# StyleMamba : State Space Model for Efficient Text-driven Image Style
Transfer

Zijia Wang Email<EMAIL_ADDRESS>Zhi-Song Liu  Imperial College
London, the United Kingdom  LUT School of Engineering Sciences, Finland  Dell
Research UK, the United Kingdom

###### Abstract

We present StyleMamba , an efficient image style transfer framework that
translates text prompts into corresponding visual styles while preserving the
content integrity of the original images. Existing text-guided stylization
requires hundreds of training iterations and takes a lot of computing
resources. To speed up the process, we propose a conditional State Space Model
for Efficient Text-driven Image Style Transfer, dubbed StyleMamba , that
sequentially aligns the image features to the target text prompts. To enhance
the local and global style consistency between text and image, we propose
masked and second-order directional losses to optimize the stylization
direction to significantly reduce the training iterations by 5$\times$ and the
inference time by 3$\times$. Extensive experiments and qualitative evaluation
confirm the robust and superior stylization performance of our methods
compared to the existing baselines.

2447

##  1 Introduction

In recent years, the intersection of computer vision and natural language
processing has led to significant advancements in multimodal perception and
understanding. One particularly intriguing area of research within this domain
is text-driven or text-guided image style transfer. This emerging field
explores the synthesis of images guided by textual descriptions, enabling the
transformation of visual content to match desired artistic styles, scenes, or
aesthetics as described in natural language. This fusion of vision and
language offers promising avenues for creative expression and content
generation.

Figure 1: Comparative results showcasing the efficacy of the StyleMamba
framework. a) highlights the rapid convergence and stylization capabilities
with fewer of epochs. b) and c) demonstrate the detailed stylization fidelity
and the transferability of various styles, including painting and lighting
styles, as well as some complicated styles like environmental style. Finally,
d) shows the great content preservation ability of StyleMamba with competitive
style transfer performance compared with other style transfer models. All
results reflect the superiority of StyleMamba .

Unlike image-driven style transfer, one notable advantage of text-driven style
transfer is its enhanced flexibility and interpretability. Texts provide a
more abstract and semantically rich representation of desired styles or
attributes than explicit reference images. This abstraction enables users to
express complex artistic concepts and emotions that are challenging to convey
through images alone. Existing text-driven approaches, e.g., Clipstyler [19],
DiffusionCLIP [18] and TxST [25], show tremendous promising results in visual
diversity. The key idea is to align the text and image embeddings [31, 41],
such that the stylized image can be faithful to the direction of the text.
However, they need to consider the nuances of style-specific text description
and visual style consistency. More importantly, they require hundreds of
training iterations and considerable GPU resources, which is inefficient for
real applications.

Thus, we introduce StyleMamba , a framework to incorporate a conditional State
Space Model [9] into the AutoEncoder systems which can use text prompts to
supervise the style fusion process. It can significantly improve the training
converge with $10\sim 20\times$ speedup. Furthermore, we explore the Masked
and Second-order directional loss to better align text and style. Figure 1
showcases the superiority of StyleMamba in performing style transfer,
illustrating its capability to rapidly assimilate and apply complex styles to
various content images with remarkable fidelity. Through a series of
comparisons, StyleMamba demonstrates enhanced performance in not only fewer
epochs (Figure 1 a) but also finer detail preservation (Figure 1 b) and
adherence to diverse artistic prompts(Figure 1 c), ranging from mimicking the
thick textures of Picasso’s oil paintings to capturing the essence of
atmospheric and lighting conditions. In Figure 1 d, these examples highlight
StyleMamba ’s better balance between content preservation and style transfer
ability, enabling more generalized applications. Our contributions can be
summarized below:

  * •

We propose a simple framework, StyleMamba , incorporating the conditional
Mamba into AutoEncoder to achieve fast text-driven style transfer (Section
3.1).

  * •

Our proposed framework achieves better stylization via novel Masked
directional loss and Second-order relational loss (Section 3.2), which can
speed up overall stylization and better grasp global and local style
consistency without compromising the contents.

  * •

We empirically show the efficiency of the proposed StyleMamba over the
existing state-of-the-art techniques in both quantitative and qualitative
measurements. Owing to the simplistic nature of StyleMamba , we further
experiment with video style transfer and multiple style transfer to show the
versatility of our framework.

##  2 Related Work

Image-driven style transfer. Style transfer aims to fuse desirable style to
the content images such that the resultant image can preserve the contents
while showing the desired style, i.e., color and textures. Early approaches
such as Gatys et al.’s work [7] on neural style transfer laid the groundwork
by formulating style transfer as an optimization problem, leveraging deep
convolutional neural networks to separate content and style representations
within images. Building upon this foundation, subsequent research [22, 15, 12,
28, 24, 43] has explored various techniques to improve the efficiency,
stability, and fidelity of style transfer algorithms. For instance, Li et al.
[15] introduced adaptive instance normalization (AdaIN), which dynamically
adjusts the statistics of intermediate feature maps to better match the style
of a reference image. Other approaches have incorporated perceptual loss
functions inspired by human visual perception to preserve semantic content
during style transfer better. Additionally, recent works [2, 11, 14, 27] have
explored conditional and controllable style transfer, allowing users to
specify desired style attributes or manipulate style transfer outcomes with
good visual quality.

Text-image multimodality. Benefiting from the study of LLMs and VLMs, it
becomes popular to use contrastive learning [3, 4] to align general texts and
images. CLIP [31] is one of the pioneering works that can use natural language
supervision for image representation. The key idea is to utilize pre-trained
language and image encoders to project texts and images onto the compressed
domain and maximize cosine similarity between paired text-image data. To
further improve the generalization and text-image alignment, SigLIP [41]
proposes to replace conservative learning with pairwise sigmoid loss. It can
scale up the batch size and end up with memory-efficient and superior text-
image alignment. Both CLIP and SigLIP offer zero-shot capacities that can be
used for many text-image multimodal tasks. For example, [32] proposes a text-
conditional image generation that can convert texts to desirable images. The
Stable Diffusion Model [33] trains a CLIP-based autoencoder that can take text
embeddings as conditions to further improve image quality. A similar concept
can also be applied to video generation [26, 23], video retrieval [39], 3D
generation [36, 20], 3D editing [16] and image captioning [8].

Text-driven style transfer. Different image-driven style transfer, text-driven
style transfer uses text prompts to guide the stylization process. Clipstyler
[19] is the first work to utilize CLIP to project both stylized images and
texts to the latent space. It can use pairwise cosine similarity to maximize
their distance. It further inspires several works on improving quality or
speeding up the training iterations. For example, DiffusionCLIP [18] combines
the state-of-the-art pre-trained diffusion model [33] and CLIP to learn
stylization in the latent space. Gatha [17] proposes to modify the global
directional loss between texts and images as a relational loss, which is
calculated based on a style tensor that can question whether the generated
image is truly aligned with the target texts. One of the key disadvantages of
the aforementioned methods is the online training process. It usually takes up
to 150 iterations (approximately 10 minutes on one standard GPU) to obtain one
good stylized image. Recent works either speed up the whole training times
[35] or perform offline contrastive training [6, 25, 38, 40].

##  3 Approach

The overall flow of the proposed StyleMamba is shown in Figure 2. The input
content image X is converted to latent representations $F_{e}(\textbf{X})$
using a pretrained Variational AutoEncoder (VAE) $F_{e}(\cdot)$from stable
diffusion [33], while the style text $t$ is converted into embeddings
$F_{T}(t)$ using pretrained text encoder $F_{T}(\cdot)$ in SigLIP [41]. They
are then fused in the proposed style fusion module to get a new feature map M.
Guided by the text-to-image style loss, the decoder $F_{d}(\cdot)$ decodes the
M into a stylized image Y. The training process is guided by the content loss
($\mathcal{L}_{\text{content}}$) based on VGG [34] or lpips [42], the second
order loss ($\mathcal{L}_{\text{so}}$), the masked directional loss
($\mathcal{L}_{\text{md}}$) and global directional loss
($\mathcal{L}_{\text{dir}}$).

Figure 2: Workflow overview of StyleMamba framework. The process begins with a
content image and a style prompt (e.g., “Paul Gauguin style”). An encoder
converts the content image into a latent representation, which undergoes style
fusion with features derived from the style prompt. This fusion is facilitated
by the Style Fusion Module, incorporating masked and second-order directional
losses to guide the text-to-image stylization. The result is a stylized image
Y, which closely adheres to the style prompt while preserving content
integrity.

###  3.1 Overall framework

The whole architecture consists of three components: i) Auto Encoder, which is
used to encode the content image X and reconstruct the stylized Image Y; ii)
Style Fusion Module, which works as a fuser to selectively attend the
$F_{T}(t)$ and $F_{e}(\textbf{X})$ to derive M; iii) SigLIP Module, which is
used to encode the style features contained in $F_{T}(t)$ and to act as a
’North Start’ to guide the whole process.

Auto Encoder. The proposed masked directional loss, coupled with second-order
loss, needs the powerful content-preserving ability of the Auto Encode, while
the better reconstruction ability is where the pretrained VAE from the Stable
Diffusion Model [33] excels, as it is adept at encoding the content image into
a rich, compressed latent space while ensuring that the essential features are
retained for accurate reconstruction. Moreover, the absence of shortcuts
between the encoder and decoder in this architecture means that the
stylization process relies entirely on the latent representation, fostering a
more profound style integration without residual information from the input
image. This approach aligns with the Information Bottleneck principle,
striking an optimal balance between data compression and preserving relevant
content features for the task. Therefore, the pretrained VAE in Stable
Diffusion Model [33] is chosen here to be $F_{e}(\cdot)$.

