Thermodynamic equilibrium condition and model behaviours of sessile drop

Transcription

1 Thermodynamic equilibrium condition and model behaviours of sessile drop Pierre Letellier 1,2, Mireille Turmine 1,2 1- CNRS, UPR15, Laboratoire Interfaces et Systèmes Electrochimiques, F , Paris, France 2- UPMC, Université Paris6, UPR 15, Laboratoire Interfaces et Systèmes Electrochimiques, F-75005, Paris, France Phone number: Fax number: Short title: Thermodynamic equilibrium and sessile drop model ABSTRACT The relationships used to express the equilibrium of drops put on a solid and to account for their behaviour are examined. We established, first of all, the relation of the thermodynamic equilibrium of the sessile drop. Then, we demonstrated how the latter allows creating models of behaviour of the system by fixing formally the pressure difference, P, between the drop and its environment. We showed that the Wenzel s relation results from the adoption of a model which refers to the behaviour of a spherical drop of liquid and that the Young-Laplace relation corresponds to a hypothetical state of this model. This approach is not unique. We showed that the adoption of a model which formally fixes an internal pressure of the drop identical to that of its environment (Isobaric Sessile Drop Model leads to a kind of Cassie- Baxter's relation, with whom is associated a hypothetical state, characterized by the 2 SV relation. A general discussion is proposed on the meaning 1 cos of the interfacial tensions according as they are involved in the models or in the relation of thermodynamic equilibrium and on the contribution of the modelling of the behaviours in the understanding of the wetting phenomena in particular for the Cassie-Baxter / Wenzel transition. Keywords: Young-Laplace, Wenzel, Cassie-Baxter, transition, sessile drop, equilibrium, contact angle, thermodynamics. 1. Introduction The behaviour of a liquid drop put on a solid substrate is generally described using the Young-Laplace relationship introduced in 1805 [1]. We have to admit that more than 200 years later, this relation continues not to completely satisfy the scientific community and that, from time to time, some researchers [2-5] question its foundations, its validity and the consequences which it imposes. The fact that a scientific unanimity could not emerge over a so long period [6, 7] shows that this relation is not as obvious as many let it believe, and that its demonstration is debatable on some points. The purpose of this work is to point them out, to separate the points 2-129

2 which correspond to the thermodynamic equilibrium condition of the system and those come within choice, conventions or hypotheses. Generally, it is admitted that a liquid drop of low volume, put on a substrate adopts the shape of a spherical cap of radius of curvature, r. The equilibrium position of the drop is characterized by the contact angle θ. The liquid volume, V, the area of the liquid-vapour interface, A, and the area of the solid-liquid interface, A, are expressed according to r and θ as follow, π 2 3 V r (1 cos (2 cos 3 2 A 2πr (1 cos A πr 2 (1 cos (1 cos This modelling of the drop, which supposes interfaces of perfect geometry, (plane, spherical cap cannot correspond to the reality. Despite all the care that can be taken for its realization, the interface between a solid substrate and a liquid or a gas could never be a perfect plan. It is the same for the liquid-vapour interface which is supposed to lose its shape near the triple contact line between the liquid, the solid and the vapour [8-10] or under the effect of the gravity [11]. We first of all will establish the strict equilibrium condition of the drop put on a substrate. 2. Thermodynamic equilibrium of a sessile drop Consider a liquid drop of volume V put on a solid substrate. In order to generalize this issue, we suppose that the system, at equilibrium, has three interfaces I, I and I SV SV whose areas are, et, respectively, without specifying their possible relationships with the geometric area A, A and A SV. It is agreed that the pressure in the drop, P d, can be different from the surrounding pressure, P. The consideration of these variables is a choice. To find the equilibrium condition of the drop, the borders of the system are virtually moved. The sum of the works generated by this transformation is null. An increase (dv of the drop volume is considered at constant contact angle (θ. This operation results in the shift of the system borders (figure 1. For each interface, the work involved is equal to the product of the interfacial tension by the variation of the corresponding area. d liquid P vapor (1 dv P d d d SV solid Figure 1: displacement of the borders of a sessile drop. For a volume increase of dv, the areas of the interfaces vary of dλ, dλ and dλ SV. The equilibrium condition of the drop is written as SV SV 0 P dv PdV d d d (2 d 2-130

