What is spinodal decomposition?

J estr J estr Jestr

Journal of Engineering Science and Technology Review 1 (2008) 25- 27 Journal of Engineering Science and Technology Review 1 (2008) 25- 27 Journal of Engineering Science and Technology Review 1 (2008) 25- 27 Lecture Note

Lecture Note

Note What is Lecture spinodal decomposition? What is spinodal decomposition? , 1 decomposition? What spinodal E. P.isFavvas* and A. Ch. Mitropoulos2

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,1 2 E. P. Favvas* and A. Ch. Mitropoulos ,1 2

E. P. Favvas* and“Demokritos”, A. Ch. Mitropoulos Institute of Physical Chemistry, NCSR 153 10 Ag.Paraskevi, Attikis,Greece. 2 1 Department of of Petroleum Technology, Kavala Institute of Technology, 65 404 St.Lucas, Kavala, Greece. Institute Physical Chemistry, NCSR “Demokritos”, 153 10 Ag.Paraskevi, Attikis,Greece. 2 1Institute of Physical Chemistry, NCSR “Demokritos”, 153 10 Ag.Paraskevi, Attikis,Greece. Department of Petroleum Technology, Kavala Institute of Technology, 65 404 St.Lucas, Kavala, Greece. 2 Department of Petroleum Technology, Kavala Institute of Technology, 65 404 St.Lucas, Kavala, Greece. Received 17 January 2008; Accepted 20 February 2008 ___________________________________________________________________________________________ Received 17 January 2008; Accepted 20 February 2008 Received 17 January 2008; Accepted 20 February 2008 ___________________________________________________________________________________________ 1

___________________________________________________________________________________________ Abstract

Abstract Abstract: Phase separation may occur in a way that the growth is not in extent but in amplitude. Only in the unstable region such a Abstract procedurePhase is thermodynamically feasible. In a that phase unstable region is amplitude. defined byOnly the spinodal. When region a system Abstract: separation may occur in a way thediagram growth the is not in extent but in in the unstable suchhas a crossed thisisseparation locus, phase spontaneously without the presence of nucleation step. This process known as spinodal Abstract: Phase mayseparation occur feasible. in occurs a way In that growth is notthein extent but in aamplitude. in the unstableis region a has procedure thermodynamically a the phase diagram unstable region is definedOnly by the spinodal. When a such system decomposition andphase commonly results to a high of two phases. The Cahn-Hilliard describes thespinodal kinetics procedure is this thermodynamically feasible. In phaseinterconnectivity diagramwithout the unstable region is adefined by the When a system has crossed locus, separation occurs spontaneously thethe presence of nucleation step.spinodal. Thisequation process is known as of the In this note bothresults processes and spinodal) are two depicted schematically. crossed thisprocess. locus, phase separation occurs spontaneously without the presence ofphases. a nucleation step. This process is known as spinodal decomposition and commonly to a (nucleation high interconnectivity of the The Cahn-Hilliard equation describes the kinetics decomposition and commonly to a high(nucleation interconnectivity of the two Cahn-Hilliard equation describes the kinetics of the process. In this noteresults both processes and spinodal) are phases. depictedThe schematically. of theKeywords: process. Inspinodal this notedecomposition, both processes nucleation (nucleationand andgrowth. spinodal) are depicted schematically. ___________________________________________________________________________________________ Keywords: spinodal decomposition, nucleation and growth. Keywords: spinodal decomposition, nucleation and growth. ___________________________________________________________________________________________

