Abstract
Appropriate combinations of laser beams can be used to trap and manipulate small particles with optical tweezers as well as to induce significant optical binding forces between particles. These interaction forces are usually strongly anisotropic depending on the interference landscape of the external fields. This is in contrast with the familiar isotropic, translationally invariant, van der Waals and, in general, Casimir–Lifshitz interactions between neutral bodies arising from random electromagnetic waves generated by equilibrium quantum and thermal fluctuations. Here we show, both theoretically and experimentally, that dispersion forces between small colloidal particles can also be induced and controlled using artificially created fluctuating light fields. Using optical tweezers as a gauge, we present experimental evidence for the predicted isotropic attractive interactions between dielectric microspheres induced by lasergenerated, random light fields. These lightinduced interactions open a path towards the control of translationally invariant interactions with tuneable strength and range in colloidal systems.
Introduction
The familiar isotropic dispersion forces between neutral objects arise from random electromagnetic waves generated by equilibrium quantum and thermal fluctuations^{1,2,3,4}. Depending on the context, these forces are known as nonretarded van der Waals–London, Casimir–Lifhsitz and, more generally, Casimir forces^{1,2,3,4}. The interplay between Casimir forces and electrical doublelayer forces, which forms the basis of the famous DerjaguinLandauVerweyOverbeek (DLVO) theory^{1} describing the forces between charged surfaces in a liquid medium, plays a key role in the colloidal behaviour observed in biological fluids (for example, proteins, biopolymers and blood cells), foodstuffs (for example, dairy, thickeners, emulsions and creams) or suspensions (for example, pharmaceuticals, slurries, paints and inks)^{5,6}. Colloids have also been shown to be extremely well suited for the study of phenomena such as crystallization, the glass transition, fractal aggregation and solid–liquid coexistence^{7,8,9}. External control of isotropic interactions in colloidal systems is therefore of key importance. Temperaturesensitive swelling of microgel particles offers control over soft repulsive forces, but the process is slow, shows hysteresis^{10} and the properties of the colloids are altered while swelling. In some cases, magnetic and dielectric dipolar forces can be induced by external fields but these interactions are strongly anisotropic, leading to the formation of chains or anisotropic domains^{11}. Other ways to control colloidal interactions usually involve the change of composition: by adding and removing electrolytes, the range of electrostatic repulsions can be tuned, and by dissolving macromolecules of appropriate size, attractive depletion forces can be induced^{8,9}. However, despite their widespread and successful use, these strategies are still tedious and slow, and do not provide the level of control over interaction forces that, as discussed here, could be achieved by using external laser fields.
Intense light fields can be used to trap and manipulate small particles^{12,13,14} as well as to induce significant optical binding (OB) forces^{13,15,16}, which, in general, are not translationally invariant, showing a strong anisotropy that depends on the interference landscape of the external fields^{16}. Here we show that artificially generated random fields with appropriate spectral distribution can provide control over attractive and repulsive isotropic (and translationally invariant) interactions with tuneable strength and range. In contrast with Casimir interactions, where the forces are dominated by the material’s response at low frequencies, our results open a new way to explore the peculiar optical dispersion of small particles and artificial metamaterials by selecting the spectral range of the random field. As an example, we predict that the interactions between semiconductor particles with relatively high refractive index can be tuned from attractive to strongly repulsive when the external frequency is tuned near the first magnetic Mie resonance^{17}.
Using optical tweezers as a gauge, we present experimental evidence for the predicted isotropic attractive interactions between dielectric microspheres induced by lasergenerated, quasimonochromatic random light fields. We note that isotropic optical forces between particles act instantly and can therefore also be applied dynamically. This can potentially be useful to anneal defects in periodic structures such as photonic crystals, to increase the effective temperature by optically shaking particles or to stabilize nonequillibrium phases such as supercooled liquids and, in general, to control the selfassembly and phase behaviour of colloidal particle assemblies on nano and mesoscopic length scales^{6,7}.
