1]Instituto de Física, Universidade Federal do Rio Grande do Sul, 91501-970,
Porto Alegre,RS, Brazil

#  Charge fluctuations in charge regulated systems: dependence on statistical
ensemble

Amin Bakhshandeh<EMAIL_ADDRESS>Yan Levin<EMAIL_ADDRESS>[

###### Abstract

We investigate charge regulation of nanoparticles in concentrated suspensions,
focusing on the effect of different statistical ensembles. We find that the
choice of ensemble does not affect the mean charge of nanoparticles, but
significantly alters the magnitude of its fluctuation. Specifically, we
compared the behaviors of colloidal charge fluctuations in the semi-grand
canonical and canonical ensembles, and identified significant differences
between the two. The choice of ensemble – whether the system is isolated or is
in contact with a reservoir of acid and salt – will, therefore, affect the
Kirkwood-Shumaker fluctuation-induced force inside concentrated suspensions.
Our results emphasize the importance of selecting an appropriate ensemble that
accurately reflects the experimental conditions when studying fluctuation-
induced forces between polyelectrolytes, proteins, and colloidal particles in
concentrated suspensions.

###### keywords:

Charge Regulation, Colloidal Particles, Kirkwood-Shumaker force

##  1 Introduction

It is a great pleasure to contribute this paper to the special issue of EPJE
that honors many contributions of Philip (Fyl) Pincus in the field of Soft
Matter Physics [1, 2, 3, 4, 5, 6, 7, 8]. Much of Fyl’s work has been focused
on understanding complicated effects resulting from Coulomb force in condensed
matter systems. We hope that Fyl will find the following paper of interest.

Electrostatic interactions are ubiquitous in physics, chemistry, and biology
[9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. They play a vital role in the
stability and function of biological molecules [20] and the structure and
dynamics of electrolyte solutions. The magnitude of the force between charged
particles is heavily dependent on the dielectric constant of the solvent. In
fact, water is essential for many biological processes [21, 22, 23], and its
unique properties largely stem from its strong electrostatic interaction with
the charged solutes. Due to the high dielectric constant of water [24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34], electrostatic interactions between charged
atoms and molecules are greatly reduced, resulting in a favorable solvation
free energy. The high dielectric constant of water also causes the
dissociation of molecules into ions [35], resulting in electrolyte solutions.
The formation of hydrogen bonds between water and biomolecules helps to
stabilize their structures and facilitate their interactions with other
molecules in biological cells.

In colloidal science, the charging process of colloidal particles in an
electrolyte solution is called charge regulation [36, 37, 38, 39, 40, 41, 42,
43, 44, 45, 46, 47, 48, 49]. The main reason for this phenomenon is that
macromolecules, in general, contain functional acidic and basic groups that
can become protonated or deprotonated, depending on the ionic strength, pH,
and solute volume fraction. In equilibrium, the charge of macromolecules will
fluctuate around some mean value. The correlated fluctuations of charge can
result in an effective attractive Kirkwood-Shumaker (KS) force between two
like-charged macromolecules close to their isoelectric point [50, 51]. A
theoretical prediction, which was confirmed by Timasheff et. al. [52] using
light-scattering techniques.

The Kirkwood-Shumaker model has been successfully applied in many areas of
biophysics, such as protein-protein interactions, protein aggregation, and
protein crystallization. In particular, it has been used to explain the
phenomenon of liquid-liquid phase separation in protein solutions [53]. The KS
interaction is also relevant for understanding the behavior of
polyelectrolytes in solutions, for which the long-range Coulomb interaction
between charged macromolecules plays a crucial role [54]. The KS model,
however, has limitations, particularly when applied to systems with highly
charged macromolecules or in the presence of multivalent ions. In such cases,
other factors, such as ion correlation effects, must be taken into account
[55]. Despite these limitations, the KS model remains a valuable tool for
understanding the long-range interactions between macromolecules in solutions.
It provides a useful framework for interpreting experimental data and can
guide the development of new theoretical models. Since the KS force arises
from correlated fluctuations of the macromolecular charge, it is of
fundamental importance to understand how much this charge fluctuates around
its mean value for different experimental conditions.

Most experimental systems consist of an isolated suspension at some fixed
volume fraction of solute. On the other hand, to perform simulations one often
uses a semi-grand canonical approach, in which suspension is effectively
separated from the reservoir of acid and salt by a semi-permeable membrane,
which allows for a free exchange of ions, but prevents the passage of
macromolecules [59, 40, 56, 46]. It is intuitive that in such open systems,
macromolecular charge will fluctuate more than in closed (isolated) systems.
The goal of the present paper is to quantify this difference.

##  2 Non-interacting systems

We start by studying an ideal system in which electrostatic interactions are
turned off. The model consists of a nanoparticle of radius $a$ and $Z$
(negative) active surface sites. Each active site can adsorb one proton to
become protonated. A nanoparticle is placed at the center of a spherical cell
of radius $R$, such that $\eta=a^{3}/R^{3}$, where $\eta$ is the volume
fraction of suspension. The cell also contains noninteracting point ions: Na+,
Cl- and H+. The proton H+ can associate with an adsorption site, resulting in
a free energy gain of $-\ln(\text{K}_{eq}/\Lambda^{3})$, where the equilibrium
constant $\text{K}_{eq}$ is the partition partition function of a bound state.
In this ideal model, the ionic charge is used just to distinguish cations from
anions and to preserve the overall charge neutrality inside the cell.

###  2.1 Canonical Theory

The free energy of protons inside an isolated (canonical) cell can be written
as,

$\displaystyle\beta\mathcal{F}(n)=-n\ln\left(\frac{\text{K}_{eq}}{\Lambda^{3}_{\text{H}^{+}}}\right)+n\ln\left(\frac{n}{Z}\right)+\left(Z-n\right)\ln\left(1-\frac{n}{Z}\right)+$
(1)
$\displaystyle\left(N_{t}-n\right)\ln\left(\frac{\Lambda^{3}_{\text{H}^{+}}\left(N_{t}-n\right)}{V}\right)-\left(N_{t}-n\right),$

where $N_{t}$ is total number of protons inside the system, of which $n$ are
in a bound state with the surface groups. $V$ is the free volume of the cell
and $\text{K}_{eq}$ and $\Lambda_{\text{H}^{+}}$ are the equilibrium constant
and the de Broglie thermal wavelength, respectively. The first term in the
expression above is the chemical energy of association between proton and an
active site. Second and third terms are the entropic contributions of the
bound protons, while the last two terms are the entropic contributions of free
protons. Minimizing Eq. 1 with respect to $n$, we obtain the equilibrium
(average) number of protonated sites:

