and the transverse one T(γ; k, ω) given by Eq. 12. Here, μ is the reciprocal of the penetration depth of the surface velocity field defined by

$$ {\mu}^2={k}^2+ i\omega \rho /\eta . $$

(14)

In the classical dispersion relation, transversal waves are coupled with longitudinal ones and propagation characteristics of both waves depend on ε^∗(ω), although the dependence of transversal wave is much weaker than that of longitudinal ones. It is known that the coupling between transversal and dilational waves becomes more effective at resonance for which the frequencies of two modes are close to each other, which generally occurs when the ratio ε/γ ≈ 0.16 [23]. As shown later in Fig. 19, ε/γ changed from approximately 0.05 to 0.45, and good coupling [4] was observed at ε/γ values where the damping coefficient became a maximum.

Furthermore, ε^∗(ω) is related to the variation of surface tension δγ with surface area δA and is defined as follows:

$$ {\varepsilon}^{\ast}\left(\omega \right)\equiv A\frac{\delta \gamma}{\delta A}. $$

(15)

Because of the dissipative effect within the adsorbed film, ε^∗(ω) can be written as a complex number ε^∗(ω) = ε(ω) + iωκ(ω), where ε(ω) and κ(ω) are the surface dilational elasticity and viscosity, respectively. The complex modulus ε^∗(ω) contains two-dimensional isotropic compression (dilation), in-plane share and out-of-plane (transversal) share contributions. Because the order of the wavelength is sufficiently large compared with that of the amplitude of the surface wave of the surfactant aqueous solution, the out-of-plane share contribution is negligible [4, 36, 45]. Furthermore, the in-plane share contribution is also much smaller than the dilational one and negligible when the adsorbed film is fluid-like, such as a gaseous or an expanded film state.

For pure liquids without surface active substances, the surface dilational viscoelasticity term also disappears (ε = κ = 0) and when the complex frequency of the transversal wave is expressed as ω + iΓ, we obtain two limiting solutions of the dispersion equation at a wavenumber of k; one is Kelvin’s law for the frequency

$$ {\omega}^2\approx \frac{\gamma }{\rho }{k}^3. $$

(16)

and the other is Stokes’ law for the damping caused by the viscous friction with the bulk liquid:

$$ \varGamma \approx \frac{2\eta }{\rho }{k}^2. $$

(17)

Surface rheological fitting analysis

The spectrum obtained from SQELS method was compared with the theoretical power spectrum equation of Kramer [46] for transversal wave in the presence of air given by

$$ {P}_{\mathrm{T}}\left(k,\omega \right)=\frac{k_{\mathrm{B}}T}{\pi \omega}\operatorname{Im}\left[\frac{L\left({\varepsilon}^{\ast };k,\omega \right)}{D\left(k,\omega \right)}\right], $$

(18)

where k_B is the Boltzmann constant. Equation 18 corresponds to the inverse Fourier transform of the auto correlation function of the surface wave. Because side peaks originating in a ripplon are generally broadened owing to instrumental considerations in the SQELS technique [12], some calibration processes are required. In our case, we determined the wavenumber of the ripplon k_rip and the instrumental resolution β on the basis of the SQELS results and the reference values of pure water. The spectrum from a pure water surface should be free from surface viscoelasticity, and thus, instrumental parameters such as k_rip and β can be determined. The reference values of fitting parameters of pure water at T = 298.15 K were ρ = 997.047 kg m⁻³, η = 0.89 mPa ∙ s, and γ = 71.96 mN m⁻¹, respectively. The SQELS spectrum from pure water provided a larger width than that theoretically calculated from Eq. 17 and it was fitted by k_rip and β with ε = κ = 0 in Eq. 18. Hence, the frequency ω and damping coefficient Γ simultaneously satisfied Kelvin’s law of Eq. 16 and Stokes’ law of Eq. 17, respectively, for the various wavenumbers k_rip from 70 to 320 cm⁻¹.

Because the instrumental resolution was dependent on the wavenumber, the uncertainty of the wavenumber δk_rip was taken into account by the following equations [47]:

$$ {P}_{\mathrm{exp}}\left(k,\omega \right)=A\int \rho \left(k,{k}^{\prime}\right){P}_{\mathrm{T}}\left({k}^{\prime },\omega \right)\mathrm{d}{k}^{\prime }, $$

(19)

and

$$ \rho \left({k}^{\prime }-k\right)=\frac{1}{\sqrt{\uppi}\beta}\exp \left[-\frac{{\left({k}^{\prime }-k\right)}^2}{\beta^2}\right]. $$

(20)

where A is an arbitrary constant and β is the uncertainty of wavenumber of ripplon δk_rip. Equations 19 and 20 imply that the experimentally obtained spectrum P_exp(k, ω) is a convolution of a theoretical function and a Gaussian function of wavenumber.

In principle, the surface rheological fitting analysis by Eqs. 10 to 14 and Eqs. 18 to 20 yields the surface dilational viscoelasticity ε^∗(ω) of the surfactant aqueous solutions. However, to determine the surface dilational viscoelasticity more accurately, the equilibrium surface tension γ of the aqueous solutions measured separately from the SQELS studies was used as a fixed parameter in Eq. 18. Thus, the surface dilational elasticity ε and viscosity κ were determined as the main and sub fitting parameters in Eq. 18, respectively. Furthermore, Cicuta et al. [12] indicated that this treatment provides a general justification for the correct recent practice of SQELS experiments, which is very different from earlier ones where the four quantities of surface dilational elasticity, surface dilational viscosity, transverse share elasticity (surface tension), and transverse share viscosity were usually chosen as physical parameters [20, 48, 49]. The surface rheological fitting analysis was performed by a fitting program coded in FORTRAN that was constructed with support from Ohmasa et al. [47]. The values of ε and κ were respectively determined to be in the range of 0 ≤ ε(ω) ≤ 100 mN m⁻¹ and −1000 ≤ κ(ω) ≤ 1000 nNs m⁻¹. This is because the shape of the power spectrum of scattered light tends to become independent of the surface dilational viscoelasticity as it becomes larger [12].

The SQELS measurements were performed at fixed wavenumbers for all surfactant solutions: k_rip ≈ 200 cm⁻¹ and δk_rip ≈ 20 cm⁻¹, which were determined from the SQELS measurements of pure water.

