# Interplay between self-assembly and phase separation in a polymer-complex
model

Tianhao Li  Department of Chemistry, Princeton University, Princeton, NJ
08544, USA  W. Benjamin Rogers  Martin A. Fisher School of Physics, Brandeis
University  William M. Jacobs<EMAIL_ADDRESS>Department of Chemistry,
Princeton University, Princeton, NJ 08544, USA

###### Abstract

We present a theoretical model for predicting the phase behavior of polymer
solutions in which phase separation competes with oligomerization.
Specifically, we consider scenarios in which the assembly of polymer chains
into stoichiometric complexes prevents the chains from phase-separating via
attractive polymer–polymer interactions. Combining statistical associating
fluid theory with a two-state description of self-assembly, we find that this
model exhibits rich phase behavior, including re-entrance, and we show how
system-specific phase diagrams can be derived graphically. Importantly, we
discuss why these phase diagrams can resemble—and yet are qualitatively
distinct from—phase diagrams of polymer solutions with lower critical solution
temperatures.

##  I Introduction

Predicting the phase behavior of biopolymer systems is challenging due to the
diversity of conformational and oligomeric states that biopolymers can
populate [1, 2]. In this context, oligomers refer to small molecular clusters
that form due to attractive interactions among biomacromolecules, including
proteins [3, 4] and nucleic acids [5, 6]. The interplay between
oligomerization and phase separation is complex and can give rise to
qualitatively different behaviors in different scenarios. In some cases, phase
separation correlates with oligomer stability, either because phase separation
is enhanced by the presence of oligomers [3, 7, 8, 9, 4, 10] or because phase
separation promotes oligomer formation in condensed phases [1, 11]. Yet in
other cases, phase separation appears to compete with oligomer formation,
meaning that the presence of stable oligomers tends to inhibit demixing into
condensed biopolymer-rich phases [12].

Here we explore this latter scenario—competition between the self-assembly of
oligomers and phase separation—using a thermodynamically consistent mean-field
model. We consider polymer solutions in which multiple polymer chains can
assemble into a stoichiometric complex, which is an oligomer comprising a
precise number of chains in a well-defined geometry [13]. A theoretical
challenge is that this scenario cannot be described in terms of pairwise
interactions alone, since the assembly of discrete stoichiometric complexes
implies that spatial correlations among multiple polymer chains cannot be
ignored [14, 15]. We therefore propose a coarse-graining strategy in which a
two-state model of stoichiometric complex self-assembly is coupled to a mean-
field model of interacting, yet spatially uncorrelated, polymer chains within
the framework of statistical associating fluid theory (SAFT) [16].

This theoretical approach treats self-assembly and phase separation in a self-
consistent framework [17, 18], allowing us to predict the effects of self-
assembly on the equilibrium phase behavior of associating polymer solutions
[19]. In the present article, we systematically develop this theoretical
approach and study the implications of the resulting model under wide-ranging
scenarios. In a closely related article [20], we separately provide validation
of our theoretical predictions in a particular DNA-based experimental system,
which utilizes the programmability of Watson–Crick base-pairing [21] to tune
both the thermodynamic stability of a designed stoichiometric complex and the
phase behavior of the DNA oligonucleotides. Taken together, the results of
these studies demonstrate the validity of the assumptions employed in our
theoretical approach and confirm the quantitative accuracy of our model’s
predictions.

The key theoretical results of our approach are presented in this paper as
follows. In Secs. II.1 and II.2, we describe the mean-field model and our
numerical strategy for calculating its phase behavior in various parameter
regimes. Then in Secs. II.3 and II.4, we investigate two scenarios in which
the excluded volume of a stoichiometric complex is either the same as or
greater than the total excluded volume of its constituent chains. Both of
these scenarios exhibit re-entrant phase behavior, in which a constant-
concentration polymer solution transitions from one phase to two coexisting
phases and back to a single phase as a single control parameter is tuned
monotonically. This control parameter represents an experimentally accessible
quantity such as, but not limited to, temperature, ionic strength, pH, or a
co-solvent concentration. Importantly, although many features of the phase
behavior resemble qualitative aspects of phase diagrams with a lower critical
solution temperature (LCST), we find that competition between self-assembly
and phase separation does not result in a critical point of this type.
Instead, we find that phase coexistence can terminate at a first-order
transition due to the competition between self-assembly and phase separation.
Our approach shows how qualitatively different phase diagrams can arise by
altering the dependence of key dimensionless parameters on experimentally
controllable variables. A more thorough discussion of the implications of
these results is provided in Sec. III.

##  II Theoretical Model and Results

###  II.1 Two-state self-assembly and phase separation

We first develop a mean-field model of coupled self-assembly and liquid–liquid
phase separation (Fig. 1a). We consider a polymer solution consisting of a
single polymeric species in an implicit solvent. Polymers are coarse-grained
into chains of $N_{\text{p}}$ “blobs,” each of which represents a Kuhn segment
[22] of the polymer with excluded volume $v_{0}$. This coarse-graining allows
us to treat polymers as freely jointed chains, meaning that the relative
orientations of adjacent blobs are uncorrelated. The total excluded volume of
a free polymer is $v_{\text{p}}=N_{\text{p}}v_{0}$. To account for “specific”
short-ranged attractive interactions, such as hybridization of DNA oligomers
[21] or screened electrostatic interactions between charged sidechains of
polypeptides [23], we introduce $N_{\text{p}}$ distinct binding sites on each
polymer. Each blob can accommodate a single coarse-grained binding site within
this framework, since the orientations of monomers within a Kuhn-segment blob
are highly correlated and thus not independent of one another [22]. Specific
interactions between pairs of binding sites therefore represent coarse-
grained, effective interactions between two Kuhn-length groups of monomers.

Figure 1: Schematic of the theoretical model. (a) Biopolymers can either
assemble into stoichiometric complexes or form a condensed phase via phase
separation. Polymers are represented as freely jointed chains of blobs (yellow
circles), each of which contains a single binding site (blue dots). Here we
consider stoichiometric complexes consisting of $n=4$ polymers. (b) The model
involves two independent free-energy parameters, $\epsilon$ and $\Delta G$,
which govern the pairwise association between binding sites on blobs and the
assembly of chains into stoichiometric complexes, respectively. Associative
interactions between blobs are treated using statistical associating fluid
theory (SAFT), while the two-state model of stoichiometric complex assembly is
treated as an equilibrium chemical reaction between the polymer (P) and
complex (S) states.

This modeling approach relies on three key assumptions. First, we assume that
each binding site can participate in at most one specific attractive
interaction at a time. Second, we assume that the assembly of $n$ polymers
into a stoichiometric complex is highly cooperative, such that partially
assembled intermediates involving fewer than $n$ polymers are vanishingly rare
at equilibrium (i.e., “two-state” self-assembly). Third, we assume that all
binding sites engage in specific interactions within a stoichiometric complex
in the assembled state. This final assumption can be relaxed, leaving
additional binding sites on assembled complexes available for interactions
with binding sites either on other complexes or on free polymers [20], but we
do not consider this possibility in the present work.

Taken together, these assumptions imply that self-assembly and phase
separation are mutually exclusive (Fig. 1a): Associative interactions between
available binding sites are required to drive phase separation, and yet these
binding sites are completely saturated by the formation of a complete
stoichiometric complex. We note that, in practice, satisfying the second,
“two-state” assumption typically requires a stoichiometric mixture of $n$
different polymers with distinct binding sites, such that each polymer species
is used exactly once in constructing a unique stoichiometric complex [24, 25].
The model that we describe here treats all polymers as being indistinguishable
for simplicity, which means that the specific interactions between free
polymers represent effective interactions that average over all possible pairs
of binding sites.

