Elasticity of two-dimensional crystalline monolayers of fatty acid salts at an air–water surface

Abstract

The elastic properties of organic–inorganic two-dimensional crystals floating at the water surface have been fully characterized by grazing incidence X-ray diffuse scattering and high-resolution diffraction. We show that the strong interaction between the organic molecules and the inorganic divalent cations is enough for these nm thick crystals to behave like true solids, with a residual tension of 1 × 10⁻⁴–10⁻³N m⁻¹. Their bending rigidity is renormalized as κ(q) ∝ q^−η_k with η_k = 0.25 ± 0.07 and a microscopic value ≈ 100 k_BT at q = 1 × 10⁹ m⁻¹. The in-plane elastic constants behave like q^η_u with η_u = 1.41 ± 0.2, obeying the scaling relation η_u = 2 − 2η_k. These results are consistent with a long-range phonon-mediated interaction between out-of-plane fluctuations but the values of the exponents differ from those generally obtained in numerical simulations.

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1 Introduction

Two-dimensional solid membranes are expected to exhibit very unusual elastic and mechanical properties like vanishing elastic moduli, diverging bending rigidity at long wavelength or a negative Poisson ratio.¹ These properties have in particular been recently invoked to explain how graphene² could be stabilized by gentle crumpling in the third dimension though the Landau–Peierls instability does not allow two-dimensional crystals to exist.³Graphene is a striking example of a perfect one-atom thick crystal, but layered materials like nacre can also be found in nature whose remarkable elastic and mechanical properties do not result only from the strength of their bonds but rather from the synergy between the organic and inorganic parts in a composite structure.⁴ Amphiphilic molecules coupled to divalent cations at the water surface are the thinnest of such organic–inorganic stratified structures, where both the van der Waals forces between the organic moieties and electrostatic interactions between headgroups and cations contribute to build strong self-assembled two-dimensional structures. Whereas Langmuir films of simple amphiphiles usually make mesophases which do not exhibit the elasticity of solid membranes,⁵ the coupling to the cations Mg²⁺, Mn²⁺ (Fig. 1), Cd²⁺ and Pb²⁺ leads to nanometre thick structures, freely floating at the water surface which exhibit a remarkably high crystallinity.^6–8 Interestingly, only minute concentrations of ions are necessary to get the two-dimensional composite crystals, e.g. only 1.4 Mg²⁺ ion per behenic acid molecule at pH 10.5⁹ in which the fatty acid and ionic lattices are commensurate (the ionic lattice is a 2 × 2 superstructure of the organic lattice in the case of Mg²⁺ or a 1 × 2 in the case of Mn²⁺⁷).

Fig. 1 Schematic view of the scattering geometry with the Langmuir trough. Inset: Brewster angle microscopy image of behenic acid on a MnCl₂ subphase (5 Mn²⁺ per headgroup) at pH 7.5.

One of the most interesting theoretical predictions concerning crystalline or “tethered” membranes is the existence of a long-range interaction between out-of-plane fluctuations (capillary waves) due to the in-plane elasticity (phonons)^10,11 which is not present in usual, fluid membranes. This interaction leads to a renormalization of the bending rigidity κ(q) ∝ q^−η_k (see ref. 1) which is accompanied by a softening of elastic constants λ(q), µ(q) ∝ q^η_u at large distances (λ and µ are the Lamé coefficients). Simple integration shows that the roughness exponent ζ defined as 〈z²〉 ∝ L^2ζ, where z is the transverse fluctuation of the membrane of size L is such that 2ζ = 2 − η_k. Rotational invariance yields the scaling relation:

More precisely, this equation has to be obeyed in order to preserve the form of the strain when renormalizing the in-plane and normal components of the fluctuations.¹² In most of the numerous numerical simulations of the flat phase of tethered membranes, the results are in the range ζ = 0.58–0.65 which implies that η_k ≈ 0.7–0.84 and η_u ≈ 0.32–0.6.^11,13 Apart from graphene,³ only two kinds of experimental realization of solid membranes have been reported so far. Light scattering and small angle X-ray scattering of the red blood cell cytoskeleton yielded ζ = 0.65¹⁴ and a fractal dimension of 2.15 was obtained by freeze fracture electron microscopy and light scattering in the case of graphitic oxide sheets.¹⁵ In none of these experiments was the scaling relation eqn (1) verified as η_u could not be measured.

