Bidet-Ildei Christel
email: christel.bidet@univ-poitiers.fr

Meugnot Aurore

Beauprez Sophie-Anne

Gimenes Manuel

Toussaint Lu

Christel Bidet-Ildei

Short-term upper limb immobilization affects action-word understanding

Keywords: short-term immobilization, action word understanding, motor imagery

The present study aimed to investigate whether well-established associations between action and language can be altered by short-term upper limb immobilization. The dominant arm of righthanded participants was immobilized for 24 hours with a rigid splint fixed on the hand and an immobilization vest restraining the shoulder, arm and forearm. The control group did not undergo such immobilization. In two experiments, participants had to judge whether a verb involved movements of the hands or feet. In Experiment 1, the response times for controls were shorter for hand-action verbs than for foot-action verbs, whereas there was no significant difference in the immobilized group. Experiment 2 confirmed these results with a pre/post test procedure. Shorter response times were shown for hand-action verbs than for foot-action verbs in the pre and post-tests for the control group and in the pre-test for the immobilized group (i.e., before immobilization). This difference was not observed for participants undergoing 24 hours of hand immobilization, who showed little progress in assessing hand-action verbs between pre-test and post-test. Moreover, participants with the highest motor imagery capacities clearly demonstrated shorter response times in Experiment 2 for both hand-action and foot-action verbs, regardless of hand immobilization. Overall, these findings demonstrate for the first time that short-term sensorimotor deprivation can affect action verb processing. We discuss our results in light of the embodiment view, which considers that cognition is grounded in sensorimotor experiences.

Introduction

Consistent with the embodied view of cognition [START_REF] Barsalou | Grounded cognition[END_REF]Pulvermüller, 2013b), lots of studies have demonstrated a clear crosstalk between action-words processing and action [START_REF] Fischer | Embodied language: a review of the role of the motor system in language comprehension[END_REF][START_REF] Jirak | Grasping language--a short story on embodiment[END_REF] for reviews). For example, behavioural experiments using a priming or a dual-task procedure have demonstrated that the presentation of an action word or an action sentence influences the characteristics of the following movement [START_REF] Boulenger | Crosstalk between language processes and overt motor behavior in the first 200 msec of processing[END_REF][START_REF] Glenberg | Grounding language in action[END_REF][START_REF] Liepelt | Bidirectional semantic interference between action and speech[END_REF] or the subsequent judgements of human movements (Bidet-Ildei, Gimenes, [START_REF] Bidet-Ildei | Sentence plausibility influences the link between action words and the perception of biological human movements[END_REF][START_REF] Bidet-Ildei | Reading action word affects the visual perception of biological motion[END_REF][START_REF] Springer | Language-induced modulation during the prediction of others' actions[END_REF][START_REF] Springer | Action semantics modulate action prediction[END_REF]. Otherwise, clinical reports have shown that the integrity of motor areas is essential for action word processing (Bak, O'Donovan, Xuereb, Boniface, & Hodges, 2001;Boulenger et al., 2008;Cardona et al., 2013;Cotelli et al., 2006;Peran, Demonet, Pernet, & Cardebat, 2004;Peran et al., 2003). A dominant hypothesis that explains the link between action and action word processing is based on the notion that both activities activate similar brain networks, including the mirror neuron system [START_REF] Aziz-Zadeh | The human mirror neuron system and embodied representations[END_REF][START_REF] Aziz-Zadeh | Congruent embodied representations for visually presented actions and linguistic phrases describing actions[END_REF][START_REF] Pulvermüller | Brain mechanisms linking language and action[END_REF]Pulvermüller, , 2013a;;Pulvermuller & Fadiga, 2010;[START_REF] Tettamanti | Listening to action-related sentences activates fronto-parietal motor circuits[END_REF]. This motor activation would be somatotopic in primary motor cortex and pre-motor cortex [START_REF] Hauk | Somatotopic representation of action words in human motor and premotor cortex[END_REF] which is in accordance with the evocation of common sensorimotor representations [START_REF] Andres | Contribution of motor representations to action verb processing[END_REF][START_REF] Bidet-Ildei | Are judgments for action verbs and point-light human actions equivalent?[END_REF][START_REF] Willems | Neural evidence for the interplay between language, gesture, and action: a review[END_REF] and with the use of implicit motor imagery [START_REF] Aziz-Zadeh | The human mirror neuron system and embodied representations[END_REF][START_REF] Willems | Neural dissociations between action verb understanding and motor imagery[END_REF] during action words processing. Interestingly, the relationship between action and action-words appears early in childhood as evidenced by the recruitment of motor areas in 4-5 years-old children when they have to process action verbs [START_REF] James | Auditory verb perception recruits motor systems in the developing brain: an fMRI investigation[END_REF]. Moreover, some studies have shown that new associations between words and action can appear rapidly with training (e.g., [START_REF] Fargier | Learning to associate novel words with motor actions: language-induced motor activity following short training[END_REF][START_REF] Kiefer | Experience-dependent plasticity of conceptual representations in human sensory-motor areas[END_REF]. For example, [START_REF] Fargier | Learning to associate novel words with motor actions: language-induced motor activity following short training[END_REF] have shown that one day of training is sufficient to associate correctly observed actions to novel verbal stimuli and to detect motor activation when these new verbal stimuli are processed.

Altogether, these experiments argue in favour of embodied cognition and demonstrated that "concepts are grounded in perception and action" [START_REF] Fargier | Learning to associate novel words with motor actions: language-induced motor activity following short training[END_REF]. However, an open issue is to determine whether well-established associations can be altered by a quick update of sensorimotor representations. Recently, short-term upper limb immobilization in healthy subjects was proposed as a new paradigm to explore how experience affects brain functions. In this context, some studies have shown that a brief period of upper limb immobilization (12 to 48 hours) is sufficient to induce cortical plastic changes and functional or behavioural consequences [START_REF] Bassolino | Functional effect of shortterm immobilization: kinematic changes and recovery on reaching-to-grasp[END_REF][START_REF] Facchini | Time-related changes of excitability of the human motor system contingent upon immobilisation of the ring and little fingers[END_REF][START_REF] Huber | Arm immobilization causes cortical plastic changes and locally decreases sleep slow wave activity[END_REF][START_REF] Meugnot | The embodied nature of motor imagery processes highlighted by short-term limb immobilization[END_REF][START_REF] Moisello | Short-term limb immobilization affects motor performance[END_REF][START_REF] Toussaint | Short-term limb immobilization affects cognitive motor processes[END_REF]. For example, it has been demonstrated that short-term arm immobilization directly affects motor performance by decreasing endpoint accuracy and modifying the kinematics associated with arm movement [START_REF] Bassolino | Functional effect of shortterm immobilization: kinematic changes and recovery on reaching-to-grasp[END_REF][START_REF] Moisello | Short-term limb immobilization affects motor performance[END_REF]. Similarly, 48 hours, and even 24 hours, of hand immobilization affected the cognitive representations of upper limb movements, as confirmed by delayed response times in a hand mental rotation task, which is known to trigger sensorimotor memories of hand movements [START_REF] Meugnot | The embodied nature of motor imagery processes highlighted by short-term limb immobilization[END_REF][START_REF] Toussaint | Short-term limb immobilization affects cognitive motor processes[END_REF]. These behavioural changes are associated with a decrease in cortical excitability [START_REF] Facchini | Time-related changes of excitability of the human motor system contingent upon immobilisation of the ring and little fingers[END_REF] and a decrease in both somatosensory and motor-evoked potentials elicited from sites over the contralateral sensorimotor cortex [START_REF] Huber | Arm immobilization causes cortical plastic changes and locally decreases sleep slow wave activity[END_REF]. This selective impairment of limb immobilization, which was observed in the contralateral hemisphere but not the ipsilateral hemisphere, has been confirmed in behavioural studies. For example, less accurate pointing movements have been observed for only the immobilized arm [START_REF] Huber | Arm immobilization causes cortical plastic changes and locally decreases sleep slow wave activity[END_REF]. Moreover, [START_REF] Meugnot | The embodied nature of motor imagery processes highlighted by short-term limb immobilization[END_REF] reported that a decrease in input-output signal processing due to left hand immobilization specifically affected the cognitive representation of left hand movements without affecting right hand movements. Finally, upper limb immobilization specifically affected the upper limb representations without affecting the cognitive representations of lower limb movements [START_REF] Meugnot | Selective impairment of sensorimotor representations following short-term upper-limb immobilization[END_REF]. Although these types of experiments are rarely reported in the literature, they clearly demonstrate that sensory motor deprivation over a brief period directly affects movement production and cognitive motor processing (i.e., motor imagery) associated with a slowdown of sensorimotor processes. Specifically, short-term limb immobilization may provide an ideal method for temporally disturbing the sensorimotor system to assess its contribution to the processing of mental phenomenon other than action (e.g., motor simulation, [START_REF] Toussaint | Short-term limb immobilization affects cognitive motor processes[END_REF].

