G Castanet

D V Antonov

P A Strizhak

S S Sazhin
email: s.sazhin@brighton.ac.uk

Effects of water subdroplet location on the start of puffing/micro-explosion in composite fuel-water droplets

Keywords:

A new approach to the verification of the predictions of the earlier developed model of puffing/microexplosion is suggested, based on the implementation in the numerical code of the analytical solution to the heat transfer equation within a composite droplet. Verification is based upon a comparison between the predictions of the numerical code with this model and those of a fully numerical solution to the equation (the heat transfer module available from COMSOL software). The agreement between the predictions of both codes supports both approaches to the problem. The model is generalised to consider the shifting of the water subdroplet away from the centre of the fuel droplet. This generalisation is based on the numerical solution to the heat transfer equation in the composite droplet taking into account this shift. The start of puffing/micro-explosion in the generalised model is related to the time instant when the temperature at the point of the water/fuel interface closest to the surface of the fuel droplet reaches the water nucleation temperature. The shift is characterised by the distance between the centres of the water subdroplet and the fuel droplet. It is shown that even if this distance is half of the maximal possible distance, the errors in times to puffing/micro-explosion obtained using the original and generalised models do not exceed 5%, which can be tolerated in most practical applications.

Introduction

The importance of puffing (swelling and break-up of droplets into several small droplets) and micro-explosion (break-up of droplets producing a cloud of aerosols) in composite fuel-water droplets has been widely discussed [START_REF] Ogunkoya | Performance, combustion, and emissions in a diesel engine operated with fuel-in-water emulsions based on lignin[END_REF] . Numerous experimental studies of the phenomena (e.g. [START_REF] Cen | Experimental investigation on the sputtering and micro-explosion of emulsion fuel droplets during impact on a heated surface[END_REF][START_REF] Avulapati | Puffing and micro-explosion of diesel-biodiesel-ethanol blends[END_REF][START_REF] Moussa | Parametric study of the microexplosion occurrence of w/o emulsions[END_REF] ) were complemented by extensive developments of the models of various levels of complexity.

The most complex models of this phenomenon are based on Direct Numerical Simulation (DNS) [START_REF] Shinjo | Modeling temperature distribution inside an emulsion fuel droplet under convective heating: a key to predicting microexplosion and puffing[END_REF][START_REF] Shinjo | Physics of puffing and microexplosion of emulsion fuel droplets[END_REF][START_REF] Fostiropoulos | Numerical investigation of heavy fuel oil droplet breakup enhancement with water emulsions[END_REF] . Several simplified models (e.g. [START_REF] Girin | Dynamics of the emulsified fuel droplet micro-explosion[END_REF][START_REF] Sazhin | A simple model for puffing/micro-explosions in water-fuel emulsion droplets[END_REF][START_REF] Fostiropoulos | A simple model for breakup time prediction of water-heavy fuel oil emulsion droplets[END_REF][START_REF] Antonov | Modeling the micro-explosion of miscible and immiscible liquid droplets[END_REF][START_REF] Sazhin | A new approach to modelling micro-explosions in composite droplets[END_REF] ) are focused on the investigation of specific aspects of the phenomena and essentially complement the approaches based on DNS. The model developed by the authors of [START_REF] Sazhin | A new approach to modelling micro-explosions in composite droplets[END_REF] assumes that a spherical water subdroplet is positioned exactly in the centre of a spherical fuel droplet, as illustrated in Fig. 1 .

Puffing/micro-explosion in this model is initiated when the temperature at the fuel-water interface becomes equal to the water nucleation temperature, which takes into account the superheating of water which retards initiation of the phonomena. This model uses the analytical solution to the transient heat transfer equation in a fuel-water droplet, with the Robin boundary condition at the droplet surface. This solution was implemented into a numerical code and used at each time step. The effects of evaporation, using the model developed by Abramzon and Sirignano [START_REF] Abramzon | Droplet vaporization model for spray combustion calculations[END_REF] , and swelling were also considered.

This model was extended using a non-self-consistent approach to the case of moving droplets to consider the effects of movement on the values of Sherwood and Nusselt numbers but not on the internal liquid recirculation inside droplets [START_REF] Antonov | Puffing and micro-explosions in rapeseed oil/water droplets: the effects of coal microparticles in water[END_REF] . This extended model proved to be effective in many applications including our recent investigations of puffing/micro-explosion in two and three droplets in a raw, one behind the other [START_REF] Antonov | Puffing/micro-explosion of two closely spaced composite droplets in tandem: experimental results and modelling[END_REF][START_REF] Antonov | Temperature measurements in a string of three closely spaced droplets before the start of puffing/micro-explosion: experimental results and modelling[END_REF] . At the same time, it has several important weaknesses.

Firstly, the verification of the model was limited by the development of two separate numerical codes, using Wolfram Mathematica v 12.1 and Matlab R2020a, in which the analytical solution to the heat transfer equation in a fuel-water droplet was implemented. Although these codes predicted the same results, both results could be wrong if there is something wrong with the analytical solution.

Nomenclature

English symbols B M(T )

Spalding mass (heat) transfer number [ Heating rate Secondly, the sensitivity of the results to the shifting of the water subdroplet away from the centre of the fuel droplet has not been investigated, and if, for example, the prediction of the model can be applied only to cases when this shift is less than 1%, this model would not be applicable to most practical engineering problems.

[W] r Radial coordinate [m] R Distance from the droplet centre [m] R d Droplet radius [m] S L /L max [-] Sc Schmidt number [-] Sh Sherwood number [-] t Time [s] T Temperature [K] X Molar fraction [-] Y Mass fraction [-] z Axial coordinate [m]
Thirdly, no quantitative estimates of the effect of this shift on predicted and observed times to puffing/micro-explosion were made. It was only observed that in most cases these times predicted by the model described in [START_REF] Sazhin | A new approach to modelling micro-explosions in composite droplets[END_REF] , or its extended version, are longer than those observed experimentally. This was related to the shift of the water subdroplet from the centre of the fuel droplet (cf. Fig. 1 ). However, no quantitative estimates have been made.

The aim of this paper is to address these three issues. As in our previous papers, the problem of the formation of composite droplets will not be considered (e.g. [START_REF] Roy | Manufacture and properties of composite liquid marbles[END_REF][START_REF] Roy | Osmotic evolution of composite liquid marbles[END_REF] ).

A new approach to the verification of the predictions of the extended version of the model developed in [START_REF] Sazhin | A new approach to modelling micro-explosions in composite droplets[END_REF] (hereafter called the Centre Model (CM)) is presented in Section 2 . The basis of this verification is a comparison of the predictions of CM with those of a numerical code for solving the same heat transfer problem in the fuel-water droplet. In Section 3 , the Centre Model is generalised to consider a shift of the water subdroplet away from the centre of the fuel droplet. This generalisation (hereafter referred to as the Shift Model (SM)) is based on the numerical solution to the heat transfer equation in a fuel-water droplet. In the same section, the errors made using the Centre Model when the water subdroplet is not perfectly centred are investigated. The predictions of the Shift Model are compared with the results of experimental observations in Section 4 . The most important findings of the paper are summarised in Section 5 .

The verification of the centre model

The verification of the Centre Model (CM) was performed by comparing its predictions with those of the modified version of this model (Modified Centre Model, MCM) in which the analytical solution to the heat transfer equation in a fuel-water droplet is replaced with its numerical solution. The latter was obtained using the heat transfer module of COMSOL Multiphysics which contains numerical tools for the simulation of heat conduction and convection based on the finite element methods [START_REF] Antonov | Puffing/micro-explosion of two closely spaced composite droplets in tandem: experimental results and modelling[END_REF][START_REF] Antonov | Temperature measurements in a string of three closely spaced droplets before the start of puffing/micro-explosion: experimental results and modelling[END_REF][START_REF] Castanet | Evaporation of closely-spaced interacting droplets arranged in a single row[END_REF][START_REF] Castanet | Investigation of heat and mass transfer between the two phases of an evaporating droplet stream using laser-induced fluorescence techniques: comparison with modeling[END_REF] . The description of the heat transfer inside a composite fuel-water droplet relied on several simplifications. One of these is that the shear force due to the ambient air flow and the internal circulation it can induce in the droplet were not considered. This approximation allowed us to solve a problem of pure heat conduction inside a composite droplet. In the case of a perfectly centred water subdroplet, the modelling approach for both CM and MCM was based on the same set of equations as in [START_REF] Sazhin | A new approach to modelling micro-explosions in composite droplets[END_REF] . The main equations of this problem are briefly summarised below considering a spherically symmetric problem:

∂T ∂t = 1 R 2 ∂ ∂R κ R 2 ∂T ∂R , ( 1
)
where R is the distance from the centre of the droplet, T temperature, t time,

κ = κ w = k w / ( c w ρ w ) κ f = k f / c f ρ f when when R ≤ R w R w < R ≤ R d (2)
κ w( f ) , k w( f ) , c w( f ) , and ρ w( f ) are the water (fuel) thermal diffusivity, thermal conductivity, specific heat capacity, and density, respectively. Both CM and MCM are based on the resolution of Eq. ( 1) with the following initial condition:

T | t=0 = T w 0 ( R ) T f 0 ( R ) when when R ≤ R w R w < R ≤ R d (3)
and boundary conditions at the fuel-water interface and the droplet surface:

T | R =R - w = T | R =R + w , k w ∂T ∂R R =R - w = k f ∂T ∂R R =R + w , ( 4
)
h ( T eff -T ( R d ) ) = k f ∂T ∂R R = R - d , ( 5
)
where h is the convection heat transfer coefficient and T eff is an "effective" ambient gas temperature which takes into consideration the effects of droplet evaporation:

T eff = T g + L ˙ m d 4 π R 2 d h . ( 6
)
Condition ( 5) is the Robin boundary condition.

