Yang Si

Chloé Grazon

Gilles Clavier

Jutta Rieger

Jean-Frédéric Audibert

Bianca Sclavi
email: sclavi@lbpa.ens-cachan.fr

Rachel Meallet-Renault
email: rachel.meallet-renault@u-psud.fr

Rachel Méallet-Renault

Rapid and accurate detection of Escherichia coli growth by fluorescent pHsensitive organic nanoparticles for high-throughput screening applications

Keywords: bacteria, fluoresceinamine, organic nanoparticles, rapid detection, high-throughput screening (HTS), pH detection

Rapid detection of bacterial growth is an important issue in the food industry and for medical research. Here we present a novel kind of pH-sensitive fluorescent nanoparticles (FANPs) that can be used for the rapid and accurate real-time detection of Escherichia coli growth. These organic particles are designed to be non-toxic and highly water-soluble. Here we show that the coupling of pH sensitive fluoresceinamine to the nanoparticles results in an increased sensitivity to changes in pH within a physiologically relevant range that can be used to monitor the presence of live bacteria. In addition, these FANPs do not influence bacterial growth and are stable over several hours in a complex medium and in the presence of bacteria.

The use of these FANPs allows for continuous monitoring of bacterial growth via real-time detection over long time scales in small volumes and can thus be used for the screening of a large number of samples for high-throughput applications such as screening for the presence of antibiotic resistant strains.

Introduction

Rapid detection of bacterial growth is important for medical diagnostics to identify bacterial contamination in medical settings (Barker et al., 2010;Palavecino et al., 2006;[START_REF] Washington | Principles of Diagnosis[END_REF] and in the food industry in order to prevent contamination of food, air, and water [START_REF] Farris | Luminescence Techniques for the Detection of Bacterial Pathogens[END_REF]Ko et al., 2010;Wang et al., 2012).

The classic techniques used to detect bacterial growth rely on plating the sample on agar and counting of colonies after 37-48 hours or more; however this is timeconsuming and requires large amounts of samples and materials and may miss most types of bacteria (Lazcka et al., 2007;Nayak et al., 2009;[START_REF] Pourciel-Gouzy | The 17th Euopean Conference on Solid-State Transducers[END_REF].

Alternative methods have been developed to reduce the amount of time and sample necessary for a reliable measurement, such as the polymerase chain reaction (PCR).

Although PCR and its variants can detect bacteria within a relatively short time (Gopinath et al., 2014;Noble and Weisberg, 2005), issues like primer design and PCRinhibitory effect of complex food matrices, in addition to its high cost, make PCR detection difficult to be a routine procedure (Gunaydin et al., 2007). Another effective method is enzyme-linked immunoabsorbent assay (ELISA). This technology uses an antigen-antibody reaction to detect unique microorganism. Different technologies have been developed to enhance ELISA measurements (Chunglok et al., 2011;Jain et al., 2012;[START_REF] Seo | [END_REF]. These methods are highly sensitive and provide specific detections. However, this approach requires considerable amounts of expensive reactants such as antibodies and a high level of antigen for reliable detection (Mouffouk et al., 2011). Microfluidic devices are also being developed for the rapid detection of small amounts of bacteria (Chang et al., 2015), however they require relatively expensive laboratory instruments and experienced operators. Biosensing methods based on nanostructures have shown great potential for bacteria detection, such as gold nanoparticle for E.coli detection (Hassan et al., 2015), magnetic nanoparticles system (Wan et al., 2014), and fluorescent silica nanoparticles (Chen et al., 2015). For these nanostructures, the potential toxicity effect on bacteria is an important issue to be considered (Adams et al., 2006;Wang et al., 2011). Polymeric organic nanoparticles, however, provide a promising platform to solve this problem. Moreover, different functional molecules can easily be immobilized on their surface to enhance their selectivity and sensitivity.

The growth of bacteria is often associated with a decrease in the pH of the growth medium due to a release of acidic metabolites such as acetic acid, lactic acid and CO2 (John et al., 2003;[START_REF] Young | [END_REF]. Different kinds of pH sensors have therefore been used to measure the growth of bacteria based on this principle (Agayn and Walt, 1993;Badugu et al., 2008;Wang, F. et al., 2014;Wang, X. et al., 2013). Unfortunately, the useful life-span of these sensors is limited due to photobleaching and instability in a complex medium. More importantly, some dyes were developed for single assay-based measurements and are often unsuitable for real-time detection applications or to accommodate a large number of samples for high-throughput screening.

While high-throughput screening of bacterial growth has been conducted extensively to search for novel compounds with antibacterial activity (Zlitni et al., 2009), such efforts have been limited by the sensitivity and accuracy of bacterial resistance detection. Traditional optical density based (OD) measurements are limited by cells forming chains, clumps, filaments, or aggregates and are difficult to perform in complex growth media which can result in light scattering or absorption and interfere the detection by OD (Chalova et al., 2003;[START_REF] Gasol | [END_REF]. Optical density was found to be linearly correlated with culture biomass between 0.01-0.3, but there might be variations depending on the growth phase and at higher densities the OD deviates from linearity (Martens-Habbena and Sass, 2006). Furthermore, OD measurements cannot distinguish between live, but slow growing, and dead cells. Screening of bacterial metabolism signatures can provide unambiguous evidence of the existence of live bacteria (Huang et al., 2011) and can offer a more reliable and sensitive assay for the presence of live microorganisms and the detection of antibiotic resistant strains.

In this work, we report the synthesis and full characterisation of two highly watersoluble, biocompatible, stable and bright fluoresceinamine-based pH-sensitive nanoparticles. The addition of these nanoparticles to a growth medium containing bacteria allows for a rapid, accurate, reproducible and highly sensitive detection of bacterial growth.

Compared with the existing methods the one described here is based on nanoparticles that are more sensitive and more stable in common growth media, are easily synthesized with at a low cost and can be used in any laboratory without additional expensive instruments or experienced operators. Moreover, these particles allow for continuous monitoring for real-time detection over long time scales and can be used for screening a large number of samples for high-throughput applications. ) was prepared with a Milli-Q system (Millipore).

Experimental Methods

Synthesis of fluoresceinamine (FA) fluorescent pH-sensitive nanoparticles

The synthesis process of macroRAFT agents and polystyrene nanoparticle 1 (PSNP1) and polystyrene nanoparticle 2 (PSNP2) are described in Supplemental Information (S1.1 S1.2 Table S1 Scheme S1). In a typical experiment, 0.3 mL of PSNPs (0.1 mg/mL) obtained above were diluted in 4 mL of distilled water in the dark. In parallel, depending on the number of the AA units, 0.5 (PSNP1) or 1 (PSNP2) equivalents of fluoresceinamine (FA 3.35 or 6.7 mg) were dissolved in 0.4 mL ethanol and the solution was added to the PSNPs solution. The mixture was then stirred at 4°C.

A solution of EDC (4 equiv., 15 mg) in water (1 mL) was then added to the previous mixture. After two hours, ethanolamine (2 equiv., 2.3 µL) was added in order to quench the reaction. The mixture was stirred at 4°C overnight in the dark. Finally, the solution was transferred into a cellulose ester dialysis membrane (MWCO: 300 kDa, Spectrapor)

and dialyzed against water in the dark for 7 days. Two different fluoresceinamine (FA) based fluorescent nanoparticles were thus obtained: FANP1 and FANP2.

pH measurements

pH measurements were carried by using a glass electrode connected to a PHM210 standard pH meter from Meterlab.