Phenomenological fitting analysis

From the phenomenological fitting analysis of the SQELS spectrum, both the angular frequency ω and damping coefficient Γ, ω + iΓ, of ripplons were evaluated. In our experimental setup, the central peak and two side peaks could be accurately described by a Gaussian function and by a Voigt function, respectively. The experimentally obtained spectrum P_exp(ω) is a convolution of Lorentzian function and Gaussian function of frequency given by

$$ {P}_{\mathrm{exp}}\left(\omega \right)=A\int \rho \left(\omega -{\omega}^{\prime}\right){P}_P\left(k,{\omega}^{\prime}\right)\mathrm{d}{\omega}^{\prime }, $$

(21)

where P _P (k, ω^′) is the phenomenological power spectrum equation:

$$ {P}_P\left(k,{\omega}^{\prime}\right)=\Gamma \left\{\frac{1}{{\left[\left({\omega}^{\prime }-{\omega}_{\mathrm{AO}}\right)+{\omega}_{\mathrm{rip}}\right]}^2+{\Gamma}^2}+\frac{1}{{\left[\left({\omega}^{\prime }-{\omega}_{\mathrm{AO}}\right)-{\omega}_{\mathrm{rip}}\right]}^2+{\Gamma}^2}\right\}, $$

(22)

and

$$ \rho \left(\omega -{\omega}^{\prime}\right)=\mathrm{F}.\mathrm{T}.\left[\exp \left\{-\frac{{\left(\left(\omega -{\omega}_{\mathrm{AO}}\right)-{\omega}^{\prime}\right)}^2}{2{\beta}^2}\right\}\right], $$

(23)

where ω_AO is the frequency shift introduced by the AO modulator and F.T. means Fourier transformation. In Eq. 23, the resolution β was the value of δω_rip calculated using δk_rip by

$$ \delta {\omega}_{\mathrm{rip}}=\frac{3}{2}\sqrt{\frac{\gamma }{\rho }}{k_{\mathrm{rip}}}^{\frac{1}{2}}\delta {k}_{\mathrm{rip}}. $$

(24)

Equation 24 is derived from Kelvin’s law given by Eq. 16.

Results and discussion

SQELS and ECW studies of short-chain phospholipid mixtures (DC₈PC+DC₆PC)

The aqueous lipid solutions were used to form monolayers because both lipids have short chains and are thus soluble in water. Their surface tension was measured as a function of the total concentration of the two lipids at fixed mixing ratios and the surface concentrations were evaluated by thermodynamic equations as described in our recent paper [24]. Figure 3a shows the SQELS spectrum at given surface mole fractions of DC₈PC, defined by ${X}_2^{\mathrm{H}}={\varGamma}_{\mathrm{DC}8\mathrm{PC}}^{\mathrm{H}}/\left({\varGamma}_{\mathrm{DC}6\mathrm{PC}}^{\mathrm{H}}+{\varGamma}_{\mathrm{DC}8\mathrm{PC}}^{\mathrm{H}}\right)$ at 35 mNm⁻¹. As mentioned in the experimental section, the angle of the incident light was fixed and accordingly the wavenumber of the detected transversal wave was fixed, which means that the ripplon frequency is proportional to the ripplon wave velocity. Figure 3b shows that the ripplon (angular) frequency ω_rip = 2πf was approximately 250 kHz and decreased by 35% compared with that of pure water surface owing to the adsorption of the lipid mixtures at approximately 2.8 μmol m⁻² and γ = 35 mNm⁻¹ [24]. Conversely, the damping coefficient of the adsorbed film was almost the same as that of pure water surface (dotted line), as shown in Fig. 3c.

Fig. 3

SQELS results of the DC₆PC-DC₈PC mixtures at a surface tension of 35 mN m⁻¹. a SQELS spectra at ${X}_2^{\mathrm{H}}$ = (1) 0, (2) 0.1, (3) 0.2, (4) 0.3, (5) 0.6, (6) 0.8, and (7) 1. b ripplon frequency f and c damping coefficient Γ plotted against surface mole fraction of DC₈PC ${X}_2^{\mathrm{H}}$

The dilational elasticity at approximately 250 kHz from the SQELS is plotted against ${X}_2^{\mathrm{H}}$ together with that at 0.2 kHz from the ECW in Fig. 4, where we note the following points. The former seems to pass through a very shallow minimum; however, the latter evidently passes through a maximum [24]. There exists a clear difference in the dilatational elasticity between the longer and shorter chain lipids at 0.2 kHz; however, this difference almost disappears at approximately 250 kHz. Moreover, the maximum, which appeared at 0.2 kHz was not observed at 250 kHz. The van der Waals interactions among the hydrophobic chains in the oriented adsorbed layer were stronger for DC₈PC than for DC₆PC and a staggered structure of the adsorbed film was indicated by examination of the minimum of the excess Gibbs energy of adsorption g^{H, E} (dashed curve in Fig. 4) around ${X}_2^{\mathrm{H}}=0.3$ [24]. Hence, molecular interactions in the adsorbed films clearly influenced the relaxation process on the order of several milliseconds (ECW); however, these interactions were less influential on the order of several hundreds of microseconds (SQELS).

Fig. 4

Comparison between the results of SQELS and ECW for the DC₆PC-DC₈PC mixtures. Dilational elasticity ε vs. ${X}_2^{\mathrm{H}}$ at surface tension of 35 mN m⁻¹. f (kHz) = 40 (SQELS, ○) and 0.2 (ECW, ●). Dashed curve represents excess Gibbs energy of adsorption g^{H, E}

We next examined whether the relaxation processes owing to the staggered structure formation was of the order of milliseconds or not. The dilational viscoelasticity was determined as a function of frequency by ECW. The storage and loss modulus, ε and ωκ, are plotted against the frequency at four surface mole fractions in Fig. 5a. The storage modulus increased with increasing frequency; however, the loss modulus showed a maximum from 150 to 300 Hz. When the frequency of surface deformation is comparable to the time scale of the dynamic process, the storage modulus curve has an inflection and the loss modulus has a maximum around the characteristic frequencies, as schematically illustrated in Fig. 5b [3, 4]. Therefore, we confirmed that the characteristic frequencies for the mixed adsorbed films of the DC₈PC+DC₆PC system likely lie in the range from 150 to 300 Hz. At 200 Hz, i.e., the maximum of the storage modulus, which appears at ${X}_2^{\mathrm{H}}$ = 0.5 in Fig. 4, is between 150 and 300 Hz. This surface mole fraction is closed to that at the minimum excess Gibbs energy attributable to the staggered structure of the adsorbed film. We concluded that the characteristic frequency of the staggered structure formation is of the order of a few hundred Hz and the characteristic time scale for the relaxation of the staggered structure is of the order of milliseconds. Ravera et al. demonstrated that the storage modulus owing to diffusion of molecules between the bulk and surface shows an inflection and the corresponding loss modulus shows a maximum at a very low frequency of the order of 10⁻² Hz [3]. Conversely, relaxations such as surfactant reorientation and aggregation in the surface have much higher characteristic frequencies. For example, for adsorption of tetraethylene glycol monodecyl ether (C₁₀E₄), the characteristic frequency for diffusion was approximately 0.4 Hz, while that for surfactant reorientation was suggested to be 800 Hz. Judging from these observations, the staggered structure formation was minimally influential on the dilational elasticity at 250 kHz, which is in accord with our observations in Fig. 4 that the dilational elasticity does not change greatly with the surface composition at 250 kHz.

