---
abstract: 'Based on the notion of a construction process consisting of the stepwise addition of particles to the pure fluid, a discrete model for the apparent viscosity as well as for the maximum packing fraction of polydisperse suspensions of spherical, non-colloidal particles is derived. The model connects the approaches by and and is valid for large size ratios of consecutive particle classes during the construction process. Furthermore, a new general form of the well-known equation allowing for the choice of a second-order coefficient for the volume fraction ($\phi^2$-coefficient) is proposed and then applied as a monodisperse reference equation in the course of polydisperse modeling. By applying the polydisperse viscosity model to two different particle size distributions (<span style="font-variant:small-caps;">Rosin-Rammler</span> and uniform distribution), the influence of polydispersity on the apparent viscosity is examined. The extension of the model to the case of small size ratios as well as to the inclusion of shear rate effects is left for future work.'
author:
- Aaron Dörr
- Amsini Sadiki
- Amirfarhang Mehdizadeh
title: A discrete model for the apparent viscosity of polydisperse suspensions including maximum packing fraction
---

Introduction {#chap:Grundlagen}
============

Multiphase flow is a very important but unexploited field of research according to the variety of unsolved questions. In both nature and technology multiphase flow is rather the rule than the exception. The field of applications includes spray, bubbly flows, process and environmental engineering, combustion, rheology of blood and suspension multiphase systems as well as electro- and magnetorheological fluids. At this point of time, researchers neither agree on the cause of various effects nor on their theoretical description. In this paper, focus is put on suspensions. Depending on the length scale of observation as well as on the effects to be described, different approaches may be chosen. On the scale of individual particles in a microscopic approach, besides the strongly restricted possibilities for exact calculation [@1906_Einstein], <span style="font-variant:small-caps;">Stokes</span>ian dynamics ([@1989_Russel; @1993_Brady]) and Lattice-<span style="font-variant:small-caps;">Boltzmann</span> methods ([@2005_Hyvaeluoma; @2005_Kehrwald]) are employed, for instance. These methods provide a means to investigate the mechanisms occurring in suspensions and to explain the origin of macroscopically observable effects. However, resolving individual particle requests a large computational effort and is therefore not applicable for engineering purposes. Increasing the observation length scale thus leads to reduced computational effort but also to loss of information because of coarser sampling. The <span style="font-variant:small-caps;">Euler-Lagrange</span>-method uses groups of particles—so-called parcels—to represent the particle phase within the fluid carrier phase whereas the <span style="font-variant:small-caps;">Euler-Euler</span>-method considers both phases as interacting continuous media (see e.g. [@2005_Chrigui]). Both methods are frequently used to investigate transport, dispersion and reaction processes in dilute and dense suspension systems. In case a pure macroscopic description of the suspension is sufficient, one may model certain flow parameters of the suspension as a whole so that only, for instance, the volume fraction of the particle phase has to be determined during the computation to provide a basis for the calculation of macroscopic suspension properties such as the apparent viscosity. Here, micromechanics models ([@2004_Pabst; @2002_Torquato]) are well-suited, especially the notion of construction processes employed in the present work.

In the following, we intend to describe the apparent viscosity as well as the maximum packing fraction of disperse systems by means of the volume fractions of the particle phase. Especially, we consider polydisperse suspensions as they are the most general case of dispersions. The apparent viscosity is a macroscopic quantity resulting from the presence of the particles. This quantity was first described by [@1906_Einstein] for the dilute limit, that is small volume fractions of the particle phase. Later, various attempts to extend the validity of the viscosity relation were made as reported in ([@2004_Pabst; @2009_Mendoza]). Taking into account that the apparent viscosity is a macroscopic quantity, individual particles are not considered, rather the particle size is represented by particle size classes. The suspension is assumed to be built up from a finite number of such classes in the course of a construction process that allows for the calculation of the macroscopic suspension properties and thus needs to be described in detail. Since the suspension shows in general non-Newtonian behavior,the influence of the shear rate on the apparent viscosity will thus also be modeled in a future study.

The present paper is organized as follows. In section \[sec:basics\] some basics on apparent viscosity are provided along with an overview of recent work. Section \[chap:viskogleichung\] is dedicated to a generalization of the viscosity correlation for monodisperse suspensions. In section \[chap:Grundmodell\] a construction process approach used to describe polydisperse suspension viscosity by monodisperse viscosity correlations is presented. Accordingly a model for the maximum packing fraction of polydisperse systems is outlined. The resulting model is then applied to two different particle size distributions demonstrating the effect of polydispersity on the apparent viscosity. Section \[chap:conclusions\] is devoted to conclusions.

Basics of apparent viscosity and review of recent works {#sec:basics}
=======================================================

Following the common quasi-single phase formulation we assume the rheological behavior of suspensions to be described by a stress tensor of the form [@2004_Pabst] $$\Tij = \Tij(\rho,T,\Sij)
\label{eq:allgT}$$ where $T$ and $\Sij$ denote the temperature and the symmetric part of the velocity gradient tensor $\Sij=\left[\nabla\ui + (\nabla\ui)^T\right]$, respectively. We confine ourselves to the case of incompressible flow with a constant deformation history (so-called viscometric flow, see [@2000_Boehme]). Within the framework of continuum mechanics ([@1977_Truesdell; @2008_Irgens]) these presumptions are met by the constitutive relation of the so-called generalized ian fluid $$\Tij = -p \mathbf{I} +2\eta(T,\gpunkt)\,\Sij
\label{eq:Newtonallg2}$$ with the generalized shear rate $$\gpunkt {\mathrel{\mathop:}=}\sqrt{2\,{\mathrm{tr}\,}\Sij^2}
\label{eq:gpunktallg}$$ which is equivalent to the second invariant of the symmetric part of the velocity gradient tensor $\Sij$. The quantity $\eta$ expresses the apparent viscosity to be determined.

Throughout the present work, we will disregard the existence of single particles but represent the particles summarily by the properties of so-called *particle size classes*. The presence of particles within the flow increases the viscous dissipation compared with the pure fluid phase with viscosity $\eta_0$ which leads to the observability of the apparent viscosity $\eta$. In order to isolate the influence of the particle phase on the apparent viscosity one defines the *relative viscosity* $\eta_r$ as $$\eta_r{\mathrel{\mathop:}=}\frac{\eta}{\eta_0}
\label{eq:defetar}$$ The apparent viscosity is mainly dependent on the *volume fraction* $\phi$ $$\phi{\mathrel{\mathop:}=}\frac{V_{\text{particle}}}{V_{\text{fluid}}+V_{\text{particle}}}
\label{eq:defphi}$$ of the particle phase, where $V_{\text{particle}}$ and $V_{\text{fluid}}$ denote the volumes of the particle and fluid phase, respectively. The volume fraction is sometimes called packing fraction as well. From experiments[@1997_Gondret] it is well-known that the relative viscosity monotonously increases with increasing volume fraction and exhibits a singular behavior at a value $\phi<1$. The point where the relative viscosity diverges is commonly denoted as the *maximum packing fraction* $\phi_c$. By experiment, it has been found that for monodisperse suspensions $\phi_c$ coincides with the so-called *random close packing* state of the suspension which equals $$\phi_c\approx0.64
\label{eq:phic064}$$ for spheres (see [@2004_Pabst]). A value for the maximum packing fraction of course also exists in the case of polydisperse systems but differs from the monodisperse case in general. We will propose a model for the polydisperse maximum packing fraction in [section \[sec:Packungsdichte\]]{}.

Suspensions having a fixed particle size distribution but a variable volume fraction of the particle phase behave quantitatively similar to the monodisperse suspension described above. The quantitative changes concern the maximum packing fraction $\phi_c$ and the slope of the curve $\eta_r=\eta_r(\phi)$. These effects have been examined systematically in [@1999_Luckham], see [Figure \[fig:1999\_Luckham\_01\]]{}.

![Dependence of the relative viscosity on the volume fraction and the width of the particle size distribution from the experiment [@1999_Luckham]: narrow (ND), moderately broad (BD1) and very broad (BD2) distributions[]{data-label="fig:1999_Luckham_01"}](1999_Luckham_01.pdf)

Therein the relative viscosity is depicted as a function of the volume fraction for three distributions of various width. Two trends can be observed. On the one hand, the viscosity of a polydisperse system with a broad size distribution always lies below the viscosity of a system with a narrow distribution, thus below the viscosity of a monodisperse system as well (see also [@1997_Gondret; @1997_Greenwood; @2001_He]). On the other hand, the maximum packing fraction increases with increasing polydispersity, that is with the broadening of the particle size distribution.

As already mentioned, the fundamental work on the apparent viscosity of disperse systems has been written by (see [@1906_Einstein], correction [@1911_Einstein]). Therein, the equation $$\eta\Delta\ui=-\frac{1}{\rho}\nabla p
\label{eq:Stokes}$$ is solved in a three-dimensional dilatational flow around a spherical particle at rest. Afterwards, the solution is transferred to the case of a suspension with a finite number of particles (volume fraction $\phi$). The dissipation change due to the presence of the particles leads to the well-known relation $$\eta_r=1+2.5\phi
\label{eq:Einstein}$$ Though fluid inertia is neglected in the equation , the [equation ]{} serves a an exact limit for dilute suspensions, that is for $\phi\to0$. The relation is commonly considered as the first-order series expansion of every correlation between viscosity and volume fraction [@2004_Pabst]. We denote the $\phi-$coefficient as *first-order intrinsic viscosity* ${\left[\eta_{1}\right]}$ and analogously ${\left[\eta_{m}\right]}$ as *$m$th-order intrinsic viscosity*, so that we can write down the series expansion of the relative viscosity in the form $$\begin{split}
\eta_r & \approx1+{\left[\eta_{1}\right]}\phi+{\left[\eta_{2}\right]}\phi^2+\dotsb+{\left[\eta_{n}\right]}\phi^n+\dotsb \\
 &=1+\sum_{m=1}^{\infty}{\left[\eta_{m}\right]}\phi^m \\
\end{split}
\label{eq:nEntwicklung}$$ This representation will be used later in this work. In the literature a great number of equations of the form $$\eta_r=\eta_r(\phi)
\label{eq:H}$$ are provided. Tables \[tab:etaint2\] and \[tab:Korrelationen\] summarize the most important correlations considering neither their derivation nor the underlying models. Most of the equations are empirical or semi-empirical, only few are exact. The reason for the existence of such a great number of different correlations is that relations of the form  do not cover the entire parameter space governing the physical problem. In [@2004_Pabst] this is expressed by the formulation $$\eta_r=\eta_r(\phi,\,\text{all other details of microstructure})
\label{eq:etarpabst}$$ Clearly the parameter space must be confined to allow for useful modeling and thus the range of validity has to be confined a priori. In the present context, the correlations , denoted as *viscosity relations* in the following, can be classified in two groups:

#### Series expansions with respect to the volume fraction for $\boldsymbol{\phi\ll1}$ according to the equation (\[eq:nEntwicklung\]):\

While the relation  with the intrinsic viscosity ${\left[\eta_{1}\right]}=2.5$ is commonly accepted as the first order series expansion of the relative viscosity $\eta_r$ of suspensions with spherical particles, there is no unique value of ${\left[\eta_{2}\right]}$ because of a strong case-sensitivity of this parameter. In [Table \[tab:etaint2\]]{} some examples taken from the literature are listed.

