F R Smith

D Brutin
email: david.brutin@univ-amu.fr

Wetting and spreading of human blood: recent advances and applications

Keywords: Drop, evaporation, interfaces, complex fluid, rheology

Investigation of the physical phenomena involved in blood interactions with real surfaces present new exciting challenges. The fluid mechanical properties of such a fluid is singular due its non-Newtonian and complex behaviour, depending on the surrounding ambient conditions and the donor/victim's blood biological properties. The fundamental research on the topic remains fairly recent; although it finds applications in fields such as forensic science, with bloodstain pattern analysis, or biomedical science with the prospect of disease detection from dried blood droplets. In this paper, we review the understanding that has been achieved by interpreting blood wetting, spreading and drying when in contact, ex-vivo, with non-coated surfaces. Ultimately, we highlight the applications with the most up to date research, future perspectives, and the need of advancing further in this topic for the benefit of researchers, engineers, bloodstain pattern analysts, and medical practitioners.

Introduction

The interaction of blood with foreign, non-coated, surfaces is an emerging research subject that finds various applications in fields such as forensic science or biomedical discipline. Although blood flow circulation is a major subject that was thoroughly studied, interest in blood exvivo, i.e. outside the body, and its interaction with a noncoated surface, is less significant. However the properties of blood as a complex, biological, colloidal fluid, differ from those of more commonly studied, pure or complex, fluids, stimulating a recent curiosity concerning the behaviour of blood with real porous and non-porous surfaces. While applied studies on the topic had already been made, in 2011 the work on the evaporation of blood droplets by Brutin et al. gave a fundamental insight into blood drying pattern dynamics [START_REF] Brutin | Pattern formation in drying drops of blood[END_REF]. This work was later completed by recent studies whose major findings are presented in table 1. When blood is in contact with a substrate the two main interface transfers are first the wetting, and then the evaporation. While wetting accounts for the equilibrium of the various applied forces at the triple line (meeting point of blood, surface, and air), evaporation describes the phase change taking place after as recalled by the review of Sefiane et al. of 2011 [START_REF] Sefiane | Wetting 580 and phase change: Opportunities and challenges[END_REF]. These processes are dependent on variables such as ambient temperature, humidity, 25 and pressure,wettability and roughness of the surface, time elapsed since blood deposition, volume and blood's clotting response. Indeed, when blood is outside the body, its properties change rapidly due to platelet activation leading to the coagulation cascade response. The impor-30 tance of understanding this problem for forensic science and biomedical applications, appears to be crucial since it is used as a tool to obtain evidence in crime scene reconstitution or in medical diagnosis. The National Institute of Standards and Technology (NIST) has pointed out, in a 35 very recent report, the urge of using valid scientific methods before presenting evidence in courtrooms [START_REF] Lund | Likelihood ratio as weight of forensic evidence: A closer look[END_REF]. This illustrates the demand that exists concerning this research. To respond, we investigated the current understanding of blood physical and mechanical properties, and the most up 40 to to date advances in the field. In this paper, we present first the characteristics of blood, then its spreading behaviour over dry porous and non-porous surfaces, followed by its drying dynamics, and finally we review the physics of impacting droplets. At last, we discuss the implications that the understanding of blood behaviour on real surfaces has for applications, and the future challenges that research is confronted with. Fluid dynamic model to estimate the point of origin of blood spatters, taking into account gravity and drag [START_REF] Laan | Bloodstain Pattern Analysis: implementation of a fluid dynamic model for position determination of victims[END_REF]. 2. Blood properties

Biological composition

Blood is categorised as a body fluid that accounts for roughly 7% of the human body weight. It has two main constituents: plasma, the fluid medium and blood cells, the colloids. Plasma is a water based solution, composed for 92% of water, which dissolves and transports organic and inorganic molecules, and for 8% of dissolved solutes [START_REF] Bruce | Molecular Biology of the Cell[END_REF]. The solutes are mainly sodium electrolytes; nutrients and organic wastes are found as well in diverse amounts, and finally the proteins, which account for 7-9% of the plasma. These proteins are albumins (80%), globulin (16%) and fibrinogen (4%) that is really important in blot clot formation. Fibrinogen is a globulin of very high molecular weight synthesised exclusively from the liver that can be precipitated easily. During coagulation, the fibrin coming from fibrinogen forms a web leading to the clot formation. The blood cells are divided into three categories, the red blood cells (RBCs) that transport oxygen and carbon dioxide, the white blood cells (WBCs) (neutrophils and monocytes, eosinophils, basophils and lymphocytes), and the platelets, which play an important role in the clotting response. Indeed platelets prevent bleeding by clumping and clotting vessel injuries. Thus all these constituents are main parameters to understand this complex, non-Newtonian, fluid, since an excess or a lack of these components would alter its wetting and its spreading.

Physical and rheological properties

Blood has a small density variation between 1020 and 1060 kg.m -3 , due to the differences in the individuals haematocrit level, which is defined as the ratio of RBCs volume to whole blood volume. The surface tension of blood is known to be similar to that of water, as shown by the work of Brutin et al. where they found θ = 69.8 mN.m -1 as average statistical surface tension. Studies, such as the one of Chao et al. of 2014, account for the shear thinning properties of blood, since its viscosity de-85 creases to a constant value at high shear rates (η = 4.8 mPa.s), although it spreads like Newtonian fluids of similar viscosity. Rheological properties of a liquid are determined by the liquid viscosity, µ, and the applied shear rate, γ, according to the following relationship: µ = k γn-1 , 90 where n<1 corresponds to pseudoplastic fluids. The is in an agreement with previous measurements of blood rheology [START_REF] Chao | Influence of haematocrit level on the kinetics of blood spreading on thin porous medium during dried blood spot sampling[END_REF]. As described by Baskurt et al., the apparent viscosity depends on the existing shear forces, and is determined by its biological properties: haematocrit, plasma 95 viscosity, RBCs aggregation, and the mechanical properties of RBCs [START_REF] Baskurt | Blood rheology and hemodynamics[END_REF]. Alteration of those properties, such as a noteworthy modification of the haematocrit value, accounts for the heamorheological variations. Additionally, the viscoelastic properties of blood come from the high 100 deformability of the RBCs.

Spreading of complex fluids

For a lot of applications (like painting, coating...), a smooth deposition is critical. However, only a few studies were published on the understanding of the influence of 105 complex fluids on the dynamics of spreading. The most obvious way to improve spreading for aqueous solutions is to add surfactants to decrease the liquid-vapour interfacial tension and increase the initial spreading coefficient. Rafaï et al. in 2002 [24], firstly studied the super-spreading of 110 aqueous surfactants drops on hydrophobic surfaces. The super-spreading is due to a large affinity of the surfactant molecules for the solid substrate. If the surfactant molecules are transported rapidly and efficiently over the (small) height of the droplet in order to saturate the solid 115 surface, this entails a large surface tension gradient over a small distance, and consequently a large Marangoni force. This, in turn, leads to the linear time evolution of the radius. The difference with the 'classical' surfactants observed here is then either a difference in affinity for the solid substrate or a difference in the transport efficiency to that same substrate, a question that remains to be answered. In 2013, Bouzeid and Brutin [START_REF] Bouzeid | Influence of relative humidity on spreading, pattern formation and adhesion of a drying drop of whole blood[END_REF] studied the influence of relative humidity (RH) on the spreading behaviour and pattern formation of a human blood droplet.

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The drops of blood of same volume are deposited on an ultra-clean glass substrate and evaporated inside a humidity chamber with a controlled range of RH between 13.5% and 78.0% . Their experiments show that the contact angle decreases as a function of the RH which influences the 130 final deposition pattern at the end of the evaporation process (Fig. 1). Due to the effect of RH on the contact angle of the drop of blood, initial evaporative rate is dependent of RH values. They observed cracks pattern at the end of the drying process which is due to the competition between the drying regime and the gelation inside the drop of blood. Indeed, at first, the Marangoni convection inside the droplet induces the transport of particles towards the rim, and thus favours evaporation at the triple line. This corresponds to a convective evaporation. Later, once the particles concentration reaches a critical point, gelation occurs, and evaporation occurs through the porous media. The transition between the purely convective evaporation regime and the gelation regime appears always at 65% of the total drying time. Thus, controlling evaporative rate by evaporating drops of blood at different RH levels impacts strongly the wettability properties and the final pattern of drying drops of blood. The influence of RH on the final pattern of a dried drop of human blood is of huge importance for biomedical applications where drops of blood the Marangoni flow motion inside a drop of human whole blood using a digital camera [START_REF] Brutin | Pattern formation in drying drops of blood[END_REF]. They showed that the motion of RBCs at the edge of the drop is the main mechanism of blood drop deposition and early evaporation, since Marangoni convection favours RBCs displacement towards 170 the rim and evaporation at the triple line. The mechanisms involved in the blood drop evaporation are confirmed through the evaporation mass flux, which is measured to be in agreement with pure water drop evaporation. Other biological elements (white blood cells, pro-175 teins) are transported to the edge of the drop with the RBCs and contribute to the pattern formation and the deposit wettability with the substrate and the mechanical properties. The blood evaporation dynamic is comparable to the evaporation of pure fluids of similar mass 180 concentration in colloids. Totally different patterns are formed according to whether the person is healthy or suffers from anaemia or hyperlipidaemia. A few years later, in 2014, Sobac and Brutin evidenced the influence on the final pattern of the wetting of blood on a substrate [START_REF] Sobac | Desiccation of a sessile drop of blood: Cracks, folds formation and delamination[END_REF].

