toltxlabel *[si:]charged-si Physikalisches Institut, Albert-Ludwigs
Universität Freiburg, Hermann-Herder-Straße 3, D-79104 Freiburg im Breisgau,
Germany Research Group Simulations of Energy Materials, Helmholtz-Zentrum
Berlin, Hahn-Meitner-Platz 1, D-14109 Berlin, Germany Research Group
Simulations of Energy Materials, Helmholtz-Zentrum Berlin, Hahn-Meitner-Platz
1, D-14109 Berlin, Germany

# Tuning the collapse transition of weakly charged polymers by ion-specific
screening and adsorption

Richard Chudoba  Institut für Physik, Humboldt-Universität zu Berlin,
Newtonstraße 15, D-12489 Berlin, Germany<EMAIL_ADDRESS>Jan Heyda  Department of Physical Chemistry, University of Chemistry and
Technology, Prague, Technická 5, CZ-16628 Praha, Czechia<EMAIL_ADDRESS>Joachim Dzubiella  Physikalisches Institut, Albert-Ludwigs Universität
Freiburg, Hermann-Herder-Straße 3, D-79104 Freiburg im Breisgau, Germany
<EMAIL_ADDRESS>
###### ABSTRACT

The experimentally observed swelling and collapse response of weakly charged
polymers to the addition of specific salts displays quite convoluted behavior
that is not easy to categorize. Here we use a minimalistic implicit solvent /
explicit salt simulation model with a focus on ion-specific interactions
between ions and a single weakly charged polyelectrolyte to qualitatively
explain the observed effects. In particular, we demonstrate ion-specific
screening and bridging effects cause collapse at low salt concentrations
whereas the same strong ion-specific direct interactions drive re-entrant
swelling at high concentrations. Consistently with experiments, a distinct
salt concentration at which the salting-out power of anions inverts from the
reverse to direct Hofmeister series is observed. At this, so called
‘isospheric point’, the ion-specific effects vanish. Furthermore, with
additional simplifying assumptions, an ion-specific mean-field model is
developed for the collapse transition which quantitatively agrees with the
simulations. Our work demonstrates the sensitivity of the structural behavior
of charged polymers to the addition of specific salt and shall be useful for
further guidance of experiments.

##  1 Introduction

Biopolymers as well as functional synthetic polymers in solution are subject
to a complex set of interactions that are decisive for their molecular
structure and function, e.g., as for folded versus unfolded states for
proteins, or, more general, collapsed versus swollen states of a polymer.1
Typically those solvated polymers are weakly charged and constitute relatively
heterogeneous, partially polar, partially hydrophobic, macromolecules. Hence
their polymeric structure is greatly affected by the solvation environment.
Apart from the solvent type (e.g., water), the polymer conformation can be
conveniently controlled by the concentration and type of added cosolutes. In
particular, ion-specific, also called Hofmeister effects2, 3 of salts on
biopolymeric structure and phase behavior have been studied and highlighted in
the recent years in an overwhelming amount of literature.4, 5, 6, 7, 8

The Hofmeister effects on the polymeric structure and the phase behavior are
very complex and polymer specific. In order to increase the accessibility to
the problem, often model systems are employed. For instance, the poly-N-
isopropylacrylamide (PNIPAM) homopolymer or elastin-like polypeptides (ELPs),
which both exhibit hydrophobic collapse transitions at their respective lower
critical solution temperatures (LCST).9, 10, 11 These collapse transitions are
analogous to cold-denaturation of proteins, in which ion-specific interactions
(Hofmeister effects) play an prominent role.

PNIPAM is neutral polymer with LCST at
$32\text{\,}\mathrm{\SIUnitSymbolCelsius}$. Charge groups can be introduced
into the polymer structure by copolymerization, typically with weak or strong
acids or bases.12, 13, 14 The electrostatic repulsion between charged monomers
then leads to an increase of LCST in pure water and a strong response to added
salt at millimolar salt concentration when charge screening sets in.

ELPs constitute a particularly nice model system as they can be expressed by
bacteria in a strictly monodisperse fashion with different repeating
sequences.15, 16, 17, 18 It is possible, for instance, to synthesize
pentapeptide sequences of the generic form (VPGXG)_n, where X stays for any
amino acid except proline.19, 20 One can thus easily modulate the hydrophilic,
hydrophobic, and charged character of the ELPs20 together with their LCST.18,
21 Salt-specific effects were studied not only for few neutral ELPs (e.g.,
V-120),17, 22, 23 but also for the weakly positively charged KV6-120,24, 8 and
the weakly negatively charged DV2F-64.9

For completely neutral polymers, e.g., pure PNIPAM or neutral ELPs, only two
regimes of salt action were reported in the literature,17, 25 see Figure 1
(a). (I) Linear and rather steep decrease of the cloud point temperature (an
approximate but direct measure of the LCST) with salt concentration was
observed for most strongly hydrated salts, such as sulfates, acetates,
fluorides, and chlorides. (II) Initial weak increase of the cloud point
temperature at low salt concentrations, typically below
$1\text{\,}\mathrm{\textsc{m}}$, with subsequent decrease at higher
concentrations. Only recently a third regime, which was predicted by computer
simulations26 and is characterized by strongly attractive bridging
interaction, was found. Guanidinium thiocyanate, one of the most potent
protein denaturants, steeply decreases the cloud point temperature below
$1.5\text{\,}\mathrm{\textsc{m}}$, and the temperature grows even more rapidly
above $1.5\text{\,}\mathrm{\textsc{m}}$ salt.23 The relation of this to the
phenomenon of cononsolvency in solvent mixtures is under discussion.27 Charged
ELPs possess a more complex behavior due to the ion-specific screening at low
salt concentrations,24, 8, 9 cf. Figure 1 (b), similar to the stability
behavior of charged proteins.4

It is now of fundamental importance to characterize the regimes in which
specific salt-effects operate. This would allow to build a predictive model,
which may be applied for determination of protein stability, but also for
designing biomaterials of desired properties in their native environments.
Computer simulations have already illuminated the action of varying solvents
and cosolutes in coarse-grained simulations28, 29, 30, 31, 32, 33, 26, 34 or
all-atom simulations35, 21, 36, 37. Typically the results are interpreted by
theories for chain structure and swelling,38 e.g., due to a counterion
condensation at highly charged polymers39, 40, 41, 42, 43 or specific steric
interactions.44, 45, 46, 47, 48, 49 Yet, a quantitative description of ion-
specific counterion condensation relies on all-atom simulations.50, 51 Driven
by increasing interest in cononsolvency and the action of mixed solutes on the
polymer structure, a whole body of mean-field theories has been developed
especially for the description of the coil-to-globule transition in mixed
solvents.26, 52, 53, 54, 55, 56, 57, 58, 27

Recently, we have developed a thermodynamic approach for neutral polymers
(such as PNIPAM, or neutral ELPs) within the framework of preferential
binding,59, 8 which helps interpreting the salt-specific thermodynamic
fingerprints of cosolute-polymer interactions.60 A similar model was used and
extended to include non-specific screening effects for weakly charged
polymers.61 In this work we aim to fully understand the complex data for
charged elastin24, 8, 9 (selected data shown in Figure 1 (b)). For this
purpose we devise a generic simulation model of a charged polymer with
different charge fractions in uni-univalent salt solutions. The ion-
specificity of salts steps in by systematically varying the polymer-ion
interactions, i.e., from more repulsive, mimicking strongly hydrated ions
(extreme ‘kosmotropes’), to very attractive, mimicking weakly hydrated ions
(extreme ‘chaotropes’).3 Our previous simulations of neutral polymers
exhibiting simple upper CST (UCST) have already demonstrated a non-trivial
response of the polymer, where collapse by depletion, swelling through weak
attraction, and re-entrant collapse by strong attractions have been
observed.26 As indicated by experiments, the situation becomes more complex in
charged systems, where screening and synergistic cation and anion interactions
with the polymer are expected. In this work, we therefore complement the
simulations with a new theory that extends our Flory-type mean-field model for
neutral polymers with approaches for ion-specific screening.61, 62

Figure 1: The change of the LCST of ELPs plotted versus the concentration of
various sodium salts (with anions as given in the legend at the bottom) added
to the polypeptide solution, as investigated experimentally.17, 24 (a)
Uncharged ELP V5A2G3-120. (b) Weakly positively charged ELP KV6-112.

