Emergence of Strong Nonlinear Viscoelastic Response of

Jan 23, 2018 - The influence of the surface pressure was also tested at 10 mN/m (Figures S3 and S4). If the viscoelastic response of the system is lin...
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Cite This: Langmuir 2018, 34, 2489−2496

Emergence of Strong Nonlinear Viscoelastic Response of Semifluorinated Alkane Monolayers Salomé Mielke,† Taichi Habe,†,⊥ Mariam Veschgini,† Xianhe Liu,‡ Kenichi Yoshikawa,§ Marie Pierre Krafft,‡ and Motomu Tanaka*,†,∥ †

Institute of Physical Chemistry, University of Heidelberg, 69120 Heidelberg, Germany Institut Charles Sadron (CNRS), University of Strasbourg, 67034 Strasbourg, France § Faculty of Life and Medical Sciences, Doshisha University, 610-0321 Kyotanabe, Japan ∥ Institute for Advanced Study, Kyoto University, 606-8501 Kyoto, Japan ‡

S Supporting Information *

ABSTRACT: Viscoelasticity of monolayers of fluorocarbon/ hydrocarbon tetrablock amphiphiles di(FnHm) ((CnF2n+1CH2)(Cm−2H2m−3)CH−CH(CnF2n+1CH2)(Cm−2H2m−3)) was characterized by interfacial dilational rheology under periodic oscillation of the moving barriers at the air/water interface. Because the frequency dispersion of the response function indicated that di(FnHm) form two-dimensional gels at the interface, the viscosity and elasticity of di(FnHm) were first analyzed with the classical Kelvin−Voigt model. However, the global shape of stress response functions clearly indicated the emergence of a nonlinearity even at very low surface pressures (π ≈ 5 mN/m) and small strain amplitudes (u0 = 1%). The Fourier-transformed response function of higher harmonics exhibited a clear increase in the intensity only from odd modes, corresponding to the nonlinear elastic component under reflection because of mirror symmetry. The emergence of strong nonlinear viscoelasticity of di(FnHm) at low surface pressures and strain amplitudes is highly unique compared to the nonlinear viscoelasticity of other surfactant systems reported previously, suggesting a large potential of such fluorocarbon/hydrocarbon molecules to modulate the mechanics of interfaces using the self-assembled domains of small molecules.

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INTRODUCTION Microscopic and mesoscopic structural patterns,1,2 such as stripes and bubbles, can be found in various systems, including noble gas molecules,3 ferrofluids,4 block copolymers,5 and surfactants6 on substrates. These equilibrium patterns exhibit the modulation of order parameters, resulting from the interplay of various molecular interactions. The characteristic length scale of patterns (stripe width and domain size) under equilibrium is determined by the competition between interactions among homologous species and the line tension γ. Surfactants with perfluorocarbon blocks have been used for designing new types of colloidal suspensions, targeting versatile medical applications.7,8 Different from the zigzag conformation formed by hydrocarbon chains, perfluorocarbon chains form helices of six turns because of the larger van der Waals radius of fluorine, 1.35 Å, and behave like a stiff rod. 9 When the surfactants with perfluorocarbon chains are confined at the air/water interface, dipolar interactions between terminal CF3 groups immersed in the medium with a low dielectric constant (air) tend to elongate the domains, competing against the line tension that tends to minimize the length of the domain boundary. © 2018 American Chemical Society

Previously, we reported that lipids with semifluorinated tails form stripe patterns at the air/water interface near the gas− liquid coexistence phase, whereas the same lipids that only possess hydrocarbon chains form circular domains.10 Oelke et al. reported that perfluorinated lipids form circular, solid domains when they are incorporated into phospholipids with hydrocarbon chains.11 Although the size of domains formed by lipids with hydrocarbon chains is heterogeneous because of coalescence of small domains, the domains of perfluorinated lipids seemed highly uniform, showing almost no sign of coalescence. This finding suggested that the interactions between fluorocarbon domains are predominantly repulsive. In fact, the potential of the mean force acting between fluorocarbon domains suggested that the fluorocarbon domains form a hexagonal order and the lateral correlation between the domains can reach over a distance that is 8 times larger than the domain size itself.11,12 The equilibrium size of domains (0.3−2 μm) exhibited a distinct dependence on the length of Received: November 21, 2017 Revised: January 11, 2018 Published: January 23, 2018 2489