$n^{*}=\frac{1}{2}\left(\frac{{V}}{{\text{K}_{eq}}}+{N_{t}}+{Z}-\sqrt{\left(\frac{{V}}{{\text{K}_{eq}}}+{N_{t}}+\text{Z}\right)^{2}-4{N_{t}}\text{Z}}\right).$
(2)

It is important to stress that the number of associated protons is not fixed,
but fluctuates around the value $n^{*}$, with the average colloidal charge
given by $Q^{*}=-(Z-n^{*})q$, where $q$ is the proton charge. The charge
fluctuation is characterized by $\langle(\Delta Q)^{2}\rangle=\langle
Q^{2}\rangle-\langle Q\rangle^{2}=q^{2}(\langle n^{2}\rangle-\langle
n\rangle^{2})$. The probability that $n$ active sites are protonated is
proportional to $\mathrm{e}^{-\beta\mathcal{F}(n)}$. Expanding the free energy
$\mathcal{F}(n)$ around the equilibrium $n^{*}$ up to second order and using
the saddle point approximation, the fluctuation of charge in canonical
ensemble is determined to be:

$\sigma_{t}^{2}\equiv\frac{\langle(\Delta
Q)^{2}\rangle}{q^{2}}=\frac{n^{*}\left(n^{*}-N_{t}\right)\left(n^{*}-Z\right)}{N_{t}Z-n^{*2}}.$
(3)

###  2.2 Semi-grand Canonical Theory

Titration simulations are usually performed in a semi-grand canonical ensemble
in which microions are free to exchange with the reservoir, while
nanoparticles are confined to stay within the system [40]. In a real
experimental system such setup requires a semipermeable membrane that
separates system from the reservoir of acid and salt. Since the counterions
are free to diffuse into reservoir, their efflux will result in an electric
field across the membrane that will oppose the flow. The concentration of ions
inside the system will, in general, be different from the concentrations in
the reservoir. When performing semi-grand canonical simulations, it is
important to keep in mind that the simulation cell is at a different
electrostatic potential than the reservoir. This electrostatic potential
difference is known as the Donnan potential. Often, Donnan potential is
implicitly taken into account when performing semi-grand canonical simulations
of charged system by forcing the insertion and deletion moves to be done in
cation-anion pairs, which effectively cancels the Donnan potential. However,
presence of the Donnan potential is often neglected when performing constant
pH titration simulations [57, 58].

Neglecting the Coulomb and steric interactions between the ions, the ideal
partition function for a system containing a nanoparticle inside a spherical
cell connected to a reservoir of acid and salt at concentrations $C_{H}$ and
$C_{s}$, respectively, can be written as:

$\displaystyle\Xi=\sum_{N_{\text{Na}},N_{\text{H}}N_{\text{Cl}}}^{\infty}\sum_{n=0}^{Z}\frac{Z!}{n!(Z-n)!}\exp\left[-\beta\left(-(Z-n)q\phi-n\ln\frac{\text{K}_{eq}}{\Lambda^{3}}-n\mu_{\text{H}}\right)\right]$
(4)
$\displaystyle\frac{1}{N_{\text{Na}}!}\frac{1}{N_{\text{Cl}}!}\frac{1}{N_{\text{H}}!}\left(\frac{V\mathrm{e}^{-\beta(q\phi-\mu_{\text{Na}})}}{\Lambda^{3}}\right)^{N_{\text{Na}}}\left(\frac{V\mathrm{e}^{\beta(q\phi+\mu_{\text{Cl}})}}{\Lambda^{3}}\right)^{N_{\text{Cl}}}\left(\frac{V\mathrm{e}^{-\beta(q\phi-\mu_{\text{H}})}}{\Lambda^{3}}\right)^{N_{\text{H}}},$

where $N_{\text{H}}$, $N_{\text{Cl}}$, $N_{\text{Na}}$ are the number of
particle of specie H, Cl, Na inside the system; and V, $\mu_{\text{H}}$,
$\mu_{\text{Na}}$, $\mu_{\text{Cl}}$, $\Lambda$ are the free volume, chemical
potential of H, Na, Cl and the thermal de Broglie wavelength, respectively. In
general, we allow the system to be at a different potential from the
reservoir, this accounts for the presence of the electrostatic potential
$\phi$ in the partition function. The two summations in Eq.4 are decoupled and
can be performed separately. The partition function for protonation of surface
sites reduces to:

$\sum_{n=0}^{Z}\frac{Z!}{n!(Z-n)!}\exp\left[-\beta\left(-(Z-n)q\phi-n\ln\frac{\text{K}_{eq}}{\Lambda^{3}}-n\mu_{\text{H}}\right)\right]=\left(\mathrm{e}^{\beta
q\phi}+\mathrm{e}^{\ln\frac{\text{K}_{eq}}{\Lambda^{3}}+\mu_{\text{H}}}\right)^{Z},$
(5)

while the partition function for free ions is:

$\displaystyle\sum_{N_{\text{H}},N_{\text{Na}},N_{\text{Cl}}=0}^{\infty}\frac{1}{N_{\text{Cl}}!}\frac{1}{N_{\text{H}}!}\frac{1}{N_{\text{Na}}!}\left({C_{\text{Na}}V\mathrm{e}^{-\beta
q\phi}}\right)^{N_{\text{Na}}}\left({C_{\text{Cl}}V\mathrm{e}^{\beta
q\phi}}\right)^{N_{\text{Cl}}}\left({C_{\text{H}}V\mathrm{e}^{-\beta
q\phi}}\right)^{N_{\text{H}}}=$ (6)
$\displaystyle\exp\left({C_{\text{H}}V\mathrm{e}^{-\beta
q\phi}}\right)\exp\left({C_{\text{Na}}V\mathrm{e}^{-\beta
q\phi}}\right)\exp\left({C_{\text{Cl}}V\mathrm{e}^{\beta q\phi}}\right),$

where we have used $\beta\mu_{i}=\ln\left(C_{i}\Lambda^{3}\right)$ for ideal
chemical potential inside the reservoir.

