aff1]Faculty of Mathematics and Informatics, Konstantin Preslavsky University
of Shumen,
115 ”Universitetska” str., 9712 Shumen, Bulgaria. aff3]Institute of
Statistics, Universidad de Valparaíso, Valparaíso, Chile. aff4]Department of
Applied Statistics and Linz Institute of Technology, Johannes Kepler
University, Altenbergerstrasse 69, 4040 Linz, Austria.

[cor1]Corresponding author<EMAIL_ADDRESS>
# Logarithm of ratios of two order statistics
and regularly varying tails

Pavlina K. Jordanova  Milan Stehlík  [  [  [

###### Abstract

Here we suppose that the observed random variable has cumulative distribution
function $F$ with regularly varying tail, i.e. $1-F\in RV_{-\alpha}$,
$\alpha>0$. Using the results about exponential order statistics we
investigate logarithms of ratios of two order statistics of a sample of
independent observations on Pareto distributed random variable with parameter
$\alpha$. Short explicit formulae for its mean and variance are obtained. Then
we transform this function in such a way that to obtain unbiased,
asymptotically efficient, and asymptotically normal estimator for $\alpha$.
Finally we simulate Pareto samples and show that in the considered cases the
proposed estimator outperforms the well known Hill, t-Hill, Pickands and
Deckers-Einmahl-de Haan estimators.

##  1 HISTORY OF THE PROBLEM

The usefulness of regularly varying (RV) functions in economics seems to be
discussed for the first time during modeling of the wealth in our society by
Pareto distribution, called to the name of Vilfredo Pareto (1897). J. Karamata
(1933) provides their definition and integral representation. Later on the
Convergence to types theorem, proved by R. A. Fisher, L. H. C. Tippett (1928),
and B.V. Gnedenko (1948) plays a key role for their future applications. It is
well known that this class of distributions describes very well the domain of
attraction of stable distribution (see Mandelbrot (1960) [15]) and max-domain
of attraction of Fr$\acute{e}$chet distribution (see M. Fr$\acute{e}$chet
(1927)). Laurens de Haan (1970) and co-authors [3, 4, 5] develop the main
machinery for working with cumulative distribution functions(c.d.fs.) with
such tail behaviour. Let us remind that the c.d.f. $F$ has regularly varying
right tail with parameter $\alpha>0$, if

$\lim_{t\to\infty}\frac{1-F(xt)}{1-F(t)}=x^{-\alpha},\quad\forall x>0.$

After their works the topic spread over the world very fast and many
estimators of the index of regular variation are proposed, see e.g. Hill
(1975) [10], Pickands (1975)[19] and Deckers-Einmahl-de Haan (1989) [6],
t-Hill (Stehlik and co-authors (2010) [24, 9, 25], and Pancheva and Jordanova
(2012) [11, 14]), among others.

Here we show the usefulness of functions of two central order statistics in
estimating the parameter of regular variation. Under very general settings we
show that the logarithm of the fraction of two specific central order
statistics is an weakly consistent and asymptotically normal estimators of the
logarithm of the corresponding theoretical quantiles. Then we use these
functions and obtain our estimator for $\alpha$. Its main advantage is that it
is very flexible and provides a useful accuracy given mid-range and small
samples. Pareto case, considered in Section 3 motivates our investigation.
First we define a biased form of the estimator. Then using results about order
statistics, which could be seen e.g. in Nevzorov (2001) [18] we obtain
explicit formulae for its mean and variance. This allows us to define unbiased
correction which is asymptotically efficient. Then we prove asymptotic
normality and obtain large sample confidence intervals. Our simulation study
depicts the advantages of the considered estimators over Hill, t-Hill, and
Deckers-Einmahl-de Haan estimators. The paper finishes with some conclusive
remarks.

Trough the paper we assume that
$\mathbf{X}_{1},\mathbf{X}_{2},...,\mathbf{X}_{n}$ are independent
observations on a random variable(r.v.) $\mathbf{X}$, and denote by
$\mathbf{X}_{(1,n)}\leq\mathbf{X}_{(2,n)}\leq...\leq\mathbf{X}_{(n,n)}$ the
corresponding increasing order statistics.

$H_{n,m}=1+\frac{1}{2^{m}}+\frac{1}{3^{m}}+...+\frac{1}{(n-1)^{m}}+\frac{1}{n^{m}},\quad
n=1,2,...,$

denotes the $n$-the Generalized harmonic number of power $m=1,2,...$, and
$H_{n}=H_{n,1}$, $n=1,2,...$ is for the well-known $n$-th harmonic number.

The main object of interest in this point are the statistics

$Q_{k,s}:=\frac{log\frac{\mathbf{X}_{(ks,(s+1)k-1)}}{\mathbf{X}_{(k,(s+1)k-1)}}}{H_{ks-1}-H_{k-1}},\quad
Q_{k,s}^{*}=\frac{log\frac{\mathbf{X}_{(ks,(s+1)k-1)}}{\mathbf{X}_{(k,(s+1)k-1)}}}{log(s)},\quad
s=2,3,...$

The estimator $Q_{k,3}^{*}$ it is obtained in Jordanova et al. [13] via
quantile matching procedure. About the last procedure see e.g. Sgouropoulos et
al. (2015) [21].

Along the paper $\stackrel{{\scriptstyle d}}{{\to}}$ means convergence in
distribution.

##  2 GENERAL RESULTS

In 1933 - 1949 Smirnoff [22] shows that in case of central order statistics,
and more precisely for $k,n$ and $p$ such that $\frac{k}{n}\to p\in(0,1)$ and
$\sqrt{n}\left(\frac{k}{n}-p\right)\to\mu\in(-\infty,\infty)$, the asymptotic
distribution of
$\frac{\sqrt{n}[X_{(k,n)}-F^{\leftarrow}(p)]}{\sqrt{\frac{p(1-p)}{f^{2}[F^{\leftarrow}(p)]}}}$
is a standard normal. Moreover it seems that he has a similar results about
bivariate order statistics. It could be seen e.g. in Arnold et al. (1992) [1],
p. 226, Mosteller (1946) [16] p.338, Nair [17], p.330, or Wilks [27] among
others. The multivariate delta method is a very powerful technique for
obtaining confidence intervals in such cases. In the next theorem we apply
them and obtain the limiting distribution of the logarithmic differences of
central order statistics.

Smirnoff’s theorem. Assume for $n\to\infty$, $\frac{k_{1}(n)}{n}\to
p_{1}\in(0,1)$, $\frac{k_{2}(n)}{n}\to p_{2}\in(0,1)$,
$f\left[F^{\leftarrow}(p_{1})\right]\in(0,\infty)$, and
$f\left[F^{\leftarrow}\left(p_{2}\right)\right]\in(0,\infty)$. Then

$\sqrt{n}\left(\begin{array}[]{c}X_{(k_{1},n)}-F^{\leftarrow}(p_{1})\\\
X_{(k_{2},n)}-F^{\leftarrow}(p_{2})\\\
\end{array}\right)\stackrel{{\scriptstyle
d}}{{\to}}\left(\begin{array}[]{c}\theta_{1}\\\ \theta_{2}\\\
\end{array}\right),\quad\left(\begin{array}[]{c}\theta_{1}\\\ \theta_{2}\\\
\end{array}\right)\in N(0,V)$

where the covariance matrix

$V=\left(\begin{array}[]{cc}\frac{p_{1}(1-p_{1})}{f^{2}[F^{\leftarrow}(p_{1})]}&\frac{p_{1}(1-p_{2})}{f^{2}[F^{\leftarrow}(p_{1})]f^{2}[F^{\leftarrow}(p_{2})]}\\\
\frac{p_{1}(1-p_{2})}{f^{2}[F^{\leftarrow}(p_{1})]f^{2}[F^{\leftarrow}(p_{2})]}&\frac{p_{2}(1-p_{2})}{f^{2}[F^{\leftarrow}(p_{2})]}\\\
\end{array}\right).$

We apply this theorem together with the Multivariate delta method and obtain
asymptotic normality of the estimators, discussed in this paper.

Theorem 1. Consider a sample of $n=(s+1)k-1$, $s=2,3,...$ independent
observations on a r.v. $X$ with c.d.f. $F$ and p.d.f. $f=F^{\prime}$. If there
exists $0<f\left[F^{\leftarrow}(\frac{1}{s+1})\right]<\infty$ and
$0<f\left[F^{\leftarrow}(\frac{s}{s+1})\right]<\infty$, then for $k\to\infty$

$T_{k,s}:=\sqrt{(s+1)k-1}\left[log\left(\frac{X_{(ks,(s+1)k-1)}}{X_{(k,(s+1)k-1)}}\right)-log\left(\frac{F^{\leftarrow}(\frac{s}{s+1})}{F^{\leftarrow}(\frac{1}{s+1})}\right)\right]\stackrel{{\scriptstyle
d}}{{\to}}N\left(0;V\right).$ (1)

The variance $V$ in (1) is
$V=\frac{1}{(s+1)^{2}}\left(\frac{s}{a_{F,s}^{2}}-\frac{2}{a_{F,s}b_{F,s}}+\frac{s}{b_{F,s}^{2}}\right)$,
where
$a_{F,s}=F^{\leftarrow}(\frac{1}{s+1})f\left[F^{\leftarrow}(\frac{1}{s+1})\right]=\frac{1}{\\{log[F^{\leftarrow}(p)]\\}^{\prime}|_{p=\frac{1}{s+1}}}$,
and
$b_{F,s}=F^{\leftarrow}(\frac{s}{s+1})f\left[F^{\leftarrow}(\frac{s}{s+1})\right]=$
$\frac{1}{\\{log[F^{\leftarrow}(p)]\\}^{\prime}|_{p=\frac{s}{s+1}}}.$

Proof: We will apply the Theorem of Smirnoff for $p_{1}=\frac{1}{s+1}$ and
$p_{2}=\frac{s}{s+1}$ and Multivariate delta method.

By assumptions the conditions
$f\left[F^{\leftarrow}\left(\frac{1}{s+1}\right)\right]\in(0,\infty)$,
$f\left[F^{\leftarrow}\left(\frac{s}{s+1}\right)\right]\in(0,\infty)$ are
satisfied. And for $k\to\infty$ we have $\frac{sk}{(s+1)k-1}\to\frac{s}{s+1}$,
$\frac{s}{(s+1)k-1}\to\frac{1}{s+1}$,
$\sqrt{(s+1)k-1}\left(\frac{sk}{(s+1)k-1}-\frac{s}{s+1}\right)\to 0$ and
$\sqrt{(s+1)k-1}\left(\frac{k}{(s+1)k-1}-\frac{s}{s+1}\right)\to 0$, therefore
the Smirnoff’s theorem on the joint asymptotic normality of the order
statistics, says that

$\sqrt{(s+1)k-1}\left(\begin{array}[]{c}X_{(k,(s+1)k-1)}-F^{\leftarrow}(\frac{1}{s+1})\\\
X_{(ks,(s+1)k-1)}-F^{\leftarrow}(\frac{i}{s+1})\\\
\end{array}\right)\stackrel{{\scriptstyle
d}}{{\to}}N\left[\left(\begin{array}[]{c}0\\\ 0\\\
\end{array}\right);D\right],\quad k\to\infty,$

where the asymptotic covariance matrix of this bivariate distribution is

$D=\frac{1}{(s+1)^{2}}\left(\begin{array}[]{cc}\frac{s}{f^{2}\left[F^{\leftarrow}(\frac{1}{s+1})\right]}&\frac{1}{f\left[F^{\leftarrow}(\frac{1}{s+1})\right]f\left[F^{\leftarrow}(\frac{s}{s+1})\right]}\\\
\frac{1}{f\left[F^{\leftarrow}(\frac{1}{s+1})\right]f\left[F^{\leftarrow}(\frac{s}{s+1})\right]}&\frac{s}{f^{2}\left[F^{\leftarrow}(\frac{s}{s+1})\right]}\\\
\end{array}\right)$

and the asymptotic correlation between these two order statistics is
$\frac{1}{s}$.

Consider the function $g(x,y)=log\left(\frac{y}{x}\right)$. For $x>0$ and
$y>0$ it is continuously differentiable.