Style Fusion Module. Inspired by DiT [29], we propose a conditional State
Space Model for style fusion. Concretely, the Style Fusion Module in the
StyleMamba framework effectively combines the textual style features with the
content image features by incorporating elements of Adaptive Layer Norm
(AdaLN) and Mamba process [9]. Mamba’s selective process accelerates the style
transfer process by focusing only on relevant features, which speeds up
training and inference. AdaLN’s ability is to adaptively adjust style
characteristics from textual descriptions using learnable shifting and scaling
values. The AdaLN approach allows for the adjustment of the normalized content
features to resonate with the stylistic elements described in the text,
achieving adaptive style manipulation. Specifically, the Style Fusion Module
incorporates AdaLN by adjusting feature distributions based on the style
features derived from the text, ensuring the content image’s integrity while
infusing the desired style. The normalization of content feature
$F_{e}(\textbf{X})$ is depicted mathematically as:

$\textbf{M}=(LN(F_{e}(\textbf{X})+\alpha_{1}\cdot
SSM(LN(F_{e}(\textbf{X})))\cdot\mu_{1}+\sigma_{1}))+\alpha_{2}+\sigma_{2}$ (1)

In Equation 1, $LN$ denotes Layer Normalization, and $SSM$ represents the
State Space Model. In our case, we use Mamba [9], $\alpha$ and $\sigma$ are
scaling and shifting parameters, respectively, which are learned from the
style features. The Mamba, with its selective state space model, allows for
content-dependent transformation, selectively attending to relevant
information and dynamically adjusting to the content based on the current
context. This allows for efficient and adaptive style transfer while
maintaining content consistency.

SigLIP Module. For the SigLIP Module model section, the introduction of the
pretrained SigLIP [41] is central to enhancing the style fusion process, which
is critically dependent on the rich semantic encoding capabilities of the
pretrained CLIP models. SigLIP achieves better zero-shot classification
accuracy on ImageNet [5] than CLIP [31]. SigLIP can better generalize to
unseen image categories based on natural language descriptions. To demonstrate
its superior performance over CLIP, we conduct ablations in the Experiment
section.

###  3.2 Proposed Text-driven style losses

Global directional loss. In the proposed net framework, the global directional
loss $\mathcal{L}_{\text{dir}}$ is the key guidance in aligning the
transformation of the image content with the specified text-driven style. Like
the direction loss in [19], it operates by computing the cosine similarity
between the direction vectors of text and image features, ensuring that the
stylization process is coherent with the textual description. Formally, this
loss is defined as:

$\begin{matrix}\\!\begin{aligned}
&\mathcal{L}_{\text{dir}}=1-\frac{\mathbf{T}_{\text{dir}}\cdot\mathbf{I}_{\text{dir}}}{\|\mathbf{T}_{\text{dir}}\|\|\mathbf{I}_{\text{dir}}\|}~{}\text{,
where}\\\
&\mathbf{T}_{\text{dir}}=F_{T}(t)-F_{T}(t_{src}),\quad\mathbf{I}_{\text{dir}}=F_{I}(\textbf{Y})-F_{I}(\textbf{X})\end{aligned}\end{matrix}$
(2)

where $F_{T}$ and $F_{I}$ are the text encoder and image encoder of the
SigLIP, respectively, then $t$ and $src$ are style text and source text (here
we define it as "a plain photo"), $\mathbf{T}_{\text{dir}}$ and
$\mathbf{I}_{\text{dir}}$ are the direction vectors for the text and image
modalities, respectively.

Masked directional Loss. As shown in [10], randomly masking patches can speed
up the reconstruction process and serve as a self-supervised training paradigm
for image reconstruction tasks. In StyleMamba , the mask can work efficiently
to enforce the convergence regarding style loss. Furthermore, we consider that
the style in the images is locally consistent and independent from the
contents. Hence, we mark partial contents to enforce the style similarity
close to the complete image results. Therefore, we extend the directional loss
with a masking step to incorporate robustness against partial visibility of
style features. Given the content image X and style text $t$, the stylized
image Y is generated and subsequently masked randomly by 50% using $16\times
16$ patches to obtain the masked image $Z$. The features of $Z$, $F_{I}(Z)$,
and the content features $F_{I}(\textbf{X})$ are then computed to calculate
the masked directional loss. The proposed masked directional loss
$\mathcal{L}_{\text{md}}$ is defined as:

$\mathcal{L}_{\text{md}}=\frac{\|F_{I}(Z)-F_{I}(\textbf{X})\|}{\|F_{T}(t)-F_{T}(t_{\text{src}})\|}$
(3)

Equation 3 leverages the reconstruction challenge as a self-supervised signal,
enhancing the model’s ability to maintain style fidelity even when significant
portions of the image are masked, drawing insights from the effectiveness of
high masking ratios in self-supervised learning settings.

Figure 3: Illustration of the proposed second-order directional loss. It shows
how $L_{so}$ allows for rapid adjustments in the direction of stylization.
Notably, it facilitates refined stylistic shifts, ensuring a swift and
coherent transition towards the desired visual style. Figure 4: Qualitive
comparison with SOTA algorithms. We show three cases of text-guided style
transfer. For reference, we use the text to retrieve a reference image for
style comparison and the input to InstantStyle.

Second-order directional loss. However, to speed up the convergence speed, the
second-order loss $\mathcal{L}_{\text{so}}$ introduces a novel approach by
considering the progression of the stylized image across successive epochs. As
shown in Figure 3, the second-order directional loss ($L_{so}$) is designed to
speed up the optimization of the alignment between the generated image and the
textual style description. It achieves this by considering the changes in
stylization direction across successive epochs, ensuring a coherent and swift
transition towards the desired visual style. This is quantified by the squared
norm of the difference between the image features at consecutive epochs,
relative to the text direction vector, and is weighted by a dynamic term, (
$\alpha_{\text{shift}}$ ), to adjust the influence of the style shift on the
loss function. Concretely, given a content image X, style text $t$, and
stylized images $\textbf{Y}(i)$ and $\textbf{Y}(i+1)$ obtained at epochs $i$
and $i+1$, the second-order directional loss $\mathcal{L}_{\text{so}}$ is
formulated as follows:

$\mathcal{L}_{\text{so}}=\left\|\frac{F_{I}(\textbf{Y}(i+1))-F_{I}(\textbf{Y}(i))}{F_{T}(t)-F_{T}(t_{\text{src}})}\right\|^{2}\cdot\alpha_{\text{shift}},$
(4)

which measures the change in stylization from epoch $i$ to $i+1$ relative to
the change in text features. Specifically, $\alpha_{\text{shift}}$ is a
dynamic weighting term that could be formulated as:

$\alpha_{\text{shift}}=\alpha\cdot\left(1-e^{-\beta\|\mathbf{F}_{I}(\textbf{Y(i)})-\mathbf{F}_{I}(\textbf{X})\|}\right)$
(5)

where $\alpha$ and $\beta$ are hyperparameters that control the sensitivity of
the loss to the magnitude of the style shift. Concretely, $\alpha$ serves as
an upper limit or a scaling factor for the weight. It can be used to adjust
the maximum influence the style shift weight can have on the second-order
loss. $\beta$ is a scaling hyperparameter that controls the sensitivity of the
weight to changes in the style feature difference. A larger $\beta$ would make
the weight more responsive to smaller differences in style features. The
exponential term ensures that the weight increases as the difference between
the target and source style features grows, but it is bound to prevent the
loss from escalating excessively. When the style feature difference is large
(meaning the current style is far from the target), this term is close to 0,
making $\alpha_{shift}=\alpha$, which in turn makes the weight larger,
allowing the second-order loss to have a smaller impact on the training
update, and vice versa. Furthermore, to enhance the training stability and
ensure fast updates, a threshold is applied to the direction loss
$\mathcal{L}_{\text{dir}}$. The second-order loss $\mathcal{L}_{\text{so}}$ is
employed only if $\mathcal{L}_{\text{dir}}$ falls below a predefined threshold
$\theta$, allowing for the refinement of style transfer in later stages of
training. Including the aforementioned masked directional loss, the overall
proposed text-driven style loss is defined as follows.

$\mathcal{L}_{\text{style}}=\mathcal{L}_{\text{dir}}+\mathcal{L}_{\text{md}}+\mathbb{I}(\mathcal{L}_{\text{dir}}<\theta)\cdot\mathcal{L}_{\text{so}}$
(6)

where $\mathbb{I}(\cdot)$ is the indicator function, which is 1 if the
condition is true, and 0 otherwise.

Content losses. The VGG content loss [28] ($L_{\text{vgg}}$) is used to
supervise content similarity. It is defined as the sum of squared errors
between the feature maps of the content and the stylized image, extracted from
various layers of the VGG network. Meanwhile, we also use the LPIPS [42] loss
$L_{\text{lpips}}$ to align the stylized results more closely with human
perceptual judgments.

Total loss. We incorporate the content loss and the style losses as
$L_{total}=\alpha L_{\text{style}}+\beta L_{\text{lpips}}+\gamma
L_{\text{vgg}}$, where $\alpha,\beta,\gamma$ are the weights that balance the
contribution of each loss component to the total loss. This formulation
ensures that the stylization aligns with the textual description while
preserving the integrity of the content image.

##  4 Experiments

$\bullet$ Datasets. To evaluate our StyleMamba framework, we utilized two
distinct datasets: COCO[21] and WikiArt[1]. It is important to note that these
datasets were not employed during the training phase of our model. Instead,
they are used in the testing and inference stages, providing a diverse range
of images and artistic styles to assess the effectiveness of our text-driven
image style transfer approach. We use the COCO dataset as content images while
applying various textual styles to evaluate the model’s performance in
capturing and rendering complex artistic styles.

$\bullet$ Parameter setting. For our experiments, we adapted the pretrained
VAE from Stable Diffusion [33], opting to keep the encoder fixed while only
training the decoder. To ensure stable training, we introduced a two-stage
learning rate strategy for the content loss weight. Initially, we set the
weight at 9000, which is then reduced to 150 after 5 epochs. This approach was
taken to prevent early overfitting to content features and to gradually refine
the content alignment as the training progressed. Additionally, our training
procedure included a masking component, where we randomly masked 50% of the
patches. With these settings, we trained our models using the Adam optimizer
with a starting learning rate of $5\times 10^{-4}$, halving it at epoch 10,
and terminating the training at 20 epochs.

