This is the accepted manuscript made available via CHORUS. The article has been published as: Similarity of the Signatures of the Initial Stages of Phase Separation in Metastable and Unstable Polymer Blends Amish J. Patel, Timothy J. Rappl, and Nitash P. Balsara Phys. Rev. Lett. 106, 035702 — Published 20 January 2011 DOI: 10.1103/PhysRevLett.106.035702 http://dx.doi.org/10.1103/PhysRevLett.106.035702 Similarity of the Signatures of the Initial Stages of Phase Separation in Metastable and Unstable Polymer Blends Amish J. Patel,1 Timothy J. Rappl,1 and Nitash P. Balsara1, 2, ∗ 1Department of Chemical Engineering, University of California, Berkeley, California 94720 2Materials Sciences Division and Environmental Energy Technologies Division, Lawrence Berkeley National Laboratory, University of California, Berkeley, California 94720 Time-resolved small angle neutron scattering was used to probe the initial stages of liquid-liquid phase separation in both, critical and off-critical binary polymer blends and the critical (qc) and most probable (qm) wave-vectors were identified for several quench depths. For the critical blend, the Cahn-Hilliard-Cook theory provides a framework for analyzing the data and explains the observed decrease in qm with time. For the off-critical blend, qm is independent of quench time, regardless of whether the quench is metastable or unstable. PACS numbers: 64.75.Va, 64.60.Q-, 82.60.Nh, 64.75.-g The standard phase diagram of binary liquid mixtures contains two curves: the binodal curve which is the lo- cus of the compositions of the coexisting phases and the spinodal curve which is the thermodynamic limit of the stability of the homogeneous phase. According to classi- cal treatments, the transformation of a homogeneous bi- nary liquid into a phase separated mixture after a quench from the single phase region of the phase diagram to the two-phase region occurs by two mechanisms: nucleation and growth, which occurs if the quench lies between the binodal and the spinodal curves (the metastable region), and spinodal decomposition if the quench lies within the spinodal curve (the unstable region) [1–3]. However, sub- sequent theoretical treatments that have examined the crossover from nucleation to spinodal decomposition sug- gest that the situation may be more complex [4, 5]. In this letter, we present time-resolved small angle neu- tron scattering (SANS) data from two polymer blends that are quenched from the one-phase to the two-phase region: a critical blend that is quenched directly into the unstable region and an off-critical blend quenched to various quench depths, both in the metastable and the unstable regions. The data were obtained during the ini- tial stages of phase separation, before coarsening sets in. SANS results from both the blends have been presented in two separate papers [6, 7]. The main purpose of this letter is to combine the data presented in these two pa- pers to illustrate that: 1) For the off-critical blend, there are no qualitative differences between quenches in the metastable and unstable regions. 2) Quenches for the critical and off-critical blends display differences in the time evolution of the most probable wave-vector. This raises questions concerning the role of the spinodal in demarcating distinct phase separation mechanisms. Both polymer blends are made up of high molecular weight liquid polyolefins: deuterated polymethylbutylene (dPMB) and hydrogenous polyethylbutylene (hPEB). The methods used to synthesize and characterize these nearly-monodisperse homopolymers are described in ref. [8]. The weight-average molecular weights, Mw, of the polymers that constitute the off-critical blend are 153 kg/mol (dPMB) and 197 kg/mol (hPEB). The radii of gyration, Rg, of both chains are 15.4 ± 1.0 nm. The re- sults reported here are for a blend with dPMB volume fraction, φdPMB = 0.20. For the critical blend, Mw of the polymers are 153 kg/mol (dPMB) and 131 kg/mol (hPEB) and Rg of both chains are 14.0 ± 1.0 nm. The critical composition, based on the Flory-Huggins theory [9, 10], is φdPMB = 0.493. All polymers are highly entan- gled, resulting in extremely slow phase separation, which can be tracked by time-resolved SANS. The azimuthally averaged coherent scattering