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# Clique numbers of graphs and irreducible exact $m$-covers of $\mathbb{Z}$

Hao Pan  Department of Mathematics, Shanghai Jiaotong University, Shanghai
200240, People’s Republic of China<EMAIL_ADDRESS>and  Li-Lu Zhao
Department of Mathematics, Nanjing University
Nanjing 210093, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.

For each $m\geq 1$, we construct a graph $G=(V,E)$ with $\omega(G)=m$ such
that

$\max_{1\leq i\leq k}\omega(G[V_{i}])=m$

for arbitrary partition $\\{V_{1},\ldots,V_{k}\\}$ of $V$, where $\omega(G)$
is the clique number of $G$ and $G[V_{i}]$ is the induced graph of $G$ with
the vertex set $V_{i}$. Using this result, we show that for each $m\geq 2$
there exists an exact $m$-cover of $\mathbb{Z}$ which is not the union of two
1-covers.

###### 2000 Mathematics Subject Classification:

Primary 05C30; Secondary 11B25, 05C90, 05C20

##  1\. Introduction

In his proof of the existence of irreducible exact $m$-covers of $\mathbb{Z}$
(the notions will be introduced soon), Zhang proved the following graph-
theoretic result [21, Lemma 2]:

######  Theorem 1.1.

For every $m\geq 1$, there exists a graph $G=(V,E)$ satisfying the following
properties:

$\omega(G)=m$, where $\omega(G)$ is the clique number of $G$, i.e., the
maximal order of the complete subgraphs of $G$ . And if the vertex set $V$ is
arbitrarily split into two non-empty subsets $V_{1}$ and $V_{2}$, then

$\omega(G[V_{1}])+\omega(G[V_{2}])>\omega(G),$

where $G[V_{i}]$ denotes the induced subgraph of $G$ with the vertex set
$V_{i}$.

In this paper, our main purpose is to give an extension of Zhang’s result as
follows:

######  Theorem 1.2.

For every $m\geq 1$ and $k\geq 2$, there exists a graph $G=(V,E)$ with
$\omega(G)=m$ satisfying the following property:

If the vertex set $V$ is arbitrarily split into $k$ subsets
$V_{1},V_{2},\ldots,V_{k}$, then

$\max_{1\leq i\leq k}\omega(G[V_{i}])=\omega(G).$

For an integer $a$ and a positive integer $n$, let $a(n)$ denote the residue
class $\\{x\in\mathbb{Z}:\,x\equiv a\ ({\rm{mod}}\ n)\\}$. For a finite system
$\mathcal{A}=\\{a_{t}(n_{t})\\}_{t=1}^{s}$, define the covering function
$w_{\mathcal{A}}$ over $\mathbb{Z}$ by

$w_{\mathcal{A}}(x):=|\\{1\leq t\leq s:\,x\in a_{t}(n_{t})\\}|.$

If $w_{\mathcal{A}}(x)\geq m$ for each $x\in\mathbb{Z}$, we say that a system
$\mathcal{A}$ is an $m$-cover of $\mathbb{Z}$. In particular, we call
$\mathcal{A}$ an exact $m$-cover provided that $w_{\mathcal{A}}(x)=m$ for all
$x\in\mathbb{Z}$. The covers of $\mathbb{Z}$ was firstly introduced by Erdős
[4] and has been investigated in many papers (e.g., [8, 10, 22, 12, 1, 15, 16,
19, 2, 6]).

Suppose that $\mathcal{A}_{1}$ is an $m_{1}$-cover and $\mathcal{A}_{2}$ is an
$m_{2}$-cover, then clearly $\mathcal{A}=\mathcal{A}_{1}\cup\mathcal{A}_{2}$
forms an $(m_{1}+m_{2})$-cover. Conversely, Porubský [11] asked whether for
each $m\geq 2$ there exists an exact $m$-cover of $\mathbb{Z}$ which cannot be
split into an exact $n$-cover and an exact $(n-m)$-cover with $1\leq n<m$.
Choi gave such a example for $m=2$:

$\mathcal{A}=\\{1(2);0(3);2(6);0,4,6,8(10);1,2,4,7,10,13(15);5,11,12,22,23,29(30)\\}.$

In [21], using Theorem 1.1, Zhang gave an affirmative answer to Porubský’s
problem. This shows that the results on $m$-covers of $\mathbb{Z}$ is
essential. In [20], Sun established a connection between $m$-covers of
$\mathbb{Z}$ and zero-sum problems in abelian $p$-groups. For more related
results, the readers may refer to [14, 18, 17]

On the other hand, for each $m\geq 2$, Pan and Sun [9, Example 1.1]
constructed an $m$-cover of $\mathbb{Z}$ (though not exact) which even is not
the union of two $1$-covers! As an application of Theorem 1.2, we have a
common extension of the above two results:

######  Theorem 1.3.

For each $m\geq 2$, there exists an exact $m$-cover of $\mathbb{Z}$ which is
not the union of two $1$-covers.

We shall prove Theorem 1.2 in the next section, and the proof of Theorem 1.3
will be given in Section 3.

##  2\. Proof of Theorem 1.2

######  Lemma 2.1.

Suppose that $G=(V,E)$ is a connected simple graph and $v_{0}$ is a vertex of
$G$. Then there exists an oriented graph $\overrightarrow{G}$ arising from
$G$, which satisfies that:

(i) $\overrightarrow{G}$ doesn’t contains any directed cycle.

(ii) For any vertex $u\in V\setminus\\{v_{0}\\}$, there exists a directed path
of $\overrightarrow{G}$ from $v_{0}$ to $u$.

###### Proof.

We use induction on $|V|$. There is nothing to do when $|V|=1$ or $2$. Now
assume that $|V|>0$ and our assertion holds for any smaller value of $|V|$.
Let $V^{\prime}=V\setminus\\{v_{0}\\}$ and $G^{\prime}=G[V^{\prime}]$. Suppose
that $v_{1},\ldots,v_{s}\in V^{\prime}$ are all vertex adjacent to $v_{0}$ in
$G$. By the induction hypothesis, there exists an oriented graph
$\overrightarrow{G^{\prime}}$ obtained from $G^{\prime}$, satisfying the
properties (i) and (ii) for the vertex $v_{1}$. Now we direct the edge
$v_{0}v_{i}$ from $v_{0}$ to $v_{i}$ for $1\leq i\leq k$, and preserve the
direction of each edge in $\overrightarrow{G^{\prime}}$. Thus we obtain an
oriented graph $\overrightarrow{G}$. Clearly $\overrightarrow{G}$ doesn’t
contain any directed cycle since $v_{0}$ can’t lie in any directed cycle. And
for any $u\in V\setminus\\{v_{0},v_{1}\\}$, since there exists a directed path
of $\overrightarrow{G^{\prime}}$ from $v_{1}$ to $u$, the property (ii) is
also satisfied. ∎

######  Lemma 2.2.

For every $k\geq 1$, we can construct a $k$-chromatic graph without any
triangle.

###### Proof.

The reader may refer to [7] (or [3, Chapter 5, Exercise 23]) for the
construction of such graph. In fact, with help of his probabilistic method,
Erdős [5] proved that there exist the graphs having arbitrarily large girths
and chromatic numbers. ∎

###### Proof of Theorem 1.2.

Let $K=(V_{K},E_{K})$ be a $(k+1)$-chromatic graph without any triangle. Let
$u_{0}$ be a vertex of $K$. Then there exists an oriented graph
$\overrightarrow{K}$ arising from $K$, which satisfies the properties (i) and
(ii) of Lemma 2.1 for the vertex $u_{0}$. Let $n=|V_{K}|$ and suppose that
$u_{0},u_{1},\ldots,u_{n-1}$ are all vertices of $K$. For $1\leq i\leq n-1$,
let $l_{i}$ denote the length of the longest directed path from $u_{0}$ to
$u_{i}$ in $\overrightarrow{K}$. By the property (ii) of Lemma 2.1, these
$l_{i}$ are well-defined. Let $l=\max_{1\leq i\leq n-1}l_{i}$, and for $1\leq
j\leq l$ let

$D_{j}=\\{1\leq i\leq n-1;\,l_{i}=j\\}$

In particular, we set $D_{0}=\\{0\\}$. For $1\leq i\leq n-1$, let

$A_{i}=\\{0\leq i^{\prime}\leq
n-1:\,\overrightarrow{u_{i^{\prime}}u_{i}}\text{ lies in
}\overrightarrow{K}\\},$

where we denote by $\overrightarrow{xy}$ the directed edge from $x$ to $y$. In
particular, we set $A_{0}=\emptyset$.

######  Lemma 2.3.

For $1\leq j\leq l$, we have

$\bigcup_{u_{i}\in D_{j}}A_{i}\subseteq\bigcup_{0\leq j^{\prime}\leq
j-1}D_{j^{\prime}}.$ (2.1)

###### Proof.

Assume on the contrary that there exist $u_{i}\in D_{j}$ and $i^{\prime}\in
A_{i}$ such that $u_{i^{\prime}}\not\in\bigcup_{0\leq j^{\prime}\leq
j-1}D_{j^{\prime}}$. From the definition of $D_{j^{\prime}}$, we know that
there exists a path from $u_{0}$ to $u_{i^{\prime}}$ with the length at least
$j$. If $u_{i}$ doesn’t lie in this path, then we get a path from $u_{0}$ to
$u_{i}$ with the length at least $j+1$, since the direction of the edge
$\overrightarrow{u_{i^{\prime}}u_{i}}$ is from $u_{i^{\prime}}$ to $u_{i}$. On
the other hand, if $u_{i}$ lies in this path, then clearly we get a directed
cycle from $u_{i}$ to $u_{i^{\prime}}$, next to $u_{i}$. This also leads to a
contradiction with the property (i) of Lemma 2.1. ∎

######  Lemma 2.4.

$D_{j}\not=\emptyset$ for each $1\leq j\leq l$.