Results
General theory of randomlightinduced interactions
Early work by Boyer^{18} derived Casimir interactions between small polarizable particles from classical electrodynamics with a homogeneous and isotropic classical random electromagnetic field having the spectral density of quantum blackbody radiation including the zeropoint radiation field. Here we extend these ideas to external artificial random fields with arbitrary spectral density, obtaining an explicit expression for the interactions between two arbitrary dielectric objects, which allows a compact description of randomlightfieldinduced interaction forces from dipolar (atomic or nanometre scale) to macroscopic objects. As a limiting case, when the spectral density of the random field corresponds to that of quantum blackbody radiation, we recover the exact trace formulae for Casimir interactions between arbitrary compact objects^{19,20}. Related trace expressions have also been obtained to describe nonequilibrium Casimir interactions between objects held at different temperatures^{21,22}.
We first analyse the connection between randomlightinduced interaction forces and Casimir interactions between two arbitrary objects. We assume that object/particle A is at the origin of coordinates and particle B is displaced at a distance r along the positive z axis in an otherwise transparent and nondispersive homogeneous medium with real refractive index . We consider that the particles are illuminated by a quasimonochromatic random field of frequency ω, which can be described as a superposition of plane waves with random phases and polarizations, propagating in all directions. Each particle can be seen as made of discretized N_{A} and N_{B}, identical cubic elements of volume v and relative permittivity , which act as small polarizable units with an induced dipole proportional to the polarizing field, that is, , where α(ω) is the polarizability given by with . In the presence of a fluctuating polarizing field, E_{inc}(r,t), the induced dipoles are fluctuating quantities and the timeaveraged force on particle B along the z axis may be written as^{18,23}
The total force can be seen as the sum of different contributions. Although for random illumination there is no net force on an isolated particle, the scattered field by B can be reflected back by particle A, leading to a series of multiple scattering events that give rise to a net interaction force between them. An additional contribution arises from the correlations between the induced dipoles. At a first sight, one could think that the incoming exciting fields on the two objects will be completely uncorrelated. However, in a random, statistically stationary and homogeneous electromagnetic field, the fields at two distant points are correlated, with a crossspectral density of the correlations identical to that of blackbody radiation^{24} (see Supplementary Note 1). In the absence of absorption (when the relative permittivity and are real numbers), the sum of the two contributions lead to a conservative interaction force, which can be expressed in terms of the T matrix^{25} of each individual object (see Supplementary Notes 2 and 3)
where [u_{E}(ω)dω]=U_{E}(ω) is the energy of the fluctuating electric field per unit of volume and k=n_{h}ω/c is the wave number (c is the speed of light in vacuum). is the Green tensor connecting the two objects and ‘Tr’ stands for the trace of the matrix. The dependence of the interaction on distance is completely contained in , whereas all the shape and material dependence is contained in the T matrices. The connection with Casimir interactions can be made through Boyer’s approach: When the objects are in equilibrium with a quantum black body radiation, the energy density U_{E}(ω) corresponds to the electric quantum zeropoint fluctuations (at zero temperature and positive ω) given by^{26,27} U_{E}(ω)=u_{E}(ω)dω=ℏk^{3}/(4π^{2})dω. For absorbing (emitting) particles, we must include an additional contribution to the total force coming from the fluctuating dipoles and the corresponding radiated fields^{28} (linked through the fluctuationdissipation theorem). Interestingly, in equilibrium, this additional contribution conspires with the force due to the field fluctuations, to give a total interaction potential, which is exactly given by equation (3), now including light absorption and emission (that is, ), and we recover the exact Casimir interaction between arbitrary compact objects^{19,20}. In contrast with the traditional Casimir forces, equation (3) opens the path towards complete control and tunability of isotropic dispersion forces between compact bodies by tailoring the spectral density of artificially generated random fields.
