# On the irreducibility of extended Laguerre Polynomials

Anuj Jakhar  ,  Srinivas Kotyada  and  Arunabha Mukhopadhyay  Department of
Mathematics, Indian Institute of Technology (IIT) Madras  Department of
Mathematics, The Institute of Mathematical Sciences (IMSc) Chennai  Department
of Mathematics, The Institute of Mathematical Sciences (IMSc) Chennai
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###### Abstract.

Let $m\geq 1$ and $a_{m}$ be integers. Let $\alpha$ be a rational number which
is not a negative integer such that $\alpha=\frac{u}{v}$ with
$\gcd(u,v)=1,v>0$. Let $\phi(x)$ belonging to ${\mathbf{Z}}[x]$ be a monic
polynomial which is irreducible modulo all the primes less than or equal to
$vm+u$. Let $a_{i}(x)$ with $0\leq i\leq m-1$ belonging to ${\mathbf{Z}}[x]$
be polynomials having degree less than $\deg\phi(x)$. Assume that the content
of $(a_{m}a_{0}(x))$ is not divisible by any prime less than or equal to
$vm+u$. In this paper, we prove that the polynomials
$L_{m,\alpha}^{\phi}(x)=\frac{1}{m!}(a_{m}\phi(x)^{m}+\sum\limits_{j=0}^{m-1}b_{j}a_{j}(x)\phi(x)^{j})$
are irreducible over the rationals for all but finitely many $m$, where
$b_{j}=\binom{m}{j}(m+\alpha)(m-1+\alpha)\cdots(j+1+\alpha)~{}~{}~{}\mbox{ for
}0\leq j\leq m-1$. Further, we show that $L_{m,\alpha}^{\phi}(x)$ is
irreducible over rationals for each $\alpha\in\\{0,1,2,3,4\\}$ unless
$(m,\alpha)\in\\{(1,0),(2,2),(4,4),(6,4)\\}.$ For proving our results, we use
the notion of $\phi$-Newton polygon and some results from analytic number
theory. We illustrate our results through examples.

###### Key words and phrases:

Irreducibility, Polynomials, Newton Polygons

###### 2010 Mathematics Subject Classification:

11C08, 11R04

– – Dedicated to Professor Michael Filaseta

##  1\. Introduction and statements of results

Let $m$ be a positive integer and $\alpha$ be an arbitrary real number. The
generalized Laguerre polynomials are defined by

$L_{m,\alpha}(x)=\sum\limits_{j=0}^{m}\frac{(m+\alpha)(m-1+\alpha)\cdots(j+1+\alpha)}{(m-j)!j!}(-x)^{j}.$

The classical Laguerre polynomials correspond to $\alpha=0.$ They appear in
various branches of Mathematics and Mathematical Physics. In a series of
papers, I. Schur (cf. [29] -[32]) obtained irreducibility results for
polynomials associated with $L_{m,\alpha}(x)$ by considering prime ideal
factorization of the principal ideals generated by the coefficients of the
polynomials. Later, Coleman [3] and Filaseta [5] established Schur’s results
by invoking the main Newton polygon result of Dumas. It is interesting to note
that Filaseta and Trifonov [7] showed the irreducibility of Laguerre
polynomials corresponding to $\alpha=-2m-1$ which in turn confirm a conjecture
of Grosswald on the irreducibility of Bessel polynomials. The irreducibility
of generalized Laguerre polynomials and their Galois groups were widely
studied by many authors for several values of $\alpha$ (cf.
[6],[8],[9],[10],[23] -[28]).

In this paper, let $m\geq 1$ and $a_{m}$ be integers. Let $\alpha$ be a
rational number which is not a negative integer such that $\alpha=\frac{u}{v}$
with $\gcd(u,v)=1,v>0$. Let $\phi(x)$ belonging to ${\mathbf{Z}}[x]$ be a
monic polynomial which is irreducible modulo all the primes less than or equal
to $vm+u$. Let $a_{i}(x)$ with $0\leq i\leq m-1$ belonging to
${\mathbf{Z}}[x]$ be polynomials having degree less than $\deg\phi(x)$. Assume
that the content of $(a_{m}a_{0}(x))$ is not divisible by any prime less than
or equal to $vm+u$. Then, we define the generalized $\phi$-Laguerre
polynomials by

$L_{m,\alpha}^{\phi}(x)=\frac{1}{m!}(a_{m}\phi(x)^{m}+\sum\limits_{j=0}^{m-1}b_{j}a_{j}(x)\phi(x)^{j}),$

where
$b_{j}=\binom{m}{j}(m+\alpha)(m-1+\alpha)\cdots(j+1+\alpha)~{}~{}~{}\mbox{ for
}0\leq j\leq m-1.$

With this definition, the aim of this paper is to prove the following results.

######  Theorem 1.1.

The polynomials $L_{m,\alpha}^{\phi}(x)$ are irreducible over the rationals
for all but finitely many positive integers $m$.

######  Theorem 1.2.

The polynomials $L_{m,\alpha}^{\phi}(x)$ are irreducible over the rationals
for $\alpha\in\\{0,1,2,3,4\\}$ unless
$(m,\alpha)\in\\{(1,0),(2,2),(4,4),(6,4)\\}.$

######  Remark 1.3.

The above theorem gives rise to a wide class of irreducible polynomials whose
irreducibility do not seem to follow from the known irreducibility criterion
(cf. [2], [12], [13], [14], [15], [16] and [17]).

######  Remark 1.4.

Theorem 1.2 can be extended for other values of $\alpha$, even for rational
values of $\alpha.$

It may be pointed out that in Theorem 1.2, the assumption that “the content of
$(a_{m}a_{0}(x))$ is not divisible by any prime less than or equal to $vm+u$”
cannot be dispensed with. This can be seen, for example taking
$\phi(x)=x^{2}-x+17$ and $m=2$, as displayed in the tables below.

Table 1.  The content of $a_{0}(x)$ is divisible by $2$. $\alpha$ |  $a_{2}$ |  $a_{1}(x)$ |  $a_{0}(x)$ |  $L_{m,\alpha}^{\phi}(x)$
---|---|---|---|---
1 |  3 |  1 |  -4 |  $\frac{3}{2}(\phi(x)+4)(\phi(x)-2)$
2 |  1 |  1 |  -4 |  $\frac{1}{2}(\phi(x)-4)(\phi(x)+12)$
3 |  1 |  1 |  -10 |  $\frac{1}{2}(\phi(x)-10)(\phi(x)+20)$
4 |  3 |  1 |  -6 |  $\frac{3}{2}(\phi(x)+10)(\phi(x)-6)$

Table 2.  $a_{2}$ is divisible by $2$. $\alpha$ |  $a_{2}$ |  $a_{1}(x)$ |  $a_{0}(x)$ |  $L_{m,\alpha}^{\phi}(x)$
---|---|---|---|---
1 |  6 |  2 |  1 |  $3(\phi(x)+1)^{2}$
2 |  4 |  1 |  -1 |  $2(\phi(x)+3)(\phi(x)-1)$
3 |  2 |  1 |  -5 |  $(\phi(x)+10)(\phi(x)-5)$
4 |  18 |  1 |  -1 |  $(\phi(x)-1)(9\phi(x)+15)$

We can also notice that Theorem 1.2 may not hold if $a_{n}$ is replaced by a
(monic) polynomial $a_{n}(x)$ with integer coefficients having degree less
than $\deg\phi(x)$. Consider $\phi(x)=x^{2}-x+5$ which is irreducible modulo
$2$ and $3$. Take $a_{2}(x)=x-3=a_{1}(x)=a_{0}(x)$. Then the polynomial
$\frac{1}{2}({a_{2}(x)}\phi(x)^{2}+2(2+\alpha)a_{1}(x)\phi(x)+(2+\alpha)(1+\alpha)a_{0}(x))$
has $3$ as a root for each $\alpha\in\\{0,1,2,3,4\\}$.

As an application, we now provide an example to show that Theorem 1.2 leads to
classes of irreducible polynomials.

######  Example 1.5.

Consider $\phi(x)=x^{2}-x+17$. It can be easily checked that $\phi(x)$ is
irreducible modulo $2,3,5,7,11$ and $13$. Let $m$ and $\alpha$ be fixed
integers such that $m\in\\{3,5,7,8,9,10,11,12\\}$, $\alpha\in\\{0,1,2,3,4\\}$.
Let $a_{m}$ be an integer and $a_{i}(x)\in{\mathbf{Z}}[x]$ be polynomials each
having degree less than $2$ for $0\leq i\leq m-1$. Assume that the content of
$(a_{m}a_{0}(x))$ is not divisible by any prime less than or equal to $13$.
Then by Theorem 1.2, the polynomial $L_{m,\alpha}^{\phi}(x)$ is irreducible
over ${\mathbf{Q}}$.

We use the concept of $\phi$-Newton polygon as introduced by Ore [22], few
results on primes from analytical number theory to prove our theorems.

##  2\. $\phi$-Newton polygon.

In this section, we shall recall the notion of $\phi$-Newton polygon and state
a result which will be used in the proof of the theorems.

For a prime $p$, $v_{p}$ will denote the $p$-adic valuation of ${\mathbf{Q}}$
defined for any non-zero integer $b$ to be the highest power of $p$ dividing
$b$. We shall denote by $v_{p}^{x}$ the Gaussian valuation extending $v_{p}$
defined on the polynomial ring ${\mathbf{Z}}[x]$ by

$v_{p}^{x}(\sum\limits_{i}b_{i}x^{i})=\min_{i}\\{v_{p}(b_{i})\\},b_{i}\in{\mathbf{Z}}.$

The $\phi$-Newton polygon with respect to $p$ is defined as below.

