QCM-D Investigations of Anisotropic Particle Deposition Kinetics: Evidences of the Hydrodynamic Slip Mechanisms

Deposition kinetics of positively charged polymer microparticles, characterized by prolate spheroid shape, at silica and gold sensors was investigated using the quartz microbalance (QCM) technique. Reference measurements were also performed for positively charged polymer particles of spherical shape and the same mass as the spheroids. Primarily, the frequency and bandwidth shifts for various overtones were measured as a function of time. It is shown that the ratio of these signals is close to unity for all overtones. These results were converted to the dependence of the frequency shift on the particle coverage, directly determined by atomic force microscopy and theoretically interpreted in terms of the hydrodynamic model. A quantitative agreement with experiments was attained considering particle slip relative to the ambient oscillating flow. In contrast, the theoretical results pertinent to the rigid contact model proved inadequate. The particle deposition kinetics derived from the QCM method was compared with theoretical modeling performed according to the random sequential adsorption approach. This allowed to assess the feasibility of the QCM technique to furnish proper deposition kinetics for anisotropic particles. It is argued that the hydrodynamic slip effect should be considered in the interpretation of QCM kinetic results acquired for bioparticles, especially viruses.


Basic Considerations
The QCM sensor load for nano-and microparticle size range, typically equal to 20-2000 mg m -2 1 , is considerably lower than the quartz crystal surface density equal to 5 10 6 mg m -2 (assuming the crystal thickness of 0.2 cm). In consequence, particle deposition occurs under the small-load (Sauerbrey) regime [2][3][4] where the complex QCM response is given by where f  is the frequency shift (a real number), The force is opposite in direction to the net force acting on particles in the vicinity of the sensor. It consist for rigid particles of the hydrodynamic, the inertia and the specific surface contributions such as electrostatic and van der Waals forces. It should be mentioned that the specific forces have both the perpendicular and tangential components, the former fix the equilibrium particle distance from the surface, whereas the latter govern the rigidity of the particle contact with the surface. For larger particles the inertia and hydrodynamic forces may exceed the specific ones that can promote their rocking and sliding motion.
In order to calculate the hydrodynamic force one should first know the ambient flow in the vicinity of the oscillating sensor considered to be a flat plate. As shown in Ref. 5 the laminar flow relative to the plate V  is given by where z is the distance perpendicular to the plate (see Figure S1), t is the time, 0 V = m a  , ω=2πfF n0 is the angular velocity of the sensor, m a is the amplitude of oscillations, is the plate tangential velocity and is the hydrodynamic boundary layer thickness also referred to as the flow penetration depth [3][4][5] , and v is the kinematic viscosity of the liquid. For distances from the plate much smaller than the penetration depth, where z/ < 1, the flow pattern simplifies to S-4 ( ) This is a simple shear flow with the rate given by Figure S1. Schematic representation of the flow pattern near a spheroid particle in the vicinity of a oscillating solid boundary (QCM sensor) It should be observed that the relaxation time of establishing the flow pattern given by Eq.(S4) can be calculated as Therefore, the flow at the distance z < δ is quasi-stationary, characterized by negligible explicit time dependence.
Knowing the flow pattern, the parallel component of the hydrodynamic force on a single spheroidal particle in the vicinity of the sensor (see Figure S1) can be calculated as 6  is the density of the liquid, b is the shorter semi-axis of the spheroid, U is the tangential component of the particle velocity due to hydrodynamic forces and torques and c F is the correction function accounting for the presence of the wall depending on the surface to surface distance h and the spheroidal particle shape defined by the parameter / ab For prolate spheroids c F also depends on the orientation of the spheroid longer axis against the flow direction, governed by the angle  .
In the general case, the hydrodynamic force can only be calculated by elaborated numerical techniques such as the Lattice Boltzmann method 7,8 or the immersed boundary method (IBM) 9,10 . However, useful analytical solutions can be derived for some limiting cases discussed below.

