S Nouira
email: samia.nouira@ensam.eu

T Hassine
email: tarek.hassine@eniso.u-sousse.tnj.fitoussijoseph.fitoussi@ensam.eu

• J Fitoussi

• M Shirinbayan

• F Gamaoun

• A Tcharkhtchi

Non-isothermal crystallization kinetics and its effect on the mechanical properties of homopolymer isotactic polypropylene

Keywords: Polypropylene, Crystallization, Mechanical characteristics

The non-isothermal crystallization of the isotactic Polypropylene (iPP) was studied using differential scanning calorimetry and polarizing optical microscopy. Jeziorny's model and Ozawa's theoretical approaches were applied to study the nonisothermal kinetics of the iPP. Jeziorny's approach proved to be the most relevant model to the present material. Simultaneously, the activation energy was calculated with Kissinger's method and Vyazovkin's iso-conversional approach. Indeed, the latter provides an activation energy varying between 100 and 176 kJ/mol. Furthermore, the WAXD scans indicated the presence of a single crystalline form in the used material. To study the effect of the microstructure on the mechanical properties of the iPP, multiscale tensile tests were carried out for different film microstructures. At macroscopic scale, the increase in the diameter of spherulites, inevitably accompanied by a rise in the crystallinity rate, induces the growth of rigidity, brittleness, and elastic limit. Moreover, the results of the in-situ micro-tensile tests present the evolution of spherulites during loading.

Introduction

Over the past years, plastic implementation has been used in multiple applications. The replacement of metal parts with polymeric ones in new technological devices and machines has become increasingly significant [START_REF] Nassiri | Polymerization of propylene with Ziegler-Natta catalyst: optimization of operating conditions by response surface methodology (RSM)[END_REF]. Many researchers have found thermoplastics attractive thanks to their mechanical and physical properties as well as their potential to be recycled. In particular, polypropylene with different tacticity has been used due to the high variability in their stereoregularity. We can find syndiotactic, atactic and isotactic polypropylene.

The most commercial one is the isotactic Polypropylene (iPP), thanks to its good regularity and high tendency towards crystallization. Its usage has to expand in the automotive and mechanical fields [START_REF] Kunimitsu | High strength fiber obtained from a high stereoregularity metallocene catalyst-synthesized polypropylene[END_REF]. It is a semicrystalline polymer, it has a relatively low density and a low price, excellent moisture, ease of processability, high toughness and chemical resistance [START_REF] Ding | Crystalline morphology and mechanical properties of isotactic polypropylene composites filled by wollastonite with β-nucleating surface[END_REF]. Its favorable mechanical and physico-chemical properties contribute to its high rate of use and enhance the control of its properties by varying the processing parameters [START_REF] Shyr | Crystallisation and spherulite morphology of polylactide stereocomplex[END_REF].

In the iPP, the pendent groups are on the same side of the carbon, which results in helicoidal determination, folded to form thin and ordered structures. Depending on the mechanical and thermal conditions, the material states range from practical amorphism when melted to high crystallinity (ordered structure with different crystalline forms). We can find four forms in an iPP: monoclinic (α), hexagonal (β), triclinic (γ), and mesophase. The latter has been reported under critical conditions [START_REF] Gallegos-Medrano | Influence of chain length, particle size, and thermal treatment of dicarboxylic acid-functionalized titanium dioxide filler in polypropylene[END_REF]. It is well known that parameters like the cooling rate influence the microstructure of semi-crystalline polymers, so that the control of crystallization during the solidification process of a thermoplastic material is of major importance to achieve optimal properties of a cured polymer [START_REF] Lotz | Structure and morphology of poly(propylenes): a molecular analysis[END_REF][START_REF] Brückner | Polymorphism in isotactic polypropylene[END_REF].

In general, crystallization studies have been limited to idealized conditions, hence neglecting in the external factors like the cooling rate and the thermal gradient [START_REF] Stern | Relationship between the structure and mechanical properties of polypropylene: Effects of the molecular weight and shear-induced structure[END_REF]; but in real situations, industrial processes take place in continuously changing conditions, thus making the study of a polymer microstructure more complex [START_REF] Carotenuto | Non-Isothermal Crystallization Kinetics of an Ethylene-Vinyl-Acetate. II. Time-Temperature-Crystallinity-Superposition[END_REF][START_REF] Gradys | Crystallization of polypropylene at various cooling rates[END_REF].

In fact, in order to properly describe the usual processing conditions and the solidification phase during manufacturing, efforts have been focused on understanding the nonisothermal crystallization -by changing the cooling rates -to describe the behavior of semicrystalline polymers for various temperatures [START_REF] Wellen | Melting and crystallization of poly(3-hydroxybutyrate)/carbon black compounds. Effect of heating and cooling cycles on phase transition[END_REF][START_REF] Keridou | Non-Isothermal Crystallization Kinetics of Poly(4-Hydroxybutyrate)[END_REF]. Consequently, it is important to study the kinetics and melting behavior after solidification to generate microstructural knowledge that can be investigated to operate industrial manufacturing processes to have final products with improved properties.

Most global crystallization models, whether under isothermal or non-isothermal conditions, have been based on work carried out between 1930 and 1940s by Kolmogorov, Avrami and Evans. These models have determined the relative transformation rate (Χ) which corresponds to the fraction of a considered volume, which will be transformed into a crystalline volume as a function of time or temperature. Avrami's model has been used universally to describe polymer crystallization kinetics due to its simple application. In many cases, isothermal models are experimentally available only on a narrow temperature range, which does not correspond to industrial conditions [START_REF] Wellen | Nonisothermal cold crystallization of poly(ethylene terephthalate)[END_REF]. Therefore, non-isothermal modeling is primordial to understand the behavior crystallization of semi-crystalline polymers. Different modifications of Avrami's equation have been made to study the non-isothermal crystallization kinetics of semi-crystalline polymers. We can use Jeziorny's model, Nakamura's model, Ozawa's model and Mo's model to describe this phenomenon [START_REF] He | Isothermal and nonisothermal crystallization kinetics of fully bio-based polyamides[END_REF].

A lot of studies have investigated the crystallization kinetics of the iPP. Mubarak et al. [START_REF] Mubarak | Modeling of non-isothermal crystallization kinetics of isotactic polypropylene[END_REF] investigated the nonisothermal crystallization studies of the iPP by comparing Nakamura's model and Ozawa's model. They suggested that for a wide range of cooling rates Nakamura's model is not successful in describing the non-isothermal crystallization kinetics of the iPP and they concluded that Ozawa's model could best describe it. Layachi et al. [START_REF] Layachi | Non-isothermal crystallization kinetics and nucleation behavior of isotactic polypropylene composites with micro-talc[END_REF] studied also the non-isothermal crystallization kinetics of the iPP by comparing Avrami's model modified by Jeziorny, Ozawa's model and Mo's model. They suggested that Jeziorny and Ozawa's approaches could clearly describe the kinetics of the iPP.