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They showed that all of the mechanisms are linked, and the change of wettability highly affects the deposit shape. The evaporation of a drop of blood with an initial angle of 90 • revealed the formation of a complex morphology due to the presence of an elastic buckling instability. In 190 this situation, a shell forms during the drying due to the accumulation of particles at the free surface, and is deformed by buckling, leading to the formation of folds. The final shape is no longer axisymmetric. The different behaviours encountered were assimilated to a stability dia-195 gram obtained for physical suspensions, and a good level of agreement was observed. The results presented are notably close to those obtained after the evaporation of drops of bovine serum, suspensions of colloidal particles and suspensions of polymers. The practical application of pattern 200 formation in drying drops of whole blood requires a complete understanding of the involved mechanisms such as spreading, wetting, evaporation heat and mass transfer, and cracks formation.

Patterns in dried pools

205

Although the drying of gels, or colloidal Newtonian suspensions has been largely studied, research on the drying of blood pools remains scarce. Unlike droplets, in the case of a pool, gravitational forces are more important than surface tension forces; this induces the flattening of the liquid 210 on the top part. One of the first considerations to observe concerning drying blood is that the properties of the blood change when the blood is ex-vivo. Visually the change of colour from bright red to dark brown due to the oxidation of the ox-haemoglobin (HbO 2 ) to methaemoglobin (met-Hb) and haemachrome (HC), is a first indicator. Another important step when bleeding occurs, is haemostasis, which involves coagulation corresponding to the change of blood from a liquid into a gel [START_REF] Gale | Continuing Education Course #2: Current Understanding of Hemostasis[END_REF]. In 2016, Laan et al. investigated the topic, and observed the different phases that a drying pool of blood goes through and arbitrarily called them coagulation stage, gelation stage, rim desiccation stage, centre desiccation stage, and final desiccation stage [START_REF] Laan | Morphology of drying blood pools[END_REF]. Comparatively to a drying droplet, their experiment showed that again 23% of the original mass remains after drying, which corresponds to the colloids, mainly RBCs, present in blood. When whole blood is deposited on a foreign surface, at first the RBCs are homogeneously distributed. Then rapidly coagulation takes places. During this phase the fibrin coming from the fibrinogen forms a web leading to the clot formation. The evaporation that follows leads to the displacement of the cells not trapped yet in the web towards the pool's rim. The drying front starts at the rim of the pool and goes towards the centre of the pool in a non-uniform way. The stress induced by the water evaporation leads to the noteworthy cracks observed on dried blood pools. In 2017, Ramsthaler et al. compared the drying processes of extra-corporal blood without any additives to blood that had received anticoagulation therapy for small volume droplets by studied the wipeability of bloodstains. As a result, they found that the time of the drying process was delayed more efficiently by anticoagulation therapy than it was by antithrombotic drugs or platelet aggregation inhibitors, meaning that the last part of the coagulation process is of high importance on the drying dynamics compared to the primary stages of the haemostasis when platelet activation and aggregation take place [START_REF] Ramsthaler | Effect of anticoagulation therapy on drying times in bloodstain pattern analysis[END_REF].

Drop impact on solid surfaces

Impact theory

When a drop of blood is released onto a dry solid surface from different heights, it impacts with energy due to the velocity of the fall. Understanding the correlation between a blood drop pattern and its velocity of fall is important to the forensic community as it could answer some crime scene reconstruction questions. In 2014, Laan et al. [START_REF] Laan | Maximum Diameter of Impacting Liquid Droplets[END_REF] described the phenomenon, and how inertia forces favour spreading, but adhesion and viscous forces counter this spreading so that the drop reaches a maximum spreading diameter commonly named D max , as shown on figure 2. It normally remains self-pinned to the surfaces before evaporation and drying mechanisms take place. The following resulting outcome of the drop stain depends upon the drop, the surface and the surrounding properties. To define the impact dynamics, the dimensionless Weber and Reynolds numbers are commonly used since they describe the liquid properties and the impact. The Weber number (We = ρD 0 U 2 /σ), is the ratio between capillary and inertial forces whereas the Reynolds number (Re = ρD 0 U/η) is the ratio between the viscous and the inertial forces. ρ, 270 η are respectively the blood density and viscosity. σ is the blood/air surface tension. U stands for the impact velocity of the droplet, and D 0 for the initial drop diameter. One of the latest model describing the maximum spreading diameter of blood droplet, is the model of 2016 of Lee 275 et al. [START_REF] Lee | 565 Universal rescaling of drop impact on smooth and rough surfaces[END_REF], who presented a universal solution taking into account that energy conservation is the only physical principal needed to describe the impact behaviour of droplets, according to the following equation:

(β 2 max -β 2 0 ) 1/2 Re -1/5 = W e 1/2 A + W e 1/2 , ( 1
)
280 with β max the maximum spreading ratio of the droplet (the ratio of the maximum spreading diameter by the initial diameter of the given droplet), β 0 the maximum spreading ratio at zero impact velocity for the same surface (the ratio of the maximum spreading diameter by the initial diam-285 eter of the droplet at 0 m.s -1 ), and A = 7.6, the fitting constant.

Surface influence

Interest for various industrial applications, led many researches to study the influence of the surfaces prop-290 erties on the impact outcomes of impacting droplets of various Newtonian fluids, as thoroughly described in the recent review of Josserand and Thoroddsen [START_REF] Josserand | Drop Impact on a Solid Surface[END_REF]. Meanwhile, in 2016, Kim al. investigated the influence of the target properties on the trajectory reconstruction of blood 295 droplets impacting different surfaces using glass, polycarbonate bare, aluminium mirror and cardstock [START_REF] Kim | How important is it to consider target properties and hematocrit in bloodstain pattern analysis? For[END_REF]. To quantify the influence, they correlated the Ohnesorge number (Oh = √ W e/Re), which relates the energy of the impact to the resistant work of surface tension and viscous forces, to the spreading ratio. They found that an increase in surface roughness reduces the spreading upon drop impact, which led them to conclude that the stain size cannot directly be related to an impact velocity, which supports the model of Lee et al. that takes into account the initial spreading ratio of the liquid at a zero impact velocity for a given surface, implying that this ratio is substrate dependent. This suggests that not only the spreading could be influenced by the target properties, but the outcome, and whether splashing can be observed or not. Recently Smith et al. investigated the influence of the surface properties on the splashing threshold of blood drip stains [START_REF] Smith | Roughness Influence on Human Blood Drop Spreading and Splashing[END_REF].

In their study they defined the splashing threshold as the minimum drop velocity inducing the drop to break-up and to have at least one secondary droplet. The experimental results showed that an increase in surface roughness leads to an earlier splashing threshold when comparing the Weber number of their impacting drops with the average surface roughness (Sa). Besides, they showed that roughness modifies the deformation of the drip stains (fig. 3) by presenting a map of the different impact outcomes of the blood droplets at different velocities and various surfaces.