In the present study, we investigated whether perturbation of sensorimotor representations related to short-term limb immobilization can affect action verbs processing in healthy participants.

For this purpose, we used a semantic decision task implying action verbs.

Experiment 1

In Experiment 1, we aimed to demonstrate that upper limb immobilization affects the performance of a semantic decision task consisting of action verbs that mainly involve movements of the hands or feet. We compared performances on a single session performed after 24 hours of right hand immobilization or by participants who did not undergo the immobilization procedure.

For each verb that appeared on the computer screen, participants were asked to answer "hand" or "foot", when they considered that action was mainly performed by upper or lower-limbs, respectively. We made the assumption that the effector-induced effects of short-term upper limb immobilization (Meugnot, Agbangla, & Toussaint, 2015;[START_REF] Meugnot | The embodied nature of motor imagery processes highlighted by short-term limb immobilization[END_REF] should manifest specifically for judgements of hand-action verbs (e.g., write) without disrupting judgements of foot-action verbs (e.g., jump). More specifically, we hypothesized that judgement of hand action verbs should be slowed in the immobilized group related with a specific slowdown of sensorimotor processes.

Method

Participants

Thirty-six right-handed French-speaking university students (18-26 years old, mean age = 20.3 years; 20 males, 16 females) participated in the experiment. They were separated randomly into 2 groups of 18 participants: a control group (mean age = 19.6; 11 males, 7 females) and an immobilized group (mean age = 21.0; 9 males, 9 females). Each participant was healthy, had normal or corrected-to-normal vision and had no history of motor or neurological disorders. The participants were naïve of the experiment's aims before testing.

Task and procedure

All participants performed a semantic decision task that involved action verbs. They were seated in front of the computer screen and had to decide whether an action verb described movements performed mainly with the hands (e.g., write, cut, clap) or with the feet (e.g., pedal, run, jump). Forty-four action verbs (22 hand-action verbs and 22 foot-action verbs) were selected from the French lexical database Lexique 2.0 (New, Pallier, Ferrand & Matos, 2001). All action verbs were in the infinitive form (see Appendix A .3 and A .4 for more details). Hand-action verbs and footaction verbs were matched for relevant lexical variables including word frequency, number of letters and number of syllables (Table 1).

Table1. Mean values of action verb frequency (FQ), number of letters (LETT) and number of syllables (SYLL) for hand-and foot-action verbs. ANOVAs are reported in the last column. and 22 foot-action verbs which were different than those used in the training phase). Each trial began when a fixation cross was displayed in the centre of the screen for 500 ms. Then, an action verb was presented and remained visible until the participant gave a verbal response ("main" or "pied", French words meaning "hand" and "foot", respectively). A blank screen appeared during the 500-ms period that preceded the next trial. The participants had to respond as fast and as accurately as possible. The E-prime 2.0 software package (Psychology Software Tools Inc., Pittsburgh, USA)

Hand

was used to present the action verbs and to record the participant's response times via a microphone connected to the computer. For each trial, the experimenter wrote and assessed the correctness of the participant's answers on a sheet.

The semantic decision task was performed in a single session for each participant. For the immobilized group, a rigid splint (model DONJOY "Comfort Digit", DJO, Surrey, UK) fixed on the hand and an immobilization vest restraining the shoulder, arm and forearm (model DONJOY "Immo Axmed") were used to ensure the participants kept their right upper limb at rest as much as possible during 24 hours. The participants were instructed to not remove the splint during the immobilization period; however, they could remove the immobilization vest for the night. To quantitatively verify whether participants followed the instructions, we monitored the physical activity of both hands through actimeters (pounds/min) placed on each hand. During the 24-hour delay, an average of 422 ±199 pounds/min (3.2 ± 1.5 kg/s) were recorded for the right immobilized hand and 2018 ±620 pounds/min (15.3 ± 4.7 kg/s) were recorded for the left non-immobilized hand (see [START_REF] Toussaint | Short-term limb immobilization affects cognitive motor processes[END_REF], for a similar procedure). An ANOVA performed on the actimeter values showed that the level of activity was significantly higher for the left hand than for the right hand, F(1,17)=139.3, p=0001, η² p =.89. The immobilization group performed the semantic decision task immediately after the splint and the immobilization vest were removed. The other group served as the control (i.e., without immobilization). At the end of the task, participants who have been immobilized were asked to perform some movements of flexion/extension to reactivate their sensorimotor system [START_REF] Bassolino | Functional effect of shortterm immobilization: kinematic changes and recovery on reaching-to-grasp[END_REF].

Data analysis

Accuracy and response times were recorded for each trial. Unreliable trials due to hesitations or microphones failures have been excluded of the analysis (less than 2 % of all trials).

Only data from correct responses were used to analyze response times. For all conditions, means and standard deviations were reported in Appendix A .1 . Accuracy and response times analyzes were performed with generalized linear mixed-effects models using the lme4 package [START_REF] Bates | Fitting Linear Mixed-Effects Models Using lme4[END_REF] in the R environment (R version 3.3.0, [START_REF] Core | R : A language and environment for statistical computing[END_REF]. In all models, participants and items were specified as random-effects factors. Two fixed-effects factors were included (group, action verbs and their interaction). The p-values were obtained using the lmerTest package [START_REF] Kuznetsova | lmerTest: Tests for random and fixed effects for linear mixed effect models[END_REF].

Results

The accuracy analyzes showed no significant main effects for neither group nor action verbs as well as no significant interaction between groups and action verbs. On average, the response accuracy was 98% ±2.3% (mean ± standard deviation).

The analyzes performed for response times showed no significant main effects. However, the interaction between groups and action verbs was significant : F(1, 4362.8) = 6.7, p = .01.

Contrast analyzes revealed that response times were faster for the hand-action verbs than for the foot-action verbs for the control group (p=.05), whereas no difference was observed for the immobilized group (p=.73) (Figure 1). Given that effects appeared only in response times, it excluded an explanation in terms of speedaccuracy trade-off.

Discussion

The objective of this experiment was to assess whether short-term upper limb immobilization can affect action verb processing. We made the assumption that differences would be observed between groups for judgements regarding hand action verbs. The results showed an interesting within-group difference: subjects who did not undergo immobilization were faster at judging hand-action verbs than foot-action verbs, whereas this difference was not present when the participants had been immobilized during the previous 24 hours. Before considering the effect of immobilization on action verb judgement, we propose to examine the difference observed between hand-action and foot-action verbs in control group in the following section.

At first level, the advantage obtained in response times for hand-action verbs could be simply due to the response modality. Participants had to respond "hand" ("main" in French) or "foot" ("pied" in French) in a microphone. Several works have shown that times in word naming task vary substantially as a function of onset phoneme (e.g., [START_REF] Spieler | Bringing Computational Models of Word Naming down to the Item Level[END_REF]) and some works have shown a nonplosive phoneme advantage (e.g., [START_REF] Ferrand | Comparing word processing times in naming, lexical decision, and progressive demasking: evidence from chronolex[END_REF]. However, time differences are very small (about 4.5 ms) and they could not explain in totality time differences obtained in our experiment (about 40 ms). Moreover, this explanation can hardly account for the advantage for hand-action verbs before immobilization that disappears after immobilization.

