Impinging jet study of the deposition of colloidal particles on synthetic polymer (Zeonor)
Introduction
Studying interactions of colloidal particles at the solid/liquid interface with respect to the adsorption/desorption phenomena is important for many scientific and industrials processes [1], [2], [3], [4], [5], [6], [7]. This knowledge is significant for better understanding of filtration processes, water treatment, microfluidic devices construction, paper making, polymer characterization, biofouling of membranes and artificial organs [8], [9], [10], [11] etc. For detailed and exact understanding of the latter mentioned phenomena well defined hydrodynamic and physicochemical conditions during experiments are required. These conditions are met in impinging jet technique, where deposition of colloidal particles is well controlled by hydrodynamic conditions during deposition process [2].
The most important sphere–wall interactions are those leading to the deposition of a spherical particle onto the surface. The flow near the collector can be decomposed into a local stagnation–point flow and a local simple–shear flow. Theoretical background is well described in our previous paper [12].
The connective–diffusion equation can be written as follows:where J is the particle flux vector, Q is a source term, D the diffusion tensor and u the velocity which can consist of several contributions e.g. uhydr, ucoll and uext. Hydrodynamic velocity is usually obtained by solving the Navier–Stokes equation neglecting the presence of the dispersed phase. Because of the latter assumption results apply only to dilute dispersions:where v is the fluid flow velocity, is the fluid density, p is the pressure, is the dynamic viscosity, Fext is the body volume force exerted on the fluid and t is the time. For the incompressible fluids Eq. (3) is complemented by the continuity Eq. (4):Analytical solution of Eqs. (3), (4) is available for a given initial velocity field and specified boundary conditions for simple geometries. In our case a stagnation–point flow collector was used which in experimental configuration is called “impinging jet” [13]. The fluid velocity components (radial and normal) in the vicinity of the stagnation point can be described [13]:where r is the radial distance from the symmetry axis, z is the distance from the adsorbing interface, αr is a flow intensity parameter and um is the mean linear velocity given in (8). The latter parameter αr is dependent on the cell geometry and the flow intensity, which is usually expressed by the Reynolds number (Re):where Q is the volume flow, rc is the capillary radius of the impinging jet, is the kinematic viscosity of the fluid.For our experimental set–up the ratio of the distance between the confining plate and the collector to the radius of the jet (h/rc) was 1.7. For this geometry, values of αr for a Reynolds number of more than 50 can be expressed as follows [14]:where u is the mean velocity of the jet.
Number of particles depositing per unit area per unit time on the wall is expressed as a particle flux j. This flux is expressed in praxis in the form of dimensionless Sherwood number Sh (10):where D0 is the diffusion constant, and n0 is the number concentration of particles. By use of Smoluchowski-Levich approximation for the case when there is no energy barier between the particle and the collector (so called fast deposition) and by neglecting gravity forces particle flux is given as:where z is the dimensionless distance between the sphere center and the wall, and Pe is the Peclet number.In the case when the energy barier is present, the deposition rate is reduced and for the particle flux we obtain:where αd is called the deposition efficiency. When electrostatic repulsion is acting between particles and the surface αd < 1 while for electrostatic attraction αd > 1 [10]. When only the Van der Waals forces are acting αd ≈ 1. Eq. (12) describe the initial deposition rate on a base surface. As deposition proceeds, deposited particles slow down the deposition. In the absence of particle detachment [10]:where nt is the number of deposited particles at time t, n∞ they steady state number, and τbloc is the characteristic time required to reach the steady state, referred to as the blocking time. The blocking time can be expressed as [15]:where γ is the blocking coefficient which represents a normalized area per deposited particle that effectively blocks further deposition, jc is the rate of particle deposition on a bare surface near to stagnation point given as [10]:The most stable equillibrium contact angle of liquid droplet on solid smooth and hoterogenous surface (θ) can be described by Young equation, which corresponds to minimal energy state among the three phases [16]:Where γSG, γSL and γLG are the surface tensions of solid- gas, solid- liquid and liquid- gas interfaces. In the case of the heterogenous rough surface observed contact angles differs from smooth surface and are described by Wenzel and Cassie-Bexter [17], [18]. If the liquid is in contact everywhere with the solid surface, system is in Wenzel regime and following equation can be used:Otherwise when the liquid drop is in conctact only with the top protrusions on the surface (no liquid penetration into a solid surface is observed) Cassie-Baxter aproach for determination of apparent equilibrium contact angle (i.e. θ∗) can be applied [19]:where Φs is the portion of solid region, that is in touch with the liquid droplet area and θ is the equlibrium contact angle on a smooth surface. If we suppose, that there system is in ideal Cassie regime, the surface textures is much smaller than the droplet size and the three-phase contact line (CL) constrain is small or inconsiderable, we can calculate total surface free energy change for very small displacements of contact line (dR) after and before the movement [19]:where dEsurface and dEline represent surface and line energies. dEsurface can be defined asThe total line energy change (Eline) can be calculated as follows:Therefore the dependence of the texture size (r) on the most stable apparent equilibrium contact angle (θ∗) using the Young equation can be determinated by extented Cassie – Bexter equation in the case, that the changes of total surface free energy are minimalized [19]:where the τ means the contact line tension at three-phase interface.
Section snippets
Methods
For impinging jet experiments was used setup original designed by Dąbroś, van de Ven and Adamczyk [3], [13], [15]. This setup is sometimes mentioned as radial impinging-jet cell (RIJ) [3]. The volumetric flow rate Q was controlled by adjustment of the vertical position of the colloidal dispersion level and by changing the inner diameter of inlet and outlet capillary. Deposition process was followed by microscope SM 5 (Intraco Micro, Czech Republic) with magnification 200 × for all experiments at
Surface energy and topology characterization
As mentioned in the introduction, magnitude of a surface free energy and of equilibrium contact angles of wetting liquids on collector substrate surfaces play important role during particle deposition process. In our study we have used polystyrene latex particles for deposition. Polymer substrate under study (Zeonor) wetting characteristics of both virgin and embossed samples are summarized in Table 2. It is evident, that for water contact angles of wetting of virgin Zeonor surfaces was ranging
Conclusions
There were performed an impinging jet deposition experiments on synthetic polymer (Zeonor) original and by micro-embossing modified substrates with exactly defined topology as confirmed by AFM and SEM. Deposition experiments were performed at ambient temperature and at selected flow regime of Re = 10. As a particles deposited the PS 1,1 μm diameter particles (Sigma–Aldrich) were used having negative charge of -20 mV as observed by zeta potential experiments. There was found gradual increase of
Conflict of interest
None declared.
Acknowledgements
Financial support from the Operational Program Research and Development for Innovations – European Regional Development Fund (grants CZ.1.05/3.1.00/14.0302 and CZ.1.05/2.1.00/03.0058) is gratefully acknowledged. Special thanks to Mgr. K. Šafářová, Ph.D. and Ing. O. Tomanec for SEM and AFM measurements.
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