As mentioned before, the mechanical properties of thermoplastic polymers significantly depend on the microstructure parameters, such as the type of spherulites and the degree of crystallinity. Like any transformation phase, crystallization abides by the laws of thermodynamics [START_REF] Lorenzo | Non-isothermal crystallization of polymers[END_REF][START_REF] Wang | Reinforcing and toughening isotactic polypropylene through shear-induced crystallization and β-nucleating agent induced crystallization[END_REF]. The size of crystals depends on the control of thermal heat processing. Several researchers confirmed that a slow-cooled rate released for a fully developed crystalline structure would reveal better mechanical properties [START_REF] Huan | Markedly improving mechanical properties for isotactic polypropylene with large-size spherulites by pressure-induced flow processing[END_REF][START_REF] Pavlov | Investigation of the effect of spherulite size on the strength and deformation characteristics of isotactic polypropylene films[END_REF][START_REF] Nitta | Influence of structural organization on tensile properties in mesomorphic isotactic polypropylene[END_REF][START_REF] Doyle | On the effect of crystallinity on the elastic properties of semicrystalline polyethylene[END_REF][START_REF] Fatahi | Correlation between different microstructural parameters and tensile modulus of various polyethylene blown films[END_REF]. In order to better predict the mechanical properties of semicrystalline thermoplastics, it seems important to quantify and understand the effect of structural parameters such as the degree of crystallinity and the crystal size on the mechanical properties [START_REF] Yasin | Optimization of Mechanical and Thermal Properties of iPP and LMPP Blend Fibres by Surface Response Methodology[END_REF], including resilience, toughness and Young's modulus. Mahmood et al. [START_REF] Mahmood | Influence of structure gradients in injection moldings of isotactic polypropylene on their mechanical properties[END_REF] and Dietz [START_REF] Dietz | Effect of cooling on crystallization and microstructure of polypropylene[END_REF] studied the effect of the internal structure of an injection-molded sample on the mechanical properties of the iPP. They showed an interrelation between the degree of crystallinity and the local cooling rates, and so the effect on the toughness and storage modulus of the polymer. In order to better analyze the microstructure under tensile tests, the tensile deformation mechanisms of the polypropylene were imaged by an optical microscope with an original micro-tensile machine.

In this work, the non-isothermal crystallization kinetics of the iPP have been studied. Then, different methodologies were employed to evaluate the non-isothermal crystallization behavior, such as Differential Scanning Calorimetry (DSC) and polarized optical microscopy. Specific theoretical approaches were also employed to model the non-isothermal kinetics of the iPP. Generally, the obtained results of various methods are controversial, especially when theoretical approximations and experimental conditions are not clear. Finally, the mechanical characteristics such as resilience and Young's modulus were analyzed for polymer films with different microstructures.

Experimental procedure

Materials

In the present study, a commercial iPP homopolymer, provided by Chemieuro with the commercial name PPH 4060, was used. It exhibited a flow index of 3 g 10 min -1 and a melting density of 0.905 gr cm -3 . The percentage of isotacticity of the used polymer was estimated at 98% utilizing a Perkin Elmer FTIR Frontier spectrometer from 4000 to 400 cm -3 in ATR (Attenuated Total Reflectance), equipped with a diamond/ ZnSe crystal.

Preparation of polypropylene films

For crystallization studies, polymer films with different thicknesses were prepared. A process of molding and heating in a thermocompression was performed. For each desired thickness, a wedge was placed between two sheets of Kapton. After that, a pressure of 200 bars was applied at a temperature of 210 °C for 5 min.

Differential scanning calorimetry

Calorimetric data of non-isothermal crystallization were obtained using a DSC machine (Q1000 V9.0 Build 275 TA instruments). Before testing, indium was used to calibrate the temperature scale and the melting enthalpies. Nitrogen gas was purged into the DSC cell with a constant flow rate of 50 ml min -1 , to avoid material oxidation. The sample weight was 6 ± 0.5 mg using a balance Mettler Toledo with 0.01 mg sensitivity.

Specimens were crimped in non-hermetic aluminum pans, while sealed and heated at a rate of 20 °C min -1 to 210 °C for 3 min. This procedure was used to delete the thermal history of the polymer. For non-isothermal crystallization studies, previously heated samples were cooled down at various cool

Evaluation of tensile properties

Macroscopic tensile tests

The mechanical characteristics such as resilience and Young's modulus were analyzed for different microstructures of the semi-crystalline polymer. The tensile properties were measured with 1BB specimens using Instron 5966 with loading capacity of 10 KN and a loading rate of 10 mm min -1 .

In-situ tensile tests

The machine of micro-traction, presented in Fig. 1, was used to follow the evolution of the morphology at the same zone during the tensile deformation by pulling from two sides to establish a chronology of the deformation mechanisms. The images were captured during testing at every second to present the change in the microstructure.

Experimental results and discussions

Non-isothermal crystallization analysis

From non-isothermal crystallization experiments, and at each cooling rate, dynamic DSC data for the crystallization exotherms as a function of temperature can be obtained, as reported in Fig. 2.

As expected, when the cooling rate increases, the crystallization exotherm becomes broader and the peak of crystallization temperature, T 0, shifts progressively to lower temperatures. Some characteristic parameters are represented in Table 1 for each cooling rate ɸ. As expected, all the characteristic temperature values decrease with the rising cooling rate: The starting crystallization process will be completed first. Based on the assumption that the evolution of crystallinity is linearly proportional to the evolution of heat release during crystallization, the transformation ratio of the material from the melting state to the crystal one, namely relative crystallinity, can be determined using DSC data [START_REF] Ray | Clay-Containing Polymer Nanocomposites: From Fundamentals to Real Applications[END_REF] as a function of time or temperature for the different cooling rates and according to the following expression:

where T onset denotes the beginning of the crystallization phenomena, T ∞ holds for the final temperature of the crystallization exotherm, ΔH c represents the enthalpy of the crystallization released with an infinitesimal dT temperature, T represents the crystallization temperature at time t, T onset and T∞ correspond to the overall enthalpy of crystallization during the transformation phase. Relative crystallinity can be represented as a function of time using the following relation:

(1)

X(t) = ∫ T T onset ΔH c (t) dt dt ∫ T ∞ T onset ΔH c (tot) dt dt (2) t = Tonset -T ϕ
where ɸ is the cooling rate. This relationship is valid when the sample probably has the same thermal history as the DSC furnace [START_REF] Righetti | Crystallization of Polymers Investigated by Temperature-Modulated DSC[END_REF]. Figure 3 represents the variation in the relative degree of crystallinity of the iPP, where X(T) can be converted to X(t) using Eq. ( 2) at different cooling rates. The obtained curves reveal a similar sigmoidal shape: the three regimes during crystallization phenomena [START_REF] Lorenzo | Non-isothermal crystallization of polymers[END_REF][START_REF] Buzarovska | Crystallization of polymers (2nd edition) Volume 2: Kinetics and mechanisms[END_REF][START_REF] Zhang | Melt Crystallization Behavior and Crystalline Morphology of Polylactide/Poly(εcaprolactone) Blends Compatibilized by Lactide-Caprolactone Copolymer[END_REF]. The beginning of the transformation at a low slope for high-temperature values corresponds to the nucleation and the formation of a sufficient number of crystallites. During this period, the relative crystallinity does not change. The second stage corresponds to the phase of the growth of spherulites. The rate of their growth is directly related to the rate of cooling. This period is more important for the greatest cooling rate. The third step corresponds to the final phase of the process [START_REF] Clark | Regime III crystallization in polypropylene[END_REF]. During this stage, the spherulites interact with each other to form solid polygons at the end of crystallization.

The quantity of non-transformed material is reduced, and the appearance of new nuclei drops, hence the growth of spherulites. We can see that the overall crystallization temperature range decreases with the increase in the cooling rate.

Jeziorny's analysis of non-isothermal crystallization

To quantitatively describe the evolution of crystallinity and polymer behavior during non-isothermal crystallization, we can apply different kinetic models, for instance Avrami's, Ozawa's and Mo's. Avrami's microkinetic approach is one of the most used theories in the field and can be employed to analyze the isothermal crystallization behavior of the polymer based on the following equation: where X(t) represents the degree of relative crystallinity at time t, Z is the crystallization rate constant depending on growth rate, and n is Avrami's crystallization exponent representing the morphology of crystals and the nature of nucleation (homogeneous or heterogenous nucleation) [START_REF] Avrami | Kinetics of Phase Change I General Theory[END_REF][START_REF] Avrami | Kinetics of Phase Change II. Transformation-Time Relations for Random Distribution of Nuclei[END_REF][START_REF] Lorenzo | DSC isothermal polymer crystallization kinetics measurements and the use of the Avrami equation to fit the data: Guidelines to avoid common problems[END_REF]. Parameters Z and n are estimated by taking the logarithmic of Eq. ( 3) so as to have a linear fitting transformation, as shown in Eq. ( 4):

(3) 1 -X(t) = exp (-Z t n )
Therefore, ln (-ln (1 -X(t))) plotted versus ln (t) yields Avrami's parameters n and Z via the slope and intercept of the linear plots, respectively. To account for the nonisothermal crystallization conditions, Avrami's approach was modified by Jeziorny, introducing the cooling rate ɸ as ln Z J = ln Z [35, 36].

