# Antichain Codes

Benjamin Gunby  Department of Mathematics, Rutgers University, Piscataway, NJ
08854, USA<EMAIL_ADDRESS>,  Xiaoyu He  Department of Mathematics,
Princeton University, Princeton, NJ 08540, USA<EMAIL_ADDRESS>,
Bhargav Narayanan  Department of Mathematics, Rutgers University, Piscataway,
NJ 08854, USA<EMAIL_ADDRESS>and  Sam Spiro  Department of
Mathematics, Rutgers University, Piscataway, NJ 08854, USA
<EMAIL_ADDRESS>
###### Abstract.

A family of sets $A$ is said to be an antichain if $x\not\subset y$ for all
distinct $x,y\in A$, and it is said to be a distance-$r$ code if every pair of
distinct elements of $A$ has Hamming distance at least $r$. Here, we prove
that if $A\subset 2^{[n]}$ is both an antichain and a distance-$(2r+1)$ code,
then $|A|=O_{r}(2^{n}n^{-r-1/2})$. This result, which is best-possible up to
the implied constant, is a purely combinatorial strengthening of a number of
results in Littlewood–Offord theory; for example, our result gives a short
combinatorial proof of Hálasz’s theorem, while all previously known proofs of
this result are Fourier-analytic.

##  1\. Introduction

In this paper, motivated by considerations from Littlewood–Offord theory, we
study the intersection of two classical combinatorial problems in the
hypercube, namely that of finding large antichains and that of finding large
distance-$r$ codes.

A family of sets $A\subset 2^{[n]}$ is an _antichain_ if $x\not\subset y$ for
any distinct $x,y\in A$. For example, the layer

$\binom{[n]}{k}=\\{x\subset[n]:|x|=k\\}$

is an antichain for all $0\leq k\leq n$, and it is a classical result of
Sperner [11] that every antichain in the hypercube $2^{[n]}$ has size at most
$\binom{n}{\lfloor n/2\rfloor}$. There are a huge number of strengthenings and
variants of Sperner’s theorem; we refer the reader to [2] for more background.

A family of vectors $B\subset\\{0,1\\}^{n}$ is called a distance-$r$ code if
the Hamming distance between any pair of vectors in $B$ is at least $r$;
identifying $\\{0,1\\}^{n}$ and $2^{[n]}$ in the natural way, we call a family
$A\subset 2^{[n]}$ a distance-$r$ code if the symmetric difference
$x\,\triangle\,y$ of any two distinct $x,y\in A$ always has size at least $r$.
One of the central problems of coding theory is to find large distance-$r$
codes with various desirable properties, and the existence of such codes has
many applications in both pure and applied problems. We refer the reader to
[6] for more on coding theory, and mention only the basic fact (as evidenced
by BCH codes) that the largest possible distance-$r$ codes in $2^{[n]}$ have
size $\Theta(2^{n}n^{-\lfloor(r-1)/2\rfloor})$.

Here, we aim to answer the following natural question: how large can the
cardinality of an _antichain code_ in $2^{[n]}$ be? After some thought, one
finds that it is difficult to do much better than taking the intersection of a
large code and a large antichain; our main result shows that such
constructions are indeed optimal.

######  Theorem 1.1.

For any fixed $r\in\mathbb{N}$, if $A\subset 2^{[n]}$ is both an antichain and
a distance-$(2r+1)$ code, then

$|A|=O\left(2^{n}n^{-r-1/2}\right).$

This result is best-possible up to multiplicative constants. Indeed, as
mentioned above, we know that there exists a distance-$(2r+1)$ code $A\subset
2^{[n]}$ with $|A|=\Theta(2^{n}n^{-r})$. By simple averaging, it is easy to
show that there exists some $x\subset[n]$ for which
$A\,\triangle\,x=\\{a\,\triangle\,x:a\in A\\}$, which is also a
distance-$(2r+1)$ code, intersects the middle layer $\binom{[n]}{\lfloor
n/2\rfloor}$ in at least $\Theta(n^{-r}\binom{n}{\lfloor
n/2\rfloor})=\Theta(2^{n}n^{-r-1/2})$ sets; then
$(A\,\triangle\,x)\cap\binom{[n]}{\lfloor n/2\rfloor}$ gives us an antichain
code of the desired size.

As mentioned earlier, the primary motivation for Theorem 1.1 comes from the
Littlewood–Offord theory of anticoncentration. In particular, Theorem 1.1 may
be viewed as a purely combinatorial abstraction of an important result of
Halász [7] that is widely used in the study of random matrices; see [5, 8] and
the many references therein. To explain this connection, we need to fill in
some background, a task to which we now turn.

Recall that the Littlewood–Offord problem asks the following: given a vector
$\mathbf{a}=(a_{1},\ldots,a_{n})$ of non-zero real numbers, estimate

$\rho(\mathbf{a})=\max_{\alpha\in\mathbb{R}}\mathbb{P}[\epsilon_{1}a_{1}+\cdots+\epsilon_{n}a_{n}=\alpha],$

where the $\epsilon_{i}$’s are independent Bernoulli random variables with
$\mathbb{P}[\epsilon_{i}=0]=\mathbb{P}[\epsilon_{i}=1]$. In their study of
random polynomials, Littlewood and Offord [9] showed that
$\rho(\mathbf{a})=O(n^{-1/2}\log n)$ for any such $\mathbf{a}$, and soon
after, Erdős [4] used Sperner’s theorem to give a simple combinatorial proof
of the sharp estimate $\rho(\mathbf{a})\leq 2^{-n}\binom{n}{n/2}=O(n^{-1/2})$.

