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Further author information: (Send correspondence to H. Xie)
J.Weng: E-mail<EMAIL_ADDRESS>
H. Xie: E-mail<EMAIL_ADDRESS>Telephone: +81-0761-51-1753

# Interactive Layout Drawing Interface with Shadow Guidance

Jiahao Weng  Japan Advanced Institute of Science and Technology, Ishikawa,
Japan  Haoran Xie  Japan Advanced Institute of Science and Technology,
Ishikawa, Japan

###### Abstract

It is difficult to design a visually appealing layout for common users, which
takes time even for professional designers. In this paper, we present an
interactive layout design system with shadow guidance and layout retrieval to
help users obtain satisfactory design results. This study focuses in
particular on the design of academic presentation slides. The user may refer
to the shadow guidance as a heat map, which is the layout distribution of our
gathered data set, using the suggested shadow guidance. The suggested system
is data-driven, allowing users to analyze the design data naturally. The
layout may then be edited by the user to finalize the layout design. We
validated the suggested interface in our user study by comparing it with
common design interfaces. The findings show that the suggested interface may
achieve high retrieval accuracy while simultaneously offering a pleasant user
experience.

###### keywords:

Educational video, slide-based video, user interface, layout design

##  1 INTRODUCTION

Online courses and academic conferences are becoming popular and prevalent
nowadays. In such online circumstances, the teachers and presenters may be
required to create well-designed slides to give a lecture or make a
presentation. The quality of the slides can affect the audience’s
comprehension [1]. However, creating aesthetically pleasing presentations is
time-consuming for inexperienced users. Although conventional software such as
Microsoft PowerPoint is quite sophisticated, it may be more difficult for
novel users to design well-designed slides than professional designers.
Although PowerPoint includes a toolkit to automatically produce layout
recommendations for users to assist them in designing slides, it has certain
limitations including the inability to give a uniform style and make
suggestions when there is too many materials on one single slide.

---
Figure 1:  The proposed interactive layout drawing interface. The left part is
the original drawing canvas, the right part is the retrieval page after user
draw the layout.

In this work, we propose an interactive drawing interface for ordinary users
by retrieving the slide layout, as demonstrated in Figure 1. Because most
academic presentations from online resource sites are in video format, it is
difficult for users to acquire references to produce aesthetically appealing
slides using the keyword-based retrieval method. Our proposed method extracts
slides from the video input using a slide detection technique and determines
the layout of the slides using a state-of-the-art image analysis approach.
Then, when users retrieve slides, we use CNN (Convolutional Neural Network) to
extract the features of the slides. Our system’s interface is divided into
three sections: the canvas section for user editing, the heatmap canvas
section for shadow guidance, and the retrieval result section for displaying
related slides with user editing. The design canvas can make it easy for users
to search and find the reference slides. The heatmap canvas can motivate and
excite unskilled users to consider how they might create the slide layout. The
retrieval result page can provide guidance to users by selecting the desired
layouts.

##  2 RELATED WORK

A drawing interface can use a free-hand drawing that is abstract and lacks
visual details. The sketch can let users express their intention clearly. In
such cases, some researches [2] analyze layout sketches to retrieve the web
page to give user references when designing, and other researches [3, 4]
analyze the sketch stroke to edit the image. Previous studies [5, 6] have also
used user sketches to provide shadow guidance to enhance user design
abilities, which can be used in motion retrieval [7] and calligraphy [8]. The
design interfaces have been proposed for cartoon image generation [9] and
facial images [3]. In this work, we aim to provide an interactive design
interface to assist users in creating the layout of their slides.

VINS [10] adopted the user interface layout as input to retrieve some designs
of mobile interfaces to give user references when designing. Other research
uses layout sketches to retrieve some example web pages that have similar
layout designs [2]. However, all of this research works only for web design.
In addition, the other approaches used the neural network [11] and the
transformer network [12] to generate the layout. In contrast to these
approaches, our work focuses on supporting users to design rather than
generating the layout automatically.

Deep learning based approaches were used extensively to analyze [13, 14, 15]
and design layouts [11, 12] . PubLayNet [14] contributed a huge dataset for
document layout analysis. In this work, we utilize the deep learning method
[13] to train a model for slides layout detection and analysis, and we apply
the neural network [16] to extract the feature of the slides layout.

##  3 SYSTEM Overview

In this work, we present an interactive layout retrieval interface and guide
users in slide design.

###  3.1 System Framework

The proposed system architecture consists of three parts: the offline
computation process for slide extraction (pre-processing); the construction
database considering slide layout features; and a layout drawing interface as
shown in Figure 2. First, we extract slides from academic conference
presentation videos and store them in the database. Then, we use the state-of-
the-art layout analysis tool [17] to analyze and label the layout of each
slide in our database. After that, we use the Convolutional Neural Network
(CNN) to extract the features of the labeled slide and store them in the
database. The proposed interface consists of three sections: the drawing
canvas section, the heat map canvas section, and the result section.

When the user draws a layout on the canvas, the client sends an HTTP request
at the same time. The request is subsequently processed by the backend, which
delivers image data to the client in the HTTP response. The data will then be
rendered by the front-end, and the slide image will be displayed in the result
area.

---
Figure 2:  The framework of our system The left part is the offline
computation process for slide extraction; the middle part is the dataset
collection; and the right part is the user interface.

###  3.2 Slides Extraction

Slides extraction is crucial in this system because it is a fundamental stage
in labeling layout, calculating distribution, and extracting the features of
the slide layout, as illustrated in the left side of Figure 2. We begin by
developing an algorithm to compare the image hash of each frame in a video of
an academic conference presentation. If the difference between two frames is
more than the threshold, the slide has changed and will be extracted.
Following extraction, we computed the distribution of all slides and built a
heat map, which is utilized to provide users with reference and inspiration.
Detectron 2 [13] is then used to train a model to analyze the slides.
Following that, we utilize deep learning based layoutparser (a tool that
includes a collection of layout data structures and constructed APIs intended
for document picture analysis tasks: picking layout components in the document
and displaying the discovered layouts) to automatically label each slide’s
layout and CNN to extract the feature. This tool uses the model developed by
Detectron 2 to extract difficult document structures using only a few lines of
code and cutting-edge deep learning models. After the pre-processing is
finished, all slide features are saved in the database and utilized for
comparison with the user input designs.

---
Figure 3:  The shadow guidance of the proposed drawing interface. Part 1 is
the heat-map of title layout, part 2 is the heat-map of text layout, part 3 is
the heat-map of figure layout, and part 4 is the heat-map of all the layout.

###  3.3 Drawing Interface

The drawing interface can assist users in quickly and easily retrieving some
related slides produced by themselves. It may also assist users in designing
the slide layout and motivate them. When users want to acquire some ideas on
how to create a slide layout, they may look at the heat-map (which indicates
the distribution of each layout by different designers based on its color
shades), which is displayed on the left side of Figure 1. Then the system will
extract the features of the user input design and compare them with all the
features stored in the database. The most similar slides will be rendered on
the web page, as shown in the right part of Figure 1. When users edit their
layouts, the reference result will change simultaneously.

As illustrated in Figure 3, the heat-map depicts the distribution of all slide
layouts in the database. It was developed to provide users with some
suggestions or inspiration for designing the layout. The heatmap was split
into three sections: title, text, and figure. Most design scenarios can
achieve this. The legend is located at the bottom of the heat-map canvas; the
deeper the color, the larger the distribution. By pressing the various
buttons, users can examine the distribution of each portion or the entire
distribution, as illustrated in Figure 1. When users complete or alter the
layout, the distribution of the heat-map changes concurrently to continually
educate users on how to create the layout.

---
Figure 4:  The retrieval results of layout design. The green box is the title
layout, the red box is the figure layout, and the blue box is the text layout.

##  4 User Study

First, we conducted an experiment to compare the conventional retrieval
interfaces with our interface. We invited 12 participants (college students
around 20-years-old; 7 males and 5 females) into 2 groups, and another 5
participants (college students around 20-years-old, 5 males) to do the
evaluation. After a brief introduction of how to use our interface, we let the
first group use the traditional interface, PowerPoint (PPT), and the second
group used our proposed interface. We first gave each group an academic paper
(a poster in the field of computer science), and let each participant design
three slides (only design the layout): introduction, method, and result. After
they finish this task, we ask another 5 participants to evaluate their design
(only evaluating the layout). The participant will assess three factors: the
content in the slides is reasonably organized, the layout is designed
aesthetically, and the slides are designed in a consistent style.

