#  $C$-open Sets on Topological Spaces

Mesfer H. Alqahtani111University College of Umluj, Mathematics Department,
University of Tabuk, Tabuk, Saudi Arabia<EMAIL_ADDRESS>
###### Abstract

An open (resp., closed) subset $A$ of a topological space $(X,\mathcal{T})$ is
called $C$-open (resp., $C$-closed) set if $cl(A)\setminus A$ (resp.,
$A\setminus int(A)$) is a countable set. This paper aims to present the
concept of $C$-open and $C$-closed sets. We first investigate their basic
properties. Then, we found some operators such as interior, closure, limit,
border, and frontier using $C$-open and $C$-closed sets. The relationships
between them are clarified and discussed. Finally, we exhibit continuous maps
and compact space defined using $C$-open and $C$-closed sets and scrutinize
their main properties.

2020 Mathematics Subject Classifications: 54A05, 54C10, 54D30
Key Words and Phrases: $C$-open set, $C$-interior points, $C$-closure points,
$C$-continuity, $C$-compactness.

##  1 Introduction

In the topological space $X$ a subset $B$ of a space $X$ is said to be a
regularly-open, called also open domain if it is the interior of its own
closure [1]. A subset $B$ is said to be a regularly-closed, called also closed
domain if it is the closure of its own interior or if its complement is an
open domain. An open (resp., closed) subset $A$ of a topological space
$(X,\mathcal{T})$ is called $F$-open (resp., $F$-closed) set if
$cl(A)\setminus A$ (resp., $A\setminus int(A)$) is finite set [4]. In this
present paper, we have concerned is to introduce and study the notions of
$C$-open and $C$-closed sets in topological spaces that are independent of
regularly-open and regularly-closed sets. In Section 2, we have defined
$C$-open and $C$-closed sets and investigated their basic properties. Then we
have shown about the relationship between them with the other kinds such as
open domain, closed domain, $F$-open, $F$-closed, closed and open sets in
topological spaces. In Section 3, we have established some operators such as
$C$-interior, $C$-closure, $C$-limit, $C$-border and $C-$frontier using
$C$-open and $C$-closed sets and the relationships between them are clarified
and discussed. In Section 4, we have displayed $C$-continuous functions and
$C$-compact sets defined using $C$-open and $C$-closed sets and examine their
basic properties. In addition, I have shown the $C$-continuious, $C$-open and
onto image of $C$-compact space is $C$-compact space. Throughout this paper,
the subset $B$ of a space $X$, I will denote the complement of $B$ in $X$ by
$X\setminus B$, the set of positive integers numbers by $\mathbb{N}$, the set
of integers numbers by $\mathbb{Z}$, the set of rational numbers by
$\mathbb{Q}$, the set of irrational numbers by $\mathbb{I}$, the set of real
numbers by $\mathbb{R},$ and usual topology in $\mathbb{R}$ by $\mathcal{U}$
[3]. Unless or otherwise mentioned, $X$ stands for the topological space
$(X,\mathcal{T}).$ We do not assume $T_{2}$ in the definition of compactness
and countable compactness. We also do not assume regularity in the definition
of $F-$Lindel$\ddot{o}$f.

##  2 Main properties of $C$-open sets

######  Definition 1.

An open subset $A$ of a topological space $X$ is called $C$-open set if
$cl(A)\setminus A$ is countable set. That is, $A$ is an open set and the
frontier of $A$ is a countable set.

######  Definition 2.

A closed subset $A$ of a topological space $(X,\mathcal{T})$ is called
$C$-closed set if $A\setminus int(A)$ is countable set. That is, $A$ is a
closed set and the frontier of $A$ is countable set.

We denoted for the collection of all $C$-open (resp., $C$-closed) subsets of a
topological space $(X,\mathcal{T})$ by $CO(X)$ (resp., $CC(X)$).

######  Theorem 3.

The complement of any $C$-open (resp., $C$-closed) subset of a topological
space $(X,\mathcal{T})$ is a $C$-closed (resp., $C$-open) set.

###### Proof.

Let $A$ be a any $C$-open subset of a topological space $(X,\mathcal{T}).$
Then $X\setminus A$ is closed and $(X\setminus A)\setminus int(X\setminus
A)=(X\setminus A)\setminus(X\setminus cl(A))=(X\setminus A)\cap
cl(A)=cl(A)\setminus A$ is countable, because $A$ is $C$-open. Therefore,
$X\setminus A$ is $C$-closed set. On the other hand, suppose that $A$ be a any
$C$-closed subset of a topological space $(X,\mathcal{T}).$ Then $X\setminus
A$ is open and $cl(X\setminus A)\setminus(X\setminus A)=(X\setminus X\setminus
A)\cap cl(X\setminus A)=A\cap cl(X\setminus A)=A\setminus(X\setminus
cl(X\setminus A))=A\setminus int(A)$ is countable, because $A$ is $C$-closed.
Therefore, $X\setminus A$ is $C$-open set.

∎

Obviously, from the definitions that any $F$-open (resp., $F$-closed) subset
of a topological space $(X,\mathcal{T})$ is $C$-open (resp., $C$-closed) set.
However, the converse always is not true. For example, the subset
$\mathbb{R}\setminus\mathbb{Z}$ is a $C$-open subset of the usual topological
space $(\mathbb{R},\mathcal{U})$, which is not $F$-open. Also by Theorem 3,
any complement of $C$-open set is $C$-closed, then $\mathbb{Z}$ is a
$C$-closed subset of the usual topological space $(\mathbb{R},\mathcal{U})$,
which is not $F$-closed, because $\mathbb{Z}\setminus
int(\mathbb{Z})=\mathbb{Z}\setminus\emptyset=\mathbb{Z}$ is not finite. It is
clear by the definitions that, any clopen (closed-and-open) subset of a
topological space $(X,\mathcal{T})$ is $C$-clopen set. Also any countable
closed set is $C$-closed. However, any countable open set may not be $C$-open.
For example, $\mathbb{R}$ with particular point topology at $1.$ We have,
$\mathbb{N}$ is a countable open set, but
$cl(\mathbb{N})\setminus\mathbb{N}=\mathbb{R}\setminus\mathbb{N}$ is
uncountable set.

In $(\mathbb{R},\mathcal{U})$ any open intervals is $C$-open set. Also any
closed intervals is $C$-closed set. It is clear by the definitions 1 and 2,
every $C$-open and $C$-closed sets are open and closed sets, respectively.
However, the converse always is not true. Here is an example of open (resp.,
closed) set which is not $C$-open (resp., $C$-closed).

######  Example 4.

Let $(\mathbb{R},\mathcal{T_{\mathbb{I}}})$ be the excluded set topological
space on $\mathbb{R}$ by $\mathbb{I}.$ Then $\mathbb{Q}$ is open in
$(\mathbb{R},\mathcal{T_{\mathbb{I}}})$. But
$cl(\mathbb{Q})\setminus\mathbb{Q}=\mathbb{R}\setminus\mathbb{Q}=\mathbb{I}$
is uncountable set. Hence, $\mathbb{Q}$ is not $C$-open set. Also,
$\mathbb{I}$ is an example of closed set which is not $C$-closed set.

There is an example of $C$-open (resp., $C$-closed) set is not an open (resp.,
closed) domain set.

######  Example 5.

Let $A=(2,5)\cup(5,9)$ is a $C$-open subset in $(\mathbb{R},\mathcal{U})$.
However, $A$ is not open domain. Moreover, $\mathbb{R}\setminus A$ is
$C$-closed set, but $\mathbb{R}\setminus A$ is not closed domain.

We provide an example of an open (resp., closed) domain set is not $C$-open
(resp., $C$-closed) set.

######  Example 6.

By Example 4, let $H=\\{1,2,3\\}$, then
$int(cl(H))=int(cl(\\{1,2,3\\}))=int(\\{1,2,3\\}\cup\mathbb{I})=\\{1,2,3\\}=H$
is open domain. But $cl(H)\setminus
H=cl(\\{1,2,3\\})\setminus\\{1,2,3\\}=(\\{1,2,3\\}\cup\mathbb{I})\setminus\\{1,2,3\\}=\mathbb{I}$
is uncountable set, then $H$ is not $C$-open set. Moreover, let
$K=\mathbb{R}\setminus\\{1,2,3\\},$ then
$cl(int(K))=cl(int(\mathbb{R}\setminus\\{1,2,3\\}))=cl(\mathbb{Q}\setminus\\{1,2,3\\})=\mathbb{I}\cup(\mathbb{Q}\setminus\\{1,2,3\\})=\mathbb{R}\setminus\\{1,2,3\\}=K$
is closed domain. But $K\setminus
int(K)=(\mathbb{R}\setminus\\{1,2,3\\})\setminus
int(\mathbb{R}\setminus\\{1,2,3\\})=(\mathbb{R}\setminus\\{1,2,3\\})\setminus(\mathbb{Q}\setminus\\{1,2,3\\})=\mathbb{I}$
is uncountable set, then $K$ is not $C$-closed set.

For a topological space $(X,\mathbb{T}),$ we always have:

$F$-open set $\Longrightarrow$ $C$-open set $\Longrightarrow$ open set

$F$-closed set $\Longrightarrow$ $C$-closed set $\Longrightarrow$ closed set

None of the above implications is reversible.

######  Theorem 7.

The finite unions of $C$-open sets is $C$-open.

###### Proof.

Suppose that $K_{i}$ be a $C$-open set for all $i\in\\{1,2,3,\dots,n\\}$. Then
$K_{i}$ is an open set and $cl(K_{i})\setminus K_{i}$ is countable for all
$i$. Since $\bigcup_{i=1}^{n}K_{i}$ is open, then we need to show the other
condition of $C$-open set. Now, we have,

$cl(\bigcup_{i=1}^{n}K_{i})\setminus\bigcup_{i=1}^{n}K_{i}=\bigcup_{i=1}^{n}(cl(K_{i})\setminus
K_{i}),$

Since the finite union of countable sets is countable. Then,
$cl(\bigcup_{i=1}^{n}K_{i})\setminus\bigcup_{i=1}^{n}K_{i}$ is countable.
Therefore, $\bigcup_{i=1}^{n}K_{i}$ is $C$-open.

∎

By Theorem 7, Theorem 3 and by Morgan’s Laws, we have the following Corollary:

######  Corollary 8.

The finite intersections of $C$-closed sets is $C$-closed.

##  3 Some operators via $C$-open and $C$-closed sets

We recall some symbol in [4]. Let $K$ be any subset of the topological space
$(X,\mathcal{T}),$ and via the concepts of $F$-open sets we denoted for the
interior by ${\rm int^{F}}(K)$, the closure by ${\rm cl^{F}}(K)$, the border
by ${\rm Bd^{F}}(K)$, the frontier by ${\rm Fr^{F}}(K)$, the exterior by ${\rm
Ext^{F}}(K)$ and the derived set by ${\rm D^{F}}(K)$.

