Time Evolution of Spinodal Interface in a Binary Polymer Mixture at the

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Geometrical Properties and Interface Dynamics: Time Evolution of Spinodal Interface in a Binary Polymer Mixture at the Critical Composition Hiroshi Jinnai,*,†,‡ Yukihiro Nishikawa,†,§ Hidetoshi Morimoto,† Tadanori Koga,†,| and Takeji Hashimoto*,†,⊥ Hashimoto Polymer Phasing Project, ERATO, Japan Science and Technology Corporation, and Department of Polymer Chemistry, Graduate School of Engineering, Kyoto University, Kyoto 606-8501, Japan Received July 29, 1999. In Final Form: January 20, 2000 The time-evolution of a three-dimensional (3D), bicontinuous interface of a phase-separated polymer mixture in the late stage spinodal decomposition (SD) process has been studied by laser scanning confocal microscopy (LSCM) and time-resolved light scattering. The time evolution of the interface between two coexisting phases developed via SD (“spinodal interface”) was quantitatively captured in 3D by using LSCM. On the basis of the differential geometry, probability densities of the local curvatures, i.e., the mean and Gaussian curvatures, of the spinodal interface have been experimentally evaluated. We found that a large portion of the interface formed in the late stage SD consists of a saddle-shaped surface, i.e., a hyperbolic surface. The probability densities were used to predict the dynamics of the spinodal interface. Two basic mechanisms are found to be important for the system to reduce interface area during the late stage SD. In addition, the probability densities of the curvatures at various times were successfully scaled by a time-dependent characteristic wavenumber, i.e., interface area per unit volume. This clearly proves that the time evolution of the spinodal interface, which characterizes the local structure of the system, is dynamically self-similar.

1. Introduction The pattern formation and dynamics of pattern during phase separation processes have been a fascinating subject of many experimental and theoretical studies over the past decades concerning nonlinear, nonequilibrium phenomena.1,2 If a binary mixture is rapidly quenched from the single-phase region to the spinodal region of the phase diagram by changing thermodynamic variables, such as temperature and pressure, the mixture gets thermodynamically unstable and separates into two phases via spinodal decomposition (SD).1-3 If the volume fractions of the two phases are equal to 0.5 (“isometric case”), the phase-separated structure is implied to be periodic and bicontinuous in aid of theories,4 experiments,5,6 and computer simulations.7-9 * To whom correspondence should be addressed. † Hashimoto Polymer Phasing Project, ERATO, Japan Science and Technology Corporation. ‡ Present address: Department of Polymer Science and Engineering, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan. § Present address: Structural Biophysics, The Institute of Physical and Chemical Research (RIKEN), 1-1-1 Kouto, Mikaduki, Sayo, Hyogo 679-5148, Japan. | Present address: Department of Chemistry, State University of New York at Stony Brook, Stony Brook, New York 11794-3400. ⊥ Department of Polymer Chemistry, Graduate School of Engineering, Kyoto University. (1) Gunton, J. D.; San Miguel, M.; Sahni, P. S. In Phase Transition and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.; Academic Press: New York, 1983; Vol. 8, p 269. (2) Hashimoto, T. In Materials Science and Technology; Cahn, R. W., Haasen, P., Kramer, E. J., Eds.; VCH: Weinheim, 1993; Vol. 12, p 251. (3) Binder, K. In Materials Science and Technology A Comprehensive Treatment; Cahn, R. W., Haasen, P., Kramer, E. J., Eds.; Phase Transformations in Materials, VCH: Weinheim, 1990; Vol. 5, p 405. (4) Furukawa, H. Phys. Rev. Lett. 1979, 43, 136. (5) Nishi, T.; Wang, T. T.; Kwei, T. K. Macromolecules 1975, 18, 227. (6) Reich, S. Phys. Lett. 1986, 114A, 90.

A considerable number of studies have been carried out to investigate the SD processes mainly in the Fourier space using scattering techniques, e.g., light scattering (LS) 10-16 and small-angle neutron scattering (SANS).17-22 As a result of these studies, a great deal of information on the time evolution of the phase-separated structures have been obtained: various characteristic lengths of the structure, e.g., characteristic length of the periodic structure, inverse of interface area per unit volume, and interface thickness, have been statistically discussed as a function of time, t. Analysis of the time evolution of such characteristic lengths gave the following notable features of the evolving structures: (i) evolution of concentration fluctuations in the early stage SD can be well described by the linearized theory,23,24 (ii) the structures grow with dynamical selfsimilarity in the late stage SD, viz., forms of the structures at various times are statistically identical while the characteristic wavelength of the periodic structure increases with t (“dynamical scaling hypothesis” proposed by Binder and Stauffer).13,16,25-31 (7) Chakrabarti, A.; Toral, A.; Gunton, J. D.; Muthukumar, M. Phys. Rev. Lett. 1989, 63, 2072. (8) Oono, Y. IEICE TRANSACTIONS 1991, E74, 1379. Shinozaki, A.; Oono, Y. Phys. Rev. E 1993, 46, 2622. (9) Koga, T.; Kawasaki, K. Physica A 1993, 196, 389. (10) Hashimoto, T.; Kumaki, J.; Kawai, H. Macromolecules 1983, 16, 641. (11) Snyder, H. L.; Meakin, P. J. Chem. Phys. 1983, 79, 5588. (12) Izumitani, T.; Hashimoto, T. J. Chem. Phys. 1985, 83, 3694. (13) Hashimoto, T.; Itakura, M.; Hasegawa, H. J. Chem. Phys. 1986, 85, 6118. Hashimoto, T.; Itakura, M.; Shimidzu, N. J. Chem. Phys. 1986, 85, 6773. (14) Sato, T.; Han, C. C. J. Chem. Phys. 1988, 88, 2057. (15) Wiltzius, P.; Bates, F. S.; Heffner, W. R. Phys. Rev. Lett. 1988, 60, 1538. (16) Hashimoto, T.; Takenaka, T.; Jinnai, H. J. Appl. Crystallogr. 1991, 24, 457. (17) Higgins, J. S.; Fruitwala, H. A.; P. E. Tomlins Br. Polym. J. 1989, 21, 247.

10.1021/la991024q CCC: $19.00 © 2000 American Chemical Society Published on Web 03/28/2000