The Jacobian of the transformation is

$J:=\left[\frac{\partial g(x,y)}{\partial x},\frac{\partial g(x,y)}{\partial
y}\right]=\left(-\frac{1}{x},\frac{1}{y}\right).$

The asymptotic mean is

$\lim_{k\to\infty}Elog\left(\frac{X_{(ks,(s+1)k-1)}}{X_{(k,(s+1)k-1)}}\right)=g\left[F^{\leftarrow}(\frac{1}{s+1}),F^{\leftarrow}(\frac{s}{s+1})\right]=log\left(\frac{F^{\leftarrow}(\frac{s}{s+1})}{F^{\leftarrow}(\frac{1}{s+1})}\right).$

Now we apply the Multivariate Delta method, which could be seen e.g. in Sobel
(1982) [23], and obtain that the asymptotic variance of $T_{k,s}$ is

$\displaystyle V:$ $\displaystyle=$ $\displaystyle J\times D\times
J^{\prime}=\left(-\frac{1}{x},\frac{1}{y}\right)\left|{}_{x=F^{\leftarrow}(\frac{1}{s+1}),y=F^{\leftarrow}(\frac{s}{s+1})}\right.$
$\displaystyle\times$
$\displaystyle\frac{1}{(s+1)^{2}}\left(\begin{array}[]{cc}\frac{s}{f^{2}\left[F^{\leftarrow}(\frac{1}{s+1})\right]}&\frac{1}{f\left[F^{\leftarrow}(\frac{1}{s+1})\right]f\left[F^{\leftarrow}(\frac{s}{s+1})\right]}\\\
\frac{1}{f\left[F^{\leftarrow}(\frac{1}{s+1})\right]f\left[F^{\leftarrow}(\frac{s}{s+1})\right]}&\frac{s}{f^{2}\left[F^{\leftarrow}(\frac{s}{s+1})\right]}\\\
\end{array}\right)\left(\begin{array}[]{c}-\frac{1}{x}\\\ \frac{1}{y}\\\
\end{array}\right)\left|{}_{x=F^{\leftarrow}(\frac{1}{s+1}),y=F^{\leftarrow}(\frac{s}{s+1})}\right.$
$\displaystyle=$
$\displaystyle\frac{1}{(s+1)^{2}}\left[\left(-\frac{1}{x},\frac{1}{y}\right)\left(\begin{array}[]{cc}\frac{s}{f^{2}(x)}&\frac{1}{f(x)f(y)}\\\
\frac{1}{f(x)f(y)}&\frac{s}{f^{2}(y)}\\\
\end{array}\right)\left(\begin{array}[]{c}-\frac{1}{x}\\\ \frac{1}{y}\\\
\end{array}\right)\right]\left|{}_{x=F^{\leftarrow}(\frac{1}{s+1}),y=F^{\leftarrow}(\frac{s}{s+1})}\right.$
$\displaystyle=$
$\displaystyle\frac{1}{(s+1)^{2}}\left[\frac{s}{x^{2}f^{2}(x)}-\frac{2}{xyf(x)f(y)}+\frac{s}{y^{2}f^{2}(y)}\right]\left|{}_{x=F^{\leftarrow}(\frac{1}{s+1}),y=F^{\leftarrow}(\frac{s}{s+1})}\right.=\frac{1}{(s+1)^{2}}\left(\frac{s}{a_{F,s}^{2}}-\frac{2}{a_{F,s}b_{F,s}}+\frac{s}{b_{F,s}^{2}}\right).$

Q.A.D.

Slutsky’s theorem about continuous functions together with the definition of
convergence in probability, application of quantile transform, and Smirnoff’s
theorem about a.s. convergence of empirical quantiles to corresponding
theoretical one, lead us to the following result. Without lost of generality
we consider only a.s. positive r.vs, however the result could be easily
transformed for $P(X<\mu)=1$ or $P(X>\mu)=1$, $\mu>0$.

Theorem 2. Assume $P(X>0)=1$. If
$f\left[F^{\leftarrow}\left(\frac{1}{s+1}\right)\right]\in(0,\infty)$,
$f\left[F^{\leftarrow}\left(\frac{s}{s+1}\right)\right]\in(0,\infty)$, then
for $s=2,3,...$

$log\left(\frac{X_{(ks,(s+1)k-1)}}{X_{(k,(s+1)k-1)}}\right)\stackrel{{\scriptstyle
P}}{{\to}}log\left[\frac{F^{\leftarrow}(\frac{s}{s+1})}{F^{\leftarrow}(\frac{1}{s+1})}\right],\quad
k\to\infty.$ (4)

##  3 PARETO CASE

In this section we assume that
$\mathbf{X}_{1},\mathbf{X}_{2},...,\mathbf{X}_{n}$ are independent
observations on a r.v. $\mathbf{X}$ with Pareto c.d.f.

$F_{\mathbf{X}}(x)=\left\\{\begin{array}[]{ccc}0&,&x\leq\delta\\\
1-\left(\frac{\delta}{x}\right)^{\alpha}&,&x>\delta\end{array}\right.,\quad\alpha>0,\quad\delta>0.$
(5)

Briefly we will denote this by $\mathbf{X}\in Par(\alpha,\delta)$. Different
generalizations of this distributions could be seen in Arnold (2015) [2]. The
number $-\alpha$ is called ”index of regular variation of the tail of c.d.f.”.
It determines the tail behaviour of the c.d.f. See e.g. de Haan and Ferreira
[5], Resnick [20], or Jordanova [12].

Denote by $\mathbf{X}\in Exp(\lambda)$, $\lambda>0$ the fact that the r.v.
$\mathbf{X}$ has c.d.f.

$F_{\mathbf{X}}(x)=\left\\{\begin{array}[]{ccc}0&,&x\leq 0\\\ 1-e^{-\lambda
x}&,&x>0\end{array}\right..$ (6)

The results in the following theorem allow us later on, in Corollaries 1 and
2, to obtain unbiased, consistent, and asymptotically efficient estimators of
the parameter $\alpha$.

Theorem 3. Assume
$\mathbf{X}_{(1,n)}\leq\mathbf{X}_{(2,n)}\leq...\leq\mathbf{X}_{(n,n)}$ are
order statistics of independent observations on a r.v. $\mathbf{X}\in
Par(\alpha,\delta)$, $\alpha>0$, $\delta>0$, and $1\leq i<j\leq n$ are
integer.

i)

Denote by $\rho$ a Beta distributed with parameters $n-j+1$, and $j-i$. Then

$log\left(\frac{X_{(j,n)}}{X_{(i,n)}}\right)\stackrel{{\scriptstyle
d}}{{=}}-\frac{1}{\alpha}\log(\rho)\stackrel{{\scriptstyle
d}}{{=}}E_{(j-i,n-i)}\stackrel{{\scriptstyle
d}}{{=}}\frac{1}{\alpha}E_{(j-i,n-i)}^{*},$

where $E_{(j-i,n-i)}$ is the $j-i$-th order statistics in a sample of $n-i$
independent observations on i.i.d. Exponential r.vs. with parameter $\alpha$,
and $E_{(j-i,n-i)}^{*}$ is the $j-i$ \- th order statistic of a sample of
$n-i$ independent observations on exponentially distributed r.v. with
parameter $1$. Its probability density function is

$f_{log\frac{\mathbf{X}_{(j,n)}}{\mathbf{X}_{(i,n)}}}(x)=\frac{\alpha(n-i)!}{(j-i-1)!(n-j)!}(1-e^{-\alpha
x})^{j-i-1}e^{-\alpha x(n-j+1)},\quad x>0.$

ii)

$\mathbb{E}\left[log\frac{\mathbf{X}_{(j,n)}}{\mathbf{X}_{(i,n)}}\right]=\frac{1}{\alpha}\left(H_{n-i}-H_{n-j}\right),$
and
$\mathbb{D}\left[log\frac{\mathbf{X}_{(j,n)}}{\mathbf{X}_{(i,n)}}\right]=\frac{1}{\alpha^{2}}\left(H_{n-i,2}-H_{n-j,2}\right).$

Proof: Let us fix $1\leq i<j\leq n$, integers. Because of $g(x)=e^{x}$ is a
strictly increasing function, it is well known that the probability quantile
transform, entails

$\left(\frac{\mathbf{X}_{(1,n)}}{\delta},\frac{\mathbf{X}_{(2,n)}}{\delta},...,\frac{\mathbf{X}_{(n,n)}}{\delta}\right)\,\,{\mathop{=}\limits^{d}}\,\,(e^{\mathbf{E}_{(1,n)}},e^{\mathbf{E}_{(2,n)}},...e^{\mathbf{E}_{(n,n)}}),$

where $\mathbf{E}_{(1,n)}\leq\mathbf{E}_{(2,n)}\leq...\leq\mathbf{E}_{(n,n)}$
are order statistics of independent identically distributed (i.i.d.) r.vs.
with $\mathbf{E}_{1}\in Exp(\alpha)$. Then, because of the multiplicative
property of the exponential distribution

$\left(\frac{\mathbf{X}_{(1,n)}}{\delta},\frac{\mathbf{X}_{(2,n)}}{\delta},...,\frac{\mathbf{X}_{(n,n)}}{\delta}\right)\,\,{\mathop{=}\limits^{d}}\,\,\left(e^{\frac{1}{\alpha}\mathbf{E}_{(1,n)}^{*}},e^{\frac{1}{\alpha}\mathbf{E}_{(2,n)}^{*}},...e^{\frac{1}{\alpha}\mathbf{E}_{(n,n)}^{*}}\right),$

where
$\mathbf{E}_{(1,n)}^{*}\leq\mathbf{E}_{(2,n)}^{*}\leq...\leq\mathbf{E}_{(n,n)}^{*}$
are order statistics of i.i.d. r.vs. with $\mathbf{E}_{1}^{*}\in Exp(1)$. See
e.g. de Haan and Ferreira [5]. Denote the logarithm with basis $e$ by log.
Because of $e>1$, $h(x)=log(x)$, is an increasing function, thus

$log\left(\frac{\mathbf{X}_{(j,n)}}{\mathbf{X}_{(i,n)}}\right)\,\,{\mathop{=}\limits^{d}}\,\,\frac{1}{\alpha}\left(\mathbf{E}_{(j,n)}^{*}-\mathbf{E}_{(i,n)}^{*}\right)\,\,{\mathop{=}\limits^{d}}\,\,\frac{1}{\alpha}\mathbf{E}_{(j-i,n-i)}^{*}.$

The last equality could be seen e.g. in de Haan and Ferreira [5] or Arnold et
al. (1992) [1].

i) Follows by the equality $P_{\frac{E_{(j-i,n-i)}^{*}}{\alpha}}(x)=\alpha
P_{E_{(j-i,n-i)}^{*}}(\alpha\,x)$, the well known relation
$-\log(\rho)\stackrel{{\scriptstyle d}}{{=}}E_{(j-i,n-i)}^{*}$ and the formula
for probability density function (p.d.f.) of order statistics of a sample of
i.i.d. r.vs. See e.g. p. 7 Nevzorov [18].

ii) The mean, and the variance of the last order statistics are very well
investigated. See e.g. Nevzorov [18], p.23. Using his results and the main
properties of the expectation and the variance we obtain:

$\displaystyle\mathbb{E}\left[log\left(\frac{\mathbf{X}_{(j,n)}}{\mathbf{X}_{(i,n)}}\right)\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left(\frac{1}{\alpha}E_{(j-i,n-i)}^{*}\right)=\frac{1}{\alpha}\mathbb{E}\left(E_{(j-i,n-i)}^{*}\right)=\frac{1}{\alpha}\left(H_{n-i}-H_{n-j}\right)$
$\displaystyle\mathbb{D}\left[log\left(\frac{\mathbf{X}_{(j,n)}}{\mathbf{X}_{(i,n)}}\right)\right]$
$\displaystyle=$
$\displaystyle\mathbb{D}\left(\frac{1}{\alpha}E_{(j-i,n-i)}^{*}\right)=\frac{1}{\alpha^{2}}\mathbb{D}\left(E_{(j-i,n-i)}^{*}\right)=\frac{1}{\alpha^{2}}\left(H_{n-i,2}-H_{n-j,2}\right).$

Q.A.D.

In the next corollary is useful when working with finite samples. We obtain
that for any $s=2,3,...$, and for fixed $k=1,2,...$ the estimators $Q_{k,s}$
are unbiased for $\frac{1}{\alpha}$. The accuracy of these estimators in that
case is explicitly calculated. However these estimators are applicable also
for large enough samples, because for $k\to\infty$ they are weakly consistent
and asymptotically efficient.

Corollary 1. Assume $n=k(s+1)-1$,
$\mathbf{X}_{(1,n)}\leq\mathbf{X}_{(2,n)}\leq...\leq\mathbf{X}_{(n,n)}$ are
order statistics of independent observations on a r.v. $\mathbf{X}\in
Par(\alpha,\delta)$, $\alpha>0$, $\delta>0$. Then, for all $s=2,3,...$, and
$k\in\mathbb{N}$,

i)

Denote by $\rho$ a Beta distributed with parameters $k$, and $(s-1)k$. Then

$Q_{k,s}\stackrel{{\scriptstyle
d}}{{=}}\frac{-\log(\rho)}{\alpha(H_{ks-1}-H_{k-1})}\stackrel{{\scriptstyle
d}}{{=}}\frac{E_{((s-1)k,ks-1)}}{H_{ks-1}-H_{k-1}}\stackrel{{\scriptstyle
d}}{{=}}\frac{E_{((s-1)k,ks-1)}^{*}}{\alpha(H_{ks-1}-H_{k-1})},$

where $E_{((s-1)k,ks-1)}$ is the $(s-1)k$-th order statistics in a sample of
$ks-1$ independent observations on i.i.d. Exponential r.vs. with parameter
$\alpha$. $E_{((s-1)k,ks-1)}^{*}$ is the $(s-1)s$ \- th order statistic of a
sample of $ks-1$ independent observations on exponentially distributed r.v.
with parameter $1$. Its probability density function is

$f_{Q_{k,s}}(x)=\frac{\alpha(H_{ks-1}-H_{k-1})(ks-1)!}{[(s-1)k-1]!(k-1)!}(1-e^{-\alpha(H_{ks-1}-H_{k-1})x})^{(s-1)k-1}e^{-k\alpha(H_{ks-1}-H_{k-1})x},\quad
x>0.$

ii)

$\mathbb{E}Q_{k,s}=\frac{1}{\alpha},$ and
$\mathbb{D}Q_{k,s}=\frac{H_{ks-1,2}-H_{k-1,2}}{\alpha^{2}(H_{ks-1}-H_{k-1})^{2}}.$

iii)

For all $\varepsilon>0$,

$P\left[\left|Q_{k,s}-\frac{1}{\alpha}\right|>\varepsilon\right]\leq\frac{H_{ks-1,2}-H_{k-1,2}}{\alpha^{2}\varepsilon^{2}(H_{ks-1}-H_{k-1})^{2}}.$

iv)

The estimator $Q_{k,s}$ is asymptotically efficient. For $k\to\infty$,

$\mathbb{D}Q_{k,s}\sim\frac{H_{ks-1,2}-H_{k-1,2}}{\alpha^{2}}\left[log\left(\frac{ks-1}{k}\right)\right]^{-2},\quad\lim_{k\to\infty}\mathbb{D}Q_{k,s}=0.$

v)