Fluorescence spectroscopy

The UV-visible spectra were measured on a Varian Cary 5000 (Palo Alto, CA USA) double beam spectrometer using 4.5mL BRAND Methacrylate (PMMA) Cuvette.

Excitation and emission spectra were recorded on a SPEX Fluoromax-3 (Horiba Jobin-Yvon). The growth of the cells was monitored every 6 minutes by reading the optical density (OD@600nm). Fluorescence was measured using a F485/14 filter for excitation and a F535/40 filter for emission. The experiments lasted overnight for ten hours.

2.5

Real-Time detection of E.coli

2.5mL of Modified M9 minimal growth medium with an initial concentration of 2x10 7 CFU/mL or 3-6x10 5 CFU/mL of E.coli bacteria was placed inside a 4.5mL

BRAND Methacrylate (PMMA) Cuvette. Different volumes of FA molecules, FANP1

or FANP2 solutions were added to this E.coli suspension to reach a final concentration of 8.2x10 -7 M. Two repeats of the same concentration were carried out during each of the experiments.

The samples were incubated at 37°C in the dark. Every 30 minutes the growth of the cells was monitored using a spectrophotometer, Varian Cary 5000 (Palo Alto, CA USA) that reads the optical density (OD@600nm) of the samples compared to a blank of Modified M9 minimal medium. The pH of the samples was measured using a glass electrode. Emission and excitation spectra of the samples were measured on a SPEX Fluoromax-3 (Horiba Jobin-Yvon).

High-through put screening of bacterial growth inhibition by antibiotics

The assay is performed in a 96 well Falcon Polystyrene Flat Bottom Plate as described above. 150µL of a bacterial solution (1:1000 dilution of an overnight culture)

were placed in each well. FANP1 and FANP2 solutions were added into the E.coli solution to reach final concentrations of 8.2x10 -8 M. Three different antibiotics (Chloramphenicol, Kanamycin, and Ampicillin) were added into the bacterial solution to reach increasing concentrations of 0.5, 1, 5, 10, 50 µg/mL. Each plate contained two repeats of the same concentration. Controls without antibiotics and one blank with only the Modified M9 minimal growth medium were also prepared.

Imaging of E.coli growth

Fluorescence images of the samples in the cuvettes and in the culture plates were taken by a smartphone (Google Nexus 5) by placing samples on a Safe Imager™ 2.0

Blue Light Transilluminator (LIfeTechnologies) (excitation at 470nm).

Results and Discussion

Synthesis of fluoresceininamine (FA) pH-sensitive nanoparticles

Fluoresceininamine was selected as a pH-sensitive molecule because it has an intense absorption band in the visible. Even though it can adopt four different forms (dianion, anion, neutral and cation) depending on the pH, it is only highly fluorescent in its dianionic form (Sjöback et al., 1995). Furthermore, the pKa of the anion/dianion couple is about 6.4, which is suitable to measure changes in pH in biological media. The grafting of amino-functionalized molecules on carboxylic groups is a standard procedure (Montalbetti and Falque, 2005). The functionalization of PSNPs with fluoresceinamine (FA) was performed at pH=8 and 4 °C. The grafting of FA on PSNP1 and PSNP2 was carried out using 0.5 and 1 equivalent of FA per acrylic acid unit, respectively (Scheme S2). In all cases, the grafting did not change the hydrodynamic diameter of the nanoparticles significantly. The hydrodynamic of the fluoresceinamine conjugated nanoparticles (FANPs) remains around 60-70 nm (S1.3 and Table S2).

Spectroscopic properties of the FANPs

Absorption and fluorescence spectroscopy analysis of FANP1 and FANP2 are shown in Figure S1, their main spectroscopic data are given in Table S2. The particle suspensions have a maximum absorption at 488 and 490 nm and a maximum fluorescence emission at 519 and 521 nm in FANP1 and FANP2 respectively, in water at pH 8. There is a slight red shift compared to the free FA dye (Sjöback et al., 1995) which may come from the change of the local environment of the dye. The fluorescence quantum yield of the nanoparticles is between 12% and 14% for both particles at pH 8. This is lower than the free fluorescein amide dye (pH > 7, ΦF = 79% (Munkholm et al., 1990)) which may be due to the formation of FA aggregates in the shell. The number of grafted fluorescein molecules is 530 and 340 for FANP1 and FANP2 respectively as estimated from the absorption spectra (Carvell et al., 1998).

The spectroscopic behaviour of these FA-grafted nanoparticles was then studied at various pH values (Figure 1 and Figure S2). The fluorescence spectra at pH=8 for FANP1 (Figure 1A) and FANP2 (Figure 1C) have a main band centred at 519 nm and 521 nm respectively. This main band was attributed to fluoresceinamide. The fluorescence spectra were then recorded at different pH (phosphate/citrate buffers 1mM in 140mM NaCl). Upon a decrease in the pH, the spectral intensity decreased both in the absorbance and fluorescence for FANPs. This behaviour is similar to that of fluorescein in water. Moreover the pKa was determined from the fluorescence spectra using the Henderson-Hasselbalch equation:

𝐼 = 𝐼 𝐴 + 𝐼 𝐵 𝐾 𝑎 10 𝑝𝐻 1 + 𝐾 𝑎 10 𝑝𝐻
Where I is the fluorescence intensity of the sample and IA and IB are the fluorescence intensities in the acidic and basic forms respectively (Valeur, 2001). The pKa for both types of nanoparticles is around 6.5 and a linear dependence of the fluorescence intensity was observed between pH 5.5 and 7.5. Both the pKa and the linear range are in agreement with the one of the free FA dye (Duong et al., 2006), therefore it seems that the grafting does not change the pH dependent fluorescent properties of FA. This is the useful pH range for the study of most biological systems. Compared to other approaches reported previously (Badugu et al., 2008;Wang et al., 2013), in this case the linearity of the signal covers a wider and more suitable range of pH values for detecting bacterial growth. The amplitude of the fluorescence intensity change between these two pH values is around 7-fold which is 3 times higher than previously reported (Wang, F. et al., 2014;Wang, X. et al., 2013). These properties will thus allow a sensitive and precise determination of the change in pH in biological systems. 49. pH values: 7.37, 7.07, 6.94, 6.41, 5.84, 4.95, 4.41, 3.85, 3.01. (D) Change in fluorescence intensity of FANP2 at 520nm as a function of pH. The data was fit using the Henderson-Hasselbalch equation with pKa=6. 53. pH values: 7.80, 7.14, 7.01, 6.55, 6.27, 5.85, 5.07, 4.57, 4.00.

Toxicity assessments of FANPs on E.coli

In order to assess the toxic effect of FANPs on E.coli, the growth rate of bacteria in the presence of FANPs was compared with the growth rate of E.coli bacteria alone.

The growth rate was measured by absorbance measurement (OD@600nm) in a 96-well plate reader. The FANPs concentrations used were 8.2x10 -7 M, 4.1x10 -7 M and 8.2x10 - 8 M.