Time Evolution of a Spinodal Interface

In the late stage of SD, local compositions of the coexisting phases have already reached equilibrium values described by the phase diagram.19 Thus, an excess free energy of the system is localized at the interface between the two phases. Structure growth is driven so as to minimize the interfacial free energy of the system and hence to decrease the interface area. Under such conditions, differential geometrical properties of the interface, e.g., the interface curvatures, interface area, and their time evolution are most important physical quantities. Note that the interface curvatures refer to the mean, H, and Gaussian, K, curvatures defined later (eq 7). Nevertheless, at this moment, little information about the geometrical properties of the interface developed in the late stage of SD (“spinodal interface”) is available, except for some pieces of information reported based on a few experimental works.32,33 Recently, some theoretical investigations start focusing on the curvatures as one of the important parameters to characterize the phase-separated structures.34,35 In the scattering experiments, a structure factor, S(q), is often used to statistically characterize the phaseseparated structure, where q is the magnitude of the scattering vector or the wavenumber of a Fourier mode of the structure. Although the interface curvatures and S(q) are implicitly related to each other, a general relationship between these two is not at all clear. Although evaluation of mean radius of the interface curvature, Rm, on the basis of theories given by Kirste and Porod36 and Tomita37 may be possible, it is often difficult to apply these theories to the scattering data.31,38,39 An alternative approach to obtain the curvature information from scattering data is to use a model first proposed by Cahn40 and later elaborated by Berk.41 The model was recently used to evaluate the area-averaged Gaussian curvature (the value averaged over all interface area), 〈K〉, of a microphase-separated bicontinuous structure of threecomponent microemulsions, i.e., an oil/water/surfactant system.42 We here note that 〈‚‚‚〉 denotes the area-averaged value. The same model was recently shown to be applicable (18) Schwahn, D.; Hahn, K.; Streib, J.; Springer, T. J. Chem. Phys. 1990, 93, 8383. (19) Jinnai, H.; Hasegawa, H.; Hashimoto, T.; Han, C. C. Macromolecules 1991, 24, 282. (20) Schwahn, D.; Janssen, S.; Springer, T. J. Chem. Phys. 1992, 97, 8775. (21) Jinnai, H.; Hasegawa, H.; Hashimoto, T.; Han, C. C. J. Chem. Phys. 1993, 99, 4845. Jinnai, H.; Hasegawa, H.; Hashimoto, T.; Han, C. C. J. Chem. Phys. 1993, 99, 8145. (22) Motowoka, M.; Jinnai, H.; Hashimoto, T.; Qiu, Y.; Han, C. C. J. Chem. Phys. 1993, 99, 2095. (23) Cahn, J. W. Acta Metall. 1962, 10, 179. (24) Cook, H. E. Acta Metall. 1970, 18, 297. (25) Binder, K.; Stauffer, D. Phys. Rev. Lett. 1974, 33, 1006. (26) Bates, F. S.; Wignal, G. D.; Koehler, W. C. Phys. Rev. Lett. 1985, 55, 2425. (27) Kyu, T.; Saldanha, J. M. Macromolecules 1988, 21, 1021. (28) Nose, T. In Space-Time Organization in Macromolecular Fluids; Tanaka, F., Doi, M., Ohta, T., Eds.; Springer: Berlin, 1989; p 40. (29) Tomlins, P. E.; Higgins, J. S. J. Chem. Phys. 1989, 90, 6691. (30) Izumitani, T.; Takenaka, M.; Hashimoto, T. J. Chem. Phys. 1990, 92, 3213. (31) Takenaka, M.; Izumitani, T.; Hashimoto, T. J. Chem. Phys. 1990, 92, 4566. (32) Jinnai, H.; Koga, T.; Nishikawa, Y.; Hashimoto, T.; Hyde, S. T. Phys. Rev. Lett. 1997, 78, 2248. (33) Jinnai, H.; Nishikawa, Y.; Hashimoto, T. Phys. Rev. E 1999, 59, R2554. (34) Kawakatsu, T.; Kawasaki, K.; Furusaka, M.; Okabayashi, H.; Kanaya, T. J. Chem. Phys. 1993, 99, 8200. (35) Matsen, M. W.; Bates, F. S. J. Chem. Phys. 1997, 106, 2436. (36) Kirste, B.; Porod, G. Kolloid-Z. Z. Polym. 1962, 184, 1. (37) Tomita, H. Prog. Theor. Phys. 1986, 75, 482. (38) La¨uger, J.; Lay, R.; Maas, S.; Gronski, W. Macromolecules 1995, 28, 7010.

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to evaluate 〈K〉 for a polymer mixture undergoing SD.43 Although the above model may be beneficial to extract information about the curvatures from the reciprocal-space data, its major drawbacks are as follows: (i) the method is applicable only to a class of structures satisfying Gaussian random field, (ii) the mean curvature, H, of the interface is assumed to be zero, and (iii) the Gaussian curvature evaluated is only an area-averaged value. Therefore, up to now, a general determination of the interface curvatures, 〈H〉 and 〈K〉, and distribution of H and K from the scattering technique is impossible. On the other hand, 3D images on phase-separated structures, if they could be obtained, should capture the local structure of the interface and hence the local curvatures and their distributions, as well as its global topology.44 Laser scanning confocal microscopy (LSCM) 45 is shown to be an excellent tool to capture the 3D interface structure of polymer blends.46 There are a few applications of LSCM to polymer systems for the structural investigation of the phase-separated structures. Verhoogt et al. first reported an application of LSCM to a polymer mixture.47 Their work, however, was instrument oriented and thus did not provide much insight into the phase-separated structure of the binary mixture itself. White and Wiltzius discussed a mechanism of the phase separation in an off-critical polymer mixture through the time evolution of size distribution of spherical domains in 3D.48 LSCM was applied to investigate the phase-separated structure of polymer mixtures at the critical composition, demonstrating that the structure is truly interconnected and periodically arranged, forming a 3D “maze”.46,49,50 These studies are considered to take full advantage of 3D reconstruction intrinsic to LSCM. We note that some other studies rather use the excellent depth resolution of confocal microscopy. Surface and bulk structures of the solvent-cast films of a polymer mixture were investigated as a function of distance from the air polymer surface.51,52 (39) The first application of the theories to the scattering profile for the estimation of the interface curvatures of the polymer mixtures in the late stage SD was reported by Takenaka et al. (ref 31). They evaluated the mean absolute radius of the interface curvature, Rm, from a relation qcRm = 1, where qc is a crossover wavenumber at which the asymptotic behavior of S(q) changes from q-n (n > 6) to q-4. This relation was experimentally supported by a combined real space analysis and reciprocal space analysis by La¨uger et al. (ref 38). However, a more rigorous evaluation of Rm using laser scanning confocal microscopy (LSCM) in three-dimensional (3D) real space illuminated that this relationship was only good qualitatively and that, quantitatively, the relation qcRm = 3.2 holds (ref 32). (40) Cahn, J. W. J. Chem. Phys. 1965, 42, 93. (41) Berk, N. F. Phys. Rev. Lett. 1987, 58, 2718. (42) Chen, S. H.; Lee, D. D.; Kimishima, K.; Jinnai, H.; Hashimoto, T. Phys. Rev. E 1996, 54, 6526. (43) Jinnai, H.; Hashimoto, T.; Lee, D.; Chen, S. H. Macromolecules 1997, 30, 130. (44) A major difficulty in taking such 3D images of the phaseseparated structure is the following: All the structures at different depth within the focal depth contribute to the observed image, which leads overlaps of the structures if they are smaller compared with the focal depth. This experimental difficulty is even emphasized if the structure gets complicated. This is not the case if the structure becomes large enough to be comparable in size with the thickness of the specimen. However, the phase separation process becomes two-dimensional (2D) in nature under such circumstances. (45) Wilson, T. In Confocal Microscopy; Wilson, T., Ed.; Academic Press: London, 1990; p 1. (46) Jinnai, H.; Nishikawa, Y.; Koga, T.; Hashimoto, T. Macromolecules 1995, 28, 4782. (47) Verhoogt, H.; van Dam, J.; Posthuma de Boer, A.; Draaijer, A.; Houpt, P. M. Polymer 1993, 34, 1325. (48) White, W. R.; Wiltzius, P. Phys. Rev. Lett. 1995, 75, 3012. (49) Ribbe, A. E.; Hashimoto, T.; Jinnai, H. J. Mater. Sci. 1996, 31, 5837. (50) Hashimoto, T.; Jinnai, H.; Nishikawa, Y.; Koga, T.; Takenaka, M. Prog. Colloid Polym. Sci. 1997, 106, 118.

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Figure 1. Schematic representation of (a) flat and (b) wavy surfaces. The wavy surface equally has concave and convex parts so that area-averaged mean, 〈H〉, and Gaussian, 〈K〉, curvatures for both surfaces are the same: 〈H〉 ) 〈K〉 ) 0. Although for the surface (b) 〈K〉 ) 0 may not be obvious, it is proven as follows. Let A be the region of the surface concerned, and the boundary of A be ∂A. The surfaces shown in the figures are such examples. According to the Gauss-Bonnet theorem, any surface bound by a closed curve satisfies the equation 2π ) ∫AK da + I∂A κg ds, + ∑ii, where κg is the curvature along ∂A and i is the exterior angle of the ith vertex of ∂A if it exists. For the case of surface b, κg is zero because each edge is a straight line, and there are four vertexes of π/2 (in radians). Thus the second and the third terms on the right-hand side are 0 and 2π, respectively, so that the term ∫AK da should be zero. ∫AK da can be written as 〈K〉A. Because A is a nonzero positive value for any surface, 〈K〉 is zero for surface b.