###### Proof.

Clearly $D_{l}\not=\emptyset$. Let $u_{i_{l}}$ be a vertex in $D_{l}$. Then
there exists a directed path in $\overrightarrow{K}$ from $u_{0}$ to
$u_{i_{l}}$ with the length $l$. Suppose that this path is

$u_{0}\to u_{i_{1}}\to u_{i_{2}}\to\cdots\to u_{i_{l-1}}\to u_{i_{l}}.$

We claim that $i_{j}\in D_{j}$ for each $1\leq j\leq l$. We use induction on
$j$. Clearly our assertion holds when $j=l$. Assume that $j<l$ and $i_{j+1}\in
D_{j+1}$. Clearly $l_{i_{j}}\geq j$ since $u_{0}\to u_{i_{1}}\to\cdots\to
u_{i_{j}}$ is a directed path with the length $j$. On the other hand, by Lemma
2.3, we have

$u_{i_{j}}\in A_{i_{j+1}}\subseteq\bigcup_{0\leq j^{\prime}\leq
j}D_{j^{\prime}}.$

Hence $l_{i_{j}}\leq j$. So $l_{i_{j}}=j$ and $i_{j}\in D_{j}$. We are done. ∎

We shall use induction on $m$ to prove Theorem 1.2. The case $m=1$ is trivial.
Now assume that $m\geq 2$ and our assertion holds for $m-1$. That is, there
exists a graph $G^{(m-1)}=(V^{(m-1)},E^{(m-1)})$ with $\omega(G^{(m-1)})=m-1$
satisfying that

$\max_{1\leq i\leq k}\omega(G^{(m-1)}[V_{i}])=m-1$

for arbitrary partition $V_{1},\ldots,V_{k}$ of $V^{(m-1)}$.

First, we shall create $n$ graphs $H_{0},H_{1},\ldots,H_{n-1}$. $H_{0}$ is a
graph only having a vertex $x_{0}$. For each $i\in D_{1}$, $H_{i}$ is one copy
of $G^{(m-1)}$. Similarly, for $2\leq j\leq l$ and every $i\in D_{j}$,
assuming $H_{i^{\prime}}$ have been created for all
$i^{\prime}\in\bigcup_{0\leq j\leq j-1}D_{j^{\prime}}$, let $H_{i}$ be

$h_{i}:=\prod_{i^{\prime}\in A_{i}}|V(H_{i^{\prime}})|$

disjoint copies of $G^{(m-1)}$, where $V(H_{i^{\prime}})$ denotes the vertex
set of $H_{i^{\prime}}$.

Next, we shall add some edges between the vertices of $H_{i}$ and the vertices
of $H_{i^{\prime}}$, for $0\leq j<j^{\prime}\leq l$, $i\in D_{j}$ and
$i^{\prime}\in D_{j^{\prime}}$. For every $i\in D_{1}$, we join $x_{0}$ and
$H_{i}$, i.e., join $x_{0}$ and all vertices of $H_{i}$. Below we shall
inductively add the edges incident with the vertices of $H_{i}$ for every
$2\leq j\leq l$ and $i\in D_{j}$. Suppose that $2\leq j\leq d$, $i\in D_{j}$
and $A_{i}=\\{i_{1}^{\prime},\ldots,i_{s}^{\prime}\\}$ with
$i_{1}^{\prime}<\cdots<i_{s}^{\prime}$. Assume that we have added the edges
between the vertices of $H_{i_{1}^{\prime}}$ and $H_{i_{2}^{\prime}}$, for
every $0\leq j_{1}^{\prime}<j_{2}^{\prime}\leq j-1$ and $i_{1}^{\prime}\in
D_{j_{1}^{\prime}},\ i_{2}^{\prime}\in D_{j_{2}^{\prime}}$. Recall that
$H_{i}$ is formed by $h_{i}$ disjoint copies of $G^{(m-1)}$. Let $\psi_{i}$ be
an arbitrary $1-1$ projection from $V(H_{i_{1}^{\prime}})\times\cdots\times
V(H_{i_{s}^{\prime}})$ to $\\{1,\ldots,h_{i}\\}$, where
$V(H_{i_{1}^{\prime}})\times\cdots\times V(H_{i_{s}^{\prime}})$ denotes the
Cartesian product of $H_{i_{1}^{\prime}},\ldots,H_{i_{s}^{\prime}}$. Then for
each $(w_{1},\ldots,w_{s})\in V(H_{i_{1}^{\prime}})\times\cdots\times
V(H_{i_{s}^{\prime}})$, we join the vertices $w_{1},\ldots,w_{s}$ to the
$\psi_{i}(w_{1},\ldots,w_{s})$-th copy of $G^{(m-1)}$ in $H_{i}$. Taking the
above processes from $j=2$ to $l$, we obtain the desired graph
$G^{(m)}=(V^{(m)},E^{(m)})$.

The remainder task is to show that $G_{m}$ certainly satisfies our
requirements. Clearly $\omega(G^{(m)})\geq m$ since $\omega(G^{(m-1)})=m-1$
and $x_{0}$ is adjacent to all vertices of at least one copy of $G^{(m-1)}$.
Let $\Omega$ be an arbitrary complete subgraph of $G^{(m)}$. We need to prove
that $\Omega$ has at most $m$ vertices. Let $U_{i}$ be the set of all vertices
of $\Omega$ lying in $H_{i}$. Notice that for distinct $i$ and $i^{\prime}$,
if there exist $w\in H_{i}$ and $w^{\prime}\in H_{i^{\prime}}$ such that
$ww^{\prime}\in E^{(m)}$, then either $i\in A_{i^{\prime}}$ or $i^{\prime}\in
A_{i}$, i.e., $u_{i}$ and $u_{i^{\prime}}$ are adjacent in the graph $K$.
Since $K$ doesn’t contain any triangle, we have
$|\\{i:\,U_{i}\not=\emptyset\\}|\leq 2$. There is noting to do if $\Omega$ is
completely contained in one $H_{i}$, since
$\omega(H_{i})=\omega(G^{(m-1)})=m-1$. Suppose that there exist distinct
$i,i^{\prime}$ such that $U_{i},U_{i^{\prime}}\not=\emptyset$. Without loss of
generality, assume that $i^{\prime}\in A_{i}$. Observe that distinct vertices
of $H_{i^{\prime}}$ are joint to distinct copies of $G^{(m-1)}$ in $H_{i}$. So
we must have $|U_{i^{\prime}}|=1$. Hence

$|V(\Omega)|=|U_{i}|+|U_{i^{\prime}}|\leq\omega(G^{(m-1)})+1=m.$

Now assume that the vertex set $V^{(m)}$ is split into $k$ disjoint subsets
$V_{1},\ldots,V_{k}$. Without loss of generality, we may assume that $x_{0}\in
V_{1}$. Let $U_{i,g}^{(t)}$ be the set of the common vertices of $V_{t}$ and
the $g$-th copies of $G^{(m-1)}$ in $H_{i}$. By the induction hypothesis, we
know that

$\max_{1\leq t\leq k}\omega(G^{(m)}[U_{i,g}^{(t)}])=\omega(G^{(m-1)})=m-1$

for every $1\leq i\leq n$ and $1\leq t\leq h_{i}$. For every $i\in D_{1}$, let
$g_{i}=1$,

$t_{i}=\min\\{1\leq t\leq k:\,\omega(G^{(m)}[U_{i,1}^{(t)}])=m-1\\}$

and arbitrarily choose a vertex $w_{i}\in U_{i,1}^{(t_{i})}$. Below we shall
determine $g_{i}$, $t_{i}$, $w_{i}$ inductively for $2\leq j\leq l$ and $i\in
D_{j}$. Assume that $j\geq 2$ and we have determined $g_{i}$, $t_{i}$, $w_{i}$
for all

$i\in\bigcup_{1\leq j^{\prime}\leq j-1}D_{j^{\prime}}.$

Then for $i\in D_{j}$, supposing
$A_{i}=\\{i_{1}^{\prime},\ldots,i_{s}^{\prime}\\}$ with
$i_{1}^{\prime}<\cdots<i_{s}^{\prime}$, let
$g_{i}=\psi_{i}(w_{i_{1}^{\prime}},\ldots,w_{i_{s}^{\prime}})$,

$t_{i}=\min\\{1\leq t\leq k:\,\omega(G^{(m)}[U_{i,g_{i}}^{(t)}])=m-1\\}$

and let $w_{i}$ be an arbitrary vertex in $U_{i,g_{i}}^{(t_{i})}$. In
particular, we set $t_{0}=1$ and $w_{0}=x_{0}$.

Now we shall color the vertices of $K$ with $k$ colors. For $0\leq i\leq n-1$,
let the vertex $u_{i}$ be colored with the $t_{i}$-th color. Since $K$ is not
$k$-colorable, there exist distinct $0\leq i,i^{\prime}\leq n-1$ such that
$t_{i}=t_{i^{\prime}}$ and $u_{i}u_{i^{\prime}}\in E_{K}$, i.e., either $i\in
A_{i^{\prime}}$ or $i^{\prime}\in A_{i}$. Without loss of generality, assume
that $i^{\prime}\in A_{i}$. Notice that $w_{i^{\prime}}\in
U_{i^{\prime},g_{i^{\prime}}}^{(t_{i})}$ and $w_{i^{\prime}}$ is adjacent to
all vertices of the $g_{i}$-th copies of $H_{i}$. Also, we have
$G^{(m)}[U_{i,g_{i}}^{(t_{i})}]$ contains an $(m-1)$-complete subgraph. Thus
we get an $m$-complete subgraph of
$G^{(m)}[U_{i,g_{i}}^{(t_{i})}\cup\\{w_{i^{\prime}}\\}]$, which is also a
subgraph of $G^{(m)}[V_{t_{i}}]$. We are done.

∎

##  3\. Proof of Theorem 1.3

For a system $\mathcal{A}=\\{a_{t}(n_{t})\\}_{t=1}^{s}$ and a graph $G=(V,E)$
with $V=\\{v_{1},\ldots,v_{s}\\}$, we say $G$ is an intersection graph of
$\mathcal{A}$ if

$a_{i}(n_{i})\cap a_{j}(n_{j})\not=\emptyset\Longleftrightarrow\text{the edge
}v_{i}v_{j}\in E$

for any $1\leq i<j\leq s$. The following result [21, Theorem 1] is due to
Zhang, although we give a slightly different proof here for the sake of
completeness.

######  Lemma 3.1.

For each graph $G=(V,E)$ with $|V|=s$, there exists a system
$\mathcal{A}=\\{a_{t}(n_{t})\\}_{t=1}^{s}$ such that $G$ is an intersection
graph of $\mathcal{A}$.

