MD Pedro Augusto

Gondim Teixeira

MD Anne-Sophie Formery

Gwenaël Balazuc

Guillaume Lux

Isabelle Loiret

Gabriela Hossu

Alain Blum

Balazuc Gwenaël

PhD Md

Gabriela Phd

Comparison between subtalar joint quantitative kinematic 4-D CT parameters in healthy volunteers and patients with joint stiffness or chronic ankle instability: A preliminary study

Keywords: Computed tomography, subtalar joint, dynamic, chronic ankle instability, joint stiffness

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

Introduction

Traumatic injury is a frequent cause of subtalar joint pathology. Complex and displaced calcaneal fractures can lead to subtalar joint stiffness, pain and disability related to secondary subtalar degenerative joint disease despite appropriate surgical fracture reduction [START_REF] Romeo | Recreational Sports Activities After Calcaneal Fractures and Subsequent Subtalar Joint Arthrodesis[END_REF][START_REF] Holm | Primary subtalar joint arthrodesis for comminuted fractures of the calcaneus[END_REF]. Subtalar joint arthrodesis is indicated in cases of persistent pain upon weight bearing, walking or performing physical activity after calcaneal fractures [START_REF] Romeo | Recreational Sports Activities After Calcaneal Fractures and Subsequent Subtalar Joint Arthrodesis[END_REF]. There is little information available on the imaging evaluation of the degree of joint amplitude loss and on the dynamic evaluation of arthrodesis effectiveness. Subtalar joint injury is relatively frequent after ankle trauma (25% of ankle sprains) [START_REF] Weindel | Subtalar instability: a biomechanical cadaver study[END_REF].

Repeated lateral ankle sprains can lead in about a third of cases to persistent pain and functional deficits, a condition termed chronic ankle instability [START_REF] Mittlmeier | Subtalar joint instability[END_REF]. Chronic ankle instability is most frequently associated with injuries to the calcaneofibular ligament (ankle), extensor retinaculum and to the cervical ligament (subtalar joint) [START_REF] Hentges | Chronic ankle and subtalar joint instability in the athlete[END_REF]. The identification and assessment of the subtalar component of chronic ankle instability is of relevance as patient with suspected subtalar joint injury require specific treatment (most frequently physiotherapy, muscle strengthening and immobilization) [START_REF] Hentges | Chronic ankle and subtalar joint instability in the athlete[END_REF]. Although clinical stress testing (internal rotation and varus with the ankle fixed in extension) and visual weight bearing evaluation of the hind foot can suggest subtalar instability, the clinical assessment of the subtalar joint remains difficult because this joint has a complex anatomy and biomechanics [START_REF] Weindel | Subtalar instability: a biomechanical cadaver study[END_REF]. Additionally, there is no recognized way to objectively assess treatment response in patients with chronic ankle instability and in this context non-invasive quantitative analysis of subtalar joint motion could prove helpful.

Imaging methods allowing a dynamic evaluation of the subtalar joint amplitude during motion could improve diagnostic accuracy and provide objective criteria for patient followup. Dynamic 4-D CT is performed by acquiring various low-dose sequential volumes with high temporal resolution of a moving joint and has been used for the dynamic evaluation of various joints such as wrist, hip, elbow and shoulder girdle [START_REF] Leng | Dynamic CT technique for assessment of wrist joint instabilities[END_REF][START_REF] Edirisinghe | Dynamic motion analysis of dart throwers motion visualized through computerized tomography and calculation of the axis of rotation[END_REF][START_REF] Demehri | Evaluation of pisotriquetral motion pattern using four-dimensional CT: initial clinical experience in asymptomatic wrists[END_REF][START_REF] Wassilew | Real time visualization of femoroacetabular impingement and subluxation using 320-slice computed tomography[END_REF][START_REF] Goh | Using the 320-Multidetector Computed Tomography scanner for four-dimensional functional assessment of the elbow joint[END_REF][START_REF] Gondim Teixeira | Musculoskeletal wide detector CT: principles, techniques and applications in clinical practice and research[END_REF]. In a recent cadaveric and healthy volunteer study Gondim Teixeira et al. showed that quantitative dynamic 4-D CT could performed on the subtalar joint suggesting a potential role of this technique in assessing joint instability and stiffness [START_REF] Gondim Teixeira | Quantitative Analysis of Subtalar Joint Motion With 4D CT: Proof of Concept With Cadaveric and Healthy Subject Evaluation[END_REF]. Increases in joint amplitudes after section of subtalar ligaments in cadavers were reported, confirming the potential of this technique in clinical practice. Recently, the use of quantitative 4-D CT has been demonstrated in the wrist, providing quantitative criteria for the differentiation of patient with and without dissociative scapholunate instability [START_REF] Abou Arab | Scapholunate instability: improved detection with semi-automated kinematic CT analysis during stress maneuvers[END_REF][START_REF] Rauch | Four-dimensional CT Analysis of Wrist Kinematics during Radioulnar Deviation[END_REF]. In a context that CT is frequently used for the evaluation of patients with hind foot trauma for fracture detection and consolidation assessment, 4-D CT could be useful to ascertain the biomechanical impact of the injuries sustained [START_REF] Berger | Application of Cinematic Rendering in Clinical Routine CT Examination of Ankle Sprains[END_REF][START_REF] Badillo | Multidetector CT evaluation of calcaneal fractures, Radiogr[END_REF][START_REF] Chen | Comparison of plain radiography and CT in postoperative evaluation of ankle fractures[END_REF].

We hypothesize that healthy volunteers, patients with clinical chronic ankle instability and subtalar joint stiffness can be differentiated based on quantitative assessment of subtalar joint amplitude with 4-D CT, which could provide additional diagnostic criteria for diagnosis and patient follow-up. Thus, four previously established 4-D CT derived parameters were compared in these three groups [START_REF] Gondim Teixeira | Quantitative Analysis of Subtalar Joint Motion With 4D CT: Proof of Concept With Cadaveric and Healthy Subject Evaluation[END_REF]. The value distribution of two angles, axial talocalcaneal (ATCA) used to assess subtalar internal-external rotation and coronal talocalcaneal (CTCA) used to assess talar tilt angles as well as two distances, medial facet gap (MFG) used to assess subtalar eversion laxity and posterior calcaneal facet uncovering (PCFU) used to assess the congruence of the posterior subtalar joint congruence were evaluated during a subtalar joint prono-supination cycle [START_REF] Weindel | Subtalar instability: a biomechanical cadaver study[END_REF][START_REF] Seebauer | Ankle laxity: stress investigation under MRI control[END_REF][START_REF] Kato | The diagnosis and treatment of instability of the subtalar joint[END_REF].

Material and methods

Study population

From November 2011 to December 2012, 38 consecutive patients were prospectively included in this ethics committee approved study. All patients were over 18 years-old, signed informed consent and underwent kinematic 4-D CT of the subtalar joint. All patients had been referred by orthopedic surgeons and rheumatologists for the evaluation of ankle pain associated with either subtalar joint stiffness or chronic ankle instability. Patients with metallic implants, inflammatory arthritis and congenital subtalar joint conditions were not included in this study. In addition, 17 healthy volunteers with no previous ankle trauma and a normal physical examination of the ankle who underwent kinematic 4-D CT of the subtalar joint were included. Physical examination was performed in all volunteers prior to CT acquisitions by a senior radiologist with at least five years of clinical experience. The evaluation of joint hyperlaxity was based on the Beighton score. One volunteer, despite an initial negative clinical history, had periosteal irregularities in the fibular insertion of the anterior talofibular ligament, which could be related to anterior ankle trauma. Two volunteers failed to perform a homogeneous prono-supination motion during the CT scan and were excluded. Two volunteers who presented clinical signs of hyperlaxity with a Beighton score higher than two were also excluded. Since one of the volunteers underwent bilateral 4-D kinematic CT, 13 ankles composed the healthy volunteer group.

Eighteen patients with signs of stiffness were also included. Patients with post-traumatic clinical limitation of the subtalar joint amplitude of motion in plantarflexion, dorsiflexion and inversion compared to the uninjured contralateral side were considered positive for subtalar joint stiffness.

