4.4 Three-phase coexistence conditions
5.1.1 Scaling of the order-parameter amplitude at the multicritical point 62
for the transition lines, we are able to access the order-parameter amplitudes in the coexisting cloud and shadow phases. All these amplitudes vanish on approach to the multicritical point. The scaling of the order-parameter amplitude as a function of
∆λ := |λ−λc| along the boundary lines of three-phase coexistence with λ ↑ λc is shown in fig. 5.4, for three cases: The amplitudes of fractionated lamellar shadows (coexisting with homogeneous clouds, on the part λ < λc of the dotted line in fig. 5.3) are marked by open diamonds, those of lamellar cloud phases (on the the dot-dashed line in fig. 5.3) by solid triangles, those of the coexisting homogeneous shadows by open squares. According to the fit performed to the latter case, the amplitudes vanish approximately linearly in ∆λat the Lifshitz point, indicating a critical exponent ψ≈1.
The same exponent is obtained from the plots of the lamellar amplitudes.
62
5.1. TRIBLOCKS WITH A SMALL NUMBER OF SEGMENTS PER BLOCK
10-4 10-3 10-2 10-1
10-5 10-4 10-3 10-2 10-1
order-parameter amplitude
τˆm /ρ0
distance to multicrit. point ∆λ = |λc - λ|
3-phase boundaries
fit 7.8(1) ∆λ0.992(2) (9 datap.) hom. shadow (λ < λc) lam. cloud (λ < λc) lam. shadow at λ < λc
Figure 5.4: Decay of the order-parameter amplitudes as a function of |λ−λc|, proceeding toward the multicritical point on the boundaries of three-phase coexistence, which are located in the regionλ < λc for triblocks withM = 3, see fig. 5.3.
Analytical determination of the critical exponent forM <7
For triblocks withM <7, the Lifshitz behavior at the multicritical point (λc, χc) implies that the wave numberkm of the global lamellar instability and of the lamellar cloud vanishes continuously on approach to (λc, Lχc), see also fig. 5.2. More specifically,km as a function of ∆λis a square root,
km∝(∆λ)1/2, λ↑λc, (5.1)
as can be derived from an expansion in ∆λof the maximumS(km2) for a λ-distribution, cf. eq. (3.37). Given that the lineχ0(λ) has nonzero slope at (λc, χc), the decay of km with ∆χ:=χ−χc occurs with exponent 1/2, too.
The continuous decay ofkm at (λc, χc) allows us to analytically extract the exponent of the order-parameter amplitude in the lamellar subsystem at (λc, χc) from the equation system (4.5) for the lamellar cloud lineχ(m) (see sec. 4.4.1). In the vicinity of (λc, χc), we can solve this system for the deviations of the sequence concentrations in the homogeneous bi-phase shadow from the global concentrations, ∆nν := n(h)ν −pν(λ), with a power-series ansatz
∆nν(∆λ) =X
j
cνj(∆λ)j. (5.2)
In the representative case ofM = 3, consistent expansion up to (∆λ)4 yields, along the line χ(m):
∆nν =−144√ 6
55 (∆λ)2+O (∆λ)3
, ν = 2,3. (5.3)
By inserting these concentration differences into an expansion of χ(m) −2L/S(k2m), which determines the order-parameter amplitude ˆτm(cf. eq. (3.48)) of the lamellar cloud phase with λ-distribution, we find the critical exponentψ= 1 for ˆτm,
ˆ
τm∝∆λ, λ↑λc. (5.4)
This result corroborates the findings for the exponent from the numerically determined values of the lamellar cloud amplitude.
CHAPTER 5. PHASE BEHAVIOR WITH FRACTIONATION
Moreover, we find that the slope of the instability lineχ0(λ) of the disordered state toward global lamellar phase separation and the slope of the lamellar cloud boundary χ(m)(λ) of the three-phase coexistence region are equal at (λc, χc):
χ(m)(λ)−χ0(λ)∝(∆λ)2, λ↑λc. (5.5) Hence, for λ . λc, there is a nearly direct transition from the disordered state to three-phase coexistence, as can be observed in fig. 5.3.
5.2 Triblocks with more segments per block and continuous-chain limit
with line styles as in fig. 5.1.The multicritical point (◦) is located at (λc, Lχc) = (−0.464 00,11.316). Crosses indicate the end points of the lines of metastable, global phase separations of the dis-ordered state, macroscopic for length scale isRchain.
