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Relationship between Characteristic Lengths and Effective Saffman Length in Colloidal Monolayers near a Water-Oil Interface

The hydrodynamic interactions (HIs) in colloidal monolayers are strongly influenced by the boundary conditions and can be directly described in terms of the cross-correlated diffusion of the colloid particles. In this work, we experimentally measured the cross-correlated diffusion in colloidal monolayers near a water-oil inter-face. The characteristic lengths of the system were obtained by introducing an effec-tive Saffman length. The characteristic lengths of a particle monolayer near a water-oil interface were found to be anisotropic in the longitudinal and transverse directions. From these characteristic lengths, the master curves of cross-correlated diffusion are obtained, which universally describe the HIs near a liquid-liquid interface.


INTRODUCTION
Confined colloidal systems are ubiquitous and important in many real scenarios. Examples of such systems include porous media 1-3 , microfluidic devices [4][5][6][7][8][9][10] , proteins in the cytoplasm 2 and cell membranes [11][12][13] . The dynamics of a confined colloidal suspension depend on the boundary conditions of the system [14][15][16][17][18][19][20] and are more complicated than those of an unbounded free 3D bulk liquid. Generally, the strength of the hydrodynamic interactions (HIs) between two tracked particles is a function of their separation distance in the liquid 3,21 . When a membrane is suspended in a liquid, the HIs in the membrane exhibit a transition from 2D to 3D in nature with an increasing distance [21][22][23] . In a quasi-2D system, such as a cell membrane, a liquid-liquid interface, or a liquid film, a characteristic length can be defined to describe the distance at which the crossover from 2D membrane dynamics to 3D bulk dynamics occurs in the liquid [23][24][25] .
When the distance is much smaller than the characteristic length, both the mass and momentum of the membrane are conserved, as in a 2D membrane 23, [26][27] , which results in a decay of the HIs with the distance as ln(1⁄ ) due to the contributions of 2D shear stress. When the distance is greater than the characteristic length, momentum is transmitted into the surrounding fluid 12,24 , which causes the HIs to decay with the distance as 1⁄ due to the contributions of 3D shear stress.
The characteristic lengths of a system depend on the boundary conditions of that system. In the Saffman model 25,28 , the characteristic length of a thin sheet of viscous fluid of viscosity suspended in an unbounded bulk liquid of viscosity is defined as λ = ⁄ , which is called the Saffman length. The characteristic length of a semiinfinite system, such as a viscous film at a water-air interface, is also taken to be equal to the Saffman length λ = ⁄ 21 . However, if the viscous film at the water-air interface is replaced by a colloidal monolayer, the characteristic length of this particle monolayer will depend on both the particle size and the Saffman length λ of the monolayer in the following form: (λ ⁄ ) 3 2 ⁄ 29 . When the colloidal monolayer is detached from the interface and placed in bulk water close to the water-air interface, the characteristic lengths in the two orthogonal directions are anisotropic; they are expressed as and ( ⁄ ) 2 3 ⁄ . However, the characteristic lengths of a particle monolayer near a water-oil interface have not been closely studied until now. We wonder whether characteristic lengths of colloidal monolayers near a water-oil interface can be achieved by the method developed in colloidal monolayers on a water-air interface 29 , and what different features will be exposed between those two systems. When a colloidal monolayer is located near a water-oil interface, momentum diffuses from the colloidal monolayer into the bulk water and bulk oil on either side. In such a scenario, this system can be regarded as similar to a 3D bulk system. However, the HIs in the monolayer will be influenced by the interface, such as that in monolayers near a water-air 4 interface 30 . Thus, it is interesting to consider which kind of HI behavior can be expected in a colloidal monolayer near a water-oil interface.