It is possible to consider that difference in response times could also be related to a higher language familiarity for hand-action verbs (e.g., [START_REF] Ando | Practice effects on reaction time for peripheral and central visual fields[END_REF][START_REF] Taniguchi | Effect of practice in bilateral and unilateral reaction-time tasks[END_REF] .

Nevertheless, given that hand-action and foot-action verbs were controlled in terms of linguistic parameters (i.e., frequency) in our experiment, we can assume that verbal experience was similar between both categories of verbs and therefore a faster response time cannot be attributed to language practice.

One other possibility is that the hand-action verbs advantage could be related to attentional mechanisms for a body-specific system (see Connell & Lynott, 2012 for a review). For example, Connell, Lynott and Dreyer (2012) reported a facilitation effect when people had to compare manipulable objects (versus non manipulable objects) with concurrent tactile stimulation of the hands, due to the fact that perceptual stimulation directed attention toward the hand. However, in our study, although participants were informed that experimental manipulations were applied to the hand, the hand-action verb advantage disappeared following 24 hours of hand and arm immobilization whereas attentional mechanisms towards the hand are probably strengthened by the difficulties link to the hand non-use. In our opinion, these results discount the hypothesis that handaction verbs advantage is due to the increase attentional demands on the hand.

According to the embodied view of cognition, an alternative explanation might be that handaction and foot-action verbs are associated with different levels of sensorimotor representations. At the neurophysiological level, this difference might be directly related to the motor homunculus of the primary motor cortex (Penfield & Boldrey, 1937). Indeed, it is acknowledged that cortical representations of body parts are not proportional to the size of the body part but rather to the complexity of movements such body parts can produce. Specifically, the hand is more represented in the sensorimotor cortex than the foot. As a previous study revealed that action word processing somatotopically activates the primary motor cortex (e.g., [START_REF] Hauk | Somatotopic representation of action words in human motor and premotor cortex[END_REF], we hypothesized that faster response times obtained for hand-action verbs might be related to the largest hand representation in the primary motor cortex. As underlined by an anonymous reviewer, if this explanation is correct, cortical somatosensory maps might be used to predict individual response times. For instance, whole-body action verbs should be identified faster than mere handor foot-action verbs. However, this explanation remains speculative and future experiments have to assess this point by combining behavioural and cerebral activation data (see [START_REF] Heim | Decreased cortical representation of genital somatosensory field after childhood sexual abuse[END_REF] for an experiment on this question).

Finally, it is possible to envisage that difference between hand and foot related verbs might be related to sensorimotor experience. This notion accords with the fact that personal motor experience influences the degree of motor activation during language understanding. Accordingly, greater activation was observed in the left premotor cortex in experienced ice hockey players who analyzed sentences that presented specific hockey actions than in non-experienced ice hockey players analysing the same sentences (Lyons et al., 2010). In the present experiment, assuming that hand movements are common in humans, we can hypothesize that sensorimotor representations associated with hand-action verbs could generate greater activity in the central nervous system than sensorimotor representations associated with foot-action verbs. This hypothesis remains to be assessed in future experiments; however, this notion is consistent with a previous study that showed motor practice directly influences motor system activation during action word processing [START_REF] Fargier | Learning to associate novel words with motor actions: language-induced motor activity following short training[END_REF].

Interestingly, in the current experiment, the effects induced by 24 hours of upper limb immobilization were manifest by the absence of a difference between processing for handaction verbs and foot-action verbs compared with controls, suggesting that short-term sensorimotor deprivation directly affects action verb processing. Although experimental evidence still needed (as done in experiment 2), it may be that hand immobilization during 24 hours slowdown the judgement of hand action verbs suggesting that well-established associations between action and action-verbs processing can be affected by peripheral sensorimotor deprivation. This notion is consistent with previous investigations that demonstrated that short-term immobilization affects sensorimotor representations [START_REF] Toussaint | Short-term limb immobilization affects cognitive motor processes[END_REF], see general discussion for more details)

relating with a decrease of cerebral activity in sensory and motor cortices controlling the immobilised limb [START_REF] Huber | Arm immobilization causes cortical plastic changes and locally decreases sleep slow wave activity[END_REF]. However, as this experiment was performed with two different groups, we cannot exclude that our results might be related to individual differences in sensorimotor experience rather than the immobilization procedure. Experiment 2 aimed to exclude this alternative hypothesis by assessing the effects of immobilization with a pre/post-test procedure.

Experiment 2

The first objective of the present experiment aimed to confirm the impact of short-term upper limb immobilization on action verb processing. Here, a pre/post-test procedure was used to ensure that the effects obtained in Experiment 1 were directly related to sensorimotor deprivation and not to individual differences as previously mentioned. In that case, we could expect that a difference in response times between the hand-action and foot-action verbs observed in Experiment 1 would be present in both the control and the immobilized groups in the pre-test, but it would not be present in the post-test for only the immobilized group. Moreover, our experimental design included a test of the individual's motor imagery capability using the Vividness of Motor Imagery Questionnaire (VMIQ, Isaac, Marks, & Russell, 1986). Previous experiments have shown that motor simulation is involved during action word processing [START_REF] Aziz-Zadeh | Embodied semantics for actions: findings from functional brain imaging[END_REF] and that the immobilization-induced effects had a greater effect on the sensorimotor representations of participants with a greater motor imagery capacity (from a first-person perspective) compared to participants who had lesser motor imagery capacities [START_REF] Toussaint | Short-term limb immobilization affects cognitive motor processes[END_REF][START_REF] Willems | Neural evidence for the interplay between language, gesture, and action: a review[END_REF]. Therefore, a second objective of this experiment was to assess the role of individual imagery capacities in the relationship between sensorimotor processes and action verb judgement.

Method

Participants

Thirty-two right-handed French-speaking university students (18-29 years, mean age = 19.75 years; 10 males, 22 females) participated in the experiment. They were assigned randomly to 2 groups of 16 participants: a control group (mean age = 18.9; 6 males, 10 females) and an immobilized group (mean age = 20.6; 4 males, 12 females). Each participant was healthy, had normal or corrected-to-normal vision and had no history of motor or neurological disorders. The participants were naïve to the aims of the experiment before testing.

Tasks and procedure

The task and procedure were similar to those used in Experiment 1, except that in Experiment 2, the semantic decision task was performed during 2 experimental sessions with a 24hr interval between sessions (pre-test and post-test). The task was performed before immobilization (pre-test) and immediately after the splint and immobilization vest were removed (post-test) for the immobilized group. The actimeters revealed that, on average, 496 ± 223 pounds/min (3.7 ± 1.7 kg/s) and 2300 ±532 pounds/min (17.4 ± 4.02 kg/s) were recorded for the right immobilized hand and left non-immobilized hand, respectively. The ANOVA performed for the actimeter data showed that the level of activity was significantly higher for the left hand than for the right hand, F(1,15)=109.4, p=000, η² p =.88.

At the beginning of the experiment, all participants completed the Vividness Movement Imagery Questionnaire (VMIQ; Isaac, Marks & Russell, 1986). The VMIQ consists of 24 items relevant to both first-person (i.e., internal imagery) and third-person perspective imagery (i.e., external imagery). The measures of imagery vividness were assessed for each imagery perspective on a 5-point Likert-type scale. High scores represent low vividness of movement imagery (for the first-person and the third-person perspectives), with the reverse being true for low scores. The mean VMIQ scores are illustrated in Table 2.

Table2. The means, standard deviations, and ranges for Scores Measuring the Vividness of Movement Imagery from a first-person perspective (i.e., internal imagery) and a third-person perspective (i.e., external imagery) for both the control and immobilized groups. ANOVAS performed on each imagery perspective separately did not reveal any significant difference between the control and the immobilized group (F(1,30)=0.39, p=.53 for the 1 st -person perspective; F(1,30)=0.04, p=.83 for the 3 rd -person perspective).

