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# Microphase separation in neutral homopolymer blends induced by salt-doping

Xian Kong<EMAIL_ADDRESS>South China Advanced Institute for Soft Matter
Science and Technology, School of Emergent Soft Matter, South China University
of Technology, Guangzhou 510640, China  Guangdong Provincial Key Laboratory of
Functional and Intelligent Hybrid Materials and Devices, South China
University of Technology, Guangzhou 510640, China  Jian Qin
<EMAIL_ADDRESS>Department of Chemical Engineering, Stanford University,
Stanford, CA 94305

###### Abstract

Microphase separation in polymeric systems provides a bottom-up strategy to
fabricate nanostructures. 1, 2, 3, 4 Polymers that are reported to undergo
microphase separation usually include block copolymers or polyelectrolytes.
Neutral homopolymers, which are comparatively easy to synthesize, are thought
to be incapable of microphase separation. Here, using a minimal model that
accounts for ion solvation, we show that microphase separation is possible in
neutral homopolymer blends with sufficient dielectric contrast, upon a tiny
amount of salt-doping. The driving force for the microphase separation is the
competition between selective ion solvation, that places smaller ions in
domains with higher dielectric constant, and the propensity for local charge
neutrality to decrease the electrostatic energy. The compromise is an emergent
length over which microphase separation occurs and ions are selectively
solvated. The factors affecting such competitions are explored, including ion
solvation radii, dielectric contrast, and polymer fraction, which point to
directions for observing this behavior experimentally. These findings suggest
a low-cost and facile alternative to produce microphase separation which may
be exploited in advanced material design and preparation.

##  1 Background

Microphase separation results from the competition among multiple length-
dependent interactions.5 A well-known example is block polymer, which is
formed by chemically linking thermodynamically incompatible components.6 The
incompatible blocks tend to separate from each other while the chemical
linkages between blocks prevent macroscopic separation, giving rise to
microphase separation.7 In the past, polymeric systems consisting of charged
homopolymers (or polyelectrolytes, PE) are also shown to be capable of
microphase separation. One example is weakly charged PEs in poor solvent. 8,
9, 10, 11, 12 The incompatibility between PE and solvent promotes a phase
separation into two macroscopic phases with high and low PE concentrations.
However, this leads to the loss in the translational entropy of counter ions,
which mostly reside in the concentrated phase. The competition between the
incompatibility-induced demixing and translational entropy loss of counter
ions, leads finally to the microphase separation. Recently, it is suggested
that microphase separation is also possible in polyelectrolytes blends, 13, 14
where the macroscopic phase separation between immiscible polyelectroytes is
suppressed by the need to minimize coulombic interaction.

The analogy between the microphase separation of polyelectrolytes and diblock
copolymer melts highlights the role of electrostatic interactions. Indeed, it
has been found that electrostatic interaction can be leveraged to manipulate
the phase behavior of polyelectrolytes solutions, 15, 16, 17 ionic polymer
blends, 18, 19, 20 neutral diblock copolymer melts, 21, 22 or charged polymer
blends 23, 24. For example, selective solvation of doped salts in
dielectrically heterogeneous copolymers has been shown to enhance the
effective Flory-Huggins parameter between two blocks of diblock copolymer, 25,
26 favoring the formation of ordered microscopic phases. Remarkably, a
“chimney” region was predicted where the solvation effects and electrostatic
correlations of ions can promote microphase formation in an otherwise fully
compatible diblock copolymer blends, that is, when the two blocks are fully
miscible.21 Similarly in polyelectrolyte, counterions of PEs are predicted to
be important in determining phase behavior, including enhancing the
compatibility between two PEs, 27, 28 narrowing the parameter space for
microphase separation, and allowing the competition between microphase and
macrophase separation, etc. 24

In this work, we show that selective ion solvation can be used not only to
tune the microphase separation of neutral block copolymers or
polyelectrolytes, but also to induce such transitions in neutral homopolymer
blends. We develop a mean-field theory for salt-doped neutral homopolymer
blends. The theory includes the Born solvation and electrostatic interactions
of doping salt ions in addition to a free energy for neat homopolymer blends.
29, 22 A key feature of the theory is the heterogeneous dielectric constant,
which depends on the local polymer fraction. With this theory, we find that
under favorable conditions, the selective solvation of doped ions in domains
with high dielectric constant causes a microscopic phase transition, in order
to reduce the loss of translational entropy of ions.

##  2 Model and Theory

We consider binary blends of homopolymers A and B doped with salts. The degree
of polymerization of the two polymers are $N_{\rm A}$ and $N_{\rm B}$. For
simplicity, we only consider the case with one type of salt containing one
cation species ($+$) and one anion species ($-$). The valencies of the cation
and anion are $z_{+}$ and $z_{-}$, respectively. In a system containing
$n_{\rm A}$ chains of polymer A, $n_{\rm B}$ chains of polymer B, $n_{+}$
cations, and $n_{-}$ anions, we can define the microscopic volume fraction
$\hat{\phi}_{i}$ (and number density $\hat{\rho}_{i}$) of each component as,

$\displaystyle\hat{\phi}_{\rm A}(\bf{r})$ $\displaystyle=v_{\rm
A}\hat{\rho}_{\rm A}({\bf{r}})=v_{\rm A}\sum_{j=1}^{n_{\rm A}}\int_{0}^{N_{\rm
A}}{\rm d}s\,\delta\left({\bf{r}}-{\bf{r}}_{j}^{\rm A}(s)\right),$ (1)
$\displaystyle\hat{\phi}_{\rm B}(\bf{r})$ $\displaystyle=v_{\rm
B}\hat{\rho}_{\rm B}({\bf{r}})=v_{\rm B}\sum_{j=1}^{n_{\rm B}}\int_{0}^{N_{\rm
B}}{\rm d}s\,\delta\left({\bf{r}}-{\bf{r}}_{j}^{\rm B}(s)\right),$ (2)
$\displaystyle\hat{\phi}_{\rm+}(\bf{r})$
$\displaystyle=v_{\rm+}\hat{\rho}_{\rm+}({\bf{r}})=v_{+}\sum_{j=1}^{n_{+}}\delta\left({\bf{r}}-{\bf{r}}_{j}^{+}\right),$
(3) $\displaystyle\hat{\phi}_{\rm-}(\bf{r})$
$\displaystyle=v_{\rm-}\hat{\rho}_{\rm-}({\bf{r}})=v_{-}\sum_{j=1}^{n_{-}}\delta\left({\bf{r}}-{\bf{r}}_{j}^{-}\right).$
(4)

Here, $v_{\alpha}$ with $\alpha\in\\{{\rm A},{\rm B},+,-\\}$ are the reference
volume s for each component. The contour curves ${\bf{r}}_{j}^{p}(s)$, with
$p\in\\{{\rm A},{\rm B}\\}$, represent the conformation of chain $j$, in which
$s$ is the contour variable for monomers. By writing this, we treat the
polymer as a continuous Gaussian chain. The positions of cation and anion are
denoted as ${\bf{r}}_{j}^{+}$ and ${\bf{r}}_{j}^{-}$, respectively. The
microscopic density is evaluated using the Dirac delta function
$\delta(\bf{r})$, which is normalized and vanishes unless the argument equals
$0$. In addition, the salt-doping level is conventionally quantified by the
ratio between the number of cations and that of monomers of polymer with high
dielectric constant, $r=n_{+}/(n_{\rm A}N_{\rm A})$.