The droplet evaporation rate ˙ m d is determined by the expression:

˙ [START_REF] Fostiropoulos | Numerical investigation of heavy fuel oil droplet breakup enhancement with water emulsions[END_REF] where

m d = -4 π R d ρ total ln ( 1 + B M ) ,
B M = ( Y v s -Y v ∞ ) / ( 1 -Y v s ) is the Spalding mass transfer number, Y v s (Y v ∞ )
is the mass fraction of fuel vapour at the droplet surface (ambient conditions), D v is the vapour/gas diffusion coefficient. The dependence of total density of the mixture of ambient gas (air) and fuel vapour ρ total = ρ g + ρ v on R was not considered.

The heat transfer coefficient h is defined as:

h = k g Nu/ (2 R d ) , ( 8
)
where Nu is the Nusselt number, estimated as: [START_REF] Fostiropoulos | A simple model for breakup time prediction of water-heavy fuel oil emulsion droplets[END_REF] is the Spalding heat transfer number, T g ( T s ) gas (surface) temperature, L specific heat of evaporation, c pv specific vapour heat capacity at constant pressure, ˙ q d heat penetrating into the droplet. When deriving [START_REF] Sazhin | A simple model for puffing/micro-explosions in water-fuel emulsion droplets[END_REF] , droplets were assumed stationary, which implies that the Spalding numbers B T and B M are related as:

Nu = 2 ln ( 1 + B T ) B T , ( 9
)
B T = c pv ( T g -T s ) L ( T s ) -( ˙ q d / ˙ m d )
B T = ( 1 + B M ) ϕ -1 , ( 11
)
where

ϕ = c pv c pg 1 Le , ( 12
)
Le = k g / c pg ρ total D v is the Lewis number.
In the CM [START_REF] Sazhin | A new approach to modelling micro-explosions in composite droplets[END_REF] , the analytical solution to the above problem was expressed as an infinite series. This was evaluated at each time step using numerical code to obtain the temperature field distribution inside the droplet. In the MCM, the finite element method was applied using COMSOL Multiphysics to resolve the same problem numerically. An unstructured mesh of about 500 triangular elements was generated for the discretisation of the space.

Both CM and MCM were used to analyse heating and evaporation of a composite n-dodecane/water droplet leading to its puffing/micro-explosion. The following input parameters were used:

• The droplet initial radius and temperature were assumed equal to 5 μm and 300 K, respectively.

• The volume fraction of water was taken equal to 15%.

• Ambient gas temperature and pressure were taken equal to 700 K and 101325 Pa, respectively. • The approximations of the thermodynamic and transport properties of n-dodecane and distilled water (based on [START_REF] Yaws | Yaws handbook of thermodynamic and physical properties of chemical compounds[END_REF] ) used in the calculations are given in Appendix B .

In the CM approach, the temperature dependence of the physical properties (namely ρ, c p and k ) was taken into account by updating their values at each time step in the analytical solution. For the MCM approach, the values of the physical properties were evaluated from the temperature field calculated at the previous time step. The variation of the properties in space was directly dealt with by the COMSOL numerical solver.

In both CM and MCM, the water nucleation temperature T N at the n-dodecane-water interface, at which puffing/micro-explosion is expected to start, was estimated as [START_REF] Sazhin | A new approach to modelling micro-explosions in composite droplets[END_REF] :

T N = T B + 12 × tanh ( ˙ T / 50) ; 0 ≤ ˙ T ≤ 300 K / s , ( 13
)
T N = 385 + 160 × tanh ( ˙ T / 10 5 ) 10 2 ≤ ˙ T ≤ 10 6 K / s , ( 14
)
T N = T B + 0 . 37 T B • ˙ T 10 /Ja HN 10 5 ≤ ˙ T ≤ 10 9 K / s , ( 15
)
where Ja HN = 626 for water, T B is the boiling temperature of water in K, ˙ T is the rate of change of the temperature at the fuel-water interface in K/s.

Plots of droplet average ( T av ) and surface ( T s ) temperatures and the temperature at the fuel-water interface ( T w ) versus time, predicted by both codes are shown in Fig. 2 a. The corresponding plots showing ˙ T w at this interface are presented in Fig. 2 b. As follows from Fig. 2 , the results obtained using the CM and MCM coincide within the accuracy of plotting. Using the results presented in Fig. 2 and Formula ( 14) , the times to puffing (times when T w = T N ) were estimated as 0.465 ms (CM) and 0.464 ms (MCM). This shows that the difference between the results is only 0.2% which allows us to conclude that these values coincide. Thus both models, CM and MCM, are verified.

Note that when preparing the plots shown in Fig. 2 , the reference molar fraction of n-dodecane vapour X v , ref was estimated as

X v , ref = X v , ∞ + 2 X v ,s 3 , ( 16
)
where X v , ∞ and X v ,s are vapour molar fractions in ambient conditions and at the droplet surface, respectively. Alternatively, the reference molar fraction of n-dodecane vapour can be estimated as [START_REF] Abramzon | Droplet vaporization model for spray combustion calculations[END_REF][START_REF] Yuen | On drag of evaporating liquid droplets[END_REF] :

X v , ref = Y v , ref M v Y v , ref M v + 1 -Y v , ref M g , ( 17
)
where Y v , ref is the reference mass fraction of n-dodecane vapour inferred from the following relation:

Y v , ref = Y v , ∞ + 2 Y v ,s 3 , ( 18
)
M v (M g ) is the vapour (air) molar mass. Y v , ∞ and Y v ,s are vapour mass fractions in ambient conditions and at the droplet surface, respectively. The same plots as in Fig. 2 , but using the previous two expressions for X v , ref [START_REF] Antonov | Temperature measurements in a string of three closely spaced droplets before the start of puffing/micro-explosion: experimental results and modelling[END_REF] and [START_REF] Roy | Manufacture and properties of composite liquid marbles[END_REF] , are presented in Fig. 3 . As can be seen in Fig. 3 , although the results obtained using Expressions ( 16) and ( 17) are slightly different, the closeness between the curves allows us to use either expression for practical applications. In the results shown in Figs. 2 and3 , the variation of the droplet radius due to fuel vaporisation and thermal expansion was ignored. R d and R w were kept constant, while the vapour mass flow rate ˙ m d was evaluated based on Expression [START_REF] Fostiropoulos | Numerical investigation of heavy fuel oil droplet breakup enhancement with water emulsions[END_REF] . In fact, the size variation due to thermal swelling and vaporisation has very limited influence on the time required to reach the nucleation temperature at the n-dodecane-water interface for the conditions used in Figs. 2 and3 . The same observation was made in [START_REF] Sazhin | A new approach to modelling micro-explosions in composite droplets[END_REF] for rather similar conditions. If required, the radii of the fuel droplets and water subdroplets ( R d and R w ) could be evaluated at each time step using the requirement of conservation of masses of ndodecane and water (see Eq. ( 13) in [START_REF] Sazhin | A new approach to modelling micro-explosions in composite droplets[END_REF] ).

Description of the shift model

The Modified Centre Model (MCM) presented in Section 2 was generalised to consider possible shifts of the location of the water subdroplet from the centre of the fuel droplet. The geometry used in this generalised Modified Centre Model, called the Shift Model T w at the n-dodecane-water interface versus time. Curves 1 and 2 refer to the results obtained using Expressions ( 16) and ( 17) , respectively. The calculations were performed using CM.