There has since been considerable interest in establishing better bounds on
$\rho(\mathbf{a})$ under stronger assumptions on the arithmetic structure of
$\mathbf{a}$. For example, Erdős and Moser [3] proved that
$\rho(\mathbf{a})=O(n^{-3/2}\log n)$ whenever all of the entries of
$\mathbf{a}$ are distinct. Sárközy and Szemerédi [10] improved this to
$\rho(\mathbf{a})=O(n^{-3/2})$, which is asymptotically best possible, and
subsequently, Stanley [12] used algebraic arguments to establish sharp bounds
for this problem. Of particular interest to us is a far-reaching
generalisation of the Sárközy–Szemerédi theorem due to Halász [7], one
formulation of which is as follows.

######  Theorem 1.2.

Let $\mathbf{a}=(a_{1},\ldots,a_{n})$ be a vector of real numbers such that
for any disjoint subsets $x,y\subset[n]$ with $|x|+|y|\leq 2r$, we have

$\sum_{i\in x}a_{i}\neq\sum_{j\in y}a_{j}.$

Then

$\rho(\mathbf{a})=O_{r}\left(n^{-r-1/2}\right).$

Note in particular that the $r=1$ case is equivalent to saying $a_{i}\neq
a_{j}$ for any $i\neq j$, so the result in this case reduces to the
Sárközy–Szemerédi theorem on the Erdős–Moser problem.

Halász’s theorem has since become a widely used tool in the study of random
matrices and random polynomials. All known proofs of Halász’s theorem use
Fourier analysis, and are very much arithmetic in nature. While searching for
an analogue of Halász’s theorem for some other anticoncentration problems (in
discussions with Berkowitz), it became clear that it would be of some help to
find a purely combinatorial proof of this result, in the spirit of Erdős’
classical arguments. Arguably, the primary motivation for Theorem 1.1 is that
Halász’s theorem is an easy corollary.

###### Proof that Thereom 1.1 implies Theorem 1.2..

First, we we may assume without loss of generality that $a_{i}>0$ for all $i$;
this follows from noting that $\rho(\mathbf{a})$ is unaltered if we replace
the $0/1$-valued Bernoulli random variables in its definition with
$-1/1$-valued Rademacher random variables.

Next, fix any real number $\alpha$ and let $A$ denote the family of subsets
$x\subset[n]$ such that $\sum_{i\in x}a_{i}=\alpha$. Note that $A$ is an
antichain, as having $x\subset y$ both in $A$ would imply

$\alpha=\sum_{i\in x}a_{i}<\sum_{i\in y}a_{i}=\alpha,$

with the second inequality using $a_{i}>0$ for all $i$. We also claim that $A$
is a distance-$(2r+1)$ code. Indeed, for any distinct $x,y\in A$ we must have

$\sum_{i\in x\setminus y}a_{i}=\sum_{j\in y\setminus x}a_{j};$

this implies, by the hypothesis on $\mathbf{a}$, that
$|x\,\triangle\,y|=|x\setminus y|+|y\setminus x|>2r$, so $A$ is a
distance-$(2r+1)$ code. Thus, the bound of Theorem 1.1 applies to $A$, and
this is equivalent to saying that the probability that
$\epsilon_{1}a_{1}+\cdots+\epsilon_{n}a_{n}=\alpha$ is $O_{r}(n^{-r-1/2})$ for
any $\alpha$; this gives the desired bound on $\rho(\mathbf{a})$. ∎

##  2\. Proof of the main result

We start with some brief comments on notation. We adopt the convention that
lower case letters $a,b,c,x,y,z$ represent subsets of $[n]$, and capital
letters $A,S$ represent families of sets, i.e., subsets of $2^{[n]}$. If
$S\subset 2^{[n]}$ and $r\geq 1$ is an integer, we write $\partial^{r}S$ for
the $r$-fold shadow of $S$, i.e., the collection of sets which can be obtained
by deleting $r$ elements of $[n]$ from some set in $S$. Similarly, we write
$\partial^{-r}S$ for the collection of sets which can be obtained by adding
$r$ elements to some set in $S$. For singletons we abuse notation by writing
$\partial^{r}x$ for $\partial^{r}\\{x\\}$ and similarly $\partial^{-r}x$ for
$\partial^{-r}\\{x\\}$. Finally, we write $x\,\triangle\,y$ for the symmetric
difference of $x$ and $y$. Throughout, we omit floors and ceilings when they
are not crucial.

Before we state and prove the main lemma that drives the proof of Theorem 1.1,
we recall one proof of Sperner’s theorem that serves as our inspiration. The
local-LYM inequality (see [1]) asserts that for any $S\subset\binom{[n]}{k}$,
we have

$|\partial S|\binom{n}{k-1}^{-1}\geq|S|\binom{n}{k}^{-1}.$

It is not hard to show using local-LYM that any antichain $A\subset 2^{[n]}$
may be ‘shifted’, by means of taking shadows, into the middle layer without
decreasing the size of the resulting family, whence we conclude that
$|A|\leq\binom{n}{\lfloor n/2\rfloor}$.

A natural approach to proving Theorem 1.1, say for antichain codes of distance
3 to be concrete, is to proceed along similar lines as above, except using the
distance condition instead of the local-LYM inequality to generate more ‘local
expansion’. Concretely, given $A\subset 2^{[n]}$ that is both an antichain and
a distance-3 code, it is easy to see for all $k$ that $|\partial A_{k}|\geq
k|A_{k}|$, where $A_{k}=A\cap\binom{[n]}{k}$; in particular, for $k\approx
n/2$, this tells us that $|\partial A_{k}|\gtrsim n|A_{k}|/2$, which is a
significant improvement over the rather modest bound $|\partial
A_{k}|\gtrsim|A_{k}|$ promised by local-LYM. It is then natural to attempt to
transform a given antichain code $A$ of distance 3, by means of taking
shadows, into a family that lives in the middle layer that is about $n$ times
larger, which would then show that $|A|\lesssim\binom{n}{\lfloor
n/2\rfloor}/n=O(2^{n}n^{-3/2})$, as desired.