To avoid the secondary crystallization and the impingement of spherulites; Jeziorny-Avrami's modified approach was used for X(t) ϵ [0.1;0.8]. Avrami's analysis results are presented in Fig. 4. A high correlation (R 2 > 0.99) can be seen indicating the efficiency of Avrami's approach modified by Jeziorny for the description of the non-isothermal crystallization kinetics of the polymer. Table 2 summarizes the main kinetic parameters deduced from Avrami and Jeziorny's analysis.

The n values are in the vicinity of 2.2-2.5, similar to most literature reports [START_REF] Zhu | Crystallization behavior of partially melting isotactic polypropylene[END_REF][START_REF] Gupta | Isothermal and non-isothermal crystallization kinetics of hydroxylfunctionalized polypropylene[END_REF][START_REF] Cho | Crystallization and melting behavior of polypropylene and maleated polypropylene blends[END_REF], and they are interpreted as two-dimensional spherulitic growth with heterogeneous nucleation. Moreover, the increase in Avami's exponent while decreasing the cooling rate is indicative of the sporadic nucleation phenomenon. However, Z J , which is the crystallization rate, goes up with the rising cooling rate, indicating the increase in the transformation rate.

(4) ln (-ln (1 -X(t))) = ln (Z) + n ln (t)

Alternatives to Avrami's approach for non-isothermal crystallization

The non-isothermal crystallization results were also compared with Ozawa's approach [START_REF] Ozawa | Kinetics of non-isothermal crystallization[END_REF]. In fact, it is a modification of Avrami's theory which directly considers the cooling rate of crystallization. The approach mentions that a nonisothermal process is a succession of infinitesimal steps of isothermal crystallization. Ozawa's equation can be obtained by applying a derivation of Evans' approach [START_REF] Evans | The laws of expanding circles and spheres in relation to the lateral growth of surface films and the grain-size of metals[END_REF] to Avrami's equation:

where R(T) denotes the cooling rate function that depends on the temperature during the thermal analysis, and m represents Ozawa's exponent.

At a fixed temperature and for different cooling rates, Ozawa's parameters are determined from the plot of ln(ln(1-X(T)) versus ln(ɸ), as reported in Fig. 5. The limit of Ozawa's method is that the linearity regression is observed for a very restricted range of cooling rates. Hence, the analysis using Ozawa's model depends on the variation between (5) 1 -X(T) = exp(-R(T)∕ m) Fig. [START_REF] Shyr | Crystallisation and spherulite morphology of polylactide stereocomplex[END_REF] Jeziorny's approach for iPP the primary and secondary crystallization processes. Most of the high crystallization temperatures vary linearly over a wide range of cooling rates as it is difficult for the polymer to enter the secondary crystallization phase. Table 3 summarizes the different Ozawa's parameters, by which the evolution of Ozawa's exponent m, obtained from the slopes of the different straight segments, does not show a clear tendency, thus a morphological interpretation would-be uncertain [START_REF] Vyazovkin | Nonisothermal crystallization of polymers: Getting more out of kinetic analysis of differential scanning calorimetry data[END_REF].

Activation energy: Kissinger's approach and iso-conversional method

The activation energy represents the minimal amount of energy required to activate the molecules of the material to accomplish a chemical transformation. Kissinger's approach [START_REF] Kissinger | Reaction Kinetics in Differential Thermal Analysis[END_REF] estimates the activation energy at several non-isothermal steps at a constant cooling rate, presented as follows:

where ɸ is the crystallization rate, T p is the peak of the exotherm of DSC data, ∆E presents the activation energy of the non-isothermal crystallization, and R = 8.31 J K -1 mol -1 is the gas constant.

According to this equation, a plot of ln T P 2 versus 1∕T P should lead to a straight line with -∆E/R as a slope. The value of the activation energy of the iPP in nonisothermal crystallization is 200 J K -1 mol -1 . The sign of energy shows the increase in the crystallization rates with the decrease in the crystallization temperature. Kissinger's approach is widely used to determine the activation energy of polymers, but it can counter several problems, as it considers the macroscopic crystallization rate, ɸ, which includes the nucleation rate and the nucleation growth, hence having different activation energy. The use of multiple cooling rates, such as the iso-conversional method, is recommended. This method can be applied to non-isothermal crystallization to evaluate the dependence of the effective activation energy on the temperature and the conversion. The iso-conversional methods, frequently used to determine the effective energy, are the differential iso-conversional methods of Friedman and the advanced Iso-conversional method of Vyazovkin et al. In this study, effective activation energy was calculated using Vyazovkin and Sbirrazzuoli approach [START_REF] Achilias | Evaluation of the Isoconversional Approach to Estimating the Hoffman-Lauritzen Parameters from the Overall Rates of Non-Isothermal Crystallization of Polymers[END_REF][START_REF] Vyazovkin | Isoconversional Analysis of Calorimetric Data on Nonisothermal Crystallization of a Polymer Melt[END_REF]. Indeed, the latter considers that the energy barrier of crystallization can vary during the non-isothermal crystallization from melting [START_REF] Keridou | Non-Isothermal Crystallization Kinetics of Poly(4-Hydroxybutyrate)[END_REF] using Eq. ( 7). ( 6) where dX dt is the instantaneous crystallization rate, as a function of time at a given degree of relative X crystallinity, and ∆E X is the activation energy related to a given conversion at different cooling rates and for a narrow temperature range.

d ln T P 2 d 1 T P = - ΔE R
According to this relation, the slopes deduced from the obtained linear plots of (dX∕dt) versus 1/T (Fig. 6) give ∆E X . The deduced values of the activation energy vary between -100 kJ mol -1 and -176.35 kJ mol -1 , which covers a wide range of activation energy. Other studies concerning the iPP have revealed a similar variation with a range from -61.3 kJ mol -1 to -210 kJ mol -1 [START_REF] Benhacine | Thermal Stability and Kinetic Study of Isotactic Polypropylene/Algerian Bentonite Nanocomposites Prepared via Melt Blending[END_REF][START_REF] Xing | Study on the Mechanism and Kinetics of Waste Polypropylene Cracking Oxidation over the Mn2O3/HY Catalyst by TG-MS and In Situ FTIR[END_REF]. The isoconversional methods take into account the variation in the activation energy with crystallization phases.

Microscopic analysis of non-isothermal crystallization

The non-isothermal crystallization analysis of the polymer (from the melting point) must be performed carefully in order to [START_REF] Brückner | Polymorphism in isotactic polypropylene[END_REF]

ln dX dt X,i = A - ΔE X RT X,i
adequately control the sample temperature. Thermal gradients in between the cooling furnace and the polymer can result in an inaccurate control of the material temperature. In addition, solidification is an exothermic process, and the heat can develop thermal gradients in the sample during the phase. As a result, the thicker the sample, the more critical this problem is. Notably, the influence of the thickness of the sample and the cooling rate on the occurrence of thermal gradients during non-isothermal crystallization is still a missing spot.

Monasse and Haudin [START_REF] Monasse | Thermal dependence of nucleation and growth rate in polypropylene by non isothermal calorimetry[END_REF] estimated that for polypropylene samples, with 200-300 μm thickness, the variation in the thermal gradient between the sample and the hot stage was negligible. Moreover, Di Lorenzo and Silvestre [START_REF] Lorenzo | Non-isothermal crystallization of polymers[END_REF] used films with 200 μm thickness and confirmed the existence of a thermal gradient in the samples. In our work, studied samples were about 80 μm and 150 μm. A cooling rate of 1 °C min -1 was used to cover a large range of experimental data.