3 SV, and are the characteristic tensions associated to the areas Λ of the interfaces I, I and I SV respectively. Consider the case of a solid having a homogeneous surface which is not necessary SV plane. This implies that d d. The result is that the thermodynamic equilibrium condition of the sessile drop is written d SV d Pd P ( (3 dv dv This differential relation can be easily integrated if, at equilibrium, the drop volume is considered as an extent of geometrical dimension equal to 3, whereas the areas of the interfaces are extents of dimension 2. That implies that if the drop volume is multiplied by a number, the areas of the interfaces will be multiplied by 2/3. The areas are Euler's functions of order 2/3 of the volume. Properties of Euler's functions 2 d V 3 dv (4 2 d V 3 dv applied to eq. 3, allow establishing 2 SV ΔP Pd P ( (5 3 V V Eq. 5 is the equilibrium condition of the sessile drop on the solid corresponding to the set of variables chosen to characterize the behaviour of the system. Geometrical areas of the interfaces can explicitly appear in this equality 2 SV ΔP Pd P A ( A (6 3V A A The fact of supposing that Λ and A are Euler s functions of the same order of the volume allows assuming that these extents are proportional. The parameters 1 and 1 are defined. A and A being the minimum area of A A the system, the values of σ and ρ are higher than one. The equilibrium condition of the sessile drop is then written 2 SV ΔP A ( A (7 3V which can be simplified by expanding the geometrical magnitudes of the 2 SV (8 system ΔP 2 1 cos r (1 cos (2 cos The principle of the thermodynamic analysis would expect that we draw the conclusions of eq. 5 by making measurements of and P for given conformations of SV the interfaces (known and in order to determine the values of ( for various liquid / solid pairs. The problem is that, in our knowledge, there is no technique of direct measurement of P. However, thermodynamics allows imagining some indirect measurements

4 Since we suppose that the pressure of the drop is different from the surrounding pressure, the physicochemical properties of the liquid and possibly those of its solutes are modified according to the laws of thermodynamics. In theory, it would thus be enough to follow the evolution of these properties to obtain the variations of P. The reality shows that this approach is unrealistic for the considered systems. To prove it, let characterize the liquid-vapour equilibrium of a liquid drop put on a solid. Let us consider a liquid i in the presence of its vapour in a closed container at the temperature T. At equilibrium, the total pressure which is applied on the pure liquid is its saturated vapour pressure, P i. If we consider a drop of the same liquid put on a solid, also in a closed container, in the presence of its vapour, the total pressure exerted on the drop is equal to its saturated vapour tension, P i. Supposing that the molar volume of i, V i, slightly varies with the pressure, we establish the following relation which connects the variations of the saturated vapour pressure with the geometrical characteristics of the drop Pi Vi 2 SV ln 2 ( (1 cos Pi Pi Pi RT r(1 cos (2 cos This equation corresponds to the Kelvin s relation for sessile drops in closed systems, the only ones for which the liquid can be strictly considered in equilibrium with its vapour [12-14]. Formally, measurements of P i and P i would allow reaching the value of the difference (γ - γ SV, once determined the geometry of the system. But, a calculation (even approximate shows that for large drops (approximately 1 µl used in the contact angle measurements, the differences between P i and P are i experimentally indiscernible, what we can check by considering a spherical drop of water of 1µL. In a closed container, the vapour tensions are linked by [15] P i Vi 2 ln Pi Pi (10 Pi RT r For these data, we calculate a difference about Pa at 298 K, taking Pi 31Pa. The conclusion is that in the range of the drop sizes commonly used to carry out contact angle measurements, the effect of curvature on the physicochemical properties of the systems are too small to be experimentally detected [16]. To obtain measurable effects, it would be necessary to consider systems of nanometric size for which we showed they follow rules of behaviour described by a non-extensive thermodynamics [17-20]. (9 The result is that the consequences of eq. 5, which characterizes the thermodynamic equilibrium of the drop put on a solid, cannot be generally checked experimentally. To get round this obstacle, using of eq. 5 was envisaged by forcing P to vary with the contact angle, for a given drop volume, according to a conventional rule. In doing so, we overstep the strict frame of the thermodynamics. A model of behaviour for a sessile drop at equilibrium is thus created. If a value of P ( is formally attributed for a given contact angle in eq. 7, the difference of the interfacial tensions SV ( is then defined. The latter becomes also conventional. x x SV 1 ΔP( r (1 cos (2 cos x x 2 (11 (1 cos 2 SV It can be different from the difference between thermodynamic extents (