######  Definition 2.1.

Let $p$ be a prime number and $\phi(x)\in{\mathbf{Z}}[x]$ be a monic
polynomial which is irreducible modulo $p$. Let $f(x)$ belonging to
${\mathbf{Z}}[x]$ be a polynomial having $\phi$-expansion111If $\phi(x)$ is a
fixed monic polynomial with coefficients in ${\mathbf{Z}}$, then any
$f(x)\in{\mathbf{Z}}[x]$ can be uniquely written as a finite sum
$\sum\limits_{i}b_{i}(x)\phi(x)^{i}$ with $\deg b_{i}(x)<\deg\phi(x)$ for each
$i$; this expansion will be referred to as the $\phi$-expansion of $f(x)$.
$\sum\limits_{i=0}^{n}b_{i}(x)\phi(x)^{i}$ with $b_{0}(x)b_{n}(x)\neq 0$. Let
$P_{i}$ stand for the point in the plane having coordinates
$(i,v_{p}^{x}(b_{n-i}(x)))$ when $b_{n-i}(x)\neq 0$, $0\leq i\leq n$. Let
$\mu_{ij}$ denote the slope of the line joining the points $P_{i}$ and $P_{j}$
if $b_{n-i}(x)b_{n-j}(x)\neq 0$. Let $i_{1}$ be the largest index $0<i_{1}\leq
n$ such that

$\displaystyle\mu_{0i_{1}}=\min\\{\mu_{0j}\ |\ 0<j\leq n,\ b_{n-j}(x)\neq
0\\}.$

If $i_{1}<n$, let $i_{2}$ be the largest index $i_{1}<i_{2}\leq n$ such that

$\displaystyle\mu_{i_{1}i_{2}}=\min\\{\mu_{i_{1}j}\ |\ i_{1}<j\leq n,\
b_{n-j}(x)\neq 0\\}$

and so on. The $\phi$-Newton polygon of $f(x)$ with respect to $p$ is the
polygonal path having segments
$P_{0}P_{i_{1}},P_{i_{1}}P_{i_{2}},\dots,P_{i_{k-1}}P_{i_{k}}$ with $i_{k}=n$.
These segments are called the edges of the $\phi$-Newton polygon of $f(x)$ and
their slopes from left to right form a strictly increasing sequence.

We shall use the following lemma[18, Theorem 1.3] in the sequel. We omit its
proof.

######  Lemma 2.2.

Let $m,k$ and $\ell$ be integers with $0\leq\ell<k\leq\frac{m}{2}$ and $p$ be
a prime. Let $\phi(x)\in{\mathbf{Z}}[x]$ be a monic polynomial which is
irreducible modulo $p$. Let $f(x)$ belonging to ${\mathbf{Z}}[x]$ be a monic
polynomial not divisible by $\phi(x)$ having $\phi$-expansion
$\sum\limits_{i=0}^{m}f_{i}(x)\phi(x)^{i}$ with $f_{m}(x)\neq 0$. Assume that
$v_{p}^{x}(f_{i}(x))>0$ for $0\leq i\leq m-\ell-1$ and the right-most edge of
the $\phi$-Newton polygon of $f(x)$ with respect to $p$ has slope less than
$\frac{1}{k}$. Let $a_{0}(x),a_{1}(x),\dots,a_{m}(x)$ be polynomials over
${\mathbf{Z}}$ satisfying the following conditions.

  * (i)

$\deg a_{i}(x)<\deg\phi(x)-\deg f_{i}(x)$ for $0\leq i\leq m$,

  * (ii)

$v_{p}^{x}(a_{0}(x))=0$, i.e., the content of $a_{0}(x)$ is not divisible by
$p$,

  * (iii)

the leading coefficient of $a_{m}(x)$ is not divisible by $p$.

Then the polynomial $\sum\limits_{i=0}^{m}a_{i}(x)f_{i}(x)\phi(x)^{i}$ does
not have a factor in ${\mathbf{Z}}[x]$ with degree lying in the interval
$[(\ell+1)\deg\phi,(k+1)\deg\phi).$

##  3\. Proof of Theorem 1.1.

Before writing the proof of the theorem, we first state two lemmas which are
needed for the proof. We skip their proofs.

The following lemma is proved in [1, Theorem 3].

######  Lemma 3.1.

Let $a$ and $b$ be fixed relatively prime integers with $b>0$, and let
$\pi(x;b,a)$ denote the number of primes $\leq x$ which are congruent to
$a\pmod{b}$. If $b\leq(\log x)^{A}$ for some fixed $A>0$ and
$x^{\frac{11}{20}+\varepsilon}\leq h\leq\frac{x}{\log x}$, then

$\pi(x;b,a)-\pi(x-h;b,a)\gg\frac{h}{\phi(b)\log x}~{}\mbox{ for every
}\varepsilon>0.$

The following lemma is an immediate consequence of Corollary 1.2 of [33].

######  Lemma 3.2.

Let $a,b,c$ and $d$ be integers with $bc-ad\neq 0$. Then the largest prime
factor of $(am+b)(cm+d)$ tends to infinity as the integer $m$ tends to
infinity.

###### Proof of Theorem 1.1.

We fix an $\alpha\in{\mathbf{Q}}$ which is not a negative integer. It is
enough to prove that
$f(x)=a_{m}\phi(x)^{m}+\sum\limits_{j=0}^{m-1}b_{j}a_{j}(x)\phi(x)^{j}$ is
irreducible over the rationals for sufficiently large $m$. Recall that
$\alpha=\frac{u}{v}$ with $\gcd(u,v)=1$ and $v>0.$

We first show that $f(x)$ does not have a non-constant factor over
${\mathbf{Z}}$ with degree less than $\deg\phi(x)$. Let $p$ be a prime number
dividing $vm+u$. Then by hypothesis, it follows that $p\nmid a_{m}$. Let $c$
denote the content of $f(x)$. As $p\nmid a_{m}$, we have $p\nmid c$. Now
suppose to the contrary that there exists a primitive non-constant polynomial
$h(x)\in{\mathbf{Z}}[x]$ dividing $f(x)$ having degree less than
$\deg\phi(x)$. Then in view of Gauss Lemma, there exists
$g(x)\in{\mathbf{Z}}[x]$ such that $\frac{f(x)}{c}=h(x)g(x)$. The leading
coefficient of $f(x)$ and hence those of $h(x)$ and $g(x)$ are coprime with
$p$. Note that $p$ divides $b_{j}$ for $0\leq j\leq m-1$. Therefore on passing
to ${\mathbf{Z}}/p{\mathbf{Z}}$, we see that the degree of $\bar{h}(x)$ is
same as that of $h(x)$. Hence $\deg\bar{h}(x)$ is positive and less than
$\deg\phi(x)$. This is impossible because $\bar{h}(x)$ is a divisor of
$\frac{\overline{f}(x)}{\bar{c}}=\frac{\bar{a}_{m}}{\bar{c}}\bar{\phi}(x)^{m}$
and $\bar{\phi}(x)$ is irreducible over ${\mathbf{Z}}/p{\mathbf{Z}}$.

Now using Lemma 2.2, we shall show that for $k\in[1,\frac{m}{2}]$ and
sufficiently large $m$, $f(x)$ can not have a factor in ${\mathbf{Z}}[x]$ with
degree lying in the interval $[k\deg\phi(x),(k+1)\deg\phi(x))$. For using
Lemma 2.2, we consider the polynomial
$g(x)=\sum\limits_{j=0}^{m}b_{j}\phi(x)^{j}$ with $b_{m}=a_{m}$. Keeping in
mind that $\alpha$ is not a negative integer implies that for each
$j\in\\{0,1,\cdots,m-1\\}$, $m-j+\alpha$ and hence $v(m-j)+u$ cannot be zero.
We assume that $g(x)$ has a factor in ${\mathbf{Z}}[x]$ with degree lying in
the interval $[\deg\phi(x),(\frac{m}{2}+1)\deg\phi(x))$ and prove our theorem
by obtaining a contradiction to Lemma 2.2.

We divide the proof into cases depending on the size of $k$ with $1\leq k\leq
m/2$.

Case (i): $m^{\frac{11}{20}+\varepsilon}<k\leq\frac{m}{2},$ where
$\varepsilon>0$ is an arbitrary small constant.

By considering $\pi(x;b,a)-\pi(x-h;b,a)$, it follows from Lemma 3.1 that for
$a$ and $b$ fixed, the interval $[x-h,x)$ contains a prime congruent to $a$
modulo $b$ if $h\geq{x^{\frac{11}{20}+\varepsilon}}$ for all sufficiently
large $x$. Taking $a=u,~{}b=v$ and $x=vm+u$, we deduce that for some integer
$j\in[0,k)$, the number $v(m-j)+u$ is prime. Call such a prime $p$, and
observe that $p\geq 2vm/3$ (since $v$ is a positive integer and $m$ is large).
It follows that $p$ does not divide $v$. Observe that

$b_{\ell}=\binom{m}{\ell}\frac{(vm+u)(v(m-1)+u)\cdots(v(\ell+1)+u)}{v^{m-\ell}}~{}~{}~{}\mbox{
for }0\leq\ell\leq m-1.$

Clearly

$v_{p}(b_{\ell})\geq 1~{}~{}~{}\mbox{ for }0\leq\ell\leq m-k.$ (3.1)

The slope of the right-most edge of the $\phi$\- Newton polygon of $g(x)$ with
respect to $p$ is

$\lambda:=\max\limits_{1\leq j\leq
m}\bigg{(}\frac{v_{p}(b_{0})-v_{p}(b_{j})}{j}\bigg{)}.$

Keeping in mind that $v_{p}(b_{m})=0$, to obtain a contradiction from Lemma
2.2 for the case under consideration, we need to show that
$\lambda<\frac{1}{k}.$ For this purpose, it is enough to show that
$v_{p}(b_{0})=1$, because it follows from (3.1), the fact $k\leq m-k$ that
$v_{p}(b_{j})\geq 1>1-(j/k)$ for $1\leq j\leq k$, and clearly
$v_{p}(b_{j})>1-(j/k)$ for $k<j\leq m$.

To show $v_{p}(b_{0})=1$, it can be easily checked using the fact $p\geq
2vm/3$ and $m$ sufficiently large that

$2p>vm+u\geq v(m-j)+u\geq v+u>-p~{}~{}\mbox{ for }~{}~{}0\leq j\leq m-1.$

As indicated earlier, none of the $v(m-j)+u$ can be zero. Hence, $p$ itself is
the only multiple of $p$ among the number $v(m-j)+u$ with $0\leq j\leq m-1$.
Since $p\nmid v$ and $b_{0}=((vm+u)(v(m-1)+u)\cdots(v+u))/v^{m}$, we obtain
$v_{p}(b_{0})=1$. Hence, for $m$ sufficiently large, $f(x)$ can not have a
factor in ${\mathbf{Z}}[x]$ with degree lying in the interval
$[k\deg\phi(x),(k+1)\deg\phi(x))$ for
$m^{\frac{11}{20}+\varepsilon}<k\leq\frac{m}{2}$.

Case (ii): $k_{0}\leq k\leq m^{\frac{11}{20}+\varepsilon}$ with
$k_{0}=k_{0}(u,v)$ a sufficiently large integer.

We wish to show that $f(x)$ does not have a factor in ${\mathbf{Z}}[x]$ with
degree lying in the interval $[k\deg\phi(x),(k+1)\deg\phi(x))$ for
$k\in[k_{0},m^{\frac{11}{20}+\epsilon}]$.

Let $z=k\sqrt{\log k}.$ We claim that there is a prime $p>z$ that divides
$v(m-j)+u$ for some $j\in\\{0,1,2....,k-1\\}$. Then $(\ref{theq1})$ follows as
before, and we will obtain a contradiction to Lemma 2.2 by showing
$v_{p}(b_{j})>v_{p}(b_{0})-(j/k)$ for $1\leq j\leq m$ as this implies the
slope of the right-most edge of the $\phi$-Newton polygon of $g(x)$ with
respect to $p$ has slope $<1/k$.