Figure 7 represents the evolution of the spherulites radius with temperature for variously spaced spherulites in the same film with 150 μm thickness. The obtained plots show a significant difference between the evolution of the different spherulite radii. This variation is caused by the gradient between the platinum hot stage and the sample during solidification. To minimize the thermal gradient, thinner films are used with 100 μm of thickness.

The diameter of the iPP spherulite can be up to 40-90 μm, so the crystals find their free 3D space to grow radially. In this situation, insignificant gradients are expected in the same sample and between the furnace and the sample even during solidification, because the heat released during crystallization is very low due to the small mass of the used film. Therefore, it can be safely assumed that the effective temperature of the sample corresponds to that recorded by the hot stage. Generally, spherulite growth rates, G, are measured under isothermal conditions, where the growing spherulite radius R depends on the crystallization time t. In this study, G is estimated by the following equation for a constant cooling rate:

The time-temperature-diameter evolution diagram of the spherulite in a sample with 100-μm thickness is presented in Fig. 8. This diagram can be useful to study the evolution between of the spherulite diameter as a function of time and temperature. This figure shows the growth of the spherulites radius of the iPP with a cooling rate of 1 °C min -1 . The range of the crystallization temperature found by optical microscopy is higher than the one defined by DSC data. Hence, the development of the spherulite radius versus temperature gives a plot that will be later fitted to a polynomial equation showing a correlation coefficient of R 2 > 0.9988. Actually, it is a third-order equation (R = 0.0466T3 -56.305T2 + 22686 T-3E + 04). Applying Eq. 8, the growth rate is then calculated as illustrated in Fig. 9.

Morphology and melting behavior after non-isothermal crystallization

The cooling rate obviously affects the final morphology, the spherulite size and even the crystallization of the iPP despite Fig. 13 WAXS intensity profiles of neat iPP for different cooling rates Fig. 14 Stress-strain curves of different films with various cooling rates its rapid crystallization. Thus, a neat iPP was subjected to heating after non-isothermal crystallization with different cooling rates to study the polymer morphology. The results of obtained endothermal peaks are presented in Figs. 10 and 11. According to these figures, it can be seen that the crystallization peak temperatures decrease with an increase in the cooling rates, and therefore the primary nucleation rises, which leads to a reduction in the spherulite size. Furthermore, the morphology depends also on the lamellar thickness.

According to the accompanying graphs in Fig. 12 and the data collected in Table 4, only the presence of the monoclinic crystalline structure α-phase can be affirmed. It is the most thermodynamically stable rather than the other three crystalline phases of the iPP, namely β, γ and smectic phases. The WAXS intensity profiles for different cooling rates are depicted in Fig. 13 while confirming the presence of the α-phase. In addition, the obtained profiles reveal the absence of a preferred orientation of the chains for different cooling rates [START_REF] Layachi | Non-isothermal crystallization kinetics and nucleation behavior of isotactic polypropylene composites with micro-talc[END_REF][START_REF] Labour | Influence of the β crystalline phase on the mechanical properties of unfilled and CaCO3-filled polypropylene I. Structural and mechanical characterisation[END_REF]. Table 4 gives the global crystallinity rate (%) using the following equation:

where ΔH f exp is the experimental crystallization enthalpy, and ΔH f 0 is the bulk enthalpy of polypropylene whose value is 177 J g -1 .

Mechanical properties after non-isothermal crystallization

At macroscopic scale

Tests were performed to evaluate the effect of the cooling rate and crystallinity on the mechanical properties of iPP polymer films. Specimens were fabricated at three different cooling rates in the hot stage under the same pressure: 60 °C min -1 (fast), 10 °C min -1 (medium), and 1 °C min -1 (slow). To have a uniform thickness for the different samples, a wedge was placed between two lamellar sheets. The crystallinity of each plate was determined using DSC.

The tensile properties were measured with 1BB specimens. Data were taken at a crosshead speed of 10 mm min -1 . In order to identify structure-property relations, the mechanical properties of thin films showing the structural state under different cooling rates were investigated. Uniaxial tensile tests were performed on 100 μm thick films prepared with linkam THM600. Young's modulus E and resilience K calculated from the stress-strain curves of these films will be discussed. The engineering stress-strain curves of different samples under a monotonic tensile test are presented in Fig. 14. Figure 15a presents Young's modulus of the iPP as a function of the crystallinity degree at different cooling rates. Each point in Fig. 15b is the average of five data. As it can be seen, the tensile properties do not significantly vary over the range of evaluated crystallinity. Despite the insignificant variation in mechanical properties, the low cooling rates show good stability in the plastic part.

A pronounced augmentation in the resilience K (Fig. 16). Thus, the changes in K reveal that this property is very sensitive to structural differences depending on the cooling rate. It is worth noting that K(ɸ) depends on the ductile behavior with a significantly higher strain in the elastic phase for the slow cooling rate, while a rather brittle behavior is observed for the fast cooling rate.

It can be suggested that the mechanical quantities determined at small stresses and at small deformations such as Young's modulus E and resilience K are influenced differently by the microstructure of the semi-crystalline polymer. This difference underlines the variety and the complexity of factors affecting the mechanical properties of the semi-crystalline polymer. The dependence of the cooling rate on the crystallinity degree measured by DSC leads to a correlation between the observed values of E and the crystallinity degree. It can also be noted that even a significant difference in crystallization kinetics, spherulite size, microstructure and degree of crystallinity between thin films obtained using different cooling rates does not definitely represent the only parameters determining the tensile modulus of the polymer. Expectedly, other features like the crystal orientation, the heterogeneity in the microstructure and the pressure during manufacturing highly affect the strength of the semi-crystalline iPP at the elastic phase with small deformations. Another type of correlation is noticed. Basically, the resilience and the strain at rupture are influenced by the degree of crystallinity and the crystalline structure. It can be denoted that the improvement of resilience corresponds to the fine structures combined with bigger spherulites. However, rapid cooling and small spherulites demonstrate a homogenous plastic deformation at the macroscopic scale and an improvement of the strain at break.

In-situ tensile behavior: evolution of spherulites

Figure 17 shows the evolution of spherulites during micro tensile tests. Up to an elongation of 3%, the deformation seems to be homogeneous in the linear part, which corresponds to the elastic zone.

After that, local deformation appears at the scale of the spherulites which propagate in perpendicular and diagonal directions.

The schematic of this phenomenon is presented in Fig. 18, which shows the ellipsoidal form of the spherulites during or after loading. After striction, microcracks appear and spread in the same stress direction. This part also highlights the fibrillary structure at the level of the spherulites.

Actually, the growth of the strain at rupture may be related to the strength of the spherulite which itself can depend on an intra/inter rupture on the spherulite, and it is in progress to be investigated in our future work by studying different morphologies related to various process conditions.

Conclusion

The non-isothermal crystallization of the iPP was analyzed in this study using calorimetric data, which permitted the prediction of crystallization parameters by employing theoretical methods via Avrami and Ozawa's models. The obtained parameters of Avrami's model estimated the variation in the crystallization rate with the crystallization temperature at different cooling rates. In addition, the isoconversional analysis gave information about the activation energy and allowed the estimation of the temperature corresponding to the maximal crystallization rate. Microscopic observation confirmed that the crystallization peak temperatures decreased with an increase in cooling rates, and consequently primary nucleation rose, leading to a reduction in the spherulite size. Furthermore, the morphology depended also on the lamellar thickness. Moreover, the multi-scale mechanical results showed that the mechanical quantities determined at small amplitudes and at small deformations, like the E moduli, the resilience and the strain at rupture, were influenced differently by the microstructure of the semi-crystalline polymer, which could be related to an inter/ intra spherulitic rupture.