___________________________________________________________________________________________ A pair of partially miscible liquids, i.e. liquids that do not mix in of all partially proportions at all liquids, temperatures, shows in do a temA pair miscible i.e. liquids that not A pair ofinpartially miscible i.e. liquidsshows thatwhere doin not perature-composition diagram a miscibility gap mix all proportions atliquids, all temperatures, a phase temmix perature-composition in all proportions all temperatures, shows a temseparation occurs. at Gibbs [1] showed that the for diagram a miscibility gapincondition where phase perature-composition diagram a miscibility gap where phase stability (oroccurs. metastability) in respect continuous changefor of separation Gibbs [1] showedtothat the condition separation occurs. Gibbs [1]derivative showed that the condition formixphase is thatmetastability) the second oftothe free energy of stability (or in respect continuous change of stability (orisbemetastability) inderivative respectthetosystem continuous changeof ing to positive. If negative, is unstable. Ifofmixzero, phase that the second of the free energy mix phase is that the second derivative of the free energy of mixthe to spinodal is defined. The free energyisofunstable. mixing,IfΔG ing be positive. If negative, the system zero,, mix ing to be positive. If negative, the system is unstable. If zero, has spinodal the following generalThe form: the is defined. free energy of mixing, ΔG , the spinodal is defined. The form: free energy of mixing, ΔGmix, has the following general has the following general form: ΔG mix = ΔH mix − TΔS mix . (1) mix mix mix Δ G = Δ H − T Δ S . (1) mix ΔG mix ΔHregular − TΔ S mix . model [2] the entropy of mixing (1) is For=the solution mix the same as for solution the ideal model mixing;[2]ΔSthe =-R(X +XlnXBis), For the regular entropy of Amixing AlnX mix For the the regular solution model [2] the entropy of components mixing is A the molar fractions of where XAasand same for X the ideal mixing; ΔS =-R(X B are AlnXA+XlnXB), mix the same themixture idealaremixing; ΔS =-R(X +XlnX AlnX A B), However, enthalpy of and BasX infor (X and X theA+X molar fractions of the components A where B=1). A the B mix and X are the molar fractions of components where X A B β is A an inmixing be written ΔH However, the enthalpy of and B inmay the mixture (XAas+X AXBβ, where B=1).=X mix +Xas However, the enthalpy and mixing B in themay mixture (XAlumping B=1). teraction parameter energy of mixing =X where β contribuis of an inbe written ΔHthe AXBβ, mix =X XBeq.(1) β, where β is an inmixing may be written as ΔH tion [3]. Under theselumping assumptions, becomes: teraction parameter theAenergy of mixing contributeraction parameter lumping the energyeq.(1) of mixing contribution [3]. Under these assumptions, becomes: tion [3]. Under these assumptions, eq.(1) becomes: mix ΔG = X A X B β + RT ( X A ln X A + X B ln X B ) . (2) mix Δ G = X X β + RT ( X ln X + X ln X ) . (2) A B B ΔG mix = X A X BAβ +B RT ( X A ln AX A + X (2) B ln X B ) .