In the experiment about user experience, we conducted a user study to verify
user experiences by asking participants to complete the questionnaire. The
questionnaire uses a 7-Point Likert Scale (1 for strongly disagree and 7 for
strongly agree). We asked the six participants who used our interface in the
first experiment to complete the questionnaire.

##  5 Results

We discuss the implementation details, the result of the layout shadow
guidance, the result of the user study, and user feedback in this section.

###  5.1 Implementation Details

In this work, the proposed interface was implemented in Python 3.8 as a
website application on Windows 11. A personal computer with AMD Ryzen5 5600X
(6-Core), 3.70 GHz, NVIDIA RTX3060 GPU, and 32GB RAM was used as the
developing environment. In addition, we used Django 3.2 to establish the
website and Vue 3.0 to render the website. We use the Detectron2 to train our
model on our dataset (443 slide images). Our prototype required 1.12s on
average for retrieving the slides.

---
Figure 5:  Examples of the designed results by participants during the user
study.

###  5.2 Layout Shadow Guidance

Figure 4 shows the examples of retrieval results. Different input layouts can
retrieve different slides. Even if users didn’t complete the whole layout, the
system will also retrieve the most similar slides for them according to their
design. The upper part of Figure 4 shows that the system still works well with
incomplete design input.

###  5.3 User Study

The questionnaire for comparing experiment uses a 5-Point Likert Scale (1 for
strongly disagreeing and 5 for strongly agreeing). The result is shown in the
Figure 7. Our interface can provide users with satisfactory slides, whilst it
can save their retrieval time. Furthermore, the design of a heat-map has been
shown to inspire users on how to design layouts. Figure 5 shows some results
designed by participants during the user study.

The result of user experience as shown in Figure 7. Although the proposed
interface had difficulty to help users design slides with aesthetic appeal, we
plan to extend our research to have the ability to provide references for the
content of slides and add the well-designed slides to our database in the
future. The user study verified that our interface can help users design more
uniform styles than the conventional interface.

Figure 6:  The results of user study. Figure 7:  The results of questionnaire.
---

##  6 Conclusion

In this work, We proposed an layout drawing interface that can provide users
with references and assist them in creating slide layouts. Using the proposed
heatmap canvas can help and motivate users to make modifications to the
layout. The users can design using the recommended references, which is
thought to inspire the users to create unique layouts. The proposed interface
can help novice users quickly find related resources and learn to construct
the layout. According to the results of user study, the proposed interface can
assist users create the slide in a consistent style and arrange the material
more rationally than the traditional interface. It is also demonstrated that
users prefer to access the reference through interactive modifications and
that the reference might inspire them during the creation of the slide layout.

In the user study, some participants expressed dissatisfaction with the
recovered presentations and claimed that the instruction falls short of their
expectations. To solve this issue, we plan to expand the constructed database
to include additional well-designed and high-quality presentations. We used
443 slides in current prototype implementation. It may be insufficient at
times for users to obtain satisfactory outcomes. We only focused on layout
design in this work, and we would like to broaden the current work to provide
the content creations on the slide layouts.

## Acknowledgements

We thank all participants in our user study. This work was partially supported
by the JAIST Research Grant and the JSPS KAKENHI Grant JP20K19845, Japan.

## References

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# The Counting function for Elkies primes

Meher Elijah Lippmann and Kevin J. McGown  California State University, Chico,
USA

###### Abstract.

Let $E$ be an elliptic curve over a finite field $\mathbb{F}_{q}$ where $q$ is
a prime power. The Schoof–Elkies–Atkin (SEA) algorithm is a standard method
for counting the number of $\mathbb{F}_{q}$-points on $E$. The asymptotic
complexity of the SEA algorithm depends on the distribution of the so-called
Elkies primes.

Assuming GRH, we prove that the least Elkies prime is bounded by $(2\log
4q+4)^{2}$ when $q\geq 10^{9}$. This is the first such explicit bound in the
literature. Previously, Satoh and Galbraith established an upper bound of
$O((\log q)^{2+\varepsilon})$.

Let $N_{E}(X)$ denote the number of Elkies primes less than $X$. Assuming GRH,
we also show

$N_{E}(X)=\frac{\pi(X)}{2}+O\left(\frac{\sqrt{X}(\log qX)^{2}}{\log
X}\right)\,.$

##  1\. Introduction

Let $E$ be an elliptic curve over $\mathbb{F}_{q}$ where $q$ is a prime power.
It is of high interest to compute $\\#E(\mathbb{F}_{q})$, where
$E(\mathbb{F}_{q})$ denotes the group of $\mathbb{F}_{q}$ points on $E$. This
is of theoretical significance in addition to its importance for cryptographic
applications. The Schoof–Elkies–Atkin (SEA) algorithm is a standard method for
accomplishing this task. For more details see [12], [7], [4, Algorithm 17.25].
One observes that the asymptotic time complexity of the algorithm depends
dramatically on the distribution of the so-called Elkies primes.

We define $t_{E}=q+1-\\#E(\mathbb{F}_{q})$; it is well known that this is the
trace of the Frobenius endomorphism on $E$. We refer to a prime $\ell$ with
$(\ell,q)=1$ as an Elkies prime if the Legendre symbol
$(\frac{t_{E}^{2}-4q}{\ell})$ equals $1$. Primes $(\ell,q)=1$ where the symbol
equals $-1$ or $0$ are called Atkin primes.

If $t_{E}^{2}-4q=0$, then there are no Elkies primes; this is the case where
$|t|=2\sqrt{q}$ and hence the curve is “strongly maximal” with respect to the
Hasse bound. However, this latter condition can only happen for supersingular
curves. We will assume throughout that $t_{E}^{2}-4q\neq 0$. Notice that if
$t_{E}^{2}-4q$ is a perfect square, then any prime $\ell$ with
$(\ell,t_{E}^{2}-4q)=1$ is an Elkies prime. We will avoid this trivial case by
assuming $t_{E}^{2}-4q$ is not a perfect square.

In [11], it is shown that the least Elkies prime is $O((\log
q)^{2+\varepsilon})$. However, it may be that explicit results are more
relevant for some applications. We give an explicit result that also offers a
small asymptotic improvement. Moreover, our proof is different from the one
appearing in [11] as we follow the approach of
Montgomery–Lagarias–Odlyzko–Bach, rather than the arguments of Ankeny.

######  Theorem 1.

Assume GRH. For $q\geq 10^{9}$, the least Elkies prime is bounded above by
$(2\log 4q+4)^{2}$. In particular, the least Elkies prime is $O((\log
q)^{2})$.

Let $N_{E}(X)$ denote the number of Elkies primes up to $X$. It is stated in
Appendix A of [11] that estimating $N_{E}(X)$ seems to be difficult (even
under GRH). We give such an estimate with error term, but will not pursue
making the constant explicit.

######  Theorem 2.

Assume GRH. We have

$N_{E}(X)=\frac{\pi(X)}{2}+O\left(\frac{\sqrt{X}(\log qX)^{2}}{\log
X}\right)\,.$

Notice that the previous theorem immediately implies (under GRH) that
$N_{E}(X)\sim\pi(X)/2$, as expected. It is noted in [13] that this corollary
follows from [11]; however, neither paper gives a bound on the error
$N_{E}(X)-\pi(X)/2$. We note that the paper [13] studies the distribution of
Elkies primes on average over all elliptic curves, which is a different
perspective than we consider here.

##  2\. A character sum estimate

We will require the next proposition in the sequel. It follows from the work
of Bach (see [1, 2, 3]) but a little effort is needed for its proof. Notice
that the modulus of the character is not required to be prime.

######  Proposition 1.

Assume GRH.

  1. (1)

When $X\geq X_{0}\geq 2$ we have

$\left|\sum_{n\leq
X}\Lambda(n)\left(1-\frac{n}{X}\right)-\frac{X}{2}\right|\leq
C_{1}(X_{0})\sqrt{X}\,.$

  2. (2)

Suppose $\chi$ is a non-principal primitive character modulo $m$. When
$X>X_{0}\geq 10$ and $m\geq m_{0}$ we have

$\left|\sum_{n\leq X}\Lambda(n)\chi(n)\left(1-\frac{n}{X}\right)\right|\leq
C_{2}(m_{0},X_{0})\sqrt{X}\log m\,,$

where values of the constants are given in Tables 1, 2, 3, generated by
Equations (1), (4), (5).