DOI: 10.1021/acs.langmuir.7b03997 Langmuir 2018, 34, 2489−2496

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Langmuir fluorocarbon chains, which could be calculated by considering the counteracting contribution of dipole repulsion and line tension.12,13 Krafft et al. reported that molecules based on the linear connection of one fluorocarbon chain and one hydrocarbon chain (the so-called “diblock” molecule, FnHm: CnF2n+1CmH2m+1) can form highly uniform, circular domains. Interestingly, these domains are formed already at low surface pressures (π = 0 mN/m), whose diameter (Φ = 30−40 nm) was about 1 order of magnitude smaller than the size of perfluorinated lipids in a hydrocarbon matrix.8 The mesoscopic domains of FnHm molecules are self-assembled into hexagonal lattices, suggesting that the interdomain interactions are predominantly repulsive. In fact, these mesoscopic domains do not coalesce even at high surface pressures.14 The lateral compression modulus of diblock monolayers amounted up to ∂π κ = A ∂A ≈ 180 mN/m.7 There have been some reports on the interfacial shear measurements on similar compounds, but they were focused on the influence of chemical connections between the fluorocarbon and hydrocarbon blocks on the shear viscosity of monolayers including no quantitative information about the structures of mesoscopic domains, nor their lateral correlation.15,16 To understand if strongly correlated monolayers of mesoscopic domains form physical gels at the air/water interface, we investigated interface rheology of “diblock” monolayers by using an interface stress rheometer.17 By systematically testing molecules with different block ratios between fluorocarbons and hydrocarbons, we demonstrated that all the diblock monolayers were predominantly elastic, forming two-dimensional (2D) gels even at π ≈ 0 mN/m. We extended our strategy to investigate the viscoelastic response of monolayers based on “tetrablock” molecules (di(FnHm) ((CnF2n+1CH2)(Cm−2H2m−3)CH−CH(CnF2n+1CH2)(Cm−2H2m−3)), Figure 1A), possessing two fluorocarbon and two hydrocarbon chains in one molecule. These “gemini” molecules also self-assemble into well-ordered circular domains (Φ ≈ 40 nm), whose structures have been

characterized by atomic force microscopy.8,18 However, there has been no study shedding light on the mechanical properties of di(FnHm) tetrablock monolayers at the interface. Our preliminary experiments demonstrated that the viscoelasticity of tetrablock monolayers was beyond the sensitivity of the interface stress rheometer (Figure S1). In this study, we therefore evaluated the viscoelastic properties of tetrablock monolayers by dilational rheology under oscillatory compression and expansion of the monolayer at the air/water interface.

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EXPERIMENTAL SECTION

Materials. The compounds tested in this study, di(F10H16), di(F10H18), and di(F10H20), were synthesized according to ref 18. Double-deionized water (Milli-Q, Molsheim) with a resistance of ρ > 18 MΩ cm was used throughout this study. Interfacial Dilational Rheology. The interfacial dilational rheology of di(F10Hm) (m = 16, 18, 20) monolayers was measured using a KSV Nima film balance equipped with two movable barriers (Figure 1A). A 40 μL portion of 1 mM di(F10Hm) dissolved in chloroform was spread on the water surface. After waiting for the evaporation of the solvent for more than 10 min, the film was compressed at a speed of 3.75 cm2/min until a surface pressure of 5 mN/m was reached. After letting the film equilibrate for 10 min, the barrier position was sinusoidally oscillated at a defined frequency (f = 1−150 mHz), and the area u(t) and the surface pressure π(t) were recorded as a function of time. The area oscillation reads as

u(t ) = A(1 + u0 sin(ωt + φu))

