# LayerDiffusion: Layered Controlled Image Editing with Diffusion Models

Pengzhi Li 1 Qinxuan Huang 1 Yikang Ding 1 Zhiheng Li 1
1 Tsinghua University
Corresponding author.

###### Abstract

Text-guided image editing has recently experienced rapid development. However,
simultaneously performing multiple editing actions on a single image, such as
background replacement and specific subject attribute changes, while
maintaining consistency between the subject and the background remains
challenging. In this paper, we propose LayerDiffusion, a semantic-based
layered controlled image editing method. Our method enables non-rigid editing
and attribute modification of specific subjects while preserving their unique
characteristics and seamlessly integrating them into new backgrounds. We
leverage a large-scale text-to-image model and employ a layered controlled
optimization strategy combined with layered diffusion training. During the
diffusion process, an iterative guidance strategy is used to generate a final
image that aligns with the textual description. Experimental results
demonstrate the effectiveness of our method in generating highly coherent
images that closely align with the given textual description. The edited
images maintain a high similarity to the features of the input image and
surpass the performance of current leading image editing methods.
LayerDiffusion opens up new possibilities for controllable image editing.

##  1 Introduction

Given a single image of your pet, it can be imagined embarking on a worldwide
journey and performing specific actions in any location. Generating such an
image is a challenging and fascinating task in image editing. It entails
preserving the specific subject’s unique characteristics in new backgrounds
and ensuring their seamless integration into the scene, harmoniously and
naturally, while simultaneously accommodating multiple editing actions.

Recently, significant progress has been made in the development of deep
learning-based large-scale text-to-image models [27, 30, 25]. These models can
generate high-quality synthetic images based on text prompts, enabling text-
guided image editing and producing impressive results. As a result, numerous
text-based image editing methods [36, 13, 10, 7, 28, 8, 35] have emerged and
evolved. However, such models cannot mimic specific subject characteristics.
Even with the most detailed textual descriptions of an object, they may
generate instances with different appearances and still struggle to maintain
background consistency. Thus, the current leading image editing methods
encounter several challenges, including rigid editing limited to specific
domain images [22, 13], the inability to simultaneously edit both the
background and specific subjects, and the requirement for additional auxiliary
input information [28, 21, 3, 5]. These issues hinder the advancement of
controllable image editing.

In this paper, we propose a semantic-based layered controlled image editing
method, which we call LayerDiffusion, to alleviate these issues. By simply
inputting textual descriptions of multiple editing actions, along with the
target image and a reference image, we can perform non-rigid editing and
attribute modification of specific subjects, generating images consistent with
the textual descriptions while maintaining the consistency of the specific
subject and background features with the input image. As shown in Fig. 1, we
can make a dog jump in a forest or a giraffe lies on a beach or modify their
shapes and attributes in the original scene.

To implement our method, we leverage the robust and high-quality image
generation capabilities of a large-scale text-to-image model [27]. Our method
comprises a well-defined sequence of steps. Initially, we utilize a mask to
eliminate interference from foreground objects effectively. Subsequently, we
apply a layered controlled optimization strategy to optimize the text
embeddings acquired from the text encoders [24], following the segmentation of
the target text. This process aims to generate image backgrounds that exhibit
a remarkable similarity to the reference images. Next, we employ a layered
diffusion training strategy to fine-tune the model, thereby augmenting its
ability to preserve the similarity between the specific subjects, backgrounds,
and input images. Finally, during the diffusion process with the fine-tuned
model, we adopt an iterative guidance strategy, where a highly constrained
text embedding is iteratively employed to denoise the images. Consequently,
this generates a final image aligning with the textual description.

We emphasize the contributions of each component in our method through
ablation studies and compare our approach with other relevant image editing
methods [17, 36, 20], clearly demonstrating superior editing quality.
Furthermore, we conduct a user study to subjectively evaluate the quality of
the images generated by our method, which aligns most closely with human
perception. We summarize our main contributions as follows:

Figure 1: Our method achieves layered image editing through text descriptions,
enabling simultaneous modifications of backgrounds and specific subjects, such
as background replacement, object resizing, and complex non-rigid changes.

  * •

``We propose LayerDiffusion. To the best of our knowledge, this is the first
image editing method that enables simultaneous editing of specific subjects
and backgrounds using a single input image.

  * •

We introduce a novel layered diffusion training framework that enables
arbitrary and controllable editing of specific subjects and backgrounds.

  * •

Experimental results demonstrate that our method generates images with highly
similar features to the input images.

##  2 Related Work

Image synthesis has recently made significant advancements [1, 12, 22, 26, 39,
2, 11, 32]. With the development of diffusion models [15, 33, 34] in image
processing tasks [16, 29, 37, 38, 6], new text-guided solutions have emerged
in the field of image editing and produced impressive results [21, 25, 40, 7,
17]. The powerful generative capabilities of diffusion models enable the
generation of numerous high-quality images. Consequently, many image editing
tasks [4, 13, 20, 36]no longer require training a large-scale text-to-image
model, as pre-trained models can be used for image editing based on textual
descriptions. Diffusion models have tremendous potential for image editing
tasks guided by text descriptions. Many studies [20, 17, 28, 36, 13] have
utilized pre-trained models as generative priors, which can be categorized
into two approaches: training-free and fine-tuned method. SDEdit [20]
introduces intermediate noise to an image, which can be augmented with user-
provided brush strokes, followed by denoising through a diffusion process
conditioned on the desired edit. P2P [13] and PnP [36] utilize cross-attention
or spatial features to edit both global and local aspects of an image by
directly modifying the text prompt. However, they often preserve the original
layout of the source image and struggle with non-rigid transformations.

Fine-tuned methods [17, 28, 18, 10] have also shown remarkable performance.
DiffusionCLIP [18] leverages the CLIP [23] model to provide gradients for
producing impressive style transfer results. Textual-inversion [10] and
Dreambooth [28] fine-tune the model using multiple sets of personalized
images, resulting in the synthesis of images depicting the same object in new
environments. Imagic [17] fine-tunes the model by optimizing text embeddings
and achieves image editing through linear interpolation of text embeddings.

Similarly, our approach leverages target text descriptions to fine-tune the
model and enable various image editing operations. Dreambooth [28] and Imagic
[17] are methods that resemble our approach. However, Dreambooth requires
multiple input images and often fails to produce satisfactory results when
dealing with a single image. Imagic, on the other hand, faces challenges in
simultaneously performing multiple editing actions, such as editing both the
background and specific subjects simultaneously. In contrast, our method
allows for simultaneous editing of specific subjects and the background using
only a single input image.

Figure 2: Our method utilizes a layered controlled optimization strategy to
refine text embeddings and a layered diffusion strategy to fine-tune the
diffusion model. During inference, an iterative guidance strategy is employed
to directly generate images aligning with the multiple editing actions
described in the input text.