To implement such an idea, we need to deal with how the shadows of the
different $A_{k}$’s overlap as we repeatedly take shadows to move $A$ into the
middle layer. For example, an estimate of the following form would be ideal:
for any $S\subset\binom{[n]}{k}$ disjoint from $A_{k}$ (i.e., thinking of $S$
as the shadow of all the $A_{\ell}$’s with $\ell>k$ in layer $k$), we have
$|\partial(S\cup A_{k})|\geq|S|+k|A_{k}|/100$. Unfortunately, this is too much
to hope for: if there are no conditions on the arbitrary set $S$, then it can
be arranged so that $\partial S$ contains the entirety of $\partial A_{k}$.
Nevertheless, the following lemma shows that something like this ideal
estimate does in fact hold when one studies the expansion of antichain codes
over (slightly) longer ranges.

######  Lemma 2.1.

Let $n,r\geq 1$ be integers with $n\geq 8r$. If $\frac{n}{2}+3r\leq
k\leq\frac{3n}{4}$, $S\subset\binom{[n]}{k}$, and
$A\subset\binom{[n]}{k-r}\backslash\partial^{r}S$ is a distance-$(2r+1)$ code,
then $|\partial^{3r}S\cup\partial^{2r}A|\geq|S|+\frac{n^{r}|A|}{4(2r)^{3r}}$.

###### Proof.

Suppose for the sake of contradiction that
$|\partial^{3r}S\cup\partial^{2r}A|<|S|+\frac{n^{r}|A|}{4(2r)^{3r}}$. Observe
that because $k\geq\frac{n}{2}+3r$, local-LYM tells us that $|\partial
B|\geq|B|$ for any family inside one of the layers
$\binom{[n]}{k},\ldots,\binom{[n]}{k-3r+1}$. Pick a uniformly random $a\in A$.
Let $b$ and $b^{\prime}$ denote a uniformly random pair of disjoint
$r$-subsets of $a$, and $c$ a uniformly random $r$-subset of $[n]\setminus a$.
Note that since $A$ is a distance-$(2r+1)$ code, $a\,\triangle\,b$ is a
uniform random element of $\partial^{r}A$, and similarly $a\,\triangle\,c$ is
a uniform random element of $\partial^{-r}A$. Also observe that $a$ is
uniquely determined by specifying either $a\,\triangle\,b$ or
$a\,\triangle\,c$.

######  Claim 2.2.

The random element $a\,\triangle\,b$ of $\partial^{r}A$ satisfies
$\mathbb{P}[a\,\triangle\,b\notin\partial^{2r}S]<\frac{1}{4}.$

###### Proof.

By assumption,

$|S|+\frac{n^{r}|A|}{4(2r)^{3r}}>|\partial^{3r}S\cup\partial^{2r}A|\geq|\partial^{2r}S\cup\partial^{r}A|=|\partial^{2r}S|+|\partial^{r}A\backslash\partial^{2r}S|\geq|S|+|\partial^{r}A\backslash\partial^{2r}S|.$

Since $A$ is a distance-$(2r+2)$ code in $\binom{[n]}{k-r}$ and $k-r\geq n/2$,
we have

$|\partial^{r}A|={k-r\choose
r}|A|\geq\left(\frac{k-r}{r}\right)^{r}|A|\geq\frac{n^{r}}{(2r)^{r}}|A|\geq\frac{n^{r}}{(2r)^{3r}}|A|.$

Combining this with the inequality above gives

$|\partial^{r}A\backslash\partial^{2r}S|<\frac{n^{r}|A|}{4(2r)^{3r}}\leq\frac{1}{4}|\partial^{r}A|.$

This proves the claim since $a\,\triangle\,b$ is a uniform random element of
$\partial^{r}A$. ∎

######  Claim 2.3.

The random element $a\,\triangle\,(b\cup b^{\prime}\cup c)$ of
$\partial^{2r}\partial^{-r}A$ satisfies $\mathbb{P}[a\,\triangle\,(b\cup
b^{\prime}\cup c)\in\partial^{2r}S]<\frac{1}{8}$.

###### Proof.

Let $P$ be the set of pairs $(x,y)\in\partial^{-r}A\times\partial^{2r}S$ for
which $y\in\partial^{2r}x$. There are a total of
$\binom{n-k+2r}{2r}|\partial^{2r}S|$ ways of picking an element
$y\in\partial^{2r}S$ and an element $x\in\partial^{-2r}y$, and of these ways
at least $\binom{k}{2r}|S|$ satisfy $x\in S$. Since $A$ is disjoint from
$\partial^{r}S$, we have that $\partial^{-r}A$ is disjoint from $S$, so

$|P|\leq\binom{n-k+2r}{2r}|\partial^{2r}S|-\binom{k}{2r}|S|\leq\binom{n/2}{2r}\cdot(|\partial^{2r}S|-|S|)<\frac{n^{3r}|A|}{4\cdot
32^{r}r^{3r}(2r)!},$

where the second inequality used $k\geq\frac{n}{2}+2r$ and $n/2\geq 4r$, and
the last inequality used $\binom{n/2}{2r}\leq(n/2)^{2r}/(2r)!$ and
$|\partial^{2r}S|\leq|\partial^{3r}S\cup\partial^{2r}A|<|S|+\frac{n^{r}|A|}{4(2r)^{3r}}$.
Also, observe that $a\,\triangle\,c$ is a uniform random element of
$\partial^{-r}A$, so $a\,\triangle\,(b\cup b^{\prime}\cup c)\in\partial^{2r}S$
if and only if $(a\,\triangle\,c,a\,\triangle\,(b\cup b^{\prime}\cup c))\in
P$. Note that $b,b^{\prime},c$ are chosen uniformly at random out of