To describe interaction in the blends, we use a minimal Hamiltonian that is
capable of reproducing experimental phase diagrams of salt-doped diblock
copolymer. 29, 30, 22 The Hamiltonian is composed of four terms and written
explicitly as,

$\displaystyle\beta\cal H=$ $\displaystyle\beta({\cal H}^{\rm id}+U^{\rm
FH}+U^{\rm B}+U^{\rm C})$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{n_{\rm
A}}\int_{0}^{N_{\rm A}}\frac{3}{2b_{\rm A}^{2}}\left(\frac{{\rm
d}{{\bf{r}}}_{j}^{\rm A}(s)}{{\rm d}s}\right)^{2}{\rm d}s$
$\displaystyle+\sum_{j=1}^{n_{\rm B}}\int_{0}^{N_{\rm B}}\frac{3}{2b_{\rm
B}^{2}}\left(\frac{{\rm d}{{\bf{r}}}_{j}^{\rm B}(s)}{{\rm d}s}\right)^{2}{\rm
d}s$ $\displaystyle+\frac{1}{v_{0}}\int{\rm d}{{\bf{r}}}\chi_{\rm
AB}\hat{\phi}_{\rm A}({{\bf{r}}})\hat{\phi}_{\rm B}({{\bf{r}}})$
$\displaystyle+\int{\rm d}{{\bf{r}}}\frac{l_{0}}{2\hat{\epsilon}_{\rm
r}(\bf{r})}\left(\frac{\hat{\phi}_{+}({{\bf{r}}})}{v_{+}a_{+}}+\frac{\hat{\phi}_{-}({{\bf{r}}})}{v_{-}a_{-}}\right)$
$\displaystyle+\frac{1}{2}\iint{\rm d}{{\bf{r}}}\,{\rm
d}{{\bf{r}}}^{\prime}\,\hat{\rho}_{\rm
q}({{\bf{r}}})g({{\bf{r}}},{{\bf{r}}}^{\prime})\hat{\rho}_{\rm
q}({{\bf{r}}}^{\prime}).$ (5)

Here, $\cal H^{\rm id}$ is the Hamiltonian of ideal Gaussian chains that
accounts for the conformational statistics 31. $U^{\rm FH}$ is the Flory-
Huggins interaction between two types of monomers, and $\chi_{\rm AB}$ is
defined on a per-reference volume ($v_{0}$) basis.

The third term is for ion solvation. The interaction between ions and polymers
is included in solvation free energy $U^{\rm B}$ approximated using the Born
solvation model,32 and no dispersion interaction among ions is considered. The
terms $l_{0}\equiv e^{2}/4\pi\epsilon_{0}k_{\rm B}T$, $v_{i}$, and $a_{i}$ are
the vacuum Bjerrum length, ion volume, and ion diameter, respectively. The
dielectric constant $\hat{\epsilon}_{\rm r}(\bf{r})$ is inhomogeneous and
depends on the polymer composition at $\bf{r}$. In this work, we use a
volumetric mixing rule for local dielectric constant, $\hat{\epsilon}_{\rm
r}(\bf{r})=(\hat{\phi}_{\rm A}({{\bf{r}}})\epsilon_{\rm r,A}+\hat{\phi}_{\rm
B}({{\bf{r}}}))\epsilon_{\rm r,B})/(\hat{\phi}_{\rm
A}({{\bf{r}}})+\hat{\phi}_{\rm B}({{\bf{r}}}))$, where $\epsilon_{\rm r,A}$
and $\epsilon_{\rm r,B}$ are dielectric constant in pure homopolymer A and B.
The ionic contributions to the dielectric constant are neglected, as we only
consider situations with dilute salt contents.

The last term $U^{\rm C}$ is the Coulombic interaction of the net charge
distribution, $\hat{\rho}_{\rm q}({\bf{r}})\equiv
z_{+}\hat{\rho}_{+}({\bf{r}})+z_{-}\hat{\rho}_{-}({\bf{r}})=z_{+}\hat{\phi}_{+}({\bf{r}})/v_{+}+z_{-}\hat{\phi}_{-}({\bf{r}})/v_{-}$,
in which $g({{\bf{r}}},{{\bf{r}}}^{\prime})$ is the Green’s function for
Poisson’s equation with an inhomogeneous dielectric constant profile,

$\displaystyle-\frac{1}{4\pi l_{0}}\nabla\cdot\epsilon_{\rm r}({\bf{r}})\nabla
g({{\bf{r}}},{{\bf{r}}}^{\prime})=\delta^{3}({{\bf{r}}}-{{\bf{r}}}^{\prime}).$
(6)

A key feature of our model related to Born and Coulomb terms is that the
dielectric constant is inhomogeneous and depends on local polymer compositions
$\phi_{\alpha}({\bf{r}})$, $\epsilon_{\rm r}({\bf{r}})=f[\phi_{\rm
A}(\bf{r}),\phi_{\rm B}(\bf{r}))]$. As will become clear in the following, the
selective ion solvation drives the microphase separation. It is therefore
necessary to point out that the Born solvation free energy of an ion of
species $\alpha$, $\sim 1/(a_{\alpha}\epsilon_{\rm r}({\bf{r}}))$, is
inversely proportional to ion radius $a_{\alpha}$ and local dielectric
constant $\epsilon_{\rm r}({\bf{r}})$. By this term alone, we may deduce that
ions prefer to stay in domains with high dielectric constant and that the
behavior of smaller ion is more susceptible to the solvation effects.

Following the standard field theory procedures, 7, 31, 22 which are detailed
in Supplementary Information, we obtain the free energy as functionals of
composition fields of all components, $\phi_{\alpha}({{\bf{r}}})$ with
$\alpha~{}\in~{}\\{\rm A,B,+,-\\}$. In the disordered phase, the system is
uniform and the composition fields are constant,
$\phi_{\alpha}({{\bf{r}}})=\bar{\phi}_{\alpha}$. Therefore the free energy of
the disordered phase can be written explicitly and we choose it as a reference
state. For any composition fluctuations around the disordered phase, we can
express the free energy change as Taylor expansion in terms of composition
differences,
${\rm\delta}\phi_{\alpha}({{\bf{r}}})\equiv\phi_{\alpha}({{\bf{r}}})-\bar{\phi}_{\alpha}$.
The expansion in Fourier space is written formally as,

$\displaystyle\Delta{\cal F}=$ $\displaystyle\ \Delta{\cal
F}^{(2)}+\Delta{\cal F}^{(3)}+\Delta{\cal F}^{(4)}$ $\displaystyle=$
$\displaystyle\
\frac{1}{2V^{2}}\sum_{{{\bf{q}}}}\sum_{\alpha,\beta}\Gamma^{(2)}_{\alpha\beta}({{\bf{q}}},-{{\bf{q}}})\delta\phi_{\alpha}({{\bf{q}}})\delta\phi_{\beta}(-{{\bf{q}}})$
(7)
$\displaystyle+\frac{1}{3!V^{3}}\\!\sum_{{{\bf{q}}}_{1},{{\bf{q}}}_{2}}\sum_{\alpha,\beta,\gamma}\Gamma^{(3)}_{\alpha\beta\gamma}({{\bf{q}}}_{1},{{\bf{q}}}_{2},-{{\bf{q}}}_{1}-{{\bf{q}}}_{2})\delta\phi_{\alpha}({{\bf{q}}}_{1})\delta\phi_{\beta}({{\bf{q}}}_{2})\delta\phi_{\gamma}(-{{\bf{q}}}_{1}-{{\bf{q}}}_{2})+\cdots,$

where $\alpha,\beta,\gamma,...$ denote species and ${{\bf{q}}}$ is wavevector.
In our convention of Fourier transform, $f({{\bf{q}}})=\int{\rm
d}{{\bf{r}}}f({{\bf{r}}})\exp(-{\rm i}{{\bf{q}}}\cdot{{\bf{r}}}).$ The vertex
function, $\Gamma^{(n)}$, is $n$-th order functional derivatives of free
energy change with respect to density deviations, evaluated at
$\phi({{\bf{r}}})=\bar{\phi}$, or equivalently
$\phi({{\bf{q}}})=\bar{\phi}\delta({{\bf{q}}})$.