(SM), is shown in Fig. 4 . The heat transfer equation ( Eq. ( 1) ) was generalised and rewritten in the cylindrical coordinate system ( r, ϑ, z), with the z-axis being the line joining the centre of the fuel droplet to that of the water subdroplet, as shown in Fig. 4 . The axisymmetric approximation was used with the symmetry condition at r = 0 :

∂T ∂r = 0 . ( 19
)
The main simplifying assumption used in the SM is that the surface temperature of the fuel droplet is uniform although it can change with time. In this case, Eqs. ( 6) -( 12) can be used without further restriction. The boundary conditions at the droplet surface ( R = R d ) are written as:

R d -R d ∂T ∂R R = R - d dz = ˙ q d 2 π R d k f (20)
∂T ∂z R = R d = 0 . ( 21
)
Equation [START_REF] Antonov | Micro-explosive droplet fragmentation of environmentally promising coal-water slurries containing petrochemicals[END_REF] indicates that temperature T s at the droplet surface is uniform. The change in this temperature with time was controlled by the average heat flux ˙ q d determined by Eq. ( 20) . Note that the spatial distribution of the heat flux at the droplet surface, required to maintain a uniform temperature at the fuel droplet surface, is not specified in Eq. ( 20) . While T s is uniform, the spatial distribution of the heat flux is not necessarily uniform. Due to the shift in location of the water subdroplet, there is no longer a spherical symmetry to the problem. The value of ˙ q d in Eq. ( 20) can be obtained from the vapour mass flow rate ˙ m d , or using the effective ambient temperature T eff :

˙

q d = | ˙ m d | c pv ( T g -T s ) B T -L ( T s ) = 4 π R 2 d • h • ( T eff -T s ) . ( 22
)
The Shift Model presented above reduces to the MCM as a limiting case when the water subdroplet is perfectly centred ( L = 0 ). In that case, (20) reduces to Eq. (5) . For the numerical resolution, difficulties are encountered in satisfying the integral boundary condition [START_REF] Castanet | Investigation of heat and mass transfer between the two phases of an evaporating droplet stream using laser-induced fluorescence techniques: comparison with modeling[END_REF] within the constraints of a uniform surface temperature ( Eq. 21 ). These difficulties, however, were overcome by using the linearity of the heat conduction problem. The approach which we used is similar to that described by Abramzon and Sirignano [START_REF] Abramzon | Droplet vaporization model for spray combustion calculations[END_REF] for solving the energy equation in the Hill vortex model (see Eqs. (40)-(44) in [START_REF] Abramzon | Droplet vaporization model for spray combustion calculations[END_REF] ). In their paper, due to the advection of a Hill vortex, a temperature field with no spherical symmetry is assumed, while a uniform temperature is imposed at the droplet surface. The heat conduction problem is decomposed into two problems, making it more straightforward to solve, with a uniform temperature imposed at the droplet surface (Dirichlet condition). See Appendix A for more details.

The shift in the position of the water subdroplet considered in the SM is described by parameter L , the distance between the centre of the water subdroplet and the fuel droplet, shown in Fig. 4 .

The maximal value of L is L max = R d0 -(R w 0 / 2) , where R d0 and R w 0 are the initial values of fuel droplet and water subdroplet radii, respectively. For our analysis, we introduced the normalised shift ( S)

S = L L max . ( 23
)
The main consequence of this shift is that the heating of the fuel-water interface in this case depends on the distance between the droplet surface and this interface. The maximal heating rate is expected at the point closest to the droplet surface. One would expect that the temperature at this point on the interface will be the first to reach the water nucleation temperature, leading to the initiation of puffing/micro-explosion at this location.

Plots of the maximal temperature at the fuel-water interface T w versus time for several S are presented in Fig. 5 a. As follows from this figure, for any given time instant T w increases with increasing S as expected.

Plots of ˙ T w versus time at the fuel-water interface, at the shortest distance from the fuel droplet surface are presented in Fig. 5 b for several values of S. As can be seen in this figure, the dependence of ˙ T w on S is more complex than that of T w . At times shorter Fig. 6. Times to puffing/micro-explosion at various S inferred from the results presented in Fig. 5 and Formulae ( 14) and ( 13) . than about 0.1 ms, ˙ T w increases with increasing S, while at times longer than about 0.1 ms, ˙ T w decreases with increasing S. At the short times, the heat diffusion inside the droplet is very limited, which leads to large temperature gradients and temporal variations in the temperature at the droplet surface and in its proximity.

Using the results presented in Fig. 5 and Correlations ( 13) -( 15) the time instants when T w = T N were obtained for various S. These time instants are known as times to puffing/micro-explosion τ p . The values of τ p for various S are presented in Fig. 6 . As follows from this figure, the values of τ p decrease by less than 1% when S increases from 0 to 0.2. This means that the model developed by the authors of [START_REF] Sazhin | A new approach to modelling micro-explosions in composite droplets[END_REF] (CM) can be safely used when S is less than 0.2. If errors up to 5% can be tolerated, then this model can be used for cases when the shifts are as high as 0.5. For all cases shown in Fig. 6 , it should be noted that τ p > 0 . 38 ms. As can be seen in Fig. 5 b, at these times ˙ T w < 10 5 K/s. Hence, Correlation [START_REF] Antonov | Puffing and micro-explosions in rapeseed oil/water droplets: the effects of coal microparticles in water[END_REF] , or even [START_REF] Abramzon | Droplet vaporization model for spray combustion calculations[END_REF] , can be used for the estimation of T N .

As a second example, plots of times to puffing/micro-explosion versus S are shown in Fig. 7 for droplets with initial radii R d0 = 0 . 85 mm and various ambient gas temperatures T g . The initial droplet temperature was assumed equal to 300 K. The volume fraction of water was assumed equal to 10% and the ambient gas pressure was fixed at 101325 Pa. As in the case of the previous figures, the transport and thermodynamic properties of n-dodecane and distilled water presented in [START_REF] Yaws | Yaws handbook of thermodynamic and physical properties of chemical compounds[END_REF] were used. When estimating the values of τ p shown in Figure 7 , the values of ˙ T w were less than 300 K/s. Thus, Correlation (13) was used for estimating T N . As follows from Figure 7 , for all temperatures τ p slowly decreases with increasing S. When S increases from 0 to S = 0 . 3 , a maximal decrease of 3% in τ p was observed for T g = 623 K. This result is consistent with the one shown in Fig. 6 . As follows from Figs. 6 and7 , the errors in estimating τ p using the Centre Model do not exceed 5% even in the case of a large shift S = 0 . 5 . In many applications, this can be tolerated in practical estimations of this parameter.

The predictions of the shift model versus experimental data

In this section, the predictions of the Shift and Centre models are compared with experimental data obtained at Tomsk National Research Polytechnical University. The experiments took place in a heated furnace where the droplets were supported by a nickelchromium alloy wire of 0.2 mm diameter as in the experimental setup described in [START_REF] Antonov | Micro-explosive droplet fragmentation of environmentally promising coal-water slurries containing petrochemicals[END_REF] . The initial droplet temperature was 300 K; the initial droplet radii were in the range 0.65 mm to 0.95 mm. The ambient pressure was atmospheric, taken equal to 101325 Pa; the gas temperatures were in the range 430 K to 850 K. The normalised shift S was inferred from direct observations with errors 5-10% when S varied from 0 to 0.75. Note that our estimations of the shift S were made at t = 0 s. However, during droplet heating, the water subdroplet could move from its start point due to the effect of natural convection, which induced circulation flows in the gas and liquid phases.

Note that in some of our experiments several water subdroplets were observed inside single fuel droplets [START_REF] Antonov | Breakup and explosion of droplets of two immiscible fluids and emulsions[END_REF] . The analysis of these cases, however, is beyond the scope of this paper.

Two examples of observations of the evolution of the shapes of composite Diesel fuel-water droplets before and during puffing/micro-explosion, using this setup, are demonstrated in Fig. 8 . In both cases, the time to puffing was considered to be the time instant when the first child droplet is separated from the parent droplet. This time instant was sometimes well separated from that when micro-explosion was observed as illustrated in Fig. 8 a. In these cases some caution would be required when the Centre Model is applied to the interpretation of experimental data as this model assumes that micro-explosion follows puffing without delay. In this case time to puffing/micro-explosion can be referred to as time to puffing.

When applying the Shift Model to the analysis of these data, we need to keep in mind that the difference between the droplet surface temperature and ambient temperature in the presence of gravity leads to the development of natural convection. This convection was considered in the expressions for the Nusselt ( Nu ) and Sherwood ( Sh ) numbers using the following formulae [START_REF] Pinheiro | Evaluation of droplet evaporation models and the incorporation of natural convection effects[END_REF] : where Gr is the Grashof number: Expressions [START_REF] Wang | Experimental study on the evaporation and micro-explosion characteristics of nanofuel droplet at dilute concentrations[END_REF] and [START_REF] Shen | Puffing and micro-explosion of heated droplets for homogeneous ethanol-propanol-hexadecane fuel and micro-emulsified ethanol-biodiesel-hexadecane fuel[END_REF] were used in both the Centre Model (CM) and the Shift Model (SM). Diesel fuel was approximated by ndodecane. Only changes in T s were considered in both CM and SM, while the changes in R d due to evaporation and thermal swelling were ignored. As follows from further analyses with CM, the effects of the change in R d on time to puffing/micro-explosion are small and can be ignored in most applications [START_REF] Wang | Experimental study on the evaporation and micro-explosion characteristics of nanofuel droplet at dilute concentrations[END_REF][START_REF] Shen | Puffing and micro-explosion of heated droplets for homogeneous ethanol-propanol-hexadecane fuel and micro-emulsified ethanol-biodiesel-hexadecane fuel[END_REF] . Hence, the assumption of constant R d is not expected to affect the comparison of CM and SM predictions with experimental results.

