Field-induced phase transitions in ferro-antiferromagnetic diblock copolymers

Abstract

We study the equilibrium properties of a model of magnetic diblock copolymer where each monomer is decorated with an Ising-like spin. Spins interact ferromagnetically within each block and antiferromagnetically across blocks, generating frustration between magnetic ordering and spatial organization. By employing a mean-field approach and Monte Carlo simulations for self-avoiding walks on the cubic lattice, we investigate the system's response to an external magnetic field. We discover a rich phase diagram that includes: a swollen phase with both filaments magnetically disordered and spatially extended; a mixed compact phase characterized by a single globule in which the two filaments are strongly intertwined; a segregated compact phase composed of two globular, magnetically ordered, and spatially separated blocks. Further, if the magnitude of the intra-block ferromagnetic interaction differs between the two blocks, we observe a hybrid segregated (``tadpole'') phase where one extended block coexists with a collapsed one. Mean-field predictions are in quantitative agreement with Monte Carlo results for the location of the phase boundaries. These findings provide a minimal statistical-mechanical framework for field-controlled self-assembly of tunable patterns by magnetically heterogeneous polymers. They may also serve as a simple platform to investigate the coupling between internal epigenetic-like states and chromatin folding.

Figures (12)

• Figure 1: (a): Cartoon of the magnetic diblock copolymer model on the square lattice. The polymer is a self-avoiding walk (SAW) $\gamma$ and is composed of two blocks $\gamma_{1}$ and $\gamma_{2}$, highlighted in yellow and pink. Each vertex (monomer) is decorated with a spin variable $S_{i}$, with possible values $S_{i} = 1$ (blue) and $S_{i} = -1$ (red). (b) Summary of the interactions among spin variables: there is an antiferromagnetic coupling between pairs of spins belonging to different blocks and a ferromagnetic one between pairs of spins in the same block.
• Figure 2: Mean field phase diagrams of the magnetic diblock copolymer model. In panels (a)-(c) three different section of the phase diagrams are shown: (a) $J_{12}-T$ plane at $H=0$ and $J_{11} = J_{22} = 1$, (b)-(c) $H-T$ plane for $J_{12} = 2$ and (b) $J_{11}=J_{22}=1$ or (c) $J_{11} = 1.0$ and $J_{22} = 0.5$. Four equilibrium phases are present: (i) the swollen phase where both blocks are extended and magnetically disordered; (ii) the mixed phase, characterized by a compact conformation where the two blocks are intertwined and have opposite magnetisation; (iii) the segregated phase, characterized by two compact globules with no inter-filaments contacts and, at $H>0$, same-sign magnetisation; (iv) the asymmetrically segregated phase (or tadpole phase) phase, that can be observed when $J_{11} \neq J_{22}$ and is characterized by the coexistence of one globular and one swollen block, depending on whichever block has the stronger ferromagnetic interaction. Dashed and solid lines denote, respectively, first-order and second-order phase transitions. The blue circle in panel (c) represents a tricritical point located at $H = 2.95$ and $T = 2.25 J$. Notably, if $J_{12}< \max (J_{11}, J_{22})$ at $H=0$, the mixed phase does not occur. Panel (d) reports cartoons of typical conformations that can be observed in the different phases, as well as a qualitative graphical representation of their contact matrix. In the swollen phase (i), there is a vanishing number of contacts on the whole chain in the MF approximation. In the mixed phase (ii), under the Hamiltonian walk approximation, contacts are uniformly distributed on the chain. In the segregated phase (iii), contacts are mainly distributed within the two diagonal blocks -- the cartoon refers to the case $H > 0$, where $m_{1} m_{2} > 0$. Finally, in the tadpole phase (iv), contacts appear only within the collapsed block.
• Figure 3: Results of MC simulations at $T = 2.0$ and varying $H$ ($J_{11} = J_{22} = 1, J_{12} = 2$). (a) Inter-filament $c_{12}$ and intra-filament average number of contacts $c_{11} + c_{22}$ per monomer. (b) Magnetisation $m$ and staggered magnetisation $m_s$ per monomer. (c) Scaled mean square radius of gyration $R_g^2/2N$ for different values of $N$. Note that $c_{12}$ drops to zero for $H > 1.1$ while, concurrently, $c_{11} + c_{22}$ increases towards a plateau (a); a similar behavior holds for $m$ and $m_s$ (b), indicating that the system is in a mixed antiferromagnetic compact phase for $H < 1.1$. No crossing appears in panel (c), suggesting that the system remains in a compact phase for all values of $H$ considered. The estimate of $H$ at the transition, $H^*$ is obtained by extrapolating the finite-size values $H_{max}(N)$, corresponding to the maximum of the variance of the inter-filament contacts, in the limit $N \to \infty$ . This gives $H^* \approx 1.0 \pm 0.1$.
• Figure 4: Results of MC simulations at fixed $T = 2.5$ and varying $H$ ($J_{11} = J_{22} = 1, J_{12}=2$). (a) Inter-filament $c_{12}$ and intra-filament average number of contacts $c_{11} + c_{22}$ per monomer. (b) Magnetisation $m$ and staggered magnetisation $m_s$ per monomer. (c) Scaled mean square radius of gyration $R_g^2/2N$ for different values of $N$. At low values of $H$, $c_{12} > 0$ and $m_s>0$, confirming the presence of a mixed compact phase (see snapshot at $H=0.0$), while the presence of a minimum in the intra-filament contacts (a) suggests the onset of a more extended phase (see snapshots at $H=1.1,1.8$). In panel (c), the appearance of crossings between the curves of $R_g^2/N$ at different values of $N$ is revealing of two consecutive transitions: the first one between a compact mixed and a swollen disordered phase, and the second one between a swollen disordered and a compact segregated phase (see snapshot at $H=2.4$). The estimates of the magnetic field values at these transitions are $H_1^* = 0.52 \pm 0.03$ and $H_2^* = 0.79 \pm 0.08$ respectively.
• Figure 5: Results of MC simulations at fixed $T = 2.5$ and varying $H$, for the asymmetric case $J_{11} \neq J_{22}$ ($J_{11} = 1, J_{22} = 0.5$ and $J_{12} = 2.0$). (a) Intra-filament and inter-filament average number of contacts per monomer $c_{11}$ and $c_{22}$, (b,c) Scaled mean square gyration radius of the second block $R_{g,2}^2/(N)$ (b)and of the first block $R_{g,1}^2/(N)$ (c) for different values of $N$. At low values of $H$, the observed large values of $c_{12}$ highlight a mixed compact phase (see snapshot at $H=0.00$). Notice that, since $J_{22} < J_{11}$, we have $c_{22} < c_{11}$ for every value of $H$ considered. The crossings observed at $0.5 < H < 0.9$ of the scaled gyration radius curves of the second block (b) pinpoint its conformational transitions from a compact to an extended phase (see snapshots at $H = 0.45, 0.80$). In contrast, panel (c) suggests that the first block remains in a compact state for all values of $H$ considered (see snapshots). We extrapolate the value of the critical field at the transition to $H^* = 1.19 \pm 0.04$.