[@lc@X r@@[.]{}l]{} Author(s) & Reference& Annotations &\
<span style="font-variant:small-caps;">Batchelor & Green</span> (1972) & [@2009_Mendoza] & ian motion neglected, random spatial particledistribution & 5&2\
<span style="font-variant:small-caps;">Batchelor</span> (1977) & [@2009_Mendoza] & ian motion included, random spatial particledistribution & 6&17\
<span style="font-variant:small-caps;">Bedeaux</span> et al. (1977) & [@2009_Mendoza] & formalism in wave number space & 4&8\
<span style="font-variant:small-caps;">Cichocki & Felderhof</span> (1991) & [@2009_Mendoza] & ian motion neglected, <span style="font-variant:small-caps;">Smoluchowski</span> equations & 5&00\
<span style="font-variant:small-caps;">Cichocki & Felderhof</span> (1991) & [@2009_Mendoza] & ian motion included, <span style="font-variant:small-caps;">Smoluchowski</span> equations & 5&91\
<span style="font-variant:small-caps;">Kim & Karrila</span> (1991) & [@2003_Cheng] & ian dynamics similar to <span style="font-variant:small-caps;">Batchelor</span> (1972) & 6&95\
<span style="font-variant:small-caps;">Krieger & Dougherty</span> (1959) & [@1959_Krieger] & second-order <span style="font-variant:small-caps;">Taylor</span> coefficient of the relation  for $\phi_c=0.64$& 5&08\
<span style="font-variant:small-caps;">Mooney</span> (1951) & [@2004_Pabst] & second-order <span style="font-variant:small-caps;">Taylor</span> coefficient of the relation  for $\phi_c=0.64$ & 7&03\

#### Correlations for the entire range of $\phi$:\

The correlations listed in [Table \[tab:Korrelationen\]]{} are intended to be valid over the entire range of $\phi$.

0.5ex

[lclr]{} Author(s) & Reference& Equation & No.\
<span style="font-variant:small-caps;">Krieger & Dougherty</span> (1959) & [@1959_Krieger] & $\eta_r= \left(1-\frac{\phi}{\phi_c}\right)^{-{\left[\eta_{1}\right]}\phi_c}$&${\refstepcounter{equation}\quad(\theequation)\label{eq:krieger1}}\label{test1}$\
<span style="font-variant:small-caps;">Mooney</span> (1951) & [@2001_He] & $\eta_r=\mathrm{exp}\left(\frac{{\left[\eta_{1}\right]}\phi}{1-\nicefrac{\phi}{\phi_c}}\right)$ &${\refstepcounter{equation}\quad(\theequation)\label{eq:Mooney}}$\
<span style="font-variant:small-caps;">Frankel & Acrivos</span> (1967) & [@2001_He] & $\eta_r=\frac{9}{8}\frac{\left(\frac{\phi}{\phi_c}\right)^{\nicefrac{1}{3}}}{1-\left(\frac{\phi}{\phi_c}\right)^{\nicefrac{1}{3}}}$ &${\refstepcounter{equation}\quad(\theequation)\label{eq:Frankel}}$\
<span style="font-variant:small-caps;">Eilers</span> (1941) & [@2004_Pabst] & $\eta_r= \left[1+\frac{1}{2}{\left[\eta_{1}\right]}\left(\frac{\phi}{1-\nicefrac{\phi}{\phi_c}}\right)\right]^2$&${\refstepcounter{equation}\quad(\theequation)\label{eq:Eilers}}$\
<span style="font-variant:small-caps;">Quemada</span> (1977) & [@1977_Quemada] & $\eta_r=\left(1-\frac{\phi}{\phi_c}\right)^{-2}$ &${\refstepcounter{equation}\quad(\theequation)\label{eq:Quemada}}$\
<span style="font-variant:small-caps;">Robinson</span> (1949) & [@2004_Pabst] & $\eta_r=1+{\left[\eta_{1}\right]}\left(\frac{\phi}{1-\nicefrac{\phi}{\phi_c}}\right)$ &${\refstepcounter{equation}\quad(\theequation)\label{eq:Robinson}}$\

They all coincide with respect to a singular behavior at the point $\phi=\phi_c$, that is when the maximum packing fraction is reached. For $\phi\to0$ the equations by and both do not reduce to the relation. Therefore, in [@2004_Pabst] a correction to the relation is proposed in the form $$\begin{split}
\eta_r=1&+\frac{9}{8}\phi_c\frac{\left(\frac{\phi}{\phi_c}\right)^{\nicefrac{1}{3}}}{1-\left(\frac{\phi}{\phi_c}\right)^{\nicefrac{1}{3}}}\cdot\\
&\cdot\left[\frac{\phi}{\phi_c}+\frac{20}{9}\left(1-\frac{\phi}{\phi_c}\right)\left(\frac{\phi}{\phi_c}\right)^{\nicefrac{2}{3}}\right]
\end{split}$$ [Figure \[fig:Korrelationen\]]{} shows the relations  to  from [Table \[tab:Korrelationen\]]{} for the parameters $\phi_c=0.64$ and ${\left[\eta_{1}\right]}=2.5$.

![Correlations between relative viscosity $\eta_r$ and volume fraction $\phi$ for $0\leq\phi<\phi_c$ according to the equations (\[eq:krieger1\]) to (\[eq:Robinson\]) with $\phi_c=0.64$ and ${\left[\eta_{1}\right]}=2.5$[]{data-label="fig:Korrelationen"}](Korrelationen.pdf)

In the next section, an attempt to generalize the viscosity correlation for monodisperse suspensions is presented.

Generalization of the viscosity correlation for monodisperse suspensions {#chap:viskogleichung}
========================================================================

As shown in the previous section, there are two main types of viscosity correlations, namely polynomial and closed correlations. Polynomial correlations are well suited for describing the low-concentration range but do not show divergence for $\phi\to\phi_c$. Closed correlations diverge for $\phi\to\phi_c$ and therefore seem to be suited for the whole concentration range, but cannot show proper asymptotic behavior for $\phi\to0$ because the second order -coefficient ${\left[\eta_{2}\right]}$ is determined a priori through the viscosity relation. So in the following section we combine the low-concentration behavior of polynomial correlations with the high-concentration behavior of a special closed viscosity correlation based on the well-known equation.

Derivation of the viscosity correlation
---------------------------------------

We derive a correlation that in addition to the properties of the closed correlations in [Table \[tab:Korrelationen\]]{} allows for choosing the second-order coefficient ${\left[\eta_{2}\right]}$. This value depends on particle interactions and may differ considerably from case to case. So the modified viscosity correlation shall include the three parameters ${\left[\eta_{1}\right]}$, ${\left[\eta_{2}\right]}$ and $\phi_c$.

Starting from the [equation ]{} $$\eta_r= \left(1-\frac{\phi}{\phi_c}\right)^{-{\left[\eta_{1}\right]}\phi_c}
\label{eq:krieger2}$$ we assume the modified correlation to be of the form $$\eta_r=A+B\left(1-\frac{\phi}{\phi_c}\right)^{-C}
\label{eq:viskansatz}$$ This ansatz has to fulfill the conditions $$\begin{aligned}
\eta_r(\phi=0)&=1\quad \text{and}\label{eq:RB1visko}\\\eta_r(\phi\to0)&=1+{\left[\eta_{1}\right]}\phi+{\left[\eta_{2}\right]}\phi^2\label{eq:RB2visko}\end{aligned}$$ Condition  implies that $$A+B=1
\label{eq:bed1}$$ The second condition  is equivalent to $$\begin{aligned}
\left.\frac{{\mathrm{d}}\eta_r}{{\mathrm{d}}\phi}\right|_{\phi=0} &= {\left[\eta_{1}\right]} \label{eq:abl1}\\[0.3cm]\left.\frac{{\mathrm{d}}^2\eta_r}{{\mathrm{d}}\phi^2}\right|_{\phi=0} &=2{\left[\eta_{2}\right]}\label{eq:abl2}\end{aligned}$$ In order to ensure the fulfillment of this condition we calculate the first two derivatives of the ansatz  with respect to $\phi$ and additionally use the [equation ]{}. We obtain $$\begin{aligned}
\frac{{\mathrm{d}}\eta_r}{{\mathrm{d}}\phi} &= \frac{BC}{\phi_c}\left(1-\frac{\phi}{\phi_c}\right)^{-C-1}\label{eq:RB2bvisko}\\[0.3cm]\frac{{\mathrm{d}}^2\eta_r}{{\mathrm{d}}\phi^2} &=\left(C+1\right)\frac{BC}{\phi_c^2}\left(1-\frac{\phi}{\phi_c}\right)^{-C-2}\label{eq:RB2cvisko}\end{aligned}$$ and evaluate the derivatives at the point $\phi=0$. It follows from the conditions  and  $$\begin{aligned}
A&=1-\frac{\phi_c{\left[\eta_{1}\right]}^2}{2\phi_c{\left[\eta_{2}\right]}-{\left[\eta_{1}\right]}}\label{eq:a}\\[0.3cm]B&=\frac{\phi_c{\left[\eta_{1}\right]}^2}{2\phi_c{\left[\eta_{2}\right]}-{\left[\eta_{1}\right]}}\label{eq:b}\\[0.3cm]
C&=\frac{2\phi_c{\left[\eta_{2}\right]}-{\left[\eta_{1}\right]}}{\phi_c{\left[\eta_{1}\right]}}\label{eq:c}\end{aligned}$$ so the final form of the modified viscosity equation according to the ansatz  is $$\boxed{
\begin{aligned}
\eta_r(\phi)=1&-\frac{\phi_c{\left[\eta_{1}\right]}^2}{2\phi_c{\left[\eta_{2}\right]}-{\left[\eta_{1}\right]}}\cdot\\ &\cdot\left[1-\left(1-\frac{\phi}{\phi_c}\right)^{\frac{2\phi_c{\left[\eta_{2}\right]}-{\left[\eta_{1}\right]}}{\phi_c{\left[\eta_{1}\right]}}}\right]
\end{aligned}
}
\label{eq:viskogleichung}$$

Special cases of the viscosity correlation
------------------------------------------

Since [equation ]{} has been derived under rather formal considerations than from a physical point of view, it has to be understood as an empirical equation to be fitted to experimental data. Because [equation ]{} reduces to the [equation ]{} for the choice $${\left[\eta_{2}\right]}=\frac{\left(1+{\left[\eta_{1}\right]}\phi_c\right){\left[\eta_{1}\right]}}{2\phi_c}
\label{eq:etaint2krieger}$$ it is of course superior to the latter for fitting purposes. As a consequence, we do not learn anything from fitting [equation ]{} to experimental results for testing.

Instead we examine if the closed correlations in [Table \[tab:Korrelationen\]]{} can be reproduced by the modified viscosity equation. From [Figure \[fig:Korrelationen\]]{} we chose two extreme examples, namely the relations by and (equations  and ). Figure \[fig:Vergleich\_Mooney\_Robinson\] shows the results.

![Comparison of equation (\[eq:viskogleichung\]) with the relations by Mooney (\[eq:Mooney\]) and Robinson (\[eq:Robinson\]); ${\left[\eta_{1}\right]}=2.5$[]{data-label="fig:Vergleich_Mooney_Robinson"}](Vergleich_Mooney_Robinson.pdf)

We find good agreement in both cases. However, it was necessary to decrease the value of the maximum packing fraction $\phi_c$ in order to reproduce the behavior of the equation because of its exponential variation compared with the algebraic variation of the [equation ]{}. The remaining correlations in [Table \[tab:Korrelationen\]]{} vary algebraically and may therefore be reproduced by the [equation ]{} with high accuracy.

Because of these reproduction properties we will in the following consider the modified viscosity [equation ]{} as a general viscosity correlation for monodisperse suspensions.