$x_{0}$ | $10^{1}$ | $2^{4}$ | $2^{5}$ | $2^{6}$ | $2^{7}$ | $2^{8}$ | $2^{9}$ | $2^{10}$ | $2^{11}$
---|---|---|---|---|---|---|---|---|---
$C_{1}$ | $0.629$ | $0.507$ | $0.048$ | $0.047$ | $0.047$ | $0.047$ | $0.047$ | $0.047$ | $0.047$
Table 1. Values of $C_{1}(X_{0})$ $x_{0}$ $m_{0}$ $C_{2}$ | $3$ | $2^{2}$ | $2^{3}$ | $2^{4}$ | $2^{5}$ | $2^{6}$ | $2^{7}$ | $2^{8}$ | $2^{9}$ | $2^{10}$ | $2^{11}$
---|---|---|---|---|---|---|---|---|---|---|---
$10$ | $3.475$ | $2.993$ | $2.379$ | $2.072$ | $1.888$ | $1.766$ | $1.678$ | $1.612$ | $1.561$ | $1.520$ | $1.487$
$2^{4}$ | $3.051$ | $2.632$ | $2.099$ | $1.832$ | $1.672$ | $1.565$ | $1.489$ | $1.432$ | $1.387$ | $1.352$ | $1.323$
$2^{5}$ | $2.621$ | $2.267$ | $1.816$ | $1.591$ | $1.455$ | $1.365$ | $1.301$ | $1.252$ | $1.215$ | $1.185$ | $1.160$
$2^{6}$ | $2.335$ | $2.025$ | $1.629$ | $1.431$ | $1.313$ | $1.233$ | $1.177$ | $1.134$ | $1.101$ | $1.075$ | $1.054$
$2^{7}$ | $2.140$ | $1.859$ | $1.501$ | $1.323$ | $1.215$ | $1.144$ | $1.093$ | $1.054$ | $1.025$ | $1.001$ | $0.981$
$2^{8}$ | $2.003$ | $1.743$ | $1.413$ | $1.247$ | $1.148$ | $1.082$ | $1.035$ | $0.999$ | $0.972$ | $0.949$ | $0.931$
$2^{9}$ | $1.908$ | $1.663$ | $1.350$ | $1.194$ | $1.101$ | $1.038$ | $0.994$ | $0.960$ | $0.934$ | $0.914$ | $0.897$
$2^{10}$ | $1.840$ | $1.606$ | $1.307$ | $1.157$ | $1.068$ | $1.008$ | $0.965$ | $0.933$ | $0.908$ | $0.888$ | $0.872$
$2^{11}$ | $1.793$ | $1.565$ | $1.276$ | $1.131$ | $1.044$ | $0.986$ | $0.945$ | $0.914$ | $0.890$ | $0.870$ | $0.855$
Table 2. Values of $C_{2}(X_{0},m_{0})$ for $\chi(-1)=-1$ $x_{0}$ $m_{0}$ $C_{2}$ | $3$ | $2^{2}$ | $2^{3}$ | $2^{4}$ | $2^{5}$ | $2^{6}$ | $2^{7}$ | $2^{8}$ | $2^{9}$ | $2^{10}$ | $2^{11}$
---|---|---|---|---|---|---|---|---|---|---|---
$10$ | $3.850$ | $3.290$ | $2.577$ | $2.221$ | $2.007$ | $1.865$ | $1.763$ | $1.686$ | $1.627$ | $1.580$ | $1.541$
$2^{4}$ | $3.455$ | $2.952$ | $2.312$ | $1.992$ | $1.800$ | $1.672$ | $1.580$ | $1.512$ | $1.458$ | $1.416$ | $1.381$
$2^{5}$ | $3.018$ | $2.582$ | $2.026$ | $1.748$ | $1.581$ | $1.470$ | $1.390$ | $1.331$ | $1.284$ | $1.247$ | $1.217$
$2^{6}$ | $2.695$ | $2.309$ | $1.819$ | $1.574$ | $1.426$ | $1.328$ | $1.258$ | $1.206$ | $1.165$ | $1.132$ | $1.105$
$2^{7}$ | $2.449$ | $2.104$ | $1.665$ | $1.445$ | $1.314$ | $1.226$ | $1.163$ | $1.116$ | $1.079$ | $1.050$ | $1.026$
$2^{8}$ | $2.262$ | $1.948$ | $1.549$ | $1.350$ | $1.230$ | $1.150$ | $1.093$ | $1.050$ | $1.017$ | $0.990$ | $0.969$
$2^{9}$ | $2.118$ | $1.830$ | $1.462$ | $1.278$ | $1.168$ | $1.094$ | $1.041$ | $1.002$ | $0.971$ | $0.947$ | $0.927$
$2^{10}$ | $2.009$ | $1.739$ | $1.396$ | $1.224$ | $1.121$ | $1.052$ | $1.003$ | $0.967$ | $0.938$ | $0.915$ | $0.896$
$2^{11}$ | $1.926$ | $1.671$ | $1.346$ | $1.184$ | $1.086$ | $1.021$ | $0.975$ | $0.940$ | $0.913$ | $0.892$ | $0.874$
Table 3. Values of $C_{2}(X_{0},m_{0})$ for $\chi(-1)=1$

###### Proof.

Let $X\geq 2$. Using the explicit formula, in the form of Lemma 1, we find

$\displaystyle\left|\sum_{n\leq X}\Lambda(n)(1-n/X)-\frac{X}{2}\right|$
$\displaystyle\leq\sqrt{X}\sum_{\begin{subarray}{c}\zeta(\rho)=0\\\
\Re(\rho)=1/2\end{subarray}}\left|\frac{1}{\rho(\rho+1)}\right|+1.84\,.$

Here we have used

$\left|\frac{1}{X}\cdot\frac{\zeta^{\prime}(-1)}{\zeta(-1)}-\frac{\zeta^{\prime}(0)}{\zeta(0)}-\frac{1}{2}\log\left(1-\frac{1}{X^{2}}\right)-\frac{1}{2x}\left(1+\frac{2}{X-1}\right)\right|\leq\frac{\zeta^{\prime}(0)}{\zeta(0)}\leq
1.84\,.$

Combining with the following (see Equations 10 and 11 from §$12$ of [5])

$\sum_{\rho}\left|\frac{1}{\rho(\rho+1)}\right|\leq\sum_{\rho}\left(\frac{1}{\rho}+\frac{1}{\overline{\rho}}\right)=\gamma+2-\log(4\pi)$

gives part (1) of the proposition with

(1) $C_{1}=\gamma+2-\log(4\pi)+\frac{1.84}{\sqrt{X_{0}}}\,.$

It remains to prove part (2).

We first deal with the case $\chi(-1)=-1$. Lemma 1 gives

$\displaystyle\left|\sum_{n\leq X}\Lambda(n)\chi(n)(1-n/X)\right|$
$\displaystyle\leq\sqrt{X}\sum_{\begin{subarray}{c}L(\rho,\chi)=0\\\
\Re(\rho)=1/2\end{subarray}}\left|\frac{1}{\rho(\rho+1)}\right|+\left|\frac{L^{\prime}(0,\chi)}{L(0,\chi)}\right|+\left|\frac{b_{\chi}}{X}\right|$
$\displaystyle+\left|\frac{\log
X}{X}+\frac{1}{2}\log\left(1+\frac{2}{X-1}\right)+\frac{1}{2x}\log\left(1-\frac{1}{X^{2}}\right)\right|\,.$