Data analysis

Accuracy and response times were recorded for each trial. Unreliable trials due to hesitations or microphones failures have been excluded of the analysis (less than 1.5 % of all trials).

Only data from correct responses were used to analyze the response times. For all conditions, means and standard deviations were reported in Appendix A .2 .

To quantify the improvement in response times between the pre-test and post-test for the judgement of hand and foot-action verbs in both the control and the immobilized groups, we computed the Index of Performance Improvement (IPI = [response time in pre-test -response time in post-test] / response time in pre-test, expressed in percentage) for each participant and each action verb. A positive value indicated that response times decreased from the pre-test to post-test (i.e., performance improvement), whereas a negative value indicated that response times increased (i.e., performance deterioration). The same procedure was followed for IPI accuracy. IPI Accuracy and IPI response times analyzes were performed with generalized linear mixed-effects models using the lme4 package [START_REF] Bates | Fitting Linear Mixed-Effects Models Using lme4[END_REF] in the R environment (R version 3.3.0, R [START_REF] Core | R : A language and environment for statistical computing[END_REF]. In all models, participants and items were specified as random-effects factors.

Two fixed-effects factors were included (group, action verbs and their interaction). The p-values were obtained using the lmerTest package [START_REF] Kuznetsova | lmerTest: Tests for random and fixed effects for linear mixed effect models[END_REF].

Results

Analyzes performed for IPI accuracy failed to show significant main effects for group and for action verbs and showed no significant interaction between groups and action verbs (ps>.38). Analyzes performed on IPI response times revealed no significant main effects. However, the interaction between group and action verbs was significant: F(1, 1323.36) = 3.71, p = .05. Contrast analyzes revealed that the IPI was lower for the hand-action verbs than for the foot-action verbs in decision task was performed (i.e., before or after right hand immobilization).

Discussion

Experiment 2 aimed to replicate and extend the results obtained in Experiment 1. By using a pre/post-test procedure, we assessed whether short-term upper limb immobilization affected action verb processing.

IPI analyzes showed that the improvement in action verb judgement observed in the posttest was lesser for the hand-action verbs in the immobilized group, whereas improvement was similar for both hand-action and foot-action verbs in the control group. Moreover, we showed that action verb processing efficiency was dependent on the individual level of motor imagery (as assessed by the first-person perspective scores of the VMIQ), whereas no correlation was observed with visual imagery capacity (assessed by the third-person perspective scores of the VMIQ).

Overall, these findings confirm the results obtained in Experiment 1 by showing that sensorimotor deprivation disturbed action verb processing. Moreover, the present findings clearly demonstrate that the short-term hand immobilization affected specifically the judgement of handaction verbs, confirming the specific effector-induced effects of short-term upper limb immobilization (Meugnot, Agbangla, & Toussaint, 2015;[START_REF] Meugnot | The embodied nature of motor imagery processes highlighted by short-term limb immobilization[END_REF]. The slowdown in sensorimotor processes induced by short-term upper limb immobilization [START_REF] Meugnot | The embodied nature of motor imagery processes highlighted by short-term limb immobilization[END_REF][START_REF] Meugnot | Functional plasticity of sensorimotor representations following short-term immobilization of the dominant versus non-dominant hands[END_REF] could explain that response times were delayed specifically for the judgement of hand-action verbs (versus foot-action verbs).

Importantly, our findings demonstrate that motor imagery processes are specifically implied during a semantic decision task with action verbs before and after immobilization. This suggests that disrupting the processing of input/output signals during a short delay does not preclude the involvement of sensorimotor processes in action verb processing. As suggested by previous works, our results argue in favour of the use of a motor action simulation during action verb understanding [START_REF] Aziz-Zadeh | Embodied semantics for actions: findings from functional brain imaging[END_REF][START_REF] Gallese | Embodied simulation: from neurons to phenomenal experience[END_REF][START_REF] Kemmerer | The Two-Level Theory of verb meaning: An approach to integrating the semantics of action with the mirror neuron system[END_REF][START_REF] Pulvermüller | Brain mechanisms linking language and action[END_REF][START_REF] Willems | Neural evidence for the interplay between language, gesture, and action: a review[END_REF]. Therefore, when the motor action simulation processes are slowdown by peripheral changes such as sensorimotor deprivation, action verb understanding are slowdown in turn in a somatotopic way (i.e., according to the immobilized body-part).

General Discussion

The current experiments were designed to explore the effects of a brief delay of sensorimotor deprivation on action verb processing. Without immobilization, we observed systematically faster response times associated with the judgements of hand-action verbs than with foot-action verbs. This difference disappeared after 24 hours of immobilization, as evidenced by less improvement in hand-action verb judgement (Experiment 2). Interestingly, Experiment 2 also showed that the understanding of action verbs was directly related to an individual's vividness of motor imagery (1 st -person perspective only) before and after immobilization.

Overall, these findings clearly corroborate previous studies that show that sensorimotor representations are directly involved in action verb processing and is associated with activation of a mirror neuron system [START_REF] Andres | Contribution of motor representations to action verb processing[END_REF][START_REF] Aziz-Zadeh | The human mirror neuron system and embodied representations[END_REF][START_REF] Aziz-Zadeh | Congruent embodied representations for visually presented actions and linguistic phrases describing actions[END_REF][START_REF] Bidet-Ildei | Reading action word affects the visual perception of biological motion[END_REF][START_REF] Bidet-Ildei | Are judgments for action verbs and point-light human actions equivalent?[END_REF][START_REF] Tettamanti | Listening to action-related sentences activates fronto-parietal motor circuits[END_REF]. Moreover, the present findings demonstrate for the first time that 24 hours of limb immobilization is sufficient to modify action verb processing, confirming the embodied view of cognition, which indicates that higher cognitive functions are directly related to sensorimotor experience [START_REF] Barsalou | Perceptual symbol systems[END_REF][START_REF] Barsalou | Grounded cognition[END_REF].

As proposed in previous studies, the effects of immobilization might be related to plastic changes in the sensorimotor areas [START_REF] Huber | Arm immobilization causes cortical plastic changes and locally decreases sleep slow wave activity[END_REF][START_REF] Moisello | Short-term limb immobilization affects motor performance[END_REF] or to difficulties in accessing or activating sensorimotor representations due to reduced processing of proprioceptive feedback from the immobilized hand [START_REF] Meugnot | Functional plasticity of sensorimotor representations following short-term immobilization of the dominant versus non-dominant hands[END_REF][START_REF] Toussaint | Short-term limb immobilization affects cognitive motor processes[END_REF]. However, as the task used in the present experiment implies at least implicitly the evocation of motor imagery, it stays difficult here to distinguish whether short-term immobilization affects action verbs processing in general or only tasks implying motor imagery. As several works evoke that motor imagery occurs automatically as a component of everyday language processing [START_REF] Pulvermüller | Brain mechanisms linking language and action[END_REF][START_REF] Zwaan | Seeing, acting, understanding: motor resonance in language comprehension[END_REF], it may be that short-term immobilization can affect all tasks implying action verbs processing but this would be assessed in future experiments.

Importantly, our findings show that short-term upper limb immobilization specifically affected the processing of hand-action verbs with less progress between pre-tests and post-tests.