Fig. 5

Frequency dependency of dilational elasticity (storage modulus) ε and loss modulus. a experimental results of the DC₆PC-DC₈PC mixtures at ${X}_2^{\mathrm{H}}$ = 0 (○), 0.2 (●), 0.5 (△), and 1 (▲). b schematic curves [Ref. 4]

Diffusion of small chemical species might show a much faster relaxation process, compared with diffusion of surfactant molecules between the bulk and surface, and surfactant reorientation and aggregation at the surface. Hence, counterions of ionic surfactants are appropriate candidates because of their typically small size. Furthermore, these molecules move back and forth between the Stern layer and diffuse layer regions in the electrical double layer. Another candidate is water molecules because these undergo cycles of hydration and dehydration around the hydrophilic groups of surfactants. For this purpose, nonionic surfactants with ethylene oxide units are appropriate. On the basis of these ideas, the SQELS method was applied to cationic surfactants with different counter ions and nonionic ones with ethylene oxide units.

SQELS studies of cationic surfactants with different counterions

Now, let us show the dilational elasticity of aqueous solutions of (A) DTABr, (B) DTABr in the presence of NaBr, and (C) DTABF₄ and DTA₂SO₄. Studies on (A) and (B) provide information about what kind of relaxation processes influence dilational elasticity at high frequencies of a few hundred kHz. The Lucassen and van den Temple (LT) and modified LT (MLT) models for the dilational viscoelasticity [17, 37,38,39] are introduced and used to investigate the kind of information that can be derived from the models. Studies on system (C) offer information about whether the degree of counter ion binding to the surfactant ion is influential or not on dilational viscoelasticity, because ${\mathrm{BF}}_4^{-}$ is more hydrophobic than Br⁻ and ${\mathrm{SO}}_4^{2-}$ is divalent. Furthermore, the distribution of Br⁻ in the electrical double layer of the adsorbed monolayers has been investigated by means of the total reflection XAFS method in our studies [33, 34].

Dodecyltrimethylammonium bromide

The surface tension γ vs surfactant concentration m₁ curves are displayed up to the concentrations just below the critical micelle concentrations in Fig. 6, and the corresponding curves of the surface density of surfactant ions ${\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$ are given in Fig. 7. Break points on the surface tension curve and corresponding discontinuous changes on the surface density curve observed in the absence of NaBr are often found for various kinds of surfactants and have been attributed to a phase transition of adsorbed films from a gaseous to an expanded state [25, 26].

Fig. 6

Surface tension γ vs. molality m₁ curves of the DTABr-NaBr mixtures at m₂ (mmol kg⁻¹) = 0 (○), 5 (●), and 10 (△)

Fig. 7

Surface density ${\varGamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$ vs. molality m₁ curves of the DTABr-NaBr mixtures at m₂ (mmol kg⁻¹) = 0 (○), 5 (●), and 10 (△)

The SQELS spectra of DTABr at various concentrations are shown in Fig. 8. All the spectra were symmetrical with respect to the top of the central peak at a frequency shift Δf = 0 and their shape changed regularly with surfactant concentration. The angular wave frequency of the side peaks ω and the damping coefficients Γ of ripplon were evaluated by phenomenological fitting with Eq. 22 and the ratio ω/ω_water and Γ/Γ_water are plotted against concentration together with those in the presence of NaBr in Fig. 9. The value of ω/ω_water clearly changed continuously but showed a break at a concentration (${m}_1^{\mathrm{br}}\Big)$, which almost coincided with the phase transition concentration determined from the surface tension measurements in the absence of NaBr. In the presence of NaBr, although a break was not observed on the surface tension curves, the ω/ω_water curves showed a break, which suggests that the surface wave properties were more influenced by the phase transition compared with surface tension [50]. Furthermore, Γ/Γ_water values pass through a maximum at a concentration ${m}_1^{\mathrm{mx}}$ slightly above ${m}_1^{\mathrm{br}}$; the combinations of $\left({m}_1^{\mathrm{br}}/\mathrm{mmol}\ {\mathrm{kg}}^{-1},{m}_1^{\mathrm{mx}}/\mathrm{mmol}\ {\mathrm{kg}}^{-1}\right)$ are roughly (2.0, 2.5), (0.75, 1.0), (0.30, 0.50) for DTABr, DTABr+NaBr (m₂/mmol kg⁻¹ = 5) and DTABr + NaBr (m₂/mmol kg⁻¹ = 10), respectively.

Fig. 8

SQELS spectra of DTABr at m₁ (mmol kg⁻¹) = (1) 0.25, (2) 0.5, (3) 0.75, (4) 1.0, (5)1.25, (6) 1.5, (7) 1.5, (8) 1.75, (9) 2.0, (10) 2.5, (11) 3.0, (12) 5.0, (13) 7.0, (14) 9.0, (15) 11, (16) 13, and (17) 15

Fig. 9

(a) Ripplon frequency ω/ω_water and (b) damping coefficient Γ/Γ_water from phenomenological fitting of the DTABr-NaBr mixtures plotted against molality m₁ at m₂ (mmol kg⁻¹) = 0 (○), 5 (●), and 10 (△)

Although it is clear from the experiments that the breaks on the ω/ω_water curves plotted against surfactant concentration or temperature are attributable to phase transitions of the interfacial films, it remains unclear whether the maximum of Γ/Γ_water stems from such a phase transition or from coupling between the in-plane (dilational and shear) and out-of-plane (capillary) waves. Kizling et al. studied the viscoelastic properties of adsorbed monolayers of dodecylammonium chloride and identified such a break on the ω curve and a maximum on the Γ curve just above ${m}_1^{\mathrm{br}}$ [51], which was essentially the same as our observations for DTABr. They claimed that ω is strongly affected by changes of surface tension and although there is a very small discontinuous change of Γ at the phase transition point, the peak of Γ is obviously rather closely related to the phase transition. Monroy et al. also reported capillary waves of the dodecylammonium chloride solution that the change of slope observed for the ω vs concentration curves followed the variation of equilibrium surface tension; thus, the break on the ω curve corresponds to the surface phase transition [23]. Furthermore, they demonstrated that two maxima appear on the Γ curve at very low concentration and at a concentration slightly above ${m}_1^{\mathrm{br}}$ corresponding to resonance between the capillary and dilational modes [23]. With respect to the Langmuir monolayer of hydrophobin, Aumaitre et al. showed that ω decreases with increasing surface concentration and forms a plateau between surface concentrations of 1.6 and 1.8 mg m⁻². This finding could indicate the coexistence of condensed and liquid domains, whereas the surface pressure remained null up to around 2.2 mg m⁻² [50]. On the basis of these findings, they emphasized that a decrease in ω does not necessarily indicate an increase of surface pressure but could also be a signature of an increase in the dilational modulus. Furthermore, they described that the damping coefficient Γ presents a peak at a concentration of the coexistence region, which corresponds to the maximum coupling between the in-plane and out-of-plane waves, that is, (ε/γ) ~0.16 [20, 23, 36, 52, 53]. Munoz et al. observed a break of ω and Γ, rather than a maximum at the temperature of the 2D liquid-solid phase transition for 1-dodecanol monolayers [54]. Considering our observation that the concentrations ${m}_1^{\mathrm{mx}}$ are different from those at the phase transition ${m}_1^{\mathrm{br}}$ and the dilational elasticity to surface tension ratio (ε/γ) is 0.15–0.16 for DTABr at ${m}_1^{\mathrm{mx}}$ (see Fig. 19), the maximum of Γ can be attributed to coupling of the in-plane and out-of-plane waves [36].