inten- sity, I, as a function of the magnitude of the scattering vector, q, was obtained by methods reported in ref. [8]. Static SANS enabled the thermodynamic characteriza- tion of our system, while time-resolved SANS enabled the study of the early stages of phase separation. We use the Flory-Huggins theory to quantify the thermodynamic properties of our blends. The T and P dependence of the Flory-Huggins interaction parameter, χ, for the system of interest and the phase diagrams of the blends used in this paper have been reported previously [6–8]. We define the quench depth, κ(T,P) = χ(T,P)/χb −1, where χb is the value of χ at the binodal. In Table 1, we list the final T and P of the quenches as well as the corresponding κ-values, for both critical and off-critical blends. TABLE I: Experimental conditions and quench depths T(◦C) P(kbar) κ T(◦C) P(kbar) κ Critical blend Off-critical blend 70 1.24 0.04 59 1.52 0.26 70 1.66 0.09 59 1.72 0.30 70 2.07 0.15 59 2.00 0.34 70 2.48 0.20 59 2.34 0.40 Off-critical blend 59 2.69 0.46 59 0.90 0.16 59 3.03 0.52 59 1.10 0.19 49 2.69 0.59 59 1.31 0.23 40 2.69 0.73 In Figure 1, we show the calculated binodal curves for both the critical and off-critical blends, as well as 2 FIG. 1: Binodal curves for the critical (in red) as well as the off-critical (in blue) blend are shown along with the locations of the various quenches studied. Inset shows the full phase diagram with both the binodal and the spinodal curves. the locations of the quenches that were investigated. [N = √ N1N2 and φc is the critical composition; Ni is the number of monomers per chain for each polymer, based on a 0.1 nm3 reference volume.] The locations of the quenches relative to the mean-field spinodal curves are shown in the inset of Figure 1. Typical data obtained from the off-critical blend are shown in Fig. 2a, where we show the SANS profiles dur- ing a quench from the one-phase region to 2.34 kbar and 59◦C, which lies in the metastable region of the phase diagram. The arrow shows the location of the critical wave-vector, qc. The SANS profiles are characterized by a rapid increase in I(t) for q-values smaller than qc as well as the appearance of a peak in I(q). In contrast, I(q > qc) does not change with time in the early stage of phase separation. We have argued that the size of the critical nucleus in the nucleation and growth regime is of order 1/qc [11]. In Fig. 2b, we show SANS profiles for the off-critical blend during a quench to 2.69 kbar and 59◦C, which lies in the unstable region of the phase diagram. There is no qualitative difference between the SANS profiles resulting from quenches in the metastable and unstable regions (compare Figures 2a and 2b). In the inset of Fig. 2b, we show the SANS profiles obtained from the critical blend during an unstable quench to 1.66 kbar and 70◦C. The arrow shows the location of the critical wave-vector, qc, obtained by well-established methods for analyzing data during spinodal decomposition [12]. Data similar to those shown in Figure 2 were obtained from the critical and off-critical blends at various quench depths by controlling the final T and P . In Fig. 3a, we show the position of the peak in I(q), qm, as a function of quench time for several quenches using the off-critical blend. There is no change (within experimental uncertainty) in qm as a function of quench time for quenches into the metastable as well as the un- stable regions. The kinetics of phase separation of the FIG. 2: Time-resolved SANS intensity, I vs q measured for the off-critical blend during the (a) 2.34 kbar (metastable) quench, and the (b) 2.69 kbar (unstable) quench (shown at t = 3, 80, 102, 128, 154 and 173 min). Inset shows I(q) for the critical blend during the 1.66 kbar quench. The arrows indicate the location of the critical wave-vector, qc. off-critical blend is thus governed by two characteristic wave vectors, qc and qp; we take qp to be the average value of qm for each of the quenches shown in Fig. 3a. Quenches of the critical blend into the unstable region of the phase diagram also result in scattering profiles with peaks. However, in this case, qm decreases monotonically with time as shown in Fig. 3b. The Cahn-Hilliard-Cook (CHC) theory predicts the time evolution of the