Representative of triblocks with M ≥7 segments per block, we analyze the phase diagram for random, continuous-chain triblocks, shown in fig. 5.5. ForM = 7, a detail of the phase diagram at the multicritical point is provided in fig. 5.6. Again, as in the
10.8
Figure 5.6: Detail of the phase diagram for triblocks Rchain, has the nonzero value 0.150π. Note that the shape of the line χ(h)(λ) still differs from the shape of χ(h)(λ) in the continuous-chain limit, cf.
fig. 5.8 below.
64
5.2. TRIBLOCKS WITH MORE SEGMENTS PER BLOCK
caseM <7, forλ > λc and increasingLχ in the two homogeneous phases, the dotted line χ(h) marks the emergence of a lamellar shadow in addition to the homogeneous bi-phase cloud. Starting from a point onχ(h), the lamellar volume fraction grows both with increasing Lχ and with decreasingλ. Upon decreasingλ, the lamellar phase takes over to be the cloud on the dot-dashed line χ(m), indicating in turn a fractionated homogeneous bi-phase shadow.
In comparison to the case M < 7, see fig. 5.1, the three-phase coexistence region is found to be larger, again restricting the predictions to values of Lχ that do not exceed considerably those of the ODT. The multicritical point is not only located at a smaller critical block correlationλcand a higher incompatibility, but is also qualitatively different: As discussed in sec. 3.3.2, the wave number of the global instability of a disordered triblock melt withλ-distribution is discontinuous at λc forM ≥7. Thus
0.00π
wave number k [1/Rchain]
block-type correlation λ
Figure 5.7:Lamellar wave number for continuous-chain triblocks. Dot-dashed, dashed (blue, green) line:
global lamellar phase and lamellar cloud (v(m) = 1) at λ < λc; dot-ted (purple) line: lamellar shadow (v(m) = 0) at λ > λc; solid (light-green) line: metastable global lamel-lae due to a second peak ofs(k2), shown at λ = −0.4625 (triangle) in the bottom inset. Circles mark the wave numbers of the coexist-ing states at the multicritical point, crosses (×, +) indicate the end points of the lines of metastable, global phase separations, cf. fig. 5.5.
when reaching λc from above, the morphology of the global ordered state changes from two homogeneous phases (critical wave number k0 = 0) to one lamellar phase with finite wave number k0,c. This feature is revealed in more detail in the plot of lamellar wave numbers in fig. 5.7. At the multicritical point, both the wave number of the global instability and the wave number of lamellae in a fractionated subsystem as a function of ∆λtend to the nonzero value k0,c = 0.326π. Due to the two peaks of the global structure function s(k2) near multicriticality (see the bottom inset in fig. 5.7), metastable global lamellae occur in a small range of block-type correlations,
−0.461 23 ≥λ > λc, where the free-energy functional’s absolute minimum indicates global macroscopic phase separation. Conversely, global macroscopic phase separation persists as a metastable state for−0.5< λ < λc(at λ=−0.5, the second derivative of s(k2) at k= 0 changes sign according to eq. (3.39)). The metastable lines, whose end points are hardly resolvable in fig. 5.5, appear in fig. 5.8 as continuations of the two parts of the lineχ0(λ) which indicate the stable structures emerging from the disordered state. Possibly due to the intersection of these metastable continuations, we find for λ < λcthat the transition lineχ0(λ) from the disordered to the global lamellar state
CHAPTER 5. PHASE BEHAVIOR WITH FRACTIONATION
lam. Figure 5.8: Blow-up of the
phase diagram for continuous-chain triblocks from fig. 5.5 around the multicritical point (λc, Lχc), with volume fraction of the lamellar phase. Addi-tional thin lines correspond to the metastable transitions to global states, as indicated in figs. 5.5 and 5.7: solid (green):
lamellae atλ > λc; dashed (red):
homogeneous phases at λ < λc.
and the lamellar cloud transition lineχ(m)(λ) to three-phase coexistence differ in their slopes at multicriticality, in contrast to the behavior at the Lifshitz points forM <7. A map of the lamellar volume fraction around the multicritical point is shown in fig. 5.8, too (note the zoom to an even smaller region than in fig. 5.3). Upon increasing Lχ from the transition lineχ(h) at λ > λc, the lamellar volume fraction grows rapidly to reach a nearly constant value of about 0.6.