In our previous paper,32 we showed quantitative measurements of area-averaged curvatures of the interface of a bicontinuous phase-separated polymer mixture on the basis of the differential geometry from the 3D LSCM images, which demonstrated that 〈H〉 = 0 and 〈K〉 < 0: The interface is, on average, hyperbolic. This intriguing result further stimulated our interest on whether the interface has a surface geometry approximately close to a group of surfaces called “periodic minimal surfaces”, mathematically predicted surfaces having least surface free energy.53 Accurate characterization of the interface of the bicontinuous structure requires measurements of distributions of local curvatures (i.e., probability densities of curvatures). Figure 1 demonstrates the significance of the curvature distribution. Part a of the figure shows a flat surface, while a wavy surface is drawn in part b. Both surfaces have the same surface geometry in terms of the area-averaged curvatures, i.e., 〈H〉 ) 0 and 〈K〉 ) 0, although the local geometrical shape is obviously different between the two surfaces. In the present paper, we will present (1) a detailed experimental evaluation of the probability densities of the curvatures and (2) a discussion on interface dynamics during the SD on the basis of the time evolution of the distributions of curvatures in section 4.4. The concept of the dynamical scaling hypothesis in terms of the local structure, i.e., geometrical properties of the interface, will be also tested.54 (51) Li, L.; Sosnowski, S.; Chaffey, C. E.; Balke, S. T.; Winnik, M. A. Langmuir 1994, 10, 2495. (52) Kumacheva, E.; Li, L.; Winnik, M. A.; Shinozaki, D. M.; Cheng, P. C. Langmuir 1997, 13, 2483. (53) Hilbert, D.; Cohn-Vossen, S. Geometry and the Imagination; Chelsea Publishers: New York, 1952. (54) The component polymers used in this work are nearly symmetric in terms of molecular weight and mobility, so that the so-called viscoelastic effect68 is believed to be insignificant and hence the phase separation behavior at a long time limit is expected to be similar to that of the simple liquid mixtures except for differences in the spatial and temporal scales (see ref 2).

Figure 2. (a) Time evolution of light scattering intensity distribution with wavenumber, q, for the DPB/PB mixture during the course of SD with ∆T ) T - Ts = 70 K (T ) 40 °C and Ts = 110 °C) or T ≡ (χ - χs)/χs = 1.1. The subscript s for T (temperature) and χ (Flory-Huggins segmental interaction parameter) represents the corresponding value at the spinodal point. (b) A double logarithmic plot of characteristic wavenumber, qm(t), determined from scattering peak position and the maximum intensity, Im(t), as a function of time, t. Reduced time, τ (≡t/tc), is also shown. tc is the characteristic time of the mixture at T ) 40 °C (tc = 860 s) determined by analyzing the early stage SD process on the basis of the Cahn’s linearized theory of SD.

2. Experimental Section 2.1. Samples. A binary mixture of deuterated polybutadiene (DPB) and polybutadiene (PB) is used. The weight-average molecular weight (Mw) and polydispersity (Mw/Mn) of DPB were, respectively, 143 × 103 and 1.12 (Mn denotes the number-average molecular weight). Mw and Mw/Mn for PB were 95 × 103 and 1.07, respectively. The 1,2-contents of PB and DPB are 20 and 32%, respectively, which were determined from 1H- and 13C-NMR. Anthracene was attached as a label to PB for a contrast enhancement in microscopy. Carboxyl groups were introduced by the three-dipole addition of p-carboxybenzhydroxamyl chloride in the presence of triethylamine.55 Anthracene was then attached to the PB using the reaction between the carboxyl group and 9-anthryldiazomethane at room temperature (ca. 3.8 anthracene molecules per PB chain on average). The critical composition of DPB is 46 vol % according to the Flory-Huggins lattice theory.56 The mixture of DPB and PB having the critical composition was used in the present study, unless otherwise stated. To prepare the test specimens, the mixtures were dissolved in benzene to form ca. 7 wt % solution and then lyophilized. The mixture after lyophilization was homogenized by mechanical mixing 57 and (55) Huigen, R. Angew. Chem., Int. Ed. Engl. 1963, 2, 565. (56) Flory, P. J. Principles of Polymer Chemistry; Cornell University: New York, 1953. (57) Hashimoto, T.; Izumitani, T.; Takenaka, M. Macromolecules 1989, 22, 2293.


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Table 1. Parameters Used for LSCM Experiments and Some Characteristic Parameters Obtained from Three-Dimensional Image Analysis t (min)

Λm(t) (µm)

φ

RI

Σ(t) (µm-1)

∆F (µm)

∆Z (µm)

L (µm)

1350 1675 2880 4610 7365 8610

8.85 ( 1.1 11.5 ( 1.5 19.3 ( 2.2 31.4 ( 3.1 47.2 ( 11.0 62.8 ( 5.3

0.52 ( 0.01 0.48 ( 0.01 0.49 ( 0.02 0.51 ( 0.01 0.53 ( 0.02 0.52 ( 0.02

0.076 0.065 0.070 0.056 0.049 0.058

0.304 ( 0.044 0.231 ( 0.006 0.147 ( 0.006 0.091 ( 0.002 0.067 ( 0.003 0.044 ( 0.001

0.87a 0.85a 1.02b 0.92b/0.85a 0.85a 2.22a

0.50a 0.50a 0.57b 0.57b/0.50a 1.00a 2.00a

0.176 0.213 0.328 0.655 0.638 0.971

a

Oil immersed lens. b Water immersed lens.

Figure 3. 2D sliced images of the DPB/PB mixture phase separated at 40 °C for 1675 min. A series of the images were taken with ∆F = 0.85 µm and ∆Z ) 0.5 µm along the depth direction at the same area. The depth of the image increases in the order of image (a) to (f) with an increment of 1.0 µm. The point marked by an arrow corresponds to the same pixel position in all images. placed between coverslips. A ring spacer of 0.2-0.5 mm thickness was used to adjust the thickness of the sample to prevent possible space-confinement effects on the growth of the phase-separated structure. The mixture was annealed at 40 °C for six different time intervals t (t ) 1280, 1675, 2880, 4610, 7365, 8610 min, all in the late stage SD) for the LSCM observation (see section 4.1). Timeevolution of light scattering intensity in the early stage SD was analyzed as described elsewhere10,12,58 in the context of Cahn’s linearized theory 40 in order to evaluate the characteristic time of the mixture at 40 °C. The characteristic time, tc, was 860 s, and hence the reduced times defined by, τ ≡ t/tc, span from 89 to 598. SANS experiments were carried out in order to estimate the spinodal temperature, Ts, of the DPB/PB mixtures. A DPB/PB mixture (8.8/91.2 vol %) was prepared in the same manner as described above. The SANS intensity distributions at 12 temperatures from 100 to 60 °C with the precision of (0.2 °C were measured. Following the conventional scheme59,60 on the basis of the random-phase approximation (RPA),61 we obtained temperature dependence of Flory’s interaction parameter, χ. The obtained χ showed a good linear relationship against the inverse of the absolute temperature: (58) Hashimoto, T.; Itakura, M.; Shimidzu, N. J. Chem. Phys. 1986, 85, 6773. (59) Sakurai, S.; Hasegawa, H.; Hashimoto, T.; Glen Hargis, I.; Aggarwal, S. L.; Han, C. C. Macromolecules 1990, 23, 451. (60) Jinnai, H.; Hasegawa, H.; Hashimoto, H.; Han, C. C. Macromolecules 1992, 25, 6078.

χ ) -3.7 × 10-3 + (1.8/T)

(1)

Since the coefficient related to the absolute temperature is positive, the DPB/PB mixture studied here exhibits an upper critical solution temperature (UCST) type phase diagram. Although χ depends on the composition for DPB/PB mixtures,21 we ignored it for a rough estimation of the spinodal temperature, Ts, at the critical composition using eq 1: Ts ≡ 110 °C. 2.2. Laser Scanning Confocal Microscopy. The phaseseparated structures of the DPB/PB mixtures were observed at room temperature by LSCM (Carl Zeiss, LSM410) with an incident laser beam of wavelength λ of 364 nm. A band-pass filter (395-440 nm) installed in front of the detector (photomultiplier) was used to detect only fluorescence from the anthracene molecules that were labeled only to the PB. The PB phase was recognized as a bright phase under the fluorescent LSCM (see Figure 3 later). An oil-immersed ×40 (Plan-Neofluar, Carl Zeiss) objective with numerical aperture (NA) of 1.3 and/or a water-immersed ×40 (C-Apochromat, Carl Zeiss) with NA ) 1.2 was used. Intensity of fluorescence from a particular point in a focal plane (x-y or lateral plane) at a given depth z, I(x,y,z), was recorded by the detector behind a pinhole (“confocal pinhole”) which efficiently excludes out-of-focus light, thus achieving an excellent depth resolution. Here, z-axis denotes the optical axis of the microscope. The laser beam was scanned in the lateral plane, giving rise to a 2D optically sliced image comprised of N2 (61) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University: New York, 1979.