######  Definition 9.

Let $K$ be a subset of the topological space $(X\mathcal{T}).$ Then,

  1. i)

The $C$-interior of $K$ is defined as the union of all $C$-open subsets of
$K$, and is denoted by $int^{C}(K)$ or the largest $C$-open set contained in
$K.$

  2. ii)

The $C$-clouser of $K$ is defined as the intersection of all $C$-closed sets
containing $K$, and is denoted by $cl^{C}(K)$.

######  Definition 10.

Let $x\in int^{C}(K)$ if and only if there exists a $C$-open set containing
$x$ and contained in $K$.

######  Theorem 11.

Let $K$ be a subset of the topological space $(X,\mathcal{T}).$ Then,

  1. i)

$int^{C}(K)\subseteq int(K)\subseteq K.$

  2. ii)

$K\subseteq cl(K)\subseteq cl^{C}(K).$

###### Proof.

  1. i)

Let $U$ be the largest $C$-open set contained in $K$, i.e. $int^{C}(K)=U$.
Since any $C$-open set is open, then $U\subseteq int(K)$. Hence
$int^{C}(K)\subseteq int(K)$. By the definition of interior of $K$, we have
$int(K)\subseteq K.$ Therefore, $int^{C}(K)\subseteq int(K)\subseteq K.$

  2. ii)

By the definition of closure, we know that $K\subseteq cl(K)$. Now, let
$cl(K)=F$ is the smallest closed set containing $K$, then $F\subseteq
cl^{C}(K)$. Suppose not, $F\nsubseteq cl^{C}(K)$, then there exist $C$-closed
set $E$ such that $cl^{C}(K)=E$ and $K\subseteq E$. But $E$ is closed, because
any $C$-closed is closed. Thus contradiction. Hence, $K\subseteq
cl(K)\subseteq cl^{C}(K).$

∎

In general $int(K)\nsubseteq int^{C}(K)$ and $cl^{C}(K)\nsubseteq cl(K)$. For
example:

######  Example 12.

By Example 4 we have the open set $\mathbb{Q}$ is neither $F$-open nor
$C$-open set. Then $int(\mathbb{Q})=\mathbb{Q}$, but
$int^{C}(\mathbb{Q})\subsetneqq\mathbb{Q}$. Therefore, $int(K)\nsubseteq
int^{C}(K).$ By Example 4 we have $\mathbb{I}$ is not $C$-closed set and
$cl^{C}(\mathbb{I})=\mathbb{R}$, but $cl(\mathbb{I})=\mathbb{I}.$ Therefore,
$cl^{C}(K)\nsubseteq cl(K)$.

######  Definition 13.

Let $K$ be a subset of the topological space $(X,\mathcal{T})$. A point $x\in
X$ is said to be $C$-limit points (or an $C$-accumulation point, or a
$C$-cluster point) of $K$ if and only if every $C$-open set $V$ containing
$x,$ contains at least one point of $K$ different from $x$. The set of all
$C$-limit points of $K$ is called the $C-$derived set of $K$ and denoted by
$D^{C}(K).$

######  Theorem 14.

Let $K$ and $H$ be subsets of a topological space $(X\mathcal{T}).$ Then we
have the following topological properties:

  1. (i)

$D(K)\subset D^{C}(K)$, where $D(K)$ is the derived set of $K.$

  2. (ii)

If $K\subseteq H,$ then $D^{C}(K)\subseteq D^{C}(H).$

  3. (iii)

$D^{C}(K)\cup D^{C}(H)=D^{C}(K\cup H)$ and $D^{C}(K\cap H)\subset D^{C}(K)\cap
D^{C}(H).$

###### Proof.

For $(i)$ it suffices to observe that every $C$-open is open. For $(ii)$
follow from Definition 13. For $(iii)$ is a modification of the standard proof
for $D$, where open sets are replaced by $C$-open sets. ∎

In general $D^{C}(K)\nsubseteq D(K)$. For example:

######  Example 15.

Let $(\mathbb{R},\mathcal{T}_{\frac{1}{2}})$ be a topological space, where
$\mathcal{T}_{\frac{1}{2}}$ is the particular point topology at $\frac{1}{2}$.
Let $K=\mathbb{R}\setminus\\{\frac{1}{2}\\}$, then
$D(\mathbb{R}\setminus\\{\frac{1}{2}\\})=\emptyset$, because
$\\{\frac{1}{2},x\\}$ is an open set for any $x\in\mathbb{R}$ and
$(\\{\frac{1}{2},x\\}\setminus\\{x\\})\cap(\mathbb{R}\setminus\\{\frac{1}{2}\\})=\emptyset.$
Since any nonempty $C$-open set in $(\mathbb{R},\mathcal{T}_{\frac{1}{2}})$ is
of the form $\mathbb{R}\setminus H$, where $H$ is any countable set and
$\frac{1}{2}\notin H.$ let $x\in\mathbb{R}$ be arbitrary, then any $C$-open
set containing $x$ contains an other point of
$\mathbb{R}\setminus\\{\frac{1}{2}\\}$ different from $x$. Hence
$D^{C}(\mathbb{R}\setminus\\{\frac{1}{2}\\})=\mathbb{R}.$ Then
$D^{C}(\mathbb{R}\setminus\\{\frac{1}{2}\\})\nsubseteq
D(\mathbb{R}\setminus\\{\frac{1}{2}\\}).$ Therefore, $D^{C}(K)\nsubseteq
D(K).$

In general $D^{C}(K\cap H)\nsupseteq D^{C}(K)\cap D^{C}(H).$ For example:

######  Example 16.

Let $K=(1,2)$ and $H=(2,3)$ be a subsets in $(\mathbb{R},\mathcal{U})$. Since
any $C$-open set in $(\mathbb{R},\mathcal{U})$ is an open interval or
$\mathbb{R}\setminus F$, where $F$ is any countable closed set, then
$D^{C}(K\cap H)=D^{C}((1,2)\cap(2,3))=D^{C}(\emptyset)=\emptyset$ and
$D^{C}(K)\cap D^{C}(H)=D^{C}((1,2))\cap D^{C}((2,3))=[1,2]\cap[2,3]=\\{2\\}$.
Then $D^{C}(K\cap H)\nsupseteq D^{C}(K)\cap D^{C}(H).$

######  Theorem 17.

Let $K$ and $H$ be subsets of a topological space $(X\mathcal{T}).$ Then we
have the following topological properties:

  1. (i)

$int^{C}(X)=X.$

  2. (ii)

$int^{C}(K)\subseteq K.$

  3. (iii)

If $K\subseteq H$, then $int^{C}(K)\subseteq int^{C}(H)$.

  4. (iv)

$int^{C}(int^{C}(K))=int^{C}(K).$

  5. (v)

$int^{C}(K\cap H)=int^{C}(K)\cap int^{C}(H).$

  6. (vi)

$int^{C}(K)\cup int^{C}(H)\subseteq int^{C}(K\cup H).$

###### Proof.

The properties $(i),(ii),(iii)$ and $(iv)$ follow from Definitions 1 and
Definition 9. To prove $(v)$, by property $(ii)$ we have $int^{C}(K)\subseteq
K$ and $int^{C}(H)\subseteq H,$ then $int^{C}(K)\cap int^{C}(H)\subseteq K\cap
H.$ As $int^{C}(K)\cap int^{C}(H)$ is $C-$open, then we have $int^{C}(K)\cap
int^{C}(H)\subseteq int^{C}(K\cap H),$ because $int^{C}(K)\cap int^{C}(H)$ is
$C$-open and $int^{C}(K\cap H)$ is the largest $C$-open set contained in
$K\cap H.$ Conversely, $(K\cap H)\subseteq K$ and $(K\cap H)\subseteq H$, by
property $(iii)$ we have $int^{C}(K\cap H)\subseteq int^{C}(K)$ and
$int^{C}(K\cap H)\subseteq int^{C}(H)$, hence $int^{C}(K\cap H)\subseteq
int^{C}(K)\cap int^{C}(H)$. Therefore, $int^{C}(K\cap H)=int^{C}(K)\cap
int^{C}(H).$ To prove $(vi)$, since $K\subseteq(K\cup H)$ and
$H\subseteq(K\cup H)$, from property $(iii)$ we have $int^{C}(K)\subseteq
int^{C}(K\cup H)$ and $int^{C}(H)\subseteq int^{C}(K\cup H).$ Therefore,
$int^{C}(K)\cup int^{C}(H)\subseteq int^{C}(K\cup H).$ ∎

In general, $int^{C}(K\cup H)\nsubseteq int^{C}(K)\cup int^{C}(H).$ For
example:

######  Example 18.

Let $K=\mathbb{Q}$ and $H=\mathbb{I}$ be a subsets in
$(\mathbb{R},\mathcal{U})$, then $int^{C}(K\cup H)=\mathbb{R}$. But
$\mathbb{Q}$ and $\mathbb{I}$ are not $C$-open sets, then
$int^{C}(\mathbb{Q})\subset\mathbb{Q}$ and
$int^{C}(\mathbb{I})\subset\mathbb{I}$. Therefore, $int^{C}(K\cup H)\nsubseteq
int^{C}(K)\cup int^{C}(H).$

######  Theorem 19.

Let $K$ and $H$ be subsets of a topological space $(X,\mathcal{T}).$ Then we
have the following topological properties:

  1. (i)

$cl^{C}(\emptyset)=\emptyset.$

  2. (ii)

$K\subseteq cl^{C}(K).$

  3. (iii)

If $K\subseteq H$, then $cl^{C}(K)\subseteq cl^{C}(H)$.

  4. (iv)

$cl^{C}(K\cup H)=cl^{C}(K)\cup cl^{C}(H).$

  5. (v)

$cl^{C}(cl^{C}(K))=cl^{C}(K).$

###### Proof.

The properties $(i),(ii),(iii)$ and $(v)$ follow from Definition 2 and
Definition 9. The property $(iv)$, follow from property $(iii)$, Definition 1,
Definition 9 and using set theoretic properties. ∎

######  Theorem 20.

Let $K$ be a subset of a topological space $(X\mathcal{T}).$ Then we have the
following topological properties:

  1. (i)

$int^{C}(K)=X\setminus cl^{C}(X\setminus K).$

  2. (ii)

$cl^{C}(K)=X\setminus int^{C}(X\setminus K).$

###### Proof.

  1. (i)

We have $int^{C}(K)=X\setminus cl^{C}(X\setminus K)$, then $X\setminus
K\subseteq cl^{C}(X\setminus K)$, thus $X\setminus cl^{C}(X\setminus
K)\subseteq K.$ Since $X\setminus cl^{C}(X\setminus K)$ is $C$-open, then
$X\setminus cl^{C}(X\setminus K)\subseteq int^{C}(K)$…(1). Now, let $V$ be any
$C$-open set contained in $K$, i.e., $V\subseteq K$ and $V$ is $C$-open, then
$X\setminus K\subseteq X\setminus V=cl^{C}(X\setminus V)$. Then
$cl^{C}(X\setminus K)\subseteq X\setminus V$, hence $V\subseteq X\setminus
cl^{C}(X\setminus K)$. That is, any $C$-open set contained in $K$ is contained
in $X\setminus cl^{C}(X\setminus K)$, which means that $int^{C}(K)\subseteq
X\setminus cl^{C}(X\setminus K)$ …(2). From (1) and (2) equality holds.

