Brownian diffusion of a partially wetted colloid

Transcription

1 SUPPLEMENTARY INFORMATION DOI: 1.138/NMAT4348 Brownian diffusion of a partially wetted colloid Giuseppe Boniello, Christophe Blanc, Denys Fedorenko, Mayssa Medfai, Nadia Ben Mbarek, Martin In, Michel Gross, Antonio Stocco & Maurizio Nobili 1. Supplementary Figures Supplementary Method Ellipsoid tracking Supplementary Discussion Contact angles of several couples of spherical particles and air-liquids interfaces Contact line contribution to the bead translational drag Contact line contribution to the ellipsoid rotational drag Contribution of the interface deformation to the ellipsoid rotational drag NATURE MATERIALS 1

2 1. Supplementary Figures microscope objective glass slide water Petri dish glass spacer metal ring Figure S1: Sketch of the cuvette used for the study of ellipsoidal particles. The container is a Petri dish, 6 mm in diameter, covered by a glass slide to avoid surface contaminations. A thick glass spacer (as a lens) is added to minimize the quantity of water involved in the experiment. Such a lens measures 58 mm in diameter and 8.2 mm in height. A metal ring encloses the lens; it has a sharp internal profile that pins the air-water interface. The deionized water is poured into the sample up to the internal edge of the ring enforcing the flatness of the air-water interface. a 5 nm 4 b 5 nm µm µm Figure S2: a. False color image of the interface around a spherical bead showing that the interface is flat and no deformations are detected to within the experimental resolution. b. False color image of the interface around a spheroidal particle with aspect ratio ϕ = 2.7. The interface is higher at the center of the spheroid (yellow zone) and lower at the tips (dark blue zone). The difference between the heights of these two zones is around 8 nm. 2

3 2. Supplementary Method 2.1 Ellipsoid tracking a y φ d i 2 2,i d 1,i φ 1 x Σd i < Σd b 1,i i 2,i y φ 2 φ 1 x Figure S3: Scheme of the bisection method. a. The orientation φ of the long main axis is in the range [φ 1, φ 2 ]. In the reported case, φ 1 = (horizontal axis) and φ 2 = π/4 (red line axis). For a generic point i in the ellipse, the distances d 1,i from axis φ 1, and d 2,i from axis φ 2 are calculated (solid blue segments). The summation over all the points i of the ellipse are compared: d 1 = i d 1,i < d 2 = i d 2,i indicating that the long ellipsis axis is closer to the axis φ 1 =. b. Iteration of the process where the farthest axis φ 2 = π/4 is replaced by the midpoint value (φ 1 + φ 2 )/2. A frame containing the image of one ellipsoid is an 8-Bit grayscale image I (x, y). In the image the particle looks like a dark ellipse over a gray-white background. The negative image is created (I 1 (x, y) = 255 I (x, y)) and a value corresponding to the actual average background intensity I bg is subtracted: I 2 (x, y) = I 1 (x, y) I bg, with I bg 2 3. We obtain a new image where the intensities I 2 (x, y) are non-null at the particle and elsewhere. The (x, y) position of each particle is then calculated by the intensity center of mass expression (eq. S1): x = xi2 (x, y) I2 (x, y) y = yi2 (x, y) I2 (x, y) (S1) Let us now consider a frame ( x, ỹ) centered in the intensity center of mass of the ellipsoid. In order to find the angle φ of the ellipsoidal long axis with respect to the axis x, the image processing software starts by counting the number of non null intensity pixels in the first ( x >, ỹ > ) and in the second quadrant ( x <, ỹ > ) of the frame. The result gives us a first information about the range in which the angle φ is: φ [, π/2] if the number of counted pixels in the first quadrant is larger than the one in the second (as in the case of Fig. S3), φ [π/2, π] in the opposite case. The process is then iterated by considering the two parts of the first quadrant separated by the first quadrant bisector. For a general iteration step the searching angular range can be written in the form [φ 1, φ 2 ]. For each pixel i of non null intensity, the distance d 1,i (d 2,i ) from the axis φ 1 (φ 2 ) is calculated. The average distances d 1 and d 2 are then given by: d 1,2 = i d 1,2,i. If d 1 < d 2, we can state that the ellipse long axis is closer to the axis φ 1. A new iteration is repeated using the same procedure in a smaller, halved range, where the farthest extreme (in this case φ 2 ) is replaced by the bisector value (φ 1 + φ 2 )/2. The iteration continues until the angular range is of the order of 1 2 rad, that is considered an acceptable resolution for φ. 3