The secondary expansion coefficient $\Gamma^{(2)}$ is a $4{\times}4$ matrix
and its inverse is the structure factor measured in scattering experiments. It
contains information about the nature of phase separation. Because of the
isotropy of the homogeneous phase, the elements of $\Gamma^{(2)}$ are
functions of $q=|{{\bf{q}}}|$, the wavenumber of density fluctuations. If we
assume the blends are incompressible, the composition fluctuations should sum
to zero, $\sum_{\alpha}\delta\phi_{\alpha}({{\bf{q}}})=0$. This means the
composition fluctuation vector $\delta\phi=[\delta\phi_{\rm A},\delta\phi_{\rm
B},\delta\phi_{\rm+},\delta\phi_{\rm-}]^{\rm T}$ is orthogonal to a
compression mode ${\bf{\varepsilon}}=\frac{1}{2}[1,1,1,1]^{\rm T}$ and only
exist in the orthogonal complement of the subspace of ${\bf{\varepsilon}}$. We
therefore span the incompressible subspace by choosing an orthonormal basis
set consisting of the following modes,
${\bf{e}}^{(1)}=\frac{1}{2}[1,1,-1,-1]^{\rm T}$,
${\bf{e}}^{(2)}=\frac{1}{2}[1,-1,1,-1]^{\rm T}$, and
${\bf{e}}^{(3)}=\frac{1}{2}[1,-1,-1,1]^{\rm T}$. After contracting the
composition fluctuations to the incompressible subspace, we reduce the
$4{\times}4$ matrix of $\Gamma^{(2)}(q)$ to a $3{\times}3$ matrix of
$\gamma^{(2)}(q)$.

$\gamma^{(2)}(q)$ should have three eigenvalues
$\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}$. In the disordered phase,
$\lambda_{1}(q)>0$, meaning that the free energy is concave up with respect to
any composition fluctuation. The stability limit (i.e. the spinodal limit) of
the disordered phase is given by the condition $\lambda_{1}(q^{*})=0$, where
$q^{*}$ is a critical wavenumber that minimizes $\lambda_{1}$. The nature of
the phase separation is indicated by the value of $q^{*}$. If $q^{*}>0$, the
phase transition is microscopic and the ordered phase has a characteristic
domain size of $D=2\pi/q^{*}$. If $q^{*}=0$, the characteristic domain size
diverges and the phase separation is macroscopic.

##  3 Results and Discussions

###  3.1 Macrophase vs microphase separation

Figure 1:  Spectral analysis of the quadratic vertex function of salt-doped
homopolymer blends, with $\chi_{\rm AB}=0$ and $r=0.01$. (a) Minimum
eigenvalue of quadratic coefficients ($\lambda_{1}$) as function of wavevector
magnitude ($q$). The critical wavevector magnitude ($q^{*}$) corresponds to
the location of the minimum of $\lambda_{1}$. The curves were shifted
vertically so that the limit values at low $q$ were 0. (b) Polymeric
components of eigenvector corresponding to $\lambda_{1}$, with
$l_{0}=15.0\,{\rm nm}$. (c) Ionic components of eigenvector corresponding to
$\lambda_{1}$, with $l_{0}=15.0\,{\rm nm}$. (d-e) Decomposition of
$\lambda_{1}$ into contributions of ideal entropy, Born solvation, and Coulomb
interaction for $l_{0}=5.0\,{\rm nm}$ (d) and $l_{0}=15.0\,{\rm nm}$ (e).

The competition between macroscopic and microscopic phase separation is
governed by the strength of electrostatic interaction. We control the
electrostatic contributions by tuning the vacuum Bjerrum length $l_{0}$. The
Bjerrum length at room temperature is about 56 nm in vacuum and 0.7 nm in
water. The (relative) dielectric constant for polymers is usually small,
between 2 to 10, which corresponds to Bjerrum length of about 1.5 to 30 nm. A
small Bjerrum length indicates strong electrostatic screening and weak
electrostatic interactions.

We first consider a symmetric case with $v_{\rm A}=v_{\rm B}=v_{0}$ and
$N_{\rm A}=N_{\rm B}=N$. Fig. 1a shows the minimum eigenvalue $\lambda_{1}$ of
$\gamma^{(2)}$ versus $q$ for different Bjerrum lengths. When $l_{0}$ is small
($l_{0}=0{\text{ or }}5.0\,{\rm nm}$), $\lambda_{1}$ increases monotonically
from $q=0$. The minimum of $\lambda_{1}(q)$ locates at $q^{*}=0$. Therefore
only macrophase separation is possible in this regime. The limiting case of
low electrostatic interaction is $l_{0}=0\,{\rm nm}$, where the electrostatic
contribution to the free energy is essentially zero. This can also be seen
from the expressions of the Born solvation energy, and Coulomb interaction
energy (eq. 2). In this limit, the ions act as non-selective neutral solvents
and it is well-known that only macroscopic separation can occur.

As the value of $l_{0}$ increases, $\lambda_{1}$ starts to change non-
monotonically with $q$ (Fig. 1a). For $q<0.02\,{\rm nm}^{-1}$, $\lambda_{1}$
remains flat, resembling the cases with $l_{0}\leq 5\,{\rm nm}^{-1}$. However,
beyond $q=0.02\,{\rm nm}^{-1}$, the value of $\lambda_{1}$ first decreases
before finally increasing unboundedly. This non-monotonic behavior results in
a finite critical wavevector $q^{*}>0$, signifying a microscopic phase
separation. The value of $q^{*}$ increases and the non-monotonic shape becomes
more pronounced as $l_{0}$ increases.

Figure 2:  Schematic illustration of the competition between macrophase
separation (a) and microphase separation (b).

To understand the nature of the instability at nonzero $q^{*}$, we examine the
components of the critical mode, i.e., the eigenvector corresponding to the
eigenvalue $\lambda_{1}$. The polymeric and ionic components of the critical
modes are plotted in Fig. 1b and c, respectively. To evaluate the volume
fractions for different components, we have used
$\delta\phi^{*}=\sum_{i=1,2,3}\varphi_{i}^{*}{\bf{e}}^{(i)}$. The results
shown in Fig. 1b are calculated from $\delta\phi^{*}$ by re-scaling each
components with corresponding bead volume, i.e.,
$\delta\rho^{*}_{\alpha}=\delta\phi^{*}_{\alpha}/v_{i}$ for $\alpha\in\\{{\rm
A,B,+,-}\\}$.

When $q$ is small ($q<0.02\,{\rm nm}^{-1}$), $\delta\rho^{*}_{\alpha}$ remain
approximately constant. $\delta\rho^{*}_{\rm A}$ and $\delta\rho^{*}_{\rm B}$
have different signs as they tend to separate from each other. The amplitude
$\delta\rho^{*}_{\rm A}$ is lower than $\delta\rho^{*}_{\rm B}$, which is
compensated by the enrichment of ions in the A domain. The difference between
cation and anion number density fluctuation mode ($\delta\rho_{\rm
q}=\delta\rho^{*}_{+}-\delta\rho^{*}_{-}$) measures the degree of net charge
separation. In the regime of $q<0.02\,{\rm nm}^{-1}$, $\delta\rho^{*}_{+}$ and
$\delta\rho^{*}_{-}$ are essentially identical, implying the absence of charge
separation. This is consistent with the expectation that charge separation at
large length scale requires high energy cost.

In the high-$q$ regime, the magnitudes of $\delta\rho^{*}_{\rm A}$ and
$\delta\rho^{*}_{+}$ increase, whereas those of $\delta\rho^{*}_{\rm B}$ and
$\delta\rho^{*}_{-}$ decrease. This suggests that more cations are distributed
in the A-rich domain while fewer anions reside in the A-rich domain. It is
energetically favorable as the small diameter of cation affords a high
(absolute value) Born solvation energy that overcompensates the loss of Born
solvation energy from the anions that transferred to the B-rich domain. As
$\delta\rho^{*}_{+}$ and $\delta\rho^{*}_{-}$ split, a net charge distribution
also develops. The length scale at which charge separation begins to appear is
about $\rm 60\,nm$, calculated from $2\pi/q^{*}$ by setting $q^{*}=0.1\,{\rm
nm}^{-1}$, which is well within the range of Coulomb interaction.

To gain more insights into the origin of charge separation, we decompose
$\lambda_{1}$ into contributions from the ideal, Born and Coulomb parts (Figs.
1d, e). The Flory-Huggins term is irrelevant because it is $q$-independent and
we set $\chi_{\rm AB}=0$. All these terms remain constant in the small $q$
regime. The ideal part is similar for cases with $l_{0}=5\,{\rm nm}$ and
$l_{0}=15\,{\rm nm}$. (Note that they are not identical, as their critical
composition fluctuations differ slightly.) In the high-$q$ regime where charge
separation takes place (Fig. 1c), we find that the decrease in the free energy
is dominated by the decrease in the Born term (Figs. 1d,e). Furthermore, the
Coulomb energy increases as the net charge developed. At even higher $q$
values, the Coulomb contributions decrease. This is because the total Coulomb
energy decreases as the length scale of charge separation decreases.

