# The stable conjugation-invariant word norm is rational in free groups

Henry Jaspars

###### Abstract.

We establish the rationality of the stable conjugation-invariant word norm on
free groups and virtually free Coxeter groups.

###### Key words and phrases:

Bi-invariant word metric, Stable word length, Context-Free Language,
Semilinear sets, Parikh’s Theorem

###### 2020 Mathematics Subject Classification:

Primary: 20F65, 20E05; Secondary: 68R15, 68Q45

##  1\. Introduction

A norm on a group $G$ is a non-negative function $\nu\colon G\to{\mathbf{R}}$
such that for all $g,h\in G$,

  1. (1)

$\nu(g)=0$ if and only if $g=1$;

  2. (2)

$\nu(gh)\leq\nu(g)+\nu(h)$.

Moreover, if $\nu(ghg^{-1})=\nu(h)$ for all $g,h\in G$, then the norm is
called conjugation-invariant. Examples of conjugation-invariant norms include
the Hofer norm on the group of Hamiltonian diffeomorphisms of a symplectic
manifold [18], the commutator length [4], and other verbal norms [6], as well
as word norms associated with generating sets invariant under conjugations
[2], which in the context of Coxeter groups are known as reflection length,
and have been studied extensively [10, 15, 17]. In a combinatorial setting,
conjugation-invariant norms also bear relation to non-crossing matchings and
enumerating unpaired nucleotides in secondary RNA structures [11].

The stable (or translation) length of $g\in G$ with respect to $\nu$ is the
limit

$\tau_{\nu}(g)\vcentcolon=\lim_{k\to\infty}\frac{\nu(g^{k})}{k}.$

In this paper we prove that the stable length with respect to the conjugation-
invariant word norm on certain virtually free groups is rational.

More precisely, throughout, let $G$ be a group normally generated by $S$. Let
$\|g\|_{S}$ denote the conjugation-invariant word norm associated with
$\bar{S}=\bigcup_{s\in S}{\rm Conj}(s)$, where ${\rm Conj}(s)$ denotes the
conjugacy class of $s$, namely,

$\|g\|_{S}\vcentcolon=\inf\\{\ell:g_{1}s_{1}g_{1}^{-1}\ldots
g_{\ell}s_{\ell}g_{\ell}^{-1}=g,\,g_{i}\in G,s_{i}\in S\\}.$

Also define the cancellation length of a word $w$ with characters in $S$ to be
the minimum number of letters which need to be deleted to obtain a word
representing the trivial element. It is known that if $G=\langle
X\,|\,R\rangle$ is a free group freely generated by $X$ or a virtually free
Coxeter group with standard Coxeter presentation, then for a word $w$ with
characters in $S$ representing an element $g\in G$, the cancellation norm of
$w$ is equal to the conjugation-invariant norm $\|g\|_{S}$, and in particular
is invariant upon the choice of $w$ representing $g$ [2, Proposition 2.E’].

The question of rationality for stable word norms over finitely presented
groups is a subject of interest in geometric group theory and analysis. In
[5], Calegari showed that the stable commutator norm over free groups is
rational using techniques from topology and geometry. It is also known that
rationality of the stable commutator norm does not hold for general finitely
presented groups [23], proving in the negative a conjecture of Gromov [13].
This motivates the analogous study of the stable conjugation-invariant word
norm, for which we prove the following result.

######  Theorem 1.1.

Let $G=\langle X\,|\,R\rangle$ be a free group freely generated by $X$ or a
virtually free Coxeter group with the standard Coxeter presentation
respectively. Let $S=X^{\pm 1}$, and further let $g_{1},\ldots,g_{n}\in G$.
Then the sequence $(\|g_{1}^{k}\ldots g_{n}^{k}\|_{S})_{k}$ is uniformly semi-
arithmetic in the sense of 2.1. Moreover,

$\tau_{S}(g_{1},\ldots,g_{n})\vcentcolon=\lim_{k\to\infty}\frac{\|g_{1}^{k}\ldots
g_{n}^{k}\|_{S}}{n}$

exists and is equal to $\frac{d}{m}\in{\mathbf{Q}}$, where $(\|g_{1}^{k}\ldots
g_{n}^{k}\|_{S})_{k}$ has period $m>0$ and difference $d\geq 0$.

######  Example 1.2.

Consider the free product of $n$ cyclic groups of order 2,
${\mathbf{Z}}/2{\mathbf{Z}}*\dots*{\mathbf{Z}}/2{\mathbf{Z}}$. Since this is a
virtually free Coxeter group admitting a standard Coxeter presentation with
generating set $X=\\{s_{1},\ldots,s_{n}\\}$ (where $s_{i}$ is the generator of
the $i$th copy of ${\mathbf{Z}}/2{\mathbf{Z}}$), Theorem 1.1 applies in this
instance. Virtually free Coxeter groups are characterised in [8, Proposition
8.8.5].

We will illustrate the result with three examples in the case where $F=\langle
a,b\rangle$ is the free group of rank 2 with free generating set $S=\\{a^{\pm
1},b^{\pm 1}\\}$.

######  Examples 1.3.

  1. (1)

For $a^{n}b^{m}\in F$, then $\|g^{k}\|=k\cdot(n+m)$, and coincides with the
standard word norm. Hence $\tau_{S}(g)=n+m$.

  2. (2)

Consider $[a,b]\in F$. Let $\psi:F\rightarrow{\mathbf{Z}}^{2}$ be the
abelianisation with $\psi(a)=(1,0)$ and $\psi(b)=(0,1)$, and for reduced
$g=s_{1}\ldots s_{n}$ with $s_{i}\in S$, let $g_{k}=s_{1}\ldots s_{k}$. Define
quasimorphism $\xi:F\rightarrow{\mathbf{Z}}$ where $\xi(g)$ is be the number
of left turns minus the number of right turns in the path
$(\psi(g_{k}))_{k=1}^{n}$ (as introduced in [3, Example 6.5]). It is immediate
that $\xi([a,b]^{k})=4k-1$, that $|\xi(gh)-\xi(g)-\xi(h)|\leq 2$, and
$|\xi(gsg^{-1})|\leq 2$ for all $g,h\in F$ and $s\in S$. Hence, if
$h=g_{1}s_{1}g_{1}^{-1}\ldots g_{\ell}s_{\ell}g_{\ell}^{-1}$, by induction,
$|\xi(h)|\leq 4\ell-2$, and hence $4k-1=|\xi([a,b]^{k})|\leq
4\|[a,b]^{k}\|_{S}-2$. Therefore, we have that $\|[a,b]^{k}\|_{S}\geq 2\lfloor
k/2\rfloor+2$.

To see that this is sharp, observe that by deleting the first $\lfloor
k/2\rfloor$ $b$’s and the final $\lfloor k/2\rfloor$ $b^{-1}$’s in the reduced
word of $[a,b]^{k}$, we obtain $b^{-\lfloor k/2\rfloor}ab^{\lfloor
k/2\rfloor}a^{-1}$ for $k$ even and $b^{-\lfloor k/2\rfloor}ab^{\lceil
k/2\rceil}a^{-1}b^{-1}$ otherwise, which both have cancellation length 2.
Therefore $\|[a,b]^{k}\|_{S}=2\lfloor k/2\rfloor+2$ and moreover
$\tau_{S}([a,b])=1$. In particular, in this case, the sequence is not
arithmetic.

  3. (3)

For $a^{m}ba^{-m},b^{-1}\in F$, since
$(a^{m}ba^{-m})^{k}(b^{-1})^{k}=a^{m}b^{k}a^{-m}b^{-k}$, it follows that
$\|(a^{m}ba^{-m})^{k}(b^{-1})^{k}\|=2\min\\{m,k\\}$ and hence
$\tau_{S}(a^{m}ba^{-m},b^{-1})=0$.

######  Remark 1.4.

These examples suggest that 1.1 cannot be strengthened, as we cannot replace
uniform semi-arithmeticity with mere arithmeticity.

######  Acknowledgements.

The author would like to offer thanks to both Jarek Kędra and Assaf Libman,
without whose indispensable mentorship, advice, time and support this paper
would not have been possible. The author would also like to thank the referee
for drawing the author’s attention to the literature on reflection lengths.