5 We shall then attempt to compare the behaviour of the real system to that of the model. This way of analyzing the evolution of the properties of the composed drops by referring to a model of behaviour is completely usual in chemistry of solutions to express, for example, the chemical potentials of the mixture constituents according to its composition. In this way, the chemical potential of a mixture constituent can be expressed by referring to the model of behaviour of an infinitely diluted solute, that it never is, or to the behaviour it would have if the mixture was ideal (model of pure component, what it is not generally. Then, the differences between the real behaviours and the model behaviours are expressed in terms of activity. Within the model behaviours, hypothetical states can be defined, as for example a standard state for the infinitely dilute solutions. In that case, the standard chemical potential is defined as that of the solute at the concentration of 1 mol L -1 (or of 1 mol kg -1 which would have the behaviour of a solute infinitely diluted in the solution, i.e., at concentration equal to zero. The adoption of two incompatible experimental conditions at the same time makes the standard state hypothetical. This definition raises no conceptual problem because the standard chemical potential is simply used as a convention of origin of chemical potentials. It is a practical notion. The study of the behaviour of sessile drops can be tackled according to a similar approach. In this work, we shall consider two different reference behaviours which differ on the convention adopted for P. We shall discuss their respective relevance. 2.1 Reference to the behaviour of the spherical liquid drop In this model, we refer, by convention, to the properties of a spherical drop of liquid in the presence of its vapour (or of another liquid. The pressure difference between the inside of the drop, P d, and the pressure of its surroundings, P, is given by the 2 Laplace relation, ΔP, r being the radius of the drop. The sessile drop is then r considered as a spherical drop severed by a horizontal plan made of the solid. In this approach, the liquid-vapour interface has a perfect shape of a spherical cap, Λ is identified with A and = 1. But, the shape of the interface I is not specified. We write: 2 2 SV 2 ( (1 cos (12 r r (1 cos (2 cos Since the way P varies with the radius of the drop is formally fixed, a new meaning are then given to the tensions associated with the interfaces I and I SV SV, and. These magnitudes are those which verify eq.12. They could be different from the thermodynamic extents of tension γ and γ SV. We shall go back over this important point in the discussion. This relation is simplified as SV cos ( (13 It links the values of the contact angles to the geometry of the solid-liquid interface Hypothetic state: Wenzel s relation, Young s relation The previous formalism can be largely simplified if we define a particular state of the system for which the value of is known and can be taken as an "origin convention" to spot the values of contact angles