The surface dilational viscoelasticity ε^∗(ω) = ε(ω) + iωκ(ω) was calculated by surface rheological fitting analysis with Eq. 18. Figure 10a shows the dilational elasticity ε(ω) and the ripplon angular frequency ω and Fig. 10b shows the dilational loss modulus ωκ(ω). The value of ε(ω) increased from zero for pure water surface to be in the range of 4–5 mN m⁻¹ and became steady in the gaseous state. In the expanded state, the value initially increased steeply in response to a steep increase in the surface density, then passed through a maximum and decreased gradually despite the continuous increase in surface density. The absolute value of ωκ(ω) was 0–10 mN m⁻¹ and its contribution to ε^∗(ω) was non-negligible compared with the elasticity ε(ω). Because the dilational viscosity has unrealistic negative values, which have been discussed from consideration of the adsorption barrier and applicability of the dispersion equation generally employed [20, 23], the dilational elasticity will be the main focus of this paper.

Fig. 10

Dilational viscoelastic properties of DTABr. (a) dilational elasticity (storage modulus) ε (○), ripplon frequency ω (●), and (b) dilational loss modulus ωκ plotted against molality m₁

The change of dilational elasticity with bulk concentration has been examined in terms of diffusion of surfactant molecules between the monolayer and the underlying aqueous phase with the use of the Lucassen and van den Tempel (LT) model [37, 55, 56]. Molecules in the adsorbed films on the aqueous surfactant solutions are dissolved or adsorbed upon compression or expansion to restore the equilibrium surface concentration. Because the shear coefficients are reasonably assumed to be negligible compared with the dilational coefficients, the complex dilational viscoelasticity ε^∗ is defined as the fluctuation of local surface tension γ _l accompanied by a change of surface area A, described by

$$ {\varepsilon}^{\ast }=A\left(\frac{\partial {\gamma}_l}{\partial A}\right)=\left(\frac{\partial {\gamma}_l}{\partial \ln A}\right). $$

(25)

When any relaxation process is either much faster or much slower than the time scale of experiments τ_exp = ω⁻¹ (in the SQELS study, this is the time scale of detected ripplon), the variation of surface tension accompanied by the surface expansion and contraction is instantaneous, and thus, the surfaces behave as if they are purely elastic. Let us designate ε^∗(ω) = ε(ω) = ε₀ in such purely elastic cases.

Purely elastic cases are realized when two conditions are fulfilled. The first condition is that reorientation of constituent molecules in the monolayer is completed on a time scale of the order of τ_exp = ω⁻¹, and thus, the equilibrium between the local surface tension and the surface density Γ should be very quickly established. The other is that diffusional exchange between the surface and bulk is negligible, and thus, the total amount of adsorbed molecules does not change on a time scale of the order of τ_exp = ω⁻¹, i.e., even a soluble monolayer behaves as if it is an insoluble one with respect to this time scale. Therefore, ε₀ can be written as

$$ {\varepsilon}_0=\left(\frac{\partial {\gamma}^{\mathrm{eq}}}{\partial \ln {A}^{\mathrm{eq}}}\right)=-\left(\frac{\partial {\gamma}^{\mathrm{eq}}}{\partial \ln {\varGamma}^{\mathrm{eq}}}\right) $$

(26)

and evaluated using the equilibrium relation between the surface tension vs surface density Γ^eq (=${\varGamma}_1^{\mathrm{H}}$ in this paper, ${\varGamma}_1^{\mathrm{H}}$ denotes ${\varGamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$ or ${\varGamma}_{\mathrm{C}10\mathrm{E}5}^{\mathrm{H}}$ in the following discussion). The variable ε₀ is commonly known as the Gibbs elasticity. There is a well-known procedure based on the Frumkin equation of state for calculating ε₀ and it often satisfactorily describes the variation of surface pressure with the bulk concentration. When the Frumkin equation was applied to the DTAB system, Stenvot et al. reported ε₀ values of 280 mNm⁻¹ at 10 mM [18], which is comparable to the highest values for DTAB near CMC in our calculations, as shown later in Fig. 11. Now, let us rewrite Eq. 25 as

$$ {\varepsilon}^{\ast }=\left(\frac{\partial {\gamma}_l}{\partial \ln {\varGamma}_l}\right)\left(\frac{\partial \ln {\varGamma}_l}{\partial \ln A}\right), $$

(27)

and assume that the equilibrium between the local surface tension and the surface density Γ is rapidly established as $\left(\frac{\partial {\gamma}_l}{\partial \ln {\varGamma}_l}\right)\approx \left(\frac{\partial {\gamma}_l}{\partial \ln {\varGamma}_l}\right)$. Whether this assumption is appropriate or not is another important issue, particularly for high-frequency experiments like SQELS. With this assumption, we obtain

$$ {\varepsilon}^{\ast }=-{\varepsilon}_0\left(\frac{\partial \ln {\varGamma}_l}{\partial \ln A}\right). $$

(28)

Fig. 11

(a) Comparisons among the dilational elasticities obtained from experiments and models for DTABr and (b) enlarged view. ε from experiments (○), Gibbs elasticity ε₀ (●), LT model ε_LT (▲), MLT model ε_MLT (△), effective LT model ε_{LT, eff} (×)

When the reorientation processes of constituent molecules in the monolayer is completed within a time scale of the order of ω⁻¹, barriers against both adsorption and desorption are absent, and the molecular exchange is controlled solely by diffusion between the subsurface and bulk solution. The factor $\left(\frac{\partial \ln {\varGamma}_l}{\partial \ln A}\right)$ can be derived from the first and second laws of Fick to give the dilational elasticity ε_LT and viscosity κ_LT from the LT model as

$$ {\varepsilon}_{\mathrm{LT}}={\varepsilon}_0\frac{1+\tau }{1+2\tau +2{\tau}^2}, $$

(29)

and

$$ \omega {\upkappa}_{\mathrm{LT}}={\varepsilon}_0\frac{1}{1+2\tau +2{\tau}^2}. $$

(30)

Here, τ is a dimensionless parameter given by

$$ \tau =\sqrt{D/2\omega}\left(\mathrm{d}{m}_1/\mathrm{d}{\varGamma}_1^{\mathrm{H}}\right), $$