##  2\. Semi-arithmetic sequences

Let $\widehat{{\mathbf{N}}}\vcentcolon={\mathbf{N}}\cup\\{\infty\\}$ with the
usual total order, where $x+\infty=\infty+x=\infty$ for any
$x\in\widehat{{\mathbf{N}}}$. Moreover, let an arithmetic sequence with
difference $d\geq 0$ have the usual meaning. Observe that the constant
infinite sequence $(a_{k})_{k}$ where $a_{k}=\infty$ for $k\geq 0$ is
arithmetic with difference $d$ for any $d\geq 0$.

######  Definition 2.1.

Let $(a_{k})_{k}$ be a sequence in $\widehat{{\mathbf{N}}}$. Then, it is:

  1. (1)

Eventually finite if $a_{k}\in{\mathbf{N}}$ for all $k$ sufficiently large;

  2. (2)

Eventually arithmetic if $(a_{k+n})_{k}$ is arithmetic for some $n\geq 0$;

  3. (3)

Semi-arithmetic with period $m>0$ if for all $n\geq 0$, the sequences
$(a_{mk+n})_{k}$ are eventually arithmetic with differences $d_{n}$;

  4. (4)

Uniformly semi-arithmetic with period $m>0$ and difference $d\geq 0$ if for
all $n\geq 0$, the sequences $(a_{mk+n})_{k}$ are eventually arithmetic with
difference $d$.

######  Remark 2.2.

Let $(a_{k})_{k}$ be a semi-arithmetic sequence with period $m$. Since, for
all $n\geq 0$, $(a_{mk+n+m})_{k}$ is a tail of $(a_{mk+n})_{k}$, it follows
that $d_{n}=d_{n+m}$ for all $n\geq 0$.

The following two results are immediate consequences of 2.1:

######  Proposition 2.3.

Let $(a_{k})_{k}$ and $(a^{\prime}_{k})_{k}$ be semi-arithmetic sequences with
periods $m$ and $m^{\prime}$ respectively. Then the sequence
$(\min\\{a_{k},a^{\prime}_{k}\\})_{k}$ is semi-arithmetic with period
$mm^{\prime}$. ∎

######  Proposition 2.4.

Let $(a_{k})_{k}$ be a eventually finite uniformly semi-arithmetic sequence
with period $m>0$ and difference $d\geq 0$. Then

$\lim_{k\to\infty}\frac{a_{k}}{k}$

exists and is equal to $\frac{d}{m}\in{\mathbf{Q}}$. ∎

######  Lemma 2.5.

Let $(a_{k})_{k}$ be semi-arithmetic. Suppose there exists $D\geq 0$ such that
$a_{k+1}\leq a_{k}+D$ for all $k\geq 0$. Then $(a_{k})_{k}$ is uniformly semi-
arithmetic.

###### Proof.

If $a_{k}=\infty$ for all $k\geq 0$, the result is immediate. Otherwise,
$(a_{k})_{k}$ is eventually finite. Hence, without loss of generality, assume
$a_{k}<\infty$ for all $k$.

Let $(a_{k})_{k}$ have period $m>0$. By 2.1, for all $n\geq 0$, the sequences
$(a_{mk+n})_{k}$ are arithmetic with some difference $d_{n}\geq 0$. By the
hypothesis, the sequence $(a_{mk+n+1}-a_{mk+n})_{k}$ is bounded above by $D$.
However, observe that $(a_{mk+n+1}-a_{mk+n})_{k}$ is eventually arithmetic
with difference $d_{n+1}-d_{n}$. Therefore, $d_{n+1}-d_{n}\leq 0$, and hence
by Remark 2.2,

$d_{0}\geq d_{1}\geq\ldots\geq d_{m}=d_{0}.$

Hence by Remark 2.2, $(d_{n})_{n}$ is constant, and $(a_{k})_{k}$ is uniformly
semi-arithmetic. ∎

##  3\. Semilinear sets

Let ${\mathbf{N}}^{X}$ denote the free abelian monoid over $X$, or
equivalently, the set of all functions $X\to{\mathbf{N}}$.

######  Definition 3.1 (Compare [16, Definition 3]).

A set $\Omega\subseteq{\mathbf{N}}^{X}$ is called linear if it is of the form
$v+M$ where $v\in{\mathbf{N}}^{X}$ and $M\subseteq{\mathbf{N}}^{X}$ is a
finitely-generated submonoid of ${\mathbf{N}}^{X}$. It called is semilinear if
it is a union of finitely many linear subsets of ${\mathbf{N}}^{X}$.

The following propositions are immediate consequences of 3.1:

######  Proposition 3.2.

Let $\phi\colon{\mathbf{N}}^{X}\to{\mathbf{N}}^{Y}$ be an homomorphism. Then
if $\Omega\subseteq{\mathbf{N}}^{X}$ is semilinear,
$\phi(\Omega)\subseteq{\mathbf{N}}^{Y}$ is also semilinear. ∎

######  Proposition 3.3.

If $\Omega_{X}\subseteq{\mathbf{N}}^{X}$ and
$\Omega_{Y}\subseteq{\mathbf{N}}^{Y}$ are semilinear, then
$\Omega_{X}\times\Omega_{Y}\subseteq{\mathbf{N}}^{X}\times{\mathbf{N}}^{Y}$ is
also semilinear. ∎

We also state the following seminal result in the study of semilinear sets.

######  Lemma 3.4 (Liu-Weiner, [16, Theorem 1]).

The family of semilinear subsets of ${\mathbf{N}}^{X}$ is closed under
intersections. ∎

######  Example 3.5.

Let $X$ be a finite set, and let $\mathbf{1}_{X}\colon X\to{\mathbf{N}}$
denote the characteristic function of $X$. Then, define the diagonal over $X$
to be the set
$\Delta_{X}\vcentcolon=\\{k\cdot\mathbf{1}_{X}:k\in{\mathbf{N}}\\}\subseteq{\mathbf{N}}^{X}$.
In particular, observe that $\Delta_{X}$ is semilinear.

######  Definition 3.6.

Let $\Omega\subseteq{\mathbf{N}}^{2}$. For $k\in{\mathbf{N}}$, let
$\Omega_{(k)}\vcentcolon=\\{\ell:(k,\ell)\in\Omega\\}$. Then the lower
envelope of the set $\Omega$ is the sequence

$\operatorname{Env}(\Omega)\vcentcolon=(\inf\Omega_{(k)})_{k},$

where, by convention, the infimum of the empty set is $\infty$.

The following proposition follows immediately from 3.6.

######  Proposition 3.7.

Let $\Omega,\Omega^{\prime}\subseteq{\mathbf{N}}^{2}$, and let
$(a_{k})_{k}=\operatorname{Env}(\Omega)$ and
$(a^{\prime}_{k})_{k}=\operatorname{Env}(\Omega^{\prime})$ respectively. Then
$\operatorname{Env}(\Omega\cup\Omega^{\prime})=(\min\\{a_{k},a^{\prime}_{k}\\})_{k}$.
∎

######  Proposition 3.8.

Let $\Omega$ be the submonoid of ${\mathbf{N}}^{2}$ generated by a finite set
$V=\\{(x_{\alpha},y_{\alpha})\\}_{\alpha\in A}\subseteq{\mathbf{N}}^{2}$. Set
$A^{\prime}\vcentcolon=\\{\alpha\in A:x_{\alpha}\neq 0\\}$ and
$\Omega^{\prime}$ be the submonoid generated by
$\\{(x_{\alpha},y_{\alpha})\\}_{\alpha\in A^{\prime}}$. Then
$\operatorname{Env}(\Omega)=\operatorname{Env}(\Omega^{\prime})$.

###### Proof.

Since $\Omega^{\prime}\subseteq\Omega$, it follows that
$\inf\Omega_{(k)}\leq\inf\Omega^{\prime}_{(k)}$. If $\inf\Omega_{(k)}=\infty$,
then clearly $\inf\Omega_{(k)}=\inf\Omega_{(k)}^{\prime}$. Otherwise, without
loss of generality, let $k\geq 1$ be such that $\inf\Omega_{(k)}<\infty$.
Then, there exists $\lambda_{\alpha}\in{\mathbf{N}}$ for all $\alpha\in A$
such that $\sum_{\alpha\in A}\lambda_{\alpha}x_{\alpha}=k$ and
$\sum_{\alpha\in A}\lambda_{\alpha}y_{\alpha}=\inf\Omega_{(k)}$. Since
$\sum_{\alpha\in A^{\prime}}\lambda_{\alpha}y_{\alpha}$ is an element of
$\Omega^{\prime}_{(k)}$, and $\sum_{\alpha\in
A^{\prime}}\lambda_{\alpha}y_{\alpha}\leq\sum_{\alpha\in
A}\lambda_{\alpha}y_{\alpha}$, by minimality, it follows that
$\inf\Omega^{\prime}_{(k)}\leq\inf\Omega_{(k)}$ for all $k$, from which
equality follows. ∎

We now state the following classical lemma.