(1)

where A is the starting molecular area in Å2 and u0 describes the strain amplitude (u0 = 0.01). ω = 2πf is the frequency of the oscillation and φu the phase shift. As di(F10Hm) monolayers show an extremely high compression modulus of κ ≈ 140 mN/m at π = 5 mN/m and collapse at π ≈ 20 mN/m (Figure 1B), only very small strain amplitudes could be applied to avoid the collapse of monolayers. In the following, we mainly focus on a strain amplitude of u0 = 1%. The influence of strain amplitudes (u0 = 0.5% and u0 = 2%) is presented in the Supporting Information (Figure S2). The influence of the surface pressure was also tested at 10 mN/m (Figures S3 and S4). If the viscoelastic response of the system is linear, the surface pressure (stress) at a frequency ω can be represented by a simple harmonics

π(t ) = π0 + π1 sin(ωt + φπ )

(2)

where π0 is the initial surface pressure, π1 the stress amplitude, and φπ the phase shift. The dynamic stress−strain relationship yields the elastic modulus E′ and the viscous modulus E″ using Euler’s equation π π E* = 1 cos φ + i 1 sin φ = E′ + iE″ u0 u0 (3) φ = φu − φπ is the phase difference between stress and strain. As observed before, the experimental setup of the oscillating barrier method involves an intrinsic phase separation φexp, which is φexp ≈ 0.28 in our case. This may cause the overestimation of the real phase shift, φ = φmeasured + φexp. Because we found that the amount of the sample deposited on the water influences the phase shift (Figure S5), one potential source for intrinsic phase separation could be the bending of the meniscus during compression and expansion. To minimize the artifacts arising from the barrier position, the spreading amount was set constant throughout the study. Nevertheless, this onset is ignored in the following discussion, because its impact on the estimation of elastic modulus and viscosity remained below 6%.

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Figure 1. (A) Experimental setup of dilational rheology measurements with oscillating barriers. The chemical structure of the tetrablock molecule is illustrated in the upper panel. (B) Isotherms of di(F10H16), di(F10H18), and di(F10H20) measured at 20 °C. The compression speed was 3.75 cm2/min.

RESULTS AND DISCUSSION Dilational Viscoelasticity. The dilational viscoelastic properties of di(F10Hm) were measured within the frequency range from 1 to 150 mHz. Figure 2A shows typical stress and 2490

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di(F10H20) (green) showed a sharp increase in φ1 up to φ150mHz ≈ 0.6 rad. This is more pronounced than the other tetrablock systems (φ1 ≤ 0.5 rad), suggesting that the viscous contribution is the most prominent for di(F10H20). Limiting the analysis up to the first mode, the viscous and elastic modulus were estimated by adopting the classical

Figure 2. (A) Surface pressure π(t) (black) and the corresponding area change u(t) (red) of a di(F10H16) monolayer. The strain amplitude u0 = 1% corresponds to the change in the area per molecule of ΔA = 0.6−0.8 Å2. The stress amplitude remains around π1 = 3−4 mN/m. (B) Phase shift φ1 between stress and strain plotted as a function of frequency. The inset shows the data in a semilogarithmic scale, revealing φ1 ≈ 0.28 rad below 10 mHz. The error bars show the standard deviations (SDs) out of more than three independent measurements.

strain signals of a di(F10H16) monolayer at 100 mHz. The oscillatory change in the area per molecule (red curve) is ΔA = 0.6−0.8 Å2 (1% area change), which is 4 orders of magnitude smaller than the area of a nanodomain, whose diameter is Φ ≈ 40 nm. Therefore, it is assumed that the global spatial distribution of nanodomains is not disturbed by the compression oscillations. Throughout the experiments, we observed no distinct drop of the surface pressure because of the loss of the material. From the global appearance of the response signal, it is obvious that the viscoelastic response of di(F10H16) is strongly nonlinear. Therefore, instead of using a commonly taken simple harmonic approximation (Figure S6), we analyzed the data with Fourier series expansion19,20 n