Nu = 2 + ( Nu 0 -2 ) F T • ln ( 1 + B T ) B T , ( 24
)
Sh = 2 + ( Sh 0 -2 ) F M • ln ( 1 + B M ) B M , ( 25
)
Nu 0 = 2 + 0 . 6 Gr 0 . 25 P r 0 . 33 , ( 26
)
Sh 0 = 2 + 0 . 6 Gr 0 . 25 Sc 0 . 33 , ( 27
)
F M(T ) = 1 + B M(T ) 0 . 7 • ln 1 + B M(T ) B M(T ) . ( 28
)
Gr = g ( 2 R d ) 3 | T g -T s | ν 2
Special care was taken to evaluate the shifting of the water subdroplet in the experiments. Planar laser-induced fluorescence (PLIF) was applied to find the position of the water subdroplet with a high degree of accuracy. As in the experiments discussed by Antonov et al. [START_REF] Antonov | Comparison of the characteristics of micro-explosion and ignition of two-fluid water-based droplets, emulsions and suspensions, moving in the high-temperature oxidizer medium[END_REF] , rhodamine B was added to water (10 0 0 μg/l) prior to the preparation of the composite droplet. Since rhodamine B is not miscible in the fuel, only the water subdroplet emits a fluorescent signal when illuminated by the laser. In addition, a shadowgraphy technique using a high-speed camera provided sharp images of the droplet edge. The centroids of the water subdroplet and the fuel droplet were determined using a homemade image analysis program based on Matlab software and its image processing toolbox. Then, the shift L between the two centroids (fuel droplet and water subdroplet) was calculated as illustrated in Fig. 9 .

PLIF visualisation of the water subdroplet was performed before heating started. This allowed us to estimate the values of L used in the Shift Model. This shift was assumed to remain unchanged until at least the time when puffing/micro-explosion occurred. In several experiments, however, some movements of the water subdroplet could be observed, which may lead to differences between the model predictions and the results of the experiments.

The observed times to puffing/micro-explosion and those predicted by the Centre Model (CM) and Shift Model (SM) for nine cases are presented in Table 1 . As follows from this table, in most cases (Cases 1, 2, 3, 5, 6, 7) the values of τ p predicted by SM are much closer to the experimental data than those predicted by CM. The deviations between the predictions of SM and the experimental results could be partly attributed to considerable uncertainty in the estimation of S using our experimental data, most probably due to movement of the water subdroplet during the experiment. There may also be several other reasons for this, including our assumption that both water subdroplet and fuel droplet are perfectly spherical. In contrast to Cases 1, 2, 3, 5, 6, and 7, for Case 4 the match between the SM predictions and the experimental results is worse than that for CM.

Using S as a tuning parameter, exact matching between the predictions of SM and experimental data was achieved: ( S= 0.96 for Table 1 Times to puffing/micro-explosion ( τp ) observed experimentally (Exp) and predicted by the Centre Model (CM) and Shift Model (SM) for nine cases with the values of R d0 , T g and S shown in the table. In all cases, the initial droplet temperature was 300 K, volume fraction of water was 10%, and ambient gas pressure was atmospheric. In these cases the predictions of CM and SM coincide and both are reasonably close to experimental data.

Case

Conclusions

An approach to the verification of the predictions of the previously developed model of puffing/micro-explosion (see [START_REF] Sazhin | A new approach to modelling micro-explosions in composite droplets[END_REF] ) is suggested. That model uses the assumption that a spherical water subdroplet is located exactly in the centre of a spherical fuel droplet. The analytical solution to the transient heat transfer equation in this droplet, with the Robin boundary condition at its surface, is implemented into a numerical code and used at each time step of the calculations. The time instant when the temperature at the fuel-water interface is equal to the water nucleation temperature is considered to be the start of puffing/micro-explosion. Verification was performed by comparing the predictions of this model with the predictions of the same model, but in which the solution to the heat conduction equation inside the composite droplet was found using the heat transfer module available from COMSOL software (a purely numerical solution to the equation). Agreement between the predictions of both codes supports the validity of both approaches to the problem.

The model suggested in [START_REF] Sazhin | A new approach to modelling micro-explosions in composite droplets[END_REF] was generalised to consider the shifting of the water subdroplet from the centre of the fuel droplet. This generalisation was based on the numerical solution to the heat transfer equation inside the composite droplet taking into account this shift with other assumptions being the same as in the model in [START_REF] Sazhin | A new approach to modelling micro-explosions in composite droplets[END_REF] . The start of puffing/micro-explosion in this generalised model is identified with the time instant when the temperature at the point of the fuel-water interface closest to the surface of the fuel droplet becomes equal to the water nucleation temperature. The effects of natural convection on droplet heating and evaporation were considered in both the original and the generalised models.

The size of the shift used in the generalised model is inferred from experimental observations. It is demonstrated that in most cases the time to puffing/micro-explosion predicted by the generalised model is closer to experimental results than that predicted by the original model. The shift is quantified by the normalised shift defined as S = L/L max , where L is the distance from the centre of the water subdroplet to that of the fuel droplet, and L max = R d0 -(R w 0 / 2) ( R d0(w 0) is the initial radius of the fuel droplet (water subdroplet)) is the maximal value of L . It was shown that for typical values of input parameters for S ≤ 0 . 2 the predictions of the original and generalised models differ by less than 1%. For S ≤ 0 . 5 this difference increases to 5% which is still acceptable in many engineering applications.

At time t 0 , the initial value of the function T 1 is obtained as: T 1 ( r, z, t 0 ) = T ( r, z, t 0 ), (31) where T ( r, z, t 0 ) is the temperature distribution in the fuel-water droplet at the last time step of the resolution. The function T 2 represents the effect of a temperature increment of 1 • C at the droplet surface. This function follows from the boundary (surface) and initial conditions: T 2 ( R = R d , t ) = 1 for t 0 ≤ t ≤ t 0 + dt (32) T 2 ( r, z, t 0 ) = 0 .

(33) Finally, the temperature is evaluated from the values of T 1 and T 2 using Eq. ( 29) . The boundary condition [START_REF] Antonov | Micro-explosive droplet fragmentation of environmentally promising coal-water slurries containing petrochemicals[END_REF] , corresponding to a uniform surface temperature, is automatically satisfied. The parameter A in Eq. ( 29) is determined due to the requirement to satisfy the boundary condition [START_REF] Castanet | Investigation of heat and mass transfer between the two phases of an evaporating droplet stream using laser-induced fluorescence techniques: comparison with modeling[END_REF] ,

A = ˙ q d 2 π R d k f I 2 - I 1 I 2 ( 34
)
I 1 = R d -R d ∂ T 1 ∂R R = R - d dz and I 2 = R d -R d ∂ T 2 ∂R R = R - d dz ( 35
)
The value of A determined by Eq. ( 34) changes with time. The two partial differential equations for T 1 and T 2 described above were solved numerically based on the finite element method using COMSOL Multiphysics. As for the numerical simulations implemented for the MCM, an unstructured mesh of about 500 triangular elements was generated for the discretisation of the space. One advantage of the proposed decomposition (29) is that functions T 1 and T 2 can be found independently of one another. If the physical properties of water and fuel (n-dodecane) do not vary with time, then the evaluation of T 2 can be performed just once at the beginning of the calculations. In the present case, to deal with the temperature dependence of the physical properties, parameters k , ρ and c in the heat transfer equation were evaluated using the temperature distribution T ( r, z, t 0 ) calculated at the previous time step. Both T 1 and T 2 were calculated as explained above.

  -] c Specific heat capacity [J/(kg K)] D Diffusion coefficient [m 2 /s] g Acceleration due to gravity [m/s 2 ] Gr Grashov number [-] h Convection heat transfer coefficient [W/(m 2 K)] k Thermal conductivity [W/(m K)] L Distance between the centres of water subdroplet and fuel droplet [m] L Specific heat of evaporation [J/kg] Le Lewes number [-] M Molar mass [kg/kmole] ˙ m d Evaporation rate [kg/s] Nu Nusselt number [-] P r Prandtl number [-] ˙ q d

Fig. 1 .

 1 Fig. 1. The position of the water subdroplet in a fuel droplet in the model developed in [12] .

Fig. 2 .

 2 Fig. 2. (a) Plots of droplet average ( T av ) and surface ( T s ) temperatures and the temperature at the n-dodecane-water interface ( T w ) versus time, (b) plots of ˙ T w at the n-dodecane-water interface versus time. The curves CM and MCM refer to the results predicted by the Centre Model and Modified Centre Model, respectively.

Fig. 3 .

 3 Fig. 3. (a) Plots of droplet average ( T av ) and surface ( T s ) temperatures and the temperature at the n-dodecane-water interface ( T w ) versus time, (b) plots of ˙T w at the n-dodecane-water interface versus time. Curves 1 and 2 refer to the results obtained using Expressions (16) and (17) , respectively. The calculations were performed using CM.

Fig. 4 .

 4 Fig. 4. Schematic presentation of the geometry used in the Shift Model.

Fig. 5 .

