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Composition fluctuations in a homopolymer-diblock copolymer mixture covering the three-dimensional Ising, isotropic Lifshitz, and Brasovskiĭ classes of critical universality
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Image of FIG. 1.
FIG. 1.

Temperature-diblock copolymer plane of the {dPB;PS} phase diagram. The individual symbols represent the following: The filled circles the Scott line; 3D Ising and isotropic Lifshitz case are separated by a gray box; the open circle the double critical point; the filled and open stars the Lifshitz line at high temperatures of disordered phases and at low temperatures of microemulsion phases, respectively; the filled triangle the disorder-microemulsion transition line; the diamonds the disorder-lamellar ordering transition line; the open triangle the transition point of the two Lifshitz lines; and the second gray box the transition from bicontinuous microemulsion to lamellar phase.

Image of FIG. 2.
FIG. 2.

Structure factor in Zimm representation for four diblock concentrations. Below is described by the Ornstein-Zernike law, above 0.04 contributions from the term becomes visible.

Image of FIG. 3.
FIG. 3.

The inverse susceptibility and square of inverse correlation length vs inverse temperature within the 3D Ising universality class. The , 0.03, and 0.04 samples follow a crossover function (solid lines) between mean-field (dashed line) and 3D Ising (dotted curve) scaling laws.

Image of FIG. 4.
FIG. 4.

Inverse susceptibility (a) and the correlation lengths (filled circles) and (open circles) (b) vs inverse temperature within the isotropic Lifshitz universality class. The theory describes the susceptibility (solid lines) and scaling laws [dashed lines in (a) and solid lines in (b)] the parameters near the critical point.

Image of FIG. 5.
FIG. 5.

The scattering profiles within microemulsion channel.

Image of FIG. 6.
FIG. 6.

Near LL inverse susceptibilities (●) and (☉). Within the temperature gap of the Lifshitz lines represents the susceptibility while at the temperature the disordered phase transforms into a droplet-microemulsion phase. At this transition line the susceptibility and correlation lengths have their largest values.

Image of FIG. 7.
FIG. 7.

The scattering function for samples in the Brasovskiĭ regime.

Image of FIG. 8.
FIG. 8.

Susceptibility of depicted in Fig. 8 vs inverse temperature. The solid lines represent a fit of theory. The deviations of experiment from fitted curve represent a transition to the ordered lamellar phase.

Image of FIG. 9.
FIG. 9.

Crossover of the critical exponents of susceptibility and correlation lengths and . The squares represent the critical exponents derived from a scaling law with the conventional reduced temperature definition. However, as the critical points (Scott line) depend on diblock concentration and are terminated at a double critical point another reduced temperature has to be defined. Within the isotropic Lifshitz regime values of both types of evaluation differ by a factor of 2.

Image of FIG. 10.
FIG. 10.

Experimental approaches towards the double critical point, the isopleths of diblock concentration approach (top), and isothermal (bottom). The isothermal approach to the critical point at and approximately represents a critical path as illustrated in the attached cartoons.

Image of FIG. 11.
FIG. 11.

Scaling behavior of the peak position according to within the disordered phase at , the microemulsion phase at , and near the disorder-microemulsion transition at where the dLL and lines meet together.

Image of FIG. 12.
FIG. 12.

Temperature dependence of the exponent . There is a clear transition at .

Image of FIG. 13.
FIG. 13.

Ginzburg number from binary blend to diblock copolymer melt.

Image of FIG. 14.
FIG. 14.

Enthalpic and entropic terms of the Flory-Huggins parameter vs diblock concentration.

Image of FIG. 15.
FIG. 15.

Phase diagram of the isotropic Lifshitz critical regime near DCP and LTP.


Generic image for table
Table I.

Sample characteristics .

Generic image for table
Table II.

Parameters derived from 3D Ising to mean-field crossover functions of susceptibility and correlation [Eq. (7)] ( and in ); in .

Generic image for table
Table III.

Critical parameters of the isotropic Lifshitz critical behavior determined from the isopleths of diblock concentration path using the modified reduced temperature field in the case of a double critical point. The parameters of the DCP are printed in bold letters. The bottom line represents data from the isothermal path at . Sample was measured with light.

Generic image for table
Table IV.

Parameters obtained from Kielhorn-Muthukumar theory for the samples. ( and in ; in ).


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Composition fluctuations in a homopolymer-diblock copolymer mixture covering the three-dimensional Ising, isotropic Lifshitz, and Brasovskiĭ classes of critical universality