##  2 Model and methods

In our model, a single polymer chain containing $N_{\mathrm{mer}}=200$ coarse-
grained monomer units is used. Monomers bearing a charge $+e$ or $-e$ are
evenly (equidistantly) distributed along the chain to obtain the desired
charge fractions $\xi$ of $0510\text{\,}\mathrm{\text{\%}}$, for cationic or
anionic chains, respectively. Furthermore, coarse-grained ions (counter-ions
of the polyelectrolyte and uni-univalent electrolyte) are explicitly resolved
in the implicit solvent environment. All beads, i.e., monomers and ions,
interact via the Lennard-Jones (LJ) pair potential, that is

$\displaystyle
U_{\mathrm{LJ}}^{\,ij}=4\epsilon_{ij}\left[\left(\frac{\sigma_{ij}}{r}\right)^{12}-\left(\frac{\sigma_{ij}}{r}\right)^{6}\right]\text{.}$
(1)

The charged particles furthermore interact through the standard Coulomb
potential. The relative permittivity of the background corresponds to water at
room temperature with a dielectric constant $\epsilon_{\mathrm{r}}=78$. This
results in the Bjerrum length
$l_{\mathrm{B}}=e^{2}/\epsilon_{\mathrm{r}}k_{\mathrm{B}}T=$0.7\text{\,}\mathrm{nm}$$
at $T=$298\text{\,}\mathrm{K}$$, where $e$ is the elementary charge and
$k_{\mathrm{B}}$ is the Boltzmann constant.

The bonds between adjacent monomers in the polymer chain are modeled with a
harmonic potential with a spring constant
$k=$100\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}\text{\,}{\mathrm{nm}}^{-2}$$
and a bond length $b=$0.38\text{\,}\mathrm{nm}$$.26 Neither angular, nor
torsional potential is applied. Non-bonded interactions between adjacent
monomers are excluded.

The non-interacting polymer chain having $N_{\mathrm{mer}}=200$ segments each
$b=$0.38\text{\,}\mathrm{nm}$$ long is considered as an ideal reference of our
simulation setup. The ideal radius of gyration equals to
$R_{\mathrm{g}}^{\mathrm{id}}=b\sqrt{N_{\mathrm{mer}}/6}=$2.19\text{\,}\mathrm{nm}$$.

The solvation characteristics of ions cannot be captured directly in the
implicit solvent model. Instead the value of the second virial coefficient
$B_{2}$ for the monomer–ion interaction (without electrostatics) can be used
as a global measure of the hydrated character and ion-specific affinity to the
polymer. Strongly hydrated ions prefer to stay in the bulk solution thus there
is a net repulsion between them and the uncharged polymer. That corresponds to
a positive $B_{2}$ value of the monomer-ion interaction. On the other hand,
weakly hydrated ions tend to preferentially adsorb to the polymer surface,
which gives a net negative $B_{2}$ value of the monomer-ion interaction. The
value of $B_{2}$ can be computed from the pairwise potential (eq 5). When the
LJ potential is employed,
$\epsilon\lesssim$0.3\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$ gives a
positive $B_{2}$ value (a net repulsion), while
$\epsilon\gtrsim$0.3\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$ yields to a
negative $B_{2}$ value (a net attraction).

The implicit-solvent simulations are symmetric regarding the positive and
negative charge. However, as known from experiments, anions exhibit greater
variety of interactions with ELPs than common cations.8 To investigate a
broader range of interaction types and yet to stay consistent with the
experiments, we varied the anion–polymer interaction strength while keeping
the strength of cation–polymer interaction unchanged. The affinity of the ions
to the polymer is controlled by $\epsilon_{\text{cation}}$ and
$\epsilon_{\text{anion}}$ in the LJ potential acting between a monomer
(regardless of its charge) and a cation, or an anion, respectively. The value
of $\epsilon_{\text{cation}}=$0.1\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$
is kept constant, corresponding to strongly hydrated cations. The value of
$\epsilon_{\text{anion}}$ varies between
$0.1\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}0.9\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$,
altering the value of the second virial coefficient of the interaction between
anions and polymer. The actual $B_{2}$ values of the interactions employed in
the simulations are summarized in LABEL:si:tab:virial.

The non-electrostatic interaction between two ions (either anions, cations or
cation-anion pairs) is determined by
$\epsilon_{\mathrm{ij}}=$0.3\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$ in
all simulations, corresponding to the zero value of the second virial
coefficient $B_{2}$ for a purely non-electrostatic interaction.

The non-bonded pair interaction between monomers reflects the quality of the
implicit solvent towards the polymer, i.e., varying solvent and polymer
temperature. Our simple polymer model without solvent and explicit
$T$-dependent interactions exhibits a UCST, while all qualitative trends
including LCST are reflected, however, in inverse manner. LJ potentials with
$\epsilon_{\mathrm{mer}}$ between
$0.1\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}0.9\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$
are used when scanning for the near-critical (collapse transition) solvent
condition.

The value of $\sigma=$0.3385\text{\,}\mathrm{nm}$$ in the LJ potential is kept
constant for all interactions,26 except for the interaction between cation and
monomer, where $\sigma=$0.4787\text{\,}\mathrm{nm}$$ corresponds to even more
strongly hydrated cations interacting with rather apolar monomers.

Langevin dynamics (LD) simulations in Gromacs 5.163 in $NVT$ ensemble are
performed. The friction constant $\gamma$ is set to
$1.0\text{\,}{\mathrm{ps}}^{-1}$. The stochastic processes correspond to the
temperature $T=$298\text{\,}\mathrm{K}$$. The time step of the velocity Verlet
integrator is set to $7.5\text{\,}\mathrm{fs}$. Together with the reduced
molar mass $8.635\text{\,}\mathrm{g}\text{\,}{\mathrm{mol}}^{-1}$ of all
coarse-grained particles, the polymer conformations are sampled effectively.
The electrostatic potential is computed using particle mesh Ewald method.64 To
efficiently employ this method in a system with a low particle density, the
value of a cut-off distance in the real space and the number of grid points in
the Fourier reciprocal space are optimized, as described in LABEL:si:tab:pme.

A single coarse-grained polymer chain and salt ions are placed into a cubic
simulation box to prepare $0.0\text{\,}\mathrm{\textsc{m}}$ (counterions
only),
$0.1\text{\,}\mathrm{\textsc{m}}0.4\text{\,}\mathrm{\textsc{m}}1.0\text{\,}\mathrm{\textsc{m}}2.0\text{\,}\mathrm{\textsc{m}}$
salt solvation environment. Periodic boundary conditions are applied. The
default volume of the simulation box $8304\text{\,}{\mathrm{nm}}^{3}$ is
reduced to one fourth when the polymer holds a compact globular conformation
and the salt concentration is $\geq$1\text{\,}\mathrm{\textsc{m}}$$. The
compositions of the systems are described in detail in LABEL:si:tab:pme. A
trajectory of $525\text{\,}\mathrm{ns}$ is obtained in every simulation run.
The initial $25\text{\,}\mathrm{ns}$ is considered as an equilibration phase
and thus excluded from the analysis. Standard Gromacs tools, namely gmx
polystat and gmx rdf, are used for the data analysis. Further statistical
evaluations are performed in R.65 The uncertainties of the quantities are
determined as a standard error obtained from ten independent parts of the
simulation trajectory.

Figure 2: A snapshot of the simulation box containing a 200-mer chain in
$0.4\text{\,}\mathrm{\textsc{m}}$ salt solution within an implicit solvent.
The neutral polymer beads are depicted in blue, while the charged ones in
purple. Univalent cations and anions of the salt are presented as green and
yellow beads, respectively.

##  3 Mean-field theory

The change of the critical conditions of a polymer in a presence of cosolute
can be described by a mean-field theory as follows. All interactions within
the theory are described by a virial expansion and the electroneutrality
condition is postulated. We choose an $NpT$ ensemble for the polymer which is
in contact with a reservoir of cosolutes at a constant chemical potential.
Cosolute molecules are allowed to exchange between the ensemble and the
reservoir. At equilibrium the Gibbs free energy is minimized in such system.
The polymer is simply modeled as an elastic, penetrable sphere that can change
its radius upon the sorption of cosolutes. (Polymer conformational degrees of
freedom are thus averaged out into an effective blob, like in the classical
Flory approaches where only the polymer size is a variable.) Hence, the total
Gibbs free energy of the polymer phase which is in a contact with a salt
reservoir containing $i$ cosolute constituents, i.e., anions and cations in
our system, can be defined as

$\displaystyle G_{\mathrm{pol}}$
$\displaystyle=F_{\mathrm{pol}}+pV_{\mathrm{pol}}-\sum_{i}\bar{\mu}_{i}N_{i}\text{,}$
(2)

where $N_{i}$ is the number of particles of the respective cosolute
constituent (in this work, only ions) contained within the polymer sphere of
the effective volume $V_{\mathrm{pol}}$, $\bar{\mu}_{i}$ is the
electrochemical potential of the respective constituent in the reservoir, $p$
is the osmotic pressure, and $F_{\mathrm{pol}}$ is the total Helmholtz free
energy of the polymer including the absorbed cosolute.

