Thermodynamic equilibrium condition and model behaviours of
sessile drop
Pierre Letellier1,2, Mireille Turmine1,2*
1- CNRS, UPR15, Laboratoire Interfaces et Systèmes Electrochimiques, F75005, Paris, France
2- UPMC, Université Paris6, UPR 15, Laboratoire Interfaces et Systèmes
Electrochimiques, F-75005, Paris, France
* e-mail: [email protected]
Phone number: +33 1 44277216
Fax number:
+33 1 44274074
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 CassieBaxter's relation, with whom is associated a hypothetical state, characterized by the
2
  ηSV   ηSL . A general discussion is proposed on the meaning
relation  LV
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
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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, ALV, and the area of the solid-liquid interface, ASL, are
expressed according to r and θ as follow,
π
V  r 3 (1  cos ) 2 (2  cos )
3
(1)
LV
A  2πr 2 (1  cos )
ASL  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 ILV, ISL and ISV
whose areas are LV , SL et SV , respectively, without specifying their possible
relationships with the geometric area ALV, ASL and ASV. It is agreed that the pressure in
the drop, Pd, 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.
P
dLV
vapor
liquid

dV
Pd
dSL
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 dΛLV, dΛSL and dΛSV.
The equilibrium condition of the drop is written as
0   Pd dV  PdV   LV d LV   SL d SL   SV d SV
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(2)
 LV ,  SL and  SV are the characteristic tensions associated to the areas Λ of the
interfaces ILV, ISL and ISV respectively.
Consider the case of a solid having a homogeneous surface which is not necessary
plane. This implies that d SL  d SV .
The result is that the thermodynamic equilibrium condition of the sessile drop is
written
LV
SL
LV d
SL
SV d
(3)
Pd  P  
 (   )
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 λ, the areas of the interfaces will be multiplied by λ2/3. The
areas are Euler's functions of order 2/3 of the volume. Properties of Euler's functions
2 LV
d LV
 V
3
dV
(4)
2 SL
d SL
 V
3
dV
applied to eq. 3, allow establishing
SL

2  LV LV
SL
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
SL

2  LV LV LV
SL
SV 
 
ΔP  Pd  P 
A

(



)
ASL 
(6)
LV
SL
3V 
A
A

The fact of supposing that ΛSL and ASL are Euler’s functions of the same order of the
volume allows assuming that these extents are proportional. The parameters

LV
LV
 1 and  
SL
SL
 1 are defined. ALV and ASL 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 LV
ΔP 
  A LV  ( SL   SV )  ASL
(7)
3V
which can be simplified by expanding the geometrical magnitudes of the
2
(8)
2  LV    SL   SV 1  cos 
system ΔP 
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
the interfaces (known  and ) in order to determine the values of (  SL   SV ) 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.
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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, Pi * . 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, Pi. Supposing that the
molar volume of i, Vi * , 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

ln * 
2  LV   ( SL   SV )(1  cos )  Pi  Pi* 
(9)
RT  r (1  cos )( 2  cos )
Pi

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 Pi and Pi * would allow reaching the
value of the difference (γSL - γSV), 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 Pi and Pi * are
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]

Pi Vi*  2  LV

ln * 
 Pi  Pi * 
(10)
RT  r
Pi



For these data, we calculate a difference about 4 10-5 Pa at 298 K, taking Pi *  31 Pa.
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].
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
(  xSL   xSV ) is then defined. The latter becomes also conventional.
 ΔP( ) r (1  cos  )( 2  cos  )

1

 xSL   xSV 
 2  LV 
(11)
 (1  cos  ) 
2

It can be different from the difference between thermodynamic extents (  SL   SV ).
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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, Pd, and the pressure of its surroundings, P, is given by the
2 LV
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, ΛLV is
identified with ALV and  = 1. But, the shape of the interface ISL is not specified. We
write:
2 LV
2

2 LV   ( λSL   λSV )(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 ISL and ISV,  λSL and  λSV .
These magnitudes are those which verify eq.12. They could be different from the
thermodynamic extents of tension γSL and γSV. We shall go back over this important
point in the discussion.
This relation is simplified as
 LV cos  ( λSV   λSL ) 
(13)


It links the values of the contact angles to the geometry of the solid-liquid interface.
2.1.1 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.
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The simplest idea is to suppose that there is a state of the solid surface for which  =
1, what means that SL = ASL. The contact angle is M. The Young Laplace relation is
then established
 LV cos  M   λSV   λSL
(14)
It is important to notice that the conditions introduced to define this state are
incompatible between them. Indeed, the condition SL = ASL imposes that the solid
surface, supposed to be at equilibrium, is perfectly smooth (because ASL 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    cos  M
(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
180
M = 105°
160
140
120
100
80
60
40
M = 75°
20

0
1
2
3
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4
5
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 θM are 75° and 105°.
For the two chosen examples (θM = 75° and θM = 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.
2.2. 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
0   LV d LV   ηSL d SL   ηSV d SV
(16)
The interfacial tension,  ηSL and  ηSV , introduced in this relation are different from
those of the previous model  λSL and  λSV , and can be also different from those
involve in eq.5,  SL and  SV .
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
0   LV ALV  ( ηSL   ηSV )  ASL
(17)
Expanding the expressions of the geometrical areas we can write
2

 LV
 ( ηSV   ηSL )
(18)
1  cos 

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.
2.2.1. Hypothetical state: a Cassie-Baxter-like
As previously, we can define a particular state of the model for which the ratio 
known. Simplest is to consider a whole of situations for which 
 is
  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.
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The only difference is that this time, the strict condition (ΔP = 0) is favoured to the
2 LV
Laplace condition ( ΔP 
).
r
The considered state is obviously hypothetical.
Thus, for the same contact angle, θM, the difference of the solid-vapour and solidliquid interfacial tensions is defined as
2 LV
 ηSV   ηSL 
(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 θM are identical to those adopted for figure 2. For
comparison, the behaviours of Wenzel are added on the same graph.
M
180
M = 105°
160
140
M = 75°
120
100
80
60
40
20

0
1
2
3
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4
5
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 θM 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 3.
M
180
160
M = 105°
140
120
M = 75°
100
80
60
40
20

0
1
2
3
4
5
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 θM 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 θM. The
hypothetical state (σ = 1, ρ = 1) then does not belong to the model. However, the
condition θ = θM 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
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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
difference in interfacial tensions,  λSV   λLV , but nothing proves that the latter is
identified with the difference in the thermodynamic variables,  SV   LV . To do this,
it would have to prove experimentally that the pressure difference ΔP is quite equal to
2 LV
, 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 ILV 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
developments, concerning the values of the interfacial tensions  λSV and  λLV 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
tensions (  SL  f ( λSV ,  λLV ) ) 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
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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 θM =
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
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supposed to carry out our calculation that areas ΛLV and ΛSL of the liquid-vapour and
solid-liquid interfaces have values different from the geometrical areas ALV and ASL,
but they are also geometrical extents of dimension 2. Nothing imposes it. This
assumption of calculation permitted to easily integrate the equilibrium condition
which was in a differential form, by using the Euler's relations. On rough and
structured interfaces, the real borders of the system cannot generally be located very
precisely. For this reason we had introduced into a former work the concept of “fuzzy
interfaces” [45] and that we proposed to examine the case of the equilibrium of a
sessile drop from the relations of the nonextensive thermodynamics [46]. The
introduction of behaviour models of sessile drop which obey this concept is obviously
possible. We had treated that which admitted that ΔP was fixed by the Laplace’s rule.
Symmetrically, we could develop an isobar model of drop.
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