The estimator $Q_{k,s}$ is weekly consistent. More precisely, for all
$\varepsilon>0$,
$\lim_{k\to\infty}P\left[\left|Q_{k,s}-\frac{1}{\alpha}\right|>\varepsilon\right]=0.$

Proof: i) and ii) follow by Theorem 1, definition of $Q_{k,s}$ and the
relations

$f_{Q_{k,s}}(x)=(H_{ks-1}-H_{k-1})f_{log\frac{\mathbf{X}_{(ks,k(s+1)-1)}}{\mathbf{X}_{(k,k(s+1)-1)}}}[x(H_{ks-1}-H_{k-1})],\quad\mathbb{E}Q_{k,s}=\frac{\mathbb{E}log\frac{\mathbf{X}_{(ks,k(s+1)-1)}}{\mathbf{X}_{(k,k(s+1)-1)}}}{H_{ks-1}-H_{k-1}},\quad\mathbb{D}Q_{k,s}=\frac{\mathbb{D}log\frac{\mathbf{X}_{(ks,k(s+1)-1)}}{\mathbf{X}_{(k,k(s+1)-1)}}}{(H_{ks-1}-H_{k-1})^{2}}$

iii) is corollary of ii) and Chebyshev’s inequality.

iv) It is well known that $\lim_{n\to\infty}[H_{n}-log(n)]=\gamma,$ where
$\gamma=-\Gamma^{\prime}(1)=\psi(1)\approx 0,5772$ is the Euler$-$Mascheroni
constant, $\Gamma(\alpha)=\int_{0}^{\alpha}x^{\alpha-1}e^{-x}dx$, and $\psi$
is the Digamma function. By ii) for any fixed $s=2,3,...$, we have

$\displaystyle\lim_{k\to\infty}\mathbb{D}Q_{k,s}$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha^{2}}\lim_{k\to\infty}\frac{H_{ks-1,2}-H_{k-1,2}}{(H_{ks-1}-H_{k-1})^{2}}=\frac{1}{\alpha^{2}}\lim_{k\to\infty}\frac{H_{ks-1,2}-H_{k-1,2}}{\left\\{H_{ks-1}-log(ks-1)-[H_{k-1}-log(k)]+log\left(\frac{ks-1}{k}\right)\right\\}^{2}}$
(7) $\displaystyle=$
$\displaystyle\frac{1}{\alpha^{2}}\lim_{k\to\infty}\frac{H_{ks-1,2}-H_{k-1,2}}{\left[log\left(\frac{ks-1}{k}\right)\right]^{2}}=\frac{1}{\alpha^{2}}\frac{\lim_{k\to\infty}H_{ks-1,2}-\lim_{k\to\infty}H_{k-1,2}}{[log(s)]^{2}}=0$
(8)

In the last equality we have used the well known solution of the Basel
problem, and more precisely the limit
$\lim_{n\to\infty}H_{n,2}=\frac{\pi^{2}}{6}.$

v) is a consequence of ii), iii) and iv). Q.A.D.

In the previous proof we have seen that for any fixed $s=2,3,...$,
$\lim_{k\to\infty}(H_{ks-1}-H_{k-1})=log(s)$. Therefore, although
$Q_{k,s}^{*}$ are biased, they are asymptotically unbiased, asymptotically
normal, weakly consistent and asymptotically efficient estimators for
$\frac{1}{\alpha}$. The next conclusions follow by the relation
$Q_{k,s}^{*}=\frac{Q_{k,s}(H_{ks-1}-H_{k-1})}{log(s)}$, and the main
properties of the mean and the variance.

Corollary 2. Assume $n=k(s+1)-1$,
$\mathbf{X}_{(1,n)}\leq\mathbf{X}_{(2,n)}\leq...\leq\mathbf{X}_{(n,n)}$ are
order statistics of independent observations on a r.v. $\mathbf{X}\in
Par(\alpha,\delta)$, $\alpha>0$, $\delta>0$.

i)

Denote by $\rho$ a Beta distributed with parameters $k$, and $(s-1)k$. Then,
for all $k\in\mathbb{N}$,

$Q_{k,s}^{*}\stackrel{{\scriptstyle d}}{{=}}-\frac{\log(\rho)}{\alpha
log(s)}\stackrel{{\scriptstyle
d}}{{=}}\frac{E_{((s-1)k,ks-1)}}{log(s)}\stackrel{{\scriptstyle
d}}{{=}}\frac{E_{((s-1)k,ks-1)}^{*}}{\alpha log(s)},$

where $E_{((s-1)k,ks-1)}$ is the $(s-1)k$-th order statistics in a sample of
$ks-1$ independent observations on i.i.d. Exponential r.vs. with parameter
$\alpha$. $E_{((s-1)k,ks-1)}^{*}$ is the $(s-1)s$ \- th order statistic of a
sample of $ks-1$ independent observations on exponentially distributed r.v.
with parameter $1$. Its probability density function is

$f_{Q_{k,s}^{*}}(x)=\frac{\alpha\,\,log(s)(ks-1)!}{[(s-1)k-1]!(k-1)!}(1-s^{-\alpha
x})^{(s-1)k-1}s^{-\alpha kx},\quad x>0.$

ii)

For all $k\in\mathbb{N}$,
$\mathbb{E}Q_{k,s}^{*}=\frac{H_{ks-1}-H_{k-1}}{\alpha\,\,log(s)},$ and
$\mathbb{D}Q_{k,s}^{*}=\frac{H_{ks-1,2}-H_{k-1,2}}{\alpha^{2}[log(s)]^{2}}.$

iii)

For all $\varepsilon>0$, and $k\in\mathbb{N}$,

$P\left[\left|Q_{k,s}^{*}-\frac{1}{\alpha}\right|>\varepsilon\right]\leq\frac{H_{ks-1,2}-H_{k-1,2}}{\alpha^{2}\varepsilon^{2}[log(s)]^{2}}.$

iv)

$Q_{k,s}^{*}$ estimator is asymptotically unbiased and asymptotically
efficient. More precisely

$\lim_{k\to\infty}\mathbb{E}Q_{k,s}^{*}=\frac{1}{\alpha},\quad\lim_{k\to\infty}\mathbb{D}Q_{k,s}^{*}=0.$

v)

$Q_{k,s}^{*}$ estimator is weekly consistent. For all $\varepsilon>0$,
$\lim_{k\to\infty}P\left[\left|Q_{k,s}^{*}-\frac{1}{\alpha}\right|>\varepsilon\right]=0.$

Applications of the previous results require knowledge about confidence
intervals. Therefore, in the the next theorem, we obtain asymptotic normality
of these estimators which allows us later on to construct large sample
confidence intervals.

Theorem 4. If $\mathbf{X}\in Par(\alpha,\delta)$, $\alpha>0$, $\delta>0$, then
for all $s=2,3,...$, and $k\to\infty$,

$\sqrt{k(s+1)-1}\left[log\left(\frac{X_{(ks,k(s+1)-1)}}{X_{(k,k(s+1)-1)}}\right)-\frac{1}{\alpha}log(s)\right]\stackrel{{\scriptstyle
d}}{{\to}}\eta_{1},\quad\eta_{1}\in
N\left(0,\frac{s^{2}-1}{s\alpha^{2}}\right),$ (9)
$\sqrt{k(s+1)-1}(H_{ks-1}-H_{k-1})\left[\alpha
Q_{k,s}-\frac{log(s)}{H_{ks-1}-H_{k-1}}\right]\stackrel{{\scriptstyle
d}}{{\to}}\eta_{2},\quad\eta_{2}\in N\left(0,\frac{s^{2}-1}{s}\right),$ (10)
$\sqrt{k(s+1)-1}\left[\alpha Q_{k,s}^{*}-1\right]\stackrel{{\scriptstyle
d}}{{\to}}\eta_{3},\quad\eta_{3}\in
N\left(0,\frac{s^{2}-1}{s[log(s)]^{2}}\right).$ (11)

Proof: In this case $F^{\leftarrow}(p)=\frac{\delta}{\sqrt[\alpha]{1-p}}$, and
$f(x)=\frac{\alpha\delta^{\alpha}}{x^{\alpha+1}}$. Therefore,
$F^{\leftarrow}\left(\frac{1}{s+1}\right)=\delta\sqrt[\alpha]{\frac{s+1}{s}}$,
$F^{\leftarrow}\left(\frac{s}{s+1}\right)=\delta\sqrt[\alpha]{s+1}$,

$f\left[F^{\leftarrow}\left(\frac{1}{s+1}\right)\right]=\frac{\alpha
s^{1/\alpha+1}}{\delta(s+1)^{1/\alpha+1}}\in(0,\infty),\quad
f\left[F^{\leftarrow}\left(\frac{s}{s+1}\right)\right]=\frac{\alpha}{\delta(s+1)^{1/\alpha+1}}\in(0,\infty),$

For $k\to\infty$ we have $\frac{s}{(s+1)k-1}\to\frac{1}{s+1}$,
$\frac{sk}{(s+1)k-1}\to\frac{s}{s+1}$,
$\sqrt{(s+1)k-1}\left(\frac{k}{(s+1)k-1}-\frac{s}{s+1}\right)\to 0$, and
$\sqrt{(s+1)k-1}$ $\left(\frac{sk}{(s+1)k-1}-\frac{s}{s+1}\right)\to 0$
therefore we can apply Smirnoff’s theorem about the joint asymptotic normality
of the order statistics and Theorem 1. In order to determine $a_{F,s}$ and
$b_{F,s}$ let us note that
$log[F^{\leftarrow}(p)]^{\prime}=\frac{1}{\alpha(1-p)}$. Therefore

$a_{F,s}=\frac{1}{\\{log[F^{\leftarrow}(p)]\\}^{\prime}|_{p=\frac{1}{s+1}}}=\frac{\alpha
s}{s+1},\quad
b_{F,s}=\frac{1}{\\{log[F^{\leftarrow}(p)]\\}^{\prime}|_{p=\frac{s}{s+1}}}=\frac{\alpha}{s+1}$

The equalities

$V=\frac{1}{(s+1)^{2}}\left(\frac{s}{a_{F,s}^{2}}-\frac{2}{a_{F,s}b_{F,s}}+\frac{s}{b_{F,s}^{2}}\right)=\frac{1}{\alpha^{2}}\left(\frac{1}{s}-\frac{2}{s}+s\right)=\frac{s^{2}-1}{\alpha^{2}s},\quad
log\left(\frac{F^{\leftarrow}(\frac{s}{s+1})}{F^{\leftarrow}(\frac{1}{s+1})}\right)=\frac{1}{\alpha}log(s).$

lead us to (9). When we multiply the numerator in (9) by
$\alpha[H_{ks-1}-H_{k-1}]$, and the denominator by $H_{ks-1}-H_{k-1}$, and use
that $D\eta_{2}=D(\alpha\eta_{1})=\alpha^{2}D\eta_{1}$ we obtain (10). If we
multiply both sides of (9) by $\frac{\alpha}{log(s)}$, and use that
$D\eta_{3}=D\left(\frac{\alpha}{log(s)}\eta_{1}\right)=\frac{\alpha^{2}}{[log(s)]^{2}}D\eta_{1}$
we obtain (11). Q.A.D.

Now we are ready to compute the corresponding confidence intervals. Let us
chose $\alpha_{0}\in(0,1)$ and denote by $z_{1-\frac{\alpha_{0}}{2}}$,
$1-\frac{\alpha_{0}}{2}$ quantile of the standard normal distribution. Using
(11), and the definition of $Q_{k,s}^{*}$ we obtain

$P\left[-z_{1-\frac{\alpha_{0}}{2}}\leq
log(s)\sqrt{\frac{s[k(s+1)-1]}{s^{2}-1}}\left(\alpha Q_{k,s}^{*}-1\right)\leq
z_{1-\frac{\alpha_{0}}{2}}\right]\to 1-\alpha_{0},\quad k\to\infty.$
$P\left[\frac{1}{Q_{k,s}^{*}}-\frac{z_{1-\frac{\alpha_{0}}{2}}}{Q_{k,s}^{*}log(s)}\sqrt{\frac{s^{2}-1}{s[k(s+1)-1]}}\leq\alpha\leq\frac{1}{Q_{k,s}^{*}}+\frac{z_{1-\frac{\alpha_{0}}{2}}}{Q_{k,s}^{*}log(s)}\sqrt{\frac{s^{2}-1}{s[k(s+1)-1]}}\right]\to
1-\alpha_{0},\quad k\to\infty.$

Therefore for any fixed $s=2,3,...$, the corresponding asymptotic confidence
intervals for $\alpha$ when $k\to\infty$ are:

$\left[\frac{log(s)}{log\frac{\mathbf{X}_{(ks,(s+1)k-1)}}{\mathbf{X}_{(k,(s+1)k-1)}}}-\frac{z_{1-\frac{\alpha_{0}}{2}}}{log\frac{\mathbf{X}_{(ks,(s+1)k-1)}}{\mathbf{X}_{(k,(s+1)k-1)}}}\sqrt{\frac{s^{2}-1}{s[k(s+1)-1]}};\frac{log(s)}{log\frac{\mathbf{X}_{(ks,(s+1)k-1)}}{\mathbf{X}_{(k,(s+1)k-1)}}}+\frac{z_{1-\frac{\alpha_{0}}{2}}}{log\frac{\mathbf{X}_{(ks,(s+1)k-1)}}{\mathbf{X}_{(k,(s+1)k-1)}}}\sqrt{\frac{s^{2}-1}{s[k(s+1)-1]}}\right].$
(12)

Simulation study

Let us now depict the rate of convergence of $1/Q_{k,s}^{*}$ for different
values of $\alpha=0.3,0.5,1,1.5$ and $s=2,3,4,5$. Figures 1-2 represent the
dependence of $1/Q_{k,s}^{*}$ and the corresponding confidence intervals, on
$k$. They are plotted via software R [26]. The real values of $\alpha$ are
plotted via straight dense line. In order to visualise the values of the
estimators $\frac{1}{Q_{k,s}^{*}}$ for any of the lines we have simulated 100
samples of $500(s+1)-1$ realizations of Pareto distributed r.v. with c.d.f.
(5), correspondingly for $\alpha=0.3,0.5,1,1.5$. Separately for any fixed
$k=1,2,...$ and $s$ the values of $\frac{1}{Q_{k,s}^{*}}$ are averaged over
these 100 samples and presented correspondingly by dense $(s=2)$, dashed
$(s=3)$, dash-dot $(s=4)$, and dotted $(s=5)$ lines. Then, for any fixed $k$,
and $s$ we have computed and plotted also $0.95$-confidence intervals (red
lines) for $\alpha$, calculated by formula (12) using the averaged values of
$1/Q_{k,s}^{*}$ instead of separate estimators $1/Q_{k,s}^{*}$. We observe
that when $\alpha$ decreases and $s$ increases, the accuracy of the estimators
improves. However, because of the sample size $k(s+1)-1$ increases with $s$ we
can not chose too big $s$ for small samples.