Figure 1. A phase diagram with a miscibility gap (lower frame) and a diagram1. of freediagram energy change (upper frame). phase diagram Figure A the phase with a miscibility gap The (lower frame) and is a the1.temperature versus the change molar fraction of a(lower component Figure A phase with a miscibility andxBa. Note diagram of thediagram free energy (upper gap frame). Theframe) phasee.g. diagram is thattemperature is symmetrical around xBof =0.5 which the the diagram ofthe thediagram free energy change (upper frame). phaseisdiagram the versus the molar fraction aThe component e.g.case xBis . of Note regular Linefraction (1)around is the this the temperature versusmodel. molar of phase xB.case Note that the solution diagram isthe symmetrical xaBcomponent =0.5boundary. which e.g. is Above the of line the thediagram twosolution liquids are miscible and thexthe system isboundary. stable (s).case Below this line line that the is symmetrical =0.5 which is the of this the B regular model. Linearound (1) is phase Above there is liquids a metastable region (m). Within thatis region the system is stable regular solution model. (1) is thethe phase boundary. this line the two are Line miscible and system stableAbove (s). Below this line to small but is the unstable to is large fluctuations. Line (2) is the the two liquids are miscible and stable (s). Below this line there is afluctuations metastable region (m).system Within that region the system is stable spinodal. Belowregion this but line systemthat is large unstable (u).system Regions (u) thereto is small a metastable (m). Within region the is (m) stable fluctuations isthe unstable to fluctuations. Line (2)and is the constitute the miscibility gap.system Within gap the to small fluctuations but unstable to large fluctuations. Line turns (2)(m) is from the spinodal. Below thisisline the isthat unstable (u).system Regions and one (u) phase to athe two-phase Temperature (Tthe the (m) upper spinodal. Below this line thesystem. system is unstable (u). Regions andconsolute (u) one c) is constitute miscibility gap. Within that gap system turns from temperature. Abovegap. this temperature thethe two(T liquids are miscible constitute the Within that gap system turns from one in all phase to miscibility a two-phase system. Temperature c) is the upper consolute changes in the free energy of mixing (ΔG), at this given temperature, in proportions. At asystem. given temperature (T) (4) miscible cuts the in phase phasetemperature. to a two-phase Temperaturethe (Ttwo ) istie theline upper consolute cthe Above this temperature liquids are all respect to (3) in the at upper Segments (ab) B are changes in xthe freeshown energybyofline mixing (ΔG), this frame. given temperature, in boundary and the spinodal at points (a, liquids c) and respectively. The temperature. Above temperature the two are miscible 2 proportions. At this a given temperature (T) the tie (b, lined), (4) cuts in theallphase andin(cd) toofamixing positive second derivative of ΔG, ∂2ΔG/x >0, changes the free energy this givenframe. temperature, in B(ab) respect tocorrespond xB are shown by line (ΔG), (3) in atthe upper Segments proportions. At a given temperature (T) the tie line (4) cuts the phase 2 2 boundary and the spinodal at points (a, c) and (b, d), respectively. The respect 2 (d) while (bd) by to negative one, ΔG/x points (b) and to(cd) xsegment shown (3) insecond the ∂upper frame. Segments (ab) B 0, boundary and the spinodal at points (a, c) and (b, d), respectively. The 2 2of ΔG, ∂2ΔG/x 2>0, 2 ∂2ΔG/x and (cd) correspond to a to positive second derivative B =0. (bd) B and (d) while segment a negative one, ∂ ΔG/x ______________ B >CBright, however. Diffusion does not take place from left to right i.e. from higher to lower concentration (downhill), but oppositely from right to left i.e. from lower to higher concentration (uphill). Note that if diffusion was taken place normally from left to right (downhill) it would result to CBleft=CBright; i.e. the system would be gone back to what is depicted on the first row. This can not be happened because is the case of a stable system; i.e. contradicts the initial statement that the system is already in the unstable region (compare also with Fig.2 to see the difference). The density is large in extent but small in degree; the arrows show the diffusion direction. As diffusion progresses the density increases until it reaches a point where becomes equal to the density of pure B (rows 3-5). Phase separation that occurs in this way is known as spinodal decomposition and the fingerprint of that mechanism is the uphill diffusion.

Figure 2 (left). The nucleation and growth process. The columns represent two adjacent local regions in the solution denoted as “left” and “right” and the rows five stages of the system denoted by numbers 1, 2, 3, 4, 5; where the first three stages show what will happen when a small fluctuation occurs and the last two when a large fluctuation takes place. Black dots represent one of the two components e.g. B. In the 1st row the solution has been brought in to the metastable region and at the moment there is not difference in the local concentrations; CBleft=CBright. In the 2nd row an infinitesimal fluctuation causes a concentration difference, such as CBleft>CBright, and therefore diffusion is expected to take place between left and right localities. In the 3rd row the expected diffusion results to CBleft=CBright. Diffusion takes place from left to right i.e. from higher to lower concentration (down-hill) and the system comes back to what is depicted on the first row. In the 4th row a large fluctuation causes the formation of a nucleus of a critical size. For presentation reasons, the figure was drawn such as all component B on the left region is spent to form that nucleus; therefore CBleft=00 the sign of D will be determined by the sign of ∂2ƒ/∂C2B. When ∂2ƒ/∂C2B>0 the solution is stable, D>0, and diffusion (if any) occurs downhill. When ∂2ƒ/∂C2B