Development of a viscosity correlation for polydisperse suspensions {#chap:Grundmodell}
===================================================================

In this section we develop the polydisperse viscosity model based on the notion of a construction process. This approach is first exactly described in [section \[sec:Aufbauprozess\]]{}. Subsequently in [section \[sec:Diskretes\_Modell\]]{} the construction process is applied to the determination of relative viscosity. Then, a model for the maximum packing fraction of polydisperse suspensions is developed in [section \[sec:Packungsdichte\]]{} to complete the viscosity model. A graphical scheme provided in Figure \[fig:Ablauf\] may serve for the reader’s guidance during the calculation.

It is important to note that the basic model developed in this section can only be applied to polydisperse suspensions with large diameter ratios of consecutive particle size classes in the construction process. Work on an extension of the model to small particle size ratios is in progress.

Starting point: The differential Bruggeman model {#sec:Bruggeman}
------------------------------------------------

The differential model (see also [@2009_Hsueh] and more detailed [@2002_Torquato]) makes it possible to derive a closed viscosity relation for the full concentration range starting from the relation. The model is also known as *Differential Effective Medium approach (DEM)*. A generalization of the DEM approach is presented in [@1985_Norris]. The model is based on the notion that an infinitesimal volume fraction of particles $\phi^*$ is added to an existing suspension with effective viscosity $\eta$ and volume fraction $\phi$. In the course of this addition it is assumed that the existing suspension can be treated as a homogeneous medium. This can only be valid if the newly added particles have a large diameter compared with the particles already present in the suspension.

We now ask for the change in effective viscosity due to the newly added volume fraction $\phi^*$. The infinitesimal volume fraction of the newly added particles in the resulting suspension is, according to [@2009_Hsueh] (compare the later relations  and ) $$\phi_{P} = \frac{\phi^*}{1 + \phi^*} = \frac{{\mathrm{d}}\phi}{1-\phi}
\label{eq:diffphistern}$$ Because of the small size of the volume fraction $\phi_{P}$ we may use the relation to describe the change in effective viscosity by $$\eta+{\mathrm{d}}\eta = \eta\left(1+{\left[\eta_{1}\right]}\phi_{P}\right)
\label{eq:etaplusdeta}$$ By inserting the [equation ]{} into the relation  we obtain $$\eta+{\mathrm{d}}\eta = \eta\left(1+{\left[\eta_{1}\right]}\,\frac{{\mathrm{d}}\phi}{1-\phi}\right) 
\label{eq:etaplusdetaeingesetzt}$$ Simplification and separation of variables yield $$\frac{{\mathrm{d}}\eta}{{\left[\eta_{1}\right]}\eta}=\frac{{\mathrm{d}}\phi}{1-\phi}
\label{eq:detasterndphi}$$ We integrate the [equation ]{} under the initial condition $\eta(\phi=0) = \eta_0$ for the pure fluid and find $$\eta = \eta_0 \left(1-\phi\right)^{-{\left[\eta_{1}\right]}}
\label{eq:roscoeohnephic}$$ The equation  is known as equation. Since the model requires a large diameter ratio of consecutively added particle classes, the suspension must therefore consist of a solid phase that can be divided into particle size classes of large diameter ratios. This structure is called hierarchical, see also [@1985_Norris] and [@2002_Torquato] So the differential model can be applied to polydisperse Suspensions.

Another important assumption of the differential model is the validity of [equation ]{}. The volume fraction of newly added spheres in [equation ]{} has to be small enough for the relation to be valid.

We note that the volume fractions $\phi^*$ and accordingly $\phi_{P}$ may be finite in principle. However, by introducing the infinitesimal increment ${\mathrm{d}}\eta$ into the differential model the volume fractions $\phi^*$ and $\phi_{P}$ are required to be infinitesimal. The advantage of this limitation is the possibility to derive the closed [equation ]{}.

Assumptions
-----------

In the following we assume the suspension to consist of a solid particle phase suspended in a fluid carrier phase with constant viscosity $\eta_0$. The particle phase consists of spheres with different diameters $d_i$ that can be categorized in a finite number $n$ of size or diameter classes. So with $i$ as the index of the size class we have $i=1\dotsc n$. The size classes shall be sorted by diameter in ascending order, so that $d_{i}<d_{i+1}$. The ratio of two consecutive diameters $$u_i{\mathrel{\mathop:}=}\frac{d_{i+1}}{d_i}
\label{eq:uidef1}$$ should be larger than 7 according to [@2010_Brouwers] so that the existing suspension behaves like a homogeneous medium towards the newly added spheres. In the completed suspension resulting from the construction process the $i$th size class occupies a volume $V_i$ while the fluid phase occupies the volume $V_f$. So the total volume $\Vges$ of the suspension is given by $$\Vges = V_f + \sum_{m=1}^n V_m
\label{eq:Vges}$$ We assume that the suspension can be entirely described by volume fractions, so that no higher moments of the so-called indicator function (see [@2002_Torquato]) have to be considered. By the way, the indicator function is defined on the entire space occupied by the suspension and is unity at all points belonging to phase 1 (e.g. the solid phase) and zero for all of the remaining points (e.g. the fluid phase). The function is commonly used for a statistical description of phase interactions by means of its moments. The lowest-order moment is given by the particle phase volume fraction. Confining ourselves to a description based on volume fractions corresponds to the assumption of an isotropic and homogeneous suspension throughout the control volume.

As already outlined in [section \[sec:basics\]]{}, the total volume fraction is defined by $$\phi=\frac{V_{\text{particle}}}{V_{\text{fluid}}+V_{\text{particle}}}=\frac{\sum_{m=1}^n V_m}{\Vges}
\label{eq:phivol}$$ Analogously it is useful to define volume fractions of single size classes, both during the construction process and in the completed suspension.

Construction process {#sec:Aufbauprozess}
--------------------

The models for the apparent viscosity and the maximum packing fraction that are developed in the following sections are based on the notion that the suspension is constructed by successive addition of new size classes. We call this process the *construction process*. In the following we generalize the considerations in [@2004_Pabst] and especially [@1985_Norris]. There are two possible approaches for the construction process shown in [Figure \[fig:Aufbauprozess2\]]{}:

Variable total volume

:   In this case the volume of the fluid phase $V_f$ is held constant during the construction process, so that the total volume of the suspension increases with each step until the suspension occupies the final volume $\Vges$. The construction process thus only consists of additions of size classes.

Constant total volume

:   In order to keep the total volume of the suspension constant throughout the construction process, it is necessary to extract a suspension volume with a size equal to the added particle volume in each construction step (see also [@1985_Norris]). So the extracted volume represents the composition of the existing suspension.\[Verschmierung\]

Though we will show in a later section that both approaches are equivalent, we will choose the constant-volume approach for reasons of simplicity.

![Scheme of the construction process for a bidisperse Suspension ($n=2$) at variable (above) and constant volume (below) including homogenization[]{data-label="fig:Aufbauprozess2"}](Aufbauprozess2){width="\columnwidth"}

The homogenization step in [Figure \[fig:Aufbauprozess2\]]{} symbolizes the fact that the existing suspension acts as a homogeneous medium towards the newly added particles because of the large diameter ratio.

### Construction process at variable total volume

In the case of variable total suspension volume the added volumes of the respective size classes are identical to the volumes of the size classes in the completed suspension at the end of the construction process. In contrast, this is not the case when the total volume is constant.

First we show explicitly the addition of the first two size classes with volumes $V_1$ and $V_2$ to the fluid phase with volume $V_f$. The total volume fraction of the particle phase after the $i$th construction step $\phi_i$ is given by (see e.g. [@2004_Pabst]) $$\begin{split}
\phi_0 &= \frac{0}{V_f+0}=0 \\
\rightarrow\,\phi_1 &= \frac{V_1}{V_f+V_1} \\
\rightarrow\,\phi_2 &= \frac{V_1+V_2}{V_f+V_1+V_2}
\end{split}
\label{eq:aufbauvariabel}$$ A general $(i+1)$th construction step can thus be written in the form $$\phi_i = \frac{\sum_{m=1}^{i}V_m}{V_f+\sum_{m=1}^{i}V_m}  \,\rightarrow\, \phi_{i+1} = \frac{\sum_{m=1}^{i+1}V_m}{V_f+\sum_{m=1}^{i+1}V_m} 
\label{eq:aufbauvariabelallg}$$ If we relate the added volume $V_{i+1}$ to the total suspension volume before the addition of $V_{i+1}$ and call this ratio $\phi_{i+1}^*$ $$\phi_{i+1}^* {\mathrel{\mathop:}=}\frac{V_{i+1}}{V_f+\sum_{m=1}^{i}V_m}
\label{eq:phiisternvar}$$ we can modify the expression for the total volume fraction of the particle phase $\phi_{i+1}$ in [equation ]{} so that $$\phi_{i+1} = \frac{\phi_i + \phi_{i+1}^*}{1 + \phi_{i+1}^*}
\label{eq:phiiphisternvariabel}$$ The concentration of the $(i+1)$th size class after the $(i+1)$th construction step is consequently given by $$\phi_{P,i+1} {\mathrel{\mathop:}=}\frac{V_{i+1}}{V_f+\sum_{m=1}^{i+1}V_m} = \frac{\phi_{i+1}^*}{1 + \phi_{i+1}^*}
\label{eq:phipplus1variabel}$$ The volume fraction $\phi_{P,i+1}$ in [equation ]{} will be important for the calculation of the apparent viscosity because it serves as an argument in the viscosity relation (compare [equation ]{}).

For a different formulation we denote the change in total volume fraction during the $(i+1)$th construction step as $$\delta\phi_{i+1}{\mathrel{\mathop:}=}\phi_{i+1}-\phi_i
\label{eq:deltaphiallg}$$ Using [equation ]{} we first obtain $$\phi_i+\delta\phi_{i+1} = \frac{\phi_i + \phi_{i+1}^*}{1 + \phi_{i+1}^*}$$ and finally under consideration of definition  $$\phi_{P,i+1} = \frac{\delta\phi_{i+1}}{1 - \phi_i}
\label{eq:deltaphivariabel}$$

### Construction process at constant total volume {#sec:Aufbaukonst}

If the total suspension volume is held constant throughout the construction process, in every construction step we have to extract as much volume as is added with the new size class. The extracted volume represents, as already explained on [page ]{}, the composition of the existing suspension and therefore contains particles of smaller size classes (see also [@1985_Norris]). This implies that the volume of the respective size class added in each construction step has to be larger than the volume of this size class contained in the completed suspension. We notice that the last size class (the largest particles) is an exception and is therefore added with its final volume.

The volume added in the $i$th construction step is thus called $V_i^*$ in order to be distinguished from the volume $V_i$ of the $i$th size class in the completed suspension. The constant total volume is still called $\Vges$.