In this paper, we report an investigation of the HIs in colloidal monolayers near a water-oil interface that was conducted by experimentally measuring the cross-correlated diffusion of the colloid particles. We find that an effective Saffman length * should be introduced in such a system. Using * , the characteristic lengths of the colloidal monolayer can be defined, and they are found to be different in the two orthogonal directions. In the longitudinal direction, the characteristic length is equal to * itself, while in the transverse direction, the characteristic length exhibits a power-law relationship with * . From these characteristic lengths, the friction coefficients of the colloidal monolayers can be estimated, and the results are consistent with those obtained from single-particle measurements. Using the characteristic lengths as scaling factors, universal curves describing the cross-correlated diffusion in a water-oil system are obtained. These universal curves are independent of the separation between the colloidal monolayer and the interface, unlike in the case of a water-air system. Such universal curves can help us to further understand the dynamics of fluids near a soft boundary.
The experimental setup is shown in Fig. 1(a). A stainless steel sample cell with two layered chambers was constructed, similar to that described in Ref. 31 . The chamber for water was 8 mm in width and 0.8 mm in depth. The chamber for oil was 15 mm in width and 1 mm in depth. An aqueous colloidal solution was introduced into the chamber for water such that it filled the chamber. Once the water-air interface was level, decalin was used to fill the oil chamber. The layers of liquid were sandwiched between two coverslips. Finally, the prepared sample was placed upside down and allowed to remain undisturbed for at least two hours. The particle solution was pinned by the edge of the stainless steel hole and remained on top of the layer of decalin. The particles sank under gravity and formed a monolayer close to the water-oil interface. The repulsive interaction of the image charges balanced the action of gravity on the particles, preventing the particles from contacting the water-oil interface. Samples of three kinds of colloid particles were used, namely, silica spheres with radii of = 0.6 μm (S1) and  Fig. 1

(b).
Under the frame-collecting rate of 14 fps, particles diffuse a short distance, which is much less than the average separation between the neighbor particles during time intervals between two adjacent frames. The closest particles in adjacent frames were identified as a same particle. Using homemade software (IDL), we obtained the trajectories of the particles by above strategy. From these particle trajectories, the single-particle diffusion and the correlated diffusion between particles were calculated to study the HIs in the colloidal monolayer.
First, from the particle trajectories, the particle displacement over a lag time τ was calculated as ∆ ⃗ ( ) = ⃗ ( + ) − ⃗ ( ). The single-particle self-diffusion coefficient ( ) was calculated from the mean square displacement using the relation 〈∆ ⃗ 2 ( )〉 = 4 ( ) . As shown in Fig. 1(c), ( ) scaled by 0 is a function of the particle area fraction , where 0 is calculated according to the Einstein-Stokes relation: 0 = 6 ⁄ . The measurements of this scaled diffusion coefficient were fit to the following second-order polynomial [31][32] : Here, the parameter is related to the local viscosity experienced by the particles in the dilute limit, and the parameter is related to the two-body interactions between the particles. When → 0, the diffusion coefficient (0) can be obtained from the formula (0) 0 ⁄ = ; this coefficient is the self-diffusion coefficient of a particle in the monolayer in the dilute limit. From (0) 0 ⁄ , the separation from the center of the colloidal monolayer to the water-oil interface was calculated according to Eq. (1) 18, 33 : where is the viscosity of the water and is the viscosity of the decalin. The fitted values of the parameters and the calculated values are shown in Table 1. (2) Figure 2 shows the cross-correlated diffusion coefficients ∥,⊥ ( ) scaled by the self-diffusion coefficient ( ) as functions of the distance , which is scaled by the particle diameter 2 , for sample S1. The area fraction for sample S1 ranges from 0.007 to 0.20. From Fig. 2, it can be seen that the scaled cross-correlated diffusion  8 ( ) and the particle diameter 2 are not good parameters with which to analyze the cross-correlated diffusion. The scaled cross-correlated diffusion coefficients ∥,⊥ ( /2 )/ ( ) for the other two samples, S2 and S3, are shown in Fig. S1 and Fig.   S2 in the Supplementary Material.
