Does coarsening begin during the initial stages of spinodal decomposition?Timothy J. Rappl and Nitash P. Balsara Citation: The Journal of Chemical Physics 122, 214903 (2005); doi: 10.1063/1.1905584 View online: http://dx.doi.org/10.1063/1.1905584 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/122/21?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Percolation-to-droplets transition during spinodal decomposition in polymer blends, morphology analysis J. Chem. Phys. 121, 1141 (2004); 10.1063/1.1760513 Determination of temperature dependent structure evolution by fast-Fourier transform at late stage spinodaldecomposition in bicontinuous biopolymer mixtures J. Chem. Phys. 116, 10536 (2002); 10.1063/1.1474583 Critical length and time scales during the initial stages of nucleation in polymer blends J. Chem. Phys. 116, 4777 (2002); 10.1063/1.1463056 Validity of linear analysis in early-stage spinodal decomposition of a polymer mixture J. Chem. Phys. 113, 3414 (2000); 10.1063/1.1287272 Spinodal decomposition of symmetric diblock copolymer/homopolymer blends at the Lifshitz point J. Chem. Phys. 110, 4079 (1999); 10.1063/1.478289

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Does coarsening begin during the initial stages of spinodal decomposition?Timothy J. RapplDepartment of Chemical Engineering, University of California, Berkeley, California 94720

Nitash P. Balsaraa!

Department of Chemical Engineering, University of California, Berkeley, California 94720 and MaterialsSciences Division and Environmental Energy Technologies Division, Lawrence Berkeley NationalLaboratory, University of California, Berkeley, California 94720

sReceived 7 March 2005; accepted 17 March 2005; published online 1 June 2005d

The initial stages of spinodal decomposition were studied by subjecting a critical blend of modelpolyolefins to a pressure quench and monitoring the evolution by time-resolved small angle neutronscattering. Contrary to the predictions of the widely accepted Cahn–Hilliard–Cook theory, wedemonstrate that coarsening of the phase-separated structure begins immediately after the quenchand occurs throughout the initial stages of spinodal decomposition. ©2005 American Institute ofPhysics. fDOI: 10.1063/1.1905584g

INTRODUCTION

Our understanding of spinodal decomposition in criticalbinary mixtures1–12 rests on a landmark publication byCahn.1 In this paper, the linearized diffusion equation wassolved to obtain the evolution of scattering intensityIsq,tdduring phase separationsq is the scattering vector andt istimed. Isq,td was found to increase with time but only forscattering vectors less than a critical scattering vectorqc.This indicates the presence of a lower cutoff for the lengthscale of the emerging phase-separated structure. In additionIsq,td contains a time-independent peak atq=qm, indicatingthat the emerging structure is periodic and characterized by adominant length scaleLm=2p /qm. In the linearized theory,this length scale is independent of time. At later times thestructure coarsens due to effects, such as interfacial tensionand hydrodynamics, which are not included in the linearizedtheory.12

Many aspects of published experimental data obtainedduring the initial stages of phase separation6–11 are in quan-titative agreement with the linearized theory.1–5 However, theimportant question of whether or not the periodic structureobtained during the initial stages of phase separation canemerge without coarsening has, to our knowledge, not beenadequately addressed experimentally. In particular, we wereunable to find any experimental data set whereinqm obtainedduring the early stages of spinodal decomposition was inde-pendent of timeswithin experimental errord. In this paper wepresent experimental data that provides a definitive answer tothe question posed in the title of the paper.

EXPERIMENT

Our system is a binary blend composed of two nearly-monodisperse high-molecular-weight liquid homopolymers:partially deuterated polymethylbutylenesdPMBd and hy-drogenous polyethylbutyleneshPEBd, synthesized and char-

acterized using the methods described in Ref. 13. The num-bers of repeat units per chain in the two components weredetermined to beNdPMB=2806 andNhPEB=2535sat ambientT andP, and based on a 100 Å3 repeat unit which is the basisfor all of the parameters reported hered, the average numberof deuterium atoms, for every five carbon atoms indPMB,was 4.5. The radii of gyrationsRgd of bothdPMB andhPEBchains are 14±1 nm.13 A blend with critical compositionbased on the Flory–Huggins theory,14–16dPMB volume frac-tion f=0.493, was studied by time-resolved small-angleneutron scatteringsSANSd. We report the azimuthally aver-aged absolute SANS intensitysafter corrections for the back-ground and empty cell scatteringd, I, as a function ofq fq=4p sinsu /2d /l, u is the scattering angle andl, thewavelength of the incident neutrons, was 1.2 nmg. Detailsregarding the instrument configurations and data reductionprocedures are similar to our previous studies.17