Note that

$\displaystyle\left|\frac{b_{\chi}}{X}\right|$
$\displaystyle\leq\frac{1}{X}\left|\frac{L^{\prime}(2,\chi)}{L(2,\chi)}\right|+\frac{1}{X}\sum\frac{1}{|\rho(\rho+1)|}+\frac{2}{X}\sum\frac{1}{|\rho(2-\rho)|}+\left|\frac{\log
2-1}{X}\right|$
$\displaystyle\leq\frac{0.87}{X}+\frac{3}{X}\sum_{\begin{subarray}{c}L(\rho,\chi)=0\\\
\Re(\rho)=1/2\end{subarray}}\left|\frac{1}{\rho(\rho+1)}\right|$

and

(2) $\displaystyle\left|\frac{L^{\prime}(0,\chi)}{L(0,\chi)}\right|$
$\displaystyle=\left|-\sum\frac{2}{\rho(2-\rho)}+\frac{L^{\prime}(2,\chi)}{L(2,\chi)}-1\right|\leq
2\sum_{\begin{subarray}{c}L(\rho,\chi)=0\\\
\Re(\rho)=1/2\end{subarray}}\left|\frac{1}{\rho(\rho+1)}\right|+1.57\,.$

For $X\geq 10$, this all leads to

$\left|\sum_{n\leq
X}\Lambda(n)\chi(n)(1-n/X)\right|\leq(\sqrt{X}+2+3/X)\sum_{\begin{subarray}{c}L(\rho,\chi)=0\\\
\Re(\rho)=1/2\end{subarray}}\left|\frac{1}{\rho(\rho+1)}\right|+2$

Using estimates from [3] we have

(3) $\sum_{\begin{subarray}{c}L(\rho,\chi)=0\\\
\Re(\rho)=1/2\end{subarray}}\left|\frac{1}{\rho(\rho+1)}\right|\leq\frac{4}{3}\sum_{\rho}\frac{1}{|\sigma-\rho|^{2}}=\sum_{\rho}\left(\frac{1}{\sigma-\rho}+\frac{1}{\sigma-\overline{\rho}}\right)\leq\frac{2}{3}(\log
m+5/3)$

and the second result follows with

(4)
$C_{2}=\frac{2}{3}\left(1+\frac{2}{\sqrt{X_{0}}}+\frac{3}{X_{0}\sqrt{X_{0}}}\right)\left(1+\frac{5}{3\log
m_{0}}\right)+\frac{2}{\sqrt{X_{0}}\log m_{0}}\,.$

Next we turn to the case of $\chi(-1)=1$. The proof is similar. We have

$\displaystyle\left|\sum_{n\leq X}\Lambda(n)\chi(n)(1-n/X)\right|$
$\displaystyle\leq\sqrt{X}\sum_{\begin{subarray}{c}L(\rho,\chi)=0\\\
\Re(\rho)=1/2\end{subarray}}\left|\frac{1}{\rho(\rho+1)}\right|+\left|\frac{1}{X}\frac{L^{\prime}(-1,\chi)}{L(-1,\chi)}\right|+\left|b_{\chi}\right|$
$\displaystyle+\log
X+\left|\frac{1}{2x}\log\left(1+\frac{2}{X-1}\right)+\frac{1}{2}\log\left(1-\frac{1}{X^{2}}\right)\right|\,,$

and we make use of

$\displaystyle\left|b_{\chi}\right|$
$\displaystyle\leq\sqrt{X}\sum_{\begin{subarray}{c}L(\rho,\chi)=0\\\
\Re(\rho)=1/2\end{subarray}}\left|\frac{1}{\rho(\rho+1)}\right|+0.57\,,$
$\displaystyle\left|\frac{L^{\prime}(-1,\chi)}{L(-1,\chi)}\right|$
$\displaystyle\leq\sum\frac{2}{|\rho(2-\rho)|}+\sum\frac{1}{|\rho(\rho+1)|}+\frac{1}{2}\left|\frac{\Gamma^{\prime}(1)}{\Gamma(1)}-\frac{\Gamma^{\prime}(-1/2)}{\Gamma(-1/2)}\right|+\left|\frac{L^{\prime}(2,\chi)}{L(2,\chi)}\right|$
$\displaystyle\leq 3\sum\frac{1}{|\rho(\rho+1)|}+0.88\,.$

For $X\geq 3$, this all leads to

$\left|\sum_{n\leq
X}\Lambda(n)\chi(n)(1-n/X)\right|\leq(\sqrt{X}+2+3/X)\sum_{\begin{subarray}{c}L(\rho,\chi)=0\\\
\Re(\rho)=1/2\end{subarray}}\left|\frac{1}{\rho(\rho+1)}\right|+\log X+1\,,$

which gives the result with

(5)
$C_{2}=\frac{2}{3}\left(1+\frac{2}{\sqrt{X_{0}}}+\frac{3}{X_{0}\sqrt{X_{0}}}\right)\left(1+\frac{5}{3\log
m_{0}}\right)+\frac{1+\log X_{0}}{\sqrt{X_{0}}\log m_{0}}\,.$

∎

##  3\. Proof of Theorem 1

In light of our assumptions (stated in Section 1) we can use the Kronecker
symbol to define a Dirichlet character

$\chi_{E}(n)=\left(\frac{t_{E}^{2}-4q}{n}\right)\,.$

Moreover, $\chi_{E}$ is a non-principal character modulo $m=|t_{E}^{2}-4q|\geq
3$ satisfying $\chi_{E}(-1)=-1$. Let $\chi$ denote the primitive character of
modulus $m^{\prime}$ that induces $\chi_{E}$. Notice that $m^{\prime}\leq
m\leq 4q$. Set $X=(2\log 4q+4)^{2}\geq 2325$. We will show that there is an
Elkies prime less than $X$. By direct computation, there is always an Elkies
prime less than $40$ when $m<2^{11}$, and hence the theorem is trivial in this
case. Henceforth, we assume $m\geq 2^{11}$.

Since our character $\chi_{E}$ of modulus $m$ is not primitive, the difference
between the sum

$\sum_{n\leq X}\Lambda(n)\chi_{E}(n)\left(1-\frac{n}{X}\right)$

and the corresponding sum where $\chi_{E}$ is replaced by the primitive
character $\chi$ is bounded (in absolute value) by

$\sum_{p|m}\sum_{\begin{subarray}{c}n\leq X\\\
n=p^{k}\end{subarray}}\Lambda(n)\left(1-\frac{n}{X}\right)\leq\sum_{p|m}\log
p\sum_{\begin{subarray}{c}n\leq X\\\ n=p^{k}\end{subarray}}1\leq\omega(m)\log
X\leq D\frac{\log m\log X}{\log\log m}$

By [10], we know that for $m\geq 3$, one has

$\omega(m)\leq 1.3841\frac{\log m}{\log\log m}$

which means we can take $D=1.3841$.

Now we begin the proof in earnest. Suppose that the least Elkies prime is
greater than $X$. That is, we have that $\chi_{E}(p)\neq 1$ for all primes
$p\leq X$. We seek a contradiction. Consider the sum

$S(X)=\sum_{n\leq X}\Lambda(n)\chi_{E}(n)\left(1-\frac{n}{X}\right)$

We write $S=S_{1}+S_{2}+S_{3}$, where

$\displaystyle S_{1}(X)$ $\displaystyle=-\sum_{n\leq
X}\Lambda(n)\left(1-\frac{n}{X}\right)$ $\displaystyle S_{2}(X)$
$\displaystyle=\sum_{p\leq
X}\Lambda(p)(\chi_{E}(p)+1)\left(1-\frac{p}{X}\right)$ $\displaystyle
S_{3}(X)$ $\displaystyle=\sum_{\begin{subarray}{c}n\leq X\\\ n=p^{k}\\\ k\geq
2\end{subarray}}\Lambda(n)(\chi_{E}(n)+1)\left(1-\frac{n}{X}\right)$

By hypothesis,

$S_{2}(X)\leq\sum_{p|m}\log
p\left(1-\frac{p}{X}\right)\leq\log(\prod_{p|m}p)\leq\log m$

Moreover, according to Corollary 4.5 of [6],

$\psi(X)-\theta(X)\leq(1+1.47\cdot
10^{-7})\sqrt{X}+1.78\sqrt[3]{X}+\leq(1+\delta)\sqrt{X}$

where $\delta=1/2$, and therefore

$|S_{3}(X)|\leq 2(\psi(X)-\theta(X))\leq 2(1+\delta)\sqrt{X}\,.$

By Proposition 1

$|S_{1}|\geq\frac{X}{2}-C_{1}\sqrt{X}$

Therefore

$\displaystyle|S(X)|$
$\displaystyle\geq\frac{X}{2}-(C_{1}+2(1+\delta))\sqrt{X}-\log m$

On the other hand,

$|S(X)|\leq C_{2}\sqrt{X}\log m+D\frac{\log m\log X}{\log\log m}$

Hence

$\frac{X}{2}\leq C_{2}\sqrt{X}\log m+(C_{1}+2(1+\delta))\sqrt{X}+\log
m+D\frac{\log m\log X}{\log\log m}$

Using $m\leq 4q$, this leads to

$\displaystyle\sqrt{X}$ $\displaystyle\leq 2C_{2}\log
4q+2(C_{1}+2(1+\delta))+\frac{2\log 4q}{\sqrt{X}}+\frac{2D\log(4q)\log
X}{\sqrt{X}\log\log 4q}$ $\displaystyle\leq 1.902\log 4q+6.094$
$\displaystyle<2\log(4q)+4.$

This gives a contradiction, thereby proving the theorem.