The competition between the Born term and the ideal term contributes to the
non-monotonic trend in $\lambda_{1}$ at large $q$ values. The value of $l_{0}$
controls the magnitude of Born and Coulomb terms. Only for sufficiently large
$l_{0}$ values, can the Born solvation term dwarfs the ideal term that causes
$\lambda_{1}$ to increase, resulting in a well-defined minimum.

The above discussions demonstrate that strong electrostatic interaction can
trigger the microphase separation in homopolymer blends, as a result of the
competition among multiple factors. The Born solvation promotes the
localization of ions inside domains with higher dielectric permittivity, which
drives phase separation so that the high permittivity domains can be formed.
When this happens, both cations and anions tend to reside inside the high-
permittivity phase, at the cost of the loss in translational entropy. One way
to alleviate this frustration is to have smaller cations reside inside the
high-permittivity domain, while allowing the anions to leak into the low-
permittivity domains. However, this scenario implies macroscopic charge
separation, which is energetically unfavorable: let the length scale for
charge separation be $D$, then the magnitude of net charge $Q$ is proportional
to $D$, and the Coulomb energy is of order $Q^{2}l_{0}/D\propto l_{0}D$, which
blows up as $D\to\infty$. The compromise leads to the emergence of a finite
domain size $D$, or $q^{*}$ value, when electrostatic interaction is
sufficiently strong. The argument is illustrated in Fig. 2. The crossover from
macrophase separation to microphase separation is identified as the Lifshitz
point. 12 The variation of the Lifshitz point and its dependence on model
parameters are explored below.

###  3.2 Lifshitz point

Figure 3:  Dependence of critical $l_{0}$ on salt-doping amounts. (a) Critical
wavevector magnitude ($q^{*}$) versus $l_{0}$ for different $r$ values.
$q^{*}$ corresponds to the location of the minimum of $\lambda_{1}$ in Fig.
1a. (b) Salt-exchange degree as quantified by
$\delta\rho^{*}_{+}-\delta\rho^{*}_{-}$ versus wavevector magnitude for
different $r$ values. (c) Decomposition of $\lambda_{1}$ into contributions of
ideal entropy, Born solvation, and Coulomb interaction for two $r$ values with
$l_{0}=9.8\,{\rm nm}$.

When the dielectric ratio $\epsilon_{\rm A}/\epsilon_{\rm B}$ is fixed, the
vacuum Bjerrum length $l_{0}$ is the primary factor determining the transition
from macrophase to microphase separation. This is demonstrated in Fig. 3a,
which shows how $l_{0}$ influences $q^{*}$, the wave number where the minimum
of $\lambda_{1}$ locates (Fig. 1a), for different salt-doping levels $r$. The
Lifshitz points, where $q^{*}$ first becomes nonzero, are located near
$l_{0}=\rm 9.7nm$. With the range of salt doping levels explored, from 0.001
to 0.05, the location of Lifshitz point barely moves, as shown by the inset of
Fig. 3a.

The weak dependence of the Lifshitz point on the doping level $r$ stems from
the insensitivity of the degree of charge separation to $r$. Fig. 3b compares
the degree of charge separation, quantified by the difference between the
cationic and anionic components of the critical mode, for different doping
levels. The results for different doping levels are nearly identical when
$q<0.1\,{\rm nm^{-1}}$, which results in nearly identical contributions from
the Born solvation term. In fact, for $q<0.1\,{\rm nm^{-1}}$, the contribution
of the Born term to $\lambda_{1}$ are almost indistinguishable for $r=0.01$
and $r=0.05$ (Fig. 3c). This is precisely the range at which $q^{*}$ rises
from 0 to finite values, which rationalizes why the location of the Lifshitz
point is insensitive to the value of $r$.

The magnitude of $q^{*}$, i.e., the characteristic domain size, does depend on
the doping level $r$ for $l_{0}>10\,{\rm nm}$. The higher the doping level,
the greater the $q^{*}$ value, as seen from Fig. 3a. Such difference is also
related to the progressively greater difference in the degree of charge
separation for larger $q^{*}$ values, shown in Fig. 3b.

###  3.3 Spinodal curves

The above sections address the conditions for microphase separation. Here we
examine the stability limit of the homogeneous phase, by evaluating the
spinodal curves, which is found by requiring that $\lambda_{1}(q^{*})=0$. Here
we recall that $\lambda_{1}$ is the minimum eigenvalue of the quadratic
expansion coefficient $\gamma^{(2)}$ and $q^{*}$ is the critical wavevector
that gives the minimum of $\lambda_{1}(q)$. The Flory-Huggins term, which was
ignored in above sections by setting $\chi_{\rm AB}=0$, contributes
$\delta\phi_{\rm A}\delta\phi_{\rm B}\chi_{\rm AB}$ to $\lambda_{1}$ for all
$q$ 22, where $\delta\phi_{\alpha}$ is the $\alpha$-component of the
eigenvector corresponding to $\lambda_{1}$. Therefore, changing $\chi_{\rm
AB}$ effectively shifts the curves of $\lambda_{1}(q)$ vertically, and the
spinodal can be readily found by requiring that the minimum of
$\lambda_{1}(q)$ vanishes.

Figure 4:  Spinodal behavior of salt-doped homopolymer blends. (a) Spinodal
$\chi_{\rm AB}$ values versus doping degree ($r$) for different $l_{0}$
values. (b) Critical wavevector magnitude ($q^{*}$) versus $r$ for different
$l_{0}$ values on the spinodal line.

The spinodal values of $\chi_{\rm AB}$ versus $r$, for several $l_{0}$ values,
are plotted in Fig. 4a. All the spinodal curves converge to $Nv_{0}\chi_{\rm
AB}=2.0$ at $r=0$, the well-known limit for symmetric binary homopolymer
blends 33. When $l_{0}=0\,{\rm nm}$, the ions became essentially non-selective
solvents. The critical composition fluctuation is proportional to
$[1,-1,0,0]^{\rm T}$ (data not shown), the same as neat symmetric homopolymer
blends. The value of $\chi_{\rm AB}$ at the spinodal increases slightly with
$r$, because of the dilution effects of non-selective solvents (the increment
is minor for the range of $r$ is narrow). For nonzero $l_{0}$ values, the
value of $\chi_{\rm AB}$ decreases with $r$, and the change is more
substantial for larger $l_{0}$. This corroborates the notion that salt-doping
can increase the effective $\chi$ parameter, as was first proposed by Wang 25.
However, Wang mainly considered the macrophase separation, whereas our focus
is the emergence of microphase separation.

This point is highlighted by the critical wavenumber at the spinodal shown in
Fig. 4b. Because the Flory-Huggins term does not alter the $q$-dependence of
$\gamma_{2}$, the information contained in Fig. 4c is the same as that in Fig.
3b. Taken together, these results suggest that the crossover between macro-
and micro-phase separation occurs slightly above $l_{0}=5\,\rm nm$.

###  3.4 Other factors

Using symmetric homopolymer blends as a model system, we have demonstrated the
possibility of microphase separation upon salt-doping and studied the
influence of salt content ($r$) and electrostatic interaction strength
($l_{0}$) in the above. In the following, we explore the influences of three
key molecular properties: ion solvation radius, dielectric contrast, and
polymer composition.

####  3.4.1 Ion solvation radius

The driving force for microphase separation in the system of interests is the
need to simultaneously lower the Born solvation free energy, reduce the
entropy loss of localized ions, and minimize the Coulomb energy. The Born
solvation energy, eq. (2), is inverse to the ion solvation radius.
Accordingly, we expect that the difference in ion solvation radii of cations
and anions to be important. The parameterization for ion radii follows our
recent study 30, in which ionic volumes $v_{+}$ and $v_{-}$ are kept constant
and ion radii $a_{+}$ and $a_{-}$ are varied.