Applications

Forensic science

Understanding the behaviour of blood on real surfaces finds a major interest in bloodstain pattern analysis, commonly called BPA, which is the forensic field that studies bloodstains found on crime scenes in order to help events reconstruction. The first works on the subject by Eduard Piotrowski, a researcher form the Institute of Forensic Medicine of Krakow, started in the late 1800s, but it took another 50 years for bloodstain evidence to be used in real cases. Since then, with a first formal bloodstain course in 1973 led by MacDonnel in the United States, 335 and the creation ten years later of the International Association of Bloodstain Pattern Analysts, this expertise was brought up to crime scene investigation with trained BPA experts, and the concern of emerging research on the topic. On crimes scenes, if a bloodshed event takes place, blood 340 traces may be found on various surfaces (floor, furniture, clothing...) giving a first piece of information. The classification of the pattern type (drip stain, cast-off pattern, drip trail, pool...) is important to correctly help reconstitution (see fig. 4). In the case of some types of spatter droplets, it 345 will impact a surface with an angle giving stains the form of ellipses. Using the stringing theory, the flight path of such stains is reconstructed, and by combining trajectories together the point of origin of the source of blood can be determined in the three-dimensional space. Nowadays, as 350 presented in the study of Joris in 2015, various automated and virtual 3D scanning methods are used during investigations [START_REF] Joris | HemoVision: An automated and virtual approach to bloodstain pattern analysis[END_REF]. A lack of adequate scientific validity is noticed in some of those methods, such as highlighted by the 2016 study of Taylor et al., where they questioned the reliabil-355 ity of pattern classification on fabrics depending on how biased was the opinion of the expert. In order to prevent doubtful BPA evidence leading to wrongful convictions, many governments currently promote the rigorous use of forensic science. To fulfil this demand, research in the do-360 main is emerging as can be illustrated by some recent studies: in 2015, Laan et al. proposed a fluid dynamic model that, unlike the stringing method, takes into account gravity effect and drag giving a more accurate estimation of path reconstruction [START_REF] Laan | Bloodstain Pattern Analysis: implementation of a fluid dynamic model for position determination of victims[END_REF]; in 2016 Geoghegan et al worked 365 on trajectory reconstruction of blood drops from expirated blood using Particle Image Velocimetry (PIV) and a numerical model [START_REF] Geoghegan | Experimental and computational investigation of the trajectories of blood drops ejected from the nose[END_REF]; in 2017, Comiskey et al. presented a theoretical numerical model describing blood spatters resulting from a blunt bullet gunshot, that they compared to 370 experimental data [START_REF] Comiskey | Hydrodynamics of back spatter by blunt bullet gunshot with a link to bloodstain pattern analysis[END_REF]. Works concerning drip stains on fabrics is thoroughly investigated as well, in 2015 by Dicken et al. on the correlation between impact velocity with the shape and the size of stain on the fabric [START_REF] Dicken | The use of micro computed tomography to ascertain the morphology of bloodstains on fabric[END_REF], in 2016 by De Castro et al. who studied the effect of prior laundering 375 on the final stain appearance [START_REF] De Castro | Drip bloodstain appearance on inclined apparel fabrics: Effect of prior-laundering, fibre content and fabric structure[END_REF], again in 2016 by Cho et al. on the differentiation between transfer patterns and spatter patterns [START_REF] Cho | Quantitative bloodstain analysis: Differentiation of contact transfer patterns versus spatter patterns on fabric via microscopic inspection[END_REF], which is a difficult question at the moment. Finally some effort is put as well onto creating a Synthetic Blood Substitute (SBS) of similar rheological 380 properties as illustrated by the research of Totesbury of 2017 [START_REF] Stotesbury | The application of silicon sol-gel technology to forensic 590 blood substitute development: Mimicking aspects of whole human blood rheology[END_REF], in order to provide tools to help research into understanding bloodstain pattern mechanisms, as working with fresh blood without any additives is a complex problem that encounter researchers in the field. digital polymerase chain reaction, also called ddPCR on 390 the maternal peripheral blood [12][13]. In Hudecova study, peripheral maternal blood is collected in EDTA tubes, and then centrifuged twice. For the purpose of the study, samples are frozen and then shipped onto dry ice. Placental DNA is extracted from the maternal blood according to a DNA purification protocol. The authors designed a ddPCR assay to detect two sequence variants based on the maternal mutational status. All ddPCR analyses were carried out on a Droplet Digital PCR system (bio-Rad) which generates up to 20 000 reactions droplets within a single 400 reaction well. Out of the 20 000 droplets generated only 200 to 2000 evidence a positive result. The results must be analysed automatically using software and compared to computer simulations in order to define the diagnostic accuracy. In 216, Laux et al. demonstrated how ultrasonic shear reflectometry could be used, as a non-intrusive tool, to determine the gelation time and the total desiccation time. The only assumption is on the consequences of ultrasonic waves on the cellular matter such as RBCs membranes, which are very fragile vesicles. So far, there is nothing in the literature about the effect of the ultrasonic wave on the blood components. In 2017, Ahmed et al. partly reviewed the situation of wetting and spreading of blood, focusing on porous substrates. They presented two situations of complete and partial wetting cases and detailed the different stages encountered. The main and historical application remains the storage and transport of blood samples using the DBS sampling technique used by the National Health Service (NHS) in the UK. In their review of 2016, Chen et al conclude that the research of medical diagnosis based on blood drop pattern is still at a very early stage. Indeed, significant efforts have to be done before reaching the biomedical application for both humans and livestock. The basis of this technique relies on the postulation that diseases modify the blood properties, such as viscosity, viscoelasticity and cracks formation after drying. For a biomedical device using dried blood drops analysis to be developed and commercialise, improvement of the fundamental knowledge of spreading, wetting, and drying of blood is required. At least one decade will be 430 needed before the appearance of a mobile analysis device using this technology.

Conclusions and prospectives

In order to use the interaction of ex-vivo blood with real surfaces for application such as forensic science or 435 biomedical diagnosis, an extensive understanding of this complex phenomenon is mandatory. The recent studies on the topic opened a new outlook on this physical and fluid mechanical problem by focusing on blood interactions with non-coated surfaces (i.e surfaces without any 440 anticoagulant or chemicals that could react with blood). Blood is, first, a non-Newtonian fluid, meaning that predicting its mechanical behaviour is complex, as it will depend on the applied shear stress, secondly, blood is a biological liquid that coagulates and ages when outside the 445 body. This invites researchers and scientists to take up the challenge that is to related the mechanical properties of blood with the potential applications and to seek deeper. In order to use the morphological analysis of dried drops of blood for biomedical purposes, we need to be able to 450 predict the spreading diameters of blood droplets created from a finger. This is a fast and cheap blood collection method that medical practitioners are trying to use more and more, but some studies reported that blood test results might vary from drop to drop in fingerprick tests.

455

Hence we have to be able to control and predict spreading and wetting of blood onto various, mainly porous, substrates. Moreover, we have to be able to predict as well the time needed to obtain a complete evaporation, since the last phase of drying no more produces important changes 460 but slowly deforms the cracks/plaques, meaning that a visual criteria to disease detection could no longer be used. Ultimately we have to understand the link between blood pattern and potential diseases. Similarly in BPA investigation, deeper correlation between observed patterns and 465 physical comprehension of blood mechanical properties are crucial. Nowadays forensic experts are trying to improve their procedures by introducing new technologies and recent findings. Moreover, this field of expertise being fairly recent, many issues that would in appearance seem simple, 470 still remain. They are thus trying to find answers to questions such as: how long as this pool of blood been drying for? Was this pattern projected or transferred onto the fabric? From what height was this droplet falling? What volume of blood is present in this droplet/pool? Thus in 475 order to prevent wrongful convictions, or better medical diagnosis, it is urgent to improve our knowledge concerning blood behaviour.

Acknowledgment

This work received a financial grant from the French 480 Agence Nationale de la Recherche in the frame of the project ANR-13-BS09-0026. This research was also supported by the Institut Universitaire de France (IUF) and has received funding from Excellence Initiative of Aix-Marseille University -A*MIDEX, a French "Investissements d'Avenir" programme. It has been carried out in the framework of the Labex MEC.

Figure 1 :

 1 Figure 1: Top-view images of drop deposits left after complete evaporation of sessile drops of whole blood (same scale). All experiments are performed for a same drop volume (V = 14.2 µL) and increasing RH (microscope ultraclean glass substrate, room temperature: 23.8 • C, room pressure: 1005 hPa), reproduced from [2] with permission from Elsevier, copyright 2013.

150

  are drying in an open atmosphere.

4 .

 4 Patterns in dried drops and pools 4.1. Patterns in dried drops Martusevich and Bochkareva presented in 2007 some first results concerning the morphology of dried blood serum 155 droplets with viral hepatitis[23]. Out of the 58 people included in the study, 32 were healthy, 14 patients had viral hepatitis B and 12 patients had viral hepatitis C. The authors established that the presence of viral hepatitis in patient's organism significantly changes the result of free 160 and initiated blood serum crystallization. They concluded on the feasibility of single out the biological fluid crystallogenesis characteristics according to virus type. Before 2011, published studies were performed on blood serum. Brutin et al. achieved in 2011 the first observations of165

Figure 2 :

 2 Figure 2: (a)-(d) High-speed camera recordings of a single blood droplet impacting at 2 m.s -1 on a stainless-steel substrate. (a) At 0.2 ms prior to droplet impact, the droplet is spherical. (b) At 0.6 ms after droplet impact on the surface, a thin lamella spreads outwards due to the inertial forces. (c) At 2.4 ms after droplet impact, the lamella increases in size. Spreading of the lamella slows down until it reaches its maximum size (Dmax), while there is a buildup of liquid in the outer rim. (d) At 4 ms after droplet impact, the bulk of the liquid is distributed over the entire stain. Reproduced from[START_REF] Laan | Maximum Diameter of Impacting Liquid Droplets[END_REF] with permission from APS, copyright 2014.

Figure 3 :

 3 Figure 3: Deformation pattern of three blood drip stains of same initial volume (V = 13.8 * 10 -3 mL) impacting at U = 2.6m.s -1surfaces of increasing roughness. Reproduced from[START_REF] Smith | Roughness Influence on Human Blood Drop Spreading and Splashing[END_REF] with permission from APS, copyright 2017.