RESULTS AND DISCUSSION

The equilibrium thermodynamic properties of blends ofthe dPMB andhPEB polymers used in this study have beenthoroughly investigated.13,17 In Fig. 1 we show the phasebehavior of our critical blend using a temperature–pressuresT–Pd phase diagram. The curve shows the Flory–Hugginsprediction for the variation of the critical temperature withpressure. SANS data obtained from blends above the criticalcurve were independent of timesafter an initial relaxationprocess due to the finite time that is required for dense poly-mer samples to respond to changes in temperature andpressure18d. The circles in Fig. 1 show the locations whereIsq=0d obtained in the one-phase region diverges, based onextrapolations of the measured 1/Isq=0d versus 1/T data atfixed pressure.19 The agreement between theoryscurved andexperimentscirclesd in Fig. 1 indicates a thorough under-standing of the equilibrium properties of our blend. Theblend was homogenized at the start of each experiment byheating to 115 °C andP=0.03 kbar. The blend was cooledunder isobaric conditions to 67.5 °C and then subjected to anadElectronic mail: nbalsara@berkeley.edu

THE JOURNAL OF CHEMICAL PHYSICS122, 214903s2005d

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isothermal pressure quench to locations indicated by thefilled squares in Fig. 1. The SANS profiles were monitoredthroughout the quenching process. In all cases, no evidenceof phase separation was observed prior to the pressurequenchst=0−d. We thus define time zerost=0d as the time atwhich the pressure quench was initiated.

Typical Isq,td data obtained during the initial stages ofphase separation are shown in Fig. 2 where the symbols rep-resent the data obtained atP=1.24 kbar and selected times.As expected, the data in Fig. 2 show the emergence of thespinodal peak. The well-established Cahn–Hilliard–CooksCHCd theory2,3 provides a unified approach for interpretingIsq,td data obtained after quenching the sample in the vicin-ity of the critical point.

Isq,td = ITsqd + fI0sqd − ITsqdgexpf2Rsqdtg, s1d

where I0sqd is the initial scattering intensity function andITsqd is the terminal scattering intensity function, obtained in

the limits t→0 andt→`, respectively, andRsqd is a kineticparameter that is related to the growthfif Rsqd is positiveg ordecayfif Rsqd is negativeg of I0sqd towardsITsqd. The curvesin Fig. 2 are examples of the CHC fits through theP=1.24 kbar data withI0sqd, ITsqd, and Rsqd as fittingparameters.20 It is clear that the time dependence ofIsq,td iswell described by the CHC theory for all scattering vectors.Similar agreement was seen at other quench depths.

Data showing the time dependence of the SANS inten-sity at a particular scattering vector are shown in Fig. 3 for aquench toP=2.48 kbar andq=0.051 nm−1. The curve inFig. 3 shows the CHC fit through the data. There is a smallbut systematic deviation between the CHC fit and the earlytime data, seen more clearly in the inset of Fig. 3. However,it is clear from Figs. 2 and 3 that the CHC fits capture theessential features of the data. We thus focus on the qualita-tive behavior of the fitted parameters,I0sqd, ITsqd, andRsqd.

The ITsqd, Rsqd, and I0sqd data for different quenchdepths are presented in Figs. 4sad, 4sbd, and 4scd, respec-tively. For theP=0.83 kbar quench, we obtain positive val-ues for I0sqd and ITsqd and negative values forRsqd at allaccessibleq values. This is the expected behavior forquenches within the one-phase region.18 More interesting be-havior is seen for the other quenches located within the two-phase regionsFig. 1d. In the inset of Fig. 4sad we show thefull range of ITsqd obtained atP=1.24 kbar. We find thatITsqd is a positive, monotonically decreasing function ofqfor largeq, a negative, monotonically decreasing function ofq for small q, and that these regimes are separated by asingularity spoled in ITsqd at q=0.028 nm−1. Similar singu-larities were obtained for all of the quenches within the two-phase region as shown in Fig. 4sad. This singularity inITsqdis an unambiguous signature ofqc.

