Loïc Iapteff

Julien Jacques

Benoit Celse

Victor Costa
email: victor.costa@ifpen.fr

Reducing the number of experimental points to fit kinetic models: A Bayesian approach

Hydrocracking is a crucial refinery process that transforms heavy molecules (i.e. vacuum gas oil (VGO)) into lighter and highly-valued products such as naphtha, kerosene and diesel. It is a two-step process. The hydrotreatment (HDT) reactor uses a more robust catalyst, which essentially serves to remove heteroatoms from the VGO feed in order to satisfy product quality constraints and avoid poisoning of the more delicate zeolite-based HCK catalysts. The second, hydrocracking (HCK) reactor uses a commercial zeolite catalyst with a carefully selected balance of acid and metallic sites. For hydrotreatment simulation, the kinetic model is decomposed in several ODE (Ordinary Differential Equation). Catalyst vendors develop more and more catalysts. For each new catalyst (new generation), the kinetic parameters must be refitted. This task is costly and time consuming.

In this paper, in order to reduce the required number of experimental points, a Bayesian transfer approach is proposed to fit the parameters of catalyst (n+1), using the past knowledge of catalyst (n) to add more information. A method for the choice of the prior is proposed and can be used for any type of parametric model. This approach is applied and shows an improvement in the prediction performance and robustness 1 compared to a classical fitting method. In our case, only 10 pilot plant points on catalyst (n+1) are requested to refit a HDN kinetic model.

Introduction

In a classical hydrocracker, a mixture of hydrocarbon feed and hydrogen is heated and injected into a reactor vessel containing a hydrotreating catalyst. This catalyst accelerates the reactions that remove sulfur and nitrogen from the hydrocarbon and open up and saturate aromatics rings. The entire output from this reactor is then injected into a second reactor containing a hydrocracking catalyst, which helps the reactions that crack apart the hydrocarbons while saturating them with hydrogen. The resulting mix of converted and unconverted hydrocarbon is then separated. Unconverted hydrocarbon can then be recycled to the hydrocracking step for further conversion, sent to a second hydrocracking vessel, or sent to another conversion unit as feed (e.g., an FCC). Diesel range material can also be drawn off at the separation steps to maximize diesel production, or it can be processed further (through recycling or second-step hydrocracking) to maximize naphtha production. Some hydrocrackers are single-stage units with just one reactor that is usually filled with hydrocracking catalyst, but the rest of the process is the same. The HCK is generally run at high temperatures (up to 420°C) and at high hydrogen pressures (>90 bar) on a bi-functional catalyst.

The hydrocracker is particularly valuable in a refinery that is trying to maximize diesel production and reduce residual fuel oil. The hydrocracker yields a high volume of kerosene and diesel of good quality (high cetane and low sulfur). However, its volume yield of naphtha is low and of low quality (low RON).

Hydrotreating before the hydrocracking reactor ensures better operation of the hydrocracking catalyst because the catalyst is sensitive to these heteroatoms. For example, neutralisation of acid sites by basic nitrogen compounds or poisoning of metal sites by sulphur or metals. Without pretreatment, hydrocracking catalysts would deactivate quickly and would have to be replaced more often.

The most widely used method for modeling hydrotreating problems are kinetic-rate models. These models provide a mathematical representation of the chemical reactions taking place. They are commonly based on differential equations. Experimental data are used to fit the model parameters. The advantage of this method is that it offers a good understanding of the process. Many research has been carried out on kinetic model construction for hydrotreating, and reviews 1,2 on kinetic modeling for these processes are proposed. The kinetic modeling can be divided in two different approaches: the detailed kinetic modeling and the lumping approach. Due to the many compounds that occur in the feedstock and the very complex reactions in the process, a detailed kinetic model is very difficult to develop. The lumping approach is the most used. It groups the oil mixture into different components according to their chemical properties. Becker et al. [START_REF] Becker | Hydrotreatment modeling for a variety of VGO feedstocks: A continuous lumping approach[END_REF][START_REF] Becker | A continuous lumping model for hydrocracking on a zeolite catalysts: model development and parameter identification[END_REF] used continuous lumping for hydrotreating and hydrocracking modeling. The authors reported that the model gives good conversion and yield structure predictions. Sánchez et al. [START_REF] Sánchez | Kinetic model for moderate hydrocracking of heavy oils[END_REF] proposed a kinetic model for hydrocracking of heavy oils using the lumping approach. The products are defined as 5 different lumps which are unconverted residue (> 538 °C), vacuum gas oil (343-538 °C), distillates (204-343 °C), naphtha (Initial Boiling Point-204 °C) and gases. The model is reported to provide a good prediction with an average absolute error of less than 5 percents. Catalyst vendors develop more and more catalysts. For each new catalyst (new generation), the kinetic parameters must be refitted.

For each catalyst generation, the following workflow is used to fit the developed kinetic The purpose of this article is to reduce the requested time for a catalyst of generation (n+1) by using the information of catalyst of generation (n).

The aim of our work is then to improve the quality of our hydrodenitrogenation (HDN) model using Transfer Learning. To define Transfer Learning, let's start by defining a domain as D = (X , P (X )) with X a feature space and P (X ) its probability distribution, and an associated task T = (Y, f ) with f the function used to predict y ∈ Y given x ∈ X . The target data is the data from the phenomenon to be modeled, while the source dataset refers to a linked dataset, used to improve the modeling of the target. We name D s and T s the domain and task of source data and D t and T t the domain and task of the target data.

Transfer Learning is intended to improve the learning of the target predictive function f t using the knowledge in D s and T s , where D s ̸ = D t , or T s ̸ = T t . Reviews 8-10 on Transfer Learning can be read for those interested in a complete analysis of the state of the art in this domain.

The work presented in this paper is an extension of the method proposed in a previous work [START_REF] Iapteff | Reducing the number of experiments required for modelling the hydrocracking process with kriging through Bayesian transfer learning[END_REF] for the modeling of the density of the hydrocracking diesel cut using a kriging model (gaussian process). The aim of the current paper is to use the information from the previous generation catalyst (n) to reduce the required time to fit the current catalyst generation (n+1) model. This method relies on Bayesian statistics, [START_REF] Robert | The Bayesian choice: from decision-theoretic foundations to computational implementation[END_REF] which is based on the Bayes rule (Equation (1)):

π(θ|y) = π(θ)f (y|θ) π(θ)f (y|θ)dθ , (1)
where f (y|θ) is a parametric model with parameters θ, π(θ) is the prior distribution and π(θ|y) is the posterior distribution. Bayesian inference assumes external knowledge about the parameters, without having seen the data. This external information is contained in the prior distribution π(θ) and have to be chosen by the user. In this work, we propose to solve the transfer learning problem by choosing a prior depending on the source model parameters.

Section 2 shows the material and the developed methods to tackle the problem.

Section 3 shows the obtained results.

Section 4 concludes the work.

2 Material and Methods

Data Presentation

In this paper, we focus on the hydrodenitrogenation (HDN) modeling. For our application, two datasets are available. The first data set corresponds to the previous catalyst generation (n). The second data set corresponds to the current catalyst generation (n+1). The aim is to fit the denitrogenation model of catalyst(n+1).

The dataset from the pilot plants using a catalyst (n) is called the source dataset. A first sorting is done in order to eliminate observations with values outside predefined intervals.

In addition, an outlier detection is carried out using the Local Outlier Factor method. [START_REF] Breunig | LOF: identifying density-based local outliers[END_REF] The source dataset is then composed of 61 observations and will be used as a basis to obtain the prior knowledge.

The dataset from the pilot plants using a new catalyst (n+1) is called the target dataset.

It is composed of 126 observations. The same methodology is applied to remove outliers.

The variable of interest is the nitrogen content (N ) after the hydrotreating stage (Nslip).

In order to model its value, an ODE-based kinetic model with a structure showing to be efficient for the modeling of hydrodenitrogenation 14 is used and the influential features are presented in Table 1. This model is very simple but enough to show the methodology.

These features are all quantitative variables, including the output to be predicted. The features space is the same for both datasets, and the variability is similar for the source and the target dataset. Nitrogen content after hydrotreating (in ppm).

The variable of interest that we want to model using other features.

[

3;400ppm]

The experimentation to obtain the observations were carried out using 32 different feed-stocks for the source dataset and 20 for the target dataset. The features correlations are quite similar for both datasets (Figure 1). The higher correlation between the feedstock features for the target dataset than for the source dataset shows that the target feedstocks, in addition to being fewer in number, are more similar to each other. Other features have low correlations, with an absolute value of less than 0.4, and are comparable for both datasets.