This confirms that 24 hours of sensorimotor deprivation has an effect on specific body parts representation [START_REF] Meugnot | Selective impairment of sensorimotor representations following short-term upper-limb immobilization[END_REF]. As participants had their arms splinted, it is possible that the effect observed could be simply related to a grabbing attention for hand movements which could cause an interference (Connell & Lynott, 2012). Even this hypothesis can be properly assessed only by the use of a more implicit task (e.g., lexical decision task) or by the adding of an attention load condition (e.g., memorise hand-action verbs), this attention explanation is less probable because the experimental manipulation was known by both groups of participants and a pure attentional explanation fails to account for the facilitation obtained in the control group. More probably, the slowdown observed for the judgement of hand-action verbs after immobilization could be related with the slowdown of sensorimotor processes attributed to the decrease in input-output signal processing for the immobilized right hand (Meugnot, Agbangla, Almecija, & Toussaint, 2015;[START_REF] Meugnot | The embodied nature of motor imagery processes highlighted by short-term limb immobilization[END_REF][START_REF] Meugnot | Functional plasticity of sensorimotor representations following short-term immobilization of the dominant versus non-dominant hands[END_REF][START_REF] Toussaint | Short-term limb immobilization affects cognitive motor processes[END_REF]) could be behind the limited improvement in judging hand-action verbs from pre-tests to post-tests observed in the immobilized group.

Notably, action verb understanding was directly related to an individual's vividness of motor imagery before and after the immobilization. This finding confirms that imagery capabilities are directly implicated in action verb processing (see [START_REF] Aziz-Zadeh | Embodied semantics for actions: findings from functional brain imaging[END_REF][START_REF] Kemmerer | The Two-Level Theory of verb meaning: An approach to integrating the semantics of action with the mirror neuron system[END_REF][START_REF] Pulvermüller | Brain mechanisms linking language and action[END_REF][START_REF] Willems | Neural evidence for the interplay between language, gesture, and action: a review[END_REF] for reviews). Interestingly, these correlations exists only with the first-person perspective suggesting that action verb processing is specifically related with motor (not visual) imagery competencies [START_REF] Hostetter | Visible embodiment: gestures as simulated action[END_REF]. Moreover, this pattern persists after 24 hours of immobilization suggesting that motor imagery is systematically involved in semantic decision tasks that require action verb processing (Pulvermuller, 2005 ;see Willems, Toni, Hagoort, & Casasanto, 2010 for contreversials evidences).

However, as previous works have shown that sensorimotor disorders can affect the use of firstperson perspective in case of co-speech action gestures [START_REF] Humphries | A third-person perspective on co-speech action gestures in Parkinson's disease[END_REF], future researches will need to determine whether correlations between action verb processing and motor imagery capabilities persist with a longer period of immobilization that could induce stronger sensorimotor deficits.

In conclusion, the present work shows for the first time that short-term limb immobilization can affect action verb processing, the quick update of sensorimotor representation having direct consequences on the well-established associations between language and action. Moreover, our study confirms by using an original methodology (i.e., short-term upper limb immobilization in healthy participants) that processing action verbs activates somatotopic representations in the sensorimotor cortex [START_REF] Hauk | Somatotopic representation of action words in human motor and premotor cortex[END_REF]. This means that a specific re-education of language could be envisaged after sensorimotor deprivation. Importantly, a novel contribution of our approach is that our work assesses a novel paradigm (i.e., short-term immobilization) to study crosstalk between the motor system and action verb processing.

Table A . 3 : List of the hands-action verbs used in the semantic decision task. English translation of each verb appears in brackets.

Hands-action verbs

Figure 1 .

 1 Figure 1. Response times (ms) according to group (control / immobilized) and action verb (hand-

Figure 2 .

 2 Figure 2. Index of Performance Improvement (%) according to group (control / immobilized) and

-action verbs Foot-action verbs ANOVA (by item)

	The semantic decision task was divided into two phases. During the first training phase,
	participants were shown 3 hand-action verbs and 3 foot-action verbs in a random order. No time
	constraint was imposed during the training phase. During the second experimental phase, the
	participants were shown 132 randomly presented action verbs (3 trials each of 22 hand-action verbs
	FQ	42.9	43.4	F(1,42) = 0.01, p =.98
	LETT	6.95	7.32	F(1,42) = 1.02, p =.38
	SYLL	2.41	2.59	F(1,42) = 1.20, p =.28

FUNDING

This work was supported by a grant from the National Centre for Scientific Research (CNRS, France).

ETHICAL CONSIDERATION

All aspects of this study were performed in accordance with the ethical standards set out in the 1964 Declaration of Helsinki and its later amendments. The study was approved by the local ethics committee of the institution-Centre for Research in Cognition and Learning (CeRCA)-and was

the immobilized group (p=.008), whereas no significant difference was observed in the control group (p=.74) (Figure 2). Finally, we examined whether performance in the semantic decision task (for the handaction and foot-action verbs) depended on imagery capacity as evaluated by the VMIQ (Isaac et al., 1986). We examined pairwise Pearson's correlations between the response times recorded in the semantic decision task and the first-person perspective imagery scores, as well as between the response times and the third-person perspective imagery scores for the control group and the immobilized group in the pre-test and post-test procedures (see Table 3).

Table3. Correlation coefficients (and p-values) between response times recorded for the hand-action and foot-action verbs and the imagery scores (VMIQ) for the control and immobilized groups in the pre-tests and post-tests. An asterisk * indicates a significant correlation. As illustrated in Table 3, significant correlations were found for only first-person perspective scores, which were observed for both groups in the pre-test and post-test procedures. The positive correlation coefficients indicated that participants who were faster in performing the semantic decision tasks were the participants who reported lower first-person perspective imagery scores (i.e., those with a high capability for imaging from an internal perspective) whenever the semantic conducted in accordance with national norms and guidelines for the protection of human subjects (CPP, N°2014-A00053-44). All participants gave their informed consent prior to their inclusion in the study and received €20 for their participation. All authors declare that they have no conflict of interest.

Group Session

APPENDIX

Table A .1 : Means (Standard Deviations) for accuracy (in %) and response times (in ms) as a function of hand-action and foot-action verbs for both groups (control and immobilized).

Table A .2 : Means (Standard Deviations) for accuracy (in %) and response times (in ms) as a function of hand-action and foot-action verbs and tests (pre-test and post-test) for both groups (control and immobilized).

Table A .4 : List of the feet-action verbs used in the semantic decision task. English translation of each verb appears in brackets.

Feet-action verbs


Yves Jannot

Benjamin Rémy

Alain Degiovanni Lemta

Measurement of thermal conductivity and thermal resistance with a Tiny Hot Plate

Keywords: measurement technique, thermal conductivity, stationary hot plate, quadrupolar model

A new thermal conductivity and thermal resistance measurement method is presented. This stationary method is dedicated to heterogeneous materials and systems to which transient measurement methods cannot be applied. The experimental device is detailed and an exact heat transfer model based on integral transforms is developed. The estimation method both with the uncertainty evaluation process is presented. Experimental results obtained with materials which thermal conductivities are known validate the method. An application to the measurement of the thermal resistance of a heterogeneous material (double glazing insert) is presented.

NOMENCLATURE

Introduction

Energy control requires to the development of more and more complex materials and systems which are difficult to characterize. It may concern thin thermal insulating materials for buildings, porous membranes for fuel cells or spacers for double-glazing. In all these applications concerning heterogeneous materials, classical transient techniques (flash method [START_REF] Matteis | Thermal diffusivity measurements of metastable austenite during continuous cooling[END_REF], hot wire [START_REF] Franco | An apparatus for the routine measurement of thermal conductivity of materials for building application based on a transient hot-wire method[END_REF], hot strip [START_REF] Jannot | Simplified estimation method for the determination of thermal effusivity and thermal conductivity with a low cost hot strip[END_REF], or hot disc [START_REF] Gustafsson | Transient plane source techniques for thermal conductivity and thermal diffusivity measurements of solids materials[END_REF]) cannot be successfully used. This is particularly the case with materials for which it is not possible to define an equivalent homogeneous material and thus a thermal diffusivity. The guarded hot plate method [START_REF] Zhou | Thermal conductivity of boron nitride reinforced polyethylene composites[END_REF] needs quite large samples that may be difficult to obtain for some specific materials. It is thus necessary to consider another approaches, less precise but allowing a thermal characterization for these types of materials. Among these approaches, a Tiny Hot Plate device using a Peltier element as a fluxmeter is proposed. The development of this technique is based on the works performed by Batsale and Degiovanni [START_REF] Batsale | Mesure de résistance thermique de plaques minces à l'aide d'une miniplaque chaude[END_REF]. An application for the measurement of the thermal resistance of spacers for double glazing will be presented.