2,3 In Fig. 4sbd we see thesecond unambiguous signature ofqc; Rsqd→0 atqc, the scat-tering vector corresponding to the singularity inITsqd. Forq.qc, Rsqd,0 fRsqd for all quenches exceptP=2.48 kbarhave been truncated by theRsqd axis scale21g whereas forq,qc, Rsqd is a positive, peaked function that decays to zeroboth atq=qc and atq=0 stheoreticallyd.

In Fig. 4scd we show I0sqd versus q for all of the

FIG. 1. T–P phase diagram of a criticaldPMB/hPEB blend calculated bythe Flory–Huggins theoryscurved. Circles: experimentally determined criti-cal points. Squares: location of experimental quenches.

FIG. 2. SANS intensity vsq at selected times during the 1.24 kbar quench.The solid curves represent CHC fits.

FIG. 3. SANS intensity vs time forP=2.48 kbar andq=0.051 nm−1. Thesolid curve is the CHC fit through the data. Inset shows early time results forthe same quench.

214903-2 T. J. Rappl and N. P. Balsara J. Chem. Phys. 122, 214903 ~2005!

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quenches and includeIsq,t=0d from Fig. 2 for reference.Since all of the quenches start atP=0.03 kbar andT=67.5 °C, in theory,I0sqd must be within experimental errorof Isq,t=0d. For the quench into the one-phase regionsP=0.83 kbard, we find an excellent agreement betweenI0sqd

and Isq,t=0d. At larger quench depth, we see an excellentagreement betweenI0sqd andIsq,t=0d in the low and highqlimits. However, systematic departures betweenI0sqd andIsq,t=0d are seen at intermediateq values. This is a reflec-tion of the fact that the CHC theory is unable to accuratelycapture the full time dependence of the SANS profiles duringthe initial stages of spinodal decompositionse.g., inset ofFig. 3d. It is worth noting, however, that the deviation be-tween I0sqd and Isq,t=0d is relatively small, maximum,600 cm−1 for P=2.48 kbar andq=0.051 nm−1, the condi-tions depicted in Fig. 3. This is comparable in magnitude tothe uncertainty in our estimate ofI0sqd, which we assume tobe the standard deviation between the scattering data and theCHC fits, 500 cm−1 for the data set in Fig. 3.

According to the CHC theory, the variation ofqc withquench depth for a critical polymer blend, is given by

fqcRgg2 = 3fxsT,Pd/xcsPd − 1g, s2d

wherexsT,Pd is the value of the Flory–Huggins interactionparameter at the quench conditions andxcsPd is the value ofthe Flory–Huggins interaction parameter at the critical pointfor the quench pressure.22 In Fig. 5 we compare the experi-mentally determined quench depth dependence ofqc fbasedon ITsqd andRsqd data given in Figs. 4sad and 4sbd, respec-tivelyg with theoretical predictions by plottingfqcRgg2 versusxsT,Pd /xcsPd. The agreement between theory and experi-ment is excellent, especially when one considers the fact thatthis agreement is obtained without any adjustable param-eters.

The agreement between the data and the CHC theoryseen in Figs. 2, 3, and 5 leaves no doubt that the data pre-sented here were acquired during the initial stages of spin-odal decomposition. We are thus poised to answer the ques-tion posed in the title. TheIsqd data obtained in the vicinityof the spinodal peakse.g., Fig. 2d were fit to a quadraticfunction of q and the locations of the peaks thus obtained,qm, are shown in Fig. 6. The scattering profiles fort,14 min did not contain peaks and are thus not representedin Fig. 6. We find that for all quenches in the two-phase

FIG. 4. CHC fitting parameters for all five quenches, the open symbolsrepresent two-phase quenches.sad ITsqd vs q. sbd Rsqd vs q. scd I0sqd vs q,including Isq,t=0d from Fig. 2 for reference. Inset insad shows the fullrange ofITsqd vs q for P=1.24 kbar.

FIG. 5. Quench depth dependence of the normalized critical scattering vec-tor, qcRg, shown asfqcRgg2 vs xsT,Pd /xcsPd. The solid line represents thetheoretical prediction of the CHC theory.

214903-3 Initial stages of spinodal decomposition J. Chem. Phys. 122, 214903 ~2005!

