Thermodynamics of Condensed Phases
2.2 THERMODYNAMICS OF CERAMICS AND GLASSES .1 Phase Equilibria in Ternary Component Systems
2.2.2 Interfacial Thermodynamics
Thermodynamics plays a fundamental role in practically all of the processes that are described in this book. Phase equilibria, which has been the focus of this chapter so far, is but one of these areas. Let us diverge from phase equilibria for a bit, and discuss a thermodynamic topic that will be of use in many of the subsequent chapters:inter- facial energy. Interfacial energies will be developed using thermodynamic arguments primarily for liquids, but we will see that the results are general to solids as well. This point will be illustrated by application of interfacial energies to an important process in the densification of commercial materials called sintering. Later on, we will see that it has enormous utility in the description of many surface-related phenomena in materials science, such as fracture, adhesion, lubrication, and reaction kinetics.
THERMODYNAMICS OF CERAMICS AND GLASSES 183
M
M M M
M M M
M M M M M
M M
M Melting point of oxide M Melting point of metal B Boiling point of metal
B B
C + O2 = CO2
temperature, °C
0
−100
−200
−300
−400
−500
−600
−700
−800
−900
−1000
−1100
−1200 0
0 K
200
pO2, (atm) 10−100 10−80 10−60 10−50 10−42 10−38 10−34 10−30 10−28 10−26 10−24 10−22 10−20 10−18 10−16 10−14 10−12 10−10 10−8 10−6 10−4 10−3 10−2 10−1 1
pCO/pCO2 pCO/pCO2 pH2/pH2O
1014
pH2/pH2O 1013
1012 1013 1012 1011 1010 109 108 107 106 105 104 103 102 10 1 10−1 10−2
1011 1010
109 108 107 106 105 104 103 102 10 1 10−1 10−2 10−3
400 600 800 1000 1200 1400 1600 H
C
∆G° = RT In pO2 (kJ)
10−4 10−3 10−5 10−6 10−7 10−8
10−5 10−6 10−7 10−8 10−9 10−10 10−12
10−14 10−4
2C + O 2= 2CO
2Mn + O2 = 2MnO 3/2Fe + O2 = 1/3Fe3
O4
Si + O2 = SiO2 Ti + O2 = TiO2
4/3Al + O2 = 2/3Al2 O3
2Ca + O2 = 2CaO 2Mg + O2 = 2MgO
6Fe + O2 = 2Fe3O4 2Co + O2 = 2CoO
2H2 + O2 = 2H2O
2Fe + O2 = 2FeO 2Ni + O2 = 2NiO
4Ag + O2 = 2Ag2O
2CO + O2 = 2CO2 4Cu + O2 = 2Cu2O
4/3Cr + O2 = 2/3Cr2 O3
Figure 2.26 Ellingham– Richardson diagram for some common metal oxides. Reprinted, by permission, from D. R. Gaskell, Introduction to the Thermodynamics of Materials, 3rd ed., p. 370. Copyright1973 by Taylor & Francis.
2.2.2.1 Surface Energy. Asurfaceis an inhomogeneous boundary region between two adjacent phases. As shown in Figure 2.27, atoms on the surface of a phase are necessarily different than those in the bulk. In particular, they have fewer nearest neighbors than the bulk, and they may be exposed to constituents from an adjacent phase. This generally means that less energy is required to remove an atom from a surface than to remove it from the bulk. Therefore, the potential energy of surface atoms is higher than bulk atoms. In turn, work is required to move atoms from the bulk to the surface. When this is done, new surface is created, and the surface area of the phase increases. The reversible work required to form the new surface,dWs, is
Figure 2.27 Schematic representation of surface and bulk atoms in a condensed phase. From W. D. Kingery, H. K. Bowen, and D. R. Uhlmann,Introduction to Ceramics. Copyright1976 by John Wiley & Sons, Inc. This material is used by permission of John Wiley & Sons, Inc.
proportional to the surface area,dA, that is created:
dWS =γ dA (2.59)
The proportionality constant, γ, is called thesurface energy. When the bulk phase in question is a liquid, the surface energy is often called surface tension. Rearranging Eq. (2.59) gives:
γ = dWS
dA (2.60)
We can see from this relation that surface energy is work per unit area, so it should have units of J/m2 in SI and ergs/cm2 in cgs. Often you will see values of surface energy expressed in units of dyne/cm, where 1 dyne=1 erg/cm.