(31)

where D represents the diffusion coefficient of surfactant in the bulk solution [37]. This is the equation of the LT model. Because the characteristic frequency of diffusion ω_dif and thus the characteristic diffusion time τ_dif are given by [23]

$$ {\omega}_{\mathrm{dif}}={\tau_{\mathrm{dif}}}^{-1}=\left(D/2\right){\left(\mathrm{d}{m}_1/\mathrm{d}{\varGamma}_1^{\mathrm{H}}\right)}^2, $$

(32)

where τ expresses the ratio of the time scale of the relevant experiment ω⁻¹ to that of the diffusion of surfactant molecule ω_dif⁻¹ as

$$ \uptau =\sqrt{\omega_{\mathrm{dif}}/\omega }=\sqrt{\tau_{\mathrm{exp}}/{\tau}_{\mathrm{dif}}}. $$

(33)

Furthermore, when an adsorption barrier is taken into account in the LT model (the modified LT model, MLT), Eq. 29 is converted to

$$ {\varepsilon}_{\mathrm{MLT}}={\varepsilon}_0\frac{1+\left(1+2\upalpha \right)\tau +2{\alpha}^2{\tau}^2}{1+2\left(1+\upalpha \right)\tau +2\left(1+{\alpha}^2\right){\tau}^2}, $$

(34)

where $\alpha =\omega /\left({k}_{\mathrm{d}}/{\Gamma}_1^{\mathrm{H}}\right)$ and k_d is the rate constant of desorption in the presence of adsorption-desorption barriers [17].

Summarizing the limiting properties of ε_LT and ε_MLT, we obtain

• Case 1: Diffusion requires a much longer time than the experiment time scale (τ_dif ≫ τ_exp) or the experiment frequency is much higher than diffusion (ω ≫ ω_dif): τ → 0, ε_LT → ε₀

• Case 2: The opposite of case 1, (τ_dif ≪ τ_exp) or (ω ≪ ω_dif):

$$ \uptau \to \mathrm{large},{\varepsilon}_{\mathrm{LT}}\to 0 $$

• Case 3: The adsorption barrier is very low $\left[\left({k}_{\mathrm{d}}/{\Gamma}_1^{\mathrm{H}}\right)\gg \omega \right]$:

$$ \left(\upalpha \right)\to 0,{\varepsilon}_{\mathrm{MLT}}\to {\varepsilon}_{\mathrm{LT}} $$

• Case 4: The experimental frequency is very high $\Big[\left({k}_{\mathrm{d}}/{\Gamma}_1^{\mathrm{H}}\right)\ll \omega$]:

$$ \upalpha \to \mathrm{large},{\varepsilon}_{\mathrm{MLT}}\to {\varepsilon}_0. $$

Now, let us examine whether the LT and MLT models can describe the dilational elasticity given in Fig. 10a. Figure 11 shows the ε and ε₀ vs concentration curves, where ε₀ was evaluated from the equilibrium surface pressure vs mean area curve. Because ε is slightly larger than ε₀ in the low concentration range (see Fig. 11b), neither ε_LT nor ε_MLT can trace ε entirely in principle. Nevertheless, we tried to apply Eqs. 29 and 34 to the experimental ε vs concentration data using the diffusion coefficients of the surfactant in the aqueous phase D = 6 × 10⁻¹⁰m² s⁻¹ and the rate constant of desorption k_d = 1.68 × 10⁻⁴mol m⁻² s⁻¹ [57,58,59]. The results are also given in Fig. 11; ε_LT was lower than ε₀ and became close to ε. Thus, there may be some contributions from surfactant ion diffusion from the bulk to the subsurface on the dilational elastic properties. However, ε_LT remained much higher than ε and could not follow the ε values even in the high concentration region, where the effects of diffusion are expected to be more influential. Furthermore, the ε_MLT values were almost equal to ε₀ and the presence of a barrier to the adsorption-desorption process of surfactant ions has little influence on the dilational elastic behavior (i.e., case 4). These findings indicate that at the higher ripplon frequency detected by the present SQELS method, that is, ω of 200–400 kHz (see Fig. 10), the relaxation processes that influence the dilational properties are neither diffusion of surfactant from the bulk phase to the subsurface nor an adsorption-desorption of surfactant ions between the adsorbed layer and subsurface. The actual relaxation processes occur at the interfacial region.

Because the chemical structure of surfactants and counterions is very simple, relaxation caused by conformational changes is unlikely. Furthermore, the lateral diffusion of surfactant molecules in a monolayer is known to be very slow, e.g., the diffusion coefficient is of the order of 10⁻¹¹ m² s⁻¹ [1]. Thus, let us assume some other possible diffusion processes are related to the counterions with an effective diffusion constant D_E within the scheme of the LT model as

$$ {\varepsilon}_{\mathrm{LT},\mathrm{eff}}={\varepsilon}_0\frac{1+{\tau}_{\mathrm{eff}}}{1+2{\tau}_{\mathrm{eff}}+2{\tau_{\mathrm{eff}}}^2}, $$

(35)

where

$$ {\tau}_{\mathrm{eff}}=\sqrt{\omega_{\mathrm{dif},\mathrm{eff}}/\omega }, $$

(36)

and

$$ {\omega}_{\mathrm{dif},\mathrm{eff}}={\tau_{\mathrm{dif}.\mathrm{eff}}}^{-1}=\left({D}_{\mathrm{E}}/2\right){\left(\mathrm{d}{m}_1/\mathrm{d}{\varGamma}_1^{\mathrm{H}}\right)}^2. $$

(37)

In this case, D_E is likely related to the diffusion processes of chemical species other than surfactant ions. These processes do not occur between the subsurface and bulk, but rather within the interfacial region. It is clear in Fig. 11 that the value of ε_{LT, eff} traces the experimental value of ε more closely than those of ε₀, ε_LT and ε_MLT. However, there is a small deviation of ε_{LT, eff} from ε at lower concentrations. The D_E value of the best fit was 7.12 × 10⁻⁸ m² s⁻¹. It should be noted that this value is almost hundred times as high as D and comparatively close to the diffusion coefficient of bromide ions in aqueous solution 2.1 × 10⁻⁹ m² s⁻¹ and water 2.30 × 10⁻⁹ m² s⁻¹ [60].

In our previous studies on the solvation structure of bromide ions in adsorbed DTABr monolayers by means of the total reflection XAFS, we demonstrated that there are essentially two kinds of bromide ions; ions of one group are hydrated by six water molecules (designated as free-Br), which exist in the diffuse layer. The second group includes those that are dehydrated by approximately three water molecules from free-Br and form ion pairs with cationic head groups of the surfactant monolayer (designated as bound-Br) existing in the Stern layer. In addition, the ratio of bound-Br defined by χ = bound-Br/(free-Br + bound-Br) can be approximated by $\upchi =0.27{\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$ in the expanded film of DTABr for pure DTABr and also in the presence of added NaBr of 10 mmol kg⁻¹ [61]. The lateral diffusion of surfactant molecules in the monolayer is very slow compared with the time scale of detecting ripplons of ω⁻¹ [1]. Thus, the surface dilation by capillary waves continually produces regions with a higher local surface density (HLSD) of a lower local surface tension and regions with a lower surface density (LLSD) of a higher local surface tension, relative to the average surface density ${\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$. The counter ions in the LLSD regions prefer to migrate into the diffuse layer; those in the diffuse layer diffuse onto the surfactant cationic head groups in the HLSD region. This effect may enhance the diffusion of counterions in the electrical double layer and raise the D_E value from the usual value of 2.1 × 10⁻⁹ m² s⁻¹ to the observed value of 7.12 × 10⁻⁸ m² s⁻¹. Therefore, it is likely that a kind of Marangoni effect is taking place in the interfacial region on a very short time scale, which is different from the two typical types of mechanism related to the Marangoni effect (namely, adsorption/desorption of surfactants from/to bulk solution and the lateral diffusion within the adsorbed layer).

