In this paper, we propose a scheme for the hybridization of domain modeling and theoretical knowledge in nanotechnology with Artificial Intelligence(AI) techniques and evaluate the success of its application to predict the relationship between nanosurface morphology and wettability. We utilize domain knowledge consisting of two parts. The first part is a mathematical modeling based on the inverse Fourier transform for the generation of rough surfaces with Gaussian or non-Gaussian height distributions, characterized by their first moments (Rms, skewness, kurtosis) and the correlation lengths along x and y-axes. The second part lies in the assumption that the Wenzel scenario for wetting of rough surfaces holds where the critical parameter for contact angle determination is the roughness ratio , defined as the ratio of true (active) area of the solid surface to the apparent (projected) area. By creating diferent types of surfaces with a variety of input parameters, we create a database linking surface roughness parameters to the ratio . This database is used to train Machine Learning (ML) models and validate them appropriately. Specifically, we train deep, feed-forward neural networks and random forest models and validate them on a separate (held-out) test dataset. We investigate systematically the amount of input data needed to get accurate predictions on the test data. We also evaluate the importance of diferent input roughness parameters with respect to their efects on surface wettability. To this end, we study the weights that the learning AI models assigned to roughness parameters through training and discuss the findings with respect to experimental expectations.
Nanostructuring plays a fundamental role in nanotechnology since
it enables new properties and functionalities of material surfaces.
In order to provide a quantitative link between the geometry of
nanostructure morphologies and the induced surface properties, we
ifrst need to find the proper mathematical tools to describe surface
morphology. To this end, several parameters and metrics have been
proposed for the quantitative characterization of nanostructure
morphology. Some of them are more closely linked to the
fabrication process, while others more directly fit to the critical property
of a targeted application. For example, surfaces with stochastic
morphologies (rough surfaces) are widely used in the strong
modification of the wetting behaviour of materials. According to the first
scenario (Wenzel model) for the impact of surface roughness on
wetting and contact angle, the critical parameter of surface
nanoroughness is the roughness ratio [
Up until now, theoretical modelling and experiments have been
used to quantify these links [
During the last decade, several nanotechnology areas have started
to benefit from AI techniques aiming for example, to predict the
properties of new nanomaterials, to enhance microscopy results,
to accelerate simulations, to link manufacturing conditions with
nanostructure morphology and then with the properties and
performance of the nanostructured devices [
The overall aim of this paper is to propose a hybridization scheme to facilitate the synergy of data-centric methods with domain modelling and theoretical knowledge concerning the link of the morphology of nanostructured surfaces with their properties and functionalities. The key ideas of this approach are: a) to use properly designed modeling results to train and validate AI methods along with experimental data when they are available and b) to exploit the ability of ML techniques to reverse input and output so that the targeted design of a nanostructured product dictates the choice of nanostructure geometry and manufacturing conditions. We will focus on the first idea and we will apply it to evaluate the success of AI techniques hybridized with domain modeling results to predict the relationship between nanosurface morphology and wettability. More specifically, the contributions of the paper can be outlined as follows: • An implementation of domain modeling results in AI techniques is realized for training and validation. • A comparison of diferent AI models is performed based on the physical modeling results. • A study on the numerosity of required data (rough surfaces), to train suficiently accurate AI methods. • An estimate of the relative importance of input roughness parameters on wetting behavior, supported by a discussion to feed domain decisions and evaluations.
The paper is structured as follows. We begin, in section 2, with a presentation of the related recent work and the need to investigate further the predictive capability of AI techniques in the nanotechnology applications and specifically wetting behavior. The mathematical modeling methodology for the generation of rough surfaces to train AI models as well as the AI techniques used in the paper is the subject of the section 3. Section 4 presents the results of AI techniques and their comparison. The paper closes with the summary in the final section 5. 2
The applications of AI to physical problems and especially materials
science have been studied in many contexts during the last decade,
creating a novel research framework termed ”data-driven materials
science” (Teng Zhou et al., 2019 [
Regarding the prediction of wetting behaviour, Amir Kordijazi
et al., (2020) used ML techniques to predict the water contact
angle on surfaces of ductile iron. They used a set of experimental
measurements with input parameters the material composition,
droplet size, the surface grit size and roughness and the time of the
exposure to the liquid. The authors also evaluated the importance
of each input parameter on the value of contact angle and they
justified the primary role of surface roughness determined by the
grit size. However, it was not specified which aspect of surface
roughness is more critical. Given that the roughness of surfaces is
a complex multifaceted phenomenon characterized by a plethora
of parameters, it is worth questioning the relative importance of
roughness parameters on surface wetting behavior. In literature,
one can find interesting results coming from both experimental and
computational approaches exploring the impact of surface
roughness parameters on contact angle and hysteresis [
In our work, we follow the data-driven approach of the 4th paradigm of science endowed by theoretical and computational modelling knowledge of the 2nd and 3rd paradigms. The aim is to investigate the prediction performance of these AI models on the efects of roughness parameters on the wetting behavior of solid surfaces. We assume that the Wenzel model assumption holds: the contact angle of droplets posed on rough surfaces is determined by the roughness ratio and especially the full active surface area. A hybridization framework is implemented, in which simulated rough surfaces with a wide spectrum of parameters and appearances are used to train and evaluate the AI models and explore their performance versus the simulation cost. We also use the capability of the developed AI models to reveal each roughness parameter importance on the observed wetting behavior. 3 3.1
In this section, we describe the methodology we used for generating
simulated rough surfaces with similar characteristics with a large
variety of experimental ones. These surfaces will be used to enrich
the dataset for training and validating the ML models. We begin
by describing the methodology for generating Gaussian and
nonGaussian surfaces with controlled spatial correlations.