Principle and experimental device

Figure 1 describes the principle of the experiment. The heating element is composed of a 39 x 39 x 0.16 mm 3 plane resistance inserted between two Kapton foils. Its electric resistance is Re = 40,5 Ω. All others elements exhibit a square geometry (a = 40 mm).

Temperatures T 0 , T 5 and T air are measured through K-type thermocouples with a 0.025 K resolution. Standard deviations in the temperature measurement can be reduced in steady-state regime by increasing the number of measurements. Tension applied through the heating element is measured with a 0.01 V accuracy. Contact resistances between the different elements are reduced using a thermoconductive grease and a loading system. The experimental device is presented in Figure 2 The principle of the experiment is to apply a tension U through the heating element to obtain air 5 T T =

. This allows us to get rid of the lateral heat losses in the two copper elements above the sample. The dissipated flux Φ 9↑ on the upper part of the heating element is measured through a Peltier element, which is used and calibrated as a fluxmeter. It allows an estimation of the value of the dissipated Φ 9↓ on its bottom face through the following relation:

φ 9↓ = φ 9 -φ 9↑ (1)
With :

Re U 2 9 = φ Temperature 0
T is fixed through a thermostat or a Peltier element. This device measures the thermal resistance between the thermocouples 0 T and 5 T :

S e Rc 2 R 2 R Cu th th λ + + = (2)
The measurement of the thermal conductivity is only possible in the case

Rc 2 R 2 S e Cu th + > λ
and is given by:

) T T ( S e 0 5 9 - φ = λ ↓ (3)
The different contact resistances have been estimated by applying a tension of 10V through the heating element and by measuring the temperatures T 0 and T 5 without any sample. Considering λ Copper = 400 W.m -1 .K -1 , we obtain r c = 10 -4 K.W -1 .m -2 . Thus, this method can be only applied for samples with e > 0,01 λ, in order to limit the errors caused by the contact resistance to a value lower than 2%. This limits the using of this method to low or middle conductivity materials (see § 4.2). Another method to get free of the contact resistances consists in making two measurements on the same material (1 and 2) with two different thicknesses:

S e Rc 2 R 2 R 1 Cu th 1 th λ + + = (4) S e Rc 2 R 2 R 2 Cu th 2 th λ + + = (5)
It can be deduced:

( )

2 th 1 th 1 2 R R S e e - - = λ (6)
Where:

2 9 0 5 2 th T T R         Φ - = ↓ and 1 9 0 5 1 th T T R         Φ - = ↓
This method is particularly interesting for thin samples (see § 4.2). Even though this condition is checked, the value given by relations (3) or ( 6) is affected by convective effects through the lateral faces of the sample that may affect the input or output fluxes through the sample, which are different from the Φ 9↓ value used for the estimation of λ . A modelling of the system taking into account these heat losses allows us to reduce the estimation errors.

Quadrupolar model

The system can be modelled through the quadrupole technique [START_REF] Maillet | Thermal quadrupoles[END_REF] by taking into account the lateral heat losses with a fin's model. The temperature in each stack element can be obtained by solving the following equation:

( ) 0 T T S P h x T air 2 2 = - λ - ∂ ∂ (7)
Where: h global heat transfer coefficient P perimeter λ thermal conductivity S cross section Front and Back faces temperatures of each element are linearly linked in the Laplace domain through the following quadrupole relation: The measured parameters are: T 0 , T 5 , φ 9 et φ 9↑ .

      φ -       =       φ - + + i air i 1 i
Setting θ = T -T air , one can write (cf. electric schemes in figure 3) :

[ ][ ][ ][ ][ ][ ][ ][ ][ ]       φ θ       =       φ θ =       φ θ ↓ 0 0 90 90 90 90 0 0 9 9 D C B A 1 M 1 RC 2 M 2 RC 3 M 3 RC 4 M 4 RC 5 M (9)
and: ( )

[ ] [ ][ ] [ ]       φ θ       =       φ =       φ θ ↓ 5
0 C B D A B D 0 90 90 90 90 9 90 9 90 = θ - - φ - θ ↓ (11) ( ) 0 C B D A B D
= θ - - φ - θ - - φ ↓ ↓ (13)
The accuracy of λ obtained from this equation is better than the approximated value given by relation (3).

Validation of the lateral heat losses calculation by the fin model

In the proposed model, the fin's approximation is applied to take into account the lateral heat exchanges. Under this condition, no 3D resolution methods are required to solve the problem. Before validating this approximation, we will first show that it is an important point to take into account the lateral exchanges in the conductivity estimation.

Error caused by neglecting the lateral heat losses

To show the significant effect of the lateral exchanges in the thermal conductivity measurement, the lower part of the apparatus has been thermally simulated with the help of the software COMSOL (see Figure 4). To simplify the analysis, thermal contact resistances have been neglected. Table 1 give the characteristics of the whole stack elements. To neglect the contact resistances, thicknesses have been chosen with e/λ > 0.01. . For different values of the heat reansfer coefficient h , the conducitivity value h has been calculated from the classical relation:

) T T ( S e 0 5 estimated - φ = λ (14)
The results are represented in figure 5. We can notice that the lateral heat losses lead to an under-estimation of the thermal conductivity λ. The error, which can be larger than 10%, mainly depends on the ratio λ 2 e h . This result can be easily explained. Indeed, the lateral heat flux dΦ p at a given location z can be written as:

( ) [ ] dz T z T P h d air s p - = φ
(15) T s (z) being the surface temperature of the sample. As a consequence, the temperature difference through a given element dz' at location z' is:

      φ - φ λ = ∫ ' z 0 p d S ' dz dT (16)
or for a sample with a thickness e:

      φ - φ λ = ∆ ∫ ∫ ' z 0 p e 0 d S ' dz T (17)
Assuming in first approximation Ts(z) as being linear:

( )

5 5 0 s T z e T T z T + - = (18)
and taking into account the fact that T 5 = T air , we find:

T 6 e S P h S e T 2 ∆ λ + λ φ = ∆ (19)
The imposed flux allowing to keep ∆T constant is:

        λ - ∆ λ = φ 6 e S P h 1 T e S 2 (20)
And consequently, relation (14) becomes:

        λ - λ = λ 6 e S P h 1 2 real estimated (21)
By taking into account the geometry of our stack (P/S 100 m -1 ):

        λ - λ = λ 2 real estimated e h 7 , 16 1 (22)
Figure 5 clearly shows that the approximating relation ( 22) is not precise enough for particular cases.

Fin's Model

For simplifying the calculations, an axial symmetry is assumed in our system. The square section can be viewed as a circular section by keeping the P/S ratio as constant and fixing the radius R of the circular section equal to a half-side a/2 of the square section. In that case, the axisymetric 2D heat transfer equation is written:

0 z T r T r r r 1 2 2 = ∂ ∂ +       ∂ ∂ ∂ ∂ (23) ( ) air T T h r T - = ∂ ∂ λ - if r = R (24) 0 r T = ∂ ∂ if r = 0 (25) Let: air T T - = θ and ( ) dr r r T R 2 R 0 2 ∫ = θ
Integrating relation (23) over [0, R] and using relations (24) and (25) yields:

0 R h 2 dz d R 2 2 = θ λ - θ (26)
θ R being the surface temperature in r = R.

The classical fin's approximation consists in working the mean radial temperature

θ = θ R : 0 R h 2 dz d 2 2 = θ λ - θ (27)
We find again the equation [START_REF] Maillet | Thermal quadrupoles[END_REF].