Let us investigate whether the addition of inorganic electrolyte influences the dilational elasticity and the diffusion coefficient. The dilational elasticity ε and the best fit of ε_{LT, eff} using Eq. 35 are plotted against the surface density in Fig. 12. Some distinctive features should be noted. First, the D_E values for the best fit of ε_{LT, eff} were 7.12 × 10⁻⁸ m² s⁻¹, 1.06 × 10⁻⁸ m² s⁻¹, and 1.99 × 10⁻⁸ m² s⁻¹ at m₂/mmol kg⁻¹= 0, 5, and 10, respectively. Second, the $\varepsilon\ \mathrm{vs}\ {\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$ curve shifted to the right as the NaBr concentration was increased. Third, the maximum values of ε are very close to each other but the surface density at the maximum ε, ${\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H},\max }$, increases as m₂ increases.

Fig. 12

Dilational elasticity vs. surface density ${\varGamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$ of DTABr. ε at m₂ (mmol kg⁻¹) = 0 (●), 5 (▲), and 10 (■). ε_{LT, eff} at m₂ (mmol kg⁻¹) = 0 (×), 5 (*), and 10 (+). Solid curves represent a guide to the eyes for ε_{LT, eff}

Figure 12 also shows that ε_{LT, eff} fits ε very well at m₂/mmol kg⁻¹= 10 but the deviation of ε_{LT, eff} from ε becomes larger with decreasing m₂. Assuming that diffusion of counterions (ε_{LT, eff}) and the barrier against the diffusion of counterions (ε_{LT, bar}) mainly contributes to ε, that is, ε ≈ ε_{LT, eff} + ε_{LT, bar}, Fig. 12 suggests that at a given value of ε, ε_{LT, eff}(0) < ε_{LT, eff}(5) < ε_{LT, eff}(10), and thus ε_{LT, bar}(0) > ε_{LT, bar}(5) > ε_{LT, bar}(10). At a given ε, the values of ${\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$ increase with increasing m₂, the counterion binding χ goes up according to $\upchi =0.27{\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$, and thus, the surface charge density goes down. This effect leads to a decrease of the adsorption-desorption barrier against diffusion of counter ions. Therefore, it is likely that ε_{LT, bar}(0) > ε_{LT, bar}(5) > ε_{LT, bar}(10) and D_E(0) > D_E(5) & D_E(10), although a quantitative analysis was not realized. To understand this feature, the $\omega\ \mathrm{vs}\ {\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$ and ε vs ω relations were constructed and are shown in Fig. 13a, b. Figure 13b shows that all the dilational elasticity vs ${\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$ curves, the adsorption-desorption barrier, and the diffusion coefficients were dependent on the concentration of the added salt. However, the value of the dilational elasticity itself was primarily determined by the ripplon frequency for a given surfactant DTABr even in the presence of added inorganic salt, at least, up to m₂/mmol kg⁻¹= 10.

Fig. 13

a Ripplon frequency ω vs. surface density ${\varGamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$ and b dilational elasticity ε vs. the ripplon frequency ω of DTABr at m₂ (mmol kg⁻¹) = 0 (○), 5 (●), and 10 (△)

In Fig. 14, τ_eff defined by Eq. 36 is plotted against ${\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$: it goes through unity at around the surface density of ${\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H},\max }$, where ε passes through a maximum, for all cases of m₂/mmol kg⁻¹= 0, 5, and 10. When τ_eff = 1, the characteristic frequency of the counterion diffusion in the surface region ω_{dif, eff} is equal to the ripplon frequency ω. Therefore, it can be said that because the ripplon frequency at the maximum ε is approximately 300 kHz from Fig. 13b, the diffusion of counterions between the Stern and diffuse layers takes place on a time scale of 3.3 μs.

Fig. 14

τ_eff vs. surface density ${\varGamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$ at m₂ (mmol kg⁻¹) = 0 (○), 5 (●), and 10 (△)

Dodecyltrimethylammonium ions with Br ⁻, ${BF}_4^{-}$, and ${SO}_4^{2-}$

The ε, ε₀, ε_{LT, eff} vs ${\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$ curves are shown for DTABF₄ in Fig. 15a and DTA₂SO₄ in Fig. 15b. For ${\mathrm{BF}}_4^{-}$, the value of ε_{LT, eff} deviates from ε at the lower ${\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$ region, as was observed for Br⁻ (see Fig. 12). However, for ${\mathrm{SO}}_4^{2-}$, the value of ε_{LT, eff} was larger than that of ε at both lower and higher values of ${\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$ and was smaller in the intermediate ${\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$ range. The D_E/m²s⁻¹ values were $4.8\times {10}^{-9}\left({\mathrm{SO}}_4^{2-}\right)<7.12\times {10}^{-8}\left({\mathrm{Br}}^{-}\right)<4.4\times {10}^{-7}\left({\mathrm{BF}}_4^{-}\right)$. Because the D_E value is very different for different counterions, even for the same DTA⁺ cation, and remains much higher than the diffusion coefficient of the DTA⁺ ion, 6 × 10⁻¹⁰m² s⁻¹, we conclude that the present SQELS study at high ripplon frequencies of 200–400 kHz provides information on both surfactant and counterion diffusion.

