Wave Number Qm Research Articles

Kinetics of microstructure formation of high-pressure induced gel from a whey protein isolate

The kinetic process of pressure-induced gelation of whey protein isolate (WPI) solutions was studied using in situ light scattering. The relationship of the logarithm of scattered light intensity (I) versus time (t) was linear after the induced time and could be described by the Cahn-Hilliard linear theory. With increasing time, the scattered intensity deviated from the exponential relationship, and the time evolution of the scattered light intensity maximum Im and the corresponding wavenumber qm could be described in terms of the power-law relationship as Im~fβ and qm~f−α, respectively. These results indicated that phase separation occurred during the gelation of WPI solutions under high pressure.

Kinetics of Phase Separation in Poly(ɛ-caprolactone)/Poly(ethylene glycol) Blends

The poly(ɛ-caprolactone)/poly(ethylene glycol) (PCL/PEG) blends reveal a miscibility window of upper critical solution temperature (UCST) character. The kinetics of liquid–liquid phase separation (LLPS) for the blends of PCL/PEG is investigated by time-resolved small angle light scattering (TRSALS). The time evolution of scattering profile is analyzed by linear Cahn–Hilliard theory for early stage of spinodal decomposition (SD). The evolution of the maximum intensity Im(t) and the corresponding wavenumber qm(t) obey the power-law scheme (Im(t)∼tβ and qm(t)∼t−α). A relation of β=3α in late stage is obtained almost the same scaling exponents with β≅1 and α≅1/3 for various quenching depths. The α≅1/3 implied that a coarsening mechanism at the late stage of phase separation may proceed with Ostwald ripening or Brownian coalescence process. Besides, the intermediate and late stages of SD can be scaled into a universal from represented well by Furukawa’s structure factor. The percolation to cluster transition is accompanied with α∼0.13→1/3 from intermediate to late stage of SD for the off-critical mixture of PCL/PEG (4/6) blend. In this study, the experimental result demonstrates that the crystallization is a viable mechanism to lock phase-separated structure of the blends. The competition between phase separation and crystallization has been suggested to determine the final morphology.

Attenuation of concentration fluctuations after a quench to a temperature in the single-phase region

Attenuation of concentration fluctuations in a polystyrene/poly(2-chlorostyrene) blend after a quench from a temperature in the spinodal region to one in the single-phase region was studied by using the time-resolved light scattering technique and a scanning electron microscope. The characteristic wave number qm, where the maximum of the scattered light intensity was located, move towards smaller values during the attenuation process. The corresponding growth of the wave length of the dominant concentration fluctuation was clearly observed by the electron microscope. The scattered light intensities at small wave numbers were found to increase before they attenuated. In the later period the power-law behavior qm∼t−0.3 was observed. The exponent was smaller than the value (0.5) obtained on the basis of the non-linear theory by Akcasu and collaborators.

Early Stage Spinodal Decomposition in Polymer Solution under High Pressure

Early stage spinodal decomposition (SD) of polymer solutions (polypropylene/trichlorofluoromethane) induced by pressure jump was investigated with a light scattering instrument designed for polymer solutions under high pressure and high temperature. The changes in the scattered intensity with time at the early stage SD were found to be approximated by Cahn linearized theory or Cahn−Hilliard−Cook theory. The analyses with the theories yielded the characteristic parameters such as the interdiffusion coefficient Dapp and the characteristic wavenumber qm(0). Unlike other experiments on the polymer solutions, the pressure dependence of the parameters was found to obey the mean field behavior. This may be because the quench depth conditions attained by the pressure jump were much deeper than those attained in other experiments.

Heterogeneous percolation-to-cluster transition in phase separation of an off-critical polymer mixture

Phase separation kinetics of an off-critical polymer mixture involving percolation-to-cluster transition (PCT) was investigated by time-resolved light scattering and by transmission optical and electron microscopies. The process involving a splitting of a single peak in the light scattering profile into a double peak was observed for the first time after the onset of PCT. Each peak changes with time (t) according to different evolution modes, e.g., the first peak wave number qm,1 evolved according to the power law of percolated structures (qm∼t−α; α=0.9), while the time evolution of the second peak wave number qm,2 is very slow, which corresponds to the growth of cluster of the droplets after PCT. Furthermore, the real space pictures and the light scattering analysis reveal that a cluster of the small spherical droplets and the locally percolating domains coexist for a very long time. The percolating domains are larger than the small droplets, minor in volume fraction, and formed in bulk and they are developed into sheet-like grains with their surfaces parallel to the glass surface.