###### Proof.

We use induction on $s$. The cases $s=1$ and $s=2$ are trivial. Assume that
$s>2$ and our assertion holds for $s-1$. Suppose that
$V=\\{v_{1},\ldots,v_{s}\\}$. Let $V^{\prime}=V\setminus\\{v_{s}\\}$ and
$G^{\prime}=G[V^{\prime}]$. Let
$\mathcal{A}^{\prime}=\\{a_{t}^{\prime}(n_{t}^{\prime})\\}_{t=1}^{s-1}$ be a
system such that $G^{\prime}$ is an intersection graph of
$\mathcal{A}^{\prime}$. Let $p_{1},\ldots,p_{s-1}$ be some distinct primes
greater than $\max\\{n_{1}^{\prime},\ldots,n_{s-1}^{\prime}\\}$. For each
$1\leq t\leq s-1$, let $n_{t}=n_{t}^{\prime}p_{t}$ and $a_{t}$ be an integer
such that $a_{t}\equiv a_{t}^{\prime}\ ({\rm{mod}}\ n_{t}^{\prime})$ and
$a_{t}\equiv 1\ ({\rm{mod}}\ p_{t})$. Let $n_{s}=p_{1}\cdots p_{s-1}$ and
$a_{s}$ be an integer such that

$a_{s}\equiv\begin{cases}1\ ({\rm{mod}}\ p_{t})&\text{if the edge
}v_{t}v_{s}\in E,\\\ 0\ ({\rm{mod}}\ p_{t})&\text{if the edge
}v_{t}v_{s}\not\in E\end{cases}$

for $1\leq t\leq s-1$. Since $a_{i}(n_{i})\cap a_{j}(n_{j})\not=\emptyset$ if
and only if $(n_{i},n_{j})\mid a_{i}-a_{j}$, it is easy to see that $G$ is an
intersection graph of the system $\mathcal{A}=\\{a_{t}(n_{t})\\}_{t=1}^{s}$. ∎

Suppose that $G=(V,E)$ is an intersection graph of
$\mathcal{A}=\\{a_{t}(n_{t})\\}_{t=1}^{s}$. By the Chinese remainder theorem,
for a subset $I\subseteq\\{1,\ldots,k\\}$, if $a_{i}(n_{i})\cap
a_{j}(n_{j})\not=\emptyset$ for any $i,j\in I$, then $\bigcap_{i\in
I}a_{i}(n_{i})\not=\emptyset$. Hence we have

$\omega(G)=\max\\{w_{\mathcal{A}}(x):\,x\in\mathbb{Z}\\},$

by recalling that $w_{\mathcal{A}}(x)=|\\{1\leq i\leq s:\,x\in
a_{s}(n_{s})\\}|$.

###### Proof of Theorem 1.3.

Let $G=(V,E)$ be the graph satisfying the properties in Theorem 1.2 for $k=2$.
Assume that $|V|=s$. By Lemma 3.1, there exists a system
$\mathcal{A}=\\{a_{t}(n_{t})\\}_{t=1}^{s}$ such that $G$ is an intersection
graph of $\mathcal{A}$. We claim that for any partition
$\\{\mathcal{A}_{1},\mathcal{A}_{2}\\}$ of $\mathcal{A}$,

$\max_{i=1,2}\omega_{\mathcal{A}_{i}}=\omega_{\mathcal{A}},$

where

$\omega_{\mathcal{A}}=\max\\{w_{\mathcal{A}}(x):\,x\in\mathbb{Z}\\}.$

In fact, letting $V_{i}\subseteq V$ be the set of vertices concerning those
arithmetic progressions in $\mathcal{A}_{i}$, we have $G[V_{i}]$ is an
intersection graph of $\mathcal{A}_{i}$. Hence

$\max_{i=1,2}\omega_{\mathcal{A}_{i}}=\max_{i=1,2}\omega(G[V_{i}])=\omega(G)=\omega_{\mathcal{A}}.$

Since $\omega(G)=m$, $w_{\mathcal{A}}(x)\leq m$ for every $x\in\mathbb{Z}$. So
we may choose integers $b_{1},\ldots,b_{r}$ such that
$\mathcal{B}=\mathcal{A}\cup\\{b_{j}(N)\\}_{j=1}^{r}$ forms an exact
$m$-cover, where $N$ is the least common multiple of $n_{1},\ldots,n_{s}$. If
$\mathcal{B}$ is arbitrarily split into $\mathcal{B}_{1}$ and
$\mathcal{B}_{2}$, then

$\max_{i=1,2}\omega_{\mathcal{B}_{i}}\geq\max_{i=1,2}\omega_{\mathcal{B}_{i}\cap\mathcal{A}}=\omega_{\mathcal{A}}=\omega_{\mathcal{B}}.$

Hence there exists an integer $x$ such that $w_{\mathcal{B}_{1}}(x)=m$ or
$w_{\mathcal{B}_{2}}(x)=m$. Without loss of generality, assume that
$w_{\mathcal{B}_{1}}(x)=m$. Then
$w_{\mathcal{B}_{2}}(x)=w_{\mathcal{B}}(x)-w_{\mathcal{B}_{1}}(x)=0$, whence
$\mathcal{B}_{2}$ is not a $1$-cover. ∎

##  4\. A Further Remark

We may consider a general problem. Let $\mathscr{H}$ be a set of graphs such
that for any $G\in\mathscr{H}$, all induced subgraphs of $G$ are also
contained in $\mathscr{H}$. Suppose that $\psi$ be a projection from
$\mathscr{H}$ to $\mathbb{N}=\\{0,1,2,\ldots\\}$. We may ask whether for every
$m\geq 0$ and $k\geq 2$, there exists a graph $G=(V,E)\in\mathscr{H}$ with
$\psi(G)=m$ satisfying that

$\psi(G)\in\\{\psi(G[V_{1}]),\psi(G[V_{2}]),\psi(G[V_{k}])\\}$

for any $k$-partition $\\{V_{1},V_{2},\ldots,V_{k}\\}$ of the vertex set $V$.

Let $l(G)$ denote the length of the longest path of $G$. Then we have the
following negative result for $l(\cdot)$.

######  Theorem 4.1.

Let $G=(V,E)$ be a graph having at least one edge. Then there exists a
partition $\\{V_{1},V_{2}\\}$ of the vertex set $V$ such that

$l(G[V_{1}])<l(G)$

and $V_{2}$ is an independent set.

###### Proof.

Suppose that $l=l(G)$ and

$\displaystyle L_{1}=x_{1,1}-x_{1,2}-\cdots-x_{1,l}$ $\displaystyle
L_{2}=x_{2,1}-x_{2,2}-\cdots-x_{2,l}$ $\displaystyle\qquad\cdots\cdots$
$\displaystyle L_{t}=x_{t,1}-x_{t,2}-\cdots-x_{t,l}$

are all paths of $G$ with the length $l$. Below we shall construct some sets
$U_{i}$ and $I_{i}$. Let $U_{1}=\\{x_{1,1}\\}$ and

$I_{1}=\\{1\leq i\leq t:\,U_{1}\cap L_{i}=\emptyset\\}.$

For $j\geq 2$, if $I_{j-1}\not=\emptyset$, then let $i^{\prime}=\min I_{j-1}$,
$U_{j}=U_{j-1}\cup\\{x_{i^{\prime},1}\\}$ and

$I_{j}=\\{1\leq i\leq t:\,U_{j}\cap L_{i}=\emptyset\\}.$

Of course, if $I_{j-1}=\emptyset$, then stop this process. Suppose that we
finally get the vertex set $U_{s}$. Assume that
$U_{s}=\\{x_{i_{1},1},x_{i_{2},1},\ldots,x_{i_{s},1}\\}$ where
$1=i_{1}<i_{2}<\cdots<i_{s}$. Let $V_{2}=U_{s}$ and $V_{1}=V\setminus V_{2}$.
First, we claim that $V_{2}$ is an independent set. Assume on the contrary
that there exist $1\leq a<b\leq s$ such that $x_{i_{a},1}$ and $x_{i_{b},1}$
are adjacent in $G$. By the construction of $U_{s}$, we have $x_{i_{a},1}$
doesn’t lie in the path $L_{i_{b}}$. Thus

$x_{i_{a},1}-x_{i_{b},1}-x_{i_{b},2}-\cdots-x_{i_{b},l}$

forms a path with the length $l+1$. It is impossible since $l(G)=l$. Second,
by noting that $I_{s}=\emptyset$, we have $V_{2}\cap L_{i}\not=\emptyset$ for
any $1\leq i\leq t$. Hence $l(G[V_{1}])<l$ since $L_{1},\ldots,L_{t}$ are all
paths of $G$ with the length $l$. ∎

######  Acknowledgment.

The authors thank Professor Zhi-Wei Sun for his useful suggestions. And the
first author thanks Professor Yu-Sheng Li for his helpful discussions.

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# Transition of a binary solution into an inhomogeneous phase

Yu.M. Poluektov<EMAIL_ADDRESS>National Science Center
“Kharkov Institute of Physics and Technology”, 61108 Kharkov, Ukraine  A.A.
Soroka  National Science Center “Kharkov Institute of Physics and Technology”,
61108 Kharkov, Ukraine

###### Abstract

The thermodynamics of phase transitions of binary solutions into spatially
inhomogeneous one-dimensional states is studied theoretically with taking into
account nonlinear effects. It is shown that below the spinodal decomposition
temperature there first occurs a phase stratification, and at a lower
temperature, after reaching the maximum possible stratification, there occurs
a second-order transition into the phase with concentration waves. The
contribution of inhomogeneity to the entropy and heat capacity of phases is
calculated and the phase diagram in the coordinates temperature – average
concentration is constructed.
Key words: binary solution, concentration, phase transition, stratification,
concentration waves, phase diagram, free energy, spinodal, entropy, heat
capacity

###### pacs:

05.70.– a, 05.70.Fh, 64.60.– i, 64.70.– p, 64.70.Kb, 64.75.+ g, 68.35.Rh,
81.30.Dz, 81.30.Hd

##  I Introduction

The transition of binary solutions into a spatially inhomogeneous state is
observed in many systems, such as solid solutions AGK , 3He – 4He solutions BE
and solutions of other liquids. A loss of stability (spinodal decomposition)
has been studied experimentally and theoretically for a long time in binary
solutions of normal liquids and polymers SS . At present, more and more
attention is paid to the study of inhomogeneous media with a complex internal
structure, since it is in such systems that there exists a real prospect of
obtaining new materials with unique properties. Therefore, the study of the
problem of loss of stability by many-particle physical systems and their
transition into spatially inhomogeneous states is of actual practical
importance. Such studies are also important from the point of view of
constructing a general theory of phase transitions into inhomogeneous states.
Although a large number of works have been devoted to the phenomenon of
spinodal decomposition, the theory of not only dynamic phenomena but also
transitions to equilibrium states arising due to a loss of stability, all the
more taking into account nonlinear effects, still cannot be considered
complete.

