Lecture 35: Introduction to Surface Thermodynamics

MIT 3.00 Fall 2002 242 c W.C Carter ° Lecture 35 Introduction to Surface Thermodynamics Last Time Electrochemical Potential Nernst Equation Estimate of the Excess Energy Associated with Surfaces In the treatment of the equilibrium of phases the effect of the surface that separates the various phases was neglected. In other words, no distinction was made between systems that had an abundance of surface and those that do not—there was no distinction made between: β β α α Figure 35-1: Including the effect of interfaces and surfaces. The treatment of equilibrium up until now treated these two systems as being alike even though one obviously has much more surface (and thus any energy associated with that surface) than the other. Consider, as an example, that an atom on a surface as having a 50% higher energy than those in the bulk, then there will be an extra energy associated with the surface of a sphere. MIT 3.00 Fall 2002 243 c W.C Carter ° To estimate how much energy is associated with the surface let: surface area ≡ surface area of α-phase = 4πRs2 4 volume ≡ volume of α-phase = πRs3 3 The energy of the system is: U XS N surf U surf + N bulk U bulk = N total (35-1) Letting the energy of an atom on the surface be half again that of the bulk: 3 U surf ≈ U bulk 2 Ã U XS (assumption) 1 N surf 1+ 2 N total = (35-2) ! (35-3) If Ω is the volume per atom and RA is the radius of an atom, then (Surface Area)RA Ω (Volume) N bulk = Ω N surf = (35-4) Letting Rs be the radius of the sphere: µ U XS = 3RA 1+ 2RS ¶ (35-5) How small does the sphere need to be in order that the excess is about 1%? RS ≈ 150RA (for excess of about 1 percent) (35-6) This is pretty small, but important important in many systems. Gibbs Treatment of the Interfacial Energy Note the atoms at the surface were treated as being somehow different than those of the bulk. This idea can be extended rigorously to treat the interface as a relatively thin layer. This thin layer will be treated as a separate “quasi-two-dimensional phase.” The original Gibbs idea was as follows: Suppose the composition is different between to phases (but dictated by µαi = µβi , of course). There is no requirement that the concentration should be uniform in the vicinity of the interface: Composition MIT 3.00 Fall 2002 244 c W.C Carter ° α−phase β−phase Cβ Cα z, spatial coordinate Figure 35-2: Illustration of the composition in the vicinity of a real interface. Composition The Gibbs treatment extends each of the homogeneous phases up to a mathematical surface: α−phase β−phase Cα Cβ mathematical interface z, spatial coordinate Figure 35-3: Gibbs idealization of the interface as a mathematical dividing surface. Subtract the real system in Figure 35-2 from its idealization in Figure 35-3 to define an excess quantity associated with the mathematical surface “The Gibbs Surface” which has no volume associated with it, but excess extensive quantities. If the following illustration represents a thermodynamic system: O-phase ∧ n → δn I-phase Figure 35-4: Illustration of an interface separating two phases. The extra surface represents the motion normal to itself and is used in the derivation of Equation 35-13. For the surface phase: dU surf = T dS surf + γdA + C X i=1 where A is the surface area of the interface. µi dNi surf (35-7) MIT 3.00 Fall 2002 245 c W.C Carter ° What is γ? Obviously, Ã γ= ∂U surf ∂A ! constS ,constNi surf (35-8) But, it also represents the work done to increase the surface area d(γA) = F dl = γwdl = F dl F =⇒ γ = w (35-9) where a force F is applied to a surface of width w to extend its length l. w F Figure 35-5: The mechanical implications of surface tension. This is the surface tension, it has units of force per unit length or, equivalently, surface energy per area. It is the energy associated with creating surface.36 Consider the entire system: δU total = δU inside + δU surf + δU outside = T δS inside − P δV inside + C X µi δNiinside i=1 + T δS surf + γδA + C X (35-10) µi δNi surf i=1 + T δS outside − P δV outside + C X µi δNioutside i=1 Because the surface represents an object that can resist pressure, it can no longer assumed that the pressures on the inside and the outside phase are equal. 36 Note: in solids, another term can arise which is associated with stretching the existing surface—this is called the surface stress. MIT 3.00 Fall 2002 246 c W.C Carter ° For equilibrium dU = 0, along with the temperature and chemical potentials being uniform: γδA − P δV inside − P δV outside (35-11) Imagine that the surface moves normal to itself (such as in Figure 35-4) Then the following relations hold: δV inside = Aδn̂ δV outside = −Aδn̂ and δA = (κ1 + κ2 )Aδn̂ = (κ + κ )δV inside 1 (35-12) (35-13) 2 1 1 =( + )δV inside R1 R2 where K1 and K2 are the “curvatures” of the surface in each of two perpendicular planes, where the axis of intersection is normal to the surface. κmean ≡ κ1 + κ2 (35-14) is called the “mean curvature” of the surface 1/R1 = K1 , and 1/R2 = K2 where R1 and R2 are the radii of curvature. Putting equations 35-12, 