######  Lemma 3.9 (Frobenius, [22, Theorem 3.15.2]).

Let $x_{1},\ldots,x_{n}\in{\mathbf{N}}$, and let $g=\gcd(x_{1},\ldots,x_{n})$.
Then there exists $N\geq 0$ such that for all $k\geq N$ with $g\ |\ k$, there
exists $\lambda_{1},\ldots,\lambda_{n}\in{\mathbf{N}}$ such that

$k=\sum_{i}\lambda_{i}x_{i}.\qed$

We arrive at the main result of this section, whose proof is adapted and
simplified from [12, Theorem, p. 329].

######  Lemma 3.10.

Let $\Omega\subseteq{\mathbf{N}}^{2}$ be the monoid generated by a finite set
$V\subseteq{\mathbf{N}}^{2}$. Then $\operatorname{Env}(\Omega)$ is uniformly
semi-arithmetic.

###### Proof.

Let $V\vcentcolon=\\{(x_{i},y_{i}):1\leq i\leq n\\}$, and let
$(a_{k})_{k}\vcentcolon=\operatorname{Env}(\Omega)$. By 3.8, we may assume
that $x_{i}\neq 0$ for all $i$, and as the result is trivial if $n=0$, we may
assume $n\geq 1$. By reordering the generating set, we may also assume that

$\frac{y_{1}}{x_{1}}\leq\frac{y_{i}}{x_{i}}$

for all $1\leq i\leq n$.

First, consider the case when $\gcd(x_{1},\ldots,x_{n})=1$. By Lemma 3.9,
there exists $N\geq 0$ such that for all $k\geq N$, $a_{k}<\infty$. Let $k\geq
0$ be such that $a_{k}<\infty$, and choose $\lambda_{i}\in{\mathbf{N}}$ for
all $1\leq i\leq n$ such that $\sum_{i}\lambda_{i}x_{i}=k$ and
$\sum_{i}\lambda_{i}y_{i}=a_{k}$. Then

(3.1)
$a_{k}=\sum_{i}\lambda_{i}y_{i}=\sum_{i}\lambda_{i}x_{i}\cdot\frac{y_{i}}{x_{i}}\geq\sum_{i}\lambda_{i}x_{i}\cdot\frac{y_{1}}{x_{1}}=k\cdot\frac{y_{1}}{x_{1}}.$

Define $\lambda^{\prime}_{i}$ by

$\lambda^{\prime}_{i}\vcentcolon=\begin{cases}\lambda_{i}+1&i=1\\\
\lambda_{i}&i>1.\end{cases}$

Then $\sum_{i}\lambda^{\prime}_{i}x_{i}=k+x_{1}$ and
$\sum_{i}\lambda^{\prime}_{i}y_{i}=a_{k}+y_{1}$. By the minimality of $a_{k}$,

(3.2) $a_{k+x_{1}}\leq a_{k}+y_{1}.$

Let $N\leq\ell<N+x_{1}$. Consider the sequence

$(b_{j})_{j\geq
0}\vcentcolon=\left(a_{\ell+x_{1}j}-(\ell+x_{1}j)\cdot\frac{y_{1}}{x_{1}}\right)_{j\geq
0}.$

Observe that $(b_{j})_{j}$ is decreasing, since by (3.2)

$b_{j+1}-b_{j}=a_{\ell+x_{1}j+x_{1}}-a_{\ell+x_{1}j}-y_{1}\leq 0$

for all $j\geq 0$. Moreover, by (3.1), $b_{j}\geq 0$. Therefore $(b_{j})_{j}$
is eventually constant. Hence there exists a constant $c$ such that for all
$j$ sufficiently large,

$a_{\ell+x_{1}j}=c+y_{1}j.$

Therefore $(a_{\ell+x_{i}j})_{j}$ is eventually arithmetic with difference
$y_{1}$. Since $N\leq\ell<N+x_{1}$ was arbitrary, it follows that
$(a_{k})_{k}$ is uniformly semi-arithmetic with period $x_{1}$ and difference
$y_{1}$.

In the general case, where $\gcd(x_{1},\ldots,x_{n})=g>1$, set
$x^{\prime}_{i}\vcentcolon=x_{i}/g$, let $\Omega^{\prime}$ be the submonoid
generated by $V^{\prime}\vcentcolon=\\{(x_{i}^{\prime},y_{i}):1\leq i\leq
n\\}$. Since $\gcd(x_{1}^{\prime},\ldots,x_{n}^{\prime})=1$ and let
$(a_{k}^{\prime})_{k}\vcentcolon=\operatorname{Env}(\Omega^{\prime})$. Then
$(a_{k})_{k}$ is uniformly semi-arithmetic with period $x_{1}^{\prime}$ and
difference $y_{1}$. Clearly,

$a^{\prime}_{k}=\left\\{\begin{array}[]{ll}a_{k/g}&\text{if }g\ |\ k\\\
\infty&\text{otherwise.}\end{array}\right.$

Hence, if $g\ |\ k$, then
$(a_{k+x_{1}j})_{j}=(a^{\prime}_{k/g+x_{1}^{\prime}j})_{j}$ is eventually
arithmetic with difference $y_{1}$. Otherwise, $a_{k+x_{1}j}=\infty$ for all
$j\geq 0$, and hence $(a_{k+x_{1}j})_{j}$ is an arithmetic sequence in
$\widehat{{\mathbf{N}}}$. It follows that $(a_{k})_{k}$ is uniformly semi-
arithmetic with period $x_{1}$ and difference $y_{1}$. ∎

######  Corollary 3.11.

Let $\Omega\subseteq{\mathbf{N}}^{2}$ be semilinear. Then
$\operatorname{Env}(\Omega)$ is semi-arithmetic.

###### Proof.

Let $M$ be a finitely generated submonoid of ${\mathbf{N}}^{2}$. By Lemma
3.10, $(a_{k})_{k}=\operatorname{Env}(M)$ is a uniformly semi-arithmetic
sequence in ${\mathbf{N}}$. It can be seen that if
$v=(x_{0},y_{0})\in{\mathbf{N}}^{2}$, then for the linear set $v+M$,

$\operatorname{Env}(v+M)_{k}=\begin{cases}\infty&\text{if }k<x_{0}\\\
a_{k-x_{0}}+y_{0}&\text{otherwise.}\end{cases}$

Therefore $\operatorname{Env}(v+M)$ is uniformly semi-arithmetic. Since a
semilinear subset of ${\mathbf{N}}^{2}$ is the union of finitely many linear
sets, the result follows by repeated application of 3.7 and 2.3. ∎

Observe that, even if $(a_{k})_{k}$ and $(a^{\prime}_{k})_{k}$ are uniformly
semi-arithmetic, is is possible that the sequence
$(\min\\{a_{k},a^{\prime}_{k}\\})_{k}$ is non-uniformly semi-arithmetic. Thus,
3.11 cannot be improved to uniform semi-arithmeticity.

##  4\. Formal Languages

Let $X^{*}$ denote the free monoid generated by a set $X$, or equivalently,
the set of all finite sequences of elements in $X$ with concatenation as
monoidal structure. Call the elements of $X^{*}$ words in the alphabet $X$,
and denote the empty word by $\varepsilon$. For $w\in X^{*}$, define length
function $|\cdot|\colon X^{*}\rightarrow{\mathbf{N}}$ to be the unique
homomorphism defined on generators by $|x|=1$ for all $x\in X$. Observe that

(4.1) $|w_{1}^{k_{1}}\ldots w_{n}^{k_{n}}|=\sum_{i=1}^{n}k_{i}\cdot|w_{i}|.$

A subset ${\mathcal{L}}\subseteq X^{*}$ is called a language on the alphabet
$X$. Given languages ${\mathcal{L}},{\mathcal{M}}\subseteq X^{*}$ let
${\mathcal{L}}\cdot{\mathcal{M}}$ denote the language

${\mathcal{L}}\cdot{\mathcal{M}}\vcentcolon=\\{uv\,:\,u\in{\mathcal{L}},v\in{\mathcal{M}}\\}.$

Write ${\mathcal{L}}^{n}$ for the $n$-fold concatenation
${\mathcal{L}}\cdot\ldots\cdot{\mathcal{L}}$, where by convention
${\mathcal{L}}^{0}\vcentcolon=\\{\varepsilon\\}$. Further define the Kleene
star of ${\mathcal{L}}$ to be

${\mathcal{L}}^{*}\vcentcolon=\bigcup_{n\geq 0}{\mathcal{L}}^{n}.$

######  Definition 4.1.

Let $X,Y$ be disjoint finite alphabets. Then the projection homomorphism of
free monoids is the homomorphism

$\pi_{X}\colon(X\sqcup Y)^{*}\to X^{*}$

defined on generators by $\pi_{X}(x)=x$ for all $x\in X$ and
$\pi_{X}(y)=\varepsilon$ for all $y\in Y$.