 5 Fig. 5. (a) Plots of the maximal temperature at the n-dodecane-water interface ( T w ) versus time for several values of S, (b) plots of ˙ T w at the n-dodecane-water interface, at the shortest distance from the fuel droplet surface, versus time for several values of S. Calculations were based on Expression (18) . The same values of input parameters as in Figs. 2 and 3 were used.

Fig. 7 .

 7 Fig. 7. Times to puffing/micro-explosion versus S at various gas temperatures T g for droplets with initial radii R d0 = 0 . 85 mm.

Fig. 8 .

 8 Fig. 8. Typical video frames showing the evolution of shapes of Diesel fuel-water droplets before and during puffing/micro-explosion. (a) Droplet with initial radius R d0 = 0 . 90 mm and S = 0 . 75 , introduced into a gas at temperature T g = 473 K. (b) Droplet with R d0 = 0 . 65 mm and S = 0 . 52 , introduced into a gas at temperature T g = 850 K. In all cases, the water volume fractions were 10%, the initial droplet temperatures were 300 K, and ambient gas pressure was atmospheric (assumed equal to 101325 Pa).

  T s , P r and Sc are Prandtl and Schmidt numbers, respectively, ν is the kinematic ambient gas viscosity, g is the acceleration due to gravity, B M ( B T ) are Spalding mass (heat) transfer numbers.

Fig. 9 .

 9 Fig. 9.A typical image of a fuel-water droplet obtained by shadowgraphy with a high-speed camera (left); image of a water subdroplet obtained using the PLIF technique with water seeded with rhodamine B (middle). Once the centres of the fuel droplet and the water subdroplet were identified, the distance L was estimated as shown in (right).

  S= 0.75 for Case 2, S= 0.71 for Case 3, S= 0.35 for Case 4, S= 0.77 for Case 5, S= 0.78 for Case 6). For Cases 8 and 9, the water subdroplet was almost exactly in the centre of the fuel droplet.

		R d0 (mm)	T g (K)	S	τp (s) (Exp)	τp (s) (CM)	τp (s) (SM)
	1	0.90	473	0.75	7.02	8.5	7.74
	2	0.65	850	0.52	1.02	1.50	1.24
	3	0.86	523	0.71	5.11	5.70	5.13
	4	0.82	523	0.72	5.11	5.30	4.72
	5	0.88	523	0.71	5.14	5.90	5.29
	6	0.86	523	0.71	4.90	5.80	5.10
	7	0.83	523	0.72	3.74	4.30	3.74
	8	0.95	430	< 0 . 01	14.40	14.50	14.50
	9	0.83	623	< 0 . 01	3.52	3.60	3.60
	Case 1,

Acknowledgments

The authors are grateful: to the Université de Lorraine and the Institut Carnot Icéel for the CALICO (Combustion d'Aérosols Liquides Complexes) grant which supported G. Castanet (who contributed to the development of the model, the simulations and preparation of the text of the paper); for a scholarship from the President of the Russian Federation (Grant SP-447.2021.1) which supported D. Antonov (who performed the experiments, and applied the model to their analysis); to the National Research Tomsk Polytechnic University (project VIU-ISHFVP-60/2019) which supported P.A. Strizhak (who contributed to the analysis of the results); and to the Russian Science Foundation (Grant 21-19-00876 ), which supported S.S. Sazhin (who contributed to the development of the model, analysis of the results and preparation of the text of the paper).

Declaration of Competing Interest

The work described has not been published previously and it is not under consideration for publication elsewhere. Its publication is approved by all authors and tacitly or explicitly by the responsible authorities where the work was carried out. If accepted, it will not be published elsewhere in the same form, in English or in any other language, including electronically without the written consent of the copyright-holder.

CRediT

Appendix A

Assuming that the temperature distribution inside the droplet is known at a certain time t 0 , the temperature at the next time step (t 0 + dt ) is presented as a linear combination of two functions T 1 and T 2 :

Functions T 1 and T 2 are both obtained by solving the heat transfer equation. They both satisfy the conditions for axi-symmetry [START_REF] Castanet | Evaporation of closely-spaced interacting droplets arranged in a single row[END_REF] . T 1 is the solution to a modified problem where the surface temperature does not vary during the time step between t 0 and t 0 + dt :

The properties of components used in calculations. T is in K.


Introduction

Modalities, whether in formal semantics [START_REF] Portner | Modality[END_REF] or in Gosselin's cognitive semiotics (2010), are most often considered at the level of utterances, or even lexemes. Furthermore, they are systematically considered as a primarily linguistic phenomenon, whether they are semantically inscribed in certain lexemes (such as the verbs "must", "can", etc.), or inferred from uttered propositions. At this level of analysis, the complexity of modalities is already evident: the parameters required for its determination are numerous, cover all the traditional fields of linguistics, and are determined by different elements that interact with each other in the sentence. They are part of what can be called a "system", following Larsen-Freeman & Cameron (2008):

In the abstract and as a broad definition, a system is produced by a set of components that interact in particular ways to produce some overall state or form at a particular point in time (Larsen-Freeman & Cameron, 2008, p. 26) Our purpose here is to explore the complexity of this system, and to reveal aspects of it that are not yet sufficiently taken into account in standard analyses. A system can be considered as "complex" when it meets several criteria. The first criterion is the presence and interaction of agents who are heterogeneous by their nature, within the system: One way in which complex systems often differ from simple systems is in having many different types of elements or agents: i.e. they are "heterogeneous". […] The complexity of a complex system arises from components and subsystems being interdependent and interacting with each other in a variety of different ways (Larsen-Freeman & Cameron, 2008, p. 28).

The second criterion is as follows: to be considered as complex, a system must have an evolutionary and dynamic nature: it is characterised by a change comparable to a flow:

In the type of complex systems that we are concerned with, everything is dynamic: not only do the component elements and agents change with time, giving rise to changing states of the system, but the ways in which components interact with each other also change with time (Larsen-Freeman & Cameron, 2008, p. 29).

Finally, the third and last criterion states that within a complex system, this constant change is not linear, due to the heterogeneous nature of its components and the interactions that bind them:

In a non-linear system, the elements or agents are not independent, and relations or interactions between elements are not fixed but may themselves change (Larsen-Freeman & Cameron, 2008, p. 31).

We will work on a short conversation held via chat, a form of written, synchronous, and online communication: users are simultaneously present in a virtual chat room and produce written utterances that constitute their turns of speech. This corpus will be explored from the three angles mentioned above. First, we will show that modality is a complex system composed of heterogeneous subsystems and elements. Chat conversations make this heterogeneity very visible, since from their appearance in the 1970s, and certainly to compensate for the lack of a mimo-gestural system in writing, they are filled with iconic pictograms representing mimics or gestures. They are intended to indicate the speaker's emotion 1 , and thus bear modality. We will

1
The speaker is the instance that takes responsability for the propositional content of utterances. This instance is built in and by utterances and is committed to the "truth" of what is said (the term "truth" should not be understood here in its usual sense in formal logic, but rather as an implicit intersubjective agreement concerning the truth values of the propositions uttered -on this subject see the work of Berrendonner, 1981 or Anscombres, therefore present into more details what we mean by "modality" in the first part, then we will describe this iconic semiotic subsystem, which bears modalities and does not belong to a linguistic system per se. Then, we will study the evolutionary dynamics of modalities as the conversation progresses and show, amongst other things, that modalities interact with each other to form a global and ever-changing frame.

Modalities as a heterogeneous system

What we mean by "modality"

There are several ways of conceiving the notion of modality, which has its source in Aristotle's early work on logic and is historically linked to the notions of possibility and necessity. Nowadays, two major approaches address this notion. The first one, formal semantics and the semantics of possible worlds (see for example [START_REF] Portner | Modality[END_REF][START_REF] Portner | Mood[END_REF], is based on a purely descriptivist approach to linguistic meaning: in this context, a linguistic utterance always bears, at a more or less deep level, an absolute and verifiable truth about the world. This position is directly derived from analytical philosophy and formal logic: behind every linguistic utterance, there is somewhere a proposition or a system of propositions (called "conversational background" in Kratzer, 1977, for example), which can be said to be true or false. In this perspective, modality is understood as "the linguistic phenomenon whereby grammar allows one to say things about, or on the basis of, situations which need not be real" (Portner 2009, p.1). This approach aims to describe everything related to the expression of possibility and necessity, for example, and especially the tense and mode systems of verbs (see for example [START_REF] Mari | Modalités et temps : des modèles aux données[END_REF].

For our part, we follow the second approach, which adopts a "broad" interpretation of modality, as defined, for example, by Gosselin (2010), who considers that modality is the "validation [or invalidation] of a predicated representation" (Gosselin 2010, p.50, personal translation).

This approach is in line with the work of [START_REF] Bally | Linguistique Générale et Linguistique Française[END_REF]Berrendonner (1981, p. 59). In addition to the famous "modus / dictum" distinction (the dictum being the "representational" part of the utterance, and the modus the behaviour that relates to it), Bally introduces the notion of modal subject, an instance that reacts to the enunciation of a truthconditional representation: when I say something, I automatically position myself in a certain way with respect to the truth of what I say.

