Spinodal Decomposition as an Interesting Example of the Application of Several Thermodynamic Principles Daryl G. Clerc and David A. ~ l e a r y ' Washington State University, Pullman, WA 99164-4630 Most students of undergraduate physical chemistry encounter a discussion of multiphase systems after being introduced to the fundamentals of solution thermodynamics, such as Raoult's Law, activity, chemical potential, Gibbs' phase rule, etc. The focus is often placed upon the interpretation of the relevant phase diagrams where the axes used are temperature and composition ( 1 3 ) .The discussion can be extended to include free energy versus composition phase diagrams (4). Examination of such diagrams for a series of temperatures is particularly instructive for the case where a solution undergoes ~ h a s seoaration e won a change in temperature. ~n&& of the iwo methods by which this ohase seoaration can occur-soinodal decomoosition and nucleation and growth-provides a n opportunity to bring together a variety of thermodynamic concepts in a n attempt to explain a physically observable phenomenon.
values of G are known or measurable. Instead, this plot reflects the behavior of a particular extensive thermodynamic property of a system if that system obeys the accepted laws of thermodynamics. For any composition, xo, the free energy is minimized with respect to fluctuations i n the composition. Specifically, if the single-phase system a t composition no decomposes into a two-phase system where the composition of the two phases are y and z , the free energy would increase by the amount indicated in Figure 1. If the system in Figure 1is brought to a lower temperature where phase separation occurs, the free energy versus composition curve will qualitatively look like that shown in Figure 2. Upon cursory examination of Figure 2, students often assume that the composition of the individual phases in the two-phase region is determined by the minima in Figure 2. Using introductory concepts and equations from undergraduate thermodynamics, we will demonstrate in the following sections that the composition of the two phases is
Free Energy versus Composition The Gibbs'free energy, G, versus composition curve for a system of two components miscible in all proportions is shown schematically in Figure 1.The components could be gases, liquids, or solids. Such a plot does not imply that
'Current address: Department of Chemistry, Gonzaga University Spokane, WA 99258.
Figure 1. Gibbs'free energy as a function of composition for a system of two components that are miscible in all proportions. Temperature is fixed. ComDositlon,,x is stable with resDect to decomDosition into two phases oi composibon y and z
Figure 2. Gibbs'free energy as a function of composition for a system that undergoes phase separation over the composition range a < x, < 6. Temperature is fixed. The straight line is the only such common tangent that can be drawn (see text).
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Journal of Chemical Education
not determined by the minima but rather by the common tangent drawn a s shown in Figure 2. Free Energy and Chemical Potential The chemical potential of a component in a system is defined a s the partial derivative of the Gibbs' free energy with respect to the number of moles of that component.
sponding molar Gibbs'free energy of solution isg'. The tanThe molar Gibbs' free energy gent slope a t (x6.g') is -A&+ of solution a t the intercept corresponding to the A component is g, =g' + &.When eq 7 is evaluated a t xfi, the intercept a t x~ = 0 equals pAin the solution. Similarly, if eq 7 is evaluated for the B component, the intercept a t XB = 1 equals pg.
where G is the Gibbs' free energy of the system, and n, is the number of moles of component i . In a multicomponent system of 2 or more phases, the phase equilibrium criterion is that the chemical potential of a given component i must be equal in all phases where i is present.
Partial Molar Volume A similar geometric interpretation for partial molar volume is rriven bv Moore (5).Here aeain the i m ~ o r t a ndiffert ence be'tween partial molar volume and molar free enerw must be em~hasized:Volume can be measured. and free energy cannot:~or the present discussion, it needonly be pointed out that for the composition xfr, the chemical potentials of A and B in solution are determined by the XB = 0 and XB = 1intercepts of the tangent a t xfi.
where pi represents the chemical potential of the ith component, and a, b, y, ...represent the different phases where i is present a t equilibrium. Graphical Representation
The chemical potential of a component can be conveniently displayed using a graphical representation. Following the presentation of Moore (51,the chemical potential of a component in a two-component solution can be represented using the method of intercepts. The solution may be solid, liquid, glass, or polymer. The molar free energy of the solution, g, is defined as
where nA is the number of moles of A, and nB is the number of moles of B. Differentiation of G with respect to nAyields the chemical potential, FA.
Slope Having established that the intercepts of tangents to a g versus composition plot are equal to the chemical potentials of the individual components in a solution, we now turn our attention to the slope of this same plot. We begin with the Gibbs equation for the differential in the Gibbs' free energy again for a single-phase, two-component system.