385 6 .

 6 2. BiomedicalIn 2017, Ragni M. considered the work of Hudevoca and al. (2017), which describes a simple and non-invasive device for prenatal detection of haemophilia by droplet

Figure 4 :

 4 Figure 4: Picture of an IABPA bloodstain analysis training course, with the tools used to reproduced various specific patterns.

Table 1 :

 1 References of special interest studies published over the last three years


Sergey Semenov

Florian Carle

Marc Medale

David Brutin

3D unsteady computations of evaporative instabilities in a sessile drop of ethanol on a heated substrate

Keywords: numbers: 68.03.Fg, 07.05.Tp, 47.55.pf, 47.20.Ma

Droplets are ubiquitous and have been studied for century. However, the flow pattern and instabilities occurring during evaporation are still under investigations and their origin is still debated. In this letter, we are comparing an ethanol drop evaporating onto a heated substrate under weightlessness conditions and with pinned contact line with a 3D unsteady computation of thermo-convective instabilities to determine with accuracy the type of instabilities. Our one-sided model, devoid of fitting parameters, demonstrates quantitative agreement with experimental data and confirms that experimentally observed instabilities are driven by thermocapillary stress, and not by the gas convection. By creating creating a numerical infrared image, we can conclude with certitude that the experimentally observed thermo-convective instabilities in evaporating sessile drops of volatile liquids, which in infrared spectrum look similar to hydrothermal waves (HTWs), are actually nothing else than unsteady Bnard-Marangoni instability.

Evaporating sessile drops of various liquids are widely encountered in nature and have a variety of industrial and biomedical applications: heat exchangers [? ], nanoparticles deposition (coffee-ring effect) [? ], spraying of herbicides and pesticides on hydrophobic leaves [? ], inkjet printing [? ? ], blood analysis [? ? ? ]. As a result, evaporating sessile drops constitute an interest for both academic community and industry. This problem is rich with numerous physical phenomena: diffusive and convective vapour transport, kinetics of vapour molecules transfer across liquid-gas interface (Hertz-Knudsen-Langmuir equation), evaporative interface cooling, radiative heat transfer, Marangoni flows and instabilities, particles sedimentation and deposition, adsorption of chemicals on interfaces, adsorbed precursor films, contact line instabilities and pinning/depinning processes. Many of these phenomena can be visualized with the aid of modern optical instruments. For example, field of vapour concentration around the droplet can be observed with the aid of digital holographic interferometry [? ]. Observations in infrared (IR) spectrum give visual information about thermal processes in a droplet [? ? ? ? ? ? ], and allow estimating temperature field on its surface [? ? ? ? ]. Among all observable phenomena, a particular interest has been given to the one distinctive type of spontaneously developed thermo-capillary instabilities, so called hydrothermal waves (HTWs). Conventional HTWs are observed in thin liquid layers whose surface is subject to a lateral temperature gradient [? ? ? ? ? ]. In sessile droplets, however, HTWs are driven by evaporation, which generates these temperature gradients naturally. In droplets they have been observed, for the first time, in IR spectrum in volatile liquids (methanol,ethanol, on heated substrates by Sefiane et al.

[? ] and later by few other researchers [? ? ? ? ]. According to Smith and Davis [? ? ] (see also [? ] and p. 115 in [? ]), HTWs are distinguished from other thermo-capillary instabilities by the following attributes: they appear only as a secondary unsteady thermo-convective instability in a basic shear flow (primary thermo-capillary flow) directed along the longitudinal temperature gradient at the liquid surface; the mechanism of HTWs propagation does not require any deflection of the free surface of the liquid layer.

It is still not clear what exactly is observed in sessile droplets: are we observing hydrothermal waves or unsteady Benard-Marangoni (BM) instabilities, or a combination of both? In order to answer this question, it is necessary to understand underlying hydrodynamics and heat transfer, identify major sources of IR radiation contributing to the experimentally observed IR image. In the present work, we study this problem numerically using a one-sided model solving 3D unsteady hydrodynamic and heat transfer equations in a sessile drop of ethanol on a heated substrate. We assume a pinned contact line and a spherical-cap shape of the liquid-gas interface. The computed temperature field is used for the numerical reconstruction of the top-view IR image of a semitransparent (in IR spectrum) droplet, which is then validated against the experimental IR image. The presented numerical model gives the idea of internal 3D flow structure in the droplet, reveals the driving mechanism and energy sources of the observed thermo-convective instability, and thus clarifies its nature. For our calculations, we will use a Cartesian system of coordinates (x, y, z) with the origin at the geometrical center of the droplet-substrate contact area, and with z-axis directed perpendicular to the liquid-solid interface and pointing into liquid phase. Let x, ŷ and ẑ be the constant unit vectors in direction of x, y and z axes, respectively. Earlier [? ] we have presented a semi-analytical formula predicting the mass flow rate, J (in kg/s), of unsteady diffusion-limited evaporation for a non-isothermal pinned sessile droplet, taking into account Stefan flow in the gas:

J Stef an (L, T av , θ(t), t) = 2πD * ef f LF (θ(t)) ×   1 + LF (θ(t)) πD * ef f t   ρ * g ln ρ * g -ρ v,∞ ρ * g -ρ * v,sat , (1)
where

D * ef f = A T av T ∞ T av + T ∞ 2, (2)
A = (Dp g ) ref p g T 3/2 ref , (3)
T av = 1 Γ Γ T dΓ, (4)
ρ * g = p g M * g /(RT av ), (5)
F (θ) =            0.6366 θ + 0.09591 θ 2 -0.06144 θ 3 / sin θ, 0 ≤ θ < π/18 0.00008957 + 0.6333 θ + 0.116 θ 2 -0.08878 θ 3 + 0.01033 θ 4 / sin θ, π/18 ≤ θ ≤ π (6) ρ v,sat (T ) = p ref v,sat M v RT exp - Λ * R 1 T - 1 T ref sat , (7)
ρ * v,sat = ρ v,sat (T av ) , (8)
L is contact radius, T av and T ∞ are correspondingly average temperature of the liquid-gas interface Γ (see Eq. ( 4)) and the ambient one, t is time, θ(t) is timedependent contact angle, F (θ) is function of contact angle, derived by Picknett and Bexon [? ] and equivalent to the one derived later by Popov [? ], asterisk stands for a dependence on T av : D * ef f is effective diffusion coefficient for vapour in ambient gas, p g is ambient gas pressure, (Dp

g ) ref = 1.337 Pa•m 2 /s at T ref = 298 K for ethanol vapour in air [? ], ρ *
g (see Eq. ( 5)) and

M * g = M air (1 -X * v ) + M v X *
v are correspondingly density and molar mass of the gas at Γ, where M air and M v are molar masses of correspondingly air and vapour (0.046 kg/mol for ethanol) and

X * v = ρ * v,sat RT av / (M v p g ) is vapour molar fraction, ρ v,sat (T ) is local saturated vapour density at Γ (Clausius-Clapeyron equation, see Eq. (7)), R is the universal gas constant, Λ * = c 1 (1 -T av /T c ) c2 is latent heat of vaporization with c 1 = 55789 J/mol, c 2 = 0.31245 and T c = 514 K for ethanol, p ref v,sat = 13838 Pa at T ref sat = 308.15 K, ρ v,∞
is vapour density in the ambient gas far away from the droplet. Equation (1) has been used in a two-parametric approximation, j approx , for a non-isothermal vapour flux distribution along the droplet surface (see [? ]):

j approx = j * i (r, θ) [1 + B(T -T av )] + ∆h∇ Γ • j τ , (9) where r = x 2 + y 2 , (10) j τ = -D(T )∇ Γ ρ v,sat (T ) + ρ v,sat (T ) u τ , (11)
B and ∆h are two parameters,

∇ = x ∂ ∂ x + ŷ ∂ ∂ y + ẑ ∂ ∂ z
is the nabla-operator, ∇ Γ = (I -nn) • ∇, where I is the identity tensor and n is unit normal vector at the liquidgas interface pointing into gas phase, j τ is tangential component of the vapour flux (see Eq. ( 11)), 23)) of the surficial vector j τ , and finally j * i (r, θ) is an isothermal version of the vapour flux distribution along the droplet surface:

D(T ) = AT 3/2 is local diffusion coefficient, u τ = (I -nn) • u is tangen- tial component of the velocity vector, ρ v,sat (T ) is local saturated vapour density, ∇ Γ • j τ is a surficial divergence (see Eq. (
j * i (r, θ) = j * 0 χ -λ (1 -ω) (12)
for 0 ≤ θ ≤ π/2 and 0 ≤ r < L, where

j * 0 = J Stef an (L, T av , θ(t), t) G(θ) 2πL 2 F (θ) , (13)
G(θ) = 0.008348 θ 4 -0.1026 θ 3 + 0.001815 θ 2 + 0.4491 θ + 0.6368, (14)
χ = 1 -(r/L) 2 , (15)
λ = (π -2θ)/(2π -2θ), ( 16)