Several combinations of ion solvation radii are explored, and the critical
wavenumbers are plotted vs $l_{0}$ in Fig. 5a. The symmetric case with
$a_{+}=a_{-}=0.1\,{\rm nm}$ is indifferent to the selective cation or anion
solvation and charge separation cannot occur, so there is no microphase
separation for all $l_{0}$ values. As $a_{-}$ increases, the discrepancy
between cation and anion size grows, and the selective solvation of cations in
the high-permittivity domain is stronger. We found that the value of $l_{0}$
at the Lifshitz point decreases from $\sim$20 nm to $\sim$7 nm as $a_{-}$
increases from $\rm 0.2\,nm$ to $\rm 0.8\,nm$, which supports our argument
that a large difference between ion radii promotes microphase separation.

Additionally, when the solvation radii are doubled while the ratio
$a_{+}/a_{-}$ is kept constant, the microphase separation is found to
disappear. Doubling both ion solvation size act effectively halves the Born
term. This reduction cannot be offset by simply doubling $l_{0}$, as doubling
$l_{0}$ also increases energy cost from the Coulomb term, which weakens the
energy gained from the Born term, making microphase separation impossible.

Fig. 5b presents the spinodal curves for different combinations of ion
solvation radii. The trends of these curves are similar: the value of
$\chi_{\rm AB}$ decreases as $l_{0}$ increases. There is, however, a weak
increment when $l_{0}$ is small. This is the entropy regime of salt-doping,
where adding ions stabilizes the homogeneous phase in order to achieve higher
translational entropy 22. Because no microphase separation is expected in the
entropy regime, we shall focus on the solvation regime below, where adding
ions de-stabilizes the homogeneous phase. The decrease of $\chi_{\rm AB}$ with
increasing $l_{0}$ depends only on the solvation size of ions and is
insensitive to the ratio of the ion solvation radii. Smaller ions decreases
$\chi_{\rm AB}$ more effectively, which is consistent with the findings of
Nakamura et al. 26. We note that such dependence is irrespective of the nature
of phase separation, macroscopic or microscopic, consistent with the findings
in Fig. 4.

Figure 5:  Effects of ionic solvation radii on the phase behavior of salt-
doped homopolymer blends. Dependence of critical wavevector magnitude $q^{*}$
(a) and spinodal $\chi_{\rm AB}$ on $l_{0}$ (b) on $l_{0}$ for different
combinations of ionic solvation radii ($a_{+}$ and $a_{-}$). Doping degree is
$r=0.01$. The unit of ion solvation radius is nm.

####  3.4.2 Dielectric contrast

Figure 6:  Effects of dielectric constant of polymer on the phase behavior of
salt-doped homopolymer blends. Dependence of critical wavevector magnitude
$q^{*}$ (a) and spinodal $\chi_{\rm AB}$ on $l_{0}$ (b) on $l_{0}$ for
different combinations of polymer dielectric constant ($\epsilon_{\rm A}$ and
$\epsilon_{\rm B}$). Doping degree is $r=0.01$.

The effects of dielectric contrast is analogous to that of the difference in
ion solvation radii. The values of $q^{*}$ and the spinodal curves are shown
in Fig. 6a, for different sets of dielectric permittivities. In the absence of
dielectric contrast ($\epsilon_{\rm A}=\epsilon_{\rm B}=1$ in Fig. 6a), no
microscopic separation is found. Because the net charge density vanishes, the
ions act effectively as neutral solvents.

The microphase separation is possible with sufficient dielectric contrast.
When we fix the dielectric constant of one polymer ($\epsilon_{\rm B}=1$), and
increase $\epsilon_{\rm A}$ from 2 to 4, the value of $l_{0}$ at the Lifshitz
point decreases from $\sim$7.5 nm to $\sim$3.7 nm (Fig. 6a). However, the
spinodal $\chi_{\rm AB}$ does not change significantly from $\epsilon_{\rm
A}=2$ to $\epsilon_{\rm B}=4$ (Fig. 6b). The quantitative dependence should be
sensitive to the average rule chosen for the dielectric permittivity of
mixtures22.

With a constant $\epsilon_{\rm A}/\epsilon_{\rm B}$, increasing both
dielectric constants shifts the value of $l_{0}$ at the Lifshitz point to
larger values. Both Born and Coulomb terms scale inversely with local
dielectric constant. Doubling both dielectric constants reduces both terms by
a factor of two. This is equivalent to reducing the Bjerrum length by a factor
of two. As shown in Fig. 6a, when we change the dielectric constant from
$(\epsilon_{\rm A},\epsilon_{\rm B})=(4,1)$ to $(8,2)$, the critical $l_{0}$
changes from $\sim$3.7 nm to $\sim$7.5 nm. Further change to $(16,4)$ gives a
critical $l_{0}$ of $\sim$15.0 nm.

####  3.4.3 Polymer composition

Figure 7:  Effects of polymer fraction on the phase behavior of salt-doped
homopolymer blends. Dependence of critical wavevector magnitude $q^{*}$ (a)
and spinodal $\chi_{\rm AB}$ on $l_{0}$ (b) on $l_{0}$ for different fractions
of A-polymer $f_{\rm A}$. Doping degree is $r=0.01$.

Fig. 7a shows that a higher fraction of high-permittivity component ($f_{\rm
A}$) requires a larger $l_{0}$ to attain microphase separation. This is
related to the translational entropy of ions. When macroscopic phase
separation occurs, ions are distributed more preferably in the A-rich phase. A
small $f_{\rm A}$ means the space available for ions is more restricted and
hence a large loss of ionic translational entropy. The free energy gain is
then more likely to drive a transition from macrophase separation to
microphase separation.

A similar trend is found for how $f_{\rm A}$ affects the spinodal curves (Fig.
7b). When $l_{0}=0\,{\rm nm}$, spinodal $\chi_{\rm AB}$ values are the same as
neat polymer blends, 33 which is symmetric from $f_{\rm A}=0.5$. As $l_{0}$
increases, the value of $\chi_{\rm AB}$ at the spinodal decreases as expected
but the changing rate is steeper for smaller $f_{\rm A}$ values. When $f_{\rm
A}$ is large, the average dielectric constant in homogeneous phase is closer
to $\epsilon_{\rm A}$, and the gain in the Born energy from ion localization
is smaller, weakening the driving force for phase separation (either
macroscopic or microscopic).

###  3.5 Why microphase separation induced by salt-doping seldom seen
experimentally?

The region of microphase separation identified in our theory is rather broad
(Fig. 4). However, such salt-doping-induced microphase separation in neutral
homopolymer blends has not been reported. This may be attributed to the over-
estimation of the solvation free energy by the simple Born expression. In
reality, many factors can mitigate ion solvation free energy such as ion-
paring, ion-clustering, and composition fluctuations 22. Other treatments of
ion solvation such as dipolar SCFT, 34, 35, 36 liquid state theory corrected
SCFT, 19, 21, 18 or classical density functional theory 37, 38, 39, 40 may
help improve the accuracy of theoretical predictions but will not change our
conclusions qualitatively. Formally weakening the Born solvation term implies
that a higher $l_{0}$ value is required to induce microphase separation. To
fulfill this condition, working with low-permittivity polymers is desirable.
It remains to be seen, if tuning the ion solvation raii, dielectric contrast,
and blending composition, as we explored above, can lead to experimental
realization of microphase separation in doped polymer blends.

##  4 Conclusions

We developed a weak segregation theory for salt-doped neutral homopolymer
blends. The model contains terms that describe the solvation free energy of
ions and Coulombic interaction in addition to standard terms for neutral
polymer blends, which has previously been used to analyze experimental phase
diagrams 29, 30. Our main result is that microphase separation may be induced
when selective solvation is sufficiently strong.

The microphase separation permits local charge separation, with cations
preferentially residing in the high-permittivity domains, whereas anions
resides in the low-permittivity domains. The net result is that the Born
solvation free energy is lower, ion entropy loss is reduced, and the Coulomb
energy is minimal. The threshold value of $l_{0}$ at the crossover from the
macro- to micro-phase separation is not sensitive to the amount of salt added.
However, the doping level changes the critical $q^{*}$ significantly, which
provides a facile means to tune the domain size of microphases.

We further probed how three key material properties, i.e., ion solvation
radius, dielectric contrast, polymer fraction affect our results on microphase
separation. It is found that larger solvation radius difference, larger
dielectric contrast, and lower high-permittivity polymer composition, all
favor the formation of microphases. Such observations may facilitate the
experimental exploration of salt-induced microphase separation in polymer
blends.