We implement these assumptions within a mean-field theory by combining
statistical associating fluid theory (SAFT) [16] with a variation of the
Flory–Huggins model of a polymer solution [22]. In accordance with the second
key assumption of two-state self-assembly, chains can either exist in the free
polymer state (P) or as part of a stoichiometric complex (S) consisting of $n$
chains. The excluded volume of a complex is $v_{\text{s}}$, which we relate to
the excluded volume of a polymer blob via $N_{\text{s}}\equiv
v_{\text{s}}/v_{0}$ for notational convenience. Importantly, because our
partitioning of chains into either P and S states allows us to treat the
configuration of an assembled complex independently from that of individual
polymers, it is not necessarily the case that $v_{\text{s}}=nv_{\text{p}}$. We
shall consider both scenarios, where the excluded volume is either conserved
or not conserved, in Secs. II.3 and II.4, respectively.

The free-energy density, $f$, comprises four terms,

$f=f^{\text{id}}+f^{\text{ev}}+f^{\text{assoc}}+f^{\text{int}},$ (1)

accounting for the ideal (id), excluded-volume (ev), associative (assoc), and
internal (int) contributions, respectively. Each term has units of energy per
volume. The ideal contribution depends on the concentrations (i.e., number
densities) of polymers in both the P and S states,

$\beta
f^{\text{id}}v_{0}=\frac{\phi_{\text{p}}}{N_{\text{p}}}(\ln\phi_{\text{p}}-1)+\frac{\phi_{\text{s}}}{N_{\text{s}}}(\ln\phi_{\text{s}}-1),$
(2)

where $\phi_{i}\equiv v_{i}\rho_{i}$ is the volume fraction and $\rho_{i}$ is
the concentration (or number density) of state $i\in\\{\text{P},\text{S}\\}$.
To avoid discussing explicit temperature dependence at this point, we use the
inverse temperature $\beta\equiv 1/k_{\text{B}}T$, where $k_{\text{B}}$ is the
Boltzmann constant and $T$ is the absolute temperature, to express all free
energies as dimensionless quantities. We employ a mean-field formula for the
excluded volume contribution that accounts for the fact that assembled
complexes are essentially rigid bodies, whereas free polymers are freely
jointed chains,

$\displaystyle\beta f^{\text{ev}}v_{0}$
$\displaystyle=\frac{\phi_{\text{s}}}{N_{\text{s}}}+\frac{1-\phi_{\text{s}}}{N_{\text{s}}}\ln(1-\phi_{\text{s}})+\frac{\phi_{\text{p}}}{N_{\text{p}}}\left[1-\ln(1-\phi_{\text{s}})\right]$
$\displaystyle\qquad+\phi_{0}\ln\left(\frac{\phi_{0}}{1-\phi_{\text{s}}}\right),$
(3)

where $\phi_{0}\equiv 1-\phi_{\text{s}}-\phi_{\text{p}}$ is the volume
fraction not occupied by either free polymers or stoichiometric complexes. A
derivation of this formula is presented in Appendix A, and a comparison to the
Flory–Huggins model is provided in Appendix B.

Following the two-state assumption, the associative contribution only accounts
for interactions among binding sites on P-state polymers. The free-energy
density is given by Wertheim’s thermodynamic perturbation theory [14], taking
the P/S mixture without attractive interactions [i.e., Eqs. (2) and II.1)] as
the reference state [15],

$\beta f^{\text{assoc}}v_{0}=\phi_{\text{p}}\left(\ln
X_{\text{p}}-\frac{X_{\text{p}}}{2}+\frac{1}{2}\right),$ (4)

where $X_{\text{p}}$ is the fraction of binding sites on P-state polymers that
are not associated. $X_{\text{p}}$ is determined by the chemical equilibrium
expression

$X_{\text{p}}+\phi_{\text{p}}e^{-\beta\epsilon}X_{\text{p}}^{2}=1,$ (5)

where $\epsilon$ is the association free energy between a pair of binding
sites (Fig. 1b). Eq. (4) can be derived by minimizing the Helmholtz free
energy of a mixture of reactive binding sites; Eq. (5) follows directly from
the derivation of $f^{\text{assoc}}$. We refer the reader to Ref. [26] for an
elegant derivation and a thorough discussion. Multiplying Eq. (5) by
$\phi_{\text{p}}$, we can easily identify this chemical equilibrium condition
as a mass-action expression relating the volume fraction of blobs that are not
associated, $X_{\text{p}}\phi_{\text{p}}$, and the volume fraction of blobs
that are associated, $(1-X_{\text{p}})\phi_{\text{p}}$, subject to an
equilibrium constant $\exp(-\beta\epsilon)$. Association does not occur in the
limit $\beta\epsilon\rightarrow\infty$. Importantly, Eqs. (4) and (5) assume
that the association states of different binding sites are uncorrelated. This
requirement is consistent with our coarse-grained treatment of association, in
which we assume that each Kuhn-segment blob accommodates a single binding
site.

By contrast with the associative interactions, the internal contribution to
the free-energy density only depends on the concentration of S-state
complexes,

$\beta
f^{\text{int}}v_{0}=\frac{\phi_{\text{s}}}{N_{\text{s}}}\left[\beta\Delta
G+n(1-N_{\text{p}})\right],$ (6)

where $\Delta G$ represents the free-energy change (excluding translational
entropy) associated with the self-assembly “reaction”
$n\text{P}\rightleftharpoons\text{S}$ (Fig. 1b). This reaction is at
equilibrium when

$n\mu_{\text{p}}=\mu_{\text{s}},$ (7)

where the P and S-state chemical potentials are $\mu_{\text{p}}\equiv\partial
f/\partial\rho_{\text{p}}$ and $\mu_{\text{s}}\equiv\partial
f/\partial\rho_{\text{s}}$, respectively. Intuitively, Eq. (6) says that the
contribution to the free energy due to polymer–polymer interactions within an
S-state complex is proportional to the concentration of S-state complexes,
$\rho_{\text{s}}=\phi_{\text{s}}/N_{\text{s}}v_{0}$. We motivate the
particular definition of $\beta\Delta G$ established by Eq. (6) by discussing
a limiting case at the end of this section.

This treatment allows us to study the consequences of stoichiometric complex
self-assembly within a mean-field theory. In reality, both self-assembly and
association among free polymers are driven by the same physical interactions
among binding sites (Fig. 1a). These interactions among binding sites can be
assumed to be essentially pairwise in nature. However, the probability that a
specific pair of binding sites is associated is highly correlated with the
binding status of proximal binding sites on the same chains when self-assembly
takes place. Such correlations are challenging to capture within mean-field
models via thermodynamic perturbation theory. By contrast, including a
discrete S state via the two-state approximation and the equilibrium condition
Eq. (7) allows us to account for these correlations, while still treating the
associative interactions between binding sites on P-state polymers as
uncorrelated in Eqs. (4) and (5).

Nonetheless, the common physical origin of polymer association and self-
assembly implies that the association free energy, $\epsilon$, and the
assembly free energy, $\Delta G$, are coupled parameters that must change in
tandem when a control parameter is tuned. To be precise, $\Delta G$ depends on
both the sum total of the $nN_{\text{p}}/2$ binding interactions with strength
$\epsilon$ within a stoichiometric complex, as well as entropic considerations
related to the geometry and flexibility of the stoichiometric complex. In
Secs. II.3 and II.4, we show how system-specific relationships between
$\epsilon$ and $\Delta G$ can be used to derive a phase diagram as a function
of a single control parameter.