ω =              H 3 -4 + 5 (1 -θ/θ HE ) 2 , 0 ≤ θ ≤ θ HE -E + λ 2 E E 2 + (1 -λ 2 E ) (H + E) 2 , θ HE < θ ≤ π/2 (17) λ E = (2θ -2θ HE ) / (π -2θ HE ) , (18)
θ HE = 0.7864

H + 0.9103, ( 19
) E = -2.679 H + 0.7265, (20)
H = 0.26 1 -χ 0.7 . ( 21
)
Defining liquid-gas interface Γ : z = f (x, y), we can represent j τ as a function v of only two coordinates:

j τ (x, y, z) = j τ (x, y, f (x, y)) ≡ v (x, y) . ( 22
)
Then, provided that n z = 0, term ∇ Γ • j τ in Eq. ( 9) can be expressed as:

∇ Γ • j τ = a • (v) x + b • (v) y , (23)
where symbols (•) x and (•) y denote partial derivatives with respect to x and y, and vectors a and b are given as follows:

a = (I -nn) • x = 1 -n 2 x , -n x n y , -n x n z , b = (I -nn) • ŷ = -n y n x , 1 -n 2 y , -n y n z , ( 24
) 1 1+(f x ) 2 +(f y ) 2
being the components of the normal vector n. Expression ( 23) is ready to be implemented in the program code. We use Eq. ( 9) in our numerical model as a boundary condition for the local normal vapour flux, j, at the liquid-gas interface: j = j approx . All input data, which are required for the computation of j approx , such as T av , θ(t), j τ , are directly available from the numerical model in the course of computation. Our one-sided model solves Navier-Stokes, continuity and heat transfer equations in the droplet bulk:

ρ ∂ u ∂ t m + [(u -w) • ∇] u = ∇ • σ, (25)
∇ • u = 0, ( 26
)
ρc p ∂ T ∂ t m + (u -w) • ∇T = ∇ • (k∇T ), ( 27
)
where u and T are respectively velocity and temperature fields, w is the mesh velocity, subscript m means that time derivative is taken at a fixed mesh node (fixed mesh coordinates), ρ, c p and k are respectively density, specific heat capacity at constant pressure and thermal conductivity of the droplet phase, σ = -pI + π is total stress tensor, p is hydrodynamic pressure and π = µ ∇u + (∇u) T is viscous stress tensor. Radiative, conductive and convective heat transfer into gas phase is neglected. Substrate is considered isothermal with constant temperature T s . Initial conditions: T | t=0 = T s , u| t=0 = 0, p| t=0 = p g + 2γ sin θ 0 /L, where γ is liquid-gas interfacial tension and θ 0 is initial contact angle. Boundary conditions: T | z=0 = T s , u| z=0 = 0. Liquid-gas interface Γ is considered as a boundary, moving in direction of normal vector n with velocity u Γ . Provided that local evaporation rate, j, is known (see Eq. ( 9)), we can define the following boundary conditions at Γ:

-k∇T • n = jΛ * /M v , (28)
ρ (u • n -u Γ ) = j, (29)
σ • n = -γ (∇ • n) Γ n + dγ dT ∇ Γ T , (30)
where (∇ • n) Γ is the divergence of vector n at Γ, that is curvature of Γ, which is equal to 2 sin θ/L. Expression for u Γ is derived based on total evaporative mass flux J = Γ jdΓ and two assumptions: pinned contact line (L = const) and spherical-cap shape of the droplet in the course of evaporation:

u Γ = - J πρL 3 (1 + cos θ) 2 sin θ z. (31)
In order to compare our numerical model with real experimental data, we have selected one particular experiment from the series of our parabolic-flight experiments [? ].

In this experiment a sessile drop of ethanol with pinned contact line evaporated in microgravity conditions from a thin heated substrate with actively imposed constant temperature T s . The IR camera "VarioCam R hr head" recorded droplet's top view in the Mid-Wavelength IR range (MW: 7.5-14 µm) to capture thermo-convective instabilities (Fig. 1c). Based on the selected experiment, we have used the following input data for our numerical model:

p g = 792 mbar, T s = 307.05 K, T ∞ = 297.55 K, L = 2.95 mm, θ 0 = 29.2 • , ρ v,∞ = 0, ρ = 772.24 kg/m 3 , µ = 1.095 mPa•s, k = 0.14 W/(m•K), c p = 2602.3 J/(kg•K), γ = 20.62 mN/m, d γ d T = -82 µN/(m•K).
Parameters B = 0.33 K -1 and ∆h = 0.019 mm (see Eq. ( 9)) were obtained by fitting local evaporation rate, j, computed with our previously developed non-isothermal two-sided axisymmetric numerical model [? ] which also used the above mentioned input data.

In the present numerical study we use a commercial software COMSOL Multiphysics R . All equations are discretized with the finite element method (FEM) using second order shape functions. Generated computational mesh contains 16547 tetrahedral elements, which results in 187050 degrees of freedom. Computation ran about 9 days on a cluster with 48 cores. Below we present computer simulation results for our unsteady one-sided 3D numerical model. Computation starts with the initial condition of uniform temperature field in the droplet bulk. Due to consumption of latent heat by the theoretically imposed evaporation at the liquid-gas interface, droplet surface cools down. At the same time temperature near the contact line is maintained high due to heat conduction from the substrate through a thin layer of ethanol. This creates a vertical temperature gradient in the droplet bulk and a horizontal one on the droplet surface near the contact line. These temperature gradients promote the development of thermocapillary Benard-Marangoni instability. After a period of about 15.4 seconds, when intensive transitional phenomena finished, one can observe a dynamic multicellular thermo-convective pattern: Fig. 2 shows the temperature (color) and velocity (arrows) fields at the droplet surface (b) and in the vertical cut plane (a) after 15.40 seconds of the evaporation process which represents about 85% of the total time of evaporation. The dynamics of this instability well resembles the experimentally observed one (a video link is available in the caption of Fig. 1). In order to get a better visual comparison of this numerical result with the experiment, we did a simplified computation of the IR image based on the computed 3D temperature field. We took into account the semitransparency of liquid ethanol in the Mid-Wavelength (MW) bandwidth: according to the spectrophotometer measurements by Brutin et al. [? ] ethanol's absorption coefficient in the MW range is a = 1.85 mm -1 . Using this value, one can compute the intensity of IR radiation arriving to the camera installed above the droplet, which then can be converted into an equivalent blackbody temperature, T IR,num (x, y) (subscript num stands for "numerical"). This computation is done by summing the IR radiation coming from the substrate surface (assumed to be a gray body) with the integral of IR radiation sources distributed across the thickness of a semi-transparent droplet:

T 4
IR,num (x, y) = εT 4 s e -af (x,y) + f (x,y) 0 T 4 (x, y, z)a e -a(f (x,y)-z) dz, (32)

where f (x, y) represents the local droplet thickness. Emissivity of a bare substrate surface, ε = T 4 IR,exp /T 4 s,exp , is estimated by correlating the experimental equivalent blackbody temperature (T IR,exp (x, y) from IR camera) with corresponding real temperature, T s,exp , obtained through thermocouple measurements.

Numerical model does not confirm the existence of any base shear flow in the droplet phase, which indicates that this is not HTWs. Instead, it possesses the features of Benard-Marangoni instability: vertical upward liquid flow under the local temperature maxima, and downward flow under the local temperature minima. Thus, we conclude that the experimentally observed thermoconvective pattern is nothing else than unsteady Benard-Marangoni instability.

Finally, let us discuss the dynamics of the observed pattern. The 3D instabilities develop in several stages. In the first one, an unsteady axisymetric base flow sets up in a torus roll adjacent to the triple line. It rotates upwards along the interface as it is driven by thermocapillary forces and superimposes to Stefan flow in the droplet. In a second stage this torus roll is destabilised and brakes into BM cells resulting in a fully 3D unsteady pattern that populates the whole droplet. Then in a third stage the BM cells concentrate in the neighbouring of the triple line where thermocapillary forces remain intense whereas they continuously weaken in the upper central region. The unsteady pattern that arises from this point results from the fact that BM cells have an intrinsic wavelength that is roughly twice the liquid thickness [? ] (λ ∼ 2 f(r)). So the number of BM cells that matches the circumference is N (t) ∼ 2π(L-f (r)) 2f (r) . But, as evaporation proceeds in the pinned mode, the droplet thickness decreases and N(t) increases leading to a continuous azimuthal movement in order to dynamically equidistribute cells along the droplet perimeter. This last dynamics has somehow been erroneously attributed to HTWs, but our comparison of experiments and simulation confirms the nature of these instabilities as BM.