6 The simplest idea is to suppose that there is a state of the solid surface for which = 1, what means that = A. The contact angle is M. The Young Laplace relation is then established SV cos M (14 It is important to notice that the conditions introduced to define this state are incompatible between them. Indeed, the condition = A imposes that the solid surface, supposed to be at equilibrium, is perfectly smooth (because A is the minimum smooth surface. As a result the pressure difference between the drop and the solid has to be equal to zero. As the value of the surrounding pressure is equal to that of the solid, the consequence is that the pressure of the drop is strictly equal to the surrounding pressure. Thus, the Young-Laplace relation corresponds to a state of the drop which answers simultaneously two irreconcilable conditions on the pressures. This relation is purely conventional and cannot claim to describe a real situation. It characterizes a "hypothetical state" the vocation of which is to serve as condition of origin to spot the values of the contact angles. It is a practical notion. So, by introducing the Young-Laplace relation into eq.13, we obtain cos cosm (15 This equality is similar to that suggested by Wenzel. As a result, the Wenzel relation is characteristic of a model of behaviour of the sessile drop referred to the behaviour of the spherical drop, taking as convention of origin, the hypothetical state expressed by the Young-Laplace relation. The interest of this relation is mainly the consequences supposed by this expression within the framework of the model. The parameter being higher than 1, the system is going to evolve very differently with the values of as the angle M is lower or upper to 90. As the values of grow, if M is lower, the system will become more and more wetting, if it is upper it will become less and less wetting. It is what we illustrated on figure 2. M M = 105 M =

7 Figure 2: Variation of contact angles for a solid-liquid system with the value of ρ according to the Wenzel s relation. Values of the contact angles θ M are 75 and 105. For the two chosen examples (θ M = 75 and θ M = 105, the model predicts for high values of ρ that a complete wetting can be reached, or on the contrary, a situation of non-wetting can be obtained. We can naturally conceive other behaviour models for the sessile drop, by adopting other conventional rules of variation of P with the contact angle. In particular, nothing obliges, a priori, that one gives a particular importance to this parameter in the establishment of the equilibrium condition of the drop. Consequently, it seemed to us interesting to develop a behaviour model based on a conventional value of P equal to zero and to compare it in its consequences to that of Laplace. In our knowledge, this way was never explored Reference to the behaviour of isobaric sessile drop model (ISDM Let us take as particular case of the equilibrium relation depicted by eq.5, that in which the pressure in the drop is equal to its environment, i.e. an isobaric sessile drop. In such a case, a behaviour model is created (ISDM for which the following equality is established SV SV 0 d d d (16 The interfacial tension, and those of the previous model involve in eq.5, and SV. SV, introduced in this relation are different from and SV, and can be also different from those Following the same approach as previously, for a drop put on a solid whose surface is not obligatorily smooth ( 1, and whose liquid-vapour interface can have not the shape of a perfect spherical cap ( 1, we obtain SV 0 A ( A (17 Expanding the expressions of the geometrical areas we can write 2 SV ( 1 cos (18 This relation links the values of the contact angles to the sessile drop geometry. It simultaneously includes the deformations of the liquid-vapour and solid-liquid areas. On the formal level, it is equivalent to eq.12, for the preceding model Hypothetical state: a Cassie-Baxter-like As previously, we can define a particular state of the model for which the ratio is known. Simplest is to consider a whole of situations for which 1. Among those, there is a particular case for which ρ =1 and σ =1. In this state, it is simultaneously supposed that the solid-liquid interface is perfectly smooth and that the liquid-vapour interface has a hemispherical shape. This state has strictly the same geometrical characteristics as those adopted to demonstrate the Young s relation

8 The only difference is that this time, the strict condition (ΔP = 0 is favoured to the 2 Laplace condition ( ΔP. r The considered state is obviously hypothetical. Thus, for the same contact angle, θ M, the difference of the solid-vapour and solidliquid interfacial tensions is defined as SV 2 (19 1 cos M This relation has a formal significance similar to the Young-Laplace relation. It does not have the role to replace it. This relation is not opposed to Young-Laplace relation. It cannot claim to describe a real behaviour. Consequently, eq. 19 can be used as origin convention to spot the values of contact angles. The following relation is then shown cos cos M 1 (20 whose form is like Cassie-Baxter relation. To avoid any ambiguity, by convention we will name it Cassie-Baxter like relation. Consequently, the form of Cassie-Baxter relation can be found from a behaviour model of the sessile drop referred to that of the isobaric drop (ISDM whose hypothetical state is characterized by eq.19. The demonstration of eq. 20 which presents a form similar to that of Cassie-Baxter does not require supposing air trapped in anfractuosities of the surface. It is a direct consequence of ISDM model. Its main interest is to reveal the same parameter ρ as that met in Wenzel relation. It is then easy to compare the variations of contact angles with ρ according to the two models. Thus, figure 3 is an illustration of the effect of a change in values of ρ on the contact angle by taking the particular case of the drops for which the liquid-vapour interface is supposed to be a spherical cap (σ = 1, but where ΔP = 0. The values of θ M are identical to those adopted for figure 2. For comparison, the behaviours of Wenzel are added on the same graph. M M = 105 M =