4 3. Supplementary Discussion 3.1 Contact angles of several couples of spherical particles and air-liquids interfaces In order to generalize our measurements we investigated the contact angle of different beads trapped at several air liquids interfaces. The table below resumes all the contact angles obtained. Hydrophilic Silica Commercial silica Silica.5% DMOAP (1 min) Silica.5% DMOAP (2h) Silica 5% DMOAP (2h) Silica 1 mm PFOTS (2h) Water Mineral oil Ethylen glycol Hexanol Lutidine Decalin F- Decalin NS NS 65-7 NM NM NM NM NM NS 9-11 NM NM NM NM NM NS NS NS PS 54-6 NM NM NM NM NM NM PMMA 7 NM NM NM NM NM NM Table S1: Contact angles for various combinations of particle surfaces and fluid interfaces. : particle totally immersed in fluid; NM: not measured; NS: Non stable due to strong Oxygen absorption. A large part of examined combinations of particle-interface have a zero contact angle, meaning that the particle is fully immersed in the liquid. Among the studied particles-interface combinations, silica beads at the air-water interface appear to be the only ones supporting a wide range of contact angles. PS and PMMA at the air water interface, and silica at the air hexanol interface, present finite contact angles. The measurements of their dynamics have been included in Fig. 2 of the main text. 3.2 Contact line contribution to the bead translational drag Let us consider a top view of a spherical bead; we focus on the contact line and on the interface at its vicinity. A cylindrical coordinate system (r, φ, z) is defined (Fig. S4). r is the radial distance from the normal z to the interface, centered on the bead. φ is the angular position with respect to an arbitrary axis w over the interface. A fluctuation i can occur on a contact line segment at a random position. Let us consider a segment having a length λ i and centered at the angular position 4

5 top view r z φ w Figure S4: Cylindrical coordinate system (r, φ, z), shown in top view of the particle at the interface. The vertical axis z is perpendicular to the interface and centered on the bead. The angle φ is defined with respect to the arbitrary axis w, that lies on the interface. The radial coordinate r is the distance between a generic point and the center of the bead. φ i. Two types of fluctuations can be involved: in the case of a moving line, a local displacement of the contact line on the particle surface (Fig. 3b in the main text); in the case of a pinned line, a local capillary deformation of the interface profile (Fig. 3c in the main text). Both fluctuations lead to a variation of the slope of the interface at the contact line, and give rise to a net lateral force FL,i = σ LV λ i (1 cos α i ) on the particle. We have denoted α i the slope of the interface due to a fluctuation, defined as the angle between the tangent to the interface at the particle and the horizontal plane (i.e. the far field orientation of the interface) (Fig. 3 in the main text) and with σ LV the liquid vapor surface tension. With reference to the arbitrary direction w (Fig. S4) the net force F L,i is written F L,i = F L,i cos φ i. (S2) In practice, the line fluctuation translates in an extra random kick to the particle, in addition to the ones provided by molecular collisions. The random force FL,i depends on the random variables α i and λ i. Let us consider now the effect of a large number of fluctuations on the motion of the bead. The total force on the bead is obtained as the sum of all the contributions: n F L = (S3) i=1 where n is the number of fluctuating segments along the contact line. Since there are no preferred position along the contact line, F L has a null ensemble average F L =. The squared ensemble average reads: F 2 L = F L,i n F L,i F L,j = FL,iF L,j i,j=1 n cos φ i cos φ j = i,j=1 F L,i 2 n cos 2 φ i = σlv 2 λ 2 (1 cos α) 2 n 2 i=1 (S4) 5

6 where we use the fact that FL,i and F L,j are uncorrelated and we introduced the effective parameters λ and α through the relationship: F L,i2 = σ 2 LV λ 2 (1 cos α) 2. (S5) Considering the length of the contact line 2πa, with a = R sin θ, and λ the characteristic lateral length of each fluctuation, the number of possible fluctuations at a given time is n = 2πa /λ. (S6) The mean square value of the force writes thus: F 2 L = σ 2 LV πr sin θ(1 cos α) 2 λ. (S7) F L is a fluctuating force with a null ensemble average and a non-null squared ensemble average. It adds to the random force usually considered in pure hydrodynamics approaches, due to the collisions between the molecules of the surrounding fluids and the particle itself. To relate a fluctuating force F to the particle viscous dissipation γ we use the fluctuation-dissipation theorem [1, 2]: γ = 1 2k B T + F ()F (t ) dt. (S8) The theorem states that if the power injected by the fluctuating force increases, the dissipated power due to friction has also to increase, as the final energy of the particle remains fixed by the equipartition theorem to k B T/2 per degree of freedom. In our case the random force at a given time t comes from two different contributions: F (t) = F H (t) + F L (t) (S9) where F H is the hydrodynamic term due to molecular collisions and F L has been introduced in eq. S3 as the random contribution of the contact line. The term in the integral in eq. S8 is developed: F ()F (t ) = [F H () + F L ()] [F H (t ) + F L (t )] = F H ()F H (t ) + F H ()F L (t ) + F L ()F H (t ) + F L ()F L (t ). (S1) Assuming no correlation between both forces leads to canceling the mixed terms. Justification of this assumption is twofold. In the case of a moving line, the contact line fluctuations are provided by tangential motions of water molecules, with respect to the particle surface, while F H comes from normal molecular collisions. In the case of a pinned line, fluctuations of the interface slope at the contact line due to capillary waves involve molecules far from the particle. Substituting the expression S1 in eq. S8, we note that the two random forces F H and F L contributes separately to the total friction γ. Thus, the friction can be expressed in the form: γ = γ H + γ L (S11) 6