Real Space Analysis of the Structural Evolution of a Polymer Blend via Spinodal Decomposition

Transmission electron microscopy (TEM) and laser scanning confocal microscopy (LSCM) are used to visualize the morphological evolution during spinodal decomposition (SD) of a binary polymer blend of poly(styrene-ran-butadiene) and polybutadiene with near critical composition. The mechanically homogenized samples were exposed to a temperature jump, and the time evolution of the characteristic wavenumber qm was recorded by time-resolved light scattering. TEM and LSCM micrographs at various stages are also presented to clarify the growing bicontinuous morphology. A real space analysis of the early stage of the spinodal decomposition is presented for the first time. Two-dimensional Fourier transformation was applied to the real space images to obtain information about the characteristic wavenumber, and the results were compared to the predictions obtained by Cahn's linearized theory.

Time Evolution of Shear-Induced Structures in Semidilute Polystyrene Solutions

We investigated time evolution of structures induced by imposing shear flow on semidilute solutions of high molecular weight polystyrene (PS) by means of both flow light scattering and rheology. PS with weight-average molecular weights of 5.48 × 106 and 2.89 × 106 were dissolved in dioctyl phthalate (DOP). After imposing shear flow with a shear rate γ̇ > γ̇c (the critical shear rate for shear-induced concentration fluctuations), the unique butterfly-type scattering pattern appeared. The pattern had a scattering maximum along the flow direction. The wave number qm and the intensity Im at the scattering maximum decreased and increased, respectively, with time and eventually reached steady state values. The rheological experiments that were carried out on the same solution under the same shear flow revealed two stress overshoots. The second overshoot was found to be related to the development of butterfly pattern and its time change, i.e., to the formation and growth of shear-induced structures.

Time-resolved small-angle neutron scattering study of spinodal decomposition in deuterated and protonated polybutadiene blends. II. Q-dependence of Onsager kinetic coefficient

The spinodal decomposition (SD) of a critical mixture of deuterated and protonated polybutadiene of nearly equal chain lengths was investigated. This mixture has an upper critical solution temperature type phase diagram and the spinodal temperature at the critical point is 99.2 °C. Phase separation was induced by quenching a single-phase specimen at an initial temperature, Ti (=102.3, 123.9, and 171.6 °C), to a final temperature, Tf (=−7.5, 1.1, and 10.5 °C). The subsequent SD was followed by time-resolved small-angle neutron scattering. The Onsager coefficient, Λ(q;Tf), as a function of wave number q and Tf, derived from experimental growth rates, R(q;Tf), of the Fourier mode of concentration fluctuations and estimation of ST(q;Tf), was compared to the reptation model theories of Pincus and Binder. Experimental Λ(q;Tf) was found to give a q-dependence greater than that given by the theories. Here, ST(q;Tf) denotes the virtual structure factor at Tf inside the spinodal region. The reduced wave number Qm(τ) and intensity S̃m(τ) at the peak of the scattering structure factor in the early and intermediate stages of SD were found to be scalable in terms of a reduced time τ when Ti was fixed and Tf was varied, but not when Tf was fixed and Ti was varied. The failure of the scaling law in the latter instance may be attributed to the fact that the concentration fluctuation at the onset of SD has a different memory of the thermal concentration fluctuation in the single-phase region depending on Ti, which affects the subsequent SD over an extended period of time.