Now we consider the construction process in detail. The first addition implies a simple change in the total volume fraction of the particle phase $\phi_i$: $$\phi_0 = \frac{0}{\Vges}=0 \quad\longrightarrow \quad\phi_1 = \frac{V_1^*}{\Vges}
\label{eq:aufbaukonst1}$$ The more complex second construction step is given by $$\phi_1 = \frac{V_1^*}{\Vges} \quad\longrightarrow\quad \phi_2 = \frac{\left(V_1^*-V_2^*\frac{V_1^*}{\Vges}\right)+V_2^*}{\Vges}
\label{eq:aufbaukonst2}$$ In the [equation ]{} the term in brackets represents the volume of the first size class still present after the second construction step. Therein the term $V_2^*V_1^*/\Vges$ describes the loss of volume of the first size class due to the necessary extraction of volume. If we introduce the notation $$\phi_k^* = \frac{V_k^*}{\Vges}
\label{eq:phikstern}$$ we may analogously to the [equation ]{} express the concentration of the $(i+1)$th size class in the existing suspension by $$\phi_{P,i+1} = \phi_{i+1}^*
\label{eq:phipplus1konstant}$$ Unlike in the case of variable total volume we obviously do not need to distinguish between the volume fractions $\phi_{P,i+1}$ and $\phi_{i+1}^*$. By rearranging of the expressions in the [equation ]{} we find $$\begin{aligned}
 \phi_1 = \underbrace{\phi_1^*}_{{=\mathrel{\mathop:}}\dphi_1^1}\quad\text{and} \label{eq:aufbaukonst2dim1}\\
\phi_2 = \underbrace{\phi_1^*(1-\phi_2^*)}_{{=\mathrel{\mathop:}}\dphi_1^2}+\underbrace{\phi_2^*}_{{=\mathrel{\mathop:}}\dphi_2^2}
\label{eq:aufbaukonst2dim2}\end{aligned}$$ In the equations  and  we have introduced a notation for the volume fractions of the individual size classes during the construction process. The representation $\dphi_k^i$ refers to the volume fraction of the $k$th size class after the $i$th construction step, that is after the addition of the $i$th size class. So $i$ means an index and no exponent. This should cause no confusion because the volume fraction will always occur linearly in all of the following expressions.

It can easily be shown, using the equations  and , that the total volume fraction after the third construction step is given by $$\label{eq:phi3konstant}
\begin{split}
\phi_3 &= \phi_2(1-\phi_3^*)+\phi_3^* \\
             &= [\phi_1^*(1-\phi_2^*)+\phi_2^*](1-\phi_3^*)+\phi_3^* \\
             &= \underbrace{\phi_1^*{(1-\phi_{2}^*)}{(1-\phi_{3}^*)}}_{{=\mathrel{\mathop:}}\dphi_1^3} + \underbrace{\phi_2^*{(1-\phi_{3}^*)}}_{{=\mathrel{\mathop:}}\dphi_2^3}+\underbrace{\phi_3^*}_{{=\mathrel{\mathop:}}\dphi_3^3}
\end{split}$$ The underlying pattern can be recognized clearly and so we may generalize intuitively: $$\phi_{i+1} = \phi_i{(1-\phi_{i+1}^*)}+\phi_{i+1}^*
\label{eq:phiiplus1allg}$$ Table \[tab:Aufbau\] schematically outlines the construction process.

0.5ex

  ---------- ----------- --------------- ------------- -------------- ---------- --------------
                                                                                 
                                                                                 
    $ 1 $     $\phi_1 $   $ \dphi_1^1 $                                          
     $2$      $\phi_2$     $\dphi_1^2$    $\dphi_2^2$                            
     $3$      $\phi_3$     $\dphi_1^3$    $\dphi_2^3$   $\dphi_3^3$              
   $\vdots$   $\vdots$      $\vdots$       $\vdots$       $\vdots$     $\ddots$  
     $n$      $\phi_n$     $\dphi_1^n$    $\dphi_2^n$   $ \dphi_3^n$   $\dotso$   $\dphi_n^n $
  ---------- ----------- --------------- ------------- -------------- ---------- --------------

By introducing the difference $\delta\phi_{i+1}$ from the [equation ]{} into the [equation ]{}, we find $$\begin{aligned}
\phi_i + \delta\phi_{i+1} &= \phi_i{(1-\phi_{i+1}^*)}+\phi_{i+1}^* \quad\text{sowie} \\
\phi_{P,i+1}&= \phi_{i+1}^* = \frac{\delta\phi_{i+1}}{1-\phi_i} \label{eq:deltaphikonstant}\end{aligned}$$ which is formally identical with the [equation ]{}. Attention should be paid to the fact that the volume fractions $\phi_{i+1}^*$ used in the cases of variable and constant volume, respectively, are not identical. The proof of the equivalence of the equations  and  is provided in appendix \[sec:Beweisderaequivalenz\] for clarity. From this proof we draw the conclusion that the construction processes at variable and constant total volume are equivalent. The only difference within the framework of our dimensionless representation, which is a representation by volume fractions, lies within the [equation ]{}. We choose the case of constant total volume because of the intuitive meaning of volume fractions originating from the constant volume $\Vges$. Considerations involving volumes may therefore easily be transferred to the notion of volume fractions. In the case of variable total volume the total volume contrarily depends on the amount of added volumes.

##### Representation of the quantities by the volume fractions $\boldsymbol{\dphi_k}$\

Up to now, the equations contain quantities influenced by the construction process. However, in the case of calculation we only know the volume fractions $\dphi_k^n$ of the size classes in the completed suspension. It is therefore useful to express the quantities describing the construction process by the composition of the completed suspension. For simplification we define $$\dphi_k {\mathrel{\mathop:}=}\dphi_k^n
\label{eq:defdphi}$$ These volume fractions in the completed suspension are given by $$\dphi_k = \frac{V_k}{V_f+\sum_{m=1}^{n}V_m} = \frac{V_k}{\Vges}
\label{eq:dphiivol}$$ From the equations  to  we deduce a general expression for $\dphi_k^i$: $$\begin{aligned}
{2}
\dphi_k^i =& \phi_k^* \prod_{m=k+1}^i {(1-\phi_{m}^*)} &\quad\text{für } k<i \label{eq:dphikhochia}\\
\dphi_k^k =&  \phi_k^* &\quad\text{für } k=i \label{eq:dphikhochib}\end{aligned}$$ We recall that there are no volume fractions with $k>i$ (compare [Table \[tab:Aufbau\]]{}). If we succeed in calculating the volume fractions $\phi_m^*$, we are in a position to calculate the volume fractions of the individual size classes $\dphi_k^i$ and the total volume fractions after each step $\phi_i$ from the equations  and . Per definition we have $$\phi_i = \sum_{k=1}^i \dphi_k^i
\label{eq:summedphi}$$ (consider the [equation ]{}as an example). Writing down the volume fractions of the last size classes in the completed suspension and using the definition  as well as the equations  and  reveals the possibility to calculate the volume fractions $\phi_k^*$: $$\begin{aligned}
\dphi_n &= \phi_n^* \label{eq:dphin1}\\
\dphi_{n-1} &= \phi_{n-1}^*{(1-\phi_{n}^*)}\label{eq:dphin2}\\
\dphi_{n-2} &= \phi_{n-2}^*{(1-\phi_{n-1}^*)}{(1-\phi_{n}^*)}\label{eq:dphin3}\\
\dphi_{n-3} &=\dotso \nonumber\end{aligned}$$ So we find by recursive insertion that $$\begin{aligned}
\phi_n^* &= \dphi_n  \label{eq:phistern1}\\
\phi_{n-1}^* &= \frac{\dphi_{n-1}}{1-\dphi_n}\label{eq:phistern2}\\
\phi_{n-2}^* &= \frac{\dphi_{n-2}}{1-\dphi_n-\dphi_{n-1}}\label{eq:phistern3}\\
\phi_{n-3}^* &=\dotso \nonumber\end{aligned}$$ This may be generalized in the form $$\boxed{
\phi_{P,k}=\phi_k^* = \frac{\dphi_k}{1-\sum_{m=k+1}^n \dphi_m}}
\label{eq:phimstern}$$ We now introduce the abbreviation $$\sigma_k{\mathrel{\mathop:}=}1-\sum_{m=k}^n \dphi_m
\label{eq:defsigma}$$ into the [equation ]{} and find $$1-\phi_k^* = \frac{1-\sum_{m=k}^n \dphi_m}{1-\sum_{m=k+1}^n \dphi_m} = \frac{\sigma_{k}}{\sigma_{k+1}}
\label{eq:einsminusphistern}$$ Therefore, we may write $$\begin{aligned}
\dphi_k^i & =\phi_k^* {(1-\phi_{k+1}^*)}{(1-\phi_{k+2}^*)}\dotsm{(1-\phi_{i}^*)} \nonumber\\
 & = \phi_k^* \frac{\sigma_{k+1}}{\sigma_{k+2}} \frac{\sigma_{k+2}}{\sigma_{k+3}} \frac{\sigma_{k+3}}{\sigma_{k+4}} \dotsm \frac{\sigma_{i}}{\sigma_{i+1}} \nonumber \\
 & = \phi_k^* \frac{\sigma_{k+1}}{\sigma_{i+1}} \label{eq:dphikhochi2}\end{aligned}$$ instead of the expression . Using the result  and the definition  it follows from the [equation ]{} that $$\boxed{
\dphi_k^i = \frac{\dphi_k}{1-\sum_{m=i+1}^n \dphi_m}}
\label{eq:dphikhochi3}$$ Combination of the equations  und  finally yields $$\phi_i = \frac{\sum_{m=1}^i \dphi_m}{1-\sum_{m=i+1}^n \dphi_m}
\label{eq:phiikonstant}$$ So we have represented all of the quantities occurring in the construction process by the volume fractions in the complete suspension.

A discrete model for the relative viscosity {#sec:Diskretes_Modell}
-------------------------------------------

As we have already noted, the differential model lacks any information about the volume fraction of the individual particle size classes. For that reason we have described the construction process of the suspension in a discrete form in [section \[sec:Aufbauprozess\]]{}. In preparation for the development of the discrete viscosity model we need to make a connection between the model and the maximum packing fraction.

### Introduction of the maximum packing fraction into the differential Bruggeman model

The [equation ]{} diverges as the total volume fraction $\phi$ approaches unity. In a real suspension the achievable value of $\phi$ is limited by the maximum packung fraction. In order to introduce the maximum packing fraction into the differential model we proceed in a way proposed in [@2009_Hsueh]. A quite similar way can be found in [@2009_Mendoza]. In both publications it emerges that the notion of maximum packing fraction is introduced under little convincing considerations.

The approach followed in [@2009_Hsueh] consists of modifying [equation ]{} by using the maximum packing fraction $\phi_c$ in the description of the volume fraction $\phi_P$. Therefore, it is supposed without derivation that $\phi_P$ may be written as $$\phi_{P} = \frac{{\mathrm{d}}\phi}{1-\frac{\phi}{\phi_c}}
\label{eq:diffphisternphic}$$ In combination with the [equation ]{} one finds $$\eta+{\mathrm{d}}\eta = \eta\left(1+{\left[\eta_{1}\right]}\,\frac{{\mathrm{d}}\phi}{1-\frac{\phi}{\phi_c}}\right) 
\label{eq:etaplusdetaphic}$$ Rearrangement of the [equation ]{} analogously to the [equation ]{} yields $$\frac{{\mathrm{d}}\eta}{{\left[\eta_{1}\right]}\eta}=\frac{{\mathrm{d}}\phi}{1-\frac{\phi}{\phi_c}}
\label{eq:detasterndphiphic}$$ which can be integrated under the initial condition $\eta(\phi=0) = \eta_0$ for the pure fluid. This results in the relation  $$\eta = \eta_0 \left(1-\frac{\phi}{\phi_c}\right)^{-{\left[\eta_{1}\right]}\phi_c}
\label{eq:krieger}$$ In [@2009_Mendoza] the introduction of a so-called effective volume fraction leads to the equation $$\eta = \eta_0 \left(1-\frac{\phi}{\phi_c}\right)^{-{\left[\eta_{1}\right]}}
\label{eq:Mendoza}$$ The expressions  and  obviously differ with respect to their exponents only. Both results show divergence for $\phi\to1$ but lack a physical rationale for the respective approaches. Therefore, we will state a different principle in the next section.