This work focused on the competition of microphase and macrophase separation
and the stability limit of the homogeneous phase. The complete phase diagrams
for salt-doped polymer blends, including the standard set of microphases (BCC,
hexagonal, gyroid, etc.) 7, 29, 22 will be presented in the future.

This research has been supported by the Recruitment Program of Guangdong
(grant no. 2016ZT06C322). J.Q. is supported by the National Science Foundation
CAREER Award through DMR-1846547.

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# Noise Audits Improve Moral Foundation Classification

Negar Mokhberian♥ Frederic R. Hopp♠ Bahareh Harandizadeh♥ Fred Morstatter♥
Kristina Lerman♥ ♥Information Sciences Institute,
University of Southern California
{nmokhber, harandiz, fredmors<EMAIL_ADDRESS>♠Amsterdam School of
Communication Research,
University of Amsterdam
<EMAIL_ADDRESS>
###### Abstract

Morality plays an important role in culture, identity, and emotion. Recent
advances in natural language processing have shown that it is possible to
classify moral values expressed in text at scale. Morality classification
relies on human annotators to label the moral expressions in text, which
provides training data to achieve state-of-the-art performance. However, these
annotations are inherently subjective and some of the instances are hard to
classify, resulting in noisy annotations due to error or lack of agreement.
The presence of noise in training data harms the classifier’s ability to
accurately recognize moral foundations from text. We propose two metrics to
audit the noise of annotations. The first metric is entropy of instance
labels, which is a proxy measure of annotator disagreement about how the
instance should be labeled. The second metric is the silhouette coefficient of
a label assigned by an annotator to an instance. This metric leverages the
idea that instances with the same label should have similar latent
representations, and deviations from collective judgments are indicative of
errors. Our experiments on three widely used moral foundations datasets show
that removing noisy annotations based on the proposed metrics improves
classification performance.111Our code can be found at:
https://github.com/negar-mokhberian/noise-audits

###### Index Terms:

crowd-sourcing, annotation, ambiguity, subjective annotations, noisy
annotations

##  I Introduction

Moral foundations theory (MFT) [1, 2] suggests that the moral values expressed
in opinions, thoughts, and cultures can be explained by five universal, but
contextually variable moral foundations. These foundations are typically
described along bipolar dimensions: care vs. harm, fairness vs. cheating,
authority vs. subversion, loyalty vs. betrayal, and purity vs. degradation.
MFT was first introduced in social psychology and has found many applications
in political science and the social sciences. For example, moral foundations
motivate behaviors such as charitable donations [3], violent protests [4] and
social homophily [5].

The broad adoption of MFT was driven, in part, by advances in natural language
processing (NLP), which enabled researchers to quantify moral values expressed
in text, including news [6, 7, 8], political speech [9], and social media
discussions [10], at scale. Early works relied on lexicons that defined words
associated with moral virtues and vices to classify moral foundations from
text [1]. However, by neglecting semantic context in sentences, lexicon-based
approaches fail to capture the nuances of moral expression [11]. To address
this challenge, more recent approaches use large language models to capture
the moral context of text [12, 13]. These approaches leverage a text corpus
manually annotated for moral values to train language models to recognize
examples of moral language. Several such ground truth data sets exist [10, 7,
14, 15, 8].

(a) This tweet organically contains different dimensions of morality. It
contains elements of injury and police action at the same time.

(b) This instance has been annotated with “degradation”, and “non-moral” but
these labels are not related to it.

(c) This instance’s moral labels are subjective, depending on whether the
annotators support the police actions or are against it.

Figure 1: Examples of high-entropy instances in MFTC dataset. The text of the
tweet and the number of annotators selecting each moral foundation as the
label are shown.

A major challenge when constructing ground truth data for training moral
foundation classifiers is the subjectivity of individual moral judgments,
which are prone to bias and noise [16]. Figure 1 illustrates this challenge
using real examples from the Moral Foundation Twitter Corpus (MFTC) [10]. The
first tweet, “#freddiegray required spinal surgery after the police beat him.
freddie was unarmed. freddie died today,” expresses a range of moral concerns
(Fig. 1(a)). It was labeled as harm, cheating, betrayal and non-moral by two
annotators each, and authority and subversion by one annotator each. Text can
also be mislabeled due to errors or ambiguity. This is illustrated by the
second example “no justice, no peace.” (Fig. 1(b)). This instance was
annotated with degradation, and non-moral, but these labels are not related to
the tweet. The third example illustrates the subjectivity of moral judgments
(Fig. 1(c)): “#freddiegray I couldn’t be happier with arrests of those 6
officers. how scared was freddie gray it makes me sick! who r the thugs now?!”
The moral labels assigned to this instance are subjective and depend on
whether the annotators support police actions or not. As a result of these
factors, individual moral labels will be noisy. To partly control for the
variability of judgments, researchers use the label chosen by the majority of
annotators as the correct ground truth label for each instance [10, 17].
However, the question of how annotation noise affects the performance of
models learned from data has been underexplored [18, 19].

In this paper we present and compare two approaches for auditing and removing
noise from annotations used to train moral foundations classifiers. The first
approach identifies difficult instances in the ground truth data. We propose
entropy as a measure of annotator disagreement in §III-A and use it to
identify instances with little agreement that will degrade classifier
training. For each instance, we calculate entropy based on how many annotators
have selected each of the labels (examples of such distributions are shown in
Fig. 1 for three instances). By removing instances on which the annotators
disagree, we hope to create better data for training classifiers.

Our second method identifies annotations that deviate from collective
judgments of all annotators. We propose the silhouette coefficient in latent
space as a measure of label quality, which leverages the idea that similarly-
labeled instances should have similar latent representations. By removing
annotations that deviate substantially from collective judgments, we hope to
improve the quality of ground truth data.

We evaluate both approaches on three large datasets with moral annotations
[10, 7, 14] (more information of the datasets in §IV-A). We show that training
models on ground truth data from which noisiest annotations have been removed
improves morality classification. This does not stem simply from having less
data: compared to a model trained on data from which the same number of
instances were removed at random, removing the noisiest instances from data
significantly improves classification performance.

Our work primarily focuses on noise audit of subjective annotations to
identify mislabeled or difficult instances in moral foundations
classification. This could be applied in any subjective labeling setting.
Identifying moral foundations in text is a representative example of a
subjective task that we have selected for this paper. An important enabler for
our noise audit is access to the individual judgments of annotators. Hence, we
encourage the crowd-sourced annotation builders to include fine-grained
details of individual annotator judgments on instances instead of the common
practice [20] of reporting an aggregated judgment (e.g., majority label).
Including the individual judgments gives the opportunity to refine the
dataset, enhancing its utility on learning tasks.

##  II Related Work

Researchers have tried to incorporate individual annotator’s perspectives of
the subjective tasks [18]. Using multi-annotator models, they have shown that
training a separate model for each annotator and then aggregating to a
majority vote performs better than aggregating labels in the data prior to
training. However, their methodology is not practical in the cases where the
data have been crowd-sourced and there are many annotators because 1) there
are not usually enough data points per annotator to train separate modules,
and 2) it is not cost-efficient to train many separate modules.

Other earlier works have studied the difficulty of data points through
information theory [21] and disagreements in annotations [22]. A leading way
to understand the difficulty of data instances is to leverage training
dynamics [23, 24]. In other words, by observing how a classifier performs on
an specific instance through epochs. Perhaps the best exemplar of this
approach is seen in [19], which assesses each point based on the model’s
_confidence_ (average probability assigned to the correct label during
training epochs) and _variance_ (variance in probability assigned to the
correct label during training epochs). While training dynamics offer a way to
measure training difficulty on an instance, their main drawback is dependency
on the design of the model and on training. However, we try to address this
issue by proposing metrics that are only depended on the annotated dataset and
training is not a requirement for their calculation. We apply our proposed
metrics to refine moral foundations datasets before training. To the best of
our knowledge no prior work has analysed how difficult instances of moral
foundations affect classification.