At constant temperature, pressure, and total number of moles ( n +~ng), this can be rewritten as
For convenience, and with no loss of generality, the total number of moles will be set equal to 1.Henceforth, G (with units of energy) a n d g (with units of energyimol) will have the same numerical value. Recalling eq 9 and the definition of the molar free energy, g, G n~ + n~
g=-
Applying the chain rule to eq 4, we get
Using the definition of mole fraction and evaluating
Substituting eq 6 into eq 4, we get eq 7
From this expression, the chemical potential of each component in a two-component system is determined a s shown in Figure 3. For a given composition, xfi, the corre-
Figure 3. The Gibbs' molar free energy of a two-component solution as a function of composition at a fixed temperature. The chemical potential of each component at a given composition is determined by the intercepts of the tangent drawn at that composition (see text). Volume 72 Number 2
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i t is immediately seen that the slope o f g versus x~ is equal to the difference i n the chemical potential of the two components in solution ( ~ -g FA).This will become important when we consider phase separation, for example, in partially miscible liquids.
XB
Free Energy and Phase Separation Figure 1 is a schematic representation of the change in free energy of a solution as the composition is varied. Because the free energy is an extensive pmperty, it is more meanmghl to plot the molar free energy (Gln,~) versus composition. This type of graph applies to any two-component system where the two components are miscible in all proportions. Examples of such systems include wateriethanol, copperlnickel, and heliumlneon. We cannot measure the Gibbs'free energy directly, but rather we measure changes in the Gibbs' f k e energy. For the preparation of the solutions described above, the change i n the Gibbs' free energy i n going from pure components to the mixed solution is identified a s AG,
Constraints Recall that the slope of g versus xg is equal to pB-FA. If the system bifurcates into two distinct phases, which we will label alpha ( a ) and beta (PI, then the condition for phase equilibrium imposes the following constraints.
I n the ideal limit, i t is equal to
Phase Stability Dividing by the total number of moles gives the molar AG,i, and converts nAtoXA and ng to xg. This is a common expression found in most physical chemistry textbooks. It is always negative, but more importantly, the curvature is always concave upward! It can easily be shown that a 2 ~ 8 0 ~ u t i od2~Gm, n -
ax:
-
ax:
RT XAXB
,
close to 0 or 1, a single-phase solution results where the composition is trivially determined by how much B was added to A. For xg = 0.5, the concavity is negative, and phase separation will result. The compositions of the two resulting phases are of interest here.
Subtracting eq 13 from eq 14, we get
In other words, the slope of g versus XB must be the same for each phase. There are an infinite number of pairs of points on the g versus xB plot t h a t have equal tangent slopes. However, because eq 7 must hold, the tangent slopes for each phase must also have the samey intercepts. Iftwo lines have the same slope andy intercept, they must coincide. Hence the composition of the two phases (for a given temperature and pressure) are fixed a t points a and b in Figure 2. This construction is known as the common tangent approach. Some texts mistakenly identify the minima as corresponding to the phase compositions (8,9). Illustration of Gibbs'Phase Rule Phase separation also provides a nice illustration of the famous Gibbs' phase rule F=C-P+2
(12)
which is a condition for phase stability. If G,,I,,~,. (or AG,,) versus xg i s concave downward, the solution is unstable with respect to separation into two distinct phases. In order to gain physical insight as to why the concavity of G (dropping the solution subscript for economy) versus composition determines phase stability, consider again Figure 1. In order for single-phase composition xo to spontaneously unmix or decompose into the two phases of composition y and z, the Gibbs' free energy must increase by +AG. At constant temperature and pressure the condition for spontaneous change is negative AG, so this unmixing will not occur. Conversely, if the G versus composition curve i s concave downward, then the sign of AG for the process of a single phase decomposing into two phases will be negative. For a given system, a2G/axf < 0 is an unphysical portion of the G versus composition plat. Amore common encounter with a n unphysical portion of a thermodynamic plot occurs with the van der Waals equation of state. For temperatures below the critical point, the portion of the van der Waals curve where increased pressure results in increased volume is unphysical. The system responds by undergoing phase separation!
where F is the number of degrees of freedom; C i s the number of components; and P is the number of phases. At fixed temperature and pressure, F = C - P. For a single-phase, two-components system, F = 1. Hence in regions 1and 3 of Figure 4, one of the composition variables must be specified in order to completely describe the system. In
Composition Having established that phase stability requires positive concavity i n the G versus composition plot, and that negative concavity will result in phase separation, we now look more closely a t the compositions of the resulting phases when separation occurs. A schematic representation of G versus composition for a system t h a t undergoes phase separation is shown i n Figure 2. Examples of such systems include waterltriethylamine and waterinicotine (6, 7). For 114
Journal of Chemical Education
Figure 4. This is the same plot as Figure 2 except that the common tangent has been removed, and the plot is divided into three regions. Regions 1 and 3 are single-phase regions, and region 2 is a twophase region.