Figure 1: Pareto case: Dependence of $1/Q_{k,s}^{*}$ and the corresponding
confidence intervals(12) on $k$, for different values of $\alpha$ and
$s=2$(solid lines), $s=3$(dashed lines), $s=4$(dash-dot lines), $s=5$(dotted
lines).

Figure 2: Pareto case: Dependence of $1/Q_{k,s}^{*}$ and the corresponding
confidence intervals(12) on $k$, for different values of $\alpha$ and
$s=2$(solid lines), $s=3$(dashed lines), $s=4$(dash-dot lines), $s=5$(dotted
lines).

If we compare these results with those about the well known Hill [10],
t-Hill[24, 11], Deckers-Einmahl-de Haan [6, 7], or Pickands [19] estimators,
described very well in Embrechts et al. [8], we observe that in this case
$1/Q_{k,s}^{*}$ estimator have better properties, especially given a small
sample.

##  4 CONCLUSIONS

The paper points out good properties of couples of central order statistics
for obtaining consistent and asymptotically normal estimators of the parameter
of regular variations of the tail of the c.d.f. of the observed r.v. We
consider more thoroughly Pareto case, where we transform the logarithm of the
fraction of the order statistics in such a way that to obtain at least
asymptotically unbiased and asymptotically efficient estimator. However our
results about the general case show that an analogous approach could be
applied in many other cases of distributions with regularly varying tails of
the c.d.f. For example: Fréchet, Pareto, Log-logistic, Hill-horror among
others. The biggest advantage of the proposed estimators is that they can be
very useful for working with relatively small samples.

##  5 ACKNOWLEDGMENTS

The authors are grateful to the bilateral projects Bulgaria - Austria,
2016-2019, Feasible statistical modelling for extremes in ecology and finance,
BNSF, Contract number 01/8, 23/08/2017.

## References

  * [1] Arnold, B.C., Balakrishnan, N., Nagaraja, H.N.: A first course in order statistics. 54, SIAM (1992)
  * [2] Arnold, B.C.: Pareto distributions. Second Edition, Chapman and Hall$/$CRC Press Taylor & Francis Group, Boca Raton, London, New York (2015)
  * [3] de Haan, L.: On Regular Variation and Its Application to the Weak Convergence of Sample Extremes, Mathematical Centre Tract, 32, Mathematics Centre, Amsterdam, Holland (1970)
  * [4] de Haan, L., Stadtmüller, U.: Generalized regular variation of second order, Journal of the Australian Mathematical Society, 61(3),381–395 (1996)
  * [5] de Haan, L. and Ferreira, A.: Extreme Value Theory: An introduction, Springer Series in Operations Research and Financial Engineering, Springer, New York (2006)
  * [6] Dekkers, Arnold LM, Einmahl, John HJ, de Haan, Laurens: A moment estimator for the index of an extreme-value distribution, The Annals of Statistics, 1833–1855 JSTOR (1989)
  * [7] Einmahl, J.H.J., Fils-Villetard, A., Guillou, A.: Statistics of extremes under random censoring, Bernoulli, 14(1), 207–227 Bernoulli Society for Mathematical Statistics and Probability (2008)
  * [8] Embrechts, P., Klüppelberg, Cl., Mikosch, Th.: Modelling extremal events: for insurance and finance, Springer Science & Business Media, 33, (2013)
  * [9] Fabián, Zd., Stehlík, M.: On robust and distribution sensitive Hill like method, IFAS res. report.43 (2009)
  * [10] Hill, Bruce M and others: A simple general approach to inference about the tail of a distribution, The Annals of Statistics, 5(3), 1163–1174 Institute of Mathematical Statistics(1975)
  * [11] Jordanova, P.K., Pancheva, E.I.: Weak asymptotic results for t-hill estimator, Comptes rendus de l’académie bulgare des sciences, 65(12), 1649–1656 (2012)
  * [12] Jordanova, P.K.: Tails and probabilities for $p$-outliers, Submitted (2019) url=https://arxiv.org/pdf/1902.03810.pdf
  * [13] Jordanova, P.K., Petkova, M.P.: Measuring heavy-tailedness of distributions, AIP Conference Proceedings, 1910(1), 060002 AIP Publishing (2017)
  * [14] Jordanova, P., Stehlík, M., Fabián, Zd., Strelec, L.: On Estimation And Testing For Pareto Tails, Pliska Studia Mathematica Bulgarica, 22(1), 89–108 IMI-BAS, Bulgaria (2013)
  * [15] Mandelbrot, B.: The Pareto-Levy law and the distribution of income, International Economic Review, 2(1), 79–106 University of Pennsylvania (1960)
  * [16] Mosteller, F.: Qn some useful ïnefficients̈tatistics, Ann. Math. Statist., 17(4), 377-408 The Institute of Mathematical Statistics (1946)
  * [17] Nair, N.U., Sankaran, P.G., Balakrishnan, N.: Quantile-based reliability analysis, Birkh$\ddot{a}$user, Basel, (2013)
  * [18] Nevzorov, V.B.: Records: mathematical theory, American Mathematical Society, United States (2001)
  * [19] Pickands III, J.: Statistical inference using extreme order statistics, The Annals of Statistics, 119–131 JSTOR (1975)
  * [20] Resnick, S.I.: Extreme Values, Regular Variation and Point Processes, Springer Series in Operations Research and Financial Engineering, Springer-Verlag, New York (1987)
  * [21] Sgouropoulos, N., Yao, Q., Yastremiz, Cl.: Matching a distribution by matching quantiles estimation, Journal of the American Statistical Association, 110(510), 742–759 Taylor & Francis (2015)
  * [22] Smirnov, N.V.: Limit distributions for the terms of a variational series, Trudy Matematicheskogo Instituta Imeni VA Steklova, 25, 3–60 Russian Academy of Sciences, Steklov Mathematical Institute of Russian Academy of Sciences (1949)
  * [23] Sobel, M.E.: Asymptotic confidence intervals for indirect effects in structural equation models, Sociological methodology, 13, 290-312 (1982)
  * [24] Stehlík, M., Potockỳ, R., Waldl, H., Fabián, Zd.: On the favorable estimation for fitting heavy tailed data, Computational Statistics, 25(3), 485–503 Springer (2010)
  * [25] Stehlík, M., Fabián, Z., Střelec, L.: Small sample robust testing for Normality against Pareto tails, Communications in Statistics - Simulation and Computation, 41(7), 1167-1194, (2012)
  * [26] R: The R Project for Statistical Computing, https://www.r-project.org/ (2019-03-28 2019)
  * [27] Wilks, S.S.: Order statistics, Bull. Amer. Math. Soc., 54(1), Part. 1, 6–50 Russian Academy of Sciences, Steklov Mathematical Institute of Russian Academy of Sciences (1948)


# Harnessing an elastic flow instability to improve the kinetic performance of
chromatographic columns

Fabrice Grittia,, Emily Chenb,, Sujit S. Dattab,
aWaters Corporation, Instrument/Core Research/Fundamental,
34 Maple Street, Milford, MA, 01757, USA
bDepartment of Chemical and Biological Engineering,
Princeton University,
41 Olden Street, Princeton, NJ, 08544, USA Co-first author; Corresponding
author<EMAIL_ADDRESS>author.Corresponding author:
<EMAIL_ADDRESS>
###### Abstract

Despite decades of research and development, the optimal efficiency of slurry-
packed HPLC columns is still hindered by inherent long-range flow
heterogeneity from the wall to the central bulk region of these columns. Here,
we show an example of how this issue can be addressed through the
straightforward addition of a semidilute amount (500 ppm) of a large,
flexible, synthetic polymer (18 MDa partially hydrolyzed polyacrylamide, HPAM)
to the mobile phase (1% NaCl aqueous solution, hereafter referred to as
“brine”) during operation of a 4.6 mm $\times$ 300 mm column packed with 10
$\mu$m BEHTM 125 Å Particles. Addition of the polymer imparts elasticity to
the mobile phase, causing the flow in the interparticle pore space to become
unstable above a threshold flow rate. We verify the development of this
elastic flow instability using pressure drop measurements of the friction
factor versus Reynolds number. In prior work, we showed that this flow
instability is characterized by large spatiotemporal fluctuations in the pore-
scale flow velocities that may promote analyte dispersion across the column.
Axial dispersion measurements of the quasi non-retained tracer thiourea
confirm this possibility: they unequivocally reveal that operating above the
onset of the instability improves column efficiency by significantly reducing
peak asymmetry. These experiments thereby provide a proof-of-concept
demonstration that elastic flow instabilities can be harnessed to mitigate the
negative impact of trans-column flow heterogeneities on the efficiency of
slurry-packed HPLC columns. While this approach has its own inherent
limitations and constraints, our work lays the groundwork for future targeted
development of polymers that can impart elasticity when dissolved in commonly
used liquid chromatography mobile phases, and can thereby generate elastic
flow instabilities to help improve the resolution of HPLC columns.

Keywords: Column efficiency; Viscoelastic mobile phase; Flow instability;
Axial and transverse dispersion; Partially hydrolyzed polyacrylamide.

##  1 Introduction

Previous knowledge pertaining to the simulation of analyte dispersion along
random sphere packings has demonstrated that the minimum reduced plate height
(RPH) of an infinite diameter column—characterized by the absence of border
and wall effects—should be around 0.9 when the particles are fully porous, 0.7
for superficially porous particles, and 0.5 for non-porous particles [1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. From an experimental viewpoint, it is
noteworthy that minimum RPHs close to 1.0 have already been observed, but
under impractical conditions that cannot be implemented in production by
manufacturers of modern HPLC and UHPLC columns. For instance, this was
achieved by 1) slow and gentle dry-packing of very large 200-220 $\mu$m non-
porous particles in a 2.4 cm inner diameter (i.d.) $\times$ 16.6 cm long
column [14]; 2) slurry-packing of capillary columns for a column-to-particle
diameter ratio (or bed aspect ratio) smaller than 7 (12 $\mu$m i.d. capillary
column packed with 5.0 $\mu$m porous particles [15], 21 $\mu$m i.d. capillary
column packed with 5.0 $\mu$m FPPs [16], 3 mm i.d. column packed with 480
$\mu$m glass beads [17, 18]); 3) 4.6 mm i.d. $\times$ 10 cm long column packed
with 2.5 $\mu$m superficially porous particles [19]; and 4) 75 $\mu$m i.d.
$\times$ 1 m long capillary column packed with 2 $\mu$m BEH Particles at very
high pressures (30 000 psi), using highly concentrated slurry suspension (200
g/L) and by applying sonication (80 kHz) [20]. In all of these cases, the bed
structure is found to be quasi-uniform across the entire column diameter (no
or little wall effects) but either the consolidated bed happens to be
unstable, the column efficiency is too low (too large particle size), or the
packing protocol is cumbersome and non-reproducible. To date, all modern HPLC
columns massively adopted by the pharmaceutical industry are slurry-packed at
very high pressure with 2 to 5 $\mu$m particles: their minimum RPHs are
systematically reported in a narrow range from 1.7 to 2.1 for fully porous
particles.

Such a significant performance gap between the idealized and practical RPH
(from 0.9 to 1.7-2.1 for porous particles) has been unequivocally explained
from the flow unevenness of the mobile phase across the column i.d. or the so-
called long-range eddy dispersion in chromatography [21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 15, 31, 32, 33, 34] [Figure 1(a)]. The transverse flow
heterogeneity in packed columns is directly related to the variation of the
local bed density which itself results from the radial distribution of shear
stress applied to the particles during slurry-packing and bed consolidation
[35, 36, 37, 38]. The stress experienced by the particles in the wall region
happens to be higher than that in the bulk central region of the column
because of the presence of the fixed column wall [36]. This effect was
directly observed by focused ion beam scanning electron microscopy for 2
$\mu$m BEH-C18 Particles slurry-packed in 2.1 mm i.d. $\times$ 50 mm stainless
steel column [31] and indirectly revealed by local electrochemical detection
across the outlet column diameter [39, 40, 27, 41].

To date, research efforts to bridge this performance gap have only concerned
small scale (nanoflow) empty channels and open tubular capillary columns by
enhancing the transverse transport of the analyte over the surface
perpendicular to main flow direction. Indeed, the RPH associated with long-
range eddy dispersion is inversely proportional to the transverse dispersion
coefficient [42, 8, 12], where transverse dispersion is only generated by
natural diffusion and transverse convection induced by external stationary
phase obstructions in packed columns [5]. Solutions based on the application
of turbulent flow at high Reynolds numbers [43, 44, 45, 46, 47, 48, 49, 50],
alternating current electro-osmetic flow (AC-EOF) [51, 52, 53, 54], and
acoustic streaming [55] have been explored in separation systems. However,
they cannot be scaled to large-scale systems such as 4.6 mm i.d. HPLC columns
packed with sub-10 $\mu$m particles and they may require active electrical
supply in the case of acoustics and AC-EOF. Therefore, a complete passive
solution applicable to large-volume chromatographic systems is still needed.

