# Nonequilibrium free energy during polymer chain growth

Michael Bley<EMAIL_ADDRESS>Applied Theoretical Physics
– Computational Physics, Physikalisches Institut, Albert-Ludwigs-Universität
Freiburg, Hermann-Herder Strasse 3, D-79104 Freiburg,Germany  Joachim
Dzubiella<EMAIL_ADDRESS>Applied Theoretical
Physics – Computational Physics, Physikalisches Institut, Albert-Ludwigs-
Universität Freiburg, Hermann-Herder Strasse 3, D-79104 Freiburg,Germany
Cluster of Excellence livMats@FIT - Freiburg Center for Interactive Materials
and Bioinspired Technologies, Albert-Ludwigs-Universität Freiburg, Georges-
Köhler-Allee 105, D-79110 Freiburg, Germany

###### Abstract

During fast diffusion-influenced polymerization, nonequilibrium behavior of
the polymer chains and the surrounding reactive monomers has been reported
recently. Based on the laws of thermodynamics, the emerging nonequilibrium
structures should be characterisable by some ’extra free energy’ (excess over
the equilibrium Helmholtz free energy). Here, we study the nonequilibrium
thermodynamics of chain-growth polymerization of ideal chains in a dispersion
of free reactive monomers, using off-lattice, reactive Brownian Dynamics
(R-BD) computer simulations in conjunction with approximative statistical
mechanics and relative entropy (Gibbs-Shannon and Kullback-Leibler) concepts.
In case of fast growing polymers, we indeed report increased nonequilibrium
free energies $\Delta F_{\rm neq}$ of several $k_{\mathrm{B}}T$ compared to
equilibrium and near-equilibrium, slowly growing chains. Interestingly,
$\Delta F_{\rm neq}$ is a non-monotonic function of the degree of
polymerization and thus also of time. Our decomposition of the thermodynamic
contributions shows that the initial dominant extra free energy is stored in
the nonequilibrium inhomogeneous density profiles of the free monomer gas
(showing density depletion and wakes) in the vicinity of the active center at
the propagating polymer end. At later stages of the polymerization process, we
report significant extra contributions stored in the nonequilibrium polymer
conformations. Finally, our study implies a nontrivial relaxation kinetics and
’restoring’ of the extra free energy during the equilibration process after
polymerization.

##  I  Introduction

Recently, we studied the nonequilibrium chain growth of single polymers by
particle-resolved, reactive Brownian Dynamics (R-BD) simulations. (Bley, Baul,
and Dzubiella, 2021) We demonstrated that fast nonequilibrium processing
pathways lead to remarkably extended structures showing enhanced scaling
exponents $\nu$ for the end-to-end distances, $R\propto N^{\nu}$, as a
function of the degree of polymerization, $N$. In particular, we found good-
solvent scaling, $\nu\approx 3/5$, although we polymerized in ideal or
$\theta$-conditions. For slow polymerization, we observed the expected
$\nu\approx 1/2$ random walk behavior for ideal or $\theta$-chains in
equilibrium. Rubinstein and Colby (2003) Different to the stretching of Rouse
chains by an external force, Liphardt (2002); Speck and Seifert (2005) the
nonequilibrium extended, and thus ’stressed’ polymer conformations resulted
from spatio-temporal correlations of both the chain and the free reactive
monomers in fast, diffusion-influenced reactions for which monomers and chain
segments had not enough time to relax during processing.

In this work, we wish to address the possible thermodynamic consequences of
these intriguing nonequilibrium structures. As is well known, the
conformational entropy controls various aspects of polymer chains and arises
from the amount of different pathways a linear polymer chain can sample for a
given end-to-end vector $\bm{R}$. de Gennes (1979); Rubinstein and Colby
(2003) External and chemical forces can cause significant modifications of the
typical Gaussian probability distributions of end-to-end distances $P(R)$ with
$R=|\bm{R}|$, and more extended chains with fixed $N$ decrease the number of
accessible realizations and thus lead to a decrease in conformational entropy
with $\Delta S_{\mathrm{conf}}<0$. Hence, out-of-equilibrium chains
temporarily store a significant amount of free energy as long as they remain
in an entropically unfavorable state. Upon relaxation back to their
corresponding equilibrium state, free energy is released from the chain(s)
through work or dissipation on different time scales and at different
magnitudes. Rouse (1953); Rubinstein and Colby (2003); Strobl (2007);
Alexander-Katz, Wada, and Netz (2009); Einert  _et al._ (2011); Latinwo,
Hsiao, and Schroeder (2014); Reiter (2020); Ramezani, Baschnagel, and Reiter
(2020); Chandran  _et al._ (2019)

In fact, it has been shown experimentally, Ramezani, Baschnagel, and Reiter
(2020) for example, that polymer melts (Polystyrene filaments) are capable in
lifting macroscopic loads under certain preparation conditions. It is thus
envisioned that elaborate synthesis and processing pathways can govern the
magnitude of the ’extra’ free energy (i.e., excess over equilibrium) of non-
equilibrated polymer chains. Chandran  _et al._ (2019) In terms of information
theory such processes encode so-called ’memory’ in polymers. Reiter (2020)
This memorized information of a chain is the stored information of in-
equilibrium unlikely or inaccessible conformations, and can be retrieved
through suited stimuli. These preparation-induced meta-stable states can
exhibit long lifetimes and require high temperatures for sufficiently fast
relaxation to return to equilibrium states.