π (t ) = π 0 +

∑ πk sin(kωt + φπ ) k

k=1

Figure 3. (A) Kelvin−Voigt model. (B−D) Dilational elastic modulus E1′ and viscous modulus E1″ for (B) di(F10H16), (C) di(F10H18), and (D) di(F10H20) measured at f = 1−150 mHz, at u0 = 1%. The error bars show the SDs out of more than three independent measurements. Solid lines coincide with the fitting results of the Kelvin−Voigt model.

(4)

where πk is the amplitude of mode k and φπk the corresponding phase shift. In the first step, we evaluated the viscoelasticity by analyzing the amplitude and phase information from only the first mode (indicated by the index 1). The phase separation between the oscillation of the area (strain) and the surface pressure (stress) E″ φ1 = tan−1 E1′ is shown in Figure 2B. It is notable that the

Kelvin−Voigt model (Figure 3A), a common model suited for predominantly elastic systems with no relaxation21−24 σ = gε + η

d ε dt

(5)

σ is the stress, ϵ the strain, g the spring constant, and η the viscosity. Under an oscillatory strain at a frequency ω, the complex dilational modulus E1* can be represented as

1

phase shift, even at the highest frequency ( f = 150 mHz), remained lower than π/2. This implies that all the tetrablock monolayers are predominantly elastic under the whole frequency range. For all three tetrablock molecules, φ increased monotonically according to the increase in frequency, reflecting an increase of the interface viscosity at higher frequencies. At low frequencies ( 100 mN/ m could be achieved only when the film was compressed to a high surface pressure, π ≥ 15 mN/m. Compared to the systems described in previous studies, di(F10Hm) monolayers show a high elastic modulus (E1′ > 120 mN/m) even at a much lower surface pressure (π = 5 mN/m). In fact, the characteristics presented in Figure 3, E′(ω) = const., E″(ω) ∝ ω, and φ ∝ ω, follow the Kelvin−Voigt model, suggesting that the relaxation of the system is very minor. To date, only a few surfactant systems form predominantly elastic 2D gels via hydrogen bonding between oligosaccharides,27,28 cross-linking of anionic moieties by divalent cations,29 and strong dipole repulsion between FnHm diblocks.17 It should also be noted that the dilational surface viscosity of di(F10Hm) monolayers, η < 0.06 mN s/m, is low compared to the values reported for other surfactants with large hydrophilic head groups,22,30−32 implying that only little energy dissipates throughout the oscillation. This can be attributed to a low friction between the monolayer and the water subphase because

di(F10Hm) molecules have no hydrophilic moiety similar to other surfactants. Such a high elasticity and low surface viscosity suggest that the mesoscopic domains of di(F10Hm) are strongly correlated with repulsive interactions. As reported previously, di(FnHm) domains do not coalesce7 because of strong dipole repulsions between CF3− termini.1,10 To gain further insight into the modulation of repulsive interactions between di(FnHm) domains by molecular structures, we are currently performing grazing incidence small-angle X-ray scattering to quantify the structure and form factors of di(FnHm) domains at the air/water interface. Nonlinear Viscoelasticity. In the next step, we included the higher modes in the Fourier series and evaluated the nonlinear viscoelasticity of di(F10Hm) monolayers. Figure 4

Figure 4. Surface pressure response of di(F10H16) plotted as a function of (A) time and (B) area per molecule. The measurements were performed at f = 1 mHz (top), 10 mHz (middle), and 100 mHz (bottom) by keeping the strain u0 = 1%. The fitting curves (red solid lines) correspond to the Fourier expansion up to the fifth mode. The offset of the sine oscillation π0 was found to depend on the previous position of the barriers and therefore varies slightly (Figure S7), which however should not influence the calculation of the viscoelastic properties.