In this letter we have reported our 3D one-sided numerical model of an evaporating sessile droplet of ethanol on a heated substrate. The model does not use any fitting parameters and demonstrates a good qualitative and quantitative comparison with the experimental data. Based on the computed temperature and flow fields we conclude that the experimentally observed thermoconvective pattern in the droplet is an unsteady Benard-Marangoni instability and is different from hydrothermal waves.

We would like to acknowledge the financial support of

  FIG. 1. Snapshot of comparison between experimental and numerical infrared images at t=15.36s: a) top left figure evidence the numerical droplet surface temperature. b) top right figure evidence the numerical droplet total radiative temperature. c) bottom central figure show the experimental infrared image obtained for an ethanol droplet under evaporation for the same conditions. Multimedia view

  FIG. 2. Snapshot of flow motion inside the droplet under evaporation from the numerical simulation at t=15.40s: a) top figure showing the thermal plume formation inside the droplet. b) bottom left figure showing the top surface temperature. c) central figure showing in a cut plane the liquid temperature and d) on the right view the vertical velocity component. Multimedia view

  the "Centre National d' Études Spatiales" research grant for post-doctorates and the European Space Agency. This work has been carried out in the framework of Labex MEC (ANR-10-LABX-0092) and A*MIDEX (ANR-11-IDEX-0001-02) projects, funded by the Investissements d'Avenir French Government program managed by the French National Research Agency.


Nick Laan
email: dr.nick.laan@gmail.com

Fiona Smith

Celine Nicloux

David Brutin

Morphology of drying blood pools

Keywords: BPA, blood pools, drying, evaporation, fluid dynamics

published or not. The documents may come   L'archive ouverte pluridisciplinaire

HIGHLIGHTS

• Blood pools are often encountered on crime scenes.

• The general knowledge concerning blood pools is very limited.

• During drying a blood pool goes through five separate stages.

• The size of a pool depends on the volume and the contact angle.

Introduction

Bloodstain pattern analysis is a forensic tool used by investigators to determine, among others, what, where and how a crime took place [1]. One of the most common types of bloodstains found on a crime scene following a deadly blood shedding event, is the blood pool (fig. 1). Ante-and postmortem it is often the case that a victim bleeds out, thus accumulating blood in one or multiple areas. Currently, when a blood pool is found, it is classified as such and an investigator can conclude that the blood donor was bleeding at that location for any reasonable period of time for the pool to be created, be it seconds, minutes or even hours. Previous studies have investigated if it was possible to determine what the volume of a blood pool was, to determine if such a loss of blood volume could constitute loss of life [2], or for other crime scene reconstruction purposes [START_REF] Lee | Estimation of original volume of bloodstains[END_REF][START_REF] Sant | Exsanguinated blood volume estimation using fractal analysis of digital images*[END_REF][START_REF] Laan | Volume determination of fresh and dried bloodstains by means of optical coherence tomography[END_REF][START_REF] Laan | Bloodstain pattern analysis: implementation of a fluid dynamic model for position determination of victims[END_REF]. However, almost no studies have been performed concerning the drying of an entire pool of blood. Such studies can be very useful for determining, e.g., the time that the blood shedding event occurred, any actions that may have occurred during the blood shedding event or the physiological state the subject was in. For example, fig. 1) shows two crime scene pictures of the same pool, 22 hours apart. In the first (top) picture, the edges and the bottom of the pool have started drying. In the second picture the pool has completely dried. Information obtained from how fast the blood dried could be crucial to determine when the pool was created. There have been several studies concerning the drying of singular blood droplets [START_REF] Brutin | Pattern formation in drying drops of blood[END_REF][START_REF] Sobac | Structural and evaporative evolutions in desiccating sessile drops of blood[END_REF][START_REF] Zeid | Influence of relative humidity on spreading, pattern formation and adhesion of a drying drop of whole blood[END_REF][START_REF] Zeid | Influence of evaporation rate on cracks formation of a drying drop of whole blood[END_REF][START_REF] Sobac | Desiccation of a sessile drop of blood: cracks, folds formation and delamination[END_REF]. To our knowledge only Ramsthaler et al. investigated the drying of blood pools [START_REF] Ramsthaler | The ring phenomenon of diluted blood droplets[END_REF]. In their study they focused on the drying and morphology of diluted blood droplets and pools to be able to distinguish between diluted and whole blood. In this paper we report on the morphology of drying blood pools. Pools of blood, obtained from healthy volunteers were deposited on linoleum surfaces. Based on our results we are able to distinguish five different stages of drying. In addition, we report the universal properties of drying blood pools, but also distinguish anomalies, which can differ between pools.

Background theory

Once bleeding occurs, blood being ex vivo, it will coagulate and dry. During the coagulation (clotting) process, fibrin strands are formed creating a solid structure of the blood, the clot. During drying water evaporates from the blood pool until only the solid matter, mainly red blood cells (RBC's), remains. Depending on the size of the pool and environmental conditions, the time the pool completely evaporates may take hours to days. On the crime scene, pools can be found in the order of millilitres to litres. We, however, focus on pools in the order of millilitres, simply because pools with a volume of several litres, without any additives like anticoagulants, would require a very large donation of a volunteer which is not a viable option. Prior to our investigation into drying blood pools we require some general knowledge about fluid dynamics of evaporating liquids.

When a droplet is deposited upon a surface it will spread. The area (A) the droplet spreads over depends on the physical properties of both the surface and liquid, where the surface tension and contact angle are the most important parameters (see supplementary materials). The surface tension (γ) is defined as the amount of energy required to increase the surface area by one square meter. In other words, increasing the area of a droplet requires energy and the higher the surface tension, the more energy this takes. As the droplet or pool lays upon a surface, the surface tension acts upon the triple-line (the line around droplet or pool where liquid, surface and air meet, see fig. 2a-b). How a droplet or pool spread upon a surface can be deduced from young's equation [START_REF] De Gennes | Capillarity and wetting phenomena: drops, bubbles, pearls, waves[END_REF]:

S = γ(cos θ -1) (1)
Here, S is the so-called spreading parameter, γ is the surface tension between liquid and gas interface and θ the contact angle between liquid and surface (see fig. 2c). When the contact angle is much smaller than 90 • (S is positive) the surface wetting, i.e., the liquid can easily spread over the surface and the surface is presumed wetting. When the contact angle is much larger than 90 • (S is negative) the liquid cannot spread over the surface easily and the surface is presumed non-wetting. With a small contact angle, the pool will cover a much larger area and have a larger perimeter, which should significantly increase the rate of evaporation. Therefore, the contact angle is a very important parameter concerning the drying of blood pools. Deegan et al. [START_REF] Deegan | Capillary flow as the cause of ring stains from dried liquid drops[END_REF] demonstrated very accurately the principles of the coffee ring effect that is observed during the drying of a droplet of a colloidal suspension. This study showed how the flow arising from the evaporating liquid induced the characteristic ring formation. A study by Brutin et al. [START_REF] Brutin | Pattern formation in drying drops of blood[END_REF] focused on the drying of sessile whole blood droplets which showed that it is very similar to the drying of a droplet of a colloidal suspension, blood being a colloidal fluid. During the drying of a blood drop the formation of cracks is observed. Moreover, the study showed that a drop of blood dries following two different regimes and goes through five different stages. The first regime (first three stages) is being driven by convection, diffusion and then gelation. At the moment the drop is deposited, RBC's are evenly distributed inside the droplet, but then the solvent starts evaporating inducing an evaporation flux at the interface and an internal flow transporting particles inside the drop. This leads to the formation of a gel once the concentration of particles is high enough. Additionally, this flow induces the formation of the so-called biological deposit on the periphery of the droplet; indeed RBC's are driven from the inner part of the droplet to its rim. Then the transition phase takes place and leads to the gelation of the entire drop. A sharp decrease in the drying rate is observed, whereas gelation is rapid.

The second regime is much slower since it is diffusive. The final two stages correspond to the drying and the formation of cracks that are nucleating and propagating. This extensive work on drying of droplets gives precious information about the process and shows accurately that desiccation starts at the periphery of the drop, and then dries towards the centre of the droplet. The work presented in this study no longer focuses on droplets but on pools. To understand the dynamics occurring during the drying of a pool, the size of the blood pool must be considered. As long as the volume is low, in the case of droplets, the surface tension forces are dominant resulting in a curved surface. In contrast, if the volume is large enough, the gravitational forces will dominate over the surface tension forces producing a flat surface on top of the pool. Similar to a droplet, a pool will have a contact angle with the substrate on the edges, but in contrast is flat otherwise (fig. 2b). The area the pool spreads over is directly dependent on the contact angle (see appendix A). In order to understand the phenomena and the dynamics driving the drying of blood pools, we performed experiments with small blood pools (about 4 ml), which were recorded by taking pictures every two minutes. Foremost, the purpose of our experiment is to identify the different drying stages of blood pools.