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Figure 4. Fluorescence LSCM images showing time evolution of bicontinuous structures in the DPB/PB mixture at critical composition. The image of 100 × 100 × 50 µm3 was displayed at (a) 1675, (b) 2880, and (c) 4860 min after the onset of SD at 40 °C. Parts a-c show a “solid model”: The images show only the PB-rich phase labeled with anthracene and the other phase (DPB-rich) is left empty. A part of the structure in (b) was removed to show a cross section of the 3D structure (colored in yellow). Parts d-f represent the interface between two coexisting phases for the 3D volumes corresponding to (a) to (c), respectively. The interface toward the DPB-rich phase is blue, while the other side is yellow. pixels2, where N is the number of pixels along the edge of the image. N ) 512 was used for most of the measurements. Images are represented by the fluorescent intensity on two-dimensional arrays of pixels (picture elements). Each pixel has 8-bit resolution for the intensity, ranging in value from 0 to 255. Size of the square pixel, L, used in the LSCM measurement is listed in Table 1. The objective was then moved along the z-axis with the precision of (0.05 µm to obtain I(x,y,z) at another depth. Thus, a series of 2D sliced images with an increment of ∆Z along the z-axis was obtained, which was later used to reconstruct the original volume object. Table 1 lists ∆Z values used in the present study. All optical slices were at least 20 µm distant from the coverslips, to avoid possible surface effects on the phase-separated structure. The depth resolution, which is the full-width-at-half-maximum (fwhm) of optical sections, ∆F, is a complicated function of λ, NA, magnification of the objective, radius of confocal pinhole, refractive index of the object, etc. (see Appendix A). The actual values of ∆F estimated for the LSCM measurements are also listed in Table 1. 2.3. Time-Resolved Light Scattering. The time evolution of the phase-separated structure was also followed in Fourier space by using time-resolved light scattering (TRLS) as detailed elsewhere.10 Two different wavelengths, i.e., 488 nm (Ar laser) and 632.8 nm (He-Ne laser), were used as incident lasers in order to expand an observable range of wavenumbers, q, which is defined by

q ≡ (4π/λLS) sin(θ/2) Here λLS and θ represent the wavelength of the incident laser and the scattering angle in the medium, respectively. Refractive indices of DPB and PB at room temperature were measured to be 1.5095 and 1.5186, respectively.

3. Results 3.1. Time Evolution of TRLS Profiles. Figure 2a shows time evolution of TRLS profiles at time t, I(q,t), of the DPB/PB mixture at 40 °C. Shortly after the onset of

SD, a scattering maximum appeared at the wavenumber, qm(t). This reflects periodicity in the concentration fluctuations developed in the system via SD. A dominant Fourier mode of the concentration fluctuations with a wavelength, Λm(t), is related to qm(t) through Λm(t) ) 2π/ qm(t). As the phase-separated structure grows with t, qm(t) shifted toward small q and the maximum intensity, Im(t), got more intense because Λm(t) and also the amplitude of the concentration fluctuations increased. In the late stage SD, the amplitude of the concentration fluctuations reaches an equilibrium value,19 while only Λm(t) increases. Figure 2b demonstrates time evolution of qm(t) and Im(t) in a double-logarithmic scale. Open symbols represent data taken by TRLS. Filled circles in Figure 2b show qm(t) calculated from the Fourier transformation of a 3D phaseseparated structure obtained from LSCM (described later in section 4.1). In the late time region, both qm(t) and Im(t) showed power law behaviors, i.e., qm(t) ∼ t-R and Im(t) ∼ tβ with the exponents R and β being equal to 1 and 3, respectively. 3.2. Phase-Separated Structure in Real Space As Observed by LSCM. Figure 3 shows a series of 2D sliced LSCM images (∆F = 0.85 µm and ∆Z ) 0.5 µm) obtained from the DPB/PB mixture at t ) 1675 min. Parts a-f show 2D slices in the same region of the specimen at different depths (depth gets deeper in alphabetical order by 1.0 µm steps). Using the excellent depth resolution of LSCM, one can actually investigate how the phaseseparated domains are spatially organized. The bright circular domain marked by an arrow, i.e., a part of the PB phase, in Figure 3a was surrounded by the dark domain, i.e., the DPB phase. It appeared isolated from the neighboring PB domains. As the focal plane got deeper, this part was found to be connected to the neighboring PB domains (see parts c and d at depths of 1.0 and 1.5 µm


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Figure 5. Comparison of scaled structure factors, F(x;t), obtained from TRLS (part a) and from LSCM (part b). F(x) shown by solid lines in parts a and b are identical and obtained from the 3D computer simulation based on the TDGL equation with hydrodynamic interactions.

relative to part a). This part of PB domain shown in part e at the depth level of 4.0 µm relative to the image in part a appeared to be a junction whose shape looked just like a “tetrapod”. The interconnected domains have interfaces which are not normally straight but curved, hence the marked region in Figure 3a became a DPB domain at 11 µm below (although not shown). Thus, the two domains are really interconnected forming a 3D maze with an average spacing of Λm(t). 3.3. Time Evolution of 3D Phase-Separated Bicontinuous Structure. The 2D LSCM slices were “binarized” in aid of computer image analysis (see Appendix B) and were stuck together to make a 3D representation of the structures (“reconstruction”). The “Marching Cubes algorithm”62 was used to model the interface between the DPB and PB components by contiguous polygons in the reconstruction process. Details of the reconstruction can be found elsewhere.62,63 The interface area was estimated by summing the area of the polygons, from which the interface area per unit volume, ∑(t), was estimated. The volume fraction of one of the phases, φ, was also measured. We note that the volume fraction of the DPB-rich phase remained 0.5 over the entire range of our annealing time in the LSCM experiments, i.e., from 1280 to 8610 min (see Table 1). In this system, φ was designed to be 0.5 by preparing the mixture with the critical composition.64 Figure 4 displays time evolution of the 3D phaseseparated structures (100 × 100 × 50 µm3 box size). The representative phase-separated structures at three different annealing times are shown in the figure, all corresponding to the structures in the late stage SD. In Figure 4, two different ways of image representation were (62) Lorensen, W. E.; Cline, H. E. Computer Graphics SIGGRAPH′87 1987, 21, 163. (63) Nishikawa, Y.; Jinnai, H.; Koga, T.; Hashimoto, T.; Hyde, S. T. Langmuir 1998, 14, 1242. (64) The compositions of DPB in the two coexisting phases were determined to be ca. 18 (DPB-poor phase) and 74 vol % (DPB-rich phase), respectively.

employed for the phase-separated structures. The upper three phase-separated structures are displayed as a “solid model” in which the blue glossy part and the transparent part represent PB and DPB domains, respectively. The bottom three represent the interface between the two coexisting phases which, respectively, correspond to the three phase-separated structures presented above: one side of the interface is blue and the other side is yellow. We note that time required for taking a series of LSCM images necessary to construct 3D volume object (e.g., the volume in Figure 4) was relatively short (ca. 6 min) so that Λm changed by, at most, ca. 0.0378 µm (dΛm(t)/dt ≡ 0.0063 µm/min) during the data acquisition of the image, which is much smaller than the resolution limit of LSCM. Note that qm(t) ∼ 103t-1 in Figure 2b in the long time limit. 4. Discussion 4.1. Scaled Structure Factor. The time evolution of the scattering intensity, I(q,t), is given by