ODE-based kinetic modeling

An ODE-based kinetic model is considered and the model structure is obtained by solving the following differential equation:

dy dt = -k 0 exp(-Ea Rg ( 1 T -1 T ref ))( ppH 2 ppH 2,ref ) m y n (1 + A 0 Res 0 )(1 + C 0 N 0 1+S 0 ) × (2) (1 -u • exp(- b R g ( 1 T - 1 T ref ))( ppH 2 ppH 2,ref ) a ( W ADT W ADT ref ) v y r ),
where k 0 and u are the kinetic pre-exponential rate constants, E a and b are the Activation Energies (cal) and R g is the universal gas constant (cal/mol/cal). In our case, y stands for nitrogen (ppm), t for the residence time (h), T for the temperature (K), ppH 2 for the partial pressure of Hydrogen, Res 0 for the Feed Resin (% m/m), N 0 for the Feed Nitrogen (ppm), S 0 for the Feed Sulfur (% m/m) and W ADT for the weighted average distillation temperature of the feedstock (K).

This rate equation can be decomposed into two terms. The first one corresponds to the removal of nitrogen-containing compounds whereas the second term takes into account the reverse reactions that might occur. Indeed, in some conditions, the aromatics hydrogenation thermodynamic equilibrium is reached. This leads to an inversion of the HDN reactions which involve a hydrogenation step.

In this model, the direct kinetic term is composed of a pre-exponential rate constant k 0 , an Arrhenius-type term (E a /R g /T ), a power law associated to hydrogen (m is the partial reaction order) and to nitrogen (n is the partial reaction order) and two inhibitors, modeled as Langmuir-type adsorption curves: resins (Res 0 ) and nitrogen over sulfur content in the feedstock N 0 1+S 0 . A 0 and C 0 are the inhibition adsorption constants to be fitted. For the second term, the inverse kinetic term is composed of a pre-exponential rate constant u, an Arrhenius-type term (b/R g /T ), a power law associated to hydrogen (a is the partial reaction order) and to nitrogen (r is the partial reaction order). An empirical term, related to the feedstock composition (W ADT ) was also added to the return term using a power law: v.

The value of the reference terms are displayed in Table 2.

For a given value of the parameters, the resolution of the differential equation is done numerically using a method of backward differentiation formula. Then, for each observation (x i ) i , a numeric estimation of the nitrogen content evolution over time is obtained. The residence time of the feed in the reactor being known (LHSV -1 ), we obtain the prediction of the nitrogen at the reactor outlet, which we note f θ (x i ) for an observation i.

θ = (k 0 , E a , m, n, a, b, A 0 , C 0 , u, r, v) such that the cost function K i=1 (f θ (x i )-y i ) 2 y i
, with K the sample size, is minimal. Moreover, boundaries are fixed for the parameters in order to keep a physical sense (Table 3).

Table 3: Boundaries of kinetic model parameters.

Param k 0 E a m n a b A 0 C 0 u r v Min 0 1 • 10 4 0,3 0,3 -10 -4 • 10 4 0 - 5
y i = f θ (x i ) + ϵ i , (3)
ϵ i ∼ N (0, σ 2 i ), σ 2 i = σ • y i .
Indeed, by noting Σ = diag(σ 2 1 , . . . , σ 2 Kt ), we obtain:

p(y|X, θ) ∼ N (f θ (X), Σ) ∝ exp(- 1 2 (y -f θ (X)) T Σ -1 (y -f θ (X))) θ M L = arg min θ 1 2 (y -f θ (X)) T Σ -1 (y -f θ (X)) = arg min θ K i=1 (y i -f θ (x i )) 2 σ 2 i = arg min θ K i=1 (y i -f θ (x i )) 2 y i
Where θ M L is the maximum likelihood estimator of (3).

Model fitting

To perform the Bayesian inference to fit the target model, we need to choose the prior distribution. The first step is therefore to fit a performing model on the source dataset to extract the knowledge we have from it. The model (3) is considered. In order to have an estimation of the parameters distribution, the optimisation of the parameters is done using a MCMC algorithm. The MCMC algorithm used is a Metropolis Hastings within Gibbs algorithm [START_REF] Tierney | Markov chains for exploring posterior distributions[END_REF] .

A flat prior is considered (constant over the parameter space), so no knowledge on the prior distribution of the parameters is brought, and thus the likelihood is optimized. All available source observations are used to fit and evaluate the model. The MCMC algorithm is used (10000 iterations). After removing a burn-in period of 2000 observations, the estimation of θs and V ar( θs ) are obtained respectively as the mean and the covariance over the remaining 8000 iterations.

The structure for the target kinetic model is the same as for the source model (equation ( 3)). The fitting is done by using the Bayesian transfer approach, by transferring the parameters knowledge using the prior. To build the prior distribution, it is assumed that the target parameters θ t follow a multivariate Gaussian distribution:

π(θ t ) ∼ N ( θs , gV ar( θs )), (4)
where θs are the estimated source parameters.

Due to the structure of the model, the variances of the parameters are really distinct from each other, which leads to a badly scaled covariance matrix. In order to avoid numerical issues, as with the inversion of the covariance matrix, parameters are normalized.

The ODE is thus rewritten, modifying each θ i by V ar( θi,s )θ i , with θ = (θ i ) i=0,...,10 = (k 0 , E a , m, n, a, b, A 0 , C 0 , u, t, v) and V ar( θi,s ) the i th diagonal element of V ar( θs ). It should be noted that this modification has no impact on the model and only affects the values taken by the parameters. In the following, this structure for the model is considered and by doing so, the diagonal of V ar( θs ) is composed of ones.

The main challenge with the Bayesian transfer approach is to choose a suitable g value to adapt the prior's impact. It should be noticed that θt → g→0 θs and θt → g→+∞ θt,ML , where θt,ML is the maximum likelihood estimator on the target sample only, without transfer. The heuristic method to chose the g value proposed for Bayesian kriging transfer [START_REF] Iapteff | Reducing the number of experiments required for modelling the hydrocracking process with kriging through Bayesian transfer learning[END_REF] is not adapted for this case. Indeed, the model is more complex and we cannot know the range in which parameters will lead to a high value of the likelihood. Two effective methods to choose it are then proposed.

Selection of the value of g by cross-validation

The first method is to use leave-one-out cross-validation to determine which value of g offers the best score. For a given sample of size n t and for each g value tested, cross-validation is performed on the training set (the given sample). n t model are thus fitted using n sample -1 observations and the score is evaluated on the remaining observations ( (ŷ i -y i ) 2 y i where i is the observation not used to fit the model).

The mean of the n t test scores (scores on remaining observations) is calculated. Finally, the value of g with the lowest averaged score is kept. The drawback of this approach is that several models have to be fitted, thus if the model is complex and the sample size is quite large, it becomes time consuming.

Selection

g selected =      min g∈G (g) i∈T rain (ŷ g i -y i ) 2 y i ≤ S exp
where S exp is the chosen expectation score and ŷg i is the prediction obtained using gV ar( θs ) as prior variance. In that way, it is ensured that the prior information does not degrade the model quality, and maximum knowledge from the prior is kept. This method implies the knowledge of the performance expectation, that is the score expected on the target dataset with a good model. If we assumed that the target model will lead to similar performances as the source model, the score obtained with the source model on the source dataset can be a good estimation. For these reason, it was used for this particular application.

Once the prior is chosen, the target Bayesian model can be fitted. In order to compare results with the current method, we also fit a model "without transfer". To optimize the parameters of the Bayesian transferred model, we first use a MCMC algorithm starting from the prior mean, which is the source parameters estimation. However, the posterior distribution is non Gaussian, with multiple modality, and a good estimation of the posterior is not attained with such an approach because the Markov Chain remains blocked in modalities close to the initialization. Therefore, another approach is considered to optimize the parameters values. A gradient descent is performed, using the L-BFGS-B algorithm [START_REF] Byrd | A limited memory algorithm for bound constrained optimization[END_REF] , in order to find parameters θt that maximize the posterior (Equation ( 5)). By doing so, we do not obtain an estimate of the posterior distribution, but an estimate of the posterior maximum, sufficient to fit the model parameters.

π(θ|X, y) ∝ p(y|X, θ)π(θ), (5)
where p(y|X, θ), Σ = diag(σ 2 1 , . . . , σ 2 Kt ) with σ 2 i = σ t • max(5, y i ) and σ t supposed to be known and equal to σs .