With the aid of Eqs. (2.1), (2.7), (2.9), and (2.11), you should be able to prove to yourself that the reversible, non-pressure–volume work,dWs, is equivalent to the free energy change,dG, so that Eq. (2.60) becomes, with proper use of partial differentials,
γ = ∂G
∂A
T ,P ,Ni
(2.61) This relationship identifies the surface energy as the increment of the Gibbs free energy per unit change in area at constant temperature, pressure, and number of moles. The path-dependent variable dWs in Eq. (2.60) has been replaced by a state variable, namely, the Gibbs free energy. The energy interpretation of γ has been carried to the point where it has been identified with a specific thermodynamic function. As a result, many of the relationships that apply toG also apply toγ:
γ =Hs−T Ss (2.62)
THERMODYNAMICS OF CERAMICS AND GLASSES 185
where the subscript s on the enthalpy and entropy indicates that these are surface properties. As mentioned earlier, the surface atoms are fundamentally different than bulk atoms, so that they have different enthalpies and entropies associated with them.
Differentiation of Eq. (2.62) with respect to temperature at constant pressure gives ∂γ
∂T
P
= −Ss (2.63)
Substitution of Eq. (2.63) in (2.62) gives γ =Hs+T
∂γ
∂T
P
(2.64) Equation (2.64) is useful from an experimental standpoint because the measurements of surface energies at various temperatures can, in principal, provide a measurement of the surface enthalpy. The surface enthalpy, Hs, can also be determined directly, because it is equivalent to the heat of sublimation or vaporization.
Since the surface energy is a direct result of intermolecular forces, its value will depend on the type of bond and the structural arrangement of the atoms. Normally, densely packed planes would have lower surface energies. Liquid hydrocarbons having only weak van der Waals forces have values of surface tension in the neighborhood of 15–30 dyn/cm, while liquids with polar forces and hydrogen bonds, like water, have surface tensions in the range of 3–72 dyn/cm. Fused salts and glasses with additional ionic bonds have surface energies from 100 to 600 dyn/cm, and molten metals have surface tensions from 100 to 3000 dyn/cm. Even higher surface energies can be found in covalent solids (see Example Problem 2.3). See Appendix 4 for values of solid and liquid surface energies for a variety of materials.
2.2.2.2 The LaPlace Equation. The concept of surface energy allows us to describe a number of naturally occurring phenomena involving liquids and solids. One such sit- uation that plays an important role in the processing and application of both liquids and solids is the pressure difference that arises due to a curved surface, such as a bubble or spherical particle. For the most part, we have ignored pressure effects, but for the isolated surfaces under consideration here, we must take pressure into account.
Consider a generic curved surface such as that found in a sphere or cylinder (see Figure 2.28). The curved surface has two principal radii of curvature,R1 andR2. The front of this surface is indicated by the line xx1, and the back is represented by the line yy1. Let us now move the surface out by a differential element, dz. The front of the new surface is now given by x˙x˙1, and the back is indicated by y˙y˙1. The work required to displace the surface this amount is supplied by a pressure difference,P. The pressure acts on an area given by (xx1)(x1y1) moving through the differential element dz, such that the total pressure–volume work associated with extending the surface throughdz is
W =P (xx1)(x1y1) dz (2.65) The pressure–volume work must be counterbalanced by surface tension forces. The work required to move against surface tension forces is best calculated by breaking it
Example Problem 2.3
An alternative method for estimating surface energies is to calculate the work required to separate two surfaces of a crystal along a certain crystallographic plane. At 0 K, this work can be approximated as the energy required to break the number of bonds per unit area, or the energy of cohesion,Ecoh. Since two surfaces are being formed in this cleavage process, the surface energy of a single surface is then
γ ≈1/2Ecoh
Let us consider, for example, separation along the (111) plane in diamond. The lattice constant for diamond isa=3.56 ˚A (see Table 1.11), so the number of atoms per square centimeter on the (111) surface is
2 atoms per plane(111)
√3a2/2 = 4
√3(3.56×10−8)2
=1.82×1015atoms/cm2
From Appendix 1, the bond energy for C–C bonds is 348 kJ/mol (83.1 kcal/mol), so that the cohesive energy is
Ecoh=(1.82×1015bonds/cm2)(348,000 J/mol)(107erg/J) 6.02×1023atoms/mol
=10,500 erg/cm2 so that the surface energy is
γ =1/2(10,500 erg/cm2)=5250 erg/cm2
into two parts,W1 andW2.W1 is the work required to move side xx1 away fromyy1 a distance(x1y1/R1) dzduring the expansion:
W1= γ (xx1)(x1y1) dz R1
(2.66) Similarly, the work required to move side xy away from x1y1 through a distance (xx1/R2) dzisW2:
W2= γ (x1y1)(xx1) dz
R2 (2.67)
Adding Eqs. (2.66) and (2.67) together to arrive at the work against surface tension, equating them with the pressure–volume work in Eq. (2.65) and simplifying leads to theLaplace equation:
P =γ 1
R1 + 1 R2
(2.68)
THERMODYNAMICS OF CERAMICS AND GLASSES 187
x x dz
y y
y1 y1
x1 R1
R2
O O′
·
·
x·1
·
Figure 2.28 An element of a curved surface with principal radiiR1 and R2. Reprinted, by permission, from J. F. Padday, inSurface and Colloid Science, E. Matijevic, ed., Vol. 1, p. 79.