In general, growing attention has been drawn to investigate nonequilibrium
polymer properties, how to conserve and control them, and their consequences
on material design. In particular, the possibility emerges to potentially
harvest the stored, extra free energy, and related stresses and memory for the
design of highly responsive, interactive, or even adaptive materials. Stuart
_et al._ (2010); Thomas  _et al._ (2011); Liu  _et al._ (2018); Walther (2020)
The anticipated wide range of new structural, dynamical, and mechanical
properties arises from the nonequilibrium competition between the time scales
of (reaction) synthesis, processing, polymer relaxation, and observation and
function. de Gennes (1982); Latinwo, Hsiao, and Schroeder (2014); Chandran
_et al._ (2019); Reiter (2020); Guérin, Bénichou, and Voituriez (2012); Katkar
and Muthukumar (2018); Chubak  _et al._ (2020); Bley, Baul, and Dzubiella
(2021) However, it is difficult from a theoretical point of view how to define
and calculate a nonequilibrium free energy of a growing polymer chain. Apart
from the general challenges to define notions like entropy and heat in
nonequilibrium, De Groot and Mazur (1962); Prigogine (1978); Seifert (2005,
2012); Evans, Cohen, and Morriss (1993); Jarzynski (1997a, b); Crooks (1999)
numerical, particle-resolved studies are hampered by the fact that one has to
gather time-dependent averages of a complex system with spatiotemporal
inhomogeneities.

In this work, we attempt to address this problem and use the model framework
of our previous study (Bley, Baul, and Dzubiella, 2021) to investigate the
thermodynamics of polymer topologies in nonequilibrium. Our model is
appropriately minimalistic (e.g., ideal gas, ideal chains, harmonic bonds)
that a nonequilibrium analysis stays relatively transparent with well known
equilibrium limits. The formation of (covalent) bonds during the
polymerization process can be characterized as an irreversible thermodynamic
process and the growing chain can be considered an entropy sink (energy is put
in or generated, while entropy decreases), which follows the second law of
thermodynamics. De Groot and Mazur (1962); Prigogine (1978); Seifert (2012) To
calculate the entropy change during the polymerization we then use the most
accepted concepts of the definition of a nonequilibrium entropy based on the
works of Gibbs Gibbs (1902) and Shannon Shannon (1948), leading to the
introduction of a concept of ’relative entropy’ (or Kullback-Leibler
divergence Kullback and Leibler (1951)) for the comparison of equilibrium to
nonequilibrium distributions. Altaner (2017)

In fact, in polymer physics, Wall derived already 80 years ago the ’Wall
equation’ for the change in entropy under certain nonequilibrium conditions,
Wall (1942) revisited later from various perspectives Dayantis (1995); Qian
(2001) and consistent with the Gibbs-Shannon and Kullback-Leibler forms. It
was recently applied, for example, for calculating the nonequilibrium entropy
of a flowing polymer melt via atomistic simulations, Edwards, Nafar
Sefiddashti, and Khomami (2021) or the kinetics and thermodynamics of
reversible polymerization in closed systems using a master equation approach.
Lahiri  _et al._ (2015) Moreover, direct experimental measurements of a
colloidal particle kept in weak nonequilibrium with a virtual double-well
potential created by a feedback trap recently demonstrated the utility of the
Gibbs-Shannon and Kullback-Leibler concepts for another stochastic molecular
system. Gavrilov, Chétrite, and Bechhoefer (2017)

Here, we employ these concepts to compare nonequilibrium polymer conformations
and the heterogeneous distributions of the surrounding free monomer
environment with the corresponding equilibrium reference states to calculate
the relative energetic and entropic differences, culminating in the
nonequilibrium extra free energy $\Delta F_{\mathrm{\rm neq}}(N)$ as a
function of the time-dependent degree of polymerization $N(t)$. For a full
understanding of the system we also consider the equilibrium entropic and
energetic cost of transferring free monomers into a polymer chain with respect
to the formed (harmonic) bonds. Our final results show significant
contributions from both nonequilibrium polymer conformations as well as free
monomer distributions to the total nonequilibrium transfer free energy (work)
in the system, and suggests a complex time dependence of this extra free
energy as well of its relaxation kinetics back to equilibrium after the
termination of the polymerization.