Let

$T=\\{v(m-j)+u:0\leq j\leq k-1\\}.$

Since $m$ is large, we deduce that each elements of $T$ is greater than or
equal to $m/2$. Also, observe that $\gcd(u,v)=1$ implies that each element of
$T$ is relatively prime to $v$. For each prime $p\leq z$, we consider an
element $a_{p}\in{T}$ with $v_{p}(a_{p})$ as large as possible. Then we
consider the set

$S=T-\\{a_{p}:p\nmid v,~{}p\leq z\\}.$

Clearly $|S|\geq k-\pi(z).$ By the well-known Chebyshev bound, we have
$|S|\geq k-\frac{2k}{\sqrt{\log k}}.$

Consider the prime $p\leq z$ with $p$ not dividing $v$, and let
$r=v_{p}(a_{p}).$ By definition of $a_{p}$, if $j>r$, then there are no
multiples of $p^{j}$ in $T$ (and, hence, in $S$). For $1\leq j\leq r$, there
are less than or equal to $[k/p^{j}]+1$ multiples of $p^{j}$ in $T$ and,
hence, at most $[k/p^{j}]$ multiples of $p^{j}$ in $S$. Therefore,

$v_{p}\bigg{(}\prod\limits_{s\in
S}s\bigg{)}\leq\sum\limits_{j=1}^{r}\bigg{[}\frac{k}{p^{j}}\bigg{]}\leq
v_{p}(k!),$

and hence

$\prod\limits_{s\in S}\prod\limits_{p\leq z}p^{v_{p}(s)}\leq k!\leq k^{k}.$
(3.2)

On the other hand using the fact that $k\leq m^{\frac{11}{20}+\varepsilon}$,
we see that

$\prod\limits_{s\in
S}s\geq\bigg{(}\frac{m}{2}\bigg{)}^{|S|}\geq\bigg{(}\frac{k^{\frac{29}{20}-\varepsilon}}{2}\bigg{)}^{|S|}.$

Recalling our bound on $|S|$, we obtain

$\displaystyle\log\bigg{(}\prod\limits_{s\in S}{s}\bigg{)}$
$\displaystyle\geq\bigg{(}k-\frac{2k}{\sqrt{\log
k}}\bigg{)}\bigg{(}\bigg{(}\frac{29}{20}-\varepsilon\bigg{)}\log k-\log
2\bigg{)}$ $\displaystyle\geq k\log
k+\bigg{(}\frac{9}{20}-\varepsilon\bigg{)}k\log k+O(k\sqrt{\log k}).$

Since $k\geq k_{0}$ where $k_{0}$ is sufficiently large and using (3.2), we
have

$\log\bigg{(}\prod\limits_{s\in S}s\bigg{)}>k\log
k\geq\log\bigg{(}\prod\limits_{s\in S}\prod\limits_{p\leq
z}p^{v_{p}(s)}\bigg{)}.$

It follows that there is a prime $p>z$ that divides some element of $S$ and,
hence, divides some element of $T$. This proves our claim.

Fix a prime $p>z$ that divides an element in $T$. Fix $j\in\\{1,2,...,m\\}$.
It only remains to show that $v_{p}(b_{0})-v_{p}(b_{j})<\frac{j}{k}$. Observe
that

$\displaystyle v_{p}(b_{0})-v_{p}(b_{j})$ $\displaystyle\leq
v_{p}((vj+u)(v(j-1)+u)\cdots(v+u))$ $\displaystyle\leq
v_{p}((vj+|u|)!)<\frac{vj+|u|}{p-1}.$

Since $p>z=k\sqrt{\log k}$ and $k\geq k_{0},$ we deduce that
$(vj+|u|)/(p-1)<j/k$ and the inequality
$v_{p}(b_{0})-v_{p}(b_{j})<\frac{j}{k}$ follows. Hence, as indicated at the
beginning of this case, we obtain a contradiction to Lemma 2.2.

Case (iii): $2\leq k<k_{0}.$
By Lemma 3.2 (with $a=v,~{}b=u,~{}c=v,$ and $d=u-v$), the largest prime factor
of the product $(vm+u)(v(m-1)+u)$ tends to infinity. Since $m$ is large, we
deduce that there is a prime $p>(v+|u|)k_{0}$ that divides $(vm+u)(v(m-1)+u).$
The argument now follows as in the previous case. In particular,

$\frac{v_{p}(b_{0})-v_{p}(b_{j})}{j}<\frac{vj+|u|}{j(p-1)}\leq\frac{v+|u|}{p-1}\leq\frac{1}{k_{0}}<\frac{1}{k}~{}\mbox{
for }1\leq j\leq m.$

Hence, in this case, we also obtain a contradiction.

Case (iv): $k=1.$

It only remain to prove that $f(x)$ does not have a factor in
${\mathbf{Z}}[x]$ with degree lying in the interval
$[\deg\phi(x),2\deg\phi(x))$. This will be achieved once we show that there
exists a prime $p>v+|u|$ such that $p|b_{j}$ for $0\leq j\leq m-1$. If $u=0$,
then it is always true for $m\geq 2$. Hence we can assume that $u\neq 0$. In
this situation, from Lemma 3.2, the largest prime factor of $m(vm+u)$ tends to
infinity with $m$. We consider a large prime factor $p$ of this product. Note
this implies $p\nmid v$. As in the previous case, we are through if $p$
divides $vm+u$. So suppose $p|m$. The binomial coefficient $\binom{m}{j}$
appears in the definition of $b_{j}$, and this is sufficient to guarantee that
$v_{p}(b_{j})\geq 1$ and $v_{p}(b_{m-j})\geq 1$ for $1\leq j\leq p-1$. On the
other hand,

$b_{j}=\binom{m}{j}\frac{(vm+u)(v(m-1)+u)\cdots(v(j+1)+u)}{v^{m-j}}.$

For $j\leq m-p$, the numerator of the fraction on the right is a product of
$\geq p$ consecutive terms in the arithmetic progression $vt+u$ with
$\gcd(p,v)=1$; thus, $v_{p}(b_{m-j})\geq 1$ for $j\geq p$. This implies that
$(\ref{theq1})$ holds with $k=1$. It follows, along the lines of the previous
two cases, that $v_{p}(b_{0})-v_{p}(b_{j})<\frac{j}{k}$ for $1\leq j\leq m$. A
contradiction to Lemma 2.2 is again obtained.

Therefore combining Cases (i)-(iv), we have shown that for
$k\in[1,\frac{m}{2}]$ and sufficiently large $m$, $f(x)$ can not have a factor
in ${\mathbf{Z}}[x]$ with degree lying in the interval
$[k\deg\phi(x),(k+1)\deg\phi(x))$. This completes the proof of the theorem. ∎

##  4\. Proof of Theorem 1.2

We shall first state two results which will be required for the proof of our
theorem.

######  Lemma 4.1.

For integers $m$ and $k$ with $m\geq k\geq 2$, let $P(m,k)$ denote the product
$(m+1)(m+2)\cdots(m+k)$. For $t\geq 1$, define $S_{t}$ as the set of pairs
$(m,k)$ for which $m\geq k\geq 2$ and the largest prime factor of $P(m,k)$ is
$k+t$. Then

$S_{1}=\\{(2,2),(7,2)\\},~{}~{}~{}S_{2}=\\{(3,3),(7,3),(5,5)\\},$
$S_{3}=\\{(3,2),(4,2),(8,2),(14,2),(23,2),(79,2),(4,4),(5,4),(6,4)\\},$

and

$S_{4}=\\{(4,3),(5,3),(6,3),(13,3),(47,3)\\}.$

It may be pointed out that, in the above lemma, for every $(m,k)\not\in
S_{1}\cup S_{2}\cup\cdots\cup S_{t-1}$ with $m\geq k\geq 2$, the largest prime
factor of $P(m,k)$ is $\geq k+t.$

The proof of the above lemma relies on a method due to Lehmer [20](for
classifying all the cases where $(m+1)(m+2)$ has all its prime factors bounded
below by a prescribed bound) and a result of Ecklund et. al. [4].

The next lemma is on solutions of some equations. The assertion $(i)$ is a
special case of Catalan’s conjecture, now Mihailescu’s Theorem when $r>1,s>1$,
see [11]. The case $r=1$ or $s=1$ is immediate. The assertions $(ii)$ and
$(iii)$ are due to Nagell [21]. For assertions $(iv)-(vi)$, see [19, Lemma 4].

######  Lemma 4.2.

Let $r>0,s>0,t>0$ be integers. The solutions of the following equations are
given by

|  Equations |  Solutions
---|---|---
$(i)$ |  $a^{r}-b^{s}=\pm 1,a,b\in\\{2,3,5\\}$ |  $3-2=1,2^{2}-3=1,5-2^{2}=1,3^{2}-2^{3}=1$
$(ii)$ |  $2^{r}+3^{s}=5^{t}$ |  $2+3=5,2^{4}+3^{2}=5^{2}$
$(iii)$ |  $2^{r}+5^{s}=3^{t}$ |  $2+1=3,2+25=27,4+5=9,8+1=9$
$(iv)$ |  $2^{r}3^{s}-5^{t}=\pm 1$ |  $2\cdot 3-5=1,2^{3}\cdot 3-5^{2}=-1$
$(v)$ |  $3^{r}5^{s}-2^{t}=\pm 1$ |  $3\cdot 5-2^{4}=-1$
$(vi)$ |  $2^{r}5^{s}-3^{t}=\pm 1$ |  $2\cdot 5-3^{2}=1,2^{4}\cdot 5-3^{4}=-1$

###### Proof of Theorem 1.2.

Since $\alpha\in\\{0,1,2,3,4\\}$, we have $u\in\\{0,1,2,3,4\\}$ and $v=1$. We
define $f(x)$ and $g(x)$ as we did in the proof of Theorem 1.1. As there
always exists a prime $p$ dividing $m+u$ except $(m,u)=(1,0)$, it follows from
the second paragraph of the proof of Theorem 1.1 that $f(x)$ can not have a
non-constant factor over ${\mathbf{Z}}$ with degree less than $\deg\phi(x).$

Therefore, it is enough to prove that $f(x)$ does not have a factor in
${\mathbf{Z}}[x]$ with degree lying in the interval
$[k\deg\phi(x),(k+1)\deg\phi(x))$ with $k\in[1,\frac{m}{2}]$ and $m\geq 2$. We
divide our remaining proof into two cases depending on whether $k\geq 2$ or
$k=1.$