  2. (ii)

Can be proved by replacing $K$ and $X\setminus K$ by $X\setminus K$ and $K$,
respectively in $(i)$ and using set theoretic properties.

∎

######  Definition 21.

Let $K$ be a subset of a topological space $(X\mathcal{T}).$ Then $C$-border
of $K$ is defined as $Bd^{C}(K)=K\setminus int^{C}(K).$

######  Theorem 22.

Let $K$ be a subset of a topological space $(X\mathcal{T}).$ Then we have the
following properties:

  1. (i)

$Bd(K)\subset Bd^{C}(K)$, where $Bd(K)$ denotes the border of $K.$

  2. (ii)

$K=int^{C}(K)\cup Bd^{C}(K).$

  3. (iii)

$int^{C}(K)\cap Bd^{C}(K)=\emptyset.$

  4. (iv)

$K$ is a $C$-open set if and only if $Bd^{C}(K)=\emptyset.$

  5. (v)

$int^{C}(Bd^{C}(K))=\emptyset.$

  6. (vi)

$Bd^{C}(Bd^{C}(K))=Bd^{C}(K).$

  7. (vii)

$Bd^{C}(K)=K\cap cl^{C}(X\setminus K).$

###### Proof.

To prove $(i)$, let $x\in Bd(K)$ be arbitrary, then $x\in K\setminus int(K)$,
then $x\notin int(K)$. By Theorem 11 part $(i)$ we have $int^{C}(K)\subseteq
int(K)\subseteq K$, then $x\notin int^{C}(K)$. Hence $x\in(K\setminus
int^{C}(K))=Bd^{C}(K)$. Therefore, $Bd(K)\subseteq Bd^{C}(K)$. The properties
$(ii),(iii)$ and $(iv)$ follow from Definition 21. To prove $(v)$, let $x\in
int^{C}(Bd^{C}(K))$ be arbitrary, then $x\in Bd^{C}(K).$ Since
$Bd^{C}(K)\subseteq K,$ then $x\in int^{C}(Bd^{C}(K))\subseteq int^{C}(K),$
then $x\in int^{C}(K)\cap Bd^{C}(K)$ which contradicts $(iii).$ For $(vi)$
from Definition 21 and property $(v)$ we have
$Bd^{C}(Bd^{C}(K))=Bd^{C}(K)\setminus
int^{C}(Bd^{C}(K))=Bd^{C}(K)\setminus\emptyset=Bd^{C}(K)$. The property
$(vii)$ follow from Theorem 20 part $(i)$ and Definition 21. ∎

In general $Bd^{C}(K)\nsubseteq Bd(K).$ For example:

######  Example 23.

By Example 4, there exists $K=\mathbb{Q}$ is an open set which is not $C$-open
set. Then $Bd(K)=\emptyset$. Since $\mathbb{Q}$ is not $C$-open set, then
$int^{C}(\mathbb{Q})\subsetneqq\mathbb{Q}.$ Hence,
$Bd^{C}(K)=\mathbb{Q}\setminus int^{C}(\mathbb{Q})\neq\emptyset.$ Therefore,
$Bd^{C}(K)\nsubseteq Bd(K).$

######  Definition 24.

Let $K$ be a subset of a topological space $(X\mathcal{T}).$ Then $C$-frontier
of $K$ is defined as $Fr^{C}(K)=cl^{C}(K)\setminus int^{C}(K).$

######  Theorem 25.

Let $K$ be a subset of a topological space $(X\mathcal{T}).$ Then we have the
following properties:

  1. (i)

$Fr(K)\subset Fr^{C}(K)$, where $Fr(K)$ denotes the frontier of $K.$

  2. (ii)

$cl^{C}(K)=int^{C}(K)\cup Fr^{C}(K).$

  3. (iii)

$int^{C}(K)\cap Fr^{C}(K)=\emptyset.$

  4. (iv)

$Bd^{C}(K)\subset Fr^{C}(K).$

  5. (v)

$Fr^{C}(K)=cl^{C}(K)\cap cl^{C}(X\setminus K).$

  6. (vi)

$Fr^{C}(K)=Fr^{C}(X\setminus K).$

  7. (viii)

$int^{C}(K)=K\setminus Fr^{C}(K).$

###### Proof.

The property $(i)$, by Theorem 11 and Definition 24, we have
$int^{C}(K)\subseteq int(K)$ and $cl(K)\subseteq cl^{C}(K),$ then
$Fr(C)=(cl(K)\setminus int(K))\subseteq cl^{C}(K)\setminus
int^{C}(K)=Fr^{C}(K).$ The properties $(ii)$ and $(iii)$ follow from
Definition 24. For $(iv)$, since $K\subseteq cl^{C}(K)$, then
$Bd^{C}(K)=(K\setminus int^{C}(K))\subseteq(cl^{C}(K)\setminus
int^{C}(K))=Fr^{C}(K).$ The property $(v)$ follow from Theorem 20 and
Definition 24. The property $(vi)$ follow from property $(v)$ and Definition
24. The property $(vii)$ follow from Definition 24.

∎

In general $Fr^{C}{K}\nsubseteq Fr(K)$ and $Fr^{C}(K)\nsubseteq Bd^{C}(K).$
For example:

######  Example 26.

By Example 4, there exists $K=\mathbb{Q}$ such that
$Fr(\mathbb{Q})=\mathbb{I}$. Since $\mathbb{Q}$ is not $C$-open set, then
$int^{C}(\mathbb{Q})\subsetneqq\mathbb{Q}$, hence
$Fr^{C}{\mathbb{Q}}=(cl^{C}(\mathbb{Q})\setminus
int^{C}(\mathbb{Q}))=\mathbb{R}\setminus int^{C}(\mathbb{Q})=\mathbb{I}\cup
H$, where $H$ is nonempty subset of $\mathbb{Q}.$ Therefore,
$Fr^{C}(K)\nsubseteq Fr(K).$ For $Fr^{C}(K)\nsubseteq Bd^{C}(K)$, let
$K=(1,2]$ be a subsets in $(\mathbb{R},\mathcal{U})$, then $Bd^{C}(K)=\\{2\\}$
and $Fr^{C}(K)=\\{1,2\\}$. Hence, $Fr^{C}(K)\nsubseteq Bd^{C}(K).$

######  Definition 27.

Let $K$ be a subset of a topological space $(X\mathcal{T}).$ Then $C$-exterior
of $K$ is defined as $Ext^{C}(K)=int^{C}(X\setminus K).$

######  Theorem 28.

Let $K$ be a subset of a topological space $(X\mathcal{T}).$ Then we have
following properties:

  1. (i)

$Ext^{C}(K)\subset Ext(K)$, where $Ext(K)$ denotes the exterior of $K.$

  2. (ii)

$Ext^{C}(K)$ is open.

  3. (iii)

$Ext^{C}(K)=X\setminus cl^{C}(K).$

  4. (iv)

$Ext^{C}(Ext^{C}(K))=int^{C}(cl^{C}(K)).$

  5. (v)

If $K\subseteq H,$ then $Ext^{C}(H)\subseteq Ext^{C}(K).$

  6. (vi)

$Ext^{C}(X\setminus Ext^{C}(K))=Ext^{C}(K).$

  7. (vii)

$int^{C}(K)\subset Ext^{C}(Ext^{C}(K)).$

  8. (viii)

$Ext^{C}(K\cup H)\subset Ext^{C}(K)\cup Ext^{C}(H).$

  9. (ix)

$Ext^{C}(K\cap H)\supset Ext^{C}(K)\cap Ext^{C}(H).$

  10. (x)

$X=int^{C}(K)\cup Ext^{C}(K)\cup Fr^{C}(K).$

###### Proof.

The property $(i)$, follow from Definition 27 and Theorem 11. The property
$(ii)$, follow from Definition 9. The property $(iii)$, follow from Theorem 20
and Definition 27. To prove $(iv)$
$Ext^{C}(Ext^{C}(K))=Ext^{C}(int^{C}(X\setminus K))=Ext^{C}(X\setminus
cl^{C}(K))=int^{C}(X\setminus(X\setminus cl^{C}(K)))=int^{C}(cl^{C}(K)).$ To
prove $(v)$ since $K\subseteq H,$ then $X\setminus H\subseteq X\setminus K,$
then $int^{C}(X\setminus H)\subseteq int^{C}(X\setminus K),$ hence,
$Ext^{C}(H)\subseteq Ext^{C}(K).$ For $(vi),$ $Ext^{C}(X\setminus
Ext^{C}(K))=Ext^{C}(X\setminus int^{C}(X\setminus
K))=int^{C}(X\setminus(X\setminus int^{C}(X\setminus
K)))=int^{C}(int^{C}(X\setminus K))=int^{C}(X\setminus K)=Ext^{C}(K).$ For
$(vii),$ $int^{C}(K)\subseteq int^{C}(cl^{C}(K))\subseteq int^{C}(X\setminus
int^{C}(X\setminus K))=int^{C}(X\setminus Ext^{C}(K))=Ext^{C}(Ext^{C}(K)).$
The property $(viii)$ $Ext^{C}(K\cup H)=int^{C}(X\setminus(K\cup
H))=(X\setminus cl^{C}(K\cup H))=(X\setminus(cl^{C}(K)\cup cl^{C}(H)))\subset
X\setminus cl^{C}(K)\cup X\setminus cl^{C}(H)=(X\setminus
cl^{C}(K))\cap(X\setminus cl^{C}(H)))=Ext^{C}(K)\cap Ext^{C}(H)\subset
Ext^{C}(K)\cup Ext^{C}(H).$ The property $(ix)$: $Ext^{C}(K\cap
H)=int^{C}(X\setminus(K\cap H))=(X\setminus cl^{C}(K\cap
H))\supset(X\setminus(cl^{C}(K)\cap cl^{C}(H)))=(X\setminus
cl^{C}(K))\cup(X\setminus cl^{C}(H)))=Ext^{C}(K)\cup Ext^{C}(H)\supset
Ext^{C}(K)\cap Ext^{C}(H).$ The property $(x)$ follow from the Definitions 24
and 27.