Interactions between dipolar electric and magnetic particles
In the limit of small dipolar particles, equation (3) leads to Renne’s result^{29} obtained from quantumelectrodynamic calculations, which is identical to Boyer’s result^{18} based on classical electrodynamics (the more familiar Casimir–Polder result is recovered in the weak scattering limit). Recent theoretical works on OB between dipolar particles under noncoherent random illumination^{30,31} and on dipolar particles near fluctuating light sources^{32,33} suggested striking similarities between dipolar optical forces in random fields and Casimir interactions. However, from a practical point of view, the creation of isotropic random light fields and the direct detection of the resulting weak OB forces between particles at room temperature is challenging. For small nonabsorbing particles, illuminated by a quasimonochromatic random field, equation (3) takes the simple form
which can be shown to be equivalent to equation (11) in ref. 31 in the absence of absorption. In the weak scattering limit
The interaction energy far from resonance is attractive but always much smaller than k_{B}T for realistic power densities, whereas at resonance the polarizability is purely imaginary (at resonance, , that is, α^{2}<0), which leads to an effective repulsion. The latter was shown to play a key role in understanding the collective behaviour of opticaltrapped neutral atoms^{34}. Plasmonic or polaritonic nanoparticles show a high real and imaginary polarizability close to a resonance, but this can lead to a significant increase of the temperature^{35}. Nonabsorbing semiconductor nanoparticles, with relatively high refractive index, or colloidal dielectric micronsized particles would offer an attractive laboratory to verify our predictions. Semiconductor particles, for example, silicon (Si) spheres with index of refraction ∼3.5 and radius ∼200 nm^{36}, present strong electric and magnetic dipolar resonances in telecom and nearinfrared frequencies (that is, at wavelengths 1.2–2 μm) without spectral overlap with quadrupolar and higherorder resonances^{17}. Assuming that the scattering by these Si particles can be described by just dipolar electric and magnetic fields, it is possible to obtain an exact closed expression for the interaction potential, equation (3), in terms of the electric and magnetic polarizabilities (see Supplementary Notes 4 and 5). Figure 1d,g illustrates how a monochromatic random illumination induces a pair potential between two identical 230nmradius Si nanospheres, which can be tuned from attractive offresonance (λ∼2 μm) to strongly repulsive when the external wavelength is tuned near the first Mie’s magnetic resonance (λ∼1.6 μm).
Interactions between micronsized dielectric particles
To derive a simplified theoretical expression for the interaction potential between micronsized dielectric particles, we approximate equation (3) at lowest order in perturbation theory (see Supplementary Note 6). In this limit, the interaction energy can be seen as given by a pairwise interaction u(r_{B}−r_{A}) summed over the volume of the spheres, that is, similar to Hamaker’s integral^{1},
with size independent coefficients as a measure of the materials contribution to the interaction potential for a given spectral component k_{n}=n_{h}ω_{n}/c of the random light field. D=r−2R denotes the gap distance between the surfaces and r is the distance between the centres of the two spheres. U_{E}(k_{n}) is the power density of the random light field for a specific wave number k_{n}. For monochromatic illumination and in the limit of kR<<1<<kD, we recover the expected results for dipolar particles^{15,31} where the interaction energy is proportional to the squared volume of the particles. Interestingly, for relatively large particle sizes (in an intermediate regime 1≪kD≪kR) we find:
that is, for relatively large particle sizes, our approach predicts that the interaction energy should scale with the particle’s radius. Otherwise equation (6) can be easily computed numerically. Figure 1h,i illustrates the predicted theoretical sensitivity of the induced pair interactions between lowindex (n=1.68) micronsized spheres to changes in the wavelength of the illuminating random field. As forces can be strongly frequency dependent, actual dispersion forces could be manipulated by tuning the spectrum of the random light field. As an example, in Fig. 1e,h we show that it is possible to tailor the range of an effective attractive dispersion interaction potential with constant depth by selecting different wavelengths and fieldpower densities (calculations are performed using equation (6)). Selecting an appropriate continuous spectrum of the random light fields it is also possible to design a nearly exponential attractive potential. Taking into account the wellknown exponential electrostatic doublelayer repulsion, it would theoretically be possible to induce an effective Morsetype interaction potential on a colloidal level, as shown in Fig. 1f,i.
