Silicon wafers (Soitec, Bernin, France) were used as substrates. Prior to the experiments, the wafers were cleaned with Piranha solution, using a 1:1 (vol:vol) mixture of hydrogen peroxide (30%, Th. Geyer, Renningen, Germany) and sulphuric acid (96%, Carl Roth, Germany). Afterwards, the wafers were rinsed with large amounts of water and dried in a nitrogen stream. Non-porous silica particles (Bangs Laboratories, Fishers, United States) with a diameter of 5 μm were used as colloidal probes. One particle was glued (UHU Endfest Plus 300, UHU, Bühl, Germany) to the end of a tipless rectangular cantilever (SD-qp-SCONT-TL, Nanosensors, Neuchatel, Switzerland) using a three-dimensional microtranslation stage. Immediately before the experiment, both, cantilever and substrate, were exposed to oxygen plasma (Diener Femto, Ebhausen, Germany) to remove all organic impurities.

Force measurements were carried out using a Cypher ES atomic force microscope (Asylum Research, Santa Barbara, United States). Before each experiment, the spring constant of the cantilever was determined as 0.012–0.020 N m⁻¹ using the method described by (Sader et al., 1999). After calibration, cantilever and substrate were completely immersed into the micellar dispersion. The temperature was set to 20.0°C via a cooler-heater sample stage and left to equilibrate for at least 30 min. The cantilever deflection was measured as a function of the z-piezo position. Using Hooke’s law, the cantilever deflection was converted into force. To convert the raw data into force F versus separation h curves (in the following described as force profiles), well-known algorithms are used (Ducker et al., 1991; Butt et al., 2005; Ralston et al., 2005). The Derjaguin approximation was used to normalise the measured force F against the effective radius R _eff. For a planar plate–sphere geometry, R _eff is simply the colloidal probe radius R. The resulting force is, therefore, in the following described as F R .

The starting point of each measurement was set to 500 nm with an approach-retraction velocity of typically 50 nm s⁻¹. For very viscous samples (pure Tween20 dispersions at all concentrations and Tween20-SDS mixtures with [c = 146 mM, X = 0.12], [c = 218 mM, X = 0.12], and [c = 224 mM, X = 0.24]), that velocity was reduced to 10 nm s⁻¹. At these velocities, the hydrodynamic force on the cantilever is negligibly small. The surfaces were assumed to be in contact once the normalised force exceeds 0.4 mN m⁻¹ (constant compliance region). For each system, at least 30 individual force profile were time averaged (binomial smooth, 10³ points) to ensure reproducibility and to substantially increase the force resolution.

The normalised interaction forces F R between two surfaces at separations h are described by the oscillatory structural force F _osc F R h = F osc R h = − A ⋅ exp − h ξ ⋅ cos 2 π λ h − h ′ . (1) with the amplitude A, the decay length ξ, and the wavelength λ. The parameter h′ corrects for the offset of the structural force. The fit parameters of the oscillatory structural force may vary so that the fit parameters are not independent of the fit region (Schön and von Klitzing, 2018). To account for this, the starting point of the fit is varied over the stable part of the first force oscillation. Error bars refer to the standard deviations of all individual fits with different starting points in the force profile.

Typically, van der Waals forces are taken into account describing the interaction forces. They are neglected in the following analysis, since they are small and short ranged and do not affect the interactions under the studied conditions. We also neglect double layer forces between the confining surfaces in this study, because they do not influence the force profile at surface separations above the first maximum of the oscillatory force profile.

The surfaces are expected to be in contact at a force of 0.4 mN m⁻¹ in order to achieve a stable contact region. In the presence of nonionic surfactants, adsorption of surfactant onto the confining surfaces may occur (Grant et al., 1998). The absolute values of the surface separation h should, therefore, be treated with care.

Figure 2 shows the force profiles between silica surfaces across nonionic Tween20 micellar dispersions measured by CP-AFM.

All data points of the individual measurements are stacked and displayed in bright colours. This unsmoothed data highlight that the transitions (also referred to as “jumps”) between stable force branches are often not clearly pronounced. Jumps in the force profile appear once the gradient in the force profile exceeds the cantilever spring constant. To increase force resolution, the stacked raw data are smoothed and displayed as larger dots (see Section 2.2.2 for details). Only stable force branches are considered while data point between them are not represented in the smoothed data.

Oscillatory structural forces emerge with increasing volume fraction of Tween20. At a volume fraction of ϕ = 0.240, almost no oscillatory structural forces could be detected which is in agreement with a former study (Christov et al., 2010). At a volume fraction of ϕ = 0.374, clear oscillatory structural forces with at least five oscillations in the force profiles are observed. These experimental data are fitted to a damped oscillatory profile (Eq. 1). The high viscosity of the dispersion enhances the thermal noise in the measurement substantially. At surface separations below h ≈ 4.8 nm, adsorbed surfactant on the silica surfaces may be confined. This layer then ruptures at F R ≈  0.8 mN m⁻¹. The fact that nonionic surfactants can adsorb onto silica surfaces has already been described in literature. Often they form hemicylinders (Grant et al., 1998).