$\binom{k-r}{r}\cdot\binom{k-2r}{r}\cdot\binom{n-k+r}{r}\geq\left(\frac{k-r}{r}\right)^{r}\cdot\left(\frac{k-2r}{r}\right)^{r}\cdot\left(\frac{n-k+r}{r}\right)^{r}\geq\frac{n^{3r}}{16^{r}r^{3r}}$
(1)

possibilities, with the last step using $k-2r\geq n/2$ and $k\leq 3n/4$. Since
at most $(2r)!$ tuples $(a,b,b^{\prime},c)$ correspond to the same pair
$(a\,\triangle\,c,a\,\triangle\,(b\cup b^{\prime}\cup c))$ (as
$a\,\triangle\,c$ determines $a$ and $c$, the only noninjectivity comes from
swapping elements between $b,b^{\prime}$), we obtain

$\mathbb{P}[a\,\triangle\,(b\cup b^{\prime}\cup
c)\in\partial^{2r}S]\leq\frac{(2r)!|P|}{|A|\cdot
n^{3r}/16^{r}r^{3r}}<\frac{1}{4\cdot 2^{r}}\leq\frac{1}{8}.\qed$

We use these two claims to show that $\partial^{2r}S$ has large distance-$2r$
‘edge expansion’ in $\binom{[n]}{k-2r}$.

######  Claim 2.4.

There are at least

$\frac{n^{3r}|A|}{2\cdot 16^{r}r^{3r}}$

pairs $(x,y)\in\binom{[n]}{k-2r}^{2}$ at distance $2r$ with
$x\in\partial^{2r}S$ and $y\not\in\partial^{2r}S$.

###### Proof.

By the two previous claims,
$\mathbb{P}[a\,\triangle\,b\notin\partial^{2r}S]<\frac{1}{4}$ and
$\mathbb{P}[a\,\triangle\,(b\cup b^{\prime}\cup
c)\in\partial^{2r}S]<\frac{1}{8}$. Thus, if we generate a random pair
$(x,y)=(a\,\triangle\,b,a\,\triangle\,(b\cup b^{\prime}\cup c))$, then $x,y$
automatically have distance $2r$, and with probability at least
$1-\frac{1}{4}-\frac{1}{8}\geq\frac{1}{2}$ they satisfy $x\in\partial^{2r}S$
and $y\not\in\partial^{2r}S$. The total number of potential pairs $(x,y)$ as
above is

$|\partial^{r}A|\cdot\binom{k-2r}{r}\binom{n-k-r}{r}={k-r\choose
d}|A|\cdot\binom{k-2r}{r}\binom{n-k-r}{r}\geq\frac{n^{3r}}{16^{r}r^{3r}}|A|,$

with this last inequality using (1); we get the desired result by multiplying
by $1/2$. ∎

We finish by enumerating in two ways the set $Q$ of pairs
$(y,z)\in\binom{[n]}{k-2r}\times\binom{[n]}{k-3r}$ where $z\in\partial^{3r}S$,
$z\in\partial^{r}y$, and $y\not\in\partial^{2r}S$. On the one hand, any pair
$(x,y)$ from the previous claim corresponds to such a pair $(y,z)\in Q$ by
taking $z=x\cap y$, and this correspondence is at most $\binom{k-2r}{r}$-to-
one. By the previous claim,

$|Q|\geq\frac{n^{3r}|A|}{2\cdot
16^{r}r^{3r}\cdot\binom{k-2r}{r}}\geq\binom{n}{r}^{-1}\frac{n^{3r}|A|}{2\cdot
16^{r}r^{3r}}.$

On the other hand, the number of ways to pick $z$ is $|\partial^{3r}S|$, the
number of ways to pick $y\in\partial^{-r}z$ is $\binom{n-k+3r}{r}$, and of
these pairs at least $\binom{k-2r}{r}|\partial^{2r}S|$ satisfy
$y\in\partial^{2r}S$. Thus,

$\displaystyle|Q|$
$\displaystyle\leq\binom{n-k+3r}{r}|\partial^{3r}S|-\binom{k-2r}{r}|\partial^{2r}S|<\binom{n/2}{r}(|\partial^{3r}S|-|\partial^{2r}S|)$
$\displaystyle\leq\binom{n/2}{r}(|\partial^{3r}S\cup\partial^{2r}A|-|S|)\leq\binom{n/2}{r}\frac{n^{r}|A|}{4(2r)^{3r}}.$

To get a contradiction, it suffices to show

$\binom{n/2}{r}\binom{n}{r}<2\cdot 2^{-r}n^{2r},$

which follows from $\binom{n/2}{r}\leq(n/2)^{r}$ and $\binom{n}{r}\leq n^{r}$.
∎

Our main result follows quickly from Lemma 2.1.

###### Proof of Theorem 1.1.

Recall that our goal is to prove that if $A\subset 2^{[n]}$ is both an
antichain and a distance-$(2r+1)$ code for some $r\geq 1$, then
$|A|=O_{r}\left(2^{n}n^{-r-1/2}\right)$.

Let $A_{k}=A\cap\binom{[n]}{k}$ and let $S_{k}\subset\binom{[n]}{k}$ consist
of the sets $x\subset[n]$ which are contained in some element of $A$. Since
$A$ is an antichain, we have
$A_{k+2r}\subset\binom{[n]}{k+2r}\setminus\partial^{r}S_{k+3r}$. Since
$S_{k}\subset\partial^{3r}S_{k+3r}\cup\partial^{2r}A_{k+2r}$, by Lemma 2.1 we
find for all $n/2\leq k\leq 3n/4-3r$ that

$|S_{k}|\geq|S_{k+3r}|+\frac{n^{r}|A_{k+2r}|}{4(2r)^{3r}}.$

By applying this result inductively, we find for all $k\geq n/2$ that

$|S_{k}|\geq\frac{n^{r}}{4(2r)^{3r}}\sum_{\ell\in L_{k}}|A_{\ell}|,$ (2)

where $L_{k}$ is the set of those $k\leq\ell\leq 3n/4-3r$ such that
$\ell\equiv k+2r\mod 3r$.