$\bullet$ Metrics and evaluation. To assessour textimage model’s effectiveness
in content consistency and style alignment, we employed three metrics: 1\.
CLIP score: This is quantified by computing the cosine similarity between the
features extracted by the CLIP (SigLIP) model from the text and the image. A
higher CLIP similarity score indicates better style alignment. 2\. SSIM: It is
used to measure the similarity between the content of the original image and
the stylized image. A higher SSIM index signifies a better preservation of
content. 3\. VGG Loss: It captures the content differences between content and
stylized images by examining the feature responses at various layers of the
VGG network. A lower VGG content loss implies better performance.

###  4.1 Comparison with the state of the art

| CLIP score $\uparrow$ | SSIM $\uparrow$ | VGG loss $\downarrow$ | Aesthetic score $\downarrow$
---|---|---|---|---
Clipstyler[19] | 0.180 | 0.988 | 2.106 | 5.379
DiffusionCLIP [18] | 0.327 | 0.979 | 1.272 | 5.742
Diffstyler [13] | 0.351 | 0.982 | 1.182 | 4.660
TxST [25] | 0.409 | 0.990 | 0.975 | 4.552
ZeCon [40] | 0.417 | 0.971 | 1.218 | 5.327
Our final model | 0.492 | 0.996 | 0.547 | 4.172

Table 1: Comparison with Sate-of-the-art methods. StyleMamba is consistently
better than other text-guided style transfer models regarding the CLIP score,
SSIM, VGG loss and Aesthetic score.

To show the superiority of our proposed StyleMamba , we compare it with
several state-of-the-art approaches, including DiffusionCLIP [18], Diffstyler
[13], TxST [25] and ZeCon [40]. We first quantitively compare the performance
of these models, the experiment is done on 10 random content images and 10
text prompts of various descriptions. Among them, TxST claims to be real-time
stylization without online optimization, and ZeCon claims to be a faster
version of Clipstyler [19]. The CLIP score of StyleMamba is 0.492, the highest
in the group, indicating proficiency in generating semantically coherent
images in response to textual descriptions. StyleMamba also shows higher
values on SSIM and VGG content loss. These metrics underscore the model’s
ability to retain the structural integrity of the content images. Lastly,
measured by the Aesthetic differences metric [30], shows the lowest score
among the compared models, suggesting that StyleMamba is capable of producing
images that are aesthetically appealing to humans. Overall, our proposed
StyleMamba effectively produces images that are visually aligned with text
prompts, structurally similar to reference images and superior in aesthetic
quality. A qualitative comparison is also conducted based on three cases as
shown in Figure 4. The retrieved image is derived based on the prompt text,
which is used as a reference for the style transfer performance and the input
to InstantStyle [37]. Apparently, StyleMamba can achieve consistently better
results in all cases regarding content preservation and style transfer
ability. Concretely, StyleMamba outperforms Clipstyler, DiffusionCLIP,
Diffstyler and ZeCon regarding both style transfer and content preservation
capabilities. Compared with InstantStyle ([37]), StyleMamba shows competitive
style transfer ability without taking in a reference style image, while the
content preservation ability is better than InstantStyle.

Model | Training time | Inference Time | Model Param.
---|---|---|---
Clipstyler[19] | 403 sec | 4 sec | 28M
DiffusionCLIP [18] | 293 sec | 56 sec | 214 M
Diffstyler [13] | - | 42 sec | 207M
TxST [25] | - | 27 sec | 51M
ZeCon [40] | - | 36 sec | 67M
Ours | 23 sec | 2 sec | 17M

Table 2: Computational complexity comparison of different models. We record
the training time, inference time, and the number of model parameters.

In Table 2, we show the computation complexity among different approaches. We
record both computation time and model parameters to demonstrate the
efficiency of our proposed StyleMamba . Note that Diffstyler, TxST and Zecon
require long offline training hours to achieve real-time stylization. The
approximate training time could be hundreds of hours. We can see that ours
achieves $10\sim 20\times$ speedup compared to Clipstyler and DiffusionCLIP on
training time. Due to the proposed fast Mamba based style fusion module,
StyleMamba also achieves a faster inference process, approximately 2$\times$
faster than Clipstyler.

Figure 5: Speed comparison between StyleMamba and Clipstyler. The graph
depicts the style loss and corresponding intermediate results over time,
illustrating the convergence efficiency of each method. The result of
StyleMamba at epoch 15 shows a more detailed stylization result than
Clipstyler’s 100 epoch result.

###  4.2 Ablation study

Style Fusion module. To test the effectiveness of the style fusion module, we
compare two settings: ours with the Mamba module, and the cross-attention
module used in [25]. Table 3 shows the experiment results. The combination of
SigLIP and Mamba achieves the best results regarding the CLIP similarity and
the VGG content loss. We can also observe that using our proposed style fusion
module achieves better stylization and runs faster than the cross-attention
module, approximately $3\times$ speedup. This is due to the efficiency of
Mamba’s sequential process over the cross-attention’s global correlation. It
can also be observed from the model complexity in Table 2.

Modules | CLIP score $\uparrow$ | SSIM $\uparrow$ | Running time$\downarrow$
---|---|---|---
Clipstyler | 0.180 | 0.988 | 4 sec
CLIP+cross-attention | 0.387 | 0.979 | 7 sec
SigLIP+cross-attention | 0.391 | 0.987 | 7 sec
CLIP+Mamba | 0.422 | 0.990 | 2 sec
SigLIP+Mamba | 0.452 | 0.991 | 2 sec

Table 3: Ablation study on Style fusion module. We compare different Style
Fusion modules used in the StyleMamba . The combination of SigLIP and Mamba
achieves the best results regarding style transfer accuracy and speed.

Text-to-image models. In Section 3, we propose to replace CLIP [31] with
SigLIP [41] because SigLIP generally performs better in text-to-image
alignment. Hence, it has the potential for a better understanding of style
representation in the images. To demonstrate its efficiency, we randomly
select 10 content images and 10 text prompts to test different models during
the training and testing phases on the performance of text-image similarity,
content preservation, and style transfer in the generated images. The average
results are displayed in Table 4.

Training & evaluation | CLIP score $\uparrow$ | SSIM $\uparrow$ | VGG loss $\downarrow$
---|---|---|---
CLIP & CLIP | 0.276 | 0.989 | 3.920
CLIP & SigLIP | 0.066 | 0.989 | 3.920
SigLIP & CLIP | 0.325 | 0.990 | 3.899
SigLIP & SigLIP | 0.127 | 0.990 | 3.899

Table 4: Ablation study on text-to-image models. To measure the alignment
between texts and images, we use CLIP and SigLIP for model training and
evaluation. The CLIP score is calculated by using corresponding evaluation
models.

In Table 4, we show the training and evaluation combinations of using
different text-to-image models. It shows that the experiments with the SigLIP
model during evaluation (rows 3 and 4) outperform the corresponding
experiments using the CLIP model (rows 1 and 2). Specifically, the CLIP
similarity score increases from 0.276 to 0.325 and from 0.066 to 0.127, when
comparing the CLIP testing model against the SigLIP testing model. This
suggests that the SigLIP model is more effective at capturing and translating
stylistic elements from text to image. Moreover, comparing rows 1 and 3, we
can see that using SigLIP to train the model can significantly improve the
CLIP score, which indicates that SigLIP is better at understanding the style
representation and aligning the stylized image closer to the style prompts.
This is also indicated by comparing rows 2 and 4.

Content Encoder $F_{e}$. Furthermore, we explored the synergistic effects of
combining different text-to-image models with VGG and stable diffusion
autoencoders (SD VAE) [33]. As shown in Table 5, the results indicated that SD
VAE outperforms VGG AE in conjunction with the same CLIP model, yielding a
higher CLIP similarity score and lower VGG content loss. Specifically, we can
see improvements with SD VAE (row 2 > row 1, and row 4 > row 3), suggesting
that SD VAE is more effective in capturing and preserving style and contents.
Further, the SigLIP model consistently delivered higher CLIP scores than CLIP
models (1&2, 3&4), affirming the superiority of SigLIP in image-text
alignment.

Modules | CLIP score $\uparrow$ | SSIM $\uparrow$ | VGG loss $\downarrow$
---|---|---|---
SigLIP+VGG AE | 0.322 | 0.990 | 3.874
SigLIP+SD VAE | 0.349 | 0.985 | 2.759
CLIP+VGG AE | 0.126 | 0.789 | 3.852
CLIP+SD VAE | 0.140 | 0.785 | 2.937

Table 5: Ablation study on Encoder model. To better preserve the contents, we
test using different pretrained autoencoder (VGG [34] and SD VAE [33]) for
style transfer.

Loss functions $\mathcal{L}_{\text{style}}$. As shown in Table 6, to assess
the impact of various losses on the stylization process, we conducted a series
of experiments by systematically integrating different loss functions and
evaluating their influence on stylization. For comparison, the baseline model
only utilizes the global directional loss ($\mathcal{L}_{\text{dir}}$) and VGG
content loss ($\mathcal{L}_{\text{vgg}}$) in row 1. Subsequently, in row 2, we
enhanced the loss function framework of Clipstyler by incorporating the
proposed masked directional loss ($\mathcal{L}_{\text{md}}$), aimed at
refining the model’s ability to capture directional attributes within the
content image. This addition sought to bolster the CLIP score, a metric
indicative of style-content alignment. Further augmentation, in row 3,
involved the amalgamation of LPIPS ($\mathcal{L}_{\text{lpips}}$) with the
masked directional loss($\mathcal{L}_{\text{md}}$), facilitating simultaneous
improvements in both content preservation, as reflected by the content loss
metric, and style alignment, as indicated by the improved CLIP score. Finally,
in row 4, we integrated a second-order loss $\mathcal{L}_{\text{so}}$,
implemented at an interval of every 5 epochs. It is pertinent to note that
narrower intervals were avoided to prevent potential instability in the
convergence pattern. The experimental outcomes, as presented in Table 6, were
generated under a consistent training duration of only 50 epochs across all
models.