A phenomenological approach to the description of spinodal decomposition was
developed by Cahn and Hilliard CH1 ; CH2 ; CH3 , and also by Hillert MH . A
somewhat different formulation of the theory was proposed in AGK . The aim of
this work is to study the thermodynamics of the phase transition of a binary
solution into spatially inhomogeneous states with taking into account
nonlinear effects. The developed approach is basically close to the works CH1
; CH2 ; CH3 ; MH ; AGK , but differs in the implementation of the method.
Since in practice the average concentration in a system is initially
determined by its composition, in the proposed approach the average
concentration is considered to be a fixed parameter.

Based on the choice of a specific type of the free energy in the form of an
expansion in powers of the concentration fluctuations, the character of phase
transitions of binary solutions with different average concentrations into
spatially inhomogeneous states is studied. The stratification of a solution
and its subsequent transition into the state with a concentration wave are
investigated. Thermodynamic potentials of inhomogeneous phases are calculated
and the type of transitions between phases is analyzed. When calculating the
entropy and heat capacity, the contribution of all possible inhomogeneous
states determined from the solution of the nonlinear equation for
concentration is taken into account.

##  II General thermodynamic relations

We will consider a solution of particles of two types, the number of which is
$N_{1}$ and $N_{2}$, occupying a constant volume $V$. We restrict ourselves to
the isotropic approximation, assuming that in a spatially homogeneous state
the thermodynamics of such a solution can be described by the free energy
which is a function of temperature $T,$ volume $V$ and the number of particles
of each component: $F=F(T,V,N_{1},N_{2})$. Sometimes it is more convenient to
pass to the total number of particles $N=N_{1}+N_{2}$ and the concentration of
one of the components $c_{1}\equiv\overline{c}=N_{1}/N$. As is known LL , in a
spatially homogeneous state the free energy for the one-component case can be
represented in the form $F=N\varphi(n,T)$. In the case of two components, a
similar relation takes the form $F=N\varphi(n,T,c)$. In a spatially
inhomogeneous state, we will consider the concentration $c=c({\bf r})$ as a
continuous function of spatial coordinates. The total density $n$ is assumed
to be constant. However, it should be noted that in the case when the masses
of atoms or their radii differ greatly, then the observed effects can be
noticeably influenced by the change in density, which we do not take into
account for now. In this case, the total free energy will be written in the
form

$\begin{array}[]{l}\displaystyle{\overline{F}=n\\!\int\\!\varphi\big{(}T,n,\overline{c}\,;\delta
c({\bf r}),\nabla\delta c({\bf r})\big{)}\,d{\bf r}.}\end{array}$ (1)

Thermodynamic potentials of this type are called potentials with incomplete
thermodynamic equilibrium. This means that temperature and density are assumed
to be equilibrium and independent of coordinates, and the order parameters, in
our case the deviation of the concentration from the equilibrium mean value
$\delta c({\bf r})=c({\bf r})-\overline{c}$, should be found as a result of
varying the free energy (1) with respect to the concentration from the
condition $\delta\overline{F}=0$, which gives the equation

$\begin{array}[]{l}\displaystyle{\frac{\partial\varphi}{\partial\delta
c}-\nabla\frac{\partial\varphi}{\partial\nabla\delta c}=0.}\end{array}$ (2)

Usually, the number of particles in a system is assumed to be specified, since
it is determined by the composition of a sample; therefore, the average
concentration $\overline{c}=N_{1}/N$ will be considered to be a fixed
parameter. By virtue of the requirement of conservation of the total number of
particles, there should hold the conditions

$\begin{array}[]{l}\displaystyle{\frac{1}{V}\\!\int\\!\delta c({\bf
r};T,n,\overline{c})\,d{\bf
r}=0,\qquad\overline{c}=\frac{1}{V}\\!\int\\!c({\bf
r};T,n,\overline{c})\,d{\bf r}.}\end{array}$ (3)

The solutions of Eq. (2), depending on temperature, density and the average
concentration $\delta c=\delta c({\bf r};T,n,\overline{c})$, should be
substituted into the expression for the free energy (1). As a result, one
obtains the formula for the equilibrium free energy

$\begin{array}[]{l}\displaystyle{F=n\\!\int\\!\varphi\big{(}T,n,\overline{c}\,;\delta
c({\bf r};T,n,\overline{c}),\nabla\delta c({\bf
r};T,n,\overline{c})\big{)}\,d{\bf r}.}\end{array}$ (4)

In the following we choose the free energy density in the form

$\begin{array}[]{l}\displaystyle{\varphi=\frac{K}{2}\big{(}\nabla\delta
c\big{)}^{2}+\tilde{\varphi}(T,n,\overline{c};\delta
c)+\varphi_{0}(T,n,\overline{c}),}\end{array}$ (5)

where in the expansion in powers of the concentration fluctuations we restrict
ourselves, as is customary in the phenomenological theory of phase transitions
LL , to terms no higher than the fourth power including the cubic term, so
that

$\begin{array}[]{l}\displaystyle{\tilde{\varphi}(T,n,\overline{c};\delta
c)=a_{2}\frac{(\delta c)^{2}}{2}+a_{3}\frac{(\delta
c)^{3}}{3}+a_{4}\frac{(\delta c)^{4}}{4}.}\end{array}$ (6)

The last term $\varphi_{0}(T,n,\overline{c})$ in (5) describes the
contribution of the homogeneous state to the total free energy. The linear
term in (6) drops out due to the conservation condition (3). The coefficients
in the expansion (5), (6) depend, generally speaking, on temperature, density,
and also the average concentration $a_{i}=a_{i}(T,n,\overline{c})$,
$K=K(T,n,\overline{c})$. To ensure stability of the homogeneous state in the
region of high temperatures, the coefficients $K$ and $a_{4}$ should be
considered positive, and the coefficient $a_{3}$ can be either positive or
negative. Let us assume that the coefficient $a_{2}$ changes sign at a certain
temperature, becoming negative at $T<T_{p}$. This spinodal decomposition
temperature is also a function of the average concentration
$T_{p}=T_{p}(\overline{c})$. In the following we will use the linear
approximation
$a_{2}=a_{0}\tau,\,\,\tau\equiv(T-T_{p})\big{/}T_{p}\,\,(a_{0}>0)$. In the
phenomenological theory of phase transitions, such a dependence is used within
the framework of the mean field theory near the phase transition temperature.
We will not introduce this restriction in the proposed model. Above $T_{p}$ at
$\tau>0$, the spatially homogeneous state, which we will call the H -phase,
remains stable for an arbitrary concentration. In the temperature range $T\leq
T_{p}$ of interest to us, the dimensionless relative temperature varies in the
range $-1<\tau\leq 0$. The use of such a simple approximation, as we shall
see, leads to reasonable results and is consistent with the derivation from
the microscopic theory. Note, however, that at low temperatures close to
$T=0$, or which is the same to $\tau=-1$, the applicability of the classical
description may be violated, since here it is necessary to take into account
quantum effects. In this paper we restrict ourselves to only the classical
consideration. For the free energy density of the form (5), (6), equation (2)
gives

$\begin{array}[]{l}\displaystyle{K\Delta\delta c=a_{2}\delta c+a_{3}(\delta
c)^{2}+a_{4}(\delta c)^{3}.}\end{array}$ (7)

Among all possible solutions of this nonlinear equation, the physically
realizable ones are only those for which $0\leq c({\bf r})\leq 1$.

##  III Spatially inhomogeneous states

Let us consider the transitions of a binary solution into spatially
inhomogeneous states, assuming that the concentration can depend only on one
spatial coordinate $x$, which varies within the limits $-L/2\leq x\leq L/2$.
In this one-dimensional case the total free energy (1) is written in the form

$\begin{array}[]{l}\displaystyle{\overline{F}=nA\\!\int\\!dx\left[\frac{K}{2}\left(\frac{d\,\delta
c}{dx}\right)^{2}+a_{2}\frac{(\delta c)^{2}}{2}+a_{3}\frac{(\delta
c)^{3}}{3}+a_{4}\frac{(\delta c)^{4}}{4}\right],}\end{array}$ (8)

where $A$ is the area of a sample, the normal to which is parallel to the axis
$x$. Equation (7) in this case takes the form

$\begin{array}[]{l}\displaystyle{K\frac{d^{2}\delta c}{dx^{2}}=a_{2}\delta
c+a_{3}(\delta c)^{2}+a_{4}(\delta c)^{3}.}\end{array}$ (9)

The first integral of this equation is

$\begin{array}[]{l}\displaystyle{\frac{K}{2}\left(\frac{d\,\delta
c}{dx}\right)^{2}=C-U(\delta c),}\end{array}$ (10)

where $C$ is the constant of integration, and

$\begin{array}[]{l}\displaystyle{U(\delta c)=-a_{4}\frac{(\delta
c)^{4}}{4}-a_{3}\frac{(\delta c)^{3}}{3}-a_{2}\frac{(\delta
c)^{2}}{2}.}\end{array}$ (11)

Equation (10) is similar to the equation of motion of a material point in an
external field $U(\delta c)$ (11) with energy $C$. In further analysis it will
sometimes be more convenient to pass to dimensionless quantities, which we
define by the relations:

$\begin{array}[]{ccc}\displaystyle{c=\sqrt{g}\,\tilde{c},\qquad
x\equiv\xi_{0}\tilde{x},\vspace{3mm}}\\\
\displaystyle{g\equiv\frac{a_{0}}{a_{4}}>0,\quad
b\equiv\frac{a_{3}}{g^{1\\!/2}a_{4}}=g^{1\\!/2}\frac{a_{3}}{a_{0}},\vspace{2mm}}\\\
\displaystyle{\varepsilon\equiv\frac{C}{a_{4}g^{2}}=\frac{C}{a_{0}g}.}\end{array}$
(12)

Here is introduced the correlation length $\xi_{0}$:

$\begin{array}[]{l}\displaystyle{\xi_{0}^{2}\equiv\frac{K}{ga_{4}}=\frac{K}{a_{0}}.}\end{array}$
(13)

In this notation Eq. (10) and the “field” (11) take the form

$\begin{array}[]{l}\displaystyle{\frac{1}{2}\\!\left(\frac{d\,\delta\tilde{c}}{d\tilde{x}}\right)^{\\!2}+\tilde{U}(\delta\tilde{c})=\varepsilon,}\end{array}$
(14)
$\begin{array}[]{l}\displaystyle{\tilde{U}(\delta\tilde{c})=-\frac{(\delta\tilde{c})^{4}}{4}-b\frac{(\delta\tilde{c})^{3}}{3}-\tau\frac{(\delta\tilde{c})^{2}}{2}.}\end{array}$
(15)

