Chapter 3 THERMODYNAMICS OF INTERFACIAL SYSTEMS A. The thermodynamics of simple bulk systems 1. Thermodynamic concepts Thermodynamics is in general useful for providing over-arching “rules” that govern the descriptions of macroscopic systems in terms of their properties and their interactions with other systems. The systems usually encountered in textbooks for scientists and engineers are pieces of infinite systems, as pictured schematically in Fig. 3-1(a). Their “boundaries” are completely conceptual and serve only to delineate the extent of the system. Furthermore, it is assumed in formulating thermodynamic descriptions that (a) (b) Fig. 3-1: (a) Thermodynamic systems as finite pieces of an infinite homogeneous system, (b) Simple-compressible system as a fluid in a piston-cylinder arrangement. the systems are in internal equilibrium so that, in the absence of external fields, they are homogeneous with respect to their relevant intensive properties. Specifically, since they are assumed to be in internal thermal, mechanical and diffusional equilibrium, they are uniform with respect their temperature, pressure and composition. This requirement poses problems for solids, in which full internal equilibrium often does not exist. In fluids, molecular mobility often (but not always) guarantees internal equilibrium, whereas in solids, non-equilibrium structures, and hence non-uniform stress fields and compositions, can be frozen in place over time scales of practical interest. (This is clearly the case for many of the purpose-built micro or nano constructions of interest in nanoscience and nanotechnology.) “Textbook chemical thermodynamics” thus focuses on fluids, and the systems commonly encountered are fluid masses delineated by boundaries that are 108 INTERFACES & COLLOIDS close to, but not coincident with their actual physical boundaries, as exemplified by the gas contained within the piston-cylinder arrangement of Fig. 3-1(b). Since the physical interfacial region between the fluid and the cylinder wall is only a few Å thick, essentially all of the mass of the fluid is captured by drawing the model system boundary just far enough away from the wall to exclude all the inhomogeneity of the interfacial layer. 2. The simple compressible system In the absence of external fields, the systems of chemical thermodynamics are subject only to the single work mode of compression (“p-V work”), and energy added to the system as work is given by: "W = #pdV . (3.1) This defines a simple-compressible system, and the only purely physical processes to which it is subject are those of compression-expansion and heating-cooling. (Mixing or de-mixing can always be deconstructed into ! compression-expansion processes.) The list of allowable processes can be extended, however, to include ones with respect to which the system is not necessarily in equilibrium, viz., chemical reactions or phase changes. The conditions of internal physical equilibrium can be envisioned to continue to obtain for individual phases while the system is undergoing one or more of the above physicochemical processes.1 The first step in describing purely physical processes consists of writing down the appropriate expression for work. The next step makes use of the State Postulate, which asserts the existence of the internal energy, U, as a property of the system, expressible for a closed system (a system of given mass or set of mole numbers) as a function of two independent relevant variables (one more than the number of independent work modes), e.g., using the independent variables T and V: U = U(T,V). (3.2) The First Law of thermodynamics asserts that any change in system energy is given by: dU = dQ + dW, (3.3) where Q is heat, defined as energy transfer to the system unaccounted for in macroscopic evaluations of work. The First Law statement is a useful, as opposed to a trivial, statement of the conservation energy because it can be shown that the heat effect Q can be measured independently by calorimetry. For systems not undergoing physicochemical processes (phase changes or chemical reactions), the heat effect is associated with a temperature change of the system, i.e. dQ = CdT, where C is the heat capacity of the system 1 Prigogine, I., and Defay, R., Chemical Thermodynamics, Longmans Green and Co., London, 1954. INTERFACIAL THERMODYNAMICS 109 dependent on the thermodynamic state of the system and the nature of the process (constant V, constant p, etc.) during which heat is added. Appropriate scales for both temperature and heat capacity (for reference substances) were established by making use of the mechanical equivalence of heat. Putting the above relationships together allows the development of an explicit expression for the heat effect, Q, accompanying any such process, e.g. % $U ( % $U ( "Q = "U #"W = ' * dT + ' * dV + pdV & $T )V & $V )T # "p & = CV