######  Definition 4.2.

Given a word $w\in X^{*}$, a subword is a subsequence of $w$. Let
$\mathcal{P}(w)$ denote the finite language consisting of the subwords of $w$.
Observe that, for words $w_{1},\ldots,w_{n}\in X^{*}$,

(4.2) $\mathcal{P}(w_{1}\ldots
w_{n})=\mathcal{P}(w_{1})\cdot\ldots\cdot\mathcal{P}(w_{n}).$

######  Definition 4.3.

Let ${\mathcal{L}}\subseteq X^{*}$ be a language. The cancellation length
relative to ${\mathcal{L}}$ of $w\in X^{*}$ is defined as

$\|w\|_{{\mathcal{L}}}\vcentcolon=\inf\\{|w|-|u|:u\in\mathcal{P}(w)\cap{\mathcal{L}}\\}.$

Equivalently, $\|w\|_{{\mathcal{L}}}$ is the infimum of all $\ell$ such that
it is possible to delete $\ell$ characters from $w$ to yield an element of
${\mathcal{L}}$.

######  Definition 4.4 (Compare [14, Section 3.1.1], [1, Theorem 3.1]).

The collection of regular languages over the alphabet $X$ is the smallest
collection of languages satisfying the following:

  1. (1)

If ${\mathcal{L}}\subseteq X^{*}$ is finite, then ${\mathcal{L}}$ is regular;

  2. (2)

If ${\mathcal{L}}$ and ${\mathcal{M}}$ are regular, then
${\mathcal{L}}^{*},{\mathcal{L}}\cup{\mathcal{M}}$ and
${\mathcal{L}}\cdot{\mathcal{M}}$ are also regular.

######  Definition 4.5 (Compare [14, Section 5.12], [7, Section 19]).

A context-free grammar (CFG) is a quadruple $G=(V,X,R,v_{0})$, where
$X\subseteq V$ are finite sets of characters, $v_{0}\in V\setminus X$ and
$R\subseteq(V\setminus X)\times V^{*}$ is a finite set. Let
${\mathcal{L}}\subseteq V^{*}$ be the smallest language containing the word
$v_{0}$ such that, if $w_{0}vw_{1}\in{\mathcal{L}}$ with $w_{0},w_{1}\in
V^{*}$ and $v\in V\setminus X$, and if $(v,v^{\prime})\in R$, then
$w_{0}v^{\prime}w_{1}\in{\mathcal{L}}$. The language
${\mathcal{L}}^{\prime}={\mathcal{L}}\cap X^{*}$ is called the context-free
language (CFL) generated by $G$.

We will now state some closure properties of CFLs.

######  Theorem 4.6 ([14, Theorem 7.27]).

Let ${\mathcal{L}}\subseteq Y^{*}$ be a CFL, and let $\phi\colon X^{*}\to
Y^{*}$ be a monoid homomorphism. Then $\phi^{-1}({\mathcal{L}})$ is also a
CFL. ∎

######  Theorem 4.7 ([14, Theorem 7.30]).

If ${\mathcal{R}}\subseteq X^{*}$ is regular and ${\mathcal{L}}\subseteq
X^{*}$ is a CFL then ${\mathcal{R}}\cap{\mathcal{L}}$ is also a CFL. ∎

######  Definition 4.8.

Let $\Phi\colon S^{*}\rightarrow G$ be the canonical projection with
$\Phi(s)=s$ for all $s\in S$. The cancellation language, ${\mathcal{W}}(G,S)$,
is defined to be

${\mathcal{W}}(G,S)=\Phi^{-1}(1)\subseteq S^{*}.$

We now state the following classical theorem.

######  Theorem 4.9 (Muller-Schupp, [19, Theorem III], [9, Section 6.1]).

The language ${\mathcal{W}}(G,S)$ is a CFL if and only if $G$ is a virtually
free group. ∎

######  Definition 4.10.

Let $X$ be a finite set. The Parikh homomophism is the abelianisation map,
denoted

$\psi_{X}\colon X^{*}\to{\mathbf{N}}^{X}.$

It follows directly from the definition that for any $w\in X^{*}$,
$\psi(w)(x)$ is equal to the number of occurrences of the letter $x$ in $w$.
Therefore, for any $w\in X^{*}$,

(4.3) $|w|=\sum_{x\in X}\psi_{X}(w)(x).$

######  Remark 4.11.

Considering that ${\mathbf{N}}^{X\sqcup
Y}\cong{\mathbf{N}}^{X}\times{\mathbf{N}}^{Y}$, the Parikh homomorphism
$\psi_{X\sqcup Y}\colon(X\sqcup Y)^{*}\to{\mathbf{N}}^{X\sqcup Y}$ has the
form

$\psi_{X\sqcup Y}=(\psi_{X}\circ\pi_{X},\psi_{Y}\circ\pi_{Y}).$

An important result in the theory of CFLs is Parikh’s theorem.

######  Theorem 4.12 (Parikh, [20]).

Let ${\mathcal{L}}\subseteq X^{*}$ be CFL. Then $\psi_{X}({\mathcal{L}})$ is
semilinear. ∎

######  Definition 4.13.

Let ${\mathcal{L}}_{1},\ldots,{\mathcal{L}}_{n}\subseteq X^{*}$ be languages,
and further let $Y=\\{y_{1},\ldots,y_{n}\\}$ be a set disjoint from $X$.
Define the enumeration language of
${\mathcal{L}}_{1},\ldots,{\mathcal{L}}_{n}$ in $(X\sqcup Y)^{*}$ to be

${\mathcal{E}}=({\mathcal{L}}_{1}\cdot\\{y_{1}\\})^{*}\cdot\ldots\cdot({\mathcal{L}}_{n}\cdot\\{y_{n}\\})^{*}.$

For simplicity, let us identify $\mathbf{N}^{Y}$ with $\mathbf{N}^{n}$.
Observe that ${\mathcal{E}}$ is the disjoint union

(4.4)
${\mathcal{E}}=\bigsqcup_{\mathbf{k}\in\mathbf{N}^{Y}}{\mathcal{E}}_{\mathbf{k}},$

where for every $\mathbf{k}=(k_{1},\ldots,k_{n})\in{\mathbf{N}}^{Y}$,

${\mathcal{E}}_{\mathbf{k}}=({\mathcal{L}}_{1}\cdot\\{y_{1}\\})^{k_{1}}\cdot\ldots\cdot({\mathcal{L}}_{n}\cdot\\{y_{n}\\})^{k_{n}}$

is the preimage of $\mathbf{k}$ with respect to
$\psi_{Y}\circ\pi_{Y}\colon{\mathcal{E}}\to{\mathbf{N}}^{Y}$. Moreover,

(4.5)
$\pi_{X}({\mathcal{E}}_{\mathbf{k}})={\mathcal{L}}_{1}^{k_{1}}\cdot\ldots\cdot{\mathcal{L}}_{n}^{k_{n}}.$

######  Remark 4.14.

Enumeration languages can be considered as graded unions of combinatorial
cubes, in the sense of [21, Section 3].

##  5\. Multivariate stability

The purpose of this section is to prove the main result of the paper.

######  Theorem 5.1.

Let ${\mathcal{L}}\subseteq X^{*}$ be a CFL, and let $w_{1},\ldots,w_{n}\in
X^{*}$. Then the sequence $(\|w_{1}^{k}\ldots
w_{n}^{k}\|_{{\mathcal{L}}})_{k}$ is uniformly semi-arithmetic in the sense of
2.1. In particular, if $(\|w_{1}^{k}\ldots w_{n}^{k}\|_{{\mathcal{L}}})_{k}$
is eventually finite, and has period $m>0$ and difference $d\geq 0$, then

$\lim_{k\to\infty}\frac{\|w_{1}^{k}\ldots w_{n}^{k}\|_{{\mathcal{L}}}}{k}$

exists and is equal to $\frac{d}{m}\in{\mathbf{Q}}$.