The net charge inside the cell is:

$\langle\mathcal{Q}_{net}\rangle=-\frac{1}{\beta}\frac{d\ln\Xi}{d\phi}=-\frac{\mathrm{e}^{\beta\phi}{Zq}}{{C}_{\text{H}}\text{K}_{eq}+\mathrm{e}^{\beta\phi}}+q\left({C}_{\text{H}}+{C}_{\text{Na}}-{C}_{\text{Cl}}\mathrm{e}^{2\beta\phi}\right)V\mathrm{e}^{-\beta\phi}.$
(7)

In the thermodynamic limit, the cell must be charge neutral, which means that
$\langle\mathcal{Q}_{net}\rangle=0$. This condition determines the Donnan
potential $\phi_{d}$. The charge of a nanoparticle at equilibrium in a semi-
grand canonical system will then be:

$\langle
Q\rangle=-\frac{\mathrm{e}^{\beta\phi_{d}}{Zq}}{{C}_{\text{H}}\text{K}_{eq}+\mathrm{e}^{\beta\phi_{d}}}.$
(8)

The fluctuation in the net charge inside the system can be obtained from the
second derivative of the partition function with respect to $\phi$. These
fluctuations decouple into those due to fluctuation of colloidal charge and of
free ions in the bulk. The fluctuation in the charge of the nanoparticle is
calculated to be:

$\sigma_{t}^{2}\equiv\frac{\langle(\Delta
Q)^{2}\rangle}{q^{2}}=-\frac{1}{q^{2}\beta}\frac{d\langle
Q\rangle}{d\phi_{d}}=\frac{C_{\text{H}}\text{K}_{eq}Z\mathrm{e}^{\beta
q\phi_{d}}}{\left(\mathrm{e}^{\beta
q\phi_{d}}+C_{\text{H}}\text{K}_{eq}\right)^{2}}.$ (9)

Comparing the charge fluctuations in canonical and semi-grand canonical
systems, we see that the fluctuations of colloidal charge in the semi-
grandcanonical system are much larger than in a canonical system. To compare
the two ensembles, in the canonical system we put exactly the same number of
protons and ions into the cell as the averages obtained in the semi-grand
canonical system. See, for example, the values of $\sigma_{t}^{2}$ in Tables 1
and 3, respectively for the two systems. To go beyond the ideal models,
requires Monte Carlo simulations methods. Below, we briefly present the semi-
grand canonical [59, 40], and canonical simulations methods that can be used
to explore charge regulation and fluctuations in the two ensembles [60, 58].

##  3 Titration Algorithms

When performing canonical simulations, the system can only exchange heat with
the surrounding environment, while the number of protons and other ions is
conserved inside the simulations cell. On the other hand in semi-grand
canonical simulations, the ions and protons can be exchanged with the
reservoir, and the average concentrations of ions inside the simulation cell
are determined by the thermodynamic equilibrium – equivalence of
electrochemical potentials in the system and the reservoir. Furthermore, as
was discussed previously, the simulation cell is at a different electrostatic
potential from the reservoir. The potential difference between the cell and
the reservoir is the Donnan potential. In systems with finite volume fractions
of nanoparticles, proteins, or polyelectrolytes, this potential difference can
not be ignored and must be taken into account when performing simulations.

###  3.1  Semi-grand canonical Monte Carlo method

In semi-grand canonical Monte Carlo (sGCMC) simulations we need to perform
protonation/deprotonation moves as well as insertion/deletion moves into/from
the cell. Since the simulation cell is at a different electrostatic potential
than the reservoir, upon entering the cell an ion acquires an additional
electrostatic energy $q_{i}\phi_{D}$, where $\phi_{D}$ is the Donnan
potential. Taking this into account the usual grand-canonical acceptance
probabilities for addition and removal of ions are modified to:

$\begin{split}\phi_{add}=\min\left[1,\frac{Vc_{i}}{N_{i}+1}\mathrm{e}^{-\beta\left(\Delta
U-\mu_{ex}+q_{i}\varphi_{D}\right)}\right],\\\
\phi_{rem}=\min\left[1,\frac{N_{i}}{Vc_{i}}\mathrm{e}^{-\beta\left(\Delta
U+\mu_{ex}-q_{i}\varphi_{D}\right)}\right].\end{split}$ (10)

For reaction moves, proton can enter from the reservoir and react with an
adsorption site resulting in its protonation. Alternatively a protonated site
can become deporonated, with the proton moving to reservoir. Again, when a
proton moves into or out of the system, the Donnan potential must be taken
into account. The acceptance probabilities for protonation and deprotonation
moves can then be written as:

$\begin{split}\phi_{p}=\min\left[1,\mathrm{e}^{-\beta\left(\Delta U+\Delta
F_{p}+q\varphi_{D}\right)}\right],\\\
\phi_{d}=\min\left[1,\mathrm{e}^{-\beta\left(\Delta U+\Delta
F_{d}-q\varphi_{D}\right)}\right],\end{split}$ (11)

where $\beta\Delta
F_{p}=-\ln(\text{K}_{eq}/\Lambda^{3}_{\text{H}})-\mu_{\text{H}}$ is the
chemical free energy change due to removal of proton from the reservoir and
its reaction with an isolated adsorption group. The chemical potential of
proton in reservoir is
$\beta\mu_{\text{H}}=\ln(c_{\text{H}}\Lambda^{3}_{\text{H}})+\beta\mu_{ex}$,
where $\mu_{ex}$ is the excess chemical potential of ions in the reservoir.
The deprotonation energy is then $\Delta F_{d}=-\Delta F_{p}$. Since the
Donnan potential is not known a priori, it is convenient to perform
insertion/deletions moves using cation-anion pairs [59, 40]. This way the
Donnan potential cancels from the acceptance probabilities. Similarly, a
protonation/deprotonation move can be combined with an insertion/deletion of
an anion into the the cell. The acceptance probabilities for such pair
protonation/deprotonation moves become:

$\begin{split}\bar{\phi}_{p}=\min\left[1,\frac{c_{\text{H}}\text{K}_{eq}Vc_{\text{Cl}}}{(N_{\text{Cl}}+1)}\mathrm{e}^{-\beta\left(\Delta
U-2\mu_{ex}\right)}\right],\\\
\bar{\phi}_{d}=\min\left[1,\frac{N_{\text{Cl}}}{c_{\text{H}}\text{K}_{eq}Vc_{\text{Cl-}}}\mathrm{e}^{-\beta\left(\Delta
U+2\mu_{ex}\right)}\right],\end{split}$ (12)

where $N_{\text{Cl}}$ is the number of anions inside the cell, $V$ is the free
volume, $\Delta U$ is the difference of energy for the pair move. For
simplicity, here we consider that all ions are monovalent and are hard spheres
of the same radius, so that $\mu_{ex}$ is the same for all the ions.

The algorithm described above can be applied to any system in which reactions
take place. We start by studying an ideal non-interacting system described in
Section 2.2. In the Table 1, we compare the results of simulations with the
theory for a nanoparticle of radius 80 Å with $Z=600$ surface active groups,
inside a simulation cell of radius $R=150$Å. The equilibrium constant for
sites is taken to be $K_{eq}=1216092$ Å3. The reservoir contains acid at
concentration $C_{H}$ and salt at concentration $C_{s}=10$ mM. The mean charge
of a nanoparticle and its fluctuation calculated using both theory and
simulations are presented in Table 1.