scattering profile of unstable sys- tems [12–15]: I(q,t) = IT (q)+[I0(q)−IT (q)] exp[2R(q)t]. In ref. [7], we showed that our measurements are in excellent agreement with the CHC equation, enabling the determination of I0(q), IT (q), and R(q) for each of the quenches. This provides the basis for under- standing the time evolution of qm. At short times, I(q,t) ≈ I0(q) + 2R(q)[I0(q) − IT (q)]t. Hence, we ex- pect a peak in I(q) to emerge at the value of q, for which the initial rate of increase of I(q,t), denoted by J(q) = 2R(q)[I0(q) − IT (q)] is maximum. While IT (q) has a pole at q = qc, R(qc) = 0 and J(q) is a contin- 3 FIG. 3: SANS peak position, qm, as a function of quench time for (a) off-critical and (b) critical blends. Solid curves are single-parameter exponential fits to the data with τ = 109, 89, 65 and 56 min for κ = 0.04, 0.09, 0.15 and 0.2 respectively. The typical uncertainty in qm is also shown. Inset shows the initial rate of increase in the scattering intensity, J(q), for the 1.66 kbar quench. The arrow indicates the location of qc. uous function of q. The pole at q = qc suggests the possibility of a maximum in J(q) in the vicinity of qc. This is indeed the case for all the quenches studied. We show J(q) for the 1.66 kbar quench in the inset of Fig. 3b. There is good agreement between the location of the maximum in J(q) and qc, shown by an arrow. At long times, the exponent in the CHC equation governs the behavior of I(q), resulting in a peak at a q-value, cor- responding to the maximum in R(q), which we call qp. Thus qm(t = 0) ≈ qc and qm(t →∞) = qp. The simplest function that captures the evolution of qm(t) from qc to qp is: qm(t) = qp + (qc − qp) exp(−t/τ). This functional form is used to fit the data from the critical blend using τ as the only adjustable parameter, and the fits are in good agreement with the data (Fig. 3b). As was the case for the off-critical blend, phase separation kinetics of the critical blend are also governed by two character- istic scattering vectors, qc and qp. While the methodolo- gies for determining qc and qp in critical and off-critical blends are different, the physical significance of the wave- vectors is the same: 1/qc represents the length scale of the smallest structures that grow during phase separation while 1/qp represents the length scale that dominates the phase separated structure formed during the early stages. Thus, the CHC theory provides a framework for an- alyzing I(q,t) from critical blends, and explains the de- crease in qm. However, there is no framework for analyz- ing time-resolved scattering data from nucleating blends. While we do not have a theoretical basis for the time in- variance of qm for off-critical quenches, a possible expla- nation might be that qm represents the average distance between nucleating centers that does not change as the nuclei grow during the early stages of phase separation. In Figures 4a and 4b, we plot qpRg and qcRg as a function of the quench depth, κ, for the critical and off- critical blends. The dependences of both characteristic wave-vectors on quench depth are similar for the crit- ical and off-critical blend except for the fact that the off-critical data are shifted to the right along the κ-axis. FIG. 4: Dimensionless characteristic wave-vectors, (a) qm, and (b) qc as a function of the quench depth, (χ − χb)/χb for the critical (open squares) and off-critical (filled circles) blends. The insets show qm and qc as a function of an alterna- tive definition of quench depth (χ−χb)/χs. The dashed lines mark the position of the spinodal for the off-critical blend. Typical uncertainties are also shown. 