represents a typical surface pressure response of a di(F10H16) monolayer measured at f = 1 mHz (top), 10 mHz (middle), and 100 mHz (bottom), plotted as a function of time (Figure 4A) and area per molecule (Figure 4B). The nonlinearity of response curves can be characterized from the flat portions in panel (A) diverging from a sinusoidal curve and the kinks in panel (B) showing a difference from a smooth Lissajou curve. To gain more understanding of the origin of the nonlinearity and quantitatively understand the flat portions in the surface pressure oscillation and Lissajou plots, the data were fitted with a Fourier series expansion up to the fifth mode (red lines in Figure 4A). Figure 5A represents the real part of the Fourier spectrum of the response curve of di(F10H16) measured at f = 100 mHz. The spectrum clearly indicates that the odd modes have a much larger proportion than the even modes. In fact, the amplitudes of the even modes, ∼0.3 mN/m, are close to the instrumental noise level, suggesting that the system is highly isotropic. The inset of Figure 5A illustrates the imaginary part of the Fourier spectrum, showing no distinct feature in higher Fourier modes. Figure 5B represents the amplitudes of the higher, odd Fourier 2492

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obtained at 5 mN/m, yielding a THD of 13.0 ± 0.7% (Figure S4). For di(F10H16), at f = 10 mHz we studied the influence of strain amplitude (u0 = 0.005 and u0 = 0.02) on the nonlinearity of the system, revealing that the THD remained almost comparable (Figure S2). At u0 = 0.02, we found the emergence of the second mode, suggesting that the effect of friction plays an increased role at higher strain amplitudes. In the case of Gibbs monolayers, it has been claimed that the nonlinear response is originated from the change in the surface concentration of the surfactants.21,33 In fact, many experimental studies tried to avoid the emergence of nonlinear effects by keeping the strain amplitude very low.20,30,34 To date, several groups have reported the nonlinear dilational stress response of Langmuir monolayers.19,20,30−32,35−39 For example, Makino and Yoshikawa demonstrated that the nonlinear viscoelasticity of dioleoylphosphatidylcholine monolayers is influenced by the strain frequency, temperature, and the subphase composition.19 Other studies showed the influence of substitutional impurities, such as anesthetics.35 Vrânceanu et al. reported that the THD of lipid monolayers doped with cholesterol did not depend on the frequency,31 which agrees well with our observation (Figure 5C). It should be noted that the nonlinear viscoelasticity reported in most of the previous reports was found at much higher surface pressures (π ≥ 15 mN/m) and strain amplitudes (u0 ≥ 5%). At low surface pressures (π = 3.5−7.5 mN/m) and small strains (u0 = 1%), the response function was almost linear (THD ≈ 0). Why does such a high nonlinearity of di(F10Hm) monolayers emerge even at a very low strain amplitude (u0 = 1%)? Some studies suggested that nonlinear effects observed in dilational rheology measurements might result from the adsorption of surface active species from the subphase, which can be understood by the model of van den Tempel and Lucassen.21,38 In our case, however, this scenario can be excluded, as di(F10Hm) molecules are not only highly hydrophobic but also oleophobic. Other papers explain the nonlinearity by lateral diffusion of the surfactants in the monolayer40 or changes in the interfacial microstructure,30,41,42 but these scenarios do not apply to our system, neither.7,8 Currently, we attribute the emergence of strong nonlinear response to the highly ordered arrangement of di(F10Hm) domains because the fact that only odd modes appear in the Fourier spectra (Figure 5A) strongly suggests that the system should be isotropic.30,43 As presented in Figure 3, the tetrablock monolayers are predominantly elastic: in the linear analysis, the dilational elastic modulus E′1 shows no frequency dependence, whereas the dilational viscous modulus E″1 is linearly proportional to the frequency. As a nonlinear elastic component should be odd under reflection because of the mirror symmetry, the first nonlinear term should be cubic, σE = gϵ + g′ϵ3. This seems plausible, as we found that the signals from even Fourier components were close to the noise level (Figure 5A). Indeed, the fact that no higher Fourier modes could be detected in the imaginary part (Figure 5A, inset) further verifies that the nonlinearity only appears in the elastic component, as the real part determines the elasticity and the imaginary part the viscosity of the system. Figure S10 represents the phase shifts of modes m = 1, 3, and 5 plotted as a function of frequency. The phase of the first mode φ1 increases linearly with the frequency, where the slope is corresponding to the 2D viscosity η of the system. On the other hand, the phase shifts of the third and fifth mode, φ3 and φ5, do not follow such a tendency, indicating

