Designing a fully-entropic phase separation in associative polymers

Lorenzo Rovigatti

Physics Department, Sapienza University of Rome

XVI International Workshop on Complex Systems, March 13th, 2023

Entropy as a driving force for phase separation

  • Crystallisation in hard spheres [figure by Pusey et al, Phil. Trans. R. Soc. 2009]
  • Gas-liquid phase separation due to depletion interactions
  • Athermal liquid-crystalline phases [figure by Dussi & Dijkstra, Nat. Commun. 2016]

Associative polymers

  • Polymers decorated with attractive monomers
  • Diluted $\to$ single-chain nanoparticles (SCNPs)[1]
  • Concentrated $\to$ transient polymer networks (useful as e.g. model biosystems[2])
  • Strong but exchangeable bonds $\to$ covalent adaptable networks (e.g. vitrimers[3])
[1] A. P. P. Kröger and J. M. J. Paulusse, J. Control. Release (2018)
[2] J.-M. Choi, A. S. Hoblehouse, and R. V. Pappu Annu. Rev. Biophys. (2020)
[3] W. Denissen, J. M. Winne, and Filip E. Du Prez, Chem. Sci. (2016)

What contributes to the thermodynamics of associative polymers?

  • I will focus on models that can form both intra- and inter-molecular bonds
  • At low concentrations polymers don't meet: intra-molecular bonds are more likely
  • Forming an intra-molecular bond costs entropy
  • All things being equal, inter-molecular bonds are combinatorially favoured
Overall, the number of inter-molecular bonds increases with density

The thermodynamics of associative polymers - results from literature

  • According to theory, "simple" associative polymers do not phase separate[1]
    • The conversion of intra- to inter-molecular bonds is continuous
    • SCNPs start to bind to each other and form transient polymer networks
  • Experiments[2] and simulations[3] confirm this picture

[1] A. N. Semenov and . Rubinstein, Macromolecules (1998)
[2] D. E. Whitaker, C. S. Mahon, and D. A. Fulton Angew. Chem. (2013)
[3] M. Formanek et al., Macromolecules (2021)

Associative polymers models

The goal: control the thermodynamics by manipulating the sequence
  • Equispaced reactive monomers along the chain
  • Three models: one, two or three types of (alternating) reactive monomers
  • Only like-type reactive monomers can bind to each other

The low-density fully-bonded limit

  • In all three cases the energy is the same
  • To satisfy all bonds, polymers have to "fold"
  • Closing a loop costs entropy: the longer the loop, the higher the cost
  • The largest the distance between "compatible" monomers, the longer the loop
  • The cost per reactive site is $\approx -4\, k_B$ for $(AAAA)_6$ and $\approx -5\, k_B$ for $(ABCD)_6$

Effective potentials from two-chain simulations

  • Only intra-molecular bonds $\to$ strong repulsion
  • Intra- and inter-molecular bonds $\to$ from weakly repulsive to strongly attractive
  • In the $(AAAA)_6$ system, $\beta V \approx 0$ $\to$ attraction $\approx$ repulsion
All bonds are formed $\to$ the only contribution is entropic

Equations of state in the fully-bonded limit

  • The $(AAAA)_6$ behaves like an ideal gas in a large density range
  • Both "designed" systems display a non-monotonic equation of state
Signature of phase separation?

Signature of phase separation!

  • In the $(AAAA)_6$ system, the interfaces melt
  • In the designed systems, the density remains heterogeneous

Phase diagram of the (ABAB)$_6$ model

Direct-coexistence simulations
$T$ is fixed, $\epsilon$ is the depth of the attraction between reactive monomers

Robustness checks

Phase separation is retained under extensive variations of the polymer architecture

Conclusions & perspectives

  • We find a way to controllably tune the behaviour of associative polymers
  • The resulting phase behaviour is fully determined by entropy
  • Future work: experimental realisation (with DNA?) and dynamics

Thank you!

Questions?

Reference: L. Rovigatti and F. Sciortino, Phys. Rev. Lett. (2022)

Model & methods - details

  • Implicit-solvent, coarse-grained description: polymer connectivity (Kremer-Grest) + attraction between the reactive monomers
  • Molecular dynamics simulations to compute two-body (effective interactions) and many-body (e.g. pressure) quantities
  • A three-body potential to enforce a single-bond-per-site constraint and enable a bond-swapping mechanism to speed-up the dynamics

The three-body potential

Configuration
V2 $-\epsilon$ $-2\epsilon$ $-\epsilon$
V3 $0$ $\lambda\epsilon$ $0$
V2 + V3 $-\epsilon$ $-2 \epsilon + \lambda\epsilon$ $\epsilon$

The value of $\lambda$ controls the behaviour

  • $\lambda \geq 1$ $\to$ single-bond-per-patch
  • $\lambda = 1$ $\to$ free swapping!
F. Sciortino, Eur. Phys. J. E (2017), L. Rovigatti et al., Macromolecules (2018)

Weak dependence of the # of bonds and $\beta V$ on the spacing

Here $M$ is the number of inert monomer separating two nearby attractive monomers

The bond-bond autocorrelation function decays to $\approx 0$


Probability that a bond existing at $t = 0$ is still there at $t = 0$ (two-chain simulations)