7 where the term γ H (γ L ) depends just on the integral of the force F H (F L ). The term γ H has been already discussed and predicted by purely hydrodynamic theories, accounting the partial immersion in fluid under different approximations [3, 4, 5]. The second term in eq. S11 is γ L = 1 2k B T + F L ()F L (t ) dt (S12) where F L is defined in eq. S3. Also in this case the random nature of the force makes the analytic solution of eq. S12 extremely difficult to obtain. However, the integral can be safely approached by taking into account a characteristic fluctuation time τ and dimensional arguments. An approximated form for γ L is γ L 1 2k B T F L() 2 τ. (S13) Substituting eq. S7 in eq. S13, γ L is finally written as a function of the parameters α, λ and τ: γ L 1 2k B T σ2 LV πr sin θ (1 cos α) 2 λτ. (S14) 3.3 Contact line contribution to the ellipsoid rotational drag Let s start by considering a flat interface around the ellipsoid. We will discuss in section 3.4 the contribution of the interface deformation on the ellipsoid viscous drag. A random force F L,i in the interface plane on a line segment i exerts a torque on the ellipsoids given by: M L,i = r i F L,i (S15) denoting r i the radius vector of the segment i (Fig. S5). Under the same assumptions as in paragraph 3.2, the total random torque M L = n i=1 M L,i along the vertical axis, due to interface fluctuations, is associated to the particle rotational drag γ R,L, via Langevin equation and the fluctuation-dissipation theorem: γ R,L 1 2k B T M L() 2 τ. (S16) In order to obtain an expression of M L () 2, we write explicitly the modulus M L,i as: M L,i = FL,ir i sin ν i (S17) where ν i is the angle between the random force and the vector r i, as depicted in Fig. S5; both of them are a function of φ i. The force FL,i as reported in the main text writes: FL,i = σ LV λ i (1 cos α i ) (S18) where λ i is a characteristic length of a fluctuation i, α i is the slope of the deformation at the triple line and σ LV is the air-water surface tension. The quantities r i and ν i are function of the angle φ i and of the aspect ratio ϕ. We parameterize with p the contact line as an ellipse in a Cartesian coordinate 7

8 top view y ~ b r i ν F ~ a L i φ i x ς i ς* i Figure S5: Top view of the contact line of a spheroidal particle. The contact line is an ellipse of axes (ã, b) in a Cartesian coordinate system (x, y). A random force F L,i acts on a generic point of the contact line, at a distance r i from the center of the particle. This force results in a torque: M L,i = FL,i r i sin ν i, where ν i is the angle between the vectors F L,i and r i, for a given fluctuation i. Inset: detail of the line force on the contact line. ς i is the angle between r i and the horizontal axis. ςi is the angle between FL,i and the horizontal. The angle ν i is given by ςi ς i. system (x, y), with axes ã = a sin θ and b = b sin θ. θ is the contact angle and a, b the main axes of the ellipsoid. The relation with the parameter p is given by x = ã cos p, y = b sin p. (S19) The distance r of a generic point of the contact line from the center (fig. S5) has a modulus: r 2 = x 2 + y 2 = ã 2 cos 2 p + b 2 sin 2 p = b 2 ( ϕ 2 cos 2 p + sin 2 p ). (S2) The orientation of the vector r i, with respect to the horizontal axis x, is provided by the angle ς: tan ς = y x = 1 tan p. ϕ (S21) To obtain the orientation ς of the random force F L,i, we assume that F L,i is normal to the contact line in the considered point. Considering m the angular coefficient of the tangent of the ellipse at the segment i, we have: tan ς = 1 m = ã2 y b2 x = ã sin p b cos p = ϕ tan p. (S22) The angle ν between the two vectors is thus ν = ς ς. (S23) We consider now the total random torque, given by all the possible fluctuations at a fixed time. The random origin of the fluctuations translates in a null ensemble average M L = (S24) but in a non null squared ensemble average n ML 2 = FL,iF L,jr i r j sin ν i sin ν j = FL 2 n ri 2 sin 2 ν i i,j=1 i=1 (S25) 8