E.coli bacteria with different concentrations FANP1 or FANP2 were incubated in the plate reader at 37°C in the dark overnight. A control with only bacteria was also prepared. Every 6 minutes the growth rate of the cells was monitored by measuring the optical density at 600nm by the plate reader (Figure 2, Figure S3). For an incubation of over ten hours the growth curves of bacteria with FANPs at different concentrations were practically the same as the growth curve of bacteria alone. This means that FANPs have no toxicity effect on bacterial growth in a long time scale independently of the kind of shell present on the FANPs.

Real-time detection of E.coli growth

Growth of E.coli bacteria with glucose results in the production of CO2 and acetic acid and, therefore, in a decrease in pH of the growth medium. These FANPs have been shown to be photostable and the pH-dependent fluorescence change reversible (Detailed information in Supplemental Information S1.5 and S2.2). These unique features are very important for continuous monitoring for real-time detection of bacterial growth on long time scales. We thus tested whether they could be used to precisely monitor the pH change due to bacterial growth. In order to measure the change in fluorescence intensity of FANPs in the presence of E.coli bacteria, the growth medium with an initial pH of 7.0 and an initial E.coli bacterial concentration of 2x10 7 CFU/mL(OD=0.25) was placed inside a cuvette in the presence of FANPs and were incubated at 37°C in the dark. Every 30 minutes the growth of the cells was monitored by measuring the optical density at 600 nm using a spectrophotometer (Figure 3C, Figure S7C). When the bacteria began to grow, the pH of the growth medium started to decrease. The pH of the samples was monitored at several time intervals with a pH meter. When the bacterial concentration increased to 1.1x10 8 CFU/mL (OD=1.1), the pH of the growth medium decreased to 5.8.

At the same time, the emission spectra of the samples were measured. As bacterial growth increased the fluorescence intensity of both FANPs decreased significantly (Figure 3A and Figure S7A).

The fluorescence intensity of FANP2 for example decreased more than 80% during bacterial growth. Thanks to the calibration curve (Figure 1D) the pH can be evaluated by recording the change in fluorescence intensity of FANP2 in the growth medium (Figure 3B). It was found that these pH values were in full agreement with those measured by a pH meter (Figure 3C). Incubating FANP1 with E.coli gave similar results as those obtained with FANP2 (Figure S7). In conclusion, the pH-sensitive nanoparticles can be used to precisely monitor the pH variation due to bacterial growth (Figure 3, Figure S7).

Sensitivity assessments

A successful pH nano-sensor for the detection of E.coli growth should be both rapid and sensitive to small changes in bacterial concentrations. We therefore performed assays in order to assess the sensitivity of FANPs compared with the one of the same concentration of free fluoresceinamine (FA) molecules using a lower starting concentration of bacteria. FANP1, FANP2 or FA molecules were added to the growth medium with an initial concentration of E.coli bacteria of 1x10 5 CFU/mL (estimated OD= 0.001, measured OD=0.003-0.006) inside a cuvette and were incubated at 37°C in the dark. Every 30 minutes the growth of the cells was monitored by using a spectrophotometer (Figure 4). At the same time, the emission spectra of the samples were measured. The corresponding decrease in the percentage of the fluorescence intensity of FA molecules, FANP1 and FANP2 as a function of time is shown in Figure 4. This difference is probably due to the high density of the sensing dyes at the surface of the nanoparticles. In solution a proton changes the fluorescence of one FA molecule whereas when a FA molecule is quenched by a proton within a FANP, it can in turn also act as a quencher, through energy transfer, to its unprotonated neighboring FAs. Indeed the emission spectrum of unprotonated FA (donor) overlaps with the absorption spectrum of protonated FA (acceptor) (Figure S14). Thus a single proton is likely to result in the quenching of several grafted FA (compared to only one in solution). This amplification can explain why FANPs are more sensitive than FA molecules to changes in pH.

Furthermore, the FANP1 appeared to be more sensitive than FANP2 in detecting the growth of E.coli bacteria. Indeed, during the first hour of bacterial growth, the fluorescence intensity of FANP1 continually decreased by almost 20% which is twice as much compared with FANP2 (Figure 4). Until 3 hours the fluorescence intensity of FANP1 decreased more than 30%, and at the end of the experiment it had decreased up to 40% (Figure 4, Figure S8E,F). The difference between the two FANPs is probably due to the different number of FA grafted on FANP1 and FANP2. Indeed FA is more concentrated in FANP1 than FANP2 (530 FA per FANP1 and 340 FA per FANP2-Table S2). Thus the mean distance between two grafted FA is shorter in FANP1 compared to FANP2. As discussed above, FRET is likely to occur between unprotonated FA (donor)

and protonated FA (acceptor). Energy transfer is distance dependent, thus it will be more efficient in FANP1 than FANP2.

As mentioned above, the growth of the cells was monitored using a spectrophotometer that reads the optical density (OD@600nm) of the samples every 30 minutes (Figure 4). From the time at which the bacteria started to grow until two hours of growth, it was found that this small amount of bacteria couldn't be measured accurately the change in optical density. In addition, there were large fluctuations in the signal in the first hour of growth, and as the bacterial number increased, the measurements of spectrophotometers became more reproducible. However, as shown in Figure 4, the fluorescence intensity measurements from the FANP1 and FANP2

particles are very reliable and reproducible from the very beginning until the end of the experiments. In addition a similar experiment with bacteria at an initial OD of 0.00001 incubated with FANP2 particles was performed (Supplemental Information S2.3).

Figure S13 shows that the measurements of OD from spectrometers are not sensitive enough in the early stages of growth, however as long as there is metabolism of E.coli bacteria, the pH-sensitive nanoparticles are able to detect it. These results show that these pH-sensitive nanoparticles are not only more sensitive than a molecular sensor such as the FA molecules but also more sensitive than typical OD measurements.

Screening of cell growth in a 96 well plate

Since the FANPs can be used to measure the change in pH resulting from the metabolic activity of live bacteria, they can be used to discriminate whether the bacteria have died or have a decreased growth rate due to the fluorescence intensity change caused by metabolism. This approach can thus be more sensitive compared with typical OD meters and can be performed in complex media. We next tested the potential ability of FANPs for high-throughput screening of E.coli bacteria.

In order to measure the fluorescence evolution of FANPs when incubated with smaller volumes of bacteria, E.coli were incubated in a 96-well plate with different concentrations (8.2x10 -7 M, 4.1x10 -7 M and 8.2x10 -8 M) of FANP1 or FANP2 and placed in a plate reader at 37°C in the dark overnight. A control with bacteria without nanoparticles was also prepared. The fluorescence intensity of each well was monitored every 6 minutes by using a F485/14 filter for excitation and a F535/40 filter for emission (Figure S9).

The fluorescence of FANPs was measured during the growth of E.coli for over ten hours. The fluorescence intensity of FANP1 decreased simultaneously with E.coli growth (Figure S9A). Comparison of the three different concentrations of FANP1 used

shows that the response is independent of the amount of FANP1 present (Figure S9B).

Using FANP2 to measure the growth of E.coli gave very similar results as those obtained with FANP1 (Figure S9C,D). The correlation between the OD600 and the fluorescence of FANPs is shown in Figure S10. One can see that the change in fluorescence of FANPs as a function of the growth of bacteria shows a good correlation with the OD (r 2 =0.94 for FANP1 and 0.93 for FANP2) over 15 hours of continuous measurements which is a significantly improved correlation compared with a previous report using acridine orange (Pak et al., 2013).