Time-resolved small-angle neutron scattering study of spinodal decomposition in deuterated and protonated polybutadiene blends. I. Effect of initial thermal fluctuations

Time-resolved small-angle neutron scattering (SANS) experiments have been performed on the self-assembling process of a binary mixture of deuterated polybutadiene and protonated polybutadiene at the critical composition. This mixture has an upper critical solution temperature type of phase diagram with the spinodal temperature at 99.2 °C. Specimens held in the single-phase state at an initial temperature (Ti) were quenched to a point inside the spinodal phase boundary at a final temperature (Tf) to induce phase separation via spinodal decomposition (SD). In order to examine the effect that thermal concentration fluctuations have on SD, three different initial temperatures, Ti=102.3 °C, 123.9 °C, and 171.6 °C, were chosen while Tf was fixed at −7.5 °C. The time-dependent SANS structure factor, S(q,t;Tf), showed clear scattering peaks corresponding to the early and intermediate stages of SD. The time changes in the wave number qm(t;Tf) and the intensity Sm(t;Tf) at the peak of S(q,t;Tf) followed different paths depending on the initial temperature. This fact evidences a definite effect of thermal concentration fluctuations on SD (i.e., a significant ‘‘memory’’ effect). A critical test of the linearized Cahn–Hilliard–Cook theory led to the conclusion that this theory can describe satisfactorily the early stage SD in the deep-quench region.

Slow spinodal decomposition in binary liquid mixtures of polymers. V. Molecular weight dependence in the later stage of unmixing

The unmixing process via spinodal decomposition of poly(styrene-ran-butadiene) (SBR)/polyisoprene (PI) mixtures was investigated by the time-resolved light-scattering method to see how it is affected by the temperature of phase separation T and by the molecular weight of PI with that of SBR fixed. Time changes in the wave number qm(t;T) and intensity Im(t;T) at the peak of the scattering function were followed to explore the coarsening process in the later stage of spinodal decomposition. For each mixture the observed changes at different T were reduced to a master curve when plotted against a reduced time τ. This finding is consistent with the scaling postulate by Langer et al. and Chou and Goldburg. The master curves for mixtures with different PI molecular weights were divergent, showing the phenomenon called the N branch. However, it was possible to bring them to a master curve when τ was multiplied by a shift factor depending on the average number of entanglements per chain. The dependence was weaker than that predicted theoretically by Onuki.

Scattering studies of self-assembling processes of polymer blends in spinodal decomposition. II. Temperature dependence

Our previous work on time evolution of the interfacial structure for a near critical mixture of polybutadiene and polyisoprene undergoing the spinodal decomposition (SD) [T. Hashimoto, M. Takenaka, and H. Jinnai, J. Appl. Crystallogr. 24, 457 (1991)] was extended to explore the behavior as a function of temperature T, again using the time-resolved light scattering method. The study involved the investigation of the time evolutions of various characteristic parameters such as the wave number qm(t;T ) of the dominant mode of the concentration fluctuations, the maximum scattered intensity Im(t;T ), the scaled structure factor F(x;T ), the interfacial area density Σ(t;T ), and the characteristic interfacial thickness tI(t;T ) from the early-to-late stage SD, where t refers to time after the onset of SD and x refers to the reduced scattering vector defined by x=q/qm(t;T ); q is the magnitude of the scattering vector. The results confirm the model previously proposed at a given T over a wider temperature range corresponding to the quench depth ΔT=T−Ts =5.5–34.5 K, or εT=(χ−χs)/χs =4.50×10−2 to 2.79×10−1, where Ts is the spinodal temperature, and χ and χs are the Flory interaction parameters at T and Ts, respectively. This blend is noted to have a phase diagram of the lower critical solution temperature type.

Phase separation in mixtures of poly(ethylene glycol monomethylether) and poly(propylene glycol) oligomers

AbstractTime‐resolved light scattering was employed to investigate kinetics of phase separation in mixtures of poly (ethylene glycol monomethylether) (PEGE)/poly (propylene glycol) (PPG) oligomers. Phase diagrams for PEGE/PPG of varying molecular weights were established by means of cold point measurements. The oligomer mixtures reveal an upper critical solution temperature (UCST). Several temperature quench experiments were carried out with a 60/40 PEGE/PPG blend by rapidly quenching from a single phase (69°C) to two‐phase temperatures (66–61°C) at 1°C intervals. As is typical for oligomer mixtures, the early stage of spinodal decomposition (SD) was not detected. The kinetics of phase decomposition was found to be dominated by the late stage of SD. Time‐evolution of scattering intensity was analyzed in accordance with nonlinear and dynamical scaling theories. The time dependence of the peak intensity Im and the corresponding peak wavenumber qm was found to follow the power‐law {Im(t)˜ tα, qm(t)˜ t‐β} with the values of α = 3 ± 0.3 and β = 1 ± 0.2, which are very close to the values predicted by Siggia. This process has been attributed to a coarsening mechanism driven by surface tension. In the temporal scaling analysis, the structure function reveals university with time, suggesting self‐similarity. Phase separation dynamics in 60/40 PEGE/PPG resembles the behavior predicted for off‐critical mixtures.