##  4\. Proof of Theorem 2

We have

$\displaystyle N_{E}(X)$ $\displaystyle=\frac{1}{2}\sum_{\ell\leq
X}\left[\left(\frac{t_{E}^{2}-4q}{\ell}\right)+1\right]-a_{E}(X)$
$\displaystyle=\frac{\pi(X)}{2}+\frac{1}{2}\sum_{\ell\leq
X}\left(\frac{t_{E}^{2}-4q}{\ell}\right)-a_{E}(X)$

In the above

$a_{E}(X)=\begin{cases}1/2&\ell\mid
t\,,\;\operatorname{char}(\mathbb{F}_{q})<X\\\ 1&\ell\nmid
t\,,\;\operatorname{char}(\mathbb{F}_{q})<X\\\ 0&\text{otherwise}\end{cases}$

Using the notation from the previous section, it suffices to bound the sum

$\sum_{\ell\leq X}\chi_{E}(\ell)\,.$

Partial summation gives

$\sum_{\ell\leq X}\chi(\ell)=\frac{1}{\log X}\sum_{\ell\leq
X}\chi(\ell)\Lambda(\ell)+\int_{2}^{X}\left(\sum_{\ell\leq
t}\chi(\ell)\Lambda(\ell)\right)\frac{dt}{t(\log t)^{2}}\,.$

Exercise 7.4.9 of [9] leads to a bound of the form:

(6) $\left|\sum_{\ell\leq X}\chi(\ell)\Lambda(\ell)\right|\leq C\sqrt{X}(\log
mX)^{2}\,,\;t>2\,.$

This leads to

$\displaystyle\left|\sum_{\ell\leq X}\chi(\ell)\right|$
$\displaystyle\leq\frac{C\sqrt{X}(\log mX)^{2}}{\log
X}+C\int_{2}^{X}\frac{(\log tm)^{2}dt}{\sqrt{t}(\log t)^{2}}\,.$

Using $X<m$ the second term is bounded by $4C(\log
m)^{2}\int_{2}^{x}t^{-1/2}(\log t)^{-2}dt$. Moreover, we have

$\int_{2}^{X}\frac{dt}{\sqrt{t}(\log t)^{2}}\leq 2\frac{\sqrt{X}}{\log X}\,.$

This yields

$\displaystyle\frac{C\sqrt{X}(\log mX)^{2}}{\log X}+\frac{8C\sqrt{X}(\log
m)^{2}}{\log X}$ $\displaystyle=C((\log mX)^{2}+8(\log
m)^{2})\frac{\sqrt{X}}{\log X}$ $\displaystyle\leq 9C(\log
mX)^{2}\frac{\sqrt{X}}{\log X}$

Putting this all together shows

$\left|N_{E}(X)-\frac{\pi(X)}{2}\right|\leq(9C/2)\left(\frac{\sqrt{X}(\log
4qX)^{2}}{\log X}\right)+1\,,$

which completes the proof.

##  5\. Appendix: An explicit formula

Although various explicit formulas associated to Dirichlet $L$-functions are
certainly well-known, we could not find in the literature this particular
explicit formula, with all terms rendered visible, for an arbitrary Dirichlet
character. As this is essential for the derivation of our results, we give a
brief argument together with relevant residue calculations. The first of the
three equations that follow appears in [8].

######  Lemma 1.

Suppose $X\geq 1$. We have

$\displaystyle\sum_{n\leq X}^{\prime}\Lambda(n)\left(1-\frac{n}{X}\right)$
$\displaystyle=\frac{X}{2}-\sum_{\rho}\frac{X^{\rho}}{\rho(\rho+1)}+\frac{1}{X}\frac{\zeta^{\prime}(-1)}{\zeta(-1)}-\frac{\zeta^{\prime}(0)}{\zeta(0)}$
$\displaystyle-\frac{1}{2}\log\left(1-\frac{1}{X^{2}}\right)-\frac{1}{2x}\log\left(1+\frac{2}{X-1}\right)\,.$

Suppose $\chi$ is a primitive Dirichlet character modulo $m$. If
$\chi(-1)=-1$, then we have

$\displaystyle\sum_{n\leq X}\Lambda(n)\chi(n)\left(1-\frac{n}{X}\right)$
$\displaystyle=-\sum_{\rho\text{ of
}L_{\chi}}\frac{X^{\rho}}{\rho(\rho+1)}-\frac{L^{\prime}(0)}{L(0)}+\frac{b_{\chi}}{X}+\frac{\log
X}{X}$ $\displaystyle+$
$\displaystyle\frac{1}{2}\log\left(1+\frac{2}{X-1}\right)+\frac{1}{2x}\log\left(1-\frac{1}{X^{2}}\right)\,,$

where $b_{\chi}$ is the constant term in the Laurent series expansion of
$L^{\prime}(s,\chi)/L(s,\chi)$ at $s=-1$. If $\chi(-1)=1$, then we have

$\displaystyle\sum_{n\leq X}\Lambda(n)\chi(n)\left(1-\frac{n}{X}\right)$
$\displaystyle=-\sum_{\rho\text{ of
}L_{\chi}}\frac{X^{\rho}}{\rho(\rho+1)}+\frac{1}{X}\frac{L^{\prime}(-1)}{L(-1)}-b_{\chi}-\log
X$
$\displaystyle-\frac{1}{2}\log\left(1-\frac{1}{X^{2}}\right)-\frac{1}{2x}\log\left(1+\frac{2}{X-1}\right)\,,$

where $b_{\chi}$ is the constant term in the Laurent series expansion of
$L^{\prime}(s,\chi)/L(s,\chi)$ at $s=0$.

###### Proof.

We begin by proving the first equation that involves the zeros of $\zeta(s)$.
We are interested in the sum

(7) $g(X):=\sum_{n\leq X}\Lambda(n)(X-n)\,.$

Along these lines, we consider the function

$\displaystyle G(s):=\sum_{n=1}^{\infty}\frac{\Lambda(n)(X-n)}{n^{s}}$
$\displaystyle=X\sum_{n}\frac{\Lambda(n)}{n^{s}}-\sum_{n}\frac{\Lambda(n)}{n^{s-1}}$
$\displaystyle=X\frac{\zeta^{\prime}(s)}{\zeta(s)}-\frac{\zeta^{\prime}(s-1)}{\zeta(s-1)}\,.$

We evaluate the following integral in two different ways:

$\mathcal{I}(X):=\frac{1}{2\pi
i}\int_{\sigma-i\infty}^{\sigma+i\infty}\left\\{-G(s)\right\\}X^{s}\frac{ds}{s}\,.$

Integrating the Dirichlet series for $G(s)$ term-by-term and using

$\int_{\sigma-i\infty}^{\sigma+i\infty}y^{s}\;\frac{ds}{s}=\begin{cases}1&y>1\\\
1/2&y=1\\\ 0&0<y<1\end{cases}$

yields $\mathcal{I}(X)=g_{0}(X)$. Here $g_{0}(X)=g(X)$ when $X$ is non-
integral. Next we evaluate the integral using the Residue Theorem. Standard
arguments show that $\mathcal{I}(X)$ is the sum of the residues in the half-
plane where $\Re(s)<\sigma$. The residues of the integrand at $s=1$ and $s=2$
are $X^{2}$ and $-X^{2}/2$, respectively. The residue at $s=0$ is

$\frac{\zeta^{\prime}(-1)}{\zeta(-1)}-X\frac{\zeta^{\prime}(0)}{\zeta(0)}\,.$

Given a non-trivial zero $\rho$, the residues at $s=\rho$ and $s=\rho+1$ are
$-X^{\rho+1}/\rho$ and $X^{\rho+1}/(\rho+1)$, respectively. The residues
coming from the trivial zeros of $\zeta(s)$, occur at $s=-2k$ and $s=-2k+1$,
for $k\geq 1$, and are equal to $\frac{X^{-2k+1}}{2k}$ and
$\frac{X^{-2k+1}}{-2k+1}$ respectively. Next we sum over $k$ and use the
following:

(8) $\frac{1}{2}\log\left(1-\frac{1}{X^{2}}\right)=-\sum_{k\geq
1}\frac{X^{-2k}}{2k}$ (9)
$\frac{1}{2}\log\left(1+\frac{2}{X-1}\right)=\sum_{k\geq
1}\frac{X^{-2k+1}}{2k-1}$

Putting this all together and dividing by $X$ gives the result.