The solution of equation ( 27) (or ( 7)) can be represented (cf. [START_REF] Zhou | Thermal conductivity of boron nitride reinforced polyethylene composites[END_REF]) by: After a second order asymptotic expansion of sh(α e), equation (31) exhibits the same form as equation ( 21), which thus appears to be a simplifying form of equation (31). Nevertheless, let notice that the relation (31) is only valid under the assumption θ = θ R . In particular, Figure 5 showed that this approximation is not always true in our facility.

      φ θ       =       φ

Exact Solution through integral transforms

To obtain the exact solution of the set of equations ( 23), ( 24) and (25), integral transforms can be used. In our case, the adapted integral transform is the Hankel transform with :

      ω R r J n 0
being the eigenfunctions and n ω the eigenfunctions defined as the roots of the following equation:

( ) ( ) n 0 n 1 n J Bi J ω = ω ω (32)
with J 0 and J 1 being the zeroth and first order Bessel functions of the first kind and

λ = R h Bi .
The integral transform is defined by: dr R

r J r ~R 0 0 ∫       ω θ = θ (33)
with the associated norm:

( ) ( ) [ ] n 2 1 n 2 0 2 R 0 n 2 0 n J J 2 R dr R r J r N ω + ω = ∫       ω θ = (34)
Applying the integral transform, equation (23

) becomes: 0 R dz d 2 2 n 2 2 = θ ω - θ (35)
(same form as the fin's equation)

The solution is given by: ( ) ( )

      ω +       ω = ω θ R z sh B R z ch A z ,
n n 1 2 0 R 0 n 0 0 0 J R dr R r J r ~ωω θ = ∫       ω θ = θ if z = 0 (37) ( ) n n 1 2 e R 0 n 0 e e J R dr R r J r ~ωω θ = ∫       ω θ = θ if z = e (38)
Constants A and B can be determined from boundary conditions (37-38):

( ) ( )                   ω θ +       ω       ω             ω θ - θ ω ω = ω θ R z ch R e sh R z sh R e ch J R z , ~n 0 n n n 0 e n n 1 2 n (39)
Finally, the solution is given by:

( ) ( ) ∑       ω ω θ = θ n n n 0 n N R r J z , z , r (40)
As the eigenvalues are function of the Biot number, they are different for each layer of the stack and require the computation of a large number of eigenvalues at each simulation for solving the heat transfer through the whole stack. The fin's approximation already gives quite satisfactory results. Thus, the idea in the following section will consist in using a simplifying solution to improve the classical fin's approximation.

Fin model correction

Equation (26) that can be also written as:

0 R h 2 dz d R ( ) ( ) ( ) n 0 n n n R J N ) z , z ω ∑ ω θ = θ (43)
In the more general case, the difficulty of this correction comes from the fact that it is a function of the z coordinate. To validate the proposed method, the θ θ R ratio will be calculated in z = e/2 by keeping either 6 terms or 100 terms in the series. Conversely, if we consider only the first term of the series, we find:

Bi 2 2 0 R ω = θ θ (44)
The main interest of this correction is that it is very easy to implement in practice substituting To validate this method, the values of the imposed flux Φ determined by COMSOL and the temperature difference T 5 -T 0 = 5 °C have been considered as experimental results. This data are then been used to estimate the conductivity λ by an inverse method based on the quadrupole model taking into account the fin's effect. The estimated value are finally compared with the nominal values used in COMSOL simulations. The obtaining results for different values of the thermal conducitivty λ, thickness e and heat transfer coefficient h are given in Table 2.

Validation of the quadrupolar model with the corrected fin model

λ real = 1 W.m -1 .K -1 e = 20mm λ real = 2 W.m -1 .K -1 e = 40mm Φ COMSOL W.
We can notice that the modified model using the first 6 terms of the series calculated with the approached function given in Appendix, allow us in all cases an estimation of the thermal conductivity λ better than 1% compared to its nominal value. The one term model is of course less precise but also very easy to use and gives quite satisfactory results (maximum error of about 2%). These results must be compared with those obtained by the classical fin's model that yields to large errors (> 10 %) in different cases.

Estimation errors on the thermal conductivity

In practice, it is difficult to precisely evaluate the heat transfer coefficient h and the contact resistances Rc between the sample and copper elements. In the next sections, we are going to give an estimation of the uncertainties caused by these ill-known parameters.

Experiment with a single sample

The assumption of a convective heat transfer coefficient included between 5 and 10 W.m -2 . K - 1 and contact resistances Rc between 5.10 -5 m 2 . K.W -1 and 2.10 -4 m 2 . K.W -1 is made. Through the quadrupole model taking into account the lateral heat losses (100 terms), tension required to obtain T 5 = T air is calculated for each configuration and the two following "extreme" conditions:

1. h = 10 W.m -2 .K -1 et rc = 2.10 -4 m 2 .K.W -1 2. h = 5 W.m -2 .K -1 et rc = 5.10 -5 m 2 .K.W -1
We perform for each values two estimations of λ (λ 1 and λ 2 ) by an inverse method based on the quadrupole model (relation (13)) with a correction of the lateral heat losses (6 terms) and using a fixed (mean) values for h = 7.5 W.m -2 . K -1 and rc = 10 -4 m 2 . K.W -1 . The error on the estimated value of λ caused by uncertainties on h and rc is then defined from the two differences |λ -λ 1 | et |λ -λ 2 |. The corresponding results are represented in the figure 7.

Precision improvement with two samples of different thickness

It is possible to get rid of the uncertainties on h or rc by making two experiments with two different thicknesses samples. The 3 unknown parameters λ, h and rc are obtained using relation (13) (in the single sample experiment, h and rc values are fixed to mean nominal values in the estimation of λ). * Case of conducting materials ( λ > 2 W.m -1 .K -1 ) In this case, errors introduced by contact resistance and the heat transfer coefficient (h value between 5-10 W.m -2 .K -1 ) has no effect on the estimation (see Table 2). The h value is then fixed to a mean value (h = 7.5 W.m -2 .K -1 ) and λ and rc are estimated by an inverse method from relation (13) written for each measurement. * Case of insulating materials (e/λ > 0.1 m 2 .K.W -1 ) In this case, contact resistance errors can be neglected (lower than 1%) and rc is fixed to a mean value (rc = 10 -4 m 2 .K.W -1 ). λ and h are estimated by an inverse method from two equations given by relation (13) written for each experiment.

* Simulation results

For a conducting material with λ = 10 W.m -1 .K -1 , h = 10 W.m -2 .K -1 and rc = 2.10 -4 m 2 .K.W -1 and for two different thicknesses e 1 = 1 cm and e 2 = 2 cm. The results obtained from each simulation (setting h = 7.5 W.m -2 .K -1 and rc = 10 -4 m 2 .K.W -1 ) are λ 1 = 8.33 W.m -1 .K -1 and λ 2 = 9.07 W.m -1 .K -1 . Using the two experiments simultaneously and setting h = 7.5 W.m -2 .K -1 , we obtain: λ 12 = 9.91 W.m -1 .K -1 and rc = 1.96 m 2 .K.W -1 .

For an insulating material with λ = 0.025 W.m -1 .K -1 , h = 5 W.m -2 .K -1 and rc = 2.10 -4 m 2 .K.W -1 and for two different thicknesses e 1 = 2 cm and e 2 = 5 cm. The results obtained for each independent experiment (setting h = 7.5 W.m -2 .K -1 and rc = 10 -4 m 2 .K.W -1 ) are λ 1 = 0.0272 W.m -1 .K -1 and λ 2 = 0.0298 W.m -1 .K -1 respectively. Using the two experiments simultanesously and setting rc = 10 -4 m 2 .K.W -1 , we obtain: λ 12 = 0.0250 W.m -1 .K -1 and h = 5.00 W.m -2 .K -1 .