Methods and Materials

To follow the drying of a blood pool we required the environment to be monitored and as constant as possible. Therefore each blood pool was created in a glovebox (Jacomex T-Box, V=700L). The humidity and temperature were recorded during drying by means of a hydrometer (Teslo AG, 175-H2, Datalogger, Germany). The temperature was constant at 22±0.5 • Celsius during all of the experiments. Within the glovebox, a camera (Nikon D200, resolution: 2592x3872 pixels or Nikon D300s resolution: 2848x4288 pixels both with a 60mm 1:2.8 lens) was suspended directly above the blood pool. The camera's enabled us to take a single picture, every two minutes, with a resolution of roughly 26 to 30 pixels per millimetre. For each blood pool created, blood of a healthy volunteer was drawn by a certified nurse in a 4.5 ml evacuated blood collection tube (VenoSafe, Terumo, France). Immediately after blood collection the tube was emptied above the target substrate (linoleum) creating a pool of blood of roughly 4 ml each time.

Moreover, blood pools were created by slowly dripping blood directly from the tube/needle connected to the arm to imitate the most realistic blood shedding event possible. Accordingly, blood pools created had a starting weight 3.5 g < m i < 7 g. No differences were observed in the drying process of the pools between the two methods of blood deposition. Of several blood pools the substrate was on top of a balance (Mettler Toledo, ML802, Switzerland) to determine exactly the mass (m) of the blood pool during the entire drying process (one measurement every minute). By means of a reference length next to the pool it was possible to determine the area of the pool, by using a program written in Matlab.

Results and Discussion

In fig. 3 we show a time-lapse of a drying blood pool deposited on a linoleum surface (see supplementary movie for the complete time-lapse). First of all the simple observation of the pictures obtained in the experimental conditions described previously allowed the identification of five distinct phases (fig. 3): (I) coagulation stage, (II) gelation stage, (III) rim desiccation stage, (IV) centre desiccation stage, (V) final desiccation stage (see supplementary materials for a point-wise summary). (I) When blood is deposited upon a surface, the RBC's are evenly distributed throughout the pool, which then will sediment. At this point the blood has a dark red colour and starts to coagulate [START_REF] Riddel | Theories of blood coagulation[END_REF]. It is possible that due to wetting and capillary action, the area of the blood pool increases as the blood spreads slowly over the surface during the initial 15 to 30 minutes. During this first stage there is a change in colour from dark red to lighter red, mainly due to coagulation. (II) As fluid evaporates from the blood pool red blood cells, that are not constricted in the fibrin web, are transported to the rim of the stain and deposited, due to flow caused by evaporation [START_REF] Deegan | Capillary flow as the cause of ring stains from dried liquid drops[END_REF]. The transition from the fluidal to a gel state is referred to as the gelation front.

The second stage starts when the gelation rim is created around the pool. The gelation front propagates inwards, towards the centre of the stain, as the pool continues to dry. (III) The third stage starts as soon as the rim turns black and starts to crack, indicating that the rim is desiccating. The transition from the red to black colour is referred to as the drying front. During this stage both the gelation and drying front propagate towards the centre of the stain. (IV) Once the gelation front reaches the centre of the stain, the entire stain has gelified and evaporation of fluid is mainly driven by the porous media drying dynamics [START_REF] Brutin | Pattern formation in drying drops of blood[END_REF]. The drying front and cracks propagate towards the middle of the stain. (V) Finally, the drying front reaches the centre of the pool. The pool has almost completely desiccated. During this last stage, the entire pool is black in colour. As the last liquid evaporates, the remains contract and the cracks reach the middle of the stain. Accordingly, flakes are separated and partially or completely detach from the surface. We have observed the five stages described above for every pool we created. However, it should be clear that a pool does not dry in a uniform manner. Instead, one part of the blood pool may be fully desiccating (left side fig. 4) while another part is still in a gel-like state (right side fig. 4). Consequently, the centre of the pool, i.e., the location where all cracks come together, does not necessarily has to be the geometrical centre of the pool. Furthermore, the duration of any one stage and complete desiccation can differ between blood pools, depending on the humidity, temperature, shape and size of the pool and the kind of surface. During the drying of the blood pool, the mass (m) of the pool was measured with a balance, every minute (fig. 5). In the first and second stage the pool loses roughly 40% of its mass. Liquid is evaporating and the height of the pool diminishes over time. A linear function was fitted to the data points for the first six hours of drying, with fitting parameters A the initial droplet mass equal to 4.39 g and B the mass loss per hour equal to 403±3 mg/h. It is clear that the mass loss during the first six hours scales linear with time. Only once the drying front is significantly formed, does the decrease in mass stop scaling linear with time. This effect can be explained as follows. The pool is pinned to the surface, i.e., the contact line cannot move. As liquid from the pool evaporates, the volume diminishes, but because the contact line is pinned, the pool can only decrease in height. Moreover, to compensate for the volume loss at the contact line, there is a flow from the middle of the pool towards the rim. This flow causes particle (RBC's) transport throughout the pool which are deposited at the contact line creating a characteristic rim around the pool. As the pool dries, the contact angle should change which in turn should decrease the evaporation rate [START_REF] Hu | Evaporation of a sessile droplet on a substrate[END_REF]. We, however, have not observed any change in evaporation rate due to this effect. As long as the area of the pool does not change, the evaporation rate does not change. Only once a critical amount of liquid has been depleted from the pool, in this case more than 55%, does the evaporation rate change. At this point the liquid and gel areas of the pool are diminishing, accordingly the evaporation rate diminishes as well, explaining the deviation from the linear trend after six hours in fig. 5. The largest visual change happens during stage III where the entire stain transforms from a liquid/gel to a solid. Finally, 21% of the initial mass remains in this case. Multiple blood pools were created and recorded with varying haematocrit values and under different humidities. The mass of those pools are shown in fig. 6b(inset). Not surprisingly, it is clear that the larger the mass of a pool, the longer it takes for the pool to dry. The mass decreases linearly with time until roughly 50% to 70% of the mass has been depleted, at which point the slope diminishes and the mass becomes constant as the last liquid evaporates from the pool. Moreover, the left-over mass is dependent on both the initial volume of the pool and the haematocrit value of the blood, which is in accor-dance with the findings of [START_REF] Laan | Volume determination of fresh and dried bloodstains by means of optical coherence tomography[END_REF][START_REF] Laan | Bloodstain pattern analysis: implementation of a fluid dynamic model for position determination of victims[END_REF]. As the pools were deposited under various environmental humidities; these findings indicate that the drying speed, i.e., the slope of the linear part of the drying curves, becomes steeper with decreasing humidity. In other words, the higher the humidity, the longer the blood pool takes to dry. However, a more in-depth investigation is required to quantify how the drying speed depends on the humidity and other factors, as temperature, contact angle and size of the pool might influence the drying speed a lot. To have a better insight into the drying of the pools, we normalized the mass according to:

m norm = m -m f m i -m f (2)
Here m i is the initial mass, m f is the final mass and t f is the final drying time defined as the point in time where the change in mass is less than 0.01 g/h. In fig. 6a, the normalized mass was plotted as a function of the normalized time to show that independent of humidity, mass, haematocrit value, or total drying time the mass diminishes similarly for every pool. We show that this dimensionless rescaling is valid for a range of masses (3.5 g < m i < 7 g) and a range of humidities (22% < H < 57%). This rescaling is the first step towards a unified theory concerning the drying of blood pools and maybe even pools in general.

Although the literature concerning the drying and evaporation of blood pools is poor, there has been extensive studies concerning the drying of gels [START_REF] Sherwood | The drying of solidsi[END_REF][START_REF] Moore | The mechanism of moisture movement in clays with particular reference to drying-a concise review[END_REF][START_REF] Macey | Clay-water relationships and the internal mechanisms of drying[END_REF][START_REF] Dwivedi | Drying behaviour of alumina gels[END_REF], which show considerable similarities. The drying process of gels has three distinct drying stages. During the first stage the evaporation rate is constant and the volume decrease of the gel is equal to the volume of liquid lost by evaporation [START_REF] Moore | The mechanism of moisture movement in clays with particular reference to drying-a concise review[END_REF][START_REF] Macey | Clay-water relationships and the internal mechanisms of drying[END_REF]. Once a critical point is reached, the volume of the gel stops to decrease and cracking may occur. According to a study of Dwivedi [START_REF] Dwivedi | Drying behaviour of alumina gels[END_REF], the evaporation rate of water from alumina gels during this first stage is comparable to the evaporation rate of pure water. Subsequent to the first stage, gels undergo two so-called 'falling rate' stages where the evaporation rate decreases towards zero. During the first falling rate stage the liquid flows through partially empty pores, followed by a second falling rate stage, where the liquid diffuses its vapour to the surface. Comparably, drying blood exhibits a similar behaviour to water and a drying gel. After blood pool creation, blood dries with a constant rate of evaporation (fig. 5). Subsequently, the drying rate decreases and cracking occurs, similar to a gel. Consequently, drying blood has analogous characteristics to that of 13 both water and a gel. The size of the pool is dependent on two important parameters. The first one is the volume of the blood, namely, the larger the volume of the blood, the larger the area will be the blood spreads over (fig. 7). The second parameter is the contact angle θ, i.e., the wettability of the surface the blood lies on, which was calculated using the approach described in the supplementary materials (Eq. S6). If a surface is wetting (θ < 90 • ) then the blood spreads over a much larger area then if the surface was non-wetting (θ > 90 • ). Accordingly, the larger the contact angle between blood and surface, the smaller the area covered by the blood. This is reflected well by our results (fig. 7) which show that blood pools with a larger contact angle deviate from the average (dashed line), i.e., they have a smaller area. These factors are very important for the drying dynamics. Namely, a higher volume blood pool takes a longer time to dry, whereas an increase in area and a lower contact angle can both speed up the drying process. It is possible that during stage II, due to clotting, serum is forced out of the main pool, also known as serum separation [1]. In this study, we did not encounter any serum separation from our blood pools. This is not a surprise considering the findings of Ramsthaler et al. [START_REF] Ramsthaler | The ring phenomenon of diluted blood droplets[END_REF] who reported no serum separation unless the volume of the pool was larger than 10 ml, while in our study all pools were smaller than 7 ml.