Twenty patients with signs or clinical history of ankle trauma with persistent pain (at least one month) were evaluated. Chronic ankle instability diagnosis was confirmed when these patients had at least one injured ankle ligament on MRI or CT arthrography. Ligament sprains on MRI (T2 signal hyperintensity and thickening without clear loss in continuity), partial tears (partial loss in ligament continuity) and full tears (complete loss in ligament fiber continuity) on MRI or CT arthrography were evaluated. The lateral and medial collateral ligament complexes, as well as the cervical and interosseous subtalar ligaments were considered in the analysis. Among these patients, 10 were excluded due to unclear ankle trauma history, previous surgery, no ligament anomalies found on imaging, no imaging available and failure to perform a homogeneous prono-supination motion during the CT scan (one patient). Thus the chronic ankle instability group was composed of 10 ankles. Table 1 shows the clinical characteristics and imaging findings these patients.

Image acquisition

Acquisitions were performed using a wide detector CT scanner with 320-detector rows (Aquilion ONE, Canon Medical Systems, Otawara, Japan) using intermittent sequential mode (no table feed), 24 cm FOV, 512x512 matrix, 0.5 mm slice thickness, tube rotation 0.35 seconds. Tube output was 80 kVp and 65 mAs (22 mA effective). Z-axis coverage varied from 8-12 cm depending on the patient morphology. Images were reconstructed with half reconstruction using bone and soft-tissue kernels. Adaptive iterative reconstruction (AIDR 3D, Canon Medical Systems, Otawara, Japan) in the standard level was used for all acquisitions.

The images were acquired during a prono-supination cycle of the subtalar joint. All acquisitions were unilateral. The patient was placed supine on the scanner table. The examination began in the maximum pronation position and ended when the foot returned to the starting point after supination, lasting 10-15 seconds. An elastic band was used to guide the movement (Fig. 1). A training session was conducted before CT acquisitions to ensure a controlled and homogeneous movement of patients. The volume CT dose index (CTDI vol ) and the dose length product (DLP) of patient acquisitions were estimated based on a phantom 16 cm in diameter. Effective doses were calculated using a conversion coefficient of 0.0002 [START_REF] Saltybaeva | Estimates of effective dose for CT scans of the lower extremities[END_REF].

Image analysis and post processing

All measurements in pathologic patients were performed independently by two readers with four (A-S.F.) and two (G.B.) years of experience in musculoskeletal radiology. Reference measures in normal volunteers were obtained by a third reader (P.T.) with 12 years of experience in musculoskeletal radiology. This reader also evaluated the MRI and CT arthrography studies in patients with suspected chronic ankle instability performed with a conventional acquisition protocol. Images were post-processed in a display console workstation V7.0 (Canon Medical Systems, Otawara, Japan) using the 4-D Ortho clinical application tool. Image post-processing was as follows. All the volumes of a kinematic acquisition were loaded and multiplanar and 3-D volume rendered images were displayed.

The examiner then placed a seed point on the medullary cavity of the talus and the registration algorithm was executed. After registration, all motion was displayed with respect to the talus, which remained static, serving as a reference for plane selection. The axial plane was defined by the long axis of the talus, coronal plane was defined by the bimalleolar axis and the sagittal plane was defined by the long axis of the tibia. Then, the examiner placed two reference markers for distance measurements and four for angular measurements over the cortical bone on the multiplanar reformats in a single acquisition volume. These markers were plotted automatically by the software to all acquisition volumes. Four different measurements were performed to describe the talocalcaneal relationship during prono-supination according to the previous work of Gondim Teixeira et al. [START_REF] Gondim Teixeira | Quantitative Analysis of Subtalar Joint Motion With 4D CT: Proof of Concept With Cadaveric and Healthy Subject Evaluation[END_REF] (Fig. 2):

-The coronal talocalcaneal angle (CTCA). The reference markers were place in a bi-malleolar coronal plane with the ankle in neutral position. A line was constructed by placing two markers at the medial and lateral extremes of the talar dome. A second line was constructed in the calcaneus by placing one marker at the most lateral point of the sustentaculum tali and another on deepest portion of Gissagne's angle. The angle formed between these lines was measured [START_REF] Seebauer | Ankle laxity: stress investigation under MRI control[END_REF][START_REF] Louwerens | Stress radiography of the talocrural and subtalar joints[END_REF][START_REF] Cordova | Mechanical joint laxity associated with chronic ankle instability: a systematic review[END_REF].

-The axial talocalcaneal angle (ATCA). The reference markers were placed in the axial plane passing through the long axis of the talus. At the level of the talus, the slice depicting the summit of the talar head and on the postero-medial process of the talus (Stieda process) was selected. A line was constructed by placing one marker on each of these two points. At the level of the calcaneus, two markers were placed in a slice depicting the middle of the calcaneocuboid articular facet and another on midpoint of the posterior calcaneal surface. A line between two markers was constructed. The angle formed between these lines was measured [START_REF] Weindel | Subtalar instability: a biomechanical cadaver study[END_REF].

-Medial facet gap (MFG). The reference markers were place with the ankle in neutral position. The slice depicting the most lateral point on the medial facet of the talus without partial volume effect was selected. In this slice, markers were placed on the medial facets of the talus and the calcaneus so that the shortest distance between these two bones was measured [START_REF] Cordova | Mechanical joint laxity associated with chronic ankle instability: a systematic review[END_REF].

-Posterior calcaneal facet uncovering (PCFU). This measurement was performed in a sagittal plane with the ankle in neutral position was selected. The reference markers were placed using the volume depicting full ankle supination. The markers were placed at the extremes of the largest distance between the center of Gissagne's angle (identified based on bone contours and the location of the nutrient foramina) and the anterior border of the posterior facet of the talus [START_REF] Kato | The diagnosis and treatment of instability of the subtalar joint[END_REF].

Once the land marks for marker placement were established, the post-processing time per data set including data loading, bone locking and performing the four measurements described lasted roughly 15 minutes.

Statistics

Statistical analysis was carried out using the R software (Version 3.0.0, The R Foundation for Statistical Computing, Vienna, Austria) with the pROC and IRR packages [START_REF] Robin | pROC: an open-source package for R and S+ to analyze and compare ROC curves[END_REF]. The changes in ratio and absolute values of the measures acquired were compared. Ratios were calculated by dividing the values obtained by the maximum pronation value. Ratios were calculated because the direct comparison of absolute values is imprecise due to technical factors, such as minor variations in reference position and differences in bone size between subjects. For each subject, four measurements were performed two angles (CTCA and ATCA) and two distances (MFG and PCFU). For each measurement a series of ratios was available (e.g. one value for each phase of the dynamic acquisition). For each measurement series acquired, the mean, standard deviation (SD), coefficient of variation (CV), maximal values, minimal values and amplitude variation were analyzed to assess subtalar joint motion during pronosupination. Intraclass correlation coefficients for the ratios of each of the parameters evaluated were calculated to assess measurement inter-observer variability. ICC values of 0-0.20 were considered slight agreement, 0.21-0.40 fair, 0.41-0.60 moderate, 0.61-0.80 substantial, and 0.81-1 was accepted as excellent agreement. The results were presented on average ± standard deviation. The Wilcoxon test was used to assess the statistical significance of the parameters evaluated between the subgroups evaluated (healthy volunteers versus patients with subtalar joint stiffness and healthy volunteers versus patient with chronic ankle instability). A Bonferroni correction was applied to account for multiple comparisons and a P value < 0.003 was considered as the threshold for statistical significance. ROC analysis was performed for all parameters (mean, SD, CV and amplitude) that showed a statistically significant differences between volunteers/patients with joint stiffness and volunteers/patients with joint instability.

Results

There were four men and eight women in the healthy volunteer group (Male/female ratio = 0.5) with a mean age of 32.7 ± 7.2 years, ranging from 24 to 50 years. Mean values of CTCA, ATCA, MFG and PCFU in healthy volunteers were 23.6 ± 8.3°, 14.8 ± 6.6°, 7.1 ± 4.4 mm and 16.3 ± 1.8 mm respectively. Among healthy volunteers CV values were higher in angular than in distance parameters (36% and 45.8% for CTCA and ATCA respectively compared to 20% and 11.4% for MFG and PCFU respectively) (Fig. 3).