##  3 Method

###  3.1 Preliminaries

Stable Diffusion Models (SDM) [27] is a publicly available text-to-image
diffusion model trained on the LAION-5B [31] dataset. Instead of directly
operating in the image space, SDM is based on the latent diffusion method,
which means the forward and reverse diffusion sampling are operated in the
latent space. Given the trained autoencoder, the image $\boldsymbol{p}$ is
convert to low-dimensional latent variable $\boldsymbol{x}$ at each timestep
t. SDM also introduces an important modification in the form of text-based
conditioning. During the denoising process, SDM can be conditioned on an
additional input vector, which is typically a text encoding produced by a pre-
trained CLIP text encoder $\mathcal{P}$. Specially, the $\mathcal{P}$ extract
words from a given text prompt $\boldsymbol{y}$ and convert them into tokens,
denoted by $\boldsymbol{e}=\tau_{\phi}(\boldsymbol{y})$. These tokens are
further transformed into text embeddings, which are used to condition the
neural network during the training process:

$\min_{\theta}\mathbb{E}_{\boldsymbol{x}_{t},\boldsymbol{x}_{0},\boldsymbol{\epsilon}\sim\mathcal{N}(0,\boldsymbol{I})}\left[\left\|\boldsymbol{\epsilon}_{\theta}\left(\boldsymbol{x}_{t},t,\tau_{\phi}(\boldsymbol{y})\right)-\boldsymbol{\epsilon}\right\|_{2}^{2}\right],$
(1)

Consequently, SDM facilitates the generation of images based on textual input
by employing reverse diffusion sampling in the latent space. Instead of
relying on $\boldsymbol{\epsilon}_{\theta}(\boldsymbol{x}_{t},t)$, the model
utilizes a text-conditioned neural network denoted as
$\boldsymbol{\epsilon}_{\theta}(\boldsymbol{x}_{t},t,\tau_{\phi}(\boldsymbol{y}))$.
We implement the proposed approach in this work by fine-tuning this pre-
trained model.

###  3.2 Layered Diffusion

Our approach leverages target text descriptions to facilitate a wide range of
image editing actions, including object size, property modifications, and
background replacement while preserving specific subject details closely tied
to the original image. To achieve this, we fine-tune a state-of-the-art
diffusion model [27]. Furthermore, we introduce a layered editing method for
the background and specific foreground objects. As illustrated in Fig. 2, our
method begins by separating the background. We apply a layered controlled
optimization strategy to refine the segmentation text embeddings acquired from
the text encoders, which come from the target text.Then we identify the
optimal text embedding that aligns with the desired target background in
proximity to the target text embedding. Subsequently, we employ a layered
diffusion strategy to fine-tune the diffusion model. This approach enhances
the model’s capability to maintain similarity between specific subjects,
backgrounds, and input images, allowing for finer control and precision in
image editing through parameter adjustments. During the inference stage, we
utilize an iterative guidance strategy to directly generate images that align
with the multiple image editing actions described in the input text without
the text embedding interpolation. Each step of the process is outlined in
detail below.

####  3.2.1 Layered controlled optimization

Due to the potential interference of multiple text descriptions, optimizing
text embeddings can be unstable during image editing. As a result, previous
methods for image editing have often struggled to effectively modify selected
object property and backgrounds simultaneously.

To this end, we aim to separate the background and foreground to reduce
interference between different textual information. The target text $T$ is
first fed into the Stable Diffusion model [27] to obtain the target image
$O_{t}$. Then $T$ is decomposed into $T_{a}$ and $T_{b}$, which describe
object properties and background separately and sent to the text encoder [24]
to output the corresponding text embeddings
$\boldsymbol{e}_{a}\in\mathbb{R}^{C\times N}$ and
$\boldsymbol{e}_{b}\in\mathbb{R}^{C\times N}$, where $C$ is the number of
tokens, and $N$ is the token embedding dimension. However,
$\boldsymbol{e}_{a}$ and $\boldsymbol{e}_{b}$ are in the distant embedding
space, so we cannot directly perform linear interpolation on them. To make
$\boldsymbol{e}_{a}$ and $\boldsymbol{e}_{b}$ match our input image background
as much as possible and be in a close embedding space, we freeze the
parameters of the diffusion model and optimize $\boldsymbol{e}_{a}$ and
$\boldsymbol{e}_{b}$ simultaneously using the diffusion model objective [15].
In fact, we can optimize the initial text embedding to make it closer to the
target image (modify the background) space or reference image (modify object
properties) space. This process is controlled by the object mask $M$ and can
be represented as follows:

$\begin{bmatrix}\boldsymbol{\hat{e}}_{a},\boldsymbol{\hat{e}}_{b}\end{bmatrix}=\arg\min\mathbb{E}_{\boldsymbol{x}_{t},\boldsymbol{\epsilon}\sim\mathcal{N}(0,\boldsymbol{I})}\left[\left\|{M}*(\boldsymbol{\epsilon}-f_{\theta}\left(\boldsymbol{x}_{t},t,\begin{bmatrix}\boldsymbol{e}_{a},\boldsymbol{e}_{b}\end{bmatrix})\right)\right\|^{2}\right],$
(2)

where $M$ is computed by Segment Anything Model (SAM) [19], and
$\boldsymbol{x}_{t}$ is the noisy version of the input image, and $f_{\theta}$
means the forward diffusion process using pre-trained diffusion model. The
optimized text embeddings make it meaningful to modify the linear
interpolation weights of $\boldsymbol{\hat{e}}_{a}$ and
$\boldsymbol{\hat{e}}_{b}$ as follows:

$\boldsymbol{e}_{opt}=\alpha*\boldsymbol{\hat{e}}_{a}+(1-\alpha)*\boldsymbol{\hat{e}}_{b},$
(3)

according to the experimental analysis of text embedding interpolation in
Imagic [17], we tend to set the weight $\alpha$ that describes object
properties to 0.7.

Figure 3: Given a complex text description, the original image (left) is
capable of performing multiple editing actions and maintaining similar
characteristics of a specific subject. Note that the mask in the bottom left
corner is used to change the size of the selected object.

####  3.2.2 Model fine-tuing

We obtain new text embeddings $\boldsymbol{e}_{opt}$ by linearly interpolating
multiple optimized text embeddings. Due to the limited number of optimization
steps, the resulting embeddings may not lead to a consistent representation of
the selected objects or background in the input image. Therefore, we propose a
layered diffusion strategy to optimize model parameters while freezing the
optimized text embeddings $\boldsymbol{e}_{opt}$. This enables the model to
fit the desired image at optimized text embedding points. To achieve the
arbitrary modification and combination of foreground object properties and
backgrounds, we employ SAM [19] to derive $M_{t}$ (object) and $1-M_{t}$
(background) from $O_{t}$ and subsequently obtain $M_{r}$ (object) and
$1-M_{r}$ (background) from the reference image $O_{r}$. The aforementioned
can be achieved by optimizing the following equations:

$\mathcal{L}_{obj}=\mathbb{E}_{\boldsymbol{x}_{t},\boldsymbol{\epsilon}\sim\mathcal{N}(0,\boldsymbol{I})}\left[\left\|{M}_{t}*(\boldsymbol{\epsilon}-f_{\theta}\left(\boldsymbol{x}_{t},t,e_{opt})\right)\right\|^{2}\right],$
(4)
$\mathcal{L}_{bg}=\mathbb{E}_{\boldsymbol{x}_{t},\boldsymbol{\epsilon}\sim\mathcal{N}(0,\boldsymbol{I})}\left[\left\|(1-{M}_{r})*(\boldsymbol{\epsilon}-f_{\theta}\left(\boldsymbol{x}_{t},t,e_{opt})\right)\right\|^{2}\right],$
(5)

The total loss can be represented as follows:

$\mathcal{L}_{total}=\lambda_{1}\mathcal{L}_{obj}+\lambda_{2}\mathcal{L}_{bg},$
(6)

This approach enables us to manipulate the foreground object and background
independently, allowing for precise control over the final output image.