###### Proof.

Let $U\subseteq{\mathbf{N}}^{2}$ be defined by

$U\vcentcolon=\\{(k,|w_{1}^{k}\ldots
w_{n}^{k}|-|u|):u\in\mathcal{P}(w_{1}^{k}\ldots
w_{n}^{k})\cap\mathcal{L},\,k\in\mathbf{N}\\}.$

Observe that by 4.3,

$U_{(k)}=\\{|w_{1}^{k}\ldots w_{n}^{k}|-|u|:u\in\mathcal{P}(w_{1}^{k}\ldots
w_{n}^{k})\cap\mathcal{L}\\},$

and hence by 3.6,

(5.1) $\operatorname{Env}(U)=(\|w_{1}^{k}\ldots
w_{n}^{k}\|_{{\mathcal{L}}})_{k}.$

Let $Y=\\{y_{1},\ldots,y_{n}\\}$ be disjoint from $X$, and consider the
enumeration language

(5.2)
${\mathcal{R}}=(\mathcal{P}(w_{1})\cdot\\{y_{1}\\})^{*}\cdot\ldots\cdot(\mathcal{P}(w_{n})\cdot\\{y_{n}\\})^{*}\subseteq(X\sqcup
Y)^{*}$

in the sense of 4.13. By 4.4, ${\mathcal{R}}$ is a regular language. With
$\pi_{X}\colon(X\sqcup Y)^{*}\to X^{*}$ in the sense of 4.1, set

(5.3) ${\mathcal{M}}\vcentcolon={\mathcal{R}}\cap\pi_{X}^{-1}({\mathcal{L}}).$

Theorems 4.6 and 4.7 imply that ${\mathcal{M}}$ is a CFL in $(X\sqcup Y)^{*}$.
By Theorem 4.12,

$M\vcentcolon=\psi_{X\sqcup Y}({\mathcal{M}})\subseteq{\mathbf{N}}^{X\sqcup
Y}$

is a semilinear set.

Consider the diagonal $\Delta_{Y}\subseteq{\mathbf{N}}^{Y}$ as defined in
Example 3.5. Then by Example 3.5 and 3.3, ${\mathbf{N}}^{X}\times\Delta_{Y}$
is a finitely-generated linear subset of ${\mathbf{N}}^{X\sqcup Y}$. By Lemma
3.4,

$M_{\Delta}\vcentcolon=M\cap({\mathbf{N}}^{X}\times\Delta_{Y})$

is semilinear in ${\mathbf{N}}^{X\sqcup Y}$. Now define a monoid homomorphism

$\displaystyle\xi\colon{\mathbf{N}}^{X\sqcup Y}$
$\displaystyle\to{\mathbf{Z}}^{2}$ $\displaystyle(f,\mathbf{k})$
$\displaystyle\mapsto\big{(}k_{1},\sum_{i=1}^{n}k_{i}\cdot|w_{i}|-\sum_{x\in
X}f(x)\big{)}$

where we identify ${\mathbf{N}}^{X\sqcup Y}$ with
${\mathbf{N}}^{X}\times{\mathbf{N}}^{Y}$.

Claim. $\xi(M_{\Delta})=U$.

Proof of claim. Observe that $(k,\ell)\in\xi(M_{\Delta})$ if and only if

$\exists\,u^{\prime}\in{\mathcal{M}}:\psi_{X\sqcup
Y}(u^{\prime})\in{\mathbf{N}}^{X}\times\Delta_{Y}\text{ and }\xi(\psi_{X\sqcup
Y}(u^{\prime}))=(k,\ell).$

Resolving coordinates, and by both Remark 4.11 and the definition of $\xi$,
this is equivalent to

$\exists\,u^{\prime}\in{\mathcal{M}}:\sum_{i=1}^{n}k\cdot|w_{i}|-\sum_{x\in
X}\psi_{X}(\pi_{X}(u^{\prime}))=\ell\text{ and
}\psi_{Y}(\pi_{Y}(u^{\prime}))=k\cdot\mathbf{1}_{Y}.$

By (4.1), (4.3) and (4.4) this is equivalent to

$\exists\,u^{\prime}\in{\mathcal{M}}:|w_{1}^{k}\ldots
w_{n}^{k}|-|\pi_{X}(u^{\prime})|=\ell\text{ and
}u^{\prime}\in{\mathcal{R}}_{k\cdot\mathbf{1}_{Y}}.$

Since ${\mathcal{M}}={\mathcal{R}}\cap\pi_{X}^{-1}({\mathcal{L}})$ and by
(4.2) and (4.5), this is equivalent to

$\exists\,u\in\mathcal{P}(w_{1}^{k}\ldots
w_{n}^{k})\cap{\mathcal{L}}:|w_{1}^{k}\ldots w_{n}^{k}|-|u|=\ell$

which is equivalent to $(k,\ell)\in U$, completing the proof of the claim. ∎

Finally, since $M_{\Delta}$ is semilinear in ${\mathbf{N}}^{X\sqcup Y}$ and
$\xi$ is a homomorphism of monoids, it follows that $\xi(M_{\Delta})$ is the
union of finitely many linear subsets of ${\mathbf{Z}}^{2}$. Since
$\xi(M_{\Delta})=U$ is contained in ${\mathbf{N}}^{2}$, this implies that
$U\subseteq{\mathbf{N}}^{2}$ is semilinear. Hence, by 5.1 and 3.11,
$(\|w_{1}^{k}\ldots w_{n}^{k}\|_{{\mathcal{L}}})_{k}$ is semi-arithmetic.

For any $k\geq 0$ the word $w_{1}^{k}\ldots w_{n}^{k}$ can be obtained from
$w_{1}^{k+1}\ldots w_{n}^{k+1}$ by deleting $D\vcentcolon=\sum_{i}|w_{i}|$
symbols. By the definition of $\|\cdot\|_{{\mathcal{L}}}$,

$\|w_{1}^{k+1}\ldots w_{n}^{k+1}\|_{{\mathcal{L}}}\leq\|w_{1}^{k}\ldots
w_{n}^{k}\|_{{\mathcal{L}}}+D.$

Hence, by Lemma 2.5, $(\|w_{1}^{k}\ldots w_{n}^{k}\|_{{\mathcal{L}}})_{k}$ is
a uniformly semi-arithmetic sequence. The final statement of the theorem
follows from 2.4. ∎

###### Proof of Theorem 1.1.

Let $\Phi\colon S^{*}\to G$ be the canonical projection as defined in 4.8. Now
consider some $g_{1},\ldots,g_{n}\in G$ and choose $w_{1},\ldots,w_{n}\in
S^{*}$ such that $\Phi(w_{i})=g_{i}$. Since the conjugation-invariant length
is equal to the cancellation length,

$\|g_{1}^{k}\ldots g_{n}^{k}\|_{S}=\|w_{1}^{k}\ldots
w_{n}^{k}\|_{{\mathcal{W}}(G,S)},$

and in particular, by Theorem 5.1, $(\|g_{1}^{k}\ldots g_{n}^{k}\|_{S})_{k}$
is uniformly semi-arithmetic sequence $(a_{k})_{k}$ with period $m>0$ and
difference $d\geq 0$. Since $\|g\|_{S}<\infty$ for all $g\in G$, the sequence
$(\|g_{1}^{k}\ldots g_{n}^{k}\|_{S})_{k}$ is eventually finite, and hence

$\lim_{k\to\infty}\frac{\|g_{1}^{k}\ldots g_{n}^{k}\|_{S}}{k}$

exists and converges to $\frac{d}{m}$. ∎

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# A simple function for calculating the interaction between a molecule and a
graphene sheet

Xiongce Zhao<EMAIL_ADDRESS>Joint Institute for Computational Sciences
and Center for Nanophase Materials Sciences, Oak Ridge National Laboratory,
Oak Ridge, Tennessee 37831, USA  Current address: NIDDK, National Institutes
of Health, Bethesda, MD 20892, USA

###### Abstract

We present a novel potential model for calculating the interaction between a
molecule and a single graphene sheet. The dispersion/repulsion, induction,
dipole-quadrupole, quadrupole-quadrupole interactions between a fluid molecule
and a graphene sheet are described by integrated functions that are only
dependent on the separation between the molecule and the graphene along its
planar normal. The derived potential functions are in excellent agreement with
the computationally demanding atom-explicit summation method. Typical errors
of the integrated potential are less than 2% in the energy minimum compared
with the exact atom-explicit summation. To examine the practical effectiveness
of the newly developed functions, Monte Carlo simulations were performed to
model the adsorption of two representative gases in graphene sheets using both
the integrated and atom-explicit potentials. The integrated potential results
in same adsorption isotherms and density profiles for the adsorbed phase while
it only requires negligible computing time compared with that using the atom-
explicit method. The newly developed potential functions provide a simple and
accurate approach to calculating the physical interaction between molecules
and graphene sheets.