### Introduction of the maximum packing fraction into the discrete construction process

It would formally be possible to transfer the modification  to the discrete construction process, that is the volume fractions , and employ the result for the viscosity calculation. In the following it will be explained why this approach cannot be valid in general. Partially anticipating the later viscosity calculation, we raise the two following points.

Firstly, the construction process described in [section \[sec:Aufbauprozess\]]{} is by no means dependent on the particle geometry. This is emphasized by the notion of homogenization between two construction steps. In contrast, the maximum packing fraction is strongly influenced by the particle geometry. So it would be artificial to introduce this quantity into the description of volume fractions during the construction process.

Secondly, the consideration of the volume fractions during the construction process is independent of the physical quantity that is calculated (here: the viscosity). It does not make any difference whether one calculates the viscosity or, for instance, the electric conductivity (or both at the same time). In both cases the construction process is constituted by the same volume fractions. Only through the employed relation between the volume fractions and the change in the physical quantity of interest parameters like the maximum packing fraction are included. This will be the approach followed during the later viscosity calculation.

The above considerations imply that the differential model only allows for the derivation of the equation  because as a consequence of this differential approach the right-hand side of the [equation ]{} may only consist of a linear expression (the relation) that cannot contain the maximum packing fraction. So the approaches presented in [@2009_Hsueh] and [@2009_Mendoza] are formally possible but physically questionable.

##### Notation for the maximum packing fraction in the construction process\
 {#sec:NotationPackung}

At this point it is necessary to introduce a distinct notation for the maximum packing fraction in order to avoid misinterpretations. The calculation of the maximum packing fraction will be conducted in [section \[sec:Packungsdichte\]]{}. We have to distinguish between $\phi_c$ as a parameter within the correlations listed in [Table \[tab:Korrelationen\]]{} and the modeled maximum packing fraction that may change during the construction process as well as under the influence of the shear rate. We choose the following notations referring to [@2010_Brouwers]:

[lcX]{} $\varphi_{Tk}^i$ & : & We denote as $\varphi_{Tk}^i$ the maximum packing fraction of a polydisperse suspension consisting of $k$ size classes after the $i$th construction step ($i$th line in [Table \[tab:Aufbau\]]{}).\
$\varphi_c$ & : & We write $\varphi_c{\mathrel{\mathop:}=}\varphi_{T1}^i$ for the monodisperse packing fraction. The monodisperse maximum packing fraction $\varphi_c$ is constant throughout the construction process and thus carries no upper index $i$.\

This notation is visualized in [Table \[tab:Packung\]]{}(compare [Table \[tab:Aufbau\]]{}).

0.5ex

  ---------- ----------- -------------- ------- ------------------ ------- ------------------- ------- ---------- ------- -------------------
                                                                                                                          
                                                                                                                          
     $1$      $\phi_1 $   $ \varphi_c$                                                                                    
     $2$      $\phi_2$    $\varphi_c$    $\to$   $\varphi_{T2}^2$                                                         
     $3$      $\phi_3$    $\varphi_c$    $\to$   $\varphi_{T2}^3$   $\to$   $\varphi_{T3}^3$                              
   $\vdots$   $\vdots$      $\vdots$                 $\vdots$                   $\vdots$                                  
     $n$      $\phi_n$    $\varphi_c$    $\to$   $\varphi_{T2}^n$   $\to$   $ \varphi_{T3}^n$   $\to$   $\dotso$   $\to$   $\varphi_{Tn}^n $
  ---------- ----------- -------------- ------- ------------------ ------- ------------------- ------- ---------- ------- -------------------

After each construction step the maximum packing fraction is newly calculated according to the new composition of the suspension. This is conducted within a recursive process (arrows in [Table \[tab:Packung\]]{}). Starting from the monodisperse maximum packing fraction $\varphi_c$ all previously added size classes are taken into account and so for each step the value $\varphi_{Ti+1}^{i+1}$ is calculated. We will need this value for the [equation ]{} still to derive.

### Transition to a discrete relative viscosity model

We now build on the description of the discrete construction process given in [section \[sec:Aufbauprozess\]]{} and focus on the effect of the construction process on the relative viscosity. When during the construction process the $(i+1)$th size class is added, the total volume fraction of the particle phase changes from $\phi_i$ to $\phi_{i+1}$. This is associated with a change in apparent viscosity from $\eta_i$ to $\eta_{i+1}$. To emphasize the analogy to the differential model, we temporarily confine ourselves to the linear relation as a description of the viscosity change and subsequently in [section \[sec:hoeher\]]{} we will extend the model to higher orders.

According to sections \[sec:Aufbaukonst\] and \[sec:Bruggeman\] at constant total volume the volume fraction of the newly added $(i+1)$th size class related to the resulting suspension volume is $$\phi_{P,i+1}=\phi_{i+1}^* = \frac{\delta\phi_{i+1}}{1-\phi_i}
\label{eq:diskretphistern}$$ So the new apparent viscosity is given by $$\eta_{i+1} = \eta_i \left( 1+{\left[\eta_{1}\right]} \phi_{P,i+1} \right)
\label{eq:etasterniplus1}$$ In the [equation ]{} we have used the volume fraction $\phi_{P,i+1}$ related to the suspension volume after the $(i+1)$th construction step according to the notion of the construction process. The volume fraction $\phi_{i+1}^*$ may not be used in general because the volume fractions $\phi_{P,i+1}$ and $\phi_{i+1}^*$ only coincide in the case of constant total suspension volume (see [section \[sec:Aufbaukonst\]]{}). All the following relations containing $\phi_{P,i+1}$ may thus be used both at variable and constant total volume.

In the latter case, the [equation ]{} can be cast in a different form employing the volume fractions $\dphi_k$ from the [equation ]{}: $$\eta_{i+1} = \eta_i \left( 1+{\left[\eta_{1}\right]}\, \frac{\dphi_{i+1}}{1-\sum_{m=i+2}^n \dphi_m} \right)
\label{eq:etasternmitdphi}$$ This equation is a recursive formula that can alternatively be written in an explicit form. However, we note that a numerical evaluation has to be conducted in a recursive way because the explicit formula offers no simplification compared with the recursive formula. The explicit form of the [equation ]{} is given by $$\eta_{i+1} = \eta_0\prod_{m=1}^{i+1}  \left( 1+{\left[\eta_{1}\right]}\, \frac{\dphi_{m}}{1-\sum_{l=m+1}^n \dphi_l} \right)
\label{eq:expetasternmitdphi}$$ with $\phi_0=0$. For $i+1 = n$ we find the apparent viscosity  of the complete suspension (it is well known that $\sum_{l=n+1}^{n} x = 0$ for all $x$, which may be important for a numerical implementation).

##### Extension of the discrete model to higher order terms of $\boldsymbol{\phi_{P}}$\
 {#sec:hoeher}

The differential [equation ]{} contains only the first-order relation because of the infinitesimal character of ${\mathrm{d}}\phi$. In the case of the discrete model represented by [equation ]{} it is not necessary to confine oneself to linear terms. Therefore, we are allowed to describe the modification of the apparent viscosity more accurately by higher order terms of $\phi_{P,k}$. So the [equation ]{} can—according to the general expression $$\eta_{i+1} = \eta_0\prod_{m=1}^{i+1}  \left[ 1+{\left[\eta_{1}\right]}\,\phi_{P,m} +{\left[\eta_{2}\right]}\left(\phi_{P,m}\right)^2+ \dotsb\right]\label{eq:allgetasterniplus1}$$ —be extended to $$\boxed{
\begin{aligned}
\eta_{i+1} =\,\, &\eta_0\prod_{m=1}^{i+1}  \Bigg[ 1 +{\left[\eta_{1}\right]}\, \frac{\dphi_{m}}{1-\sum_{l=m+1}^n \dphi_l} +\\
 &+ {\left[\eta_{2}\right]} \left( \frac{\dphi_{m}}{1-\sum_{l=m+1}^n \dphi_l}\right)^2 + \dotsb\Bigg]
\end{aligned}
}
\label{eq:allgexpetasternmitdphiphic}$$ which according to the [equation ]{} may alternatively be written as $$\boxed{
\begin{aligned}
\eta_{i+1} =\,\, &\eta_i \Bigg[ 1+{\left[\eta_{1}\right]}\, \frac{\dphi_{m}}{1-\sum_{l=m+1}^n \dphi_l} +\\
&  + {\left[\eta_{2}\right]} \left( \frac{\dphi_{m}}{1-\sum_{l=m+1}^n \dphi_l}\right)^2 + \dotsb\Bigg]
\end{aligned}}
\label{eq:rekursetasternmitdphi}$$

##### Comparison with relations present in the literature\
 {#sec:VergleichLiteratur}

In the following we draw a connection between the [equation ]{} and relations existing in the literature. Selecting the coefficients from the expansion of a viscosity relation (, for instance) at the point $\phi=0$ for the coefficients ${\left[\eta_{k}\right]}$ in the [equation ]{} and formally considering an infinite number of terms, we may substitute the bracketed term in the [equation ]{} with the viscosity relation itself. This step requires the introduction of the maximum packing fraction. Using the [equation ]{} we find $$\boxed{
\begin{aligned}
\eta_{i+1} =\,\, &\eta_0\prod_{m=1}^{i+1} \left\{\rule{0cm}{7ex}1-\frac{\varphi_{Tm}^{m}{\left[\eta_{1}\right]}^2}{2\varphi_{Tm}^{m}{\left[\eta_{2}\right]}-{\left[\eta_{1}\right]}}\cdot\right.\\
&\left.\rule{0cm}{7ex}\cdot\left[1-\left(1-\frac{\phi_{P,m}}{\varphi_{Tm}^{m}}\right)^{\dfrac{2\varphi_{Tm}^{m}{\left[\eta_{2}\right]}-{\left[\eta_{1}\right]}}{\varphi_{Tm}^{m}{\left[\eta_{1}\right]}}}\right]\right\}
\end{aligned}
\label{eq:allgvisko}}$$ In the [equation ]{} $\varphi_{Tm}^{m}$ refers to the maximum packing fraction after the $m$th construction step that will be calculated in [section \[sec:Packungsdichte\]]{}. This notation has already been outlined in [section \[sec:NotationPackung\]]{}. The general recursive representation of the [equation ]{} is given by $$\boxed{
\eta_{i+1}=\eta_i\,\eta_r(\phi_{P,i+1},\varphi_{Ti+1}^{i+1})
}
\label{eq:etamitvarphi}$$ and more specifically using the viscosity relation  $$\boxed{
\begin{aligned}
\eta_{i+1} =\,\, & \eta_i\left\{\rule{0cm}{8ex}1-\frac{\varphi_{Ti+1}^{i+1}{\left[\eta_{1}\right]}^2}{2\varphi_{Ti+1}^{i+1}{\left[\eta_{2}\right]}-{\left[\eta_{1}\right]}}\cdot\right.\\
&\left.\rule{0cm}{8ex}\cdot\left[1-\left(1-\frac{\phi_{P,i+1}}{\varphi_{Ti+1}^{i+1}}\right)^{\dfrac{2\varphi_{Ti+1}^{i+1}{\left[\eta_{2}\right]}-{\left[\eta_{1}\right]}}{\varphi_{Ti+1}^{i+1}{\left[\eta_{1}\right]}}}\right]\right\}
\end{aligned}
}
\label{eq:rekursallgvisko}$$ Despite the use of the variable maximum packing fraction the [equation ]{} corresponds to the so-called model [@1968_Farris] also referred to in [@2010_Brouwers] and [@1990_Cheng]. So we have found an interesting connection between the differential model and the model which is drawn by the discrete construction process. Remarkably, the approach followed in [@1990_Cheng] includes a viscosity relation which depends on the particle size class, that is the particle radius. Although this additional degree of freedom is useful for fitting purposes, it is inconsistent with the assumptions of the models by and , which only consider a relative but not an absolute influence of the particle radius. Therefore, we will not use this approach in the current work.