I(q,t) ) C〈η(t)2〉Λm(t)3S(q,t) ∼ C〈η(t)2〉qm(t)-3S(q,t) (2) where C is a proportionality constant. 〈η(t)2〉 is the mean squared refractive index fluctuations proportional to the squared amplitude of concentration fluctuations, and S(q,t) is the scattering structure factor at a given t during SD which characterizes the form of the phase-separated structure. We now define a scaled structure factor, F(x,t), by

F(x,t) ≡ I(x,t)qm(t)3 ∼ C〈η(t)2〉S(x,t)

(3)

x ≡ q/qm(t)

(4)

with

In the late stage SD, since 〈η(t)2〉 becomes a constant equilibrium value, F(x,t) does not depend on t, if S(x,t)

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becomes a time-independent universal scaling function, S(x), which depends only on the form of the phaseseparated structure. Consequently, in the late stage SD, F(x,t) can be rewritten as F(x) (“universal scaled structure factor”), if this criterion is satisfied. F(x,t) experimentally obtained from TRLS (152.9 min < t < 1069 min or 11 < τ < 75 is shown in Figure 5a. The scaled structure factor from LSCM (1280 min < t < 4610 min or 89 < τ < 322) (shown in Figure 5b) was calculated in the following way: I(qx,qy;t) was first obtained from the 3D image by taking the square of the magnitude of its 3D Fourier transformation where qi (i ) x and y) denotes x and y axes in Fourier space. |q| ≡ 2πj/NL (j ) 0, 1, 2, ‚‚‚, N). I(qx,qy;t) thus obtained was then orientationally averaged in the qx-qy plane to calculate I(q,t). The abscissa has been scaled by qm(t) and the ordinate by Im(t) to obtain F(x,t). Demonstrating in Figure 5, F(x,t) fell on to a single master curve both for the TRLS and LSCM results, suggesting that the form of the phase-separated structures at various times can be scaled with the single length parameter, Λm(t): The structures grow with self-similarity. The solid lines in Figure 5 show the F(x) obtained from a 3D computer simulation based on the timedependent Ginzburg-Landau (TDGL) equation with incorporating hydrodynamic interactions.9,65,66 F(x) values both from TRLS and LSCM were in excellent agreement with the theoretical structure factor over a large intensity scale, as large as 5 orders of magnitude, and a wide spatial scale, as wide as 2 orders of magnitude. This result demonstrates that F(x) from the two independent methods (TRLS and LSCM) are quantitatively identical, hence, ensuring that the LSCM captures the real structural entity. Moreover, the TDGL theory gives a physical basis to the experimentally obtained bicontinuous phaseseparated structure developed via SD. Furthermore, the unique scaled structure factor manifests the spongelike real-space structure shown in Figure 4. As presented in Figure 2b, qm(t) and Im(t) exhibited a power law behavior, i.e., qm(t) ∼ t-R and Im(t) ∼ tβ. According to eq 2, Im(t) can be written as

Im(t) ≡ I(qm(t),t) ∼ C〈η(t)2〉qm(t)-3S(qm(t),t) ) C〈η(t)2〉qm(t)-3 (5) Note that S(qm(t),t) ) 1 was used in eq 5. In the late stage SD, 〈η(t)2〉 becomes constant, thus Im(t) ∼ qm(t)-3, i.e., Im(t) ∼ tβ ∼ qm(t)-3 ∼ t3R. We should have the following relation between the exponents on Im(t) and qm(t) (section 3.1):

β ) 3R

(in the late stage SD)

(6)

We had R ) 1 and β ) 3 (at τ > 20), which fulfills the above relation, over the entire time range of our LSCM experiments as demonstrated in Figure 2b. It is important to point out that the exponent R ) 1 is also predicted from the computer simulation9 work based on the TDGL equation with the hydrodynamic interactions.65 We note that the TDGL simulation quantitatively describes the phase separation behavior of simple liquid mixtures as well as polymer mixtures67 free from the viscoelastic effects.68 Therefore, the DPB/PB mixture used in the present study can serve as a model for binary liquid mixtures.67 4.2. Measurements of Distribution of Curvatures of Spinodal Interface. 4.2.1. Principle. The 3D inter(65) Kawasaki, K.; Ohta, T. Physica A 1983, 118, 175. (66) Koga, T.; Jinnai, H.; Hashimoto, T. Physica A 1999, 263, 369.

Figure 6. (a) Schematic diagram of a surface, expressed in a parametric form p(u,v), and “sectioning planes” which define p(0,v) and p(u,0). p(0,0) is a point of interest (POI) at which the local curvatures are measured. (b) Definition of the sign of a principal curvature, K, used in this study. The sign of K is taken as positive if the interface is concave toward the PB-rich phase.

faces obtained by LSCM during the late stage SD were subjected to an analysis of probability density distributions of the local curvatures. The curvature of a site on an arc is equal to the reciprocal of the radius of a circle best fit to the curve at that site (“osculating circle”). Although detailed numerical procedures and its precision will be detailed elsewhere,69 we describe principle of the measurements below. Figure 6a schematically shows a surface on which the local curvatures are measured. Note that the curvature is defined to be positive throughout this work, if the center of the osculating circle locates in PB domain, as shown in Figure 6b. The surface is expressed in a parametric form as p(u,v) ) (x(u,v),y(u,v),z(u,v)) around a point of interest (POI) at which the curvatures are measured. A vertex of triangles, whose assembly forms a surface or interface, was chosen as a POI. The choice of a curvilinear coordinate (u,v) is arbitrary and does not have to be orthogonal. The POI is expressed as p(0,0). The local mean curvature, H, and the Gaussian curvature, K, can be calculated from the differential geometry53

H≡

κ1 + κ2 EN + GL + 2FM ) 2 2(EG - F2) K ≡ κ1κ2 )

LN - M2 EG - F2

(7)

where κ1 and κ2 are the principal curvatures at the POI. The parameters in eq 7 are given by (67) Hashimoto, T.; Koga, T.; Jinnai, H.; Nishikawa, Y. Nuovo Cimento Soc. Ital. D 1998, 20, 1947. (68) Doi, M.; Onuki, A. J. Phys. II 1992, 2, 1631. (69) Nishikawa, Y.; Koga, T.; Jinnai, H.; Hashimoto, T. Submitted.


E ) pu‚pu,

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F ) pu‚pv, G ) pv‚pv, M ) puv‚e, N ) pvv‚e (8) L ) puu‚e,

where e is a unit normal vector defined by e ) pu × pv/|pu × pv|. The subscripts of p represent the partial derivatives with respect to u or v, pu ≡ ∂p/∂u and puv ≡ ∂2p/∂u∂v. The surface is first “sectioned” by a plane that contains e. An intersection between the plane and the surface is defined as p(u,0). Parameters E and L in eq 8 can be estimated from the p(u,0). Subsequently, the surface is cut by another plane which defines p(0,v). Now F, G, and N are estimated. The parameter M, however, remains unsolved since determination of complete functional form of the surface around the POI, i.e., p(u,v), involves twodimensional spline fitting,70 which came out technically very difficult. Thus, we rearranged eq 7 to eliminate M:

f(i,H,K) ≡ 0 ) 4Fi2{LiNi - K(EiGi - Fi2)} {EiNi + GiLi - 2H(EiGi - Fi )} (9) 2

Here subscript i denotes ith set of the curvilinear coordinates (u,v) defined by the two sectioning planes. Since two unknown quantities, H and K, are invariant for any choice of the coordinates, the two equations obtained from eq 9 for two sets of the curvilinear coordinates give a set of simultaneous equations for H and K. In reality, several sets of f(i,H,K) are used to solve the two unknown quantities by using nonlinear regression fitting (“sectioning-and-fitting method, “SFM”).69 As the SFM provides local surface curvatures at a given point on the surface, the curvature distributions for H and K can be obtained by conducting the above measurements at many points on the surface. A joint probability density of H and K, P(H,K;t), at a given time t after the onset of SD is calculated as

P(H,K;t) )

∑j δA(j,t)