The proof for $L(s,\chi)$ is similar, with the main difference being that
$L(s,\chi)$ does not have a pole at $s=1$. The residues arising from non-
trivial zeros $\rho$ of $L(s,\chi)$ are exactly as described in above.

When $\chi(-1)=-1$, the trivial zeros of $L(s,\chi)$ occur $s=-1,-3,-5,\dots$.
There is a double-pole at $s=0$ whose residue equals

$-X\frac{L^{\prime}(0)}{L(0)}+b_{\chi}+\log X\,,$

where $b_{\chi}$ is the constant term in the Laurent series expansion of
$L^{\prime}(s-1,\chi)/L(s-1,\chi)$ at $s=0$. Having already dealt with $s=0$,
the remaining residues arising from the trivial zeros occur at $s=-2k+1$ and
$s=-2k$ for $k\geq 1$, where the residues are equal to
$\frac{X^{-2k+2}}{2k-1}$ and $\frac{X^{-2k}}{-2k}$ respectively. Summing these
residues over $k\geq 1$ gives the result.

When $\chi(-1)=1$, the trivial zeros of $L(s,\chi)$ occur
$s=0,-2,-4,-6,\dots$; the argument is essentially the same, but the details
are different. The main thing to point out is that our definition of
$b_{\chi}$ is slightly different in this case; here we denote by $b_{\chi}$
the constant term in the Laurent expansion of $L^{\prime}(s)/L(s)$ at $s=0$.
We include table of the residues in all three cases, so that the reader can
verify the details. ∎

| $0$ | $1$ | $2$ | $\rho$ | $\rho+1$ | $-2k$ | $-2k+1$
---|---|---|---|---|---|---|---
$\zeta(s)$ | $\displaystyle\frac{\zeta^{\prime}(-1)}{\zeta(-1)}-X\frac{\zeta^{\prime}(0)}{\zeta(0)}$ | $X^{2}$ | $\displaystyle\frac{X^{2}}{2}$ | $\displaystyle-\frac{X^{\rho+1}}{\rho}$ | $\displaystyle\frac{X^{\rho+1}}{\rho+1}$ | $\displaystyle\frac{X^{-2k+1}}{2k}$ | $\displaystyle\frac{X^{-2k+1}}{-2k+1}$
$\begin{subarray}{c}L(s,\chi)\\\ \chi(-1)=-1\end{subarray}$ | $\displaystyle-X\frac{L^{\prime}(0)}{L(0)}+b_{\chi}+\log X$ | $0$ | $0$ | $\displaystyle-\frac{X^{\rho+1}}{\rho}$ | $\displaystyle\frac{X^{\rho+1}}{\rho+1}$ | $\displaystyle\frac{X^{-2k}}{-2k}$ | $\displaystyle\frac{X^{-2k+2}}{2k-1}$
$\begin{subarray}{c}L(s,\chi)\\\ \chi(-1)=1\end{subarray}$ | $\displaystyle\frac{L^{\prime}(-1)}{L(-1)}-Xb_{\chi}-X\log X$ | $0$ | $0$ | $\displaystyle-\frac{X^{\rho+1}}{\rho}$ | $\displaystyle\frac{X^{\rho+1}}{\rho+1}$ | $\displaystyle\frac{X^{-2k+1}}{2k}$ | $\displaystyle\frac{X^{-2k+1}}{-2k+1}$
Table 4. Residues of $-G(s)X^{s}/s$

Finally, we will require the following lemma to deal with the quantity
$b_{\chi}$ showing up in the explicit formula.

######  Lemma 2.

If $\chi(-1)=-1$, then

$b_{\chi}=\frac{L^{\prime}(2,\chi)}{L(2,\chi)}+\sum_{\rho\text{ of
}L_{\chi}}\frac{1}{\rho(\rho+1)}-\sum_{\rho\text{ of
}L_{\chi}}\frac{2}{\rho(2-\rho)}+\log 2-1\,.$

If $\chi(-1)=1$, then

$b_{\chi}=\frac{L^{\prime}(2,\chi)}{L(2,\chi)}-\sum_{\rho\text{ of
}L_{\chi}}\frac{2}{\rho(2-\rho)}\,.$

###### Proof.

Consider Equation 17 on page 83 of [5] at $s=-1$ and $s=2$. Subtracting these
and appealing to known properties of the gamma function, the results follows.
∎

## References

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# Simultaneous depletion and adsorption in polymer solutions near a solid wall

Suzanne Lafon<EMAIL_ADDRESS>Laboratoire de
Physique des Solides, Université Paris Saclay, CNRS, 91405 Orsay, France
Tiago Outerelo Corvo  Laboratoire de Physique des Solides, Université Paris
Saclay, CNRS, 91405 Orsay, France  Marion Grzelka  Laboratoire Léon Brillouin,
CEA Saclay, 91191 Gif-sur-Yvette, France  Arnaud Hélary  Laboratoire Léon
Brillouin, CEA Saclay, 91191 Gif-sur-Yvette, France  Philipp Gutfreund
Institut Laue-Langevin, 71 rue des Martyrs, 38000 Grenoble, France  Liliane
Léger  Laboratoire de Physique des Solides, Université Paris Saclay, CNRS,
91405 Orsay, France  Alexis Chennevière  Laboratoire Léon Brillouin, CEA
Saclay, 91191 Gif-sur-Yvette, France  Frédéric Restagno  Laboratoire de
Physique des Solides, Université Paris Saclay, CNRS, 91405 Orsay, France

###### Abstract

Polymer solutions exhibit peculiar properties near surfaces: polymer chains
either adsorb onto or can be repelled by the wall. Only a few techniques are
able to probe their structure in the vicinity of solid substrates, because of
the small length scales over which liquids are influenced by the wall. In this
paper, we use neutron reflectivity measurements at the interface between a
polystyrene semi-dilute solution in a good solvent and a smooth sapphire
surface. We show that polymer chains are globally depleted from the solid
surface, but contrary to what is generally assumed, this does not prevent some
chains to still adsorb on the wall. We also observe that the Newtonian flow of
the solution has a negligible effect on the size of the depletion layer, which
is a hypothesis often made but rarely measured in the literature.

††preprint: APS/123-QED

##  I Introduction

Molecular structure of liquids near solid interfaces is a complex topic. In
the close vicinity of the surface, or in confined geometries, the presence of
the wall can modify their behavior. A first historical effect of the role of
the walls on fluid properties is capillary condensation, where the walls
affect the phase transition pressure of the fluid [1, 2, 3]. Besides, walls
have strong implications in biological flows [4] and nanofluidics [5], where
recent fundamental advances have evidenced the role of quantum mechanics in
liquid-solid friction [6, 7]. We are now able to probe the liquid/solid
interfaces down to the nanometric scale thanks to the development of
techniques of high precision: volume and near-field laser velocimetry [8, 9,
10, 11, 12], Surface Force Apparatus [13], Atomic Force Microscopy [14], Total
Internal Reflection Fluorescence Microscopy [15], nanofluidics in nanochannels
[16] completed by numerical simulations [17, 5, 18].