##  I Introduction

Graphene, or individual layers of graphite in which each carbon atom is bonded
to three other carbon atoms, has been the subject of considerable research
interest recently geim07 . Being a representative two dimensional crystal,
graphene has many peculiar properties such as low dimensionality, surface
homogeneity, structure stability, conductivity, and charge transport ability.
It is deemed as one of the most promising materials for future applications in
various fields such as electronics, novel materials, sensors, biodevices etc.
Lee2008 ; Ponomarenko2008 ; wang2008 ; li2008a ; li2008b ; ruoff08 ; geim07 ;
schedin07 ; fasolino07 ; dikin07 ; abanin06 ; novoselov05 ; zhang05 .

In the investigations of fundamental properties of graphene, it is often of
great interest to understand the interaction of molecules with graphene on a
molecule level. For example, graphene sheets have been used as high-
sensitivity sensors schedin07  or background membranes Meyer2008  in studying
the behavior of individual molecules. In such applications, interaction
between the target molecule and the graphene is one of the key properties that
needs to be understood in order to design effective devices. Therefore, a
simple and reliable approach to estimating the graphene-fluid interaction is
essential. Furthermore, computer simulation, which is becoming a useful
partner of experiments in studying fluid-solid interactions, requires
realistic and computationally affordable method with sufficient accuracy to
calculate the molecule-solid interactions, such as that between molecules and
graphene sheets. However, to our knowledge, a simple and accurate function
describing the interaction between a molecule and a single graphene sheet is
still not available. In this paper, we attempt to develop a novel potential to
solve this problem.

There are two general approaches to calculating the interactions between a
molecule and a crystal solid like graphene. One can use an atom-explicit
potential to describe the interaction between each atom in the molecule and
each atom in crystal, calculating the total potential energy by summing up all
pairs of atoms. Dispersion, repulsion, electrostatic and induction
interactions can all be computed in this way. But such an approach is
computationally costly, especially for large systems involving significant
number of solid atoms. Alternatively, one can develop integrated potential
functions that accounts for all the solid atoms in an effective way, making
use of the periodicity and homogeneity in the distribution of atoms in the
crystal. This has been a popular and useful approach for modeling solid-fluid
interactions for many systems, including graphite and metals steele73 .
Integrated potentials are easy to program and very efficient computationally.

In this paper the second approach was employed to derive a new set of
effective potential force fields for molecules on a graphene sheet. The types
of solid-fluid interactions of interest include the dispersion and repulsive
(Lennard-Jones, or LJ), the induction due to the dipole in the molecule and
the polarizability of carbon atoms in graphene, and the interaction between
multipoles in the molecule and the permanent quadrupole in graphene. The
potential functions derived in this work are extensions of a previous work by
us zhao05  for modeling interaction between gases and a semi infinite
graphitic surface. The accuracy of the derived potentials are evaluated by
comparing the energies calculated from the integrated expressions and those
from atomistic summation approach. The effectiveness of the potentials are
demonstrated by using the derived formulas in simulating the adsorption of two
representative gases into slits composed of single graphene sheets.

In Section II, the functions are derived. Section III and IV are examinations
of the new potentials via comparison with atomistic summation methods. Section
V is concluding remarks.

##  II Potential Development

We assume that a molecule is located right above the center of a single
graphene sheet. The origin of the system was set as the mass center of the
molecule. If we only consider the pairwise interactions, the total potential
energy between the molecule and the graphene can be expressed by a summation

$U=\sum_{i}^{\infty}u(r_{i}),$ (1)

where $r_{i}=(x_{i},y_{i},z)$ denotes the position of a carbon atom in
graphene relative to the molecule, $u(r_{i})$ is the pairwise potential
between the molecule and carbon atom $i$ in graphene, which is only dependent
on the separation between them, $r_{i}=(x_{i}^{2}+y_{i}^{2}+z^{2})^{1/2}$. For
a graphene sheet of infinite size along its lateral directions ($x$ and $y$),
$U$ only depends on the separation between the molecule and graphene along its
surface normal $z$, and Eq.(1) becomes

$U(z)=\sum_{i}^{\infty}u(x_{i},y_{i},z).$ (2)

For example, the Lennard-Jones interaction between the molecule and a carbon
atom is given as

$u(r)=4\varepsilon_{\mathrm{sf}}\left[\left(\frac{\sigma_{\mathrm{sf}}}{r}\right)^{12}-\left(\frac{\sigma_{\mathrm{sf}}}{r}\right)^{6}\right],$
(3)

where $\varepsilon_{\mathrm{sf}}$ and $\sigma_{\mathrm{sf}}$ are the LJ
potential parameters for the cross interaction between the molecule and a
graphene carbon atom. Substituting Eq.(3) into Eq.(2) gives

$U(z)=\sum_{i}^{\infty}4\varepsilon_{\mathrm{sf}}\left[\frac{\sigma_{\mathrm{sf}}^{12}}{(x_{i}^{2}+y_{i}^{2}+z^{2})^{6}}-\frac{\sigma_{\mathrm{sf}}^{6}}{(x_{i}^{2}+y_{i}^{2}+z^{2})^{3}}\right],$
(4)

If we assume that the graphene sheet is homogeneous and continuous along its
$x$ and $y$ directions, $U(z)$ can be approximated by an integral over $x$ and
$y$

$U(z)\\!=\\!4\varepsilon_{\mathrm{sf}}d_{\mathrm{s}}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\left[\frac{\sigma_{\mathrm{sf}}^{12}}{(x^{2}\\!+\\!y^{2}\\!+\\!z^{2})^{6}}\\!-\\!\frac{\sigma_{\mathrm{sf}}^{6}}{(x^{2}\\!+\\!y^{2}\\!+\\!z^{2})^{3}}\right]\\!dxdy,$
(5)

where $d_{\mathrm{s}}$ is the density of carbon atoms in a unit graphene area
$dxdy$. The integration can be simplified by letting $x^{2}+y^{2}=S$, which
gives $dxdy=\pi dS$. Then Eq.(5) reduces to

$U(z)=4\pi\varepsilon_{\mathrm{sf}}d_{\mathrm{s}}\\!\\!\int_{0}^{\infty}\\!\\!\left[\frac{\sigma_{\mathrm{sf}}^{12}}{(S+z^{2})^{6}}-\frac{\sigma_{\mathrm{sf}}^{6}}{(S+z^{2})^{3}}\right]dS.$
(6)

Eq.(6) leads to an integrated potential for the LJ interaction between a
molecule and an infinitely large graphene sheet

$U_{\mathrm{LJ}}(z)=2\pi\varepsilon_{\mathrm{sf}}\sigma_{\mathrm{sf}}^{2}d_{\mathrm{s}}\left[\frac{2}{5}\left(\frac{\sigma_{\mathrm{sf}}}{z}\right)^{10}-\left(\frac{\sigma_{\mathrm{sf}}}{z}\right)^{4}\right].$
(7)

The derived expression Eq.(7) for the LJ interaction between a molecule and a
graphene is analogous to a potential by Steele steele73 . The interaction is
only a function of the distance of the molecule from the graphene along its
planar normal, $z$, thus the numerical calculation is substantially
simplified. It is noteworthy to point out that unlike the potential for
graphite steele73  the new function for graphene, Eq.(7), does not contain any
empirical term that requires fitted parameters. This is due to the fact that
no approximation was involved in the derivation to account for the semi-
infinite layers of graphene for a graphite surface. Only assumption here is
that the atoms in graphene sheet are continuous, otherwise the derivation is
strictly exact.