Copyright1969 by John Wiley & Sons, Inc.
The Laplace equation in this form is general and applies equally well to geometrical bodies whose radii of curvature are constant over the entire surface to more intri- cate shapes for which the Rs, are a function of surface position. In the instance of constant radii of curvature across the surface, Eq. (2.68) reduces for several common cases. For spherical surfaces,R1=R2=R, whereRis the radius of the sphere, and Eq. (2.68) becomes:
P = 2γ
R (2.69)
For a cylindrical surface, R1 (or R2) is infinity, so the remaining radius, R, is the radius of the cylinder and:
P = γ
R (2.70)
Finally, for a planar surface, both radii of curvature become infinity, and:
P =0 (2.71)
The pressure difference may also be numerically zero in the instance where the two principal radii of curvature lie on opposite sides of the surface, such as in the case of a saddle.
2.2.2.3 The Young Equation. The principle of balancing forces used in the deriva- tion of the Laplace equation can also be used to derive another important equation in surface thermodynamics, theYoung equation. Consider a liquid droplet in equilibrium
Liquid gLS
gL
gS q
Solid
Figure 2.29 Schematic diagram of liquid droplet on solid surface. From Z. Jastrzebski,The Nature and Properties of Engineering Materials, 2nd ed., Copyright1976 by John Wiley &
Sons, Inc. This material is used by permission of John Wiley & Sons, Inc.
with its own vapor and a flat, solid surface at constant temperature, as shown in Figure 2.29. The liquid–solid, liquid–vapor, and solid–vaporinterfacial surface ener- gies are defined asγLS,γLV, andγSV, respectively. Technically, the liquid–vapor and solid–vapor interfacial energies should be for the liquid and solid in equilibrium with their respective vapors, which are probably not the same. In practice, however, the vapor is usually a nonreactive gas, so γLV and γSV become the surface tension and solid surface energy in the gas of interest and are simply labeled γL andγS, respec- tively, as shown in Figure 2.29. These are both approximations. The surface tensions and surface energies of some common liquids and solids are listed in Appendix 4.
At equilibrium an angle θ, called the contact angle, is formed at the three-phase solid–liquid–gas junction. The contact angle can have values from zero to 180◦. When θ =0, the liquid completely spreads of the solid surface, forming a thin monolayer (see Figure 2.30). Such a condition is termed wetting. The other extreme is called nonwetting, and it occurs when the entire liquid droplet sits as a sphere on the solid.
All values of contact angle in between these two extremes are theoretically possible.
Obviously, the chemical nature of the liquid, solid, and vapor determines the extent to which the liquid will wet the solid, and temperature has an important influence.
Keep in mind that the contact angle under consideration here is anequilibrium contact angle. There are alsodynamic, receding, andadvancing contact angles associated with droplets as they spread and move on substrates. For now, the term contact angle will refer to the equilibrium condition.
The three interfacial surface energies, as shown at the three-phase junction in Figure 2.29, can be used to perform a simple force balance. The liquid–solid interfa- cial energy plus the component of the liquid–vapor interfacial energy that lies in the same direction must exactly balance the solid–vapor interfacial energy at equilibrium:
γLcosθ+γSL =γS (2.72)
This equation is calledYoung’s equation; it is named after Thomas Young, who first pro- posed it in 1805. The derivation presented here in terms of force balances is simplistic, but there are more rigorous thermodynamic arguments to support its development.
In practice, the contact angle can be experimentally determined in a rather routine manner, as can the liquid surface tension and even the solid surface energy. The interfacial energy for the liquid–solid system of interest, γSL, can then be calculated using Young’s equation. Alternatively, ifγSL,γL, and γS are known as a function of temperature, the contact angle can be predicted at a specified temperature.