We show here that the out-of-plane fluctuations of the above mentioned organic–inorganic monolayers are governed by bending rigidity over a large q range, a unique feature at the water surface which can be tuned by changing the concentration in cations. Using grazing incidence diffuse scattering to measure η_k and grazing incidence diffraction to measure η_u, we show that their bending rigidity is renormalized and that the exponents are consistent with the scaling relation eqn (1).

2 Experimental

Behenic acid [CH₃(CH₂)₂₀COOH] (Fluka, purity > 99%) was dissolved in chloroform (Merck, analytical grade) and spread on a solution of MgCl₂ or MnCl₂ (Sigma purity 99.99%) in ultra-pure de-ionized water from a Milli-Q 185 + Millipore system contained in a home-made Teflon trough mounted on an active antivibration system for the X-ray experiments. Concentrations of 1.88 × 10⁻⁵ mol l⁻¹MgCl₂ (20 ions per behenic acid molecule), 1.88 × 10⁻⁶ mol l⁻¹MgCl₂ (2 ions per behenic acid molecule) and 8.46 × 10⁻⁵ mol l⁻¹MnCl₂ (90 ions per behenic acid molecule) were used, above the threshold for fast crystal formation which is 1.4 ions per behenic acid molecule for MgCl₂ at pH = 10.5 (fixed using KOH) and 7 ions per behenic acid molecule for MnCl₂ at pH = 7.5 (fixed using NaHCO₃).⁹ After solvent evaporation, the monolayer was gently compressed to ≈ 22 Ų molecule⁻¹ in order to maximize the coverage by crystals but stay at zero surface pressure (measured using a Wilhelmy balance) in order to avoid exerting constraints on them. During X-ray measurements, the Langmuir trough was translated by 0.5 mm every minute in order to avoid radiation damage. The experiments reported here used an 8.00 keV X-ray beam (wavelength λ = 0.155 nm) at the Troika II beamline of the European Synchrotron Radiation Facility (ESRF). The scattering geometry is described in Fig. 1. The monochromatic incident beam was first extracted from the polychromatic beam of the undulator source using a two-crystal diamond (111) monochromator. Higher harmonics were eliminated using two palladium coated glass mirrors, also used to set the grazing angle of incidence θ_in = 1.7 mrad below the critical angle for total external reflection θ_c (which occurs for X-rays since the refractive index of the matter is less than 1). The effective penetration length of the evanescent wave was ≈ 4.6 nm. The grazing incidence diffraction (GID) experiments were performed either using Soller slits in front of the gas-filled position sensitive detector (PSD) (low resolution mode Δq_|| = 6.43 × 10⁷ m⁻¹ for structure determination) or using a Ge (111) crystal analyzer (high-resolution mode Δq_|| = 1.47 × 10⁶ m⁻¹ for Bragg peak analysis). For diffuse scattering measurements, the resolution was set by 100 µm slits placed in front of the PSD and 600 mm from it, giving an in-plane resolution Δq_|| = 6.7 × 10⁶m⁻¹. Extreme care was taken to minimize the background. The incident and scattered beams travelled through vacuum paths and the measurements were performed under a water saturated helium atmosphere. The residual background was recorded and subtracted by lowering the trough and scanning around the direct beam.¹⁶

3 Results and discussion

The diffuse scattering data were analyzed using for the differential scattering cross-section (intensity scattered per unit solid angle Ω in the direction k_sc per unit of incident flux in the direction k_in):q_|| and q_z are the in-plane and vertical components of the wavevector transfer. A is the illuminated area and r_e ≈ 2.818 × 10⁻¹⁵ m is the classical electron radius which is the scattering length of electrons for elastic photon scattering. tⁱⁿ (resp.: t^sc) is the Fresnel transmission coefficient between air and water, for the grazing angle of incidence θ_in (resp.: the scattering angle θ_sc). The coefficient tⁱⁿ represents a good approximation to the actual field at the interface while t^sc describes how the scattered field propagates to the detector.¹⁷t^sc is responsible for the peaks at low θ_sc in Fig. 2 (the so-called Yoneda peak). cos²ψ (see Fig. 1) is a polarisation factor (here close to 1). 〈z(0)z(r_||)〉 is the height–height correlation function, and 〈z²〉 ≡ 〈z(0)²〉 the surface r.m.s. roughness. δρ(z) is the electron density profile of the film of thickness l minus the electron density of waterρ_w. Eqn (2) is finally multiplied by the incident flux and integrated over the detector solid angle to get the scattered intensity. The approximation which is valid for small q_z²〈z²〉 values was used to perform numerical calculations with

〈z(q_||)z(−q_||)〉 = k_BT/[Δρg + γq_||² + κ(q_||)q_||⁴] (4)

Fig. 2 Scattered intensity along the position sensitive detector as a function of the scattering angle θ_sc for ψ = 0.08°. Behenic acid on a 1.88 × 10⁻⁵ mol l⁻¹MgCl₂ subphase (20 ions per headgroup) at pH = 10.5 (filled squares). Behenic acid on a 8.46 × 10⁻⁶ mol/l MnCl₂ subphase (90 ions per headgroup) at pH = 7.5 (filled circles). Best fits are indicated by solid lines. Inset: corresponding electron density profiles. Behenic acid on a 1.88 × 10⁻⁵ mol l⁻¹MgCl₂ subphase (20 ions per headgroup) at pH = 10.5.