The electrochemical potential $\bar{\mu}_{i}$ can be expressed with the
explicit electrostatic term as

$\displaystyle\bar{\mu}_{i}=\mu_{i}+z_{i}e\phi\text{,}$ (3)

where $\mu_{i}$ is a chemical potential of the $i$-th constituent, having a
charge $z_{i}e$, which is located in an electrostatic field of a potential
$\phi$. If the electroneutrality condition inside the effective volume of the
polymer is assumed, the electrostatic part of the potential does not have to
be considered, as no net charge is exchanged between the polymer spherical
region and the reservoir. Thus we will use only the chemical potentials
$\mu_{i}$ of the constituents. The electroneutrality is described with the
condition

$\displaystyle\xi
N_{\mathrm{mer}}z_{\mathrm{mer}}+\sum_{i}N_{i}z_{i}=0\text{,}$ (4)

where $N_{\mathrm{mer}}$ is the total number of monomers in a polymer chain,
and $\xi$ is the fraction of charged segments, each bearing the charge
$z_{\mathrm{mer}}e$.

The interactions between particles are described by a virial expansion up to
the fourth order in our model. Note, the expansion at least up to the third
order is required to guarantee a physically relevant solution for the negative
second virial coefficient. The second viral coefficient $B_{2}^{ij}$ between
the particles $i$ and $j$ can be easily evaluated from the Lennard-Jones pair
potential $U^{ij}_{\mathrm{LJ}}$,

$\displaystyle B_{2}^{ij}=-2\pi\int_{0}^{\infty}r^{2}\left[\exp(-\beta
U^{ij}_{\mathrm{LJ}})-1\right]\differential{r}\text{.}$ (5)

We approximate all three body interactions using the third virial coefficient
$B_{3}^{ijk}=2.00\,\sigma^{6}$, consistently with our previous work.26 The
four body interactions are modeled analogously with the fourth virial
coefficient of hard spheres with a diameter $\sigma$,
$B_{4}^{ijkl}=2.64\,\sigma^{9}$.

The polymer chain itself is modeled as an elastic (entropic) spring66, 53

$\displaystyle\beta
F_{\text{chain}}=\frac{4}{9}(\alpha^{2}+\alpha^{-2})\text{,}\quad\alpha=R_{\mathrm{pol}}/(b\sqrt{\tfrac{N}{6}})\text{,}$
(6)

where $\alpha$ is an elongation of the chain relative to the ideal chain
having $N$ segments of a length $b$, and $\beta=(k_{\mathrm{B}}T)^{-1}$ with
the Boltzmann constant $k_{\mathrm{B}}$ and temperature $T$. To relate the
polymer radius $R_{\mathrm{pol}}$ and volume $V_{\mathrm{pol}}$, a spherical
shape of the polymer is assumed. The total Helmholtz free energy of the
polymer is then given as

$\displaystyle\begin{split}\beta F_{\mathrm{pol}}=\beta
F_{\text{chain}}+V_{\mathrm{pol}}\left[\sum_{i}\rho_{i}\ln\rho_{i}+\sum_{j,k}B_{2}^{jk}\rho_{j}\rho_{k}+{}\right.\\\
\left.\frac{1}{2}\sum_{j,k,l}B_{3}^{jkl}\rho_{j}\rho_{k}\rho_{l}+\frac{1}{3}\sum_{j,k,l,m}B_{4}^{jklm}\rho_{j}\rho_{k}\rho_{l}\rho_{m}\right]\text{,}\end{split}$
(7)

where $\rho$ is the particle density inside the effective polymer volume and
the index $i$ stands for the cosolute constituents and the indices $j,k,l,m$
stand for the chain segments and/or the cosolute constituents.

The reservoir is modeled as a real gas which follows the fourth order virial
expansion

$\displaystyle\begin{split}\beta
F_{\mathrm{out}}=V_{\mathrm{out}}\left[2\rho_{\mathrm{out}}\ln\rho_{\mathrm{out}}+4B_{2}^{\mathrm{}}\rho_{\mathrm{out}}^{2}+{}\vphantom{\frac{8B_{3}^{\mathrm{}}}{2}}\right.\\\
\left.\frac{8B_{3}^{\mathrm{}}}{2}\rho_{\mathrm{out}}^{3}+\frac{16B_{4}^{\mathrm{}}}{3}\rho_{\mathrm{out}}^{4}\right]\text{,}\end{split}$
(8)

where $\rho_{\mathrm{out}}$ is the density of (uni-univalent) salt in the
reservoir. The osmotic pressure $p$ and the chemical potentials $\mu_{i}$ in
the reservoir can consequently be expressed as

$\displaystyle p$
$\displaystyle=\left.\partialderivative{F_{\mathrm{out}}}{V_{\mathrm{out}}}\right|_{N_{j},T}\text{,}$
(9) $\displaystyle\mu_{i}$
$\displaystyle=\left.\partialderivative{F_{\mathrm{out}}}{{N_{i}}}\right|_{N_{j\neq
i},V,T}\text{,}$ (10)

respectively, for any given cosolute concentration $\rho_{\mathrm{out}}$.

In order to find the equilibrium we numerically minimize the Gibbs free energy
of the polymer $G_{\mathrm{pol}}$ (eq 2) with respect to its effective volume
$V_{\mathrm{pol}}$ and the number of cosolute particles $N_{i}$. Note that the
theory can model both UCST and LCST polymers, depending on the input, in
particular, $T$-dependence of the viral coefficients.

Knowing the equilibrium concentration of the ions inside the polymer phase,
the Donnan potential can be evaluated using eq 3 directly, for either cation,
or anion. The chemical part $\mu_{i}$ of the electrochemical potential
$\bar{\mu}_{i}$ of the respective cosolute $i$ inside the polymer phase can be
expressed from eq 7, computing the derivative

$\displaystyle\mu_{i,\mathrm{in}}$
$\displaystyle=\left.\partialderivative{F_{\mathrm{pol}}}{N_{i}}\right|_{N_{j\neq
i},V_{\mathrm{pol}},T}\text{.}$ (11)

Detailed evaluation of the Donnan potential is provided in the SI.

The mean effective volume of the polymer can be obtained as a Boltzmann weight
average. The fluctuations in the radius of the polymer $(\langle
R_{\mathrm{pol}}^{4}\rangle/\langle R_{\mathrm{pol}}^{2}\rangle^{2}-1)$ are
computed as a function of the solvent quality ($\epsilon_{\mathrm{mer}}$). At
the critical solution temperature (CST) the fluctuations maximize.67 The
$n$-th moment of $R_{\mathrm{pol}}$ was computed using the Boltzmann
distribution

$\displaystyle\langle
R_{\mathrm{pol}}^{n}\rangle=\frac{\iint_{R,\rho}R_{\mathrm{pol}}^{n}\exp(-\beta
G_{\mathrm{pol}})\mathop{\mathrm{d}\\!}\rho_{\mathrm{in}}\mathop{\mathrm{d}\\!}R_{\mathrm{pol}}}{\iint_{R,\rho}\exp(-\beta
G_{\mathrm{pol}})\mathop{\mathrm{d}\\!}\rho_{\mathrm{in}}\mathop{\mathrm{d}\\!}R_{\mathrm{pol}}}\text{,}$
(12)

where $\rho_{\mathrm{in}}$ is the salt concentration inside the polymer phase,
maintaining the electroneutrality condition, and $R_{\mathrm{pol}}$ is the
effective polymer radius ($V_{\mathrm{pol}}=4/3\pi R_{\mathrm{pol}}^{3}$).

Note that our mean-field theory can be considered valid only close to the
coil-to-globule equilibrium when the polymer chain forms a rather compact
conformation. The further assumptions are that the fraction of charged
monomers has to be low and the salt concentration relatively high, i.e., the
neutral reference state is only slightly perturbed.

A similar thermodynamic description has been recently developed for microgels,
albeit a different procedure to find the thermodynamic equilibrium was
employed.62 Salt-specific effects on weakly charged copolymers can be captured
qualitatively by a simplified theory which treats the ion-specific and non-
specific electrostatic contributions on a simple additive level, as
demonstrated on PNIPAM copolymers.61 In contrast, the theory developed in this
work includes the ion-specific and non-specific interactions in a self-
consistent way.

The equations are solved using numeric solvers of Wolfram Mathematica 11.2
software package.68

##  4 Results and discussion

###  4.1 Solvent quality near critical conditions

In order to find conditions close to the swollen-to-collapse transition, a
solvent quality parameter, i.e., an effective interaction between monomers
$\epsilon_{\mathrm{mer}}$, is varied from
$0.1\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}0.9\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$.
Polymer chain conformation changes from an extended coil to a collapsed
globule in this solvent quality range, as shown in Figure 3.