Before proceeding, it is instructive to consider the behavior of this model in
three limiting scenarios. If $\beta\epsilon$ is finite and $\beta\Delta
G\rightarrow\infty$, then self-assembly cannot occur, and the model reduces to
that of an associative polymer solution in which the polymers obey the
statistics of freely jointed chains [22, 19]. This polymer solution has a
critical point at a dimensionless associative free energy
$(\beta\epsilon)_{\text{c}}$ and polymer volume fraction $\phi_{\text{c}}$.
This critical point is commonly described as an upper critical solution
temperature (UCST), since phase separation only occurs at temperatures below
$T_{\text{c}}=\epsilon k_{\text{B}}^{-1}(\beta\epsilon)_{\text{c}}^{-1}$ if
$\epsilon$ is taken to be a temperature-independent constant [22].
Alternatively, if $\beta\epsilon$ is finite and $\beta\Delta
G\rightarrow-\infty$, then the solution is a fluid of S-state complexes in
implicit solvent. The model reduces to a regular solution [27] with excluded
volume interactions only in this case, since we have assumed that there are no
associative interactions among completely assembled complexes. Lastly, if
$\beta\epsilon\rightarrow\infty$ and $\beta\Delta G$ is finite, then there are
no associative interactions among free polymers, and the model reduces to a
concentration-dependent description of two-state self-assembly. In this case,
Eq. (7) simplifies to an expression of chemical equilibrium for a reaction
involving ideal gases in the dilute limit, $\phi_{\text{p}}\ll 1$ and
$\phi_{\text{s}}\ll 1$,

$\frac{\phi_{\text{s}}}{\phi_{\text{p}}^{n}}=e^{-\beta\Delta G}.$ (8)

This limiting scenario justifies our expression for the internal free energy
in Eq. (6).

###  II.2 Phase-coexistence calculations and master phase diagrams

Figure 2: Schematic of phase-coexistence calculations. (a) For a given chain
concentration $\rho$, we first calculate the equilibrium composition by
minimizing the free energy with respect to the stoichiometric complex (S)
fraction (left). This calculation satisfies the condition of chemical
equilibrium, $\mu_{\text{s}}=n\mu_{\text{p}}$, where $n=4$ in this example
(right). The equilibrium free energy is indicated by the orange diamond ($\rho
N_{\text{p}}v_{0}=0.093$) assuming $N_{\text{p}}=6$, $N_{\text{s}}=24$,
$\beta\epsilon=-4.025$, and $\beta\Delta G=-27$ (corresponding to point d in
Fig. 3a). (b) Performing chemical equilibrium calculations at various total
concentrations results in a free-energy profile. Points on this profile are
colored according to the equilibrium fraction of chains that assemble into
stoichiometric complexes. Common-tangent constructions (red dashed lines) are
then used to identify coexistence regions (shaded regions). Red points
indicate the concentrations and free energies of the coexisting phases. In
this example, there are two distinct coexistence regions at different ranges
of the total concentration. (c) The common-tangent construction ensures that
the coexisting phases have the same chemical potential, $\mu\equiv\partial
f/\partial\rho$, and the same grand potential, $f-\mu\rho$. As an illustrative
example, we show the grand potential at the coexistence chemical potential of
the first coexistence region defined in panel b. The equilibrium compositions
indicate that a dilute phase of mostly assembled complexes coexists with a
condensed phase of mostly free chains.

Our aim is to compute the phase behavior of the mean-field model introduced in
the previous section. This model is completely specified by the five
independent parameters $n$, $N_{\text{p}}$, $N_{\text{s}}$, $\beta\epsilon$,
and $\beta\Delta G$, which are all considered to be constants when carrying
out phase-coexistence calculations. We shall assume that the number of chains
per stoichiometric complex is $n=4$ throughout this work, and then consider
representative scenarios for various combinations of the remaining four
parameters. Although the free-energy density, $f$, is a function of both
$\phi_{\text{p}}$ and $\phi_{\text{s}}$, the concentrations of free polymers
and stoichiometric complexes are related via the chemical equilibrium
constraint, Eq. (7). The total polymer concentration,
$\rho\equiv\rho_{\text{p}}+n\rho_{\text{s}}$, is therefore the sole
independent variable when performing phase-coexistence calculations. For
convenience, we present our results in terms of a dimensionless total
concentration $\rho N_{\text{p}}v_{0}$, which is equal to the volume fraction
if all polymers are in the P state, since this dimensionless quantity is
bounded by 0 and 1. We present the composition of a solution with fixed total
concentration in terms of the fraction of polymers in the S state, i.e., the
“S fraction,” which is equal to $n\rho_{\text{s}}/\rho$. The S fraction is
also bounded by 0 and 1.

For a given set of parameters, we determine the concentrations and
compositions of coexisting phases by computing and analyzing a free-energy
landscape as a function of $\rho$. This calculation is carried out in two
steps. In the first step, we determine the equilibrium S fraction and the
corresponding free-energy density at a fixed value of $\rho$ by solving Eq.
(7). In cases where there are multiple solutions to this chemical equilibrium
equation, we choose the one with the lowest free-energy density; this choice
is equivalent to performing a global minimization of $f$ as a function of the
S fraction at constant $\rho$. An example calculation showing the equivalence
between constrained free-energy minimization and chemical equilibrium is
illustrated in Fig. 2a.

In the second step, we repeat this chemical-equilibrium calculation for many
values of the total concentration in order to construct the equilibrium free-
energy landscape $f(\rho)$ numerically. We then use the common-tangent
construction [27] to identify one or more coexistence regions, where two total
concentrations have the same chemical potential $\mu\equiv\partial
f/\partial\rho$ and osmotic pressure $P\equiv f-(\partial
f/\partial\rho)\rho$, if any such regions exist. An example landscape with two
coexistence regions is illustrated in Fig. 2b. In practice, we compute the
convex hull of a free-energy landscape computed on a grid of discrete total
concentration points, $\\{(\rho,f(\rho))\\}$ [28]. Any point $(\rho,f(\rho))$
that is not part of the convex hull must be within a coexistence region, in
accordance with the common-tangent construction; adjacent grid points that are
not part of the convex hull must be within the same coexistence region. In
this way, we are able to identify the number of coexistence regions and obtain
estimates of the total concentrations of the coexisting phases for each
region. We then refine these estimates to compute the total concentrations and
compositions of each pair of coexisting phases $\alpha$ and $\beta$. This
final refinement step is achieved by solving the equal-chemical-potential,
$\mu^{(\alpha)}=\mu^{(\beta)}$, and equal-osmotic-pressure,
$P^{(\alpha)}=P^{(\beta)}$, conditions to numerical precision. The results of
these calculations are most easily visualized by examining the grand-potential
density, $f-\mu\rho$, with $\mu$ chosen to be equal to the chemical potential
of the two coexisting phases. The grand-potential density is equal to the
osmotic pressure at equilibrium. For example, the case shown in Fig. 2c
corresponds to the lower-concentration coexistence region found in Fig. 2b, in
which a dilute solution consisting primarily of assembled stoichiometric
complexes coexists with a condensed phase of disassembled polymers stabilized
by their associative interactions.

In order to summarize the dependence of the phase behavior on the parameters
$\beta\epsilon$ and $\beta\Delta G$, we introduce the concept of a master
phase diagram. Specifically, for fixed choices of $N_{\text{p}}$ and
$N_{\text{s}}$, we determine the number of distinct coexistence regions that
exist over the complete range of total concentrations, $0\leq\rho
N_{\text{p}}v_{0}\leq 1$, as a function of $\beta\epsilon$ and $\beta\Delta
G$. Example master phase diagrams are shown in Figs. 3a and 5a, and are
discussed in the following sections. In this representation, the boundaries
between parameter regimes in the $\beta\epsilon$–$\beta\Delta G$ plane that
have differing numbers of coexistence regions indicate lines of either first-
order or second-order transitions where coexistence regions terminate. In
other words, crossing one of these boundaries on the master phase diagram
means that one or more coexistence regions either appear or disappear within
some range(s) of total concentration as $\beta\epsilon$ and/or $\beta\Delta G$
are tuned.

Master phase diagrams are useful for understanding how different relationships
between the association and assembly free energies can result in qualitatively
different phase diagrams when calculated as a function of a single control
parameter. For example, consider a biopolymer solution in which temperature
controls both $\beta\epsilon$ and $\beta\Delta G$. Depending on the precise
chemical details of the biopolymers, changing temperature traces out a curve
in the $\beta\epsilon$–$\beta\Delta G$ plane (see, e.g., Figs. 3a and 5a).
Computing the phase behavior as a function of the total concentration at
points along this temperature-parameterized curve therefore allows us to
construct a conventional temperature–concentration phase diagram (see, e.g.,
Figs. 3b and 5b). The topology of this conventional phase diagram is
determined by the way in which the temperature-parameterized curve intersects
the boundaries between parameter regimes with different numbers of coexistence
regions in the $\beta\epsilon$–$\beta\Delta G$ plane. Importantly, this
approach allows us to disentangle the predictions of the mean-field model,
which depend on the free-energy parameters $\beta\epsilon$ and $\beta\Delta
G$, from system-specific parameterizations of these quantities.