Suppose first that $k\geq 2$ and $m\geq 2.$ In the notaion of Lemma 4.1,
observe that $P(m-k+u,k)$ divides $b_{j}$ for $0\leq j\leq m-k.$ Since
$m-k+u\geq m-k\geq k,$ Lemma 4.1 implies that $P(m-k+u,k)$ has a prime divisor
$p\geq k+u+1$ unless

$(m-k+u,k)\in S=S_{1}\cup S_{2}\cup\cdots\cup S_{u},$ (4.1)

where $S_{i}$ is as given in Lemma 4.1 for $1\leq i\leq 4$. We first assume
that $(m-k+u,~{}k)\notin S$ and fix a prime $p$ with $p\geq k+u+1$ as above.
Since $p$ divides $P(m-k+u,k)$, we have $p$ divides $b_{j}$ for $0\leq j\leq
m-k.$ We wish to point here that if there exists a prime $p\geq u+2$ with
$p|b_{j}$ with $0\leq j\leq m-1$, then all the following arguments will work
even for $k=1.$ By hypothesis $p\nmid b_{m}$. For using Lemma 2.2, we are left
with verifying that the slope of the right-most edge of the $\phi$-Newton
polygon of $g(x)$ with respect to $p$ has slope $<\frac{1}{k}$ for $k\geq 2$.
The slope of the right-most edge is given by

$\max\limits_{1\leq j\leq
m}\bigg{\\{}\frac{v_{p}(b_{0})-v_{p}(b_{j})}{j}\bigg{\\}}.$

Observe that

$\frac{b_{0}}{b_{j}}=\frac{(j+u)(j-1+u)\cdots(1+u)}{\binom{m}{j}}.$ (4.2)

It follows that

$v_{p}(b_{0})-v_{p}(b_{j})\leq v_{p}((j+u)(j-1+u)\cdots(1+u))\leq
v_{p}((j+u)!).$

If $j\leq k,$ then $p\geq k+u+1$ implies that $v_{p}(b_{0})-v_{p}(b_{j})\leq
0.$ If $j>k,$ then using the simple inequality $(j+\alpha)/(k+\alpha)<j/k$, we
obtain

$v_{p}(b_{0})-v_{p}(b_{j})\leq
v_{p}((j+\alpha)!)<\frac{j+\alpha}{p-1}\leq\frac{j+\alpha}{k+\alpha}<\frac{j}{k}.$

We combine the above to deduce, as desired, that the right-most edge has slope
$<\frac{1}{k}.$ Hence, using Lemma 2.2, $f(x)$ can not a factor in
${\mathbf{Z}}[x]$ with degree lying in the interval
$[k\deg\phi(x),(k+1)\deg\phi(x))$ with $k\in[2,\frac{m}{2}]$, unless
$(m-k+u,k)\in S.$ Now suppose $(m-k+u,k)\in S$. The following table provides
us all the exceptional cases corresponding to each $u$ with $2k\leq m$.

$u$ |  $(m,k)$
---|---
1 |  $(8,2)$
2 |  $(7,2),(8,3)$
3 |  $(6,2),(7,3),(7,2),(13,2),(22,2),(78,2)$
4 |  $(5,2),(6,2),(12,2),(21,2),(77,2),(6,3),(12,3),(46,3)$

For all these pairs $(m,k)$ except for $(m,k,u)\in\\{(6,2,4),(6,3,4)\\}$, we
now provide a prime number $p$ (see the following tables) such that $p$
divides $b_{j}$ for $0\leq j\leq m-k$ and the right-most edge has slope
$<\frac{1}{k},$ and hence we are done for all these cases in view of Lemma
2.2.

$u$ |  $k$ |  $m$ |  $p$
---|---|---|---
$1$ |  $2$ |  $8$ |  $7$
$2$ |  $2$ |  $7$ |  $7$
$2$ |  $3$ |  $8$ |  $7$
$3$ |  $2$ |  $6$ |  $5$
$3$ |  $3$ |  $7$ |  $7$
$3$ |  $2$ |  $7$ |  $7$
$u$ |  $k$ |  $m$ |  $p$
---|---|---|---
$3$ |  $2$ |  $13$ |  $13$
$3$ |  $2$ |  $22$ |  $7$
$3$ |  $2$ |  $78$ |  $7$
$4$ |  $2$ |  $5$ |  $11$
$4$ |  $2$ |  $6$ |  $--$
$4$ |  $2$ |  $12$ |  $11$
$u$ |  $k$ |  $m$ |  $p$
---|---|---|---
$4$ |  $2$ |  $21$ |  $7$
$4$ |  $2$ |  $77$ |  $7$
$4$ |  $3$ |  $6$ |  $--$
$4$ |  $3$ |  $12$ |  $11$
$4$ |  $3$ |  $46$ |  $23$

Now we deal with the case when $k=1$, $m\geq 2$. We divide this case into five
sub-cases according to values of $u$. We first recall, as pointed out in the
previous case, that if there exists a prime $p\geq u+2$ such that $p|b_{j}$
for $0\leq j\leq m-1$, then we are done in this case in view of Lemma 2.2.

Subcase (i) $u=0$. In this situation, note that $k+u+1=2$ and there is clearly
a prime $p\geq 2$ dividing $m$ for $m\geq 2$ and hence, in view of Lemma 2.2,
$f(x)$ can not have a factor in ${\mathbf{Z}}[x]$ with degree lying in the
interval $[\deg\phi(x),2\deg\phi(x))$. So, we are done.

Subcase (ii) $u=1$. This subcase follows by observing that there always exists
a prime $p\geq 3(=u+2)$ dividing $m(m+1)$ for $m\geq 2$.

Subcase (iii) $u=2$. Observe that $m(m+2)$ always divides $b_{j}$ for $0\leq
j\leq m-1$ for $m\geq 3$. Note that $m=2$ is an exceptional case. Keeping in
mind Lemma 4.2, one can easily verify that there always exists a prime $p\geq
5$ dividing $m(m+2)$ unless $m\in\\{4,6,16\\}$.

Let $m\in\\{4,6,16\\}$. We set $p$ = $2$ or $3$ according as $m\in\\{4,16\\}$
or $m=6$. One can easily check that $p$ divides $b_{j}$ for $0\leq j\leq m-1$.
Keeping in mind Equation (4.1), one can check that the right-most edge of the
$\phi$-Newton polygon of $g(x)$ with respect to $p$ has slope $<1$. Thus we
are done in view of Lemma 2.2.

Subcase (iv) $u=3$. Note that $m(m+3)$ always divides $b_{j}$ for $0\leq j\leq
m-1$ for $m\geq 2$. Keeping in mind Lemma 4.2, it can be easily seen that
there always exists a prime $p\geq 5$ dividing $m(m+3)$ unless
$m\in\\{3,6,9,24\\}$.

If $m\in\\{3,6,9,24\\}$, then one can check that $3$ divides $b_{j}$ for
$0\leq j\leq m-1$ and the right-most edge of the $\phi$-Newton polygon of
$g(x)$ with respect to $3$ has slope $<1$. So, using again Lemma 2.2, we are
done.

Subcase (v) $u=4$. Using Lemma 4.2, it can be verified that there always
exists a prime $p\geq 7$ dividing $m(m+4)$, and hence $b_{j}$, for $0\leq
j\leq m-1$ unless $m\in\\{2,4,5,8,12,16,20,32,36,60,96,320\\}.$

Consider $p=3$ if $m\in\\{2,8,12,32,36,96\\}$, $p=2$ if $m=16$, and $p=5$ if
$m\in\\{5,20,60,320\\}.$ One can check that $p$ divides $b_{j}$ for $0\leq
j\leq m-1$ and the right-most edge of the $\phi$-Newton polygon of $g(x)$ with
respect to $p$ has slope $<1$. Therefore, using Lemma 2.2, we are done.

This completes the proof of the theorem. ∎

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# Correction of Wall Adhesion Effects in Batch Settling of Strong Colloidal
Gels

D. R. Lester111ph: +61 3 9252 6195<EMAIL_ADDRESS>R. Buscall  CSIRO
Mathematics, Informatics and Statistics, PO Box 56, Highett, Victoria 3190,
Australia  MSACT Research and Consulting, Exeter, United Kingdom  Particulate
Fluids Processing Centre, Dept. of Chemical and Biomolecular Engineering,
University of Melbourne, Victoria 3010, Australia

###### Abstract

The batch settling test is widely used to estimate the compressive rheology of
strongly flocculated colloidal suspensions, in particular the compressive
yield strength and hydraulic permeability. Recently it has been discovered
(D.R. Lester et al, J. Rheol., 58(5):1247–1276, 2014) that wall adhesion
effects in these tests may be significantly greater than previously
appreciated, which can introduce unbounded errors in the estimation of these
rheological functions. Whilst a methodology to solve the underlying static
problem and correct for wall adhesion effects has been developed, this method
is quite complex and unwieldy, involving solution of a 2D hyper-elastic
constitutive model for strong colloidal gels. In this paper we develop a
highly simplified 1D visco-plastic approximation to the hyper-elastic model
which admits analytic expressions for the equilibrium solids concentration
profile and bed height. These expressions facilitate robust estimation of the
compressive yield and wall adhesion strength via nonlinear regression of
experimental data in the presence of small measurement errors.

###### keywords:

colloidal gel, compressive rheology, wall adhesion

††journal: Journal of Non-Newtonian Fluid Mechanics

##  1 Introduction

Solid-liquid separation of strongly flocculated colloidal gels is dictated by
the compressive rheology of these materials, where the relevant rheological
functions (compressive yield stress, hydraulic permeability) govern a wide
range of separation processes, from sedimentation and continuous thickening
through to centrifugation and pressure filtration. Central to characterization
of compressive rheology over a range of solids volume fraction ($\phi$) is the
batch settling test, the transient behaviour of which pertains to the
hydraulic permeability $R(\phi)$, whilst equilibrium behaviour is dictated by
the compressive yield stress $P_{y}(\phi)$. To extract these rheological
functions, a one-dimensional (1D) vertical momentum balance is typically used
[18, 25, 15, 20, 22, 10] to model sedimentation and consolidation of the
suspension toward equilibrium, yielding an inverse problem regarding
estimation of compressive rheology from experimental data.

Whilst several transient hydrodynamic mechanisms compromise the assumption of
1D settling behaviour, including inherent Rayleigh-Taylor instabilities within
the suspension and velocity gradients arising from no-slip wall conditions, at
equilibrium the major challenge arises from adhesion between colloidal
particles and the container wall. This attractive force manifests as the wall
adhesion (shear) strength $\tau_{w}(\phi)$, which, similar to the bulk shear
yield strength, is a strongly increasing function of the solids concentration
$\phi$. As equilibrium data (either in the form of solids volume fraction
profiles or a series of bed heights) is used to determine the compressive
yield stress, which in turn facilitates deconvolution of the hydraulic
permeability from transient data, it is of paramount importance that this
equilibrium state is modelled correctly.