Figure 3: Mean radius of gyration $R_{\mathrm{g}}$ of a 200-mer as a function
of the solvent quality, expressed by the monomer interaction strength
$\epsilon_{\mathrm{mer}}$. Various charged fractions $\xi$ are distinguished
by the symbol shape and color. Values for the chains in a pure solvent are
shown with filled symbols while empty symbols are used for chains in
$0.1\text{\,}\mathrm{\textsc{m}}$ solution of a repulsive salt. Lines serve as
a guide to the eye. The radius of gyration of the ideal chain
$R_{\mathrm{g}}^{\mathrm{id}}$ is shown for comparison.

The radius of gyration of an uncharged polymer chain ($\xi=0.00$) at
$\epsilon_{\mathrm{mer}}\simeq$0.4\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$
almost exactly corresponds to the radius of gyration of the ideal, non-
interacting polymer chain $R_{\mathrm{g}}^{\mathrm{id}}$. The radius of
gyration increases with the charge fraction of the polymer and the transition
is shifted to higher $\epsilon_{\mathrm{mer}}$ values. However, a small
concentration of salt ($0.1\text{\,}\mathrm{\textsc{m}}$) already suppresses
this trend due to the charge screening, as shown in Figure 3. Thus we choose
$\epsilon_{\mathrm{mer}}=$0.4\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$ to
describe the solvent quality for uncharged and charged polymers close to the
critical conditions, with the exception of strongly charged ($\xi=0.10$)
anionic polymer where the value
$\epsilon_{\mathrm{mer}}=$0.5\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$ is
used instead. Note that the critical value of $\epsilon_{\mathrm{mer}}$
depends on the actual chain size and the charge fraction. The value approaches
$0.3\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$ for an infinitely long
uncharged chain.

###  4.2 Salt interaction with an uncharged ELP

Figure 4: (a–c) Mean radius of gyration $R_{\mathrm{g}}$ obtained from
simulations of 200-mer as a function of salt concentration. The polymer bears
various charge fractions $\xi$ of positively charged monomers: (a) $0.00$, (b)
$0.05$, and (c) $0.10$, respectively. Symbols and colors in the plots indicate
the various strength of the interaction between the anions and the monomers,
$\epsilon_{\text{anion}}$. Lines serve as a guide for the eye. The mean radius
of gyration of an ideal chain $R_{\mathrm{g}}^{\text{id}}$ is indicated. (d–f)
An effective radius of the polymer $R_{\mathrm{pol}}$ computed using the mean-
field theory for the very same systems (column-wise). The same color code is
used for the strength of the anion-polymer interaction. (g–i) The standard
free energy of the coil-to-globule transition, $\Delta_{\text{coil-
glob}}G^{\circ}$, computed from the simulations. An increase in the energy of
the transition corresponds to the elevation of LCST, proportional to $\Delta
T_{\mathrm{LCST}}$ in Figure 1.

First, we investigate the uncharged ELP and relate qualitatively the
simulation results to the known experimental data. Two different regimes,
determined by the salt type, can be observed when the mean radius of gyration
of the uncharged 200-mer is plotted as a function of the salt concentration,
Figure 4 (a). The repulsive salts
($\epsilon_{\text{anion}}=\text{$0.1\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}0.3\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$}$)
induce a collapse through repulsion as the salt concentration increases. The
salts with an attractive anion
($\epsilon_{\text{anion}}=\text{$0.7\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}0.9\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$}$)
cause swelling of the polymer coil by attraction. Both regimes are described
in detail for non-ionic cosolutes in our previous work,26 with the typical
conformations of the polymer chain illustrated in Figure 5 (a, d),
respectively.

The change in the radius of gyration can be also interpreted as a change of
the standard free energy of the coil-to-globule transition

$\displaystyle\Delta_{\text{coil-
glob}}G^{\circ}=-k_{\mathrm{B}}T\ln\frac{p_{\text{glob}}}{p_{\text{coil}}}\text{,}$
(13)

where $p$ is the probability of the respective state. The probabilities can be
determined from the distribution of the radius of gyration. The value of the
ideal radius of gyration can be used as a reasonable dividing boundary between
the collapse and swollen states. As we see, the standard free energy of the
coil-to-globule transition changes when the salt is added to the polymer,
Figure 4 (g). The shift in the free energy has to be compensated by the change
in the quality of the solvent to maintain the critical conditions. The quality
of the solvent has to improve as the free energy decreases to reestablish the
population of extended states, and vice versa. For polymers exhibiting a LCST,
like ELPs, a decrease in the free energy of the coil-to-globule transition
manifests as an LCST decrease. Polymers exhibiting an UCST, as our simple
polymer model, show the opposite trend at the critical temperature.

The standard free energy of the coil-to-globule transition of the uncharged
polymer obtained from simulation Figure 4 (d) can be qualitatively compared
with the experimental LCST data of uncharged ELP, as shown in Figure 1 (a).
The anion transients from a strongly hydrated (little polymer affinity) to a
weakly hydrated (large polymer affinity) as the value of
$\epsilon_{\text{anion}}$ increases. Comparing to the charged polymer
discussed later, only ion-specific effects can be observed both in the
simulation and in the experiment.

Figure 5:  Simulation snapshots of typical polymer chain conformations under
various conditions: uncharged 200-mer in $2\text{\,}\mathrm{\textsc{m}}$ salt
solution (a, d), highly charged 200-mer in $2\text{\,}\mathrm{\textsc{m}}$ (b,
e) and $0.1\text{\,}\mathrm{\textsc{m}}$ (c, f) salt solution. The salt
consists of a very repulsive cation and a repulsive (a–c), or attractive (d–f)
anion, respectively. Neutral and positively charged monomers are depicted in
blue and purple, respectively. Green cations and yellow anions are are shown
within a distance of $0.7\text{\,}\mathrm{nm}$ from the polymer chain.

Thiocyanate SCN- is a weakly hydrated anion thus relatively high
$\epsilon_{\text{anion}}\sim$0.9\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$
is its appropriate parametrization in our simple model. The simulations
predict a shift towards the swollen state of the uncharged polymer as the
concentration of the salt increases, which agrees with the LCST elevation in
the experimental results. Iodide I- is another weakly hydrated anion (from the
observed behavior
$\epsilon_{\text{anion}}\sim$0.5\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}0.7\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$)
having the same but weaker effect on uncharged ELP both in simulations and
experiments. Bromide has the slightest effect of the salts experimentally
studied and the anion with
$\epsilon_{\text{anion}}\sim$0.3\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$
exhibits the similar effects in the simulation. Moving towards the strongly
hydrated anions in the Hofmeister series, chloride Cl- and fluoride F-, the
LCST drops significantly in the experiments. The same transition is observed
in the simulations containing anions with
$\epsilon_{\text{anion}}\sim\text{$0.1\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}0.3\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$}$.
It can thus be concluded from the simulations and the experimental data that
the salting-out (globule stabilization) power of anions follows the direct
Hofmeister series for the uncharged ELP.

###  4.3 Salt interaction with a positively charged ELP

Now we turn to a cationic ELP. For this, a polymer chain containing 200
monomers of which 20 are positively charged (a charge fraction $\xi=0.10$) is
simulated. Its radius of gyration $R_{\mathrm{g}}$ and the standard free
energy of the coil-to-globule transition as a function of salt concentration
are shown in Figure 4 (c, i). Two distinct phenomena can be observed,
depending on salt concentration. Screening of the electrostatic repulsion
between the charged monomers by the ions onsets at low salt concentration
($\lesssim$0.1\text{\,}\mathrm{\textsc{m}}$$). The polymer rapidly collapses
from an extended coil ($R_{\mathrm{g}}\simeq$4.7\text{\,}\mathrm{nm}$$) to a
more compact conformation as the salt is added and the Debye length,
$\kappa^{-1}=(8\pi l_{\mathrm{B}}N_{\mathrm{A}}I)^{-1/2}$, decreases. Here,
$N_{\mathrm{A}}$ is the Avogadro number and $I$ is molar ionic strength. This
screening strengthens with the increasing salt concentration in the polymer
phase. Weakly hydrated ions with large polymer affinity, that is, having
greater $\epsilon_{\text{anion}}$, are more attracted to the polymer and cause
stronger screening.