As for Berrendonner, he proposes to redefine what is meant by "truth" when analysing linguistic utterances. In particular, he shows that even assertions are based on a relative truth, shared by a community of speakers (this is what he calls "on-vérité" or 'we-truth'). In this context, he considers that it is more appropriate to speak about the "validation" or "invalidation" of a content than about "truth" and "falseness".

Modality is therefore understood as the positioning of an enunciative instance (the modal subject) with respect to the relative "truth" of the utterance. This view of modalities is much broader than the first approach, and is not limited to the analysis of phenomena explicitly manifested in language, such as temporal or modal morphemes. In this perspective, modalities can be defined according to nine parameters: "validation instance, direction of adjustment, strength of validation, level in the syntactic hierarchy, scope in the logical structure, enunciative commitment, relativity, temporality, marking" (Gosselin 2010). These parameters concern most fields of classical linguistics: syntax, semantics, pragmatics, and enable to define the traditional categories of modality: alethic, epistemic, appreciative, axiological, boulic, and deontic.

Different "subsystems" to express modalities

Below is an exchange, taken from a chat corpus. The speakers (<L1> and <L2>) mention a poker game. The ten turns of speech are indicated using T1, T2, etc.; the rest of the conversation is authentic and appears as such in the chat software. We will not analyse here all the modalities of exchange, due to a lack of space, but we will propose some observations that will allow us to further broaden the notion of modality. First, it is possible to identify two ways of expressing modalities, as already described elsewhere (see Gosselin 2010). Some modalities are marked: they are semantically coded and integrated in certain linguistic forms, for example verbs. Thus, in T4, the verb "devoir" 'must' semantically bears an epistemic modality: the speaker indicates that he is not sure that he is "at -100$" but that it is probable. Other modalities are rather inferred and implicit: in T2, a fixed phrase such as "m'en parles pas " 'tell me about it' triggers an inferential process that leads to the conclusion that L2 has lost, but also that he is disappointed (appreciative modality). In addition to these two systems of meaning -semantics and pragmatics, which are well described in the literature on modalities -there are other lesser-studied elements in digital writings: pictograms, here emoticons (T1, T3, T8 and T9), which also bear modalities (on this subject see Halté 2018[START_REF] Halté | Iconicité et signification modale : l'émoticône, de l'icône du corps au geste énonciatif[END_REF]. They constitute a new "subsystem" for the expression of modality. Indeed, we believe that beyond linguistic phenomena, other utterances belonging to various semiotic systems ( for example, gestures in oral language, or pictograms in digital writings, which can be seen as forms of written gestures, see Halté, to appear), allow the speaker to validate or invalidate predicated representations.

Iconic modalities

In addition to the already complex system of expressed or inferred linguistic modalities, iconic modalities are therefore added in digital interactions in writing. These modalities are identified and interpreted according to an iconeme: a minimal unit of iconic meaning, based on the recognition of the resemblance to an object. The emoticons of this excerpt 2 are composed of two minimal units: on the one hand, the colon represents the eyes and constitutes what can be called a "positional" iconeme (its only purpose is to help understand how the icon is spatially organised); and, on the other hand, the mouth shapes, which are modal iconemes since they are the sole bearer of modality. This can be illustrated with the following test, which simply consists in switching these icons:

(1) T8[15:00] <L2> mais bon en ce moment ça vient pas trop :S 'but right now it's not coming too much :S'

Replacing the "S", an icon representing a mouth twisted with disgust, by a closing parenthesis changes the orientation of the modality (here appreciative), which shifts from undesirable to desirable:

(2) T8[15:00] <L2> mais bon en ce moment ca vient pas trop :) 'but right now it's not coming too much :)'

The modalities borne by modal iconemes interact with the modalities borne by the linguistic part of utterances, and produce effects (irony, emphasis, empathy, etc.) depending on whether they are in opposition or in agreement with the latter (see [START_REF] Yus | Cyberpragmatics: Internet-mediated communication in context[END_REF].

To sum up: the interpretation of linguistic modality is based on nine parameters; modalities can be marked or inferred; and finally, modalities can to be borne by different semiotic systems interacting more or less linearly with each other as the conversation unfolds. Rather than using the term modality, we propose that from now on we talk about "complex modal system", and we will now try to describe its evolution over the course of a conversation. Indeed, it seems plausible to imagine that all these elements would set up a global modal configuration, which would condition the uttered contents and the successive behaviours adopted by speakers as their conversation unfolds.

Dynamics of modalities in conversation

Monological modal sequences

We will defend here the idea that modality is not only reaction of a modal subject to a content that he/she utters, but rather a coconstruction based on conversational parameters: sequences of turns of speech, enunciation situation, knowledge shared by the interlocutors, etc. In this perspective, following [START_REF] Bres | Dialogisme : du principe à la matérialité discursive[END_REF] in particular, we consider that a modality can, not only relate to a content uttered by the speaker (monological modality), but also to a content uttered by the interlocutor (dialogical modality). Our purpose here is to describe sequences of monological modalities, then of dialogical ones, and finally that of an exchange, which will help to show the dynamic, changing and non-linear aspect of the complex modal system that we are trying to describe here.

At a first level, modalities can be distributed and interact within an utterance produced by the speaker. It is sometimes possible to identify modalities very locally, at the level of one or more lexemes, but these local modalities generally contribute to the construction of a global modality, which is interpreted at the level of utterances. This is the case, for example, in T4, which is globally uttered in an epistemic manner:

(3) T4 [14:59] <L2> bah je dois en être à -100$ 'well I must be at -100$'

Here, the epistemic modality, which is semantically borne by the verb "devoir" 'must', is combined with an appreciative modality borne by "bah" 'well' (which has also an epistemic value here). In T6, the speaker combines an alethic modality, by presenting a content as an objective truth using an assertion centred around the verb "être" 'to be', and an appreciative modality borne by the emoticon:

(4) T6 [14:59] <L2> mais bon c'est que des gains que j'ai perdu :) 'but I only lost money I had won :) (lit: but it is only I who lost money I had won)'

The appreciative modality, unlike other modalities, can easily be combined with all the other modalities.

Modalities can also explicitly show their fundamentally interactional nature. Thus, in T1, the question mark, with an epistemic modality, and the "sticking one's tongue out" emoticon indicating that the utterance previously produced is a provocation have one thing in common: they make explicit the fact that these modalities are addressed to others.

(5) T1[14:54] <L1> alors t as gagné ?

'so you won?' T1 [14:54] <L1> :p

The question mark indicates a real question, asked to the interlocutor; and the "sticking one's tongue out", which relates to the previously uttered proposition (it indicates that in some way, the speaker expects a negative answer to the question asked), can only be explicitly addressed to the interlocutor. The epistemic modality, which expresses doubt or questioning, necessarily triggers an attempt by the latter to answer this doubt or question, whether it is explicitly (as is the case here) or implicitly addressed to the interlocutor. These sequences of monological modalities come into contact, at a second level, with modalities assumed by the interlocutor: they thus become dialogical modalities.

Dialogical modal sequences

Let's look at an example taken from our conversation: (6) T8 [15:00] <L2> mais bon en ce moment ca vient pas trop :S 'but right now it's not coming too much :S' T9 [15:00] < L1> :( In T8, L2 produces an utterance with a modality that gives an appreciation (positive or negative), materialised by the emoticon ":S", which indicates disgust. T9 is produced by L1 and only consists of an emoticon representing a mimic of sadness. It is clear that the modality expressed by L2 in T8, which is an appreciative modality oriented towards the "undesirable" pole (see Gosselin 2010 for more data on appreciative modalities), determines which modalities can be used in T9. The question is how. Two hypotheses can be suggested:

• By producing T9, L1 implicitly takes up the content (or dictum, which is therefore not modalised) produced by L2 in T8 and modalises it in turn;

• By producing T9, L1 assigns an appreciative modality to a complete and already modalised utterance, produced in T8 by L2.

We believe that the second option is the least costly and most defensible. It is impossible to imagine that the emoticon in T9 would relate to anything other than T8, that is to say, a complete utterance, which is therefore already modalised by L2. We agree here with Bres & Nowakowska's proposition when they define dialogism in the following way:

(a) I will consider as dialogical an utterance (or fragment of an utterance) in which the modalisation of E1 applies to a dictum presented as already having the status of utterance (i.e. e), i.e. a dictum that has been modalised by another enunciator, whom I refer to as e1 (Bres 1999: p. 72, personal translation).

(b) We assume that dialogical utterances differ from monological utterances in the following way: in monological utterances, deictic and modal actualisation applies on a dictum; in dialogical utterances, this operation is not carried out on a dictum, but on (what is presented as) an already actualised utterance (Bres and Nowakowska 2006: p. 29, personal translation).

The framework set by Bres & Nowakowska changes the scope of application of modalities, as defined by Gosselin (2010). Indeed, this view on dialogism presupposes that it is possible to have modalities on something other than a predicated representation: an already modalised utterance. Monological modality thus sets a kind of modal background, to which other modalities can be added -dialogical ones -which are expressed later. These dialogical modalities, which apply to a previously fixed modal domain, have effects: it is because L1 expresses in T9 an affective modality along the same lines as that expressed in T8 that T9 is interpreted as an expression of empathy.

