Capillary Force on a Micrometric Sphere Trapped at a Fluid Interface Exhibiting Arbitrary Curvature Gradients

Blanc, Christophe; Fedorenko, D.; Gross, Michel; Martin, Ivar; Abkarian, Manouk; Gharbi, Mohamed Amine; Fournier, Jean‐Baptiste; Galatola, P.; Nobili, Maurizio

doi:10.1103/physrevlett.111.058302

Cited by 45 publications

(56 citation statements)

References 22 publications

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“…In experiment, microparticles migrate along curvature gradients to sites of high curvature, as has now been observed for microcylinders [2], microspheres [11], and microdisks [12]. Theoretically, the curvature capillary energy driving this migration is simply the sum of the surface energies and pressure work for particles at the interface.…”

Section: Introductionmentioning

confidence: 88%

“…We apply the approach to derive the curvature capillary energy for spheres with equilibrium contact angles. While the curvature capillary energies for this scenario have been derived previously and reported to be quadratic in the deviatoric curvature of the interface [11,[21][22][23], we find that this term has prefactor zero. We identify the source of the discrepancy between our result and that published previously.…”

Section: Introductionmentioning

confidence: 88%

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Curvature capillary migration of microspheres

2015

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We address the question: How does capillarity propel microspheres along curvature gradients? For a particle on a fluid interface, there are two conditions that can apply at the three phase contact line: Either the contact line adopts an equilibrium contact angle, or it can be pinned by kinetic trapping, e.g. at chemical heterogeneities, asperities or other pinning sites on the particle surface. We formulate the curvature capillary energy for both scenarios for particles smaller than the capillary length and far from any pinning boundaries. The scale and range of the distortion made by the particle are set by the particle radius; we use singular perturbation methods to find the distortions and to rigorously evaluate the associated capillary energies. For particles with equilibrium contact angles, contrary to the literature, we find that the capillary energy is negligible, with the first contribution bounded to fourth order in the product of the particle radius and the deviatoric curvature. For pinned contact lines, we find curvature capillary energies that are finite, with a functional form investigated previously by us for disks and microcylinders on curved interfaces. In experiments, we show microsphere migrate along deterministic trajectories toward regions of maximum deviatoric curvature with curvature capillary energies ranging from 6 × 10 3 − 5 × 10 4 kBT . These data agree with the curvature capillary energy for the case of pinned contact lines. The underlying physics of this migration is a coupling of the interface deviatoric curvature with the quadrupolar mode of nanometric disturbances in the interface owing to the particle's contact line undulations. This work is an example of the major implications of nanometric roughness and contact line pinning for colloidal dynamics.

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Section: Introductionmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 88%

Curvature capillary migration of microspheres

2015

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“…Using the above expressions h 0 ∝ r 2 and η ∝ r −2 , and letting R out → ∞, these authors find in (24) of [1] the relation E out = −E in . This leads them to the conclusion that the trapping energy vanishes at second order, E = E in + E out = O(∆c 4 ). Yet this argument is flawed by the fact that E out is calculated with the near-field deformation (1) which is not correct at R out .…”

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confidence: 99%

Comment on “Curvature capillary migration of microspheres” by N. Sharifi-Mood, I. B. Liu, K. J. Stebe, Soft Matter, 2015,11, 6768