###  II.3 Self-assembly with conserved excluded volume

We first consider the scenario where the self-assembly of $n$ polymers into a
stoichiometric complex conserves the excluded volume, such that
$v_{\text{s}}=nv_{\text{p}}$. Here we compute phase diagrams for an example
case with $N_{\text{p}}=6$ blobs per polymer chain. Conservation of excluded
volume thus implies that $N_{\text{s}}=nN_{\text{p}}=24$, since we assume that
$n=4$ chains comprise each stoichiometric complex.

The master phase diagram for this scenario is shown in Fig. 3a. The model
predicts five parameter regimes in the $\beta\epsilon$–$\beta\Delta G$ plane:
two distinct regimes, (1) and (3), in which only one phase exists at all
concentrations; two distinct regimes, (2) and (5), in which a single
coexistence region exists over a finite range of concentrations; and one
regime, (4), in which two different coexistence regions exist over different
concentration ranges. We discuss the physical origin and significance of these
various regimes in the remainder of this section.

Figure 3: Phase-behavior predictions when self-assembly conserves the excluded
volume of the constituent chains. (a) The master phase diagram predicted by
our model exhibits rich phase behavior. The parameter space is shaded
according to the number of distinct coexistence regions at a given
$\beta\epsilon$ and $\beta\Delta G$. In the unshaded regions, the system is
homogeneous at all concentrations. Gray shaded regions indicate that phase
separation occurs in a single coexistence region, while the gold region
indicates the existence of two distinct coexistence regions within different
concentration ranges. These regions are bounded by dashed curves (i), (ii),
and (iii), as described in the text. Solid blue curves show where the dilute
phase (of the lower-concentration coexistence region, if there are two) has
the indicated total concentration. The path taken by a hypothetical control
parameter, which relates $\beta\epsilon$ to $\beta\Delta G$, is shown by a
solid red line; labeled points correspond to the free-energy profiles in
panels c–h. (b) The phase diagram mapped out by the hypothetical control
parameter shown in panel a. Points representing the concentrations of
coexisting phases are colored according to the equilibrium composition of
stoichiometric complexes. The coexistence regions terminate at the points
(iv), (v), and (vi), as described in the text. (c)–(g) Representative free-
energy profiles (points) along the path taken by the control parameter shown
in panel a. For comparison, we also show the free-energy profiles of pure
chains (100% P, solid blue curves) and pure complexes (100% S, solid green
curves). Red dashed lines indicate common-tangent constructions, red points
indicate coexisting concentrations, and gray shaded regions indicate
coexistence regions. (h) A representative free-energy profile in two-phase
region (2) of the master phase diagram, panel a.

####  II.3.1 Phase diagram construction as a function of a control parameter

As discussed in Secs. II.1 and II.2, the free-energy parameters
$\beta\epsilon$ and $\beta\Delta G$ must be related to one another due to
their common dependence on the interactions between pairs of binding sites
(Fig. 1a). We therefore construct a conventional concentration-dependent phase
diagram by tracing a path through the $\beta\epsilon$–$\beta\Delta G$ plane
that depends on a control parameter. For example, if the control parameter is
temperature, then the resulting conventional phase diagram depicts the phase
behavior in the temperature–concentration plane. We show a possible
parameterization of $\beta\epsilon$ and $\beta\Delta G$ in Fig. 3a, which
results in the conventional phase diagram shown in Fig. 3b. This example curve
is chosen because it passes through four of the five regimes on the master
phase diagram. Importantly, this example also satisfies the relation
$\partial\beta\epsilon/\partial\beta\Delta G>0$, since both free energies
should typically be expected to increase or decrease in tandem. Specific
points along the parameterized curve are labeled c–g in Fig. 3a and are
indicated on the conventional phase diagram in Fig. 3b.

The appearance and disappearance of various coexistence regions in Fig. 3b is
determined by the intersections between the control-parameter curve and the
boundaries of the distinct regimes in Fig. 3a. Starting from low values of the
control parameter, the polymer solution is initially in a one-phase region at
all concentrations at point (c). Upon crossing line (ii) in Fig. 3a, two
distinct coexistence regions appear simultaneously at point (vi) in Fig. 3b.
These coexistence regions imply that a polymer solution at a total
concentration within a shaded region will phase separate at equilibrium into
two coexisting phases with different concentrations and compositions, as
indicated by the equilibrium S fraction shown on the coexistence-region
boundaries in Fig. 3b. Thus, while the number of coexistence regions is the
same at points (d) and (e), phase coexistence in the lower-concentration
coexistence region is qualitatively different at these two points.
Specifically, the dilute phase at point (d) is dominated by assembled
stoichiometric complexes, while the dilute phase at point (e) is primarily
composed of free polymers. The control-parameter curve then crosses line (iii)
in Fig. 3a, which indicates the termination of the high-concentration
coexistence region at point (v) in Fig. 3b, leaving only the lower-
concentration coexistence region at point (f). Finally, the control-parameter
curve crosses line (i) in Fig. 3a, corresponding to point (iv) in Fig. 3b, so
that the polymer solution exists in a single phase at all concentrations once
again at point (g).

Both the higher-concentration and the lower-concentration coexistence regions
in Fig. 3b exhibit re-entrance, since a polymer solution at constant total
concentration can pass from a single phase to two phases and back to a single
phase as the control parameter is varied monotonically from point (c) to point
(g). However, re-entrance only occurs over a finite range of total
concentrations. To determine whether re-entrance will occur at a specific
total concentration, we also plot curves of constant dilute-phase
concentration on the master phase diagram in Fig. 3a. These curves indicate
the conditions in the $\beta\epsilon$–$\beta\Delta G$ plane at which the
dilute phase has the indicated total concentration, and thus where the binodal
intersects a line of fixed total concentration in a control parameter versus
total concentration phase diagram (e.g., Fig. 3b). Because the example
control-parameter curve shown in Fig. 3a crosses the curve of a 1% dilute-
phase concentration twice but does not cross the curve of a 0.1% dilute-phase
concentration, we can easily predict that re-entrance will occur for $\rho
N_{\text{p}}v_{0}=1\%$ but not at $\rho N_{\text{p}}v_{0}=0.1\%$ in Fig. 3b.
These examples demonstrate how the essential features of the conventional
phase diagram can be inferred graphically by comparing a control-parameter
curve, which is specific to a particular system, with the master phase
diagram, which depends only on the parameters $N_{\text{p}}$, $N_{\text{s}}$,
and $n$ in this model.

####  II.3.2 Free-energy landscapes

Representative free-energy landscapes at various points in the
$\beta\epsilon$–$\beta\Delta G$ plane, corresponding to each labeled point on
the control-parameter curve as well as the additional labeled point (h), are
shown in Fig. 3c–h. Equilibrium landscapes are indicated by points whose
coloring reflects the composition, as in Fig. 3b. Coexistence regions and tie-
line constructions are indicated for each equilibrium landscape. For
comparison, we also show free-energy landscapes computed assuming a
nonequilibrium composition of either 100% S or 100% P.

We find that we can predict the qualitative phase behavior at each point along
the control-parameter curve in Fig. 3b by examining the relationship between
these nonequilibrium landscapes, despite the fact that the composition is
never exactly 100% S or 100% P due to the entropy of mixing. In Fig. 3c, the
equilibrium landscape is close to the 100% S curve, which is always below the
100% P curve. Thus, the polymer solution primarily consists of assembled
stoichiometric complexes in a single phase at all concentrations. In Fig. 3d,
the lower-concentration coexistence region is approximately given by a common-
tangent construction between the 100% S curve (on the left) and the 100% P
curve (on the right). However, the lower-concentration coexistence region in
Fig. 3e and the sole coexistence region in Fig. 3f are approximately given by
a common-tangent construction using the 100% P curve alone, since this curve
is nonconvex due to the associative contribution to the free energy, Eq. (4).
This nonconvexity in the 100% P curve disappears at high $\beta\epsilon$,
resulting in no phase separation in Fig. 3g. By contrast, the higher-
concentration coexistence region in Figs. 3d and 3e and the sole coexistence
region in Fig. 3h are approximately given by a common-tangent construction
between the 100% P curve (on the left) and the 100% S curve (on the right).