As described above, the amount of FANP used in the cuvette experiments for sensitivity assessments was 8.2x10 -7 M. Screening of bacterial growth in a 96 well plate with different FANP concentrations (8.2x10 -7 M, 4.1x10 -7 M and 8.2x10 -8 M) shows that even after a ten-fold decrease in the concentration of FANP (8.2x10 -7 M8.2x10 -8 M), one can still detect bacterial growth rapidly and accurately. Because of their stability and sensitivity these FANPs could be used as a novel indicator for screening of potential growth inhibition experiments.

High-throughput screening of bacterial growth inhibition by antibiotics

After demonstrating that FANPs can be used for measurement of E.coli bacteria growth in small volumes, we wanted to determine whether this approach could be used for high throughput experiments such as the detection of antibiotic resistance. In order to compare the results obtained by the change in OD with the change in fluorescence of FANPs (8.2x10 -8 M) the effects of several common antibacterial drugs (Chloramphenicol, Kanamycin, and Ampicillin) on bacteria growth and metabolism were measured. The concentration of antibiotics used were 0.5, 1, 5, 10, 50 µg/mL, which cover the range of the known recommended concentrations of these antibiotics.

The inhibitory effect of these drugs on the growth of E.coli as measured by the increase of the OD or the decrease in fluorescence of FANP1 is shown in Figure 5.

When increasing the concentration of antibiotics, the growth rate of E.coli bacteria decreases as measured by OD (Figure 5A,C,E). At the same time, the fluorescence intensity of FANP1 decreases at a slower rate when increasing the dose of the drugs (Figure 5B,D,F).

Moreover, from the growth curve, one can notice that after about ten hours of incubation bacterial growth reached the stationary phase, as measured by the OD, indicating that cell growth slowed down and that bacteria grew and died at the same rate, resulting in a constant number of bacteria. However, by measuring the OD it is not possible to measure the presence of bacterial growth once the population has reached stationary phase. On the other hand, the increase in the change in fluorescence at these later time points allows for a measurement of the amount of ongoing metabolism of E.coli. The same results were obtained for FANP2 (Figure S11). As anticipated, reducing the dose of antibiotics reduces its inhibitory effect, and the changes in cell metabolism can be measured reliably and accurately by using FANPs. In order to test whether these FANPs could be used to monitor bacterial growth on a solid support, they were embedded in agar with growth medium, as used traditionally to follow the formation of bacterial colonies. As shown in Figure 6B and Figure S12, the pH-sensitive cell culture plates initially display a homogeneous yellow-green fluorescence, indicating a starting pH value of 6.8. After an overnight growth of bacteria it can be seen that significant acidification of the whole agarose layer has occurred. Even if the initial localizations of bacterial solutions are different, the protons produced by the bacteria during growth have caused a global influence on the particles in the agarose as the fluorescence intensity of the whole surface is strongly reduced. This result indicates that these particles could be used to screen for bacteria that grow selectively on solid media.

Conclusions

The novel pH-sensitive nanoparticles described herein have distinct features that make them suitable for the development of biosensors including brightness, stability in growth medium, non-toxicity, high water solubility, high sensitivity, and ease of bioconjugation and incorporation into a biocompatible matrix. They can rapidly and accurately detect bacterial growth by detecting the change of pH resulting from cellular metabolism. Due to the brightness of these nanoparticles, they could be used to develop new systems for quantitative biosensor applications for microfluidic devices. One of the important advantages of this approach is that a change in pH can be easily monitored either by a simple change in fluorescence at a specific wavelength or even by eye, for a qualitative result in either a liquid or solid support. These FNAPs can be easily synthesized with low cost and can be used in any laboratory without additional expensive instruments or operators. Moreover, these particles allow for continuous monitoring for long time scale experiments and can be used to monitor a large number of samples for high-throughput screening applications. For example they can be used to measure the metabolic activities of bacteria as an indicator of their susceptibility to antibacterial drugs, as shown here, or for the screening and discovery of novel growth inhibitors.

These particles could also be used to study the change in metabolism in different biological systems such as during the formation of biofilms [START_REF] Vroom | [END_REF], in the event of skin infections [START_REF] Valdés | Molecular Fluorescence[END_REF] or even to detect the presence of tumour cells during surgery due to their acidic environment (Pellegrini et al., 2014).

grant for Y.S. was provided by the MENESR (Ministère de l'Education Nationale, de l'Enseignement Superieur et de la Recherche) of France. 49. pH values: 7.37, 7.07, 6.94, 6.41, 5.84, 4.95, 4.41, 3.85, 3.01. D: Change in fluorescence intensity of FANP2 at 520nm as a function of pH. The data was fit using the Henderson-Hasselbalch equation with pKa=6. 53. pH values: 7.80, 7.14, 7.01, 6.55, 6.27, 5.85, 5.07, 4.57, 4.00.

Figure 1 .

 1 Figure 1. Fluorescence emission spectra of FANP1 (A) and FANP2(C) as a function of pH

Figure 2 .

 2 Figure 2. Growth curves of E.coli bacteria alone and of E.coli bacteria incubated with FANP2 particles at 8.2x10 -7 M, 4.1x10 -7 M and 8.2x10 -8 M. The error bars indicate the difference between the three replicates in the experiment.

Figure 3 .

 3 Figure 3. (A) Change in the fluorescence emission spectra of FANP2 with bacterial growth (λexc=495nm; Modified M9 Minimal Medium). (B) Change in fluorescence intensity of FANP2 at 520 nm and corresponding pH of the growth medium calculated by the Henderson-Hasselbalch equation. (C) Realtime bacterial growth curve (dotted line) and the corresponding pH evolution as measured by a pH meter (grey line) and the pH change calculated by the Henderson-Hasselbalch equation from the change in fluorescence (black crosses).

Figure 4 .

 4 Figure 4. Measurement of E.coli bacteria growth incubated with FA molecules (black lines), FANP1 (light grey lines) and FANP2 (dark grey lines) using either the measurement of OD@600nm (dotted lines) or the % decrease in fluorescence intensity (full lines) as a function of time for FA molecules, FANP1 and FANP2. The two lines represent the replicates within the experiment.

Figure 5 .

 5 Figure 5. The inhibitory effect of three antibiotics on the growth of bacteria: Chloramphenicol (A,B), Kanamycin (C,D), and Ampicillin (E,F) at different concentration (0.5, 1, 5, 10, 50 µg/mL). The change in the growth of E.coli was measured by an increase of the OD (A, C, E) and by a decrease of fluorescence of FANP1 (B, D, F). The error bars indicate the difference between the three replicates in the experiment.

Figure 6 .

 6 Figure6. A: a0, c0: FANP1 and FANP2 in growth medium with an initial concentration of E.coli of 3x10 5 CFU/ml (OD=0.003). b0 and d0: FANP1 and FANP2 in growth medium. a1, b1, c1, d1 are fluorescence images of a0, b0, c0, d0 after one night incubation (37°C). B: Fluorescence images of bacterial growth on FANP2 agarose (a0, b0, c0), compared to fluorescence images of FANP2 agarose alone (d0). a1, b1, c1, d1 are fluorescence images of a0, b0, c0, d0 after one night incubation (37°C).

Figure 1 .