Fig. 1

 1 Fig. 1 Procedure of in-situ tensile tests

Fig. 2

 2 Fig. 2 Non-isothermal crystallization exotherms of neat iPP

Fig. 5

 5 Fig. 5 Plots of ln(-ln(1-X)) versus log(ɸ) for iPP at different crystallization temperatures

Fig. 6 Fig. 7

 67 Fig.[START_REF] Lotz | Structure and morphology of poly(propylenes): a molecular analysis[END_REF] Plots of ln(dX/dt) versus 1/T at constant cooling rates and different degrees of crystallinity (yellow, grey, orange, blue and green correspond to 0.8, 0.7, 0.6, 0.5 and 0.4 values, respectively)

Fig. 8 Fig. 9

 89 Fig. 8 Time-temperaturediameter diagram of spherulite in a sample with 100-μm thickness

Fig. 10 Fig. 11

 1011 Fig.[START_REF] Gradys | Crystallization of polypropylene at various cooling rates[END_REF] Optical micrographs taken at the end of exothermal peaks of iPP performed at cooling rates of 1 °C min -1 (a), 5 °C min -1 (b) and 30 °C min -1 (c)

Fig. 15

 15 Fig. 15 Tensile modulus of iPP as a function of crystallinity degree at different cooling rates

Fig. 17 Fig. 18

 1718 Fig. 17 Aggregate scale MOP under micromechanical tensile test

Table 1

 1 T 0 crystallization temperature, and T onset and T ∞ temperatures.of neat iPP at various ɸ cooling rates

	Cooling rates (°C min -1 )	T c (°C)	T onset (°C)	T ∞
	30	108.47	115.10	88
	20	111.1	117.85	97
	10 5	115.28 119.5	120.33 125.76	107 111	Fig.

[START_REF] Ding | Crystalline morphology and mechanical properties of isotactic polypropylene composites filled by wollastonite with β-nucleating surface[END_REF]

Avrami's plots: Evolution of relative crystallinity as a function temperature

Table 2

 2 Non-isothermal crystallization kinetics parameters for iPP using DSC results

	Cooling rate (°C min -1 ) Z J	n	R 2
	30	1.1	2.24	0.999
	20	1.09	2.25	0.999
	10	1.08	2.37	0.999
	5	0.84	2.38	0.998

Table 3

 3 Ozawa's parameters fitted for non-isothermal crystallization

	kinetics of iPP
	T/°C	Log ( R(T))	m	R 2
	113	2.08	2.04	0.9398
	115	2.19	2.44	0.9850
	116	2.23	2.64	0.9942
	118	2.58	3.50	0.9975
	120	2.13	3.57	0.9738

Table 4

 4

	Crystallization fraction of iPP for different cooling rates	Cooling rate (°C min -1 )	X c (%)
		5	55.7
		10	52.5
		20	44.2
		30	38.7

Declarations

Conflict of interest There is no conflict and interest for this paper. All of persons had same contribution on this paper.

Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Christophe Muth-Seng
email: christophe.muth-seng@ensam.eu

Maxime Huneidi

Cécile Heidsieck

Sébastien Laporte

Jean-Yves Le Coz

Olivier Gille

Wafa Skalli

Personalised gravitational loading of the cervical spine from biplanar X-rays for asymptomatic and clinical subjects in neutral standing position

Keywords:

Background: As a leading cause of disability with a high societal and economic cost, it is crucial to better understand risk factors of neck pain and surgical complications. Getting subject-specific external loading is essential for quantifying muscle forces and joint loads but it requires exertion trials and load cells which are uncommon in clinical settings. Methods: This paper presents a method to compute the gravitational loading at four levels of the cervical spine (C3C4, C4C5, C5C6, C6C7) in neutral standing position from biplanar radiographs exclusively. The resulting load was decomposed in local disc frames and its components were used to compare different populations: 118 asymptomatic subjects and 46 patients before and after surgery (anterior cervical discectomy and fusion or total disc replacement). Comparisons were performed at C6C7 and the upper level adjacent to surgery. Findings: Significant changes in gravitational loading were observed with age in healthy subjects as well as in patients after surgery and have been associated with changes in posture. Interpretation: This approach quantifies the influence of postural changes on gravitational loading on the cervical spine. It represents a simple way to obtain necessary input for muscle force quantification models in clinical routine and to use them for patient evaluation. The study of the subsequent subject-specific spinal loading could help further the understanding of cervical spine biomechanics, degeneration mechanisms and complications following surgery.

Introduction

Neck pain has been identified as one of the three most common musculoskeletal disorders [START_REF] Trinh | Acupuncture for neck disorders ( review )[END_REF] and as the fourth leading cause for disability in the US [START_REF] Murray | The state of US health, 1990-2010: burden of diseases, injuries, and risk factors[END_REF]. Its annual prevalence has been reported to range between 15 and 50% [START_REF] Binder | Musculoskeletal disorders neck pain search date may 2007 musculoskeletal disorders neck pain[END_REF][START_REF] Fejer | The prevalence of neck pain in the world population: a systematic critical review of the literature[END_REF][START_REF] Fernández-De-Las-Peñas | Prevalence of neck and low back pain in community-dwelling adults in spain: A population-based national study[END_REF][START_REF] Hogg-Johnson | The burden and determinants of neck pain in the general population. results of the bone and joint decade 2000-2010 task force on neck pain and its associated disorders[END_REF] and, with the soaring ubiquity of portable electronics and changes in lifestyle, is expected to keep rising [START_REF] Côté | The burden and determinants of neck pain in workers[END_REF][START_REF] Vasavada | Gravitational demand on the neck musculature during tablet computer use[END_REF]. It entails major societal and economic expenses in both direct [START_REF] Dieleman | US spending on personal health care and public health, 1996-2013[END_REF] and indirect costs [START_REF] Côté | The prevalence and incidence of work absenteeism involving neck pain. A cohort of ontario lost-time claimants[END_REF].

Surgery as a treatment to alleviate neck pain has been on a rising trend [START_REF] Marquez-Lara | Current trends in demographics, practice, and in-hospital outcomes in cervical spine surgery: A national database analysis between 2002 and 2011[END_REF][START_REF] Wang | Trends and variations in cervical spine surgery in the United States: medicare beneficiaries, 1992 to 2005[END_REF]. While anterior cervical discectomy and fusion (ACDF) appears to be the gold standard, total disc replacement (TDR) has become a motion-preserving alternative. Complications following surgery, such as adjacent segment disease (ASD), have been reported for both though [START_REF] Bevevino | Adjacent segment disease: incidence and recommended treatment[END_REF][START_REF] Bible | Anterior cervical discectomy and fusion: surgical indications and outcomes[END_REF][START_REF] Kong | Prevalence of adjacent segment disease following cervical spine surgery a PRISMA-compliant systematic review and metaanalysis[END_REF].

The burden neck pain represents to patients' quality of life and expenses makes it crucial to further prevention. Though some risk factors have been identified, including muscular dysfunction [START_REF] Alpayci | Decreased neck muscle strength in patients with the loss of cervical lordosis[END_REF][START_REF] Cheng | Altered co-contraction of cervical muscles in Young adults with chronic neck pain during voluntary neck motions[END_REF][START_REF] Falla | Muscle pain induces task-dependent changes in cervical agonist/ antagonist activity[END_REF][START_REF] Fernández-De-Las-Peñas | Cervical muscle co-activation in isometric contractions is enhanced in chronic tension-type headache patients[END_REF] and sagittal posture [START_REF] Lau | Relationships between sagittal postures of thoracic and cervical spine, presence of neck pain, neck pain severity and disability[END_REF] for neck pain and spine curvature for ASD [START_REF] Katsuura | Kyphotic malalignment after anterior cervical fusion is one of the factors promoting the degenerative process in adjacent intervertebral levels[END_REF][START_REF] Xiong | 0.5-to 1-fold intervertebral distraction is a protective factor for adjacent segment degeneration in single-level anterior cervical discectomy and fusion[END_REF], a deeper understanding of cervical spine biomechanics and degeneration is still needed.