9 Figure 3: Variation of contact angles for a solid-liquid system with the value of ρ according to eq. 20 (with σ = 1. Values of angles θ M are 75 and 105. Behaviours of Wenzel are depicted in dashed lines. Contrary to the Wenzel s approach, the values of contact angles increase with ρ, even for initially wetting situations. For non wetting situations, the variations of contact angles with ρ are less important than for the Laplace model. In the described case on figure 3, for the same value of ρ = 4, the application of Wenzel relation supposes a perfectly non-wetting behaviour whereas this model calculates an angle of 140. This shows that, for Cassie Baxter-like behaviour, the perfectly non-wetting state is reached only in an asymptotic way with the variable ρ. This thought can be widened by considering a deformation of the liquid-vapour interface. We reported on figure 4, the case where σ =1.6 with the same data as those adopted for figure M M = 105 M = Figure 4: variation of contact angles for a liquid-solid system against the value of ρ according to eq. 20 taking σ = 1.6. Values of angles θ M are 75 and 105. Wenzel behaviours are depicted by the dashed lines. The consideration of the deformation of the liquid-vapour interface results in a decrease in the contact angle values without modifying the direction of the evolution. It is possible in this case to obtain values of contact angles lower than θ M. The hypothetical state (σ = 1, ρ = 1 then does not belong to the model. However, the condition θ = θ M for σ/ρ = 1 is recovered. 3. Discussion This study shows that for lack of being able to directly measure the pressure difference between the drop and its environment, the interpretation of the experimental data on the liquid drops put on solid substrates, was generally carried out, since nearly 200 years, through the distorting prism of the model of Laplace. The consequence is that the latter is intuitively accepted like having to describe the behaviour of real systems without always dissociating the thermodynamics from the 2-137

10 modelling. What poses a number of problems of consistency. Let us take for example the Young-Laplace relation. It comes in fact from three successive operations: i establishment of a general equilibrium condition of the drop (eq. 5, ii the conventional introduction of the Laplace s rule (eq. 13 and, iii the creation of a hypothetical state implying simultaneously two incompatible conditions on the pressure. It is consequently normal that the Young equation presented as an equilibrium condition of the sessile drop for a real system causes a number of discussions and reserves. A very important point is to notice that the use of this law gives access to the SV difference in interfacial tensions,, but nothing proves that the latter is SV identified with the difference in the thermodynamic variables,. To do this, it would have to prove experimentally that the pressure difference ΔP is quite equal to 2, that is not only a convention, and, that ρ =1, which is not easily possible. r Especially since, for many systems, we have the experimental certainty that it cannot be so, mainly because interface I does not have the shape of a spherical cap. It is noticed on many pictures of drops put onto structured solids [21, 22], that the shape of the interface is not regular any more, that it cannot be any more likened to a spherical cap, and sometimes the drop is anisotropic [23]. In these conditions, it is unlikely that the behaviour of the real system can be described by the Laplace model and that the use of Young-Laplace or Wenzel relations could make sense, or by that of Cassie-Baxter by retaining the Laplace s assumption, but supposing that a part of the drop is in contact with air trapped in anfractuosities of the substrate. The consequence of this is that interpretations or SV developments, concerning the values of the interfacial tensions and deduced from the Young-Laplace relation are based on a core data whose thermodynamic significance is not guaranteed. This is the case of estimation of solid surface energies [24, 25] or of studies of Neumann et al. who proposed relations between the surface SV tensions ( f (, leading to the concept of equation of state [26]. This does not by any means question the descriptive and predictive qualities of these studies, but raises the question of the thermodynamic validity of the data that they considered. The first conclusion of this discussion is that nothing proves that the pressure difference between a drop and its environment must follow one or the other of the conventions examined in this work. The internal pressure of the drop is not a controllable variable of the system. Its value is adjusted with nature of the liquid and the solid and with the topology of the interfaces. More generally, one can wonder about the need for introducing models into the analysis of the sessile drops behaviours knowing that the attribution of the laws of the model to the real system is an operation more than chancy. It is not because a contact angle characteristic of a wetting situation (θ < 90 decreases when a roughness appears on a solid substrate that one can affirm that ΔP is strictly described by the Laplace rule, or, if it increases that the pressure difference is strictly null (model ISDM. One can just conclude that the behaviour of the real system can be described by one or/and another model, by adjusting the parameters σ and ρ. The establishment of model behaviours can however be fruitful in the field of the phenomena understanding because it can generate new lines of thinking, which can be 2-138