The use of the reduced concentration $\tilde{c}$, along with the physical
concentration $c$, is convenient because the dependence on the positive
parameter $g$ dropped out of the formula (15). As a result, the form of the
field (15) is determined by the single dimensionless parameter
$b=a_{3}\big{/}a_{4}g^{1\\!/2}$ and also by the dimensionless temperature
$\tau$. For $b=0$ the field (15) does not depend at all on material constants,
but the parameter $g$ determines the region of permissible change in the
reduced concentration

$\begin{array}[]{l}\displaystyle{0\leq\tilde{c}\leq\frac{1}{\sqrt{g}}.}\end{array}$
(16)

The solution of Eq. (14) in general form is given by the formula

$\begin{array}[]{l}\displaystyle{\int\\!\frac{d\,\delta\tilde{c}}{\sqrt{\varepsilon-\tilde{U}(\delta\tilde{c})}}=\pm\sqrt{2}\,\tilde{x}.}\end{array}$
(17)

An analysis of possible solutions of Eq. (17) may be more convenient in terms
of the total concentration. In this representation the field (15) takes the
form

$\begin{array}[]{l}\displaystyle{\tilde{U}(\tilde{c})=-\frac{1}{4}\left(\tilde{c}-\tilde{\overline{c}}\right)^{2}\left[\tilde{c}^{\,2}+2\\!\left(\\!-\tilde{\overline{c}}\,+\frac{2}{3}b\right)\\!\tilde{c}+\tilde{\overline{c}}^{\,2}\\!-\\!\frac{4}{3}b\,\tilde{\overline{c}}\,+2\tau\right].}\end{array}$
(18)

In order to demonstrate the application of the proposed method for describing
phase transitions into spatially inhomogeneous states, in this work we
restrict ourselves to considering the particular case $b=0$. The conditions to
which the crystal symmetry must conform for this purpose in the case of solid
solutions are analyzed, for example, in AGK . The analysis of the more general
case $b\neq 0$ does not present any fundamental difficulty and can be carried
out in a similar way. For $b=0$, the field (18) takes the form

$\begin{array}[]{l}\displaystyle{\tilde{U}(\tilde{c})=-\frac{1}{4}\left(\tilde{c}-\tilde{\overline{c}}\right)^{2}\left[\left(\tilde{c}-\tilde{\overline{c}}\right)^{2}+2\tau\right].}\end{array}$
(19)

If $\tau>0$, the field (19) vanishes at $\tilde{c}=\tilde{\overline{c}}$ and
takes negative values at all other concentrations. Thus, in this temperature
range the field (19) has no minimum and, therefore, only the homogeneous state
is possible. If $\tau<0$, then the function $\tilde{U}(\tilde{c})$ has the
minimum $\tilde{U}(\tilde{\overline{c}})=0$ and the maximums
$\tilde{U}(\tilde{c}_{m1})=\tilde{U}(\tilde{c}_{m2})=|\tau|^{2}\\!\big{/}4$ at

$\begin{array}[]{l}\displaystyle{\tilde{c}_{m1}=\tilde{\overline{c}}\,-\\!\sqrt{|\tau|},\qquad\tilde{c}_{m2}=\tilde{\overline{c}}\,+\\!\sqrt{|\tau|}.}\end{array}$
(20)

At concentrations

$\begin{array}[]{l}\displaystyle{\tilde{c}_{01}=\tilde{\overline{c}}\,-\\!\sqrt{2|\tau|},\qquad\tilde{c}_{02}=\tilde{\overline{c}}\,+\\!\sqrt{2|\tau|}}\end{array}$
(21)

the field vanishes
$\tilde{U}(\tilde{c}_{01}\\!)=\tilde{U}(\tilde{c}_{02}\\!)=0$. Taking into
account the condition (16) for concentration, there are two qualitatively
different possibilities, which are shown in Fig. 1. At concentrations
$0<\overline{c}\leq 1/2$ and temperatures
$\tau<-\,\overline{c}^{\,2}\\!\big{/}g$ (Fig. 1a), as well as
$1/2\leq\overline{c}<1$ and $\tau<-(1-\overline{c})^{2}\\!\big{/}g$ (Fig. 1b),
the maximum possible value of the parameter $\varepsilon=\varepsilon_{m}$ is
less than the value of the field at the maximum. Here, as will be shown, there
are only solutions in the form of concentration waves. At $0<\overline{c}\leq
1/2$, $\tau\geq-\,\overline{c}^{\,2}\\!\big{/}g$ and $1/2\leq\overline{c}<1$,
$\tau\geq-(1-\overline{c})^{2}\\!\big{/}g$ (Fig. 1c), the maximum possible
value of the parameter $\varepsilon=\varepsilon_{m}$ coincides with the value
of the field at the maximum. In this case the solution with the minimum energy
at $\varepsilon=\varepsilon_{m}$ has the form of a kink (31).

Figure 1:  The form of the field $\tilde{U}(\tilde{c})$ at different values of
average concentration and temperature. (a) $0<\overline{c}\leq 1/2$,
$\tau<-\,\overline{c}^{\,2}\\!\big{/}g$, (b) $1/2\leq\overline{c}<1$,
$\tau<-(1-\overline{c})^{2}\\!\big{/}g$ (W -phase), (c) $0<\overline{c}\leq
1/2$, $\tau\geq-\,\overline{c}^{\,2}\\!\big{/}g$; $1/2\leq\overline{c}<1$,
$\tau\geq-(1-\overline{c})^{2}\\!\big{/}g$ (K -phase).

Thus, along with the homogeneous H -phase, the stable inhomogeneous K \- and W
\- phases can exist in binary solutions.

##  IV Free energy parameters

Further consideration will be based on the expansion of the free energy in
powers of the concentration fluctuations (8). To estimate the phenomenological
parameters entering in (5), (6) and find their dependence on the average
concentration, temperature and density, we use the well-known representation
of the free energy AGK ; MT , which in the considered case of an isotropic
medium can be represented as

$\begin{array}[]{l}\displaystyle{\overline{F}=\frac{n^{2}}{2}\\!\int\\!d{\bf
r}d{\bf r}^{\prime}V\big{(}|{\bf r}-{\bf r}^{\prime}|\big{)}c({\bf r})c({\bf
r}^{\prime})+nT\\!\int\\!d{\bf r}\Big{[}c({\bf r})\ln c({\bf
r})+\big{(}1-c({\bf r})\big{)}\ln\big{(}1-c({\bf
r})\big{)}\Big{]},}\end{array}$ (22)

where $V\big{(}|{\bf r}-{\bf r}^{\prime}|\big{)}$ is the interaction potential
between particles. Expanding (22) in small fluctuations $\delta c({\bf
r})=c({\bf r})-\overline{c}$ and comparing with (4) – (6), we obtain the
following expressions for the expansion coefficients:

$\begin{array}[]{ccc}\displaystyle{a_{2}=nV_{0}+\frac{T}{\overline{c}\,(1-\overline{c})},\qquad
a_{3}=-\frac{(1-2\overline{c})}{2\,\overline{c}^{\,2}(1-\overline{c})^{2}}\,T,\qquad
a_{4}=\frac{(1-3\overline{c}+3\overline{c}^{\,2})}{3\,\overline{c}^{\,3}(1-\overline{c})^{3}}\,T,\vspace{2mm}}\\\
\displaystyle{K=-\frac{2\pi}{3}\,n\\!\int\\!dr\,r^{4}V(r),\qquad\varphi_{0}(T,\overline{c})=\frac{\overline{c}^{\,2}n}{2}\,V_{0}+T\big{[}\,\overline{c}\ln\overline{c}+(1-\overline{c})\ln(1-\overline{c})\big{]},}\end{array}$
(23)

where $V_{0}=\\!\int d{\bf r}V(r)$. When calculating the gradient
contribution, we used the expansion $\delta c({\bf
r}+{\boldsymbol{\uprho}})\approx\delta c({\bf r})+\rho_{i}\nabla_{i}\,\delta
c({\bf r})+\frac{1}{2}\rho_{i}\rho_{k}\nabla_{i}\nabla_{k}\,\delta c({\bf r})$
in the first term in (22), and also took into account that in an isotropic
medium

$\begin{array}[]{ccc}\displaystyle{\int\\!d{\boldsymbol{\uprho}}\,\rho_{i}V(\rho)=0,\qquad\int\\!d{\boldsymbol{\uprho}}\,\rho_{i}\rho_{k}V(\rho)=\delta_{ik}\,\frac{1}{3}\int\\!d\rho\,\rho^{2}V(\rho).}\end{array}$
(24)

As is seen, the coefficient $a_{2}$ can change sign at a certain temperature,
becoming negative if only $V_{0}<0$, so that the possibility of a loss of
stability is determined by the character of the interaction between atoms. In
the potential of interparticle interaction, as a rule, it is possible to
distinguish the region of repulsion at small distances and the region of
attraction at large distances. For many model potentials it is assumed that
the repulsion at small distances becomes infinitely large, which turns out to
be inconvenient in theoretical studies. However, it is quite admissible and
natural to consider effective potentials that remain finite at small distances
YP1 ; PS . For further estimates we will use the modified Sutherland potential

$\begin{array}[]{ll}\displaystyle{V(r)=\left\\{\begin{array}[]{l}\,\,\,\,v_{0},\qquad\qquad\quad
r<r_{0},\vspace{3mm}\\\
\displaystyle{-\,v_{m}\\!\left(\frac{r_{0}}{r}\right)^{\\!6},\quad\,\,\,\,r>r_{0},}\end{array}\right.}\end{array}$
(25)

where $v_{0}>0$ and $v_{m}>0$. For this potential we obtain

$\begin{array}[]{l}\displaystyle{V_{0}=\frac{4\pi}{3}r_{0}^{3}(v_{0}-v_{m}),\qquad
K=-\frac{2\pi}{15}\,nr_{0}^{5}(v_{0}-5v_{m}).}\end{array}$ (26)

For a sufficiently deep well in the potential (25) $v_{m}>v_{0}$, there holds
the condition $V_{0}<0$. In this case it is also ensured the condition of
stability of the homogeneous state $K>0$. The spinodal decomposition
temperature is then determined by the formula

$\begin{array}[]{l}\displaystyle{T_{p}=n|V_{0}|\,\overline{c}\,(1-\overline{c}).}\end{array}$
(27)