Maximizing the posterior corresponds to solve the Equation ( 6):

θt = arg max θ exp(- 1 2 (y -f θ (X)) T Σ -1 (y -f θ (X)))exp(- 1 2 (θ -θs ) T gV ar( θs ) -1 (θ -θs )) = arg min θ 1 2 (y -f θ (X)) T Σ -1 (y -f θ (X)) + 1 2 (θ -θs ) T gV ar( θs ) -1 (θ -θs ) (6)
To ensure that the global minimum (or at least a good local minimum) is found, several initializations are considered. 100 distinct initializations are randomly sampled using the distribution N ( θs , 10 × gV ar( θs )).

For the sake of calculation time, the same initializations are used for the "without transfer" approach. As a consequence, we do not cover the space of possible values for the parameters.

Thus we indirectly constrain the "without transfer" model parameters to stay close to the estimated source parameters values. It is possible that a combination of parameters far from the initialization leads to an higher likelihood value. This option was not explored.

This would mean a better "without transfer" model on the training set, but it does not imply a better model for the test set, particularly with a small training set. Actually, it provides beneficial information here. With few observations and initialization far from source parameters estimation, we can build a "without transfer" model that perform better on the training set but is worse on the test set compared to the "without transfer" model we use. The comparison between the Bayesian transferred and the classical fitted model therefore remains relevant.

Results and Discussion

Source modelling

Firstly, the model fitted on the source dataset is studied. The parity plots of the results are presented in Figure 2. The fitted model offers satisfying results.

f θs (x i + (DeltaT i , 0, 0, 0, 0, 0, 0)) = y i , (7)
with the first element of x i being the temperature of the reactor. For example, a Delta T of 5 • C means that with a value of reactor temperature 5 • C higher than the actual value, the nitrogen content estimation will be the actual value. It is an interesting measure because the reactor temperature furnace is the only operating condition that can easily be changed by the refiner. A similar conclusion to the analysis of parity graphs is reached: satisfying results are obtained. The quantified summary of the performances obtained with the source model is as follows:

• 61 i=1 (f θ (x i )-y i ) 2 y i = 2.34,
• 54% of the observations have an absolute Delta T lower than 2 • C.

The model obtained is considered of satisfying quality and it will be considered as the model to transfer to the target dataset.

Since it is not possible to plot the corresponding 12-dimensional Gaussian prior, Figure 4 shows the marginal histograms and the marginal distribution of each dimension of θs . The correlation between source parameters over the 8000 iterations is also plotted (Figure 5).

It can be noticed that for some parameters the Gaussian distribution does not represent well the stationary distribution of the Markov Chain (parameters a, u and t for example)

and another prior might be more appropriate for these parameters. Furthermore, due to the bounds, for some parameters there is a spike on the bar graph for the boundary values (parameters m, a, b and v) and then the Gaussian prior does not fit perfectly once again.

However, the choice to maintain this Gaussian prior is made to conserve the covariance information between all parameters.

Before fitting a new target model, we are interested in the results of the source model applied to the target data. The results obtained are, as expected, not good and the Nitrogen prediction is higher than the Nitrogen experiments because the catalyst (n+1) is more active than the catalyst (n) (Figure 6).

Similar performance is obtained with a Delta T (defined in equation 7) higher than 10 degrees Celsius (Figure 7).

Target modelling

Different target samples of different sizes are considered, randomly sampled, in order to evaluate the Bayesian transfer, to compare it to the "without transfer" approach, and to study the impact of the scalar g. For each sample size n t ∈ {5, 10, 15, 20}, 10 samples are drawn. A "without transfer" model and Bayesian transfer models with different value for g (g ∈ {1, 10, 100, 1000, 10000}) are fitted for each sample. Due to the Bayesian definition, with an high g value the prior is neglected and the score obtained with the Bayesian transfer and that obtained without transfer are similar. This is the case on both the training set (the size 15 target sample in the example of Figure 8)

and the test set (the entire target dataset) (see for example g = 10 4 ). For the training set, when the value of g decreases, the training score monotonously increases to tend to the score using source model when g is near to zero. Concerning the test set, the evolution of the score as a function of g is not monotonous. A well chosen g value increases model quality and conversely a badly chosen g value decreases it. It is thus crucial to define a method to chose a good g value which is a trade off between the train and the test score. In our case, a g value around 10 is optimal.

To deepen the example of this sample, the parity plots for the "without transfer model" and the Bayesian models with a bad value of g and a well chosen g value are plotted (Figure 9).

As 15 observations are sufficient to fit a satisfying model without transfer for this application, similar results are obtained with the Bayesian transfer method as shown in Figure 9. Nevertheless, a slight improvement is obtained with Bayesian transfer with a good choice for the g value (center graphs in Figure 9). If less observations are available from the target dataset, the classical approach can lead to really bad models. For example, the size 10 random sample presented in Figure 10 leads to poor results with the classical approach, but by using the Bayesian transfer model with a well chosen g value, the results are satisfying and thus a great improvement is provided. We see that the Bayesian transfer, with a correct choice of the g value, leads to an improvement of the model's prediction performance. Thus, two methods are proposed to define the optimal g value to chose.

Selection of the g value by cross-validation

The first method is to use leave-one-out cross-validation to determine which value of g offers the best score.

In order to compare the Bayesian transfer models with the classical fitted models, this method is tested on the different sample sizes and the averaged scores and the minimummaximum score interval (the minimum and the maximum score obtained over the 10 random samples) are studied for each sample size (Figure 11). The method leads to an improvement of the score with the Bayesian transfer, especially for small designs, and thus to a good choice for the g value. Moreover, the Bayesian transfer model is less affected by the design quality, as it leads to a smallest maximum score even if the sample size is low. This method is then more robust to a bad design choice than classical method. It shows that using only 10 pilot plant points is enough to refit the HDN model. However, the main drawback of this method is that several models have to be fitted, thus if the model is complex and the sample size is quite large, it may become time consuming.

Selection of the g value by "bound on training score" The second approach to chose the value of g, which we call "bound on training score", is based on the score obtained for the different g values on the training dataset. This method is tested on the different samples (Figure 12). As for the first method, it leads to a good choice for the g value and thus to an improvement of the score with the Bayesian transfer and a smallest maximum score. The scores are slightly less good than those with the cross validation method (Figure 13), but has the advantage of being less time consuming. Again only 10 pilot points are requested to fit the kinetic model.

Conclusion

In this paper, a methodology to fit a kinetic model for the hydrotreating process for the new catalyst generation (n+1) catalyst using few observations is proposed. To add some information and then to decrease the number of requested pilot plant points, data of previous catalyst generation (n) are used. A Bayesian transfer method, which leads to more robust models, is proposed. It is less impacted by the design quality when compared to the classical model. Also, the predictions performance is improved, especially for small designs where it could not be possible to fit a satisfying model without such an approach. In our case, only 10 pilot points are enough to refit a HDN kinetic model. To obtain a good predictive model, a good choice of the prior variance and thus of the g value is crucial. Two methods for choosing it efficiently are proposed and tested. The cross-validation method is recommended for the best performance, but it may be time consuming depending on the size of the training set.

The "bound on training score" method is a good alternative. This work is performed on a particular application but the transfer method developed can be used for any kinetic or any parametric model. It thus can be used in many other areas, enabling the valorization of past knowledge for many applications.

Sadighi et al. 6

 6 developed a 6-lump kinetic model including a catalyst decay for a commercial VGO hydrocracker. The 6 lumps are light naphtha (40-90 °C), heavy naphtha (90-150 °C), kerosene (150-260 °C), diesel (260-380 °C), Vaccum Gas Oil and unconverted oil (> 380 °C).

models 7 : 1 . 2 .

 712 Definition of an experimental design 3 Execution of the pilot plant tests 3. Validation of the pilot plant tests 4. Kinetic parameters estimation Overall, this is very time consuming and costly. The definition of an experimental design is usually a function of the allocated experimental time and the project priorities. As for the tests themselves, they can last for 4 to 6 weeks each, producing 4 to 8 experimental points. A full campaign for a catalyst generation can involve 5 to 10 tests during 6 to 12 months. Once the test is finished and the analytical results are available, a fine analysis (trend evolution, outlier detection, comparison with previous tests, ...) is required to process and validate the results. This may take up to 3 months. At the end of the line, the raw validated data undergoes further treatment by the kinetic model engineer. A model is then defined and the parameters are fitted. This stage normally requires 3 months of work. As a rule of thumb, we consider that a new generation of catalysts arrives every couple of years.