Experimental observation of random light interactions
Experimentally, the manipulation of relatively large, micronsized dielectric particles requires highintensity laser fields: in conventional optical tweezing experiments, the focused laser intensity I_{0}=P/A required to tightly trap a micronsized polystyrene sphere (refractive index n∼1.6) is on the order of mW μm^{−2} and the corresponding power density is U_{E}=n_{h}/c·I_{0}. Therefore, to create random light fields with a comparable power density, the surface area A cannot exceed some tens of micrometres squared, for an incident laser power P on the scale of Watts. To this end, we have designed a miniaturized sample cell that allows for the simultaneous creation of a random light field in a small cavity by illumination with a strong green laser λ=532 nm and the observation of OB forces between two isolated microspheres (Fig. 2). We use the technique of timeshared optical tweezers^{37,38,39} in combination with umbrella sampling^{40}, to probe the particle pair interaction potential U(r) of two melamine microspheres (n=1.68) as a function of their centretocentre separation distance r. The principle of this method is to trap, at the same time, two particles in two identical optical tweezers that are separated by a distance r_{0} using a nearinfrared laser beam with a wavelength of 785 nm. By monitoring the relative motion of the trapped particles with a digital camera, information about the pair potential U(r) around r_{0} is gained. In contrast to vacuum, the interactions between particles suspended in water commonly involve both van der Waals and screened electrostatic repulsive interactions (DerjaguinLandauVerweyOverbeek DLVO interactions). The latter ensures the stability against particle coagulation. Equally, in our measurements, this repulsive part dominates at very short distances and in turn this allows us to probe the superimposed lightinduced attractions, without particles sticking together irreversibly (see Supplementary Fig. 1). We follow the approach of Grier and coworkers^{41}, to obtain an autocalibrated measurement of the colloidal interaction potential by analysing the differences between the distributions of the particle positions in the presence and the absence of optically induced forces (see Supplementary Methods and Supplementary Fig. 2). Thus, in our experiments, by turning on and off the random light field, we obtain directly the lightinduced interaction potential without an explicit measurement of the complete pair interaction potential. Because of the finite width of the laser traps, the particles sample only a small range of U around r_{0}. To determine the potential over a wider range of distances, we repeat the measurements for different relative positions of the optical traps (umbrella sampling). Typically, we start at an average trap separation that is near contact and increase r_{0} in steps of Δ=40 nm until six different trap positions are scanned. Δ is chosen to be smaller than the width of the trap potential 2σ≈140 nm, to have sufficient overlap between neighbouring trap separations.
The final results for U(D=r−2R), shown in Fig. 3, are obtained by averaging over 16 independent experimental runs carried out under the same conditions. We find clear evidence for OB in a random light field. Moreover, we are able to quantify the potential U(D=r−2R) as well as its dependence on the incident light power. Varying the laser power from 5.0 to 1.0 W weakens the attractive potential. For an estimate of the contact potential U_{0}≡U(D=0), we adjust the prefactor in equation (6) for a best fit to the data as shown in Fig. 3. The overall agreement between experiments and theory is remarkable, except for the predicted oscillation of the interaction potential at large interparticle distances. We believe the lack of clear oscillations can be explained by the approximations made when deriving equation (6) in the first Born (Rayleigh–Gans) approximation. The predicted oscillations are due to constructive and destructive interferences when summing over equidistant pairs of elementary dipolar scatters in the particle. However, for higher refractive indexes these interferences are smeared out, as equidistant pairs of dipoles positioned at different distances from the particle surface do not contribute in the same way anymore. An additional contribution that might affect the comparison between theory and experiment is the absence of a full 4π isotropic illumination in the experiment due to the absence of incoming photons with momentum near parallel to the mirror.
In Fig. 4, we display results for the contact potential U_{0} as a function of laser power P for the three different particle sizes. From the slope of the linear fits (dashed lines) we can extract the normalized contact potential U_{0}/P, which does not depend on the laser power. U_{0}/P increases with particle size and the data set is consistent with the linear increase as predicted by equations (6) and (7). Finally, we attempt a quantitative comparison between the experimental results and the theoretical predictions for the contact potential U_{0}. We note that such a comparison is based on a number of uncertainties related to the approximations made in the theory as well as the experiment. From equation (6), we compute the theoretically predicted values for melamine microspheres with an index of refraction of n_{P}=1.68 in water (n_{h}=1.33) as a function of particle size. For the crosssectional area of the lightfilled cavity, we use A≃0.004 mm^{2}. Moreover, we take the light power in the cavity equal to the incident laser power for a vacuumwavelength λ_{0}=532 nm. Under these assumptions, the theory predicts . This result matches the experimental value (Fig. 4, inset). Given the approximations made, such a near quantitative agreement might be somewhat fortuitous but nonetheless the overall agreement between experiment and theory supports our findings.