Previous studies of the wall adhesion strength [25] showed that for the
electrolytically coagulated colloidal suspensions tested, $\tau_{w}$ was small
in comparison to the compressive yield strength $P_{y}(\phi)$, and so wall
adhesion effects in wider columns are negligible, as reflected by the 1D
equilibrium momentum balance (assuming $\phi$ is constant radially),

$\frac{dP_{y}}{d\phi}\frac{d\phi_{\infty}}{dz}-\Delta\rho
g\phi_{\infty}+\frac{2\tau_{w}(\phi)}{R}=0,$ (1)

where $z$ is the vertical bed depth (downwards from the suspension/supernatant
interface), $\phi_{\infty}$ is the equilibrium solids concentration,
$\Delta\rho$ the interphase density difference, $g$ gravitational acceleration
constant, and $R$ is the radius of the batch settling column. As the apparent
wall adhesion strength $\tau_{w}(\phi)$ typically appears [30, 6, 1] to be of
the same order as the bulk suspension shear yield strength $\tau_{y}(\phi)$,
and the ratio of shear to compressive yield strength
$S(\phi):=\tau_{y}(\phi)/P_{y}(\phi)$ appears to vary over the range 0.001-0.2
[4, 5, 9, 34, 7], the 1D approximation at equilibrium appears to be generally
sound for all but narrow settling columns. Hence wall adhesion effects have
been largely ignored over the past half century.

Recently, it has been established [21] that wall adhesion effects may be much
more significant and prevalent than previously appreciated, especially for
colloidal suspensions flocculated with high molecular-weight polymer
flocculants which exhibit shear/compressive strength ratios of order 0.1.
These suspensions, along with many coagulated systems, are both strongly
cohesive and adhesive, and so readily adhere to settling column walls. For
such materials it has been shown that impractically wide settling columns
($R\sim$ 1[m]) are required to render wall adhesion effects negligible, whilst
for narrower columns, wall adhesion effects can introduce very serious errors
in estimates of the compressive yield strength. When wall adhesion effects are
significant the particulate network experiences a combination of shear and
compressive stresses. From (1), the wall adhesion stress acts to counteract
the interphase gravitational force up to the point that the entire suspension
weight can be supported by wall shear stress alone. This state corresponds to
the critical solids concentration $\phi_{c}$

$\tau_{w}(\phi_{c})\approx\frac{1}{2}\Delta\rho g\phi_{c}R,$ (2)

which represents an asymptotic limit for the equilibrium solids volume
fraction profile $\phi_{\infty}(z)$. As such, the criterion upon which the
relative magnitude of wall adhesion effects should be judged is not simply the
ratio $S(\phi)$, but rather the magnitude of the wall adhesion strength
$\tau_{w}(\phi)$ with respect to the wall shear stress $\frac{1}{2}\Delta\rho
g\phi R$, where over a range of $\phi$ the former term must be much smaller
than the latter for the 1D approximation to hold.

Hence wall adhesion effects can be highly significant for both polymer
flocculated and electrolytically coagulated strong colloidal gels, and for
other types of gel when the particle size is small, especially in the nano-
scale range, since the strength ratio $S(\phi)$ is expected to increase with
decreasing particle size. In general is it almost impossible to detect the
presence or magnitude of wall adhesion errors from the equilibrium solids
volume fraction profile alone. The only clear-cut case is that where the
asymptotic limit
$\lim_{z\rightarrow\infty}\phi_{\infty}(z)\rightarrow\phi_{c}$ is approached,
and in this case errors arising from wall adhesion effects are unbounded as
the corresponding predicted compressive yield stress diverges. As such, there
is a clear and direct need to develop methods to detect and correct for wall
adhesion effects in equilibrium batch settling data.

In a previous study [21] we developed and validated a mathematical model of
the suspension equilibrium stress state in the presence of wall adhesion, and
developed error estimates for the uncorrected compressive yield strength.
Analysis of the governing multidimensional force balance and suspension
behaviour under arbitrary tensorial loadings raised fundamental questions
regarding the constitutive modeling of strongly flocculated colloidal
suspensions, particularly the relative validity and utility of both visco-
plastic and visco-elastic rheological models for strong colloidal gels. A
solution method was developed based upon the 2D static problem relating to the
equilibrium stress balance which facilitated accurate estimates of the
compressive and shear yield strengths from equilibrium solids volume fraction
profiles. Whilst successful, this method is somewhat unwieldy in that it
involves iterated numerical solution of the governing hyper-elastic
constitutive model which is both computationally intensive and algorithmically
complex.

What is required is a more simple and robust means of accurately estimating
the compressive and shear yield strengths from equilibrium batch settling
data, such that analysis of experimental data can be performed routinely. In
this paper we develop such a method using a highly simplified 1D analytic
solution for the equilibrium solids concentration profile based upon a visco-
plastic constitutive model. The validity of this approximate solution is
tested and justified against both the full hyper-elastic solution and
experimental data, and methods for estimation of the compressive rheological
functions from either solids volume fraction profile data or multiple
equilibrium height data are developed and tested.

In the following Section we briefly review the equations governing multi-
dimensional batch settling along with the visco-elastic and visco-plastic
constitutive models for the the rheology of strong colloidal gels. In Section
3 we develop and validate a simplified 1D model based on the visco-plastic
constitutive model, and in Section 4 this model is applied to equilibrium
solids volume fraction profile data to validate the method. This method is
then extended to equilibrium height data in Section 5, and a sensitivity
analysis is performed on this method prior to conclusions being made in
Section 6.

##  2 Batch Settling Modelling of Strong Colloidal Gels

###  2.1 Governing Equations

Prior to the development of simplified models for solution of the equilibrium
batch settling problem in Section 3, we briefly review governing equations and
constitutive approaches. Although this study is focussed on the batch settling
experiment under equilibrium conditions, the integral hyper-elastic
constitutive model is couched in terms of the deformation history of the
suspension, and so this model requires evolution of the initial condition
$\phi=\phi_{0}<\phi_{g}$ (where $\phi_{g}$ is the suspension gel point) to the
equilibrium state. Under the assumption that during sedimentation the bulk
suspension velocity $\mathbf{q}:=\mathbf{v}_{s}\phi+\mathbf{v}_{f}(1-\phi)$ is
zero (where $\mathbf{v}_{s}$, $\mathbf{v}_{f}$ respectively are the solid and
fluid phase velocities), the inter-phase force balance over the particulate
network [23] simplifies to

$\frac{\partial\phi}{\partial
t}+\nabla\cdot\frac{(1-\phi)^{2}}{R(\phi)}\left(\nabla\cdot\Sigma^{N}+\Delta\rho\mathbf{g}\phi\right)=0,$
(3)

where $\Sigma^{N}=-p_{N}\mathbf{I}+\bm{\sigma}_{N}$ is the network stress
tensor, defined [2] as the difference between the total suspension stress
$\Sigma$ and the fluid stress $\Sigma^{f}$:

$\Sigma^{N}:=\Sigma-\Sigma^{f}.$ (4)

At long times the batch settling transient solution (3) converges to the
equilibrium state $\phi_{\infty}$, where the solid and fluid fluxes decay to
zero, yielding a balance between the network stress gradient and gravitational
force as

$\nabla\cdot\Sigma^{N}+\Delta\rho\mathbf{g}\phi_{\infty}=0.$ (5)

Given appropriate boundary conditions, this equation is capable of describing
the batch settling experiment under equilibrium conditions in the presence of
wall adhesion effects. To close both the transient (3) and equilibrium (5)
equations, an appropriate constitutive model for strong colloidal gels is
required for the network stress tensor $\Sigma^{N}$.

###  2.2 Visco-plastic Constitutive Model

Strong colloidal gels are mostly commonly modelled as visco-plastic materials
(such as Bingham or Herschel-Bulkley models) as they are generally brittle in
shear, with typical yield strains of the order 0.01-1%. For the present
purposes, the yield strain can be taken to mean the apparent yield strain,
given by the ratio of yield stress to linear shear modulus. Similarly,
compression is modelled as a self-limiting visco-plastic process in terms of
the compressive yield stress $P_{y}(\phi)$. Whilst these idealized models only
capture the gross rheology, they are simple and tractable, and allow the
suspension network pressure $p_{N}$ and deviatoric stress
$\tau_{N}=|\bm{\sigma}_{N}|$ to be quantified directly in terms of the
compressive and shear yield strengths as

$\displaystyle p_{N}\approx P_{y}(\phi),\,\,\frac{D_{s}\phi}{Dt}\geqslant 0,$
(6)
$\displaystyle\tau_{N}\approx\left(\frac{\tau_{y}(\phi)}{\dot{\gamma}}+\eta(\phi,\dot{\gamma})\right)\dot{\gamma}\,\,\,\text{for}\,\,\tau_{N}\geqslant\tau_{y}(\phi),$
(7)

where $\frac{D_{s}}{Dt}$ is the material derivative with respect to the solid
phase, where $\tau_{N}$ is the 2nd invariant of $\bm{\sigma}_{N}$,
$\eta(\phi,\dot{\gamma})$ is the apparent suspension viscosity (which is
typically non-Newtonian), and $\dot{\gamma}$ is the rate of shear strain.

###  2.3 Hyper-elastic Constitutive Model

However, as recent studies [31, 24, 12, 29, 16, 11, 27, 28, 26, 8, 32, 17, 33,
14] have established, the true nature of yield of these materials is more
somewhat more complicated than is captured by visco-plastic constitutive
models, and rather strong colloidal gels act as visco-elastic materials which
undergo rapid strain hardening in compression and can undergo rapid strain
softening and/or hardening in shear. As such, a more consistent and accurate
way to describe strong colloidal gels is as visco-elastic media with strongly
nonlinear shear modulus $G$ and bulk modulus $K$. Under this hyper-elastic
formulation, the network pressure $p_{N}$ and deviatoric stress
$\bm{\sigma}_{N}$ are given by the integral evolution equations [21]

$\displaystyle p_{N}=\int_{\phi_{0}}^{\phi(t)}K(\phi)d\ln\phi,$ (8)
$\displaystyle\bm{\sigma}_{N}=\int_{-\infty}^{t}\frac{\partial
G(\phi,\gamma,t-s)}{\partial s}\bm{\Phi}(s)ds,$ (9)

where $\bm{\Phi}$ is the deviatoric component of the Hencky strain tensor
$\mathbf{H}$ for the solid phase. Due to the integral nature of the hyper-
elastic constitutive model, the initial state $\phi(\mathbf{x},0)=\phi_{0}$
must be evolved via the transient equation (3) toward the equilibrium state
$\phi_{\infty}(\mathbf{x})$. Conversely, the visco-plastic model does not
require the full deformation history, and so acts directly on the equilibrium
force balance (5).

