A single-chain nanoparticle-based mean-field theory for associative polymers

TL;DR

The paper develops a mean-field theory for associative polymers with alternating sticker types by treating fully bonded single-chain nanoparticles (SCNPs) as a reference state and adding a bond-swapping free-energy cost that converts intra- to inter-chain bonds. A chain-level mass-action law $\frac{q}{(1-q)^2}=\frac{4c v_L}{2^{3ν+1}L}$ relates intra- and intermolecular bonding, and the total free energy $βf(c)=βf_{\rm ig}+βf_{\rm ex}+βf_{\rm bond}$ yields phase diagrams that reproduce known numerical results: increasing sticker diversity (higher $m$) promotes phase separation, while a single sticker type does not. The theory provides scaling relations for critical parameters, showing $B^{\rm crit}_{2,m}\propto m^{3ν-1}$ and revealing how chain architecture (Gaussian vs real chains) and spacer length $l$ affect phase behavior. Overall, the work connects microscopic sticker topology to macroscopic phase separation through a transparent, quantitative framework that aligns with simulations and offers guidance for designing SCNP-based materials.

Abstract

Associative polymers are a class of polymers containing attractive stickers that can reversibly bind to each other. Their fully-bonded state gives rise, in dilute conditions, to a fluid phase of so-called single-chain nanoparticles (SCNPs). These constructs have been used in a wide range of applications, from the design of new materials (e.g. biomolecular condensates) to drug-delivery vectors. The thermodynamic properties of SCNPs sensitively depend on the number of different sticker types, since numerical simulations show that a continuous transition to a network of chains upon increase of polymer concentration in the single sticker-type case can be replaced by an abrupt network formation (via a first-order phase transition) in the multiple sticker-type case. We present here a theory that, using the SCNP fluid as the reference system, quantifies the free energy change associated with transferring an intra-molecular bond to an inter-molecular bond, elucidating the impact on the phase separation process of the sticker topology. Despite its simplicity, the theory highlights which microscopic assumptions (looping statistics, chain-level excluded volume) are most relevant for accurately capturing the thermodynamics of these systems. Our results match available numerical predictions obtained via coarse grained simulations of these systems, highlighting in particular the sensitivity of the phase behaviour on the sequence of the stickers along the chain.

Figures (6)

• Figure 1: (a) Segment of an associative polymer chain with $l=10$, $m=2$ and $L=ml=20$. (b-c) Fully-bonded configurations of (b) $m = 1$ and (c) $m = 2$ associative polymers with $\mathcal{M} = 24$ stickers and $l = 10$ inert monomers between stickers. The $m=2$ polymer presents a more compact structure because of the alternating looping scheme. Note that in the (a-b-c) panels, for clarity, inert monomers and stickers are shown with smaller and larger diameters, respectively, than their actual physical size. (d) Cartoon of the "chemical reaction" between two $A_{i1}$ units (forming two intra-polymer bonds) and one $A_{i2}$ unit (forming two inter-polymer bonds).
• Figure 2: The compressibility factor of fully-bonded systems made of $m = 1$ and $m = 2$ chains, when only intra-chain bonding is allowed (dashed lines). The solid lines have been drawn by considering a second-order virial expansion, fixing the value of the $B_2$ to that extracted from two-body effective interactions (black and red lines, see text for details), or to the second virial coefficient of single monomers, $B_2^{\rm mon} \approx 2.2 a^3$.
• Figure 3: Mass action laws from MD simulations of a polymeric sequence with $l=18$ inert monomers and two additional stickers, one at each end. To reproduce the ideal chain behaviour, the non-bonded monomer-monomer interaction has been set to zero. The values shown have been computed running 8 simulations for each state point and calculating the average value $<q/(1-q)^2>$ for each state point. The error bars correspond to the standard error. Dashed lines are one-parameter fits to Eq. \ref{['MAL']}, which yield $3\nu_{\textrm{real}}=2.25$ and $3\nu_{\textrm{ideal}}=1.56$, while solid lines are parameter-free curves obtained by plugging the theoretical values $3\nu^*_\textrm{real}=2.225$ and $3\nu^*_{\textrm{ideal}}=1.5$ in Eq. \ref{['MAL']}. In these calculations we have chosen $\beta \epsilon=6$ ($\epsilon$ indicates the attractive potential depth). Only fully-bonded configurations have been included in the analysis.
• Figure 4: (a) Binodal lines for the $m=1$ and the $m=2$ system of a solution of ideal SCNPs. Here $\mathcal{M}=24$, $a=1$, $l=10$ and $N=\mathcal{M}l=240$. (b) Binodal lines for the $m=1$ and $m=2$ system of real SCNPs (including excluded volume interactions), for the same parameter values. The dashed lines, which match the corresponding system colour, indicate the value of the chain-chain second virial coefficient, as extracted from simulations (see the Methods).
• Figure 5: Dashed lines are the effective interactions between fully-bonded SCNPs with $\mathcal{M} = 24$, $l = 10$, and two types of sticker sequence, $m = 1$ and $m = 2$, as a function of the chain-chain centre of mass separation distance $R$, when only intra-chain bonding is allowed. Solid lines are best fits to Gaussian functions, which are then used in Eq. \ref{['eq:B2']} to compute the second virial coefficients, yielding $B_2 = 8200\, a^3$ and $B_2 = 5400\, a^3$ for the $m = 1$ and $m = 2$ systems, respectively.