In later sections we will use the result  as a general formula for the apparent viscosity of hierarchical polydisperse suspensions (compare [section \[sec:Aufbauprozess\]]{}). It is important to note that the extension of the [equation ]{} to  has been conducted by means of considering the complete expansion of relation , suggesting the admissibility of arbitrary volume fractions of the individual size classes which is uncertain regarding the notion of the construction process. Since it is not possible to state a limiting value of $\phi_{P,i+1}$ for the validity of the result , we will though use this equation at least as a reasonable approximation also for higher values of $\phi_{P,i+1}$.

To sum up, we state that the [equation ]{} resulting form the construction process is at least justified for the first two orders of $\phi_{P,i+1}$, while the model (corresponding to the [equation ]{}) provides a useful but possibly unjustified extrapolation for high values of $\phi_{P,i+1}$.

Determination of the maximum packing fraction {#sec:Packungsdichte}
---------------------------------------------

In the following we will present two models for computing the polydisperse maximum packing fraction. The first model has been proposed by (see also [@2010_Brouwers]). Although this model is insufficient for reasons to be given in [section \[sec:DasModellvonFurnas\]]{}, we will use the underlying considerations for developing an improved model in [section \[sec:DasneueModell\]]{}. The approaches differ with respect to the underlying treatment of the particle size distribution.

During the calculation we will use the quantities occurring in the construction process according to the tables \[tab:Aufbau\] and \[tab:Packung\]. For instance, after the third construction step the values $\varphi_{T2}^3$ and $\varphi_{T3}^3$ are being calculated starting from the monodisperse maximum packing fraction $\varphi_{T1}^3=\varphi_c$ and considering the existing volume fractions $\dphi_1^3$, $\dphi_2^3$ and $\dphi_3^3$.

According to [section \[sec:NotationPackung\]]{} we denote as $\varphi_{Tk}^i$ the maximum packing fraction after the $i$th construction step so far containing $k$ size classes while the completed suspension consists of $n$ size classes.

### The Furnas model {#sec:DasModellvonFurnas}

The model is based on the assumption that the state of maximum packing fraction is constructed successively from size classes with decreasing diameter. At first, the larger spheres fill the entire available suspension volume with the monodisperse maximum packing fraction $\varphi_c$. So the total volume fraction is $$\varphi_{T1}^i = \varphi_1^i = \varphi_c
\label{eq:FurnasT0}$$ Subsequently, the remaining volume fraction $(1-\varphi_c)$ is filled by the spheres with the next smaller diameter, also to a realizable part of $\varphi_c$. So the additional volume fraction of small spheres $\varphi_{T1}^i$ is given by $$\begin{split}
\varphi_{T1}^i &= \varphi_1^i+\varphi_2^i\\ 
&= \varphi_c + (1-\varphi_c)\varphi_c = 1 - (1-\varphi_c)^2
\end{split}
\label{eq:FurnasT1}$$ Generalizing these considerations to a number of $k$ size classes, one finds $$\varphi_{Tk}^i = 1 - (1-\varphi_c)^{k}
\label{eq:FurnasTn}$$ According to [@2010_Brouwers] this approach requires a minimal size ratio of 7 to 10 between consecutive size classes in order to avoid interactions between the packings of the individual size classes.

##### Discussion of the Furnas model\

The advantages of the model are its simple derivation, the applicability to an unlimited number of size classes and the simplicity of the result. Remarkably, the maximum packing fraction does not depend on the volume fractions of the individual size classes but only on their number. This is the crucial disadvantage of the model. As an illustration, imagine a given polydisperse suspension with a total volume fraction of $\phi_i<\varphi_{Ti}^i$. In the first case, we add a large amount of small spheres, in the first case a small amount. The model would yield the same value of the maximum packing fraction for both cases, which runs contrary to intuition. In this model the maximum packing fraction is calculated without effectively considering the particle size distribution in the suspension.

### An improved model for the maximum packing fraction {#sec:DasneueModell}

We will derive the new model in a way that at first the vivid cases of bi- and tridisperse suspensions are treated explicitly in order to prepare the subsequent general derivation. The general formulae will contain the bi- and tridisperse suspensions as special cases.

##### Description of the volume fractions\

A consequent calculation of the maximum packing fraction requires the consideration of the individual particle size classes’ volume fractions. This can be achieved by retaining the particle size distribution existing in the fluent suspension state during the composition of the maximum packing fraction. This assumption can be expressed in the form $$\frac{\varphi_{k+1}^i}{\varphi_k^i} = \frac{\dphi^i_{k+1}}{\dphi^i_k}
\label{eq:verhaeltnis}$$ where $\varphi_k^i$ in the state of maximum packing fraction corresponds to $\dphi^i_k$ in the fluent state, that is the volume fraction of the $k$th size class. Introducing the abbreviation $$\phi^{i}_k {\mathrel{\mathop:}=}\sum_{m=1}^{k}\dphi^i_m=\dphi^i_1+\dphi^i_2+\ldots+\dphi^i_k\label{eq:phi_idef}$$ we conclude that $$\begin{aligned}
\phi^{i}_1 &= \dphi^i_1 \label{eq:phi_1ident}\quad\text{as well as}\\
\phi^{i}_k &= \phi^{i}_{i-1}+\dphi^i_k \label{eq:phi_irekurs}\end{aligned}$$

##### Derivation for bidisperse systems\

Bidisperse (or bimodal) suspensions contain two different particle sizes. Given the ratio $$\frac{\varphi_2^i}{\varphi_1^i} = \frac{\dphi^i_{2}}{\dphi^i_1} = \frac{\dphi^i_{2}}{\phi_1^i}
\label{eq:alpha1a}$$ we ask how the state of maximum packing fraction can be achieved retaining the volume fraction ratio . There are tow possibilities differing with respect to the value of $\dphi^i_2$. Both situations are visualized in [Figure \[fig:Bimodal\]]{}.

#### Situation 1\

The volume fraction $\varphi_2^i$ of the large spheres lies below the monodisperse packing fraction $\varphi_c$ and the remaining interstice $1-\varphi_2^i$ is filled to an amount of $\varphi_c$ with small spheres, compare [Figure \[fig:Bimodal1\]]{}. So the total particle volume fraction is $$\varphi_{T2}^{i\eins} = \varphi_1^i+\varphi_2^i = \varphi_2^i + (1-\varphi_2^i)\varphi_c
\label{eq:IchT2}$$ (compare [equation ]{}, where restrictively $\varphi_2^i = \varphi_c$). Using the [equation ]{} we find from the [equation ]{} $$\varphi_2^i = \frac{\varphi_c}{\varphi_c + \frac{\phi_1^i}{\dphi^i_2}}
\label{eq:phi2bi}$$ and furthermore $$\label{eq:phiT1sit1}
\begin{split}
\varphi_{T2}^{i\eins} &= \varphi_1^i+\varphi_2^i =\left(\frac{\phi_1^i}{\dphi^i_2}+1 \right)\frac{\varphi_c}{\varphi_c + \frac{\phi_1^i}{\dphi^i_2}}\\
&= \frac{\varphi_c\phi_2^i}{\varphi_c\dphi^i_2 + \phi_1^i}
\end{split}$$ In the last step of the [equation ]{} we have used the identity . The limitation for the validity of the [equation ]{} follows from the realizability condition   (arbitrarily, we assign the equal sign to the situation 2 in order to avoid an ambiguous definition of the case $\varphi_2^i=\varphi_c$\[fuss\]) which in combination with the [equation ]{} yields $$\dphi^i_2 < \frac{\phi_1^i}{1-\varphi_c}
\label{eq:alphagrenz1sit1}$$ as the condition for the validity of situation 1.

#### Situation 2\

In this situation the volume fraction $\varphi_2^i$ of the large spheres equals the monodisperse maximum packing fraction $\varphi_c$ (it is not possible to exceed the value of $\varphi_c$ in a monodisperse loading) and so we have $\varphi_2^i = \varphi_c$, see [Figure \[fig:Bimodal2\]]{}. Using the [equation ]{} we immediately find $$\begin{aligned}
 \varphi_1^i&=\varphi_2^i\frac{\phi_1^i}{\dphi^i_2}\quad\text{and} \label{eq:phi1sit2}\\
\varphi_{T2}^{i\zwei} &= \varphi_1^i+\varphi_2^i = \varphi_c\left(1+\frac{\phi_1^i}{\dphi^i_2}\right) = \varphi_c\frac{\phi_2^i}{\dphi^i_2}
\label{eq:phiT1sit2}\end{aligned}$$ It follows from the fundamental condition $\varphi_{T2}^{i\zwei}<1$ and the [equation ]{} that $$\dphi^i_2>\frac{\varphi_c\phi_1^i}{1-\varphi_c}
\label{eq:alpha1schwach}$$ whereas the realizability condition $\varphi_1^i\leq(1-\varphi_2^i)\varphi_c$ under consideration of $\varphi_2^i=\varphi_c$ and the [equation ]{} yields $$\dphi^i_2\geq\frac{\phi_1^i}{1-\varphi_c}
\label{eq:alphagrenz1sit2}$$ The condition includes the [inequality ]{}. So the [equation ]{} serves as a condition for the validity of situation 2.

#### Summary for bidisperse systems\

The inequalities  and cover all possible values of $\dphi^i_2$. On the basis of a given volume fraction $\dphi^i_2$ one has to decide whether situations 1 or 2 are present. Subsequently the respectively valid equations  or  may be used to calculate the maximum packing fraction $\varphi_{T2}^i$.

The relations  and  are—if notation is adapted—identical to the equations (3) and (4) presented in [@1997_Gondret].

##### Derivation for tridisperse systems\

The following considerations refer to tridisperse systems, that is systems with three particle size classes. The pattern underlying the derivation will later be turned into a general formulation for an arbitrary number of size classes.

With tridisperse systems we also distinguish between two situations because, regardless of the number of size classes, there are always two situations. This relies on the geometric fact that no particle class except the one with the largest particle diameter can reach the monodisperse packing fraction. If hypothetically a particle class with a smaller diameter reached the monodisperse maximum packing fraction, the larger particles would not fit into the interstices between the smaller particles and could thus not contribute to this state of maximum packing fraction.

