4.4 Three-phase coexistence conditions
5.2.1 Scaling of the order-parameter amplitude at the multicritical point 68
to three-phase coexistence,χ(h) andχ(m), are single-valued as a function of λaround the multicritical point (despite the discontinuity of the wave numberk0 at the critical correlationλc). In this case, the numerical results along both linesχ(h) andχ(m) can be used to determine the critical exponent ψ for the decay of the order-parameter amplitude (see fig. 5.11). Evaluating the amplitude of the lamellar cloud and shadow, respectively, we find ψ = 0.5 along both lines consistently. In order to validate this
10-4 10-3 10-2 10-1
10-6 10-5 10-4 10-3 10-2 10-1
order-parameter amplitude
τˆm /ρ0
distance to multicrit. point ∆λ = |λc - λ|
fit 0.2516(1) ∆λ0.50071(6) (22 dp.)
hom. cloud (λ > λc)
fit 0.3393(5) ∆λ0.4989(2) (12 datap.) lamellar shadow (λ > λc)
Figure 5.12:Order-parameter am-plitude of homogeneous bi-phase cloud and of lamellar shadow on the lineχ(h)for continuous-chain tri-blocks, with enhanced expression for the lamellar free energy, cf. fig. 5.9.
result, we analyze in fig. 5.12 the scaling of the order-parameter amplitude on approach to the multicritical point for the same system, but with the enhanced theory, which had led to a different location of the transition lines, cf. figs. 5.9 and 5.5. The amplitudes, in this case evaluated on the lineχ(h), of both lamellar shadow and homogeneous bi-phase cloud vanish forλ↓λcwith an exponent of ψ= 0.5, which corroborates the findings for continuous-chain triblocks with the simplified theory. We conclude that the critical exponent of the order-parameter amplitude for triblocks withM ≥7 isψ= 0.5, distinct from the exponentψ= 1 for the systems withM <7, cf. eq. (5.4).
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5.3. SEQUENCE FRACTIONATION FOR TRIBLOCKS
5.3 Sequence fractionation for triblocks
In this section, we return to a central aim of this work, viz., the analysis of the sequence fractionation or partitioning between coexisting subsystems of different morphologies.
To this end, we can exploit the detailed information on the three-phase region obtained from the solution of the coexistence conditions, cf. sec. 4.4.
λ-distribution
ABA + BAB AAB + BBA
AAA + BBB
λ = 0.2 λ = 0 λ = -0.2 λ = -0.4 λ = λc λ = -0.5 fractionation:
Figure 5.13:Sequence distribution triangle for random continuous-chain triblocks at various block-type correlations, with the enhanced analytical expression for the lamellar free-energy, cf. the phase diagram in fig. 5.9. The triangle is spanned by the unit vectorsnν (not drawn) pointing from the center (+, origin of the coordinate system) to the vertices, each labeled by one sequence classν. Within the triangle, one sequence distribution {pν} is represented as a point. The vector p of this point relative to + determines each concentration pν via p·nν = 3 (pν−1/3)/2, i.e., the larger the projection of the vector ponto nν, the higher is the concentrationpν. The origin corresponds to equal concentrations of all sequence classes (pν =1/3,ν = 1,2,3). Distributions defined byλlie on the (red) curve, withλranging from
−1 at the triangle’s bottom left vertex to +1 at its top. Solid symbols on this curve mark the sequence distribution of the cloud phase(s) at the boundary line(s) of three-phase coexistence.
Off-curve solid symbols mark the distributions of the coexisting shadow phases. Open symbols display the distributions of the coexisting subsystems at equal volume fractions (v(m)= 0.5).
For triblocks, the optimization of the subsystems’ distributions proceeds via selection out of three sequence classes, cf. eq. (3.42). Therefore, a distribution triangle allows for a compact map of coexisting distributions in systems characterized by different values of λwithin one diagram. In this way, fig. 5.13 depicts sequence fractionation starting from several global distributions of continuous-chain triblocks. The data
CHAPTER 5. PHASE BEHAVIOR WITH FRACTIONATION
have been obtained with the enhanced theory for the lamellar subsystem, cf. fig. 5.9.
Each vertex of the triangle represents one of the sequence classes, one point within the triangle one AB exchange-symmetric sequence distribution (for details of the encoding see the figure caption). The sets for three supercritical values of the block-type correlation,λ > λc,λ= 0.2 (diamonds), λ= 0 (circles), andλ=−0.2 (up triangles), visualize the following fractionation mechanism: On the curve of λ-distributions, the solid symbol marks the sequence distribution of the homogeneous cloud phase(s) on the boundaryχ(h) of fractionated three-phase coexistence. A solid symbol of the same shape and color to the bottom right of the curve marks the distribution of the coexisting lamellar shadow phase (with zero volume fraction). The finite deviation of the lamellar shadow’s sequence distribution from theλ-distribution indicates that the transition to three-phase coexistence is discontinuous. Upon increasing incompatibility, the lamellar phase’s volume fraction increases (cf. fig. 5.8), and its sequence distribution departs ever more from theλ-distribution (the open symbols to the bottom right of theλ-curve display lamellae at 0.5 volume fraction). Sequence class 2 (AAB/BBA) substantially accumulates in the lamellar phase, also class 3 (ABA/BAB). Moreover, since the ratio of these two sequence concentrations differs from theλ-defined ratio p2(λ)/p3(λ), the fractionated sequence distribution in the lamellar phase does not ensue from merely expelling homopolymers into the coexisting homogeneous phases at a constant concentration ratio of the two other sequence classes. (The shifted concentrationratios become visible in the detailed distribution chart for the case λ= 0, in fig. 5.14.) As the volume fraction of the homogeneous, initial cloud phase(s) decreases, their distribution (at volume fraction 0.5 marked by open symbols to the top left) deviates increasingly
from the λ-curve, showing in turn a particular depletion in AAB/BBAsequences.
For λ=−0.4, the reentrant behavior of the three-phase boundary line, cf. fig. 5.9, gives rise to two sets of coexisting distributions (down triangles). Upon increasingLχ in the two homogeneous phases, the three-phase region appears with a lamellar shadow (nearly on the curve ofλ-distributions, shift to the bottom hardly visible), which grows with Lχ in volume fraction until it becomes a lamellar cloud (now the symbol on the curve of λ-distributions) coexisting with two homogeneous shadows (triangle shifted slightly to the top). The homogeneous shadows are enriched in homopolymers and depleted from alternating sequences. Upon further increasingLχ, the global lamellar phase gives way to a three-phase coexistence again. The topmost triangle represents the distribution of the homogeneous bi-phase shadow at this reentrance. Forλ≤λc, the lamellar cloud line is the only three-phase boundary. The topmost symbols for the critical and subcritical correlationsλ=λc and λ=−0.5 show the distributions of the coexisting homogeneous shadows which deviate markedly from the λ-distributions.
A detailed sequence distribution diagram for the three-phase coexistence atλ= 0 is displayed in fig. 5.14. The representation of all six species’ concentrations additionally visualizes the segregation within a sequence class into A- andB-rich subspecies between the two homogeneous phases. Using again the two homogeneous phases’ A B exchange symmetry in a globally symmetric distribution, only the distribution of the A-rich, homogeneous phase is shown; the chart for the B-rich phase is obtained by exchanging letters A and B in the key. Consistent with fig. 5.13, fractionation is seen to be effective already on the boundaryχ(h): The sequence class concentrations of the emerging lamellae can be immediately distinguished from the λ-distribution
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