Würger

2016

Soft Matter

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In a recent paper, Sharifi-Mood et al. study colloidal particles trapped at a liquid interface with opposite principal curvatures c1 = −c2. In the theory part, they claim that the trapping energy vanishes at second order in ∆c = c1 − c2, which would invalidate our previous result [Phys. Rev. E, 2006, 74, 041402]. Here we show that this claim arises from an improper treatment of the outer boundary condition on the deformation field. For both pinned and moving contact lines, we find that the outer boundary is irrelevant, which confirms our previous work. More generally, we show that the trapping energy is determined by the deformation close to the particle and does not depend on the far-field. PACS numbers:Colloidal particles trapped at a curved liquid interface are subject to capillary forces that do not depend on their mass or charge but on geometrical parameters only. In Ref.[1], Sharifi-Mood et al. provide an interesting analysis of the role of contact line pinning. Regarding the trapping energy, however, these authors assert that it vanishes at second order, contrary to previous work, and they state that "the origin of the discrepancy is an inappropriate treatment of the contour integral" in [2]. The present comment intends to refute this claim of [1], to unambiguously determine the trapping energy, and to clarify the role of the far-field.Previous works [2-4] rely on the assumption that curvature-induced forces arise from the interface close to the particle and that the far-field is irrelevant. Thus the profile of the bare interface is taken in small-gradient approximation, h 0 = ∆c 4 cos(2ϕ)r 2 , which is valid only at distances shorter than the curvature radius R c = 1/∆c. Adding a particle modifies the profile as h = h 0 +η, where the deformation field η = ∆c 12 cos 2ϕ r 4 0is determined from the contact angle at the particle surface. By the same token, the trapping energy comprises only near-field contributions, and is given by the boundary term along the contact line of radius r 0 ,Since a similar line integral of η∇η is cancelled by the area change due to displacement of the contact line on the particle surface, one obtains the curvature-dependent energy E = E in [2], which was confirmed in [3,4]. Sharifi-Mood et al. attempt to go beyond the nearfield approach and to evaluate the term arising at the outer boundary,where, in the simplest geometry, R out (ϕ) is the distance of the container wall from the particle. Using the above expressions h 0 ∝ r 2 and η ∝ r −2 , and letting R out → ∞, these authors find in (24) of [1] the relation E out = −E in . This leads them to the conclusion that the trapping energy vanishes at second order, E = E in + E out = O(∆c 4 ). Yet this argument is flawed by the fact that E out is calculated with the near-field deformation (1) which is not correct at R out . (Moreover, h 0 ∝ r 2 is valid at distances within the curvature radius only [5].) In the following we evaluate E out with the the correct deformation field η, which satisfies appropriate conditions at the outer bou...

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“…A widely used approach to calculate a minimum energy surface is by means of the Surface Evolver program. 42 But several other approaches, both theoretical and numerical, have been used for studying the fluid-fluid interface shape in different physical problems, e.g., menisci shapes and capillary interactions, [43][44][45][46][47][48][49][50][51][52] droplet shapes, [53][54][55][56][57] diffuse interfaces, [58][59][60] or fluid-fluid interfaces in contact with deformable solids. [61][62][63] In this article, we introduce a new numerical method to obtain the minimum-energy shape of a fluid-fluid interface.…”

Section: Introductionmentioning

confidence: 99%

The equilibrium shape of fluid-fluid interfaces: Derivation and a new numerical method for Young’s and Young-Laplace equations

Soligno

Dijkstra

Roij

2014

The Journal of Chemical Physics

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Articles you may be interested inMany physical problems require explicit knowledge of the equilibrium shape of the interface between two fluid phases. Here, we present a new numerical method which is simply implementable and easily adaptable for a wide range of problems involving capillary deformations of fluid-fluid interfaces. We apply a simulated annealing algorithm to find the interface shape that minimizes the thermodynamic potential of the system. First, for completeness, we provide an analytical proof that minimizing this potential is equivalent to solving the Young-Laplace equation and the Young law. Then, we illustrate our numerical method showing two-dimensional results for fluid-fluid menisci between vertical or inclined walls and curved surfaces, capillary interactions between vertical walls, equilibrium shapes of sessile heavy droplets on a flat horizontal solid surface, and of droplets pending from flat or curved solid surfaces. Finally, we show illustrative three-dimensional results to point out the applicability of the method to micro-or nano-particles adsorbed at a fluid-fluid interface. C 2014 AIP Publishing LLC.[http://dx

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Capillary Force on a Micrometric Sphere Trapped at a Fluid Interface Exhibiting Arbitrary Curvature Gradients

Cited by 45 publications

References 22 publications

Curvature capillary migration of microspheres

Curvature capillary migration of microspheres

Comment on “Curvature capillary migration of microspheres” by N. Sharifi-Mood, I. B. Liu, K. J. Stebe, Soft Matter, 2015,11, 6768

The equilibrium shape of fluid-fluid interfaces: Derivation and a new numerical method for Young’s and Young-Laplace equations

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