Figure 1 :

 1 Figure 1: Correlation matrix for hydrotreating source and target datasets. The correlation coefficient use is the Pearson's product-moment coefficient. The size and color of the squares both represent the value of the correlation coefficient. The larger the square the higher the absolute value of the coefficient.

  function in θ is equivalent to maximize the likelihood of the statistical model (3):

  of the value of g by "bound on training score" The second approach to chose the value of g, which we call "bound on training score", is based on the score obtained for the different g values on the training dataset. This method relies on the score we wish to obtain on the whole dataset. With g near to 0, the model is close to the source model and the score is far from expectation. With an high value of g, the model moves closer to the target model without transfer and the score monotonically reaches the score without transfer as the value of g increases. The aim is to keep a maximum of prior information without getting a model leading to bad prediction on the training set. Then, we look for the smallest g value among the tested value G for which the training score is lower than a fixed value for the expectation score:

Figure 2 :

 2 Figure 2: Parity plot of the fitted source model with 61 observations. The dashed lines represent the intervals for which the prediction error is less than 15% and 30% of the actual value. On the right, a zoom for nitrogen values below 100 ppm is drawn.

Figure 3 :

 3 Figure 3: Delta T for the fitted source model with the 61 observations. The dashed lines represent the intervals for which the delta T is less than 5 and 2 and 1 • C.

Figure 4 :

 4 Figure 4: Univariate chosen prior and MCMC sample for the different parameters. Values are hidden for confidentiality reasons.

Figure 5 :

 5 Figure 5: Correlation between source parameters over the 8000 iterations of the MCMC algorithm, burn in period being removed.

Figure 6 :

 6 Figure 6: Parity plot of source model applied to target dataset. The dashed lines represent the intervals for which the prediction error is less than 15% and 30% of the actual value. On the right, a zoom for nitrogen values below 100 ppm is drawn.

Figure 7 :

 7 Figure 7: Delta Temperature for the source model applied to the target dataset. The dashed lines represent the intervals for which the delta T is less than 1 and 2 and 5 • C.

Figure 8

 8 Figure 8 illustrate the impact of the value of g. It shows, for a sample size n t = 15, the averaged score obtained over the 10 samples, with different g value for the Bayesian transfer, on the training and on the test set (continuous line). The dashed lines are the score with a non transfer model fitted on the training data set (blue) and tested on the test set (orange). Without transfer, the training score is very low (around 1.6) but the test score is high (around 3.4) because no a priori information is added to the fit.

Figure 10 :

 10 Figure 10: Parity plot for a random sample of size 10. Left: Without transfer, Right: Bayesian transfer with g=10.

Figure 11 :

 11 Figure 11: Comparison of classical fitted model and Bayesian model, g chosen with the cross validation method, for different size of the training sample. The score K i=1 (ŷ i -y i ) 2 max(5,y i ) evolution is plot according to training sample size. The mean and the minimum-maximum score range over the 10 samples are plotted. On the bottom, a zoom is applied on the score. Blue continuous line: average score with Bayesian Transfer, Orange continuous line: average score without transfer, Blue dashed line: Min/Max score with transfer, Orange dashed line: Min/Max score without transfer 25

Figure 12 :Figure 13 :

 1213 Figure 12: Comparison of classical fitted model and Bayesian model, g chosen with the bound on training score method, for different size of the training sample. The score K i=1 (ŷ i -y i ) 2 max(5,y i ) evolution is plot according to training sample size. The mean and the minimum-maximum score range over the 10 samples are plotted. On the bottom, a zoom is applied on the score. Green continuous line: average score with Bayesian Transfer, Orange continuous line: average score without transfer, Green dashed line: Min/Max score with transfer, Orange dashed line: Min/Max score without transfer 26

Table 1 :

 1 Features description Range.

	Feature	Description

Table 2 :

 2 References values of the kinetic model.

	Symbol	Value
	R g	1.987215583 cal/mol/K
	T ref	649.15 K
	ppH 2,ref	32.5 bar
	W ADT ref	643.15 K
	The objective is to optimize parameters


Marc Ponçot

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Thomas Schenk

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Complementarities of high energy WAXS and Raman spectroscopy measurements to study the crystalline phase orientation in polypropylene blends during tensile test

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

true mechanical behavior of materials in a Representative Volume Element (RVE) where microstructural analyses by the both experimental techniques are obtained. Evolutions of the macromolecular chains orientation have been obtained live at the macromolecular and crystalline cell scales. Experimental results show both that over the course of adding charges in the iPP matrix or increasing the true strain rate, this major micromechanism of deformation of semi-crystalline polymers would be less and less important until its complete disappearance in case of iPP/EPR filled by µ-talc particles. Moreover, correlations made between both techniques evidence similar results over a wide range of true strains. However, measurements diverge at lower and higher strains due to singularities of each both techniques which are discussed in terms of experimental protocols and materials microstructure modifications (transient mesophase, volume damage). Finally, the determination of the true intrinsic mechanical behaviors show similar stress hardening slopes for the three studied iPP blends which means that they all present an highly fibrillar microstructure. The differences of macromolecular chains orientation levels between each material is then discussed in terms of analysis volume of both techniques which could be responsible of their lack of accuracy in case of filled polymers.

Keywords: Isotactic polypropylene; Ethylene propylene rubber; µ-talc; Wide angle X-ray scattering;

Raman spectroscopy; Uniaxial tensile test; Chains orientation; Transient mesophase; Volume damage.

Introduction

Relationships between mechanical properties and microstructure of materials, especially for solid polymers, represent the major key to better understand, control and predict their final inuse behavior. Numerous experimental works have been devoted to characterize the microstructural deformation mechanisms of polymers involved in uniaxial tension and many elementary descriptive models have been already proposed and discussed by the past [START_REF] Peterlin | Molecular model of drawing polyethylene and polypropylene[END_REF][START_REF] Schultz | Polymer Materials Science[END_REF][START_REF] Haudin | Plastic Deformation of Amorphous and Semi-Crystalline Materials[END_REF][START_REF] Friedriech | Advances in Polymer Science, Crazing in Polymers[END_REF]. They depend on the initial microstructures of polymers which are mainly induced by the formulation and the conformation processes. As a result, mechanical behaviour presents significant differences if the polymer matrix is amorphous [START_REF] Boyce | An experimental and analytical investigation of the large strain compressive and tensile response of glassy polymers[END_REF][START_REF] Bartczak | On the plastic deformation of the amorphous component in semicrystalline polymers[END_REF][START_REF] Steenbrink | Void growth in glassy polymers[END_REF][START_REF] Ponçot | True intrinsic mechanical behavior of semi-crystalline and amorphous polymers: Influences of volume deformation and cavities shape[END_REF], semi-crystalline (Ward, 1971;[START_REF] Samuels | Plastic Deformation of Polymers[END_REF][START_REF] G'sell | Microstructure transformation and stress-strain behavior of isotactic polypropylene under large plastic deformation[END_REF][START_REF] Amornsakchai | On the influence of initial morphology on the internal structure of highly drawn polyethylene[END_REF][START_REF] Séguela | On the natural draw ratio of semi-crystalline polymers: review of the mechanical, physical and molecular aspects[END_REF][START_REF] Ponçot | True intrinsic mechanical behavior of semi-crystalline and amorphous polymers: Influences of volume deformation and cavities shape[END_REF][START_REF] Pawlak | Cavitation during deformation of semicrystalline polymers[END_REF][START_REF] Chaudemanche | Evolution of the Raman backscattered intensity used to analyze the micromechanisms of deformation of various polypropylene blends in situ during an uniaxial tensile test[END_REF] or filled with mineral and/or rubber particles [START_REF] Bucknall | Rubber-toughening of plastics, part 1: Creep mechanisms in HIPS[END_REF]Pukansky et al., 1994;[START_REF] Plummer | Deformation and entanglement in semi-crystalline polymers[END_REF][START_REF] Van Der Wal | Polypropylene-rubber blends: 5. Deformation mechanism during fracture[END_REF][START_REF] Nitta | Morphology and mechanical properties in blends of polypropylene and polyolefin-based copolymers[END_REF][START_REF] Omiya | Microstructural observation and simulation of micro damage evolution of ternary polypropylene blend with ethylene-propylene-rubber (EPR) and talc[END_REF]Ponçot, 2009;[START_REF] Chaudemanche | Evolution of the Raman backscattered intensity used to analyze the micromechanisms of deformation of various polypropylene blends in situ during an uniaxial tensile test[END_REF]. Chain orientation, shear banding, and volume damage (voids resulting from crazing or nodules/particles decohesion) are the main involved deformation micromechanisms whatever the initial microstructure of the considered material. The prevalence of one mechanism regarding the others induces such kind of mechanical properties differences. To investigate such mechanism of deformation of polymer blends involved during stretching, the need of fast and non-destructive techniques combining commodity in use is growing among the researchers community [START_REF] Martin | Study of the Crystalline Phase Orientation in Uniaxially Stretched Polypropylene by Raman Spectroscopy: Validation and Use of a Time-Resolved Measurement Method[END_REF][START_REF] Martin | Real-time Raman spectroscopy measurements to study the uniaxial tension of iPP: a global overview of microstructural deformation mechanisms[END_REF][START_REF] Ponçot | Study of the dimensional instabilities of laminated polypropylene films during heating treatments[END_REF]Chaudemanche et al., 2013). In this paper, two of them are performed simultaneously during uniaxial tensile tests to study the micromechanisms of deformation of different polypropylene blends (neat isotactic polypropylene, high impact polypropylene, and high impact polypropylene filled by 7%wt of µ-talc particles). They are high energy wide angle X-rays scattering -HE-WAXS (P07 beamline, PETRA III, DESY synchrotron, Hamburg, Deutschland) and Raman spectroscopy. HE-WAXS is a very suitable experimental technique to characterize the crystalline structure of semi-crystalline polymers. In case of neat polypropylene and its blends, crystallinity changes, phase transitions, crystalline macromolecular chains orientation, development of smectic mesophases during uniaxial tensile tests have been already explored (Davies et al., 2010;[START_REF] Bao | Deformation-induced morphology evolution during uniaxial stretching of isotactic polypropylene: effect of temperature[END_REF]. Although this method is really sensitive, it is not so simple to perform such experiments since the use of specific beamline at a synchrotron is needed. This paper