In addition to the renormalized bending rigidity, it has been taken into account in eqn (4) that in our system capillary waves will always be limited by gravity at very long wavelengths (Δρ is here the density difference between water and air) and also that the crystals might have a residual tension γ. Eqn (3) was carefully checked in the case of a constant κ for which an analytical expression for 〈z(0)z(r_||)〉 is available.

Experimental results for the diffuse scattering are represented on Fig. 2 and 3. In Fig. 3 numerical integration was performed between θ_sc = 12.6 × 10⁻³ and θ_sc = 21.1 × 10⁻³ rad, whereas the z-dependence of the scattered intensity along the position sensitive detector for q_|| = 5.65 × 10⁷ m⁻¹ (ψ = 0.08 degrees) is given in Fig. 2. As can be seen on Fig. 3, the scattering is dramatically enhanced when the organic–inorganic crystals are present at the surface. Such an enhanced scattering was previously observed in the case of ferric stearate, but on compressed films exhibiting no crystallinity.¹⁸ Starting at large q_|| values where all curves merge, the scattering from the water surface increases with the q_||⁻² power law characteristic of capillary waves limited by surface tension. All curves for the organic–inorganic two-dimensional crystals first increase with a much larger exponent, close to 4. This is to our knowledge the first time such a clear signature for bending limited fluctuations is observed at the air–water interface.

Fig. 3 Diffuse scattering differential cross-section dσ/dΩ as a function of the wavevector transfer q_||. Pure water (empty squares). Behenic acid on a 1.88 × 10⁻⁵ mol l⁻¹MgCl₂ subphase (20 ions per headgroup) at pH = 10.5 (filled squares). Behenic acid on a 1.88 × 10⁻⁶ mol l⁻¹MgCl₂ subphase (2 ions per headgroup) at pH = 10.5 (empty circles). Behenic acid on a 8.46 × 10⁻⁶ mol l⁻¹MnCl₂ subphase (90 ions per headgroup) at pH = 7.5 (filled circles). Best fits are indicated by solid lines. The long-dashed line has η_k = 0[(q_||⁻⁴) power law] and the short-dashed line has η_k = 0.65.

All fits of the diffuse scattering curves or Bragg peak profiles were performed by systematically exploring all parameter values on a grid, therefore avoiding local χ² minima. The error bars were estimated by changing the value of a given parameter and fitting the other parameters until χ² reaches its minimum value plus 1. Results of the fit are given in Table 1. Interestingly, all curves can be fitted with similar values of η_k. We also give in Table 1κ(q_|| = 1 × 10⁹ m⁻¹) which can be understood as the “microscopic” bending rigidity as 1 × 10⁹ m⁻¹ is smaller than 2π/(intermolecular distance) and η_k is small (therefore the exact q_|| value is unimportant). κ(q_|| = 1 × 10⁹m⁻¹) is of the order of 100 k_BT, a value which lies in between the bending rigidities of lipid bilayers in the fluid phase (20–30 k_BT) and in the gel phase (300 k_BT)¹⁹ and is therefore not extremely large. This is indeed the in-plane cohesion and elasticity which is responsible for the large effects at small q. In all cases η_k is different from zero and also from 0.65 found in most numerical simulations of tethered membranes.¹¹ The effect of renormalization is sizeable; for example we have for the Mg²⁺ case κ(q_|| = 1 × 10⁹m⁻¹) = 110 k_BT and κ(q_|| = 1 × 10⁸m⁻¹) = 170 k_BT. In contrast to the exponent and microscopic rigidity, very different values of the tension are obtained for the different curves. In particular, the value of the tension for Mg²⁺ with 2 ions per headgroup is larger than that for 20 ions per headgroup which is consistent with the idea of a lower cohesion when less cations are present in the membrane. Following this line of interpretation, cohesion is also lower for Mn²⁺ at pH = 7.5 than for Mg²⁺ at pH = 10.5.