dT + T% ( dV , (3.4) $ "T ' ! V # "U & where the heat capacity at constant volume, Cv = % ( , and the form of the $ "T 'V coeffici!ent in the second term of Eq. (3.4) is obtained by ordinary thermodynamic reductions; in this case: # "U & # "p & ! % ( = T% ( ) p. (3.5) $ "V 'T $ "T 'V One may proceed from this point to the expression for entropy change, dS, for a quasi-static process, viz. dQ/T, substituting from Eq. (3.4): ! "Qrev Cv $ #p ' dS = = + & ) dV , (3.6) T T % #T (V and thence to the Helmholtz free energy function: F = U –TS: dF = d(U " TS) = "SdT + #w = "SdT " pdV . (3.7) ! The Second Law of Thermodynamics states that for the spontaneity of a proposed process in an isolated system, dS ≥ 0, or dF ≤ 0 for systems constrained to constant T and V. Independent variables T and p may be ! chosen instead of T and V, in which case it is convenient to introduce the system enthalpy, H = U + pV and the Gibbs free energy (or free enthalpy), G = H - TS. The enthalpy is a useful function for describing heat and work effects accompanying constant pressure processes, and the Gibbs free energy is a useful function for describing system equilibria and stability with respect to processes at constant temperature and pressure. In this set of variables, the descriptive equations become: % $V ( % $V ( "W = #p' * dT # p' * dp (3.8) & $T ) p & $p )T ! 110 INTERFACES & COLLOIDS % $V ( "Q = CpdT # T' * dp (3.9) & $T ) p dH = d(U + pV) = CpdT + Vdp (3.10) ! "qrev Cv $ #p ' Cp $ #V ' dS = = + & ) dV = *& ) dp (3.11) T T % #T (V T % #T (V ! dG = d(H " TS) = "SdT + Vdp (3.12) In all cases, mathematical reductions have been made which put the ! expressions in a form such that the coefficients can be evaluated from 2 qua!nt ities obtainable in the laboratory. For systems modeled as “simple- compressible,” the list of such quantities includes only: • volumetric data (p-V-T equations of state) • calorimetric data (heat capacities, latent heats) • composition (whose changes allow one to follow a physicochemical process) From the above type of development, much can be done toward describing the behavior of real systems. In particular, expressions for heat and work effects are made available, and expressions for the “driving forces,” whose sign and magnitude determine the spontaneity of various processes, are derived. B. The simple capillary system 1. The work of extension In constructing a model for systems that includes their interface(s) with adjacent systems, it is first recognized that if it is to be a piece of an infinite system, the piece must include one or more interfaces within its conceptual boundaries. Since the latter are finite, the interfacial area in the system is also finite,3 and the extensive properties of the system can no longer be obtained by simply multiplying the corresponding intensive properties by the system mass. Such a system will be termed in general an interfacial system, and if the bulk phase states involved are fluids, it is a capillary system. As stated earlier, because of their greater simplicity, it is useful to first set forth the thermodynamic description applicable to capillary systems and to point out and discuss later the ramifications of extending the description to fluid-solid interfacial systems. 2 Recall that absolute entropy, S, is obtainable (using the Third Law of Thermodynamics) calorimetrically or spectroscopically. 3 Hill, T. L., Thermodynamics of Small Systems, Part I, W. A. Benjamin Publ., New York (1963). INTERFACIAL THERMODYNAMICS 111 The simplest model one may use is that of the simple capillary system, pictured in Fig. 3-2. It consists of three parts in internal equilibrium: two portions of bulk phase, of volumes V′ and V″, which are “simple compressible,” and the interface itself, of area A, regarded mechanically as a membrane of zero thickness in uniform, isotropic tension. It is not possible to choose any simpler model, such as the interfacial layer by itself, because the latter is not “autonomous,” i.e. it is inextricably connected to at least small adjacent portions of the bulk phase on either side.4 Fig. 3-2: The simple capillary system. Work may be done on a simple capillary system in accord with: dW = "p# d V # " p# # d V ## + $dA, (3.13) where the last term on the right is the “work of area extension.”5 In writing Eq. (3.13), it is clear that the membrane model of Young has been incorporated into the simple capillary system model. Recall that the ! difference between p′ and p″ is related to the interfacial tension and the local curvature of the interface, κ, in accord with the Young-Laplace Equation, Eq. (2.29): p" " - p" = # $. (3.14) The pressure difference is usually not significant for purposes of computing work unless the surface is of high curvature (Rm = 1/κ ≤ 1 µm). One thus generally uses the approximation of a flat surface system, so that: ! dW " #pdV + $dA.