(a) MFTC dataset

(b) MFNC dataset

Figure 2: Examples of moral judgments with low silhouette coefficients.
Judgments A, B are on the same text but from different annotators. In the
language model latent space this text is very close to other judgments of
Degradation in the left figure and to Authority in the right figure. In
judgment A the annotator has assigned it a label that does not match the
labels of similar texts. Because it is far from other texts with the same
label, but close to texts that have been assigned different labels, judgment A
will get a low silhouette coefficient. Judgment B is close to the other
judgments of the same label so it will get high silhouette coefficient. Our
methodology suggests removing judgment A from the training data but to keep
judgment B on the same text. In judgment C the annotator has also selected a
label that is different from the label of the other similar texts, so we
suggest to remove judgment C.

##  III Methodology

In this section we propose two metrics entropy and silhouette for identifying
noise in instance-level and judgment-level respectively. We use entropy (see
§III-A) to find out annotator disagreements on a given instance. An instance
is a piece of text that has been shown to annotators to collect their
judgments. The disagreement on an instance is high when disparate judgments
have been collected from annotators; on the other hand, the disagreement is
low when all annotators agree on the same label. We refine the datasets by
removing the high-entropy instances (instances with high disagreement on their
assigned labels) as a pre-processing step before training a classifier.

In addition, we propose using the silhouette coefficient (see §III-B) as a
fine-grained metric to quantify how a single judgment differs from other
instances of the same-label judgments. Note that unlike entropy, this metric
takes into account the text of the instance. Removing judgments we deem noisy
with this metric increases the inter-annotator agreement on a given instance.

Unlike prior work in machine learning [23, 25, 26, 19], our measures are
calculated before the model training starts and do not depend on the training
dynamics. We provide the details of one the training dynamics metrics in
§III-C and in the experiments (§IV-D) will monitor their improvement as we
filter data based on our metrics.

###  III-A Entropy at the Instance Level

We use entropy of annotations to quantify diversity of labels gathered for a
text. For a text $t_{i}$ and its multi-label annotations
$<l_{1}:c_{i1},...,l_{N}:c_{iN}>$ in which $l_{j}$ is a member of all the
labels $L=\\{l_{1},l_{2},...,l_{N}\\}$ and $c_{ij}$ is the count of annotators
who have assigned $l_{j}$ to $t_{i}$, we calculate the entropy as:

$entropy(t_{i})=-\sum_{j=1}^{N}{P(l_{j},t_{i})logP(l_{j},t_{i})}$

in which

$P(l_{j},t_{i})=\frac{c_{ij}}{\sum_{j=1}^{N}c_{ij}}.$

If all annotators agree on the same label, the entropy is zero. At the other
extreme, if every annotator gave the instance a different label, entropy has
its maximum value.

###  III-B Silhouette Coefficient at the Judgment Level

For a given judgment $x$ = ($t_{i}$, $l_{j}$, $a_{k}$) in which instance
$t_{i}$ has been assigned a label $l_{j}$ by annotator $a_{k}$, the silhouette
coefficient [27] is defined as a combination of its distance from the same-
label (intra-cluster) judgments and from other-label (inter-cluster)
judgments. Let $l_{j}$ be a member of the set of all labels
$L=\\{l_{1},l_{2},...,l_{N}\\}$. We consider all the instances assigned to the
same label to be a cluster. The intra-cluster measure $a(x)$ is defined as the
average dissimilarity of $text_{i}$ to all other texts labeled with $l_{j}$.
The inter-cluster metric is defined as:

$b(x)=min_{y\in L,y\neq l_{j}}\bar{d}(t_{i},T_{y}),$

where $\bar{d}(t_{i},T_{y})$ is the average dissimilarity of $t_{i}$ to all
texts in the dataset labeled with $y$. Finally, $a(x)$ and $b(x)$ are
aggregated as:

$silhouette(x)=\frac{b(x)-a(x)}{max\\{a(x),b(x)\\}}$

To calculate the dissimilarities ($d$), we represent the text instances in a
latent space using a language model and calculate the distance of vectors
corresponding to texts.

The silhouette coefficient thus captures the consistency of an annotator’s
label of a specific text with other same-label texts. If the instance is too
different from the content of other examples with the same label, it will have
a low silhouette coefficient, and we consider it to be noise. Training the
model on this judgment may confuse it and reduce classification performance.
Hence, we suggest removing this judgment from the ground truth corpus before
training. However, other high-silhouette judgements for $t_{i}$ may be
preserved and $t_{i}$ can be part of the filtered dataset using other
judgements of it.

###  III-C Model’s Training Dynamics

Recent work by [19] uses signals from epochs during training (training
dynamics) as a proxy for exploratory data quality estimation. They define
conﬁdence of the model on instance $i$ as the average of model’s probability
of its true label ($y_{i}^{*}$) across epochs:

${\hat{\mu}}_{i}=\frac{1}{E}\sum_{e=1}^{E}p_{\theta^{(e)}}(y_{i}^{*}|t_{i})$

where $p_{\theta^{(e)}}$ indicates to the model’s probability with parameters
$\theta^{(e)}$ at the end of the epoch $e$. Their experiments show that the
examples with low confidence are likely to be mislabeled.

##  IV Experiments

###  IV-A Data

(a) Moral Foundation Twitter Corpus (MFTC)

(b) Moral Foundation News Corpus (MFNC)

(c) Moral Foundation Reddit Corpus (MFRC)

Figure 4: Measures of disagreement. The left column shows the distributions of
the silhouette metric for the three datasets. The silhouette coefficient is
calculated on annotations. The right column show the distributions of entropy
metric for instances, colored by the majority-vote of moral foundation
assigned to the instance.

In this paper we focus on three large moral foundations annotated textual
datasets. In these datasets the annotations of many texts are very diverse.
This diversity can appear because of the difficulty of the labeling task,
subjectivity of moral judgments, the bias based on personal beliefs, ambiguity
of texts, and errors made by annotators.

####  IV-A1 The Moral Foundations Twitter Corpus (MFTC) [10]

_The Moral Foundations Twitter Corpus (MFTC)_ is a textual multi-label dataset
containing tens of thousands of tweets related to various social movements,
with each tweet annotated with categories of moral foundations. For annotating
MFTC, several human annotators were trained to manually annotate 35k tweets.
The tweets are drawn from 7 socially relevant topics: All Lives Matter (ALM),
Black Lives Matter (BLM), the Baltimore protests, the 2016 Presidential
election, hate speech and offensive language, Hurricane Sandy, and #MeToo. For
each tweet several annotators selected as many moral foundations as they saw
relevant, which resulted in a multi-label dataset. Note that for each tweet,
the number of annotators that selected each class is known. There are in total
eleven labels (ten moral foundations and one “non-moral” class indicating the
text does not have moral relevance).

####  IV-A2 The Moral Foundations News Corpus (MFNC) [8]

In the MFNC [8], a crowd of 854 annotators was drawn from the general United
States population using the crowd-sourcing platform Prolific Academic
(PA222https://www.prolific.ac/). Sampling was designed to match annotator
characteristics to the US general population in terms of political affiliation
and gender, thereby lowering the likelihood of obtaining annotations that
reflect the moral intuitions of only a small, homogeneous group (see
supplemental materials in [7] for detailed information on annotators) .
Fifteen randomly selected news documents (from among 2,995 articles total)
were assigned to each annotator. The selected corpus consisted of online
newspaper articles discussing a wide range of sociopolitical topics from 11
prominent, U.S. news outlets. 557 annotators completed all assigned tasks.
Each annotator underwent an online training explaining the purpose of the
study, the basic tenets of MFT, and the annotation procedure. Annotators were
instructed that they would be annotating news articles, and that for each
article they would be (randomly) assigned one of the five moral foundations.
Next, using a digital highlighting tool, annotators were instructed to
highlight all portions of a news article that they understood to reflect their
assigned moral foundation. In total, 63,958 annotations (i.e., textual
highlights) were produced by the 557 annotators. Note that the coding task of
the MFNC differed from the MFTC task in important ways: First, annotators were
assigned to focus on the presence of one (randomly assigned) moral foundation
per article, rather than assigning portions of the article to any moral
foundation. Second, annotators labeled the holistic presence of a moral
foundation rather than differentiating whether a foundation was upheld (e.g.
care) or violated (e.g. harm). Third, annotators were free to highlight
portions of a news article, in contrast to labeling the entire coding unit
with a moral foundation.