##  II Model and simulations

Our chain polymerization study is based on reactive Brownian dynamics (R-BD)
computer simulations as described in detail previously. Bley, Baul, and
Dzubiella (2021) All simulations have been performed with the LAMMPS package
Plimpton (1995) using a standard overdamped BD algorithm for the translational
dynamics. In our model, we consider $M(t)=M_{0}-N(t)-1$ free reactive monomers
and $N(t)$ polymer monomers in a fixed volume $V$ at time $t$. $N(t)$ is the
time-dependent degree of polymerization. The initial amount of free monomers
at time $t=0$ is $M(0)=M_{0}=1000$ and in addition we consider one polymer
seed $N(0)=N_{0}=1$, cf. Fig. 1(a). All monomers (free and polymer) are not
interacting, so that we consider the free monomers as an ideal gas, and the
polymers as ideal chains in our study.

The free monomers can diffuse and react with one reactive polymer end – called
in the following the active center (AC) – to form and grow a single linear
chain of $N(t)>1$ polymer monomers at time $t$, cf. Fig. 1(b). The diffusion
coefficient for all monomers (free and bound within the chain) is set to
$D_{0}=\sigma^{2}/\tau_{\mathrm{B}}$. The size $\sigma=1.0$ defines our unit
for length. The Brownian time $\tau_{\mathrm{B}}$ fixes our unit of time. We
employ an integration time step of $\Delta t=10^{-5}\tau_{\mathrm{B}}$. We
simulate at fixed temperature $T$, and our energy unit is the thermal energy
$k_{\mathrm{B}}T$, with $k_{\mathrm{B}}$ being the Boltzmann constant. Our
statistical ensemble is thus canonical with constant $V$, $T$, and the total
amount of all (free and polymer) monomers $M_{0}+1$.

The bonds between two chain monomers $i$ and $j$ in the polymer (see again
Fig. 1(b)) are described through a harmonic bond potential of the form

$u_{ij}(b_{ij})=\frac{K_{\mathrm{bond}}}{2}(b_{ij}-b_{0})^{2},$ (1)

where $b_{ij}=|\bm{b}_{ij}|$ is the magnitude of the bond vector,
$K_{\mathrm{bond}}=20\varepsilon\sigma^{-2}$ is the spring constant, and
$b_{0}=1.0\sigma$ is the zero-temperature bond length. We fix the energy to
$\varepsilon=1k_{\mathrm{B}}T$. No other bonded and non-bonded interactions
are considered here, as in the ideal-chain Rouse model. Rouse (1953); Doi and
Edwards (1986) Note, however, that there are two important differences to the
Rouse model: i) we interpret our bonded potentials as energetic, not entropic,
thus they store potential energy. ii) we employ a non-vanishing, zero-
temperature bond extension $b_{0}$ with a relatively stiff spring constant.
Hence, our chain model is actually closer to a freely jointed chain (FJC)
model Rubinstein and Colby (2003); Strobl (2007) (with segment length $b_{0}$)
than to the Rouse chain. Rouse (1953); Doi and Edwards (1986)

The polymer grows with a prescribed reaction probability if a free gas monomer
is close to the AC, depicted as yellow bead in Fig. 1(b). For simulating the
chain growth for the reaction $N\rightarrow N+1$, the algorithm checks every
$(\lambda\Delta t)^{-1}$ integration time steps with a probability
$p_{\mathrm{react}}=1.0$ if a reaction is possible. Here, $\lambda$ has the
unit per time (i.e., is a rate) and is the reaction propensity following the
Doi scheme. Doi (1975); Erban and Chapman (2009) The parameter $\lambda$ thus
controls the speed of the polymerization. A cut-off protocol method de Buyl
and Nies (2015) as implemented in the LAMMPS software package Plimpton (1995)
checks if a bond formation between the closest free monomer found within a
spherical reactive volume of radius $R_{c}=\sqrt[6]{2}\sigma\approx
1.122\sigma$ around the AC is possible. Once the a new bond has been formed,
the newly added monomer becomes the AC, and the previous AC is deactivated for
any further reaction.

We consider in our study two reaction speeds: ’slow’ and ’fast’, as
characterized in detail previously. Bley, Baul, and Dzubiella (2021) The fast
reaction has a reaction propensity $\lambda\Delta t=10^{-3}$
($\lambda\tau_{\mathrm{B}}=10^{2}$), while for slow simulations $\lambda\Delta
t=10^{-5}$ ($\lambda\tau_{\mathrm{B}}=1$). For fast growth simulations, more
than 4000 different trajectories of a total simulation length of
$5\tau_{\mathrm{B}}$ have been collected and analyzed, whereas for the slow
growth 350 simulations of length $250\tau_{\mathrm{B}}$ were considered.