∎

In general $Ext(K)\nsubseteq Ext^{C}(K),$ $Ext^{C}(K\cup H)\nsupseteq
Ext^{C}(K)\cup Ext^{C}(H)$ and $Ext^{C}(K\cap H)\nsubseteq Ext^{C}(K)\cap
Ext^{C}(H).$ For example:

######  Example 29.

By Example 4, there exists $K=\mathbb{I}$, such that
$Ext(K)=int(\mathbb{R}\setminus\mathbb{I})=int(\mathbb{Q})=\mathbb{Q}$. But
$Ext^{C}(K)=int^{C}(\mathbb{R}\setminus\mathbb{I})=int^{C}(\mathbb{Q})\subsetneqq\mathbb{Q}$,
because $\mathbb{Q}$ is not $C$-open set. Hence, $Ext(K)\nsubseteq
Ext^{C}(K).$ For $Ext^{C}(K\cup H)\nsupseteq Ext^{C}(K)\cup Ext^{C}(H)$, let
$K=(-\infty,2)$ and $H=(0,\infty),$ then $Ext^{C}(K\cup
H)=Ext^{C}(\mathbb{R})=\emptyset$ and $Ext^{C}(K)\cup
Ext^{C}(H)=(2,\infty)\cup(-\infty,0).$ Hence, $Ext^{C}(K\cup H)\nsupseteq
Ext^{C}(K)\cup Ext^{C}(H)$. For $Ext^{C}(K\cap H)\nsubseteq Ext^{C}(K)\cap
Ext^{C}(H)$, let $K=(-\infty,2]$ and $H=[2,\infty),$ then $Ext^{C}(K\cap
H)=Ext^{C}(\\{2\\})=\mathbb{R}\setminus\\{2\\}$ and $Ext^{C}(K)\cap
Ext^{C}(H)=(2,\infty)\cap(-\infty,2)=\emptyset$, hence, $Ext^{C}(K\cap
H)\nsubseteq Ext^{C}(K)\cap Ext^{C}(H).$

##  4 $C$-continuity and $C$-compactness

In this section, we define $C$-continuous functions and $C$-compact sets via
the concepts of $C$-open and $C$-closed spaces and investigate their master
properties.

######  Definition 30.

A function $h:(X,\mathcal{T})\to(Y,\mathcal{P})$ is said to be $C$-continuous
if $h^{-1}(U)$ is $C$-open in $X$ for all open sets $U$ in $Y.$

######  Definition 31.

Let $(X,\mathcal{T})$ and $(Y,\mathcal{P})$ be topological spaces and
$h:(X,\mathcal{T})\to(Y,\mathcal{P})$ be a $C$-continuous functions. $h$ is
$C$-open if and only if for all open sets $U\subseteq X,$ $h(U)$ is $C$-open
set in $Y.$ $h$ is $C$-closed if and only if for all closed sets $V\subseteq
X,$ $h(V)$ is $C$-closed set in $Y.$

######  Definition 32.

A bijection function $h:(X,\mathcal{T})\to(Y,\mathcal{P})$ is said to be
$C$-hmoeomrphism if and only if $h$ and $h^{-1}$ are $C$-continuous.

######  Theorem 33.

Any $C$-continuous function is continuous.

###### Proof.

Let $h:(X,\mathcal{T})\to(Y,\mathcal{P})$ be a $C$-continuous function. Let
$U$ be any open set in $Y$, then by $C$-continuity $h^{-1}(U)\in CO(X)$. Since
any $C$-open set is open set, then $h^{-1}(U)\in\mathcal{T}$. ∎

######  Theorem 34.

Any $F$-continuous function is $C$-continuous.

###### Proof.

Obvious, because any $F$-open set is $C$-open. ∎

It is clear, for any topological space $(X,\mathcal{T})$ we have have.

$F$-continuous function $\Longrightarrow$ $C$-continuous function
$\Longrightarrow$ continuous function.

None of the above implications is reversible.

Here an example of continuous function is neither $C$-continuous nor
$F$-continuous.

######  Example 35.

Let $(\mathbb{R},\mathcal{T_{\mathbb{I}}})$ be the excluded set topological
space on $\mathbb{R}$ by $\mathbb{I}.$ Then the identity function
$id:(\mathbb{R},\mathcal{T_{\mathbb{I}}})\to(\mathbb{R},\mathcal{T_{\mathbb{I}}})$
is continuous function, which is neither $C$-continuous nor $F$-continuous,
because $\mathbb{Q}\in\mathcal{T_{\mathbb{I}}}$ is open and
$id^{-1}(\mathbb{Q})=\mathbb{Q}$ is neither $C$-open nor $F$-open set, because
$cl(\mathbb{Q})\setminus\mathbb{Q}=\mathbb{R}\setminus\mathbb{Q}=\mathbb{I}$
is uncountable set.

Also, there is an example of $C$-continuous function which is not
$F$-continuous function.

######  Example 36.

Let $(\mathbb{R},\mathcal{C})$ be the co-countable topological space (see
[3]). Let $id$ be the identity function such that
$id:(\mathbb{R},\mathcal{C})\to(\mathbb{R},\mathcal{C})$ and let $K$ be any
open set in $(\mathbb{R},\mathcal{C})$, then $K$ is $C$-open, because
$cl(K)\setminus K=\mathbb{R}\setminus K$ is countable. Hence,
$id:(\mathbb{R},\mathcal{C})\to(\mathbb{R},\mathcal{C})$ is $C$-continuous
function. However, $\mathbb{R}\setminus\mathbb{Z}$ is open set such that
$id^{-1}(\mathbb{R}\setminus\mathbb{Z})=\mathbb{R}\setminus\mathbb{Z}$ is not
$F$-open set, because
$cl(\mathbb{R}\setminus\mathbb{Z})\setminus(\mathbb{R}\setminus\mathbb{Z})=\mathbb{R}\setminus(\mathbb{R}\setminus\mathbb{Z})=\mathbb{Z}$
is not finite set.

######  Definition 37.

Let $X$ be a topological space, then $X$ is $C$-compact (resp.
$C$-Lindel$\ddot{o}$f) if and only if any open cover of $X$ has a finite
(resp. countable) subcover of $C$-open sets.

######  Definition 38.

Let $X$ be a topological space, then $X$ is $C$-countably compact if and only
if any countable open cover of $X$ has a finite subcover of $C$-open sets.

######  Theorem 39.

Any $C$-compact space is compact space.

###### Proof.

Obvious, because any $C$-open set is open. ∎

######  Corollary 40.

Any $C$-Lindel$\ddot{o}$f (resp., $C$-countably compact) space is
Lindel$\ddot{o}$f (resp., countably compact) space.

######  Theorem 41.

Any $F$-compact space is $C$-compact space.

###### Proof.

Obvious, because any $F$-open set is $C$-open. ∎

######  Corollary 42.

Any $F$-Lindel$\ddot{o}$f (resp., $F$-countably compact) space is
$C$-Lindel$\ddot{o}$f (resp., $C$-countably compact) space.

######  Theorem 43.

Let $h:(X,\mathcal{T})\to(Y,\mathcal{P})$ is $C$-continuous, onto, $C$-open
function and $(X,\mathcal{T})$ is $C$-compact, then $Y$ is $C$-compact.

###### Proof.

Let $\\{V_{\alpha}:\alpha\in\Lambda\\}$ be any open cover of $Y$. Since $h$ is
$C$-continuous, then $h^{-1}(V_{\alpha})$ is $C$-open in $X$ for each
$\alpha\in\Lambda$. Since $Y\subseteq\bigcup_{\alpha\in\Lambda}V_{\alpha}$,
then $X=h^{-1}(Y)\subseteq
h^{-1}(\bigcup_{\alpha\in\Lambda}V_{\alpha})=\bigcup_{\alpha\in\Lambda}h^{-1}(V_{\alpha})$
), that is means $\\{h^{-1}(V_{\alpha}):\alpha\in\Lambda\\}$ is an open cover
of $X$ (because any $C$-open set is open). Then, by the $C$-compactness of
$X$, there exist $\alpha_{1},\alpha_{2},\cdots\alpha_{n}\in\Lambda$ such that
$h^{-1}(V_{\alpha_{1}})\cup h^{-1}(V_{\alpha_{2}})\cup\dots\cup
h^{-1}(V_{\alpha_{n}})=X$, then $h[h^{-1}(V_{\alpha_{1}})\cup
h^{-1}(V_{\alpha_{2}})\cup\dots\cup h^{-1}(V_{\alpha_{n}})]=h(X)$, then
$h(h^{-1}(V_{\alpha_{1}}))\cup h(h^{-1}(V_{\alpha_{2}}))\cup\dots\cup
h(h^{-1}(V_{\alpha_{n}}))=Y$, then $V_{\alpha_{1}}\cup
V_{\alpha_{2}}\cup\dots\cup V_{\alpha_{n}}=Y$. Since $h$ is $C$-open, then
$\\{V_{\alpha_{1}}\cup V_{\alpha_{2}}\cup\dots\cup V_{\alpha_{n}}\\}$ is a
finite subcover of $C$-open sets for $Y$. Therefore, $(Y,\mathcal{P})$ is a
$C$-compact space. ∎

A subset $B$ of a space $X$ is $C$-compact if and only if $B$ is a $C$-compact
topological space with the subspace topology.

######  Theorem 44.

Let $h:(X,\mathcal{T})\to(Y,\mathcal{P})$ is $C$-continuous function and
$(X,\mathcal{T})$ is $C$-compact, then $Y$ is compact.

###### Proof.

Using the same proof of Theorem 43. ∎

######  Theorem 45.

Let $h:(X,\mathcal{T})\to(Y,\mathcal{P})$ is $F$-continuous, $F$-open, onto
function and $(X,\mathcal{T})$ is $F$-compact, then $Y$ is $C$-compact.

###### Proof.

The proof very clear, by Theorem 41 any $F$-compact space is $C$-compact, and
Theorem $13$ in [4] any $F$-continuious, $F$-open and onto image of
$F$-compact space is $F$-compact space. ∎

######  Theorem 46.

Let $h:(X,\mathcal{T})\to(Y,\mathcal{P})$ is $C$-continuous, $C$-open, onto
function and $(X,\mathcal{T})$ is $C$-Lindel$\ddot{o}$f (resp., $C$-countably
compact) space, then $Y$ is $C$-Lindel$\ddot{o}$f (resp., $C$-countably
compact).

###### Proof.

Using the same proof of Theorem 43. ∎

######  Theorem 47.

Let $h:(X,\mathcal{T})\to(Y,\mathcal{P})$ is $F$-continuous, onto, $F$-open
function, and $(X,\mathcal{T})$ is $F$-Lindel$\ddot{o}$f (resp., $F$-countably
compact) space, then $Y$ is $C$-Lindel$\ddot{o}$f (resp., $C$-countably
compact).

###### Proof.

Obviously, by Corollary 42 any $F$-Lindel$\ddot{o}$f (resp., $F$-countably
compact) space is $C$-Lindel$\ddot{o}$f (resp., $C$-countably compact) space,
and Theorem $15$ in [4] any $F$-continuious, $F$-open and onto image of
$F$-Lindel$\ddot{o}$f (resp., $F$-countably compact) space is
$C$-Lindel$\ddot{o}$f (resp., $C$-countably compact) space. ∎

######  Theorem 48.

Let $h:(X,\mathcal{T})\to(Y,\mathcal{P})$ is $C$-continuous and onto function
and $(X,\mathcal{T})$ is $C$-Lindel$\ddot{o}$f (resp., $C$-countably compact)
space, then $Y$ is Lindel$\ddot{o}$f (resp., countably compact).

###### Proof.

Using the same proof of Theorem 43. ∎

Obviously, for any topological space $(X,\mathcal{T})$ we have have.

$F$-compactness $\Longrightarrow$ $C$-compactness $\Longrightarrow$
compactness.

None of the above implications is reversible. Here an example of compact space
is neither $C$-compact nor $F$-compact spaces.

