Trapping energy of a spherical particle on a curved liquid interface

Leandri, J.; Würger, Aloïs

doi:10.1016/j.jcis.2013.04.024

Cited by 26 publications

(44 citation statements)

References 35 publications

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“…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: 44%

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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: 44%

“…Without loss of generality, we focus on interfaces with zero mean curvature as the role of finite mean curvature gradient has been addressed in the literature [12,22].…”

Section: Theorymentioning

confidence: 99%

Curvature capillary migration of microspheres

2015

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“…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,…”

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confidence: 94%

“…Previous works [2][3][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.…”

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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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“…As in electrostatics, the height field can be written as a superposition of monopole (isotropic u B ln(r)), dipole (u B r À1 cos f), quadrupole (u B r À2 cos(2f)) and higher terms. 16,22,23 When a spherical particle binds at a planar interface, no deformation appears unless an applied force acts on the particle (e.g., from gravity). 15,24,25 For particles with an anisotropic shape, however, the constant-contact angle condition leads to deformations even without any applied force.…”

Section: Introductionmentioning

confidence: 99%

Deformation of the contact line around spherical particles bound at anisotropic fluid interfaces

Şenbil

Dinsmore

2017

Soft Matter

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a When a particle adsorbs at a liquid interface, the 3-phase contact line geometry depends on the shape of the particle and of the liquid interface. The shape of the contact line is the key to controlling capillary forces among particles, and is therefore a useful means to direct assembly of interfacial particles. We measured the shape of the contact line around millimeter-sized PDMS-coated glass spheres at water/air interfaces with anisotropic shapes. We studied the advancing and receding conditions separately. We focused on interfaces with a cylindrical shape, where the predominant deformation of the meniscus and the contact line both have quadrupolar cos(2f) symmetry. We related the measured magnitude of the quadrupolar deformation to the applied vertical force on the sphere and the interface's deviatoric curvature, D 0 . For modest curvature (D 0 o 0.1 Â sphere radius), our results agree with the theoretical prediction for free particles. At higher curvature, the measurements exceed the theory. The theory appears to apply even when there is contact-angle hysteresis, as long as the measured contact angle is used rather than the equilibrium (Young-Dupré) angle. The magnitude of the quadrupolar deformation depends on the applied force. Together, these results show the range of validity of the theory.

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Trapping energy of a spherical particle on a curved liquid interface

Cited by 26 publications

References 35 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

Deformation of the contact line around spherical particles bound at anisotropic fluid interfaces

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