We fix the volume of the periodically repeated cubic simulation box to
$V=(15\sigma)^{3}$. In all our polymerization simulations the initial density
of free monomers $\rho_{0}=M_{0}/V\simeq 0.297\sigma^{-3}$ with $M_{0}=1000$
is used. We also conduct equilibrium simulations of chains with fixed $N=10$,
$20$, $50$, $100$, $150$, $250$, $500$ and $1000$ and the corresponding
monomer number $M=M_{0}-N-1$ for $50$ up to $5000~{}\tau_{\mathrm{B}}$ to
compare one-to-one to the nonequilibrium situation characterized by $M(t)$ and
$N(t)$.

Figure 1: (a) Initial reference state for polymerization: Free (non-bonded)
monomers (green) constitute a reactive ideal gas of initial density
$\rho_{0}=(M_{0})/V$. The yellow bead is the polymerization seed, i.e., the
initial active center (AC) $N(0)=1$ for polymerization towards $N>1$. (b)
Intermediate polymerized state: A few free monomers of (a) became part of the
polymer of $N(t)$ monomers with an AC (yellow) of a reactive radius $R_{c}$
and $N-1$ reacted chain beads (blue) connected through harmonic bonds (gray)
with bond vectors $\bm{b}_{ij}$ (pink).

##  III Nonequilibrium polymerization thermodynamics

Our thermodynamic analysis of the changes of the free energy in the system
during polymerization is based on the following considerations. At first, we
consider an unpolymerized equilibrium reference state (Fig. 1(a)), which has
an equilibrium (Helmholtz) free energy $F_{0}$. We define the (total)
nonequilibrium free energy difference to a polymerized state with $N(t)>1$
chain monomers and free energy $F(N)$, as in Fig. 1(b), as

$\begin{split}\Delta F(N)=F(N)-F_{0}=\Delta F_{\rm eq}(N)+\Delta F_{\rm
neq},\end{split}$ (2)

where $\Delta F_{\rm eq}(N)=F_{\rm eq}(N)-F_{0}$ is the equilibrium work of
forming the polymer state. Hence, we expect $\Delta F(N)\rightarrow\Delta
F_{\rm eq}(N)$ for very slow, quasi-stationary polymerization. The true
nonequilibrium contributions are in $\Delta F_{\rm neq}=\Delta F-\Delta F_{\rm
eq}$. Further, we assume we that we can split $\Delta F_{\rm neq}$ into a
’polymer’ part and a ’monomer gas’ part, via

$\Delta F_{\rm neq}=\Delta F_{\rm pol}+\Delta F_{\rm gas},$ (3)

as discussed in detail in the forthcoming sections. The nonequilibrium free
energy $\Delta F_{\rm neq}$ we interpret as the ’extra’ or ’stored’ free
energy in nonequilibrium, i.e., the free energy in excess over equilibrium. As
motivated in the introduction, the contributions in Eq. (3) we will calculate
in terms of Gibbs-Shannon and Kullback-Leibler concepts from nonequilibrium
distributions.

According to the additive assumptions of Eq.(2), the explicit consideration
and evaluation of $\Delta F_{\rm eq}$ is in principle not necessary in our
work, since $\Delta F_{\rm neq}$ will be evaluated independently as the
deviation from the (in simulation calculated) equilibrium distributions.
However, we inspect $\Delta F_{\rm eq}$ also in detail in to understand the
thermodynamics of the modelled process in a more comprehensive fashion. Note
that expression (2) can in principle also be considered as a function of time
$t$ instead of $N$, because $N(t)$ can be directly accessed from simulations
(or from experiments) or modelled through a reaction rate law, if available.
Bley, Baul, and Dzubiella (2021)

###  III.1 Equilibrium transfer free energy: $\Delta F_{\rm eq}$

With $\Delta F_{\rm eq}$ we denote the equilibrium free energy difference of
transferring a free monomer from the bulk of accessible volume $V$ (and
density $\rho(t)=M(t)/V$) into the polymer chain of length $N$. We approximate
$\Delta F_{\rm eq}$ by the following expression derived from equilibrium
partition sums (see section S1 in the supplementary material),

$\Delta F_{\rm eq}(N)\simeq-
k_{\mathrm{B}}T\left[(N-1)\ln\left(\frac{v_{0}}{V}\right)+\ln\frac{M_{0}!}{N!M!}\right].$
(4)

Using the Stirling approximation $\ln(n!)\simeq n\ln(n)-n$, and assuming $N\gg
1$ and $M_{0}\gg N$, this can be written in a more familiar way, as:

$\Delta F_{\rm eq}\simeq-
Nk_{\mathrm{B}}T\left[\ln\left(\rho\frac{v_{0}}{N}\right)-1\right].$ (5)

Thus, it describes essentially the free energy cost of confining $N$ free
ideal particles of bulk density $\rho$ within the polymer chain with an
effective density $N/v_{0}$. The nature and definition of the volume $v_{0}$
will be motivated and explained in the following. In our evaluation we use the
form (4) and not the more approximate expansion (5), because we are not always
strictly in the Stirling approximation regime.