By decreasing the size of $A$ by at most a factor of 2, we may assume
$A_{\ell}=\emptyset$ for all $k<n/2$. Similarly by decreasing $A$ by a factor
of at most $3r$, we may assume there exists some $i$ such that
$A_{\ell}=\emptyset$ for all $\ell\not\equiv i\mod 3r$. By standard estimates
for the binomial coefficients, we have that $\sum_{\ell\geq
3n/4-3r}{n\choose\ell}\leq 1.9^{n}$ for $n$ sufficiently large. Thus if $k\geq
n/2$ is the smallest integer with $k\equiv i-2r\mod 3r$, these assumptions and
(2) imply that for $n$ sufficiently large, we have

$\frac{n^{r}}{4(2r)^{3r}}\left(|A|-1.9^{n}\right)\leq|S_{k}|\leq{n\choose
k}=O(2^{n}n^{-1/2}).$

Rearranging this gives us the desired bound on $|A|$. ∎

## Acknowledgements

The second author was supported by the NSF Mathematical Sciences Postdoctoral
Research Fellowships Program under Grant DMS-2103154, the third author was
supported by NSF grants CCF-1814409 and DMS-1800521, and the fourth author was
supported by the NSF Mathematical Sciences Postdoctoral Research Fellowships
Program under Grant DMS-2202730. We are grateful to Noga Alon and Ross
Berkowitz for stimulating conversations.

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# Silver-Platinum nanoparticles and nanodroplets supported on silica surfaces:
structure and chemical ordering

F. Ait Hellal  C. Andreazza-Vignolle  P. Andreazza  J. Puibasset

###### Abstract

Stable and metastable metallic nanoparticles exhibit unique properties
compared to the bulk, with potentially important applications for catalysis.
This is in particular the case for the AgPt alloy that can exhibit the ordered
L11 structure (alternation of pure Ag and Pt (111) planes) in nanometer size
particles. However, for such small systems, the interfaces play an important
role. Therefore, the support used to elaborate the nanoparticles in ultrahigh
vacuum experiments may influence their properties, even in the case of weakly
interacting substrates like amorphous carbon or silica. This work focuses on
the AgPt nanoparticles deposited on silica, and investigates the effect of the
support disorder and roughness on the structure and chemical ordering, in
particular at the interface with the substrate, by Monte Carlo calculations of
the atomic density profiles with semi-empiric potentials.

###### keywords:

metallic nanoparticle , AgPt nanoalloy , chemical ordering , density profile ,
Monte Carlo simulation

††journal: Journal of Alloys and Compounds

organization=ICMN, CNRS, Université d’Orléans,addressline=1b rue de la
Férollerie, CS 40059, city=Orléans, postcode=45071 cedex 02, country=France

##  1 Introduction

Supported metallic nanoparticles (NPs) are catalyst models for structure and
chemical ordering studies [1, 2, 3, 4, 5]. Choosing an amorphous substrate
like amorphous carbon or silicium oxide, as silica, has the advantage of
minimizing the interactions with the NP, hence preserving the structure and
morphology it would adopt in vacuum in the absence of interactions. This
strategy is used in experiments where the NPs are grown on amorphous supports
in ultrahigh vacuum conditions [6, 7, 8]. It has however been observed that
this type of support may still influence the NP structure and morphology [9,
8]. Although the expected effects are less spectacular than for crystalline
supports like MgO which strongly interact with NPs [10, 11, 12, 13, 14, 15,
16], theoretical works have recently focused on the effect of weak surfaces
[17, 18, 19].

Among the possible effects of the substrate, chemical ordering is particularly
relevant for catalysis, since NPs offers the unique opportunity to possibly
drastically minimize the required amount of active matter by an optimal
organization of the chemical species at the NP external surface.

The silver-platinum alloy exhibits interesting features [20, 21, 22, 23], in
particular an ordered L11 structure (alternation of pure Ag and Pt (111)
planes) that has been observed in nanometer size particles [24, 25], and
stimulated theoretical studies [26, 27, 28]. It has also important
applications as catalyst in fuel cells, where chemical ordering strongly
influences its catalytic efficiency [29, 30, 31, 32]. Therefore, assessing the
structure of the supported AgPt nanoalloy is a relevant issue.

The elaboration process strongly influences the structure of the NP which is
not necessarily at equilibrium (the system remains trapped in local minima,
corresponding to metastable states) [25, 33]. In simulations, the situation is
even worse since the limited capabilities of computers impede to reach the
experimental timescale, and thus strongly limits the possibilities of atomic
reorganization. To circumvent the problem, it is possible to increase the
metal mobility by increasing the temperature [27]. This is why, besides the
solid NP, we will also consider the corresponding liquid droplet at 1200 K and
1500 K. Beyond the fact that this forces atomic mobility towards equilibrium,
the disappearance of the L11 chemical order in the core due to the thermal
agitation should leave room for the observation of a possible remnant chemical
ordering at the interfaces, in particular that with the silica support. It is
emphasized that, in contrast to simulations, increasing the temperature of Ag-
based nanoparticles like AgPt up to the liquid phase is experimentally
difficult both in ultrahigh vacuum conditions, due to sublimation before
melting, and at ambient pressure, due to the contamination by the atmosphere.

This paper is divided as follows: We first present the numerical model and the
methods to characterize the structure and chemical ordering in terms of atomic
density profiles. Then the results follow, with a focus on the effect of the
temperature on the structure and chemical ordering, as well as the influence
of the disorder and roughness of the support.

##  2 Numerical Details and Methods

Molecular model: We perform Monte Carlo (MC) simulation of supported AgPt
nanoparticles on various silica substrates that mimic the supports used to
elaborate the systems in ultrahigh vacuum experiments. The system is described
at the atomic level, with semi-empirical interatomic potentials. The
simulations are performed in the canonical ensemble, including random
displacements and atomic exchanges between the metallic species Ag and Pt.
These exchanges are particularly useful at low temperature where the energy
barrier associated to atomic diffusion in the core of the NP is too high to
allow chemical reorganization.