#### Situation 1\

The volume fraction $\varphi_2^i$ of the large spheres is smaller than the monodisperse maximum packing fraction $\varphi_c$ and so the remaining interstice $1-\varphi_2^i$ is filled by the medium and smallest particles to an amount of $\varphi_{T2}^i$ (equations  or ) which is the *bidisperse* maximum packing fraction calculated previously. The volume fraction of the medium and small spheres is thus given by $$\varphi_1^i+\varphi_2^i = (1-\varphi_3^i)\varphi_{T2}^i
\label{eq:phi1phi2mitphi3}$$ Rearranging the [equation ]{} and using the [equation ]{} we find $$\begin{split}
\varphi_1^i+\varphi_2^i &= \varphi_3^i\frac{\dphi^i_1}{\dphi^i_3}+\varphi_3^i\frac{\dphi^i_2}{\dphi^i_3}\\
 &= \phi_2^i\frac{\varphi_3^i}{\dphi^i_3}
\end{split}
\label{eq:phi1phi2mitalpha1}$$ It is now possible to insert the [equation ]{} into the relation which yields $$\varphi_3^i = \frac{\varphi_{T2}^i}{\varphi_{T2}^i + \frac{\phi_2^i}{\dphi^i_3}}
\label{eq:phi3tri}$$ From the equations  and  we can deduce the total volume fraction of spheres $$\label{eq:phiT2sit1}
\varphi_{T3}^{i\eins} = \varphi_1^i+\varphi_2^i+\varphi_3^i = \frac{\varphi_{T2}^i\phi_3^i}{\varphi_{T2}^i\dphi^i_3 + \phi_2^i}$$Using [equation ]{}, we can write the realizability condition $\varphi_3^i<\varphi_c$ in the form $$\dphi^i_3 < \frac{\varphi_c\phi_2^i}{\varphi_{T2}^i(1-\varphi_c)}
\label{eq:alphagrenz2sit1}$$ The inequality serves as a condition for the validity of situation 1.

#### Situation 2\

Analogously to the bidisperse case, situation 2 is characterized by a volume fraction $\varphi_3^i \equiv \varphi_c$ of the largest spheres. Under consideration of the [equation ]{} (see also [equation ]{}) we have $$\label{eq:phiT2sit2}
\begin{split}
\varphi_{T3}^{i\zwei}&=\varphi_1^i+\varphi_2^i+\varphi_3^i = \varphi_c\left(\frac{\dphi^i_1}{\dphi^i_3}+\frac{\dphi^i_2}{\dphi^i_3}+1\right)\\
&= \varphi_c \frac{\phi_3^i}{\dphi^i_3}
\end{split}$$ for the total volume fraction of spheres in situation 2 of a tridisperse suspension. The bound for $\dphi^i_3$ follows from the condition $\varphi_1^i+\varphi_2^i \leq (1-\varphi_c)\varphi_{T2}^i$ in the form $$\dphi^i_3 \geq \frac{\varphi_c\phi_2^i}{\varphi_{T2}^i(1-\varphi_c)}
\label{eq:alphagrenz2sit2}$$ corresponding to [inequality ]{} as expected.

#### Summary for tridisperse systems\

By means of the inequalities  and one decides whether the situations 1 or 2 are present. Afterwards, $\varphi_{T3}^i$ is calculated according to the equations  or ).

##### Generalization for polydisperse systems\

Now we will generalize the model for polydisperse systems which corresponds to deriving expressions for the total volume fraction $\varphi_{Tk+1}^i$ in the situations 1 and 2 and for the limiting value of $\dphi^i_{k+1}$.

#### Situation 1\

In the polydisperse case the interstices between the largest particles are filled by the smaller particle size classes with the packing fraction $\varphi_{Ti}^k$ which is determined by the volume fractions $\dphi^i_1$ to $\dphi^i_k$. So the volume fraction of the largest particles is $\varphi_{k+1}^i$ and the one of all the smaller particles is $(1-\varphi_{k+1}^i)\varphi_{Tk}^i$. The sum of both fractions yields the maximum packing fraction, that is $$\varphi_{Tk+1}^{i\eins} = \varphi_{k+1}^i + (1-\varphi_{k+1}^i)\varphi_{Tk}^i
\label{eq:phiTiplus1urspr}$$ So we may express the volume fractions of all the small particles by $$\sum_{m=1}^k \varphi_m = (1-\varphi_{k+1}^i)\varphi_{Tk}^i
\label{eq:summephik}$$ Equation allows for representing all the occuring volume fractions by the fraction $\varphi_{k+1}^i$ of the largest particles, so we can restate the sum in the [equation ]{} with the help of the [equation ]{}: $$\sum_{m=1}^k \varphi_m = \varphi_{k+1}^i {\frac{1}{\dphi^i_k}} \sum_{m=1}^k \dphi^i_m = \varphi_{k+1}^i \frac{\phi^{i}_k}{\dphi^i_{k+1}}
\label{eq:summephik2}$$ It immediately follows from the equations  and that $$(1-\varphi_{k+1}^i)\varphi_{Tk}^i = \varphi_{k+1}^i \frac{\phi^{i}_k}{\dphi^i_{k+1}}$$ that can be rearranged into $$\varphi_{k+1}^i = \frac{\varphi_{Tk}^i\dphi^i_{k+1}}{\varphi_{Tk}^i\dphi^i_{k+1} + \phi^{i}_k}
\label{eq:phiiplus1}$$ Additionally, under consideration of the [equation ]{} the following relation holds for both the situations 1 and 2: $$\begin{split}
\varphi_{Tk+1}^i &= \varphi_{k+1}^i + \sum_{m=1}^k \varphi_m = \varphi_{k+1}^i \left(1+\frac{\phi^{i}_k}{\dphi^i_{k+1}}\right)\\
 &= \varphi_{k+1}^i\frac{\phi^{i}_{k+1}}{\dphi^i_{k+1}}
\end{split}
\label{eq:phiTiplus1allg}$$ Inserting the [equation ]{} into , we find the result for the maximum packing fraction in situation 1: $$\boxed{
\varphi_{Tk+1}^{i\eins} = \frac{\varphi_{Tk}^i\phi^{i}_{k+1}}{\varphi_{Tk}^i\dphi^i_{k+1} + \phi^{i}_k}
\label{eq:phiTiplus11}
}$$ The limiting value for $\dphi^i_{k+1}$ follows analogously to the bi- and tridisperse cases from the requirement $\varphi_{k+1}^i<\varphi_c$ and is thus given by $$\boxed{
\dphi^i_{k+1} < \frac{\varphi_c\phi^{i}_k}{\varphi_{Tk}^i(1-\varphi_c)}
\label{eq:alphagrenzsit1}
}$$

#### Situation 2\

In situation 2 the largest particles occupy a volume fraction equal to the monodisperse maximum packing fraction, so $\varphi_{k+1}^i=\varphi_c$. Therefore, by means of the [equation ]{} we may write $$\boxed{
\varphi_{Tk+1}^{i\zwei} = \varphi_c \frac{\phi^{i}_{k+1}}{\dphi^i_{k+1}}
\label{eq:phiTiplus12}
}$$ for the maximum packing fraction in situation 2. The limiting value of $\dphi^i_{k+1}$ in situation 2 is determined by the condition $$\sum_{m=1}^k \varphi_m \leq (1-\varphi_c)\varphi_{Tk}^i
\label{eq:forderung2}$$ and combined with  and  takes the form $$\boxed{
\dphi^i_{k+1} \geq \frac{\varphi_c\phi^{i}_k}{\varphi_{Tk}^i(1-\varphi_c)}
\label{eq:alphagrenzsit2}
}$$

#### Properties of the bounds\
 {#Schranken}

The bounds  and  can alternatively be derived by equaling the prescriptions  and . This underlines the consistency between the situations 1 and 2 because there is a continuous transition. This is equivalent to choosing the smallest value out of the two calculated maximum packing fractions because for all values of $\dphi_{k+1}^i$ the situation that is present is always the one with the smaller maximum packing fraction. Thus we are lead to the prescription $$\boxed{
\varphi_{Tk+1}^i=\text{min}\left[\varphi_{Tk+1}^{i\eins},\varphi_{Tk+1}^{i\zwei}\right]
}
\label{eq:minphiT1}$$

Scheme for the viscosity estimation
-----------------------------------

[Figure \[fig:Ablauf\]]{} shows the procedure for the calculation of maximum packing fraction and relative viscosity after the $i$th construction step. The complete suspension shall contain $n$ particle size classes.

![Scheme for the calculation of maximum packing fraction and relative viscosity after the $i$th construction step for $n$ particle size classes with references to the respective equations[]{data-label="fig:Ablauf"}](Ablauf.pdf)

Influence of polydispersity on relative viscosity
-------------------------------------------------

In this section we examine the change in relative viscosity due to the number of particle size classes, provided that the size distribution is given. As already noted in [section \[sec:Aufbauprozess\]]{}, the model developed so far and presented in the scheme on [page ]{} is only valid for diameter ratios between consecutive size classes of about 7 to 10. A special choice of the diameter ratio—in the following we will choose $u_i=10$—results in a logarithmic sampling of the continuous size distribution. Therefore, the sampling points, that is the first diameter $d_1$, have to be adequately chosen in order to reproduce the characteristics of the distribution. If the volume fractions $\dphi_k$ are not equally distributed among the different diameter values, the choice of the sampling points thus depends on the number of size classes. This will be the case in the following examination concerning the <span style="font-variant:small-caps;">Rosin-Rammler</span> distribution. Subsequently, we will consider the case of a uniform distribution.

### Rosin-Rammler distribution

The <span style="font-variant:small-caps;">Rosin-Rammler</span> distribution is frequently used to describe the drop size distribution of sprays. Its cumulative distribution function $R_{\text{CDF}}$ is given by $$R_{\text{CDF}}(d)=1-{\mathrm{e}^{-\left(\frac{d}{X}\right)^q}}
\label{eq:CDF}$$ where $d$ is the particle diameter as well as $X$ and $q$ are model parameters. By differentiation of  the probability density function $R_{\text{PDF}}$ (we do not rigorously distinguish between probability and relative frequency) can be deduced: $$R_{\text{PDF}}(d)=\frac{q}{d}\left(\frac{d}{X}\right)^q{\mathrm{e}^{-\left(\frac{d}{X}\right)^{q-1}}}
\label{eq:PDF}$$ The function $R_{\text{PDF}}(d)$ describes the relative frequency of occurrence of a particle diameter between $d$ and $d+{\mathrm{d}}d$ and is normalized to unity. So we find the volume fraction occupied by the particles with diameters between $d$ and $d+{\mathrm{d}}d$ through multiplication of the total volume fraction $\phi$ by $R_{\text{PDF}}(d)$ (it is $\phi=\phi_n=\sum_{m=1}^{n}\dphi_m$ for a number of $n$ size classes). [Figure \[fig:Verteilung\_01\]]{} shows the probability density function $R_{\text{PDF}}(d)$ for a special choice of the parameters $X$ and $q$ in [equation ]{}.

![Probability density function by Rosin-Rammler (equation (\[eq:PDF\])) with parameters  and  and sampling for bi- and tridisperse representations ($n=2$: $[d_1,d_2]=[5,50]\,\text{\textmu{}m}$ and $n=3$: $[d_1,d_2,d_3]=[1,10,100]\,\text{\textmu{}m}$, respectively)[]{data-label="fig:Verteilung_01"}](Verteilung_01.pdf)

With a diameter ratio of $u_i=10$, the distribution only allows for useful sampling at no more than three points because of its asymptotic declination. The samplings for two and three particle size classes are shown in [Figure \[fig:Verteilung\_01\]]{}. We proceed as follows: In order to distribute the total volume fraction $\phi$ among the size classes we divide the individual values of the PDF at the sampling points by their sum and multiply them by $\phi$. So we achieve the volume fractions to be distributed according to the continuous size distribution. Of course, it is not necessary to choose sampling points for the monodisperse suspension.

We now evaluate the assignment between the diameter $d_k$ and the relative volume fraction $\dphi_k$ given by the <span style="font-variant:small-caps;">Rosin-Rammler</span> distribution  for different values of $\phi$. Thereby we apply the [equation ]{} for the viscosity as well as the equations  and  for the maximum packing fraction. We therefore assume the relation  to be valid and so we de facto apply the -Modell (see [section \[sec:VergleichLiteratur\]]{}).

The results are presented in [Figure \[fig:Verteilung\_02\]]{}.