Figure 6: Radial distribution functions of ions around a center of mass of a
cationic polymer. Distributions of anions are shown in solid lines while the
distribution of cations in dashed lines. (a) Various salt types at the
isospheric point (a positively charged 200-mer bearing a charge fraction
$\xi=0.05$ in $1.0\text{\,}\mathrm{\textsc{m}}$ salt solution). The cation is
always strongly repulsive while the quality of the anion changes from
repulsive
($\epsilon_{\text{ion}}=$0.1\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$) to
attractive
($\epsilon_{\text{ion}}=$0.9\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$). (b)
and (c) Effect of the salt concentration on the radial distribution function
in a solution with positively charged polymer bearing a charge fraction
$\xi=0.10$. Salt with a repulsive
($\epsilon_{\text{anion}}=$0.1\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$)
and attractive
($\epsilon_{\text{anion}}=$0.9\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$)
anion is depicted in panel (b) and (c), respectively.

However, the collapse of the polymer chain beyond the size of the uncharged
polymer in a salt-free solution, as in the case of
$\epsilon_{\text{anion}}=$0.9\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$,
cannot be explained by the charge screening alone and another effect has to be
taken into account. As the electroneutrality condition is fulfilled inside the
polymer volume, the concentration of the strongly attractive anions is
significantly higher than the concentration of the repulsive cations. The
average $B_{2}$ of the ion-mer interactions is thus shifted towards more
negative, i.e., attractive, values, entering the bridging adhesion regime.26
The radial distribution functions of anions and cations in Figure 6 (c) prove
the greater abundance of strongly attractive anions compared to cations at low
salt concentrations. Typical conformations of the polymer chain are shown in
Figure 5 (c, f) for the repulsive, and attractive salt, respectively. As salt
concentration increases ($\gtrsim$1\text{\,}\mathrm{\textsc{m}}$$), the salt
specific effects begin to contribute to the interactions, resembling the case
of the uncharged polymer, however with a notably different polymer size, cf.
Figure 5 (a–b, d–e).

When the standard free energy of the coil-to-globule transition is treated as
a function of salt concentration, it monotonically decreases for more
repulsive ions with little affinity to the polymer. A different behavior is
observed for the attractive ions: After an initial depression caused by the
screening and bridging effects at low salt concentrations, the function rises
because the swelling by attraction mechanism is introduced at high salt
concentration. However, in the case of highly charged polymer chain
($\xi=0.10$), the screening-bridging mechanism dominates in the whole
investigated concentration range (up to $2\text{\,}\mathrm{\textsc{m}}$). Thus
the salting-out power of ions is reversed when compared to the uncharged
polymer.

The same simulation set is performed for a 200-mer bearing a lower charge
fraction, $\xi=0.05$. Two salt concentration ranges can be distinguished based
on the radius of gyration and the free energy of the coil-to-globule
transition, Figure 4 (b, h). When the salt concentration is less than
$1\text{\,}\mathrm{\textsc{m}}$, the screening and bridging mechanism
dominates and the anions follow the reverse series regarding the salting-out
power. At concentrations higher than $1\text{\,}\mathrm{\textsc{m}}$, the
specific ion effects are the driving force and the direct Hofmeister series of
anions is obeyed.

The dividing point which separates the concentration ranges can be called the
isospheric point as the radius of gyration of the polymer chain is the same
regardless of the anion type. The isospheric point moves to the higher salt
concentrations as the charge fraction of the polymer chain increases. For the
uncharged polymer ($\xi=0.00$) it is trivially located at zero salt
concentration, while for the highly charged polymer ($\xi=0.10$) at
$2\text{\,}\mathrm{\textsc{m}}$ salt concentration. The $R_{\mathrm{g}}$ at
the isospheric point corresponds approximately to the $R_{\mathrm{g}}$ of the
uncharged polymer in a salt free solvent in our setup, i.e., close to its
radius of gyration under the critical conditions.

Though the size of the polymer at the isospheric point is the same, the
concentration of the salt inside the polymer volume varies greatly depending
on the salt type. Radial distribution functions of cations and anions around
the polymer’s center of mass at the isospheric point of the weakly charged
polymer are shown in Figure 6 (a). The concentration of the anions—and
synergetically also of the cations—increases inside the effective polymer
volume as the affinity of the anions to the polymer increases. The attractive
anions even make the concentration of the highly repulsive cation higher
around the polymer compared to the bulk solution. Note, the salts with a
highly attractive anion, however, do not intersect the isospheric point and
the corresponding radius of gyration is achieved at a higher salt
concentration, see LABEL:si:fig:thermodynamics-high. Very weakly hydrated
ions, such as perchlorate ClO4-, are experimentally known to deviate in a
similar manner.24

When we compare the data from simulation, Figure 4 (e), to the experiments,
Figure 1 (b), a very good qualitative agreement can be found. When a small
amount of salt is added, the LCST decreases rapidly in experiments and the
order of anions follows the reverse series at low salt concentration, in
accordance with the simulation predictions. Crossing the isospheric point at
$\sim$0.45\text{\,}\mathrm{\textsc{m}}$$, the anion order inverts and the
direct Hofmeister series is recovered.

###  4.4 Theoretical predictions

For a better interpretation and categorization of the simulation results, we
apply the mean-field theory (Section 3) for the very same systems.
Minimization of the Gibbs free energy of the system provides the effective
volume of the polymer in the thermodynamic equilibrium with the salt
reservoir, or equivalently the effective radius $R_{\mathrm{pol}}$ if a
spherical geometry is assumed. The polymers’ effective radii in the
thermodynamic equilibrium are depicted in Figure 4 (d–f) for an uncharged,
weakly charged, and highly charged polymer chain, respectively. The results
will not change significantly if a mean volume is used instead of the
equilibrium volume.

The theory matches the simulation results almost quantitatively when the
effective radius (Figure 4 (d–f)) and the radius of gyration, (Figure 4 (a–c))
are compared. The theory correctly predicts salting-out power of anions and
its concentration dependence for the uncharged polymer. In the cases of weakly
and highly charged polymer chains, the location of the isospheric points
matches the simulation data very well. Below the isospheric point, the
salting-out power of anions forms the reverse series, while above the direct
series is obeyed, which is in agreement with both the simulations and the
experimental data.

The osmotic pressure of the salt inside the effective volume of the polymer at
the isospheric point is the same, regardless of the salt type, as can be
deduced from the mechanical (pressure) equilibrium. The osmotic pressure of
the reservoir depends only on the salt concentration in our model. The counter
pressure originates from the polymer chain and the osmotic pressure of ions.
The pressure of the polymer chain is determined by its spatial extension, thus
it does not differ at the isospheric point. Hence the osmotic pressure of the
ions inside the effective polymer volume has to be the same, independent of
the salt type. Note, however, the net salt-effect is a sum of two large
contributions (see eq 7), namely of an ideal gas and of an excess interaction,
which significantly vary with the salt type (i.e., with
$\epsilon_{\text{anion}}$).

The salt concentration at the isospheric point $c_{\mathrm{iso}}$ scales
linearly with the polymer’s charge fraction $\xi$, provided the charge
fraction is low ($\xi\lesssim 0.05$), see Figure 7. As the polymer’s charge
fraction increases, the isospheric point weakly deviates from the original
linear trend and it is also gradually smeared out to a concentration range, as
discussed in detail in the SI. Very weakly hydrated ions, e.g., perchlorate
ClO4-, do not exhibit the isospheric point in experiments.24 Our theoretical
model predicts such behavior for salts having
$\epsilon_{\text{anion}}\gtrsim$1.1\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$.

Figure 7: The isospheric point as a function of the polymer charge fraction
$\xi$ predicted by the mean-field theory. The isospheric point is computed as
an intersection of $R_{\mathrm{g}}(\xi,c)$ curves of salts with
$\epsilon_{\text{anion}}=\text{$0.1\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}0.9\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$}$,
respectively (blue line with symbols). The dashed line shows linear asymptotic
scaling at low charged fractions ($\xi\leq 0.05$). Figure 8: Fluctuations in
the effective radius of the polymer chain $R_{\mathrm{pol}}$ as a function of
the solvent quality at various salt concentrations (from
$0.4\text{\,}\mathrm{\textsc{m}}3.0\text{\,}\mathrm{\textsc{m}}$), as derived
from the mean-field theory. The solvent quality is expressed as a strength of
LJ interactions between mers, $\epsilon_{\mathrm{mer}}$. The top row (a–c)
depicts the uncharged 200-mer ($\xi=0.00$), while the bottom row (d–f) shows
fluctuations of the positively charged 200-mer with a charge fraction
$\xi=0.05$. The left column (a, d) corresponds to a strongly hydrated
(repulsive) salt
($\epsilon_{\text{ion}}=$0.1\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$), the
middle column (b, e) to a rather indifferent salt
($\epsilon_{\text{ion}}=$0.5\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$), and
the right column (c, f) to a weakly hydrated (attractive) salt
($\epsilon_{\text{ion}}=$0.9\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$). The
dashed lines indicate the maxima.