Interatomic potentials: The many-body metal-metal interaction derives from the
tight binding scheme in the second moment approximation (TBSMA) [34]. The
ordered AgPt L11 structure being stabilized by the contribution of the second
neighbors, an additional Gaussian term has been developed by Front and Mottet
to reproduce the main structural properties of AgPt alloys [26, 27]. They have
in particular determined the most stable structures of TOh AgPt NPs smaller
than 7 nm. In this study we will focus on the NP with 1289 atoms (3.4 nm)
which exhibits a rich structure depicted in Fig. 1.

Figure 1:  Atomic configuration of the most stable TOh AgPt NP comprising 1289
atoms. Grey: silver; Magenta: platinum. Left: front view. Right: sectional
view.

The metal-support interaction being weak (van der Waals like), a simple
Lennard-Jones potential has been used, that has been previously developed for
pure silver and platinum NPs as well as AgPt nanoalloys [35]. This potential
has been parameterized to reproduce the aspect ratio (defined as H/D where H
is the height and D the diameter) of experimentally deposited NPs (the
parameters are given in Table 1).

Table 1: Metal-silica support interaction, described by the Lennard Jones potential $V(r)=4\epsilon_{\rm M-S}\left[\left(\sigma_{\rm M-S}/r\right)^{12}-\left(\sigma_{\rm M-S}/r\right)^{6}\right]$ where M is the metal (Ag or Pt) and S is one of the silica species (Si or O) [35]. metal |  $\epsilon_{\rm M-Si}$ (meV) |  $\sigma_{\rm M-Si}$ (nm) |  $\epsilon_{\rm M-O}$ (meV) |  $\sigma_{\rm M-O}$ (nm)
---|---|---|---|---
Ag | 13 | 0.329 | 80 | 0.278
Pt | 18 | 0.324 | 110 | 0.273

Supports: We consider two silica supports to evidence the effect of the atomic
disorder and roughness that can be observed in experimental supports like
oxidized silicon wafers. As a perfectly flat and ordered surface we use the
(100) quartz surface. It is emphasized that this surface undergoes a $1\times
2$ reconstruction with a top layer twice as dense as the bulk quartz (see Fig.
2a) [36]. To model a disordered substrate we simply cut and relax a slab of
amorphous silica (a-SiO2) (see Fig. 2b). More details in the method and the
potentials used can be found in Ngandjong et al. [18, 19]. It is mentioned
that, despite the fact that the amorphous silica surface is hydroxylated, the
hydrogen species are not explicitly taken into account in the interactions.

Figure 2:  Side (left) and top (right) views of the two silica substrates. a)
The (100) quartz surface (with the $1\times 2$ reconstruction visible on the
top view). b) The amorphous SiO2 substrate (the surface hydroxyls are visible
on the top view). The lateral dimension is 6.8 nm.

Chemical ordering and density profiles: The objective is to measure the effect
of the substrate on the chemical ordering in the nanoalloy. This is done
simply by comparing layer by layer with the equilibrium structure of the free
NP (Fig. 1) since the geometric structure is marginally affected by the weak
interaction with the substrate. However, the time scale involved in
experiments being inaccessible to simulations, the intrinsic mobility of Ag
(due to its lower cohesion) is largely underestimated in the calculations. The
introduction of MC exchanges between Ag and Pt greatly solves the problem and
allows chemical rearrangement, but possibly misses some facets of the complex
cross-diffusion of Ag and Pt in the NP. We therefore consider the effect of
temperature, ranging from the solid up to the liquid state (above
approximately 1200K for Ag3Pt) [27]. In this case, the NP structure is fully
disordered (droplet) but may exhibit partial chemical ordering, in particular
close to the interfaces (with vacuum and substrate).

In the liquid case, the structure of the nanodroplet is characterized by
averaging the atomic density profiles of the metal. Since our objective is to
determine the aspect ratio ($H/D$) we focus on two density profiles, along the
$z$ axis (perpendicular to the surface) and along the radial coordinate
$r_{cyl}$ (in the plane parallel to the substrate, see Fig. 3). So, from the
local atomic density $\rho(r_{cyl},\theta,z)$ we construct the two quantities:

$\displaystyle\xi_{r}(z)=\int\rho(r_{cyl},\theta,z)r_{cyl}dr_{cyl}d\theta$
$\displaystyle\xi_{z}(r_{cyl})=\int\rho(r_{cyl},\theta,z)r_{cyl}dzd\theta.$

These density profiles have the dimension of the inverse of a distance, and
their integrals give the total number of atoms in the NP.

Figure 3:  Snapshot of a AgPt droplet on the amorphous silica at $T$ = 1500 K.
The aspect ratio is defined as $H/D$ where $H$ is the height of the droplet
above the substrate and $D$ its diameter. b) Simple model of a supported and
truncated liquid drop with uniform density in the core (<$R_{1}$) and a thin
skin with smoothly varying density between $R_{1}$ and $R_{2}$. $r_{cyl}$,
$\theta$ and $z$ are the cylindrical coordinates with the origin at the center
of the droplet; $r_{sph}$ is the spherical radial distance from the origin.

In order to determine the aspect ratio from the data, these profiles are
fitted on a simple model of a truncated sphere of uniform density $\rho_{0}$
of radius $R_{1}$ with a skin of thickness $R_{2}-R_{1}$ where the density
decreases from $\rho_{0}$ down to zero. We define the droplet radius as
$R=0.5(R_{1}+R_{2})$ and the aspect ratio is $H/D=(h+R)/2R$ (see Fig. 3). In
practice, $\xi_{z}(r_{cyl})$ mostly depends on the droplet radius, while
$\xi_{r}(z)$ is sensitive to $h$.