####  3.2.3 Iterative guidance strategy

We first represent the diffusion process of a pre-trained model as follows:

$\begin{array}[]{lcr}I_{T},I_{T-1},\ldots,I_{0}.\
I_{t-1}=D(I_{t}|y)\end{array}$ (7)

where $D$ represent an update process:
$\mathcal{I}\times\mathcal{C}\rightarrow\mathcal{I}$,
$\mathcal{I}\in\mathbb{R}^{H\times W\times C}$ is the image space, and
$\mathcal{C}$ is the condition space, and $y\in\mathcal{C}$ is a text prompt.
From ${T}$ to ${0}$, $I_{T}$ gradually changes from a Gaussian noise
distribution to a desired image by $y$. Nonetheless, due to the significant
gap between the initial image and the desired image in our task, applying the
base generative diffusion process with fine-tuned models under condition
$y(i.e.,e_{opt})$ may still result in failures in modifying object properties
in sometimes, such as modifications of actions.

This issue in image editing is due to the lack of a strong constraint
corresponding to the text description of the edited attributes in the
diffusion process. The network bias leads the diffusion model to favor object
properties in the initial image. To address this, we strengthen the object
properties by utilizing the decomposed $\boldsymbol{\hat{e}_{a}}$ in the
diffusion process. Specifically, we perform the following approach:

$I_{t-1}=\begin{cases}D(I_{t}|\boldsymbol{\hat{e}_{a}}),&\text{if
$t$\%$2=0$}\\\ D(I_{t}|\boldsymbol{e_{opt}}),&\text{otherwise}\end{cases}$ (8)

###  3.3 Implementation details

We adopt the stable diffusion text-to-image model [27] as the baseline for our
method. Specifically, we utilize the publicly available v1.4 version of the
model, which was pre-trained on the LAION-5B dataset [31] and built upon the
latent diffusion model. We first fine-tune the text embeddings with a learning
rate of 1e-3 using Adam [9] and perform 500 steps in most of our experiments.
Subsequently, we fine-tune the diffusion model itself, using a learning rate
of 2e-6 and executing 250 steps. We employ an iterative guidance strategy
throughout the diffusion process, starting from random noise. This iterative
process consist of 50 iterations by default, resulting in more refined
results. For one image, it takes about 2 minutes to run on a single NVIDIA
A100 GPU.

##  4 Experiments

###  4.1 Qualitative Evaluation

We extensively evaluate our approach using images from various domains and
categories. Our method involves a simple text prompt-based editing process,
allowing for tasks such as background replacement and object property
modification. The images utilized in our experiments are copyright-free on the
Internet. We employ a layered editing strategy to ensure robustness and
controllability in the editing process. This approach enables multiple editing
actions simultaneously on the images, demonstrating excellent editing
controllability. The probabilistic diffusion model also motivates us to test
our method under different random seeds. By employing our layered diffusion
strategy, we can generate images that closely match the provided text
descriptions while preserving the critical attributes of the original image in
most cases. Our method produces multiple editing results from a single text
prompt, providing users with a selection of options to choose from.

In Fig. 3, we present some edited images. These images preserve the distinct
characteristics of the input image, and they are altered based on text prompts
to accommodate a range of editing actions that go beyond mere background
replacement and property modification. Our method can execute precise editing
actions on the images by leveraging reference background or foreground
objects. For instance, we can alter foreground objects based on reference
foreground object maps or implement background modifications guided by
reference background maps. More results can be found in the supplementary
material.

Figure 4: We present several edited images and compare them with similar image
editing algorithms, such as SDEdit [20], Imagic [17], and PnP [36]. Our method
generates the best results.

| Settings |  | CLIP score
---|---|---|---
| $\mathcal{L}_{obj}$ | $\mathcal{L}_{bg}$ |
(a) | $\times$ |  | 0.28
(b) |  | $\times$ | 0.32
(c) |  $\lambda_{1}$ = 1 |  | 0.29
(d) |  $\lambda_{1}$ = 3 |  | 0.32
(e) |  $\lambda_{1}$ = 1 |  $\lambda_{2}$ = 3 | 0.28
(f) |  |  | 0.35

| Settings |  |  |  | CLIP score
---|---|---|---|---|---
| L-c-o | Fine-tune | I-g | $\alpha$ |
(g) | $\times$ |  |  |  | 0.33
(h) |  | $\times$ |  |  | 0.32
(i) |  |  | $\times$ |  | 0.29
(j) |  |  |  | = 1 | 0.32
(k) |  |  |  | = 0 | 0.29
(f) |  |  |  |  | 0.35

Table 1: Quantitative results with different settings. We report the CLIP
score [14] over 300 images.

###  4.2 Comparisons

We primarily compare our proposed image editing method with previous text
prompt methods, such as SDEdit [20], Imagic [17], and PnP [36]. It is worth
noting that Imagic [17] necessitates fine-tuning of both the network and text
embeddings, while our method adopts a similar fine-tuning approach.

As shown in Fig. 4, non-rigid editings, such as jumping and rotation, have
significant challenges in image editing tasks. This complexity leads to the
failure of both PnP [36] and SDEdit [20] in performing editing actions.
Additionally, Imagic [17] tends to produce overfitting of the original image
and text embeddings during training, thereby making accurate image editing
difficult, especially when modifying text prompts go beyond attribute editing
and involve adding other editing actions, such as foreground-background
editing simultaneously. In contrast, our approach adopts a layered strategy
that allows for the simultaneous execution of multiple editing actions. As a
result, our method achieves impressive results in real image editing tasks.
The last two columns of Fig. 4 show the edited results generated by employing
different random seeds. Our method outperforms others in multitask editing
performance.

In Fig. 4, we generate a reference background image from the diffusion model
[27], and our layered diffusion approach allows us to make the edited image as
close as possible to the reference background image. We can also choose our
reference image as long as it is close to the perspective of the original
image. We show more results in the supplementary material.

Text-based image editing methods are a relatively new direction, and there is
currently no standard benchmark for evaluating our approach. Although Imagic
[17] propose the TEdBench, it includes only a single non-rigid edit, which is
also not fully applicable to our approach. To further assess the quality of
our generated results, we utilize the TEdBench dataset to generate over 300
images per method for a preliminary evaluation. The supplemental material
includes the used text prompts. We use a CLIP-based text-image clip score [14,
23], which measures the cosine similarity between the text prompt and the
image embeddings. As our method aims to maintain proximity between selected
object features before and after non-rigid editing, the CLIP score does not
effectively demonstrate the superiority of our approach (see Fig. 5 (f)-(g)
and Tab. 1 (f)-(g)). However, it still partially reflects the state of our
image editing, such as changes in the motion of the selected object. Tab. 1
provides an approximate representation of the CLIP scores for several methods,
and our approach achieves the highest score.

