Sedimentation of a Charged Spherical Particle in a Viscoelastic Electrolyte Solution

Abstract

When a charged particle translates through an electrolyte solution, the electric double layer around it deforms in response to the fluid motion and creates an electric force opposite the direction of motion, decreasing the settling velocity. This is a multidisciplinary phenomenon that combines fluid mechanics and electrodynamics, differentiating it from the classical problem of an uncharged sedimenting particle. It has many applications varying from mechanical to biomedical, such as in drug delivery in blood through charged microparticles. Related studies so far have focused on Newtonian fluids, but recent studies have proven that many biofluids, such as human blood plasma, have elastic properties. To this end, we perform a computational study of the steady sedimentation of a spherical, charged particle in human blood plasma due to the centrifugal force. We used the Giesekus model to describe the rheological behavior of human blood plasma. Assuming axial symmetry, the governing equations include the momentum and mass balances, Poisson's equation for the electric field, and the species conservation. The finite size of the ions is considered through the local-density approximation approach of Carnahan-Starling. We perform a detailed parametric analysis, varying parameters such as the ζ potential, the size of the ions, and the centrifugal force exerted upon the particle. We observe that as the ζ potential increases, the settling velocity decreases due to a stronger electric force that slows the particle. We also conduct a parametric analysis of the relaxation time of the material to investigate what happens generally in viscoelastic electrolyte solutions and not only in human blood plasma. We conclude that elasticity plays a crucial role and should not be excluded from the study. Finally, we examine under which conditions the assumption of point-like ions gives different predictions from the Carnahan-Starling approach.

Figure 1

Schematic of the spherical particle sedimenting through a viscoelastic fluid under the effect of body force, , which in this study is a centrifugal force. Here, Ω represents the area that the viscoelastic material occupies. A_s and S_outer denote the axis of symmetry and the far-field boundary, respectively.

Figure 2

Effect of the g-factor, , on (a) the particle velocity and (b) the electric force for different values of ζ and Φ_B = 0. The particle is sedimenting in blood plasma which follows the Giesekus model using the parameters of Table 2.

Figure 3

Contours of T_zz for (a) , (Wi = 0.58) and (b) , (Wi = 23.232), when ζ = 3 and Φ_B = 0. Τhe extent of the EDL is indicated by the white dashed lines. The particle is sedimenting in blood plasma which follows the Giesekus model using the parameters of Table 2.

Figure 4

Distortion of the dimensionless counterion concentration profiles for different Weissenberg numbers: (a) , (Wi = 2.56), (b) , (Wi = 9.16), and (c) , (Wi = 23.232) when ζ = 3 and Φ_B = 0. The particle is sedimenting in blood plasma which follows the Giesekus model using the parameters of Table 2.

Figure 5

Electric potential and the corresponding electric field lines around the particle for different Weissenberg numbers: (a) , (Wi = 2.56), (b) , (Wi = 9.16), and (c) , (Wi = 23.232), when ζ = 3 and Φ_B = 0. The particle is sedimenting in blood plasma which follows the Giesekus model using the parameters of Table 2.

Figure 6

Effect of the Weissenberg number on the particle velocity for for different values of ζ and Φ_B = 0. The particle is sedimenting in a viscoelastic material that follows the Giesekus model using the parameters of Table 2. Only the relaxation time varies here.

Figure 7

Effect of the Zeta potential on the particle velocity for (a) , (Wi = 0.528) and (b) , (Wi = 5.28). The particle is sedimenting in blood plasma which follows the Giesekus model using the parameters of Table 2.

Figure 8

Effect of the zeta potential on the (a) electric force, (b) viscoelastic drag, and (c) form drag for , (Wi = 5.28). The particle is sedimenting in blood plasma which follows the Giesekus model using the parameters of Table 2.

Figure 9

Steric effect on ionic concentration for , (Wi = 5.28) when ζ = 3, (a) cations and (b) anions. The left-hand side of each panel corresponds to Φ_B = 0.2, and the right-hand side corresponds to Φ_B = 0. The particle is sedimenting in blood plasma which follows the Giesekus model using the parameters of Table 2.

Figure 10

Contours of T_rr for , (Wi = 0.528) and for two different values of zeta potentials: (a) ζ = 2 and (b) ζ = 4. The particle is sedimenting in blood plasma which follows the Giesekus model using the parameters of Table 2.

Figure 11

Contours of T_rz for , Wi = 0.528 and for two different values of zeta potential: (a) ζ = 2 and (b) ζ = 4. The particle is sedimenting in blood plasma which follows the Giesekus model using the parameters of Table 2.

Figure 12

Contours of the trace of the conformation tensor for , Wi = 0.528 and for two different values of zeta potential: (a) ζ = 2 and (b) ζ = 4. The particle is sedimenting in blood plasma which follows the Giesekus model using the parameters of Table 2.

Figure 13

Effect of the ratio of the diameter to the Debye length at ζ = 4 for (a) , (Wi = 0.528) and (b) , (Wi = 5.28). The particle is sedimenting in blood plasma which follows the Giesekus model using the parameters of Table 2.

Figure 14

Effect of the ratio of the particle diameter to the Debye length on (a) the form drag (left) and on (b) the viscoelastic drag (right) for , Wi = 0.528 with ζ = 4. The particle is sedimenting in blood plasma which follows the Giesekus model using the parameters of Table 2.

Figure 15

Contours of the pressure field for (a) K = 0.09, (b) K = 2, and (c) K = 4. The values of the other dimensionless numbers correspond to , (Wi = 0.528) with ζ = 4. The particle is sedimenting in blood plasma which follows the Giesekus model using the parameters of Table 2.

Figure 16

Contours of T_rz for (a) K = 0.5 and (b) K = 1. The values of the other dimensionless numbers correspond to (Wi = 0.528) and ζ = 4. The particle is sedimenting in blood plasma which follows the Giesekus model using the parameters of Table 2.

Figure 17

Contours of T_rz for (a) Φ_B = 0 and (b) Φ_B = 0.2. The values of the other dimensionless numbers correspond to K = 1 for , (Wi = 0.528) with ζ = 4. The particle is sedimenting in blood plasma which follows the Giesekus model using the parameters of Table 2.

Figure A1

(a) Effect of the Reynolds number on the particle velocity for K = 2 and different values of zeta potential. The range of and the other dimensionless numbers is summarized in Table A1. (b) Mesh convergence for the Newtonian material for ζ = 4 and for three different meshes (Table A2) and (c) mesh convergence validation for the viscoelastic material for ζ = 3 and for three different meshes (Table A2).