The interobserver variability of the measurements performed in the patients evaluated was considered excellent with ICC values of 0.92 in patients with chronic ankle instability. In this subgroup angular measurement presented excellent while distance measurement presented substantial interobserver variabilities (ICC = 0.94 for ATCA/CTCA and 0.72 and 0.75 for MFG and PCFU respectively). The interobserver variability was also considered excellent with ICC values of 0.81 in patients with joint stiffness. In this subgroup CTCA and PCFU presented an excellent interobserver variability (0.93 and 0.9 respectively) while it was substantial with ATCA (0.79) and moderate with MFG (0.44). The mean CTDI volume, DLP and effective dose of the studies performed were 20.6 ± 5.6 mGy, 168.1 ± 43.7 mGy.cm and 0.067 ± 0.01 mSv, respectively.

Patients with subtalar joint stiffness

There were eight men and ten women among patients with subtalar joint stiffness (Male/female ratio = 0.8). The age in this group was higher than that of the healthy volunteers with a mean age of 52.8 ± 11.8 years ranging from 26 to 71 years. Mean values of all parameters evaluated in this subgroup and the P values with respect to healthy volunteer data are presented in table 2. Compared to normal volunteers there was a general tendency towards reduction in the parameters evaluated in patients with subtalar joint stiffness. With both angles and MFG, amplitude and CV values had a statistically significant difference between volunteers and patients with subtalar joint stiffness for both readers (P <0.0011)(Fig. 4)(Supplementary material 1). If the CV and amplitude values of CTCA, ATCA and MFG were considered there was a 54-67% value reduction in patients with subtalar joint stiffness with respect to healthy volunteers. As expected, maximum values of were also lower in patients with joint stiffness compared to healthy volunteers, with the exception of MFG values for reader 1. However, these differences were only significant with PCFU.

After ROC analysis, for both readers, PCFU yielded the best sensitivity for subtalar joint stiffness identification. Sensitivity was 92% with PCFU maximal values (cut off 17mm) and CV (6% cut off) and of a 100% using PCFU amplitude (1.6 mm). Both angular measurements (CTCA and ATCA) yielded the best specificity values for both readers varying from 74-94%.

CTCA CV yielded the highest sensitivity for both readers (85-94% with a 28% cut off). The diagnostic performance of maximum values, CV and variation amplitude for all angles and distances measured is presented on Table 3.

Patients with Chronic ankle instability

There were four men and six women among patients with chronic ankle instability (Male/female ratio = 0.66) with a similar age then healthy volunteers mean age of 32.6 ± 9 years ranging from 21 to 47 years. There are no records of surgical treatment following imaging for these patients in our database. Mean values of all parameters evaluated in this subgroup and the P values relative to healthy volunteer data are presented in table 4. The only measurement that yielded statistically significant differences with respect to the healthy volunteer group for both readers was PCFU maximum values. There was a 25%-44% decrease in maximal values PCFU for readers 1 and 2 respectively (P varied from < 0.001) in patients with chronic ankle instability. The PCFU CV values increased in patients with chronic ankle instability compared to healthy volunteers for both readers (29%-48% increase for readers 1 and 2 respectively)(Fig. 5)(Supplementary material 2). These differences, however, were not statistically significant (P = 0.31 and 0.006 for readers 1 and 2 respectively). With ROC analysis performed with PCFU derived parameters, maximal values yielded the best sensitivity 92-95% and the best specificities 80-92% (cut off 17mm) for the identification of chronic ankle instability.

For reader one, mean and maximum MFG were higher (23.4% and 16.6% respectively) in patients with chronic ankle instability compared to healthy volunteers. Reader two data showed a slight decrease in mean and maximum MFG values in chronic ankle instability (-5.3% and -5.7%). These differences, however, were not statistically significant (P > 0.016).

Discussion

The differentiation between healthy volunteers and patients with subtalar joint stiffness and chronic ankle instability was possible with kinematic 4-D CT. There were significant changes in all parameters evaluated (ATCA, CTCA, MFG and PCFU) between volunteers and patients with joint stiffness (P varying from <0.0001 to 0.0015). In patients with chronic ankle instability PCFU values were significantly different than those of healthy volunteers (P values varying from <0.0001 to 0.001). Measurement ratio interobserver reproducibility in pathologic patients was considered to be excellent (ICC > 0.81) which is a considerable advantage of this semi-automatic method compared to manual measurements [START_REF] Forsberg | Quantitative analysis of the patellofemoral motion pattern using semi-automatic processing of 4D CT data[END_REF]. The interobserver reproducibility of each individual measurement was considered substantial to excellent (ICC varying from 0.72 to 0.94) with the exception of MFG in patients with joint stiffness which was considered moderate (ICC = 0.4). This finding could be related to difficulties in marker placement in patients with calcaneal morphologic changes secondary to fractures. Moreover, the effective dose was lower than < 0.1 mSv. These results indicate that kinematic 4-D CT can have a potential role in the evaluation of subtalar joint pathology providing means for an objective joint amplitude assessment which could improve the accuracy in identifying subtalar involvement in patients with chronic ankle instability and a more precise assessment of subtalar joint stiffness.

In patients with subtalar joint stiffness, there was a significant reduction in the variation amplitude of all measurements performed (p < 0.02). CTCA, ATCA and MFG were the best measurements for the identification of subtalar joint stiffness with a reduction of more than 50% of the CV and variation amplitude with respect to healthy volunteers. The performance of kinematic 4-D CT for the identification of patients with clinical joint stiffness was also good. PCFU related parameters yielded the best sensitivities (92-100%) and the best specificity was reached when CV and variation amplitude of angular measurements (CTCA and ATCA) were assessed (72-94%). Thus, Kinematic 4-D CT could be an interesting option to objectively confirm and grade subtalar joint stiffness. This method could also be interesting for the evaluation of the effectiveness of orthopedic constraint devices and surgical arthrodesis on subtalar joint motion by detecting and quantifying residual micro motion.

Although PCFU analysis yielded statistically significant results for the differentiation between healthy volunteers and patients with chronic ankle instability (p < 0.001) with high sensitivity and specificity values (92% and 95% respectively). This data should be interpreted with caution. There was a reduction in mean and maximal PCFU in this patient group compared to healthy volunteers, which could seem paradoxical. Patients with chronic ankle instability usually present with pain and proprioception issues with apprehension during subtalar joint motion [START_REF] Pisani | Chronic laxity of the subtalar joint[END_REF][START_REF] Vaseenon | Comparison of two manual tests for ankle laxity due to rupture of the lateral ankle ligaments[END_REF]. This could explain at least partially the reduction in PCFU values compared to normal volunteers. Although statistically significant for reader 2 only (P = 0.006), both readers identified an increase in the CV of PCFU in patients with instability compared to healthy volunteers (24%-48%). These findings should be confirmed by larger studies but may suggest a change in the respective motion between the talus and calcaneus in patients with joint instability even if maximal joint amplitudes remain normal or decreased, a phenomenon similar to carpal bone hysteresis [START_REF] Berdia | The hysteresis effect in carpal kinematics[END_REF].

In a previous cadaveric study on subtalar joint 4-D CT it was suggested that MFG assessment would be better suitable for the evaluation of subtalar joint instability as this parameter presented the highest increase before and after resection of subtalar joint ligaments (cervical and interosseous) [START_REF] Gondim Teixeira | Quantitative Analysis of Subtalar Joint Motion With 4D CT: Proof of Concept With Cadaveric and Healthy Subject Evaluation[END_REF]. In the present study, the mean and maximum MFG were higher in patients with chronic ankle instability compared to normal volunteers only for one of the readers and these differences were not statistically significant (P > 0.015). This could be at least partially explained by the fact that the injuries sustained by the evaluated patients were likely less severe than the full tearing of the cervical and interosseous ligaments created in the prior cadaveric study. Additionally, the small number of patients in the chronic instability group could have also contributed to this discrepancy.

Various limitations of this study must be acknowledged. The subtalar joint has not been evaluated under weight-bearing conditions. This could lead to an underestimation of the diagnostic performance of kinematic 4-D CT for the identification of subtalar joint pathology, especially in patients with chronic ankle instability. A device allowing kinematic ankle imaging under stress is currently being developed. Such a device is could increase the diagnostic performance of this method [START_REF] Gondim Teixeira | State-of-the-Art Subtalar Joint Kinematic and Stress Imaging with Emphasis on 4-D CT: Where Do We Stand and Where Are We Going[END_REF]. Another limitation is the size of the study population in general but particularly in the chronic ankle instability group. As the nonsurgical confirmation of chronic ankle instability is challenging, we chose to use strict clinical and imaging criteria, which led to a 50% reduction of the initial patient population. Thus, studies with larger patient groups are needed to confirm the presented findings. Some patients did not succeed in performing adequate subtalar joint prono-supination which may suggest the need for a more aggressive guidance of patient motion. The ligament lesions visualized on MRI and CT arthrography were not surgically confirmed as the patients included were not managed surgically. This evaluation, however, was performed by an experienced musculoskeletal radiologist based on previously described imaging criteria.