Table 1:  Nanoparticle charge $Q$ and its fluctuation $\sigma^{2}$ obtained from simulations (s) and theory (t) for different concentrations of acid in the reservoir $C_{\text{H}}$ The concentration of salt in the reservoir is fixed at $C_{s}=10$ mM. The average number of ions of each type present inside the simulation cell after equilibration is also provided. C${}_{\text{H}}$[M] | $Q^{s}/q$ | $Q^{t}/q$ | $\sigma_{s}^{2}$ | $\sigma_{t}^{2}$ | $N_{\text{H}}$ | $N_{\text{Na}}$ | $N_{\text{Cl}}$
---|---|---|---|---|---|---|---
$10^{-3}$ | $-195.4$ | $-195.7$ | 88.3 | 131.8 | 424 | 203 | 28
$10^{-4}$ | $-417.9$ | $-418.7$ | 98.9 | 126.4 | 185 | 426 | 12
|  |  |  |  |  |  |

We see that while the effective charge is in perfect agreement with
simulations, the fluctuations show significant deviations. The reason for this
is that the pair insertion moves restrict the charge fluctuations inside the
cell. Indeed, if we perform simulations using individual insertions, Eq. (11)
with the Donnan potential fixed at the value predicted by the theory, we
obtain exactly the same numbers of ions inside the simulation cell and the
same colloidal charge as found using the pair insertion algorithm, Eq. (12).
Thus, as expected, the Donnan potential leads to an overall charge neutrality
on average. Furthermore, individual insertion algorithm with Donnan potential,
results in colloidal charge fluctuations very similar to the ones predicted by
the theory, see Table 2.

Table 2:  Colloidal charge and its fluctuations obtained using individual insertion algorithm (o) with the Donnan potential fixed at the value predicted by the theory (t). Compare the fluctuations obtained using the individual insertions algorithm $\sigma_{o}^{2}$, with the ones obtained using the pair insertion method $\sigma_{s}^{2}$ presented in Table 1. All other parameters are the same as in the Table 1. C${}_{\text{H}}$[M] | $Q^{o}/q$ | $Q^{t}/q$ | $\sigma_{t}^{2}$ | $\sigma_{o}^{2}$
---|---|---|---|---
$10^{-3}$ | $-196.1$ | $-195.7$ | 131.8 | 135.4
$10^{-4}$ | $-418.8$ | $-418.7$ | 126.4 | 131.0
|  |  |  |

Clearly, restricting the insertion moves to keep the system charge neutral at
each Monte Carlo step, strongly affects the fluctuations of colloidal charge.
With these insights, we are now ready to explore isolated canonical systems.

###  3.2  Canonical reactive Monte Carlo method

When performing a canonical simulation, the number of protons and ions inside
the simulation cell is fixed. However, the protons can either be in a bound
state or free, see Fig. 1. The average number of bound protons will determine
the equilibrium charge of nanonparticles.

Figure 1: (a) Protonation and (b) deprotonation moves in canonical ensemble:
($i$) initial state,($f$) final state.

In a protonation move, a proton that is initially in the bulk, moves to the
adsorption site. The probability for initial and final configuration are then
proportional to:

$\displaystyle\Pi_{i}\sim\frac{V^{N}}{\Lambda^{3N}N!}\mathrm{e}^{-\beta
U_{N}},$
$\displaystyle\Pi_{f}\sim\frac{V^{N-1}}{\Lambda^{3(N-1)}\left(N-1\right)!}\mathrm{e}^{-\beta
U_{N-1}+\ln\text{K}_{eq}/\Lambda^{3}},$ (13)

where $N$ is the number of free protons and $U_{N}$ is the total electrostatic
energy of the system. Similarly for deprotonation move

$\displaystyle\Pi_{i}\sim\frac{V^{N}}{\Lambda^{3N}N!}\mathrm{e}^{-\beta
U_{N}},$
$\displaystyle\Pi_{f}\sim\frac{V^{N+1}}{\Lambda^{3(N+1)}\left(N+1\right)!}\mathrm{e}^{-\beta
U_{N+1}-\ln\text{K}_{eq}/\Lambda^{3}},$ (14)

Using the usual detailed balance argument, the acceptance probabilities for
deprotonation and protonation moves can now be written as:

$\displaystyle\text{P}_{d}=\min\left[1,\frac{V}{\text{K}_{eq}\left(N_{\text{H}^{+}}+1\right)}e^{-\beta\Delta
U}\right],$
$\displaystyle\text{P}_{p}=\min\left[1,\frac{\text{K}_{eq}N_{\text{H}^{+}}}{V}e^{-\beta\Delta
U}\right],$ (15)

We can now check the consistency of the two simulation methods. We first run
the semi-grand canonical simulation to determine the number of ions, free
protons, and the nonoparticle charge inside the simulation cell for a given
concentration of acid and salt in the reservoir. We can then strip all the
associated protons and put them into the bulk of the simulation cell, so that
all sites are again deprotonated. We then run canonical reactive Monte Carlo
algorithm, Eq. (3.2) to determine the equilibrium number of protonated sites
and the equilibrium colloidal charge. The data is presented in Table 3. While
the charge of the nanoparticle calculated using canonical simulation is in
perfect agreement with the results of the semi-grand canonical algorithm, the
fluctuation in the charge is significantly lower than what is observed in an
open system. Furthermore, we see that for a canonical ideal system, both the
nanoparticle charge and its fluctuation are in excellent agreement with the
predictions of the theory. With the insights gained from studying non-
interacting ideal systems, we are now ready to explore charge fluctuations in
non-ideal systems, with ions of finite size interacting through Coulomb
potential.

Table 3: Colloidal charge and its fluctuations for ideal system calculated
using canonical simulations (s) compared with the predictions of theory (t),
Eqs.(2) and (3). The number of ions and protons inside the simulation cell is
exactly the same as are the averages found using sGCMC simulations, see Table
1

.