Fig. 15

Dilational elasticity vs. surface density ${\varGamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$ curves of a DTABF₄, b DTA₂SO₄. ε from experiments (○), Gibbs elasticity ε₀ (◇), and effective LT model ε_{LT, eff} (▲). Solid curves are guide to the eyes for ε_{LT, eff}

The ε vs ${\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$ curves are compared with each other in Fig. 16. It is clear that the maximum value of ε, ε^max, and the value of ${\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$ at ε^max, ${\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H},\max }$, increased in the order of ${\mathrm{BF}}_4^{-}>{\mathrm{Br}}^{-}>{\mathrm{SO}}_4^{2-}$. This order is the opposite to ordering of the magnitude of the dehydration free energy [62], i.e., ∆_dehydG/kJ mol⁻¹ is: $190\ \left({\mathrm{BF}}_4^{-}\right)<315\left({\mathrm{Br}}^{-}\right)<1080\left({\mathrm{SO}}_4^{2-}\right)$. The counterion binding to the surfactant ions generally becomes stronger as ∆_dehydG decreases; hence, the tendency to escape from the Stern layer to the diffuse layer is accompanied by monolayer compression and this tendency is expected to become small. This expectation agrees with the findings in Fig. 16 that ${\mathrm{BF}}_4^{-}$ gives the highest ε^max and that D_E increases in the order: $4.8\times {10}^{-9}\left({\mathrm{SO}}_4^{2-}\right)<7.12\times {10}^{-8}\left({\mathrm{Br}}^{-}\right)<4.4\times {10}^{-7}\left({\mathrm{BF}}_4^{-}\right)$, as evaluated from the ε_{LT, eff} fitting. Fig. 17a indicates that the dependence of ε on ω is clearly different for the different counterions. Conversely, the dependence is almost the same for Br⁻ even in the presence of added salt, as shown in Fig. 13b. This finding confirms that the dilational elasticity estimated by SQELS at 200–400 kHz reflects the diffusion of counterions in the interfacial region and is determined not only by the perturbation (ripplon) frequency but also by the strength of binding of the counterion to the surfactant ion. The τ_eff vs ${\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H}}$ curves given in Fig. 17b show that at ${\Gamma}_{{\mathrm{DTA}}^{+}}^{\mathrm{H},\max }$, τ_eff passes through unity, and thus, the characteristic frequency of the diffusion ω_{dif. eff} is equal to the ripplon frequency ω. The results in Fig. 17a, give ω_{dif. eff} values of approximately $300\ \mathrm{kHz}\ \left({\mathrm{Br}}^{-}\right),310\ \mathrm{kHz}\ \left({\mathrm{BF}}_4^{-}\right),320\ \mathrm{kHz}\ \left({\mathrm{SO}}_4^{2-}\right)$. Thus, the time scale of counterion diffusion is estimated to be in the range of 3.3 – 3.1 μs.

Fig. 16

Dilational elasticity ε vs surface density ${\varGamma}_1^{\mathrm{H}}$ curves of DTA₂SO₄ (○), DTABr (▲), DTABF₄ (◇), C₁₀E₅ (■)

Fig. 17

a Dilational elasticity vs ripplon frequency and b τ_eff vs surface density for DTA₂SO₄ (○), DTABr (▲), and DTABF₄ (◇)

SQELS studies of nonionic surfactants with oxyethylene units (C₁₀E₅)

In the above section with respect to cationic surfactants, we demonstrated that relaxation processes relevant to high frequency ripplons are not related to diffusion of surfactant ions but rather to diffusion of counter ions in the interfacial region. Although the transfer of counterions between the Stern layer and diffuse layer generally accompanies hydration-dehydration of counterions, it is practically impossible to treat the influence of counterions and water on dilational elasticity separately. To know how water molecules influence the dilational elasticity in the high frequency regions, the SQELS studies were adapted to the adsorbed monolayers of nonionic surfactants with oxyethylene units, CH₃(CH₂)₉(OCH₂CH₂)₅OH, abbreviated as C₁₀E₅.

The ε vs ${\Gamma}_{{\mathrm{C}}_{10}{\mathrm{E}}_5}^{\mathrm{H}}$ plot of this system shows a notable difference from those of the ionic surfactants, as shown in Fig. 16. For example, ${\Gamma}_{{\mathrm{C}}_{10}{\mathrm{E}}_5}^{\mathrm{H},\max }$ is approximately 1 μmol m⁻² and much smaller than the saturation adsorption (~ 3.3 μmol m⁻²). Additionally, the value of ε decreases slowly over a wide range of ${\Gamma}_{{\mathrm{C}}_{10}{\mathrm{E}}_5}^{\mathrm{H}}$ above ${\Gamma}_{{\mathrm{C}}_{10}{\mathrm{E}}_5}^{\mathrm{H},\max }$ as compared with the behavior of the ionic surfactants, suggesting a different relaxation mechanism for the C₁₀E₅ surfactants. The ε values are plotted against ${\Gamma}_{{\mathrm{C}}_{10}{\mathrm{E}}_5}^{\mathrm{H}}$ together with curves of ε⁰,ε_LT, ε_MLT, and ε_{LT, eff} in Fig. 18, where D/m² s⁻¹ = 4.30 × 10⁻¹⁰ and k_d/mol m⁻²s⁻¹ = 4.25 × 10⁻⁴ were used [56, 57]. It should be noted that ε_LT based on the diffusion coefficient of C₁₀E₅ in aqueous solution D is almost equal to ε₀ and does not trace ε even at higher surface densities where the diffusion of surfactant molecules is expected to be influential. The ε_MLT values calculated from Eq. 34 coincide with ε₀ perfectly as expected at high frequencies for case 4. These observations derive from the conclusion that the relaxation affecting the dilational elasticity is not attributable to diffusion of C₁₀E₅ molecules but to some processes relevant to the water molecules.

Fig. 18

Comparisons among the dilational elasticities obtained from experiments and models for C₁₀E₅. ε from experiments (○), Gibbs elasticity ε₀ (●), LT model ε_LT (▲), MLT model ε_MLT (△), effective LT model ε_{LT, eff} (×)

Because even ε_{LT, eff} largely deviates from ε over a wide range of ${\Gamma}_{{\mathrm{C}}_{10}{\mathrm{E}}_5}^{\mathrm{H}}$, the examination on the basis of the diffusion models is not realistic. As a possible qualitative explanation, the cross-sectional area of even a non-hydrated hydrophilic group is approximately 0.23 nm² and considerably larger than that of a hydrocarbon chain (0.18 nm²). Thus, the hydrated hydrophilic groups are expected to be sitting closer to each other than the hydrophobic chains even for small ${\Gamma}_{{\mathrm{C}}_{10}{\mathrm{E}}_5}^{\mathrm{H}}$. When regions with a higher local surface density (HLSD) of lower local surface tension and those with a lower surface density (LLSD) of higher local surface tension, relative to the equilibrium situation, are created by the capillary wave, the possible relaxation processes include distortion of the hydration sphere and/or hydration-dehydration of the hydration sphere caused by diffusion of water molecules. It seems probable that at a lower ${\Gamma}_{{\mathrm{C}}_{10}{\mathrm{E}}_5}^{\mathrm{H}}$, only the distortion responds to ripplon perturbation. However, at a higher ${\Gamma}_{{\mathrm{C}}_{10}{\mathrm{E}}_5}^{\mathrm{H}}$, the results suggest that the hydrophilic part of the monolayer is dehydrated to some extent [63], hydration-dehydration processes are essential for inducing perturbations in ripplons. Assuming that the ripplon frequency at the surface density ${\Gamma}_{{\mathrm{C}}_{10}{\mathrm{E}}_5}^{\mathrm{H},\max }$ giving ε^max corresponds to the characteristic frequency for diffusion of water, ω_{dif. eff}, as is the case for an ionic surfactant, the frequency is estimated to be approximately 310 kHz. Because the ripplon frequency above ${\Gamma}_{{\mathrm{C}}_{10}{\mathrm{E}}_5}^{\mathrm{H},\max }$ decreases to approximately 210 kHz at the saturated adsorption, we expected that the dehydration-hydration caused by diffusion of water molecules has a strong influence on ε. This expectation is consistent with the fact that ε_{LT, eff} becomes comparatively close to ε at high ${\Gamma}_{{\mathrm{C}}_{10}{\mathrm{E}}_5}^{\mathrm{H}}$.