Gaussian surfaces: We produce three-dimensional Gaussian
surfaces by inputting the Rms of the height distribution and the
correlation lengths along the x and y axes ( and ) of the surface
(Table 1). (Figure 1a and 1b). The heights of the generated surfaces
are calculated on a square lattice × points and area × .
Therefore, the spacing in x and y direction corresponds the ratio:
. The methodology for simulating the Gaussian surfaces is
ba−s1ed on the work of Garcia et.al. [
Gaussian Rms, , active area
Non-Gaussian Rms, , , skewness, kurtosis active area Table 1: Input parameters used for generating the active area of Gaussian and non-Gaussian surfaces. The input (roughness) parameters for the Gaussian consist of the Rms heights and the correlation lengths () in x and y directions. For the Non-Gaussian surfaces the inputs include additionally the skewness and kurtosis. noise distribution with mean value zero and standard deviation equal to the input Rms value. By applying the Gaussian filter , described in the following equation (Eq. 1) to the distribution we add the desired correlations along x and y axis.
2 2 = (−(2 2 + 2 2 )) (1)
Then, we take the Inverse Fourier Transform of the product of
Fourier transforms of and and multiply with normalization
factors to generate correlated isotropic and anisotropic Gaussian
surfaces After producing the surface, their Rms, and are
compared with the inputs to check possible divergence. The divergence
is related with limitations imposed by the discrete sampling and
ifnite range of surfaces. In such case, the algorithm of the surface
generation is repeated until the input parameters are converged.
Non-Gaussian surfaces: The method we used to model non-Gaussian
surfaces is based on the work of Yang et al. [
Subsequently, the active area was measured by integrating the secant of the angle between the surface normal and its z-direction normal.
∫ ∫ = ∫ ∫ ( ) (2)
Where, is defined as the angle that the z-axis makes with the normal vector of the diferential surface dA. In this section, we outline the AI methods we utilized in this work. We begin by overviewing the main points describing the methods and then we align these descriptions with our work.
A physical model is a domain-driven model that is created through functions that follow underlying physical laws to predict a property. When using those models, the relation between the input parameters and the output value is already known and used for predictions. A ML model is a data-driven model that is used to find an appropriate (originally unknown) function that reflects the connection of an input (given) property to an output property that we want to predict. Even though the actual relation is unknown, given suficient values for input and output values from experiments, we can create a ML (statistical) model that approximates the relation. Along with pre-existing human expertise, the approximations could add value to the manufacturer that aims to predict physical properties, without the need to exhaustively perform experiments, given the cost of such experiments in time and money.
In this work, we are using three ML models: 1) Linear
Regression 2) Random Forests 3) Neural Networks. We try these diferent
families of learning, since this problem has no prior indication of
the underlying input-output relation, which may afect the method
selection. For example, linear regression models will assume a
linear relationship between the input and the output. On the other
hand, Random forests and Neural network models use diferent
methodologies to approximate/learn non-linear relationships
between (even high dimensional) input parameters and a predicted
output property. We stress that there is no single, overall better
method for all estimation problems. This statement is better known
as the "no-free-lunch theorem" [
Linear regression models [
Random Forests [
Neural Networks (NNs) are models that learn complex functions,
combined in a non-linear manner over several layers. As such, NNs
have been shown to be excellent approximators of many families
of functions [
NN models with many layers are called Deep Neural Networks. Such a network may contain millions of parameters, identifying the function the network represents. No matter whether we have a deep network or not, these parameters are optimized to minimize the estimation error. In other words, we take each training example, provide it as input to the network and get its output prediction. Based on the real value it should have output, we change (i.e. optimize) the parameters of the network to minimize the prediction error. A number of optimization methods can be used to infer these parameters from the training data, the most well-known being back-propagation.