The self-diffusion coefficient for a single particle is written as where is a dimensionless constant. represents the friction experienced by one particle. For a particle in a colloidal monolayer, normally such a friction include two terms, Γ (0) and Γ (1) ,  Table I.
Because Γ (0) represents the friction experienced by a single particle in dilute limit and only Γ (1) represents the friction contributed by the interaction between the particles in the monolayer 29 . Considering that ∥,⊥ reflects the strength of HIs between the particles, the change of ∥,⊥ induced by is contributed only by the change of Γ (1) , and has nothing to do with Γ (0) . After all, Γ (1) is a function of n, while Γ (0) is a constant and independent of . Thus, the effective diffusion coefficient D s ′ ( ), which only relates to the friction coefficient Γ (1) , is a more suitable scaling factor than ( ) is .
The plots of ∥,⊥ / in Fig. 2 should thus be replaced with plots of ̃∥ ,⊥ = ∥,⊥ ′ ⁄ . In the insets of Fig. 4(a, b), the plots of ̃∥ ,⊥ ( /2 ) increase with increasing Γ (1) , showing an influence of the concentration , as expected. Moreover, new characteristic lengths should be determined to normalize the distance to obtain a universal function of cross-correlated diffusion. As shown in Fig. 4(a, b) layer with a high Γ (1) . For the other two samples, we obtained similar results using the same method (see Fig. S3 and Fig. S4 in the supplementary material). The scaled cross-correlated diffusion coefficients ̃∥ ,⊥ ( ∥,⊥ ′ ) for all three samples with different separations ⁄ collapse to single master curves, as shown in Fig. 4(c, d). The universal master curves ̃∥ ,⊥ ( ∥,⊥ ′ ) for the different samples indicate that the HIs in a particle monolayer near a water-oil interface can be described by the same rule, even for monolayers containing particles of different sizes or lying at different separation distances from the interface.
It is noted that a bit of curvature shows at small r in Fig. 2(b) and Fig. 4(b). When the distance r is very small, the magnitude of r is close to the size of particles, i.e. particles cannot be regarded as mass points anymore, as the theoretical derivation assumed. It may cause the curvature at small r.
The universal master curves can be obtained only with suitable parameters ‖,⊥ . As length scales, ‖,⊥ can be regarded as characteristic lengths of the system. The question then becomes how these characteristic lengths ‖,⊥ should be interpreted. In the Saffman model 25 , the characteristic length of a membrane is λ = ⁄ , which depends on the viscosity of the membrane. Based on the concept of λ , we define an effective Saffman length * for a particle monolayer near a water-oil interface, expressed as * ≡ Γ (1) 6 ⁄ . The effective Saffman length is the ratio of the friction coefficient Γ (1) of the particles in the monolayer to the traditional friction coefficient 6 of the particles in a bulk liquid. In the present system, the effective coefficient * is written as * = Γ (1) (6 • ( + ) 2 ⁄ ) ⁄ . The characteristic length ‖ in the longitudinal direction is taken to be equal to * , as shown in Eq. (3): For given values of ‖ , corresponding friction coefficient values, denoted by Γ ‖ (2) , can be obtained from the definition of * . The superscript and subscript of Γ ‖ (2) suggest that this friction coefficient can be estimated from two-particle diffusion measurements in the longitudinal direction. Meanwhile, Γ (1) can also be obtained from singleparticle diffusion measurements, in accordance with the formulas ′ ( ) = Γ (1) ⁄ and 1 ( ) As shown in Fig. 5(a), the Γ ‖ (2) values obtained from ‖ agree with the Γ (1) values obtained from ′ ( ), which justifies the rationality of the relationship ‖ = * . However, in the transverse direction, the equivalent of Eq. (3), ⊥ = * , does not work as the characteristic length. If ⊥ = * is assumed, then the friction coefficient Γ ⊥ (2) obtained from ⊥ is not a linear function of Γ (1) . In fact, Γ ⊥ (2) will exhibit a power-law relationship with Γ (1) , expressed as Γ ⊥ (2)~( Γ (1) ) 2 3 ⁄ . Such a wrong dependence of Γ ⊥ (2) on Γ (1) given by the assumption ( ⊥ = * ) is shown in Fig. 5(b), as a comparison of the correct dependence in Fig. 5(a).