Further studies are also necessary to compare the diagnostic performance of stress MRI and kinematic 4-D CT.

In conclusion, Kinematic 4-D CT is an objective and reproducible tool for the evaluation of in vivo subtalar joint motion. This method could provide objective criteria subtalar joint assessment with a potential impact on patient management as there were significant changes in the quantitative parameters evaluated in patients with subtalar joint stiffness and patients with chronic ankle instability with respect to healthy volunteers with high sensitivities and specificities. Although further studies in larger patient populations and with weight bearing simulation are needed, particularly in patients with chronic ankle instability, kinematic 4-D CT shows potential for the objective evaluation of patients with subtalar joint pathology.

Tables

Fig. 1 -Fig. 2 -

 12 Fig. 1 -Series of photographs demonstrating patient positioning and motion in a prono-

Fig. 3 -

 3 Fig.3-Graphic demonstrating the ratio variation of the coronal talocalcaneal angle (CTCA),

Fig. 4 -

 4 Fig.4-66 year-old male with a history of proximal tibial fracture treated surgically

Fig. 5 -

 5 Fig. 5 -25 year-old female with a history of multiple ankle sprains and persistent ankle pain.

Table 1 -

 1 Clinical and demographic characteristics of patients in the chronic ankle instability group.

	Patients Sex Age	Clinical findings	Imaging findings
	1	F	42	Ankle sprain 7 months prior,	ATFL and CFL sprain on MRI
				instability sensation and pain
	2	F	21	Prior history of severe ankle trauma,	ATFL full tear and DL sprain on MRI
				persistant pain
	3	F	47	Severe ankle trauma 2 year prior,	ATFL full tear on CT arthrography
				persistant pain
	4	M	33	Multiple ankle sprains, persistant	ATFL full tear on MRI
				pain
	5	M	34	Multiple ankle sprains, recent acute	ATFL full tear, CFL sprain, DL partial tear,
				sprain	cervical ligament sprain
	6	M	21	Multiple prior sprains, persistant	ATFL partial tear, Anterior tibiofibular
				pain	ligament sprain on MRI
	7	F	42	Ankle sprain 6 months prior,	ATFL full tear, CFL and DL sprain on MRI
				persistant pain
	8	M	28	Ankle sprain 6 months prior,	ATFL full tear, DL partial tear, posterior
				instability sensation and pain	tibiofibular ligament sprain on CT
					arthrography
	9	F	25	Multiple sprains, persistant pain	ATFL full tear on MRI
	10	F	31	Multiple prior ankle sprains, recent	ATFL and CFL full tear on MRI
				acute sprain
	ATFL = Anterior talofibular ligament
	CFL = Calcaneofibular ligament
	DL = Deltoid ligament


Anne Cordier

Hana El Khal

Elisabeth Siebert

Marlu César

Marlu C Steil

H El Khal

On the Role of the Pore Morphology on the Electrical Conductivity of Porous Yttria-Stabilized Zirconia

Keywords: porous Zirconia, ionic conductivity, Archie's law, tortuosity

Yttria Stabilized Zirconia ceramics with well-controlled porosity, pore size and shape were prepared using well-calibrated poly-methyl-methacrylate (PMMA) micro-beads (MB) as a poreforming agent. The microstructure was observed by Scanning Electron Microscopy. Impedance spectroscopy was used to evaluate the effect of pore morphology (pore size, pore size distribution, pore shape and interconnectivity) on the electrical properties of YSZ ceramics. Archie's law based analyzes to express the dependence of conductivity on porosity have shown that Archie's law is independent of pore size for a pore diameter of between 1 and 7 μm. The Bruggeman model could be used to predict the bulk conductivity if the porosity was less than 25 %, thus showing that the impedance response included the effect of sinuousness and constriction induced by pores.

Therefore, the tortuosity factor calculated from the bulk conductivity was higher than that predicted by the Bruggeman model for porosities greater than 25 % and spherical pores wide (> 20 μm).

Another point relates to the comparison between tortuosity factors obtained for pore samples fabricated with pore-forming PMMA or by sub-sintering.

Introduction

The addition of porosity in a material results in an alteration of its properties, like thermal conductivity, mechanical or electrical properties. The porosity can act in different ways depending on the volume fraction of porosity, the pore size, the pore size distribution, the pore shape and interconnectivity. Consequently, deep studies using model materials have to be conducted to understand and predict the effects of the porosity on the final properties of the material. Different processing techniques [START_REF] Ishizaki | Porous Materials: Process Technology and Applications[END_REF] have been developed for the fabrication of porous ceramics. Among these techniques, the use of additional organic particles as a pore-forming agent makes it possible to produce samples with controlled porosity and pores of the same size, homogeneously distributed throughout the material.

Yttria Stabilized Zirconia (YSZ) has found a wide range of applications such as thermal insulators, biomaterials, membranes or filters, catalysts and as solid electrolyte of gas sensors, electrolysis cells or solid oxide fuel cells [START_REF] Verweij | Nanocrystalline and nanoporous ceramics[END_REF]. For these last applications, a good ionic conductivity at an intermediate temperature is necessary. However, it has demonstrated that the presence of porosity in YSZ decreases drastically its electrical conductivity [START_REF] Steil | Densification of yttria stabilized zirconia: impedance spectroscopy analysis[END_REF]. In most studies [START_REF] Steil | Densification of yttria stabilized zirconia: impedance spectroscopy analysis[END_REF][START_REF] Gibson | Sinterability of commercial 8 mol % yttriastabilized zirconia powders and the effect of sintered density on the ionic conductivity[END_REF][START_REF] Kleitz | Microstructure blocking effects versus effective medium theories in YSZ[END_REF][START_REF] Kleitz | Model for ion-blocking at internal interfaces in zirconias[END_REF][START_REF] Costa | Synthesis of lanthanum beat-alumina powders by the polymerci precursor technique[END_REF][START_REF] De Florio | Sintering of zirconia-yttria ceramics studied by impedance spectroscopy[END_REF][START_REF] Van Herle | Microstructure-conductivity correlation for hot-pressed 8YSZ powders[END_REF][START_REF] Pérez-Coll | Influence of porosity on the bulk and grainboundary electrical propeties of Gd-doped ceria[END_REF] on the role of porosity on the transport properties of solid electrolytes, porosity is a consequence of the sintering process, i.e. the porosity is conversely related to the densification process. As a result, pore size, pore size distribution and interconnectivity are not well controlled and may change from a work to another. Consequently, even if all studies agree with the fact that improving the densification of the sample decreases the resistance, it is very difficult to explain in which proportion each microstructure defect such as grain boundaries, insulating inclusions or cracks is responsible for the observed effect.

In this work, porous YSZ model samples were prepared using different pore volumes of wellcalibrated poly-methyl-methacrylate (PMMA) micro-beads (MB). MB were used as a poreforming agent to obtain porous YSZ samples with a small pore size distribution and a well-defined pore size. High purity YSZ powder with a good ability for sintering was used. The pellets were sintered at high temperature and during a long time in order to limit the influence of the grain boundaries, residual porosity or insulator inclusions. The prepared samples were characterized by impedance spectroscopy in order to clearly evidence the role of the pore size on the electrical properties. In a previous work [START_REF] Steil | Densification of yttria stabilized zirconia: impedance spectroscopy analysis[END_REF], we have used Archie's law [START_REF] Archie | The ellectrical resistivity Log as an aid in determining some reservoir characteristics[END_REF] to express the porosity dependence of the ionic conductivity of YSZ with varying degrees of densification. This empirical formula uses an exponent m, which reflects the connectivity between the conducting phase (bulk) and the non-conducting phase (porosity) (see Eq.( 3)).