We have to stress that the five stages of blood pool drying reported in this study are not the same stages as reported for single sessile droplets [START_REF] Brutin | Pattern formation in drying drops of blood[END_REF]. The drying stages observed in our experiments are very similar to those of drying blood droplets, however there are several distinct differences. First of all, the droplets reported by [START_REF] Brutin | Pattern formation in drying drops of blood[END_REF][START_REF] Zeid | Influence of evaporation rate on cracks formation of a drying drop of whole blood[END_REF][START_REF] Sobac | Desiccation of a sessile drop of blood: cracks, folds formation and delamination[END_REF] contain anti-coagulant and therefore do not clot, while clotting is a very important parameter, as it may inhibit particle transport due to evaporative flow. On crime scenes blood almost always clots once it has left the body and the drying dynamics could change a lot if the blood did not clot, e.g., serum separation would not occur at all. In addition, during clotting (stage I) the colour of blood turns from a dark red to a bright red, something which was not observed for non-coagulating blood droplets.

Secondly, pools are much larger than droplets which change the evap-oration dynamics and the final appearance of the remains. Drying blood droplets show mobile fragmented cracking patterns at the corona and fine cracking patterns at the middle and periphery of the droplet [START_REF] Brutin | Pattern formation in drying drops of blood[END_REF]. The cracking patterns of blood droplets have been extensive investigated and resemble the cracking patterns of colloidal gels [START_REF] Zeid | Influence of evaporation rate on cracks formation of a drying drop of whole blood[END_REF][START_REF] Sobac | Desiccation of a sessile drop of blood: cracks, folds formation and delamination[END_REF][START_REF] Pauchard | Patterns caused by buckle-driven delamination in desiccated colloidal gels[END_REF], i.e., the peripheral cracks of a blood droplet divide the remains in polygonal shaped cells. However, the cracking patterns of an entire (coagulating) blood pool are quite dissimilar from a single blood droplet. With blood pools we observe long elongated cracking patterns which propagate towards the middle of the stain, that turn black when completely desiccating. Once more, the size and thickness of the pool and coagulation of the blood can be the main reasons why blood pools show very different cracking patterns from those of single blood droplets. The cracks of a blood pool are similar to those observed by Pauchard et al.

for a colloidal suspension (with a low ionic strength I = 0.4 mol/l) [START_REF] Pauchard | Influence of salt content on crack patterns formed through colloidal suspension desiccation[END_REF] and for a drying suspension of latex particles (0.1 µm) resulting from directional growth of fractures [START_REF] Pauchard | Morphologies resulting from the directional propagation of fractures[END_REF]. Cracks closely follow the drying front and the width between cracks is directly related to the thickness of the sample, with a prefactor depending on gel thickness, physicochemical properties, adhesion onto substrate and desiccation conditions [START_REF] Pauchard | Morphologies resulting from the directional propagation of fractures[END_REF][START_REF] Lazarus | From craquelures to spiral crack patterns: influence of layer thickness on the crack patterns induced by desiccation[END_REF]. Finally, a single blood droplet only shows a gelation front which partly propagates from the rim towards the middle. In contrast, a drying blood pool shows both a gelation front and a drying front that is created at the rim and completely propagates towards the middle of the stain. It is specifically the evolution of the drying fronts that define our different stages.

This study showed clearly that the drying dynamics of a pool of blood could be identified. However many more parameters would need to be investigated in further studies. Such parameters would be the influence of humidity and temperature, but as well the influence of the substrate which would mainly change the contact angle, and thus drying rate since it would change the spreading of the pool. Moreover it would be interesting to consider more the influence of the shape of the pool. Finally, size of the pool is very important. In these experiments, each pool was in the order of 4 ml, while in practice they might be much larger or even much smaller. For the latter, it is necessary to distinguish when a volume of liquid is considered a pool or a droplet. We suggest that a pool should be defined as having a flat surface. Accordingly, in our experiments given a contact angle of roughly 45 • , the minimum volume of a pool should be much more than 170 µl (see supplementary materials, Eq. S11), which is the case in our experiments.

Conclusion

In this study, for the first time the drying dynamics of pools of whole blood were investigated, for a range of 3.5 to 4.5 ml. We were able to distinguish five different drying stages, each with their own characteristics. The mass of a blood pool diminishes in a very reproducible manner, first linearly in time and then approaches a constant value. Additionally, we were able to collapse all mass curves onto a single curve by normalizing the mass and time of drying. The general knowledge concerning blood pools within the field of bloodstain pattern analysis is very limited at most. This work is a step forward in the classification and characterization of blood pools within this field. Prospectively, the results of this study may be used for crime scene reconstruction or for future investigations into determining the time the blood shedding event occurred. Finally, we verified that the size of the blood pools is directly related to its volume and the wettability of the surface. This result could be used to estimate the original volume of a dried blood pool, to answer the question if the amount of blood could constitute loss of life. We anticipate this study to be of considerable importance for forensics and bloodstain pattern analysis as a whole.

Acknowledgments

Of the Institut de Recherche Criminelle de la Gendarmerie Nationale we would like to thank the bloodstain pattern group for helping in this investigation, the nurses from the infirmary for helping us with the blood collection and all volunteers for giving their blood. This work received a financial grant from the French Agence Nationale de la Recherche in the frame of the project ANR-13-BS09-0026. Also, this work has been carried out in the framework of the Labex MEC (ANR-10-LABX-0092) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the "Investissements dAvenir" French Government program managed by the French National Research Agency (ANR).

[1] S. H. James, P. E. Kish, Sutton, Principles of Bloodstain Pattern Analysis, Theory and Practice, CRC, 2005.

[2] H. F. Bartz, Estimating original bloodstain volume: the development of a new technique relating volume and surface area, Ph.D. thesis, Department of Biology, Laurentian University, Sudbury, Ontario (2003).

Figure 1 :

 1 Figure 1: Picture of a real pool of blood found on a crime scene, (top) before the body was removed and (bottom) 22 hours later. The yellow liquid is serum which was separated during clotting and the black mass in the top picture is a large formed clot.

Figure 2 :

 2 Figure 2: A schematic representation of the cross-section of a) a single droplet, b) a pool and c) three droplets on surfaces varying in wettability.

Figure 3 :

 3 Figure 3: time-lapse of a drying pool of blood from a healthy person, at 21C with a relative humidity of 32%. For a movie of the drying blood pool, see supplementary material. Note that the elapsed time between pictures is different for each stage. The time of drying for each picture is shown in fig. 5.

Figure 4 :

 4 Figure 4: close-up view of a drying pool, with several defined properties of the pool. The yellow dashed and dotted lines represent the gelation and drying front, respectively.

Figure 5 :

 5 Figure5: the mass of a blood pool as a function of time with some pictures of the pool at several moments in time. The vertical dashed lines represent the time at which the pictures in fig.3where taken. The red line is the mass of the pool, the dashed line a linear fit to the data, from t=0 till t=8.2 h, the time the pool transitioned into stage III. The percentages are the amount of mass left at that specific point in time.

Figure 6 :

 6 Figure 6: a) the normalized mass as a function of normalized time of the blood pools and b) the mass of the pools as a function of time (inset), for different humidities and haematocrit values.

Figure 7 :

 7 Figure 7: the area A and as a function of the volume V of the blood pools. The colours represent the contact angle of the blood with the surface, red θ = 37 • , orange 40 • < θ < 45 • , green 45 • < θ < 50 • , blue 50 • < θ < 60 • , black θ = 65 • . Errors in volume are smaller than the size of the symbols.