Hence the visco-plastic constitutive model represents an simplified
approximation to the full hyper-elastic constitutive model which is less
computationally intensive, but it also suffers from the drawback that the
stress-strain relationship is ill-defined prior to yield. In the case of
multi-dimensional batch settling, this corresponds to a statically
indeterminate equilibrium stress state which requires special treatment to
resolve.

###  2.4 Closure of Viscoplastic Model

To resolve this statically indeterminate state, the assumption of negligible
normal stress differences

$\displaystyle N_{1}$ $\displaystyle:=\Sigma^{N}_{zz}-\Sigma^{N}_{rr}=0,$ (10)
$\displaystyle N_{2}$
$\displaystyle:=\Sigma^{N}_{rr}-\Sigma^{N}_{\theta\theta}=0,$ (11)

was invoked [21] to close the equilibrium stress state, and it was found that
under this closure the visco-plastic solution closely approximates the full
hyperelastic solution. Whilst this closure may not be appropriate in general,
i.e. strong colloidal gels may exhibit significant normal stress differences
in other applications, these materials under static equilibrium do not exhibit
such effects. In conjunction with this closure, the assumption of axisymmetry
within a cylindrical batch settling column simplifies the network stress
tensor to

$\Sigma^{N}=\left(\begin{array}[]{ccc}\Sigma^{N}_{rr}&\Sigma^{N}_{r\theta}&\Sigma^{N}_{rz}\\\
\Sigma^{N}_{r\theta}&\Sigma^{N}_{\theta\theta}&\Sigma^{N}_{\theta z}\\\
\Sigma^{N}_{rz}&\Sigma^{N}_{\theta z}&\Sigma^{N}_{zz}\\\
\end{array}\right)=\left(\begin{array}[]{ccc}p_{N}&0&\tau_{N}\\\ 0&p_{N}&0\\\
\tau_{N}&0&p_{N}\\\ \end{array}\right),$ (12)

where $p_{N}=\frac{1}{3}\text{tr}(\Sigma^{N})$,
$\tau_{N}=|\bm{\sigma}^{N})|=|\Sigma^{N}_{rz}|$ are the isotropic and
deviatoric network stresses respectively. Under these simplifications, and
ignoring the angular components $\theta$, the equilibrium network force
balance reduces to

$\displaystyle\frac{\partial p_{N}}{\partial
r}+\frac{\partial\tau_{N}}{\partial z}$ $\displaystyle=0,$ (13)
$\displaystyle\frac{1}{r}\frac{\partial}{\partial
r}\left(r\tau_{N}\right)+\frac{\partial p_{N}}{\partial z}+\Delta\rho
g\phi_{\infty}$ $\displaystyle=0,$ (14)

which applies over the suspension domain
$\mathcal{D}:\\{r,z\\}\in[0,R]\times[0,h_{\infty})$, where $h_{\infty}$ is the
equilibrium suspension height. If the initial solids volume fraction
$\phi_{0}$ is less than the suspension gel point $\phi_{g}$, then at
equilibrium the network pressure $p_{N}$ is everywhere equal to the
compressive yield strength

$p_{N}(\mathbf{x})=P_{y}(\phi(\mathbf{x})_{\infty})\forall\mathbf{x}\in\mathcal{D}.$
(15)

The hyperbolic system (13), (14) is also subject to the boundary conditions

$\displaystyle p_{N}|_{z=0}$ $\displaystyle=0,$ (16)
$\displaystyle\tau_{N}|_{z=0}$ $\displaystyle=0,$ (17)
$\displaystyle\tau_{N}|_{r=0}$ $\displaystyle=0,$ (18)
$\displaystyle\tau_{N}|_{r=R}$ $\displaystyle=\tau_{y}(\phi|_{r=R}),$ (19)

where the critical boundary condition (19) may be recast via (15) as

$\tau_{N}|_{r=R}=F_{1}(p_{N}|_{r=R}),$ (20)

and the function $F_{1}$ reflects the relationship between the compressive
yield strength $P_{y}(\phi)$ and shear yield strength $\tau_{y}(\phi)$ at a
given volume fraction $\phi$.

Based upon physical arguments and experimental data, [3] proposes a
relationship for the ratio $S(\phi)=\tau_{y}(\phi)/P_{y}(\phi)$ between the
shear and compressive yield strength of a particulate gel, which rapidly
decreases from around 1 near the gel point $\phi_{g}$ to the asymptotic value
$S_{\infty}$ with increasing $\phi$. For a compressive yield strength function
of the form

$P_{y}(\phi)=k\left(\left(\frac{\phi}{\phi_{g}}\right)^{n}-1\right),$ (21)

the asymptotic value is

$S_{\infty}=\kappa n\gamma_{c}.$ (22)

where $\gamma_{c}$ is the critical shear strain and $\kappa$ the ratio of
shear to compressive moduli, which is related to the the Poisson ratio $\nu$
as

$\kappa=\frac{2}{3}\left(\frac{1-\nu}{1-2\nu}\right),$ (23)

where $\nu=3/8$, $\kappa=5/3$ for systems bound by central forces as per
Cauchy’s relationships. For a compressive yield strength with the functional
form (21), $S(\phi)$ is then

$S(\phi)=\left(\left(\frac{1}{S_{\infty}}-1\right)\left(1-\left(\frac{\phi}{\phi_{g}}\right)^{-n}\right)+1\right)^{-1},$
(24)

and $F_{1}$ in the wall boundary condition (19) simplifies to

$\displaystyle
F_{1}(p_{N})=\frac{p_{N}}{\left(\frac{1}{S_{\infty}}+1\right)\left(\frac{p_{N}}{p_{N}+k}\right)+1}.$
(25)

The hyperbolic system (13) (14) manifests as a pair of characteristics which
propagate down the bed at approximately 45 degrees to the container wall and
are reflected off the container wall and symmetry axis as per the reflection
conditions (18), (19). In terms of the shear stress distribution, these
characteristics manifest as $C^{1}$ shocks (i.e. non Lifshitz-continuous) in
the shear stress distribution, as shown in Fig 1(a), which are clearly non-
physical. These artifacts are a direct result of the hyperbolic nature of the
visco-plastic constitutive model, and call into question its validity for
multi-dimensional static problems.

|  |
---|---|---
(a) | (b) | (c)
Figure 1: Comparison of (a) network shear stress $\tau_{N}$, (b) network
pressure $p_{N}$ and solids volume fraction $\phi_{\infty}$ distributions for
visco-plastic constitutive model.

Conversely, the stress distribution for the hyperelastic solution (see [21]
for details) shown in Fig. 2(a) has the same gross form as the visco-plastic
solution, but the $C^{1}$ shocks are smoothed out, which appears to be a more
physically realistic prediction. Whilst the shear stress distribution appears
to be markedly different between the visco-plastic and hyper-elastic models,
the corresponding pressure and solids volume fractions are remarkably similar
(Fig. 1(b)-(c), Fig. 2(b)-(c)). As such, despite the $C^{1}$ shocks in the
shear stress distribution, the visco-plastic constitutive model generates a
highly accurate (within 1-2% relative error) approximation to the equilibrium
solids volume fraction distribution.

|  |
---|---|---
(a) | (b) | (c)
Figure 2: Comparison of (a) network shear stress $\tau_{N}$, (b) network
pressure $p_{N}$ and solids volume fraction $\phi_{\infty}$ distributions for
hyper-elastic constitutive model.

##  3 Analytic 1D Approximation

Whilst the visco-plastic model provides a good approximation to the full
hyper-elastic solution, and may be solved numerically in a straightforward
manner, no analytic solution is known for the hyperbolic system (13), (14),
hence computationally intensive numerical iteration is still required to fit
experimental data. As we seek a simplified analytic solution which can be
employed routinely to analyse equilibrium batch sedimentation data, an
accurate 1D approximation to the multi-dimensional problem is sought. To
develop such an approximation, we consider the cross-sectional average

$\bar{f}:=\frac{2}{R^{2}}\int_{0}^{R}rf(r)dr,$ (26)

of (14), yielding the averaged force balance

$\frac{d\bar{p}_{N}}{dz}+\frac{2}{R}\tau_{y}(\phi_{\infty}|_{r=R})-\Delta\rho
g\bar{\phi}_{\infty}=0.$ (27)

This force balance is very similar to (1) proposed by Michaels and Bolger
[25], but it differs in that (27) is exact (up to the assumption that
$\tau_{y}\approx\tau_{w}$), where in general
$\bar{\phi}_{\infty}\neq\phi_{\infty}|_{r=R}$. We express the deviation of
$\phi$ from its cross-sectional average $\bar{\phi}_{\infty}$ as

$\phi_{\infty}=\bar{\phi}_{\infty}-\delta\phi_{\infty},$ (28)

and determine the magnitude of $\delta\phi_{\infty}$ as follows. As the shear
yield strength $\tau_{y}(\phi)$ is significantly smaller than the compressive
yield stress $P_{y}(\phi)$ for volume fractions $\phi$ greater than the gel
point, then typically $\tau_{N}\ll p_{N}$, and so from (14) and boundary
conditions (18), (19), $p_{N}$ only varies radially very weakly, as is shown
in Fig. 1 (b). Furthermore, as $P_{y}(\phi)$ is a strongly increasing (roughly
power-law) function of $\phi$, then $\phi_{\infty}$ varies even more weakly
with radius, hence $\delta\phi_{\infty}$ is considered negligible, as is
reflected in Fig. 1 (c). As such, the approximation

$\phi_{\infty}|_{r=R}\approx\bar{\phi}_{\infty}\approx
P_{y}^{-1}(\bar{p}_{N}),$ (29)

is invoked in (27), yielding an integral equation for the average network
pressure as a function of bed depth $z$ in terms of the rheological functions
$P_{y}(\phi)$, $\tau_{y}(\phi)$ as

$z\approx\int_{0}^{\bar{p}_{N}}\frac{dp^{\prime}}{\Delta\rho
gP_{y}^{-1}(p^{\prime})-\frac{2}{R}F_{1}(p^{\prime})}.$ (30)

Figure 3: Approximation of the function $F_{1}(p)$ (solid) for various values
of $S_{\infty}$ with $S_{\infty}\left(\frac{p}{k}+1\right)$ (solid).