First, the present study aimed to investigate the influence of the pore size and interconnectivity on the value of m. Then, the tortuosity factor (τ) [START_REF] Copper | Quantifying the transport properties of solid fuel cell electrodes[END_REF], which is a measure of the degree of transport reduction caused by the pore morphology, was determined as a function of the pore size and porosity. The suitability of this method was compared to the models quantifying the tortuosityporosity relationship. To the best of our knowledge, no study addresses the effect of the pore morphology and interconnectivity on the electrical properties of YSZ ceramics.

Theoretical section

The electrical conductivity of YSZ is commonly characterized by impedance spectroscopy [START_REF] Steil | Densification of yttria stabilized zirconia: impedance spectroscopy analysis[END_REF].

The obtained impedance diagrams are composed of two semi-circles in the Nyquist plot, in the frequency range of 10 7 to 10 Hz. The high frequency (HF) semi-circle is related to the intrinsic properties of the material, denoted bulk response, while the low frequency (LF) semi-circle corresponds to the microstructure defects (MD) properties. MD are supposed to be pores, grain boundaries, cracks or insulating inclusions [START_REF] Steil | Densification of yttria stabilized zirconia: impedance spectroscopy analysis[END_REF][START_REF] Kleitz | Model for ion-blocking at internal interfaces in zirconias[END_REF][START_REF] Badwal | Yttria-zirconia: effect of microstructure on conductivity[END_REF][START_REF] Kleitz | Dessemond in Science and Technology of Zirconia[END_REF]. The apparent conductivity of the porous samples is calculated according to the following relationship:

σ = (1)
where R is the resistance determined by the intersection of the semi-circle on the real axis of the Nyquist plot, l is the sample thickness and S its surface area. Regarding the Nyquist plot, it can be distinguished two apparent resistances, the bulk resistance RB, and the MD resistance RMD, deduced from the intersection of the HF and the LF semicircle on the abscise axis, respectively.

In solid ionic conductors, a fraction of mobile charge carriers can be blocked at internal interfaces [START_REF] Kleitz | Model for ion-blocking at internal interfaces in zirconias[END_REF]. This phenomenon is described as a blocking process at the internal interface which can be characterized by an adimensional parameter called the "blocking factor αr". This parameter is defined as the ratio of the blocking resistance RMD to the total resistance [START_REF] Steil | Densification of yttria stabilized zirconia: impedance spectroscopy analysis[END_REF][START_REF] Kleitz | Model for ion-blocking at internal interfaces in zirconias[END_REF][START_REF] Khal | Effect of porosity on the electrical conductivity of LAMOX materials[END_REF]:

α = (2)
Most studies on solid electrolytes [START_REF] Steil | Densification of yttria stabilized zirconia: impedance spectroscopy analysis[END_REF][START_REF] Khal | Effect of porosity on the electrical conductivity of LAMOX materials[END_REF][START_REF] Roberts | Grain consolidation and electrical conductivity in porous media[END_REF][START_REF] Pérez-Coll | Influence of porosity on the bulk and grainboundary properties of Gd-doped ceria[END_REF][START_REF] Marrero-López | Influences of the microstructure on the bulk and grain boundary conductivity in apatite-type electrolytes[END_REF] have shown that the porosity dependence of the ionic conductivity of ceramic materials could be described with Archie's law. This dependence was first adopted to measure the conductivity of sands consisting of an insulating matrix filled with a liquid electrolyte and then it was extrapolated to all types of rocks [START_REF] Archie | The ellectrical resistivity Log as an aid in determining some reservoir characteristics[END_REF]. Archie's empirical formula is expressed as:

σ = σ°(1 -p) (3)
where σ is the apparent conductivity of the porous material, σ is the intrinsic electrical conductivity of the dense conductive material, m is an empirical exponent that describes the connectivity between the conducting phase (bulk) and the non-conducting phase (porosity) and p is the pore volume fraction in the sample which is related to the volume fraction of YSZ phase (v) in porous sample according to:

p = 1 -v (4)
Considering the porous conductive material as a tortuous percolating ionic phase, the apparent conductivity can also be expressed as a function of the tortuosity factor (τ), which is defined according to:

σ = σ° (5)
The tortuosity factor reflects the geometrical effects such as sinuousness and connectivity of the conductive path (tortuosity) as well as constrictive effect in the microstructure since cross-section areas for conductivity are not constant due to the existence of pores. In the literature [START_REF] Copper | Quantifying the transport properties of solid fuel cell electrodes[END_REF][START_REF] Shen | Critical review of the impact of tortuosity on diffusion[END_REF][START_REF] Berg | Re-examining Archie's law: Conductance description by tortuosity and constriction[END_REF],

several theoretical and empirical models have been proposed to describe the dependence of the tortuosity on the porosity. The theoretical Bruggeman correlation, for spherical pores homogeneously distributed in a conductive matrix, is the most commonly cited as a porositytortuosity relationship as discussed by Tjaden et al [START_REF] Tjaden | On the origin and application of the Bruggeman correlation for analyzing transport phenomena in electrochemical systems[END_REF]. This correlation is expressed as:

τ = (1 -p) (6)
Combining Eqs. ( 5) and ( 6), we find that the variation in the conductivity of the porous medium as a function of porosity obeys the following relationship:

σ = σ (1 -p) . (7)
This means that, for random porous media, with small spherical pores that are small in comparison to the sample size, the conductivity obeys Archie's law with an exponent m equal to 1.5.

Other empirical relationships that relate the tortuosity factor of the ionic conductive phase to its volume fraction have been suggested [START_REF] Copper | Quantifying the transport properties of solid fuel cell electrodes[END_REF][START_REF] Shen | Critical review of the impact of tortuosity on diffusion[END_REF]. Among them, the logarithmic function has been proposed:

τ = 1 -C ln(1 -p) ( 8
)
where C is an adjustable parameter which is dependent on the pore geometry. For a liquid electrolyte containing spherical isolating particles randomly distributed within the electrolyte, Weissberg [START_REF] Weissberg | Effective diffusion coefficient in porous media[END_REF] derived Eq. ( 8) with C = 0.5 from a theoretical treatment assuming that the particles were spherical and may overlap. This leads to:

τ = 1 -0.5 ln(1 -p) (9)
This relationship was successively used for spherical glass beads of different mean sizes in a liquid electrolyte [START_REF] Barrande | Tortuosity of porous particle[END_REF].

Assuming that the YSZ matrix can be identified to the liquid electrolyte and that the spherical pores can be identified to the spherical particles, Eq. ( 9) can be tentatively used to describe the behavior of porous YSZ.

It should be noted that in the porosity range corresponding to the study on porous materials, i.e. 0 < p <0.4, Eqs. ( 6) and [START_REF] De Florio | Sintering of zirconia-yttria ceramics studied by impedance spectroscopy[END_REF] are almost equivalent as emphasized by Copper [START_REF] Copper | Quantifying the transport properties of solid fuel cell electrodes[END_REF] and Tjaden et al [START_REF] Tjaden | On the origin and application of the Bruggeman correlation for analyzing transport phenomena in electrochemical systems[END_REF]. Moreover, for low porosity (p → 0), Eqs. ( 6) and ( 9) are equivalent and lead to the following expression:

τ = 1 + 0.5 p (10)
which is that of Maxwell-Garnett established for dilute inclusions [START_REF] Maxwell | A treatise on electricity and magnetism[END_REF]. This expression is also equivalent to that proposed by Mizusaki at al. [START_REF] Mizusaki | Simple mathematical model for the electrical conductivity of highly porous ceramics[END_REF] for spherical pores assuming a homogenous distribution of pores within the ceramics material and making p → 0. This means that for low porosities, all the porosity-tortuosity correlations are equivalent to the linear relationship given by Eq. ( 10) and established by Maxwell-Garnett.

Experimental part

Sample preparation

The starting YSZ powder was the 8 mol % Y2O3 doped ZrO2 commercial powder (Tosoh TZ-8Y) with a specific surface area equal to 16 m 2 .g -1 . The poly-methyl-methacrylate micro-beads (PMMA MB) with different particle sizes (6, 10, 30, 50 and 100 µm from MICROBEADS® and 3, 1.5, 0.4, 0.15 µm from Soken Chemical and Engineering Co., Ltd) were used as pore forming agent. YSZ powder was composed of agglomerates from 15 µm to 140 µm in diameter, formed of 90 nm particles. This powder was firstly ground for 5 hours with a TURBULA® mixer using zirconia balls and ethanol as the milling medium. The suspension was then dried under ambient air and temperature and crushed in an agate mortar. The ground powder and different volume percentages [START_REF] Copper | Quantifying the transport properties of solid fuel cell electrodes[END_REF][START_REF] Khal | Effect of porosity on the electrical conductivity of LAMOX materials[END_REF][START_REF] Mizusaki | Simple mathematical model for the electrical conductivity of highly porous ceramics[END_REF][START_REF] Khabbazi | Determining the impact of rectangular grain aspect ratio on tortuosity-porosity correlations of two-dimensional stochastically generated porous media[END_REF], 50 or 65 vol. % ) of PMMA MB powder were mixed for 5 hours using zirconia balls and a minimum amount of isopropanol in the TURBULA mixer and then dried under ambient conditions.