For LJ particles, Eq. (7) should be a sufficient approximation for calculating
the molecule-graphene potential. However, for molecules that has partial
charges or permanent multipole moments, additional polar interactions have to
be included. For this purpose, similar procedures are applied to derive
integrated forms for the induction and multipolar interactions between the
molecule and a graphene sheet. We note that the above derivation procedure is
applicable as long as the pairwise potentials are dependent only of $r$ by an
inverse power $r^{-n}$ with $n>1$. This condition is satisfied if we use point
or angle-averaged dipoles and point quadrupoles for the molecule-graphene
interaction. For example, the angle-averaged dipole-induced dipole interaction
between a polar molecule and a carbon atom in the graphene is given by
maitland81

$u_{\mu}(r)=-\frac{\mu_{\mathrm{f}}^{2}\alpha_{\mathrm{s}}}{(4\pi\varepsilon_{0})^{2}r^{6}},$
(8)

where $\mu_{\mathrm{f}}$ is the permanent dipole moment of the molecule,
$\alpha_{\mathrm{s}}$ is the isotropic polarizability of a carbon atom in
graphene, and $\varepsilon_{0}$ is the vacuum permittivity.

The angle-averaged dipole-quadrupole interaction is given by maitland81

$u_{\mu\Theta}(r)=-\frac{\mu_{\mathrm{f}}^{2}\Theta_{\mathrm{s}}^{2}}{kT(4\pi\varepsilon_{0})^{2}r^{8}},$
(9)

where $\Theta_{\mathrm{s}}$ is the permanent quadrupole moment on each carbon
atom in graphene, $k$ is Boltzmann’s constant and $T$ is the absolute
temperature.

Finally, the angle-averaged quadrupole-quadrupole interaction for the molecule
and carbon atom is given by maitland81

$u_{\Theta\Theta}(r)=-\frac{14\Theta_{\mathrm{f}}^{2}\Theta_{\mathrm{s}}^{2}}{5kT(4\pi\varepsilon_{0})^{2}r^{10}},$
(10)

where $\Theta_{\mathrm{f}}$ is the permanent quadrupole moment of the
molecule.

Substituting Eqs. (8)$-$(10) into Eq.(2) and integrating, we obtain the
following expressions. The integrated induction potential is

$U_{\mu}(z)=-\frac{\pi
d_{\mathrm{s}}\mu_{\mathrm{f}}^{2}\alpha_{\mathrm{s}}}{2(4\pi\varepsilon_{0})^{2}}\frac{1}{z^{4}}.$
(11)

The integrated dipole-quadrupole potential is

$U_{\mu\Theta}(z)=-\frac{\pi
d_{\mathrm{s}}\mu_{\mathrm{f}}^{2}\Theta_{\mathrm{s}}^{2}}{3kT(4\pi\varepsilon_{0})^{2}}\frac{1}{z^{6}}.$
(12)

The integrated quadrupole-quadrupole potential is

$U_{\Theta\Theta}(z)=-\frac{7\pi
d_{\mathrm{s}}\Theta_{\mathrm{f}}^{2}\Theta_{\mathrm{s}}^{2}}{10kT(4\pi\varepsilon_{0})^{2}}\frac{1}{z^{8}}.$
(13)

Likewise, the potential functions in Eq.(11)$-$(13) are only dependent on the
separation between the multipole and graphene along its planar normal $z$. By
using Eq.(7) and Eqs.(11)$-$(13), one can calculate the interaction potential
energy between a polar molecule and a graphene sheet without using the costly
pairwise atomistic summation.

##  III Comparison with atomistic potentials

To examine the accuracy of the derived functions, we calculated the potential
energies of several representative nonpolar and polar molecules interacting
with a single graphene sheet using the integrated functions Eqs.(7) and
(11)$-$(13), to compare with those from atomistic summation of potentials Eqs.
(3) and (8)$-$(10). The graphene sheet in the atomistic summation was modeled
as a square about 10 nm in a side, containing 3680 carbon atoms. Trial
calculations indicate that such a system size contains more than enough carbon
atoms to approximate a graphene sheet that is infinite along its lateral
directions. A single molecule was placed over the center of the graphene sheet
and its interaction potential energies were computed by summing up
interactions from each carbon atom.

Table 1:  The Lennard-Jones and multipole parameters for the example molecules studied.  |  $\varepsilon_{\mathrm{f}}$ (kcal/mol) |  $\sigma_{\mathrm{f}}$ (Å) |  $\mu_{\mathrm{f}}$ (Debye) |  $\Theta_{\mathrm{f}}$ (10-20C Å2)
---|---|---|---|---
CH4 | 0.2378 | 3.527 | $-$ | $-$
H2O | 0.1554 | 3.152 | 2.351 | $-$
Cl2 | 0.7101 | 4.115 | $-$ | 10.79

The LJ and multipole potential parameters for the example molecules chosen are
shown in Table 1. The nonpolar molecule studied is methane, modeled by a
single LJ site jiang93 . The representative polar molecules selected are water
and chlorine. Water is modeled by the SPC/E potential Berendsen87 , with a
strong dipole moment of 2.351 Debye. Please note that the LJ parameters of
water in Table 1 are for the O atom. The chlorine molecule is modeled by a
single LJ site plus a quadrupole moment of 10.79$\times 10^{-20}$ C Å2 rowley
. The purpose of the calculation was to study the accuracy of the integrated
potential compared with atomistic summation. Therefore, the potential models
for the molecules were selected arbitrarily. The values of potential
parameters for carbon atoms in graphene are $\sigma_{\mathrm{s}}$=3.40 Å,
$\varepsilon_{\mathrm{s}}$=0.05569 kcal/mol, $d_{\mathrm{s}}$=0.382 Å-2 zhao05
, $\alpha_{\mathrm{c}}$=1.76 Å3 bates77 ,
$\Theta_{\mathrm{s}}$=$-3.03\times$10-20 C Å2 whitehouse93 . The cross
interaction parameters $\varepsilon_{\mathrm{sf}}$ and $\sigma_{\mathrm{sf}}$
were calculated using the Lorentz-Bertholet rules.

Figure 1:  Interaction of methane with a single graphene sheet calculated
using the atom-explicit pairwise summation of Eq.(3) (line) and the integrated
function Eq.(7) (symbols).
Figure 2:  Interaction of a water molecule with a single graphene sheet
calculated using the atomistic summation (lines) and integrated functions
(symbols). Atomistic summations: LJ interaction Eq.(3) (solid line), LJ plus
induction interactions Eq.(3)$+$Eq.(8) (long dashed line), and LJ plus
induction plus dipole-quadrupole interactions Eq.(3)$+$Eq.(8)$+$Eq.(9) (dashed
line). Integrated functions: LJ interaction Eq.(7) (circle), LJ plus induction
interactions Eq.(7)$+$Eq.(11) (square), and LJ plus induction plus dipole-
quadrupole interactions Eq.(7)$+$Eq.(11)$+$Eq.(12) (diamond).
Figure 3:  Interaction a chlorine molecule with a single graphene sheet
calculated using the atomistic summation (lines) and integrated functions
(symbols). Atomistic summations: LJ interaction Eq.(3) (solid line), LJ plus
quadrupole-quadrupole interactions Eq.(3)$+$Eq.10 (long dashed line).
Integrated functions: LJ interaction Eq.(7) (circle), LJ plus quadrupole-
quadrupole interactions Eq.(7)$+$Eq.(13) (square).

In Figs. 1 to 3 we compare the potential energies calculated from the
integrated potentials and the atom-explicit potentials at 300 K, for CH4, H2O,
and Cl2. It is seen that the integrated expressions are in excellent agreement
with the atomistic summation results. If we take the atomistic summation
results as the standard, the typical errors in the integrated potential are
less than 2% at the energy minimum. The integrated functions always give a
slightly more attractive potential compared to the atom-explicit methods,
possibly due to the fact that we treat the graphene carbon atoms as continuum
in the integrated potential.

The integrated potentials are most accurate when the graphene sheet is
infinitely large. However, we also carried out series of calculations using
graphene sheets of various sizes, to test the applicability of the derived
functions to the finite-size graphene. We found that the integrated potentials
are in excellent agreement with the atomistic potentials as long as the
square-shaped graphene sheet contains more than $\sim$500 carbon atoms (or a
graphene sheet of $\sim$38$\times$38 Å2). This suggests that the derived
potentials can also be readily used in estimating the interaction of molecules
with finite graphene sheets that are nanometer-scale in size.