 1 Figure 1. Fluorescence emission spectra of FANP1 (A) and FANP2(C) as a function of pH (λexc=495nm; phosphate/citrate buffer 1mM and 140mM NaCl). B: Change in fluorescence intensity of FANP1 at 520nm as a function of pH. The data was fit using the Henderson-Hasselbalch equation with pKa=6.49. pH values: 7.37, 7.07, 6.94, 6.41, 5.84, 4.95, 4.41, 3.85,

Figure 2 .

 2 Figure 2. Growth curves of E.coli bacteria alone and of E.coli bacteria incubated with FANP2 particles at 8.2x10 -7 M, 4.1x10 -7 M and 8.2x10 -8 M. The error bars indicate the difference between the three replicates in the experiment.

Figure 3 .

 3 Figure 3. A: Change in the fluorescence emission spectra of FANP2 with bacterial growth (λexc=495nm; Modified M9 Minimal Medium). B: Change in fluorescence intensity of FANP2 at 520nm and corresponding pH of the growth medium calculated by the Henderson-Hasselbalch equation. C: Realtime bacterial growth curve (dotted line) and the corresponding pH evolution as measured by a pH meter (grey line) or the corresponding pH change calculated by the Henderson-Hasselbalch equation from the change in fluorescence (black crosses).

Figure 4 .

 4 Figure 4. Measurement of E.coli bacteria growth incubated with FA molecules (black lines), FANP1 (light grey lines) and FANP2 (dark grey lines) using either the measurement of OD@600nm (dotted lines) or the decrease (%) of fluorescence intensity (full lines) as a function of time for FA molecules, FANP1 and FANP2. The two lines represent the replicates within the same experiment. Each experiment was repeated twice.

Figure 5 .

 5 Figure 5. The inhibitory effect of three antibiotics on the growth of bacteria: Chloramphenicol (A,B), Kanamycin (C,D), and Ampicillin (E,F) at different concentration (0.5, 1, 5, 10, 50 µg/mL). The change in the growth of E.coli was measured by an increase of the OD (A, C, E) and by a decrease of fluorescence of FANP1 (B, D, F).

Figure 6 .

 6 Figure6. A: a0, c0: FANP1 and FANP2 in growth medium with an initial concentration of E.coli of 3x10 5 CFU/ml (OD=0.003). b0 and d0: FANP1 and FANP2 in growth medium. a1, b1, c1, d1 are fluorescence images of a0, b0, c0, d0 after one night incubation (37°C). B: Fluorescence images of bacterial growth on FANP2 agarose (a0, b0, c0), compared to fluorescence images of FANP2 agarose alone (d0). a1, b1, c1, d1 are fluorescence images of a0, b0, c0, d0 after one night incubation (37°C).

Screening of bacterial growth in a 96 well plate

	plate reader (Perkin Elmer Victor3 1420 Multilabel Plate Counter) at 37°C in the dark.
	Live E.coli bacterial cells (K-12, BW25113) were used for the following
	experiments. Bacterial cultures were prepared overnight from stock cultures inoculated
	in Luria Broth (LB) growth medium.
	The overnight culture of bacteria was diluted 1:1000 in the Modified M9 minimal
	growth medium. (Detailed composition of the M9 medium is in Supplemental
	Information S1.4). 150µL of this bacterial solution were placed in each well of a 96 well
	Falcon Polystyrene Flat Bottom Plate. Different volumes (from 2 to 10 µl) of the stock
	FANP1 and FANP2 solutions were added into the E.coli solution to reach final
	concentrations of 8.2x10 -7 M, 4.1x10 -7 M and 8.2x10 -8 M. Each plate contained three
	repeats of the same concentration. One control with only bacteria and one blank with
	only the Modified M9 minimal growth medium were also prepared. 70 µL of mineral
	oil were added to each well in order to avoid evaporation. Samples were incubated in a

Acknowledgements

Financial support for this work was provided by the CNRS (Centre National de la Recherche Scientifique) of France -Interface Physique Chimie Biologie Soutien à la prise de risque-and by Institut d'Alembert (FR 3242 CNRS-ENS de Cachan). A PhD


Anas El Maakoul

Benjamin Remy

Alain Degiovanni

Modeling of thermal contacts with heat generation: Application to electrothermal problems

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

Introduction:

Modeling thermal contacts problems with heat generation is both complex and difficult. This difficulty comes mainly from modeling the interfacial conditions, and until now, to the authors knowledge, there is no model or method that received unanimous acceptance by the heat transfer community.

In the literature, there are numerous studies where different models are proposed and applied to various applications of thermal contacts with heat generation. Depending on the application, two mains categories can be identified: Contacts with friction (molding, machining, …) and electrothermal contacts (electrical conductors, spot welding process …).

A description of representative studies that appear in the literature now follows. For the first category of contacts with friction, several research works dealt with machining and molding [START_REF] Kato | Energy partition in conventional surface grinding[END_REF][START_REF] Chandraeker | Thermal aspects of surface finishing processes[END_REF][START_REF] Grzesik | The role of coatings in controlling the cutting process when turning with coated indexable inserts[END_REF][START_REF] Grzesik | A computational approach to evaluate temperature and heat partition in machining with multilayer coated tools[END_REF], while in others, the authors studied dry contact under perfect contact conditions [START_REF] Kulchytsky-Zhyhailo | Axi-symmetric contact problem with frictional heating for thermally nonlinear sliders[END_REF][START_REF] Kulchytsky-Zhyhailo | A simplified solution for three-dimensional contact problem with heat generation[END_REF][START_REF] Pauk | Periodical contact problems for a half-space involving frictional heating[END_REF][START_REF] Pauk | Plane contact problem for a layer involving frictional heating[END_REF][START_REF] Grylitsky | Some quasistationary contact problems for half-space involving heat generation and radiation[END_REF][START_REF] Yevtushenko | Axi-symmetrical transient contact problem for sliding bodies with heat generation[END_REF] or Imperfect contact conditions [START_REF] Mazo | Etude des conditions de liaison thermique dans le cas du frottement sec métalplastique[END_REF][START_REF] Kulchytsky-Zhyhailo | Thermoelastic contact problems with frictional heating and convective cooling[END_REF][START_REF] Komanduri | Analysis of heat partition and temperature distribution in sliding systems[END_REF][START_REF] Komanduri | Thermal analysis of dry sleeve bearings-a comparison between analytical, numerical (finite element) and experimental results[END_REF]. The heat generated through friction was evaluated differently depending on the authors. Some distributed the heat sources [START_REF] Barber | Distribution of heat between sliding surfaces[END_REF][START_REF] Kounas | The distribution of friction heat between a stationary pin and a rotating cylinder[END_REF], some located the heat generation on the contact plane [START_REF] Levytskyi | Plane contact problem with heat generation account of friction[END_REF], others located it on one of the friction surfaces [START_REF] Ciavarella | Interaction of thermal contact resistance and frictional heating in thermoelastic instability[END_REF]. Other authors, based on the approach used by Bardon [19], considered interfacial heat generation using the notion of 'part of the generated heat that causes a spike in temperature at the interface' or 'heat generation factor' [START_REF] Chantrenne | A microscopic thermal model for dry sliding contact[END_REF][START_REF] Bauzin | Simultaneous estimation of frictional heat flux and two thermal contact parameters for sliding contacts[END_REF][START_REF] Laraqi | Contact temperature and flux partition-coefficient of heat generated by dry friction between 2 solids-new approach to flux generation[END_REF][START_REF] Guillot | Experimental study of thermal sliding contact with friction: application to high speed machining of metallic materials[END_REF][START_REF] Chantrenne | Study of a macroscopic sliding contact thermal model from microscopic models[END_REF]. As for the second category of electrothermal contacts, heat is generated in the matter of the materials in contact. There are various models for this category, such as with or without contact resistance and with or without interfacial heat generation [START_REF] Han | Resistance spot welding: a heat transfer study[END_REF][START_REF] Cho | A study of the thermal behavior in resistance spot welds[END_REF][START_REF] Khan | Numerical simulation of resistance spot welding process[END_REF][START_REF] Wang | Modeling dynamic electrical resistance during resistance spot welding[END_REF][START_REF] Le Meur | Measurement of contact parameters at electrode/sheet interface during resistance spot welding process[END_REF][START_REF] Le Meur | Microscopic analysis of interfacial electrothermal phenomena--definition of a heat generation factor[END_REF][START_REF] Feulvarch | Resistance spot welding simulation: a general finite element formulation of electrothermal contact conditions[END_REF][START_REF] Rogeon | Characterization of electrical contact conditions in spot welding assemblies[END_REF][START_REF] Rogeon | A microscopic approach to determine electrothermal contact conditions during resistance spot welding process[END_REF].