![Relative viscosity as a function of the volume fraction for various numbers of size classes $n$ from equation (\[eq:rekursallgvisko\]) with parameters $\varphi_c=0.64$, ${\left[\eta_{1}\right]}=2.5$ and ${\left[\eta_{2}\right]}=6.17$; Rosin-Rammler size distribution (samplings shown in Figure \[fig:Verteilung\_01\]); maximum packing fraction $\varphi_c$, $\varphi_{T2}^2$ and $\varphi_{T3}^3$[]{data-label="fig:Verteilung_02"}](Verteilung_02.pdf)

It is clearly visible that the relative viscosity decreases as the number of particle size classes increases. The strongest influence can be noticed during the transition from a monodisperse to a bidisperse suspension while adding a third size class only causes relatively small changes. The curves diverge at the respective values of the final maximum packing fraction $\varphi_{Tn}^n$ which are also depicted in [Figure \[fig:Verteilung\_02\]]{}.

### Uniform distribution

The uniform size distribution is characterized by the equality of all the volume fractions $\dphi_k$, so one has $\dphi_k=\phi/n$ for each size class. Analogously to the case of the <span style="font-variant:small-caps;">Rosin-Rammler</span> distribution, we calculate the relative viscosity as a function of the total volume fraction $\phi$ using the [equation ]{} and the maximum packing fraction applying the relations  and . The results are depicted in [Figure \[fig:Verteilung\_03\]]{}, whereas the behavior qualitatively coincides with the results shown in [Figure \[fig:Verteilung\_02\]]{}.

![Relative viscosity as a function of the volume fraction for various numbers of size classes $n$ from equation (\[eq:rekursallgvisko\]) with parameters $\varphi_c=0.64$, ${\left[\eta_{1}\right]}=2.5$ und ${\left[\eta_{2}\right]}=6.17$ for a uniform distribution[]{data-label="fig:Verteilung_03"}](Verteilung_03.pdf)

The relative viscosity decreases at constant $\phi$ and increasing $n$. We confine ourselves to at most four size classes because the trend is already clearly visible with that number.

### Range of influence of polydispersity

Table \[tab:Verteilung\] shows the viscosity decrease as a function of the polydispersity by means of different values of $\phi$ related to the monodisperse viscosity.

1ex

  Distribution                Ratio in                $\phi=0$   0.1    0.2    0.3    0.4    0.5    0.6
  -------------- ----------------------------------- ---------- ------ ------ ------ ------ ------ -----
                  $\frac{\eta_r(n=2)}{\eta_r(n=1)}$    100.0     99.1   95.4   87.0   71.1   43.6   7.1
                  $\frac{\eta_r(n=3)}{\eta_r(n=1)}$    100.0     98.7   94.0   83.9   66.1   38.2   5.5
                  $\frac{\eta_r(n=2)}{\eta_r(n=1)}$    100.0     99.0   95.1   86.9   71.9   46.2   8.6
                  $\frac{\eta_r(n=3)}{\eta_r(n=1)}$    100.0     98.6   93.6   83.4   65.7   38.6   5.9
                  $\frac{\eta_r(n=4)}{\eta_r(n=1)}$    100.0     98.5   92.9   81.7   63.0   35.5   5.1

The numerical values for the <span style="font-variant:small-caps;">Rosin-Rammler</span> distribution rarely differ from the values for the uniform distribution. At a total volume fraction of $\phi=0.1$ the deviation amounts to approximately one percentage point, for $\phi=0.2$ partially to more than five percentage points. If, for instance, the derivation shall be kept below one percentage point, it is necessary to calculate the viscosity using the polydisperse formulae at volume fractions higher than 0.1, provided that the particle size distribution is broad enough. A polydisperse calculation only makes sense if the size distribution allows for at least two size classes with significant volume fractions having a diameter ratio of 7 to 10. For example, the inclusion of a fourth size class with a particle diameter of $d_4=1000\micm$ in the case of the <span style="font-variant:small-caps;">Rosin-Rammler</span> distribution in [Figure \[fig:Verteilung\_01\]]{} has no effect on the relative viscosity because the relative frequency is approximately equal to $9\times10^{-16}$ and therefore negligible.

Consideration of particle deformation {#sec:Deformation}
-------------------------------------

In [@2009_Hsueh] an approach to the consideration of deformable particles is developed. Thereby, the particle deformability is represented by a particle viscosity $\eta_p$. Using the so-called modified <span style="font-variant:small-caps;">Eshelby</span> model (see also [@2002_Torquato]) as well as the elastic-viscous analogy the derivation of the equation $$\eta_r=\frac{2\eta_p+3\eta_0+3\phi\left(\eta_p-\eta_0\right)}{2\eta_p+3\eta_0-2\phi\left(\eta_p-\eta_0\right)}
\label{eq:etarhsueh}$$ is outlined. This equation is confined to long-range particle interactions and is thus invalid for dense suspensions. However, as presented in [@2009_Hsueh], the first-order series expansion of the relation  can be used for the differential model (see [section \[sec:Bruggeman\]]{}), analogously to the use of the relation in [equation ]{}. So for the differential approach we have $$\eta+{\mathrm{d}}\eta = \eta\left[1+2.5\,\phi_{P}\left(\frac{\eta_p-\eta}{\eta_p+1.5\eta}\right)\right] 
\label{eq:betarhsueh}$$ To be consistent with the [equation ]{}, one has to set ${\left[\eta_{1}\right]}=2.5$. For $\eta_p\to\infty$ (rigid particles) the [equation ]{} reduces to the relation . In the course of the discrete construction process higher order terms out of the series expansion of the [equation ]{} can be considered. However, the limited validity of this equation for larger values of $\phi_{P}$ must be regarded. If we include terms up to the second order we can express the relative viscosity in the discrete construction process by $$\begin{split}
\eta_{i+1} = &\eta_i \Big[ 1+2.5\,\phi_{P,i+1}\left(\frac{\eta_p-\eta_i}{\eta_p+1.5\eta_i}\right)\\ 
&+2.5\,\phi_{P,i+1}^{2}\left(\frac{\eta_p-\eta_i}{\eta_p+1.5\eta_i}\right)^2\Big]
\end{split}
\label{eq:betahsueh}$$ The [equation ]{} allows for the effect of the particle viscosity $\eta_p$ on the relative viscosity of polydisperse systems to be described approximately. Since in this work we do not focus on particle deformation, we simply state the result  without further validation.

Conclusions {#chap:conclusions}
===========

In the present work we provided a model for the relative viscosity of polydisperse suspensions of spherical non-colloidal particles. Using monodisperse viscosity correlations, we described polydisperse suspensions by means of a construction process consisting of successive additions of particle size classes.As a starting point, we proposed a generalized form of the well-known equation that allows for the choice of the second order intrinsic viscosity ${\left[\eta_{2}\right]}$. This modified equation can be used to approximate the various monodisperse viscosity relations existing in the literature and can therefore be regarded as a generic relation.Later, we described the construction process in detail applying a dimensionless way of description based on volume fractions. This rigorous description served as a basis for the calculation of the relative viscosity during the construction process. Starting from the model, we finally arrived at the model, connecting two approaches commonly regarded as uncorrelated.As an entirely new component, we introduced the polydisperse maximum packing fraction into the model. Here, we followed a physically consistent approach in contrast to the approaches presented in the literature. Consistently with the relative viscosity calculation, we derived a formalism to determine the polydisperse maximum packing fraction by means of a common construction process.The entire formalism for calculating the relative viscosity—whether including the maximum packing fraction or not—is depicted on page . We evaluated the model in the case of two different particle size distributions in order to observe the influence of polydispersity represented by the number of particle size classes.Additionally, we revealed a possible approach for integrating particle deformability, represented by a particle phase viscosity, into the viscosity model using a result from the literature.So far, our model is only valid for large diameter ratios of consecutive size classes during the construction process. An attempt to generalize the model to the case of small size ratio as well as the shear rate dependence of the relative viscosity will be presented in a future work.

Proof of equivalence of the construction processes at variable and constant volume {#sec:Beweisderaequivalenz}
==================================================================================

The Definition  implies for both cases—variable and constant total volume—the corollary $$\sum_{m=1}^i \delta\phi_m = \phi_i
\label{eq:summedeltaphi}$$ Obviously both approaches must result in the same total volume fraction $\phi_n$ at the end of the construction process. So we ask if all partial sums $\phi_i$ ($i\leq n$) are identical in both cases. As a consequence, the values of $\delta\phi_{i+1}$ for $i=1\ldots n$ would coincide, too. The validity of these identities is important for the derivations in [section \[sec:Diskretes\_Modell\]]{} because it ensures that the viscosities calculated in both cases are equal. This is a prerequisite for a consistent model. In the following we show the equivalence of the two approaches.

To distinguish between the volume fractions $\phi_{i+1}^*$ occurring in each case we introduce the notations $\phi_{i+1}^{*\variabel}$ for the case of variable total volume and $\phi_{i+1}^{*\konstant}$ for the case of constant total volume. In a first step we show that the volume fraction $\phi_i$ coincide for $i=1\ldots n$, so that $$\phi_i^{\variabel} = \phi_i^{\konstant}
\label{eq:ident1}$$ According to the [equation ]{}, the volume fraction $\phi_i^{\variabel}$ is given by $$\phi_i^{\variabel} = \frac{\sum_{m=1}^{i}V_m}{V_f+\sum_{m=1}^{i}V_m}
\label{eq:phiivariabel}$$ while the volume fraction $\phi_i^{\konstant}$ from the [equation ]{} is $$\phi_i^{\konstant} = \frac{\sum_{m=1}^i \dphi_m}{1-\sum_{m=i+1}^n \dphi_m}
\label{eq:phiikonstant2}$$ Insertion of the relation  into the [equation ]{} yields after an intermediate step $$\begin{split}
\phi_i^{\konstant} &= \frac{\sum_{m=1}^i V_m}{V_f+\sum_{m=1}^{n}V_m - \sum_{m=i+1}^{n}V_m}\\
 &= \frac{\sum_{m=1}^{i}V_m}{V_f+\sum_{m=1}^{i}V_m} = \phi_i^{\variabel}
\end{split}
\label{eq:ident1gezeigt}$$ and so the validity of the proposition  has been proved. It follows directly the coincidence of the differences $\delta\phi_{i+1}$ in the [equation ]{}. In a second and last step we proof that $$\phi_{P,i+1}^\variabel=\frac{\phi_{i+1}^{*\variabel}}{1+\phi_{i+1}^{*\variabel}} = \phi_{i+1}^{*\konstant}=\phi_{P,i+1}^\konstant
\label{eq:ident2}$$ Using the definition  we may write the left-hand side of the [equation ]{} in the form $$\frac{\phi_{i+1}^{*\variabel}}{1+\phi_{i+1}^{*\variabel}} = \frac{V_{i+1}}{V_f+\sum_{m=1}^{i+1} V_m}
\label{eq:linksident2}$$ Rearranging of the right-hand side of the representation  and using the equations  and  we find $$\begin{split}
\phi_{i+1}^{*\konstant}&=\frac{\dphi_{i+1}}{1-\sum_{m=i+2}^n \dphi_m}\\ 
&= \frac{ V_{i+1}}{V_f+\sum_{m=1}^{n}V_m - \sum_{m=i+2}^{n}V_m}\\
&= \frac{V_{i+1}}{V_f+\sum_{m=1}^{i+1} V_m} = \frac{\phi_{i+1}^{*\variabel}}{1+\phi_{i+1}^{*\variabel}}
\end{split}
\label{eq:rechtsident2}$$ So the proof of the proposition  is complete.

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