####  II.3.3 Parameter-regime boundaries on the master phase diagram

The nature of the boundaries between regimes in the master phase diagram can
be understood by examining the behavior of the system at points where the
coexistence regions terminate in Fig. 3b. Point (iv) in Fig. 3b is a critical
point, since the derivatives $\partial\mu/\partial\rho$ and
$\partial^{2}\mu/\partial\rho^{2}$ vanish. As noted at the end of Sec. II.1,
this critical point is typically referred to as an upper critical solution
temperature (UCST). Line (i) in Fig. 3a is therefore a line of critical points
that bound the lower-concentration coexistence region. Since the two phases
that merge at point (iv) are composed mostly of free polymers, line (i) is
nearly independent of $\beta\Delta G$. The value of $\beta\epsilon$ along this
line is thus approximately equal to the critical association free energy of an
associating polymer solution in the absence of self-assembly (i.e.,
$\beta\Delta G\rightarrow\infty$).

By contrast, point (v) in Fig. 3b is a first-order transition, where the
composition changes discontinuously from mostly assembled stoichiometric
complexes to mostly free polymers upon a change in $\beta\epsilon$ and/or
$\beta\Delta G$. This transition occurs when the free energy of a 100% P
solution drops below that of a 100% S solution at $\rho N_{\text{p}}v_{0}=1$,
which we can see by comparing the free-energy landscapes shown in Figs. 3e and
3f. We can therefore predict the line of first-order transitions indicated by
line (iii) in Fig. 3a by equating the free energies of 100% P and 100% S
solutions at $\rho N_{\text{p}}v_{0}=1$. To the left of line (iii), both
coexisting phases have total concentrations less than one, as evidenced by the
example coexistence region shown in Fig. 3h. To the right of line (iii), the
higher-concentration coexistence region does not exist.

####  II.3.4 Physical origin of re-entrance

We now focus on the physical origin of the re-entrant behavior that occurs at
relatively low values of the total concentration in Fig. 3b. In essence, re-
entrance manifests because a solution of free polymers phase separates at high
values of the control parameter, but a single-phase solution of predominantly
assembled stoichiometric complexes becomes stable at low values of the control
parameter. Re-entrance therefore results from mutually exclusive competition
between polymer phase separation and stoichiometric complex self-assembly.
More precisely, re-entrance occurs when the control-parameter curve crosses a
line of constant dilute-phase concentration (e.g., $\rho
N_{\text{p}}v_{0}=1\%$) twice in Fig. 3a. These constant dilute-phase
concentration curves are bounded by lines (i) and (ii), so this form of re-
entrance can only appear in a limited portion of the
$\beta\epsilon$–$\beta\Delta G$ plane. While the upper bound of the lower-
concentration coexistence region with respect to the control parameter is set
by point (iv) in Fig. 3b, the lower bound on this lower-concentration
coexistence region is determined by point (vi). This lower bound corresponds
to the intersection between line (ii) and the control-parameter curve in Fig.
3a.

Figure 4: Coexistence regions terminate at a first-order phase transition in
the conserved-excluded-volume scenario. (a) Here we examine the nature of the
transition from a one-phase region to two two-phase regions along line (ii) in
Fig. 3a, which corresponds to point (vi) in this zoomed-in portion of the
phase-diagram shown in Fig. 3b. (b) The concentration-dependent chemical
potential, $\mu$, at fixed $\beta\epsilon=-2.303$ and variable $\beta\Delta
G$. Coexisting phases are indicated by circles and connected by dashed lines.
The transition from two coexistence regions ($\beta\Delta G\gtrsim-16.015$) to
zero coexistence regions ($\beta\Delta G\lesssim-16.02$) does not involve a
critical point, which would imply a concentration where
$\partial\mu/\partial\rho=0$. Instead, the two coexistence regions move
towards one another as $\beta\Delta G$ is decreased and vanish at the
azeotrope concentration $\rho N_{\text{p}}v_{0}\simeq 0.271$ (dashed line),
where $\partial\mu/\partial\rho$ jumps discontinuously. (c) When no
coexistence regions are present ($\beta\Delta G=-16.03$), the composition is a
continuous function of $\rho$. Yet when two coexistence regions are present
($\beta\Delta G=-16$), the composition changes discontinuously across each
coexistence region. (d) The first-order nature of this transition at finite
$\beta\Delta G$ (dashed line), corresponding to line (ii) in Fig. 3, is clear
from the discontinuity in the composition at the azeotrope concentration.
Moving from left to right across this plot is analogous to moving through
point (vi) in Fig. 3b at constant $\rho$ from below to above.

Yet unlike point (iv) in Fig. 3b, point (vi) is not a critical point. Instead,
point (vi) indicates a first-order transition where the composition jumps from
a high value to a low value of the equilibrium S fraction as the control
parameter is increased. The distinction between the behavior at point (vi) and
a lower critical solution temperature (LCST) can be understood by examining
the chemical potential in the vicinity of the transition (Fig. 4a). In Fig.
4b, we show the chemical potential as a function of the total concentration at
fixed $\beta\epsilon$. As we tune $\beta\Delta G$ near the transition point,
we see that the two coexistence regions converge at a crossover concentration
(vertical dashed line in Fig. 4b). However, the derivative of the chemical
potential at the crossover concentration does not vanish, as would be required
for a second-order transition. By contrast, the equilibrium S fraction at the
crossover concentration changes discontinuously, as shown in Fig. 4c–d. Since
the composition is determined from a first derivative of the free energy, this
behavior is consistent with a first-order transition.

We propose that this behavior can be understood intuitively by drawing an
analogy to a positive azeotrope [29]. In a binary mixture of two fluids, a
positive azeotrope occurs at a specific composition at which the two fluids
share the same boiling point; thus, raising the temperature of the mixture at
the azeotrope composition causes the liquid mixture to transform directly into
a mixture of gases with the same composition. Qualitatively, the free-energy
landscape of a fluid mixture with an azeotrope is analogous to that of the
nonequilibrium 100% S and 100% P curves shown in Fig. 3c–d, whereby the
“liquid-like” 100% S curve is stable below a transition point $(\beta\Delta
G,\beta\epsilon)$ and the “vapor-like” 100% P curve is stable above. We
therefore propose that the crossover concentration shown in Fig. 4b is
analogous to an azeotrope composition, even though there is technically only
one equilibrium free-energy landscape as a function of the total concentration
in our model due to the chemical equilibrium constraint, Eq. (7). Finally, we
note that line (i) terminates at line (ii) in Fig. 3a for a similar reason:
For sufficiently negative $\beta\Delta G$, the 100% S free-energy landscape
curve becomes more stable than the 100% P curve at all concentrations.
Consequently, the nonconvex portion of the free-energy landscape, which
describes a polymer solution composed of mostly free polymers, becomes
metastable to the left of line (ii) in Fig. 3a.

####  II.3.5 Dependence of the master phase diagram on $N_{\text{p}}$ and $n$

Up to this point, we have discussed the master phase diagram assuming the
specific parameter choices $N_{p}=6$ and $n=4$. These calculations can easily
be repeated using alternative parameters. In this way, we find that the
topology of the master phase diagram remains unchanged as long as $n>1$ and
the excluded volume is conserved, such that $N_{\text{s}}=nN_{\text{p}}$.
However, changing the parameters $N_{\text{p}}$ and $n$ alter the positions of
the phase boundaries in the $\beta\epsilon$–$\beta\Delta G$ plane, as we
discuss next.