Figure 5: We present the edited images with different settings. For each
setting, we show two generated images using different random seeds. (f)
illustrates the final edited results.

###  4.3 Ablation Study

In this section, we present a comprehensive analysis of the three modules
employed in our method. We utilize the TEdBench [17] dataset and generate over
300 images using 20 different random seeds. As an auxiliary objective
evaluation metric, we employ the text-image CLIP score [14], which is
presented in Fig. 7. Furthermore, we present the specific performance of each
component in Tab. 1. As mentioned previously, the CLIP score may not fully
capture the suitability of our method as it primarily focuses on the alignment
between images and text. For instance, the results of (b), (g), and (i) show
high CLIP scores, but their object features significantly differ from the
reference images.

As shown in Fig. 5 and Tab. 1, (a) does not utilize $\mathcal{L}_{obj}$,
resulting in a background that matches the reference image, while the
properties of the foreground objects differ substantially. On the other hand,
(b) demonstrates that $\mathcal{L}_{bg}$ preserves a more similar background.
(c), (d), and (e) analyze the impact of different weights assigned to the two
losses, which affect the similarity of the background and foreground objects.
In this paper, we mostly set $\lambda_{1}$ to 2 and $\lambda_{2}$ to 1, except
when $\lambda_{1}$ is set to 3 for smaller foreground objects. (g), (h), and
(i) validate the effectiveness of each of the three modules in our method. (g)
enhances the similarity of the background, (h) controls the global features,
and (i) significantly increases the percentage of image generation results
that satisfy the description text, rising from 43$\%$ to 81$\%$.

Figure 6: We present several failure cases, including artifacts on faces and
significant disparities in the camera angles of the images.

###  4.4 User Study

Furthermore, we conduct a user study to evaluate and compare the subjective
perception of our method with several other approaches. To ensure a fair
comparison, we randomly select ten generated images and utilize two random
seeds to generate our results. We present two discriminant conditions for each
image: background similarity and action similarity. We then ask 20
participants to rate the resulting images on a scale ranging from 1 to 5, with
a rating of 5 indicating a very good match and 1 indicating a very poor match.
The histogram on the right-hand side of Fig. 7 shows the average scores.
Remarkably, our method achieves optimal subjective performance compared to the
other methods.

Figure 7: We compare several image editing methods using the CLIP and
subjective user perception scores. Our method achieves a relatively higher
score.

##  5 Limitations

While our approach demonstrates superior performance in achieving controlled
image editing and accomplishes remarkable results in tasks involving multiple
editing actions, it is essential to acknowledge three significant challenges.
(1) Dealing with fine-grained tasks is still challenging for our method while
we rely on a pre-trained text-to-image diffusion model and the problem of
overfitting that occurs during model fine-tuning. Fig. 6 demonstrates that our
method will produce artifacts when confronted with textures with intricate
details or facial features. (2) As shown in the Fig. 6, another challenge
arises when there is a notable disparity in camera angles between the input
reference image and the desired edited image, leading to the creation of
visually inconsistent scenes. This limitation can be mitigated by
incorporating additional descriptions about the camera position in the target
text. (3) We need to fine-tune the model to accommodate the reference image.
Appropriately fine-tuning specific parameters may be required for
unconventional or atypical manifestations to generate good results in
sometimes.

##  6 Conclusion

We propose LayerDiffusion, a semantic-based layered image editing method that
simultaneously edits specific subjects and backgrounds using a single input
image. LayerDiffusion preserves the unique characteristics of the subjects
while integrating them seamlessly into new scenes. Extensive experimentation
demonstrates that our method generates images closely resembling the feature
of the input images, surpassing existing approaches in editing quality and
controllability. User studies confirm the subjective perception of the
generated images, aligning with human expectations. Our contributions include
introducing LayerDiffusion as the first method for simultaneous editing of
specific subjects and backgrounds. We develop a layered diffusion training
framework for controllable image editing, which opens up new possibilities for
text-guided image editing tasks. We may focus on preserving complex textures
and facial features in the future.

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Figure 8: We present more edited results. Each triplet consists of the
original image on the left, the edited result on the right, and a small
reference image enclosed within a red box.

##  Appendix A More Results

We present additional image editing results in Figure 1, demonstrating the
versatility of our approach across various image types.

##  Appendix B More Ablation Study

###  B.1 Training

We conduct an analysis of the number of layered controlled text embeddings
optimization steps, as well as the fine-tuning steps. When the number of
optimization steps for text embeddings is too low, the similarity of image
features is relatively low. Conversely, when the number of steps is
excessively high, the quality of generated images deteriorates. Similarly,
when the number of fine-tuning steps is insufficient, the generated images
exhibit lower feature similarity. Conversely, with an excessively high number
of fine-tuning steps, the content of the images gradually deviates from the
textual cues. After extensive experimentation, we determine that the optimal
configuration for the number of layered controlled optimization steps is 500,
while the number of fine-tuning steps is set to 250.

Figure 9: We employ two random seeds to show the image editing results as the
number of steps gradually increase.

###  B.2 Text embeddings interpolation

We first review the process of linear interpolation of the optimized text
embedding in the main paper:

$\boldsymbol{e}_{opt}=\alpha*\boldsymbol{\hat{e}}_{a}+(1-\alpha)*\boldsymbol{\hat{e}}_{b},$
(9)

This linear interpolation is analyzed a bit in Imagic [17]. But our approach
differs from it in that we are interpolating two optimized text embeddings
while preserving the features of both text embeddings. Next, we will analyze
the choice of the value of $\alpha$. In Fig. 11 and Fig. 10 , we set $\alpha$
to 1 and 0. Fifty different sets of results are generated using different
random seeds under the same training parameters, and the image editing results
that satisfy the text prompt are less. In Fig. 12, we also set $\alpha$ to
0.3, which is the opposite of our setting in the experiment, and the image
editing results that satisfy the text prompt are also less. This indicates
that an appropriate value of a can improve the image editing results that
satisfy the text prompt.

We add a strong constraint on the object properties, which also substantially
increases the validity of our generated results. In Fig. 13, we remove this
module and the number of valid editing results is drastically reduced compared
to our final results in Fig. 14.

Figure 10: We show the results of 50 randomly generated edited image with
$\alpha$ set to 1, where the red boxes indicate the images that satisfy the
text prompt. Figure 11: We show the results of 50 randomly generated edited
imagewith $\alpha$ set to 0, where the red boxes indicate the images that
satisfy the text prompt. Figure 12: We show the results of 50 randomly
generated edited image with $\alpha$ set to 0.3, where the red boxes indicate
the images that satisfy the text prompt. Figure 13: We present 50 sets of
randomly generated results without employing an iterative guidance strategy.
The red boxes indicate the edited images that satisfy the text prompt. Figure
14: In our final experiment, we include all the utilized modules and
parameters, resulting in a collection of image editing outputs. The number of
edited images satisfying the text prompt within the red boxes is remarkably
large.


# Evaporation of liquid coating a fiber

M. Corpart  Université Paris-Saclay, CNRS, Laboratoire de Physique des
Solides, 91405, Orsay, France.  J. Dervaux  Université de Paris, Laboratoire
Matière et Systèmes Complexes, UMR 7057 CNRS, F-75013 Paris, France  C.
Poulard  Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides,
91405, Orsay, France.  F. Restagno  Université Paris-Saclay, CNRS, Laboratoire
de Physique des Solides, 91405, Orsay, France.  F. Boulogne  Université Paris-
Saclay, CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France.