In the derivation of $v_{0}$, we follow earlier work on the partition sum of a
slightly extensible FJC model, i.e., a bonded polymer like in our simulation
model with relatively stiff harmonic bonds. Glatting, Winkler, and Reineker
(1993); Manca  _et al._ (2012); Fiasconaro and Falo (2019) The $N-1$ bond
degree of freedoms of the polymer chain decouple and per bond one can define
the internal partition sum $v_{0}$ (in units of volume), according to

$v_{0}=4\pi\int_{0}^{\infty}\mathrm{d}b\,b^{2}\exp\left[-(b-b_{0})^{2}/(2\sigma_{\mathrm{b}}^{2})\right],$
(6)

where $\sigma_{\mathrm{b}}=(2K_{\mathrm{bond}}\beta)^{-1/2}$ is the
temperature-dependent standard deviation of the Gaussian probability
distribution, and the factor 2 in front of the simulation spring constant
reflects the action of the two neighboring potentials. Evaluating Eq. (6) in
the limit of large and stiff bonds ($b\simeq b_{0}\gg\sigma_{\mathrm{b}}$), we
find the known result Glatting, Winkler, and Reineker (1993); Fiasconaro and
Falo (2019)

$v_{0}\approx 4\pi b_{0}^{2}\sqrt{2\pi}\sigma_{\mathrm{b}},$ (7)

which essentially describes bond vector rotations on a sphere of radius $4\pi
b_{0}^{2}$ as in the FJC model, Strobl (2007) and, additional small
longitudinal vibrations of size $\sqrt{2\pi}\sigma_{\mathrm{b}}$.

For the estimate of $\Delta F_{\rm eq}$, Eq. (4), from simulations, we assume
that the stiff spring limit Eq. (7) applies, but with values of $b_{0}$ and
$\sigma_{\rm b}$ replaced by average bond lengths $\bar{b}$ and their standard
deviations $\bar{\sigma}_{\rm b}$ in the simulations. The reason is that it is
a priori unclear if the assumptions leading to Eq. (7) are valid in
nonequilibrium and the bond values remain the same as predicted by theory. The
resulting bond values are summarized in Table 1. As we see, the values hardly
depend on the process (equilibrium versus slow versus fast polymerization),
demonstrating that our statistical mechanics model is meaningful to estimate
the equilibrium free energy $F_{\rm eq}$ contribution during the
polymerization process. The average simulated bond length is indeed close to
$b_{0}$ and around $\bar{b}\simeq 1.04\sigma$, while the width is
$\sigma_{\mathrm{b}}\simeq 0.16\sigma$, highly consistently comparing to the
stiff spring limit
$\sigma_{\mathrm{b}}=(2K_{\mathrm{bond}}\beta)^{-1/2}=(1/\sqrt{40})\sigma\simeq
0.16\sigma$.

Table 1: Fitted mean bond lengths $\bar{b}$ and standard deviations $\bar{\sigma}_{\mathrm{b}}$ for selected chain lengths $N$ for equilibrium systems (eq), and slow and fast growth. $N$ | $\bar{b}_{\mathrm{eq}}\sigma^{-1}$ | $\bar{\sigma}_{\mathrm{b,eq}}\sigma^{-1}$ | $\bar{b}_{\mathrm{slow}}\sigma^{-1}$ | $\bar{\sigma}_{\mathrm{b,slow}}\sigma^{-1}$ | $\bar{b}_{\mathrm{fast}}\sigma^{-1}$ | $\bar{\sigma}_{\mathrm{b,fast}}\sigma^{-1}$
---|---|---|---|---|---|---
20 | 1.04 | 0.16 | 1.04 | 0.15 | 1.03 | 0.16
50 | 1.04 | 0.16 | 1.04 | 0.16 | 1.04 | 0.16
100 | 1.04 | 0.16 | 1.04 | 0.16 | 1.05 | 0.16
150 | 1.04 | 0.16 | 1.05 | 0.16 | 1.05 | 0.16
200 | 1.04 | 0.16 | 1.04 | 0.16 | 1.05 | 0.16
250 | 1.04 | 0.16 | 1.04 | 0.15 | 1.05 | 0.16
Figure 2:  (a) Equilibrium transfer free energy $\Delta F_{\rm eq}(N)$, (b)
entropic contribution $T\Delta S_{\rm eq}(N)$, and (c) internal energy $\Delta
U_{\rm eq}(N)$ per bond as a function of the chain length $N$. The black
symbols represent results from non-growing, equilibrium chains, and the fast
($\lambda\Delta t=10^{-3}$) and slow reactions ($\lambda\Delta t=10^{-5}$) are
depicted as red and blue lines. The gray dash-dotted line in (c) represents
the $(N-1)k_{\mathrm{B}}T/2$ contribution, as estimated from the statistical
mechanics model, Eq. (9).