Discussion
In this study, we introduce the concept of randomlightfieldinduced forces as a tool to control translationally invariant interactions between small particles in two or three dimensions with tuneable strength and range. Such external control over isotropic interactions in colloidal systems is unprecedented and of key importance^{7,8,9}. Other ways to control colloidal interactions, such as adjusting the solvent quality, fieldinduced dipolar interactions or doublelayer screening, are limited in scope and precision or involve the change of composition^{8,9,10,11}. Nonetheless, these strategies are widely applied even though they do not provide the level of control over interaction forces that, as discussed here, can be achieved by using externally applied random light fields. Moreover, lightinduced forces between particles act instantly on time scales of Brownian motion. This rapid response opens the path for new applications in threedimensional colloidal systems: randomlightcontrolled forces of sufficient strength can induce phase separation and aggregation, but when applied dynamically could also be targeted to stabilize nonequillilibrium phases or to anneal defects in periodic structures such as photonic crystals. In summary, the approach presented here will open the pathway for a new field of fundamental research on isotropic optical forces and at the same time it will equip researcher with a powerful tool to control and manipulate the selfassembly and phase behaviour of colloidal particle assemblies on nano and mesoscopic length scales^{6,7}.
Methods
The creation of artificial random light fields
A focused green laser beam (Coherent VerdiV5, λ=532 nm, calculated 1/e^{2} beamwaist 2w_{0}=25 μm) with a laser power of up to 5 W is focused with an f=75 mm lens on the back surface of a glass capillary filled with a dense amorphous solid composed of PMMA (polymethylmethacrylate) beads, diameter ∼0.4 μm, with a layer thickness of ∼20 μm. This first layer scatters >99% of the incident power and thereby creates a random light field in the sample cavity of thickness t≃15 μm (inset, Fig. 2). The latter is filled with a very dilute suspension of melamine microspheres (Microparticles GmbH, Germany) with a diameter of 2 μm≤2R≤4 μm and refractive index n=1.68, dispersed in aqueous solution and sealed with ultravioletcurable glue. To screen the electrostatic repulsions, we add 2.7 mM KCl. This results in a Debye screening length of λ_{D}≃6 nm. The thickness of the layer containing the melamine microspheres is controlled by adding a small amount of t=2R=15 μm silica spheres that serve as spacers. Both layers are separated by the outer glass wall of the capillary with a thickness of h=20±2 μm (CM Scientific). A dichroic mirror with a reflectivity >99% at λ=532 nm (for incident angles 0°–45°, custom made by Asphericon GmbH, Jena, Germany) is placed on the opposite side of the cell. The mirror is transparent for wavelengths above λ∼650 nm, allowing both the trapping and the visual observation of the melamine microspheres. The clear layer sandwiched between the turbid layer and the mirror (cavity) thus has a thickness of h+t≃35 μm. The thin slab geometry together with the scattering in the turbid layer and multiple reflections in the cavity assure that in the centre of the sample cell we generate a fairly isotropic and quasimonochromatic random light field. The volume speckle inside the cavity displays intensity fluctuations on a typical length scale^{42} of λ/2∼200 nm. In contrast with the anisotropic static light speckle pattern used to generate random potentials landscapes in the quest for Anderson localization of matter waves^{43}, we generate random temporal fluctuations of the light fields by rapidly scanning the green laser focal spot over the surface of the turbid layer. We set the scan distance to ±7 μm at an oscillation frequency of 500 Hz using a galvano mirror. The oscillation frequency is chosen in a way that light field fluctuates randomly much faster than τ_{B}=R^{2}/(6D_{0})=776 ms, the time scale for Brownian motion of the smallest melamine microspheres under study. Here, D_{0} denotes the Brownian diffusion coefficient.
It has been recently reported that the collective motion of a large set of microspheres under the influence of both Brownian and selfinteracting optical forces becomes active and their dynamical quantities are no longer representative of thermodynamic equilibrium^{44}. In our case, however, the rapidly fluctuating speckle is not coupled to the motion of the melamine spheres. Other recent experimental studies have investigated small particles in one and twodimensional spatially modulated light fields^{45,46,47}. Such static or slowly varying light fields, however, influence the behaviour of small particles in a fundamentally different way than dispersion forces. They result in a quasistatic potential energy landscapes similar to the case of optical traps. The same strategies have already been exploited earlier to create energy landscapes with crystalline or quasicrystalline order with the aim to manipulate or influence the spatial distribution of small particles in two dimensions^{14,48}. In our case, however, the mean light power is spatially invariant on Brownian and longer times scales, which, in full analogy to dispersion forces, leads to translationally invariant interactions in contrast to these previous approaches.