This has opened plenty of interesting questions in many fields and, in
particular, these results have challenged the validity of the no-slip boundary
condition at small scales [19]. In the case of polymer solutions, the bulk
concentration is not always maintained near the surface: the species which has
the strongest affinity with the wall is over-concentrated at the liquid/solid
interface. If this species is the solvent, we call this phenomenon depletion:
there is a lower concentration of polymer near the surface compared to the
bulk. For neutral species, the size of this depletion layer is given by the
correlation length of the solution [20, 21, 22, 23]. For charged species and
charged surfaces, the depletion layer is also related to electrostatic
interactions, and thus can be much thicker [24, 15, 25]. In the reverse
situation, when the polymer/wall interaction is more favorable than the
solvent/wall one, not only the polymer is over-concentrated near the
interface, but in addition, some polymer chains may adsorb onto the surface,
leading to a so-called Guiselin pseudo-brush[26]. This is due to the strong
entropic nature of polymers: a single chain has multiple adsorption points
with the surface, and thus the probability for a chain to desorb is the
probability that all these adsorption points to desorb, which is extremely low
and therefore adsorption of long polymer chains is almost irreversible.

Several experimental measurements have allowed to indirectly measure the
presence of a depletion layer when a semi-dilute polymer solution is put onto
a repulsive solid surface. These measurements are based on the increase of the
fluid mobility close to the surface. Among these experiments, we can cite,
direct pressure drop measurements in porous media [27, 28, 29], microfluidics
experiments [16], surface forces apparatus [30, 24] or rheology [31, 32]. In
general, it is difficult to directly measure the concentration of polymer
close to the interface. The first observation of depletion of a polymer
solution near a solid surface has been made by Allain et al. in 1982 [20] with
evanescent waves. Since then, different techniques have been used to directly
measure depletion, such as neutron reflectivity [33] and ellipsometry [25].
Finally, the effect of flow on depletion is poorly studied because of the
difficulty of the experiments, but for now, what is reported is a thickening
of the depletion layer at high flow rates [34, 35] for rigid polymers.

In this paper, we use neutron reflectivity [36, 37, 38] to directly measure
depletion near a smooth sapphire surface for non-charged semi-dilute
polystyrene solutions. We observe an intriguing behavior: the polymer chains
are depleted from the wall, but some of them still adsorb on the solid
surface. We discuss this observation in terms of comparative interactions.
Finally, we show that the Newtonian flow rate has no effect on the
concentration profile near the solid wall.

##  II Results and Discussion

As a first result, we plot in Fig. 1 (left) the reflectivity curves of two
static solutions at different bulk volume fractions $\phi_{\text{b}}$ of
polymers and the same molar mass $M_{n}=195$ kg/mol. The data are plotted in
the Porod represention ($RQ^{4}$ vs $Q$) where $Q$ is the scattering vector
and $R$ is the reflectivity signal, to reveal differences between the two
curves.

The observed differences between these two curves can stem from the difference
in coherent neutron scattering length density of the solutions and/or a change
in the polymer segment density profile. As shown in figure 1, the computed
Fresnel reflectivity does not allow to describe the experimental data, which
highlights that near surface polymer concentration is not equal to the bulk
concentration. In order to get a quantitative description of the interfacial
structure, the data were fitted using the refnx Python module [39] assuming an
exponential evolution of the volume fraction profile $\phi(z)$ as proposed by
de Gennes [40]:

$\phi(z)=\phi_{\text{b}}+(\phi_{\text{w}}-\phi_{\text{b}})\text{e}^{-z/d}$ (1)

where $z$ is the distance from the sapphire surface, $\phi_{w}$ and
$\phi_{\text{b}}$ are the surface and bulk volume fractions, respectively, and
$d$ is the typical size over which the polymer concentration differs from the
bulk one. $\phi_{\text{w}}>\phi_{\text{b}}$ corresponds to adsorption while
$\phi_{\text{w}}<\phi_{\text{b}}$ corresponds to depletion.

The resulting neutron scattering length density (SLD) profiles and polymer
segment density profiles are plotted in Fig. 1. In the vicinity of the
interface, data fitting shows unambiguously a polymer depletion which
characteristic size decreases from $d=109\pm 18$ Å for $\phi=3~{}\%$ to
$d=65\pm 12$ Å at $\phi=6~{}\%$. These characteristic distances can be
compared to the bulk blob size $\xi$ measured using Small Angle Neutron
Scattering (see Supp. Mat. Fig. S1). We found that the characteristic
depletion distance $d$ is twice larger than $\xi$ and is in agreement with the
bulk scaling law $\xi\propto\phi^{-3/4}$ [41].

This is in good agreement with theoretical predictions which state that the
size of the depletion layer is the blob size [21, 22, 23] and consistent with
previous measurements at liquid/air interface [33]. This is schematically
illustrated in Fig. 1, right.

Figure 1: Effect of bulk volume fraction $\phi_{b}$. Left: reflectivity
curves. Solid lines correspond to the Fresnel reflectivity if the solution was
homogeneous until the solid surface. Dashed lines correspond to fits from
which the SLD profiles (second graph) and thus the volume fraction profiles
(third graph) are extracted. Right: cartoon of the interface. The size of the
depletion layer $d$ is typically the size of the blob $\xi$.

The observed depletion layer leads to the conclusion that polystyrene chains
are less attracted by the sapphire compared to the solvent. In Figure 1, one
can observe that the polymer volume fraction at the interface can be larger
than zero. It means that some monomers are in contact with the surface but the
lone segment density profile does not allow to conclude if they are physically
adsorbed. In order to probe adsorption, we have compared the reflectivity
curves of the solvent either on a clean sapphire and on a sapphire that has
previously been in contact with the dPS/DEP solution ($\phi=6$ %, $M_{n}=1.56$
Mg/mol). The resulting neutron reflectivity (NR) curves are plotted in Fig. 2
(top). One can see that at large $Q$ values, the profiles are significantly
different. If there was pure depletion of dPS from the interface, the two
profiles would be the same. Our measurement suggests that some dPS chains
remain adsorbed on the interface. However, due to the low contrast between the
adsorbed chains and the solvent, it is impossible to extract quantitative
information from this measurement.

To confirm this hypothesis, we have conducted X-ray reflectivity on
air/sapphire interface for a sapphire which has been in contact with the
polystyrene solution during one hour and then thoroughly rinsed with DEP and
dried. The corresponding reflectivity curve is plotted in Fig. 2 (bottom) and
show a clear Kiessig fringe between $Q=0.15$ Å-1 and $Q=0.3$ Å-1. A single
layer model with roughness at both interfaces fits well the data and confirms
the presence of a dry adsorbed layer of thickness $h_{\text{dry}}=24$ Å. This
value allows us to estimate a surface volume fraction of $0.7~{}\%$ and a
maximum swollen thickness of about $320$ nm (assuming an Alexander - de Gennes
brush [42], see Supp. Mat. section 4), resulting in an extremely low
concentration gradient, which further confirms that it is nearly impossible to
fit the NR reflectivity data of Fig. 2 (top).

Figure 2: Top: Neutron reflectivity curves of DEP on clean sapphire (blue
diamonds, fit is the blue dashed line) and on sapphire which has been
incubated with dPS/DEP solution prior to the experiment (orange squares).
Bottom: X-ray reflectivity curve of the sapphire surface that has been
incubated with PS/DEP solution ($\phi=6\%$, $M_{n}=708$ kg/mol), and then
rinsed and dried. The black line is a fit with a rough interfacial layer.
Inset: Electronic density as a function of the distance from the interface.
Corresponding cartoons are plotted on the right.

It is quite surprising to see both depletion and adsorption as they are
usually excluding scenari. The measurement of depletion is rather robust since
the neutron flux of the ILL is strong enough to give precise measurements, and
we could not fit any adsorption profile on our data while depletion profiles
were easily fitted with good $\tilde{\chi}^{2}$ values (between $0.8$ and
$3.3$). As for adsorption, both the qualitative NR measurements and the
quantitative X-Ray ones leave little doubts on the presence of remaining
chains on the surface. In addition, Barraud et al. [24] have mentioned an
adsorption/depletion situation for a charged polymer on a metallic surface:
they observed a depletion layer of the polyelectrolyte above its own adsorbed
layer. However, in their case, adsorption was the result of the favorable
electrostatic attraction between the metallic surface and the chains, and
depletion was the consequence of the electrostatic repulsion between the
adsorbed layer and the bulk chains. These arguments do not apply to our
neutral system.