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aims to compare results obtained by HE-WAXS with an experimental technique easier to perform at the laboratory scale or at the industry. Raman spectroscopy is well-suitable to determine the vibrational frequencies of the covalent chemical bonds of polymers and allow analysis of molecular species at different scales, from the chemical repetitive unit to the conformational architecture of the macromolecular chain [START_REF] Koenig | Spectroscopy of Polymers[END_REF]. The Raman spectra of isotactic polypropylene have been the object of several investigations and the assignment of Raman scattering bands are rather well established [START_REF] Tadokoro | Normal vibrations of the polymer molecules of helical conformation: V. Isotactic polypropylene and its deuteroderivatives[END_REF][START_REF] Fraser | The vibrational spectrum of polypropylene[END_REF][START_REF] Chalmers | Raman spectra of polymorphs of isotactic polypropylene[END_REF]. [START_REF] Martin | Study of the Crystalline Phase Orientation in Uniaxially Stretched Polypropylene by Raman Spectroscopy: Validation and Use of a Time-Resolved Measurement Method[END_REF][START_REF] Martin | Real-time Raman spectroscopy measurements to study the uniaxial tension of iPP: a global overview of microstructural deformation mechanisms[END_REF] have defined a specific spectral criterion which enables to characterize the macromolecular chains orientation of the crystalline phase of a neat iPP using a combination of different polarization geometries for both the incident and scattered radiations [START_REF] Purvis | Molecular orientation in poly(ethylene terephthalate) by means of laser-Raman spectroscopy[END_REF][START_REF] Cambon | Raman-laser study of isotactic polypropyleneuniaxial drawing deformation[END_REF][START_REF] Tanaka | Review Polarised Raman spectroscopy for the study of molecular orientation distributions in polymers[END_REF].

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 4 2. Experimental part

Materials

Isotactic polypropylene (iPP)

The semi-crystalline isotactic polypropylene that was investigated in this work was provided by Arkema (ref. 3050 MN1). The average molecular weights are n M = 75,900 g.mol -1 and w M = 262,000 g.mol -1 (Ponçot, 2009). The density determined by hydrostatic weighting is ρ = 0.946 g.cm -3 . The crystallinity index is X c = 50 %vol. The glass transition temperature T g (determined by dynamic mechanical analysis) and the melting temperature T m (determined by differential scanning calorimetry) are 10°C and 167°C, respectively. Wide angle X-rays scattering analyses revealed the exclusive presence of the α-monoclinic crystal of iPP. The cell parameters are a = 6.666 Å, b = 20.78 Å, c = 6.495 Å, α = 90°, β = 99.6° and γ = 90° [START_REF] Natta | Structure and properties of isotactic polypropylene[END_REF]Turner-Jones et al. (1994)]. Forming of the material is made by injection-molding of 4 mm thick plates.

Isotactic polypropylene/ethylene-propylene rubber (iPP/EPR)

The second polymer investigated is an isotactic polypropylene / ethylene-propylene rubber (iPP/EPR) supplied by Total Petrochemicals (ref. PPC 3650). The EPR content is 17 %wt.

The final blend presents the following average molecular weights: n M = 69,000 g.mol -1 and w M = 599,000 g.mol -1 . As previously evidenced with scanning electron microscopy and transmission electron microscopy analysis, EPR nodules are homogeneously dispersed in the iPP matrix. They consist in "core-shell"-like particles of about 10 µm in diameter enclosing crystalline polyethylene lamellae (as revealed by transmission electron microscopy using ruthenium oxide coloration (Ponçot, 2009)). Then, differential scanning calorimetry analysis, performed at the standard heating rate of 10 °C.min -1 , reveals two melting peaks; a first one at 115 °C with a very low enthalpy (inferior to 1 J/g) which corresponds to the melting of polyethylene crystals contained in the EPR nodules (Ponçot, 2009) and a second one at 167 °C attributed to the melting of the α-iPP crystalline phase. The glass transition of the iPP matrix is still close to 10 °C while the EPR elastomeric phase exhibits a glass transition at -45 °C (as revealed by dynamic mechanical analysis). Further WAXS analyses established that the average crystallinity index of iPP matrix is 45 %vol. (Ponçot, 2009). The samples were processed by injection moulding into 4 mm-thick plates. The iPP/EPR previously described has been then modified by the addition of 7 %wt. µ-talc platelets as mineral fillers. The aspect ratio of these particles is about 5.5 with an average length of 25µm and an average thickness of 5µm. Neither chemical nor mechanical treatments are applied to these particles before their mixing with the iPP/EPR matrix. Differential scanning calorimetry analysis reveals the two previously identified peaks at 115 °C and 164

°C attributed to the melting of the polyethylene crystals present in the EPR nodules and α-iPP crystals, respectively. The presence of a third small peak located at 146 °C, usually attributed to the melting of β-iPP crystals, inform us about the fact that µ-talc particles promotes their creation i) due to their nucleation effect (Lotz et al., 1996) more than the one determined in the case of the iPP/EPR. Samples were processed by injection molding into 5 mm-thick plates.

Experimental setup

Mechanical testing equipment

The video-controlled mechanical testing method used in this paper is the VidéoTraction ™ system [START_REF] G'sell | Video-controlled tensile testing of polymers and metals beyond the necking point[END_REF][START_REF] G'sell | Experimental characterization of deformation damage in solid polymers under tension and its interrelation with necking[END_REF][START_REF] Addiego | Characterization of volume strain at large deformation under uniaxial tension in high-density polyethylene[END_REF]Ponçot, 2009, Ponçot et al., 2013). For different deformation paths, it gives access to the true mechanical behaviour of polymers by taking into account of the development of the polymer necking. Video measurements are performed locally at the centre of the sample in a specific Representative Volume Element (RVE) where plastic deformation appears. To ensure the localization of the necking zone at the centre of the sample, width of the dumbbell-shaped specimen is slightly reduced by milling (reduction of 6 % of the specimen width). The axial true strain in the RVE, ε 33 , is obtained by a polynomial interpolation of partial strains measured from the displacement of axial markers using Lagrange polynomial interpolation and following Hencky' strain theory. For uniaxial tensile testing, transverse strains in the RVE, ε 11 and ε 22 , are equal considering the realistic hypothesis that the strain field is transversally isotropic in the centre of the neck [START_REF] Bucknall | Rubber-toughening of plastics, part 1: Creep mechanisms in HIPS[END_REF][START_REF] Naqui | Review tensile dilatometric studies of deformation in polymeric materials and their composites[END_REF]Pukansky et al., [START_REF] G'sell | Experimental characterization of deformation damage in solid polymers under tension and its interrelation with necking[END_REF][START_REF] Addiego | Characterization of volume strain at large deformation under uniaxial tension in high-density polyethylene[END_REF]Ponçot, 2009). Lastly, the volume strain in the RVE, ε v , is computed as the trace of the true strain tensor:

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 6 1994; G'sell
ε v = ε 11 + ε 22 + ε 33 (1)
The axial true stress (Cauchy stress) is determined in the same RVE as the load per unit actual cross-section:

) 2ε exp( S F σ 11 0 33 - = , ( 2
)
where S 0 is the initial cross-section of the tensile specimen.