Experimental results

Using relation (13), λ 1 has been estimated setting h = 5 W.m -2 .K -1 , rc = 5.10 -5 m 2 .K.W -1 and λ 2 has been obtained setting h = 10 W.m -2 .K -1 , rc = 2.10 -4 m 2 .K.W -1 . Glass Sample:

Experiment 1: e = 5.92 mm

: λ 1 = 1.013 W.m -1 .K -1 λ 2 = 1.075 W.m -1 .K -1
Experiment 2: e = 2.07 mm : λ

1 = 1.008 W.m -1 .K -1 λ 2 = 1.185 W.m -1 .K -1 Experiment 1 + Experiment 2: λ = 1.015 W.m -1 .K -1 , rc = 5.01.10 -5 m 2 .K.W -1
The result does not depend on h for values included in the range 5 W.m -2 .K -1 < h < 10 W.m -2 .K -1 . An independant measurement obtained from a differential calorimeter and a flash experiment gives: a = 5.00.10 -7 m 2 .s -1 , ρc = 2.00.10 6 J.kg -1 leading to: λ = 1.00 W.m -1 .K -1 The difference between the two results is about of 1,5%. Titanium Sample: e = 3.10 mm : λ 1 = 4.84 W.m -1 .K -1 λ 2 = 9.15 W.m -1 .K -1

The Hot-disk and Hot plane probe lead to the following value: λ = 6.60 W.m -1 .K -1 High density Polyethylene Sample: e = 7.88 mm : λ

1 = 0.396 W.m -1 .K -1 λ 2 = 0.407 W.m -1 .K -1
The manufacturer value is: λ = 0.42 W.m -1 .K -1 Expended Polystyrene sample with 38 kg.m -3 density: e = 18.00 mm : λ 1 = 0.0334 W.m -1 .K -1 λ 2 = 0.0367 W.m -1 .K -1 PVC Sample: e = 5.88 mm : λ 1 = 0.1808 W.m -1 .K -1 λ 2 = 0.1858 W.m -1 .K -1

Double glass window spacers

Working in a steady-state regime allows us to measure the thermal resistances of heterogeneous materials. An example is given by the thermal characterization of double glass window spacers (see Figure 8).

To perform this measurement, the spacers have been split into six 40 mm long pieces gathered to obtain a 40 mm x 40 mm sample with thicknesses included between 15 and 20 mm (air layer thickness of the double glass). Table 3 gives the spacers properties and the thermal resistance by unit length (r thm in m.K.W -1 ). For comparison, a theoretical value (r thc in m.K.W -1 ) obtained from a parallel model is given. As properties of each layer (λ and e) are badly known (see Table 4), this last value must be considered as a simple order of magnitude.

Conclusion

A simple and easy to use apparatus allowing the measurement of the thermal conductivity of insulators or low conductivity materials (thermal conductivity included between 0.025 down to 5 W.m -1 . K -1 ) as well as the thermal resistance of heterogeneous systems (from 2.10 -1 to 10 - 3 m 2 .K.W -1 ) has been presented. The main interest of this facility stays in its ability to perform measurement on sample with small dimensions (40 mm x 40 mm width and 1 to 30 mm height). A precise and theoretical study has been carried out on the effects of the lateral heat losses (heat transfer coefficient h) and thermal contact resistance rc and allowed us to set up an optimal estimation procedure taking into account their effects. The theoretical model proposed in this study is based on the fin's approximation but has been extended in the cases of large Biot number (>2), which allows us to keep the quadrupole approach and to develop a accurate and simple estimation procedure. Nevertheless, when the sample is available with a single thickness, it is not possible to get totally free from h and rc parameters (the thermal conductivity being correlated to heat losses and contact resistances). We have given in this study the resulting uncertainties versus the thermal conductivity and the thickness of the sample. In the case of double thicknesses samples, it is possible to get almost totally free from h and rc values by making two different experiments on two different thicknesses samples. Finally, different experimental results have been presented. βi (Bi = 0)

0

θ

  can be removed from these two equations to lead to the following equation where the only unknown quantities are the sample thermal conductivity λ , the heat transfer coefficient h and the copper-sample contact resistances. Indeed, ↓ Φ 9 is a known quantity that can be obtained from the flux ↑ Φ 9 given by the Peltier element (used as a fluxmeter) and the relation (1). If temperatures 5 T and a air T are closed then the flux ↑ Φ 9 is always very low and the fluxmeter calibration has no significant influence on the measurement.

Table 1 :

 1 Details of the simulated device

Figure 5 :

 5 Figure 5 : Ratio of estimated and real values of the thermal conductivity

  θ , 0 Φ and e Φ represent the mean temperatures and heat fluxes in z = 0 and in z = e, respectively.

Figure 6 :ω

 6 Figure 6 : Relative difference between the values of λ estimated obtained by simulation with FEMLAB and calculated with relation (45)

Table 2 :

 2 Values of the thermal conductivity estimated with COMSOL and with the fin corrections.

	h	W.m	-2 .K -1	5		10	5	10
					λ real = 0,1 W.m	-1 .K -1	λ real = 0,1 W.m	-1 .K -1
					e = 1mm		e = 5mm
	Φ COMSOL	W.m	-2	498.1	497.0	98.0	96.4
	λ without convection				0.0996	0.0995	0.0980	0.0965
	λ fin				0.0999	0.0999	0.1001	0.1005
	λ fin corr. 1 term	W.m	-2 .K -1	0.0999	0.0998	0.0997	0.0991
	λ fin corr. 6 terms				0.0999	0.0999	0.0999	0.1000
	λ fin corr. 100 terms				0.0999	0.0999	0.0999	0.1000
					λ real = 0.1 W.m	-1 .K -1	λ real = 0.1 W.m	-1 .K -1
					e = 10mm	e = 20mm
	Φ COMSOL	W.m	-2	46.53	44.07	19.32	16.44
	λ without convection				0.0931	0.0882	0.0773	0.0654
	λ fin				0.1010	0.1031	0.1046	0.1123
	λ fin corr. 1 term	W.m	-2 .K -1	0.0993	0.0980	0.0995	0.0981
	λ fin corr. 6 terms				0.1000	0.1000	0.1004	0.1006
	λ fin corr. 100 terms				0.1000	0.1000	0.1004	0.1005
					λ real = 0.2 W.m	-1 .K -1	λ real = 0.5 W.m	-1 .K -1
					e = 4mm		e = 10mm
	Φ COMSOL	W.m	-2	248.0	246.5	245.6	241.9
	λ without convection				0.1984	0.1972	0.4912	0.4838
	λ fin				0.1999	0.2000	0.4999	0.5006
	λ fin corr. 1 term	W.m	-2 .K -1	0.1998	0.1995	0.4995	0.4991
	λ fin corr. 6 terms				0.1999	0.1999	0.4997	0.4997
	λ fin corr. 100 terms				0.1999	0.1999	0.4998	0.4997

Table 3 :

 3 Spacers characteristics

		Sample1	Sample 2	Sample 3
	Layer 1	Aluminium	Plastics	Plastics
		e = 0.3 mm	e = 1 mm	e = 1 mm
	Layer 2	Air	Air	Air
		E = 6 mm	E = 4.45 mm	e= 5 mm
	Layer 3	Aluminium	Plastics	Plastics
		e = 0.3 mm	e = 1 mm	e = 1 mm
	Layer 4		Aluminium	Steel (inox)
			e = 0.04 mm	e = 0.1 mm
	r thm (m.K.W -1 ) r thc (m.K.W -1 )	0.126 0.129	2.32 2.01	6.37 7.12

Table 4 :

 4 Thermal conductivity of the layers

	Material	Aluminium	Air	Plastics	Steel (inox)
	λ (W.m -1 .K -1 )	200	0.025	0.3	20

Appendix 2

Relation between U Peltier and (T 5 -T air )

In practice, it is difficult to precisely measure the temperature T air in a location where convection is weak (low heat transfer coefficient between air and thermocouple) and where radiation takes place between one side of the thermocouple and walls submitted to large temperature gradients and in the other side with a cooling surface. A more precise estimation of T air can be obtained through the measurement of the tension of the Peltier element.

= -

The error on the difference T 5 -T air is less than 0.036K that is more precise than a direct measurement of the temperature in a motionless air.