From a chemical point of view, the surface of the sapphire displays hydroxyl
groups with a surface density in the range of $1-10$ OH/nm2 [43, 44]. Contrary
to PS (hydrogenated or deuterated), DEP is able to make hydrogen bonds with
these exposed OH groups, which is likely to favor the DEP-surface interaction
compared to the dPS-surface interaction which further corroborates the
depletion scenario. However, adsorption of PS chains onto sapphire surfaces
has already been mentionned in the literature for PS melts [45], which means
that PS can also interact favorably with the surface. Indeed, PS chains are
able to adsorb onto exposed hydroxyl groups through $\pi$-H interaction with
their phenyl groups [46, 47]. The fact that we globally see depletion suggests
that this interaction is weaker than the H-bonds between DEP and hydroxyl
groups, but it might not exclude some PS chains to still adsorb.

Finally, we have studied the effect of a Poiseuille flow on the concentration
profiles of the dPS/DEP solutions near the sapphire surface. The results are
plotted in Fig. 3, for a solution with a $6~{}\%$ volume fraction and a $1.56$
Mg/mol molar mass. The flow rate is characterized by the Weissenberg number
Wi, which is the dimensionless number comparing the typical relaxation time of
the polymer solution ($\tau$) and the typical time scale of the flow
($\dot{\gamma}_{\text{max}}^{-1}$): Wi$=\dot{\gamma}_{\text{max}}\tau$. In our
geometry, the maximum shear rate $\dot{\gamma}_{\text{max}}$ is given by
$6Q/(\ell h^{2})$ with $Q$ the flow rate imposed by the pump, and $\ell$ and
$h$ the width and the height of the cell, respectively. As for the typical
relaxation time, we use the reptation time (also called terminal relaxation
time) $\tau_{\text{rept}}$ of the solution. Oscillatory rheology (described in
Supp. Mat. Fig. S5) gave $\tau_{\text{rept}}=0.23\pm 0.01$ s.

Figure 3: Effect of the flow on the reflectivity profiles. Left: reflectivity
curves and corresponding SLD profiles in the inset. Right: Size of the
depletion layer extracted from the fits as a function of the Weissenberg
number Wi. These measurements have been done with $1.56$ Mg/mol dPS/DEP at a
volume fraction $\phi=6\>\%$.

The reflectivity curves are very similar for both the static solution and the
flowing ones, and the fit yields the same SLD profiles, with
$\tilde{\chi}^{2}$ between $1$ and $2$. The size of the depletion layer does
not vary significantly with the Weissenberg number up to Wi $=0.01$. Reaching
higher values of Wi is challenging because it requires both a highly viscous
liquid (high $\tau$) and a strong flow rate $Q$, and we are limited by either
the upper limit of flow rates accessible by the pump or by the total amount of
solution we have, which conditions the emptying time of the syringe.

The flow rate can have an effect on both the depletion layer and on the
adsorbed chains. Previous works on the effect of the flow rate on the size of
the depletion layer report interesting behaviors. For rigid rodlike particles,
Ausserré et al. have shown that the depletion size $d$ increases with the flow
rate due to hydrodynamics lift [34]. For dilute polymer solutions, de Pablo et
al. have predicted a decreasing $d$ with the flow rate at moderate flow rates
and an increasing $d$ at large flow rates [35], which corroborates the
experiment done by Ausserré et al.. They have used a dumbbell model, which
aligns with the flow at low flow rates and starts to rotate quickly at high
flow rates, so that the volume occupied by the dumbbell is larger and thus it
flows further away from the wall. On the contrary, in our experiments, the
dispersed polymers are flexible and therefore this description does not apply
to our system. The depletion size might have been changed if the flow would
have an effect on the blob size of the solution. Here, we see that up to
Wi$=0.01$, no effect of the flow is visible and thus the size of the blobs
remains constant. As for the adsorbed layer, Korolkovas et al. [48] have used
neutron reflectivity to study the effect of flow rate on the interface between
a dPS/DEP solution and PS brushes in a cone-plate rheometer. They have shown
that the thickness of the brush decreases when increasing the Weissenberg
number above $1$, which in their case is $\dot{\gamma}\tau$, with
$\dot{\gamma}$ the applied shear rate. The grafting densities of their brushes
varied between $0.04$ and $0.4$ nm-2. In our case we are at Wi$\leq 0.01$ and
we have a density of adsorbed chains of $0.002$ nm-2, and therefore we do not
see an effect of the flow even if there is adsorption.

In conclusion, we have shown directly using neutron reflectivity that dPS/DEP
solutions near a sapphire surface exhibits depletion. Interestingly, we show
that depletion of polymer chains does not prevent some chains to adsorb onto
the sapphire surface, probably due to a favorable interaction between the
phenyl groups of the chains and the exposed hydroxyl groups of the solid
surface. In addition, we show that the flow rate has no effect on the
depletion layer up to Weissenberg number Wi $\approx 0.01$. These results give
a precious insight into the interfacial structure of a semi-dilute polymer
solution flowing onto smooth surfaces. Further studies with other solvents and
at higher Weissenbeg number would help to understand the scope of these
results.

##  III Experimental method

### Solutions

The solutions are made of fully deuterated polystyrene (dPS) (Polymer Source,
$\text{{\char 208\relax}}=1.17$ $M_{n}=195$ kg/mol or $M_{n}=1.56$ Mg/mol) in
hydrogenated diethylphtalate, DEP (Sigma Aldrich), which is a good solvent for
PS (see Appendix 1). The volume fractions $\phi$ ($3$ and $6$ $\%$) are chosen
to be in the semi-dilute regime $\phi>\phi^{*}$ with $\phi^{*}\approx
N^{-4/5}$ the overlap volume fraction, which is about $0.24\%$ for the $195$
kg/mol polystyrene, and $0.06$ for the $1.56$ Mg/mol one. Solutions are
homogenized under gentle stirring during two weeks prior to the experiment.
The wall of interest is a polished sapphire surface from Fichou. The cell is
drawn in the Supp. Mat. (Fig. S2). The top surface is a PTFE surface of size
$4$x$8$ cm. The gap $h$ is $1$ mm, controlled by a rectangular Viton frame.
Between each experiment, the cell is rinsed with clean toluene (Sigma
Aldrich). The sapphire is dried with nitrogen and put under a UV-ozone lamp
(ProCleaner${}^{\text{TM}}$ Plus, Bioforce Nanosciences) during at least $30$
minutes.

### Neutron reflectivity

The flow is applied through a Chemyx Fusion 6000 syringe pump with $50$ mL
steel syringes. The flow rate can be varied between $10^{-4}$ and $270$
mL/min, both in injection and withdrawal. Measurements are done either with a
constant flow or with an alternating injection/withdrawal flow, in which case
the acquisition of reflected neutrons is synchronized with the frequency of
the flow. We call the latter procedure ”stroboscopic measurement”. The
experiment has been conducted on D17 [49] at Institut Laue-Langevin (ILL). We
use Time-Of-Flight (TOF) reflectivity at two angles of incidence $0.5\degree$
and $2.5\degree$, with acquisition times of $30$ and $60$ minutes
respectively. The Scattering Length Densities (SLD) of the substrate and the
solvent have been measured beforehand and are $5.77$ $10^{-6}$ Å -2 and $1.62$
$10^{-6}$ Å -2 respectively (see Supp. Mat. Fig. S3), in good agreement with
theoretical values. The abrupt transition of SLD between the sapphire and the
interfacial liquid is modeled by a rough interface.

### X-ray reflectivity

The X-ray reflectivity (XRR) measurements were performed on a Xeuss 2.0
instrument (Xenocs) with a Cu k-$\alpha$ source of wavelength $1.54$ Å and a
Pilatus 1M 2D detector (Dectris). The experiment is conducted under vacuum,
with a sample-detector distance of $1.214$ m. We use two collimation slits set
to $0.5$ mm x $1$ mm and $0.3$ mm x $1$ mm (height x width).

###### Acknowledgements.

This work was supported by the ANR-POILLU program (Grant No. ANR-19-CE06-007).
We thank O. Tessier for machining the cell. We thank the ILL for beamtime
(DOI: 10.5291/ILL-DATA.9-11-2020).

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