Comparing rows 1 and 2, we can see that masked directional loss markedly
enhances the style similarity. Rows 2 and 3 show that adding LPIPS loss can
significantly improve content preservation, with approximately 2.2 reductions
in VGG loss. The last row is the complete losses used in StyleMamba , we can
see that adding second-order directional loss can further improve the style
similarity without compromising the content information.

Loss terms | CLIP score $\uparrow$ | SSIM $\uparrow$ | VGG loss $\downarrow$
---|---|---|---
$\mathcal{L}_{\text{dir}}+\mathcal{L}_{\text{vgg}}$ (baseline) | 0.329 | 0.986 | 0.624
baseline+$\mathcal{L}_{\text{md}}$ | 0.422 | 0.889 | 2.779
baseline+$\mathcal{L}_{\text{md}}+\mathcal{L}_{\text{lpips}}$ | 0.428 | 0.985 | 0.593
baseline+$\mathcal{L}_{\text{md}}+\mathcal{L}_{\text{lpips}}+\mathcal{L}_{\text{so}}$ | 0.475 | 0.986 | 0.572

Table 6: Ablation study on loss functions. To compare different loss terms, we
gradually add more loss terms to evaluate the performance. The final model
combining $\mathcal{L}_{\text{md}}$, $\mathcal{L}_{\text{lpips}}$ and
$\mathcal{L}_{\text{so}}$ achieves the best results in terms of the style
similarity and content similarity.

To assess the training efficiency of the proposed StyleMamba compared to
Clipstyler, we plot the progression of style loss over 100 epochs (Figure 5).
The red line in the plot indicates that StyleMamba begins to exhibit signs of
convergence by epoch 15 while Clipstyler achieves similar style loss at epoch
100. Approximately, we achieve 7$\times$ speedup. Qualitatively, we show the
enlarged patches generated by the StyleMamba and Clipstyler, and we can see
that ours can better transfer the “bee hives” patterns to the content image.

Figure 6: StyleMamba in the wild. Our better balance between content
preservation and style transfer ability can enable many applications with high
content preservation requirements, while with good style transfer ability.

###  4.3 Style transfer in the wild

To test the generalization of our proposed StyleMamba , we show six real
applications in Figure 6, which showcases the versatile capabilities of the
proposed network across a range of creative domains, demonstrating its robust
generalization ability. Figure 6 a shows the multiple style transfer. Given
the two text prompts, “cyberpunk” and “impressionism”, we see a coherent
stylized result that can reflect both style features. Figure 6 b highlights
the network’s application in product design. Knowing the mask of the contents,
we can customize the appearance of the desktop by providing any text prompts.
Figure 6 c shows that the proposed StyleMamba can understand the contents from
sketches and generate desirable colors and textures to complete the painting,
indicating its ability for painting assistance. This ability could speed up
the process of art creation, providing artists with a powerful tool to
visualize and experiment with different color palettes and textural effects.
The network’s prowess extends to user interface (UI) design, where in Figure 6
d, it reimagines the main page of ECAI 2024 with a watercolor painting style.
This adaptation is not just cosmetic but conceptual, showcasing how a digital
platform can embody the aesthetics of traditional art, potentially making
digital experiences more organic and visually engaging. Then the cinematic
style transformation in Figure 6 e highlights StyleMamba ’s capacity to alter
not just the mood but the narrative tone of a scene. It adeptly applies a
"dark tone" to a typical urban setting, demonstrating the potential to guide
the viewer’s emotional response and enrich tvisual media storytelling. .
Lastly, Figure 6 f explores the frontier of fashion design. Here, StyleMamba
demonstrates a keen eye for the transposition of iconic art onto wearable
designs. The network takes the quintessential patterns of a painting and
coherently wraps them around a three-dimensional garment, bridging the gap
between classical art and modern apparel. This instance exemplifies how
StyleMamba could be a transformative tool for fashion designers, enabling them
to create innovative prints and textiles inspired by a vast array of visual
art forms.

These six applications not only affirm the adeptness of StyleMamba in
understanding and executing complex visual prompts and illuminate its
potential as a catalyst in creative industries, from enriching visual
storytelling to revolutionizing product and fashion design.

###  4.4 Limitations and discussion

As shown in Figure 7, StyleMamba has limitations on understanding content-
guided or less commonly used texts. For example, it can not understand “Pixar
styles” for face editing (Figure 7 a) or “watermelon” (Figure 7 b) for shape
transformation. Moreover, a controled style transfer can not be done with
StyleMamba , while the CLIP doesn’t have the segmentation capcity.

These limitations highlight areas for future research and development, such as
improving the model’s ability to handle diverse facial features and expanding
its understanding of novel and abstract concepts for more accurate and varied
style transfers.

Figure 7: Failure examples. StyleMamba has some limitations on face images and
object manipulation.

##  5 Conclusion

This paper presented StyleMamba , an innovative framework for text-driven
image style transfer that utilizes a conditional State Space Model integrated
within an AutoEncoder architecture. Our framework demonstrates a significant
advancement in the field by reducing the number of training iterations and
also the inference time and training time required for each epoch, as well as
enhancing the efficiency of the stylization process without sacrificing the
quality of the stylized images. The results from our extensive experiments
illustrate that StyleMamba outperforms existing baselines in terms of speed,
stylization accuracy, and content preservation. The introduction of novel loss
functions, masked and second-order directional losses, have been particularly
effective in achieving high-quality stylization that aligns closely with
textual descriptions. This capability allows for greater flexibility and
creative expression in the applications of style transfer, making StyleMamba a
valuable tool for both artistic endeavors and practical applications like UI
design, clothes design, etc. In future work, we aim to explore the integration
of more diverse linguistic inputs and expand the model’s ability to handle a
broader range of visual styles in a finer, controlled way.