###### Abstract

We investigate theoretically and numerically the diffusion-limited evaporation
of a liquid deposited on a fiber in two configurations: a sleeve and a
axisymmetric barrel-shaped droplet. For a sleeve, the local flux depends on
both the aspect ratio and the smallest length of the problem. By using
analytical calculations and 3D finite elements simulations, we predict a
divergence of this flux further localized at the edge as the aspect ratio
increases. The evaporation of axisymmetric drops on a fiber is studied with
numerical simulations. For sufficiently large volumes, we evidence that the
evaporation rate is almost independent of the wetting properties of the
liquid, even for small contact angles, and that the droplets evaporate as
spheres of the same volume.

##  1 Introduction

The coating of liquid on a solid is a common operation in everyday life as
well as in industrial processes (e.g. lubrication, painting). The first
theoretical description of the deposited layer has been made almost
simultaneously by Landau and Levich [1] and Derjaguin [2]. The coated
thickness $h$ is found to be proportional to the radius $a$ of the fiber and
depends in particular of the entrainment velocity $V$, the liquid surface
tension $\gamma$ and the fluid viscosity $\eta$ through the so-called
capillary number $\textrm{Ca}=\eta V/\gamma$, which compares the velocity to a
fluid intrinsic velocity. Thus, the thickness $h$ can be written as
$h=a\,f(\textrm{Ca})$, where $f$ is a function which can be simplified under
certain conditions on the capillary ranges. Coating of complex fluids also
received some attention, such as non-Newtonian fluids or water-surfactant
solutions [3, 4].

Due to the fiber curvature, the liquid film is unstable; a mechanism that was
first characterized by Plateau [5] and rationalized by Lord Rayleigh in the
19th century [6]. The liquid surface tension leads to the minimization of its
surface area by breaking the film into a series of regularly spaced droplets.
The characteristic growing time of the Rayleigh-Plateau instability
$\tau=12\eta a^{4}/(\gamma h^{3})$ depends significantly on the coating
thickness. For non-Newtonian fluids, or for colloidal suspensions, the
Rayleigh-Plateau instability can be delayed or suppressed [7, 8, 9].

Carroll [10] described the shape of axisymmetric drops on a fiber, which
depends on the wetting properties of the material and the droplet volume with
respect to the fiber size. Based on Carroll’s results, the drop height and the
length of the wetted area can be used to characterize the wettability of the
liquid on the fiber [11, 12, 13, 14]. Beyond the axisymmetric shape, the
equilibrium conformation of a drop on a fiber can also be a clam-shell: a drop
sitting on a curved surface. Experimental observations have evidenced that
both barrel and clam-shell conformations can coexist and Chou et al. studied
in detail the phase diagrams of droplet-on-fiber with or without gravity [15].

The aim of the paper is to describe the different evaporation regimes of the
fiber coatings when the evaporation is dominated by the diffusion of water in
the atmosphere. Despite the fact that this is a model system with many
industrial applications, there is a lack of theoretical rationalization on
this subject.

In 1918, Langmuir [16] explained for the first time that the total diffusive
evaporation flux over a spherical droplet of size $R$ is not proportional to
$R^{2}$, which means that, due to the curvature of the surface of the droplet,
the evaporation flux is not proportional to the surface of the drop. The case
of a sessile drop evaporation (see [17, 18] for a review) is more complex
because of the diverging evaporative flux at the triple line [19, 20, 21] and
several studies have been devoted to the nature of the substrate. For
instance, the case of a drop on a tilted surface [22], on crossed fibers [23],
on superhydrophobic surfaces [24] or with complex wetting patterns [25] can be
cited. The situation of a droplet on a curved surface such as a convex or
concave surface [26, 27] has been studied as well, the latter corresponding to
a clam-shell on a fiber.

In this paper, we discuss the two limit cases of the diffusive evaporation of
a liquid deposited on a fiber. In a first part, we study the evaporation of a
liquid cylinder deposited on a fiber that we call the sleeve configuration.
This configuration corresponds to a deposited layer observed at a time much
smaller than the Rayleigh-Plateau characteristic time or when the instability
is inhibited. To do so, we will develop an analytical model to calculate the
evaporation, valid for small and large aspect ratios. These predictions are
supplemented by a full 3D numerical simulation using finite element method. In
the second part, we study the evaporation of a droplet on a fiber which
corresponds to destabilized coating on a fiber. More precisely, we restricted
the study to the axisymmetric barrel situation which is the common case, in
particular for small contact angles of the liquid on the fiber. We first
numerically calculate the shape of a droplet around a fiber using Surface
Evolver [28], a surface minimization algorithm, whose results can be compared
to the prediction by Carroll [10] and Chou et al. [15] in absence of gravity.
By using finite element computations, we calculate the evaporation flux around
the droplet, and show that except for very small liquid volume, the
evaporation rate of the droplets is not significantly affected by the presence
of the fiber, whatever the liquid contact angle is.

##  2 Sleeve

We consider a sleeve of liquid of length $2L$ and of radius $a$ as depicted in
figure 1(a), such that the liquid cylinder has the same radius as the fiber. A
natural geometrical parameter is $\lambda=L/a$.

Figure 1: (a) Notations used to describe a sleeve of length $2L$ on a fiber of
radius $a$. The dimensionless length is defined as $\lambda=L/a$. Boundary
conditions used to solve equation 1 as well as the coordinate system are also
represented schematically. Vapor concentration field for (b) $\lambda=5\cdot
10^{-3}$ and (c) $\lambda=500$, obtained from COMSOL for sleeves of radius
$a=125$ µm.

We assume that the liquid evaporation occurs in a diffusion-limited process in
the stationary regime. Thus, the concentration field $c$ in the gas phase is
the solution of the Laplace equation $\triangle c=0$, which reads in
cylindrical coordinates

$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial c}{\partial
r}\right)+\frac{\partial^{2}c}{\partial z^{2}}=0.$ (1)

In this geometry, the boundary conditions are (a) a saturated vapor
concentration $c_{\rm sat}$ in the vicinity of the interface, i.e.
$c(r=a,z)=c_{\rm sat}$ for $|z|<L$, (b) the absence of evaporative flux at the
surface of the fiber, i.e. $\left.\frac{\partial c}{\partial
r}\right|_{r=a}=0$ for $|z|>L$, and (c) a constant concentration far from the
liquid, $\lim\limits_{r\to\infty}c=c_{\infty}$.

We provide in Supplementary Materials the derivation of analytical solutions
to equation 1 in the limits of small and large aspect ratios $\lambda$. The
difficulty associated with this linear boundary-value problem (BVP) resides in
the discontinuity of the boundary conditions (BC) at the inner boundary $r=a$.
More precisely, two different types of BCs, namely a Dirichlet-type BC and a
Neumann-type BC are applied on disjoint complementary subdomains of the
cylindrical surface located at $r=a$. This difficulty prevents the use of the
classical Fourier-Hankel analysis of the problem and, instead, the BVP is
reduced to a set of dual integral equations. Following the theory of Sneddon
[29], these coupled integral equations are then reduced to a single integral
equation with a weak, logarithmic singularity that is solved analytically in
the asymptotic cases $\lambda\ll 1$ and $\lambda\gg 1$. From the resulting
concentration field $c(r,z)$, we can compute the local evaporative flux $j(z)$
defined as $j(z)=-{\cal D}\left.\frac{\partial c}{\partial r}\right|_{r=a}$,
where $\cal D$ is the diffusion coefficient of the vapor in the gas phase.