×

∑j δA(j,t|H - (∆H/2) e H(j,t) < H + (∆H/2) and K - (∆K/2) e K(j,t) < K + (∆K/2)) (10) H(j,t) and K(j,t) are the mean and the Gaussian curvatures at the jth POI. The term δA(j,t|H - (∆H/2) ‚‚‚) denotes the area on the jth POI which satisfies its local curvatures given by H - (∆H/2) e H(j,t) < H + (∆H/2) and K - (∆K/2) e K(j,t) < K + (∆K/2). ∆H and ∆K are the class interval of H and K. According to the definition of the joint probability density, the front factor 1/∆H∆K∑jδA(j,t) in eq 10 was introduced for the normalization condition, ∑H∑KP(H,K;t)∆H∆K ) 1, which can be replaced to ∫∫P(H,K;t) dH dK ) 1, in the limit of ∆H f 0 and ∆K f 0. Local area assigned to the jth POI, δA(j,t), was given by

1 3

∑i Atri(i;j,t)

∫P(H,K;t) dK 1

) ∆H

×

∑j δA(j,t)

(

∑j δA j,t|H -

(12a)

∆H 2

(11)

Here Atri(i;j,t) represents the area of ith triangles sharing the jth POI. The factor 1/3 is necessary to avoid overcounting the total area by three times since there are three vertexes for each triangle. Similar to eq 10, marginal probability densities of the mean curvature, PH(H;t), and (70) de Boor, C. A practical guide to splines; Splinger-Verlag: New York, 1978.

e H(j,t) < H +

)

∆H 2

(12b)

and

PK(K;t) ≡

∫P(H,K;t) dH 1 ∆K

∑j δA(j,t)

(

∑j δA j,t|K -

(13a)

×

∆K 2

e K(j,t) < K +

)

∆K 2

(13b)

Normalization of PH(H;t) and PK(K;t) are ∑HPH(H;t)∆H ) 1 and ∑KPK(K;t)∆K ) 1, which ∫PH(H;t) dH ) ∫PK(K;t) dK ) 1 in the limit of ∆H f 0 and ∆K f 0, respectively. 4.2.2. Measurements and Time Evolution of the Distribution of Local Curvatures. Smoothness of the reconstructed interface significantly affects accuracy of the measured curvature distributions. It is of particular importance to have a similar smoothness for 3D images at all phase separation times for a quantitative comparison. Thus, we introduce a roughness index, RI, defined by

RI ) 〈A∆〉1/2

∆H∆K

δA(j,t) )

PH(H;t) ≡

) 2

1

the Gaussian curvature, PK(K;t), defined below, are calculated from P(H,K;t)

|〈κ1〉 + 〈κ2〉| 2

(14)

where 〈A∆〉 expresses an average area of each triangle, 〈A∆〉 ≡ ∑iA∆i/N∆, and A∆i and N∆ are the area of ith triangle and the total number of triangles, respectively.63 RI gives a measure of roughness of the interface constructed by a characteristic length of the triangles, 〈A∆〉1/2, relative to absolute values of the radius of the averaged curvature, 1/〈κi〉| (i ) 1, 2). The smaller the RI is, the smoother the constructed interface becomes.71 The values RI at various t are listed in Table 1.72 Figure 7 shows bird’s-eye views of P(H,K;t) at three representative times. PH(H;t) and PK(K;t) are also displayed. P(H,K;t) at different t have common characteristic functional forms: Data only exist in a region of K e H2. The solid line in the H-K plane in Figure 7 shows the relation, i.e., K ) H2. This relation is required because the principal curvatures, which can be calculated from H and K via κi ) H ( (H2 - K)1/2 (i ) 1, 2), should be real numbers. Each data point in the H-K plane represents a local shape of the interface at the POIs. For example, data on the line of K ) H2 except for the origin correspond to spheres that are a representative of elliptic surfaces (K > 0). Cylinders, a simple model of the parabolic surface, locate along the line of K ) 0 except for the origin. Lamellae, which also (71) The square root of the area of a triangle and the average curvature have dimensions of length and reciprocal length, respectively, and hence the index is a dimensionless quantity. (72) Precision checks using various kinds of model surfaces, e.g., spheres, cylinders, periodic minimal surfaces, etc., have been done, from which we concluded that the SFM is capable of measuring the interface curvatures within (6% error at RI ≈ 0.06 for any type of surfaces.

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Figure 7. Bird’s-eye view of the joint probability densities, P(H,K;t), for the DPB/PB mixture at three representative phase separation times, i.e., at (a) 1675, (b) 2880, and (c) 4860 min. PH(H;t) and PK(K;t) are also shown on the walls. The solid parabolic line in the H-K plane shows the trajectory for K ) H2.

Figure 9. Scaled probability densities (a) P ˜ H(H ˜ ;t) and (b) P ˜ K(K ˜ ;t) for the DPB/PB mixture in the late stage SD.

Figure 8. Time evolution of PH(H;t) in (a) and that of PK(K;t) in (b) and (c) for the DPB/PB mixture in the late stage SD. Note scales for both ordinate and abscissa are different between (b) and (c).

belong to the parabolic surface, locate at the origin. In the case of the DPB/PB mixture, most of the measured points were in the region of K < 0 (hyperbolic interface).

P(H,K;t) got sharper, i.e., the width became narrower and the maximum of the distribution increased with time (see also PH(H;t) and PK(K;t)). Since the curvature is an inverse of the radius of the osculating circle, this result indicates a natural trend that the average absolute radius of curvature for the spinodal interface gets larger as the bicontinuous phase-separated structure grows. Figure 8 shows PH(H;t) (part a) and PK(K;t) (parts b and c highlighting PK around a wide |K| range and over a small range of |K|, respectively) as a function of t for the phaseseparated structure of the DPB/PB mixture at 40 °C. PH(H;t) at various t showed a symmetrical distribution


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around H ) 0, revealing that the spinodal interface curves equally inward and outward to one of the phase-separated domains. An area-averaged mean curvature stays zero at all times (see Appendix C), which implies that the system tends to take an evolution path of low free energy in the phase-separation process. PK(K;t) distributes mostly (up to 93%) in the K < 0 region over the whole range of t: Several percent of the POIs had positive Gaussian curvatures. fwhm of PH(H;t) decreased and thus the peak height of PH(H;t) got increased with time. Similarly, PK(K;t) got narrower as time increased by reducing number of data points at large absolute values of K; the peak height got higher just as PH(H;t) did. 4.3. Test of Dynamical Scaling in Terms of Scaled Probability Density Distributions of Local Curvatures. In our previous paper,32 we reported that the dynamical scaling law in terms of local structure is valid by exploring the time evolution of the area-averaged curvatures. However, as mentioned in the introductory part of the present paper, a more rigorous test of the validity requires critical examination of the probability densities distribution of local curvatures scaled by a characteristic length parameter. As such a length parameter, we chose the inverse of the interface area per unit volume at time t, ∑(t)-1, because it is a quantity intrinsically related to the interface. In addition to this reason, ∑(t) can be directly and accurately obtained from the 3D LSCM images. The scaled probability density ˜ ;t) and P ˜ K(K ˜ ;t), are, distributions of H and K, P ˜ H(H respectively, defined by

P ˜ H(H ˜ ;t) ) ∑(t)PH(H;t) and

˜ ;t) ) ∑(t)2PK(K;t) P ˜ K(K

(15a)

with

H ˜ ) H∑(t)-1

and

K ˜ ) K∑(t)-2

(15b)

˜ ;t) and P ˜ K(K ˜ ;t) satisfy the normalization conditions, P ˜ H(H ˜ ;t) dH ˜ ) ∫P ˜ K(K ˜ ;t) dK ˜ ) 1. The scaled probability ∫P ˜ H(H densities thus evaluated at various t are presented in Figure 9. The results clearly reveal that the dynamical scaling law holds for the probability density distributions of local curvatures in the late stage SD within the experimental accuracy. 4.4. Dynamics of the Spinodal Interface. It is critical to investigate which part of the interface is less stable compared with the other parts and hence will change more rapidly in the course of the phase separation. It is also a fundamental question how the interface changes its shape with time. Time-resolved LSCM observations at exactly the same volumes of the (bicontinuous) structure with a sufficiently small time interval are required to clarify those questions. Although we do not have such LSCM data at this moment, it is interesting to consider the dynamics of the spinodal interface in terms of time change in curvatures. Figure 10 presents such attempt, in which difference of the probability densities at two different phaseseparation times (1675 and 2880 min, or τ ) 117 and 201) with the time interval, ∆t ) 2880 - 1675 ) 1205 min, or ∆τ ) 201 - 117 ) 84, i.e., ∆P(H,K;∆t) ≡ P(H,K;t)2880) - P(H,K;t)1675) is shown. If ∆t gets infinitely small, this quantity is related to ∂P(H,K;t)/∂t. As time increased from t ) 1675 min, the probability density in the region drawn by yellow to red decreased (∆P(H,K;∆t) < 0). On the other