Numerical modelling can be a precious tool in that regard for assessing the influence of geometry, loading, for implant design and patient evaluation. While advancements have been made towards personalised geometry [START_REF] John | Cervical spine morphology and ligament property variations: a finite element study of their influence on sagittal bending characteristics[END_REF][START_REF] Laville | Parametric and subject-specific finite element modelling of the lower cervical spine. Influence of geometrical parameters on the motion patterns[END_REF][START_REF] Nikkhoo | Development and validation of a geometrically personalized finite element model of the lower ligamentous cervical spine for clinical applications[END_REF] and numerical implant evaluation [START_REF] Gandhi | Biomechanical analysis of the cervical spine following disc degeneration, disc fusion, and disc replacement: a finite element study[END_REF][START_REF] Mackiewicz | Comparative studies of cervical spine anterior stabilization systems -finite element analysis[END_REF][START_REF] Rousseau | Influence of the geometry of a ball-andsocket intervertebral prosthesis at the cervical spine: a finite element study[END_REF], obtaining subjectspecific loading remains challenging. Musculoskeletal models can actually yield personalised spinal loading but they require exertion trials including external loading and other measurements, such as motion capture and EMG [START_REF] Alizadeh | An electromyography-assisted biomechanical cervical spine model: model development and validation[END_REF][START_REF] Moroney | Analysis and measurement of neck loads[END_REF][START_REF] Mortensen | The inclusion of hyoid muscles improve moment generating capacity and dynamic simulations in musculoskeletal models of the head and neck[END_REF]. This experimental pre-requisite makes them difficult to implement into a clinical setting.

Recent developments of low-dose biplanar X-ray full-body acquisitions [START_REF] Dubousset | EOS: a new imaging system with low dose radiation in standing position for spine and bone; joint disorders[END_REF] in standing position with associated 3D reconstructions [START_REF] Humbert | 3D reconstruction of the spine from biplanar X-rays using parametric models based on transversal and longitudinal inferences[END_REF][START_REF] Nérot | A 3D reconstruction method of the body envelope from biplanar X-rays: evaluation of its accuracy and reliability[END_REF][START_REF] Rousseau | Reproducibility of measuring the shape and three-dimensional position of cervical vertebrae in upright position using the EOS stereoradiography system[END_REF] allows subject-specific postural analysis in clinical routine. Particularly, reconstruction of the external envelope associated with density models allows an estimation of mass and the centre of mass of each body segment [START_REF] Amabile | Determination of a new uniform thorax density representative of the living population from 3D external body shape modeling[END_REF]. Computation of the barycentre of the body segment above a given joint allows estimation of the gravitational load taking into account subject-specific posture [START_REF] Amabile | Alignment of centers of mass of body segments with the gravity line[END_REF]. The clinical relevance of this barycentremetric approach has been assessed in the study of idiopathic scoliosis [START_REF] Thenard | Analysis of center of mass and gravity-induced vertebral axial torque on the scoliotic spine by barycentremetry[END_REF].

Using a similar approach, the present study deals with obtaining subject-specific gravitational loading of the cervical spine from biplanar X-rays for both asymptomatic and clinical subjects. Decomposition of the resulting load should help quantify significant differences between age groups and between healthy/clinical subjects, providing insight regarding the mechanical effects of age, pathology and treatment.

Methods

Populations

The control population was constituted from data of 118 healthy subjects (57 females, 61 males) of mean age 42.0 y.o. (SD: 11.5) acquired during previous studies [START_REF] Heidsieck | Effect of postural alignment alteration with age on vertebral strength[END_REF]. Clinical exams were performed to ensure that the subjects presented no symptomatic lower limb nor spine pathology that could affect the spinal alignment and patient with a frontal Cobb angle over 20 • were excluded.

The clinical population was made up of 46 patients (29 females, 16 males) of mean age 51.4 y.o. (SD: 11.5) experiencing neck pain and treated by surgery. Thirteen underwent TDR, including four with a CP-ESP prosthesis (FH Orthopedics) and nine using a Baguera prosthesis (SpineArt SA, Geneva, Switzerland), and the remaining 33 underwent ACDF. Their data were collected pre and post-surgery (follow-up ranged from 6 weeks to 2 months) in routine clinical practice.

Data acquisition was performed after subjects' written consent and approval by an ethics committee (Comité de Protection des Personnes CPP N • 2010/113, N • 06036 for control population, and N • 2014/89 for patients).

Populations under study were sorted into three age groups: younger than 40 y.o, between 40 and 60 y.o. and over 60 y.o., which will be referred to as G1, G2 and G3 respectively. Further details on the populations are available in Table 1.

3D reconstructions and barycentremetry

Each subject underwent biplanar X-rays in free standing position [START_REF] Steffen | 3D postural balance with regard to gravity line: an evaluation in the transversal plane on 93 patients and 23 asymptomatic volunteers[END_REF] acquired using an EOS system (EOS imaging SA., Paris, France). This allowed 3D reconstructions of the spine from C1 to L5 [START_REF] Humbert | 3D reconstruction of the spine from biplanar X-rays using parametric models based on transversal and longitudinal inferences[END_REF][START_REF] Rousseau | Reproducibility of measuring the shape and three-dimensional position of cervical vertebrae in upright position using the EOS stereoradiography system[END_REF] and of the body envelope from head to toe using a validated semi-automatic method [START_REF] Nérot | A 3D reconstruction method of the body envelope from biplanar X-rays: evaluation of its accuracy and reliability[END_REF], briefly reminded hereafter. Manual positioning of a few anatomical landmarks allowed morphing of a reference average template (either male or female). The resulting initial model was then retroprojected on each X-Rays and manually adjusted to fit the subject's contours.

From the 3D model of the spine, a plane was assessed at four intervertebral disc (IVD) levels (C3C4, C4C5, C5C6, C6C7) using the adjacent endplates, and a local frame was computed for each based on the vertebral frames as defined by [START_REF] Rousseau | Reproducibility of measuring the shape and three-dimensional position of cervical vertebrae in upright position using the EOS stereoradiography system[END_REF]. Details of the method and the evaluation of its reproducibility are available in Supplementary Data.

The IVD planes were used to intersect the model of the body envelope. For each of them, the volume of everything above it was computed. Combined with a generic uniform density model for the head and neck [START_REF] Dempster | Properties of body segments based on size and weight[END_REF], body segment mass and barycentre were estimated; yielding the mechanical loading on each IVD due to gravity (Fig. 1a).

Said gravitational loading was decomposed from the local IVD frame (Fig. 1b) into force components: posteroanterior shear (Fx), mediolateral shear (Fy) and compression (Fz) and moment components: mediolateral (Mx), flexion-extension (My) and axial torsion (Mz). Posteroanterior and mediolateral moment arms from the disc centre to the centre of mass were measured, as well as the disc tilt angle with regards to the horizontal plane.

A reproducibility study (20 subjects, 2 operators, 2 times per operator) yielded standard deviations of uncertainty on IVD plane orientation of 1.3 • , 1.6 • and 1.5 • in the frontal, sagittal and axial planes respectively. These uncertainties were used as random errors in a Monte-Carlo analysis repeated 2000 times for a healthy subject to quantify its impact on the components of the gravitational force.

Analysis and statistical tests

A linear correlation study was carried out for the asymptomatic populations to assess the influence of age, weight and height on headneck mass, C6C7 disc angulation and posteroanterior and mediolateral moment arms to the disc centre.