From the free energy, Eq. (4), the equilibrium entropy change follows from
$\Delta S_{\rm eq}=-\partial\Delta F_{\rm eq}/\partial T$, and we obtain

$\Delta S_{\rm eq}(N)=-\Delta F_{\rm eq}(N)/T+(N-1)k_{\mathrm{B}}/2.$ (8)

The difference in internal energy $\Delta U_{\rm eq}(N)=\Delta F_{\rm
eq}(N)+T\Delta S_{\rm eq}(N)$ is then given by

$\Delta U_{\rm eq}(N)=(N-1)k_{\mathrm{B}}T/2.$ (9)

This result for the energy is understandable in terms of $N-1$ independent
harmonic oscillators which act as linearly vibrating diatoms. McQuarrie (1976)
The vibrational degree of freedom partitions $k_{\mathrm{B}}T/2$ into the
kinetic energy which cancels in the difference $\Delta F_{\rm eq}$ with one
translational degree of freedom of the ideal monomer gas. Equally, one
$k_{\mathrm{B}}T/2$ goes in the potential energy which remains as net effect.
For evaluating $\Delta U(N)$ from the simulations, we simply average the
internal bond energy for a fixed $N$ over nonequilibrium trajectories. In that
sense, $\Delta U(N)$ is an average in nonequilibrium Altaner (2017) in our
polymerization simulations and should be denoted as $\Delta U_{\rm neq}(N)$.
However, we will see that is has negligible nonequilibrium contributions and
we can safely interpret it as $\Delta U_{\rm eq}(N)$.

Fig. 2(a) shows the resulting curves from the simulation averages for $\Delta
F_{\rm eq}(N)$ per bond (i.e., divided by $N-1$). Entropic and energetic
contributions are presented in panels (b) and (c), respectively. We see that
the free energy cost $\Delta F_{\rm eq}(N)$ per formed bond is on the order of
several $k_{\mathrm{B}}T$, mostly contributed ($\gtrsim 90\%$) from entropy as
seen from panel (b). Hence, the formation of a new harmonic bond leads to a
significant loss of entropy for the added particles to the polymer chain from
the ideal gas state, since the particle’s accessible volume and thus
configurational freedom decreases drastically from $V$ to $v_{0}$. This
penalty per bond increases with $N$ (or, with time) because the density of the
free monomer gas decreases with polymerization in our canonical ensemble. The
contribution of the internal energy $\Delta U_{\rm eq}$ (Fig. 2c) is close to
$k_{\mathrm{B}}T/2$ in very good agreement with our statistical predictions
above. It is thus only a minor contribution to the free energy change. (We
observe a slight but systematic increase of $\Delta U_{\rm eq}(N)$ per bond in
the simulations, possibly because for shorter chains small finite length
effects play a role in the equi-partitioning among the vibrational modes. Note
also that for larger $N\gtrsim 100$ the equilibrium simulation saturates at
values about 5% larger than $k_{B}T/2$. This is very probably due to the
approximations made when going from Eq. (6) to Eq. (7) and the $k_{B}T/2$
limit is not exact.)

Importantly, the free energy cost estimated here for all processes is
essentially the same, in particular the differences between fast and slow
polymerization reactions with respect to equilibrium are negligibly small on
the $k_{B}T$ scale. Hence, nonequilibrium effects are not significant in our
evaluation of $\Delta F_{\rm eq}(N)$ which validates our treatment to estimate
a true equilibrium contribution of the free energy of polymerization. We will
in the following better understand where the nonequilibrium contributions in
the polymer are, when we discuss $F_{\rm pol}$, i.e., the nonequilibrium free
energy stored in the polymer conformations.

Note that our model reaction would be characterized as ’endogenic’ and
’endothermic’, Atkins and de Paula (2010) because $\Delta F_{\rm eq}>0$ and
$\Delta U_{\rm eq}>0$, respectively, i.e., the reaction is not spontaneous and
leading to higher system energies. In our model it could be simply made ’exo-’
(spontaneous, and producing energy) by shifting the bonding energy by a
constant to very favorable negative values. The following analysis on the
nonequilibrium contribution $\Delta F_{\rm neq}$ would not be influenced by
such a global energetic shift.

An interesting final remark in this section concerns the statistical $1/N!$
’Gibbs factor’ in Eq. (4). We include the latter (as in the conventional
Boltzmann statistics of classical gases McQuarrie (1976)) because we assume in
our system that the $N$ polymer beads are indistinguishable. Hence, in our
interpretation, an exchange of two beads does not yield a different microstate
we want to consider. In other words, upon particle exchange in the polymer the
system information (relative entropy) stays the same in our interpretation.
The analysis framework and main conclusions of our study on the behavior of
$F_{\rm neq}$ do not change by neglecting the Gibbs factor. (The consequence
would be the absence of the $N$-factor in the logarithm in Eq. (5) and thus a
systematic change and shift of the $N$-dependence of $\Delta F_{\rm eq}$).