Figure 10. Difference of the probability densities between the two separate experimental times, 1675 and 2880 min, ∆P(H,K;∆t) ≡ P(H,K;t)2880) - P(H,K;t)1675). The region colored by red (blue) corresponds to that having the largest negative (positive) value, in which the probability densities decreased (increased) most dramatically during the time interval ∆t ) 1205 min.

hand, data close to the origin of the H-K coordinate represent, ∆P(H,K;∆t) > 0 (drawn by blue to violet), indicating that the probability density got bigger with time in this region. This means that the portion of the interface at t ) 1675 min having ∆P(H,K;∆t) < 0 decreases its frequency and will be transformed to the interface (at t ) 2880 min) whose curvatures correspond to ∆P(H,K;∆t) > 0. Figure 11 shows a part of the spinodal interface of the DPB/PB mixture at t ) 1675 min (18.33 µm3 in volume, corresponding to [1.6Λm(t)]3). The phase-separated domains facing to the bright and dark sides of the interface are DPB and PB domains, respectively. Only the unstable parts of the spinodal interface corresponding to negative ∆P(H,K;∆t) are shown by dots. H and K are independently displayed in parts a and b of Figure 11, respectively, because two different kinds of curvatures can be obtained at a single POI. Pseudocolors from red to blue indicate curvatures for H (in µm-1) and for K (in µm-2) (see Figure 6b for a definition of the sign of the curvatures used in this work). The part of the interface marked by a circled A in Figure 11 is an elliptic surface characterized by the positive Gaussian curvature. Corresponding mean curvature was negative, demonstrating that the elliptic interface was curved inward relative to the DPB-rich domain. We now consider the Laplace equation73 (73) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces, 6th ed.; Wiley: New York, 1997.

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Figure 11. Portion of the spinodal interface of the DPB/PB mixture at t ) 1675 min (18.33 µm3 box size). Dots correspond to the portion of the interface having ∆P(H,K;∆t) < 0. Pseudocolors from red to blue indicate (a) the local mean curvatures in units of µm-1 and (b) the local Gaussian curvatures in units of µm-2 for the interface at t ) 1675 min. The bar shows 5 µm. The interface labeled by A displays a portion which demonstrates that a concave/convex surface becomes a flat surface with time. Circles labeled by B and C show “bridges” connecting the two adjacent domains. These parts will break up in the course of the SD process to reduce interface area.

∆p ) γ(κ1 + κ2) ) 2γH

(16)

where γ and ∆p are the interfacial tension and the pressure difference between both sides of the interface, respectively. The above equation indicates that there is a pressure difference between the curved surfaces when H * 0. Note that the principal curvature and the pressure difference are defined to be positive if the center of the osculating circle locates in the PB phase and the pressure of the PB phase is larger than that of the DPB phase, respectively.74 Although the Laplace equation is originally developed for static interface in an equilibrium and the spinodal interface is in nonequilibrium state, we try to apply the Laplace equation in order to explain the interface dynamics. The part of the interface marked by the circled A had negative H and hence ∆p was also negative. This means that the pressure of the DPB phase was larger than that of the PB phase. In order for the interface to relax such local pressure difference, it will change the local shape of the interface in such a way that |H| decreases, e.g., an elliptic surface changes into a plane surface. The two principal curvatures are zero for the plane and ∆p is therefore zero (no pressure difference across the interface). Besides the pressure difference being released by this mechanism, the interface area can also be reduced. (74) This definition can be understood from the following simple example: A PB droplet in DPB matrix. In such a case, H is positive from our definition and the pressure inside the droplet is larger than that in the matrix. (75) Rayleigh, L. Proc. R. Soc. London 1879, 29, 71. (76) Tomotika, S. Proc. R. Soc. London, Ser. A 1935, 150, 322. (77) These two polymers have narrow molecular weight distributions, a well-defined phase diagram, and low glass transition temperatures so that at working temperatures and in the time scale of our observation they behave like liquids. Moreover, they have nearly symmetrical properties in terms of molecular weight, monomeric frictional coefficients, segment length, and segmental densities. They can be considered as a large molecule analogue of small molecules at large spatio-temporal scales.

There is different kind of the interfaces having negative ∆P(H,K;∆t). They are distinguished by circled letters B and C. These parts of the interface have a constricted (or squeezed) shape, characterized by negative Gaussian curvatures, i.e., a hyperbolic surface. The middle of the constricted part B (“bridge”) had negatively large K (K ∼ -0.3 µm-2). H at the corresponding portion showed a relatively large positive number (∼0.4 µm-1): The combination of these H and K numbers means that (1) the principal curvatures have opposite signs and that (2) the radius of the principal curvature whose center of the osculating circle locates in the PB domain is smaller than that of the other principal curvature. It is known that a cylindrical thread of fluid is unstable to surface waves whose length exceeds the circumference of the cylinder. This phenomenon was first studied by Lord Rayleigh.75 Tomotika later extended the theory in the case of one fluid in a second immiscible one.76 The capillary wave fluctuations are brought about in the thread. The fluctuations then build up the capillary pressure gradients that drive the thread-forming fluid from a thinner part of the thread to the thicker part. The thread is eventually broken up into a number of spherical droplets aligned along the thread. Similar capillary instability and break up may occur in the portion marked by the circled C. The bridge connecting the thick parts of the domain would be pinched off (just like the interface marked by the circled C) due to the capillary pressure gradients and the resulting local flow of the PB. The experimental observation that the hydrodynamic effect is significant in the ordering process (see section 4.1) supports the interface dynamics proposed here. Once the breakup of the bridge occurs, it is expected that it will split into a pair of elliptic surfaces, i.e., corns facing each other. The Gaussian curvature would become suddenly positive. Therefore, the curvature distribution


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Figure 12. 2D optically sliced image of the DPB/PB mixture phase-separated at 40 °C for 1675 min (part a). ∆F ∼ 0.85 µm. Part b shows the same area as part a after the image processing, contrast variance enhancement (CVE), and 3D median filtering. The binarized image, which was used in the 3D reconstruction process, is shown in part c. Parts d-f show histograms of pixel intensities of the images a-c, respectively. The threshold used in binarization was 128 as indicated by an arrow in part e. The bar in part a, corresponding to 20 µm, is also applicable to parts b and c.

should be a sensitive measure to examine the mechanism of the phase separation process. More rigorous observations of the motion of interface deserve future work. 5. Conclusions Time evolution of interface between the two coexisting phases developed during the late stage of spinodal decomposition (SD) was studied for a model mixture77 of a deuterated polybutadiene (DPB)/polybutadiene (PB) mixture having the critical composition by a combination of laser scanning confocal microscopy (LSCM) and timeresolved light scattering (TRLS). It was found that the phase separation process of the DPB/PB mixture quantitatively agreed with predictions from 3D computer simulations based on the time-dependent Ginzburg-Landau equation with hydrodynamic interactions, which ensures that the DPB/PB mixture can be considered as a model system for soft condensed matter systems. Namely, the DPB/PB blend has universality with other binary liquid systems such as simple liquid mixtures. Probability densities of local interface curvatures, i.e., the mean and Gaussian curvatures, were quantitatively measured as a function of time, t. They clearly showed that the large portion of the interface is hyperbolic surface. The concept of the dynamical scaling hypothesis was examined for the local structure of the system that is characterized by the curvatures as well as for the global structure quantified by the wavelength of the dominant mode of concentration fluctuations: The probability density distributions of local curvatures at different times were scaled by a time-dependent characteristic wavenumber, i.e., interface area per unit volume. The scaled probability density distributions fell onto a master curve, which demonstrates that the time evolution of the interface developed during SD is dynamically self-similar. Dynamics of the interface in the late stage SD was qualitatively studied in terms of the time change in local

