Passive Sedimentation Control in Containers Using Marangoni Forces
When fluid-filled containers are stored for long periods of time with negligible motion, sedimentation and gravitational settling of particles can occur. This article discusses a passive sedimentation control mechanism that is driven by Marangoni stress, which is induced in enclosed geometries when their walls are lined with gas-filled hydrophobic microcavities.
The proposed concept has significant practical applications in the pharmaceutical industry, as it would allow medicines and active fluids to be stored for long periods of time using such containers and eliminate concerns related to sedimentation. Although additional research and development are required to arrive at a practical, optimized, commercial design, this preliminary work outlines the mathematical development and computational verification of the concept.
Sedimentation and Marangoni Forces
During gravitational settling of particles, a definitive vertical concentration gradient gradually develops. Undesired sedimentation occurs in unstable colloidal dispersions, which can then form loosely joined masses of fine particles of either aggregates or agglomerates (due to interparticle attractions) as the particles assemble. Although aggregation is a reversible process, agglomeration is not. Both should be avoided if possible.
If the walls of a container are lined with air-filled (or inert gas–filled) hydrophobic microcavities, a Marangoni force will appear because of the dependence of surface tension on concentration. This Marangoni force will propel the particles from the low-
surface-tension region to the high-surface-tension region. The force will then act as a negative feedback and prevent sedimentation, agglomeration, and aggregation of particles by mildly, yet continuously, remixing the content.
The use of walls lined with air-filled hydrophobic microcavities and superhydrophobic surfaces is not new; however, its application has so far been limited to being a passive method for drag reduction in which the air-liquid interface, owing to the trapped air of the nanocavity, translates into the appearance of slip velocity in the superhydrophobic surface.
Self-Sustained Sedimentation Control
The Marangoni effect is the mass transfer along an interface between two fluids due to a gradient of the surface tension. In the case of temperature dependence, this phenomenon is generally referred as thermocapillary convection.
The explanation of this phenomena is straightforward: A liquid with a high surface tension pulls more strongly on the surrounding liquid than one with a low surface tension, and therefore the presence of a gradient in surface tension will naturally cause the liquid to flow away from low-surface-tension regions toward high-surface-tension regions. The surface tension gradient can be created by a thermal gradient, owing to its temperature dependence, or by a concentration gradient. The concentration dependence provides the possibility to use walls lined with air-filled (or inert gas–filled) hydrophobic microcavities as a self-controlled and completely passive method to prevent sedimentation, agglomeration, and cluster formation in enclosed geometries. As mentioned, during gravitational settling, a vertical concentration gradient starts to develop, with the higher concentration at the bottom (sedimentation). Thus, if container walls are lined with such microcavities, the concentration gradient can trigger a capillary motion that will act as a feedback, preventing or mitigating the sedimentation process. As the concentration gradient (sedimentation rate) increases, the capillary flow becomes stronger. (Figure 1 illustrates this idea.)
Methods
First, consider a fully developed, two-dimensional flow between parallel plates that are separated by a distance, b, and a length, h, and lined with air-filled hydrophobic microcavities. When precipitation and gravitational settling start, a vertical concentration gradient appears and, as a result, a Marangoni stress is generated across the wall because of the free interfaces introduced by the microcavities (as described previously). The maximum velocity, vmax, attainable by this Marangoni flow—which might be used as a rough estimation of the capability for resuspension of particles—occurs at the wall, and can be estimated using the following equation:
\(\text{Equation 1:}\ V_{max}\approx\frac{b}{ 2\mu}\frac{dσ}{dz} \)
where b is the distance between plates, μ is the dynamic viscosity of the liquid, and dσ/dz is the surface tension gradient. In our case, because the surface tension gradient is driven by a concentration gradient,\(\nabla_ZC\), equation 1 can be rewritten as follows:
\(\text{Equation 2:}\ V_{max}\approx\frac{bσ_c}{ 2\mu}\nabla_zC \)
where σc is the surface tension coefficient with concentration.