###  III.2 Polymer conformational entropy in nonequilibrium: $\Delta F_{\rm
pol}$

Per construction and our interpretation, the term $\Delta F_{\rm eq}$ solely
contains the changes in equilibrium free energy originating from the internal
bond confinement and longitudinal distributions while conserving full
rotational freedom per segment. Nonequilibrium effects, however, originate
from rotational constraints of the bonds, which do appear in nonequilibrium.
These rotational correlations lead to deviations from ideal chain statistics,
as for the FJC model, in particular, leading to chain elongation and the
previously reported remarkable self-avoiding walk scaling for ideal chains.
Bley, Baul, and Dzubiella (2021)

The enhanced size scaling in nonequilibrium polymerization leads to shifted
probability distributions $P(R(N))$ for the end-to-end distances $R(N)$.
Examples for emerging probability distributions $P(R(N))$ for equilibrium and
slow and fast reaction conditions for $N=50$ and $N=150$ are presented in Fig.
3. The distributions for fast reactions are indeed substantially shifted to
more extended nonequilibrium conformations for both chain lengths.

Figure 3:  (a) and (b) Normalized distributions of end-to-end distances
$P(R(N))$ from simulation (symbols) and Gaussian fits (solid lines). The
equilibrium case is compared with the theory (gray dashed line) for ideal
chains Rubinstein and Colby (2003); Strobl (2007) at chain lengths $N=50$ and
$150$, respectively. Note that we plot the distribution with the appropriate
spherical weight, $4\pi r^{2}P(R)$, so that the quality of the fits of the
tails of the distributions is visible.

The distributions $P(R)$ can be used for calculating the relative entropy
$\Delta S_{\rm pol}$ (also, Kullback-Leibler divergence) Kullback and Leibler
(1951); Altaner (2017) for accessing the $\Delta F_{\rm pol}$-term of Eq. (2).
While we employ the end-to-end distance, we believe a more accurate treatment
would need to address all individual microstates, i.e., the (translational and
rotational) distributions of every individual bond vector and evaluate them
using the Kullback-Leibler relative-entropy form.Qian (2001) This is, however,
statistically very challenging. We chose therefore the end-to-end distance
following the arguments of Dayantis.Dayantis (1995) He showed that as long as
the bond microstates are behaving homogeneous, i.e., have all the same
distributions for a given end-to-end distance, then our Eq.(10) should be
accurate. However, our chain has some spatial anisotropy because it is
stretched more at the growing front than at the relaxing other end. In other
words, for a fixed end-to-end distance the microstates are not equally
probably but biased by the slightly stretched growing end. Thus, we expect
deviations in a more accurate treatment, and likely the real contributions are
somewhat larger than currently calculated.

Given these considerations, the relative entropy for the polymer conformations
in our work is defined as

$\Delta S_{\rm pol}=-4\pi
k_{\mathrm{B}}\int_{0}^{\infty}\mathrm{d}R\,R^{2}\;P(R(N))\ln\left[\frac{P(R(N))}{P_{0}(R(N))}\right],$
(10)

and we use the definition of a nonequilibrium free energy Qian (2001); Altaner
(2017)

$\Delta F_{\rm pol}(N)=-T\Delta S_{\rm pol}$ (11)

which is the (purely entropic) free energy difference associated with the
nonequilibrium deviation from the equilibrium distribution. Dayantis (1995)
The latter we denote as $P_{0}(R)$. Rubinstein and Colby (2003); Strobl (2007)
In other words, Eq. (11) vanishes in equilibrium, that is, it accounts only
for the deviations of information from equilibrium. We obtain continuous
functions for $P(R(N))$ from fit functions of the simulated histograms (see S2
in the supporting information for the functions and fit parameters). Fig. 3
demonstrates that Gaussian fits also describe well the nonequilibrium
distributions, also accounting for the tails of $4\pi R^{2}P(R)$.

Fig. 4(a) presents the resulting $\Delta F_{\rm pol}(N)$ (filled symbols) for
the two reaction speeds. As expected, the nonequilibrium free energy
contribution to Eq. (2) is negligibly small for slow reactions, as provided
already by the agreement of the $P(R)$ for equilibrium and slow reactions
(Fig. 3). Fast reactions, however, lead to an almost linear dependency $\Delta
F_{\mathrm{pol}}(N)\propto N$ with $\Delta F_{\mathrm{pol}}(N)\approx
1.0~{}k_{\mathrm{B}}T$ for long chains ($N\approx 250$). This indicates that
the more extended chain stores some extra free energy, which can be released
when the chain relaxes to equilibrium. A scaling analysis put forward in our
previous paper Bley, Baul, and Dzubiella (2021) suggested that the relative
lengths of polymer segments that can relax during polymerization is decreasing
with polymerization time, scaling with $\propto\sqrt{1/N}$. Hence, we expect
$\Delta F_{\rm pol}(N)$ to continuously grow with chain length.