curvatures. The probability densities of local curvatures at two discrete times, i.e., 1675 and 2880 min, were compared to elucidate the parts of the interface whose probability densities decreased with time. Two basic processes were found to occur: (1) The elliptic interface will change to the flat surface. (2) The hyperbolic interface characterized by the negatively large Gaussian curvature, which has a “bridge” shape, will be pinched off and relaxed to interface with smaller local curvatures. Both processes are to reduce the interface area. Acknowledgment. Thanks are due to Dr. Yoshitsugu Hirokawa for his assistance in anthracene labeling of polybutadiene and to Ms Yuko Kanazawa for synthesizing the polymers used in this work. We thank Dr. Charles C. Han and Messrs. Masaki Hayashi and Kenji Sato for their assistance with the SANS measurements and data analysis. This work is partially supported by the Ministry of Education, Science, Sports and Culture, Japan (Grantin-Aid No. 11305067). Appendix A The size of the confocal pinhole has a strong influence on the resolution of the optical sectioning as well as on the signal intensity. In general, the resolution of the optical sectioning is described by the full width at half-maximum (fwhm), ∆F, of the weighting function of the incident beam intensity along the z-axis (optical axis) at the focal plane. The following formula gives an excellent fit to the measured data,78

∆F )

(

1.41

){

λem λexc

(

)}

noλexc 2no 2 NA P P+ exp 2 M NA λexcM NA

(A1)

λexc and λem are the wavelength of excitation and that of emitted fluorescence light, respectively. NA is the nu-

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Jinnai et al.

Figure 14. Time evolution of the reduced radius, 〈R(t)〉/Λm(t) (O, left axis), and the reduced wavenumber, ∑(t)/qm(t) (0, right axis), evaluated by the real-space analysis. The solid line represents the value obtained from LS and the computer simulation.82 Dotted and dash-dot lines show the average values of 〈R(t)〉/Λm(t) (∼0.283 ( 0.007) and ∑(t)/qm(t) (∼0.451 ( 0.012), respectively.

Figure 13. (a) Time evolution of the area-averaged mean, 〈H(t)〉, and Gaussian, 〈K(t)〉, curvatures. The dashed line shows 〈H(t)〉 ) 0. The solid line going through 〈K(t)〉 is a guide for the eye. (b) Time evolution of characteristic wavenumbers related to the interface, -〈K〉1/2 and ∑(t).

merical aperture of the objective, and M is the magnification of the objective. no is the refractive index of the object. P denotes a digital pinhole parameter of the LSM410 instrument, which is related to the diameter of the projection of the pinhole in the object plane, d, by d ) 1.75P/M (in µm). The formula contains a linear term and an exponential term with P, each giving an approximation for very high and low pinhole parameters, respectively. Appendix B An accurate determination of the interface involves “binarization” of optically sliced 2D images obtained by LSCM with a proper threshold before stacking them to construct a 3D structure. Raw 2D slices sometimes suffer from spatial intensity variation, i.e., inhomogeneous illumination and high-frequency noise due to statistical counting errors during the imaging process. Figure 12 shows a “binarization” process of one of the optically sliced images that were obtained for the specimen phase separated at 40 °C for 1675 min (40 µm away from the cover slips). Bright and dark phases correspond to the PB and DPB phases, respectively. Parts a-c show the raw LSCM sliced image, the corresponding image after the image processing, and the binarized image, respectively. In the image processing, the “contrast variance enhancement (CVE)” method79,80 was used. The CVE is a technique that makes all local regions of an output image have an equal variance in terms of their image intensities. (78) Wilhelm, S. About the 3D image quality of a confocal laser scanning microscope. Diploma Thesis, Fachhochschule, Ko¨ln, 1994.

This turned out to be quite effective in reducing a large regional contrast variation which was mainly caused by the inhomogeneous illumination due to the various kinds of aberrations of the objective, a slight misalignment of the confocal pinhole, etc. Subsequently, a median filter,81 a class of nonlinear spatial filter which is designed to remove outlier pixels whose intensities are completely inconsistent with their surrounding values, was applied. The filter places a box surrounding a particular pixel and performs a sorting of the intensities within the box. It then replaces the intensity of the pixel with the median value. As clearly shown in part b, the image contrast was successfully enhanced and noises were also reduced after these image processing procedures. The histogram, H(I), which is the number of pixels that have the corresponding gray value, I, was used to determine an appropriate threshold for the binarization. The histogram of the raw LSCM image (Figure 12d) showed a bimodal intensity distribution with the two peaks corresponding to the bright and dark regions in the image. A major fraction of the pixel intensities distributed over the range from 0 to 200. The image processing described above made the two peaks alienate, and now the intensity in the 2D contrast-enhanced image distributes over the entire 256 gray scale values as shown in Figure 12e. The binarized image (Figure 12c) was produced by choosing the intensity 128 as the threshold (see Figure 12e). A trajectory of the isointensity line of the threshold value corresponds to the interface between the two coexisting phases in the binarized image. Appendix C Area-averaged mean and Gaussian curvatures, 〈H(t)〉 and 〈K(t)〉, were estimated from the following definition

〈H(t)〉 ≡

and

∑j H(j,t)δA(j,t) ∑j δA(j,t)


〈K(t)〉 ≡

∑j K(j,t)δA(j,t) ∑j δA(j,t)

Langmuir, Vol. 16, No. 9, 2000 4393

(C1)

Figure 13a shows time evolution of the area-averaged curvatures, 〈H(t)〉 and 〈K(t)〉. 〈H(t)〉 hardly showed time dependence. It is essentially zero at all times. 〈K(t)〉 at various t remains negative, demonstrating that the interface is hyperbolic regardless of t. 〈K〉 decreased with t, reflecting that the area-averaged mean absolute values of radius of curvature defined as 〈R(t)〉 ≡ (|〈R1〉| + |〈R2〉|/2 with |〈Ri〉| ≡ (|1/〈κi〉| (i ) 1 or 2) increased as the bicontinuous structure grew. In Figure 13b, the inverse values of the two length parameters characterizing the (79) Andrews, H. C. Appl. Opt. 1976, 15, 495. (80) Harris, J. L. Appl. Opt. 1977, 16, 1268. (81) Chen, H.; Swedlow, J. R.; Grote, M.; Sedat, J. W.; Agard, D. A. In Handbook of Biological Confocal Microscopy, 2nd ed.; Pawley, J. B., Ed.; Plenum: New York, 1995; p 197. (82) Koga, T.; Kawasaki, K.; Takenaka, M.; Hashimoto, T. Physica A 1993, 198, 473.

local structure of the spinodal interface were plotted. Power law behavior was found: -〈K(t)〉1/2 ∼ t-1 and ∑(t) ∼ t-1. It is worth while to note that qm(t), characterizing the global feature of the phase-separated structure, exhibited the same exponent, i.e., qm(t) ∼ t-1 (see sections 3.1 and 4.1), in the late stage SD (hydrodynamic regime). Figure 14 shows a semilogarithmic plot of the reduced mean radius of curvature, 〈R(t)〉/Λm(t) as a function of time. A relation of 〈R(t)〉/Λm(t) ) 0.283 ( 0.007 holds over the time range of our LSCM experiments, which demonstrates that R(t) has exactly the same growth rate as Λm(t). This is consistent with the concept of dynamical self-similarity. It is intriguing that the two length parameters, 〈R(t)〉 and Λm(t), characterizing the local and the global features of the phase-separated structure, respectively, are coupling. The time evolution of another dimensionless quantity, ∑(t)/qm(t), is also plotted in Figure 14. ∑(t)/qm(t) reached a plateau value of 0.451 ( 0.012 (dash-dotted line), in good agreement with the value found for the PB/PI mixture (PI denotes polyisoprene) and the TDGL computer simulation of late stage SD,82 0.5 (solid line). LA991024Q

Time Evolution of Spinodal Interface in a Binary Polymer Mixture at the

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