$N_{\text{H}}$ | $N_{\text{Na}}$ | $N_{\text{Cl}}$ | $Q^{s}/q$ | $Q^{t}/q$ | $\sigma_{s}^{2}$ | $\sigma_{t}^{2}$
---|---|---|---|---|---|---
424 | 203 | 28 | $-195.4$ | $-195.5$ | 17.4 | 17.6
185 | 426 | 12 | $-418.6$ | $-418.6$ | 4.2 | 4.1
|  |  |  |  |  |

##  4 Systems with Coulomb interactions

We now consider a nanoparticle of radius of 80 Å with $Z=600$ active negative
point sites distributed uniformly on its surface [39], inside a spherical cell
of radius $R=150$ Å. The intrinsic pKa of surface groups is set to
pK${}_{a}\equiv-\log_{10}$Ka=5.4, corresponding to carboxylic acid. Recall
that acid dissociation constant is $\text{K}_{a}=1/\text{K}_{eq}$. All ions
are modeled as hard spheres of radius $r=2$ Å, with point charge at the
center. Water is treated as a uniform medium of dielectric constant
$\epsilon=78$. The electrostatic energy now includes Coulomb ion-ion, ion-
site, and site-site interactions. Again, we first run the sGCMC simulations to
determine the average number of ions, the number of free protons, the mean
charge of the nanoparticle, and its fluctuation $\sigma^{2}$, for a given
concentration of acid and salt in the reservoir, Table 4. It is interesting to
compare the results obtained using the sGCMC simulations with the recently
developed theory that allows us to accurately predict the effective colloidal
charge in concentrated suspensions [39].

The effective charge (number of deprotonated groups) predicted by the theory
is [39]:

$Q^{t}=-\frac{Zq}{1+\text{K}_{eq}c_{a}e^{-\beta(q\phi_{0}-\phi_{disc}-\mu^{ex}-\mu_{sol})}},$
(16)

where $\mu_{sol}$ the electrostatic solvation free energy of a charged site:

$\beta\mu_{sol}=\frac{\lambda_{B}}{2}\int_{0}^{\infty}\frac{k-\sqrt{\kappa^{2}+k^{2}}}{k+\sqrt{\kappa^{2}+k^{2}}}e^{-2kr_{ion}}dk,\
$ (17)

and $\phi_{0}$ is the mean-field electrostatic potential at the surface
determined from the solution on non-linear Poisson-Boltzmann equation.
$\phi_{disc}$ is the correction due to the discreteness of surface groups:

$\beta\phi_{disc}=-\frac{\lambda_{b}MQ^{t}}{qa\epsilon_{w}\sqrt{Z}},$ (18)

where $M$ is the Madelung constant for hexagonal crystal state of the one
component plasma [39]. The excess chemical potential of ions in the reservoir
$\mu_{ex}$ can be approximated as the sum of the mean spherical approximation
(MSA) chemical potential and the Carnahan-Starling expression for the excluded
volume contribution [61, 62, 63, 64, 65, 66, 67, 68, 69], which are:

$\beta\mu_{MSA}=\frac{\lambda_{B}\left(\sqrt{1+2\kappa d}-\kappa
d-1\right)}{d^{2}\kappa},\qquad\beta\mu_{CS}=\frac{8\eta-9\eta^{2}+3\eta^{3}}{\left(1-\eta\right)^{3}},\qquad$
(19)

where $\eta=\frac{\pi d^{3}}{3}c_{t}$, $d$ is the ionic diameter,
$c_{t}=c_{s}+c_{a}$ is the total concentration of salt and acid,
$\lambda_{B}=q^{2}/\epsilon_{w}k_{B}T$ is the Bjerrum length, and
$\kappa=\sqrt{8\pi\lambda_{B}c_{t}}$ is the inverse Debye length.

At the same level of approximation, the fluctuation of surface charge can be
written as:

$\sigma_{t}^{2}=\frac{Z~{}\text{K}_{eq}c_{a}\mathrm{e}^{\left(-\beta[\phi_{0}-\phi_{disc}-\mu^{ex}-\mu_{sol}]\right)}}{\left(1+\text{K}_{eq}c_{a}\mathrm{e}^{\left(-\beta[\phi_{0}-\phi_{disc}-\mu^{ex}-\mu_{sol}]\right)}\right)^{2}},$
(20)

Comparison of theory with the sGCMC results are shown in Table 4. We see that
the theory again agrees well with the colloidal charge calculated in
simulations, but the fluctuations in the nanoparticle charge differ
significantly from the predictions of the theory. This is similar to what was
found in the ideal case when comparing theory with sGCMC simulations that used
pair insertions. The pair insertion algorithm restricts charge fluctuations
inside the simulation cell, affecting also the fluctuations of nanoparticle
charge.

Table 4: Colloidal charge and its fluctuations for interacting system: theory (t) and simulations (s). The Table also presents the average number of ions and free protons inside the cell. The same numbers are used to perform canonical simulations, results of which are presented in the Table 5. $C$ M | $Q^{s}/q$ | $\sigma_{s}^{2}$ |  $N_{\text{H}}$+ |  $N_{\text{Na}}$+ |  $N_{\text{Cl}}$- | $Q^{t}/q$ | $\sigma^{2}_{t}$
---|---|---|---|---|---|---|---
$10^{-5}$ | $-66.7$ | 29.0 | 533 | 112 | 45 | $-65.4$ | 58.2
$10^{-5.5}$ | $-106.6$ | 39.4 | 493 | 145 | 39 | $-102.9$ | 85.1
$10^{-6}$ | $-160.6$ | 52.4 | 439 | 195 | 34 | $-152.9$ | 113.8
|  |  |  |  |  |  |

We next run the canonical reactive Monte Carlo algorithm, Eq. (3.2). The
simulation cell contains the same number of ions and protons as was obtained
using sGCMC simulations previously. After the simulation reaches equilibrium,
we see that the average charge of a nanoparticle calculated using canonical
simulation is in perfect agreement with the results of the semi-grand
canonical algorithm. On the other hand, canonical fluctuations of the
nanoparticle charge are almost 4 orders of magnitude lower than what was found
for an open system, see Table 5.

Table 5:  $Q$ and its fluctuations in the canonical ensembles for an interacting system. Note that the fluctuations are four orders of magnitude smaller than observed in the sGCMC case for exactly the same parameters, compare with Table 4. $N_{\text{H}}$+ |  $N_{\text{Na}}$+ |  $N_{\text{Cl}}$- | $Q^{s}/q$ | $\sigma^{2}$
---|---|---|---|---
533 | 112 | 45 | $-67.11$ | 0.11
493 | 145 | 39 | $-106.04$ | 0.04
439 | 195 | 34 | $-161.01$ | 0.01
|  |  |  |

##  5 Conclusions

In this paper we investigated charge regulation in isolated (canonical) and
open (semi-grand canonical) systems. An open system can exchange heat, ions,
and protons with an external reservoir, while a closed system can only
exchange heat. In both cases, equivalence of ensembles extends to the
prediction of the effective charge of nanoparticles – both ensembles predict
exactly the same charge. On the other hand, the fluctuations of colloidal
charge are very different, with canonical fluctuations 4 orders of magnitude
smaller than the semi-grand canonical ones. Since the Kirkwood-Shumaker force
depends on charge fluctuations, its manifestation should be very different in
the two ensembles, in particular for concentrated suspensions. At infinite
dilution, we expect the difference between the two ensembles to vanish. In the
future work, we will use the simulation methods discussed in this paper to
explicitly calculate Kirkwood-Shumaker force between nanoparticles in
different ensembles.