Nonionic BrijL23 micelles show slower formation and decomposition kinetics compared to Tween20 micelles (Patist et al., 2002). BrijL23 micelles are, therefore, considered as more stable. Force profiles across dispersions of BrijL23 micelles are shown in Supplementary Figure S3. There, the transitions between stable force branches are more pronounced.

Interaction forces between silica surfaces across mixed Tween20-SDS micelles are shown in Figure 3. The plots are arranged in a manner that: 1) the micelle fractional surface charge β increases from left to right; 2) the volume fraction ϕ increases from top to bottom.

Oscillatory structural forces are observed across all these systems. The magnitude and range varies according to the micelle fractional charge β and volume fraction ϕ. At the smallest volume fraction (ϕ = 0.190) and smallest ratio of anionic surfactants in system (X = 0.12), only one oscillation occurs (Figure 3A). The oscillatory structural force becomes stronger due to two reasons: higher micellar surface charge and higher volume fraction. In consequence, the oscillatory structural force is most pronounced in Figure 3 1) showing at least four distinct force oscillations up to surface separation above 40 nm.

Additional SDS in the micelles enhances the electrostatic repulsion between individual micelles and also amplifies the interaction between the micelles and the negatively charged confining surfaces. As a result and unlike for pure nonionic Tween20 micelles, no adsorbed surfactant layer is detected. Moreover, the oscillatory structural forces across mixed micellar dispersions are better defined compared to the pure Tween20 micelles, discussed in Figure 2.

The characteristic intermicellar distances in bulk D* are extracted from SANS measurements (Ludwig et al., 2021). These values are compared to the characteristic intermicellar distance under confinement extracted as wavelength λ of the oscillatory structural force.

Figure 4 shows the characteristic intermicellar distances obtained in bulk and under confinement as a function of the micelle volume fraction ϕ. Distances are normalised with respect to the effective micelle diameter 2r _eff for better comparison.

The characteristic intermicellar distances in bulk D * ( 2 r eff ) − 1 of uncharged Tween20 micelles vary only little with the volume fraction ϕ (open, black diamonds). At small volume fractions ϕ of uncharged micelles, no pronounced ordering is obtained. The onset of a structure factor S(q) in such systems, however, suggests that some of the micelles are within the pair-potential of neighbouring micelles. As a result, the characteristic intermicellar distance approaches a threshold which is just slightly larger than the diameter of a single micelle (see Supplementary Figure S5 for more details).

At higher ϕ, the values of nonionic micelles approach the scaling behaviour for charged particles since at close packing geometries (ϕ ≈ 0.52–0.74) the interparticle distances of charged and uncharged particles will not differ. In the same way, it is reasonable that the characteristic distance between micelles at ϕ ≈ 0.35 changes only slightly by introducing surface charges to the micelles. At ϕ ≈ 0.20, the intermicellar distance subsequently increases with increasing ratio of SDS, i.e., with increasing fractional surface charge β (indicated by the arrow in Figure 4). The electrostatic repulsion between micelles increases with the amount of surface charges. As a result, the micelles form a specific ordering when the micelles exceed a certain amount of surface charge - in this study mixed Tween20-SDS micelles with X = 0.24 and X = 0.35.

At such an ordering, the characteristic intermicellar distances D* for charged particles only depends on the particle number density n _p as D * ∝ n p − 1 / 3 considering simple packing arguments. The normalisation to the effective micelle diameter 2r _eff transfers this into a dependency on the volume fraction ϕ: D * 2 r eff = f ϕ − 1 / 3 (2) The type of particle packing determines the value of the pre-factor f. For a simple cubic packing D * = n p − 1 / 3 = ( 4 / 3 π ) 1 / 3 r eff ϕ − 1 / 3 , so that f = 0.806 in Eq. 2. Experimentally f = 0.718 is found for the ordering of microemulsion droplets (Lindner and Zemb, 2002). In our study, the charged micelles follow approximately the inverse cubic root scaling law [Eq. 2 with f = 0.718, dashed line in Figure 4 (mixed Tween20-SDS micelles) and S7 (a) (pure SDS micelles)].