Force on rigidly fixed particles
For a stationary spherical particle where U = 0, with its center located at the distance z = b + h much smaller than the penetration depth (see Figure S1) the hydrodynamic force calculated using Eqs.(S6,S9) is given by where mp is the particle mass, f  , p  are the fluid and particle densities, respectively, is the universal hydrodynamic function pertinent to a spherical particle in a simple shear flow 11,12 . For a particle touching the sensor where h = 0, 8  On the other hand, in Ref. 15 the hydrodynamic force on the ensemble of Np particles forming a rigid layer at the sensor was calculated applying the multipole expansion method.
The numerical calculations were interpolated by the following analytical expression where N =Np/Si is the surface concentration of particles, It was also shown in Ref. 15 that the correction function is independent of the structure of the particle layer for the coverage up to 0.85.
In consequence the hydrodynamic force on a single particle in the layer is given by the formula ( ) Interesting analytical solution was also derived in Ref. 4 in the opposite limit where the hydrodynamic boundary layer becomes much smaller than the particle radius, which is the case for larger particles and large oscillation frequency (overtone number). Under such conditions one can assume that the particle attached to the interface effectively oscillates in a stagnant fluid. In consequence, to calculate the hydrodynamic force, one can use the known solution discussed in Ref. 16 . In consequence, as shown in Refs. 4,5 the hydrodynamic force on the particle is given by   = One should remember that in the case of a fixed particle, the net force consist of the hydrodynamic and the inertia contributions, i.e., S14)

S-7
In Ref. 12 the translational and the rotational motion of a neutrally buoyant spherical particle immersed in a simple shear flow s V = G z near a solid wall was analyzed. The exact numerical solutions obtained in Ref. 12 yielded the particle tangential velocity as a function of h One can predict from Eqs.(S15,S16) that the normalized slip velocity of the particle relative to the ambient shear flow defined as ( V  -U) / V  is equal to 0.526, 0.43, and 0.255 for h/b = 0.005, 0.02 and 0.1. This indicates that even at distance between the particle surface and the solid wall (sensor) equal to 0.5% of b the hydrodynamic forces is not sufficient to fully immobilize the particle. It is interesting to mention that such an effect was numerically confirmed in Ref. 9 .
Using Eqs.(S9,S15) the hydrodynamic force in the case of a neutrally buoyant particle can be expressed as * 38 Because of the quasi-stationary motion assumption, Eq.(S17) remains exact if the particle semi-axis (radius) b remains is smaller than the penetration depth .
Considering the above results one can predict that the force exerted on the neutrally buoyant particle moving near a boundary in shear flow is considerably lower than that the force exerted on a stationary one. For example, at the distance h equal to 10% of the particle radius, the force is more than four time smaller.

S-8
In Ref. 18 analogous problem of a free spherical particle motion parallel to a solid wall in a quiescent fluid was considered. The exact numerical solutions were interpolated as where 0 i UV = is the uniform translation velocity of the particle far from the wall and 4 ( / ) F h b is the universal correction function, which can be approximated as 17 One can calculate from Eqs.(S18,S19) that in this case the normalized particle slip velocity relative to the bulk 00 ( ) / U U U − is equal to 0.736, 0.672, and 0.543 for h/b = 0.005, 0.02 and 0.1. This indicates that the hydrodynamic coupling with the wall is much stronger compared to the particle motion in shear flow.
Using Eqs.(S18,S19) one can formulate the following expression for the hydrodynamic force on a freely oscillating particle in the case where  < 1 2 * 4 The above derived expressions for the hydrodynamic force enable to explicitly calculate the frequency and dissipation shifts using the constitutive Eq.(S3) by noting that the excess tangential force acting on the sensor * i F  is opposite to the net force acting on the particle.
Thus, in the case of stationary (fixed) spheres, the frequency and dissipation shifts due to a layer of Np particles of equal size calculated from Eqs.(S12,S14) are FN is the correction function, which is expected to be close to unity for the entire range of particle coverage.
Using Eq.(S17) one can derive analogous dependencies for the layer of neutrally buoyant particles and () 1 / bh For the opposite case where () 1 / bh + one obtains from Eq.(S20) the following expression for the frequency and dissipation shifts due to a layer of Np single particles FN is the correction function, which is expected to be close to unity for the entire range of particle coverage.

Modeling Adsorption Kinetics of Particles in the QCM cell.
Particle deposition kinetics under convective-diffusion transport conditions at solid substrates (for example QCM sensor) can be theoretically described using a hybrid approach exploiting the convective-diffusion equation 17 where n is the number concentration of particles, t is the time, D is the translation diffusion coefficient, k is the Boltzmann constant, T is the absolute temperature, F is the external force where m is the primary minimum distance.
In order to explicitly calculate particle deposition kinetics from Eq.(S31) one should also know the blocking function, which can be conveniently acquired from the random sequential adsorption (RSA) modeling 17 For the side-on adsorption of prolate spheroidal particles, the blocking function at the jamming limit assumes an analogous form where the dimensionless constant C  varies between 2.8 -3.2 for spheroids 21

S-13
It was shown in Ref. 17  where Le is the electric double layer thickness, dp is the particle characteristic dimension, o  is electrostatic energy at contact and ch  is the characteristic interaction energy.
Consequently, one can calculate the jamming coverage for interacting particles referred to as the maximum coverage) from the relationship