Acknowledgments

This work was partially supported by the CNPq, the CAPES, and the National
Institute of Science and Tech- nology Complex Fluids INCT-FCx.

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11institutetext: LIPN, Université Sorbonne Paris Nord, France 22institutetext:
Dept. of Computer Science, University College London, UK

# Multi-modalities and non-commutativity/associativity in functorial linear
logic: a case study

Carlos Olarte 11 Elaine Pimentel 22

##  1 Introduction

Similar to modal connectives, the exponential $\mathop{!}$ in intuitionistic
linear logic ($\mathsf{ILL}$) is not canonical [3], in the sense that if
$i\not=j$ then $\hbox{$\mathop{!}^{i}$}F\not\equiv\hbox{$\mathop{!}^{j}$}F$.
Intuitively, this means that we can mark the exponential with labels taken
from a set $I$ organized in a pre-order $\preceq$ (i.e., reflexive and
transitive), obtaining (possibly infinitely-many) exponentials
($\mathop{!}^{i}$ for $i\in I$). Also as in multi-modal systems, the pre-order
determines the provability relation: for a general formula $F$,
$\hbox{$\mathop{!}^{b}$}F$ implies $\hbox{$\mathop{!}^{a}$}F$ iff $a\preceq
b$.

The algebraic structure of subexponentials, combined with their intrinsic
structural property allow for the proposal of rich linear logic based
frameworks. This opened a venue for proposing different multi-modal
substructural logical systems, that encountered a number of different
applications.

There are, however, two main differences between multi-modalities in normal
modal logics and subexponentials in linear logic.

i. substructural behaviour. Subexponentials carry the possibility of having
different structural behaviors;

ii. nature of modalities. Normal modal logics start from the weakest version,
assuming only axiom $\mathsf{K}$, then extensions are considered, by adding
other axioms. Exponentials in linear logic“take for granted” the behaviors
expressed by axioms $\mathsf{T}$ and $\mathsf{4}$.

Regarding (i), originally [11], subexponentials could assume only weakening
and contraction axioms, while in [5, 6], non-commutative systems allowing
commutative subexponentials were presented

$\mathsf{C}:\;\;\hbox{$\mathop{!}^{i}$}F\mathbin{-\circ}\hbox{$\mathop{!}^{i}$}F\otimes\hbox{$\mathop{!}^{i}$}F\qquad\mathsf{W}:\;\;\hbox{$\mathop{!}^{i}$}F\mathbin{-\circ}1\qquad\mathsf{E}:\;\;(\hbox{$\mathop{!}^{i}$}F)\otimes
G\equiv G\otimes(\hbox{$\mathop{!}^{i}$}F)$

Finally, in [2] associativity was added to the scene, where exponentials could
assume or not the axioms

$\mathsf{A1}:\;\;\hbox{$\mathop{!}^{i}$}F\otimes(G\otimes
H)\mathbin{-\circ}(\hbox{$\mathop{!}^{i}$}F\otimes G)\otimes
H\qquad\mathsf{A2}:\;\;(G\otimes
H)\otimes\hbox{$\mathop{!}^{i}$}F\mathbin{-\circ}G\otimes(H\otimes\hbox{$\mathop{!}^{i}$}F)$

Concerning (ii), Guerrini et al. [4] unified the modal and $\mathsf{LL}$
approaches, with the exponentials assuming only the linear version of
$\mathsf{K}$, with the possibility of adding modal extensions to it. This
discussion was brought to multi-modal case in [8], where subexponentials
consider not only the structural axioms for contraction and weakening, but
also the subexponential version of axioms
$\\{\mathsf{K},\mathsf{4},\mathsf{D},\mathsf{T}\\}$:

$\mathsf{K}:\;\;\hbox{$\mathop{!}^{i}$}(F\mathbin{-\circ}G)\mathbin{-\circ}\hbox{$\mathop{!}^{i}$}F\mathbin{-\circ}\hbox{$\mathop{!}^{i}$}G\qquad\mathsf{D}:\;\;\hbox{$\mathop{!}^{i}$}F\mathbin{-\circ}\hbox{$\mathord{?}^{i}$}F\qquad\mathsf{T}:\;\;\hbox{$\mathop{!}^{i}$}F\mathbin{-\circ}F\qquad\mathsf{4}\>;\;\hbox{$\mathop{!}^{i}$}F\mathbin{-\circ}\hbox{$\mathop{!}^{i}$}\hbox{$\mathop{!}^{i}$}F$

In this work, we intend to join these two studies. This means that
$\mathord{?}^{i}$ can behave classically or not, model associative and
commutative systems or not, but also with exponential behaviors different from
those in $\mathsf{LL}$. Hence, by assigning different modal axioms one
obtains, in a modular way, a class of different substructural modal logics.

##  2 Non-commutative, non-associative linear logic ($\mathsf{acLL}_{\Sigma}$)

In the following, we will briefly describe the system
$\mathsf{acLL}_{\Sigma}$. For more details, please refer to [2].

Connectives. First of all, non-commutativity implies that the left residual
$\mathrel{\mbox{$\circ\\!-$}}$ should be added to the language of
(propositional intuitionistic) linear logic with subexponentials
($\mathsf{SELL}$), which contains the connectives
$\otimes,\mathbin{\&},\oplus,1,0,\mathbin{-\circ},\hbox{$\mathop{!}^{l}$},\hbox{$\mathord{?}^{l}$}$.
The subexponentials are organized as follows.

######  Definition 1 (Signature)

Let
$\mathcal{A}=\\{\mathsf{C},\mathsf{W},\mathsf{A1},\mathsf{A2},\mathsf{E}\\}$.
A subexponential signature is given by a triple $\Sigma=(I,\preccurlyeq,f)$,
where $I$ is a set of indices, $(I,\preccurlyeq)$ is a pre-order, and $f$ is a
mapping from $I$ to $2^{\mathcal{A}}$. Finally, every signature $\Sigma$ is
assumed to be upwardly closed w.r.t. $\preceq$, that is, if $i\preceq j$ then
$f(i)\subseteq f(j)$ for all $i,j\in I$.