Furthermore, it is possible not only to infer modalities, but also to make implicit contents bear modalities, still within the context of dialogue. Thus, in the sequence from T1 to T3:

(7) T1[14:54] <L1> alors t as gagné ?

'so you won?' The surprise emoticon that appears at the very beginning of T3 brings an epistemic modality to an implicit content. Indeed, in T2, L2 produces "pffff m'en parles pas " 'pfff tell me about it'. This utterance, consisting of a fixed phrase preceded by an onomatopoeic interjection indicating disappointment, cannot be interpreted literally, but it triggers an inferential process leading to a conclusion that can be glossed as follows: "I have lost and I am disappointed". The epistemic modality, borne by the surprise emoticon, relates to this implicit conclusion and not to the literally uttered proposition "ne m'en parle pas" (lit: don't talk about it to me). The rest of the conversation develops based on this epistemic modality, and constitutes in turn a request for explanation that L2 indeed provides.

Conclusion

While the definition of modality as the "validation of a predicated representation" appears as quite simple and easily isolable in the context of monological utterances, the analysis of dialogical utterances shows that the notion needs to be extended, or at least to be considered at different levels of conversation: we are dealing with a real modal system, which is rather a frame in which predicated, and sometimes already modalised representations appear. We propose that, in addition to the modalities studied at the level of utterances or turns of speech, research should also include the study of modalities at the level of exchanges, as a complex system: a non-linear and unpredictable succession of validations or invalidations of explicit and implicit contents and/or already modalised contents, marked by signs belonging to different semiotic systems, which determine subsequent turns of speech.

2

  To be read by tilting your head to the left, to recognise:


Shaban Shahzad
email: shahzad@insa-toulouse.fr

Ahmed Toumi

Jean-Paul Balayssac

Anaclet Turatsinze

Design of smart cementitious composites based on multi-walled carbon nanotubes (MWCNTs) using probe ultrasonicator for dispersion

Introduction

Most of the civil infrastructures including highways, residential and commercial buildings, and dams are exposed to the severe and changing environment, chemical and physical agents, and thermal stresses. In addition, cementitious materials are under selfdegradation due to aging, shrinkage cracking and construction defects. So, in the recent developments, monitoring of concrete structure is of prime concern to avoid catastrophic impact on humanity. For this purpose, traditional non-intrinsic based sensors such as strain gauges, optical fibers, shape memory alloys, and piezoelectric materials were used to monitor the behavior of the structure under external loading. Although these non-intrinsic sensors are easy to install, they are very sensitive, and can provide misleading information due to electromagnetic interference and lower sensitivity to the response. Moreover, non-intrinsic sensor and cementitious composites, being different materials, poses serious concern over the long term monitoring and high maintenance cost [START_REF] Abedi | A review of intrinsic self-sensing cementitious composites and prospects for their application in transport infrastructures[END_REF].

In the past few decades, the development of a cementbased intrinsic piezo-resistive sensor has reshaped the field of Structural Health Monitoring (SHM) in the construction industry. In line with the piezo-resistivity phenomenon, cement-based sensors can be used to monitor the health of the structures (mechanical strain or stress variation) by measuring the variation in the concrete electrical resistivity with external loading. However, the cementitious materials offer high electrical resistance. So, to develop a smart cementitious material, inclusion of conductive material is essential to reduce the electrical resistance of the composite material.

Various types of conductive materials are used to prepare the smart cementitious materials including steel fibers, carbon fibers, carbon nanotubes (CNTs), slag, and carbon powder. Since after the discovery of carbon nanotubes by Iijima in 1991 [START_REF] Iijima | Helical microtubules of graphitic carbon[END_REF], they have been extensively used in smart cementitious composites due to their high electrical conductivity.

Due to the large specific surface area of CNTs, they develop strong Van der Waals attractive forces resulting in the agglomeration of CNTs in aqueous medium [START_REF] Peigney | Specific surface area of carbon nanotubes and bundles of carbon nanotubes[END_REF]. Moreover, increase in the concentration of CNTs within the solution resulting in the formation of CNTs agglomeration to greater extent [START_REF] Imani Yengejeh | Carbon nanotubes as reinforcement in composites: A review of the analytical, numerical and experimental approaches[END_REF], [START_REF] García | Fabrication and Nanocompression Testing of Aligned Carbon-Nanotube-Polymer Nanocomposites[END_REF]. So, optimum concentration and suitable method of dispersion is very important for uniform dispersion of CNTs.

There are various dispersion techniques which are being used by different researchers for the uniform dispersion of CNT within the solution first, and then in the cementitious matrix. Broadly, they are categorized into physical, chemical, and mechanical methods of dispersion which depend on non-covalent and covalent functionalization, and mechanical mixing respectively [START_REF] Vaisman | The role of surfactants in dispersion of carbon nanotubes[END_REF].

Current study investigated the effect of P-CNT and A-CNT dispersed by probe sonicator for dispersion on various properties of cementitious composites. Moreover, effect of various concentrations on the sensitivity of selfsensing response is also studied.

Experimental program

2-1 Materials

Portland cement (CEM I 52.5R, provided by Lafarge Holcim) and normalized sand was used to prepare mortar. The conductive material used was multi-walled carbon nanotubes (MWCNTs) provided by Nanocyl, and having an average diameter of 9.5 nm with average length of 1.5 𝜇m. The resistivity of used CNTs is 10 -4 Ω.cm, with specific surface area of 250-300 m 2 /g. Master Glenium 27 from BASF was used as superplasticizer.

2-2 Preparation of specimens

Pristine and Annealed CNT are used for the preparation of specimens. Annealed CNTs are prepared by heating the CNT at 500 °C for 30 minutes. During the process of annealing, CNTs are modified in terms of increase in crystallinity, reduction of wall defects and removal of impurities such as metal oxides [START_REF] Kim | Annealing effect on disordered multi-wall carbon nanotubes[END_REF]- [START_REF] Che | The Fabrication and Properties of a Bendable High-Temperature Resistance Conductive Pitch-Based Carbon/CNT Film Nanocomposite[END_REF]. This modification in structure of CNT through annealing improves the hydrophilicity of CNT.

After preparation of annealed CNT, both annealed CNT and Pristine CNT are mixed with water separately. Then the solution is sonicated using the ultrasonicator provided by Hielscher Ultrasound Technology. The time of sonication depends on volume of solutions and the ratio of CNT. So, after extensive experimentation, it is observed that the solution required 1 minute/liter of sonication based on the maximum content of CNT (1 minute of sonication is required to homogenize the 1 liter of water solution containing 20g CNTs) to get the homogenized mixture. If sonication is continued beyond this time, solution containing CNT is losing its workability very fast and it starts to get stiffen as time of sonication is increased. So, 1 minute/liter rate of sonication was found to be optimum.

Seven different types of mortar mix with composition as shown in Table 1 were prepared using a mixer conforming to EN 196-3 [START_REF]2005+A1:2008 -Methods of testing cement. Determination of setting times and soundness[END_REF]. Initially, cement and sand were homogeneously blended. Then, the dispersed CNT solution and superplasticizer were added gradually to the dry mixture and mixed until the dry mixture had become a fresh mixture. Prismatic specimens of 40 x 40 x 160 mm 3 for resistivity and flexural tests. After conducting the flexure test on prismatic specimens, cubes of 40 mm from same samples were prepared for compressive test. Fresh mixture was poured into the mold in two layers followed by compaction using a vibrating table. After the preparation of the specimens, they were placed in curing chamber at 20 °C and relative humidity of 100%. Detail of the samples cast can be seen in the Table 1. A nomenclature has been adopted for the ease of referencing each mix. For example, in CNT0.5A, CNT refers to carbon nanotubes, 0.5 refers to the dosage of CNT, and A refers to annealed

2-3 Test Method

2-3-1 Resistivity test

To test the efficiency of the different solutions of CNT dispersion, the electrical resistivity of mortar specimen was measured using uni-axial method of measurement [START_REF] Azarsa | Electrical Resistivity of Concrete for Durability Evaluation: A Review[END_REF] as shown in Fig. 1. Potentiostat/galvanostat provided by Gamry Instrument was used and galvanostatic pulse method was followed to measure the electrical resistance of the material. The direct current was injected from the electrodes, instantaneous change in the voltage was measured between two current peaks, and corresponding resistance was calculated using Ohm's law. By measuring the contact area and length of the specimen, resistivity is calculated using the following equation. Where, I is the current injected, V is the voltage measured and R is the electrical resistance. ρ is the electrical resistivity, A is the contact area of specimen and electrodes and L is the distance between two electrodes.

2-3-2 Physical & Mechanical test

In order to analyze the mechanical properties of cementitious material, 40 mm cubes and prisms (40 mm x 40 mm x 160 mm) were prepared. The compressive and flexural tests were conducted at 28-days of curing in humidity chamber.