For the different types of interactions between a molecule and graphene, it is
interesting to see that, for example from Fig. 2, the induction and dipole-
quadrupole interactions for strongly polar molecules such as water are non-
negligible compared with the LJ energy. On the other hand, based on the Cl2
example, the quadrupole-quadrupole interaction is usually not significant,
even for molecules with a strong quadrupole moment like chlorine.

##  IV Applications in adsorption simulation

One of important applications of the potential functions developed in this
work is molecular simulation of adsorption of molecules in pores composed of
single graphene sheets. In turn, such simulations serve as a verification of
the accuracy and effectiveness of the integrated functions in practical
applications. For this purpose, we chose to simulate the adsorption of two
typical nonpolar and polar gases, methane and water, confined in single
graphene sheets. The simulations were performed using the grand canonical
Monte Carlo (GCMC) method, the detailed description of which can be found in
literature allen .

The potentials parameters describing the interactions of water and methane
with graphene are given in Table 1. The methane is modeled as a simple one-
site LJ particle, with a cutoff of 9 Å. The LJ interaction between water
molecules is also modeled by a cutoff of 9 Å, without long-range correction
applied, as suggested by the original literature Berendsen87 . The
electrostatic interaction between water molecules are modeled by the partial
charges distributed on the three charge sites on the water model, without long
range correction. The interaction between the molecules and graphenes is
calculated by either the atomistic summation approach or integrated functions
for comparison.

The GCMC cell for adsorption simulations is a rectangular box. Two graphene
sheets are placed in the cell with their planes parallel to the $x-y$ plane of
the box. The distance between the two graphene layers along the $z$ direction
is 15 Å. Again, this separation is chosen arbitrarily. Periodic boundary
conditions are applied in all three directions. Therefore, the height of
simulation box in the $z$ direction is 30 Å. The typical box sizes in the $x$
and $y$ directions are set as 32 and 34 Å respectively, leading to a cell
volume of about 33 nm3. The types of move attempted during a GCMC simulation
were selected randomly with probability of 0.40, 0.40, 0.10, and 0.10 for
displacements, rotations, creations, and deletions of a fluid molecule
respectively. For simulation of the spherical methane, the rotation moves were
merged to displacement moves. Each simulation included equilibration of
2$\times 10^{6}$ MC moves and production of 2$\times 10^{6}$ MC moves.

The simulation of methane adsorption in graphene sheets were performed at 77 K
jiang93 . Previous theoretical studies predict that water only wets graphitic
carbon surfaces when temperature is above $\sim$510 K gatica04 ; zhao07 .
Therefore, we chose to perform simulations of water/graphene at 550 K. In GCMC
simulations the reduced chemical potential was varied to obtain isotherms and
density profiles of adsorbed methane or water, using either atomistic or
integrated potential for comparison.

Figure 4:  Adsorption isotherm of methane in singlet graphene sheets at 77 K.
Circles were calculated using the atomistic summation potential and squares
were calculated using the integrated potential. Adsorbed amount ($\rho^{*}$)
and chemical potential ($\mu^{*}$) are in reduced unit.
Figure 5:  Density profiles of methane adsorption in graphene sheets at 77 K
and reduced chemical potential of $-6.5$. (a) was calculated using the
atomistic summation potential and (b) was calculated using the integrated
potential. Two curves are almost identical so (a) was shifted by 0.01 in
$P(z)$ for clarity. The dashed lines at $z=\pm 7.5$ Å represent the two
graphene sheets in the system.
Figure 6:  Adsorption isotherm of water in graphene sheets at 550 K. Circles
were calculated using the atomistic summation potential and squares were
calculated using the integrated potential. Adsorbed amount ($\rho^{*}$) and
chemical potential ($\mu^{*}$) are in reduced unit.
Figure 7:  Density profiles of water adsorption in graphene sheets at 550 K
and reduced chemical potential of $-26$. (a) was calculated using the
atomistic summation potential and (b) was calculated using the integrated
potential. Two curves are almost identical so (a) was shifted by 0.005 in
$P(z)$ for clarity. The dashed lines at $z=\pm 7.5$ Å represent the two
graphene sheets in the system.

Calculated adsorption isotherms for methane in graphene sheets at 77 K are
presented in Fig. 4. The circles are simulation results computed from the
atomistic summation of Eq.(3), while the squares are from simulations using
the integrated potential Eq.(7). It can be seen that, within statistical
uncertainty, the isotherms from these two different potentials are in
excellent agreement. The first plateau in the isotherm corresponds to the
first layers of methane adsorbed on graphene sheets, and the second plateau is
from the methane adsorbed onto first layers as chemical potential increases.
This is confirmed by density profile distribution of the adsorbed phase shown
in Fig. 5. The transition from nonadsorption to the first layer occurs at
about $\mu^{*}=-8.6$, and the transition from the first to the second layer
occurs at about $\mu^{*}=-7.3$.

It is known that the isotherm shapes, especially the layering transition of
adsorbed phase calculated from simulations, are very sensitive to the solid-
fluid interactions zhao02 . A small deviation in the solid-fluid potential can
result in significant shift in the layer transition location in isotherms. The
excellent agreement in the adsorption layering transition for the two
potentials indicates that the integrated functions are robust alternatives to
the atomistic models.

One example of density profiles of methane adsorbed in graphene layers is
given in Fig. 5. Sharp peaks are observed at $z=\pm 4$, $\pm 11$ Å, which
correspond to the first adsorbed methane layers on the graphene sheets.
Additional broad peaks are observed at $z=0$, $\pm 15$ Å, representing the
adsorbed second layers between the first layers. Again, it is seen that the
density distribution of adsorbed methane in graphenes calculated using the
atomistic and integrated potentials are in excellent agreement. Note that the
density profiles calculated from the two potentials are almost identical to
each other, so that we have to present them by shifting one curve along the
vertical axis by a constant value to have a clear view.

The results for water adsorption in singlet graphenes at 550 K are shown in
Figs. 6 and 7. It is seen from the water adsorption isotherm that a continuous
wetting transition occurs at about $\mu^{*}=-33.4$. The density profile peaks
corresponding to the first adsorbed layers on the graphene sheets locate at
$z=\pm 4.4$ and $\pm 10.6$ Å. Slightly different from that of methane, the
adsorbed phase of water forms two second layers between the first layers, at
$z=\pm 1.5$ and $\pm 13.5$ Å, due to the relatively smaller size of water
molecule compared to methane. The isotherms and density profiles of water in
graphene sheets calculated from the integrated and atomistic potentials are
also in excellent agreement. This indicates that the integrated potentials for
multipolar interactions are also excellent approximations to the atomistic
approach.

We monitored the computational time required for the simulations using two
different approaches. It is found that typically the computational time for
the integrated potential is $<1\%$ of that required for the atomistic
summations.

##  V Conclusions

A set of effective potentials were derived for calculating the Lennard-Jones,
dipole-induced dipole, dipole-quadrupole, and quadrupole-quadrupole
interactions between fluid molecules and a graphene sheet. The integrated
potential functions depend only on the separation between a molecule and the
graphene surface. They are mathematically simple and easy to use in either
estimating the interaction between a single molecule and a graphene sheet or
in large-scale molecular simulations. The potential energies calculated from
the integrated potentials are in excellent agreement with the results
calculated from direct atomistic summations, while the integrated ones are
computationally negligible compared with their atomistic counterparts.
Adsorption simulations of two representative gases in singlet graphene sheets
were performed to further test the derived potential functions. The adsorption
isotherms and density profiles calculated from the derived potentials and
atomistic models are in excellent agreement, indicating that the derived
potential can predict both the equilibrium and the structural properties of
adsorbed phase with excellent accuracy. The potentials developed in this work
provide a simple, accurate, and robust method for calculating the physical
interaction of molecules and single graphene sheets.

Finally, we note that the self-consistency in the polarization of carbon atoms
by a polar fluid molecule was neglected in the derivation. However, we believe
that the effect is relatively small compared with the dispersion, induction,
and dipole-quadrupole interactions. Also, by using an integrated potential,
one assumes the graphene surface is smooth and the impact of the surface
corrugation on the molecule-solid interaction is ignored.

###### Acknowledgements.

The author thanks Peter T. Cummings for many insightful discussions. This
research was conducted at the Center for Nanophase Materials Sciences, which
is sponsored at Oak Ridge National Laboratory by the Division of Scientific
User Facilities, U.S. Department of Energy. This research used resources of
the National Energy Research Scientific Computing Center, which is supported
by the Office of Science of the U.S. Department of Energy under Contract No.
DE-AC02-05CH11231.

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