Table 1 Bending rigidity exponent η_k, surface tension γ and microscopic value of the bending rigidity κ(q_|| = 1× 10⁹m⁻¹) for organic–inorganic two-dimensional crystals

	ηk	γ/mN m^−1	κ (q|| = 1 × 10^9 m^−1)/kBT
Mg^2+ 20 ions head^−1	0.18 ± 0.06	0.45 ± 0.07	110 ± 12
Mg^2+ 2 ions head^−1	0.29 ± 0.06	2.2 ± 0.25	110 ± 11
Mn^2+ 90 ions head^−1	0.30 ± 0.08	6.5 ± 0.6	88 ± 30

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In order to further investigate the possibility that our non zero η_k could be due to long-range phonon-mediated interactions, we have performed high-resolution diffraction measurements. The diffraction data were first analyzed by correcting the apparent diffraction angle on the detector for the diffraction from the analyzer crystal: due to diffraction, ψ is shifted by arcsin(sinθ_B/ cos θ_sc) − θ_B, where θ_B is the Bragg angle of the analyzer crystal.^20,21q_|| and q_z were then calculated using this corrected value, and the signal was finally integrated along q_z. The Bragg peak profile results from elastic fluctuations. In order to obtain the experimentally measured intensity, we also need to take into account the finite size of the domains and their random orientation ϕ. We finally get for the differential scattering cross-section in the neighborhood of a reciprocal space vector:^22,23

where G is a reciprocal lattice vector, u a small displacement and L the average domain size. For a hexagonal lattice, 〈eiG·[u(r∥) − u(0)]〉 ∝ r−ηG∥ with ηG = (kBT|G|2/4π) (3µ + λ)/µ(2µ + λ), which implies that ηG ∝ G(2 − ηu). The integration on r|| giveswhere Γ is the incomplete Gamma function and 1F1 the confluent hypergeometric Kummer function used in eqn (5) for the numerical calculation of the scattering cross-section which is finally convoluted with the experimental resolution function to obtain the scattered intensity. From the fit to the experimental data, we obtain ηG and from the dependence of ηG on G, we obtain ηu.

Experimental data and best fits are given in Fig. 4 for the (01) and (02) reflections of behenic acid on a 1.88 × 10⁻⁵ mol l⁻¹MgCl₂ subphase (20 ions per headgroup) at pH = 10.5. Obviously, the η_G values which describe the long-range decay of the intensity differ with η_G = 0.29 ± 0.3 for the (01) reflection and η_G = 0.43 ± 0.1 for the (02) reflection. By fitting these values and corresponding exponents for the (01) (11̄) (2̄1) and (02) reflections for behenic acid on a 8.46 × 10⁻⁵ mol l⁻¹MnCl₂ subphase (90 ions per headgroup) at pH = 7.5 to η_G ∝ G^η_u, we find η_u = 1.35 ± 0.18 in the Mg²⁺ case and η_u = 1.47 ± 0.18 in the Mn²⁺ case. The value expected from the scaling relation η_u = 2 − 2η_k with η_k ≈ 0.25 ± 0.07 is η_u = 1.5 ± 0.14 and is in good agreement with the experimental values.

Fig. 4 Bragg singularity profiles for the (01) (empty circles) and (02) (filled circles) reflections and best fits (lines). The dashed line represents the resolution function.

4 Concluding remarks

In this paper, we have determined the exponents governing the bending rigidity η_k = 0.25 ± 0.07 and the elastic constants η_u = 1.41 ± 0.2 for organic–inorganic two-dimensional crystals. These crystals have a microscopic bending rigidity ≈ 100 k_BT at q_|| = 1 × 10⁹ m⁻¹ consistent with that of organic amphiphiles under similar conditions. Remarkably, this is the crystallinity brought by the interaction between the organic molecules and the inorganic divalent cations which leads to a strenghtening of the nm thick membranes at larger length scales. This interaction also determines the residual tension which is chemistry dependent. Whereas the exponents obey the scaling relation η_u = 2 − 2η_k, supporting our interpretation that we observe the phonon-mediated interaction between capillary waves expected in crystalline membranes, the values we find differ from most of the values found in numerical simulations (η_k ≈ 0.25 ± 0.07 instead of 0.7–0.84 for the simulated value and η_k ≈ 1.3–1.5 instead of 0.32–0.6 for the simulated value). This discrepancy, which requires further analysis, could be due to the tension absent in the simulations; electrostatic interactions could also play a role.

Acknowledgements

Funding by the ACI-nanosciences program under grant NR0106 is gratefully acknowledged.

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