35-13, and 35-14 together, the following important relation holds: γκmean = P inside − P outside (35-15) which is known as the Gibbs-Thompson equation. It relates the difference in pressure at an interface to its surface tension and curvature. Curvatures of Simple Surfaces The relation: µ δA = κmean δV = 1 1 + R1 R2 ¶ δV (35-16) was used above to relate the pressure difference across an interface. If the pressure is to be the same within each phase, then at equilibrium a surface must have constant curvature. Consider how curvature relates to some geometrical objects that have “constant mean curvature.” For the sphere: ∂A ∂A ∂RS 8πRS 2 = = = = κmean (35-17) 2 ∂V ∂RS ∂V 4πRS RS There is one radius of curvature for each perpendicular plane with an axis normal to the surface at each point. For the cylinder: ∂A ∂A ∂RC 2πL 1 = = = = κmean (35-18) ∂V ∂RC ∂V 2πRC L RC MIT 3.00 Fall 2002 247 c W.C Carter ° Fundamental relations for surfaces In this section it will be shown that there are additional solubility effects associated with interfaces. Consider: C X dU surf = T dS surf + γdA + µi dNi surf (35-19) i=1 since the dependent variables are all extensive, we can integrate (i.e. homogeneous degree 1 in all of its variables), therefore U surf = T S surf + γA + C X µi Ni surf (35-20) i=1 Taking the derivative (as was done when deriving the Gibbs-Duhem equation); dU surf = T dS surf + S surf dT + γdA + Adγ + C X µi dNi surf + i=1 C X Ni surf dµi (35-21) i=1 comparing to Equation 35-19, 0 = S surf dT + Adγ + C X Ni surf dµi (35-22) i=1 which expresses a relation between variations of the intensive degrees of freedom for a surface to remain in equilibrium. Dividing through by the total surface area (so as to normalize by the area, creating derived intensive variables) and defining S surf g S surf = (35-23) A as the entropy of the surface per area, then, g 0 = dγ + S surf dT + Γ1 dµ1 + Γ2 dµ2 + . . . ΓC dµc C X g = dγ + S surf dT + Γi dµi (35-24) i=1 where g Γi ≡ Ni surf (35-25) is the standard notation for the excess surface concentration. Holding everything (temperature, et cetera) constant except µ1 , we get a relation that expresses the relation between the change in surface tension to the change in chemical potential of an absorbing species: µ ¶ ∂γ = −Γi (35-26) ∂µi constantT,µj 6=µi This is the “Gibbs Absorption Isotherm.” MIT 3.00 Fall 2002 248 c W.C Carter ° Note that if a species absorbs to the surface Γi > 0 and the surface tension decreases as the chemical potential of that species is increased. The Conditions of Equilibrium where Several Surfaces Intersect Consider the case where three different phases make contact: Aαβ α−phase ζα A β−phase ζ−phase Aβζ Figure 35-6: Intersection, or triple line, where three phases make contact in space. Considering dU surf = γ αβ dAαβ + γ βζ dAβζ + γ ζα dAζα must be a minimum, one may derive two relations for the angles of contact: sin φαβ sin φβζ sin φζα = = γαβ γβζ γζα (35-27) which is the general equation the angles at a triple line, called Young’s equation, where γαβ φβζ φζα γζα φαβ γβζ Figure 35-7: The definitions of the terms in Young’s Equation, Equation 35-27. MIT 3.00 Fall 2002 249 c W.C Carter ° which is equivalent to the “force balance” where each of the γ are considered the forces applied to the vertex. For the special case in which one interface is constrained to be flat, as in lv γ liquid vapor φ vs γ γsl solid Figure 35-8: o ne can derive (by force balance most simply) γlv cos φ + γsl = γvs γvs − γsl cos φ = γlv (35-28) which is Young’s equation for flat surfaces. φ ≡ wetting angle 0 < φ < 180◦ (35-29) The Shapes of Things Above, γ has been assumed to be isotropic. Under this assumption, a finite (isolated) volume body will reduce its total surface energy to a minimum. The result for an isolated body for isotropic surface tension is a sphere. Figure 35-9: The minimizing surface for a fixed volume with isotropic surface tension. However, for crystals, γ is a function of the orientation of the surface γ(n̂). For example, in 2-D MIT 3.00 Fall 2002 250 c W.C Carter ° γ(θ) Figure 35-10: Example of how the surface tension might depend on the orientation of n̂ of a surface. The shape is given by the Wulff construction: Figure 35-11: Example of the Wulff construction to calculate the minimizing surface for a fixed volume with anisotropic surface tension γ(n̂). The interior envelope in the right figure is the minimizing shape in two dimensions. The Wulff construction is performed as follows:37 For each orientation n̂, draw a ray from the origin to the surface of γ(n̂). At the end of each ray, construct the perpendicular half plane. The interior of the envelope that results from all such half planes is the minimizing shape for a finite isolated volume. 37 The Wulff theorem, which establishes the Wulff construction is the minimizing surface is surprisingly hard to prove, but can be understood on the basis of what has been learned in thermodynamics course in a paper by J.W. Cahn and W.C. Carter, Crystal shapes and phase equilibria: A common mathematical basis, Met. Trans. A, pp 1431, (1996).