##  3 Results

###  3.1 Structure and chemical ordering of solid AgPt NPs on quartz

At low temperature (around room $T$ or below), the supported AgPt NP remains
essentially frozen in its initial state, preserving its highly structured
layers and chemical ordering (in particular the L11 structure) with only
scarce exchanges between Ag and Pt species. Increasing the temperature
enhances the atomic mobility and somehow mimics the experimental conditions
where moderate annealing allows the AgPt NPs to reorganize thanks to the large
mobility of Ag atoms. The optimal temperature is around 700 K, the largest
possible below the melting point of Ag for a NP of few nanometers [37]. We
first consider the AgPt NP deposited on the perfectly ordered quartz surface.
The atomic density profiles along the $z$ axis are acquired for Ag and Pt and
shown in Fig. 4.

Figure 4:  Density profiles $\xi_{r}(z)$ for each metal species in AgPt on
quartz at $T$ = 700 K. Inset: zoom of the first and last peaks.

One observes a strong layering through the whole NP showing that at 700 K the
NP remains solid during the simulation run. One can however observe some
atomic mobility at the external surface of the NP, as revealed by the small
peak C in Fig. 4 corresponding to adatoms on the top layer. An example where
such adatoms can be seen is given in Fig. 5.

Figure 5:  Snapshot of the AgPt NP on quartz at $T$ = 700 K showing adatoms on
the top layer and close to the substrate (arrows).

Despite the low atomic mobility, the MC chemical exchanges between Ag and Pt
species allow the system to reorganize. It is observed that the chemical
ordering associated to the alternating Ag and Pt planes in the core of the NP
is lost, showing that the L11 structure is destabilized by the temperature.

On the other hand, the outermost silver layer is quite robust, as can be seen
on the snapshot (Fig. 5) and the presence of essentially pure Ag peaks in the
first and last layers denoted A and B in Fig. 4. Note however that these
layers are not perfectly pure Ag anymore, revealing that some Pt atoms can
diffuse in the outer Ag shell. But, as can be seen in the inset, the peaks
associated to Pt in these layers are not centred with respect to the
corresponding Ag peaks. In each case the Pt peak is shifted towards the centre
of the NP, meaning a strong penalty for Pt at the surface. Quantitatively, the
integration of Pt peaks gives their proportion in the A and B layers. One gets
for A 3.7% Pt and for B 5.5% Pt. The free Ag surface is slightly more
favourable for Pt compared to the surface in contact with the support. This is
an interesting feature at odds with the observation that the Pt-SiO2
interaction is stronger than the Ag-SiO2 interaction (Table 1). A possible
interpretation is that the Ag layer at the interface with silica is more
constrained than the free one.

###  3.2 Structure and chemical ordering of AgPt nanodroplets on quartz

What happens above the melting point of AgPt? The structure of the NP is now
expected to be destabilized (and at equilibrium thanks to the atomic
mobility), which could influence the chemical ordering. The double objective
is thus to determine the aspect ratio of the AgPt droplet and the chemical
profiles at the interfaces. The first calculations are done well above the
melting point, at $T$ = 1500 K, for the AgPt NP deposited on the quartz
surface.

We first calculate the density profiles $\xi_{r}(z)$ and $\xi_{z}(r_{cyl})$
for all atoms in the drop without distinguishing Ag and Pt (Fig. 6). As can be
seen on $\xi_{r}(z)$, the layer structure along the $z$ axis is smoothed out
due to the thermal agitation, except close to the perfectly flat and ordered
quartz surface, where one can observe at least three layers. In the upper
region of the droplet the layering has completely disappeared and the density
profile decreases smoothly with a small tail at $z$ = 20 Å.

Figure 6:  Atomic density profiles $\xi_{r}(z)$ (left panel) and
$\xi_{z}(r_{cyl})$ (right panel) for the AgPt droplet comprising 1289 atoms on
the quartz support at $T$ = 1500 K. The Ag and Pt atoms are not distinguished
in this figure. The grayed region in the left panel for $z$<-13 Å represents
the quartz support. The smooth red lines are best fit with the liquid drop
model (Fig. 3b). Inset: density profile used for the liquid drop model.

Along the radial coordinate, $\xi_{z}(r_{cyl})$ shows an initial increase
essentially linear corresponding to the integration of a uniform density on
the surface of a cylinder. It then reaches a maximum and rapidly decreases due
to the spherical shape of the drop, with a tail due to the smooth transition
between the metal core and the surrounding vacuum.

Examination of the density profiles gives a good estimate of the drop height
and diameter, but a best fit with the liquid drop model depicted in Fig. 3b
gives a better insight (smooth solid red lines in Fig. 6). Obviously, the
layering observed in $\xi_{r}(z)$ cannot be described by the uniform density
model, but the average variations are caught. However, this reveals that the
first layers are particularly structured because of the height of the maxima
compared to the drop model curve. This is essentially because of the perfectly
ordered quartz surface. Otherwise, the model describes quantitatively the
variations of $\xi_{r}(z)$ far from the support, as well as the variations of
$\xi_{z}(r_{cyl})$ in the whole range of values. The corresponding density
profile of the liquid drop model is shown in the inset, and the values given
by the best fit are $h$ = 13 Å and $R$ = 19.5 Å, giving the aspect ratio $H/D$
= 0.83. The uniform density is $\rho_{0}$ = 0.049 atom/Å3.

The density profile in the inset shows that the thickness of the skin (5 Å)
somehow corresponds to 1 to 2 atomic diameters and is not negligible compared
to the radius of the droplet. A more refined profile can be acquired directly
during the course of the simulation by calculating a spherical radial
distribution $\rho(r_{sph})$ taking care to exclude the rays in the solid
angle defined by the intersection between the sphere and the substrate. The
result is given in Fig. 7. It confirms that the density within the core of the
droplet is essentially uniform within uncertainties, due to the low statistics
in the centre of the sphere. The average density on the plateau is in
agreement with the value previously extracted from the best fit. It also
confirms that the atomic density profile drops smoothly to zero within a skin
thickness of 5 Å.

Figure 7:  Radial (spherical coordinate $r_{sph}$) atomic density profiles for
the AgPt droplet comprising 1289 atoms on the quartz support at $T$ = 1200 K
and 1500 K. The Ag and Pt atoms are not distinguished in this figure.