## References

  * Wik [2016] K. nichol. painter by numbers, wikiart.  https://www.kaggle.com/c/painter-by-numbers, 2016.
  * An et al. [2021] J. An, S. Huang, Y. Song, D. Dou, W. Liu, and J. Luo.  Artflow: Unbiased image style transfer via reversible neural flows.  In _CVPR_ , 2021.
  * Chen et al. [2020a] T. Chen, S. Kornblith, M. Norouzi, and G. Hinton.  A simple framework for contrastive learning of visual representations.  In _ICML_ , 2020a.
  * Chen et al. [2020b] T. Chen, S. Kornblith, K. Swersky, M. Norouzi, and G. E. Hinton.  Big self-supervised models are strong semi-supervised learners.  In _NeurIPS_ , 2020b.
  * Deng et al. [2009] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei.  ImageNet: A Large-Scale Hierarchical Image Database.  In _CVPR09_ , 2009.
  * Fu et al. [2022] T.-J. Fu, X. E. Wang, and W. Y. Wang.  Language-driven artistic style transfer.  In _European Conference on Computer Vision (ECCV)_ , 2022.
  * Gatys et al. [2016] L. A. Gatys, A. S. Ecker, and M. Bethge.  Image style transfer using convolutional neural networks.  In _CVPR_ , 2016.
  * Ge et al. [2023] S. Ge, T. Park, J.-Y. Zhu, and J.-B. Huang.  Expressive text-to-image generation with rich text.  In _IEEE International Conference on Computer Vision (ICCV)_ , 2023.
  * Gu and Dao [2023] A. Gu and T. Dao.  Mamba: Linear-time sequence modeling with selective state spaces.  _arXiv preprint arXiv:2312.00752_ , 2023.
  * He et al. [2022] K. He, X. Chen, S. Xie, Y. Li, P. Dollár, and R. Girshick.  Masked autoencoders are scalable vision learners.  In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , pages 16000–16009, 2022.
  * Hong et al. [2021] K. Hong, S. Jeon, H. Yang, J. Fu, and H. Byun.  Domain-aware universal style transfer.  In _ICCV_ , 2021.
  * Huang et al. [2022] N. Huang, F. Tang, W. Dong, and C. Xu.  Draw your art dream: Diverse digital art synthesis with multimodal guided diffusion.  In _ACM MM_ , MM ’22, page 1085–1094, 2022.  ISBN 9781450392037.
  * Huang et al. [2024] N. Huang, Y. Zhang, F. Tang, C. Ma, H. Huang, W. Dong, and C. Xu.  Diffstyler: Controllable dual diffusion for text-driven image stylization.  _IEEE Transactions on Neural Networks and Learning Systems_ , pages 1–14, 2024.  10.1109/TNNLS.2023.3342645.
  * Huang et al. [2023] S. Huang, J. An, D. Wei, J. Luo, and H. Pfister.  Quantart: Quantizing image style transfer towards high visual fidelity.  In _2023 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ , pages 5947–5956, jun 2023.  10.1109/CVPR52729.2023.00576.
  * Huang and Belongie [2017] X. Huang and S. Belongie.  Arbitrary style transfer in real-time with adaptive instance normalization.  In _ICCV_ , 2017.
  * Hyung et al. [2023] J. Hyung, S. Hwang, D. Kim, H. Lee, and J. Choo.  Local 3d editing via 3d distillation of clip knowledge.  In _2023 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ , pages 12674–12684, 2023.  10.1109/CVPR52729.2023.01219.
  * Jandial et al. [2023] S. Jandial, S. Deshmukh, A. Java, S. Shahid, and B. Krishnamurthy.  Gatha: Relational loss for enhancing text-based style transfer.  In _2023 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW)_ , pages 3546–3551, 2023.
  * Kim et al. [2022] G. Kim, T. Kwon, and J. C. Ye.  Diffusionclip: Text-guided diffusion models for robust image manipulation.  In _2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ , pages 2416–2425, 2022.
  * Kwon and Ye [2021] G. Kwon and J. C. Ye.  Clipstyler: Image style transfer with a single text condition.  _arXiv preprint arXiv:2112.00374_ , 2021.
  * Lin and et al [2023] C.-H. Lin and et al.  Magic3d: High-resolution text-to-3d content creation.  In _IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2023.
  * Lin et al. [2014] T.-Y. Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ramanan, P. Dollár, and C. L. Zitnick.  Microsoft coco: Common objects in context.  In _ECCV_ , 2014.
  * Liu et al. [2021a] S. Liu, T. Lin, D. He, F. Li, M. Wang, X. Li, Z. Sun, Q. Li, and E. Ding.  Adaattn: Revisit attention mechanism in arbitrary neural style transfer.  In _ICCV_ , 2021a.
  * Liu et al. [2021b] X. Liu, C. Gong, L. Wu, S. Zhang, H. Su, and Q. Liu.  Fusedream: Training-free text-to-image generation with improved clip+gan space optimization, 2021b.
  * Liu et al. [2021c] Z.-S. Liu, V. Kalogeiton, and M.-P. Cani.  Multiple style transfer via variational autoencoder.  In _ICIP_ , 2021c.
  * Liu et al. [2023] Z.-S. Liu, L.-W. Wang, W.-C. Siu, and V. Kalogeiton.  Name your style: text-guided artistic style transfer.  In _2023 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW)_ , pages 3530–3534, 2023.  10.1109/CVPRW59228.2023.00359.
  * Luo et al. [2021] H. Luo, L. Ji, M. Zhong, Y. Chen, W. Lei, N. Duan, and T. Li.  CLIP4Clip: An empirical study of clip for end to end video clip retrieval.  _arXiv preprint arXiv:2104.08860_ , 2021.
  * Lötzsch et al. [2022] W. Lötzsch, M. Reimann, M. Büßemeyer, A. Semmo, J. Döllner, and M. Trapp.  Wise: Whitebox image stylization by example-based learning.  _ECCV_ , 2022.
  * Park and Lee [2019] D. Y. Park and K. H. Lee.  Arbitrary style transfer with style-attentional networks.  In _CVPR_ , 2019.
  * Peebles and Xie [2023] W. Peebles and S. Xie.  Scalable diffusion models with transformers.  In _2023 IEEE/CVF International Conference on Computer Vision (ICCV)_ , pages 4172–4182, 2023.  10.1109/ICCV51070.2023.00387.
  * Pressman et al. [2022] J. D. Pressman, K. Crowson, and S. C. Contributors.  Simulacra aesthetic captions.  Technical Report Version 1.0, Stability AI, 2022.  https://github.com/crowsonkb/simulacra-aesthetic-models.
  * Radford et al. [2021] A. Radford, J. W. Kim, C. Hallacy, A. Ramesh, G. Goh, S. Agarwal, G. Sastry, A. Askell, P. Mishkin, J. Clark, et al.  Learning transferable visual models from natural language supervision.  In _ICML_ , 2021.
  * Ramesh et al. [2022] A. Ramesh, P. Dhariwal, A. Nichol, C. Chu, and M. Chen.  Hierarchical text-conditional image generation with clip latents.  _ArXiv_ , abs/2204.06125, 2022.  URL https://api.semanticscholar.org/CorpusID:248097655.
  * Rombach et al. [2022] R. Rombach, A. Blattmann, D. Lorenz, P. Esser, and B. Ommer.  High-resolution image synthesis with latent diffusion models.  In _CVPR_ , pages 10674–10685, 2022.  10.1109/CVPR52688.2022.01042.
  * Simonyan and Zisserman [2015] K. Simonyan and A. Zisserman.  Very deep convolutional networks for large-scale image recognition.  In _ICLR_ , 2015.
  * Suresh et al. [2024] A. P. Suresh, S. Jain, P. Noinongyao, A. Ganguly, U. Watchareeruetai, and A. Samacoits.  Fastclipstyler: Optimisation-free text-based image style transfer using style representations.  In _2024 IEEE/CVF Winter Conference on Applications of Computer Vision (WACV)_ , pages 7301–7310, 2024.  10.1109/WACV57701.2024.00715.
  * Tang et al. [2023] J. Tang, J. Ren, H. Zhou, Z. Liu, and G. Zeng.  Dreamgaussian: Generative gaussian splatting for efficient 3d content creation.  _arXiv preprint arXiv:2309.16653_ , 2023.
  * Wang et al. [2024] H. Wang, Q. Wang, X. Bai, Z. Qin, and A. Chen.  Instantstyle: Free lunch towards style-preserving in text-to-image generation.  _arXiv preprint arXiv:2404.02733_ , 2024.
  * Wang et al. [2023] Z. Wang, L. Zhao, and W. Xing.  Stylediffusion: Controllable disentangled style transfer via diffusion models.  In _2023 IEEE/CVF International Conference on Computer Vision (ICCV)_ , pages 7643–7655, 2023.
  * Xue et al. [2022] H. Xue, Y. Sun, and et al.  Clip-vip: Adapting pre-trained image-text model to video-language alignment.  In _International Conference on Learning Representations_ , 2022.
  * Yang et al. [2023] S. Yang, H. Hwang, and J. C. Ye.  Zero-shot contrastive loss for text-guided diffusion image style transfer.  In _2023 IEEE/CVF International Conference on Computer Vision (ICCV)_ , pages 22816–22825, 2023.
  * Zhai et al. [2023] X. Zhai, B. Mustafa, A. Kolesnikov, and L. Beyer.  Sigmoid loss for language image pre-training.  In _2023 IEEE/CVF International Conference on Computer Vision (ICCV)_ , pages 11941–11952, 2023.
  * Zhang and et al [2018] R. Zhang and et al.  The unreasonable effectiveness of deep features as a perceptual metric.  _CVPR_ , 2018.
  * Zhang et al. [2022] Y. Zhang, F. Tang, W. Dong, H. Huang, C. Ma, T.-Y. Lee, and C. Xu.  Domain enhanced arbitrary image style transfer via contrastive learning.  In _ACM SIGGRAPH_ , 2022.


11institutetext:  Institute of Physics, Polish Academy of Sciences, Al.
Lotników 32/46, 02-668 Warsaw, Poland
Department of Chemistry, University of Virginia, Charlottesville, VA 22901,
USA
Institut für Physik, Johannes Gutenberg Universität Mainz, 55099 Mainz,
Germany
Leibniz-Institut für Polymerforschung, Institut Theorie der Polymere, Hohe
Str. 6, 01069 Dresden, Germany
Bulgarian Academy of Sciences, Institute of Physical Chemistry, 1113 Sofia,
Bulgaria

Drops and bubbles Interface structure and roughness Molecular dynamics and
particle methods

# Rugotaxis: Droplet motion without external energy supply

Panagiotis E. Theodorakis 11 Sergei A. Egorov 223344 Andrey Milchev
551122334455

###### Abstract

Nano-patterned substrates offer possibilities for controlling the motion of
fluids without external energy supply in novel technologies in microfluidics,
coatings, etc. Here, we report on the rugotaxial motion of droplets on
wrinkled substrates with gradient in the wavelength of the wrinkles by
exploring a broad range of parameters, such as amplitude of the wrinkles,
substrate wettability, droplet size and wavelength gradient. Adopting a
theoretical and molecular dynamics approach, we determine the Cassie–Baxter
and Wenzel states of the droplets, investigate the efficiency of rugotaxis as
a function of different parameters, and discuss additional effects, such as
pinning. We find that shallow wrinkles characterised by small wavelength
gradients, and moderate adhesion of the droplet to the substrate favour the
rugotaxis motion with growing droplet size, when pinning is avoided. We also
find that the driving force in rugotaxis is the gain in interfacial energy
between the droplet and the substrate as the droplet enters regions of denser
wrinkles (smaller wavelengths of the wrinkles).

###### pacs:

47.55.D-

###### pacs:

68.35.Ct

###### pacs:

02.70.Ns

The development of various technologies in microfluidics, microfabrication,
coatings, and biology require the motion of fluids along predetermined
trajectories [1, 2, 3, 4, 5, 6, 7, 8]. A possibility of realising such motion
is by using gradient substrates, namely substrates with gradually changing
properties in a certain direction along the substrate [9, 10, 11, 12]. For
example, by exploiting differences in tissue stiffness, cells are able to move
from softer to stiffer regions, a phenomenon known as durotaxis [13, 14].
Durotaxis is particularly appealing for nanotechnology applications, because
fluid motion is sustainable without providing external energy from a source
[15, 16, 17, 18]. In contrast, motion caused, for example, by a temperature
gradient (thermotaxis) would require external energy supply into the system to
maintain the gradient that is responsible for the fluid motion [19]. In this
regard, characteristic cases of droplet motion caused by external energy
supply are chemically driven droplets [20, 21], and droplets on vibrated
substrates [22, 23, 24] or wettability ratchets [25, 26, 27].

Hiltl and Böker have recently demonstrated in their experiments the
possibility of causing the motion of a water droplet onto a sinusoidal
wrinkle-patterned, solid substrate without using an external energy source, a
phenomenon known as rugotaxis [28]. In this case, the wavelength
characterising the wavy shape of the wrinkles decays as a function of the
position, namely, there is a wavelength gradient characterising the wrinkles
[28]. Hiltl and Böker have found that the droplet moves toward smaller wrinkle
dimensions (wrinkles described by smaller wavelength). By investigating
wrinkled substrates with wavelengths between 230 and 1200 nm, and amplitudes
ranging from 7 to 230 nm, they attributed this phenomenon to the imbalance of
the receding and advancing contact angles. In particular, contact angles
correlate with the wavelength of the wrinkles, that is, larger contact angles
for larger wavelengths, which naturally corresponds to differences of the
substrate density along the direction of the wavelength gradient and the
rugotaxial motion. The imbalance between the advancing and the receding
contact angles as the driving force for droplet motion has been theoretically
discussed in detail by Brochard [29] in the context of chemical or thermal
gradients. Recently, an asymptotic theory has been developed to match the
advancing and the receding angles to respective solutions of the problem at
the microscale, using the velocity as a small parameter of an asymptotic
expansion, which determines droplet shape and velocity as a function of the
wettability gradient [30].

