Hydrodynamic Solvent Coupling Effects in Quartz Crystal Microbalance Measurements of Nanoparticle Deposition Kinetics

Hydrodynamic coupling effects pertinent to quartz crystal microbalance (QCM) investigation of nanoparticle adsorption kinetics were evaluated using atomic force microscopy and the theoretical modeling. Monodisperse polymer particles of the size between 26 and 140 nm and the density of 1.05 g cm–3 were used. The ζ-potential of particles was opposite to the substrate ζ-potential that promoted their irreversible adsorption on the silica sensor. The experimental kinetic data were interpreted in terms of theoretical calculations derived from the hybrid random sequential adsorption model. This allowed us to determine the amount of hydrodynamically coupled solvent (electrolyte) for the absolute particle coverage range up to 0.5. The coupling function representing the ratio of the solvent to the particle volumes was also determined and used to explicitly calculate the solvent level in particle monolayers. It is shown that the solvent level abruptly increases with the particle coverage attaining values comparable with the particle size. One can expect that these results can serve as useful reference data for the interpretation of protein adsorption kinetics on rough surfaces where the presence of stagnant solvent is inevitable.


Basic characteristic of the polymer particles.
Footnotes: particle density  p = 1.05 g cm -3 , S-6

The hydrodynamic boundary layer over an oscillating plate
The fluid velocity distribution over a plate undergoing harmonic oscillations in the tangential direction characterized by the angular velocity =2πf (where f is the oscillation  frequency) in a Newtonian liquid is given by 1 (S1) where V 0 is the amplitude of plate velocity oscillations, v is the kinematic viscosity of the fluid, z is the vertical distance from the plate.
Eq.(S1) indicates that the fluid velocity exponentially decreases with the distance z proportionally to the square root of the angular velocity.
Let us calculate the distance where the flow amplitude decreases (compared to the plate velocity) by the factor denoted by f v . Using Eq.(S1) this factor can be expressed as for ln( f v -1 ) = 1 one has which is commonly considered as a hydrodynamic boundary layer thickness 1 .
In  A graphical comparison of the comparison of the polymer particle sizes used in this work with the hydrodynamic boundary layer thickness for the 1 st and the 7 th overtone is presented in Figure S4. Figure S3. The polymer particle size compared with the hydrodynamic boundary layer thickness calculated for the 1-st and the 7 th overtones, and equal to 238 and 90 nm, respectively.
From Eq.(S1) one can deduce that the flow shear rate is given by 1 Consequently, the shear rate amplitude attains the maximum value of at the plate surface and exponentially decreases with the distance from the surface with the characteristic decay rate equal to the hydrodynamic boundary layer thickness. At the distance equal to particle radius a p the shear rate amplitude normalized by the surface amplitude is given where f Sh is the dimensionless correction factor, which only depends on the ratio of the particle to the hydrodynamic boundary layer thickness. This correction factor calculated for various particle sizes as a function of the overtone number are collected in Table S3.

Estimation of the adhesion hydrodynamic and inertia forces on particles.
The basic assumption of the DLVO theory 2,3 is that the net interaction energy of particle with interfaces is a sum the electrostatic double-layer and the van der Waals contributions. Therefore, approximating the electrostatic component by the linear superposition approach (LSA) 2 one can formulate this postulate as follows where is the not energy, L e is the double layer thickness is the electrostatic energy, is the van der Waals energy, A 123 is the Hamaker constant for the interactions of the vdW  particle with the interface through the solvent (electrolyte), a p = d p /2 is the particle radius.
is the permittivity of the medium, k is the Boltzmann constant, T is the absolute temperature,  ζ p , ζ i are the zeta potentials of the particle and the interface, respectively, h= z -a p is the surface to surface distance between the particle and the interface.
It should also be mentioned that the expression for the van On the other hand, the amplitude of the inertia force exerted on an attached particle due to the sensor oscillation is given by Comparing Eq.(S7) and Eq.(S10) one can deduce that for a given system the ratio of the shearing to the DLVO force scales up as the particle size, whereas the ratio of the acceleration to the DLVO force scales up as the square of the particle size. One can, therefore, expect that the acceleration force will dominate over the DLVO and over the shearing forces for larger particle sizes. In order to quantitatively analyze this behavior, in Table S4 these forces are compared for particle sizes used in this work. Footnotes: the temperature 298 K, kinematic viscosity 0.00893 cm 2 s -1 , δ h = 238 nm, zeta potential of the sensor equal to -20 mV for pH 4 and 40 mV for PAH modified sensor in the case of the L40 particles (at pH 5.7), A 123 = 1.4 10 -20 kT, V 0 = 6 cm s -1 , frequency  5 10 6 s -1 (first overtone), F 8 =1.70.  S-12

Modeling Adsorption Kinetics of Particles in the QCM cell.
As discussed in Ref. 2

particle deposition kinetics under convective-diffusion transport
conditions under solid substrates (for example QCM sensor) can be theoretically described using a hybrid approach exploiting the convective-diffusion equation where n is the number concentration of particles, t is the time, D is the translation diffusion coefficient, F is the external force vector (e.g. the gravitation force) rand V is the unperturbed (macroscopic) fluid velocity vector.
Eq.(S11) is coupled with the surface layer transport equation where the fluid convection effects are neglected 1 where j a is the net adsorption/desorption flux, is the characteristic cross-section of the g S particle, is the macromolecule coverage, N p is the particle surface concentration, Eq.(S12) is used as the boundary condition for the bulk mass transfer equations, Eq.(S11).
Under convective transport, where the particle concentration n( a ) remains in a local equilibrium with the surface coverage, the constitutive expression for the adsorption flux, Eq.(S12) becomes 2 (S13) where K = k a / k c is the dimensionless coupling constants, K d = k d /(S g k c n b ) is the dimensionless desorption constant, and k c is the bulk transfer rate constant, known in analytical form for many types of flows and n b is the bulk number concentration of particles.
Eq.(S3) can be integrated, which yields the following dependence (S14) is the initial coverage of particles.

 
Eq.(S14) represents a general solution for particle deposition kinetics under convection driven transport. However, it can only be evaluated by numerical integration methods if the blocking function is known in an analytical form.
It is interesting to mention that in the case of particle desorption run where the bulk concentration n b vanishes, Eq.(S14) simplifies to the form assuming that the blocking function does not change much during the desorption run and is equal to B 0 one can integrate Eq.(S15) to the useful form where K a = k a /k d is the equilibrium adsorption constant.
Hence, the characteristic desorption time is given by It is useful to express Eq.(S14) in terms of the mass coverage of particles connected with the dimensionless coverage by (S18) where m 1 is the mass of a single particle.
It should be mentioned that in terms of the QCM nomenclature, the coverage Γ is the so called 'dry' mass.
Using this definition one can express Eq.(S14) in the form Often the interaction energy around the primary minimum and the barrier region can be approximated by a parabolic distribution. Consequently the kinetic adsorption constants can be approximated by 2 and D( b ) is the value of the diffusion coefficient approximated within the barrier region according to the lubrication theory by the formula .
It should be mentioned that for a barrier-less adsorption regime the kinetic adsorption constant can be calculated from the dependence 2 (S25) (