This thesis is a computational investigation of some aspects of the constant stress and pressure rheology of dense polydisperse colloidal suspensions. This work reveals the critical role of interfacial motion in suspension stress and structure.
235 7.2 (Color online) Simulation results as functions of stress Péclet. a): the shear and normal viscosities, respectively ηsandηn, and (b): the volume fraction φ. Inset: the product (ηId∞s /(Πa2) as functions of volume fraction φ. b) The peak difference ∆p = max(gcomp) − max(gext) as functions of the strain rate scale ˙γa2/d∞s.
LIST OF TABLES
INTRODUCTION
The fourth part of the thesis focuses on the constant stress and pressure rheology of colloidal suspensions without and with hydrodynamic interactions. Sather, "The hydrodynamic interaction of two inequivalent spheres moving under gravity through still viscous fluid", J.
SHORT-TIME TRANSPORT PROPERTIES OF BIDISPERSE COLLOIDAL SUSPENSIONS AND POROUS MEDIA
Introduction
Here, we present the following transport properties of bidisperse porous media: (1) translational drag coefficient, which is related to permeability, (2) translationally hindered diffusivity, and (3) rotationally hindered diffusivity. The simplicity of the SD framework unfortunately comes at the cost of the accuracy of some transport properties.
Bidisperse suspensions and porous media Static structuresStatic structures
Srα(t)= hP2[ ˆni(t)· nˆi(0)]i,i ∈α, (2.28) where P2(x) is the second-order Legendre polynomial, defining the short-time rotational self-diffusion. The movement of one species can give rise to a strong backflow that reverses the sedimentation rate of the second species, i.e. the particles move in a direction opposite to the imposed body force, especially when the body force is weak [28].
The polydisperse Stokesian Dynamics
When calculating the contributions of the three bodies to the short-term self-diffusion of the suspension, the lubrication corrections corresponding to the collective motion destroy the convergence and must be eliminated [10]. Note that the finite-size collapse of H22(q) in Fig.2.2c for sparsely scattered small Nis due to the limited number of large particles, e.g., at N = 100, there are only 11 particles large in the mix.
The pairwise additive approximation
The integrals for the PA approximations are numerically evaluated using Gauss-Kronrod quadrature over the entire integration domain. Table 2.1 shows PA approximation coefficients for suspension properties with gα β = 1 and for sedimentation rates density ratio γ =1.
Analytical results beyond the PA level Suspension propertiesSuspension properties
For the sedimentation rate coefficientSα β, the density ratio γ = 1. The integrals for the PA approximations are evaluated numerically using the Gauss-Kronrod quadrature over the entire integration domain. In this range, the presence of the second species always reduces the peak value relative to monodisperse suspensions, and the reduction increases with increasing φ, e.g., atφ=0.4, the maximum reduction is 20%. As revealed by experiments and simulations [31], the decrease in viscosity is mainly due to improved packing for polydisperse suspensions, i.e., the average particle spacing increases with λ, leading to a reduction in viscosity.
Results for porous media Permeability (mean drag coefficient)Permeability (mean drag coefficient)
The hindered diffusivity for small particles, dHD,1t, shows a moderate increase relative to monodisperse systems, similar to dts,1. For φ > 0.6, the hindered diffusivity for larger dtHD,2 particles becomes extremely sensitive to small particles. In bidisperse porous media, drHD, α for both species is very sensitive to the size ratio λ.
Concluding remarks
Woutersen, "The viscosity of bimodal and polydisperse suspensions of hard spheres in the dilute limit", J. Beenakker, "The effective viscosity of a concentrated suspension of spheres (and its relation to diffusion)", Physica A. Jeffrey, "The calculation of the low Reynolds number drag functions for two odd spheres”, Phys.
SHORT-TIME DIFFUSION OF COLLOIDAL SUSPENSIONS
Introduction
This method, commonly referred to as the δγ scheme, uses resummation techniques that compute an infinite subset of the hydrodynamic distribution series [31], including all particles in suspension. The proposed, rescaled δγ scheme can be particularly useful in the analysis of scattering experiments, where only a limited part of the hydrodynamic function can be measured due to the limited range of accessible wave vectors. Our SD simulations are described in Sec.3.4, which is followed by a discussion of static pair correlation functions in Sec.3.5.
Bidisperse hard-sphere suspensions
By comparing with our SD simulation results, we show that the rescaled, modified δγ scheme predicts the partial hydrodynamic functions of both species with surprisingly good accuracy for suspension volume fractions as high as 40%. In Section 3.7 we present our results for partial hydrodynamic functions of different suspensions, and we draw our final conclusions in Section 3.8. The solvent is assumed to be incompressible and the Reynolds number of particle motion is assumed to be very small so that the solvent velocity field v(r) and the dynamic pressure field p(r) satisfy the stationary Stokes equation with incompressibility constraint.
Short-time diffusion
In computer simulations, Sα β(q,t) and SN N(q,t) are easily extracted when all the time-dependent particle positions rkγ(t) are known, but the challenge lies in the exact calculation of the latter . D(q) = kBTH(q)·S−1(q), (3.13) to a product of the matrix H(q) of partial hydrodynamic functions Hα β(q) and the inverse partial static structure factor matrixS−1( q). The short translational self-diffusion coefficient dαs is equal to the time derivative of the mean square displacement Wα(t) = 16D.