The line of critical points (i) is nearly independent of the degree of
complexation $n$ because these critical points only depend on the properties
of free polymers, except when in close proximity to the intersection of lines
(i) and (ii). Line (i) shifts upward in the $\beta\epsilon$–$\beta\Delta G$
plane as the degree of polymerization $N_{\text{p}}$ increases, since the
critical per-binding-site association free energy becomes weaker as the number
of binding sites per polymer increases. This behavior is analogous to that of
the Flory–Huggins homopolymer model, in which the critical interaction
strength weakens as the degree of polymerization increases [22].

By contrast, the two first-order phase boundaries, lines (ii) and (iii),
depend on both $n$ and $N_{\text{p}}$. The slope of line (iii) is exactly
proportional to $nN_{\text{p}}$; this can be shown by equating the free
energies of the 100% P and 100% S solutions at $\rho N_{\text{p}}v_{0}=1$. The
slope of line (ii), as well as the difference in $\beta\Delta G$ between lines
(ii) and (iii) at constant $\beta\epsilon$, also scales roughly linearly with
$nN_{\text{p}}$. This dependence can be understood by considering the
transition between regions (1) and (4) in Fig. 3a in the strong association
limit, $\beta\epsilon\rightarrow-\infty$ [20]. In this limit, the relative
stabilities of the 100% P and 100% S solutions are primarily determined by the
balance between the association free energy in the P state, in which nearly
all binding sites are associated since $X_{\text{p}}\rightarrow 0$, and the
internal contribution to the free energy in the S state, which is nearly
proportional to $\beta\Delta G$. Consequently, $\beta\Delta G$ must scale with
the total number of associative interactions in which the complex-forming
polymers could potentially engage, which is equal to $nN_{\text{p}}$. Thus,
$nN_{\text{p}}$ sets the scale for both lines (ii) and (iii). Increasing
$nN_{\text{p}}$ tends to stretch the master phase diagram in the horizontal
direction.

###  II.4 Self-assembly with nonconserved excluded volume

Next, we consider the alternative case in which self-assembly does not
conserve the excluded volume. This scenario can be motivated by considering
the second-virial coefficient between two assembled stoichiometric complexes
in solution, which is given by $B_{\text{ss}}=N_{\text{s}}v_{0}/2$ in our
model (see Appendix B). This second-virial coefficient represents the total
volume (divided by two) that the center of mass of one stoichiometric complex
cannot access due to the presence of the other complex. If the polymers self-
assemble into a structure that has a lower density than a polymer melt, then
it is possible to have $v_{\text{s}}=2B_{\text{ss}}>nv_{\text{p}}$. For
example, this scenario is relevant in the case of nucleic-acid complexes,
because secondary-structure formation increases the persistence length of
duplexes relative to free strands [30, 31]; nucleic-acid complexes can
therefore have a relatively low density of polymer within a relatively large
pervaded volume. We therefore consider an example scenario in which
$N_{\text{s}}=200>nN_{\text{p}}=24$ in this section.

Figure 5: Phase-behavior predictions when self-assembly does not conserve the
excluded volume of the constituent chains. (a) A single one-phase region
(unshaded), where the system is homogeneous at all concentrations, lies to the
right of dashed curve (i) and above dashed curve (ii), as described in the
text. A two-phase region exists within some concentration range everywhere
else within the master phase diagram. See Fig. 3a for explanations of the
control parameter path and total concentration curves. (b) When the excluded
volume is not conserved by the two-state model of self-assembly, the lone
coexistence region terminates at a single point (iii), as described in the
text. This phase diagram is constructed analogously to that shown in Fig. 3b.
(c)–(h): Representative free-energy profiles along the path taken by the
control parameter shown in panel a, and at the additional labeled point (h) in
panel a. See the description of Fig. 3c–h for details. Because of the excluded
volume difference, the 100% S curves terminate at the total concentration
where the complexes are close-packed, $\rho N_{\text{s}}v_{0}/n=1$, where
$n=4$ in this example. Consequently, condensed phases with $\rho
N_{\text{p}}v_{0}>nN_{\text{p}}/N_{\text{s}}$ are always polymer-dominated.

####  II.4.1 Similarities and differences with the conserved-excluded-volume
scenario

The master phase diagram for this scenario is shown in Fig. 5a. The control-
parameter curve, which is the same as that in Fig. 3a, maps out the
conventional concentration-dependent phase diagram shown in Fig. 5b. In
contrast with the conserved-excluded-volume scenario, this master phase
diagram only exhibits two parameter regimes: one regime, (1), in which the
solution exists in a single phase at all concentrations; and one regime, (2),
with a single coexistence region. Yet despite these drastic changes in the
topology of the master phase diagram relative to the conserved-excluded-volume
scenario, the coexistence region in Fig. 5b strongly resembles the lower-
concentration coexistence regime in Fig. 3b for values of the control
parameter above point (c).

These differences can be understood by noting that the increased excluded
volume of the stoichiometric complex means that chains must exist in the free-
polymer state at high total concentrations. More precisely, since the volume
fraction $\phi_{\text{s}}$ cannot exceed one, the maximum total concentration
at which stoichiometric complexes can exist is given by $\rho
N_{\text{p}}v_{0}=nN_{\text{p}}/N_{\text{s}}$. This upper limit on the
concentration of stoichiometric complexes is reflected in the free-energy
landscapes shown in Fig. 5c–h, where the 100% S landscapes do not extend over
the full range of total concentrations. Consequently, there is no analog of a
higher-concentration coexistence region, as observed in the conserved-
excluded-volume case, when we consider the nonconserved scenario. However, the
phase behavior at low total concentrations, $\rho N_{\text{p}}v_{0}\ll
nN_{\text{p}}/N_{\text{s}}$, is qualitatively the same in both excluded-volume
scenarios. We also note that the boundary between the two regimes, line (i) in
Fig. 5a, is a single, continuous line of critical points. This critical line
is the same as line (i) in the conserved-excluded-volume case at large
$\beta\Delta G$, but turns into a continuous transition between assembled
stoichiometric complexes and free polymers when it bends upward near point (h)
in Fig. 5a.

####  II.4.2 Physical origin of re-entrance

Re-entrance occurs in the nonconserved-excluded-volume scenario due to the
same competition between self-assembly and polymer phase separation discussed
in the context of the conserved-excluded-volume scenario. Minor quantitative
differences, such as the maximum total concentration at which re-entrance
occurs, appear between the two scenarios due to the differences in the
excluded-volume and internal contributions to the free energy, $f^{\text{ev}}$
and $f^{\text{int}}$, in Eqs. (II.1) and 6. However, the absence of a one-
phase parameter regime at low $\beta\Delta G$ in the nonconserved case means
that the coexistence region does not terminate at low values of the control
parameter in Fig. 5b. Instead, the maximum concentration at which re-entrance
can occur tends to $\phi_{\text{s}}=1$ in the limit $\beta\Delta
G\rightarrow-\infty$. In this limit, phase coexistence occurs between a lower-
concentration phase composed entirely of assembled stoichiometric complexes
and a higher-concentration phase of free polymers, regardless of the value of
$\beta\epsilon$.

##  III Discussion

We have demonstrated how competition between self-assembly and liquid–liquid
phase separation can emerge from the same set of molecular interactions among
biopolymers in solution. Our theoretical approach captures the many-body
interactions that are inherent to this competition by coarse-graining the
conformational states of polymers into either free chains and assembled
stoichiometric complexes. The resulting mean-field model predicts that re-
entrance can emerge from the competition between self-assembly and phase
separation, both of which involve only attractive associative interactions.
Our results provide insight into the key features that control the phase
behavior of these systems, providing a unified view of different phase
diagrams that can result given a parametric dependence on temperature, ionic
strength, pH, or other experimentally controllable variables.