4 The shift is larger in the qpRg versus κ plot. In the in- sets of Figures 4a and 4b, we show the same data plotted versus κs = (χ−χb)/χs, where χs is the value of χ at the spinodal. The distinction between critical and off-critical systems is significantly reduced when κs is used to define quench depth. The data in Figures 2 and 4 indicate that the quali- tative features of the scattering profiles obtained during the initial stages of phase separation are independent of quench depth. While the spinodal may help in organiz- ing the data as we have shown in the insets of Fig. 4, it is does not demarcate different mechanistic regimes. Theoretical work of Binder and Stauffer [16] showed that the characteristic length-scale of phase separation in mix- tures of low molecular weight compounds was unaffected by the presence of the spinodal. It was argued that the spinodal, a mean-field concept, is destroyed by concen- tration fluctuations. Anticipation that the spinodal curve and mean-field behavior would be recovered in polymer blends was based on the scaling arguments of de Gennes, who showed that the Ginzburg Number, Gi, which quan- tifies the importance of fluctuations, follows a Gi ∼ 1/N scaling law [17]. More refined calculations by Wang [18] indicate that while this scaling law is correct in the large N limit, it is not obeyed for blends with N < 104. For Gi to be significantly smaller than unity (say 0.01), the values of N required are of order 104 for φ = 0.5, corre- sponding to component Mw ∼ 103 kg/mol. While exper- iments on such large systems may one day be carried out, they represent a small portion of parameter space with virtually no practical significance. There are thus com- pelling reasons for eliminating the spinodal curve from the reported phase diagrams of binary mixtures that un- dergo liquid-liquid phase separation as we have done in Figure 1. Additionally, theoretical studies that improve upon the Cahn-Hilliard treatment by using a non-linear lo- cal free energy [4, 5] report that there is no change in the mechanism of phase separation on crossing the spin- odal. Novick-Cohen uses a quartic free energy expression and reports that a parameter B, which is related to the higher derivatives of the free energy, governs the mecha- nism of phase separation [4]. The non-linear theory pre- dicts a crossover inside the spinodal from a nucleation- like mechanism near the spinodal (B >> 1) to clas- sical spinodal decomposition, deep within the spinodal (B << 1). It can be shown that for a Flory-Huggins blend, B ≈ 4.5 √ (χs −χb)/|χ−χs|. For the critical blend, χb = χs =⇒ B = 0 and thus classical spin- odal decomposition is observed. For the off-critical blend studied in this letter, the lowest value of B was 5.2, i.e. the B << 1 criterion was never reached and thus only the nucleation-like mechanism was observed. In conclusion, we have found similarities in the time- resolved scattering signatures of the initial stages of phase separation, with peaks observed in I(q) for all quenches and for both the critical and the off-critical polymer blends (Fig. 2). However, there are subtle dif- ferences, specifically in the evolution of qm with quench time (Fig. 3). For critical quenches, the CHC theory ex- plains the observed decrease in qm with time. In contrast, for the off-critical blend, qm is independent of quench time, regardless of whether the quench is metastable or unstable. The difference between critical and off-critical quenches as well as the similarities between metastable and unstable off-critical quenches can be explained in the context of theories that incorporate non-linear effects into the Cahn-Hilliard analysis. However, a theoretical frame- work for describing I(q,t) for nucleating blends, akin to the CHC theory for critical quenches, is still missing. We hope that the data in Figures 3 and 4 will guide the de- velopment of such a framework. We acknowledge the National Science Foundation (NSF, Grant No. CBET 0966632 and DMR-0966662), and Tyco Electronics for financial support, the National Institute of Standards and Technology, U.S. Department of Commerce, for providing the neutron research facilities used in this work (NSF, DMR-0454672), and Boualem Hammouda for his guidance. ∗ Electronic address: nbalsara@berkeley.edu [1] J. W. Gibbs, The Scientific Papers of J. Willard Gibbs (Dover, New York, 1961). [2] J. W. Cahn, J. Chem. Phys. 42, 93 (1965). [3] J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 31, 688 (1959). [4] A. Novick-Cohen, J. Stat. Phys. 38, 707 (1985). [5] U. Thiele, M. G. Velarde, and K. Neuffer, Phys. Rev. Lett. 87, 016104 (2001). [6] A. J. Patel and N. P. Balsara, Macromolecules 40, 1675 (2007). [7] T. J. Rappl and N. P. Balsara, J. Chem. Phys. 122, 214903 (2005). [8] N. P. Balsara, S. V. Jonnalagadda, C. C. Lin, C. C. 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