####  IV-A3 The Moral Foundations Reddit Corpus (MFRC) [14]

The MFRC consists of 16,123 Reddit comments drawn from 12 different
subreddits. Every instance has been labeled by at least three annotators from
a set of ﬁve trained annotators.

###  IV-B Metrics of Disagreement

Figure 4 shows the distributions of entropy and silhouette metrics for the
three datasets. The distribution varies for each category and each dataset.
For example, in MFTC, “non-moral” has the higher concentration of posts with
zero entropy, meaning that in many instances all the annotators agreed on the
non-moral category. On the other hand the classes purity and degradation have
few samples with zero entropy of annotations. Also, in MFNC, all the
categories have high concentration of posts with zero entropy which is because
there are many instances with single highlights in MFNC that have no overlap
with other highlights. Figure 4 also shows the distribution of silhouette
coefficient of annotator judgements for each of the moral foundation
categories. This metric has a consistently broad distribution for all
datasets, unlike entropy.

###  IV-C Experimental Settings

####  IV-C1 Classifier Model

Getting a text as input, our model’s task is to classify it to a moral
foundation label. The classification task aims to predict majority-vote for
each textual instance. Which is the label with highest number of annotators
selecting it for the instance (we select randomly if there is a tie). In MFTC
there are eleven labels (the labels from vices and virtues of the five moral
foundations plus one label denoting non-moral category). In MFNC there are
only five labels related to the main foundations regardless of the polarity.
For MFRC we convert the labels “proportionality” and “equality” to fairness
and only keep the labels from the main foundations to keep consistency with
MFNC.

We use pretrained language model RoBERTa [28] and finetune it on our datasets.
This is done by adding a multi-class classiﬁcation layer on top of the
language model and with a cross-entropy loss updating all the parameters end-
to-end for five epochs. We minimize cross-entropy with the Adam optimizer [29]
with learning rate $2\times 10^{-5}$. Our experiments use a batch size of 50.
We run each experiment with five random seeds and split data into $70\%$ train
and $30\%$ test sets. Each experiment is performed on a single GTX 1080Ti GPU.
In our implementation we use the Huggingface Transformers library [30].

####  IV-C2 Silhouette Coefficient Calculation

To calculate the silhouette coefficient of the judgments we need to use a
language model to represent the text of instances in a latent space. We use
sentenceBERT framework [31], which has been shown to be helpful for capturing
semantic textual similarity. We use the pretrained model ”all-mpnet-
base-v2”333https://huggingface.co/sentence-transformers/all-mpnet-base-v2. We
use the default parameters of sklearn implementation of the silhouette
coefficient444https://scikit-
learn.org/stable/modules/generated/sklearn.metrics.silhouette_samples.html
which uses euclidean distance as a metric to capture semantic distance of
texts.

###  IV-D Results

(a) MFTC dataset

(b) MFNC dataset

(c) MFRC dataset

Figure 5: Morality classification after de-noising annotations. Comparison of
F1 (left sub-figures), accuracy (middle sub-figures), and distributions of
model confidence on instances (right sub-figures) when removing high-entropy
instances, low-silhouette judgments, or removing randomly. Removing low-
silhouette judgments helps with model’s performance on all MFTC, MFNC, and
MFRC datasets. However, removing high-entropy instances is only effective on
MFTC and doesn’t help with learning on MFNC or MFRC due to the high ratio of
instances with zero entropy.

We study the effect of removing noisy annotations on the performance of the
trained language model, discarding from each dataset either 1) highest-entropy
instances (and all the labels assigned to them), or 2) lowest-silhouette
coefficient labels. After filtering the data with a specific ratio, we split
the dataset to train and test set. We compare performance to models trained on
data from which the same ratio of instances have been removed at random.
Figure 5 shows performance on the three datasets in terms of the F1 Macro
(left subplot), and accuracy (middle) as a function of the fraction of data
removed. These measures are aggregated over five runs with different random
seeds and calculated on held-out test data.

Discarding data at random shows a non-increasing trend in the performance of
the model. On the other hand, discarding portions of judgments with lowest
silhouette coefficients improves the performance of moral classification on
all three datasets. These results suggest that our proposed method identifies
better-quality ground truth data that helps model performance.

Although removing highest-entropy instances improves performance of the
classifier on MFTC, it does not help improve classification performance on the
MFNC dataset and offers only a small improvement on MFRC. This result can be
explained by the way each dataset has been collected. In MFTC, each tweet was
shown to several annotators and their judgments about all the labels were
collected. However, in MFNC, annotators were shown a news document and asked
to identify segments of text relevant to a specific moral foundation. For
example, annotator $a1$ was asked to highlight any part of document $d$
related to the care foundation, but annotator $a2$, $a3$ were asked to
highlight portions of $d$ relevant to authority and loyalty. Their highlighted
texts might overlap at sentence $s$, and $s$ at the end will have annotations
$<care:1,fairness:0,authority:1,loyalty:1,purity:0>$. At first glance, it
seems there is no agreement on $s$ and its entropy is very high. However, if
the annotators were given the opportunity to choose any label, we could have
seen more agreement in the distribution of labels. In crowd-sourcing moral
annotations, it helps to simplify the task to finding only one moral
foundation in a document to reduce confusion [7]. However, this results in a
proliferation of instances labeled by only one annotator. The low entropy
score will deceptively indicate low disagreement, even though the single
annotation could have been made in error.

It is important to understand how the dataset was constructed when choosing
between using instance-level entropy and judgment-level silhouette coefficient
when preprocessing the data with filtering.

The rightmost panels in Figure 5 show model confidence as a function of the
fraction of data discarded by the entropy or silhouette methods. The figures
show that the distribution of the model’s confidence on instances shifts
toward higher values when we discard labels with lowest silhouette values. As
a reminder, higher confidence means that the model is more likely to correctly
classify the instances. The same improvement occurs when we discard high-
entropy instances in MFTC. Similar to previous results, removing high-entropy
instances in MFNC or MFRC does not improve confidence. Moreover, the
distribution of confidence shifts toward lower values or stays the same when
we remove instances at random from all datasets. Prior work [19] has shown
that if a trained model has low confidence on an instance, it means that the
instance was likely mislabelled. The observed positive shift in the
distribution of confidence values suggests that the subset of the data we kept
has fewer mislabelled instances.

##  V Conclusions

In this work we show that auditing annotated data for noise can improve
morality classification. We propose two metrics—entropy and silhouette
coefficient—for refining annotated datasets in a pre-processing step, i.e.
before training a classifier. The metrics leverage annotator judgments in
order to identify instances that are difficult to label or have been
mislabeled. We show that removing these instances reduces the noise in the
ground truth data, improving classification performance of models trained on
the remaining data. We also show that refining annotations improves the
training dynamics of the model. As a result, the average of confidence
increases when the model is trained on the instances that are recognized as
less noisy with our proposed metrics.

We validated our approach on three datasets where multiple annotators were
asked to label the moral values expressed in text. Classifying morality is
inherently a subjective and difficult task, and the resulting ground truth
data from human annotation will naturally contain a large amount of
disagreement and noise, which can degrade the performance of single-task
models trained on the data. We showed that a classifier trained on refined
data, from which the potentially noisy samples have been removed, can learn
better models that more accurately recognize new instances of moral
foundations. Our approach is not specific to moral annotations and can be
applied to other datasets constructed from subjective judgments of multiple
annotators.

##  VI Limitations and Ethical Considerations

Other works [32, 33, 34, 17] have shown annotator demographic features and
annotators’ life experiences can impact their judgments. In this work we focus
on single-task classifier models that need an aggregated label (e.g. to
majority vote or averaging) for training. However, we encourage future work to
design models beyond single-task classifiers in order to overcome the need for
aggregating the labels and move to subjective models that can give predictions
based on different beliefs, demographics, and backgrounds.

Also, considering the use-case of a gathered dataset, the removal criteria
described in section III can be enhanced to make sure we are not removing the
samples from specific demographics or groups of people. A possible remedy is
to discard weighted portions of data from each demographic group in a way to
keep a more balanced subset of data or to prioritize keeping the data gathered
by minorities.

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