Importantly, we find that re-entrance in this model is not a consequence of a
lower critical solution temperature (LCST), despite qualitative similarities
between the phase diagrams predicted by our model and those involving
UCST+LCST miscibility loops [22, 32]. In the conserved-excluded-volume
scenario, the lower termination of the coexistence regions occurs at a first-
order transition, which is qualitatively different from an LCST. By contrast,
when self-assembly does not conserve excluded volume, the coexistence region
does not terminate at all in the limit of stable stoichiometric complexes.
Nonetheless, the physical origin of re-entrance is the same in both cases, and
the phase behavior at low polymer concentrations is essentially unchanged
between these two scenarios. We therefore believe that the qualitative
predictions of our model—particularly the absence of an LCST—are robust
despite the approximations invoked in our mean-field treatment. We note that
analogous behavior has also been observed in conceptually similar mean-field
models of monomeric as opposed to polymeric solutions, in which phase
separation either competes with self-assembly [17] or a unimolecular chemical
reaction [33]. In that case of unimolecular chemical reactions [33], however,
the low-temperature terminus of the coexistence region [i.e., point (vi) in
Fig. 3b] always occurs at $\phi=1$.

A key approximation in this work is the use of a two-state model of self-
assembly. In principle, the accuracy of our model could be improved by
accounting for additional intermediate states between completely dissociated
polymers and completely assembled stoichiometric complexes. Our model also
assumes that the excluded volume of the stoichiometric complexes,
$v_{\text{s}}$, is a constant. A more versatile model could account for
variations in $v_{\text{s}}$ as a function of concentration, which would
provide a better description of complexes that deform at high concentrations
in the nonconserved-excluded-volume scenario. However, we do not believe that
these modifications would affect our qualitative conclusions regarding the
nature of the various phase transitions. In particular, the first-order
transitions that we observe emerge, fundamentally, from the cooperative nature
of the self-assembly process, which implies the existence of a free-energy
barrier between the free-polymer (P) and assembled stoichiometric-complex (S)
states. We therefore believe that the phase behavior will remain qualitatively
unchanged as long as this essential feature is preserved. For the same reason,
we believe that this model can apply to a wide variety of self-assembling
polymer solutions in which self-assembly is cooperative and mutually exclusive
with phase separation. Our approach would be less appropriate for modeling the
phase behavior of polymer solutions in which less cooperative intramolecular
bonding competes with associative polymer phase separation, as is the case in
many experiments on nucleic-acid repeat sequences [34].

In conclusion, we have introduced and analyzed a broadly applicable mean-field
model of biopolymer assembly and phase separation. Our “master-phase-diagram”
approach makes it possible to derive system-specific phase diagrams in a
unified and intuitive fashion. It is also straightforward to extend our
theoretical framework to model more complex scenarios involving “hierarchical”
assembly and phase separation, such as polymer solutions in which self-
assembled complexes can themselves undergo phase separation via associative
interactions [20]. Most importantly, the simple manner in which our model can
be parameterized makes this framework useful for generating experimentally
testable predictions for complex biopolymer solutions.

###### Acknowledgements.

This work was supported by a grant from the National Science Foundation
(DMR-2143670) to WMJ, as well as support from the Human Frontier Science
Program (RGP0029) and the Smith Family Foundation to WBR. Source code is
provided at https://github.com/wmjac/
polymer-phase-sep-self-assembly.

##  Appendix A Excluded volume formulae

To derive an expression for the excluded-volume contribution to the mixing
free energy, we consider a cubic lattice with $N$ lattice sites. Each lattice
site has the same volume, $v_{0}$, as a monomer (i.e., a blob) of the polymer,
P. Each polymer chain occupies $N_{\text{p}}$ lattice sites. Each
stoichiometric complex, referred to as an “S molecule” in this section,
occupies an excluded volume of $v_{\text{s}}=N_{\text{s}}v_{0}$. We compute
the free energy of mixing, in the absence of associative interactions or
chemical reactions, in a two-step manner (Fig. A1). An empty lattice, with a
free energy equal to zero, is chosen as the reference state. At each step, the
free-energy change is purely entropic due to the increase in the number of
microstates, $\Omega_{i-1}\rightarrow\Omega_{i}$,

$\beta N\Delta f_{i}v_{0}=-\frac{\Delta
S_{i}}{k_{B}}=-\ln\left(\frac{\Omega_{i}}{\Omega_{i-1}}\right),$ (9)

where $\Delta S_{i}$ is the entropy change associated with step $i$ from state
$i-1$ to state $i$.

First, we consider the free-energy change due to the addition of S molecules
occupying a volume fraction $\phi_{\text{s}}$. Importantly, each S molecule is
considered to be a compact, rigid unit, and must therefore occupy
$N_{\text{s}}$ contiguous lattice sites. We therefore compute the approximate
entropy change associated with step 1 (Fig. A1) by coarse-graining the lattice
into “super sites” of volume $v_{0}N_{\text{s}}$, and then applying the
standard regular-solution formula [27],

$\beta\Delta
f_{1}v_{0}=\frac{\phi_{\text{s}}}{N_{\text{s}}}\ln\phi_{\text{s}}+\frac{1-\phi_{\text{s}}}{N_{\text{s}}}\ln(1-\phi_{\text{s}}).$
(10)

Second, we add P molecules occupying a volume fraction $\phi_{\text{p}}$ into
the system. Following the Flory–Huggins derivation [22], we ignore chain
connectivity when assigning polymer blobs to non-contiguous unoccupied lattice
sites. This leads to the mean-field result

$\beta\Delta
f_{2}v_{0}=\frac{\phi_{\text{p}}}{N_{\text{p}}}\ln\left(\frac{\phi_{\text{p}}}{1-\phi_{\text{s}}}\right)+\phi_{0}\ln\left(\frac{\phi_{0}}{1-\phi_{\text{s}}}\right),$
(11)

where ${\phi_{0}\equiv 1-\phi_{\text{s}}-\phi_{\text{p}}}$ represents the
volume fraction not occupied by either free polymers or stoichiometric
complexes. Summing the free-energy changes of the two steps, we find that the
total free-energy difference is

$\displaystyle\beta$
$\displaystyle\left(f^{\text{id}}+f^{\text{ev}}\right)v_{0}=\frac{\phi_{\text{s}}}{N_{\text{s}}}\ln\phi_{\text{s}}+\frac{1-\phi_{\text{s}}}{N_{\text{s}}}\ln(1-\phi_{\text{s}})$
$\displaystyle\qquad\qquad\;+\frac{\phi_{\text{p}}}{N_{\text{p}}}\ln\left(\frac{\phi_{\text{p}}}{1-\phi_{\text{s}}}\right)+\phi_{0}\ln\left(\frac{\phi_{0}}{1-\phi_{\text{s}}}\right),$
(12)

which accounts for both the ideal-gas and the excluded-volume contributions to
the free energy. Taking partial derivatives of Eq. (12) with respect to
$\rho_{\text{p}}$ and $\rho_{\text{s}}$, and subtracting the ideal chemical
potentials $\beta^{-1}\ln\phi_{\text{p}}$ and $\beta^{-1}\ln\phi_{\text{s}}$,
gives the excluded-volume contributions to the excess chemical potentials,

$\displaystyle\beta\mu^{\text{ev}}_{\text{s}}$
$\displaystyle=-\ln(1-\phi_{\text{s}})+N_{\text{s}}\ln\left(\frac{1-\phi_{\text{s}}}{\phi_{0}}\right)$
(13) $\displaystyle\quad-
N_{\text{s}}\left(1-\frac{1}{N_{\text{p}}}\right)\frac{\phi_{\text{p}}}{1-\phi_{\text{s}}},$
$\displaystyle\beta\mu^{\text{ev}}_{\text{p}}$
$\displaystyle=1-N_{\text{p}}+(N_{\text{p}}\\!-\\!1)\ln(1\\!-\\!\phi_{\text{s}})-N_{\text{p}}\ln\phi_{0}.$
(14)

Both $\beta\mu^{\text{ev}}_{\text{s}}$ and $\beta\mu^{\text{ev}}_{\text{p}}$
diverge as $\phi_{0}\rightarrow 0$. Since $\phi_{\text{s}}+\phi_{\text{p}}\leq
1$, this means that both $\beta\mu^{\text{ev}}_{\text{s}}$ and
$\beta\mu^{\text{ev}}_{\text{p}}$ diverge if $\phi_{\text{s}}\rightarrow 1$ or
$\phi_{\text{p}}\rightarrow 1$.