Statistical tests were performed for each component of mechanical loading at the C6C7 level to compare groups G1, G2, G3 of the asymptomatic population; as well as between the G2 groups of the control and clinical populations pre and post-surgery. Distribution normality was evaluated using a Shapiro-Wilk test. Then, comparisons between groups were made via either ANOVA or Kruskal-Wallis test (depending on normality) followed by a post hoc analysis using the Tukey-Kramer method. Additionally, paired comparisons for each components at C6C7 between pre and post-surgery were performed using either paired t-tests or Wilcoxon signed-rank tests (depending on normality). Finally, at each IVD level, normality and subnormality corridors were defined for each component based on the means and standard deviations (SD) from each group of the control population (median and interquartile range for non-normal distributions). Patients' results were considered normal if they were less than 1*SD away from the control mean value, subnormal if less than 2*SD away, and abnormal beyond. This analysis was performed at the C6C7 IVD and at the level upperadjacent to the operated IVD.

Every significant difference and observation on load components from either statistical or corridor analysis was complemented with a similar analysis comparing moment arms and IVD tilt angle.

Results

Reconstruction of the cervical spine and the body envelope took 15 to 20 min per subject, and the associated loading was computed in a matter of seconds. Following the sensitivity analysis, the uncertainty on the C6C7 disc plane orientation yielded 95% confidence intervals lesser than 0.1 N for all three force components projected from the body segment weight.

As results regarding mediolateral and torsion moments were less than 0.5 Nm and that the associated analyses showed no significant difference between groups, the following results will focus on the other components.

Comparing asymptomatic subjects

Average mass of the head-neck segment above C6C7 of healthy subjects was 5.0 (SD: 0.7) kg. It was correlated with subjects' height (R 2 = 0.59, p = 8.3 10 -8 ) and total body mass (R 2 = 0.57, p = 2.5 10 -21 ), but not age (Fig. 2a). Regarding C6C7 sagittal tilt angle, only a low correlation with subject's age was found (R 2 = 0.2, p = 5.7 10 -7 ) (Fig. 2b). G1 had a significantly lower sagittal tilt than G2 and G3 (p = 0.028 and 2.5 10 -6 respectively.) Average tilt angles were 19.5 • (SD: 7.7), 24.4 • (SD: 8.3) and 29.6 • (SD: 8.8) for G1, G2 and G3 respectively. Posteroanterior and mediolateral moment arms were not correlated to age, total mass or height, and were on average of 23 (SD: 14) mm and 2.7 (SD: 8) mm respectively.

All components of the loading at C6C7 were normally distributed for the control G1 and G3 groups. For the control G2 group, only the compression and mediolateral moment components were not normally distributed (p = 0.03 and 0.04 respectively). Variance analysis (Fig. 3) showed that control G1 had significantly lower posteroanterior shear than control G2 and G3 (p = 0.034 and 7.32•10 -4 respectively) as well as significantly higher compression than control G3 (p = 0.006).

The mean and SD values of each component at each IVD, which were instrumental in defining normality corridors, are available in Table 2.

Comparing control, pre and post-surgery

Statistical analysis at C6C7 for subjects between 40 and 60 y.o

Components of C6C7 loading for control-G2 were normally distributed except in compression (p = 0.031), mediolateral moment (p = 0.039) and in torsion (p = 0.35). For preoperative-G2, only the flexionextension moment was not (p = 0.067). Finally, for postoperative-G2 only the compression (p = 0.010) and torsion (p = 0.0031) components were not normally distributed.

Group comparisons (Fig. 4) found mediolateral shear was significantly closer to zero in the preoperative than in the control group (p = 0.026). This observation was confirmed by a significantly higher frontal tilt angle for control than for preop subjects (p = 0.022). Furthermore, the paired analysis highlighted a significantly lower posteroanterior shear (p = 1.31• •10 -4 ) before surgery than after. Similarly, preoperative sagittal tilt angle was found significantly lower than post-surgery (p = 4.53•10 -5 ).

Corridor analysis

Table 3 shows the evolution of the number of clinical subjects with weight components within the normality and subnormality corridors in pre and postoperative at C6C7 and at the level upper-adjacent to the surgery.

At C6C7, except in mediolateral shear, consistently more subjects were considered normal after surgery. Similarly, regarding individual patients changing corridors, up to three more of them got closer to normality than those that got further from it for posteroanterior shear, compression and flexion-moment.

At the upper level adjacent to the operation, a few more subjects got further from normality regarding effort components. Regarding the flexion-extension moment, 14 subjects (30% of the population) changed corridors towards normality, over three times as many as those that got further away from it. Anteroposterior moment arm of that level followed the same trend, with 13 subjects getting closer to normality.

Discussion

This present work aimed at developing a method to compute personalised mechanical loading of the cervical spine from medical images. From biplanar radiographs, the mass of the head-neck segment, moment arms and gravitational loading in standing position were quantified at multiple IVD levels for both asymptomatic and clinical subjects pre and postoperative. The method used was fast, only required medical images and reconstruction time could be cut down further using automation [START_REF] Gajny | Quasi-automatic 3D reconstruction of the full spine from low-dose biplanar X-rays based on statistical inferences and image analysis[END_REF]. The robustness of the method was verified with respect to the uncertainties associated with spine reconstruction.

While the presented approach yields subject-specific gravitational loadings of the cervical spine in neutral standing position, as it does not account for muscular response, it provides an incomplete spinal loading. For instance, though the external compression from this barycentremetric approach may correspond to 75% of the actual intervertebral joint compression assessed with a proprioception-based muscle model ( [START_REF] Van Den Abbeele | A subject-specific biomechanical control model for the prediction of cervical spine muscle forces[END_REF], it appears to represent less than 50% of the joint compression from other models [START_REF] Barrett | Cervical spine joint loading with neck flexion[END_REF][START_REF] Bayoglu | Twente spine model: a thorough investigation of the spinal loads in a complete and coherent musculoskeletal model of the human spine[END_REF][START_REF] Moroney | Analysis and measurement of neck loads[END_REF]. As such, the resulting loading from the present method should be used as input to employ muscle force quantification models.

Asymptomatic subjects

The mass of the head-neck segment of asymptomatic subjects above the C6C7 disc level estimated in this study are consistent with the values reported in the literature [START_REF] Clauser | Weight, Volume, and Center of Mass of Segments of the Human Body[END_REF][START_REF] Dempster | Properties of body segments based on size and weight[END_REF][START_REF] Yoganandan | Physical properties of the human head: mass, center of gravity and moment of inertia[END_REF]. Its correlation with subject weight and height could be explained by the slight increase in segment volume related to these two variables.

Correlation and statistical analyses showed a significant increase of the tilt angle of C6C7, i.e. the IVD becomes more vertical, with age. This observation is consistent with studies on the effect of age on cervical alignment in asymptomatic subjects, reporting changes in lordosis [START_REF] Gore | Roentgenographic findings of the cervical spine in asymptomatic people[END_REF][START_REF] Klinich | Cervical spine geometry in the automotive seated posture: Variations with age, stature, and gender[END_REF][START_REF] Liu | The effects of aging on the profile of the cervical spine[END_REF] and an increase in thoracic kyphosis [START_REF] Boyle | Influence of age on cervicothoracic spinal curvature: an ex vivo radiographic survey[END_REF]. Cervical spine tissue degeneration is also very common, with up to 81% of healthy subjects showing signs

Table 3

Evolution of the gravitational load components following surgery with regards to the corridors from control subjects at C6C7 and at the level upper-adjacent to the operated IVD.

+ = 8 N -= 9 N + = 5 N -= 7 N + = 5 N -= 6 N + = 14 N -= 4
N norm , N sub and N a correspond to numbers of subjects that are considered normal, subnormal or abnormal respectively. N + (respectively N -) correspond to number of subjects changing corridor by getting closer (respectively further) to the mean value of the control group.

of it within 10 years [START_REF] Okada | Aging of the cervical spine in healthy volunteers: a 10-year longitudinal magnetic resonance imaging study[END_REF] without experiencing symptoms, and has been associated with postural changes [START_REF] Gore | Roentgenographic findings of the cervical spine in asymptomatic people[END_REF].