The fluctuation of the radius of gyration of a polymer chain is maximized
under the critical conditions.67 The amount of fluctuations can be accessed
from the theory using the Boltzmann distribution of the chain’s states, eq 12.
The dependence of the fluctuations on the solvent quality is depicted in
Figure 8 for the uncharged and weakly charged polymer chains. With increasing
concentration of repulsive salt (panels a, d), the maximum in fluctuations
(i.e., the critical conditions) is shifted to a better solvent quality (lower
$\epsilon_{\mathrm{mer}}$). This corresponds to a lower LCST for ELPs. The
opposite trend is observed for the attractive salt (panels c, f) where the
solvent quality has to worsen to keep the critical conditions as the salt
concentration increases. The relative shift of the critical solvent quality
for the uncharged (c) and weakly charged (f) polymer is consistent with the
difference in the standard free energy of the coil-to-globule transition
obtained by simulations, Figure 4 (g, h).

Let us stress again that the theory assumes a spherical geometry of the
polymer and electroneutrality within this effective polymer volume. These
assumptions are well fulfilled close to the critical conditions at
$\gtrsim$0.1\text{\,}\mathrm{\textsc{m}}$$ salt concentration. For example,
our mean-field theory cannot predict the radius of gyration of a charged
polymer without salt as no explicit electrostatic interaction between charged
monomers is included. Also at low salt concentrations, the effective radius
predicted by the theory deviates significantly from the radius of gyration. In
such cases, different approaches have to be used.69, 43, 56

###  4.5 Mean-field theory predictions for a negatively charged ELP

Figure 9: (a) Mean radius of gyration $R_{\mathrm{g}}$ of the negatively
charged ELP (charge fraction $\xi=0.10$) as a function of salt concentration
$c$ from the simulations. Salts with various anions and the same highly
repulsive cation are used. Lines serve as guides to the eye. (b) A prediction
of the effective radius of the polymer $R_{\mathrm{pol}}$ computed from the
mean-field theory applied to the same systems.

Having verified the descriptive power of the mean-field theory, we use it now
to predict the effects of salts to a negatively charged polymer. The set of
salts remains the same, i.e., the cation is highly repulsive while the type of
the anion varies. The solvent quality has to worsen to
$\epsilon_{\mathrm{mer}}=$0.5\text{\,}\mathrm{\mathit{k_{\mathrm{B}}T}}$$ to
retain the critical conditions. The theory prediction is depicted in Figure 9
(b), while the simulation results which we use for verification are shown in
panel (a) of the same figure. To the best of our knowledge, no systematic
experimental data are yet available for such setup.

The simulation and theory results are in a very good mutual agreement. When
the salt is added the collapse of the polymer chain is observed at first. The
extend of the collapse is determined mainly by the (same) strongly hydrated
cation and only mildly modulated by the anion type. Both electrostatic
screening in the polymer and build-up of the osmotic pressure in the bulk
solution contribute to the collapse. As the concentration of the salt
increases, the anion-specific effects further drive the collapse or swelling,
similarly to the case of an uncharged polymer. Contrary to the cationic
polymer, neither the extent of the initial collapse dependents on the salt
(anion) type, nor the isospheric point occurs. Consequently, the salting-out
power of anions follows the direct Hofmeister series in the whole
concentration range.

##  5 Conclusions

The experimentally observed LCST response (and corresponding swelling and
collapse) of elastin-like polypeptides (ELPs) to the addition of salts
displays a rich behavior resulting from a combination of ion-specific
screening of these weakly charged polyelectrolytes and direct ion-polymer
interaction effects. We developed a minimalistic implicit-solvent CG model
that can reproduce and qualitatively explain all the experimentally observed
effects.17, 24 Employing further simplifying assumption, we have developed a
mean-field model which provides quantitative agreement with the CG computer
simulations. Here, an effective second virial coefficient $B_{2}$ represents
the quality of ions (strongly vs. weakly hydrated, or ‘kosmotropic’ vs.
‘chaotropic’ in classical terms) with respect to their affinity and
preferential interaction with the polymer.

In particular, we demonstrated that for the uncharged chain ion-specific
effects only are at work and the collapse is mediated through repulsion and
swelling by attraction regimes; the salting-out power of ions follows a direct
Hofmeister series characterized by a linear regime up to intermediate salt
concentrations. For the charged, cationic polymers we demonstrated ion-
specific screening and bridging at low salt concentration and strong ion-
specific direct interaction effects at high salt concentration. We made an
important theory-grounded observation of an isospheric point, i.e., salt
concentration at which ion-specific effects vanish and the Hofmeister series
inverts. That is, the salting-out power of ions follows a reverse series
bellow the isospheric point and a direct one above. The isospheric salt
concentration depends on the polymer charge fraction. Our observation is
supported by experimental data for ELPs17, 24, as well as for other positively
charged proteins.4

For the anionic polymers our mean-field theory predicts a non-linear and for
weakly hydrated (very attractive) anions also non-monotonic change of the LCST
with salt concentration. At low salt concentrations, the strongly hydrated
cation determines the screening and osmotic pressure in the polymer phase,
only mildly modulated by the anion type. In the limit of high salt
concentration, the ion-specific direct interactions drive the polymer
collapse, or swelling, respectively, which results in two regimes of salt
action. Consequently the direct Hofmeister series is followed in the whole
concentration range, unlike for the cationic polymer.

Our work demonstrates the sensitivity of the structural and phase behavior of
charged polymers to the addition of specific salts. Our mean-field model shall
be useful for further extrapolation and guidance of related experiments.

The authors would like to thank Won Kyu Kim for a valuable discussion about
the mean-field theory. R. C. and J. D. acknowledge funding from the Deutsche
Forschungsgemeinschaft (DFG grant DZ-74/6), Germany, for this project. J. H.
thanks the Czech Science Foundation (grant 16-24321Y) for support.

Mathematica notebook implementing the mean-field theory; simulation setup
details; trajectories and distributions of the radius of gyration; the
isospheric point / highly attractive anions; evaluation of the Donnan
potential.

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# Improving Performance of End-to-End ASR on Numeric Sequences

###### Abstract

Recognizing written domain numeric utterances (e.g., I need $1.25.) can be
challenging for ASR systems, particularly when numeric sequences are not seen
during training. This out-of-vocabulary (OOV) issue is addressed in
conventional ASR systems by training part of the model on spoken domain
utterances (e.g., I need one dollar and twenty five cents.), for which numeric
sequences are composed of in-vocabulary numbers, and then using an FST
verbalizer to denormalize the result. Unfortunately, conventional ASR models
are not suitable for the low memory setting of on-device speech recognition.
E2E models such as RNN-T are attractive for on-device ASR, as they fold the
AM, PM and LM of a conventional model into one neural network. However, in the
on-device setting the large memory footprint of an FST denormer makes spoken
domain training more difficult. In this paper, we investigate techniques to
improve E2E model performance on numeric data. We find that using a text-to-
speech system to generate additional numeric training data, as well as using a
small-footprint neural network to perform spoken-to-written domain denorming,
yields improvement in several numeric classes. In the case of the longest
numeric sequences, we see reduction of WER by up to a factor of 8.

##  1 Introduction

An ongoing challenge of ASR systems is to model transcriptions that do not
exactly reflect the words spoken in an utterance. For example, the spoken
utterance set an alarm for four fifteen is typically decoded in the written
form as set an alarm for 4:15. Numeric utterances, such as addresses, phone
numbers, and postal codes are particularly hard members of this category, due
to the inherent out-of-vocabulary (OOV) issues of long written-domain numeric
sequences. This problem arises because of a data sparsity issue, namely that
long numeric sequences are unlikely to be present in training data.

A natural solution to the data sparsity/OOV problem is to train an ASR system
in the spoken domain, and then to denorm the results back into the written
domain [1, 2, 3]. Such a solution for conventional ASR systems was presented
in [4]. In that work, the AM and PM are trained in the spoken domain. A
verbalizer weighted finite state transducer (WFST) is then inserted before the
LM to convert commonly used spoken-domain numerics to the written domain.
Finally, a class-based n-gram LM, modeled as an FST, is used to transduce
additional spoken domain entities to the written domain.

There are a few issues with the above approach for handling the numerics
problem. First, it is based on a set of rules in the verbalizer and class-
based LM, which does not scale well to changes in training data [5]. In
addition, the production FST-LM is large, which presents a challenge for
memory-constrained on-device applications [6].