Whilst this integral does not have an analytic solution for the functional
forms of $P_{y}(\phi)$, $\tau_{y}(\phi)$ given by (21), (25), an analytic
solution is generated when $F_{1}(p)$ is approximated by the simple function

$F_{1}(p)\approx S_{\infty}\left(\frac{p}{k}+1\right),$ (31)

which, as shown in Fig. 3, is a good approximation for $p/k>S_{\infty}$. Under
this approximation, (30) is then

$z\approx\int_{0}^{\bar{p}}\frac{dp^{\prime}}{\Delta\rho
g\phi_{g}\left(\frac{p^{\prime}}{k}+1\right)^{\frac{1}{n}}-\frac{2S_{\infty}}{R}\left(\frac{p^{\prime}}{k}+1\right)},$
(32)

which may be integrated analytically to yield approximate expressions for both
the average network pressure $\bar{p}_{N}$

$\bar{p}_{N}(z)\approx k\left[\frac{\Delta\rho
g\phi_{g}R}{2S_{\infty}k}+\left(1-\frac{\Delta\rho
g\phi_{g}R}{2S_{\infty}k}\right)\exp\left(-\frac{n-1}{n}\frac{2S_{\infty}}{R}z\right)\right]^{\frac{n}{n-1}}-k,$
(33)

and equilibrium solids volume fraction $\bar{\phi}_{\infty}$

$\bar{\phi}_{\infty}(z)\approx\phi_{g}\left[\frac{\Delta\rho
g\phi_{g}R}{2S_{\infty}k}+\left(1-\frac{\Delta\rho
g\phi_{g}R}{2S_{\infty}k}\right)\exp\left(-\frac{n-1}{n}\frac{2S_{\infty}}{R}z\right)\right]^{\frac{1}{n-1}},$
(34)

|  |
---|---|---
(a) | (b) | (c)
Figure 4: Comparison of typical predictions of (a) average network pressure
$\bar{p}_{N}$ and (b) average equilibrium solids volume fraction
$\bar{\phi}_{\infty}(z)$ for $R_{s}$=0.011 [m] (gray) and $R_{l}$=0.055 [m]
(black) column widths and (c) error summary for network pressure (dashed) and
solids volume fraction (solid) between 2D visco-plastic and 1D approximate
solutions of equilibrium stress state.

which satisfies
$\lim_{z\rightarrow\infty}\bar{\phi}_{\infty}\rightarrow\phi_{c}$. As shown in
Fig. 4, these expressions represent quite accurate approximations to both the
2D visco-plastic and 2D hyper-elastic solutions, with typical errors for
$\bar{\phi}_{\infty}(z)$ and $\bar{p}_{N}(z)$ less than 1% and 1-5%
respectively. These errors decay with increasing $z$ due to increasing
accuracy of the approximations (29) and (31) with $p_{N}$. These closed-form
1D approximations (33), (34) represent a significant simplification of both
the multi-dimensional visco-plastic and hyper-elastic solutions which appear
to be accurate enough for analysis of experimental data.

##  4 Fitting of Equilibrium Solids Volume Fraction Profiles

Using (33), (34), the inverse problem of estimation of the suspension rheology
from a series of equilibrium solids volume fraction profiles is now greatly
reduced to a simple nonlinear regression, where given such data and
$\Delta\rho$, $g$, $R$, the rheological parameters $\phi_{g}$, $k$, $n$,
$S_{\infty}$ which define the compressive and shear yield strengths can be
directly estimated. To test the accuracy of this method, equation (34) is
fitted to a series of equilibrium solids volume fraction profiles presented in
[21] for three different polymer-flocculated colloidal suspensions. These
suspensions consist of 4 micron calcium carbonate primary particles under
different flocculation conditions as summarized in Table 1 (see [21] for
further details). Equilibrium solids volume fraction profiles for each of
these suspensions were measured using gamma ray attenuation [19] in two
different width columns; a narrow column $R_{s}$=0.011 [m], and a wide column
$R_{l}$=0.055 [m].

|  |
---|---|---
(a) | (b) | (c)
Figure 5: Measured data (points) and model fits (1D model - solid lines, 2D model - dashed lines) of the equilibrium solids volume fraction profile $\bar{\phi}_{\infty}(z)$ for $R_{s}$=0.011 [m] (gray) and $R_{l}$=0.055 [m] (black) column widths for calcium carbonate suspensions under flocculant types and dosages (a)-(c) summarized in Table 1. Suspension | Method |  $\phi_{g}$ [-] |  $k$ [Pa] |  $n$ [-] |  $S_{\infty}$ [-] |  $k/\phi_{g}^{n}$ [Pa]
---|---|---|---|---|---|---
(a) Magnafloc 46 [g/t] | 2D | 0.0918 | 3.211 | 5.482 | 0.1572 | 1.557$\times$106
(a) Magnafloc 46 [g/t] | 1D | 0.0923 | 3.204 | 5.495 | 0.1597 | 1.556$\times$106
(b) Rheomax 30 [g/t] | 2D | 0.1042 | 0.631 | 7.031 | 0.1121 | 4.211$\times$106
(b) Rheomax 30 [g/t] | 1D | 0.1017 | 0.476 | 7.008 | 0.0966 | 4.308$\times$106
(c) Rheomax 46 [g/t] | 2D | 0.0890 | 0.1627 | 7.014 | 0.1132 | 3.805$\times$106
(c) Rheomax 46 [g/t] | 1D | 0.1150 | 0.9395 | 7.001 | 0.1159 | 3.540$\times$106
Table 1: Suspension flocculant type, dosage and fitted rheological parameters.

The measured solids volume fraction profile data and fitted profiles using
both the 1D analytic approximation (34) and 2D visco-plastic model are shown
in Fig. 5, which indicate both methods provide good fits to the experimental
data, as reflected by the fitted rheological parameters shown in Table 1. The
agreement between the fitted parameters is excellent for the Magnafloc
suspension, whereas it is only reasonable for the two Rheomax suspensions. As
shown in Table 1, these poorer fits are attributed to ambiguity in
deconvolution of $k$ and $\phi_{g}$ in the term $k/\phi_{g}^{n}$ from the
functional form of $P_{y}(\phi)$ in (21). This behaviour is typical of
estimation of the compressive yield stress in general, as the suspension gel
point $\phi_{g}$ represents an asymptotic limit of vanishing stress, and hence
it is difficult to predict from equilibrium solids profiles alone [13]. Hence,
whilst the 1D and 2D methods can generate different estimates of the
individual rheological parameters, the estimated rheological functions
$P_{y}(\phi)$, $\tau_{y}(\phi)$ are quantitatively very similar, as shown in
Fig.s 5 and 6.

The fitted rheological parameters can also be used to compare the estimated
shear yield strength (under the assumption that this is similar to the wall
adhesion strength) with _in-situ_ measurements as performed in [21]. A
comparison between the measured data and predictions from the 1D analytic and
2D numerical visco-plastic models is shown in Fig. 6. In this case, the
agreement between the 1D and 2D fitted curves for $\tau_{y}(\phi)$ is very
good for all suspensions tested. These results suggest that the 1D
approximation (34) is capable of generating accurate estimates of the both the
compressive and shear yield strength of colloidal suspensions from equilibrium
solids volume fraction profile data. Note that application of the 1D or 2D
methods to either the narrow or wide column solids volume fraction profile
data alone results in unacceptably large errors in the estimated rheological
parameters, reinforcing the notion that several columns of different width are
required to deconvolute contributions from compression and wall adhesion.

|  |
---|---|---
(a) | (b) | (c)
Figure 6: Comparison between in-situ measurements (points) and model
predictions (1D model - black lines, 2D model - gray lines) of the shear yield
stress $\tau_{y}(\phi)$ for suspensions (a)-(c) summarized in Table 1.

To gauge the sensitivity of the 1D regression method to measurement errors,
Gaussian noise with zero mean and standard deviation
$\sigma_{\bar{\phi}_{\infty}}$ is added to the experimental solids volume
fraction profile data for the Magnafloc suspension shown in Fig. 5(a). Two
different levels of Gaussian noise $\sigma_{\bar{\phi}_{\infty}}$=0.002,
$\sigma_{\bar{\phi}_{\infty}}$=0.005 are used to generate $10^{4}$ random
realizations of experimental data, and the resultant mean $\mu_{\epsilon}$ and
standard deviation $\sigma_{\epsilon}$ of the relative error of the fitted
rheological parameters $\phi_{g}$, $k$, $n$, $S_{\infty}$ are summarized in
Table 2. In both cases the mean $\mu_{\epsilon}$ is small for all of the
rheological parameters, and the standard deviation $\sigma_{\epsilon}$ is
small for all parameters except for the coefficient $k$. Again, this error
appears to be related to the difficulty in deconvoluting the group
$k/\phi_{g}^{n}$, as the relative error for this group (Table 2) is
universally small. This is also reflected in Fig. 7, where the range of
estimates of the compressive yield stress are excellent for both values of
$\sigma_{\bar{\phi}_{\infty}}$, and are much smaller than suggested by the
errors associated with the parameters $k$, $\phi_{g}$ alone.

| $\sigma_{\bar{\phi}_{\infty}}$ | $\phi_{g}$ | $S_{\infty}$ | $k$ | $n$
---|---|---|---|---|---
$\mu_{\epsilon}$ | 0.005 | -0.032% | -0.008% | 2.564% | -0.008%
$\sigma_{\epsilon}$ | 0.005 | 3.681% | 4.727% | 25.56% | 4.728%
$\mu_{\epsilon}$ | 0.002 | -0.015% | 0.004% | 0.356% | 0.004%
$\sigma_{\epsilon}$ | 0.002 | 1.483% | 1.894% | 10.05% | 1.893%
Table 2: Mean and standard deviation of relative error $\epsilon$ in estimates of rheological parameters $\phi_{g}$, $k$, $n$, $S_{\infty}$, based upon Magnafloc data for different values of solids volume fraction error $\sigma_{\bar{\phi}_{\infty}}$. |
---|---
(a) | (b)
Figure 7: (a) Compressive yield strength $P_{y}(\phi)$ curve for suspension
(a) (solid, gray), and 95% confidence intervals for
$\sigma_{\bar{\phi}_{\infty}}$=0.002 (dashed, black) and
$\sigma_{\bar{\phi}_{\infty}}$=0.005 (dashed, gray), and (b) plots of standard
deviation $\sigma_{P_{y}}(\phi)$ of the compressive yield strength for
$\sigma_{\bar{\phi}_{\infty}}$=0.002 (dashed, black) and
$\sigma_{\bar{\phi}_{\infty}}$=0.005 (dashed, gray).