In the coefficient $a_{2}=a_{0}\tau$ the proportionality parameter is

$\begin{array}[]{l}\displaystyle{a_{0}\equiv\frac{T_{p}}{\overline{c}\,(1-\overline{c})}=n|V_{0}|.}\end{array}$
(28)

Let us also present expressions for the dimensionless parameters (12):

$\begin{array}[]{ccc}\displaystyle{g\equiv
g(\tau,\overline{c})=\frac{g_{0}(\overline{c})}{(1+\tau)},\qquad
g_{0}(\overline{c})\equiv\frac{3\overline{c}^{\,2}(1-\overline{c})^{2}}{(1-3\overline{c}+3\overline{c}^{\,2})},\vspace{2mm}}\\\
\displaystyle{b\equiv-(1+\tau)^{1\\!/2}b_{0}(\overline{c}),\qquad
b_{0}(\overline{c})\equiv\frac{\sqrt{3}}{2}\frac{(1-2\overline{c})}{(1-3\overline{c}+3\overline{c}^{\,2})^{1\\!/2}}.}\end{array}$
(29)

For completeness we gave a formula for the coefficient $b$, although, as was
said, it is omitted in this work. The correlation length for potential (25) is
given by the formula

$\begin{array}[]{ccc}\displaystyle{\xi_{0}^{2}=\frac{2\pi}{3V_{0}}\\!\int\\!d\rho\,\rho^{4}V(\rho)=r_{0}^{2}\frac{(v_{m}-v_{0}/5)}{2(v_{m}-v_{0})},}\end{array}$
(30)

and is of the order of magnitude of the radius of action of the interparticle
potential $\xi_{0}\sim r_{0}$. The macroscopic description and expansion in
terms of small concentration gradients is valid if the characteristic distance
at which the concentration changes $L_{c}$ is much larger than the correlation
length $L_{c}\gg\xi_{0}$. We also note that to analyze nonlinear effects we
use the expansion in powers of concentration (8), without imposing
restrictions on the magnitude of the concentration fluctuations.

##  V Phase diagram

Bounded solutions of Eq. (17) exist in the range of variation of the parameter
$0\leq\varepsilon\leq\varepsilon_{m}$. As noted above, depending on the values
of $\tau$ and $\overline{c}$ there are two possibilities. First, at
permissible concentration values the parameter $\varepsilon$ can take its
maximum value equal to the field maximum
$\varepsilon_{m}=\tilde{U}(\tilde{c}_{m1})=\tilde{U}(\tilde{c}_{m2})=|\tau|^{2}\\!\big{/}4$
(Fig. 1c). In this case, at $\varepsilon=\varepsilon_{m}$ the solution of Eq.
(17) has the form of a kink

$\begin{array}[]{ccc}\displaystyle{\delta\tilde{c}(\tilde{x})=\tilde{c}(\tilde{x})-\tilde{\overline{c}}=\pm\sqrt{|\tau|}\,\,{\rm
th}\\!\\!\left(\\!\frac{\sqrt{|\tau|}\,\tilde{x}}{\sqrt{2}}\right).}\end{array}$
(31)

This formula describes a solution, in which with a change of the coordinate a
transition occurs from the homogeneous state with an increased concentration
to the homogeneous state where the concentration is less than the average one,
so that $c(+\infty)-c(-\infty)=2\sqrt{g|\tau|}$. The solution stratifies into
two areas, in which the concentration is above and below the average one. The
width of the transition region, where the concentration changes significantly,
is given by the formula $L_{K}=\sqrt{\frac{2}{|\tau|}}\,\xi_{0}$. At
$\tau\rightarrow 0$ the concentration difference tends to zero and the width
of the transition region tends to infinity, so that in the result the system
goes over to the spatially homogeneous state.

In the range of variation of the parameter $0<\varepsilon<\varepsilon_{m}$,
the solution has the form of a concentration wave

$\begin{array}[]{ccc}\displaystyle{\delta\tilde{c}(\tilde{x})=\tilde{c}(\tilde{x})-\tilde{\overline{c}}=\pm\,D_{-}\,{\rm
sn}\\!\left(\\!\frac{\tilde{x}}{\sqrt{2}}\,D_{+},m\right),}\end{array}$ (32)

where $\displaystyle{D_{\pm}\equiv
D_{\pm}(\tau,\varepsilon)\equiv\sqrt{|\tau|\pm\sqrt{\tau^{2}-4\varepsilon}}}$,
$m\equiv m(\tau,\varepsilon)=D_{-}^{2}\big{/}D_{+}^{2}$, ${\rm
sn}\\!\left(\\!\frac{\tilde{x}D_{+}}{\sqrt{2}},m\right)$ is the elliptic sine
function AS . The description of the distribution of atoms of a solid solution
by means of concentration waves was proposed by Krivoglaz MK  and developed in
relation to the description of ordered solutions in AGK . In the limit
$\varepsilon\rightarrow|\tau|^{2}\\!\big{/}4$ or $m\rightarrow 1$, this
solution goes over to the kink solution (31). The case $\varepsilon=0$
corresponds to the spatially homogeneous state. The solution with the maximum
possible value $\varepsilon=\varepsilon_{m}$ in the form of a kink (31)
corresponds to the minimum value of the free energy.

In another possible case, the maximum value of the parameter
$\varepsilon=\varepsilon_{m}$ at all permissible concentrations is less than
the maximum value of the field $\varepsilon_{m}<|\tau|^{2}\\!\big{/}4$ (Fig.
1a,b). At $\overline{c}\leq 1/2$
$\displaystyle{\varepsilon_{m}=\tilde{U}(0,\tau)=-\frac{\overline{c}^{\,2}}{4g}\left(\frac{\overline{c}^{\,2}}{g}+2\tau\right)}$,
and at $\overline{c}\geq 1/2$
$\displaystyle{\varepsilon_{m}=\tilde{U}\left(\frac{1}{\sqrt{g}},\tau\right)=-\frac{(1-\overline{c})^{2}}{4g}\left(\frac{(1-\overline{c})^{2}}{g}+2\tau\right)}$.
In this case, there is no solution in the form of a kink and all bounded
solutions at $0<\varepsilon\leq\varepsilon_{m}$ have the form of concentration
waves (32).

Figure 2:  Phase diagram in the coordinates
$\big{(}\overline{c},T\\!\big{/}n|V_{0}|\big{)}$. The lines $a_{1}ca_{2}$,
$a_{1}b$, $ba_{2}$ respectively correspond to the dependencies:
$T_{p}\big{/}n|V_{0}|=\overline{c}\,(1-\overline{c})$,
$T_{1}\\!\big{/}n|V_{0}|=3\overline{c}\,(1-\overline{c})^{3}\\!\big{/}(4-9\overline{c}+6\overline{c}^{\,2})$,
$T_{2}\big{/}n|V_{0}|=3\overline{c}^{\,3}(1-\overline{c})\\!\big{/}(1-3\overline{c}+6\overline{c}^{\,2})$.
Above the curve $a_{1}ca_{2}$ there is the homogeneous H -phase. Inside the
area $a_{1}ca_{2}b$ is the K -phase. Inside the area $a_{1}ba_{2}$ is the W
-phase.

As we can see, while at $\tau\geq 0$ there is only the spatially homogeneous H
-phase, at $\tau<0$ there are two qualitatively different states. In one of
them, in addition to the solutions in the form of concentration waves, there
exists the solution in the form of a kink (31), which actually corresponds to
the smallest value of the free energy. We called this state the K -phase. In
this phase the solution is stratified into two spatial regions with increased
and depleted concentrations. In the other phase all solutions have the form of
concentration waves (32), and there is no solution in the form of a kink. We
called this state the W -phase. The phase diagram in the coordinates
temperature – average concentration is shown in Fig. 2. Note that at
$\overline{c}\rightarrow 0$ we have $T_{p}\sim\overline{c}$, and at
$\overline{c}\rightarrow 1$ we have $T_{p}\sim(1-\overline{c})$, so that the
transition temperature tends to zero in both limiting cases. Here, generally
speaking, due to the proximity of $T_{p}$ to absolute zero the applicability
of the model and the used classical description may be violated.

The concentration distributions in the inhomogeneous phases during
stratification (a) and in the concentration wave (b) are shown in Fig. 3. As
is seen, transitions between the phases are accompanied by the transfer of a
significant amount of matter. In this work, however, we do not consider the
kinetics of transitions between the phases and do not estimate the
characteristic times during which such transitions occur, but we study only
thermodynamically equilibrium states.

Figure 3:  (a) The concentration distributions $c(\tilde{x})_{K}$ in the K
-phase ($\tilde{T}_{2}\leq\tilde{T}<\tilde{T}_{p}=0.24$) at temperatures
$\tilde{T}=T\\!\big{/}n|V_{0}|$ : 1 – $\tilde{T}=0.235$, 2 –
$\tilde{T}=0.215$, 3 – $\tilde{T}_{2}=0.191$. The line $c=\overline{c}=0.6$
corresponds to the H -phase at $\tilde{T}\geq\tilde{T}_{p}$. (b) The
concentration distribution $c(\tilde{x})_{W}$ in the W -phase
$(\tilde{T}<\tilde{T}_{2})$ at $\tilde{T}=0.185$.