It has been reported that the annealing of CNT at high temperature increases the porosity of CNT due to the variation in the structural configuration [START_REF] Norzilah | Surface Modification Effects on CNTs Adsorption of Methylene Blue and Phenol[END_REF]. To study the impact of CNT on the porosity of cementitious material, porosity was measured according to BS EN 1936 standard [START_REF] Standards | BS EN 1936:2006 Natural stone test methods. Determination of real density and apparent density, and of total and open porosity[END_REF].

2-3-3 Self-Sensing test under compression

Self-sensing is the reactive behavior of smart material under the change in its own condition such as stress, strain or damage [START_REF] Dong | Piezoresistive properties of cement-based sensors: Review and perspective[END_REF]. These properties are based on measuring the piezo-resistivity of the material. Under uni-axial compression loading, conductive materials start to get closer each other making the conductive path easier for the current to flow and under un-loading condition, electrical measurement come back to initial stage due to relaxation of the material. Self-sensing assessment of smart material is carried out under uniaxial loading cycle. Piezo-resistive response is measured using Wheatstone bridge (WSB) circuit which converts the variation of resistance into a variation of voltage. Alternating voltage of 20V with the frequency of 10 Hz is applied as an input. A specimen is connected to one arm of WSB circuit and potentiometers are connected to other three arms of the bridge to adjust the resistance to get balanced condition as shown in Fig. 2. Five cycles of loading/unloading were monitored with WSB circuit. Cyclic loading is applied on the specimen between 0.6 to 10 MPa at a loading rate of 0.3 MPa/s using MTS press. Schematic diagram of testing arrangement is shown in Fig. 3. Two strain gauges were mounted on the surface of the specimen to measure the deformation during the experiment. Stainless steel plates were used as electrodes to measure the change in voltage and conductive paint provided by Bare Conductive is applied on the interface of specimens and electrode to minimize the interface resistance. Fig. 3. Self-sensing test setup [START_REF] Shahzad | Cementitious composites incorporating Multi-Walled Carbon Nanotubes (MWCNTs): effects of annealing and other dispersion methods on the electrical and mechanical properties[END_REF] 3 Results & Discussions

3-1 Electrical Properties

The electrical resistivity of cementitious composites containing two different types of CNT at different concentrations were presented in Fig. 4. The resistivity was measured at different curing times to evaluate its variation with of hydration progress.

Fig. 4. Evolution of electrical resistivity with curing period for plain mortar, and mortar containing A-CNT & P-CNT

From Fig. 4, it can be observed that the resistivity responses of specimens having 0.75 % and 1 % dosages of Pristine and Annealed CNT are constant irrespective of the age of specimens. However, for 0.5 % of CNT, electrical resistivity increases with the time, but this increase is less steep as compared to plain mortar. In literature, it is explained that the percolation theory is the main reason behind the resistivity trends. Percolation threshold is the content of conductive material at which conductive filler start to form continuous conductive path within the composites that leads towards reduction in resistivity [START_REF] D'alessandro | A comparative study between carbon nanotubes and carbon nanofibers as nanoinclusions in self-sensing concrete[END_REF]- [START_REF] Galao | Strain and damage sensing properties on multifunctional cement composites with CNF admixture[END_REF].

In order to analyze the drying effect on the resistivity evolution, samples were placed in oven at 40 °C and resistivity test was performed on the samples at different duration of drying as shown in Fig. 5.

Fig. 5. Evolution of electrical resistivity with drying period (40 °C) for plain mortar, and mortar containing A-CNT & P-CNT

Cementitious materials with 0.75 % and 1 % dosages of CNT show stable resistivity response regardless of the duration of drying as shown in Fig. 5 indicating the achievement of percolation threshold at 0.75 % and beyond. However, for the plain mortar, there is large increase in the resistivity evolution as the drying period is increased giving the information that saturation degree is mainly responsible for the conduction in plain mortar.

From resistivity results, no significant difference was observed between pristine and annealed CNT even at different dosage of CNT.

3-2 Physical & Mechanical Properties

Compressive and bending tests were carried out at 28days of curing to assess the mechanical properties. In parallel, a porosity test was also carried out to determine the porosity of cementitious materials. Figs. 6, 7 and 8 show compressive, flexural strength and porosity of the cementitious composites incorporating annealed CNT and pristine CNT respectively. Compressive and flexural strength of all the composites containing CNT either annealed or pristine are less as compared to plain mortar, result in agreement with the previous findings [START_REF] Hu | Modification Effects of Carbon Nanotube Dispersion on the Mechanical Properties, Pore Structure, and Microstructure of Cement Mortar[END_REF]. From the results of compression and flexural tests, the reduction in mechanical properties is more in annealed CNT as compared to pristine CNT due to the slight increase in porosity. It is also observed that the porosity of specimens containing annealed CNT is slightly more as compared to the pristine one. This is evident that during heating of CNTs, porosity of CNT increases [START_REF] D'alessandro | A comparative study between carbon nanotubes and carbon nanofibers as nanoinclusions in self-sensing concrete[END_REF] and hence overall porosity of the specimen is increased.

So, based on the mechanical, physical and electrical properties of cementitious composites discussed earlier, it is concluded that 0.75 % content of P-CNT is the optimum content to develop smart cementitious composite to monitor the health of structures.

3-3 Self-Sensing Properties

Self-sensing response of the material under cyclic compressive loading is measured by measuring the change in the voltage (ΔV) at WSB circuit. ΔV is measured using the following equation.

∆𝑉 = 𝑉 -𝑉

! (3)
where, V is the voltage measured during loading, and Vo is the voltage at so-called balanced condition at stress of 0.6 MPa (Ideally, it should be zero to have a balanced condition, However, with one arm of WSB representing a mortar is not possible to achieve zero voltage at balance. So, the resistance from the potentiometer is adjusted in such a way to get minimum voltage at socalled balanced condition). Resistance for three potentiometers and corresponding so called balanced voltage (Vo) for each material are presented in Table 2. Specimens were partially saturated and had been kept in air dry condition for three weeks before the test. In Fig. 9, each material shows piezo-resistive response under cyclic loading (from 0.6 MPa as minimum stress to 10 MPa as peak) except plain mortar. From Fig. 9, it is observed that smart cementitious material at various concentrations of CNT show self-sensing response in line with strain. However, with the increase in the content of CNT, repeatability and sensitivity of self-sensing response is improved. ΔV for CNT1A and CNT1P is 155 mV and 225 mV respectively, while for CNT0.5A and CNT0.5P, it is less than 20mV. These results indicate the decrease in the electrical resistance with the increase in the content of CNT due to which ΔV is also increased.

Conclusion

The results presented in this paper illustrated physical and mechanical properties as well as the self-sensing response of carbon nanotubes based smart cementitious materials. Following conclusions can be drawn from the results.

• Probe sonicator is an efficient device for the dispersion of CNT either pristine or annealed. • Increase in porosity of cementitious composites containing annealed CNT is slightly more as compared to the pristine CNT. • 0.75% content of pristine CNT can be adopted to develop smart cementitious material for monitoring of the structures.

Fig. 1 .

 1 Fig. 1. Test setup for resistivity test 𝑉 = 𝐼 𝑅 (1) 𝜌 = 𝑅 𝐴 𝐿 (2)

Fig. 2 .

 2 Fig.2. Wheatstone bridge (WSB) circuit setup for self-sensing test[START_REF] Shahzad | Cementitious composites incorporating Multi-Walled Carbon Nanotubes (MWCNTs): effects of annealing and other dispersion methods on the electrical and mechanical properties[END_REF]

Fig. 6 .Fig. 7 .Fig. 9 .Fig. 8 .

 6798 Fig. 6. Evolution of compressive strength of cementitious composites at different contents of pristine CNT (P-CNT) & annealed CNT (A-CNT)

Table 1 .

 1 Mix Composition and nomenclature for mortar specimens

Designation Type of CNT Cement Sand Water Dosage of CNT Superplasticizer - - kg/m 3 kg/m 3 kg/m 3

					%
	PM	-			0.0	0.0
	CNT0.5A				0.5	1.0
	CNT0.75A	Annealed			0.75	1.5
	CNT1A	500	1600	250	1.0	2.0
	CNT0.5P				0.5	1.0
	CNT0.75P	Pristine			0.75	1.5
	CNT1P				1.0	2.0

Table 2 .

 2 Voltage and resistance at so-called balance condition of WSB circuit

	Specimen	Vo	Resistance at Potentiometer
	-	mV	kΩ
	PM	680	4020
	CNT0.5A	11.22	6.5
	CNT0.75A	10.9	1.2
	CNT1A	14.5	0.7
	CNT0.5P	10.1	7.3
	CNT0.75P	5.0	1.2
	CNT1P	10	0.6
	4

  • High electrical resistance offered by plain mortar indicate the no self-sensing response of the material under cyclic compression test. • Self-sensing test results indicate that smart cementitious materials show piezoresistive response. Moreover, with the increase in content of CNT, sensitivity, and repeatability of the measurement is improved to the significant level.

MATEC Web of Conferences 364, 05012 (2022) https://doi.org/10.1051/matecconf/202236405012 ICCRRR 2022