ABSTRACT
The properties of a contact between the particles of a solid phase depend on a combination of the properties of that solid phase, the media in the gap between the particles, and the interactions between the solid phase and the medium� To a great extent, these interactions are due to intermolecular forces acting at the interface between phases� Since surface forces are one of the main subjects discussed in modern books on colloid and surface science, we will restrict ourselves only to a brief review of the subject relevant to physical-chemical mechanics�
Before turning to a discussion on particular interactions, it is worth reviewing the basic principles of the thermodynamics of surface forces and the concept of surface (or interfacial) free energy and of surface tension in particular�
On an intuitive level, the concept of surface tension becomes apparent in the description of Dupré’s original experiment, in which one observes the stretching of a soap film, either in a bubble or on a frame (Figure 1�1)� Equilibrium is established when a force, F = 2dσ, is applied to the moving boundary, where d is the frame width, σ is the surface tension, and the numerical coefficient 2 implies that the film has two sides�
Displacing the boundary by Δl increases the film area by 2dΔl and requires that work equal to F Δl is expended (Figure 1�1)� Per unit of the newly formed surface, this work is F Δl/2dΔl = F/2d = σ� Consequently, the meaning of σ now is specific surface free energy (units of work per unit surface area)� In a thermodynamic sense, the term “free” means that this (equilibrium) process takes place at constant temperature and volume� The identity of those two notions, that is, of the surface tension and the surface free energy, is true only for “common” (unassociated) liquids� For solids, the situation is more complex [1-3]�
A rigorous definition of the specific surface free energy of a body as the excess of free energy at an interface between the condensed phase and air (or vacuum), as well as the definition of specific interfacial free energy at the interface between two condensed phases, can be given in terms of Gibbs’s thermodynamics of surface phenomena [2,4]� Often the terms “surface free energy” and “interfacial free energy” are used interchangeably, but, as the discussion that follows will show, it is best to separate them�
As shown in Figure 1�2, let us choose a prism with a unit cross section of S = 1, along a direction 1 perpendicular to the interface between phase 1 and phase 2, for example, perpendicular to the interface between a liquid (1) and its vapor (2)� In a region adjacent to the geometrical interface between those two phases (the so-called dividing surface), we can select a transition layer of finite thickness (physical discontinuity surface), with properties differing from those of both phases 1 and 2� The boundaries of this layer are positioned at distances δ′ and δ″ from the dividing surface, and consequently, the effective thickness of the discontinuity surface is δ = δ′ + δ″� Let the density of the surface free energy f (units of work per unit volume) outside the discontinuity surface be equal to f′ within phase (1) and f″ within phase (2)� The distribution of the surface free energy density along the z-axis is illustrated by a plot of f(z) in Figure 1�2 within the physical discontinuity surface f(z) and is different from both f′ and f″� In principle, the transition from f′ to f″ may be smooth, but it is more
commonly “tonguelike,” as seen in Figure 1�2� To characterize this “tongue” quantitatively as the free energy density excess over f′ and f″, one needs to extrapolate f′ and f″ to the dividing surface� According to Gibbs, this excess defines the value of the specific (per unit area) interfacial free energy, or surface free energy, in the case of the interface between a vapor and a vacuum:
s - - -d
= éë ùû + éë ùû ¢
f z f dz f z f dz( ) ( ) (1�1)
Typically, when far away from the critical conditions, δ is on the order of interatomic (intermolecular) distances, b, that is, on the order of angstroms (1 Å = 10-10 m)� As the system nears its critical point corresponding to the complete miscibility or complete mutual solubility of the constituent phases, the value of δ may reach hundreds or thousands of angstroms�
Gibbs’s approach to the determination of σ is valid for any interface: liquid/gas (L/G), liquid/ liquid (L1/L2), solid/gas (S/G), solid/liquid (S/L), or solid/solid (S1/S2)� The methods are used to either measure or indirectly estimate the interfacial free energy, which may vary significantly depending on the type of interface [3,5]�
Typical measured values for the surface and interfacial tension, σ, for liquid phases (in mN/m = dyn/cm = mJ/m2) are as follows [6-11]:
It would be worthwhile to immediately turn one’s attention to a discussion of the molecular origin of the surface free energy and the surface tension, and the principal difference of the latter from the state of a rigid solid body deformed by stretching�
To do this, let us follow Laplace and consider a spherical gas holder shell (i�e�, a balloon filled with gas) with a radius R and an excessive pressure, P, inside (Figure 1�3)� Let us position a cone of small volume with a small angle at the apex, 2ϕ, in such a way that the cone’s apex matches the center of the sphere and the base matches the surface of the shell� The length of the base is a = Rϕ, and the length of the base perimeter is 2πa = 2πrϕ, while the area is πa2 = πR2ϕ2� The pressure force exerted on this surface is F = PR2ϕ2�If the stretching elastic stresses acting in the shell are equal to σelast [N/m], then the force F is balanced by the component of a stretching stress acting in the direction of the radius R (axis of the cone), namely, σelastϕ, acting along the perimeter 2πrϕ� This force is given by F = 2πRϕ σelastϕ� The force balance relationship, F = PπR2ϕ2 = 2πRϕσelastϕ, yields the Laplace equation for a sphere
P
R = 2s
elast (1�2)
The aforementioned equation constitutes Laplace’s law for a sphere� For a cylinder, the expression is similar, except that it does not have a numerical coefficient 2�
Instead of considering a balance of forces, one may also derive Laplace’s law by using the variation of work approach� Let Δp be the excessive pressure as compared to the surrounding atmospheric pressure inside a bubble with radius r and a surface tension of σ� The change in the bubble radius in the vicinity of the equilibrium by δr is associated with the work of expansion in the system
W p V p r r p rexp = =
æ è ç
ö ø ÷ =D d D d p p D d
4 3
43 2 (1�3)
As a result, the free energy of the system is lowered by the amount of this work� At the same time, an increase in bubble size results in an increase in the surface area associated with an increase in the surface free energy, σδ(4πr2) = 8 πrσr� The overall free energy variation around the equilibrium is zero
d - p D d p sdW r p r r r= + =4 8 0 2
which readily yields the Laplace equation
Dp p
r = =s
s2 (1�4)
This “continuum” approach does not clearly reveal a molecular-level difference between an elastic deformation of a solid shell and the nonelastic behavior of a liquid surface� The essence of this difference is as follows: Stretching a solid body (within the limits of ideal elasticity) causes the stretching of all interatomic (intermolecular) bonds within the volume of the body in the direction of the applied stretching force� This process is reversible-both thermodynamically and mechanically� As a result of such stretching, an increase in dimensions (length, volume, surface area) takes place� This increase is not large, as determined by a high value of Young’s modulus, and the increase in surface area is stipulated strictly by the change in the distance between the surface atoms with their relative position remaining unchanged� This scenario is completely different from what happens in the course of the deformation of a liquid at moderate pressures, such as that caused by viscous flow� In that case, a change in the surface area neither results in a change in the intermolecular distances nor does it cause any change in volume� Instead, it is caused by molecules entering the surface from the bulk of the liquid phase� This process is associated with the restructuring of the intermolecular bonds� These two scenarios for the increase in surface area taking place in a solid body and a liquid are illustrated in Figure 1�4�
The following simple example illustrates quantitative relationships within a dense pack of spherical particles, for example, atoms� In a metal, an atom in the bulk is surrounded by 12 neighbors, while an atom within the surface layer is surrounded by only 9 neighbors� This deficiency of three interatomic bonds results in ~25% lower cohesive energy at the surface as compared to that in the bulk� The energy of cohesion in the bulk of a body is negative, because it is counted from the zero reference point when mutually attracting particles cluster together from infinity to form a solid body� The lack of any bulk energy of cohesion per unit area is exactly the positive excess associated with the surface layer, referred to as the specific surface free energy, σ�
This scheme corresponds to the well-known Stefan’s approach to the estimation of the surface energy of solid phases� The density (energy per volume) of the energy of cohesion, Gc, in the bulk of a condensed phase may be approximately estimated as the heat of sublimation per molar volume, Hsubl/Vm, or as the Young’s modulus, E, or as the so-called molecular pressure, K (see [12-14] for details)�
Within the volume of a discontinuity surface of thickness δ, which is on the order of the interatomic distance b, the lack of cohesive energy is qbGc, where q is the ratio of the so-called boundary coordination number, zb, to the bulk coordination number, z, within the bulk of a phase� The boundary coordination number zb is characterized as the number of “missing bonds” (or missing neighbors) between surface atoms� For instance, in the case of face-centered close packing, z = 12, and the value of zb for the atoms in the (111) plane is 3, with q = 1/4 (as above)� In this way, we have introduced a possibility for getting an estimate of the free surface energy of the condensed phases, σ, as qbGc� Such an approximate estimate allows a comparison with other approaches and clearly illustrates the general conclusion, namely, that for solids having high strength and high melting points, such as transition metals, oxides, carbides, and nitrides, the values of σ are the highest, (~2-3 × 103 mJ/m2), while organic liquids have low σ (tens of mJ/m2), and the values of σ for liquefied gases are even lower�
When one compares the thermodynamic description of a surface with that of a rigid shell, the following principal difference between these two cases needs to be revealed� In the case of a rigid body, the entire potential energy of elastic deformation is determined by the work performed on the body� In the case of a liquid surface, stretching is also associated with the transfer of heat� The total surface energy (“internal surface energy”), ε, at a given temperature T includes, in addition to the mechanical work, σ, also hidden heat, ε = σ + ηT, where η is specific (per unit area) surface entropy, which can be established from the temperature dependence of the surface tension, η = – dσ/dT (Figure 1�5)� For a free surface, the surface entropy is positive and, for most liquid phases, is around 0�1 mJ/m2 K�
Before we start a discussion on the thermodynamics of a contact between particles, it is worthwhile to briefly address the phenomena taking place at the three-phase contact line, and in particular, wetting and capillary forces acting within a liquid meniscus� We will also briefly summarize the principal methods of surface tension measurement�
Following Young, let us examine the contact between a solid phase and a liquid phase in ambient air, that is, a drop of liquid sitting on a solid substrate, as shown in a 2D representation in Figure 1�6� The three interfaces share a common perimeter, which is a linear boundary perpendicular to the plane of the drawing� The drop is at equilibrium when a balance is achieved between the three surface tension forces acting along the perimeter, as depicted by the vectors in Figure 1�6� The equilibrium between those three forces requires that the algebraic sum of the projections of the three vectors is equal to 0, which readily yields Young’s equation
s s q sSL LG SGcos+ =
where the contact angle, θ, is measured inside the liquid� Depending on whether θ < 90º or θ > 90º, one talks about wetting or nonwetting, respectively (Figure 1�6)�
Young’s equation can also be obtained by a variation approach, that is, by considering variations in the free energy of the system: the algebraic sum of the small variations in the three components of the interfacial energy should equal to zero when we encounter small deviations from the equilibrium� This approach is schematically summarized in Figure 1�7� By shifting the three-phase contact point A by a small distance AA′, we introduce small variations in the three components of the free interfacial energy, the sum of which is zero
d d s d s d s s s q sF S S S AA AA AAs SL SL LG LG SG SG SL LG SG= + + = + - = ¢ ¢ ¢cos 0
which readily yields the Laplace law�
We have addressed the situation of a rigid solid surface when a single contact angle is formed� In a contact involving two liquid phases and a gas phase (e�g�, a drop of fat on water surface), two contact angles are formed� For each of these angles, the equations describing the equilibrium must take into account the individual horizontal and vertical components� The same argument can also be applied to the solid-liquid-gas interface between a gas, a liquid, and a yielding material, such as a polymer with a low Young’s modulus�
It is also worth emphasizing that the interatomic bonds are not fully compensated at the threephase contact line� This results in a free energy excess and a linear tension, æ, acting along the perimeter of a three-phase boundary� This linear tension can be either positive or negative and does not exceed 10-4 dyn/cm� While the linear tension can in most cases be neglected, it plays an essential role in the case of very small droplets, particularly in nucleation�
When a three-phase contact line is formed by a solid phase and two liquid phases, a selective wetting of the solid phase by one of the liquids takes place� Usually, there is competition between the polar phase (e�g�, water) and the nonpolar phase (e�g�, hydrocarbon or “oil”) in the wetting of the polar and nonpolar solid surfaces� By convention, in selective wetting, the contact angle, θ, is measured into the more polar phase� The solid surface is referred to as hydrophilic (“oleophobic”) when it is predominantly wet by water (θ < 90º), and hydrophobic (“oleophilic”) when it is predominantly wet by a nonpolar liquid (θ > 90º), as illustrated in Figure 1�8�
The concept of oleophobicity is applicable only to the case of selective wetting by two liquids: in air, hydrocarbons exhibit good wetting toward nearly all solid materials� The terms “surface hydrophilicity” and “surface oleophilicity” were introduced by Freundlich to indicate the physical-chemical similarity between the solid and liquid phases or more precisely the affinity of a solid phase toward a liquid, whether oil or water� Both of these terms go back to a common notion of “lyophilicity,” which in literal translation from the Greek means “liking to dissolve�” In the opposite case, one would use the term “lyophobicity” to stress dissimilarity between the phases (the word itself literally means “fearing to dissolve”)�
Selective wetting (or nonwetting) is governed by the difference in the values of σSL1 and σSL2� Here, it would be worthwhile addressing the general concept of interfacial free energy, σ12, associated with the interface between any two condensed phases, including the contact between two solids�
Following Dupré’s approach, one can consider a column having a unit cross-sectional area consisting of two phases in contact with each other, phases 1 and 2 (Figure 1�9)�
Separating the phases requires that the adhesion at the interface is overcome, that is, that the work of adhesion, Wa, is consumed� As a result of the separation of phases 1 and 2, the interface with energy σ12 vanishes, and two new surfaces with surface free energies σ1 and σ2, respectively, are formed� The energy balance of this process is given by Dupré’s equation and can be written as follows:
s s s12 1 2+ = +Wa (1�5)
This rule for the case of contact between a solid and a liquid phase can be written as
s s sSL a SG LG+ = +W (1�6)
By combining Young’s equation with Dupré’s equation, one gets
Wa SG LGcos-s q s=
or
W Wa SG LG wcos= +( ) = +s q s1 (1�7)
where the work of wetting, Ww, is given by
Ww SG SG SLcos= =s q s - s (1�8)
Dupré’s scheme may be complicated by the possibility of the adsorption of liquid phase molecules on a bare solid substrate� Such adsorption can be neglected for hydrophobic solid surfaces�
Young’s and Dupré’s equations enable one to compare the values of σSL, σLG,σSG,Wa, and Ww and thus to assess the degree of physical-chemical similarity (“philicity”) of the phases in contact� The larger the similarity between the phases (e�g�, between the surface of the ionic crystal and water), the higher the value of Wa and the lower the value of σ12 = σSL� In the limiting case of completely identical phases (corresponding to continuous substance, either solid or liquid), σ12 = 0, and Wa = 2σ1 = 2σ2 = Wc1 = Wc2, where Wc is the work of cohesion inside a given condensed phase� For a liquid, Wc(L) = 2σLG� The work of cohesion corresponds to the work necessary for the formation of two identical surfaces having a unit area in a thermodynamic equilibrium process� In the opposite case of two “very dissimilar” phases in contact, such as a drop of mercury on paraffin, where the contact angle reaches 150º, the interfacial energy, σ12, is high, and the work of adhesion is low�
The ratio (σSG – σSL)/σLG may serve as a dimensionless characteristic of cohesion between phases� When the value of this ratio ranges between –1 and +1, (σSG – σSL)/σLG = cos θ, that is, it corresponds to the measured contact angle� This ratio may not assume values of less than –1, while (σSG – σSL)/ σLG > 1 yields σSG – σSL > σLG, which corresponds to a complete spreading of a liquid phase over the solid substrate, that is, an equilibrium contact angle is not established under these conditions� The work of spreading, Wsp, can be written as Wspr = σSG – σSL – σLG� For the work of spreading, one can also write that Wspr = Wa(SL) – Wc(L)� This expression is also true for the case of the spreading of one liquid over the surface of another liquid� The condition when Wa(SL) = ½ Wc(L) corresponds to the situation when θ = 90°� We will use these equations further during the discussion of the molecular dynamics of wetting�
Of special interest is the case of contact between solid and liquid metals of similar chemical nature, such as zinc and mercury and aluminum and gallium� In these cases, the work of adhesion between the solid and liquid phases remains high, but the interfacial tension, σSL, may be very low, reaching about 10% of σSG� Such lowering in the interfacial tension is the reason for liquid-metal embrittlement, which is discussed in detail in Chapter 7�
The wetting of solid surfaces by liquids is greatly influenced by the state of the solid surface and particularly by its microgeometry, that is, surface roughness� The topography of a surface can be approximated with a series of microgrooves of depth H and width d; H = (d/2) tan χ, where χ is the angle between the idealized flat surface and a side wall of the groove (Figure 1�10)�
If the surface is rough, the real surface area, Sreal, is greater than that of an idealized surface, Sideal� The ratio of the real surface area to the area of its projection onto an idealized flat surface is referred to as the coefficient of roughness, kr:
k S
S d
deal = = =
/cos cos
c c
Roughness results in an increase in the true surface area of the solid, which in turn increases the input from the solid-liquid and solid-gas interfaces into the energy of wetting� According to Derjaguin, the expression for the work of adhesion in the case of contact between a liquid and a real solid surface should be written as
W ka r SG SL LG= +( )s - s s
and the averaged (the “effective”) value of the cosine of the contact angle is
cos
( ) cos
cos
cos q s - s
s s - s s c
q cef
LG = = =k
The aforementioned equation shows that surface roughness improves the wetting of a solid surface by a liquid (the value of θef decreases) but makes nonwetting worse (the value of θef increases)� The condition where χ = θ is sufficient for wetting to turn into spreading� This effect is used in such processes as soldering and gluing: prior to applying glue or solder, the solid surfaces are treated with sand paper, which in addition to removing the impurities makes the surfaces rough�
A very important application of nonwetting is the attachment of rock particles to air bubbles in flotation� This is schematically shown in Figure 1�11 and will be discussed in more detail in Chapter 2�
Finally, let us briefly address some of the principal methods that are used to measure surface tension (free surface energy) in liquids� Nearly all of these methods rely on the Laplace equation:
P p
r = =s
s2 (1�9)
which is the main equation for capillarity, describing the capillary pressure over curved surfaces [3,5,7]�
In this method, the rise of a liquid in a capillary of radius r is examined (Figure 1�12)� The capillary rise force, Fσ= πr2pσ= πr2 2σ/r = 2πrσ, acting from the side of a concave hemispherical wetting (θ = 0) meniscus with a radius equal to the radius of the capillary, is compared to the weight of a column of liquid with height H, Fg = πr2Hρg, where g is the acceleration of gravity and ρ is the density of the liquid (corrected for the density of the ambient air)� Fluid rises in the capillary until equilibrium is established, so that Fσ= Fg, which readily yields σ
s r= 1
2 r Hg (1�10)
In the case of incomplete wetting, characterized by a nonzero value of the contact angle θ, the earlier expression is written as
s r
q = r Hg
2cos (1�11)
This precise and universal method utilizes the deformation of a relatively large drop of a liquid resting on a solid substrate under gravity, as shown in Figure 1�13� As a result of deformation,
the droplet’s surface has a nonspherical shape, and the Laplace pressure at any point on the surface is determined by main radii of curvature, r1 and r2
p
r r s s= +
æ
è ç
ö
ø ÷
1 1 1 2
(1�12)
The numerical integration of the Laplace equation enables one to estimate the value of σ at a given liquid density as a function of the droplet’s maximum diameter, dmax, and the distance from the plane of maximum diameter to the top of the droplet, H* (Figure 1�13), both of which can be directly measured� The results of a numerical integration of the Laplace equation are utilized in the software supplied with modern contact angle instrumentation and can also be found in published tables [15]� The sessile drop method can be used to determine the surface tension of melts of metals with high melting points: the shape of a molten metal droplet placed into an isolated high-temperature heater can be obtained from the x-ray image�
This method is used to determine the interfacial tension between two liquids, σL1/L2, and enables one to measure interfacial tensions as low as 10-3 mN/m� In the spinning drop method, a drop of a less dense fluid is placed into a tube containing a more dense fluid� Then, the tube is made to spin rapidly, and the centrifugal forces elongate the droplet of the less dense fluid into a cylinder along the axis of rotation (Figure 1�14)� The interfacial tension can then be determined from the geometry of the elongated drop, and in the case when the shape of the drop can be approximated by a cylinder, Vonnegut’s equation can be used
s w Dr=
4 r
(1�13)
where ω is the angular rotation speed Δρ is the density difference between the liquids r is the diameter of the elongated droplet
The spinning drop method is commonly used in petroleum-related applications for determining the interfacial tension at the crude oil/aqueous solution interface in the presence of different surfactants�
All of the methods described earlier can be classified as static, because all of the measurements are conducted under conditions of stable equilibrium� This condition corresponds to the minimum in potential energy, which in turn corresponds to the balance between the total surface energy and the energy in a field of gravitational or centrifugal force� There are also methods in which the measurements of surface tension are conducted under conditions of metastable equilibrium, corresponding to the maximum of the free energy of the system� Such methods can be classified as semistatic.