The mixture of powders was compacted into cylindrical pellets of 10 mm in diameter and ~ 5 mm in thickness on applying uniaxial pressure of 62 MPa, followed by the isostatic pressure of 80

MPa. The green pellets were sintered in two steps. First, they were heated in air at 560 °C for 2 h.

The heating rate was limited to 25 °C.h -1 to allow the evaporation of volatile components and to prevent the formation of bulk defects such as cracks and swelling. Then, they were heated at 1500 °C for 20 h with a heating rate of 100 °C.h -1 . After this heat treatment, the pellets were cooled to 20 °C with a cooling rate of 300 °C.h -1 . The pellets were machined in order to obtain accurate geometric dimensions (thickness ≈ 4 mm and diameter ≈ 7.6 mm). The relative density of the sintered pellets was calculated from the exact measurement of the geometrical dimensions and weight to the theoretical density (5.971 g.cm -3 and 1.392 g.cm -3 for 8YSZ and PMMA, respectively). The prepared samples were identified thanks to the diameter of the PMMA MB and the obtained sample density. For instance, 6-d90 means that the porous YSZ sample has a final density of 90 % and that it was prepared from PMMA MB with a diameter of 6 µm.

The effect of the PMMA balls dispersion was tested using un-milled powder. This sample with 70 % of density is denoted 6NB-d70. Additionally, two pellets (un-milled powder) were subsintered at 1200 °C during 1 and 24 h to compare the present results to those classically obtained in the literature. They are denoted SF-d60 and SF-d75 with 60 % and 75 % of density, respectively.

Sample characterization

Microstructure observations of the sample were made thanks to a Field Emission Gun Scanning Electron Microscope (SEM) (Zeiss Ultra 55) equipped with an energy and angle selective backscattering detector. Before SEM observation, the samples were fractured and coated with a 2 nm layer of Au-Pd. The estimation of the mean diameter of the grain and the pores was made thanks to the ImageJ software (National Institute of Health-USA) with more than 50 measurements.

For the electrical characterization, the surfaces of the sintered pellets were covered with Pt paint (Matalor 6926 Pt ink) and heated at 850 °C for 2 h with a heating and cooling rate of 300 °C.h -1 .

The electrical properties were measured by impedance spectroscopy using a Hewlett Packard impedancemeter (HP 4194A) in the frequency range of 15 MHz to 100 Hz with an amplitude of 0.1 V. Measurements were performed in air from 800 °C to 300 °C each 50 °C with a homemade software. The impedance diagrams were analyzed with the Z-view software (Scribner Associates Incorporated) by fitting the data using an equivalent electrical circuit consisting of two R// CPE elements in series attributed to the bulk and the MD contribution, respectively. CPE was a constant phase element with an impedance ZCPE defined as:

Z &'( = )(*+) , (11)
where Q was the pseudo-capacitance, ω the angular frequency and n the CPE exponent. This impedance was used to represent the non-ideal behavior of an RC circuit. The capacitance was calculated from Q according to the classical expression [START_REF] Mcdonald | Impedance Spectroscopy: Emphasizing Solid Materials and Systems[END_REF][START_REF] Mcnealy | On the use of the constant phase element to understand variation in grain boundary properties[END_REF]:

C = (QR / ) // (12)
All impedance diagrams were normalized to the geometric factor (k) of the sample defined as:

k = (cm ) (13)
The value of k deduced from the dimensions of the machined sample was 0.9 cm -1 .

Results

Microstructural characterization

Fig. 1 shows the SEM micrographs of porous YSZ samples made with PMMA MB. After sintering, the PMMA MB was successfully removed and porous YSZ without processing bulk defects were obtained. The microstructural properties of the different samples are reported in Table 1, which gives the porosity, the grain diameter and the pore diameter deduced from SEM micrographs analysis. As seen in Table 1, the average grain size was weakly affected by the introduction of the PMMA MB. Indeed, it was in the range of 4 to 5 µm for the porous sample made with PMMA MB which was almost similar to that of the fully dense sample (4.5 µm). The smaller values of PMMA MB diameter (<3 µm) tended to limit the grain growth and this was even more true that the proportion of MB was high. Indeed, before the sintering process, the MB are located between the grains.

Consequently, the smaller the size of the MB and the higher the proportion of pores is (see Fig. 1.a and Fig. 1.b). This may be attributed to the smaller contact area between the grains during the sintering process, which prevents the matter exchange. It results in an inter-granular porosity and in the agglomeration of the pores since the PMMA MB are difficult to be dispersed during the synthesis process.

The grain size of sub-sintered samples was much smaller, i.e. less than 200 nm for SF-d60 and less than 400 nm for SF-d75. These values slightly differed from the previous study conducted by Steil et al. [START_REF] Steil | Densification of yttria stabilized zirconia: impedance spectroscopy analysis[END_REF] but were of the same order of magnitude.

Porosity

All the porous samples prepared with PMMA MB showed a spherical pore shape, which made the measurement of the pore size easier. Moreover, the pore size distribution was very narrow.

These spherical pores were well dispersed in the YSZ matrix and porosity was only composed of the pores coming from the MB while the rest of the sample was almost fully dense. Sometimes, YSZ clusters could be observed inside the pore (see Fig. 1.e). The formation of these clusters has been explained by the rearrangement of local ceramic particles during the volatilization of PMMA particles [START_REF] Liu | Effect of mono-dispersed PMMA micro-balls as pore-froming agant on the properties of porous YSZ ceramics[END_REF]. As seen in Table 1, the porosity of YSZ samples made with PMMA MB ranged from 10 to 40%. It increased with increasing the volume percentage of PMMA MB but was independent on the particle size of the pore-forming agent when using the same amount of PMMA MB. A part of the porosity induced by the MB was eliminated during the sintering process when using the small size of PMMA MB (diameter <1µm). For the samples with the higher MB size, the porosity was found to increase linearly with the volume percent of MB. The increase was the same whatever the MB size. The final pore size was found to depend directly on the MB size (see Table 1). A The characterization of porosity of the sub-sintered samples (SF-d75 and SF-d60) was more difficult because they exhibited a percolating porosity with an irregular shape. The pore size was estimated at 200 nm for SF-d75 and around 100 nm for SF-d60. These values of pore size were of the same order of magnitude than that reported by Steil et al. [START_REF] Steil | Densification of yttria stabilized zirconia: impedance spectroscopy analysis[END_REF].

It can be concluded that the microstructure of the porous samples made with MB was very similar, independently to the pore size and pore volume fraction. It could be described as monodispersed spherical pores, almost homogeneously distributed within dense YSZ consisting in sintered grains with a constant grain size of the order of 4.5 µm. In the dense part, the contact between the grains was well defined and similar to that in the dense sample. Consequently, the variation of the electrical behavior of these samples could be attributed only to the variation of the porosity, i.e. size pore, homogeneity, proportion and not to bad contact between the grains. For low porosity (p ≤ 0.2) and small pore size (Φpore < 20 µm), the variation of RB was independent on the pore size and found to slightly increase with the porosity (Fig. 4.a). However, at higher porosity (p > 0.2), RB increased more rapidly. This increase is stronger for higher pore sizes. Typically, for a large pore of 34 µm (i.e. ~8 times the grain size), the resistance was multiplied by a factor of 3 for a porosity of 25 %. The same trend was also observed for RMD meanwhile the dispersion on the experimental points was more important (Fig. 4.b). This dispersion could be related to the greater uncertainty on the fitting data of RMD due to the small size of the MD semi-circle in comparison to the bulk one.