There is no a-priori best way to create a NN. However, standard
practices can be applied to create such an architecture [
Linear regression models are faster that both Neural networks and Random Forests but have less accuracy in cases where nonlinear inner dependencies appear in data. Both, Neural Networks (NN) and Random Forests ofer good levels of performance in different application areas. However, diferent methods ofer diferent potential for learning and approximating, coupled with diferent processing requirements (in time and memory). Random forests training costs less time (when compared to NNs in a generic setting) and after training, they can be more interpretable than the average NN. On the other hand, the accuracy a neural network can reach is higher, if we have access to the required volume and diversity of data (more data is usually needed to tune more parameters). Essentially, the selection of training data is a defining factor for ML, beyond the algorithms and the corresponding architecture. These data should satisfy three basic requirements in order to make a good approximation through a ML model. Those requirement are related to the: 1) quantity 2) diversity 3) quality of data. Regarding our work, we aimed to satisfy these three requirements while creating the training and testing datasets before using the models. By applying the methodology described in section 3.1, we generated two databases for the training and the validation of the ML models respectively. The databases consist of surfaces with diverse combinations of roughness parameters (Rms, correlation lengths, Skewness and Kurtosis) and the corresponding functional parameter (active area). The distribution of the training and validation datasets are shown in Figure 2 a) and b):
The total volume of training data-surfaces has been 3000 surfaces while for validating reached 15000. We trained each model with diferent percentages of surfaces from the train dataset to examine the efects of the training data size on model success. Also, for every percentage of the training set, we applied the training procedure 10 times with randomly selected surfaces from the database. 4.2
For the validation of the predictability of models, we evaluated the RMSRE (Root Mean Square Relative Error). While the RMSE (Root Mean Square Error) can indicate successfully the appearance of outliers, the relative value of RMSE has no units. RMSE is the squared root error of the average predicted active area from the average actual active area .
=
We trained a set of Linear Regression models as a basis comparison with the rest of models. Figure 3 shows the RMSRE of the trained models within a volume range of training data (surfaces). The linear models reached a plateau of RMSRE 9.6% using the validation dataset after approximately 100 training data points-surfaces. The model predictions diverge from the true values for high and low active area values, as seen in Figure 4.
We used random forest estimators fitting 25 regression decision trees each. Each tree averages the results to improve the predictive accuracy and control over-fitting. The decision trees used for training had a maximum depth of 25 branches with a minimum number of internal node splits of two and a mean squared error criterion for splitting. Random Forest models outperforms the Linear Regression models after approximately 120 data-surfaces (Figure 3) and achieved less than 4.0% RMSRE after 2100 training data-surfaces. The standard deviation of the RMSRE decreases as the training dataset becomes larger. Figure 5 shows that the models were able to predict the high and low active areas of the surfaces with less error as compared to the the linear regression models. However, there is still a significant amount of error for high active areas.
We used a set of neural network (NN) and a deep neural network
(DNN) models for the prediction of active areas. NN models were
trained using the adam [
For training datasets with a size of less than 300 surfaces, the NNs
performed worse as compared to the linear regression models
(Figure 3). After 300 surfaces, they outperformed the linear regression
models reaching a = 6%. It is noted that in this case as
well, the standard deviation increases with the size of the training
data. The performance of DNNs is comparable to the Random forest
model. For 2700 surfaces of training data, the DNN models show
an average error of = 2% (Figure 6). Therefore, the DNNs
generally show a significantly lower error compared to the NNs.
By taking the average of the absolute value of the weights between
the input layer and the first layer of the DNN, we can identify the
most important features of the model. Figure 7 shows that Rms has
the highest weight intensity, followed by the correlation lengths in
x and y axis ( and resp). This results is in harmony with
previous findings based on computational analysis and experimental
measurements [
In our work, we proposed a hybridization scheme that combines
data-driven methodologies (4th paradigm [
In particular, we trained Linear Regression, Random Forest, Neural Network and Deep Neural Network models with simulated nanosurfaces in order to predict the true (active) area of the surface, a critical parameter for wetting when Wenzel model is assumed. We then compared their performances in relation to the required data (simulation cost to produce rough surfaces). Random Forests and Deep Neural Networks showed the highest performance reaching 4 % of RMSRE after 1000 training data-surfaces. The models and particularly the Deep Neural Networks indicate that Rms has the highest importance in wetting behavior. The correlation lengths in the x and y axis showed lower but significant importance as well whereas skewness and kurtosis play a minor though detectable role.