11 illustrated by considering Wenzel and Cassie-Baxter relations. The demonstrations of these relations were the subject of lot of controversy until very recently [27-35]. We show that both can be regarded as the result of conventional rules adopted on ΔP. Consequently, in the same way it admits that the Wenzel behaviour results from the Laplace model, one can conceive that Cassie-Baxter behaviour ensues from the ISDM. Thus, the transition from Wenzel regime to Cassie-Baxter regime can be simply explained by a decrease of the internal pressure of the drop. This allows suggesting a physical origin to the observed transitions for the solids whose surface morphology is modified in a continuous and controlled way [36-41] or whose the external conditions are changed [42]. One can suppose that the structuring of the solid surface and the resulting deformation of the liquid-vapour interface have as a consequence a decrease of the internal pressure of the drop. This is illustrated on figure 4 by the evolution of the angle θ M = 75. For a wetting situation, a Wenzel regime is followed for values of ρ close to the unit with a decrease in the value of θ, then when the internal pressure of the drop is equal to zero, a Cassie-Baxter regime is obtained for the strong values of ρ, with an abrupt increase in the value of θ followed by a very slow evolution of the value of the contact angle with the structuring. If we stick to this approach, it is not necessary to accord a preponderant role to the surrounding gas atmosphere, although several authors [41, 43] showed that the transition from the mode of Cassie-Baxter regime to Wenzel mode was accompanied by the appearance of gas bubbles in the drop. On this point, however, it is important to differentiate the behaviour from the drop put onto structured solids (eq. 20, from the grids effects which correspond to the behaviour of liquids put onto substrates which objectively show that the drop is simultaneously on solid surfaces and on spaces filled of air. It is the case, for example, of pigeon s feathers [44]. In such case, it is certain that the presence of a heterogeneous substrate must be regarded through a suitable behaviour model. But, in our opinion, the systematic generalization of the grid effect to microstructured solids seems to us completely daring. 4. Conclusion The relations usually suggested to describe the behaviour of the drops put on substrates at equilibrium generally do not differentiate formally what is due to the equilibrium condition, and what concerns assumptions and conventions. In our opinion, that is what maintains confusion on the subject. To illustrate this remark, we presented two behaviour models, one of which (ISDM had never been presented. We could imagine other models by giving various formulations to the pressure difference (ΔP between the interior of the drop and its environment. But would that further the phenomenon understanding? We are not certain. Since one does not measure, or does not control, the value of ΔP involved in the equilibrium relation, this one cannot be exploited and one cannot reach the differences of the thermodynamic interfacial tensions. The resort to behaviour models is obviously interesting because it makes it possible to have various examples of evolution of the contact angle with the external constraints imposed on the system such as for example the interfaces deformation. However, the problem then remains to explain the evolution of the real system with the lighting of the model system. If identification is made, it must be proven. In our opinion, that is not the case today and the problem remains whole. This is all the more true as the models which we presented are very simple even simplistic. We 2-139

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