While wrinkled substrates without gradients [31, 32, 33, 34, 35, 36, 37, 38]
and substrates of similar geometries [39, 40, 41, 42, 43, 44, 45] have been
studied in different contexts, rugotaxis have remained unexplored. Inspired by
the work of Hiltl and Böker[28], we employ theoretical [46, 47] and molecular
dynamics (MD) [17] modelling to investigate the self-propelled motion of
nanodroplets on wrinkled, solid substrates with wavelength gradient
characterising the wrinkles. We show that the efficiency depends on the
particular choice of parameters for the substrate, which leads to specific
scenarios, when rugotaxis is possible. We also find that the driving force of
the rugotaxis phenomenon is the gain in the interfacial energy between the
droplet and the substrate as the droplet reaches areas of denser wrinkles.
Toward studying rugotaxis, we have also explored the transition between
Cassie–Baxter [48] (CB, a state where the liquid droplet does not fully
penetrate the grooves on rough surfaces and leaves air gaps between the
droplet and the substrate) and Wenzel [49] (W, a state where the droplet fully
penetrates the grooves) states in substrates without gradient and compared our
results with theoretical predictions [38] in order to set the stage for a
better understanding of rugotaxis.

Figure 1:  A typical snapshot of an initial configuration for the rugotaxis
study with a droplet on a wrinkled substrate with gradient in the wavelength,
$\lambda$, characterising the sinusoidal shape of the wrinkles. The dimensions
of the substrate in the $x$ and $y$ directions are $L_{x}=L_{y}=100\sigma$.
The amplitude of the wrinkles is $A=2\sigma$, the initial wavelength at the
very left side of the substrate is $\lambda_{0}=10\sigma$ and that at the
right side of the substrate is $\lambda_{E}=\sigma$. Given the linear decrease
of the wavelength in the $x$ direction, the wavelength gradient is constant
and equal to $G_{\lambda}=(\lambda_{0}-\lambda_{E})/L_{x}=0.09$, in this case.
The rugotaxial motion of the droplet takes place in the $x$ direction, along
the wavelength gradient, as indicated by an arrow. Snapshots have been
produced using VMD software [50].

Figure 1 illustrates an initial configuration of our in silico experiments for
a particular case. A liquid droplet is placed on a sinusoidal substrate with
wavelength gradient of the wrinkles in the $x$ direction. This requires that
the $x$ and $z$ coordinates of the beads be associated with the relation
$z=A\sin{(2\pi x/\lambda)}$, where $A$ is the amplitude and $\lambda$ is the
wavelength characterising the wrinkles. To implement a gradient in the
wavelength of the wrinkles, one needs to multiply the wavelength with a
coefficient that depends linearly on the position $x$ of the bead on the
substrate. Such substrate design is able to cause the rugotaxial motion of the
droplet in the $x$ direction along the wavelength gradient. Rugotaxial motion
will take place for a particular set of parameters characterising the
substrate design, such as the amplitude of the wrinkles, $A$, as well as the
initial wavelength, $\lambda_{0}$ at the one end of the substrate and the
final wavelength at the other end, $\lambda_{E}$ ($\lambda_{0}>\lambda_{E}$).
The latter wavelengths determine the wavelength gradient of the wrinkles,
expressed as $G_{\lambda}=(\lambda_{0}-\lambda_{E})/L_{x}$ ($L_{x}$ is the
linear dimension of the substrate in the $x$ direction). The adhesion of the
droplet to the substrate is controlled through the $\varepsilon_{sp}$
parameter of the Lennard-Jones potential. The details of our MD model are the
same as in Ref. [17] (see, also, Supplementary Information). In order to
suppress droplet evaporation, we consider polymer melt droplets made of
sufficiently long linear macromolecules of length $N=10$ monomers each. This
choice ensures the absence of evaporation effects, which might have otherwise
affected the outcome of our in silico experiments [17]. In our study,
different droplet sizes were considered, namely $N_{p}=100$, $600$, and
$4800$, where $N_{p}$ is the total number of polymer chains comprising the
droplet.

Figure 2:  State diagrams for droplets on wrinkled substrates without gradient
as a function of the wavelength, $\lambda$, and the substrate–droplet
attraction strength, $\varepsilon_{sp}$, for different substrate amplitudes,
as indicated. Symbols are as follows: W: Wenzel state, CB: Cassie–-Baxter
state, WCB: droplet is in the W or the CB states at different instances in
time during the simulation due to the thermal fluctuations, and F: Film
(droplet spreading due to the strong attraction between the droplet and the
substrate as the density of the wrinkles increases). Here, $N_{p}=600$.

By investigating a range of amplitudes ($A=0.5$–2.5$\sigma$), wavelengths
($\lambda=1$–10$\sigma$) and strengths of attraction
($\varepsilon_{sp}=0.1$–0.7$\varepsilon$) between the droplet and the
substrate, we have initially determined the state [e.g. Wenzel (W),
Cassie–Baxter (CB)] of the droplet on substrates without gradient by using MD
simulation. Our results are presented in the form of state diagrams in Fig. 2
for two cases, namely $A=\sigma$ and $A=2\sigma$. The cases $\lambda=\sigma$
are not shown in the figure, because they represent flat substrates and a
discussion about W and CB states would be irrelevant. Also, cases that the
attraction strength $\varepsilon_{sp}=0.1\varepsilon$ were omitted, because
the droplet would detach from the substrate due to its thermal fluctuations.
The results of Fig. 2 are in agreement with theoretical predictions [38],
which suggest that the CB state gradually appears in our diagrams in place of
the W state as the amplitude of the wrinkles increases, which translates into
the increase of the substrate roughness $A/\lambda$ for each $\lambda$.
Moreover, an increasing strength of attraction between the droplet and the
substrate favours the W state, in agreement with the theory [38]. While for
very small amplitude ($A=0.5\sigma$) the W state appears even for very small
strength of attraction between the substrate and the droplet and the CB state
is only partially present during the simulations, for $A>0.5\sigma$, the CB
state becomes stable and appears in a larger number of cases as the roughness
$A/\lambda$ increases. However, the MD results also indicate the existence of
a metastable state (WCB; Figure 2), where the droplet is either in the W or
the CB state during the simulation. In addition, we observe that the WCB
regime is generally narrow and, also, absent in particular cases as
$\varepsilon_{sp}$ increases. Our results can be summarised as follows: The
formation of the W state is favoured by a small amplitude, $A$, a large
wavelength, $\lambda$, and a large attraction between the droplet and the
substrate, $\varepsilon_{sp}$. Our results are in agreement with the
theoretical predictions and further comparisons with theory will be discussed
in the following [38].

The equilibrium contact angle [51, 52, 53, 54], $\theta_{E}$, provides the
means of theoretically computing the free energies of Wenzel (W) and
Cassie–Baxter (CB) states as a function of the substrate roughness
$r=A/\lambda$ [38]. These results are presented in Fig. 3a for four
representative values of $\theta_{E}$. The free energies are given in
dimensionless form, $F/(\gamma_{LV}\lambda\sigma)$, where $\gamma_{LV}$ is the
surface tension of the liquid–vapour boundary, and $\sigma$ is the monomer
diameter. As expected (Fig. 2a), with increasing roughness, the W state
becomes metastable, while the CB state becomes stable. For each value of
$\theta_{E}$, this transition occurs at a particular critical roughness,
$A_{c}/\lambda$. Above the line, the CB states are stable, while below the
line the W states are stable. Following this analysis, the results obtained
from the simulation are presented in Fig. 3b, which are in very good agreement
with the theoretical predictions (see Fig. S1 and discussion in Supplementary
Information). In particular, for $\theta_{E}<90^{\circ}$, only W states are
possible. In addition, as $\theta_{E}$ approaches 180∘, even a small degree of
roughness can induce the transition from W to CB states. The simulation
results indicate that the boundary between the W and CB becomes sharper for
larger values of $\theta_{E}$. Finally, the CB state is absent for
$\theta_{E}<90^{\circ}$ ($\varepsilon_{sp}>0.6\varepsilon$), in agreement with
the theoretical calculations [38] (Fig. 3a). We have also analysed the
different contributions to the free energy on the basis of a hybrid MD–DFT
approach [46, 47] (see Supplementary Information for more details on the
method and references therein). In particular, we have identified the
attractive term between the droplet and the substrate, $F_{sp}$, as the main
factor that determines the state of the droplet (W or CB). Interestingly, the
dependence of $F_{sp}$ on the attraction strength, $\varepsilon_{sp}$, shows a
faster than linear decrease as $\varepsilon_{sp}$ increases, that is when the
adhesion of the droplet to the substrate is stronger. This is due to the
increase of the substrate density as the wrinkles become denser for smaller
wavelengths, $\lambda$ (inset of Fig. 3a). Moreover, differences in $F_{sp}$
are small, at least for weaker adhesion, when the droplet is in ‘loose’
contact with the substrate.

Figure 3:  (a) Free energies of Wenzel (W) and Cassie–Baxter (CB) states of a
droplet with $N_{p}=600$ as a function of substrate roughness, $A/\lambda$,
for four values of the contact angle $\theta_{E}$, as indicated. Solid lines
correspond to stable states while dashed lines correspond to the metastable
states. Inset presents the free energy component of the interaction between
the droplet and the substrate as a function of the interaction parameter
$\varepsilon_{sp}$ for a substrate with $A=2.0\sigma$. (b) Diagram of
different states (W, CB, WCB, F) as a function of the roughness, $A/\lambda$,
and the contact angle, $\theta_{E}$ (cf. Fig. S1 of Supplementary Information
showing the theoretical prediction[38]).

We have analysed the rugotaxial motion for a wide and relevant range of
parameters. From our results, we have determined that each case is distinct.
For example, two substrates with the same gradient, $G_{\lambda}$, but
different sets of ($\lambda_{0}$, $\lambda_{E}$) can show completely different
behaviour regarding their ability to cause rugotaxial motion. Apart from these
obvious differences, certain parameter combinations can cause the droplet to
be in the W or CB states, the formation of a film, pinning, and others. In a
recent molecular dynamics study [55], it has been found that even the
slightest heterogeneity on the substrate can hinder the motion of a droplet on
a substrate (pinning). Overcoming such a pinning barrier (i.e. depinning) can
take place by increasing the interfacial energy between droplet and substrate
due to changes of the substrate properties along the direction of motion.
Thus, the difference in wettability on both sides of the pinning barrier has
been exploited [55], while in our study the droplet steps over the barrier due
to the decrease of the wavelength, which in turn results in an increase of the
interfacial energy between droplet and substrate. This further highlights that
successful and efficient rugotaxis requires an appropriate design, which takes
into account all these effects.