The characteristic interparticle distance in bulk D* is now compared to the interparticle distance under confinement λ. An existing analytical semiempirical model for the oscillatory structural force exists based on numercial results for hard-sphere fluids to describe the wavelength λ from uncharged micelles. The use of the effective diameter 2r _eff again reveals the dependency to the volume fraction ϕ as (Trokhymchuk et al., 2001): λ 2 r eff = 2 π 4.45160 + 7.10586 ϕ − 8.30671 ϕ 2 + 8.29751 ϕ 3 − 1 (3) In this study, characteristic intermicellar distance between uncharged Tween20 micelles in bulk (empty black symbols in Figure 4) and under confinement (filled black symbol in Figure 4) is smaller than predicted by Eq. 3 (dotted line in Figure 4). Interestingly, the bulk intermicellar distance is larger than under confinement ( D * > λ ). The same behaviour is obtained for pure, uncharged BrijL23 micellar dispersions (Supplementary Figure S6A). Obviously, uncharged micelles are compressible under confinement. The results from SANS and CP-AFM deviate stronger at higher volume fractions ϕ. In contrast to uncharged micelles, charged micelles obey the same intermicellar distance as in bulk and are not compressible (D* = λ). This incompressibility is observed for mixed Tween20-SDS (filled, coloured symbols in Figure 4) as well as for pure SDS micellar dispersions (Supplementary Figure S7A). The wavelengths λ of all charged micelles are described by their characteristic intermicellar distance in bulk D* with f = 0.718 (Eq. 2). This is in good agreement with a previous study on pure anionic micelles, where f = 0.698 is found (Tulpar et al., 2006). Solid colloidal particles, like silica nanoparticles, however, are typically modelled with a factor of f = 0.806, indicating perfect simple cubic packing (Tulpar et al., 2006; Zeng and von Klitzing, 2012; Ludwig et al., 2019). The reduction of the intermicellar distances in bulk and under confinements suggests a less pronounced ordering compared to solid colloidal particles. Whether this is due to surfactant micelles being a more dynamic and softer or being more elliptical compared to solid nanoparticles remains unclear.

It has to be further mentioned that the 3D intermicellar structure can only be sensed as 2D representation, both in the SANS and CP-AFM measurements. Therefore, at high volume fractions ( ϕ >  0.3), the characteristic intermicellar distance may be smaller than the effective diameter of the micelle as indicated by a value below 1 of D* (2r _eff) ⁻¹ for uncharged micelles.

In CP-AFM, the correlation length under confinement is extracted from the decay length ξ from the fit to the oscillatory structural force. In bulk, thus from SANS results, a correlation length is extracted from the full-width at half maximum of the structure peak Δq as 2 Δ q .

Figure 5 compares both correlation lengths. The analytical semiempirical model for the oscillatory structural force of uncharged micelles yields the decay length ξ depending on the effective particle radius r _eff and the volume fraction ϕ as (Trokhymchuk et al., 2001) (dotted line in Figure 5): ξ 2 r eff = 4.78366 − 19.64378 ϕ + 37.37944 ϕ 2 − 30.59647 ϕ 3 − 1 (4) In this study, the decay length ξ is almost constant for all dispersions under confinement, irrespective of the micellar surface charge z or volume fraction ϕ. The hard-sphere repulsion is the dominant parameter for the decay length in the confined dispersions and introducing surface charges does not affect these values. The correlation length of the confined dispersions (filled symbols in Figure 5) agrees well with the correlation length determined for uncharged micelles in bulk (open, black diamonds in Figure 5). The bulk correlation length, however, increases with increasing micellar surface charge. This effect of electrostatic interactions becomes especially important at lower volume fractions, i.e., when the particles are further apart from each other and is only observed between charged micelles in bulk but not under confinement.

The amplitude A of oscillatory structural forces is often related to the interparticle interactions in dispersions. It was, however, demonstrated that the interaction between confining surfaces and the particles alters the amplitude A, e.g., by varying the potential of the confining surfaces (Grandner et al., 2009; Grandner and Klapp, 2010; Ludwig and von Klitzing, 2021), meaning that the amplitude does not only depend on properties of the bulk dispersions.

Figure 6 compares the amplitude A with the strength of the intermicellar ordering in bulk calculated from the excess intensity of the structure factor peak: the maximum intensity of the structure factor is normalised to 1: S _max + S ₀ − 1. This was done, since S(q) = 1 indicates a random intermicellar distribution.

The amplitude A of the oscillatory structural forces enhances at higher surface charges of the micelles (Figure 6A). The effect of ϕ on A is almost negligible, compared to the influence of the surface charge. The amplitude from the oscillatory structural force across pure Tween20 micelles is neglected in the discussion. There, the first layer in the force profile is considered as an adsorbed layer of nonionic surfactant so that the data cannot be compared. Unlike the effect on A, micellar surface charges do almost not affect the intensity of the structure factor peak S _max + S ₀ − 1 in bulk. This parameter is dominated by the volume fraction ϕ. This means, that ordering of the micelles in bulk is due to hard-sphere repulsions, at ϕ ≳ 0.30. As a result of both values, A and S _max + S ₀ − 1, showing different trends, it is assumed that A strongly depends on the interactions of the micelles with the confining surfaces. This information cannot be obtained from bulk small-angle scattering (SANS) experiments because this parameter does not include any information on the interaction between the micelles and the confining surfaces.