Figure A1: Two-step calculation of the excluded-volume contribution to the
free energy. Starting from the reference state of an empty lattice (state 0,
left), stoichiometric complexes (S) are added in step one. Stoichiometric
complexes occupy $N_{\text{s}}$ contiguous lattice sites. The free-energy
change associated with this process is $\Delta f_{1}$, resulting in the
intermediate configuration (state 1, middle). In step two, polymers (P) are
added to reach the final configuration (state 2, right). Polymers occupy
$N_{\text{p}}$ non-contiguous lattice sites. The free-energy change associated
with this second process is $\Delta f_{2}$.

##  Appendix B Comparison with the Flory–Huggins polymer model

The excluded-volume contributions embodied in Eqs. (12), (13), and (14) differ
from those implied by the standard Flory–Huggins free-energy density [22],

$\beta
f^{\text{FH}}v_{0}=\frac{\phi_{\text{s}}}{N_{\text{s}}}\ln\phi_{\text{s}}+\frac{\phi_{\text{p}}}{N_{\text{p}}}\ln\phi_{\text{p}}+\phi_{0}\ln\phi_{0},$
(15)

which treats both the S and P species as mean-field polymers. The excluded-
volume contribution to the chemical potential of species
$i\in\\{\text{s},\text{p}\\}$ is

$\beta\mu^{\text{FH,ev}}_{i}=1-N_{i}-N_{i}\ln\phi_{0}$ (16)

in the Flory–Huggins model. Focusing on the case of the S molecule, Eq. (16)
highlights the problem with applying the Flory–Huggins model to our system:
The chemical potential of the S molecule depends on $N_{\text{s}}$ regardless
of the polymer concentration. As a result, the Flory–Huggins equation of state
of a pure solution of S molecules depends on the ratio of the excluded volume
of a rigid S molecule to the excluded volume of a polymer blob, even when no
polymer is present in the system. This is an undesirable feature, which
necessitates the use of the formulae derived in Appendix A.

It is also instructive to compare the second-virial coefficients,
$B_{\text{ss}}$, $B_{\text{sp}}$, and $B_{\text{pp}}$, of the two models. To
this end, we expand the pressure,
$P=-f+\mu_{\text{s}}\rho_{\text{s}}+\mu_{\text{p}}\rho_{\text{p}}$, to second
order in the concentrations $\rho_{\text{s}}$ and $\rho_{\text{p}}$, such that
$P=\sum_{i\in\\{\text{s},\text{p}\\}}\rho_{i}+\sum_{i,j\in\\{\text{s},\text{p}\\}}B_{ij}\rho_{i}\rho_{j}+\mathcal{O}(\rho^{3})$.
For the Flory–Huggins model, this yields the coefficients

$B^{\text{FH}}_{\text{ss}}=\frac{N^{2}_{\text{s}}v_{0}}{2},\hskip
7.5ptB^{\text{FH}}_{\text{sp}}=\frac{N_{\text{s}}N_{\text{p}}v_{0}}{2},\hskip
7.5ptB^{\text{FH}}_{\text{pp}}=\frac{N^{2}_{\text{p}}v_{0}}{2},$ (17)

whereas the coefficients derived from our excluded-volume model, Eqs. (14) and
(13), are

$B^{\text{ev}}_{\text{ss}}=\frac{N_{\text{s}}v_{0}}{2},\hskip
7.5ptB^{\text{ev}}_{\text{sp}}=\frac{N_{\text{s}}v_{0}}{2},\hskip
7.5ptB^{\text{ev}}_{\text{pp}}=\frac{N^{2}_{\text{p}}v_{0}}{2}.$ (18)

The $B_{\text{pp}}$ coefficients are the same in both expressions as expected,
since the polymers are treated equivalently in the two models. However, the
second-virial coefficients involving the S molecules differ.
$B^{\text{ev}}_{\text{ss}}$ is a more physically reasonable result than
$B^{\text{FH}}_{\text{ss}}$, since the S molecules are assumed to be compact,
rigid units. In fact, $B^{\text{ev}}_{\text{ss}}$ is the correct result for
monodisperse hard particles (such as hard spheres [35]) that exclude a volume
of $N_{\text{s}}v_{0}$. The cross term, $B^{\text{ev}}_{\text{sp}}$, is less
obvious, but we can build intuition by considering the insertion of polymers
into a low-density gas of S molecules. For simplicity, let us assume that the
S molecules are hard spheres with diameter $D\simeq
N_{\text{s}}^{1/3}v_{0}^{1/3}$. The average end-to-end distance of the freely
jointed polymer chains is $d\simeq v_{0}^{1/3}N_{\text{p}}^{1/2}$. A simple
approximation of the cross term is therefore
${B^{\text{ev}}_{\text{sp}}\approx(D+d)^{3}/2=N_{\text{s}}(1+r)^{3}v_{0}/2}$,
where $r=N_{\text{p}}^{1/2}/N_{\text{s}}^{1/3}$. Our excluded-volume model of
S/P mixtures is therefore most accurate when the S molecules are large
compared to the individual polymer chains (such as in the case considered in
Sec. II.4), while the standard Flory–Huggins model is more appropriate when
the size of an S molecule is comparable to that of an individual polymer blob.
We therefore conclude that the excluded-volume formulae Eqs. (14) and (13) are
the more appropriate choice for the model and parameter regimes considered in
this study.

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Similar to above Theorem 7.4, let $Q_{A}:=\\{$non-trivial quasitriangular
structures on $A(8n,\sigma,\tau)$ $\\}$, then we have

######  Theorem 7.6.

$Q_{A}\stackrel{{\scriptstyle
1-1}}{{\longleftrightarrow}}\\{(\beta,\delta)|\;\beta^{2n}=P_{s^{2n}},\;\delta^{2}=\frac{\tau(a,a)\tau(b,a)}{\tau(b,b)\sigma(a)}P_{s^{n}}\beta^{1+n}\\}$,
where $s=a^{2}$.

###### Proof.

Let $R$ be a non-trivial quasitriangular structure on $K(8n,\sigma,\tau)$ and
let $w^{1}(s,s)=\beta,w^{4}(a,a)=\delta$, then it can be seen that
$(\beta,\delta)$ such that the following conditions

(7.4)
$\displaystyle\beta^{2n}=P_{s^{2n}},\;\delta^{2}=\frac{\tau(a,a)\tau(b,a)}{\tau(b,b)\sigma(a)}P_{s^{n}}\beta^{1+n}.$

due to (ii),(iv) of Proposition 5.14. Conversely, if $(\beta,\delta)$
satisfies the equation (7.4), then we can use (i)-(iv) of Proposition 5.15 to
define a $R$ as follows

(7.5) $\displaystyle
w^{1}(s,s):=\beta^{2}\sigma(s),\;w^{2}(s,a)=w^{3}(a,s):=\beta,\;w^{4}(a,a):=\delta.$

Then we can see that the four tuple $(\beta^{2}\sigma(s),\beta,\beta,\delta)$
satisfies conditions (i)-(vi) of Proposition 5.14 and thus $R$ is a non-
trivial quasitriangular structure. Moreover, since Lemma 4.2, we know $R$ is
unique if it is a $\varphi$-symmetric quasitriangular structure and satisfies
$w^{2}(s,a)=w^{3}(s,a)=\beta,\;w^{4}(a,a)=\delta$. Finally, since Proposition
7.3, we know that this correspondence we have discussed is one-to-one. ∎

######  Remark 7.7.

_From the proof of the above theorem, we know that all non-trivial
quasitriangular structures on $A(8n,\sigma,\tau)$ are given by (i)-(iv) of
Proposition 5.15, where $(\beta,\delta)$ are in Theorem 7.6 and $w^{1}$ is
defined by (7.5) above and $\alpha_{11}=w^{1}(s,s),\;\gamma=\beta$._

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