Based on our previous experience with investigations on the durotaxis
phenomenon [17] and the experience gained by the current study, we have
determined that the average rugotaxial velocity, $V$, of the droplet is an
appropriate measure to assess the overall efficiency of the rugotaxis motion
in successful cases, that is cases that can uninterruptedly translocate
throughout the substrate in the direction of the wavelength gradient. In this
case, $V=L/t$, where $L$ is the distance covered by the centre of mass of the
droplet during its translocation from the very left part of the substrate to
the very right part of the substrate in the $x$ direction (Fig. 1), and $t$ is
the time required to complete the rugotaxis motion. Then, Fig. 4 presents
various characteristic examples that illustrate the dependence of the velocity
for different droplet size and different scenarios of successful rugotaxis
cases. In general, larger droplets are able to easier overcome the pinning
barriers [55], with the latter becoming more pronounced as the droplet is in
the Wenzel state (Fig. 2), which otherwise favour the rugotaxial motion.
Still, one can observe the difference in the behaviour of large droplets
($N_{p}=4800$) between substrates with $A=0.5\sigma$ and $A=2.0\sigma$, where
the increase of $\varepsilon_{sp}$ leads to different behaviour in the
rugotaxis efficiency. Moreover, we have found that $A=0.5\sigma$ leads to the
most efficient rugotaxis, particularly for smaller droplets.

We take a closer look at the rugotaxis efficiency and inspect the different
scenarios in our in silico experiments, providing, also, further insight into
the parameters determining rugotaxis’ performance (Fig. 4). We have identified
a clear distinction between substrates with small ($A=0.5\sigma$) and larger
amplitudes $A>0.5\sigma$. In particular, in the case of shallow wrinkles,
$A=0.5\sigma$, we observe two different scenarios. In the first scenario, the
droplets attain their highest velocity $V$ for small $\lambda_{0}$, namely
$\lambda_{0}=2\sigma$ and $\lambda_{E}=\sigma$, albeit $V$ tends to decrease
with growing adhesion $\varepsilon_{sp}\gtrsim 0.4$. In particular, the motion
of small droplets ($N_{p}=100$) is significantly hampered as the adhesion
strength, $\varepsilon_{sp}$, grows.

Figure 4:  The dependence of the average rugotaxis velocity, $V$, on the
attraction strength, $\varepsilon_{sp}$, for different cases of $\lambda_{0}$,
$\lambda_{E}$, $N_{p}$ and $A$, as indicated.  Figure 5:  Interfacial energy,
$E_{sp}$, and position of the mass centre of the droplet, $x_{com}$, as a
function of time, as indicated. Hence, the distance covered by the droplet
refers to its centre of mass and droplets of different size will move a
different distance in each in silico expriment. Arrows indicate the the time
that pinning occurs for different cases of $N_{p}$ as indicated. We also
indicate the unpinning of the droplet with an arrow in the case of
$N_{p}=600$. For $N_{p}=100$, the droplet remains pinned throughout the
simulation. For $N_{p}=4800$, there is no pinning. Here, $A=2\sigma$,
$\lambda_{0}=4\sigma$, $\lambda_{E}=\sigma$,
$\varepsilon_{sp}=0.4\varepsilon$. $N_{p}=100$, $600$ and $4800$, as
indicated.

In contrast (cf. Fig. 4a), larger gradients (e.g
$\lambda_{0}=6\sigma,\lambda_{E}=3\sigma$), generally lower the mean velocity,
$V$, which is found, however, to grow then steadily with increasing adhesion
$\varepsilon_{sp}$. Moreover, the larger gradient
($\lambda_{0}=6\sigma,\lambda_{E}=2\sigma$) nearly doubles $V$ with respect to
($\lambda_{0}=6\sigma,\lambda_{E}=3\sigma$) whereby for $\varepsilon_{sp}=0.6$
the speed of the largest drop attains that from the first scenario. For deeper
wrinkles, $A=2\sigma$ (Fig. 4b), the same trend is observed although pinning
effects occur more frequently so that for the case, $\lambda_{0}=2\sigma$,
$\lambda_{E}=\sigma$ (not shown here), no rugotaxis whatsoever takes place.
Apart from the smallest droplets, $N_{p}=100$, an increase in the gradient
$G_{\lambda}$ leads similarly to faster motion, which becomes more efficient
with growing drop size. However, further increase of the gradient or
$\lambda_{0}$ will hinder the motion of the droplets, independently of the
droplet size. Moreover values of $\varepsilon_{sp}=0.7\varepsilon$ (strong
adhesion of the droplet to the substrate) will always prevent the motion of
the droplet or lead to complete droplet spreading (e.g. for $N_{p}=4800$).
Still, we have observed that larger droplets overcome more easily the pinning
barriers, which is in agreement with recent simulation results [55].

As in the case of durotaxial motion [17], we have identified the change in the
interfacial energy between the droplet and the substrate as the main driving
force of rugotaxis, also determining its efficiency (Fig. 5). In our case, the
interfacial energy is simply the total pair interaction between the droplet
and the substrate beads. Crucially, for a certain droplet size this depends on
the particular parameters, $A$, $\varepsilon_{sp}$, $\lambda_{0}$, and
$\lambda_{E}$. We have also identified cases when the droplet exhibits pinning
during rugotaxis but is also able to overcome the associated energy barrier
caused by the pinning and eventually complete the rugotaxial motion. While in
such cases, rugotaxis is successful and eventually the droplet reaches the
very right end of the substrate (Fig. 5), in many other cases considered in
our study as unsuccessful rugotaxis cases, the droplet failed to overcome the
associated energy barrier. Typically, such pinning events occur at the
transition from $W$\- to $CB$-state as the drop partially climbs out of the
substrate troughs whereby the interfacial energy $E_{sp}$ rises before the
whole droplet attains a new $CB$-state where droplet beads cannot enter into
the narrowed wrinkles[55].

Figure 5 contains plots of the position of the centre of mass of the droplet,
$x_{com}$, and the interfacial energy, $E_{sp}$, as a function of the elapsed
time from the beginning until the end of the rugotaxial motion. For
$N_{p}=600$, we find that the droplet tranlocates efficiently until pinning
occurs. An important point here is that the energy, $E_{sp}$, continues to
decrease just after the pinning, which allows the droplet to move to the
right, where the underlying density (and wettability) of the substrate
increases with the wavelength gradient.

In summary, we have investigated solid, wrinkled substrates with and without
wavelength gradient in one direction, in an attempt to explore the rugotaxial
motion of droplets. To achieve our goal we have mainly employed MD simulation
based on coarse-grained force-fields, as we have done previously in the case
of durotaxis.[17] We have also provided relevant theoretical background and,
also, performed numerical calculations based on a hybrid MD–DFT approach. [46,
47, 56]

We have constructed the diagram of W and CB states for a range of parameters.
For small wrinkle amplitudes, the W state prevails for a range of
droplet–substrate adhesion strength and wrinkle wavelengths. As the amplitude
increases, the CB state sets in initially at small adhesion strengths and
wrinkle wavelengths. As the amplitude further increases, the CB state appears
for a larger range of wavelengths and adhesion strengths. Theory [38] predicts
the boundary between W and CB states, which is in agreement with the
simulation results. Based on a hybrid MD–DFT approach, we have identified the
interfacial energy-term to be mainly responsible for the different behaviour
(W or CB).

In the case of the rugotaxis MD in silico experiments, we find a range of
optimal choices of the parameters that govern rugotaxis. In the case of
shallow wrinkles with $A=0.5\sigma$, we observe the most efficient rugotaxial
motion for $\lambda_{0}=2\sigma$, which corresponds to a small wavelength
gradient. For larger amplitudes, increased wavelength gradients lead to more
efficient rugotaxis up to some moderate adhesion strength. As a result, our
results indicate that an appropriate substrate design will determine the
efficiency of the rugotaxial motion.

Based on our analysis, we have identified that changes of the interfacial
energy between the substrate and the droplet constitute the driving force of
rugotaxial motion, as has been also found previously in the case of durotaxis
[17]. Hence, the two phenomena share characteristics, but, in the case of
rugotaxis, this crucially depends on a larger number of parameters that can
lead to different scenarios, with pinning playing a crucial role. We
anticipate that our study elucidates important aspects of the rugotaxis
phenomenon and suggests design principles for nano-fabricated wirnkle-
patterned substrates that can increase the efficiency of rugotaxial motion.

###### Acknowledgements.

This research has been supported by the National Science Centre, Poland, under
grant No. 2019/35/B/ST3/03426. A. M. acknowledges support by COST (European
Cooperation in Science and Technology [See http://www.cost.eu and
https://www.fni.bg] and its Bulgarian partner FNI/MON under KOST-11). This
research was supported in part by PLGrid Infrastructure.