Stokesian Dynamics simulations
The near-field smearing correction (R2B −R∞2B) is based on exact two-body solutions with far-field contributions removed and accounts for singular HIs when the particles are in close contact. The SD method recovers exact solutions of two-particle problems and has been shown to agree well with the exact solution of three-particle problems [44]. The lubrication correction (R2B−R∞2B) for a pair of particles with radiiaαandaβ is based on the exact solution of two-body problems in Ref.
Static pair correlations
The sedimentation velocity from a finite size system with periodic boundary conditions is a superposition of the velocities from random suspensions and cubic lattices [54,55] . The main source of error for our method is from the various approximations made in the δγ scheme and its modifications, rather than the small inaccuracy of the structural input. In addition, a related line of research deals with testing and improvements of the various δγ scheme approximations (for monodisperse suspensions) [34].
Rescaled δγ scheme
The size of γ-type particles is chosen such that φγ = φ= φα+φβ, and their center-of-mass positions match those of α- and β-type particles in the bidisperse suspension (top panel). The estimate of fα in Eq. 3.33) requires an approximation of the short-time self-diffusion type dαs/d0α in the mixture. A series expansion in the inverse particle separation gives the far-field order terms of the integrand. 3.37).
Results and discussions
The results for the modified δγ scheme are shown as solid curves, and the results for the parameter-free rescaled δγ scheme with fα from Eq. The performance of the rescaled δγ scheme for size ratiosλ, 2 (and in particular for λ > 2) remains to be investigated. However, establishing an accuracy measure of the rescaled δγ scheme in the full range of suspension parameters requires direct comparison with accurate hydrodynamic calculations.
Conclusions
Ladd, “Short-time motion of colloidal particles: numerical simulation via a fluctuating lattice-Boltzmann equation,” Phys. Powell, "Computation of ewald summed far-field mobility functions for arbitrary size spherical particles in stokes flow", Phys. Brady, “A new drag function for two rigid spheres in a uniform compressible low Reynolds number flow”, Phys.
SPECTRAL EWALD ACCELERATION OF STOKESIAN DYNAMICS FOR POLYDISPERSE COLLOIDAL SUSPENSIONS
Introduction
The rest of the chapter is organized as follows: Section 4.2 establishes the basic formalism for HIs in compressible Stokes flow. In Section 4.4, we present Spectral Ewald Accelerated Stokesian Dynamics (SEASD) and its mean-field Brownian approximation, SEASD-nf, for dynamical simulations of Brownian polydisperse suspensions. In Section 4.5, we carefully discuss the accuracy and parameter choices of the SE method and the computational scaling of different SEASD implementations.
Hydrodynamic interactions in (compressible) Stokes flow The mobility and resistance formalismThe mobility and resistance formalism
The surface force density is localized on the particle surface, i.e., fj(r) = σ(r) · njδ(kr − rjk −aj), where σ is the stress tensor, nj is the surface normal of particle j, and δ(x ) is the Dirac -delta function. The velocity perturbationu0i(r) =Ui+Ωi×(r−ri)−v∞(r), where v∞(r) is the ambient current satisfying ∇· v∞ = 0, and Ui and Ωi are the linear and angular velocities, respectively of particlesi. From the Fourier transform the solution is Q(r) = 1. The above wave space sum can be split into two exponentially convergent series [30, 60] using. 4.34) As with Q(r), the fundamental pressure solutionP(r) in Eq. 4.29) can also be extended to periodic systems.
The mobility computation
The wavelet space results are then returned to real space by inverse FFTs. The first is related to the truncation of the wavespace sum (k-sum) in Eq. After converting real space H(t) to wave space ˆHq with FFTs, it produces a wave space calculation.
Dynamic simulation with Stokesian Dynamics
0=−RnfF U · UH+FH,ff +FHP, (4.55) where RnfF U is the F U coupling in Rnf and stored as a sparse matrix, FHP = FP+RnfFE·E∞ contains the interparticle forceFP and the near-field contributions of E∞. The far-field hydrodynamic force FH,ff satisfies. where SH,ff is the far-field stresslet of HIs. Due to the special coupling between the pressure moments and other force moments in compressible suspensions (Sec.4.2), the interaction contribution to the far-field pressure moment is evaluated as FH,ff and the traceless part of SH,ff is given in Eq. Note that Rnf composed of two-body problems contains both the relative and the collective motions of the particle pair and, as pointed out by Cichockiet al.[23], the smearing corrections corresponding to the collective motion can destroy the far-field asymptotes of the pair shallow.
Accuracy and performance Mobility computation accuracyMobility computation accuracy
As mentioned earlier, the SE method allows for separate controls on real-space and wavespace truncation errors and wavespace interpolation error. The total error∞,r(E) reaches a minimum at intermediateξa1 when the wavelet and real-space errors are approximately the same. The GPU results are shown in black lines and the CPU results in Fig.4.2b are reproduced in gray lines.
Static and dynamic simulation results Short-time transport propertiesShort-time transport properties
The differences between the results of SEASD and conventional SD are more significant for bidisperse suspensions. Finally, we mention in passing that shear-induced particle migration takes place in confined suspensions with a spatially varying strain rate, e.g. Poiseuille flow, and can be investigated computationally by introducing confining boundaries. In addition, SEASD and SEASD-nf results are shown with solid and open symbols, respectively.