Contexts and rules. Losing commutativity implies that contexts should be
handled by lists instead of multisets, and the lack of associativity forces
tree-shaped contexts. Rules then act deeply in these structures. For example,
the promotion and exchange rules are

$\Gamma\Rightarrow\hbox{$\mathop{!}^{i}$}F\Gamma^{{\uparrow\\!\\!({i})}}\Rightarrow
F\qquad\Gamma\mathopen{}\left\\{(\hbox{$\mathop{!}^{e}$}\Delta_{1},\Delta_{2})\right\\}\mathclose{}\Rightarrow
G\Gamma\mathopen{}\left\\{(\Delta_{2},\hbox{$\mathop{!}^{e}$}\Delta_{1})\right\\}\mathclose{}\Rightarrow
G\qquad\Gamma\mathopen{}\left\\{(\Delta_{1},\hbox{$\mathop{!}^{e}$}\Delta_{2})\right\\}\mathclose{}\Rightarrow
G\Gamma\mathopen{}\left\\{(\hbox{$\mathop{!}^{e}$}\Delta_{2},\Delta_{1})\right\\}\mathclose{}\Rightarrow
G$

where ${\uparrow\\!\\!({i})}$ denotes the _upset_ of the index $i$, i.e., the
set $\\{j\in I:i\preccurlyeq j\\}$.

###### Example 1

Let
$\Gamma=(\hbox{$\mathop{!}^{i}$}A,(\hbox{$\mathop{!}^{j}$}B,\hbox{$\mathop{!}^{k}$}C))$
be represented below left, $i\preceq j$ but $i\not\preceq k$, and
$\mathsf{W}\in f(k)$. Then
$\Gamma^{{\uparrow\\!\\!({i})}}=(\hbox{$\mathop{!}^{i}$}A,\hbox{$\mathop{!}^{j}$}B)$
is depicted below right

pt,pt$\hbox{$\mathop{!}^{i}$}A$pt,pt$\hbox{$\mathop{!}^{j}$}B$pt$\hbox{$\mathop{!}^{k}$}C$pt,pt$\hbox{$\mathop{!}^{i}$}A$pt$\hbox{$\mathop{!}^{j}$}B$

Observe that, if $\mathsf{W}\notin f(k)$, then
$\Gamma^{{\uparrow\\!\\!({i})}}$ cannot be built. In this case, any derivation
of $\Gamma\Rightarrow\hbox{$\mathop{!}^{i}$}(A\otimes B)$ cannot start with an
application of the promotion rule $\hbox{$\mathop{!}^{i}$}R$ (similarly to how
promotion in $\mathsf{ILL}$ cannot be applied in the presence of non-classical
contexts). In this case, if $A,B$ are atomic, this sequent would not be
provable.

##  3 Linear nested systems

In this section we present a brief introduction to linear nested systems. For
further details, refer to ([7, 9]).

One of the main problems of using sequent systems as a framework is that
sequents are often not adequate for expressing modal behaviors. In order to
propose a better formulation, we need a tighter control of formulas in the
context, something that sequents do not provide. Hence the need for extending
the notion of sequent systems.

######  Definition 2 (LNS)

The set $\mathsf{LNS}$ of _linear nested sequents_ is given recursively by:

(i) if $\Gamma\Rightarrow\Delta$ is a sequent then
$\Gamma\Rightarrow\Delta\in\mathsf{LNS}$

(ii) if $\Gamma\Rightarrow\Delta$ is a sequent and
$\mathcal{G}\in\mathsf{LNS}$ then
$\Gamma\Rightarrow\Delta\mathbin{\\!/\mkern-5.0mu/\\!}\mathcal{G}\in\mathsf{LNS}$.

We call each sequent in a linear nested sequent a component and slightly abuse
notation, abbreviating “linear nested sequent” to $\mathsf{LNS}$. We shall
denote by $\mathsf{LNS}_{\mathcal{L}}$ a linear nested sequent system for a
logic $\mathcal{L}$.

In words, a linear nested sequent is simply a finite list of sequents that
matches exactly the _history_ of a backward proof search in an ordinary
sequent calculus ([7]).

A further advantage of this framework is that it is often possible to restrict
the list of sequents in a $\mathsf{LNS}$ to the last 2 components, that we
call active.

######  Definition 3 (End-active)

An application of a linear nested sequent rule is _end-active_ if the
rightmost components of the premises are active and the only active components
(in premise and conclusion) are the two rightmost ones. The _end-active
variant_ of a $\mathsf{LNS}$ calculus is the calculus with the rules
restricted to end-active applications.

###  3.1 $\mathsf{LNS}$ for multi-modal $\mathsf{LL}$

In the quest for locality, [4] proposed 2-sequents systems for $\mathsf{LL}$
variants, with separate rules for the exponentials. In [8] this work was
revisited, establishing a lighter notation and extending the discussion to
multi-modalities.

$\mathsf{LNS}_{\mathsf{LL}}$ ([8]) is an end-active, linear nested system for
linear logic. In this system, the promotion rule is split into the following
local rules:

${\cal
E}\mathbin{\\!/\mkern-5.0mu/\\!}\Gamma\Rightarrow!F\vdash\Gamma\Rightarrow\mathbin{\\!/\mkern-5.0mu/\\!}\Rightarrow
F\qquad\Gamma,\mathop{!}F\Rightarrow
H\mathbin{\\!/\mkern-5.0mu/\\!}\Delta\Rightarrow G\Gamma\Rightarrow
H\mathbin{\\!/\mkern-5.0mu/\\!}\Delta,\mathop{!}F\Rightarrow G$

Observe that no checking must be done in the context in order to apply the
$\mathop{!}L$ rule: The only checking is in the $\mathop{!}R$ rule, where
${\cal E}$ should be the empty sequent or an empty list of components.

More precisely, applying the $\mathop{!}$ rule enables the creation of the
future history, in which the banged formula should be proved. The intended
interpretation of a $\mathsf{LNS}$ in $\mathsf{LL}$ is

$\begin{array}[]{lcl}\iota(\Gamma\Rightarrow
F)&:=&\bigotimes\Gamma\mathbin{-\circ}F\\\ \iota(\Gamma\Rightarrow
F\mathbin{\\!/\mkern-5.0mu/\\!}\mathcal{H})&:=&\bigotimes(\Gamma\mathbin{-\circ}F)\bindnasrepma\;\mathop{!}\iota(\mathcal{H})\end{array}$

The notion of signatures is then enhanced. We say that
$\Sigma=(I,\preccurlyeq,f)$ is a functorial signature if it is defined over
$\mathcal{A}=\\{\mathsf{C},\mathsf{W},\mathsf{A1},\mathsf{A2},\mathsf{E},\mathsf{K},\mathsf{4},\mathsf{T},\mathsf{D}\\}$
and $\mathsf{K}\in f(i),\;\forall i\in I$. If, moreover,
$\mathsf{A1},\mathsf{A2}\in f(i),\;\forall i\in I$, then the functorial
signature is called associative.