The foregoing short introduction shows that the main difficulty lies in the representation of the heat sources, particularly if we want to keep the notion of "contact resistance". One representation that can be generalized was introduced by Bardon [19] which, as mentioned previously, consists in considering interfacial heat generation and introducing a heat generation factor , the equivalent thermal circuit for this representation in shown in figure 1. In this case, and are respectively the non-perturbed thermal resistances of mediums 1 and 2, is the contact resistance and the total rate of heat generation.

In this representation, the coefficient depends not only on the spatial distribution of the heat sources but also on the thermal conductivities and of the mediums 1 and 2. Another problem with this representation is that does not permit the evaluation of the contact temperature.

In this paper, we consider the solid-solid thermal contact problem with heat generation and propose a general representation, that is easily applicable, to evaluate the heat transfer rates and contact temperature. To do so, we first need to examine in detail the notion of "contact resistance".

Concept of "contact resistance":

Let's consider the contact between two solids without heat sources (figure 2), because the solid contact spots are interspersed with gaps (that are generally filled with a fluid), the actual contact area (solid contact spots) is much smaller than the apparent area. The contact resistance is described using an axisymmetric elementary contact cell that corresponds to a reference contact area associated to a rate of heat transfer; the overall contact and heat transfer are represented by multiple elementary cells in parallel [START_REF] Degiovanni | Un modèle de résistance thermique de contact en régime permanent[END_REF][START_REF] Degiovanni | Résistance thermique de contact en régime permanent: influence de la géométrie de contact[END_REF] as shown in figure 2. Figure 3 represents schematically an elementary cell. With reference to figure 3, the rigorous approach consists in splitting the elementary cell into two parallel heat transfer regions (internal and external). In the internal region, heat flows through the asperities while in the external region heat flows though the interstitial fluid. The foregoing is presented in figure 4.a, where the two regions are separated with dashed curves.

Noting that the total heat transfer is the sum of the heat transfer rates and , we can write:

1 = 1 + 1 (1)
Where is the total thermal resistance, and are respectively the thermal resistances of the internal and external regions. Let's decompose the resistance of each region into resistances connected in series (figure 4.b and figure 5): In figure 5, it can be demonstrated (see appendix 1) that the potentials and are the identical, the same is true for the potentials and . Moreover, in most cases, the actual contact area is much smaller than the apparent area, therefore the external constriction resistances ( and ) are negligible compared to the fluid resistance and the internal constriction resistances are very close to the intrinsic constriction resistances (irrespective of the asperities and the fluid) [START_REF] Degiovanni | Thermal resistance of a multi-constrictions contact: A simple model[END_REF]. The inferred thermal circuit is shown in figure 6 where is the contact resistance. Can the preceding equivalent thermal circuit be modified to consider internal heat sources? in the case of localized sources, the problem is solved by simply applying Kirchhoff's current law, while for a distributed source (heat generation varying with position), it was demonstrated that the problem has an equivalent thermal circuit [START_REF] El Maakoul | A general approach to solve heat conduction problems with internal heat sources using resistance and quadrupole concepts[END_REF]. In the next section, we briefly present this demonstration for the special case of curvilinear coordinates.

= + + + + + (2)
= + + + + (3)

Notion of partition coefficient in a steady conduction problem with an internal source [37]:

Let's consider the case of unidirectional conduction with internal sources where the heat flux vector is perpendicular to the isothermal surfaces. The sources are uniform between two isotherms and we can define a curvilinear coordinate s orthogonal to the isotherms (figure 7).

With ( ) the volumetric heat generation rate, ( ) the isothermal section and constant thermophysical properties, we can write Fourier's equation as:

( ) = -( ) ( ) (4)
And the energy balance:

= ( ) ( ) (5)
We define: Integrating equation ( 5) from * to +, , we get:

! = " ( ) ( ) # $%& # '( )( ) = " ( ) # # '( = " ( ) # $%& # '(
+, = ! + * (6)
Integrating equation ( 4) from * to +, yields:

* -+, = +, -" )( ) ( ) ( ) # $%& # '( (7)
As demonstrated in [START_REF] El Maakoul | A general approach to solve heat conduction problems with internal heat sources using resistance and quadrupole concepts[END_REF], relations ( 6) and ( 7) correspond to the thermal circuit shown in figure 8, which shows that a medium with an internal heat source can be represented using an equivalent thermal circuit with two resistances and (1 -) , where the total rate of heat generation ! is dissipated between the two resistances. Therefore:

= " )( ) ( ) ! ( ) # $%& # '( (8)
is the partition coefficient and ! is the total rate of heat generation in the medium.

In this representation, is independent of the medium's thermal conductivity and is only a function of the source's distribution. With reference to figure 7, we consider the problem of heat generation due to Joule heating which represents the conversion from electrical to thermal energy in a current-carrying medium. An electric potential difference is imposed between * and +, , thus an electric currentflows through the medium. Considering . the electrical resistivity of the medium, the electrical resistance of an element is expressed in the same way as the thermal resistance (solution of the same problem, ∆ = 0 or ∆1 = 0, for the same volume and boundary conditions):

) = . ( )
Thus ( ) = . ( ) - 1 ( ) = . - ( ) (9)
Relation (8) becomes:

= " 2 ( ) # # '( 2 ( )
# $%& # '( . - ( ) 2 . - ( ) ( ) # $%& # '( ( ) # $%& # '( (10)
Which reduces to

= " 2 ( ) # # '( • ( ) 42 ( ) # $%& # '( 5 # $%& # '( (11)
Let

6( ) = 2 7# 8(#) # # '(
(with 6( * ) = 0), we get:

= 1 6 ( +, ) " 6( )6 9 ( ) # $%& # '( = 1 2 (12)
The partition coefficient is a constant equal to 1 2 ⁄ . This remarkable result can be demonstrated for the general electrothermal problem (steady or transient) with any geometrical configuration (see [START_REF] El Maakoul | A general approach to solve heat conduction problems with internal heat sources using resistance and quadrupole concepts[END_REF]).