We also propose to solve equation 1 by using finite element method implemented
in the proprietary software COMSOL multiphysics using Transport of Dilute
Species physics in axisymmetric 2D geometry. To compute accurately $j(z)$
while keeping the computational time reasonable, we divide the atmosphere in
three concentric domains centered on the liquid. The maximum mesh size in
these domains is chosen according to the variations of concentration: the area
near the three-phase contact is meshed with more refinement to capture
properly the divergence of the concentration gradient at the contact line, and
as the distance to the fiber increases, the maximum mesh size allowance is
increased. The size of the box describing the atmosphere is at least one
hundred times larger than the largest of the lengths of the system in order to
be considered as infinite with respect to the above boundary conditions. With
this approach, the computational time is reduced to few hours for the largest
aspect ratios. The numerical computations are performed for a vapor saturating
concentration $c_{\rm sat}=1.8\cdot 10^{-2}$ kg/m3, a vapor concentration far
from the liquid $c_{\infty}=0$ and a diffusion coefficient of vapor in air
${\cal D}=2.36\cdot 10^{-5}$ m2/s [30]. These values are chosen to be those of
water evaporating at 20 ∘C and zero relative humidity $\mathcal{R}_{\rm
H}=c_{\infty}/c_{\rm sat}$. The evaporative flux being proportional to ${\cal
D}c_{\rm sat}(1-{\cal R}_{\rm H})$, these results can be easily generalized
for any value of ${\cal R}_{\rm H}$, ${\cal D}$ and $c_{\rm sat}$. Sleeves of
different aspect ratios are generated by changing independently the length and
the radius.

###  2.1 Results

In the limit $\lambda\ll 1$, we obtained analytically the local evaporative
flux

$j_{\rm small}(z)=j_{0}^{\rm small}\left(1-\frac{z^{2}}{L^{2}}\right)^{-1/2},$
(2)

where $j_{0}^{\rm small}$ is the local flux at $z=0$ defined as

$j_{0}^{\rm small}=\frac{{\cal D}(c_{\rm
sat}-c_{\infty})}{L(1-2\gamma_{e}-\ln\left(\frac{\lambda}{4}\right))},$ (3)

with $\gamma_{e}\approx 0.577$, the Euler gamma constant.

Because the geometry is not reduced to a single lengthscale, equation 3
indicates that the flux depends both on the sleeve length $L$ and the aspect
ratio $\lambda$, which is found numerically as shown in the Figure 2a. We find
a good agreement between the numerical results and equation 3.

Equation 2 indicates that the flux diverges at the edge between the liquid and
the solid. This behavior is confirmed numerically as observed in the Figure 2b
where numerical results obtained for various $\lambda$ are compared to the
analytical prediction of equation 2 represented in solid black line. This
divergence is also obtained for sessile droplets, which indeed exhibit the
same diverging expression in the limit $\theta\to 0$ [21].

We also solve analytically equation 1 for infinitely long sleeves i.e.
$\lambda\to\infty$. Under this hypothesis, the system is invariant by
translation along the fiber axis such that the local flux is uniform. We can
write

$j^{\rm large}_{0}=\frac{{\cal D}(c_{\rm
sat}-c_{\infty})\pi}{2a\left(2-2\gamma_{e}+\ln{2}+\frac{\pi}{2}\ln{\lambda}\right)}.$
(4)

We extend this result to large but finite aspect ratios by arguing that the
local evaporative flux remains mostly uniform, such that $j_{\rm
large}(z)\simeq j^{\rm large}_{0}$.

Again, we observe a good agreement between numerical and analytical results
for the local flux at the center of the sleeve $j_{0}$ as shown in Fig. 2a. As
for small $\lambda$, because of the cylindrical geometry, numerical results
(Fig. 2a) and equation 4 indicate that the flux depends on both the sleeve
radius $a$ and the aspect ratio $\lambda$.

For $\lambda>1$, Figure 2b shows that the increase of $\lambda$ leads to a
localization of the divergence at the contact line. For the largest tested
aspect ratios, the flux per unit surface is nearly constant along the sleeve
except close to the contact line. However, as shown in the inset of Figure 2b,
we still find close to the contact line the classical minus one-half power-law
divergence even for large aspect ratio. Indeed, mathematically, this
divergence with a power $-1/2$ is the only one that is solution of equation 1
to describe the divergence of the local flux at the contact line. For
$\lambda=5\cdot 10^{3}$, numerical uncertainties prevent us from concluding on
the expression of the divergence, but we expect that the classical square root
law holds.

The total evaporative flux is defined as $Q=\int j(z)\,{\rm d}S$ where the
integral is taken over the liquid-vapor surface area. In the limit of small
aspect ratios, the total flux writes

$Q_{\rm small}=\frac{2{\cal D}(c_{\rm
sat}-c_{\infty})a\pi^{2}}{1-2\gamma_{e}-\ln\left(\frac{\lambda}{4}\right)}.$
(5)

In the limit of large aspect ratio, we neglect the contribution of the
divergence. Indeed, this divergence is localized and contributes weakly once
integrated to the total flux [31]. Thus, the total evaporative flux writes
$Q_{\rm large}=j^{\rm large}_{0}\,4\pi aL$, which gives

$Q_{\rm large}=\frac{2{\cal D}(c_{\rm sat}-c_{\infty})\lambda
a\pi^{2}}{2-2\gamma_{e}+\ln{2}+\frac{\pi}{2}\ln{\lambda}}.$ (6)

As shown in figure 3, where we plot the dimensionless total flux $Q/(2{\cal
D}(c_{\rm sat}-c_{\infty})a\pi^{2})$ as a function of the aspect ratio
$\lambda$, we observe an excellent agreement between the numerical results and
the analytical model. The good agreement between numerical results and theory
at large aspect ratio demonstrates the small effect of the edge contribution
to the local flux, which is due to the localization of the divergence when
$\lambda$ increases. Indeed, the more localized the divergence, the more
constant the local flux $j$ can be considered and the less effect the
variation of $j$ with $z$ has on the total integrated flux $Q$.

As a final note, the model described here is valid asymptotically for
$\lambda\ll 1$ and $\lambda\to\infty$ but still provides an excellent
description of both the local flux in the center of the liquid $j_{0}$ (Fig.
2a) and the total evaporative flux $Q$ (Fig. 3) for $\lambda\sim 1$.

In the next section, we present the method to perform the numerical
computation of equation 1 with the associated boundary conditions for
axisymmetric droplets on fibers. We then discuss these results and compare
them to those obtained for sleeves and spherical drops.