######  Example 49.

Overlapping Interval Topology [3]. On the set $X=[-1,1]$ we generate a
topology from sets of the form $[-1,b)$ for $b>0$ and $(a,1]$ for $a<0.$ Then
all sets of the form $(a,b)$ are also open. $X$ is compact, since in any open
covering, the two sets which include $1$ and $-1$ will cover $X$. The space
$X$ is neither $C$-compact nor $F$-compact spaces, because there exists
$\\{[-1,0.5),(-0.5,1]\\}$ is an open cover for $X$ has no finite subcover of
$C$-open sets or $F$-open sets, because $[-1,0.5)$ and $(-0.5,1]$ are neither
$C$-open nor $F$-open sets
($cl[-1,0.5)\setminus[-1,0.5)=[-1,1]\setminus[-1,0.5)=[0.5,1]$ is neither
countable nor finite set and
$cl(-0.5,1]\setminus(-0.5,1]=[-1,1]\setminus(-0.5,1]=[-1,-0.5]$ is neither
countable nor finite set).

##  5  Conclusion and other tasks

In this article, we have displayed the notions of $C$-open sets and discussed
its master properties. Then, we have defined some operators via $C$-open and
$C$-closed sets. We have disclosed the relationships between these operators
and examined their mains features. In addition, we have introduced continuous
functions and compact sets using $C$-open and $C$-closed sets and checked
their main properties.

Our next works will concentrate on studying further topological concepts by
$C$-open and $C$-closed sets on separation axioms, connectedness, and the
other topological properties.

## References

  * [1] C. Kuratowski, Topology I, 4th. ed., in French, Hafner, New york, 1958.
  * [2] R. Engelking, General Topology. PWN, Warszawa, 1977.
  * [3] L. Steen and J. Seebach, Counterexamples in Topology. Dover Publications, INC. 1995.
  * [4] M. H. Alqahtani, $F$-open and $F$-closed Sets in Topological Spaces. European Journal of Pure and Applied Mathematics, Vol. 16, No. 2, (2023), 819-832.


<EMAIL_ADDRESS>(S.A. Hosseini)

# Mandelic acid single-crystal growth: Experiments vs numerical simulations

Q. Tan  1  S.A. Hosseini  1,2  A. Seidel-Morgenstern  3  D. Thévenin  1  H.
Lorenz  3  11affiliationmark:  Laboratory of Fluid Dynamics and Technical
Flows, University of Magdeburg “Otto von Guericke”, D-39106 Magdeburg,
Germany.
22affiliationmark:  Department of Mechanical and Process Engineering, ETH
Zürich, 8092 Zürich, Switzerland.
33affiliationmark:  Max Planck Institute for Dynamics of Complex Technical
Systems (MPI DCTS), 39106 Magdeburg, Germany.

###### Abstract

Mandelic acid is an enantiomer of interest in many areas, in particular for
the pharmaceutical industry. One of the approaches to produce enantiopure
mandelic acid is through crystallization from an aqueous solution. We propose
in this study a numerical tool based on lattice Boltzmann simulations to model
crystallization dynamics of (S)-mandelic acid. The solver is first validated
against experimental data. It is then used to perform parametric studies
concerning the effects of important parameters such as supersaturation and
seed size on the growth rate. It is finally extended to investigate the impact
of forced convection on the crystal habits. Based on there parametric studies,
a modification of the reactor geometry is proposed that should reduce the
observed deviations from symmetrical growth with a five-fold habit.

###### keywords:

Mandelic acid, Phase-field model, Lattice Boltzmann method, Ventilation effect

76T20

##  1 Introduction

lbmLBMlattice Boltzmann method lbLBlattice Boltzmann nsNSNavier-Stokes
pdePDEpartial differential equation Mandelic acid is an aromatic alpha-hydroxy
acid, with formula ${\rm C}_{8}{\rm H}_{8}{\rm O}_{3}$. It is a white
crystalline powder that is soluble in water and most common organic solvents.
It has a density of 1.3 g/cm3 and molecular weight of 152.5 g/mol. It is
particularly important in the pharmaceutical industry for the organic
synthesis of pharmaceutical components. For instance an ester of mandelic acid
is essential to produce homatropine, used in eye drops as both a cycloplegic
and mydriatic substance. In addition, it is also popular in the production of
face peeling components [49], urinary tract infection treatments [5], and for
oral antibiotics [44]. In toxicological studies, the concentration of styrene
or styrene oxide is quantified by converting it into mandelic acid.

Figure 1: Molecular structure of mandelic acid enantiomers.

Mandelic acid exists in two enantiomeric forms as shown in Fig. 1, (S)- and
(R)- mandelic acid. Most practical applications require the enantiopure form
[5]. Amongst the different approaches to separate enantiomers, crystallization
process such as classical resolution and preferential crystallization
approaches are frequently used [30]. In such separation processes, the
properties of the crystalline products such as crystal size and shape are
largely determined by the growth process, which in turn depends on the
crystallization conditions. In the pharmaceutical industry, the resulting
crystal morphology is often of great importance, since it influences the rate
of dissolution and the absorption of drugs. Compressibility, hardness, and
flow properties of the drug are also strongly dependent on the crystal form
[16]. Accurate investigations regarding crystal growth are difficult because
the growth process varies greatly even under similar conditions: _crystal
growth dispersion_ is the term used to describe the fact that crystals,
although initially of same shape and size, can rapidly grow differently even
under the same growth conditions [45, 31]. The main reason for these growth
differences is probably related to minute tensions and deformations, leading
in turn to minimal structural differences [17]. Other reasons are accidental
deposits, or deposits of foreign bodies on the growing crystals’ surface,
which lead to incorporation into the crystal and ultimately different growth.
A proper understanding of growth conditions and their effect on the final
product is therefore essential to design and scale-up production units for
enantiopure substances.
A lot of experimental studies have been conducted concerning crystallization-
based enantioseparation process including growth kinetics of mandelic acid,
e.g. [1, 30, 11, 12, 13, 36, 45, 10]. However, numerical studies regarding
crystal habit and size of enantiopure S-mandelic acid remain scarce. The
phase-field method has been shown in general to be a powerful tool for
modeling structural evolution of materials and crystals. It is now widely used
for modeling solidification [4, 35] and grain growth [8, 47, 26, 50]. The
phase-field approach has also been used in the context of the lattice
Boltzmann method, now widely recognized as an efficient alternative to
classical tools, to simulate solidification processes [54, 29, 53, 38, 42, 51,
56]. This approach can reproduce numerically the solid-liquid interface
interactions and the hydrodynamic effects affecting the habits of growing
crystals [33, 15, 32, 40, 7, 48].
In this contribution, we study the growth of a single (S)-mandelic acid
crystal under different conditions (supersaturation, initial crystal size,
flow rate) with a previously developed and validated lattice Boltzmann-based
numerical model [48]. All simulations presented in this article are carried
out using the in-house solver ALBORZ [18]. The obtained results are validated
and compared with experimental data. After validating the numerical procedure
in a standalone manner via a self-convergence test, it is used to model the
growth of a single (S)-mandelic acid rhombic seed at temperature and
supersaturation corresponding to experimental settings; this provides a
further, independent validation of the numerical model. The solver is then
used to investigate the effect of different parameters such as supersaturation
and initial seed size on crystal growth. Finally, a detailed study of the
interaction between forced convection and crystal growth is presented.
Analyzing the results, a simple solution is proposed to improve symmetrical
growth under natural convection in the single-crystal cell used for all
experimental investigations.

##  2 Numerical method

###  2.1 Diffuse-interface formulation: governing equations

In the phase-field method solid growth dynamics are expressed via a non-
dimensional order parameter, $\phi$, going from (+1) in the solid to (-1) in
the pure liquid phase. The space/time evolution equations are written as [24,
2]:

$\tau_{0}a_{s}^{2}(\textbf{n})\frac{\partial\phi}{\partial
t}=W_{0}^{2}\bm{\nabla}\cdot\left(a_{s}^{2}(\textbf{n})\right)\bm{\nabla}\phi+W_{0}^{2}\bm{\nabla}\cdot\left(|\bm{\nabla}\phi|^{2}\frac{\partial[a(\textbf{n})^{2}]}{\partial\bm{\nabla}\phi}\right)\\\
+(\phi-\phi^{3})+\lambda U(1-\phi^{2})^{2},$ (1)

and:

$\frac{\partial U}{\partial
t}+\left(\frac{1-\phi}{2}\right)\bm{u}\cdot\bm{\nabla}U=D\bm{\nabla}\cdot\left(q(\phi)\bm{\nabla}U\right)-\frac{\partial\phi}{\partial
t}.$ (2)

Here $\tau=\tau_{0}a_{s}^{2}(\textbf{n})$. The coefficient $\lambda$ describes
the strength of the coupling between the phase-field and the supersaturation
field, $U$. The parameter $\tau_{0}$ denotes the characteristic time and
$W_{0}$ the characteristic width of the diffuse interfaces. In Eq. (1), the
latent heat of melting is written as $L$. The specific heat capacity $c_{p}$
is assumed to be the same in the two phases (symmetric model). The quantity
$\textbf{n}=-\frac{\bm{\nabla}\phi}{\left|\bm{\nabla}\phi\right|}$ is the unit
vector normal to the crystal interface – pointing from solid to fluid, while
$a_{s}(\textbf{n})$ is the surface tension anisotropy function. In the context
of the hexagonal mandelic acid crystal growth, this quantity is defined as:

$a_{s}(\textbf{n})=1+\epsilon_{xy}\cos(6\theta),$ (3)

rhsRHSright hand side lhsLHSleft hand side where
$\theta=\arctan(n_{y}/n_{x})$. The numerical parameter $\epsilon_{xy}$
characterizes the anisotropy strength, and is set in the present study to
$\epsilon_{xy}=0.05$ [25]. The term $(\phi-\phi^{3})$ is the derivative of the
double-well potential. The last term in Eq. (1) is a source term accounting
for the coupling between supersaturation $U$ and order parameter $\phi$.
There, $(1-\phi^{2})^{2}$ is an interpolation function minimizing the bulk
potential at $\phi=\pm 1$.
In Eq. (2), $\bm{u}$ denotes the local fluid velocity while $q(\phi)=(1-\phi)$
is a function canceling out diffusion within the solid. As a consequence,
solute transport is assumed to take place only within the fluid phase (one-
sided model). The parameter $D$ is the diffusion coefficient of S-mandelic
acid in water.