Thermal contact with heat sources:

Let's consider the circuit shown in figure 5. In the presence of heat sources in the internal and external branches, the potentials and are no longer identical and the same is true for potentials and . This prevents, in the general case, the definition of the contact resistance.

Let's consider the two following particular cases.

Fluid with a high thermal resistance:

An example is the thermal contact in a vacuum environment. For this case, the circuit in figure 5 reduces to one branch (internal) with heat sources and we can use the preceding result shown in figure 8. The thermal equivalent circuit for this case is shown in figure 10 where:

= + + And = + +
and are respectively the total thermal resistances of medium 1 and 2. is the contact temperature. Note that in the circuit of figure 10, the only approximation is that the contact is isothermal.

Figure 10: Equivalent thermal circuit of thermal contact with heat source and → ∞

Fluid with a non-negligible thermal resistance:

In this case, the solution is obtained by exploiting the linearity of the problem. The temperature is then written in the form = > + ?. For the problem in >, there is a temperature difference ( ≠ ) with no heat source, while for the problem in ?, there is a heat source without temperature difference ( = ). The solution to the problem in > is given by figure 6. The solution to the problem in ? can be simplified if we assume that the heat transfer through the external region is negligible (this hypothesis will be later verified using numerical simulations), thus, the solution to the problem in ? is given by figure 10 where = = 0. The equivalent thermal circuits for this case are shown in figure 11.

The heat transfer rates are:

= A + B ( 13
)
= A + B ( 14
)
And the contact temperature

= > + ? (15)
For electrothermal problems, we only need to use = 1/2.

Numerical simulations and comparison to the macroscopic approach: application to an elementary contact cell

In this section, the results of the analytical approach are compared to numerical simulations for the elementary contact cell described in figure 12. The height D and D are sufficiently larger than the radius E so that the constriction is fully developed; the radius F is sufficiently small as to obtain a realistic model.

The geometrical parameters are: The numerical simulations are carried out using FLUENT, the geometry is modeled exploiting its axisymmetry. The computational domains are meshed with triangle and quadrilateral elements. the conformal mesh is locally refined at the interface. For each case, a mesh sensitivity test was conducted where the optimal mesh is adopted. The energy and electric potential equations are iteratively solved. The convergence was judged by monitoring residuals and key quantities such as the temperature and potential at the interface ( and V ). The numerical results are summarized in table 1. , , , and are wall type resistances, that is, they take the form W/ where W is the wall thickness, its conductivity and the heat transfer area. and are the constrictions resistances and depend on the boundary condition at the contact interface. The classical approach [START_REF] Maillet | Thermal quadrupoles: solving the heat equation through integral transforms[END_REF] shows that there are two limiting conditions that correspond to 'uniform heat flux' and 'uniform temperature'. The analytical solutions obtained in the form of series were approximated and presented in a practical manner by various authors, here we will use the following formulas proposed in [START_REF] Maillet | Thermal quadrupoles: solving the heat equation through integral transforms[END_REF]:

E =
Uniform temperature = 1 4 ) X Y ) Z (16) Uniform heat flux = 8 3] ) X Y ) Z (17) With X Y ) Z = 1 -1.288 ) + 0.288 Y ) Z ^.N_
In the following we will show that the deviation between the analytical and numerical results comes solely from the approximate calculation of the constriction resistances. We will also use the resistances obtained using numerical simulations.

Analytical-numerical comparison

In this configuration, the interface temperature is uniform as demonstrated in [START_REF] Degiovanni | Résistance thermique de contact en régime permanent: influence de la géométrie de contact[END_REF] and confirmed by the numerical results as shown in figure 13. The numerical temperature at f = T is plotted against the radius (0 ≤ ) ≤ F). The small deviations are caused by the singularity at ) = F. The overall results for this configuration are summarized in table 2. The 'analytical-numerical' method is about using the analytical approach with the numerically obtained thermal resistances. The 'analytical-analytical' method uses the analytical approach with the constriction resistance evaluated through equations ( 16) and ( 17), the first and second lines contain respectively the results for a 'uniform temperature' and a 'uniform heat flux' interfacial boundary conditions. The detailed calculations are presented in appendix 2. 60.0 18.20 110.9 27.29

From the foregoing, it is clear that there is a good agreement between the numerical and analytical 'uniform temperature' approach, for both the heat transfer rates and interface temperatures.

. Analytical-numerical comparison

In this configuration, the temperature at f = T is not uniform (see figure 13). This is also confirmed from the preceding values for the contact resistances where the numerical contact resistance is between the two analytical contact resistances.

The results are presented in table 3, there is a perfect analytical-numerical agreement by using the numerical constriction resistances in the analytical calculations. When using the analytical constriction resistances in the analytical calculations, the maximum relative deviation is less than 2%. The results for this configuration are reported in table 4. There is a good analytical-numerical agreement. Taking the numerical results as a reference, the numerical-analytical and analyticalanalytical methods have maximum deviations of less than 1% and 4% respectively.

Conclusion:

In this work, it was shown that it is possible to use the 'contact resistance' concept to describe solidsolid contact with heat sources. This is done by using the notion of heat transfer partition coefficient.

For electrothermal problems, we demonstrated that the partition coefficient is a constant equal to ½ whatever the system's geometrical configuration.

Numerical simulations were carried out for various thermal contact configurations with different cases.

The excellent agreement between the numerical and analytical results demonstrates the validity of the proposed approach, which also allows to obtain the interface temperature using a simple thermalelectrical analogy. The deviation between Numerical and analytical results comes mainly from the calculation of the constriction resistances. To demonstrate that = and = (see figure 5), we only need to consider the hypothesis of a fully developed constriction, i.

Nomenclature

Figure 1 :

 1 Figure 1: Equivalent thermal circuit proposed byBardon [19]

Figure 2 :Figure 3 :

 23 Figure 2: Representation of contact between two solids

Figure 4 :

 4 Figure 4: a) internal ( ) and external ( ) heat transfer regions for an elementary cell; b) thermal resistances in series for each region (Eq (2-3))

Figure 5 :

 5 Figure 5: The complet equivalent thermal circuit for contact problems without heat source

Figure 6 :

 6 Figure 6: Simplified equivalent thermal circuit for contact problems without heat source

Figure 7 :

 7 Figure 7: Unidirectional conduction with internal sources

Figure 8 :

 8 Figure 8: Equivalent thermal circuit of heat conduction in a medium with an internal heat source

Figure 11 :

 11 Figure 11: Equivalent thermal circuit for thermal contacts with heat source and interstitial fluid

• Configuration 1 : 1 •

 11 10 GH ; F = 1 GH ; D = D = 20 GH Two materials are considered, Iron and Copper, the interstitial fluid is air. The relevant physical properties are listed below: • Iron: = 40 KH L M L and . = 10 LN ΩH • Copper: = 400 KH L M L and . = 2 • 10 LP ΩH • Air: = 0.025 KH L M L and . = 1.24 • 10 S ΩH We define three interface configurations: Asperities with nil thickness T = T = 0 without any interstitial fluid • Configuration 2: Asperities of thickness T = T = 0.5 GH without interstitial fluid • Configuration 3: Asperities of thickness T = T = 0.5 GH with air as interstitial fluid And four different cases: • Case 1: iron to iron contact with = = 0° and ∆V = 4.24 • 10 L Case 2: iron to copper contact with = = 0° and ∆V = 4.24 • 10 L 1 • Case 3: iron to iron contact with = 100° , = 0° and ∆V = 4.24 • 10 L 1 • Case 4: iron to copper contact with = 100° , = 0° and ∆V = 4.24 • 10 L 1

Figure 12 :

 12 Figure 12: Thermal contact with heat source in an axisymmetric elementary cell 6.1. Numerical results:

6. 3 .