####  3.1.1 The associative case.

In the presence of associativity, contexts are lists of formulas, and the tree
structure is not needed. Given an associative signature $\Sigma$, we introduce
nesting operators $\mathbin{\\!/\mkern-5.0mu/^{i}\\!}$ and their unfinished
versions $\mathbin{\\!\backslash\mkern-5.0mu\backslash^{i}\\!}$ for every
$i\in I$, and change the interpretation so that they are interpreted by the
corresponding modality:

$\displaystyle\iota(\Gamma\Rightarrow F)$
$\displaystyle:=\bigotimes\Gamma\mathbin{-\circ}F$
$\displaystyle\iota(\Gamma\Rightarrow
F\mathbin{\\!/\mkern-5.0mu/^{i}\\!}\mathcal{H})$
$\displaystyle:=\iota(\Gamma\Rightarrow
F\mathbin{\\!\backslash\mkern-5.0mu\backslash^{i}\\!}\mathcal{H}):=(\bigotimes\Gamma\mathbin{-\circ}F)\;\bindnasrepma\hbox{$\mathop{!}^{i}$}\,\iota(\mathcal{H})$

The operators $\mathbin{\\!\backslash\mkern-5.0mu\backslash^{i}\\!}$ indicate
that the standard sequent rule for the modality indexed by $i$ has been
partially processed as shown below.

The exponential rules for the end-active linear nested system
$\mathsf{LNS}_{(I,\preccurlyeq,F)}$ is given by the following rules

$\Gamma,\hbox{$\mathop{!}^{j}$}F\Rightarrow
G\mathbin{\\!\backslash\mkern-5.0mu\backslash^{i}\\!}\Delta\Rightarrow
H\Gamma\Rightarrow
G\mathbin{\\!\backslash\mkern-5.0mu\backslash^{i}\\!}F,\Delta\Rightarrow
H\qquad\Gamma\Rightarrow\hbox{$\mathop{!}^{i}$}F\Gamma\Rightarrow\mathbin{\\!\backslash\mkern-5.0mu\backslash^{i}\\!}\Rightarrow
F\qquad\mathcal{E}\mathbin{\\!\backslash\mkern-5.0mu\backslash^{i}\\!}\Gamma\Rightarrow
F\Gamma\Rightarrow F$ $\Gamma,\hbox{$\mathop{!}^{j}$}F\Rightarrow
G\mathbin{\\!\backslash\mkern-5.0mu\backslash^{i}\\!}\Delta\Rightarrow
H\Gamma\Rightarrow
G\mathbin{\\!\backslash\mkern-5.0mu\backslash^{i}\\!}\hbox{$\mathop{!}^{j}$}F,\Delta\Rightarrow
H\quad\Gamma,\hbox{$\mathop{!}^{i}$}F\Rightarrow G\Gamma\Rightarrow
G\mathbin{\\!\backslash\mkern-5.0mu\backslash^{i}\\!}F\Rightarrow$
$\Gamma\mathopen{}\left\\{\hbox{$\mathop{!}^{i}$}F\right\\}\mathclose{}\Rightarrow
G\Gamma\mathopen{}\left\\{F\right\\}\mathclose{}\Rightarrow G$

Note that the order of formulas in the context is respected in the application
of the rules.

###  3.2 The non-associative case

Since $\mathsf{LNS}$ do not preserve tree structures, the system just proposed
is not adequate for the non-associative setting. In this case, we propose a
skeleton that keeps track of the shape of the structure as follows.

Given a structure $\Gamma$, we represent by $\Gamma^{\circ}$ its underlying
tree structure, where the leaves of $\Gamma$ are substituted by empty contexts
$\\{\;\\}$ (holes). $\Gamma^{\circ}\\{\\}$ will then represent
$\Gamma^{\circ}$ with a specific position highlighted, and the usual context
substitution can be applied. That is, given an underlying structure
$\Gamma^{\circ}$, we write
$\Gamma^{\circ}\mathopen{}\left\\{F\right\\}\mathclose{}$ for the context
where some specific hole $\Gamma^{\circ}\mathopen{}\left\\{\mkern
3.0mu\right\\}\mathclose{}$ has been replaced by $F$. If
$\Gamma^{\circ}=\varnothing$ the hole is removed.

The $\mathsf{LNS}$ rules are then adapted from the rules of the associative
case, for example

$\Gamma\mathopen{}\left\\{\hbox{$\mathop{!}^{j}$}F\right\\}\mathclose{}\Rightarrow
G\mathbin{\\!\backslash\mkern-5.0mu\backslash^{i}\\!}\Gamma^{\circ}\mathopen{}\left\\{\mkern
3.0mu\right\\}\mathclose{}\Rightarrow H\Gamma\mathopen{}\left\\{\mkern
3.0mu\right\\}\mathclose{}\Rightarrow
G\mathbin{\\!\backslash\mkern-5.0mu\backslash^{i}\\!}\Gamma^{\circ}\mathopen{}\left\\{F\right\\}\mathclose{}\Rightarrow
H\qquad\Gamma\Rightarrow\hbox{$\mathop{!}^{i}$}F\Gamma\Rightarrow\mathbin{\\!\backslash\mkern-5.0mu\backslash^{i}\\!}\;\Gamma^{\circ}\Rightarrow
F\qquad\mathcal{E}\mathbin{\\!\backslash\mkern-5.0mu\backslash^{i}\\!}\Gamma^{\circ}\Rightarrow
F\Gamma\Rightarrow F$
$\Gamma\mathopen{}\left\\{\hbox{$\mathop{!}^{i}$}F\right\\}\mathclose{}\Rightarrow
G\Gamma\Rightarrow
G\mathbin{\\!\backslash\mkern-5.0mu\backslash^{i}\\!}\Gamma^{\circ}\mathopen{}\left\\{F\right\\}\mathclose{}\Rightarrow$

###### Example 2

Let $\Gamma$ be as in Example 1, and suppose that
$j\in{\uparrow^{\mathsf{4}}\\!\\!({i})},\mathsf{4}\notin f(i)$ and
$i\not\preceq k$. Below are possible configurations of $\Gamma^{\preceq i}$
after applying a $\mathsf{K}\mathsf{4}_{i}$ rule

,$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hbox{$\mathop{!}^{j}$}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
,$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

##  4 Conclusion

In this ongoing work, we propose a modular system for dealing with multi-modal
$\mathsf{LL}$ systems, where substructural and modal axioms are taken into
account.

The motivation for that is twofold: (1) to expand the role of $\mathsf{LL}$ as
a framework for reasoning about systems, as in [10, 13, 15] and, mainly, (2)
extending the computational interpretation of subexponentials [12, 14] also to
the non-associative/commutative case. We intend to pursue these directions in
the near future.

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