No such observation could be made regarding the moment arms. The present study quantifies the influence of this change in alignment on the gravitational loading of C6C7 and shows a significant increase in posterior-anterior shear and a significant decrease in compression between G1 and G3. This observation reflects C6C7 becoming more vertical with age, which tends to align the local posterior-anterior direction of the IVD with that of gravity. Though the increase in shear may be rather small -from 16.2 (SD 6.8) N to 23.4 (SD 7.1) Nit may indicate a less economical posture for aging subjects, requiring a higher muscular response and making them more susceptible to fatigue.

Control and clinical subjects

Statistical comparisons performed for subjects between 40 and 60 y. o. mainly revealed a significant difference between asymptomatic and preoperative subjects in mediolateral shear stress at C6C7. This difference was not apparent for postoperative patients, which was consistent with the frontal tilt of the disc getting closer to control values after surgery. The paired analysis showed the C6C7 became significantly more vertical, thus significantly increasing the posterior-anterior shear component between pre and postoperative. Studying the evolution of patients with regards to the normality corridors from the control group mainly showed improvement for the flexion-extension moment at the IVD upper-adjacent to the operated level, stemming from a change in gravitational moment arm.

However, these observations are limited by the absence of evaluation of the evolution of pain for the patients, as well as by the very short postoperative follow-up. Studies with much longer postoperative followups have demonstrated changes in cervical alignment in both arthrodesis [START_REF] Hyun | Clinical impact of T1 slope minus cervical lordosis after multilevel posterior cervical fusion surgery[END_REF][START_REF] Lan | Relationship between T1 slope minus C2-7 lordosis and cervical alignment parameters after adjacent 2-level anterior cervical diskectomy and fusion of lower cervical spine[END_REF] and arthroplasty [START_REF] Guérin | Sagittal alignment after single cervical disc arthroplasty[END_REF][START_REF] Kim | Effects of a cervical disc prosthesis on maintaining sagittal alignment of the functional spinal unit and overall sagittal balance of the cervical spine[END_REF][START_REF] Wang | Cervical sagittal alignment after prestige LP cervical disc replacement: radiological results and clinical impacts from a single-center experience[END_REF]. Some of these changes have been identified as risk factors for complications such as ASD [START_REF] Katsuura | Kyphotic malalignment after anterior cervical fusion is one of the factors promoting the degenerative process in adjacent intervertebral levels[END_REF][START_REF] Xiong | 0.5-to 1-fold intervertebral distraction is a protective factor for adjacent segment degeneration in single-level anterior cervical discectomy and fusion[END_REF] and pseudarthrosis [START_REF] Choi | Preoperative radiographic parameters to predict a higher pseudarthrosis rate after anterior cervical discectomy and fusion[END_REF].

Another limitation of this study is the small number of patients. Although there were 33 patients treated with arthrodesis, only a dozen underwent an arthroplasty. Further longitudinal studies performed on a larger population could investigate the gravitational loads not only at the upper adjacent level but also at the operated and lower adjacent level, both for non fusion and for fusion techniques. This would increase our understanding on how gravitational forces analysis, in relation with subject specific posture, can impact the clinical outcome.

Conclusion

In this article was described a method to obtain personalised external loading of the cervical spine in neutral standing position and its decomposition in local IVD frames using biplanar X-rays exclusively. Comparisons between populations yielded significantly different loadings among asymptomatic subjects as well as pre and postoperative patients in posterior-anterior shear and compression. The present method should be easier to implement in a clinical setting than exertion trials and provide necessary inputs to muscle force quantification models and actual joint load assessment for patient evaluation.

Declaration of Competing Interest

None declared.

Fig. 1 .

 1 Fig. 1. (a) 3D reconstruction of the spine and body envelope from stereoradiographs (b) Sagittal illustration of the decomposition of the gravity force in the local C6C7 IVD frame.

Fig. 2 .

 2 Fig. 2. Correlation study of the head-neck mass (a) and the C6C7 tilt angle (b) with subjects' age, height and total body mass. Tilt angle is measured positive clockwise to horizontal, illustrated as θ C6C7 in Fig. 1b.

Fig. 3 .

 3 Fig. 3. Distribution of the components of the gravitational loading at C6C7 for asymptomatic subjects. Shear and compression components are on the left-hand side, flexion-extension moment component on the right-hand side. Horizontal bars indicate significant differences.

Fig. 4 .

 4 Fig. 4. Distribution of the components of the gravitational loading at C6C7 for asymptomatic, pre and postoperative subjects. Shear and compression components are on the left-hand side, flexion-extension moment component on the right-hand side. Black and blue horizontal bars indicate significant differences from variance and paired analysis respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 1

 1 Control and clinical populations.

	Level	Age group	Posteroanterior shear	Mediolateral shear	Compression	Flexion-Extension moment
			Fx (N)	Fy (N)	Fz (N)	My (Nm)
		< 40 y.o.	18,5 (5,3)	-1,5 (3,0)	-37,9 (4,3)	0,67 (0,46)
	C3C4	40-60 y.o.	18,0 (4,3)	-1,1 (3,4)	-39,5 (7,6)	0,47 (0,39)
		> 60 y.o.	19,2 (6,5)	-2,1 (3,7)	-37,1 (4,9)	0,66 (0,56)
		< 40 y.o.	18,1 (5,5)	-1,8 (3,5)	-40,4 (5,4)	0,84 (0,51)
	C4C5	40-60 y.o.	18,5 (7,1)	-1,6 (3,9)	-41,0 (11)	0,61 (0,49)
		> 60 y.o.	20,0 (6,8)	-2,5 (3,5)	-38,8 (4,8)	0,83 (0,66)
		< 40 y.o.	17,2 (6,1)*	-2,1 (3,6)	¡43,0 (4,7)*	0,97 (0,58)
	C5C6	40-60 y.o.	20,0 (7,5)	-1,7 (4,2)	-42,6 (9,2)	0,78 (0,54)
		> 60 y.o. < 40 y.o.	22,8 (7,3)* 16,2 (6,8)* , **	-3,0 (3,4) -2,4 (3,6)	¡39,1 (5,4)* ¡45,5 (4,7)*	1,0 (0,74) 1,1 (0,65)
	C6C7	40-60 y.o.	20,5 (8,3)**	-2,3 (4,2)	-44,8 (9)	1,0 (0,64)
		> 60 y.o.	23,4 (7,1)*	-3,0 (3,2)	-41,2 (5,7)*	1,3 (0,84)

Numbers in bold indicate the significantly different groups, with associated p values: *0.02 ≤ p ≤ 0.05 **p ≤ 0.005.

Table 2

 2 Mean (SD) values of each component of the gravitational load for asymptomatic subjects of different age groups at each IVD level.

		Control	Clinical
	Age group	N subjects	N subjects	Surgery	Surgery level
					C3C4	C4C5	C5C6	C6C7
	G1 < 40 y.o.	65	5	1 ACDF 4 TDR			1 2	2
	G2 40-60 y.	27	33	24 ACDF	1	3	13	7
	o.			9 TDR			4	5
	G3 > 60 y.o.	26	8	8 ACDF 0 TDR			6	2
	ACDF: Anterior Cervical Discectomy and Fusion; TDR: Total Disc Replacement.

Acknowledgements

The authors would like to express their gratitude to the BiomecAM chair program of Fondation ParisTech, financed by Société Générale, Covea, Proteor and Fondation Cotrel, who supported and funded this study. They would also like to thank the staff of Hôpital Pellegrin (Bordeaux) and Hôpital privé Francheville (Périgueux) for their contribution to clinical data acquisition.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi. org/10.1016/j.clinbiomech.2022.105577.