 3 Configuration 1: Asperities with nil thickness `a = `b = c: 6.3.1. Thermal resistancesThe thermal resistances values for Iron are presented below, in M/K. For copper, the resistances can be obtained by dividing the values below by 10 ( +dd e / e+* = 10).

Figure 13 :

 13 Figure 13: Temperature profile at f = T for configuration 1, case 1 and configuration 2, case 1.

6. 4 . Configuration 2 :

 42 Asperities of thickness `a = `b = c. h ij without interstitial fluid: 6.4.1. Thermal resistancesThe thermal resistances values for Iron are reported below, in M/K. For copper, the resistances can be obtained by dividing the values below by 10 ( +dd e / e+* = 10).

6. 5 . Configuration 3 : 2 .

 532 Asperities of thickness `a = `b = c. h ij with air as interstitial fluid 6.5.1. Thermal resistances The thermal resistances values for Iron are reported below, in M/K. For copper, the resistances can be obtained by dividing the values below by 10 ( +dd e / e+* = 10). Analytical-numerical comparison Here, the analytical calculations are done using the superposition principle as shown in section 4.2 and represented in figure 11 with = 1/2.

  e. the heat flux in D and D is uniform (D and D are sufficiently larger than F). With reference to figure 5, we can write: flux) Therefore = . In a similar manner, we can demonstrate that = . heat transfer rates and the interface temperature (p a = p b = c with heat source) In the last expression, we can see that the interface temperature depends only on the rate of heat generation II. Calculations of the heat transfer rates and the interface temperature (p a ≠ p b without heat source) = -

Table 1 :

 1 Numerical resultsConfiguration 1: Asperities with nil thickness T = T = 0

		Case 1	Case 2	Case 3	Case 4
	Parameter	Iron	Iron	Iron	Iron
		Iron	Copper	Iron	Copper
	0.0424 0 -0.01595 0.01596 0.0212 55.41 0.76 Configuration 2: Asperities of thickness T = T = 0.5 GH without interstitial fluid 0.0424 0.0424 ∆V (1) 0 100 ∆ -0.02475 -0.00893 -0.01175 0.0424 100 (K) 0.02884 0.02323 0.04185 (K) 0.0072 0.0212 0.0072 V (1) 17.04 106.25 26.14 (° ) 1.26 0.76 1.26 -( ) Case 1 Case 2 Case 3 Case 4
	Parameter	Iron	Iron	Iron	Iron
		Iron	Copper	Iron	Copper
	0.0424 0 -0.01000 0.01000 0.0212 56.17 0.47 Configuration 3: Asperities of thickness T = T = 0.5 GH with air as interstitial fluid 0.0424 0.0424 ∆V (1) 0 100 ∆ -0.01541 -0.00555 -0.00732 0.0424 100 (K) 0.01793 0.01445 0.02603 (K) 0.0071 0.0212 0.0071 V (1) 17.03 106.19 26.12 (° ) 0.79 0.47 0.79 -( ) Case 1 Case 2 Case 3 Case 4
	Parameter	Iron	Iron	Iron	Iron
		Iron	Copper	Iron	Copper
	∆V (1) ∆ (K) (K) V (1) (° ) -( )	0.0424 0 -0.00993 0.00993 0.0212 56.31 0.47	0.0424 0 -0.01519 0.01800 0.0072 17.69 0.78	0.0424 100 -0.00495 0.01491 0.0212 106.31 0.47	0.0424 100 -0.00659 0.02659 0.0072 27.15 0.78
	6.2. Calculation of thermal resistances:

Table 2 :

 2 Numerical and analytical comparison for configuration 1: Asperities with nil thickness

	Method	Parameter	Case 1 Iron-Iron	Case 2 Iron-Copper	Case 3 Iron-Iron	Case 4 Iron-Copper
	Numerical Numerical-Analytical Analytical-Analytical	(HK) (HK) (° ) (HK) (HK) (° ) Eq (16) Eq (17) (HK) Eq (16) Eq (17) (HK) Eq (16) (° ) Eq (17)	-15.95 15.96 55.40 -16.06 16.06 56.1 -16.06 -16.06 16.06 16.06 56.5	-24.75 28.84 17.04 -24.74 28.81 17.02 -24.73 -24.74 28.81 28.81 17.13	-8.93 23.23 106,2 -8.91 23.21 106.1 -8.95 -9.37 23.77 22.75 106.5	-11.75 41.85 26.14 -11.74 41.80 26.11 -11.82 -12.58 41.72 40.96 26.22

Table 3 :

 3 Numerical and analytical comparison for configuration 2: Asperities of thickness T = T = 0.5 GH without interstitial fluid

	Method	Parameter	Case 1 Iron-Iron	Case 2 Iron-Copper	Case 3 Iron-Iron	Case 4 Iron-Copper
	Numerical Numerical-Analytical Analytical-Analytical	(HK) (HK) (° ) (HK) (HK) (° ) Eq (16) Eq (17) (HK) Eq (16) Eq (17) (HK) Eq (16) (° ) Eq (17)	-10.00 10.00 56.17 -10.00 10.00 56.16 -10.00 -10.00 10.00 10.00 55.07 57.27	-15.41 17.93 17.03 -15.40 17.93 17.02 -15.40 -15.40 17.93 17.93 16.69 17.35	-5.55 14.45 106.2 -5.55 14.45 106.1 -5.46 -5.63 14.54 14.36 105.0 107.3	-7.32 26.03 26.12 -7.31 26.02 26.11 -7.15 -7.46 26.18 25.86 25.78 26.44

Table 4 :

 4 Numerical and analytical comparison for configuration 2: Asperities of thickness T = T = 0.5 GH with air as interstitial fluid

	Method	Parameter	Case 1 Iron-Iron	Case 2 Iron-Copper	Case 3 Iron-Iron	Case 4 Iron-Copper
	Numerical Numerical-Analytical Analytical-Analytical	(HK) (HK) (° ) (HK) (HK) (° ) Eq (16) Eq (17) (HK) Eq (16) Eq (17) (HK) Eq (16) (° ) Eq (17)	-9.93 9.93 56.31 -9.91 9.91 55.66 -9.91 -9.91 9.91 9.91 54.57 56.75	-15.19 18.00 17.69 -15.26 17.76 16.87 -15.26 -15.26 17.76 17.76 16.54 17.20	-4.95 14.91 106.3 -4.90 14.92 105.70 -4.81 -4.98 15.00 14.84 104.6 106.7	-6.59 26.59 27.15 -6.60 26.42 25.96 -6.45 -6.76 26.58 26.27 25.63 26.29

We can also write

In the case where medium 1 and medium 2 have identical geometrical parameters:

Let's consider

In the last expression, we can see that the interface temperature does not depend on (fluid resistance).