The same analysis has been done at a lower temperature $T$ = 1200 K (see Fig.
8 for $\xi_{r}(z)$ and $\xi_{z}(r_{cyl})$ and Fig. 7 for $\rho(r_{sph})$). As
can be seen, reducing the temperature enhances the observed layering at the
quartz surface. Otherwise, the structure is not affected significantly except
for a slightly larger density in the core: $\rho_{0}$ = 0.051 atom/Å3. The
extracted values from the fit are $h$ = 13 Å and $R$ = 19.7 Å, giving an
aspect ratio $H/D$ = 0.83.

Figure 8:  Same as Fig. 6 for $T$ = 1200 K.

In order to quantify the chemical ordering at the interface due to the support
we focus only on the case of the AgPt droplet at 1200 K on quartz: the low
temperature and the strong interaction with the support is expected to enhance
the effect. Figure 9 shows that at midheight (around $z$=0) the Ag and Pt
species have equal probabilities to be at any position. However, close to the
substrate, one observes a strong chemical ordering with an almost pure Ag
layer at the interface with the quartz, the second layer being filled
essentially with Pt, the silver atoms being at the periphery. On the other
side (top of the NP), we also observe chemical ordering (silver excess at the
interface with vacuum). All this is explained by the low surface tension of Ag
which preferentially migrates to the interfaces, while Pt accumulates in
subsurface, in particular at the interface with the support, a behaviour
similar to what was observed in the solid state.

Figure 9:  Density profiles along the $z$ axis $\xi_{r}(z)$ for each metal
species in AgPt on quartz at $T$ = 1200 K. Inset: Ag concentration.

###  3.3 Effect of the support disorder and roughness on the liquid NPs

Does the strong layering observed on the quartz persist in presence of
disorder or roughness? To answer this question, we performed MC simulations of
the AgPt NP on the amorphous SiO2 surface (Fig. 2b) at $T$ = 1500 K. The
density profiles $\xi_{r}(z)$ and $\xi_{z}(r_{cyl})$ exhibit essentially the
same characteristics as for the quartz support, except for two points (see
Fig. 10)

Figure 10:  Same as Fig. 6 where the quartz support is replaced by the
amorphous silica (see Fig. 2b). Inset: density profile used for the liquid
drop model. The temperature is $T$ = 1500 K.

(i) The layering close to the support is significantly smoothed out due to the
atomic disorder of the amorphous silica surface. Note that the roughness of
this surface is quite small, but also participates to the destabilization of
the first layer. The consequence is that the density profile now closely
approaches the smooth curve given by the liquid drop model.

(ii) The best fit with the model gives the following parameters: $h$ = 14.5 Å
and $R$ = 19.5 Å, giving an aspect ratio $H/D$ = 0.87. As can be seen,
compared to quartz, the aspect ratio is slightly closer to 1, in agreement
with the fact that the surface density of the a-SiO2 is lower than that of the
quartz which exhibits a densification due to the reconstruction. Otherwise,
the density in the core of the drop is $\rho_{0}$ = 0.049 atom/Å3, a value
identical to that observed on the quartz.

Reducing the temperature favors layering along the $z$ axis (see Fig. 11). It
is however much less pronounced than on quartz. The main difference is that
here the layering has a uniform amplitude from the first layer in contact with
the substrate up to the center of the NP, while on quartz it was highly
increasing in the vicinity of the surface. This suggests that in the case of
the quartz support, the surface clearly imposes a strong ordering, while, on
the a-SiO2 support, the layering is essentially intrinsic to the metal
structure although it is of course initiated and stabilized by the surface.

Figure 11:  Same as Fig. 10 at $T$ = 1200 K.

The best fit with the drop model gives the following parameters: $h$ = 14.5 Å
and $R$ = 19.5 Å, giving an aspect ratio $H/D$ = 0.87, and $\rho_{0}$ = 0.051
atom/Å3, a value identical to that on the quartz at the same temperature.

##  4 Conclusion

The structure and chemical ordering in AgPt NPs of 3.4 nm (1289 atoms)
deposited on silica supports have been investigated by Monte Carlo
simulations. The introduction of chemical exchanges between the Ag and Pt
species allows to converge towards the chemical ordering at equilibrium even
for NPs trapped in a metastable structure in terms of atomic positions. It is
observed that silver preferentially migrates to the outer surface and at the
interface with the substrate, preserving the same stable structure as for the
free NP. Increasing the temperature to 700 K allows partial atomic mobility
without melting the NP. One observes Ag adatoms on the external surface and
close to the substrate, and a small diffusion of Pt atoms from the subsurface
layer to the surface layer. It is however observed that the Pt atoms at the
periphery always remain slightly embedded in the outermost Ag layer, which is
of importance for catalysis. Since the local structure can be strongly
influenced by the surrounding atmosphere, further studies are necessary to
quantify the catalytic efficiency of this system in real conditions. Systems
exhibiting similar structures or chemical ordering illustrate this point [38,
39, 40, 41].

Higher temperatures have been considered, in the liquid state. The layer
structure is expected to disappear, but the presence of the support is able to
stabilize the first layers. This is particularly visible for the ordered
quartz surface, but less pronounced for the disordered amorphous silica. The
determination of the aspect ratio shows that the morphology of the supported
drop follows the expected behaviour, with a lower aspect ratio on the more
attractive quartz surface. Regarding chemical ordering, the external surface
at the interface with the substrate is mostly composed of Ag species, with
statistically few Pt atoms. Here again, examination of density profiles shows
that the Pt atoms always remain slightly embedded in the outermost Ag layer.

##  5 Declaration of interests

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

##  6 Acknowledgements

F. A. H. acknowledges a grant from the education and research ministry for her
Ph.D. The authors would like to acknowledge support from the International
Research Network - IRN “Nanoalloys” of CNRS.

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