###  2.2 Lattice Boltzmann formulation

##### Flow field solver

The flow field behavior, described by the incompressible and continuity
equations, is modeled using the classical isothermal formulation consisting of
the now-famous stream-collide operators:

$f_{\alpha}\left(\bm{r}+\bm{c}_{\alpha}\delta_{t},t+\delta_{t}\right)-f_{\alpha}\left(\bm{r},t\right)=\delta_{t}\Omega_{\alpha}\left(\bm{r},t\right)+\delta_{t}F_{\alpha},$
(4)

where $f_{\alpha}$ and $\bm{c}_{\alpha}$ are the discrete populations and
corresponding particle velocity vectors, $\bm{r}$ and $t$ are the position in
physical space and time, $\delta_{t}$ is the time-step size. $F_{\alpha}$
represents contributions from external body forces defined as [14]:

$F_{\alpha}=w_{\alpha}\left(1-\frac{\delta_{t}}{2\tau_{f}}\right)\left[\frac{\bm{F}\cdot\bm{c}_{\alpha}}{c_{s}^{2}}+\frac{\left(\bm{u}\otimes\bm{F}+\bm{u}\otimes\bm{F}^{\dagger}\right)\left(\bm{c}_{\alpha}\otimes\bm{c}_{\alpha}-c_{s}^{2}\bm{I}\right)}{2c_{s}^{4}}\right].$
(5)

Here $\bm{F}$ is the external body force vector, $\bm{u}$ is the local fluid
velocity vector, $\bm{I}$ is the unit rank-two tensor, $c_{s}$ is the so-
called lattice sound speed, $w_{\alpha}$ are weights associated to each
discrete velocity and $\tau_{f}$ is the relaxation time. Here, the external
body force also includes interaction with the solid phase as [2]:

$\bm{F}=-\frac{h\tau_{f}(1+\phi)^{2}(1-\phi)\bm{u}}{4W_{0}^{2}},$ (6)

where $\phi$ is the phase indicator detailed in the next paragraphs, $W_{0}$
is the interface thickness tied to the phase-field solver and $h$ is a
dimensionless constant, chosen as $h=2.757$ following [2]. Due to the absence
of fluid velocity within the solid crystal, the fluid velocity $\bm{u}$ is
updated as:

$\bm{u^{*}}=\frac{(1-\phi)}{2}\bm{u},$ (7)

where the re-defined fluid velocity $\bm{u^{*}}$ is used in the equilibrium
distribution function [2]. The collision operator $\Omega_{\alpha}$ follows
the linear Bhatnagar-Gross-Krook approximation:

$\Omega_{\alpha}=\frac{1}{\tau_{f}}\left[f^{(eq)}_{\alpha}-f_{\alpha}\right],$
(8)

edfEDFequilibrium distribution function where $f_{\alpha}^{(eq)}$ is the
discrete isothermal defined as:

$f_{\alpha}^{(eq)}=\rho
w_{\alpha}\sum_{i}\frac{1}{i!c_{s}^{2i}}a^{(eq)}_{i}(\bm{u}):\mathcal{H}_{i}(\bm{c}_{\alpha}),$
(9)

where $a^{(eq)}_{i}$ and $\mathcal{H}_{i}(\bm{c}_{\alpha})$ are the
corresponding multivariate Hermite coefficients and polynomials of order $i$
[22]. Further information on the expansion along with detailed expressions of
the can be found in [43, 20, 18]. In the present work, an extended range of
stability is obtained by using a central Hermite multiple relaxation time
implementation; corresponding details can be found in [19]. The relaxation
time $\tau_{f}$ is tied to the fluid kinematic viscosity, $\nu$, as:

$\tau_{f}=\frac{\nu}{c_{s}^{2}}+\frac{\delta_{t}}{2}.$ (10)

It must be noted that conserved variables, i.e., density and momentum are
defined as moments of the discrete distribution function:

$\rho=\sum_{\alpha}f_{\alpha},$ (11)
$\rho\bm{u}=\sum_{\alpha}\bm{c}_{\alpha}f_{\alpha}.$ (12)

##### Advection-diffusion-reaction solver for supersaturation field

The space/time evolution equation of the supersaturation field $U$ is modeled
using an advection-diffusion-reaction -based discrete kinetic equation. It is
defined as[39, 23, 21]:

$g_{\alpha}\left(\bm{r}+\bm{c}_{\alpha}\delta_{t},t+\delta_{t}\right)-g_{\alpha}\left(\bm{r},t\right)=\delta_{t}\Omega_{\alpha}\left(\bm{r},t\right)+\delta_{t}\dot{\omega}_{\alpha},$
(13)

where $g_{\alpha}$ are the corresponding discrete populations and
$\dot{\omega}_{\alpha}$ is the source term:

$\dot{\omega}_{\alpha}=-w_{\alpha}\frac{\partial\phi}{\partial t}.$ (14)

The collision operator $\Omega_{\alpha}$ for the supersaturation field is:

$\Omega_{\alpha}=\frac{1}{\tau_{U}}\left[g^{(eq)}_{\alpha}-g_{\alpha}\right].$
(15)

where $g_{\alpha}^{(eq)}$ is the defined as:

$g^{(eq)}_{\alpha}=w_{\alpha}U\left[1+\frac{\bm{c}_{\alpha}\cdot\bm{u}}{c_{s}^{2}}\right].$
(16)

The supersaturation is the zeroth-order moment of $g_{\alpha}$:

$U=\sum_{\alpha}g_{\alpha},$ (17)

and the relaxation coefficient is tied to the diffusion coefficient of
mandelic acid in the aqueous solution, $D$, as:

$\tau_{U}=\frac{Dq(\phi)}{c_{s}^{2}}+\frac{\delta_{t}}{2}.$ (18)

##### Solver for phase-field equation

The phase-field equation is modeled using a modified scheme defined as [52,
6]:

$a_{s}^{2}(\bm{n})h_{\alpha}(\bm{r}+\bm{c}_{\alpha}\delta_{t},t+\delta_{t})=h_{\alpha}(\bm{r},t)-\left(1-a_{s}^{2}(\bm{n})\right)h_{\alpha}(\bm{r}+\bm{c}_{\alpha}\delta_{t},t)-\\\
\frac{1}{\tau_{\phi}(\bm{r},t)}\left[h_{\alpha}(\bm{r},t)-h_{\alpha}^{eq}(\bm{r},t)\right]+w_{\alpha}Q_{\phi}(\bm{r},t)\frac{\delta_{t}}{\tau_{0}},$
(19)

where the scalar function $Q_{\phi}$ is the source term of the phase-field
defined as:

$Q_{\alpha}=(\phi-\phi^{3})+\lambda U(1-\phi^{2})^{2},$ (20)

while the , $h_{\alpha}^{eq}$, is defined as:

$h_{\alpha}^{eq}=w_{\alpha}\left(\phi-\frac{1}{c_{s}^{2}}\bm{c}_{\alpha}\cdot\frac{W_{0}^{2}}{\tau_{0}}|\bm{\nabla}\phi|^{2}\frac{\partial(a_{s}(\bm{n})^{2}}{\partial\bm{\nabla}\phi}\frac{\delta_{t}}{\delta_{r}}\right).$
(21)

where $\delta_{r}$ is the grid-size. The local value of the order parameter
$\phi$ is computed as:

$\phi=\sum_{\alpha}h_{\alpha},$ (22)

while the relaxation is set to:

$\tau_{\phi}=\frac{1}{c_{s}^{2}}a_{s}^{2}(\bm{n})\frac{W_{0}^{2}}{\tau_{0}}+\frac{\delta_{t}}{2}.$
(23)

##  3 Experimental setup

Experimental data for the growth rates have been obtained in the single-
crystal growth cell [13, 3] illustrated in Fig. 2. The supersaturated aqueous
solution of mandelic acid is pumped into a constant-temperature cylindrical
crystallization cell, with solution temperatures varying between 20 and 30 ∘C.
The temperature within the cell is maintained constant via a water-based
cooling/heating system connected to a Pt-100 sensor monitoring the temperature
inside the cell. Vessel 2, denoted V2 in Fig. 2b contains a saturated solution
at temperature $T_{2}$ while vessel 1 (V1) was set to a lower temperature
$T_{1}$, corresponding to the temperature of the cell. To create the
supersaturated solution, the initially saturated solution in V2 is pumped into
V1 and cooled down to $T_{1}$ before entering the growth cell. This
effectively allows to control the supersaturation level of the incoming
solution by choosing temperature $T_{1}$. In the present case, the
supersaturation is defined as [34]:

$U=\frac{C_{sat,2}-C_{sat,1}}{C_{sat,1}}$ (24)

Figure 2: Single-crystal growth cell used for all experiments: (a) photograph;
(b) Schematic diagram of experimental arrangement for the measurement of
growth rate of a single crystal [13, 3]

.

To start the experiment, the supersaturated solution is continuously pumped
from vessel 1 to the growth cell, in which a single (S)-mandelic crystal is
glued on the pin head of a crystal holder. Then, the solution is recycled to
vessel 2 and the concentration of the solution is compensated. In that way, a
stable degree of supersaturation is guaranteed during the whole process. A
microscope with camera (Stemi2000C, company Carl Zeiss) is used to take
pictures of the single crystal at every one hour. The images are afterwards
post-processed by applying Carl Zeiss’ Axio Vision software [13]. A picture of
the single-crystal cell is shown in Fig. 2.

##  4 Simulations and analysis of the results

###  4.1 Validation

####  4.1.1 Self-convergence of the numerical solver

Based on the experiments, the evolution of mandelic acid without enantiomers
follows habits with hexagonal symmetry. First, before going into further
validation steps against experimental results, we look into the convergence
behavior of the numerical scheme. To that end growth simulations are conducted
using the hexagonal anisotropy function that will be used for the remainder of
this work, starting with a rhombic initial seed. The seed is placed at the
center of a fully periodic rectangular domain, with a length of 31 and a width
of 26 mm. The perimeter of the initial rhombic crystal is $6.9$ mm and the
initial supersaturation is set to $U=0.06$. Simulations are conducted using
four different spatial resolutions,
$\delta_{r}\in\\{0.04,0.025,0.02,0.0125\\}$ mm. Since the overall size of the
numerical domain is kept fixed, an improved spatial resolution automatically
comes with a larger number of grid points.

Figure 3: Left: $\phi=0$ iso-contour, showing the boundary of the solid
crystal (only the central part of the numerical domain is shown) after 16
hours; Right: Evolution of function $\phi$ in space along the line joining the
center of the domain at (0,0) and point (4 mm,0) for $U=0.06$ at increasing
spatial resolution (grids with 775 $\times$ 650, 1240 $\times$ 1040, 1550
$\times$ 1300, and 2480 $\times$ 2080 points, respectively).

The highest resolution simulation with 2480 $\times$ 2080 points is used as
reference to compute relative errors at the three lower spatial resolutions.
The $\mathit{l^{2}}$ relative error norm is calculated based on the
$\phi$-profiles plotted along the $x$-direction on the centerline starting
from the center of the domain (0,0) in positive $x$-direction until point (4
mm,0). The corresponding profiles along with the crystal shape obtained after
16 hours are shown in Fig. 3. The $\mathit{l^{2}}$ norm is defined as:

${\rm
E}_{\mathit{l^{2}}}=\sqrt{\frac{\sum_{i}\left(\phi_{i}-\phi_{ref,i}\right)^{2}}{\sum_{i}\phi_{ref,i}^{2}}}$
(25)

where $\phi$ represents a lower resolution and $\phi_{ref}$ denotes the
highest resolution (used as reference). The errors obtained from the different
simulations are illustrated in Figure 4.