In Fig. 2(a,b), ∥ ⁄ ( ) for sample S1 decays with the distance as ~1⁄ in the limit of large , while ⊥ ⁄ ( ) decays as ~1 2 ⁄ . These behaviors are the result of the anisotropic HIs in a colloidal monolayer near a liquid-liquid interface. In the longitudinal direction, the HIs respond to a 3D shear stress that is similar to that in a 3D bulk liquid, and in the transverse direction, the HIs respond to a 2D compressive stress 12,24 . The anisotropy of the characteristic lengths ‖ and ⊥ of a colloidal monolayer near a water-oil interface is also similar to that near a water-air interface 30 . This anisotropic behavior can be explained by analogy to the lubrication force between two solid walls 30, 34 .
In the system of a water-oil interface, the scaled correlated diffusion coefficients ̃∥ ,⊥ ( ∥,⊥ ′ ) are independent on the separation , as seen in Fig. 4(c, d), while ̃∥ ,⊥ ( ∥,⊥ ′ ) in the water-air system are dependent on the separation 30 . Comparing these two systems, the significant difference is that the viscosity and mass density of the air can be negligible comparing with oil. There is no clear influence of z on D ∥,⊥ curves in fig. 4. HIs contributed by the oil overwhelms small change of HIs resulting from the different locations of colloidal monolayers. In the colloidal monolayer near the water-air interface, the momentum (stress flux) of the monolayer diffuse only into one side of the interface, i.e. the region of water. The HIs near the water-air interface is sensitive to the thickness of the water film between the monolayer and the interface. In the water-oil system, the momentum (stress flux) of the monolayer diffuse into both sides of the interface, i.e. the regions of water and oil. Comparing with the influence of the bulk of water and oil, the influence of the thin water film between the monolayer and the water-oil interface can be negligible, since z is only of the order of μm. The diffusion of the momentum (stress flux) into the whole space causes that ̃∥ ,⊥ ( ∥,⊥ ′ ) in the water-oil system is insensitive to the separation .
The high viscosity of oil may also cause the curves collapse Fig. 2 (Sample S1) and In this work, we measured the cross-correlated diffusion in particle monolayers near a water-oil interface. An effective Saffman length of * ≡ (1) 6 ⁄ was defined for such a colloidal monolayer. On the basis of this effective Saffman length, anisotropic characteristic lengths ‖ and ⊥ were introduced in the longitudinal and transverse directions, respectively. In the longitudinal direction, the characteristic length ‖ is equal to * , which reflects the continuous nature of the system. In the transverse direction, the characteristic length ⊥ is a function of * and the particle radius , with the form ( * ⁄ ) 2 3 ⁄ , which reflects the discrete nature of the system. The anisotropy of the characteristic lengths reflects the fundamental differences in the dynamics of the colloidal monolayer in the two directions. With these two characteristic lengths, Figures   FIG. 1. (a) Schematic illustration of the experimental setup. The distance between particles is denoted by , the separation from the center of the particle monolayer to the water-oil interface is denoted by , and the parameter is the particle radius. as functions of (2 ⁄ ) for sample S1 at different particle area fractions on a loglog plot. The concentration ranges from 0.007 to 0.20, and the friction coefficient of the particle monolayer ranges from 0.06 cP μm to 3.53 cP μm. The differently colored symbols represent different values of and different viscosity coefficients, where the black squares correspond to ≈ 0.007 and the maroon inverted triangles correspond to ≈ 0.20. The dashed lines are guides for the eye, corresponding to ~1⁄ in (a) and to ~1 2 ⁄ in (b).