region 2, C remains 2, but P is now also 2, resulting i n F = 0. Hence. the com~ositionsof both phases are fured, and no further information is necessary to define the system! The ohase rule does not Drovide information about the relative amounts of each phase, only their existence. Deviations from the Single-Phase Composition We have considered in detail regions 1and 3 in Figure 4 a s well as the concave downward portion of region 2. In order to smoothly connect regions of positive concavity and negative concavity, there must be points of inflection (labeled c and d in Fig. 4). How does a solution with composition between points a and c (or d and 6) behave with respect to phase stability? This portion of t h e curve i s concave upward, and therefore we expect the single-phase solution to be stable. However, i n making this statement, we are assuming t h a t the compositions of any two new phases will be in the region between xe = 0 and point c (or d and xa = 1). \\'e regard such deviarions from the singls:-phase compos i t ~ o nto be small fluctuations. The conclusion, rherefbrv, is that in the region between a and c (or d and b), the singleDhase solution is stable with respect to small fluctuations in composition. If, however, we anow for large fluctuations, for example the formation of a second phase with composi6, we now have a process that will result in an tion a t overall decrease in the Gibbs' free energy. The amount of the is eaual to the vertical distance between the ~ - decrease - ~ point between a and c under question and a straight line connectine a and 6. All such distances constitute neeative AG. ~~~~
~~
~
-
Nucleation a n d Growth We are left with the interestine conclusion that a solution having a composition between points a and c (as well t small fluctuaas between d and bi is stable with r e s ~ e cto tions, but i t i s unstable with respect to large fluctuations. How does a solution achieve a laree - fluctuation without having first undergone ;I small fluctuntiun'!Th(! rcwlution to thii dilemma has ht,m treated heforr whtm cunsidermgthe formation of liquid droplets from a vapor phase (10). Small liquid droplets (that undergo small fluctuations in size) are unstable and evaporate. Beyond a critical size (a large fluctuation), however, the droplet is stable and continues to grow. The detailed treatment of this phenomenon is called nucleation and growth. Hence, a single-phase solution with between a and c (or d and b). will - ~ ~ ~ -~~~ - ~ comoosition - decompose into two distinct phases of composition a and 6. The mechanism for this Dhase separation is identified as nucleation and growth because of the analogy with droplet formation.
-
~
~
Spinodal Decomposition The formation of two distinct phases for compositions between c and d is straightforward. For any single-phase to
two-phase change, AG is negative. Hence i n this range, the system is unstable with respect to all fluctuations, small and large. When the decomoosition occurs for com~ositions ~~-~~~ between c and d, the mechanism is called spinodal decomposition. How might one observe such a phenomenon? Returning to the example cited already with water and nicotine, a t 220 "C, these two components are miscible in all proportions. Hence, the Gibbs' free energy versus composition plot will qualitatively look like Figure 1. There= 0.5 has fore, a solution of composition %,,,, a2G/ax2,,,,.> 0. AS the temperature is lowered to 150 "C the system bifurcates into a water-rich phase and a nicotine-rich phase: Now a2Giax2,,te,a t x,,,, = 0.5 i s negative. Neither of the two phases present has x,,,, = 0.5, despite the fact that the original single-phase solution had x,,,,, = 0.5. The overall G versus composition plot will resemble Figure 2. ~~
-
~
Morphology The moroholow -" of a solid two-ohase svstem deoends upon which mechanism is responsible for the separation. The different morphologies impart important properties to the resulting material. This is especially important in such diverse fields as glass making (11,12), metallurgy (13,141, polymer science (15,16),and statistical mechanics (17,181. Summary A discussion of phase separation serves a s a n important culmination in the study of a variety of related thermodynamic principles. Moreover, highlighting the practical conseauences of the materials resultine from such Drocesses provides both the science and engineering student with a n aooreciation of the Dower of thermodvnamics to describe .. and predict physical processes.
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Literature Cited
4. Beny. R. 8.; Ria, S. A ; Ross, J. Phyricnl Chrmisiry; Wiley: Now York. 1980; p 932. 5. Moore, W. J. PhysIFnI ChrmisLry, 4th ed.; PreniiccHall: New Ymk. 1972: p 235. See also. Earl, B. L. J Chrm Educ 1989.66,56.
BosWn. 8. Bar& R. J.: thenex, G. J.An Inlrnduclion to SnlrdSInleDiNuszon;Aeademie: 1988: p 303. 9, w e a , A, R. solid stare Ch~misrryorid Its Applim(zonr; Wiley: Chichertel 1981; p 614. Wiley: NewYork, 1982;~ 3 1 9 . 10. Adamson,A. WPhvsiml ChemislryofSur/ne~a,4thed.; 11. re19. p 622. 12. Fine. G.J. JChem. Edue 1991.6Ri91.765. k aondEngineeri?i#, 4thad.; Addison13. Van Vlack, L. H.Elemenlr o f M n l ~ a Science Wesley: Reading. MA, 1980;p 377. 14. Cahn, J. WAC!" Melaiiur 1961.9, 795. 15. Song, M.: Liang, H.; Jiang,B. Poly BUN.19W.21, 615. 16. Konno. M.: Wang.2.-Y.: Saito, S. Poivmer 1990.31,2329. 17. Debenedet6. P. G.: Raghavan. V S.: Rorick. S. S. J Ph.vs. Chem. 1991,95,4540. 18. Be~ry,R. S.; Wales, D.JPhys. Rev. la,*.1989.63, 1156.
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