9 with n the number of fluctuating segments of the triple line. The summation in eq. S25 can be written as an integral along the contact line, since the mean length λ of a fluctuation is much smaller than the contact line perimeter (λ dl): n ri 2 sin 2 ν i = 1 ri 2 sin 2 ν i dl λ i=1 where the infinitesimal arc length on the ellipse perimeter is known dl = b ϕ 2 sin 2 p + cos 2 pdp. (S26) (S27) Substituting dl in eq. S26 and using previous definitions (eqs. S2, S21, S22), we obtain n r 2 i sin 2 ν i = b λ i= 2π b3 λ 2π ( ϕ 2 cos 2 p + sin 2 p ) sin 2 ν We denote with I(ϕ) the integral: I(ϕ) = 1 ϕ 2π r 2 i sin 2 ν i ϕ 2 sin 2 p + cos 2 pdp = ϕ 2 sin 2 p + cos 2 pdp. ( ϕ 2 cos 2 p + sin 2 p ) sin 2 ν ϕ 2 sin 2 p + cos 2 pdp (S28) (S29) which is numerically solved taking into account the expression for ν (eq. S23). We can now express γ R,L (Eq. S16) as γ R,L = K l I(ϕ) (S3) where K l contains all the local terms and constants related to the fluctuations. According to the previous discussion, K l writes K l = 1 2k B T σ2 LV (1 cos α) 2 r 3 sin 3 θ [λτ(λ)] (S31) since b 3 = r3 ϕ sin 3 θ, where r is the radius of the spherical bead before the stretching process. The dependence on the aspect ratio, provided by I(ϕ), is plotted in Fig. S6. At ϕ = 1, i.e. for spherical particles, I(ϕ = 1) =. A radial random force F L,i is parallel to the radius r i. Thus, the cross product M i = r i F L,i is identically null. It follows that fluctuations of the interface at the contact line do not affect the rotation of a sphere around a vertical z axis. The random torque is instead rapidly increasing with the aspect ratio, and I(ϕ) varies over 2 orders of magnitude in the range 2 1 of aspect ratio. 3.4 Contribution of the interface deformation to the ellipsoid rotational drag The effect of an interface deformation on particle drag has been first pointed out on the translational motion at the air water interface of heavy millimetric 9

10 I(φ) aspect ra o, φ Figure S6: Numerically obtained I(ϕ) (Eq. S28) as a function of the aspect ratio. The line contribution to the rotational drag is zero for spherical particle and increases by two order of magnitude when ϕ changes from 2 to 1. beads by Petkov et al. [6]. For copper beads of radius.52 mm and density 9 times the one of water having a contact angle θ = 78, the viscous drag at the interface is measured to be 1.75 times larger than the one expected for the same bead totally immersed in water. The authors propose that such an increase in the drag felt by the particle is due to the hydrodynamic resistance induced by the curved meniscus around the heavy particle: the curved meniscus has to move together with the particle, leading to an increase of the volume displaced well beyond the one of the sole particle. Following the same approach we can evaluate the effect of the interface deformation around the ellipsoid on its rotational drag. To this scope we consider the volume of water displaced during the particle mouvement corresponding to the meniscus around the particle. From PSI measurements such a volume can be overestimated by a disk having as radius the long axis of the ellipsoid (see Fig.S7) and with an height corresponding to the height of the deformation along the ellipsoid perimeter. For the ellipsoid in Fig. S7 with ϕ = 2.7 and an amplitude of deformation of 4nm, the volume of water displaced is less than 11% of the ellipsoid volume. Such a slight increase in the volume cannot explain the huge increase of one order of magnitude of the measured rotational drag. 1

11 5 nm µm Figure S7: False color image of the typical interface deformation around a spheroidal particle. The interface is higher at the center of the spheroid (yellow zone) and lower at the tips (dark blue zone). The maximum deformation defined as the difference between the highest and lowest points of the interface is z = 8nm. The deformed region around the particle can be approximated by an isotropic disk (white dashed line). 11

12 References [1] Langevin, P. Sur la théorie du mouvement brownien. CR Acad. Sci. Paris 146, (198). [2] Landau, L. & Lifshitz, E. Cours de physique théorique: Physique statistique (mir, 1967) ch. 5. Rep. Progr. Phys 29, 255 (1966). [3] Danov, K., Aust, R., Durst, F. & Lange, U. Influence of the surface viscosity on the hydrodynamic resistance and surface diffusivity of a large brownian particle. J. Colloid Interface Sci. 175, (1995). [4] Pozrikidis, C. Particle motion near and inside an interface. J. Fluid Mech. 575, (27). [5] Fischer, T. M., Dhar, P. & Heinig, P. The viscous drag of spheres and filaments moving in membranes or monolayers. J. Fluid. Mech. 558, (26). [6] Petkov, J. et al. Measurement of the drag coefficient of spherical particles attached to fluid interfaces. J. Colloid Interface Sci. 172, (1995). 12