One potential passive solution to enhance transverse dispersion of analytes is
the use of an elastic flow instability [56, 57, 58, 59, 60, 61, 62, 63, 64]
generated by the addition of flexible high molecular weight polymers to the
mobile phase, which imparts elasticity to the fluid even in dilute amounts. As
the polymer chains traverse tortuous, curved streamlines within the porous
medium, elastic stresses can accumulate and destabilize the flow, generating
turbulent-like chaotic fluctuations despite negligible inertia (Reynolds
number, $\mathrm{Re}\ll 1$). Recent work by Browne & Datta [61] directly
observed and characterized the onset of such an elastic instability during the
flow of a dilute solution of partially hydrolyzed polyacrylamide (HPAM) in a
three-dimensional (3D) packing of consolidated glass spheres (average diameter
$\approx 330~{}\mu$m). The instability is characterized by persistent
spatiotemporal flow fluctuations above a threshold flow rate or Weissenberg
number, Wi, which describes the ratio of elastic to viscous stresses in the
fluid. Further, the onset of the instability coincides with a large increase
in the flow resistance across the medium, resulting from an increase in energy
dissipation from the elastic instability. Thus, one indicator of an elastic
instability—without having access to the pore-scale flow fields—is an increase
in the macroscopic flow resistance.

Flow fluctuations associated with the onset of an elastic instability have
tantalizing potential for use as a mechanism to enhance mass, solute, and heat
transport in low Reynolds number porous media flows where transport is
typically dominated by slow diffusion. Indeed, previous work has shown that
elastic instabilities may enhance solute mixing and increase the rate of
chemical reactions [65, 66, 64], redirect flow to lower permeability regions
in stratified porous media [63], alter transverse and axial dispersion
properties of colloids [67, 68], and improve heat transfer efficiency [69,
70]. An example is shown in Figure 1(b), which presents confocal fluorescence
images taken within a porous medium, showing enhanced solute mixing and
dispersion generated by an elastic instability. However, to the best of our
knowledge, elastic flow instabilities have yet to be explored in large-scale
porous systems such as slurry-packed chromatographic columns. Thus, research
is needed to investigate whether elastic instabilities can help improve the
resolving power of chromatography columns.

In this work, we demonstrate the first, to our knowledge, proof-of-concept
application of an elastic flow instability in the interparticle volume of
slurry-packed HPLC columns in order to increase the column’s resolving power.
The method consists of adding HPAM to the mobile phase to generate the elastic
instability. First, we measure the bulk rheological properties of a 500 ppm
solution of 18 MDa HPAM dissolved in brine to determine the bulk elastic
properties of the mobile phase and thereby predict the flow conditions
necessary for the elastic instability (Wi $>$ Wic), where Wic represents a
threshold value of the Weissenberg number. Secondly, we verify that such
elastic instabilities arise in a 4.6 mm $\times$ 300 mm column packed with
11.3 $\mu$m BEH 125 Å Particles using friction factor versus Reynolds number
measurements. Finally, we perform axial dispersion measurements of a thiourea
analyte, which demonstrate that the concurrent injection of polymer in the
mobile phase does indeed mitigate the negative impact of trans-column flow
heterogeneities on the efficiency of the HPLC columns when operating above the
onset of elastic instability. Our experiments therefore highlight the
potential of elastic instabilities for improving column performance; however,
the important constraints of detection compatibility, chemical inertness, and
chromatographic speed required by modern routine HPLC applications should be
considered in future research.

##  2 Theoretical details

###  2.1 Mobile phase rheology

We use the Carreau-Yasuda constitutive model to describe the steady-shear
rheology of the polymer solution [71]. In particular, we fit our measurements
of the dynamic shear viscosity using the relation:

$\eta(\dot{\gamma})=\eta_{\infty}+(\eta_{0}-\eta_{\infty})\left(1+\left(\frac{\dot{\gamma}}{\dot{\gamma}_{c}}\right)^{a}\right)^{\frac{n-1}{a}},$
(1)

where $\dot{\gamma}$ is the imposed shear rate, $\eta_{0}$ is the zero shear
viscosity, $\eta_{\infty}$ is the infinite shear viscosity, $\dot{\gamma}_{c}$
is the onset shear rate for shear-thinning, $a$ is a parameter ($a\simeq 2$)
that controls the transition from the low shear rate Newtonian plateau to the
shear thinning regime, and $n$ is the power law index ($0\leq n\leq 1$). At
low shear rates, the Carreau-Yasuda fluid behaves like a Newtonian fluid with
constant viscosity $\eta_{0}$; as the shear rate increases beyond
$\dot{\gamma}_{c}$, the fluid begins to shear-thin with a power law decay
described by $n$. At high shear rates, the Carreau-Yasuda fluid again behaves
like a Newtonian fluid with viscosity $\eta_{\infty}$. Physically, the shear-
thinning of the polymer solution is due to microstructural rearrangement and
alignment of the polymer chains under flow, leading to a reduction in the
solution viscosity [72, 71, 73, 74].

In addition to the shear viscosity, we simultaneously measure the first normal
stress difference, $N_{1}(\dot{\gamma})=\tau_{xx}-\tau_{yy}$, where
$\tau_{xx}$ is the streamwise normal stress in the direction of flow and
$\tau_{yy}$ is the transverse normal stress (i.e. radial direction in a
cylindrical geometry). We fit the experimental $N_{1}$ flow curves to a power
law, $N_{1}(\dot{\gamma})=K_{N_{1}}\dot{\gamma}^{n_{N_{1}}}$, where
$K_{N_{1}}$ is the material prefactor with units $\mathrm{Pa\cdot
s^{n_{N_{1}}}}$ and $n_{N_{1}}$ is the power law index.

###  2.2 Flow through porous HPLC columns

The flow is described by two key dimensionless parameters: the Reynolds number
quantifying the relative strength of inertial to viscous stresses, and the
Weissenberg number quantifying the relative strength of elastic to viscous
stresses. The former is defined as $\mathrm{Re}=\frac{\rho
u_{S}d_{p}}{\eta(1-\epsilon_{e})}$, where $\rho$ is the density of the fluid,
$u_{S}$ the average superficial linear velocity (flow rate per unit area of
the empty column), $d_{p}$ the average particle size, $\eta$ the fluid dynamic
shear viscosity, and $\epsilon_{e}\simeq$ 40% the external porosity of the
column packing. The latter is defined as $\mathrm{Wi}=\tau\dot{\gamma}$, where
$\tau$ is the longest relaxation time of the polymer solution,
$\dot{\gamma}=\frac{16}{3}\frac{u_{S}}{\epsilon_{e}d_{\mathrm{tube}}}$ is the
average shear rate in the pore space [75], and
$d_{\mathrm{tube}}=0.42\frac{\epsilon_{e}}{1-\epsilon_{e}}d_{p}$ is the
equivalent hydraulic diameter in a chromatographic column [76].

The pressure drop per unit of column length can be described using the
standard Ergun equation [77], modified to also include energy dissipated by
the elastic instability following our previous work [61, 63]:

$\frac{\Delta
P}{L}=K_{c}\frac{(1-\epsilon_{e})^{2}}{\epsilon_{e}^{3}d_{p}^{2}}\eta(\dot{\gamma})u_{S}+1.75\frac{1-\epsilon_{e}}{\epsilon_{e}^{3}d_{p}}\rho
u_{S}^{2}+\frac{\Delta P_{EI}}{L},$ (2)

where $\Delta P$ is the pressure drop along the column, $L$ its length,
$K_{c}$ an empirical constant ($\simeq$ 180 for the specific tortuosity and
surface area of consolidated random sphere packings), and $\Delta P_{EI}\geq
0$ the added pressure drop due to the elastic instability. The first and
second terms on the right-hand-side of Eq. 2 account for the resistance to
fluid flow due to dissipation by viscous and inertial forces, respectively, in
the stable laminar base case; for the range of Reynolds numbers investigated
in this work ($\mathrm{Re}\ll 1$), the second term in Eq. 2 is always
negligible, and we therefore do not consider it hereafter. The third term on
the right-hand-side of Eq. 2 generally accounts for the added resistance
imparted by the elastic flow instability; it can be calculated by considering
the viscous dissipation associated with spatio-temporal fluctuations of the
strain-rate tensor [61], and increases with increasing $u_{S}$ when Wi $>$
Wic.

Following convention, we then define the friction factor as:

$f=\frac{\Delta P}{L}\frac{d_{p}}{\rho
u_{S}^{2}}\frac{\epsilon_{e}^{3}}{1-\epsilon_{e}}.$ (3)

In the case of laminar flow ($\mathrm{Re}\ll 1$ and $\mathrm{Wi}\ll 1$) with a
Newtonian fluid ($\eta$ is shear rate independent), the combination of Eq. 3
and simplified Eq. 2 gives

$\log(f)=\log(K_{c})-\log(Re).$ (4)

Therefore, in a log-log representation, the friction factor is expected to
decrease linearly with increasing Reynolds number; any deviation from this
linearity reveals a change in flow resistance relative to Newtonian fluids
under laminar flow. This deviation can be due to e.g., a modification of the
flow path tortuosity, a change in external porosity $\epsilon_{e}$, turbulent
flow at $\mathrm{Re}>2000$, or an elastic instability at Wi $>$ Wic.

###  2.3 Resolution of HPLC columns

The resolution of a chromatographic column can be adversely impacted by the
existence of long-range velocity biases from the bulk center region to the
wall region [Figure 1(a)]. However, the transverse dispersion of analyte
across the column diameter [5] mitigates the negative impact of such radial
flow heterogeneities on the column efficiency because it permits the analyte
to be exchanged between different flow streamlines.

We quantify this intuition using a simple calculation, defining the absolute
radial position, $r$; column inner radius, $r_{c}$; dimensionless radial
position across the column, $x=\frac{r}{r_{c}}$; column plate height
associated with long-range eddy dispersion, $H_{\mathrm{eddy,long}}$;
effective diffusion coefficient of the analyte in the packed bed volume in the
absence of convection, $D_{\mathrm{eff}}$; local migration velocity of the
analyte at radial position $x$, $u_{R}(x)$; radially-averaged migration linear
velocity of the analyte, $\bar{u}_{R}$; and the local transverse dispersion
coefficient of the analyte at radial position $x$, $D_{t}(x)$. When the number
of analyte exchanges becomes very large (infinitely long column, asymptotic
dispersion regime), the column plate height is given by extending [78] the
general dispersion theory of Aris for open tubular columns to packed beds by
considering their specific radial velocity profile [79, 8, 31] [Figure 1(a)].
In particular,

$H_{\mathrm{eddy,long}}=A_{0}\frac{4r_{c}^{2}}{D_{\mathrm{eff}}}\bar{u}_{R},$
(5)

where the coefficient [8] $A_{0}=\frac{I_{1}-2I_{2}+I_{3}}{2}$ with

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{\Phi^{2}(x)}{2x\Psi(x)}\mathrm{d}x$ (6)
$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{x\Phi(x)}{2\Psi(x)}\mathrm{d}x$ (7)
$\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{x^{3}}{2\Psi(x)}\mathrm{d}x,$ (8)

$\Phi(x)=\int_{0}^{x}2x^{{}^{\prime}}\phi(x^{{}^{\prime}})\mathrm{d}x^{{}^{\prime}}$,
$\phi(x)=\frac{u_{R}(x)}{\bar{u}_{R}}$, and
$\Psi(x)=\frac{D_{t}(x)}{D_{\mathrm{eff}}}$. Hence, the plate height
$H_{\mathrm{eddy,long}}$ can be reduced by promoting transverse dispersion,
i.e., increasing $D_{t}(x)$, which appears in the denominators of the right-
hand-side integrals in Equations 6, 7, and 8, thereby reducing $A_{0}$.
Promoting transverse dispersion of the analyte is therefore a useful way to
limit the negative impact of radial flow heterogeneity on the performance of
chromatographic separation systems [80, 81, 82, 83]. This is the main
rationale of our work, where our objective is to determine whether an elastic
flow instability induced by the addition of a well-chosen polymer in the
mobile phase can help mitigate the negative impacts of radial flow
heterogeneities on column efficiency by promoting transverse dispersion.

##  3 Experimental details

###  3.1 Chemicals

We use HPLC grade water from Fisher Scientific (Fair Lawn, NJ, USA) as the
solvent; the added sodium chloride salt (NaCl) and thiourea ($>$99% purity)
are obtained from Sigma Aldrich (Milwaukee, WI, USA). The polymer additive
used is 18 MDa partially hydrolyzed polyacrylamide (HPAM, 30% hydrolysis)
obtained from Polysciences, Inc (Warrington, PA, USA).

###  3.2 Instrumentation

We acquire all chromatographic data on the Arc SystemTM (Waters, Milford, MA,
USA). The standard configuration of this instrument includes a multi-solvent
delivery system (the quaternary solvent manager, QSM), a gradient
proportioning valve (GPV), a primary and accumulator pump heads, a two-paths
flow mixer, an auto-sampler with a 30 $\mu$L sample loop, a two-column
selection valve, a two-column semi-adiabatic oven, and a single wavelength
transmission UV-detector (8 $\mu$L cell volume). In this work, we place the
QSM pump with an isocratic solvent manager (ISM) pump equipped with primary
and accumulator pump heads, but without any mixer.

The instrument is run by the EmpowerTM 3 Chromatography Data Software (Waters,
Milford, MA, USA). The extra-column volumes are 0.042 mL from the auto-sampler
needle seat to the UV cell. The offset time between the moment the
chromatogram starts being recorded and the moment the injection valve actually
switches is 0.7 s. All measurements are carried out at room temperature
(297$\pm$1 K). The wavelength of the UV detector is set at 272 nm, with a
sampling rate of 10 Hz.