End-to-end (E2E) ASR models have gained popularity in recent years. These
models, which fold the AM, PM and LMs into a single neural network, are
particularly attractive for on-device applications. E2E models, including
RNN-T, have been shown to achieve comparable performance to conventional
models at a fraction of the size [7, 6]. However, these same constraints
preclude the use of an FST denormer, and thus make spoken domain training more
difficult.

The goal of this work is to improve E2E performance on numerics by applying
several techniques. Synthetic data has been used successfully to address data
sparsity issues in image processing for scene text recognition [8], text
spotting [9], object detection [10] and speech recognition [11]. Given this
success, we first explore augmenting our training set using a text-to-speech
(TTS) system to synthesize additional written-domain numeric training data. In
doing so, we choose transcripts from several challenging numerics categories
in order to improve coverage of those cases.

Data-sparsity/OOV issues have also been addressed by neural correction models
in both in NLP [12, 13] and in speech [14, 15, 16, 17, 18]. A recent example
in ASR is [17], which proposed a neural error corrective language model
(NECLM) trained on pairs of transcripts and corresponding errorful ASR output
as a last step during decoding. This approach was applied to denorming in [18]
with a neural written-to-spoken denormalizer. We build on the work in [18] by
training a neural correction network, which is trained on written-domain
ground-truth/RNN-T hypothesis pairs and learns to correct mistakes.

Given the success of the spoken domain for numeric entities, a third avenue we
explore is to train RNN-T in the spoken domain and denorm back to written
domain. We compare both a traditional FST-based denormer as in [4] and a
neural denormer that is based on our written correction model.

We evaluate the different E2E numeric solutions on a variety of different
numeric test sets, which differ in their numeric sequence length. We find that
while written domain approaches show significant improvements, the largest
improvements are in the spoken domain, in which the OOV and data-sparsity
issues are mitigated, and particularly for long numeric sequences.
Specifically, in the written domain, the addition of TTS training data gives a
65% relative WER improvement for small numbers and a 35% improvement for large
numbers, increasing to 75% and 49% with the addition of a neural correction
model. In the spoken domain, we see our best results, where the incorporation
of a neural denorming model yields a 75% improvement on short numbers and 83%
on large numbers. On a curated eval set of specifically difficult
verbalizations, results on long utterances are even stronger, showing a factor
of 8 reduction in WER.

The rest of this paper is organized as follows. Section 2 introduces the four
variations of standard RNN-T that we worked with. This section includes our
procedure for TTS utterance generation as well as our neural correction/denorm
architecture. Section 3 describes the details of our experiments. Section 4
presents our results and analysis. Finally, Section 5 summarizes our
conclusions.

Category | Example Transcript Template | Example Numeric | Average Numeric Length
---|---|---|---
DAY | remind me on monday the $DAY | 31st | 1.8
PERCENT | turn down sound to $PERCENT | 20.22% | 2.2
POSTALCODE | how far away is $POSTALCODE | 86952 | 5.1
TIME | set second alarm for $TIME p.m. | 10:46 | 3.0
YEAR | play the top 40 from $YEAR | 1648 | 4.0
Table 1: Sample TTS Utterance Categories, with Example Synthetic Numerics

##  2 Methods

In this section, we present different ideas explored to address the long-tail
numeric issue of our RNN-T system. We give each approach a label, which we
will reference in Table 2 below.

###  2.1 TTS Training Data (W1)

To address the numeric data-sparsity issue, we generate additional training
data that represent challenging and realistic numeric sequences. To this end,
we choose numeric categories that we see often in Google Assistant traffic.
See Table 1 for sample categories. We specifically choose categories that
represent a variety of numeric sizes. To obtain transcripts in these
categories, we choose 200 unique templates for each category from anonymized
Google Assistant utterances, and inject the numeric entities by performing
weighted sampling from our production numeric WFST grammar (weighted on the
spoken domain). Two parallel transcripts are generated for each template, one
in the spoken domain and the other in the written domain. We perform this
process 100 times for each template, each time synthesizing a unique
utterance. All models below use TTS training data generated in this manner.

###  2.2 Written-Domain Neural Correction (W2)

We seek to correct written RNN-T hypotheses with a post-processing neural
correction step. Our written-domain correction model, shown in Figure 1, is an
attention-based sequence-to-sequence model, for which the input is RNN-T
hypotheses and the output is a corrected transcript. Our architecture is
adapted to the correction setting by accounting for the fact that the majority
of words in an input sentence are simply copied to the output during
correction. This is done by training an additional tagger RNN to run on the
input sequence before the sequence-to-sequence model. The tagger tags each
word in the input sequence as either trivial, in which case it is simply
copied to the output sequence, or non-trivial, in which case it is passed into
the attention model and decoder. This architecture has been reported for
contextual text normalization for text-to-speech [5], but we adapt it here to
the correction setting for ASR. While we use the model for correction, it
could also be used to re-rank an n-best list in a second-pass setting.

Figure 1: Neural Denormer Architecture. T stands for trivial. N stands for
non-trivial. S stands for start. C stands for continuation.

####  2.2.1 Encoder Layer

Suppose we seek to map between input sequence
$\mathbf{x}=\\{x_{1},\dots,x_{I}\\}$ and output sequence
$\mathbf{y}=\\{y_{1},\dots,y_{T}\\}$, where the sequence vocabulary is
composed of words. We define a BiRNN encoder such that

$h_{i}=[\overrightarrow{h}_{i};\overleftarrow{h}_{i}]$

where

$\overrightarrow{h}_{i}=\mathbf{RNN}_{\rightarrow}(\overrightarrow{h}_{i-1},x_{i})$

and

$\overleftarrow{h}_{i}=\mathbf{RNN}_{\leftarrow}(\overleftarrow{h}_{i+1},x_{i})$

where $\mathbf{h}=h_{1},\dots,h_{I}$ are hidden encoder states.

####  2.2.2 Tagging Layer

We define the tagger RNN as

$s_{i}=\mathbf{RNN}_{\text{tag}}(s_{i-1},t_{i-1},h_{i})$

where $\mathbf{s}=s_{i},\dots,s_{I}$ are hidden tagger states, with
corresponding observations, i.e., tag sequence $\mathbf{t}=t_{i},\dots,t_{I}$.

Each tag $t_{i}$ is a joined tag in the cross-product set of
$\\{\mbox{trivial},\mbox{non-
trivial}\\}\times\\{\mbox{start},\mbox{continuation}\\}$ to model whether a
word is the beginning of a new segment needed to be corrected or a
continuation of the previous segment. This refinement allows us to model
consecutive non-trivial segments.

The learning objective of the tagger is

$\operatornamewithlimits{argmax}_{\mathbf{t}}\prod_{i=1}^{I}{P(t_{i}|s_{i})}$
(1)

where $P$ is defined as a linear projection of $\mathbf{s}$ followed by a
softmax layer. We obtain alignments for training the tagger by using a
heuristic alignment algorithm that determines subsequences common to the input
and output transcripts. These common subsequences are marked as trivial.

####  2.2.3 Decoder Layer

We use the results of the tagger to extract text snippets to be corrected.
Suppose a text snippet spans from time $s$ to $e$. The input span
$\\{x_{s},\dots,x_{e}\\}$ along with the context hidden states
$\overrightarrow{h}_{s}$ and $\overleftarrow{h}_{e}$ become the input to the
next stage attentional model. We define a BiRNN encoder
$\mathbf{RNN}_{\text{mid}}$ over $\\{x_{s},\dots,x_{e}\\}$.

Finally, the attentional RNN decoder $\mathbf{RNN}_{\text{dec}}$ is defined as

$d_{i,t}=\mathbf{RNN}_{\text{dec}}(d_{i,t-1},y_{i,t-1},c_{i,t})$

where $c_{i,t}$ is the result of the attention function of $d_{i,t-1}$,
$\overrightarrow{h}_{s}$, $\overleftarrow{h}_{e}$, and
$\mathbf{RNN}_{\text{mid}}(\\{x_{s},\dots,x_{e}\\})$. The two-dimentional
indices $(i,t)$ indicate $t$ is relative to a given position $i$ ($s$, $e$) in
the input sentence. The architecture of the attention component is the same as
in [5].

The learning objective of the decoder is

$\operatornamewithlimits{argmax}_{\mathbf{y}}{\prod_{t=1}^{L(i)}{P(y_{i,t}|d_{i,t})}}$
(2)

In this way, the attention mechanism and decoder are applied only to relevant
spans of text, which decreases cost and improves accuracy. As shown in the
derivation steps, at training time the two objectives in Equation 1 and
Equation 2 translate to two cross-entropy losses that can be linearly combined
during training. At decoding time, the two models work as a pipeline, with the
attentional model only used as required by the tagging model.