All the mechanical tests were performed at room temperature (22 °C). It is also worth noting that VidéoTraction™ system enables to perform tensile testing at constant axial true strain rate. The influence of the true axial strain rate on both the mechanical behaviour and deformation mechanisms of iPP and modified iPP was also investigated. Various true strain rates (5.10 -3 s -1 , 10 -3 s -1 and 5.10 -4 s -1 ) were chosen in such a way to ensure the control loop of the testing machine actuator and to avoid any self-heating of the material during the mechanical testing [START_REF] Billon | Thermal effects during mechanical characterisation of solid polymers[END_REF]Ye, 2015].

Raman spectroscopy analysis

Raman scattering experiments were performed using a RXN1 Kaiser micro-spectrometer linked to a remote head of probe by means of two optical fibers. A 300 mW laser diode emitting a 785 nm exciting line was used which allows reducing fluorescence emission and local heating of the polymer specimen. All spectra were recorded in backscattered geometry with a working distance of 20 cm. Within such configuration, the laser waist on the specimen is about 0.16 mm 3 . The micro-spectrometer is equipped with a 1200 gr.mm -1 grating and a spectral resolution of about 1 cm -1 is achieved. Rayleigh scattered light is eliminated through a holographic Notch filter. Before each tensile test, shift in the Raman spectra is calibrated with the silicon band located at 521 cm -1 . An individual spectrum recording lasts a few seconds depending on the resolution and the true strain rate at the time of in situ analysis during stretching.

The crystalline macromolecular chains orientation is estimated from Raman spectra by performing acquisitions in particular polarization geometry of the incident and scattered radiations where both the incident and scattered radiations are polarized along the stretching direction I. Following these conditions, we demonstrated in previous papers that the ratio I 973cm-1 / I 998cm-1 of the integrated intensity of the two particular Raman scattering bands at

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998 cm -1 (rocking mode of CH 3 lateral alkyl groups in crystalline phase), enable to estimate orientation of crystalline phase chains in the tensile direction I (Ponçot, 2009;Martin, 2009;[START_REF] Martin | Study of the Crystalline Phase Orientation in Uniaxially Stretched Polypropylene by Raman Spectroscopy: Validation and Use of a Time-Resolved Measurement Method[END_REF][START_REF] Martin | Real-time Raman spectroscopy measurements to study the uniaxial tension of iPP: a global overview of microstructural deformation mechanisms[END_REF][START_REF] Ponçot | Shrinkage Study of Polypropylene Films Laminated on Steel-Influence of the Conformation Processes[END_REF][START_REF] Ponçot | Study of the dimensional instabilities of laminated polypropylene films during heating treatments[END_REF]Chaudemanche, 2011;Chaudemanche et al., 2013).

Synchrotron 2D WAXS measurements

Experiments were carried out in the PETRA III synchrotron facility at the DESY laboratory (Hamburg -Germany) on the P07 -High Energy Materials Science beamline. The beam energy was about 60 keV (λ = 0.0206 nm). The beam cross section was 150 x 150 µm². The samples were analyzed using Wide Angle X-ray Scattering (WAXS) in transmission. The scattered beam was collected with a PerkinElmer XRD1622AP digital flat panel detector. The chosen high energy resulted in a low X-ray absorption in the polymer (6.8% for a 4 mm thick sample). Coupled with the relatively short duration of the tensile tests, this leads to no measurable heating of the polymer or radiation degradation, despite the beam's high influence, characteristic of synchrotrons radiations. A one second frame was acquired every second for samples tested at a true strain rate of 10 -4 s -1 , whereas for samples tested at a 10 -3 s -1 , the acquisition time was of a 0.5 second frame every 0.5 second. The distance between the detector and the sample was about 1.36 m, allowing us to focus on low scattering angle. Thus, both the central scattering zone corresponding to amorphous material and the diffractions patterns of the crystalized polymer and mineral charges were recorded with a good precision. With our detector, at the wavelength used for the experiment, measurements were realized with a 2-θ step of 0.0084°, up to a value of 12° (corresponding to Q-vector space values of a 0.044 nm -1 step, up to 62 nm -1 ). The WAXS patterns were integrated with Fit2D to produce 1-dimensional intensity-vs-2theta graphs. The stretching direction (I) is always vertical on the acquired 2D WAXS patterns.

The orientation level of the crystallites can be formalized on a quantitative numerical basis using Hermans' orientation function /I X F which describes the orientation of a given crystallographic axis X relative to one reference direction, usually the stretching direction (I).

Its general expression is:

( ) 2 1 α cos 3 F /I X 2 /I X - > < =
(3) with:

( )

∫ ∫ ⋅ ⋅ ⋅ ⋅ ⋅ >= < π/2 0 hkl π/2 0 2 hkl /I X 2 dφ sinφ I dφ sinφ φ cos I α cos (4) M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 8
where hkl I represents the maximal intensity scattered by the (hkl) planes relative to a given crystallographic axis X as a function of the azimuthal angle φ [START_REF] Hermans | Quantitative X-ray investigations on the crystallinity of cellulose fibers. A background analysis[END_REF].

Table 1 develops the meaning of the Hermans' orientation factor values related to the specific crystallographic orientation.

( )> < /I X 2 α cos /I X F Texture 0 -1/2 I is perpendicular to X 1/3 0 Isotropic distribution of I in space 1/2 1/4 Isotropic distribution of I in the plane parallel to X 1 1 I is parallel to X

Synchrotron Wide Angles X-rays Scattering (WAXS), Raman spectroscopy and

VidéoTraction™ coupling system

The experimental set-up in Figure 1

True mechanical behavior

Influence of the polymer formulation

Since glass transition temperatures of EPR (-45 °C) and iPP (10 °C) are below the room temperature, all the specimens that were uniaxially stretched in this work are in their rubber state. Figure 2 shows the tensile curves recorded at a constant true strain rate (5.10 -3 s -1 ) for the three considered polymers, iPP, iPP/EPR and iPP/EPR/µtalc. In Figure 2 a), despite the different values of stress and strain at the yield point (σ y = 36,2 MPa and ε y = 0,09 for iPP ;

σ y = 23,2 MPa and ε y = 0,07 for iPP/EPR), it is to be noticed that the tensile curve of iPP presents a similar shape to that of iPP/EPR until the plastic plateau (which begins at ε p = 0,27 and ε p = 0,18 for iPP and iPP/EPR, respectively) including a weak softening stage for both.

Young's moduli are calculated at the lowest values of strain at the early stage of the viscoelastic domain. They are equal to 1695 MPa and 1130 MPa for iPP and iPP/EPR, respectively. The influence on mechanical toughness of the addition of the viscoelastic EPR phase into the iPP matrix is there clearly put in evidence. Then, the extents of the plastic plateau are different one to the other: the strain-hardening occurs earlier in terms of true strain value in case of iPP/EPR and shows a more important increase of the slope of the stress/strain curves at the highest values of strain. The addition of 7 %wt. of µ-talc particles into the iPP/EPR polymer blend strongly modifies the true mechanical behavior as indicated in Figure 2 a). The Young's modulus is about 2025 MPa which is higher than the one determined for iPP. Moreover, the achieved stress level in comparison to the neat isotactic polypropylene is increased. Concerning yielding, it occurs at a stress value situated between the iPP and iPP/EPR. It is noticed that the yield strain is the lowest of all the studied materials which indicates that the plastic damage takes place earlier. Finally, the magnitude of the strainhardening is similar to the one observed in case of iPP.

Influence of true strain rate

The influence of true strain rate in the range of 5.10 -4 s -1 to 1.10 -3 s -1 on the true mechanical behavior of polymers is presented in Figure 3. An increase of stress is observed up to moderate strain levels with increasing the stretching rate whatever the material. Then, the yield point is higher for neat iPP than for the blends whatever the studied strain rate. Different amplitudes of microstructural softening appear since organic and mineral charges promote microstructural damages to macromolecular rearrangements in the blends. Stronger softening is observed in case of iPP/EPR + µ-talc. Then, plasticity is characterized by a quite large plateau for all materials. Strain-hardening takes place at large strain level. However, during the strain-hardening stage, the stress-strain curves are not parallel anymore for the two filled materials, as indicated in Figure 3. This phenomenon does not enable to predict the tensile behavior at large strain level using phenomenological constitutive equations as the one proposed by G'sell and Jonas (1979) (Ponçot, 2009 ;[START_REF] Ponçot | True intrinsic mechanical behavior of semi-crystalline and amorphous polymers: Influences of volume deformation and cavities shape[END_REF].