Figure 4: Scaling of the $l^{2}$ norm of errors as obtained from the self-
convergence study. Black markers represent error data from the simulations
while the black dashed line shows the theoretical $-2$ slope.

As observed from this plot, the numerical scheme is convergent as the error
decreases with resolution. Furthermore, as expected from theoretical analyses,
a second-order convergence is obtained in space.

####  4.1.2 Validation against experimental data

Next, to showcase the ability of the model to correctly reproduce the behavior
of the real system, 2-D simulations are considered using the real reactor
geometry. The simplification to two-dimensional simulations is justified by
the fact that, in all conditions considered here, the crystal follows a
platelet growth mode indicating clear separation of scales between growth in
the axial and planar directions, in connection to a symmetry of the flow field
[15]. The geometry used for the simulations is shown in Fig. 5. First,
configurations are considered where forced convection is negligible. For all
experiments presented in this section the initial seed is a rhombic crystal.
Two different initial supersaturations are considered, namely $U=0.06$ and
$U=0.11$ for the same temperature, $T=20^{\circ}$C. The diffusion coefficient
of mandelic acid in the aqueous solution under the conditions considered here
is $D=4.273\times 10^{-4}{\rm mm}^{2}/{\rm s}$ [9] and the other physical
parameters are listed in table 1. Based on the excellent agreement observed in
the previous section, all simulations are conducted with a spatial resolution
of $\delta_{r}=0.025$ mm. The interface thickness is set to $W_{0}=0.05$ mm,
the relaxation time to $\tau_{0}=11$ s, and the coupling coefficient
$\lambda=3$ was treated as a numerical parameter, consistently with the
standard phase-field method for dendrite growth [37]. At the walls of the
reactor, zero-flux boundary conditions are applied to both the species and
phase fields via the anti-bounce-back scheme. At the inlet a constant
supersaturation is imposed. Details on the implementation can be found in
[28].

Table 1: Physical parameters for a single S-ma crystal growth[55, 46, 41] Surface energy | Melting temperature | Vometric heat capacity | Latent heat | Capillary length
---|---|---|---|---
$[\hbox{J}\hbox{m}^{-2}]$ | $[\hbox{K}]$ | $[\hbox{J}/\hbox{m}^{3}\hbox{K}]$ | $[\hbox{J}/\hbox{m}^{3}]$ | $[\hbox{m}]$
0.05 | 392 | 1.7 $\times 10^{6}$ | 6.6 $\times 10^{7}$ | 7.65 $\times 10^{-9}$

Simulation results are compared to experiments and validated both
qualitatively using the crystal shape, and quantitatively by comparing the
growth rate.

Figure 5: Reactor geometry employed for all simulations. Figure 6: Method used
to number the crystal sides and the associated normal directions [3].

To measure the crystal growth rate in both experiments and numerical
calculations, the average length quantifying the crystal size is introduced
following [3] as illustrated in Fig. 6:

  * •

Connect opposite sides via their centers.

  * •

Identify the crystal center as the intersection between those lines.

  * •

Number the different sides as shown in Fig. 6.

  * •

Compute the lengths of the corresponding normal distance from the center
identified in the previous step ($L_{1}$, $L_{2}$, $L_{3}$, $L_{4}$, $L_{5}$
and $L_{6}$).

  * •

The average normal length is simply defined as $L_{\rm
avg}=(L_{1}+L_{2}+L_{3}+L_{4}+L_{5}+L_{6})/6$.

The experiments are systematically conducted over a period of 12 hours. The
average length is computed every hour and subsequently fitted with a linear
function to extract an average growth rate $G_{th}=L_{avg}/t$, where $t$ is
the corresponding growth time during the crystallization process. The
evolution of the crystal shape in both experiments and simulations are
illustrated in Figures 7 and 8.

Figure 7: Contours of an (S)

-mandelic acid crystal vs time as obtained from (a) experiments [3] and (b) simulations. The supersaturation is $U=0.06$ in both cases. The spatial scale is the same in all images, enabling a direct comparison.

Figure 8: Contours of the (S)-mandelic acid crystal vs time as obtained from
simulations for $U=0.11$.

A visual comparison regarding crystal shape and size over time for $U=0.06$
points to a good agreement between experiments and simulations. For $U=0.11$
only numerical results are shown since experimental snapshots are not
available. To validate the results in a quantitative manner, the growth rates
corresponding to the six different sides of the crystal for both
supersaturations as obtained from experiments and simulations are compared in
Table 2.

Table 2: Comparisons between experiments and simulations for supersaturation $U=0.06$ and $U=0.11$ [3] Cases | Experiments | Simulations
---|---|---
Number | I | II | I | II
Supersaturation(%) | 0.06 | 0.11 | 0.06 | 0.11
Seed perimeter (mm) | 6.9 | 9.5 | 6.9 | 9.5
Parameter | Slope | $R^{2}$ | Slope | $R^{2}$ | Slope | Slope
Normal 1 | 0.07 | 0.99 | 0.14 | 0.98 | 0.08 | 0.14
Normal 2 | 0.09 | 0.97 | 0.14 | 0.94 | 0.08 | 0.14
Normal 3 | 0.00 | 0.10 | 0.00 | 0.50 | 0.01 | 0.02
Normal 4 | 0.06 | 0.94 | 0.06 | 0.78 | 0.08 | 0.14
Normal 5 | 0.03 | 0.92 | 0.08 | 0.86 | 0.08 | 0.14
Normal 6 | 0.00 | 0.20 | 0.09 | 0.93 | 0.01 | 0.02
Average growth rate |  |  |  |
$G_{th}$ (mm/h) | 0.06 | 0.1 | 0.057 | 0.10

$R^{2}$ is the coefficient of determination shown here to characterize the
reliability of the linear regression used to extract growth rates.
Representing the length of side $i$ measured at time $t$ in experiments as
$L_{i}(t)$ and the value of the linear function as $L_{i}^{\prime}(t)$ the
coefficient is computed as:

$R^{2}=1-\frac{A_{\rm res}}{A_{\rm tot}},$ (26)

where the residual sum of squares is:

$A_{\rm res}=\sum_{t}{(L_{i}(t)-L_{i}^{\prime}(t))}^{2},$ (27)

and the total sum of squares is:

$A_{\rm tot}=\sum_{t}{(L_{i}(t)-\overline{L_{i}})}^{2},$ (28)

where $\overline{L_{i}}$ represents the average over all data points.
A direct comparison of the growth rates for both values of $U$ confirms the
very good agreement between experimental observations and numerical
simulations. For $U=0.06$, the growth rate is numerically underpredicted by
less than 6%. At $U=0.11$, the relative difference is even reduced to 2%. This
proves the ability of the numerical model to capture the growth of
(S)-mandelic acid in a pure aqueous environment. At higher supersaturation,
$U=0.11$, the crystal experiences as expected a faster growth as compared to
the lower supersaturation case, $U=0.06$. It is interesting to take now a
closer look into the effects of supersaturation on growth rate.

####  4.1.3 Impact of supersaturation on growth rate

In order to have a better understanding of the effect of supersaturation on
crystal growth dynamics we keep a configuration similar to the previous one,
but considering many more supersaturation values,
$U\in\\{0.06,0.085,0.11,0.15,0.2\\}$. In all simulations presented in this
section, the initial seed size and geometry follow that of configuration I in
the previous section, see Table 2. The evolution of the crystal shape over
time as obtained from these simulations are shown in Fig. 9. In Fig. 9, the
facets of the crystal start deviating from a straight line resulting from the
onset of primary branching instabilities. This usually occurs when the initial
value of supersaturation is sufficiently large.

Figure 9: Boundaries of the single crystal of (S)-mandelic acid (iso-contours
of $\phi=0$) at time $t$=0 (blue), 4 (red), 8 (black), 12 hours (purple) as a
function of supersaturation: (top-left) $U=0.085$, (top-right) $U=0.11$,
(bottom-left) $U=0.15$, and (bottom-right) $U=0.2$.

Furthermore, the data representing average growth rate are listed in Table 3.

Table 3: Numerically observed averaged crystal growth rate as function of supersaturation between $U=0.06$ and $0.2$ Supersaturation(%) | 0.06 | 0.085 | 0.11 | 0.15 | 0.2
---|---|---|---|---|---
Seed perimeter (mm) | 6.9 | 6.9 | 6.9 | 6.9 | 6.9
Average growth rate |  |  |  |  |
$G_{th}$ (mm/h) | 0.057 | 0.0871 | 0.1136 | 0.1553 | 0.2093

As expected, higher supersaturations lead to faster crystal growth. It is
worth to note that the average growth rate of the crystal can be very well
approximated in the considered range as a linear function of supersaturation
with slope one. Another parameter that has been observed experimentally to
affect growth dynamics, especially during the early phase, is the initial seed
size, better quantified via its perimeter. We will look into that effect in
the next section.

####  4.1.4 Impact of initial size on growth rate

To quantify the effects of initial seed size, which is known to be sometimes
important [45], a configuration with supersaturation $U=0.06$ at
crystallization temperature $T=20^{\circ}$C has been retained. Seeds with the
same initial rhombic shape but different sizes have been simulated; the
initial seed perimeters are $P\in\\{6.9,6.96,8.4,8.8\\}$ mm, corresponding to
available experimental data. The resulting growth rate data and parameters
extracted from the experiments are listed in Table 4.

Table 4: Impact of initial seed size in experiments with supersaturation $U=0.06$ [3] Experiment Number | I(1) | I(2) | I(3) | I(4)
---|---|---|---|---
Perimeter (mm) | 6.9 | 6.96 | 8.4 | 8.8
Parameter | Slope | $R^{2}$ | Slope | $R^{2}$ | Slope | $R^{2}$ | Slope | $R^{2}$
Normal 1 | 0.07 | 0.99 | 0.09 | 0.99 | 0.09 | 0.99 | 0.10 | 0.98
Normal 2 | 0.09 | 0.97 | 0.09 | 0.99 | 0.12 | 0.98 | 0.10 | 0.97
Normal 3 | 0.00 | 0.10 | 0.01 | 0.24 | 0.01 | 0.67 | 0.02 | 0.82
Normal 4 | 0.06 | 0.94 | 0.03 | 0.92 | 0.07 | 0.94 | 0.07 | 0.85
Normal 5 | 0.03 | 0.92 | 0.03 | 0.89 | 0.06 | 0.94 | 0.05 | 0.98
Normal 6 | 0.00 | 0.20 | 0.01 | 0.45 | 0.03 | 0.85 | 0.00 | 0.10
Average growth rate |  |  |  |
$G_{th}$ (mm/h) | 0.06 | 0.06 | 0.06 | 0.07

Simulations with exactly the same configurations have been conducted. The
corresponding results are listed in Table 5.