There have been some reports on the dilational elasticity of C_iE_j surfactants. Ravera et al. demonstrated that for C₁₀E₄ surfactant adsorption, the characteristic frequency for diffusion is approximately 0.4 Hz, while that for surfactant reorientation is approximately 800 Hz [3]. This result supports our finding that the relaxation process captured by SQELS at ripplon frequencies of several hundred kHz is neither related to surfactant diffusion nor to surfactant reorientation. Sharpe and Eastoe [21] reported a dilational elasticity in the range of 20–30 mN m⁻¹ as a function of ripplon frequency from 30 to 90 kHz for C₁₀E₈ and C₁₂E₅, which roughly coincides with ε^max of the present study.

Damping coefficients

The damping coefficient evaluated from phenomenological fitting is plotted against ε/γ for DTABr at m₂/mmol kg⁻¹= 0, 5, and 10 in Fig. 19, which clearly demonstrates that Γ/Γ^max has a maximum in the range of ε/γ ≈ 0.15 – 0.2. Because ωκ was shown to take a negative value at approximately ε/γ~0.1, such a negative viscosity closely correlates with resonance between the transversal and dilational waves [64].

Fig. 19

Loss modulus ωκ and damping coefficients Γ/Γ_water plotted against ε/γ for the DTABr-NaBr mixtures at m₂ (mmol kg⁻¹) = 0 (○), 5 (●), and 10 (△)

Eastoe et al. measured the surface dilational viscoelasticity of aqueous solutions of hexadecyltrimethylammonium bromide (HTABr) with and without dodecane (C12) by the SQELS method and found that the sign of κ switched from negative to positive through the addition of C12 [27]. Monroy et al. also found negative dilational viscosities for HTABr from the ECW experiments [17] and reported that such negative dilational viscosities arise from deficiencies in the hydrodynamic description of the interface and an appropriate dispersion equation of interface is absolutely necessary [23]. The amplitude of the capillary wave τ estimated by <τ²>^1/2 =  √ (kT/(γq² + ρg) was 0.2–0.5 nm at a surface tension γ of the aqueous DTAB solution. Conversely, the interface thickness (Debye length) estimated from the Gouy–Chapmann distribution was 8–2.5 nm in the absence of NaBr and 3–2 nm at a NaBr concentration of 10 mM for the aqueous DTAB solutions. Furthermore, the thickness of the counterion distribution, including the surface roughness, was estimated to be approximately 1.1 nm from X-ray reflectivity measurements [65]. Therefore, although the amplitude of the capillary wave might be somewhat smaller than the interface thickness, there is every possibility that the capillary deformation of the interface brings the monolayer into contact with the bulk region. In those cases, an additional coupling term should be incorporated into the dispersion equation as described by Monroy et al. [23], Thominent et al. [28], and Hennenberg et al. [22]. Furthermore, the model proposed by Bonfillon and Langevin is likely more sophisticated than the usual MLT model for charged surfactants [19].

To improve our understanding of these systems further theoretical and experimental studies are needed. In particular, we need more experiments on ionic surfactants in the presence of added inorganic salts at higher concentrations than those used in the present study. Furthermore, mixtures of different ionic surfactants with hydrocarbons and different types of nonionic compounds of the form C _i E _j should be investigated and analyzed using different useful models. Our future SQELS studies will focus on these aspects.

Conclusions

The surface dilational viscoelasticities of aqueous solutions of short chain lipid mixtures, cationic surfactants with different counterions, and nonionic surfactants with oxyethylene units were investigated by measuring SQELS spectra and analyzing these data by both surface rheological and phenomenological fitting. The evaluated dilational elasticities were examined by the Lucassen-van den Temple model and its analogues.

From the ECW and SQELS studies on the lipid mixtures, we demonstrated that the molecular interaction between DC₈PC and DC₆PC in the adsorbed film clearly influenced the relaxation process on the order of milliseconds (ECW); however, these interactions were not so influential on relaxation of the order of microseconds (SQELS). The frequency dependence of the dilational viscoelasticity with the use of ECW suggested that the characteristic frequencies for relaxation from staggered structure formation were likely in the range from 150 to 300 Hz. Thus, the time scale was of the order of milliseconds.

The dilational elasticity of cationic surfactants with different counterions were evaluated from SQELS studies at 250–400 kHz and examined using the LT, modified LT, and effective LT models. Examination of the DTABr adsorbed film suggested that the main influential relaxation processes on the dilational property at this high frequency were neither related to diffusion of surfactant from the bulk phase to the subsurface nor an adsorption-desorption of surfactant ions between the adsorbed layer and subsurface. Rather, the most influential factors related to diffusion of counter ions taking place in the interfacial region owing to capillary wave dilation. This was confirmed from studies on the DTABr and NaBr system, which showed that the value of dilational elasticity itself was primarily determined by the ripplon frequency for a given surfactant, DTABr. This case was true even in the presence of added inorganic salt and the diffusion of counterions between the Stern and diffuse layers takes place on a time scale of 3.3 μs. Moreover, investigations on the DTABF₄ and DTA₂SO₄ systems revealed that the order of the change in magnitude of the diffusion coefficient in the interfacial region coincided with the degree of counterion binding to surfactant cation and dehydration energy. This result confirms the suggestion that the most influential relaxation processes on the dilational property were related to counterion diffusion in the interfacial region.

The SQELS studies of nonionic surfactants with oxyethylene units (C₁₀E₅) showed that neither the LT, modified LT, nor effective LT models could completely follow the experimental dilational elasticity. However, it was suggested that the relaxation process captured by the SQELS studies at ripplon frequencies of several hundred kHz were neither related to surfactant diffusion nor to surfactant reorientation. Rather, the possible relaxation processes were related to distortion of the hydration sphere and/or hydration-dehydration to/from hydration sphere caused by diffusion of water molecules.

Finally, negative values of surface dilational viscosity were also found. We concluded that Γ/Γ^max has a maximum at approximately ε/γ ≈ 0.15~0.2 and ωκ starts to take a negative value around ε/γ~0.1. Therefore, such negative viscosity might be closely related to the resonance between the transversal and dilational waves.

The present SQELS studies could provide information on the diffusion of counterions and water molecules in the interfacial region. The conclusions given above should be examined further using other types of ionic and nonionic surfactants, surfactant mixtures. Nevertheless, we showed that SQELS studies are capable of capturing useful and interesting information, which other methods at lower frequencies do not provide.