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A rubbery material based mainly on cis-rich polyisoprene. A lightly crosslinked network provides recoverable deformation through rubber elasticity.

Reading the lowest point of the order-disorder boundary in the supplied experimental diagram gives approximately(about $0.45$–$0.47$, allowing for the graphical resolution). The ordered region just above that minimum is lamellar.

This is a reading of the supplied polyisoprene-polystyrene phase diagram. It should not be replaced by the often quoted symmetric-chain mean-field theory value $10.5$: that refers to a different idealization and is not the minimum drawn here.

For the dry diblock copolymer, the block volume fractionis the volume assigned to the polyisoprene block divided by total polymer volume, not its mass fraction or the volume fraction of polymer in the whole solution. It controls domain curvature and topology. Similar block volumes favor alternating lamellae; a smaller minority block generally favors curved interfaces, cylinders and eventually isolated spherical domains.

The chemical block lengths are fixed. Temperature changes their volumes only through thermal expansion, so $f_{PI}$ is often approximated as temperature independent; unequal expansion coefficients can produce a small change.

Solvent requires a distinction between dry composition and swollen domain composition. The dry $f_{PI}$ remains fixed. If solvent volumes $V_{s,PI}$ and $V_{s,PS}$ partition into the two domains, an effective swollen fraction isA polystyrene-selective solvent therefore lowers $f_{PI}^{\mathrm{eff}}$, whereas a polyisoprene-selective solvent raises it. Equal proportional swelling leaves this effective ratio unchanged.

The starting point is in the lamellar region. Preferential swelling of polystyrene makes polyisoprene the increasingly small minority domain, so the relevant effective composition moves left from $f_{PI}^{\mathrm{eff}}=0.45$.

If segregation stayed near $\chi N=35$, following the horizontal path in the supplied diagram would give lamellae → hexagonally perforated layers → hexagonally arranged PI cylinders → PI spheres, before sufficient asymmetry reaches the disordered region. The perforated-layer interval is narrow, around $f_{PI}\simeq 0.36$–$0.40$ at this segregation strength; precise boundaries should not be read more accurately than the figure allows.

Solvent also changes effective segregation, so the actual trajectory need not be horizontal. If it passes closer to the order-disorder boundary, it can enter the narrow double gyroid region instead of, or after, the hexagonally perforated layer phase. Greater dilution can cause disorder before all the ordered morphologies appear. Thus a plausible alternative is lamellae → gyroid → cylinders → disorder.

The robust prediction is the change toward minority-PI curved structures, not a guarantee that every named phase occurs. A solvent partition coefficient and a concentration-dependent interaction model would be needed to locate the actual path quantitatively; applying a dry-melt diagram to a swollen material is an effective approximation.

Entropic elasticity. A rubber contains flexible polymer strands tied into a network by cross-links. Stretching biases the strands toward relatively extended configurations, reducing their configurational entropy. The restoring entropic force follows from the increase in Helmholtz free energy $F=U-TS$. For a Gaussian strand with $N_s$ statistical steps of length $b$, its deformation-dependent free energy is approximately $3k_{B}T{R}^2/(2N_s{b}^2)$. Its stiffness therefore increases with absolute temperature, even if the underlying bond stiffness changes little.

In the ideal affine-network model of rubber elasticity, the shear modulus iswhere ${\nu }_s$ is the number density of elastically active network strands. Thus heating increases the modulus approximately in proportion to $T$, provided the network density and rubbery state remain approximately unchanged. Thermal expansion, changing relaxation times or damage can modify that simple trend.

Examples are lightly crosslinked natural rubber, based on cis-rich polyisoprene, and crosslinked polydimethylsiloxane. The crosslinks prevent the long-time macroscopic flow of an unconnected chain solution.

Density estimate and the unspecified modulus convention. With room temperature $T\simeq 298\mathrm{K}$, the Boltzmann constant gives $k_{B}T\simeq 4.11\times 10^{-21}\mathrm{J}$. If the stated modulus is $G=2\times 10^5\mathrm{Pa}$, thenThe molar value counts moles of active strands. If “modulus” instead means Young's modulus $E$, a nearly incompressible isotropic rubber has $E\simeq 3G$, and the inferred value isThus the robust order-of-magnitude estimate is $10^{25}{\mathrm{m}}^{-3}$. The cross-link density inferred from rubber elasticity is usually an effective strand density; the actual junction density is ${\nu }_j=2{\nu }_s/f$ for junction functionality $f$ in an ideal network without inactive loops or dangling ends. For four-functional junctions this would be half the strand density. The paper supplies neither modulus type nor junction functionality, so a unique chemical junction count cannot be extracted.