###  2.3 Spoken Domain Training, FST Denorming (S1)

Following the success in conventional ASR models, we explore spoken domain
training for RNN-T. We train RNN-T on a spoken-domain version of our training
set, and leave the translation back to written domain to an FST denormer
derived from our production grammar [4].

In order to train this model, we require examples of utterance transcripts in
both the spoken and written domains. We gather these examples by passing
written-domain transcripts from our training set through an FST verbalizer. We
then choose a single verbalization by passing each candidate hypothesis
through a lexicon, and force-aligning the resulting phone sequences against
the phones in the utterance. For TTS training data, we simply use the spoken
domain transcript that we obtained using our verbalization grammar.

###  2.4 Neural Denorming (S2)

Since the FST-based denorming approach discussed in Section 2.3 is challenging
to put on device, we also explore a neural denorming approach. Specifically,
we adapt our written correction model from Section 2.2 to the spoken domain by
rephrasing it as a denormer which consumes spoken domain training data and
emits written domain output. The architecture of the neural denorming model is
identical to the written correction model.

|  | Written | Written TTS | Written TTS | Spoken TTS | Spoken TTS
---|---|---|---|---|---|---
|  |  |  | Neural Correction | FST Denorm | Neural Denorm
| Average Num Len | W0 | W1 | W2 | S1 | S2
VS |  | 6.7 | 6.8 | 6.5 | 7.6 | 6.9
NUMERICS | 3.0 | 6.9 | 6.6 | 6.3 | 8.3 | 7.2
SAMPLED_SHORT | 2.1 | 7.6 | 2.6 | 2.1 | 4.2 | 1.9
SAMPLED_MEDIUM | 3.3 | 11.9 | 5.7 | 5.2 | 7.6 | 4.0
SAMPLED_LONG | 7.4 | 26 | 16.9 | 13.2 | 16.8 | 4.4
TAIL_SHORT | 2.7 | 6.8 | 4.9 | 4.7 | 7.3 | 4.2
TAIL_MEDIUM | 4.3 | 15.4 | 12.9 | 11.6 | 11.6 | 6.7
TAIL_LONG | 6.9 | 92.7 | 74.8 | 50.9 | 11.9 | 11.3
Table 2: Results: WER by Model and Test Set

##  3 Experiments

###  3.1 Data Sets

Our experiments are conducted on a $\sim$30,000 hour training set consisting
of 43 million English utterances. The training utterances are anonymized and
hand-transcribed, and are representative of Google’s voice search traffic in
the United States. Multi-style training (MTR) data are created by artificially
corrupting the clean utterances using a room simulator, adding varying degrees
of noise and reverberation with an average SNR of 12dB [19]. The noise sources
are drawn from YouTube and daily life noisy environmental recordings. The real
audio test sets we report results on include $\sim$14.8K voice search (VS)
utterances extracted from Google traffic, and real-audio numeric test set of
$\sim$8.0K utterances derived from the VS distribution (NUMERICS).

In order to generate synthetic training data, we use a multi-speaker TTS
system based on the baseline architecture as described in [20], where the
Tacotron model [21] generates a mel-spectrogram conditioned on phonemes and a
64-dimensional speaker embedding learned for each speaker during training. The
predicted mel-spectrogram is then inverted to time-domain waveform with a
WaveRNN neural vocoder [22]. We again use MTR to add artificial noise to our
synthesized audio. The TTS training data consists of 84 English speakers
covering American, Australian, British and Singaporean accents with a Google
Assistant clean speaking style, comprising 370 hours of audio in total. During
inference, numeric input texts are mapped to phonemes and a speaker is
randomly selected.

This system is used to synthesize audio for templated transcripts, as
described above. In this manner we create training sets for each numeric
category, and create our final training set by grouping all categories
together. For each batch in training, we draw 10% of training examples from
this TTS set and 90% of examples from the real-audio training set described
above.

Since we know that the length of a numeric sequence exacerbates the OOV issue,
for our test sets we split numerics into three categories based on length. For
these categories, we generate transcripts using the same template phrases as
in training, but with unique verbalizations, to ensure that no phrase is used
in both training and test. We call these sets the SAMPLED_SHORT,
SAMPLED_MEDIUM and SAMPLED_LONG test sets. We are also interested in
evaluating our models on the tail of the distribution - that is, on the most
difficult numerics. To this end, for each category we maintain a list of
particularly difficult numeric verbalizations. For example, the postal code
22110 might be verbalized as double two double one oh. We call the test sets
generated with these more difficult verbalizations TAIL_SHORT, TAIL_MEDIUM,
and TAIL_LONG.

In order to ensure that our system has not overfit to a specific sort of TTS
data, we synthesize audio for our test sets using separate TTS systems. The
audio for the SAMPLED sets is created using a single-speaker WaveNet [23]
system. The audio for the TAIL sets is created using a concatenative TTS [24]
system.

###  3.2 RNN-T

All experiments use the same RNN-T architecture, following the setup in [6].
Specifically, an 8-layer unidirectional LSTM of width 2,048 with 640 recurrent
projection units is used as the encoder. A time-reduction layer with the
reduction factor $2$ is inserted after the second LSTM layer of encoder to
reduce model latency. The RNN-T decoder consists of a prediction network with
2 LSTM layers of 2,048 hidden units and a 640-dimensional projection per layer
as well as an embedding layer of 128 units. The outputs of the encoder and
prediction network are fed to a joint network that has 640 hidden units,
followed by a 4,096 wordpiece softmax output. In total, RNN-T has
approximately 114M parameters. The RNN-T model is implemented in Tensorflow
[25] and trained on $8\times 8$ Tensor Processing Units (TPU) slices with a
global batch size of 4,096 for $\sim 200K$ steps.

###  3.3 Neural Correction/Denorming

The neural denormer uses a bidirectional single-layer GRU encoder with 256
units that emits a 256-dimensional hidden state. The tagging RNN is a single-
layer GRU with 64 units. The decoder is a bidirectional single-layer GRU with
256 units. The encoder/tagger portion of the model, which runs for all input,
contains about 2M parameters, while the attention/decoder portion of the
model, which runs only for text spans marked for correction, contains about 4M
parameters. The small footprint of the neural correction model makes it
attractive for the on-device setting. The model is implemented in Tensorflow
and trained asynchronously on 12 GPUs, with a batch size of 16.

##  4 Results

Table 2 gives WER results for each of our experiments on the SAMPLED and TAIL
test sets, as well as the real-audio VS and NUMERICS test sets. We use the
labels given in Section 2 (W1, W2, S1, S2) for convenience. We use W0 to refer
to the baseline RNN-T model.

The results for the written domain models are characterized by a steep decline
in quality with increasing numeric length. While the the addition of TTS data
in W1 and the incorporation of a correction model in W2 improve over the
baseline W0, the error rates are still sometimes higher on the MEDIUM and LONG
test sets than on the non-numeric VS set.

The errors made by model tell a clear story: the written domain models
struggle to correctly format numerics, and often confuse parts of numeric
sequences for similar-sounding words. The following are representative errors
taken from the SAMPLED sets. Here, references are on the left of the arrow and
erroneous hypothesis are on the right.
W1:

$180.50 into inr $\rightarrow$ $180 - $50 in inr

house for rent 60003 $\rightarrow$ house for rent $6003

W2:

$487 / 6 $\rightarrow$ 4876

48007 to 08618 $\rightarrow$ 480-708-6618

In the spoken domain, we see a different error pattern. S1 errors show that
the FST denormer often completely or partially fails to perform denorming, for
example:
S1:

code 30441 $\rightarrow$ code 300 double 41

the 32nd door $\rightarrow$ the 30 second door

This indicates the problem with rule-based FST denormers, which often cannot
adapt well to changes in training data.

Progressing to S2, these issues largely disappear, and the errors that remain
are a blend of formatting, denorming, and other errors, in considerably
smaller numbers than for all other models. It seems that the avoidance of OOV
issues obtained by training in the spoken domain largely solves the formatting
problems experienced by the written domain models, while using a neural
denormer, which learns how to denorm from data, sidesteps the denorming errors
seen in the FST-based spoken domain model. Finally, the spoken denorming
approach does not result in a significant degradation on the real-audio VS or
NUMERIC sets.

##  5 Conclusions

In this paper, we experimented with four approaches for improving end-to-end
ASR performance on numeric utterances. We found that all approaches yield
improvements, with the largest improvements occurring when TTS training data,
spoken domain training, and neural denorming are all used together. The fact
that we see the largest improvements in the longest utterances, together with
the types of errors that seem to be avoided, suggest that the improvements are
mainly due to the alleviation of the out of vocabulary problem.

##  6 Acknowledgements

We thank Gabriel Mechali, Mark Epstein, Michael Riley, and Richard Sproat for
help and comments on this work.

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