Enhanced theory with wave-number dependence of fourth order
In this section, we refine the analysis of the fractionation scenario by accounting for the wave-number dependence of the fourth-order coefficients of eq. (3.25), which lead to the enhanced expression eq. (3.54) for the lamellar free energy instead of the simplified version eq. (3.50). Figure 5.9 shows the results of this more detailed calculation for
4 eq. (3.25) in the lamellar free energy, cf. eq. (3.54). Line styles as in fig. 5.1. Crosses indicate the end points of the line of metastable macro-scopic phase separations for λ < λc (×) and of metastable lamellae forλ > λc (+).
the phase diagram of continuous-chain triblocks. First, we find the region of global lamellar phase separation to be larger than in fig. 5.5 and, in marked contrast to the
66
5.2. TRIBLOCKS WITH MORE SEGMENTS PER BLOCK
results we could obtain with the simplified free energy, to extend into the half-plane λ > λc. The main virtue of the expression eq. (3.54) is precisely to open the possibility of global lamellae also atλ > λc, since the optimal wave numberkopt, cf. eq. (3.53), is not fixed by the global sequence distribution as km, but may change upon increasing Lχ. Second, the transition line χ(h) between global macroscopic phase separation and the three-phase region at λ > λc is located at lowerLχ than the line calculated with the simplified free energy, cf. fig. 5.5. In effect, the enhanced description of the lamellar subsystem in our fractionation scheme, which rests on a superposition of free energies, is seen to increase the tendency toward lamellar phase separation.
0.25π 0.50π 0.75π 1.00π
-0.8 -0.6 -0.4 -0.2 0
wave number k [1/Rchain]
block-type correlation λ v(m)=0 v(m)=1 km(λ)
-0.1 0 0.1
0 1 2 3 4 5 amplitude τˆm/ρ0
x[Rchain] Lχ(h) = 9.688
Figure 5.10:Wave numberkm(λ) of global lamellar instability (dashed), of lamellar cloud with v(m) = 1 (dot-dashed), and of lamellar shadow with v(m) = 0 (dotted); top right inset:
order-parameter amplitude in lamellar shadow atLχ(h)(λ=−0.2) = 9.688.
The graph of the wave numbers in fig. 5.10 suggests that on the lamellar shadow line (v(m)= 0, λ > λc), smaller wave numbers are realized than in the simplified picture, cf.
fig. 5.7. As mentioned above, the corresponding incompatibilities χ(h) are smaller, too, and accordingly, the amplitudes on this line. Indeed, the inset shows a lamellar shadow amplitude, which is still reasonably small at λ = −0.2, a value not in the ultimate vicinity of the multicritical point. The line χ(h) therefore may be regarded as consistent with the expansion in the order parameter,Lχˆτ /%0, assumed for the lamellar phase, cf.
eq. (3.25), not only in the closest vicinity of (λc, Lχc). Moreover, on the lineχ(h), the cloud is the homogeneous bi-phase, whose amplitude is calculated exactly.
At a given λ < λc, however, the global lamellar wave number is found to increase substantially in the rangeχ0 < χ < χ(m). Due to the rather large lamellar amplitudes on the part of the lineχ(m) at λ < λc, the validity of the predictions for this part of χ(m) is restricted.
The region of global lamellar phase separation atλ > λc is in fig. 5.9 bounded toward largerLχby the lineχ(m), which indicates a reentrance into the three-phase coexistence.
Numerical SCFT results, shown in fig. 5.16 below, predict instead that the lamellar cloud line lies entirely in the half-planeλ > λcand thus global lamellar phase separation is always favored at sufficiently large incompatibilities.
CHAPTER 5. PHASE BEHAVIOR WITH FRACTIONATION
10-3 10-2 10-1
10-5 10-4 10-3 10-2 10-1
order-parameter amplitude
τˆm /ρ0
distance to multicrit. point ∆λ = |λc - λ|
fit 0.1503(7) ∆λ0.5000(6) (85 dp.)
lamellar shadow (λ > λc) fit 0.259(1) ∆λ0.4963(7) (20 datap.) lamellar cloud (λ < λc)
Figure 5.11: Scaling of the lamel-lar order-parameter amplitude on ap-proach to the multicritical point along two three-phase transition lines for continuous-chain triblocks, with the simplified expression eq. (3.50) for the lamellar free energy.
5.2.1 Scaling of the order-parameter amplitude at the multicritical point