2D WAXS patterns

Combination of the VidéoTraction™ system with WAXS acquisitions allows us to follow the evolution of the crystalline phase during the uniaxial true tensile test with high time resolution. Figure 4 shows three sets of 2D WAXS patterns recorded at different true strain level for the three different polymers investigated ( T = 22 °C and ε = 5.10 -4 s -1 ). Patterns at null true strain value are indexed in terms of characteristic scattering crystallographic planes of the polypropylene monoclinic crystalline cell ((110), ( 040), ( 130), ( 111) and ( 131)). It can be noticed that the three materials show quasi-isotropic rings at the initial state for all values of the azimuthal angle (0° to 360°). Only weak variations of the intensity are observed for filled polypropylene (iPP/EPR and iPP/EPR filled with µ-talc) suggesting a slight orientation of the crystalline phase at the initial state. Additional rings are naturally present in 2D WAXS Depending on the polymer formulations the rings intensity towards the azimuthal angle does not present the same evolution as a function of true strain. In case of neat iPP and iPP/EPR, most of the initial rings change into highly illuminated arcs as a result of stretching with a phenomena more pronounced for neat iPP. Concerning iPP/EPR filled with µ-talc, only weak modifications of the rings intensity distribution towards the azimuthal angle can be observed for high true strain values superior to 0.6 and they do not result in the appearance of arcs but in a kind of an equatorial diffusion. This diffraction peak tendency to collapse is widely more developed in case of iPP, followed by iPP/EPR and finally by iPP/EPR + µ-talc.

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11

The macromolecular chains orientation of the iPP crystalline phase, as one of the main micromechanisms of deformation of uniaxially stretched semi-crystalline polymers, is focused. From 2D WAXS patterns, the Hermans'factor ܨ ሬԦ ூ ⁄ = ܨ (ସ) ூ ⁄ is then calculated.

Indeed, the normal direction of this particular crystallographic plane of the α monoclinic cell is strictly perpendicular to the ܿ Ԧ axis, the backbone chains direction. Figure 5 reports the variation of this Hermans' orientation factor as a function of the axial true strain for various true strain rates and for the three polymers that were investigated in this study.

Firstly, it has to be noticed that the three studied material show a weak initial texture induced by the forming process by injection molding ܨ( (ସ) ூ ⁄ = -0.12, ܨ (ସ) ூ ⁄ = -0.1 and ܨ (ସ) ூ ⁄ = -0.06 for iPP, iPP/EPR and iPP/EPR filled with µ-talc, respectively). Mechanical probes are then machined out in the manner that this crystalline orientation of the macromolecular chains is taken as the stretching direction. Addition of organic and mineral charges into the iPP matrix reduces the initial texture involved by the injection molding process. This is the reason why a small variation of ܨ (ସ) ூ ⁄ towards 0 (isotropic microstructure) is observed at the earliest true axial strain values in cases of iPP and iPP/EPR until the middle of the respective plastic plateaus. It could be actually attributed to a first step of reorganization of the initial crystalline texture previous to the main effect of macromolecular chains orientation induced by stretching which is then responsible of the decrease of ܨ (ସ) ூ ⁄ . This effect is not observable for iPP/EPR + µ-talc and in case of iPP/EPR stretched at the lowest true strain rate (5.10 -4 s -1 ). Influence of this last mechanical test parameter is the most evident for neat iPP: higher is the true strain rate; lower is the crystalline chains orientation all along stretching. Secondly, over the course of adding charges in the iPP matrix, Hermans' orientation factors show that this major micromechanism of deformation of semi-crystalline polymers seems less and less important until its complete disappearance in case of iPP/EPR + µ-talc. Indeed, experimental results evidence that the magnitude of the crystalline chains orientation changes a lot depending on the material formulation. ܨ (ସ) ூ ⁄ in case of the neat iPP reaches close values of -0.5 whatever the strain rate which is characteristic of a highly fibrillar microstructure along the stretching direction I.

The addition of the EPR nodules reduces much this micromechanism. Maximal value of ) which presents a similar crystalline structure to α's but with a lot of imperfections. This is the reason why the diffraction peaks of the WAXS patterns are enlarged as illustrated in Figure 9 for iPP and iPP/EPR. This diffusion phenomenon is responsible of the lack of X-rays scattering precision in addition to the over whole intensity decrease. The mesophase is weakly observable in case of iPP/EPR + µ-talc [START_REF] Liu | Influence of Trace Amount of Well-Dispersed Carbon Nanotubes on Structural Development and Tensile Properties of Polypropylene[END_REF] due to the predominance of the volume damage micromechanism of deformation in case of filled polypropylene matrixes. The macromolecules orientation keeps low or highly located in the polymer fibrils. [START_REF] Jang | Rubber toughening in polypropylene[END_REF]Wu, 1985;[START_REF] Lazzeri | Dilatational bands in rubber-toughened polymers[END_REF][START_REF] Dagli | Particle size effect in craze plasticity of high-impact polystyrene[END_REF]Zebarjad et al., 2004;[START_REF] Nitta | Morphology and mechanical properties in blends of polypropylene and polyolefin-based copolymers[END_REF] ). As a result, the iPP fibrils located between two adjacent destroyed nodules should be then composed with highly oriented macromolecular chains along the stretching direction.

ܨ (ସ) ூ ⁄ is
When iPP/EPR blends are filled with mineral particles, many authors have demonstrated that whatever the considered fillers and the cohesion level with the iPP matrix, the first micromechanism of deformation involved during stretching is cavitation due to the mineral particles decohesion with the polymer matrix (Pukanzky et al., 1994;[START_REF] Omiya | Microstructural observation and simulation of micro damage evolution of ternary polypropylene blend with ethylene-propylene-rubber (EPR) and talc[END_REF]Ponçot, 2009;[START_REF] Chaudemanche | Evolution of the Raman backscattered intensity used to analyze the micromechanisms of deformation of various polypropylene blends in situ during an uniaxial tensile test[END_REF]. In the same way as previously discussed, the iPP fibrils located between two adjacent voids should be then composed with highly oriented macromolecular chains along the stretching direction.

However, both techniques show a weak orientation effect during stretching for iPP/EPR and iPP/EPR + µ-talc (Figures 5 and6), whereas regarding the true intrinsic mechanical behaviors (relative to the iPP main matrix), similar stress hardening magnitude are observed for all three materials whatever their formulation. It can be discussed in terms of analysis volume of both techniques. In case of WAXS, the beam cross section was 150 µm x 150 µm = 0.022 mm².

For Raman spectroscopy, the volume of the waist was 0.16 mm 3 which corresponds to a diameter of about 80 µm and a circular cross section of about 0.021 mm² (Chaudemanche, 2013). These two dimensions are largely bigger than the micro-fibrils and voids as revealed Wu S., "Phase structure and adhesion in polymer blends: a criterion for rubber toughening", Polymer, 26, 1855Polymer, 26, -1863Polymer, 26, , (1985) ) Ye J., Ph.D. Thesis, Lorraine University-LEMTA-ENSEM, Nancy, France (2015)

Zebarjad S.M., Bagheri R., Lazzeri A., Serajzadeh S., "Dilatational shear bands in rubber modified isotactic polypropylene", Materials and Design, 25, 247-250, (2004)

Figures captions:

Figure 1: 1) Tensile test machine, 2) Dumbbell-shape specimen, 3) Z-motorized stage, 4)

Video camera, 5) Raman head of probe, 6) VidéoTraction™ camera, 7) X-rays monochromator. macromolecule segment of the helical helix of iPP [START_REF] Natta | Structure and properties of isotactic polypropylene[END_REF]. -We study the micromechanisms of deformation involved during uniaxial stretching of three different iPP blends (neat isotactic polypropylene, high impact polypropylene, and high impact polypropylene filled by 7%wt of µ-talc particles) at various strain rates (5.10 -3 s -1 , 10 -3 s -1 and 5.10 -4 s -1 ). -A special experimental set up was developed to perform simultaneous high energy wide angle X-rays scattering (P07 beamline, PETRA III, DESY synchrotron, Hamburg, Deutschland) and Raman spectroscopy analysis during the mechanical test. -Complementarities and singularities of both techniques to characterize the polymer microstructure are put in evidence. -Taking into account the in situ measured volume strain, the true intrinsic mechanical behaviors of blends are obtained and similar stress hardening slopes are observed which means that a similar fibrillar microstructure is reached. -Both techniques are not able to measure this high orientation level located in fibrils in case of filled iPP due to their two large volume of analysis (the volume fraction of oriented fibrils is too small in comparison of voids volume fraction, fillers volume fraction and undamaged iPP matrix fraction).