###  3.3 Columns

The research prototype column (4.6 mm $\times$ 300 mm) is home-packed with
11.3 $\mu$m fully porous BEH125 Particles prepared on-site (Waters, Milford,
MA). We determine the characteristic particle diameter, 11.3 $\mu$m, by
identifying the median of the particle size distribution measured using a
Coulter counter. The external porosity of the column is 42.3%, as determined
using permeability measurements (pure water, $T=297$ K, $\eta$=0.91
mPa$\cdot$s, $K_{c}$=180) performed at flow rates of 0.1, 0.25, 0.50, 0.75,
and 1.50 mL/min.

###  3.4 Bulk rheology of the HPAM-modified mobile phase

The polymer-amended mobile phase is a 500 ppm aqueous solution of the 18 MDa
HPAM polymer with 1 wt% NaCl added. To mix the polymer solution, we gradually
add the HPAM in stepwise increments to water while stirring gently with a
magnetic stir bar for 24 h. We then add the NaCl salt in stepwise increments
and allow the solution to mix for another 24 h. We then let the solution rest
undisturbed to allow for bubbles to dissipate. All polymer solutions are used
within 1 month of preparation. The overlap concentration describing the
concentration at which adjacent HPAM polymer coils begin to interact in
solution is $c^{*}=200$ ppm [84]. Our experimentally-chosen concentration of
500 ppm thus corresponds to the semi-dilute regime, with $c/c^{*}=2.5$.

We characterize the rheology of the polymer-amended mobile phase using a
stress-controlled Anton Paar MCR501 rheometer (Ashland, VA, USA) fitted with a
truncated cone-plate geometry (CP50-2: 50 mm diameter, 2∘, 53 $\upmu$m gap),
operating at 298 K. The measurements are presented and detailed in §4.1. We
measure steady-state flow curves for the polymer solution by ramping down over
shear rates of $\dot{\gamma}=100-0.1\>\mathrm{s^{-1}}$, with the time-per-
shear rate set by the instrument. We measure four replicates of separate
samples from the same polymer solution batch and report the standard deviation
as error bars.

###  3.5 Column friction factor and efficiency measurements

We characterize the performance of our prototype column by measuring the
friction factor $f$ and efficiency $N$ at a temperature $T=297$ K at various
stages of operation. To determine $f$, we perform pressure drop measurements
over a range of flow rates and apply Eq. 3. To determine $N$, we measure the
first ($\mu_{1}$) and second ($\mu_{2}^{{}^{\prime}}$) central moments of the
concentration distribution recorded at a detection wavelength of 272 nm (10 Hz
sampling rate) for a 5 $\mu$L volume of the small molecule thiourea injected
at a flow rate of $F_{v}=0.5$ mL/min. When the thiourea is dissolved in brine,
this flow rate corresponds to conditions of laminar steady flow in the column;
however, when the thiourea is dissolved in HPAM-amended brine (step #3 below),
this flow rate is well above the threshold of elastic instability, as
described further in §4. To account for retention and dispersion taking place
in the HPLC system (injection needle, needle seat capillary, inlet and outlet
connecting tubes, and detection cell volumes), we correct the moments
$\mu_{1}$ and $\mu_{2}^{{}^{\prime}}$ by those measured in the same conditions
but after replacing the column with a zero dead volume union connector,
$\mu_{1,ex}$ and $\mu_{2,ex}^{{}^{\prime}}$; the efficiency is thus given by
$N=\frac{\left(\mu_{1}-\mu_{1,ex}\right)^{2}}{\mu_{2}^{{}^{\prime}}-\mu_{2,ex}^{{}^{\prime}}}$.

We thereby characterize the column performance through the following series of
experiments:

  1. 1.

Equilibrate the column with a mobile phase of just brine for 1.5 h at a flow
rate of 1 mL/min, corresponding to $\simeq$18 empty column volumes. We then
increase the flow rate stepwise (0.1 mL/min step, each plateau lasting 2 min)
from 0.1 to 2.0 mL/min. This measurement thereby provides the reference flow
resistance of the column for a Newtonian fluid. We also use the thiourea
concentration measurements to determine the baseline efficiency of the
pristine column before contact with the HPAM polymer, $N=N_{\mathrm{Ref}}$.

  2. 2.

Equilibrate the column with the HPAM-amended mobile phase for 15 h at a flow
rate of 0.1 mL/min, again corresponding to $\simeq$ 18 empty column volumes.
We then wash the column abundantly with the polymer-free brine and repeat the
thiourea concentration measurements in brine to determine how the column
efficiency is altered by contact with the HPAM polymer,
$N=N_{\mathrm{Before}}$.

  3. 3.

Again equilibrate the column with the HPAM-amended mobile phase for 15 h at a
flow rate of 0.1 mL/min, again corresponding to $\simeq$ 18 empty column
volumes, and then increase the flow rate stepwise from 0.0125 to 0.025, 0.05,
0.075, 0.100, 0.125, 0.150, 0.175, 0.200, 0.225, 0.250, and 0.275 mL/min. For
each 60 min-long flow step, the flow rate increases linearly with time during
the first 5 min and is then maintained constant over the remaining 55 min in
order to reach a steady pressure drop along the column. This measurement
thereby provides the flow resistance of the column for the polymer-amended
fluid. We also use the thiourea concentration measurements, this time in the
HPAM-amended mobile phase, to determine the column efficiency in the
elastically-unstable flow state, $N=N_{\mathrm{HPAM}}$.

  4. 4.

Wash and re-equilibrate the column with the polymer-free brine for 3 h at a
flow rate of $F_{v}=0.5$ ml/min, again corresponding to $\simeq$18 empty
column volumes, until the pressure drop returns to a new steady state
pressure. The flow rate is then increased stepwise (each plateau lasting 2
min) from 0.0125 to 0.025, 0.05, 0.075, 0.100, 0.125, 0.150, 0.175, 0.200,
0.225, 0.250, and 0.275 mL/min. This measurement provides the flow resistance
of the column after using the polymer-amended fluid. We repeat the thiourea
concentration measurements in polymer-free brine to determine the column
efficiency after using the polymer-amended fluid, $N=N_{\mathrm{After}}$.
Together, these measurements characterize whether the column is modified in
any way by the polymer.

##  4 Results

###  4.1 Rheology

The results of the rheology measurements of the HPAM polymer-amended mobile
phase are shown in Figure 2. Figure 2(a) shows the steady-shear viscosity flow
curve, $\eta(\dot{\gamma})$. The polymer solution is already strongly shear-
thinning at the lowest measurable shear rates in the rheometer. The solid
curve represents the best fit of the standard Carreau-Yasuda model (Eq. 1)
with $\eta_{0}=1.47\>\mathrm{Pa\cdot s}$, $\eta_{\infty}=2.8\>\mathrm{mPa\cdot
s}$, $\dot{\gamma}_{c}=0.0033\>\mathrm{s^{-1}}$, and $n=0.05$, with $a$ set to
2. We extrapolate this fit to larger shear rates $\sim
10^{4}\>\mathrm{s^{-1}}$ to compare to shear rates typically encountered in
HPLC applications; as shown by the red line in Figure 2(a), the corresponding
polymer solution viscosity expected for our solution under typical HPLC
operating conditions lies in the high shear rate Newtonian plateau,
$\eta_{\infty}$. In particular, in the experiments using thiourea
concentration measurements to determine column efficiency, we use an imposed
constant flow rate of $F_{v}=0.5$ mL/min, which corresponds to a shear rate of
$\dot{\gamma}\approx 2000\>\mathrm{s^{-1}}$ and $\mathrm{Wi}=360$. This Wi is
well above the threshold of elastic instability, Wi${}_{c}\approx 85$,
determined from the onset of flow-thickening in the friction factor
measurements in Figure 3(b).

Figure 2(b) shows the first normal stress difference, $N_{1}(\dot{\gamma})$,
and corresponding power law fit,
$N_{1}(\dot{\gamma})=K_{N_{1}}\dot{\gamma}^{n_{N_{1}}}$ with
$K_{N_{1}}=5.3\;\mathrm{Pa\cdot s}^{n_{N_{1}}}$ and $n_{N_{1}}=0.85$. The
lower limit of the measured first normal stress difference, $N_{1}$, is
approximately 10 Pa due to the normal force sensitivity of the instrument,
limiting the range of reported shear rate values for $N_{1}$. The growth of
$N_{1}$ with increasing shear rate results from the development of elastic
stresses as polymers undergo shearing-driven alignment in the curved geometry.

We could not obtain shear relaxation time measurements for the polymer-amended
mobile phase because of instrument sensitivity limitations for HPAM
concentrations below 1000 ppm. Therefore, to estimate the characteristic
relaxation time of the polymer solution, we first calculate the Rouse and Zimm
time scales for dilute polymer solutions. The Rouse time scale, which neglects
intramolecular hydrodynamic interactions, is given by [85, 86, 87, 73]
$\tau_{R}=\frac{6}{\pi^{2}}\frac{\eta_{s}[\eta]M}{RT}=12.4$ ms, where
$\eta_{s}$ is the solvent viscosity, $[\eta]$ is the intrinsic viscosity, $M$
is the polymer molecular weight, $R$ is the ideal gas constant, and $T$ is
temperature. The Zimm time scale, which incorporates intramolecular
hydrodynamic interactions, is given by [84]
$\tau_{Z}=0.422\frac{\eta_{s}[\eta]M}{RT}=8.6$ ms. However, these time scales
describe dilute polymer solutions, whereas our polymer-amended mobile phase is
in the semidilute regime; the Rouse and Zimm time scales therefore represent
lower bounds to the true solution relaxation time. Moreover, previous shear
stress relaxation measurements [84] on a more concentrated solution of 1000
ppm HPAM in brine provide an upper-bound estimate of $\approx 200$ ms for the
true relaxation time. Hence, we expect that the solution relaxation time is
somewhere in the range $\sim 10-200$ ms; for simplicity, we use $\tau=200$ ms
in calculating the Weissenberg number, but note that the true value of Wi may
be lower by as much as a factor of 20.

###  4.2 Column flow resistance

Figure 3 summarizes our measurements of the flow resistance, quantified by the
friction factor $f$, as a function of Reynolds number Re in the lab-scale test
column for the different mobile phases used in the experiments: polymer-free
brine and 500 ppm HPAM-amended brine. Figure 3(a) shows the reference flow
resistance before the packed BEH125 Particles are exposed to the HPAM polymer
solution (step #1 in §3.5). As expected, the polymer-free brine is a Newtonian
fluid of constant viscosity and the flow is in the laminar regime (0.002
$<\mathrm{Re}<$ 0.044): the experimental (circles) and theoretical (Eq. 4 with
$K_{c}=180$, solid line) plots of $\log f$ versus log Re are nearly
indistinguishable.

We observe two key differences when using the HPAM-amended mobile phase (step
#3 in §3.5), shown by the symbols in Figure 3(b). First, the friction factor
is shifted to higher values at the lowest Re, with the permeability constant
$K_{c}$ increasing from 180 to 420 [red curve]. One possible explanation for
this shift is that the HPAM adsorbs as a $\approx 250~{}\upmu$m-thick film
uniformly over the entire surface of the BEH125 Particles, reducing the
overall packing porosity $\epsilon_{e}$ from $\approx 42\%$ to $\approx 35\%$.
If true, the internal volume of the BEH Particles would become inaccessible
and the elution times of the non-retained peaks of thiourea and chloride
anions would be as small as 3.45 min; however, we consistently measure their
mean elution time as 7.84 min, as detailed below in §4.3. The measurement thus
suggests that HPAM adsorption does not completely coat the BEH125 Particles
and render them impermeable, but instead simply constricts the narrowest
interparticle flow paths, augmenting the tortuosity of the flow-through
channels by 53%. This result is consistent with the findings of more
simplified microfluidic experiments [88]. Second, despite the shear-thinning
behavior of the bulk mobile phase [Figure 2(a)], we observe flow-thickening at
sufficiently large flow rates: As Re increases from 0.0001 to 0.0049 (Wi
increases from $\approx$9 to 200), the measured friction factor exceeds the
expected Newtonian behavior, indicated by the upper double-headed arrow in
Figure 3(b). This enhanced flow resistance does not arise due to inertial
effects, given that $\mathrm{Re}\ll 1$. Instead, it reflects the onset of an
elastic instability induced by the addition of HPAM [61]. We thus determine
the critical Weissenberg number for the onset of the elastic instability from
the onset of flow-thickening in the friction factor as
$\mathrm{Wi}_{\mathrm{c}}=85$.

Finally, repeating the flow resistance experiments after abundantly washing
the column with brine solution (step #4 in §3.5) provides a way to assess
irreversible changes to the column caused by interactions with HPAM. As shown
in Figure 3(c), the flow resistance of the Newtonian brine solution remains
shifted vertically relative to that measured prior to HPAM introduction, with
$K_{c}$ remaining equal to 420 [red curve]—suggesting that HPAM adsorption to
the external BEH surfaces is irreversible. This suggestion is corroborated by
the thiourea elution experiments described below in §4.3. Nevertheless, as
expected, the plot shows Newtonian behavior in the absence of HPAM in the
mobile phase.