Figure 2: (a) Local evaporative flux the center of the sleeve $aj_{0}$ as a
function of the dimensionless sleeve length $\lambda=L/a$ obtained from
numerical computations for various sleeves lengths $L$ and three different
radii $a$ (see caption). The black solid line corresponds to equation 3 and
the black dashed line corresponds to equation 4 both multiplied by $a$. In the
inset, the local flux at the center of the sleeve $j_{0}$ is plotted as a
function $L^{-1}$ for the three sleeves radii mentioned above. (b)
Dimensionless flux density $j/j_{0}$ as a function of dimensionless position
$z/L$ along the sleeve. The points are obtained from numerical computation for
$a=125$ µm and various sleeves lengths. The black solid line corresponds to
equation (2) nondimensionalized by equation 3. The black dashed line
corresponds to $j=j_{0}$ as predicted for infinitely long sleeve (Eq. 4). In
the inset, the flux $j$ is plotted as a function of the distance to the
contact line for $a=125$ µm and various sleeves lengths. The curves are
arbitrarily shifted for clarity. The color gradient represents the length $L$
of the sleeve on both graphics.  Figure 3:  Total dimensionless flux as a
function of the dimensionless sleeve length $\lambda=L/a$ obtained from
numerical computations for $a=125$ µm. The solid black line corresponds to
equation 5 and the dashed black line to equation 6.

##  3 Axisymmetric drop on a fiber

Due to the curvature of the substrate, a droplet of perfectly wetting liquid
put on a fiber does not necessarily adopt a sleeve morphology and a
macroscopic axisymmetric drop with a vanishing contact angle can exist [10,
11, 12, 13, 14, 15]. This morphology is called a barrel-shaped droplet and is
stable for low contact angle $\theta$ and quite large volume with respect to
$a^{3}$. For small volume and/or large contact angle a droplet on a fiber
adopt a clam-shell morphology. Here, we choose to focus on barrel shaped
droplets. To study the evaporation of such droplets, we vary independently the
drop volume $\Omega$ at constant fiber radius, $a=125$ µm, and the contact
angle $\theta$. In contrast to the sessile droplet for which the system is
completely defined by the choice of only two parameters, $\Omega$, and
$\theta$, the equilibrium profile of a drop on a fiber depends on $\Omega$,
$\theta$ and $a$. Once two dimensionless parameters, say $\theta$ and
$\Omega/a^{3}$ are chosen, the liquid adopts its equilibrium shape as
represented in figure 4a, where the profile of the drop is defined by $h(z)$.
We denote the height of the liquid at the apex $h_{0}=h(z=0)$ and the wetted
length $2L$. We define two dimensionless parameters, the dimensionless length
$\lambda=L/a$ and the drop aspect ratio $\mathcal{H}=h_{0}/L$.

The barrel shape morphology exists for a limited dimensionless parameter space
($\lambda,\mathcal{H}$), which restricts the studied range of these
parameters. Due to the complexity of the problem arising from the drop shape,
this part of the study is performed numerically.

In the next section, we present the method to get numerically the drop
equilibrium profile and then to solve equation 1 for this system.

###  3.1 Numerical procedure

In contrast to the numerical resolution of the sleeve configuration, our
numerical procedure for the drop on fibers is decomposed in two steps. First,
we used Surface Evolver [28], a computer program that minimizes the energy of
a surface subject to constraints, to obtain the meshed surface of the drop on
a fiber. The parameters are the dimensionless drop volume $\Omega/a^{3}$ and
the liquid contact angle $\theta$. The convergence of the minimization process
is testified by comparing the drop height $h_{0}$ and length $L$ (Fig. 4) with
the analytical predictions made by Carroll [10]. Illustrations of the
resulting shapes are presented in figure 4(a-c). The second step consists in
computing the vapor concentration field with COMSOL multiphysics in a similar
manner as for the sleeves except that the simulations are performed in 3D in
order to easily import the profiles obtained with Surface Evolver. With
COMSOL, the Surface Evolver meshes are converted in 2D to keep only the
boundary surface. A 3D drop is then reconstructed to ensure the compatibility
with other elements of the geometry and to mesh properly the contact between
the drop and the fiber. The boundary conditions are the same as for the
sleeve, the vapor concentration is equal to the saturating vapor concentration
near the interface and is constant far from the droplet i.e. $c(r=h,z)=c_{\rm
sat}$, $\lim\limits_{r\to\infty}c=c_{\infty}$. Initial conditions are
$c=c_{\infty}$ for the concentration in the atmosphere and $c=c_{\rm sat}$
inside the drop. There is no vapor flux normal to the surface of the fiber
$\left.\frac{\partial c}{\partial r}\right|_{r=a}=0$ for $|z|>L$ (Fig. 4a). As
for the sleeve case, the dimension of the box describing the atmosphere is
taken to be at least one hundred times larger than the longest lengths of the
system to ensure $c=c_{\infty}$ far from the droplet.

Figure 4: Surface Evolver profiles of axisymmetric drops on fiber of radius
$a=125$ µm. (a) Drop of volume $\Omega=1$ µL and contact angle
$\theta=10^{\circ}$. The wetted length is $L$ and the height of the drop
$h_{0}$ is defined between the surface of the fiber and the apex of the drop.
(b) Drop of volume $\Omega=0.01$ µL and contact angle $\theta=10^{\circ}$. (c)
Drop of volume $\Omega=1$ µL and contact angle $\theta=45^{\circ}$. (d) Vapor
concentration field from COMSOL for profile given in (a) $a=125$ µm,
$\Omega=1$ µL, $\theta=10^{\circ}$.

Figure 5: (a) Total evaporative flux of droplets on fiber as a function of
$\Omega^{1/3}$. (b) Total dimensionless evaporative flux as a function of
contact angle. The points are numerical results for evaporating drops on fiber
for different contact angles. Colors indicate the variation of the droplet
aspect ratio ${\cal H}=h_{0}/L$. The dashed black line is the evaporative flux
of a sphere (Eq. 7) in the same condition as the numerical simulations.

Figure 6: Dimensionless evaporative flux per unit surface $j/j_{0}$ as a
function of $1-(z/L)^{2}$. The points are obtained by numerical simulations
for droplets of volume $\Omega$ and contact angle $\theta$ placed on a fiber
of radius $a=125$ µm. The dimensionless wetted length $\lambda=L/a$ of the
droplets are also given and the results for sleeves with the same $\lambda$
are plotted for comparison. The colors represent the droplet aspect ratio
${\cal H}=h_{0}/L$. (a) Results obtained for $\Omega=1$ µL and contact angle
varying from $10$ to $80^{\circ}$ corresponding to $\lambda$ ranging from $5$
to $8$ and aspect ratio ${\cal H}$ between 0.5 and 0.8. (b) Results obtained
for $\theta=10^{\circ}$ and droplets volumes varying from $0.01$ µL to $1$ µL
corresponding to $\lambda$ ranging from $2$ to $8$ and aspect ratio ${\cal H}$
varying from $0.1$ for the smallest volume tested to $0.5$.

###  3.2 Results

First, we want to understand how the presence of the fiber influences the
evaporation speed. To do so, we obtained numerically the total evaporative
flux $Q$ of barrel-droplets on fibers of radius $a=125$ µm. Numerical
simulations are performed for various drop volumes $\Omega$ and contact angles
$\theta$. We recall that we limit ourselves to cases of barrel-shaped
droplets, which only exists for drops of sufficiently large volumes that are
placed on hydrophilic substrates [15].