Figure 5. (A) Real part of the Fourier-transformed response function of a di(F10H16) monolayer recorded at f = 100 mHz. The inset shows the imaginary part of the Fourier spectrum. (B) Fractions of the higher mode signals of di(F10H16) plotted as a function of frequency. (C) THD for di(F10H16), di(F10H18), and di(F10H20) plotted over the whole frequency range. Error bars are the SDs from at least three independent measurements.

modes normalized by the amplitude of the first Fourier mode for di(F10H16). The amplitude of the seventh mode signal is already in the noise range (2% of σ1). Actually, the reduced χred2 of the fit results implies that the further expansion beyond the fifth mode does not improve the fit quality (Figure S6). The datasets of the other two tetrablock molecules also showed the same tendency (Figures S8 and S9). The data shown in Figure 5 indicate that the amplitude of Fourier coefficients and thus the nonlinearity of response signals do not depend on the frequency. This was further verified by calculating the total harmonic distortion (THD)

THD =

1 π1

n

∑ πk 2 k=2

(7)

where πk is the amplitude of the k-th Fourier mode. THD can be used as an indicator for the significance of nonlinearity of the system. Figure 5C represents the THD of di(F10H16), di(F10H18), and di(F10H20) plotted as a function of frequency. As stated above, n = 5 was chosen as the maximum number of modes for Fourier expansion. The mean THD values are comparable: THDdi(F10H16) = (12.5 ± 0.9)%, THDdi(F10H18) = (11.3 ± 0.8)%, and THDdi(F10H20) = (11.6 ± 0.8)%. Finally, we confirmed that the oscillation of the area (strain) does not contain any nonlinear component by Fourier series expansion. The measurements of di(F10H16) at a surface pressure of 10 mN/m show a similar nonlinearity to the data 2493

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odd Fourier modes but not from even modes, which suggests that the nonlinearity results from the elastic part of the response function. In fact, within the framework of Landau theory, the emergence of the third and fifth harmonics can be explained by a first-order phase transition of the monolayer under the oscillatory stresses. The significance of nonlinearity was quantified using the THD, yielding a THD of ≈12% even at a low surface pressure (π = 5 mN/m) and a low strain amplitude (u0 = 1%). The emergence of highly nonlinear response at such a low frequency and amplitude regime has not been reported for other surfactant monolayers, which is attributed to the repulsion between the di(F10Hm) domains. Because the monolayers of semifluorinated surfactants open a broad range of biomedical applications, such as stable microbubbles for sonography, it is highly relevant to understand the general principle of how the self-assembled domains of such small molecules modulate the dilational viscoelasticity of interfaces.

that the viscous part of the system is only represented by the linear part (m = 1). Therefore, we can describe our data with an extended nonlinear Kelvin−Voigt model, where the nonlinear terms only appear in the elastic components π (t ) ∝ g + g ′ε 3 + g ″ε 5 + η

∂ε ∂t

(8)