Finally, the feasibility for resuspension of sediment by the induced capillary flow may be preliminarily assessed by equating equation 2 with the terminal velocity of particles, which, for a low Reynolds number less than unity, is calculated as follows:
\(\text{Equation 3:}\ V_t=\frac{g(d_p^2)}{18\mu}(\rho_\rho - \rho) \)
where g is gravity, dp is the diameter of the particle of solute, μ is the dynamic viscosity of the fluid, and \(\rho_\rho\) and \(\rho\) are the density of the particle and the density of the fluid, respectively. Equating equation 2 with equation 3, we obtain:
\(\text{Equation 4:}\ \nabla_zC = \frac{gd_p^2 }{9σ_cb} \)
which gives us the concentration gradient required to compensate the gravitational settling as a function of the diameter of the particle.
Looking at equation 4, one may think that the required concentration is reduced inasmuch that the distance between plates, b, increases. However, this is not the case because equation 2 for the capillary velocity cannot be used for any size of container. Further, in the derivation of equation 2 in the momentum equations, it was assumed that
\(\frac{{V_z}^2}{h}\lt \lt \frac{\mu \vert v_z \vert}{\rho b^2} \)
and then the validity of the velocity flow by equation 2 is given by:
\(\text{Equation 5:}\ b^2 \lt \lt \frac{4\mu^2 \rho h}{\vert σ_c \vert \nabla_zC} \)
To obtain some idea of the concentration gradient predicted by equation 4, let us assume some typical values for a NaCl
solution with densities \(\nabla\) ≈ 103 kg/m3 and \(\nabla\)p ≈ 2.17 × 103 kg/m3; μ ≈ 10–3 Pas; \(\nabla\)c = 1.2 × 10–3N/m(molarity);
Figure 2 shows the resulting curve as a function of the particle diameter. It is seen that the required concentration gradient—even for solute particles with diameters ≥10 μm—is within the range of concentration one would expect in a sedimentation. However, as mentioned previously, the calculations must be given careful attention because the results are valid only when the distance between plates satisfies the relationship given in equation 5.
Computational Simulation
To assess the capability for the proposed self-sustained control mechanism, hydrodynamic computational simulations in unsteady-state conditions were performed using Ansys Fluent computational fluid dynamics (CFD) software version 14.
Figure 3 shows schematically the problem to be considered. The NaCl solution was considered inside a rectangular box of sides l = 5 mm and b = 5 mm. The boundary conditions were as follows: The bottom and top of the box had a zero-slip condition, and the left and right sides had a Marangoni stress shear condition. However, because the Ansys Fluent CFD Marangoni stress option is only directly available for thermocapillary flow (i.e., considering thermal coefficient of surface tension), it was necessary to create a user-defined function to customize Fluent. The user-defined function read the local concentration gradient at the wall and then defined a “fictitious” thermal gradient associated at this place using the following expression:
\(\text{Equation 6:}\ \nabla_zT = \frac{σ_c }{σ_T} \nabla_zC \)
which allows reproduction of the Marangoni effect driven by local concentration gradients. To avoid any undesired collateral effect of using a fictitious thermal gradient, the fluid’s thermal expansion coefficient was set to zero. Finally, to reproduce a free surface at the top of the container, a gap of air with a thickness of 0.1 mm was introduced. The analysis was carried out with a simple algorithm and Presto for pressure discretization, and a second-order upwind scheme for momentum and energy. Relaxation factors were taken to be default values. Convergence criteria were set as 10–3 for continuity, z-momentum, and x-momentum, and as 10–6 for energy. Constant properties of water were considered, with = 103 kg/mg3 and μ = 10–3 Pas. For the thermal coefficient of surface tension, σT, and for the concentration coefficient of the surface tension for supersaturated solution of NaCl with particles near nanometric size, it was assumed that σ = 100 mN/m (mass fraction of NaCl).
Conclusion
From the simulations, it was found that the induced convective Marangoni flow could control the gravitational settlement of colloidal particles with diameters less than 1 μm or thereabouts. Figure 4 shows the final concentration profile after gravitational settling without Marangoni effect and considering several uniform initial concentrations. Likewise, Figures 5 and 6 show some sequences of the computational simulation for the concentration pro-file without and with Marangoni effect, respectively. Considering that particles before agglomeration and growth are expected to be around 1 μm or smaller, we may hypothesize that the walls of large containers can be lined with hydrophobic microcavities and the negative feedback from self-sustaining Marangoni forces can prevent sedimentation. As mentioned, with further research and development, this could offer a significant opportunity within the pharmaceutical industry for the long-term storage of medicinal products.
Acknowledgments
This research was supported by the Spanish Ministry of Economy and Competitiveness under fellowship grant Ramon y Cajal: RYC-2013-13459.