## References

  * [1] M. Srinivasarao, D. Collings, A. Philips, and S. Patel.  Three-dimensionally ordered array of air bubbles in a polymer film.  Science, 292:79–83, 2001.
  * [2] M. K. Chaudhury and G. M. Whitesides.  How to make water run uphill.  Science, 256(5063):1539–1541, 1992.
  * [3] T.-S. Wong, S. H. Kang, S. K. Y. Tang, E. J. Smythe, B. D. Hatton, A. Grinthal, and J. Aizenberg.  Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity.  Nature, 477:443–447, 2011.
  * [4] G. Lagubeau, M. Le Merrer, C. Clanet, and D. Quéré.  Leidenfrost on a ratchet.  Nat. Phys., 7:395–398, 2011.
  * [5] M. Prakash, D. Quéré, and J. W. Bush.  Surface tension transport of prey by feeding shorebirds: The capillary ratchet.  Science, 320:931–934, 2008.
  * [6] A. Darhuber and S. Troian.  Principles of microfluidic actuation by modulation of surface stresses.  Annu. Rev. Fluid Mech., 37:425–455, 2005.
  * [7] Z. Yao and M. J. Bowick.  Self-propulsion of droplets by spatially-varying surface topography.  Soft Matter, 8(4):1142–1145, 2012.
  * [8] H. Li, T. Yan, K. A. Fichthorn, and S. Yu.  Dynamic contact angles and mechanisms of motion of water droplets moving on nano-pillared superhydrophobic surfaces: A molecular dynamics simulation study.  Langmuir, 34:9917–9926, 2018.
  * [9] A. Bardall, S.-Y. Chen, K. E. Daniels, and M. Shearer.  Gradient-induced droplet motion over soft solids.  IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications), 85(3):495–512, 2020.
  * [10] G. Karapetsas, N. T. Chamakos, and A. G. Papathanasiou.  Efficient modelling of droplet dynamics on complex surfaces.  J. Phys.: Condens. Matter, 28(8):085101, 2016.
  * [11] B. Zhang, X. Liao, Y. Chen, H. Xiao, Y. Ni, and X. Chen.  Rapid Programmable Nanodroplet Motion on a Strain-Gradient Surface.  Langmuir, 35(7):2865–2870, 2019.
  * [12] J. Leng, Y. Hu, and T. Chang.  Nanoscale directional motion by angustotaxis.  Nanoscale, 12:5308–5312, 2020.
  * [13] K. A. Lazopoulos and D. Stamenović.  Durotaxis as an elastic stability phenomenon.  J. Biomech., 41(6):1289–1294, 2008.
  * [14] I. Palaia, A. Paraschiv, V. E. Debets, C. Storm, and A. Šarić.  Durotaxis of passive nanoparticles on elastic membranes.  ACS Nano, 15(10):15794–15802, 2021.
  * [15] A. S. Barnard.  Nanoscale locomotion without fuel.  Nature, 519:37–38, 2015.
  * [16] T. Chang, H. Zhang, Z. Guo, X. Guo, and H. Gao.  Nanoscale directional motion towards regions of stiffness.  Phys. Rev. Lett., 114(1):015504, 2015.
  * [17] P. E. Theodorakis, S. A. Egorov, and A. Milchev.  Stiffness-guided motion of a droplet on a solid substrate.  J. Chem. Phys., 146(24):244705, 2017.
  * [18] M. Becton and X. Wang.  Controlling nanoflake motion using stiffness gradients on hexagonal boron nitride.  RSC Adv., 6:51205–51210, 2016.
  * [19] M. Becton and X. Wang.  Thermal gradients on graphene to drive nanoflake motion.  J. Chem. Theory Comput., 10:722–730, 2014.
  * [20] F. D. Dos Santos; and T. Ondarquhus.  Free-Running Droplets.  Phys. Rev. Lett., 75(16):2972, 1995.
  * [21] S. W. Lee, D. Y. Kwok, and P. E. Laibinis.  Chemical influences on adsorption-mediated self-propelled drop movement.  Phys. Rev. E, 65(5):9, 2002.
  * [22] S. Daniel and M. K. Chaudhury.  Rectified motion of liquid drops on gradient surfaces induced by vibration.  Langmuir, 18(9):3404–3407, 2002.
  * [23] P. Brunet, J. Eggers, and R. D. Deegan.  Vibration-induced climbing of drops.  Phys. Rev. Lett., 99(14):3–6, 2007.
  * [24] P. Brunet, J. Eggers, and R. D. Deegan.  Motion of a drop driven by substrate vibrations.  Eur. Phys. J.: Spec. Top., 166(1):11–14, 2009.
  * [25] A. Buguin, L. Talini, and P. Silberzan.  Ratchet-like topological structures for the control of microdrops.  Appl. Phys. A: Mater. Sci. Process., 75(2):207–212, 2002.
  * [26] U. Thiele and K. John.  Transport of free surface liquid films and drops by external ratchets and self-ratcheting mechanisms.  Chem. Phys., 375(2-3):578–586, 2010.
  * [27] X. Noblin, R. Kofman, and F. Celestini.  Ratchetlike motion of a shaken drop.  Phys. Rev. Lett., 102(19):1–4, 2009.
  * [28] S. Hiltl and A. Böker.  Wetting phenomena on (gradient) wrinkle substrates.  Langmuir, 32(35):8882–8888, 2016.
  * [29] F. Brochard.  Motions of droplets on solid surfaces induced by chemical or thermal gradients.  Langmuir, 5(2):432–438, 1989.
  * [30] L. M. Pismen and U. Thiele.  Asymptotic theory for a moving droplet driven by a wettability gradient.  Phys. Fluids, 18(4):042104, 2006.
  * [31] E. Cerda and L. Mahadevan.  Geometry and physics of wrinkling.  Phys. Rev. Lett., 90:074302, 2003.
  * [32] A. Malijevský.  Filling, depinning, unbinding: Three adsorption regimes for nanocorrugated substrates.  Phys. Rev. E, 102:012804, 2020.
  * [33] S. Hiltl, M.-P. Schürings, A. Balaceanu, V. Mayorga, C. Liedel, A. Pich, and A. Böker.  Guided self-assembly of microgels: from particle arrays to anisotropic nanostructures.  Soft Matter, 7:8231–8238, 2011.
  * [34] S. G. Lee, H. S. Lim, D. Y. Lee, D. Kwak, and K. Cho.  Tunable anisotropic wettability of rice leaf-like wavy surfaces.  Adv. Funct. Mater., 23(5):547–553, 2013.
  * [35] K. W. Sun, M. S. Ambrosia, T. W. Kwon, and M. Y. Ha.  A hydrophobicity study on wavy and orthogonal textured surfaces.  Comput. Fluids, 140:347–356, 2016.
  * [36] J. Y. Chung, A. J. Nolte, and C. M. Stafford.  Surface wrinkling: A versatile platform for measuring thin-film properties.  Adv. Mater., 23:349–368, 2011.
  * [37] H. E. Jeong, M. K. Kwak, and K. Y. Suh.  Stretchable, adhesion-tunable dry adhesive by surface wrinkling.  Langmuir, 26:2223–2226, 2010.
  * [38] G. Carbone and L. Mangialardi.  Hydrophobic properties of a wavy rough substrate.  Eur. Phys. J. E, 16(1):67–76, 2005.
  * [39] J. A. Ko, M. Ambrosia, and M. Y. Ha.  A study of the wetting characteristics of a nano-sized water droplet on heterogeneous striped surfaces.  Comput. Fluids, 112:19–34, 2015.
  * [40] M. S. Ambrosia, J. Jang, and M. Y. Ha.  Static and dynamic hydrophobicity on a nano-sized groove/ridge surface.  Comput. Fluids, 114:75–83, 2015.
  * [41] D. M. Spori, T. Drobek, S. Zürcher, M. Ochsner, C. Sprecher, A. Mühlebach, and N. D. Spencer.  Beyond the lotus effect: Roughness influences on wetting over a wide surface-energy range.  Langmuir, 24(10):5411–5417, 2008.
  * [42] V. Kumar and J. R. Errington.  Impact of small-scale geometric roughness on wetting behavior.  Langmuir, 29(38):11815–11820, 2013.
  * [43] E. M. Grzelak and J. R. Errington.  Nanoscale limit to the applicability of Wenzel’s equation.  Langmuir, 26(16):13297–13304, 2010.
  * [44] C. Liu, I. Legchenkova, L. Han, W. Ge, C. Lv, S. Feng, E. Bormashenko, and Y. Liu.  Directional droplet transport mediated by circular groove arrays. part i: Experimental findings.  Langmuir, 36(32):9608–9615, 2020.
  * [45] C. Liu, I. Legchenkova, L. Han, W. Ge, C. Lv, S. Feng, E. Bormashenko, and Y. Liu.  Directional droplet transport mediated by circular groove arrays. part ii: Theory of effect.  Langmuir, 37(5):1948–1953, 2021.
  * [46] F. Lo Verso, L. Yelash, S.A. Egorov, and K. Binder.  Interactions between polymer brush-coated spherical nanoparticles: The good solvent case.  J. Chem. Phys., 135:214902, 2011.
  * [47] A. Milchev, S.A. Egorov, D.A. Vega, K. Binder, and A. Nikoubashman.  Densely packed semiflexible macromolecules in a rigid spherical capsule.  Macromolecules, 51:2002–2016, 2018.
  * [48] A. B. D. Cassie and S. Baxter.  Wettability of porous surfaces.  Trans. Faraday Soc., (5):546–551, 1944.
  * [49] R. N. Wenzel.  Resistance of solid surfaces to wetting by water.  Ind. Eng. Chem. Res., 28(8):988–994, 1936.
  * [50] W. Humphrey, A. Dalke, and K. Schulten.  VMD: Visual Molecular Dynamics.  J. Mol. graph., 14:33–38, 1996.
  * [51] P.-G. de Gennes, F. Brochard-Wyart, and D. Quéré.  Capillarity and wetting phenomena: Drops, bubbles, pearls, waves.  Springer, New York, 2004.
  * [52] P. G. De Gennes.  Wetting: Statics and dynamics.  Rev. Mod. Phys., 57(3):827–863, 1985.
  * [53] P. E. Theodorakis, E. A. Müller, R. V. Craster, and O. K. Matar.  Modelling the superspreading of surfactant-laden droplets with computer simulation.  Soft Matter, 11:9254–9261, 2015.
  * [54] E. R. Smith, P. E. Theodorakis, R. V. Craster, and O. K. Matar.  Moving contact lines: Linking molecular dynamics and continuum-scale modeling.  Langmuir, 34:12501–12518, 2018.
  * [55] P. E. Theodorakis, A. Amirfazli, B. Hu, and Z. Che.  Droplet control based on pinning and substrate wettability.  Langmuir, 37(14):4248–4255, 2021.
  * [56] H. Xu, L. HongLai, and H. Ying.  Dynamic density functional theory based on equation of state.  Chem. Eng. Sci., 62:3494–2016, 2007.