###  4.3 Mass transfer within the column

We evaluate the column efficiency by measuring chromatograms of the non-
retained compound thiourea. Before the packed BEH125 Particles are exposed to
the HPAM polymer solution (step #1 in §3.5), the thiourea chromatogram shows a
well-defined, approximately symmetric peak with an average retention time of
7.84 min [blue curve in Figure 4, $N_{\mathrm{Ref}}$=8050]. After the
particles are incubated with the HPAM-amended mobile phase (step #2 in §3.5),
the chromatogram—obtained again with thiourea dissolved in the polymer-free
brine mobile phase—shows the same average retention time of 7.84 min, but with
a broader peak and lower efficiency [black curve in Figure 4,
$N_{\mathrm{Before}}$=160]. The same curve, normalized by its peak intensity,
is shown by the black curve in Figure 5. Notably, the unchanged average
retention time demonstrates that the thiourea can still penetrate the same
internal mesoporous volume of the BEH125 Particles; that is, the HPAM
adsorption indicated by the flow resistance measurements described in §4.2
does not completely coat the BEH125 Particles and render them impermeable.
Instead, the broader peak shape and lower column efficiency suggests that
irreversibly-adsorbed HPAM occludes some of the mesopore openings between
particles, augmenting the tortuosity of these channels and giving rise to a
broader flow velocity distribution. At the molecular scale, the collision
events between thiourea molecules present in the external mobile phase and the
surface of the particles rarely lead to an effective absorption of thiourea
into the particles. The molecules of thiourea that elute first (reflected by
the sharp front of the black chromatogram in Figure 4) are those mostly
bouncing against the particle surfaces and transported along with fast
velocity streamlines. Those eluting last (reflected by the rear tailing of the
black peak in Figure 4) are the few molecules localized within the particles
that are slowly released and transported along with slow velocity streamlines.

Even though the column performance is irreversibly affected by HPAM
adsorption, the thiourea chromatograms measured in the HPAM-amended mobile
phase at $\mathrm{Wi}>\mathrm{Wi}_{\mathrm{c}}$ (step #3 in §3.5)
unambiguously demonstrate that the elastic instability promotes transverse
dispersion in the column. The experimental results are summarized in Figure 5;
the normalized chromatograms obtained prior to and during flow of the HPAM-
amended mobile phase are shown by the black and red curves, respectively. In
the presence of the elastic instability, the elution peak shape returns to
quasi-Gaussian (red curve), and the column efficiency more than doubles from
$N_{\mathrm{Before}}$=160 to $N_{\mathrm{HPAM}}$=420—indicating that velocity
biases are averaged by increasing the local transverse dispersion coefficient
of thiourea ($D_{t}(x)$, Eq. 5, asymptotic dispersion regime). As a final
confirmation of this point, we re-measure the thiourea chromatogram in
polymer-free brine after washing and re-equilibrating the column with just
brine (step #4 in §3.5). As shown by the blue curve in Figure 5, the
chromatogram returns to its broader peak shape with a lower efficiency
$N_{\mathrm{After}}$=180, indicating that the increase in transverse
dispersion is indeed generated by the elastic instability imparted by HPAM.

##  5 Conclusion

The fundamental physics of elastic flow instabilities generated by flexible
high molecular weight polymers has been studied for decades [56, 57, 58, 59,
60, 61, 62, 63, 64]. However, the use of such instabilities in applications
has remained limited. Recent work [64] showed that the chaotic flow generated
by an elastic instability in model 3D glass bead packings—at $\mathrm{Re}\ll
1$, where the flow is typically laminar—can dramatically enhance solute
transverse dispersion, suggesting that such instabilities can help mitigate
the negative impact of flow heterogeneities in chromatography. Here, we
experimentally demonstrated a proof of this concept—for the first time, to the
best of our knowledge. We generated the elastic instability by adding 500 ppm
of 18 MDa HPAM polymer to a brine mobile phase in a 4.6 mm i.d. $\times$ 300
mm long HPLC column packed with 11.3 $\mu$m 125 Å BEH fully porous
organic/inorganic hybrid Particles. By measuring the friction factor of the
column before, during, and after flow of the polymer-amended mobile phase, we
determined the conditions under which the elastic instability arises. Then, by
measuring the mass transfer of a non-retained small molecule analyte in the
column before, during, and after flow of the polymer-amended mobile phase, we
showed that the elastic instability causes the elution peak shape to become
sharper and the column efficiency to more than double, owing to the increase
of the transverse dispersion coefficient of the analyte across the column
diameter. Our work thus paves the way towards closing the gap between the
current efficiency of slurry-packed columns and the maximum theoretical
efficiency of ideally-packed column without wall effects, simply by adding
flexible polymers to the mobile phase.

While our study highlights the promise of elastic instabilities in improving
chromatography, this approach necessarily has limitations and challenges that
remain to be addressed in future work to enable broad implementation. Most
prominently, our measurements of the increased friction factor [red curve,
Figure 3(c)] and reduced column efficiency [Figures 4–5] indicate that the
HPAM polymer irreversibly adsorbs to the external surfaces of the packed
BEH125 Particles. Hence, more research is needed to develop “polymer
enhancers” that can routinely be added to the mobile phase to minimize flow
heterogeneities in chromatography with the following properties:

  * •

Sufficient elasticity to generate the flow instability, while still not
generating a prohibitively large pressure drop across the column, thereby
ensuring acceptable flow speeds in both preparative and analytical liquid
chromatography in addition to having a practically-feasible onset flow rate to
generate the elastic instability.

  * •

Compatibility with the use of common mobile phase compositions in RPLC, HILIC,
IEX, and SEC. In this work, for instance, no organic modifier was added to the
mobile phase while it is commonly used in RPLC and HILIC.

  * •

Minimal irreversible adsorption to the surfaces of the packed particles, and
similarly, chemically inert with respect to the analyte.

  * •

Minimal scission of the polymer chains as they are transported through the
tortuous pore space.

  * •

Does not interfere with analyte detection (via optical, mass, and fluorescence
means). Here, the HPAM polymer was suitable for optical detection wavelengths
only larger than 270 nm.

  * •

Increases transverse dispersion via an elastic instability, but does not
simultaneously increase axial dispersion to a prohibitive extent.

##  6 Acknowledgements

It is a pleasure to acknowledge funding support from the Princeton Center for
Complex Materials (PCCM), a National Science Foundation (NSF) Materials
Research Science and Engineering Center (MRSEC; DMR-2011750), as well as the
Camille Dreyfus Teacher-Scholar Program. We also acknowledge the use of
Princeton’s Imaging and Analysis Center (IAC), which is partially supported by
the PCCM.

##  7 Note

BEH, ACQUITY, UPLC, Arc, and Empower are trademarks of Waters Technologies
Corporation.