It is important to note that roughly two thirds of the incident laser light will be reflected by the turbid layer. However, the reduced incident power is compensated by multiple reflections inside the cavity, except for the small residual losses by the dichroic mirror. Therefore, we expect the laser power in the cavity to be approximately the same as the incident laser power P. Reflections in the cavity can lead to an increase of the effective surface area A as the reflected light can spread out laterally. To obtain an estimate of A, we image the residual green light, for an incident laser power of 1 W, transmitted by the dichroic mirror by increasing the exposure time and the gain setting of the camera corresponding to an increase in detection efficiency by more than four orders of magnitude. From an analysis of the slightly elliptical intensity distribution, we derive an areal crosssection of A∼π × 41 × 34 μm^{2} ∼0.004 mm^{2} based on the 1/e decay length along the major and the minor axis. This means that for a laser power of up to 5 W, we can indeed reach intensities ∼mW μm^{−2} comparable to the case of optical tweezing.
Experimental procedure
In one experimental run, we observe the Brownian motion of two micronsized melamine microspheres that are held at a mean distance r_{0} in the centre of the lightfilled cavity, both laterally and axially, using a specifically adapted Nikon Eclipse TS100 brightfield microscope composed of (i) a long working distance objective (20 × /0.42 EO Plan Apo ELWD) for the sidewise white light illumination, (ii) an oilimmersion objective for both the trapping and the observation of the particle motion using a CCD (chargecoupled device) camera and (iii) a notch filter to filter out residual stray light coming from the trapping laser as well as a dielectric mirror to couple the trapping laser into the optical path of the microscope.It is noteworthy that the brightfield illumination takes place across the first diffusing layer. For the laser trapping, we use a nearinfrared laser beam (Toptica DL 100) with a wavelength of 785 nm and a power of ∼8 mW (measured at the exit of the laser) that is rapidly switched between two positions with a galvano mirror to produce the time shared dual traps^{37,38,39}.
Using a digital camera (Prosilica GC650, Allied Vision Technologies GmbH, Germany) we record images of 120 × 120 pixels with a frame rate of 90 Hz and an exposure time of 0.3 ms at × 80 magnification. The edge length per pixel is d_{pix}≈100 nm, which provides a subpixel localization accuracy of the particle centre better than 10 nm^{41}. For a given mean separation distance r_{0}, we perform two experiments (see Fig. 2). In the first experiment, we acquire a movie of 4,000 images at a frame rate of 90 Hz under the influence of a random light field. In a subsequent reference experiment, the random light field is turned off and the measurement is repeated under otherwise identical conditions. The recorded sequence of images is analysed using a standard particletracking algorithm^{49} to obtain distributions of the centretocentre separations of the trapped particles for both experiments (see Supplementary Methods and Supplementary Fig. 2).
Additional information
How to cite this article: Brügger, G. et al. Controlling dispersion forces between small particles with artificially created random light fields. Nat. Commun. 6:7460 doi: 10.1038/ncomms8460 (2015).
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Acknowledgements
This work was supported by the Swiss National Science Foundation through project numbers 132736 and 149867, and through the National Centre of Competence in Research BioInspired Materials. J.J.S. acknowledges financial support by the Spanish MEC (grant number FIS201236113), Comunidad de Madrid (grant number S2009/TIC1476Microseres Program) and by an IKERBASQUE Visiting Fellowship (J.J.S.). We thank J.F. Dechézelles for providing us with the PMMA particles.
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Author notes
 Frank Scheffold
 & Juan José Sáenz
These authors contributed equally to this work.
Affiliations
Department of Physics, University of Fribourg, Chemin du Musée 3, Fribourg CH1700, Switzerland
 Georges Brügger
 , Luis S. FroufePérez
 & Frank Scheffold
Depto. de Física de la Materia Condensada, Instituto Nicolás Cabrera and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, Fco. Tomas y Valiente 7, Madrid 28049, Spain
 Juan José Sáenz
Donostia International Physics Center (DIPC), Paseo Manuel Lardizabal 4, DonostiaSan Sebastian 20018, Spain
 Juan José Sáenz
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Contributions
J.J.S. and F.S. conceived the study. G.B. and F.S. designed the experiment. G.B. carried out the experiments. J.J.S. derived the theory. G.B. and L.F. performed the numerical calculations. G.B., F.S., L.F. and J.J.S. analysed and interpreted the data. J.J.S., F.S. and G.B. wrote the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Frank Scheffold or Juan José Sáenz.
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Supplementary Figures 12, Supplementary Notes 16, Supplementary Methods and Supplementary References.
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