##  5 Fitting of Equilibrium Bed Height Data

In many instances the averaged solids volume fraction profile
$\bar{\phi}_{\infty}(z)$ is not available experimentally, however it is simple
to measure the equilibrium height $h_{\infty}$ over a range of experiments of
different initial suspension height $h_{0}$ and column radius $R$. Estimation
of the compressive yield strength $P_{y}(\phi)$ from equilibrium height data
alone is not new [13], and in the absence of wall adhesion effects, this
procedure is relatively straightforward. In this section we derive
relationships for the equilibrium height $h_{\infty}$ as a function of the
column radius $R$ and the linear solids volume

$m:=\phi_{0}h_{0}=\int_{0}^{h_{\infty}}\bar{\phi}_{\infty}(z)dz,$ (35)

which facilitate estimation of the rheological parameters $\phi_{g}$, $k$,
$n$, $S_{\infty}$ from a set of equilibrium height data. To derive this
relationship, we directly integrate (34) to yield

$\begin{split}m(h_{\infty})=&\frac{\phi_{g}(n-1)}{(q-1)r}\Big{[}-\,_{2}F_{1}\left(1,1,\frac{n-2}{n-1},\frac{q}{q-1}\right)\\\
&+(1+q(e^{h_{\infty}r}-1))(q+(1-q)e^{h_{\infty}r})^{\frac{1}{n-1}}\,_{2}F_{1}\left(1,1,\frac{n-2}{n-1},\frac{qe^{h_{\infty}r}}{q-1}\right)\Big{]},\\\
&\text{where}\,\,\,\,r=\frac{n-1}{n}\frac{2S_{\infty}}{R},\,\,q=\frac{\Delta\rho
q\phi_{g}R}{2S_{\infty}k},\end{split}$ (36)

and $\,{}_{2}F_{1}$ is the hypergeometric series

$\,{}_{2}F_{1}(a,b;c;z):=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}}\frac{z^{n}}{n!},$
(37)

where $(x)_{n}$ is the Pochhammer symbol

$(x)_{n}=\begin{cases}1\quad&n=0,\\\ x(x+1)\cdots(x+n-1)\quad&n>0.\end{cases}$
(38)

To test the accuracy of this method, we generated synthetic equilibrium height
data for all combinations of 5 linear solids volumes
$m=\phi_{0}h_{0}$=0.02-0.1 [m] in steps of 0.02 [m], and across 3 column radii
$R$=(0.02,0.05,0.1)[m] based upon the rheological parameters derived for
Magnafloc in Table 1. A significant number (15) of experiments are required to
generate a dense enough data set to provide reliable estimates of these
parameters, and significantly wide columns are also required to deconvolute
compression and wall adhesion effects. Note that for very dilute suspensions,
the column heights required to achieve $m=\phi_{0}h_{0}\sim 0.1[m]$ may be
very large, however as we are only interesting in equilibrium data, this
problem may be alleviated by continually topping up settled beds and decanting
supernatant until the requisite solids mass has been added, significantly
reducing the maximum required column height.

| $\sigma_{h_{\infty}}$ | $\phi_{g}$ | $S_{\infty}$ | $k$ | $n$ | $k/\phi_{g}^{n}$
---|---|---|---|---|---|---
$\mu_{\epsilon}$ | 0.5 [mm] | -3.665% | 0.041% | 10.60% | -0.001% | 0.247%
$\sigma_{\epsilon}$ | 0.5 [mm] | 17.48% | 4.041% | 74.21% | 1.044% | 6.304%
$\mu_{\epsilon}$ | 0.2 [mm] | -0.079% | 0-0.079% | 3.352% | -0.0193% | -0.051%
$\sigma_{\epsilon}$ | 0.2 [mm] | 4.964% | 1636% | 29.33% | 0.422% | 2.528%
Table 3: Mean and standard deviation of relative error $\epsilon$ in estimates
of rheological parameters $\phi_{g}$, $k$, $n$, $S_{\infty}$, based upon
Magnafloc data for different values of height error $\sigma_{h_{\infty}}$.

To estimate the impact of errors in measuring the equilibrium height
$h_{\infty}$, random Gaussian noise with zero mean and standard deviation
$\sigma_{h_{\infty}}$ was added to the generated height data over $10^{4}$
samples, and the regression estimates recorded. The mean $\mu_{\epsilon}$ and
standard deviation $\sigma_{\epsilon}$ of the relative error $\epsilon$ over
this ensemble is shown in Table 3 for two noise levels
$\sigma_{h_{\infty}}$=0.5[mm], $\sigma_{h_{\infty}}$=0.2[mm], and for the case
$\sigma_{h_{\infty}}=0$ the rheological parameters in Table 1 are recovered to
within machine precision. Whilst there appears to be only moderate skewness in
the parameter estimates, as reflected by the mean $\mu_{\epsilon}$, the spread
of errors as quantified by $\sigma_{\epsilon}$ is large, especially for the
suspension gel point $\phi_{g}$ and compressibility parameter $k$. Again, this
is reflected in the errors for the grouping $k/\phi_{g}^{n}$, which are
significantly smaller that those of $\phi_{g}$ or $k$ alone, and indicate
accurate estimates of the compressive yield strength can be made (Fig. 8)
although the individual parameter estimates may contain significant errors. In
all cases, it is critical to make accurate equilibrium height measurements, as
the difference between $\sigma_{h_{\infty}}=0.5[mm]$ and
$\sigma_{h_{\infty}}=0.2[mm]$ is quite significant.

| $\sigma_{h_{\infty}}$ | $S_{\infty}$ | $k$ | $n$
---|---|---|---|---
$\mu_{\epsilon}$ | 0.5 [mm] | 0.057% | 0.027% | -0.005%
$\sigma_{\epsilon}$ | 0.5 [mm] | 1.542% | 2.995% | 0.555%
$\mu_{\epsilon}$ | 0.2 [mm] | 0.044% | -0.078% | 0.015%
$\sigma_{\epsilon}$ | 0.2 [mm] | 0.608% | 1.189% | 0.220%
Table 4: Mean and standard deviation of relative error $\epsilon$ in estimates of rheological parameters $k$, $n$, $S_{\infty}$, based upon Magnafloc data for different values of height error $\sigma_{h_{\infty}}$, assuming gel point $\phi_{g}$ is known. |
---|---
(a) | (b)
Figure 8: (a) Compressive yield strength $P_{y}(\phi)$ curve for suspension
(a) (solid, gray), and 95% confidence intervals for $\sigma_{h_{\infty}}$=0.2
[mm] (dashed, black) and $\sigma_{h_{\infty}}$=0.5 [mm] (dashed, gray) when
$\phi_{g}$ is unknown, and (b) plots of standard deviation
$\sigma_{P_{y}}(\phi)$ of the compressive yield strength for
$\sigma_{h_{\infty}}$=0.2 [mm] and $\phi_{g}$ unknown (dashed, black),
$\sigma_{h_{\infty}}$=0.5 [mm] and $\phi_{g}$ unknown (dashed, gray),
$\sigma_{h_{\infty}}$=0.2 [mm] and $\phi_{g}$ known (solid, black),
$\sigma_{h_{\infty}}$=0.5 [mm] and $\phi_{g}$ known (solid, gray).

Often the suspension gel point $\phi_{g}$ has been obtained by other means,
which constrains the parameter estimation problem. Such information resolves
the deconvolution problem between $\phi_{g}$ and $k$, and significantly
improves the accuracy of all parameter estimates. Whilst in principle the gel
point can be estimated from equilibrium height data, estimates of $\phi_{g}$
from the column data given tend to over-estimate the gel-point, rather smaller
values of $m$ are required to extrapolate toward the zero-pressure case. The
estimates for when $\phi_{g}$ is known are shown in Table 4, and the relative
errors are significantly reduced, less than 2.5% for the 0.5[mm] error and 1%
for the 0.2[mm] error levels. These results in Fig. 8 suggest that it is
possible to quite accurately estimate the compressive yield strength from
equilibrium height data given careful experimentation and prior knowledge of
the suspension gel point.

##  6 Conclusions

Wall adhesion effects in equilibrium batch settling tests have been shown to
be more prevalent than previously thought for strongly flocculated colloidal
gels, and such artifacts can lead to unbounded errors in estimation of the
compressive rheology of these materials. Indeed, wall effects are always
likely to be important in gravity batch settling for two reasons, firstly, the
effect of shear relative to compressive strength is amplified by a geometric
factor of $2h_{\infty}/R$, and second, the suspension is usually not far from
the gel-point where the ratio $S(\phi)$ of shear (adhesive) to compressive
strength is unity, regardless of how small its asymptotic value $S_{\infty}$
is at high volume-fraction. In this study, we develop and test a 1D
approximate solution to the equilibrium stress state which represents a
significant simplification of previous multi-dimensional solution methods
using either visco-plastic or hyper-elastic constitutive models. The 1D visco-
plastic analytic solution provides accurate estimates (within several percent)
of both the 2D visco-plastic and hyper-elastic solutions, and so is suitable
for direct estimation of both the compressive $P_{y}(\phi)$ and shear
$\tau_{y}(\phi)$ yield strength from equilibrium batch settling data via
nonlinear regression.

Methods have been developed to estimate the shear and compressive rheology
from either several equilibrium solids volume fraction profiles in different
width settling columns, or a series of equilibrium height measurements over a
combination of total solids and column radii. For equilibrium solids
concentration profile data, accurate estimates of both $P_{y}(\phi)$ and
$\tau_{y}(\phi)$ are possible, given the local cross-sectionally average
equilibrium solids volume fraction $\bar{\phi}_{\infty}$ can be measured
accurately, however there does exist an outstanding issue with respect to
deconvolution of the parameter group $k/\phi_{g}^{n}$. In essence, this issue
means that whilst accurate estimates of the compressive and shear yield stress
curves can be gained, the individual parameters $k$ and $\phi_{g}$ are less
accurately estimated. In the case of equilibrium height data $h_{\infty}$, it
is necessary to undertake a significant number of tests under variable column
height and radii to provide a dense enough dataset for accurate estimation.
Prior knowledge of the suspension gel point $\phi_{g}$ and accurate
measurement of $h_{\infty}$ are required to generate accurate estimates of the
shear and compressive rheology. These results indicate that wall adhesion
effects can be routinely corrected for in equilibrium batch settling tests,
yielding accurate estimates of the shear and compressive rheology of strong
colloidal gels. The important case of centrifugal batch settling will be
considered in a subsequent paper wherein it will be shown that the errors due
to adhesion are normally much less significant, in part because then most of
the gel is not near the gel-point and second because $h_{\infty}/R$ decreases
with increasing acceleration.

##  7 Acknowledgements

This work was conducted as part of AMIRA P266G “Improving Thickener
Technology” project, supported by the following companies: Alcoa World
Alumina, Anglo American, BASF, Bateman Engineering, Cytec Australia Holdings,
Delkor, FL Smidth Minerals, Kemira, MMG, Newcrest, Outotec, Rusal, Senmin,
Shell Energy Canada, Teck Resources, Total E&P Canada and WesTech. The authors
are indebted to Jon Halewood for undertaking the flocculation, rheological and
solid volume fraction profile measurements, Andrew Chryss for design of
rheological measurements.

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