## References

  * [1] J. Delgado, A critical review of dispersion in packed beds, Heat Mass Transfer 42 (2006) 279–310.
  * [2] D. Hlushkou, , U. Tallarek, Transition from creeping via viscous-inertial to turbulent flow in fixed beds, J. Chromatogr. A 1126 (2006) 70–85.
  * [3] S. Khirevich, A. Daneyko, A. Höltzel, A. Seidel-Morgenstern, U. Tallarek, Statistical analysis of packed beds, the origin of short-range disorder, and its impact on eddy dispersion, J. Chromatogr. A 1217 (2010) 4713–4722.
  * [4] S. Khirevich, A. Höltzel, S.-M. A., U. Tallarek, Time and length scales of eddy dispersion in chromatographic beds, Anal. Chem. 81 (2009) 7057–7066.
  * [5] A. Daneyko, A. Höltzel, S. Khirevich, U. Tallarek, Influence of the particle size distribution on hydraulic permeability and eddy dispersion in bulk packings, Anal. Chem. 83 (2011) 3903–3910.
  * [6] A. Daneyko, D. Hlushkou, V. Baranau, S. Khirevich, A. Seidel-Morgenstern, U. Tallarek, Computational investigation of longitudinal diffusion, eddy dispersion, and trans-particle mass transfer in bulk, random packings of core–shell particles with varied shell thickness and shell diffusion coefficient, J. Chromatogr. A 1407 (2015) 139–156.
  * [7] F. Gritti, G. Guiochon, Mass transfer kinetics, band broadening and column efficiency, J. Chromatogr. A 1221 (2012) 2–40.
  * [8] F. Gritti, G. Guiochon, Perspectives on the evolution of the column efficiency in liquid chromatography, Anal. Chem. 85 (2013) 3017–3035.
  * [9] F. Gritti, G. Guiochon, The van deemter equation: Assumptions, limits, and adjustment to modern high performance liquid chromatography, J. Chromatogr. A 1302 (2013) 1–13.
  * [10] F. Gritti, On the relationship between radial structure heterogeneities and efficiency of chromatographic columns, J. Chromatogr. A 1533 (2018) 112–126.
  * [11] F. Gritti, M. Wahab, Understanding the science behind packing high-efficiency columns and capillaries: Facts, fundamentals, challenges, and future directions, LC-GC North Am. 36 (2018) 82–98.
  * [12] F. Gritti, A stochastic view on column efficiency, J. Chromatogr. A 1540 (2018) 55–67.
  * [13] K. Broeckhoven, G. Desmet, Approximate transient and long time limit solutions for the band broadening induced by the thin sidewall-layer in liquid chromatography columns, J. Chromatogr. A 1172 (2007) 25–39.
  * [14] P. Magnico, M. Martin, Dispersion in the interstitial space of packed columns, J. Chromatogr. 517 (1990) 31–49.
  * [15] S. Hsieh, J. Jorgenson, Preparation and evaluation of slurry-packed liquid chromatography microcolumns with inner diameters from 12 to 33 $\mu$m, Anal. Chem. 68 (1996) 1212–1217.
  * [16] R. Kennedy, J. Jorgenson, Preparation and evaluation of packed capillary liquid chromatography columns with inner diameters from 20 to 50 $\mu$m, Anal. Chem. 61 (1989) 1128–1135.
  * [17] J. Knox, J. Parcher, Effect of column to particle diameter ratio on the dispersion of unsorbed solutes in chromatography, Anal. Chem. 76 (2004) 5777–5786.
  * [18] J. H. Knox, Band dispersion in chromatography - a niew view of a-term dispersion, J. Chromatogr. A 831 (1999) 3–15.
  * [19] F. Gritti, I. Leonardis, D. Shock, P. Stevenson, A. Shalliker, G. Guiochon, Performance of columns packed with the new shell particles, kinetex-c18, J. Chromatogr. A 1217 (2010) 1589.
  * [20] J. Godinho, A. Reising, U. Tallarek, J. Jorgenson, Implementation of high slurry concentration and sonication to pack high-efficiency, meter-long capillary ultrahigh pressure liquid chromatography columns, J. Chromatogr. A 1462 (2016) 165–169.
  * [21] M. Kaminski, Simple test for determination of the degree of distortion of the liquid-phase flow profile in columns for preparative liquid chromatography, J. Chromatogr. 589 (1992) 61–70.
  * [22] T. Farkas, M. Sepaniak, G. Guiochon, Radial distribution of the flow velocity, efficiency, and concentration in a wide hplc column, AIChE J. 43 (1997) 1964–1974.
  * [23] D. Roper, E. Lightfoot, Estimating plate heights in stacked-membrane chromatography by flow reversal, J. Chromatogr. A 702 (1995) 69–80.
  * [24] H. Guo, D. Frey, Interpreting the difference between conventional and bi-directional plate-height measurements in liquid chromatography, J. Chromatogr. A 1217 (2010) 6214–6229.
  * [25] J. Baur, E. Kristensen, R. Wightman, Radial dispersion from commercial high-performance liquid chromatography columns investigated with microvoltametric electrodes, Anal. Chem. 60 (1988) 2334–2338.
  * [26] J. Abia, K. Mriziq, G. Guiochon, Radial heterogeneity of some analytical columns used in high-performance liquid chromatography, J. Chromatogr. A 1216 (2009) 3185–3191.
  * [27] F. Gritti, G. Guiochon, Relationship between trans-column eddy diffusion and retention in liquid chromatography: Theory and experimental evidence, J. Chromatogr. A 1217 (2010) 6350–6365.
  * [28] D. Astrath, F. Lottes, D. Vu, W. Arlt, E. Stenby, Experimental investigation of liquid chromatography columns by means of computed tomography, Adsorption 13 (2007) 9–19.
  * [29] F. Gritti, G. Guiochon, Impact of retention on trans-column velocity biases in packed columns, AIChE J. 56 (2010) 1495–1509.
  * [30] F. Gritti, G. Guiochon, New insights on mass transfer kinetics in chromatography, AIChE J. 57 (2011) 333–345.
  * [31] A. Reising, S. Schlabach, V. Baranau, D. Stoeckel, U. Tallarek, Analysis of packing microstructure and wall effects in a narrow-bore ultrahigh pressure liquid chromatography column using focused ion beam scanning electron microscopy, J. Chromatogr. A 1513 (2017) 172–182.
  * [32] S. Bruns, U. Tallarek, Physical reconstruction of packed beds and their morphological analysis: Core–shell packings as an example, J. Chromatogr. A 1218 (2011) 1849–1860.
  * [33] S. Bruns, J. Grinias, L. Blue, J. Jorgenson, U. Tallarek, Morphology and separation efficiency of low-aspect-ratio capillary ultrahigh pressure liquid chromatography columns, Anal. Chem. 84 (2012) 4496–4503.
  * [34] R. Shalliker, H. Ritchie, Segmented flow and curtain flow chromatography: Overcoming the wall effect and heterogeneous bed structures, J. Chromatogr. A 1335 (2014) 122–135.
  * [35] B. Yew, E. Drumm, G. Guiochon, Mechanics of column beds: I. acquisition of the relevant parameters, AIChE J. 49 (2003) 626–641.
  * [36] B. Yew, J. Ureta, R. Shalliker, E. Drumm, G. Guiochon, Mechanics of column beds: Ii. modeling of coupled stress-strain-flow behavior, AIChE J. 49 (2003) 642–664.
  * [37] R. Shalliker, B. Broyles, G. Guiochon, Axial and radial diffusion coefficients in a liquid chromatography column and bed heterogeneity, J. Chromatogr. A 994 (2003) 1–12.
  * [38] R. Shalliker, B. Broyles, G. Guiochon, Evaluation of the secondary consolidation of columns for liquid chromatography by ultrasonic irradiation, J. Chromatogr. A 878 (2000) 153–163.
  * [39] K. Mriziq, J. Abia, Y. Lee, G. Guiochon, Structural radial heterogeneity of a silica-based wide-bore monolithic column, J. Chromatogr. A 1193 (2008) 97–103.
  * [40] J. Abia, K. Mriziq, G. Guiochon, Radial heterogeneity of some analytical columns used in high-performance liquid chromatography, J. Chromatogr. A 1216 (2009) 3185–3191.
  * [41] F. Gritti, I. Leonardis, J. Abia, G. Guiochon, Physical properties and structure of fine core–shell particles used as packing materials for chromatography: Relationships between particle characteristics and column performance, J. Chromatogr. A 1217 (2010) 3819–3843.
  * [42] M. J. E. Golay, Theory of chromatography in open and coated tubular columns with round and rectangular cross-sections, In : DH Desty, ed. Gas Chromatography 1958 (Amsterdam symposium), Butterworths, London, pp. 36-55, 1958.
  * [43] J. C. Giddings, W. A. Manwaring, M. N. Myers, Turbulent-gas chromatography, Science 154 (1966) 146–148.
  * [44] V. Pretorius, T. Smuts, Turbulent flow chromatography. a new approach to faster analysis, Anal. Chem. 38 (1966) 274–281.
  * [45] J. Hart, I. Guymer, A. Jones, V. Stovin, Longitudinal Dispersion Coefficients within Turbulent and transitional Pipe Flow, In : P. Rowinski (ed.), Experimental and Computational Solutions of Hydraulic problems, Springer Berlin Heidelberg, London, pp. 133-145, 2013.
  * [46] A. Van Es, J. Rijks, C. Cramers, Turbulent flow in cappilary gas chromatography, J. Chromatogr. 477 (1989) 39–47.
  * [47] J. Knox, Evidence for turbulence and coupling in chromatographic columns, Anal. Chem. 38 (1966) 253–261.
  * [48] M. Martin, G. Guiochon, Influence of retention on band broadening in turbulent-flow liquid and gas chromatography, Anal. Chem. 54 (1982) 1533–1540.
  * [49] F. Gritti, M. Fogwill, Speed-resolution advantage of turbulent supercritical fluid chromatography in open tubular columns: Ii – theoretical and experimental evidences, J. Chromatogr. A 1501 (2017) 142–150.
  * [50] F. Gritti, Extension of golay’s plate height equation from laminar to turbulent flow i – theory, J. Chromatogr. A 1492 (2017) 129–135.
  * [51] M. Martin, G. Guiochon, Axial dispersion in open-tubular capillary liquid chromatography with electroosmotic flow, Anal. Chem. 56 (1984) 614–620.
  * [52] M. Martin, G. Guiochon, Y. Walbroehl, J. Jorgenson, Peak broadening in open-tubular liquid chromatography with electroosmotic flow, Anal. Chem. 57 (1985) 559–561.
  * [53] E. Westerbeek, P. Gelin, W. Olthuis, J. Eijkel, W. D. Malsche, C-term reduction in 3 um open-tubular high-aspect ratio channels in ac-eof vortex chromatography operation, Anal. Chem. 95 (2023) 4889–4895.
  * [54] I. Bihi, P. Gelin, I. Frankel, W. De Malsche, A numerical study of the effects of lateral flow and retention in open tubular vortex chromatography, J. Chromatogr. A [Manuscript submitted for publication].
  * [55] P. Gelin, D. Maes, W. D. Malsche, Reducing taylor-aris dispersion by exploiting lateral convection associated with acoustic streaming, Chem. Eng. J. 417 (2021) 128031.
  * [56] R. G. Larson, Instabilities in viscoelastic flows, Rheol. Acta 31 (3) (1992) 213–263.
  * [57] E. S. G. Shaqfeh, Purely Elastic Instabilities in Viscometric Flows, Annu. Rev. Fluid Mech. 28 (1) (1996) 129–185.
  * [58] P. Pakdel, G. H. McKinley, Elastic Instability and Curved Streamlines, Phys. Rev. Lett. 77 (12) (1996) 2459–2462.
  * [59] A. Groisman, V. Steinberg, Elastic turbulence in curvilinear flows of polymer solutions, New J. Phys. 6 (2004) 29.
  * [60] C. A. Browne, A. Shih, S. S. Datta, Bistability in the unstable flow of polymer solutions through pore constriction arrays, J. Fluid Mech. 890.
  * [61] C. A. Browne, S. S. Datta, Elastic turbulence generates anomalous flow resistance in porous media, Sci. Adv. 7 (2021) 11.
  * [62] S. Datta, A. Ardekani, A. B. P. Arratia, I. Bischofberger, G. McKinley, J. Eggers, J. E. López-Aguilar, S. Fielding, A. Frishman, M. Graham, J. Guasto, S. Haward, A. Shen, S. Hormozi, A. Morozov, R. Poole, V. Shankar, E. Shaqfeh, H. Stark, V. Steinberg, G. Subramanian, H. Stone, Perspectives on viscoelastic flow instabilities and elastic turbulence, Phys. Rev. Fluids 7 (2022) 080701.
  * [63] C. A. Browne, R. B. Huang, C. W. Zheng, S. S. Datta, Homogenizing fluid transport in stratified porous media using an elastic flow instability, J. Fluid Mech. 963.
  * [64] C. A. Browne, S. S. Datta, Harnessing elastic instabilities for enhanced mixing and reaction kinetics in porous media (Nov. 2023).
  * [65] A. Groisman, V. Steinberg, Efficient mixing at low Reynolds numbers using polymer additives, Nature 410 (2001) 905–908.
  * [66] T. Burghelea, E. Segre, I. Bar-Joseph, A. Groisman, V. Steinberg, Chaotic flow and efficient mixing in a microchannel with a polymer solution, Phys. Rev. E 69 (6) (2004) 066305.
  * [67] C. Scholz, F. Wirner, J. R. Gomez-Solano, C. Bechinger, Enhanced dispersion by elastic turbulence in porous media, EPL 107 (2014) 54003.
  * [68] M. Kumar, D. Walkama, A. Ardekani, J. Guasto, Stress and stretching regulate dispersion in viscoelastic porous media flows, Soft Matter 19 (2023) 6761–6770.
  * [69] B. Traore, C. Castelain, T. Burghelea, Efficient heat transfer in a regime of elastic turbulence, J. Non-Newton. Fluid Mech. 223 (2015) 62–76.
  * [70] R. Whalley, W. Abed, D. Dennis, R. Poole, Enhancing heat transfer at the micro-scale using elastic turbulence, Theor. App. Mech. 5 (3) (2015) 103–106.
  * [71] R. G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press, 1999.
  * [72] C. W. Macosko, Rheology: principles, measurements, and applications, Advances in interfacial engineering series, VCH, New York, 1994.
  * [73] M. Rubinstein, R. H. Colby, Polymer Physics, Oxford University Press, 2003.
  * [74] J. F. Ryder, J. M. Yeomans, Shear thinning in dilute polymer solutions, J. Chem. Phys. 125 (19) (2006) 194906.
  * [75] F. Gritti, K. Wyndham, Retention mechanism in combined hydrodynamic and slalom chromatography for analyzing large nucleic acid biopolymers relevant to cell and gene therapies, J. Chromatogr. A (2024) 465075.
  * [76] Q.-H. Wan, Effect of electrical double-layer overlap on the electroosmotic flow in packed-capillary columns, Anal. Chem. 69 (1997) 361–363.
  * [77] S. Ergun, Fluid flow through packed columns, Chem. Eng. Prog. 48 (1952) 89–94.
  * [78] R. Aris, On the dispersion of a solute by diffusion, convection and exchange between phases, Proc. Roy. Soc. A 252 (1959) 538–550.
  * [79] F. Gritti, G. Guiochon, Complete temperature profiles in ultra-high- pressure liquid chromatography columns, Anal. Chem. 80 (2008) 5009–5020.
  * [80] F. Gritti, M. Martin, G. Guiochon, Influence of viscous friction heating on the efficiency of columns operated under very high pressures, Anal. Chem. 81 (2009) 3365–3384.
  * [81] F. Parmentier, Effect of diffusional bridging in multicapillary packing, C. R. Chim. 23 (2020) 415–431.
  * [82] F. Gritti, D. Hlushkou, U. Tallarek, Multiple-open-tubular column enabling transverse diffusion. Part 1: Band broadening model for accurate mass transfer predictions, J. Chromatogr. A 1625 (2020) 461325.
  * [83] F. Gritti, D. Hlushkou, U. Tallarek, Multiple-open-tubular column enabling transverse diffusion. Part 1: Band broadening model for accurate mass transfer predictions, J. Chromatogr. A 1642 (2021) 462033.
  * [84] Y. Liu, Y. Jun, V. Steinberg, Concentration dependence of the longest relaxation times of dilute and semi-dilute polymer solutions, J. Rheol. 53 (5) (2009) 1069–1085.
  * [85] P. E. Rouse, A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers, J. Chem. Phys. 21 (7) (1953) 1272–1280.
  * [86] B. H. Zimm, Dynamics of Polymer Molecules in Dilute Solution: Viscoelasticity, Flow Birefringence and Dielectric Loss, J. Chem. Phys. 24 (2) (1956) 269–278.
  * [87] G. Chauveteau, M. Moan, A. Magueur, Thickening behaviour of dilute polymer solutions in non-inertial elongational flows, J. Non-Newton. Fluid Mech. 16 (3) (1984) 315–327.
  * [88] S. Parsa, E. Santanach-Carreras, L. Xiao, D. A. Weitz, Origin of anomalous polymer-induced fluid displacement in porous media, Physical Review Fluids 5 (2) (2020) 022001.
  * [89] F. Gritti, K. Horvath, G. Guiochon, How changing the particle structure can speed up protein mass transfer kinetics in liquid chromatography, J. Chromatogr. A 1263 (2012) 84–98.

Figure 1: Schematic highlighting (a) the problem associated with radial flow
heterogeneity in chromatography columns, and (b) our proposed solution using
an elastic flow instability to promote analyte transverse dispersion. (a) Plot
of the interstitial linear velocity $u(r)$, normalized by the center velocity
$u_{c}$, within a 2.1 mm i.d. $\times$ 50 mm slurry-packed column with 2
$\mu$m particles [31], as a function of the radial distance from the column
wall $r$ normalized by the particle diameter ${d_{p}}$. The data are obtained
using focused-ion-beam scanning electron microscopy and the solid curve is a
best fit. The velocity drop-off near the wall is evidence of a wall-to-center
velocity bias that negatively affects column performance. Adapted from [89]
with permission. (b) An elastic flow instability enhances transverse mixing of
solute in a 3D porous medium: fluorescence confocal micrographs of solute
(dyed) and solute-free (undyed) streams mixing. The polymer solution shows
pronounced mixing due to an elastic instability (bottom row) compared to the
case of a polymer-free solvent (top row). Images were taken at three different
longitudinal distances along the porous medium equal to 0, 10, and 24 particle
diameters. The average linear velocity is 1.7 mm/s. Reproduced with permission
from [64].

Figure 2: Polymer solution rheology. (a) Flow curve of dynamic viscosity,
$\eta(\dot{\gamma})$, for the 500 ppm HPAM in brine solution, exhibiting
strong shear-thinning behavior. Empty circles correspond to averages over 4
separate measurements and error bars reflect one standard deviation. The solid
curve is the best fit to the Carreau-Yasuda model. We show an extrapolation of
the Carreau-Yasuda model to the relevant shear rate regime encountered in HPLC
applications, suggesting that the polymer solution viscosity lies in the high
shear-rate plateau. (b) First normal stress difference, $N_{1}(\dot{\gamma})$,
versus applied shear rate. The solid curve is the best fit to a power law
model with power law exponent of $\alpha_{N_{1}}=0.85$.

Figure 3: Friction factor $f$ before, during, and after flow of the polymer-
amended mobile phase determined from measurements of the pressure drop across
the column using Eq. 3. (a) Polymer-free 1% NaCl aqueous mobile phase (brine)
before the column is in contact with the HPAM polymer. The black curve shows
the prediction for a Newtonian fluid in laminar flow ($f\sim\mathrm{Re}^{-1}$,
Eq. 4) with $K_{c}$=180. (b) Polymer-amended (500 ppm of 18 MDa HPAM) brine
mobile phase. The black curve shows the prediction for a Newtonian fluid in
laminar flow with $K_{c}$=420. The slight vertical shift from the polymer-free
case (indicated by the lower double-headed arrow) reflects irreversible
adsorption of the polymer on the solid particles. The pronounced positive
deviation from the red curve (indicated by the upper double-headed arrow)
shows evidence of added flow resistance due to an elastic flow instability.
(c) Polymer-free brine mobile phase after the column is in contact with the
HPAM polymer, again showing Newtonian laminar behavior, confirming that the
positive deviation from the red curve in (b) is due to the elastic flow
instability and that the slight vertical shift from the black curve reflects
irreversible polymer adsorption.

Figure 4: Chromatograms of thiourea before (black) and after (blue) the column
is equilibrated with the polymer-amended brine mobile phase and then washed
with the polymer-free brine. In both cases, thiourea exhibits the same overall
retention time, yet has a broader peak after the column has been equilibrated
with the polymer, indicating irreversible but not complete adsorption of the
polymer on the particle surfaces.

Figure 5: Normalized chromatograms of thiourea before (black), during (red),
and after (blue) addition of HPAM polymer to the mobile phase. The elution
curves are normalized by the peak intensity to facilitate comparison of their
shapes. When the experiment is performed with polymer-amended mobile phase at
a Weissenberg number (Wi) sufficiently large to generate an elastic
instability, the elution peak is considerably more symmetric
(red)—demonstrating that the elastic instability enhances transverse
dispersion and increases column efficiency.