Slow spinodal decomposition in binary liquid mixtures of polymers. III. Scaling analyses of later-stage unmixing

Later-stage unmixing process of a near critical mixture of polybutadiene (PB)/poly(styrene- r-butadiene)(SBR) were examined at real time t and in situ at several temperatures T by time-resolved light scattering method. The magnitude of scattering vector qm(t,T) at which the intensity becomes maximum and the maximum intensity Im(t,T) were analyzed in order to characterize the coarsening processes of the later-stage spinodal decomposition. The variations of Im and qm with t at different T’s were found to fall onto master curves on the reduced plots, thus assuring the scaling postulate that the data obtained at different t and T for given mixtures are properly scaled with the temperature-dependent characteristic wave number qm(0,T) and characteristic time tc(T).

Correlations in phonon-transmission spectra in periodic and quasiperiodic superlattices.

We study the origin of correlations in phonon-transmission spectra in Fibonacci and periodic su­ perlattices (SL's), which have recently been recognized by numerical calculations and then confirmed by phonon-imaging experiments. The structure factors describing the interference effects of phonons reflected from the interfaces of both periodic and Fibonacci SL's are calculated. It is found that the wave number Qm.n (indexed by two integers m and n) for which the largest reflection occurs in a Fibonacci SL is identical to the wave number Qm + n for which (m + n )th Bragg reflection of phonons occurs in the corresponding periodic SL. We also show more generally that major phonon-transmission dips in the Fibonacci SL occur at frequencies close to each transmission dip in the periodic SL. Two-dimensional maps for the Bragg and Bragg-like reflections of phonons in these systems indicate that the correlations are applicable, irrespective of propagation directions and modes participating in the reflection processes. Since the discovery of quasicrystals,1 very significant advance has been made in the study of systems with quasiperiodic order.2 In particular, the spectral property of one-dimensional quasiperiodic (Fibonacci) lattices has been one of the topics studied extensively.3 The interest stems partly from the fact that the problem represents, in some sense, an intermediate case between periodic and random lattices. Undoubtedly, a major impetus toward reexamining the Fibonacci lattices was given by Merlin et al.4 In their pioneering work they fabricated a semi­ conducting layered structure based on the Fibonacci se­ quence, known as a Fibonacci superlattice (SL). The measurements of x-ray scattering revealed that this artificial heterostructure indeed exhibited properties characteristic of quasiperiodic systems.4,5 Raman experi­ ments have been performed on similar Fibonacci SL'S.6-9 Recently, the phonon-transmission spectra in a Fi­ bonacci SL have also been examined both experimental­ lylO and theoretically. 11 A striking similarity has been found between the Fibonacci and periodic SL's when the major transmission dips in the former are compared with the transmission dips in the latter. (The distribution of frequency gaps in the phonon dispersion relations also ex­ hibits the same similarity.) It has been predicted numeri­ cally on the basis of the transfer-matrix method for both the fre~uency and angular dependences of the transmis­ sion.IO, 1 The predicted angular dependences of transmis­ sion spectra in these SL's are in reasonable agreement with the results of phonon-imaging experiment. 10 The purpose of the present work is to clarify the origin of these correlations found in the phonon spectra of the periodic and Fibonacci SL's. This will be made by study­ ing SL structure factors which describe the interference effects of phonons reflected at the interfaces of constitu­ ent layers. We shall find that the wave numbers of pho­ nons for which the largest reflections occur in the Fi­ bonacci SL coincide with the wave numbers satisfying the Bragg condition in the associated periodic SL. This ob­ servation leads more generally to the result that major 39 reflections in Fibonacci SL closely correlate with Bragg reflections in the periodic SL in both their magnitudes