Thus, our experimental findings strongly suggest that the emergence of strong nonlinear elastic response of di(F10Hm) monolayers could result from the repulsive interactions between domains. The flat regions in the Lissajou plots (Figure 4B) coincide with an infinite lateral compressibility, −

1 ∂A A ∂π T

( )

≈ ∞. This

clearly indicates that the system undergoes a first-order phase transition, which shares common features with the phase transition of lipid monolayers.44 Under the framework of Landau theory, the free energy of the system with a first-order phase transition is generally given as in eq 9 by using the density q as an order parameter45,46 F (q , t ) =

β γ α 2 q + q 4 + q6 + h(t )q 2 4 6

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* Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.7b03997. Measurement of shear elastic modulus G′ and shear viscous modulus G″, influence of strain amplitude on dilational interfacial rheology, analysis of nonlinear interfacial dilational rheology, influence of spreading amount on the phase separation, Fourier series fit quality of the raw data, overview of di(F10H18) and di(F10H20) data, and phase shift between stress and strain (PDF)

(9)

h(t) is an external, time-dependent force acting on the system. Note that the density q = q(r,t) is also time-dependent in our system and therefore45,46 ∂q(r , t ) ∂F(q , t ) ∝− ∂t ∂q

(10)

In case the free energy can be represented by eq 9, the external force h(t) is given then by h(t ) ∝ uq + vq3 + wq5 + s

∂q ∂t

ASSOCIATED CONTENT

S

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(11)

with u, v, w, and s being proportionality parameters. Because the area per molecule u(t) in our experiments is equivalent to the density q(t), h(t) corresponds to the stress response π(t). Eq 11 then results in a Kelvin−Voigt model including a third and fifth order nonlinearity in the elastic part as presented in the previous paragraph with eq 8. Thus, the first-order phase transition between a dispersed and assembled state of di(F10Hm) domains explains the emergence of the third and fifth harmonics in the stress response π(t) as observed in our experiments (Figure 5A).

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (M.T.). ORCID

Marie Pierre Krafft: 0000-0002-3379-2783 Motomu Tanaka: 0000-0003-3663-9554 Present Address ⊥

Kao Corporation, Wakayama 650-8480, Japan (T.H.).

Notes

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The authors declare no competing financial interest.

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CONCLUSIONS The viscoelasticity of monolayers of semifluorinated surfactants di(F10Hm) (m = 16, 18, and 20) was measured by dilational rheology at the air/water interface between f = 1 and 150 mHz. The stress response of di(F10Hm) monolayers was measured as the change in surface pressure, while keeping the strain amplitude very low (u0 = 1%). We found that di(F10Hm) monolayers are predominantly elastic (E′ > E″), implying the formation of 2D physical gels. Because the elastic modulus E′1 showed no frequency dependence and the viscous modulus E″1 linearly scales with the strain frequency, the viscoelastic response of di(F10Hm) monolayers was fitted using the Kelvin−Voigt model first. However, as the stress response exhibited a clear deviation from the sinusoidal response function of a linear viscoelastic body, we further analyzed the nonlinearity of the response function by the Fourier series expansion. We observed a clear increase in the intensity from

ACKNOWLEDGMENTS We thank the French Research Agency (grant ANR-14-CE350028-01) and the German Science Foundation (Ta259/12) for supports and Teclis Instruments (France) for technical help. M.T. and K.Y. thank MEXT (No. 16H00802 for M.T. and No. 25103012 for K.Y.) for support. M.T. thanks World Premier International Research Center Initiative (WPI), MEXT, Japan and Nakatani Foundation for supports. S.M. acknowledges the Konrad-Adenauer-Stiftung and M.V. the Deutsche Forschungsgemeinschaft (GRK1114 and EcTop2) for fellowships.

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REFERENCES

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Langmuir

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DOI: 10.1021/acs.langmuir.7b03997 Langmuir 2018, 34, 2489−2496

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DOI: 10.1021/acs.langmuir.7b03997 Langmuir 2018, 34, 2489−2496