##  VI Thermodynamics of inhomogeneous states

Let us calculate the contribution of inhomogeneity to the thermodynamic
functions of the phases and analyze the character of transitions between the
phases. Substituting the solution at a certain value of $\varepsilon$ into
(8), we find the contribution of this state to the equilibrium free energy.
For the kink in the limit $L\rightarrow\infty$, we obtain

$\begin{array}[]{ccc}\displaystyle{\frac{F}{a_{4}g^{2}N}=-\varepsilon=-\frac{\tau^{2}}{4}.}\end{array}$
(33)

For the concentration wave at a certain value of $\varepsilon$, we have

$\begin{array}[]{ccc}\displaystyle{\frac{F(\tau,\varepsilon)}{a_{4}g^{2}N}=\frac{1}{3}\\!\left[\varepsilon+\tau
D_{+}^{2}(\tau,\varepsilon)\left(1-\frac{E(m)}{K(m)}\right)\right],}\end{array}$
(34)

where $K(m),E(m)$ are the complete elliptic integrals of the first and second
kind AS . In the following, in (34) it is more convenient to express the
parameter $\varepsilon$ in terms of the parameter

$\begin{array}[]{ccc}\displaystyle{m=\frac{|\tau|-\sqrt{\tau^{2}-4\varepsilon}}{|\tau|+\sqrt{\tau^{2}-4\varepsilon}},}\end{array}$
(35)

which varies from $m=0$ at $\varepsilon=0$ to
$m=m_{*}(\tau)\equiv\displaystyle{\Big{(}|\tau|-\sqrt{\tau^{2}-4\varepsilon_{m}}\Big{)}\\!\Big{/}\\!\Big{(}|\tau|+\sqrt{\tau^{2}-4\varepsilon_{m}}}\Big{)}$
at $\varepsilon=\varepsilon_{m}$. So we find the energy of the inhomogeneous
state per one particle

$\begin{array}[]{ccc}\displaystyle{\frac{F(\tau,m)}{N}=-G\tau^{2}f(m),}\end{array}$
(36)

where the function

$\begin{array}[]{ccc}\displaystyle{f(m)\equiv\frac{(2+m)}{(1+m)^{2}}-\frac{2}{(1+m)}\frac{E(m)}{K(m)}}\end{array}$
(37)

is defined, and $G\equiv\frac{1}{3}a_{4}g^{2}$ is a constant which with taking
into account the formulas (23), (27), (29) can be presented in the form

$\begin{array}[]{ccc}\displaystyle{G=T_{p}\frac{\eta(\overline{c})}{(1+\tau)},\qquad\eta(\overline{c})\equiv\frac{\overline{c}(1-\overline{c})}{(1-3\overline{c}+3\overline{c}^{\,2})}.}\end{array}$
(38)

The form of the function (37) is shown in Fig. 4. It is positive and increases
from $f(0)=0$ to $f(1)=3/4$, and a very rapid growth occurs at $m\approx 1$,
so that here the derivative goes to infinity.

Figure 4:  Form of the function $f(m)$ (37).

The function $F(\tau,m)$ (36) determines the contribution of solution with
index $m$ to the energy. If excitations with different energies are possible
in a physical system, then all of them contribute to the total equilibrium
free energy. According to the general principles of statistical physics, the
probability of finding the system in the state with the parameter $m$ in the
interval $m\div m+dm$

$\begin{array}[]{ccc}\displaystyle{dw=\phi(T,m)\,dm}\end{array}$ (39)

is determined by means of the “one-particle” distribution function over states
$m$

$\begin{array}[]{ccc}\displaystyle{\phi(T,m)\equiv\frac{e^{-\frac{F(\tau,m)}{NT}}}{Z}=\frac{e^{B(\tau)f(m)}}{Z},}\end{array}$
(40)

where we use the notation

$\begin{array}[]{ccc}\displaystyle{B(\tau)\equiv\frac{\tau^{2}}{(1+\tau)^{2}}\,\eta(\overline{c}).}\end{array}$
(41)

The statistical integral is determined by the normalization condition
$\int_{0}^{m_{*}(\tau)}\\!\\!\phi(T,m)\,dm=1$:

$\begin{array}[]{ccc}\displaystyle{Z=\int\limits_{0}^{m_{*}(\tau)}e^{B(\tau)f(m)}dm.}\end{array}$
(42)

The average of an arbitrary function $A=A(m)$ is given by the formula

$\begin{array}[]{ccc}\displaystyle{\big{\langle}A\big{\rangle}=\int\limits_{0}^{m_{*}(\tau)}A(m)\phi(\tau,m)dm.}\end{array}$
(43)

As follows from the form of the function (37) (Fig. 4), the largest
contribution is made by the state with the maximum value $m=m_{*}$, while the
contribution of the states with smaller values of $m$ rapidly decreases.

The average value of the function (36) determines the contribution of
inhomogeneous states to the average energy

$\begin{array}[]{ccc}\displaystyle{E=\int\limits_{0}^{m_{*}}F(\tau,m)\phi(\tau,m)dm=-NG\,\tau^{2}\big{\langle}f\big{\rangle}.}\end{array}$
(44)

The contribution to the total entropy due to inhomogeneity of states is
determined through the average of the logarithm of the distribution function
LL , so that the entropy per one particle is given by the formula

$\begin{array}[]{ccc}\displaystyle{\frac{S}{N}\equiv-\big{\langle}\ln\phi\,\big{\rangle}=\ln
Z-B\big{\langle}f\big{\rangle}.}\end{array}$ (45)

The contribution of inhomogeneity to the total heat capacity can be determined
through the derivative of the entropy

$\begin{array}[]{ccc}\displaystyle{\frac{C}{N}=\frac{T}{N}\frac{dS}{dT}=\frac{2\,\tau^{3}}{(1+\tau)^{4}}\,\eta^{2}(\overline{c})\Big{(}\\!\big{\langle}f\big{\rangle}^{2}-\big{\langle}f^{\,2}\big{\rangle}\\!\Big{)}+(1+\tau)\frac{e^{Bf(m_{*})}}{Z}\\!\left[1+\frac{\tau^{2}}{(1+\tau)^{2}}\,\eta(\overline{c})\Big{(}\\!\big{\langle}f\big{\rangle}-f(m_{*})\Big{)}\right]\\!\frac{dm_{*}}{d\tau}\,,}\end{array}$
(46)

where

$\begin{array}[]{ccc}\displaystyle{\big{\langle}f^{\,n}\big{\rangle}=\int\limits_{0}^{m_{*}}\,\big{[}f(m)\big{]}^{n}\phi(\tau,m)\,dm.}\end{array}$
(47)

In particular, in the K -phase $m_{*}=1$, and in the W -phase

$\begin{array}[]{ccc}\displaystyle{m_{*}(\tau)=\frac{(\tau+1)}{\tau\\!\left(\\!\displaystyle{\frac{2}{\tau_{i}}}+1\\!\right)\\!-1},\qquad\frac{dm_{*}}{d\tau}=-\frac{2\,(\tau_{i}+1)}{\tau_{i}\\!\left[\tau\\!\left(\\!\displaystyle{\frac{2}{\tau_{i}}}+1\\!\right)\\!-1\right]^{\\!2}}.}\end{array}$
(48)

Here at $0<\overline{c}\leq 1/2$ $i=1$, and at $1/2\leq\overline{c}<1$ $i=2$,
and at that

$\begin{array}[]{ccc}\displaystyle{\tau_{1}=\frac{T_{1}-T_{p}}{T_{p}}=-\frac{\overline{c}^{\,2}}{\left[\overline{c}^{\,2}+g_{0}(\overline{c})\right]},\qquad\tau_{2}=\frac{T_{2}-T_{p}}{T_{p}}=-\frac{(1-\overline{c})^{2}}{\left[(1-\overline{c})^{2}+g_{0}(\overline{c})\right]}.}\end{array}$
(49)

The temperature dependences of the contributions of inhomogeneous states to
the entropy and heat capacity are shown in Fig. 5.

Figure 5:  Dependencies of (a) the entropy and (b) the heat capacity on the
dimensionless temperature $\tilde{T}=T\\!\big{/}n|V_{0}|$
($\overline{c}=0.3$, $\tilde{T}_{1}=0.168$, $\tilde{T}_{p}=0.21$).

As we can see, in the K -phase at $\tilde{T}_{1}\leq\tilde{T}<\tilde{T}_{p}$
the entropy and heat capacity associated with inhomogeneity are small, so that
on the scale of the figure they are practically close to zero. Upon transition
to the W -phase the entropy decreases, and the heat capacity at the transition
temperature $\tilde{T}_{1}$ increases by a jump and also decreases
monotonically with decreasing temperature. As it was noted, near absolute zero
the applicability of the used model and the classical approach may be
violated.

The existence, along with the main solutions determining the structure of the
phases, also of other solutions in the form of concentration waves with
energies greater than the energy of the main solution, will somewhat affect
the concentration distribution in the phases, but due to the randomness of
phases this will not lead to a qualitative change in the form of the
concentration distributions in the K \- and W -phases shown in Fig. 3.

It is important to note that a loss of stability of the homogeneous state is
described by the solution of the nonlinear equation in the form of a kink (31)
even near $T_{p}$. Thus, to analyze a loss of stability it is not enough
applying a linear approximation, and taking into account nonlinear effects
proves to be important even for small $\tau$.

##  VII Phase transition from the H -phase to the K -phase

As can be seen from the phase diagram in Fig. 2, with decreasing temperature
at $T=T_{p}$ the solution loses its stability and begins to stratify passing
to the K -phase. Let us consider in more detail the thermodynamics of this
transition. Near the transition from the homogeneous H -phase to the K -phase,
where $|\tau|\ll 1$ and $B\approx\eta(\overline{c})\tau^{2}<1$, from (45) and
(46) there follow formulas for the entropy and heat capacity:

$\begin{array}[]{ccc}\displaystyle{\frac{S}{N}=\frac{\big{[}\eta(\overline{c})\big{]}^{2}}{2}\big{(}f_{1}^{\,2}-f_{2}\big{)}\hskip
0.85358pt\tau^{4},}\end{array}$ (50)
$\begin{array}[]{ccc}\displaystyle{\frac{C}{N}=2\big{[}\eta(\overline{c})\big{]}^{2}\big{(}f_{1}^{\,2}-f_{2}\big{)}\hskip
0.85358pt\tau^{3}.}\end{array}$ (51)

Here $f_{n}=\int_{0}^{1}\\!\big{[}f(m)\big{]}^{n}dm$ and $f_{1}\approx 0.162$,
$f_{2}\approx 0.043$, $f_{1}^{\,2}-f_{2}=-0.017$. Thus, there occurs a smooth
phase transition from the homogeneous state to the phase where the solution is
stratified. In this case in the inhomogeneous state the entropy slightly
decreases and the heat capacity increases (Fig. 5). Only the fourth derivative
of the entropy with respect to temperature experiences a jump. The form of the
temperature dependences (50) and (51) is mainly determined by the fact that
the linear approximation of the dependence of the coefficient
$a_{2}=a_{0}(T-T_{p})/T_{p}$ in (6) on temperature was used. As is known LL ;
HS ; PP , near the phase transition temperature, fluctuations begin to play an
important role, which are not taken into account in this approach based on the
mean field theory. An effective account for fluctuations should lead to a
modification of the critical exponents and temperature dependences of
quantities near the phase transition GS ; YP2 .

Figure 6:  Evolution of the concentration distribution during the transition
of the stratified solution into the phase with concentration waves. Curve 1 is
the distribution at the transition boundary at $\tilde{T}=\tilde{T}_{1}$.
Curves 2 and 3 are the distributions in the W -phase at
$\tilde{T}<\tilde{T}_{1}$, with respectively one and two wave periods at the
sample length $L$. Parameters: $\overline{c}=0.4,\,\tilde{T}_{1}=0.191$.