Electrical characterization

Effect of the pore volume fraction and the pore size

All the samples exhibited the same activation energy (Ea = 0.96 5 0.03 eV) independently of the pore volume fraction and the pore size which indicated that the conduction mechanisms were the same that those of the dense sample and that they were not affected by the pore volume fraction and the pore size.

The parameter α characteristic of the blocking effect was almost constant and independent of the pore size and the fraction of pore volume. The mean value of α with its standard deviation is equal to 0.10 ± 0.05. This value was of the same order of magnitude than that measured with the dense sample (i.e. 0.04). It is worth mentioning that this value was lower than that measured with a sub-sintered sample. This point will be more extensively discussed in section 4.2.2.

Thus, the first conclusion is that it is possible to limit the impact of porosity on the electrical properties of YSZ sample if the pore size is controlled and in the range of 1 to 10 µm. Second, the blocking effect on porous conductive ceramics consisting of a dense matrix with isolated spherical pores is of the same origin as in the dense sample. This allows us to propose that the blocking effect results from grain boundaries and that its nature remains an open question (phase segregation [START_REF] Marrero-Jerez | Re-examining of effects of alumina on bulk and grain boundary conductivities of CGO solid electrolytes[END_REF], mis-orientation [START_REF] Ficher | The influence of grain boundary misorientation or ionic conductivity in YSZ[END_REF], space charge [START_REF] González-Romero | A molecula dynamics study of grain boundary in YSZ: Strucutre energetics and diffusion oxygen[END_REF][START_REF] Gryaznov | Numerical study of grain boundary diffusion in nanocrystalline ionic materials including blocking space charges[END_REF]).

Effect of the grain sintering

The effect of the grain sintering was studied by comparing the electrical properties of the subsintered sample with a relative density of 60% (SF-d60) and a PMMA MB sample with a pore diameter in the range of 1 to 7 µm (6-d60). Both samples had the same pore volume fraction of 0.4.

As evidenced in part 4.1, it should be recalled that the pore morphology was very different from one sample to another. Indeed, it consisted of percolating spherical pores of 4.5 µm in diameter, close to that of the grains in sample 6-d60 (Fig. 2b), whereas it consisted of percolating submicronic pores (average size ≈ 0.09 µm) with a wide size distribution in the sub-sintered SF-d60 sample (Fig. 2d).

The impedance diagrams at 350 °C for both samples are reported in Fig. 5. They consisted of two depressed semi-circles. However, the diagram shape of the sub-sintered sample was very different since it exhibited a large MD semi-circle in agreement with our previous study [START_REF] Steil | Densification of yttria stabilized zirconia: impedance spectroscopy analysis[END_REF]. This resulted in a high value of αR, i.e. αR = 0.7 for the sub-sintered sample instead of αR = 0.1 for the PMMA MB YSZ sample. Moreover, the bulk resistance RB of the sub-sintered sample was slightly higher than that of the PMMA MB sample. This different behavior clearly indicated that the nature of the MD semi-circle was not the same in both samples. Therefore, it can be suggested that the increase of the blocking factor observed in the sub-sintered samples can be attributed to the quality of grain-grain contacts and to pore morphology. Indeed, Kleitz et al. [START_REF] Kleitz | Model for ion-blocking at internal interfaces in zirconias[END_REF] have already suggested that the increase of RMD results from poor contacts between grains and they included the pores surface as the origin of the blocking effect. In this present study, we demonstrate that the pore volume fraction is a secondary parameter for RMD and that the most important parameter is the grain sintering that improves the grain-grain contacts and decreases the surface area of the pore-grain interface as already stated in [START_REF] Khal | Effect of porosity on the electrical conductivity of LAMOX materials[END_REF]. The sharp increase of RMD in the sub-sintered sample can be assigned predominantly to the quality of grain-grain contacts. Consequently, it is possible to maintain a high conductivity for the highly porous sample by improving the grain sintering and controlling the pore size in the range of 1 to 10 µm.

Discussion

Porosity dependence of bulk conductivity

We showed experimentally that for mono-dispersed spherical pores with pore sizes in the range 1 to 7 µm, i.e. same order of magnitude as the grain size, RB is independent on the pore size. Thus making it possible to establish a common law of variation of the bulk conductivity, σ 7 , as a function of the porosity in PMMA MB YSZ porous samples. The corresponding Archie's plot (log-log representation) presented in Fig. 6 shows for low porosity (0 < p ≤ 0.25), a positive linear dependence with slope value of mB = 1.6. At higher pore volume fraction, log(σ 7 ) deviates from this straight line. If we consider the entire porosity range, we obtain a higher value of m, i.e. m =1.95 but this value of m correspond to a large dispersion on the linear fit (R 2 = 0.78). This result is similar to that obtained in a previous work on porous LAMOX ceramics [START_REF] Khal | Effect of porosity on the electrical conductivity of LAMOX materials[END_REF] consisting also of spherical pores almost homogeneously distributed with a dense LAMOX matrix. The intercept at p = 0 in Fig. 6 correspond to the bulk conductivity of YSZ. It is equal to 4.10 -5 S.cm -1 at 350 °C in agreement with the value reported by Steil et al. [START_REF] Steil | Densification of yttria stabilized zirconia: impedance spectroscopy analysis[END_REF], i.e. 310 -5 S cm -1 . The obtained values of mB were compared with those of other porous solid electrolytes such as YSZ [START_REF] Steil | Densification of yttria stabilized zirconia: impedance spectroscopy analysis[END_REF], CGO [START_REF] Pérez-Coll | Influence of porosity on the bulk and grainboundary properties of Gd-doped ceria[END_REF], LAMOX [START_REF] Khal | Effect of porosity on the electrical conductivity of LAMOX materials[END_REF] and germanate apatite [START_REF] Marrero-López | Influences of the microstructure on the bulk and grain boundary conductivity in apatite-type electrolytes[END_REF] as shown in Table 2. Note that those materials corresponded to sub-sintered samples. At low porosity (0 < p ≤ 0.25), the Archie exponent was similar to that obtained in LAMOX ceramics and close to the theoretical value of m = 1.5 corresponding to the Bruggeman equation (cf. Eqs. ( 6) and ( 7)). This result demonstrates that for a porosity resulting from isolated spherical pores randomly distributed within a YSZ dense matrix, the Bruggeman approximation can be used to predict the bulk conductivity dependence on the porosity. Under those conditions, as already stated in the experimental part, the MD contribution can be attributed to grain boundary without the contribution of the pores, except from the tortuosity induced on the conductive path. It results from that a small value of α , almost independent on the porosity. At higher porosity (p ≥ 30 %), the Archie exponent became high with a value similar to that of CGO and germanate apatite but lower than that obtained with the sub-sintered YSZ. Under those conditions, the Bruggeman equation is no more valid. The bulk conductivity decreases sharply as the porosity increases above 30 %, i.e., as the pores are beginning to percolate. The high value of mB could indicate that the pore size and morphology are likely to play a role in the bulk electrical response. This result is in agreement with previous work made by Muller-Huber et al. [START_REF] Müller-Huber | The effect of a variable pore radius on formation resistivity factor[END_REF]. They have combined the results of theoretical calculations with experimental data and they have shown that pore geometry has a strong effect on electrical resistivity of carbonate rocks [START_REF] Müller-Huber | The effect of a variable pore radius on formation resistivity factor[END_REF].

It can be emphasized that this low temperature trend can be extrapolated to higher temperatures because the activation energy is similar regardless of the porosity morphology of the sample.

Tortuosity -porosity relationship

We have shown in section 5.1 that the high-frequency response included the effect of sinuousness and constriction. Therefore, the tortuosity factor (τ) was calculated from the bulk conductivity. Fig. 7 shows the evolution of the tortuosity factor as a function of the volume fraction of YSZ (1-p). This plot was established with the data corresponding to the pore sizes ranging from 1 to 7 µm, which gives a common law as already stated. The tortuosity factor was calculated from σ 7 and σ 7 = 4.10 -5 S cm -1 and compared with the theoretical values deduced from the Bruggeman approximation (Eq.6). We also added in this figure the tortuosity factor of the sub-sintered YSZ sample calculated from the data of Steil et al. [START_REF] Steil | Densification of yttria stabilized zirconia: impedance spectroscopy analysis[END_REF].