Table 5: Impact of initial seed size in simulations with supersaturation $U=0.06$ Simulation Number | I(1) | I(2) | I(3) | I(4)
---|---|---|---|---
Perimeter (mm) | 6.9 | 6.96 | 8.4 | 8.8
Parameter | Slope | Slope | Slope | Slope
Normal 1 | 0.08 | 0.08 | 0.082 | 0.082
Normal 2 | 0.08 | 0.08 | 0.082 | 0.082
Normal 3 | 0.01 | 0.01 | 0.012 | 0.016
Normal 4 | 0.08 | 0.08 | 0.082 | 0.082
Normal 5 | 0.08 | 0.085 | 0.085 | 0.082
Normal 6 | 0.01 | 0.01 | 0.012 | 0.015
Average growth rate |  |  |  |
$G_{th}$ (mm/h) | 0.0567 | 0.0575 | 0.0592 | 0.0598

Both experiments and simulations, while in fair agreement with each other,
point to the fact that the average growth rate is only slightly affected by
the initial seed size. The effect appears to be somewhat stronger for a larger
initial seed. The growth behavior over time is illustrated in Fig. 10. The
results for initial perimeters $6.9$ and $6.96$ mm cannot be differentiated
visually. For a larger initial seed, the differences between $8.4$ and $8.8$
mm can be recognized, but only a minute increase in average growth rate can be
recognized. It can be concluded that, compared to supersaturation, the
influence of initial seed size is minor. Still, a larger initial seed
corresponds to a slightly increased growth rate.

Figure 10: Numerical results concerning the effect of initial seed perimeter
on the growth rate.

All the previous results have been obtained while neglecting any convection
effect around the crystal. However, it is known that forced convection might
lead to asymmetric crystal growth. This will be explored next.

###  4.2 Ventilation effects

In the real single-crystal reactor the incoming flow of (S)-mandelic acid in
water might have a large impact on crystal growth rate and shape development.
The aim of the present section is to check and quantify this point.

##### Validation in presence of convection

First, we validate the numerical model against available experimental data
taking into account the real inflow conditions used in the reactor cell. For
this purpose, and following the experimental settings [27], a hexagonal seed
of perimeter $P=3.7$ mm is used. The initial supersaturation is $U=0.045$, the
Reynold number Re = 17.2. Here, we keep the same Reynold number as the
experiment [27]. The results, represented by the crystal shape, are compared
over time via snapshots taken every two hours over an overall growth period of
16 hours, as shown in Fig. 11.

Figure 11: Morphologies of (S)-mandelic acid crystal captured by (a) camera
during the experiments [27]; (b) simulations.

It is observed that the evolution of the crystal shape over time as obtained
from both experiment and simulation are in good agreement with each other;
they both point to a non-symmetrical growth. The constant inflow of a solution
with a higher concentration hitting the inlet-facing sides of the crystal
subjects them to noticeably larger gradients at the interface, as compared to
the other, leeward sides; this induces lower adsorption rates at the latter.
As a result the (S)-mandelic crystal grows faster on the sides facing the
inflow, leading to a steady increase of the aspect ratio, defined as the ratio
of the horizontal size of the crystal to its vertical size. This is clearly
visible in Fig. 12 where both the velocity and supersaturation fields at
$t=$16 hours are shown.

Figure 12: Non-symmetric growth of a (S)-mandelic acid crystal taking into
account convection as obtained from the simulation for $U=0.045$ after 16
hours. Flow is from left to right in the reactor.

To further illustrate the effects of hydrodynamics on the crystal habit, the
effect of the Reynolds number and of the initial orientation of the seed are
considered numerically.

##### Effect of Reynolds number

For this purpose, the supersaturation is kept constant at $U=0.045$. The
Reynolds number is the most important non-dimensional parameter of fluid
dynamics, comparing quantitatively convective effects to dissipation by
viscosity. It is defined as:

${\rm Re}=\frac{u_{\rm in}P}{\nu_{f}},$ (29)

where $P$ is the initial seed perimeter. Two different Reynolds numbers, i.e.
Re$=8.6$ and $17.2$, are considered. The obtained crystal shape and velocity
fields are shown in Fig. 13.

Figure 13: Convection effects on (S)-mandelic acid crystal growth after 10
hours for $U=0.045$ and two different Reynolds numbers. Left side: Re $=17.2$;
Right side: Re $=8.6$

. The white line represents the crystal boundary taking into account the flow
(ventilation effect), while the grey line shows the same results in the
absence of any inflow.

As expected, higher inlet velocities hitting the inflow-facing sides of the
crystal result in a faster growth in that direction; the resulting asymmetry
becomes more marked, leading to elongated crystals in the horizontal direction
for the present setup. This is particularly clear looking at Fig. 14. After 10
hours, the aspect ratio for Re$=17.2$ is already more than twice as large than
for Re$=8.6$.

Figure 14: Evolution of the aspect ratio vs time for Reynolds numbers Re$=8.6$
and $17.2$.

##### Effect of the initial orientation of the seed

To show the effect of seed orientation, four simulations have been carried out
with the same Reynolds number, Re$=17.2$, but with different initial
orientations, namely
$\theta\in\\{0,\frac{\pi}{12},\frac{\pi}{6},\frac{\pi}{4}\\}$. This choice of
tilt is motivated by the six-fold symmetry of the crystal natural habit, so
that only the 0-$\pi/3$ range is relevant. The obtained crystal habits after
10 hours along with the corresponding velocity fields are illustrated in Fig.
15.

Figure 15: Effect of initial seed orientation on (S)-mandelic acid crystal
growth after 10 hours for $U=0.045$. Top left: without rotation; Top right:
initial rotation of $\frac{\pi}{12}$ (clockwise rotation); Bottom left;
initial rotation of $\frac{\pi}{6}$; Bottom right: initial rotation of
$\frac{\pi}{4}$. The white line represents the crystal boundary taking into
account the flow, while the green line shows the same results in the absence
of any inflow.

It is seen that the initial orientation not only affects the symmetry of the
crystal, but also its average growth rate. Taking into account convective
effects and initial seed orientation, the crystal habits become highly
asymmetrical. It is also observed that a slight initial rotation in clockwise
direction can result in a final habit showing preferential counter-clockwise
orientation, due to a strong interaction with the convective flow field.

##### Improving symmetrical growth in presence of convection using a

baffle As seen from the previous simulations, the overall shape of the crystal
varies considerably as function of the Reynolds number. It was mentioned
earlier that the regularity of the crystal shape is a property of high
interest regarding the final products performance. Therefore, it is desirable
to find a simple geometrical modification to the single-crystal growth cell
leading to isotropic growth rates and a desired final aspect ratio. For this
purpose, a simple flat baffle has been placed in the simulation directly in
front of the inlet in order to prevent a direct impact of the incoming flow
onto the growing seed. Three different configurations (different position,
different size) of the baffles have been compared. The resulting
configurations are illustrated in Fig. 16; configuration 1 is the original
case, without any baffle.

Figure 16: Proposed modifications of the geometry of the single-crystal growth
cell reactor including a baffle (three different possible configurations).

To check the robustness of the proposed modification with the three different
baffles (plus the original case), two different Reynolds numbers (Re$=8.6$ or
$17.2$), and two different initial seed orientations ($\theta$=0 or $\pi/6$)
have been considered, making up for a total of ($4\times 2\times 2=$) 16
different cases. All numerical results after 10 hours of growth are shown in
Figs. 17 (for Re=$17.2$) and 18 (for Re$=8.6$).

Figure 17: Numerical prediction for the growth on (S)-mandelic acid crystal
after 10 hours for $U=0.045$ and Re$=17.2$. First column: original
configuration, without baffle; second column: baffle configuration 2; third
column: baffle configuration 3; fourth column: baffle configuration 4. Top
line (a): without any rotation of the initial seed; Bottom line (b): with
initial rotation of the seed by $\pi$/6. Figure 18: Numerical prediction for
the growth on (S)-mandelic acid crystal after 10 hours for $U=0.045$ and
Re$=8.6$. First column: original configuration, without baffle; second column:
baffle configuration 2; third column: baffle configuration 3; fourth column:
baffle configuration 4. Top line (a): without any rotation of the initial
seed; Bottom line (b): with initial rotation of the seed by $\pi$/6.

To quantify the effects of the baffles on the quality of the crystal, a
quality parameter is defined as $Q=\max(L_{i})/\min(L_{i})$ where index
$i\in\\{0,\dots,5\\}$ covers the length of all sides of the resulting crystal.
Parameter $Q$ quantifies non-isotropic growth, with $Q=1$ (the minimum value)
corresponding to a perfectly isotropic growth, while an increasing value of
$Q$ corresponds to growing non-isotropy. The values of crystal quality as
obtained from all simulations after 10 hours of growth are listed in Table 6.

Table 6: Impact of the baffles for $U=0.06$ for two different Reynolds numbers and seed orientations in Fig. 16 Q | Re=8.6; tilt=0 | Re=8.6; tilt=$\pi$/6 | Re=17.2; tilt=0 | Re=17.2; tilt=$\pi$/6
---|---|---|---|---
No Baffle | 1.28 | 1.23 | 1.625 | 1.59
Baffle 1 | 1.22 | 1.21 | 1.41 | 1.39
Baffle 2 | 1.13 | 1.16 | 1.24 | 1.21
Baffle 3 | 1.05 | 1.07 | 1.14 | 1.12

From Table 6 it is clearly observed that, while all baffles contribute to
reducing asymmetrical growth, the larger one, i.e. baffle 3, leads to the best
crystal quality in terms of symmetry for all considered conditions. It
successfully reduces deviations from perfect symmetry by about 20% for
Re$=8.6$ and 30% for Re$=17.2$. Since the complexity of the experimental setup
would not be significantly increased by adding a baffle, such a modification
is recommended for further studies.

##  5 Conclusions and perspectives

In this work, a numerical model based on lattice Boltzmann has been developed
and validated to describe crystal growth. It has been shown to correctly
capture the dynamics of (S)-mandelic acid crystal growth. The numerical
simulations were compared to experimental data from a single-crystal growth
reactor and are in very good agreement. The model was then used to investigate
the effects of important parameters such as supersaturation and initial seed
size on growth dynamics. It was depicted that higher supersaturation levels
lead to much faster growth rates; the impact of a larger initial seed crystal
is by far not as strong, but increases slightly the growth rate as well.
It was also demonstrated that hydrodynamics can have pronounced effects on
both average growth rate and habit, and may lead to a clear rupture of
symmetry. The evolution of the crystal habit was shown to change significantly
with the Reynolds number, but also with the initial orientation of the seed
with respect to the incoming flow. Finally, a simple modification of the
reactor geometry was proposed to minimize non-symmetrical growth. This will be
tested in later experiments.
While the model was successfully applied in the present study to pure
(S)-mandelic acid crystal growth under isothermal conditions, in the future,
these simulations will be extended to more complex situations involving a
mixture of both (S)- and (R)-mandelic acid and taking into account temperature
changes.

## Acknowledgments

Q.T. would like to acknowledge the financial support by the EU-program ERDF
(European Regional Development Fund) within the Research Center for Dynamic
Systems (CDS). S.A.H. acknowledges the financial support of the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) in TRR 287 (Project-
ID 422037413). The authors gratefully acknowledge the computing time granted
by the Universität Stuttgart-Höchstleistungsrechenzentrum Stuttgart (HLRS);
all calculations for this publication were conducted with computing resources
provided under project number 44216.

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