The results are shown in figure 5a where we plot $Q$ as a function of
$\Omega^{1/3}$ and in figure 5b representing the dimensionless evaporative
flux as a function of the contact angle. The total evaporative flux of a
spherical drop [16],

$Q_{\rm sphere}\>=\>(48\pi^{2})^{1/3}\ {\cal D}\ (c_{\rm sat}-c_{\infty})\
\Omega^{1/3},$ (7)

is also represented in dashed black line for comparison. We observe that the
evaporation rate of a barrel-shaped drop on a fiber is similar to that of a
sphere for all the contact angles studied except for the smallest volumes
tested. This means that varying the contact angle has a negligible effect on
the evaporative flux except for small volume drops. We also note that, for
small volumes, the deviation from the sphere is increasing as the contact
angle is getting smaller, the largest variation being observed for the 0.01 µL
drop having a contact angle of $10^{\circ}$ whose profile is given in figure
4b. From Fig. 5b, we can quantify the deviation from the sphere, which varies
from a few percent for $\Omega\geq 0.1µL$ to around 35$\%$ for
$\theta=10^{\circ},~{}\Omega=0.01$ µL

We can also note that decreasing contact angle has a small effect on the drop
aspect ratio $\mathcal{H}$ except for small volume drops. This is due to the
curvature of the substrate that allows the existence of a macroscopic drop
($\mathcal{H}\sim 1$) even for small contact angles.

To understand these observations in more details, we consider the local
evaporative flux of barrel-shaped droplets. First, we focus on the effect of
the contact angle. In figure 6a, we plot the local flux $j$ as a function of
the position along the interface $z$ for a droplets of volume 1 µL with
various contact angle corresponding to aspect ratios ${\cal H}\simeq 1$. This
figure shows that the flux $j$ diverges in close vicinity of the drop contact
line. The decrease of the contact angle has a very small effect on the local
evaporative flux and this effect is significant only near the contact line. We
also compare the results obtained for barrel drops for which
$\lambda\in[5;10]$ with those of the sleeves having the same dimensionless
length. These sleeves have an aspect ratio $\lambda>1$, which means that the
divergence of the local flux has already the localization effect described in
the previous Section2. The comparison between the drops and the sleeves
highlight the effect of the drop profile curvatures. The longitudinal
curvature of the surface is estimated by $\mathcal{H}$, with $\mathcal{H}=1$
corresponding to a spherical drop whereas $\mathcal{H}=0$ is a sleeve. Fig. 6a
demonstrates that, due to the curvature of the drop surface, the divergence of
the evaporation flux of a drop on a fiber is even more localized at the triple
line than for the sleeve. We propose a phenomenological equation for the local
flux $j(z)$ of drops on fibers defined as

$j(z)=j_{0}\left[\beta\left(1-\frac{z^{2}}{L^{2}}\right)^{-\alpha}+(1-\beta)\right],$
(8)

where $\alpha$ and $\beta$ are positive adjustable parameters. Figure 4
provided in the Supplementary Materials shows fitted data for 1 µL drops for
different contact angle. The good agreement between numerical results and
fitted curves shows that, unlike sessile drops, the local evaporative flux of
a drop on a fiber cannot be described by a simple power law. If we compare the
local flux of barrel-shaped droplets to the local flux of a spherical droplet
(Fig 2a-b in Supplementary Materials) we see that the local flux at the center
of the barrel-shaped droplets is about $1.5$ times smaller than the
evaporative flux of a sphere. Thus, barrel-shaped drops are evaporating at the
same speed as spherical droplets because the difference in local fluxes at the
center of the drop compensates the localized divergence of the local flux of
barrel-shaped droplets.

Finally, to understand the difference observed between the total evaporative
flux of a spherical drop and the one of a small volume wetting barrel-drop on
a fiber, we plot in figure 6 the local evaporative flux as a function of the
position along the interface for drops of small contact angle
$\theta=10^{\circ}$ and different volumes. This difference is significant for
small volumes and small contact angles, i.e. when $\mathcal{H}$ vanishes. In
this case, the geometry is similar to a to liquid cylinder and we obtain
results comparable to those expected for a sleeve. Nevertheless, the drop,
which has a dimensionless length $\lambda\approx 2.3$ can be well compared to
the sleeve for a greater aspect ratio, $\lambda=5$. We interpret that although
$h_{0}$ tends to 0, the liquid thickness still has a significant effect on the
evaporative flux.

##  4 Conclusion

In this paper, we studied the curvature effect on the evaporation rate of a
liquid deposited on a fiber when the evaporation is isothermal and purely
diffusive. Two particular situations where analyzed: a sleeve of liquid of
size $L$ deposited on fiber of radius $a$ and a droplet of volume $\Omega$ and
wetting contact angle $\theta$ on the same fiber. The sleeve is obtained when
a liquid fiber is coated by a liquid layer before the onset of destabilization
of the Rayleigh-Plateau instability. The droplet, and more particularly the
axisymmetric barrel shape that we studied here, is encountered in the late
stage of destabilization when the liquid coating has been destabilized in a
series of liquid pearls.

For the evaporation of the sleeve we obtained an analytical calculation of the
local evaporation rate along the sleeves and the full evaporation rate in the
case of small and large aspect ratios. For large aspect ratios, the
evaporation rate is almost uniform along the sleeve except near the edges,
where we recover the same power divergence than the one observed for sessile
droplets at low contact angles. For small aspect ratio, the role of the edges
progressively becomes more and more important, the evaporation rate is varying
significantly along the sleeve since the power law divergence invades all the
sleeve. We compared our analytical calculation to finite element computations
and shown that our asymptotic calculations captures quantitatively the
simulations even in the regime where the aspect ratio is close to one.

For the evaporation of the liquid axisymmetric barrels, we performed numerical
simulations in order to calculate precisely the effect of the fiber on the
evaporation rate. We have evidenced that, for drops of volume $\Omega\geq 0.1$
µL corresponding to dimensionless volume $\Omega/a^{3}\geq 50$, the
evaporation rate is almost independent of the wetting contact angle and that
the droplet evaporates as a sphere of the same volume. More precisely, the
evaporation flux diverges near the triple line, but due to the barrel shape,
the divergence is strongly localized close to the edges and the evaporation
rate is nearly constant. Indeed the divergence close to the edge compensates
fortunately the fact that the longitudinal curvature of the surface at the
apex, estimated by $\cal H$, is not equal to the one of a spherical drop.

This study provides precise calculations that capture the drying dynamics of
the two important morphologies on a fiber under the assumption of an
isothermal evaporation process. They both emphasize the localization of the
evaporation divergence close to the triple line, in contrast to sessile drops.
To complete this analysis of the evaporation of a liquid coated on a fiber, it
is necessary in a future work to focus more precisely on what happens after
the destabilization of the sleeve into a series of regularly spaced drops.
Indeed, additional studies would be necessary to understand and quantify the
mutual influence of the drops on each other.

## Acknowledgments

We thank Saint-Gobain and ANRT for funding this study and J. Delavoipière, M.
Lamblet, and B. Sobac for useful discussions. We are also grateful to C.
Dubois from COMSOL for providing support.

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