5

  Isotactic polypropylene/ethylene-propylene rubber filled with mineral fillers (iPP/EPR + µ-Talc)

  was specifically designed to characterize the polymer micro-structure at different scales in real time with the course of a uniaxial tensile test and is based on the system patented byDahoun et al. in 2011. It consists in the combination of a video-controlled tensile test machine (1) using the VidéoTraction™ system (6) with a Raman spectrometer and a synchrotron High Energy WAXS beam line. They are fixed up on a 3-D directions movable hexapod. Before each tensile test, laser from the Raman head of probe (5) and X-rays beam from the X-ray monochromator (7) are strictly aligned towards the RVE of the polymer specimen (2) which is gripped between the clamps of the tensile-testing machine.Since uniaxial tension is applied by the displacement of the upper grip, gradual misalignment of the RVE with respect to laser and X-ray beams is expected. Thanks to a video camera continuously pointed to the polymer specimen and linked to a z-motorized stage (3) via a control loop, both the laser and X-ray beams remain aligned with the RVE during the entire course of the tensile test.

  Figure 2 b) illustrates the evolutions of the volume strain (calculated by the Equation 1) as a function of the axial true strain corresponding to the mechanical behaviors of Figure 2 a) for all the three materials. All curves show an almost linear evolution with stretching. Mainly the magnitudes are different. The addition of EPR nodules increases the volume strain in comparison to the neat matrix.This effect is emphasized by the introduction of mineral particles of µ-talc.

Figure 6

 6 Figure6presents the variations of the ratio I 973cm-1 / I 998cm-1 as a function of the true strain for the three studied materials and for at least two true strain rates. Apart from the initial disorientation phenomenon previously observed by WAXS, a similar description of the evolutions of the I 973cm-1 / I 998cm-1 ratio as for ܨ (ସ) ூ ⁄ is available regarding the influences of the materials formulation and the true strain rate: orientation micromechanism seems to disappear adding organic and mineral fillers and in the same way the sensitivity towards the true strain rates decreases. Some differences between results of both techniques can be observed in case of iPP at the very low and high levels of true strain. The evolution of the Raman criterion keeps constant at the beginning of the stretching test whereas it was noticed a variation of the WAXS Hermans' factor. In the inverse manner, I 973cm-1 / I 998cm-1 still increases at high true strain values whereas ܨ (ସ) ூ ⁄ reaches a plateau close to -0.5. For the three materials, the initial value of I 973cm-1 / I 998cm-1 is about 2. It means that the crystalline macromolecular chains present an initial orientation due to injection molding taking as the stretching direction. Martin et al.(2013) have already showed that in case of non-textured neat iPP, the initial value of the criterion is 1.5.

  The three isotactic polypropylene blends exhibit singular true mechanical behaviors in uniaxial tension depending on the type of fillers as revealed by figures 2 and 3. The introduction of EPR nodules inside the iPP matrix reduces drastically the yield and plastic stresses in comparison to the neat polymer. Addition of µ-talc improves the viscoelastic resistance of the high impact polypropylene, whereas the plastic stress becomes and stays inferior to the neat matrix. Even the stress hardening stage is less pronounced for this last blend. It was observed in Figure2 b) that volume strain increases with the amount and the diversity of the added fillers. The iPP/EPR + µ-talc blend shows the most predominant effect.As described byPonçot (2009) and Ponçot et al.(2013), volume strain was taken into account to determine the true intrinsic mechanical behaviors of polymer materials without voids. The true stress and the true strain are recalculated using the equations previously established in order to obtain the true intrinsic stress and the true intrinsic strain which represent the true intrinsic mechanical behavior of the solid polymer without cavities. The corrected mechanical curves are then presented in Figure10. No significant change happens before the materials plasticity since volume strain keeps low. At the largest strain values, similar slopes of the stress hardening are observed between the three different polymer blends. Stress hardening is usually attributed to a highly fibrillar microstructure where macromolecules are wholly aligned along the stretching direction (I). Consequently, in both cases (iPP/EPR and iPP/EPR + µ-talc), the fibrils located between two adjacent voids should be made of highly oriented macromolecular chains of iPP. Indeed, first step of plasticity in case of polymers filled with core-shell elastomer nodules is their own destruction upon stretching what results in the development of 45° oriented dilatational shear bands as presented in Figure11

  by scanning electron microscopy observations resumed in Figure12. The volume fraction of highly oriented crystalline macromolecular chains is then really weak in comparison of those made of voids and undamaged iPP matrix. That is the reason why it cannot be detected by both used experimental techniques. of VidéoTraction™, WAXS and Raman spectroscopy especially designed to establish the narrow relationship between polymer blends microstructure evolution and mechanical behaviors and properties was used at the DESY laboratory of PETRA III synchrotron. The work focused on neat and filled iPP matrix. The modifications of the mechanical properties due to the gradual complexity of formulations (addition of EPR nodules and µ-talc particles) were explained in terms of influence of the micromechanisms of deformation during a uniaxial stretching at true strain rates. Good correlations are obtained between orientation factors determined by WAXS and Raman spectroscopy. Singularities were highlighted: WAXS is more efficient at lower strains to observe the prior reorientation of the crystalline chains, however at the time of plasticity the signal is interfered by the development of a crystalline mesophase; Raman spectroscopy is not sensitive to the reorientation phenomenon at low strain values due to the initial laser polarization in the stretching direction, however it enables to follow the orientation until the end of the mechanical test. The true intrinsic mechanical behaviors are then determined and give finally the real mechanical behavior of blends evicting the inside voids. Similar strain hardenings were obtained whatever the blend when the volume strain correction is applicate. However, macromolecular orientation levels are totally different. It was explained by the dimensions of the analysis volume of the used experimental techniques which are too large to locally measure the orientation gathered right in the micro-fibrils, "alone" matter between two adjacent voids. The more elevated is the volume strain; the lower is the measured macromolecular orientation whereas similar stress hardening slopes are obtained. M., Mechanical Properties of Solid Polymers,Wiley-Interscience, London, (1971)

Figure 2 :

 2 Figure 2: a) True mechanical behaviors and b) volume strain curves as a function of true strain of iPP, iPP/EPR and iPP/EPR + 7%wt. µ-talc at ambient temperature and constant true strain rate of 5.10 -3 s -1 .

Figure 3 :

 3 Figure 3: True mechanical behaviors of a) iPP, b) iPP/EPR and c) iPP/EPR + 7 %wt. µ-talc as a function of strain rates at the ambient temperature.

Figure 4 :

 4 Figure 4: In situ evolutions of the 2D WAXS patterns of a) iPP, b) iPP/EPR and c) iPP/EPR + 7 %wt. µ-talc stretched at 22 °C and 5.10 -4 s -1 and recorded at different true strain levels.The stretching direction (I) is vertical.

Figure 5 :

 5 Figure 5: In situ evolutions of the Hermans' factor ܨ (ସ) ூ ⁄ as a function of the axial true strain of a) iPP, b) iPP/EPR and c) iPP/EPR + 7 %wt. µ-talc stretched at 22 °C.

Figure 6 :

 6 Figure 6: In situ evolutions of the crystalline chains orientation determined by Raman spectroscopy at the ambient temperature as a function of the true strain rates in case of a) iPP, b) iPP/EPR and c) iPP/EPR + 7 %wt. µ-talc.

Figure 7 :

 7 Figure 7: Correlations between the evolutions of the normalized orientation factors determined by Raman spectroscopy and 2D WAXS at the ambient temperature as a function of the true strain rates in case of a) iPP, b) iPP/EPR and c) iPP/EPR + 7 %wt. µ-talc.

Figure 8 :

 8 Figure 8: a) α monoclinc crystals of iPP (Cylinders represent the macromolecules), b)

Figure 9 :

 9 Figure 9: Equatorial WAXS diffractograms evolution of a) iPP and b) iPP/EPR as a function of stretching.

Figure 10 :

 10 Figure 10: Comparison of the true intrinsic mechanical behaviors of iPP, iPP/EPR and iPP/EPR + 7%wt. µ-talc at ambient temperature and constant true strain rate of 5.10 -3 s -1 (Ponçot et al., 2013).

Figure 11 :

 11 Figure 11: Representation of the dilatational shear bands which is developed during uniaxial stretching of iPP/EPR blend(Ponçot, 2009). The tensile direction (I) is vertical.

Figure 12 :

 12 Figure 12: Scanning electron microscopy images of iPP/EPR and iPP/EPR + µ-talc uniaxially stretched at the ambient temperature and 5.10 -4 s -1 (Ponçot, 2009). Short dashed circle schematizes the circular cross section of the Raman laser and the solid square the beam cross section of X-rays. The tensile direction (I) is vertical.

Table 1 :

 1 Remarkable values of

	/I X F in relation to the associated
	crystalline macromolecular textures.

Acknowledgement:

The P07 -High Energy Materials Science beamline of PETRA III synchrotron facility at the DESY laboratory (Hamburg -Germany).