As expected, at low porosity value (p ≤ 25 %), the Bruggeman approximation can be used to predict the tortuosity factor of PMMA MB YSZ sample with pore size ranging from 1 to 7 µm. In this range of porosity, the PMMA MB samples correspond to the ceramics with spherical pores homogenously distributed inside YSZ matrix. On the contrary, the tortuosity factor of sub-sintered YSZ is always higher. This result confirms that the Bruggeman relation is only valid in a situation

where the isolating phase is present in a low volume fraction (p ≤ 25 %) and represented by isotropic spheres. Therefore, it performs better for PMMA MB than sub-sintered porous YSZ sample. Moreover, if the pores have a small size with a complex geometry that allows interconnection as in sub-sintered YSZ, the tortuosity factor is higher whatever the porosity range. Indeed, D. Chung et al. [START_REF] Chung | Validity of the Bruggeman relation for porous electrodes[END_REF] have suggested that inhomogeneities in particle size distribution and particle packing result in a tortuosity increase of 13-16 % with respect to a uniformly packed electrode for lithium-ion batteries, in the limit of the perfectly spherical particles.

The tortuosity -porosity dependence was fitted with a Weissberg equation. A line in Fig. 7 represents the results. The data fitting gave values of C equal to 1.3 for PMMA MB YSZ and 2.3

for the sub-sintered YSZ. These obtained results are in good agreement with the results reported by A. E. Khabbazi et al. [START_REF] Khabbazi | Determining the impact of rectangular grain aspect ratio on tortuosity-porosity correlations of two-dimensional stochastically generated porous media[END_REF]. They showed that for porous media consisting of parallelepipedal particles of low thickness, C was equal to 1.2.

Conclusion

Porous YSZ samples with monodispersed spherical pores in the range of 1 to 73 µm were successfully synthesized by using PMMA MB as a pore-forming agent.

The blocking factor of PMMA MB YSZ samples was small and independent of porosity contrary to that traditionally observed in sub-sintered YSZ samples. Therefore, the increase of the blocking factor observed in the sub-sintered samples can be attributed to the quality of grain-grain contacts and to pore morphology.

The Bruggeman model would be used to predict the bulk conductivity of porous PMMA MB YSZ and calculate the tortuosity factor if the porosity was less than 25 % and if the pore size was between 1 and 7 µm. This good agreement allowed us to conclude that the high-frequency impedance response includes the effect of sinuousness and constriction induced by pores. For sub-sintered YSZ, the tortuosity factor was always higher than predicted by the Bruggeman model, thus showing that pore morphology plays a key role on the conductivity -porosity relationship. For porosities greater than 25 % and large spherical pores (diameter > 20 µm), the tortuosity factor was always greater than that predicted by the Bruggeman model. It should be noted that the entire analysis is based on relatively low temperature data (350 °C), however the observed trends can be extrapolated to higher temperatures because the activation energy is similar regardless of the porosity of the sample.

  Fig.2.c).

Fig. 1 .

 1 Fig. 1. SEM micrographs of porous YSZ samples made with PMMA MB. a) 3-d90, b) 6-d90, c) 6-d84, d) 6-d70, e) 50-d84 and f) 100-d84. The scale is the same for all micrographs, i.e. 50µm.

Fig. 2 .

 2 Fig.2. SEM micrographs of porous YSZ samples made with PMMA MB compared to sub-sintered samples with the same porosity. a) 6-d75, b) 6-d60, c) SF-d75 and d) SF-d60.

  (a) intergranular; b) agglomerated; (c) intragranular; (d) percolation 4.1.1. Grains

decrease of about 33 %

 33 was observed between the MB size and the final pore size, regardless of the volume percentage of PMMA MB. With further increasing of the proportion of PMMA and decreasing of particle size of PMMA, pore contact (or interconnectivity) might occur, resulting in a percolating porosity throughout the ceramic sample. Typically for a MB size of 6 µm, the PMMA MB content corresponding to the percolation threshold was approximately 50 %, which was equivalent to 30 % of porosity (Fig.1.d and Fig.2.b).

Fig. 3

 3 Fig.3shows the impedance diagrams measured at 350 °C for two porosities, i.e. p = 0.25 (Fig.3.a) and p = 0.16 (Fig.3.b), as a function of the pore size. For comparison, the impedance diagram of the dense sample (p = 0) is also represented (Fig.3.c). The shape of the impedance diagrams of the porous samples was similar to that of the dense sample. It consisted of two depressed semi-circles with a large HF semi-circle (bulk) and a small LF semi-circle (MD). The depression angle of both semi-circles was very low indicating that all samples were rather homogenous in terms of conductivity, permittivity and distribution of both parameters, as reported by McNealy et al.[START_REF] Mcnealy | On the use of the constant phase element to understand variation in grain boundary properties[END_REF]. The variation of the resistivity, RB and RMD, as well as the blocking factor αR measured at 350 °C as a function of porosity is represented in Fig.4for pore sizes ranging from 1 to 73 µm. The activation energy of the total conductivity as a function of the temperature is also reported (Fig.4.d).

Fig. 3 .

 3 Fig.3. Impedance diagrams measured at 350 °C as a function of pore size. a) p = 0.25, b) p = 0.16; and the diagram of the dense sample is represented for comparison c) . The numbers on the diagrams are the logarithms of the frequency.

Fig. 4 .

 4 Fig.4. Evolution of a) RB, b) RMD, c) αR at 350 °C and d) Ea as a function of porosity for different pore sizes.

Fig. 5 .

 5 Fig.5. Impedance diagrams of PMMA MB YSZ and sub-sintered YSZ samples (p = 0.4).The numbers on the diagrams are the logarithms of the frequency.

Fig. 6 .

 6 Fig.6. Archie's plot of the bulk conductivity at 350 °C

Fig. 7 .

 7 Fig.7. Evolution of tortuosity factor as a function of PMMA MB samples at 350 °C. For comparison, we indicated the tortuosity factor of sub-sintered YSZ, obtained from ref.[START_REF] Steil | Densification of yttria stabilized zirconia: impedance spectroscopy analysis[END_REF]. The solid black and red lines represent the fitting results with the Weissberg equation.

Table 1 .

 1 Microstructural characterization of porous YSZ samples

	Notation PMMA	Ø PMMA MB	Porosity	Ø pores	Ø grains	Remarks porosity
		(Vol. %)	(µm)	(Vol. %)	(µm)	(µm)
	0-d100	0		0		4.5	Dense
	0.15-d90	25	0.15	10	0.83	3.6	inter (a) and agglo (b)
	0.4-d90	25	0.4	10	1.2	4	inter and agglo
	3-d90	12	3	10	2	5.6	intra (c)
	6-d90	12	6	10	4.5	6.3	intra
	6-d87	15	6	13	4.4	4.2	intra
	10-d87	15	10	13	7	4.5	intra
	30-d87	15	30	13	22	5	intra
	50-d87	15	50	13	34.5	4.4	intra
	6-d84	25	6	16	4.3	4	intra
	10-d84	25	10	16	7.1	4.1	intra
	30-d84	25	30	16	20	4.2	intra
	50-d84	25	50	16	34	4	intra
	100-d84	25	100	16	73	4	intra
	1.5-d82	25	1.5	18	1.52	3.8	inter and agglo
	3-d82	25	3	18	1.7	4.1	inter and intra and some
							parts agglo
	6-d75	35	6	25	4.4	3.5	intra beginning of
							interconnectivity
	10-d75	35	10	25	6.9	4.5	intra
	30-d75	35	30	25	21	4.5	intra
	50-d75	35	50	25	34	4.6	intra
	SF-d75	0		25	0.2	0.38	perco (d)
	6-d70	50	6	30	4.6	4.2	beginning of
							interconnectivity
	6-d60	65	6	40	4.5	3.7	perco
	SF-d60	0		40	0.09	0.17	perco

Table 2 .

 2 Values of mB for a given temperature (T) obtained for various porous ceramics.

	Materials	T(°C)	Porosity range (%)	mB	References
		350	10 -40	1.95
	YSZ PMMA MB	350	10 -25	1.60	Present work
	YSZ	350	3 -41	2.70	[3]
		355	4.4 -22	1.60
	La2Mo1.5W0.5O9	355	4.4 -34	2.7	[15]
	CGO	300	1 -30	1.90	[17]
	La9.6Ge5.5Al0.5O26.15	325	5 -25	1.80	[18]

Acknowledgements

The authors express their thanks to Dr. F. Charlot and J. Calabro (CMTC, INP Grenoble) for SEM observations.