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region, qm is a monotonically decreasing function of timethroughout our experimental time window.23,24 The decreasein qm is also evident when the raw SANS data are examinedvisually se.g., Fig. 2d. We expect departures between theCHC theory and experiment because the theory does notaccount for the time dependence ofqm. The lack of agree-ment betweenI0sqd and Isq,t=0d fFig. 3 sinsetd and Fig.4scdg, and the lack of a well-defined peak inRsqd fFig. 4sbdgare examples of such departures.25 One expects a linearizedtheory to apply when changes in the dependent variablesscattering intensity in this cased are small. This may be thereason for the applicability of the CHC theory in the vicinityof qc but not in the vicinity ofqm.

The answer to the question “Does coarsening begin dur-ing the initial stages of spinodal decomposition?” is thus“Yes.”

We are not aware of any experimental data set that is atodds with this conclusion. In most cases, e.g., Refs. 6, 7, and9, the authors attribute their observation of decreasingqm totheir inability to access the initial stages of the phase sepa-ration. In Ref. 11 the authors argued that they had observed aregime whereqm was time independent. However, their datafFig. 3sad in Ref. 11g show an initial increase inqm with timefollowed by a short “plateau,” and then a decrease inqm. Theauthors concluded that the initial increase inqm was due toan artifact of the quenching processsthey used temperaturequenches to initiate phase separation as opposed to the pres-sure quenches used hered, and that smaller values ofqm weredue to the finite time that their sample spent at lower quenchdepths during the thermal quench. We argue that the ob-servedqm versust plateau is simply a crossover from theartifact-dominated regime to a spinodal decomposition-dominated regime whereqm is a monotonically decreasingfunction of time.

In summary, we studied the initial stage of phase sepa-ration in a critical blend of model polyolefins subjected to apressure quench. We show that coarsening begins immedi-ately after the quench and occurs throughout the initial stagesof spinodal decomposition. Our data suggest the need for arevised nonlinear theory.

ACKNOWLEDGMENTS

Acknowledgment is made to the donors of The Petro-leum Research Fund, administered by the ACS, and the De-partment of EnergysPolymer Program of the Materials Sci-ences Division of Lawrence Berkeley National Laboratorydfor support of this research. We acknowledge the support ofthe National Institute of Standards and Technology, U.S. De-partment of Commerce, in providing the facilities used inthis work. This work utilized facilities supported in part bythe National Science Foundation under Agreement No.DMR-9986442. Certain commercial equipment, instruments,or materialssor suppliers or softwared are identified in thispaper to foster understanding. Such identification does notimply recommendation or endorsement by the National In-stitute of Standards and Technology, nor does it imply thatthe materials or equipment identified are necessarily the bestavailable for the purpose.

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s1992d.19Isq=0d was obtained by fitting 1/Isqd versus q2 to a line limited to

0.040,q,0.088 nm−1, the low q regime forl=0.6 nm. See Ref. 17 formore details.

20For each quenchIsq,td was fit to the CHC equation for times 0, t, tfinal, where tfinal=225, 93, 87, 76, and 75 min, respectively, forP=0.83, 1.24, 1.65, 2.07, and 2.48 kbar. The fitting results are insensitive tothe exact value oftfinal.

21In Fig. 4sbd we have omitted the data forRsqd,−0.025 min−1 because wedo not have adequate time resolution in this range.

22Both xsT,Pd andxcsPd are linear functions of pressure. ForT=67.5 °C,x /xc=0.127Pskbard+0.885. See Ref. 17 for more details.

23Multiple scattering becomes important for long times, especially at higherquench depth. However, as shown in Ref. 24, the value ofqm remainsunaffected.

24J. A. Silas and E. W. Kaler, J. Colloid Interface Sci.257, 291 s2003d.25In our discussion ofI0sqd, ITsqd, andRsqd, we have restricted our attention

to the qualitative features of these curves. In past analyses, e.g., Ref. 10,deviations in the measuredRsqd and that predicted by the CHC theory areattributed to theq dependence of the Onsager coefficient. We have shownthat in the present system, substantial deviations between theory and ex-periment arise due to coarsening. Quantitative comparison between themeasuredI0sqd, ITsqd, andRsqd and theory can only be made after theoriesthat incorporate both coarsening andq-dependent Onsager coefficient areavailable.

FIG. 6. SANS peak positionqm vs time for the quenches into the two-phaseregion. The typical uncertainty inqm is shown.

214903-4 T. J. Rappl and N. P. Balsara J. Chem. Phys. 122, 214903 ~2005!

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