##  VIII Phase transition between inhomogeneous phases

As it was noted earlier (Fig. 2), below the temperature $T_{p}$ after a loss
of stability of the homogeneous phase there first appears the K -phase and the
stratification of the solution occurs. The phase with stratification exists in
the temperature range $T_{1}\leq T<T_{p}$ if $\overline{c}\leq 1/2$, and
$T_{2}\leq T<T_{p}$ if $\overline{c}\geq 1/2$, where

$\begin{array}[]{ccc}\displaystyle{T_{1}=T_{p}\frac{3(1-\overline{c})^{2}}{(4-9\overline{c}+6\overline{c}^{\,2})},\qquad
T_{2}=T_{p}\frac{3\overline{c}^{\,2}}{(1-3\overline{c}+6\overline{c}^{\,2})}.}\end{array}$
(52)

At lower temperatures $0<T<T_{1}$ if $\overline{c}\leq 1/2$, and $0<T<T_{2}$
if $\overline{c}\geq 1/2$, the W -phase appears (Fig. 2). Figure 6 shows the
evolution of the concentration distribution near the transition from the
stratified state to the state with a concentration wave. With a further
decrease in temperature the concentration wave takes a shape close to
sinusoidal (Fig. 3b).

Let us consider in more detail the character of the transition from the K
-phase to the W -phase. In the region of existence of the K -phase the entropy
and heat capacity can be represented in the form

$\begin{array}[]{ccc}\displaystyle{\frac{S_{K}(\tau)}{N}=\ln
Z_{K}-\frac{\tau^{2}}{(1+\tau)^{2}}\,\eta(\overline{c})\big{\langle}f\big{\rangle}_{\\!K},}\end{array}$
(53)
$\begin{array}[]{ccc}\displaystyle{\frac{C_{K}(\tau)}{N}=\frac{2\,\tau^{3}}{(1+\tau)^{4}}\,\eta^{2}(\overline{c})\Big{(}\\!\big{\langle}f\big{\rangle}_{\\!K}^{2}-\big{\langle}f^{\,2}\big{\rangle}_{\\!K}\\!\Big{)},}\end{array}$
(54)

where

$\begin{array}[]{ccc}\displaystyle{Z_{K}=\int\limits_{0}^{1}\\!e^{Bf(m)}dm,\qquad\big{\langle}f\big{\rangle}_{\\!K}=\frac{1}{Z_{K}}\int\limits_{0}^{1}\\!f(m)\hskip
1.42262pte^{Bf(m)}dm,\qquad\big{\langle}f^{\,2}\big{\rangle}_{\\!K}=\frac{1}{Z_{K}}\int\limits_{0}^{1}\\!f^{\,2}(m)\hskip
1.42262pte^{Bf(m)}dm.}\end{array}$ (55)

In the W -phase with concentration waves the same quantities have the form

$\begin{array}[]{ccc}\displaystyle{\frac{S_{W}(\tau)}{N}=\ln
Z_{W}-\frac{\tau^{2}}{(1+\tau)^{2}}\,\eta(\overline{c})\big{\langle}f\big{\rangle}_{\\!W},}\end{array}$
(56)
$\begin{array}[]{ccc}\displaystyle{\frac{C_{W}(\tau)}{N}=\frac{2\,\tau^{3}}{(1+\tau)^{4}}\,\eta^{2}(\overline{c})\Big{(}\\!\big{\langle}f\big{\rangle}_{\\!W}^{2}-\big{\langle}f^{\,2}\big{\rangle}_{\\!W}\\!\Big{)}+(1+\tau)\frac{e^{Bf(m_{*})}}{Z_{W}}\\!\left[1+\frac{\tau^{2}}{(1+\tau)^{2}}\,\eta(\overline{c})\Big{(}\\!\big{\langle}f\big{\rangle}_{\\!W}-f(m_{*})\Big{)}\right]\\!\frac{dm_{*}}{d\tau}\,,}\end{array}$
(57)

where

$\begin{array}[]{ccc}\displaystyle{Z_{W}=\int\limits_{0}^{m_{*}}\\!e^{Bf(m)}dm,\qquad\big{\langle}f\big{\rangle}_{\\!W}=\frac{1}{Z_{W}}\int\limits_{0}^{m_{*}}\\!f(m)\hskip
1.42262pte^{Bf(m)}dm,\qquad\big{\langle}f^{\,2}\big{\rangle}_{\\!W}=\frac{1}{Z_{W}}\int\limits_{0}^{m_{*}}\\!f^{\,2}(m)\hskip
1.42262pte^{Bf(m)}dm.}\end{array}$ (58)

The parameter $m_{*}$ is defined by the formula (48). The calculation shows
that the entropy and heat capacity in the W -phase are much higher than the
values of the same quantities in the K -phase, so that in Fig. 5 where the
temperature dependences of these quantities are presented, as it was noted, on
the used scale they are close to zero. Attention is drawn to the curious fact
that when approaching zero temperature, the contribution of the concentration
wave to the heat capacity per one particle proves to be close to unity at
various average concentrations.

At the K -W transition, the entropy turns out to be continuous and the heat
capacity undergoes a jump

$\begin{array}[]{ccc}\displaystyle{\frac{\Delta
C}{N}\equiv\frac{C_{W}-C_{K}}{N}=\frac{2}{|\tau_{i}|}\frac{e^{B(\tau_{i})f(1)}}{Z_{K}(\tau_{i})}\\!\left[1+B(\tau_{i})\Big{(}\\!\big{\langle}f\big{\rangle}_{\\!K}-f(1)\Big{)}\right],\quad\,\,(i=1,2).}\end{array}$
(59)

Thus, the transition between the K \- and W \- phases is a second-order
transition. Figure 7 shows the dependence of the magnitude of the jump in the
heat capacity on the average concentration.

Figure 7:  Dependence of the jump in the heat capacity $\Delta
C=C_{W}(T_{1})-C_{K}(T_{1})$ at the transition between the K \- and W \-
phases on the average concentration.

Note that the jump in the heat capacity remains finite even in the limit of
low concentrations, although, as mentioned above, the condition of
applicability of the model and the classical description of the solution may
be violated here.

##  IX Conclusion

The thermodynamics of equilibrium phase transitions of the binary solution
into spatially inhomogeneous states is investigated theoretically taking into
account nonlinear effects. It is shown that as a result of a loss of stability
of the homogeneous H -phase with decreasing temperature, at first there occurs
the stratification of the solution for which the concentration distribution is
described by the dependence in the form of a kink (31) (the K -phase). After
reaching the maximum possible stratification at a lower temperature there
occurs a second-order phase transition into the state with a concentration
wave which is described by the periodic solution (32) (the W -phase). The
function of distribution of states in the inhomogeneous solution is introduced
and with its help the contribution of inhomogeneity to the entropy and heat
capacity is calculated. The phase diagram in the coordinates temperature –
average concentration is constructed. The proposed approach can be used to
describe transitions into spatially inhomogeneous phases in other many-
particle systems.

## List of Notations

$N$ | total number of particles
---|---
$N_{1},N_{2}$ | numbers of particles of two types
$n$ | total density of number of particles
$V$ | volume of a solution
$T$ | temperature
$\overline{c}=c_{1}$ | average concentration of the first component
$c=c({\bf r})$ | local concentration of the first component
$\delta c({\bf r})$ | deviation of the concentration from the average value
$\varphi$ | free energy per one particle
$\varphi_{0}$ | contribution of the homogeneous state to $\varphi$
$\overline{F}$ | total free energy in the state of incomplete thermodynamic equilibrium
$F$ | total free energy in the equilibrium state
$K,a_{2},a_{3},a_{4}$ | coefficients in the expansion of $\varphi$ in $\big{(}\nabla\delta c\big{)}^{2}$ and powers of $\delta c$
$T_{p}$ | temperature at which the coefficient $a_{2}$ changes sign
$\tau=(T-T_{p})\big{/}T_{p}$ | parameter of dimensionless temperature, such that $a_{2}=a_{0}\tau$
$L,A$ | length and area of a sample
$x$ | one spatial coordinate on which the concentration depends
$U(\delta c),C$ | effective field and constant of integration in the equation for $\delta c$
$\tilde{c}=g^{-1/2}c$ | reduced concentration
$\tilde{\overline{c}}=g^{-1/2}\overline{c}$ | reduced average concentration
$\delta\tilde{c}=\tilde{c}-\tilde{\overline{c}}$ | reduced deviation of the concentration from the average value
$\tilde{x}=x/\xi_{0}$ | dimensionless coordinate, where $\xi_{0}$ is the correlation length
$g,b,\varepsilon$ | dimensionless parameters, defined by the formula (12)
$\tilde{U}(\delta\tilde{c};b,\tau)$ | reduced effective field, given by the formula (15)
$\tilde{U}(\tilde{c};\tilde{\overline{c}},b,\tau)$ | reduced effective field as a function of $\tilde{c}$, given by (18) in general case
| and by (19) in the case $b=0$
$\tilde{c}_{m1},\tilde{c}_{m2}$ | maximums of $\tilde{U}(\tilde{c})$ for the case $b=0$
$\tilde{c}_{01},\tilde{c}_{02}$ | zeros of $\tilde{U}(\tilde{c})$ for the case $b=0$
$V\big{(}|{\bf r}-{\bf r}^{\prime}|\big{)}$ | interaction potential between particles
$V_{0}=\\!\int d{\bf r}V(r)$ | constant relating to the interaction potential
$v_{0},v_{m},r_{0}$ | constants of the potential $V(r)$, defined by the formula (25)
$\varepsilon_{m}$ | maximum possible value of the parameter $\varepsilon$
$L_{K}$ | width of the transition region in the kink (31)
$m(\tau,\varepsilon)$ | parameter (35) entering into the elliptic functions and integrals
$m_{*}(\tau)$ |  $m(\tau,\varepsilon_{m})$, i.e. maximum value of $m$ at given $\tau$
$\phi(T,m)$ | “one-particle” distribution function over states $m$
$f(m)$ | characteristic function (37)
$E$ | average energy contribution due to inhomogeneous states
$S$ | entropy contribution due to inhomogeneous states
$C$ | heat capacity contribution due to inhomogeneous states
$T_{1},T_{2}$ | temperatures of the phase transition between the inhomogeneous
|  K \- and W -phases, given by the formula (52)
$\Delta C$ | jump of the heat capacity at the K -W transition, given by (59)

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