Figure 4(b) now plots $\Delta F_{\rm pol}(N)(t)$ as a function of time for
fast reactions and shows that this free energy contribution increases in time.
Hence, nonequilibrium polymer chain conformations can store a significant
amount of free energy, increasing with polymerization time. However, it is not
very large in our model system within the studied observation time, just on
the order of one $k_{\mathrm{B}}T$.

Coming back briefly to the discussion whether the end-to-end distance is a
good coordinate for our evaluation. We can give some complementary,
approximative arguments why our current treatment is at least reasonably
accurate but probably a lower limit. In our previous paper, Bley, Baul, and
Dzubiella (2021) we showed that the growing bond vectors are directed and
instead of rotating full area $4\pi$ (of the unit sphere), only rotate about
80% of it (as estimated from the accessible azimuthal angles). This leads to
an entropic penalty of roughly $-k_{B}T\ln(0.8)\simeq 0.22~{}k_{B}T$ per bond.
We also demonstrated that around 5 bonds have not relaxed in the early steady
state (after one $\tau_{B}$). This would amount to about $5\times 0.22\simeq
1.1~{}k_{B}T$ entropic penalty per chain, growing in time because of less and
less relaxation of the chain in the process (see also the arguments later at
the end of section III.D). This estimate is larger than (but close to) about
0.6$k_{B}T$ which we calculate after one $\tau_{B}$ (where the system is
diffusive stationary). Hence, the conformational entropy might be indeed a bit
higher than calculated.

###  III.3 Free monomer gas translational entropy in nonequilibrium: $\Delta
F_{\rm gas}$

Figure 4:  (a) Nonequilibrium polymer conformational free energy $\Delta
F_{\rm pol}(N)$ of a growing chain as function of chain length $N$. (b)
$\Delta F_{\rm pol}(t)$ as a function of time $t$ for fast reactions.
Evaluated from Eq. (11), see details in the text.

In addition to the polymer chain, also the free monomer ideal gas exhibits
nonequilibrium spatio-temporal correlations. Bley, Baul, and Dzubiella (2021)
Due to the fast reaction at the AC, the free monomer one-body density profile
$\rho(r,t)$, where $r$ is the distance to the center-of-mass of the AC, is
non-uniform in space and time. A classical, well-known reference is the
density depletion ($\rho(r)\propto(1-R_{c}/r$)) around the spherically-
symmetric ’Smoluchowski sink’ of radius $R_{c}$ in steady-state diffusion-
controlled bimolecular reactions. von Smoluchowski (1918) The magnitude of
depletion depends on the reaction speed in diffusion-influenced reaction with
radiation (Colllins-Kimball) or Doi-type of boundary conditions. Dibak  _et
al._ (2019)

In order to study and characterize the inhomogeneous monomer distributions, we
calculated the non-equilibrium one-body density profiles at fixed $N$,
$\rho(r;N)$, as a function of the radial distance $r$ from the AC of the
growing chain. In general, the monomer distribution will be heterogeneous in
space and a function of absolute space, $\rho(\bm{r};N)$, but is difficult to
obtain for our anisotropic system – so we approximate the distribution by
considering the average, spherically symmetric profile $\rho(r;N)$ around the
AC. The profiles are averaged for fixed $N$ at the time right when the
reaction occurs. Bley, Baul, and Dzubiella (2021)

Results are presented in Fig. 5(a) and (b) for chain lengths $N=50$ and $150$,
respectively. Indeed, faster reaction rates cause a large depleted volume
around the AC for $r\lesssim 5\sigma$, qualitatively similar to the classical
Smoluchowski density holes with a size set by the reactive sink radius
according to $\rho(r)\propto(1-R_{c}/r$). The nonequilibrium depletion then
leaves a wake of holes vanishing in space and time behind the AC. Bley, Baul,
and Dzubiella (2021) In contrast, for the slow reactions much smaller
depletion regions $r\lesssim 2\sigma$ are observable, as we expect from
diffusion-influenced Doi-reactions with smaller intrinsic propensities. von
Smoluchowski (1918); Dibak  _et al._ (2019) For large distances, the
nonequilibrium profiles saturate to an almost homogeneous profile at the
simulation box boundary, being very close to the homogeneous equilibrium
profile $\rho_{0}(N)$ (independent of $r$) for fixed $N$.

Figure 5:  Nonequilibrium density profiles $\rho(r;N)$ of the free monomer gas
at fixed $N$ and radial distance $r$ from the active center $\rho_{r,N}$.
Filled symbols are data points from the simulation and empty symbols from
extrapolation (see S3 in the supporting information) for chain lengths (a)
$N=50$ and (b) $N=150$. The solid lines represent the interpolation functions
between the data points and are used for subsequent integration. The gray
dash-dotted lines represent the homogeneous (independent of $r$) equilibrium
density $\rho_{0}(N=50)\approx 0.28~{}\sigma^{-3}$ and $\rho_{0}(N=150)\approx
0.22~{}\sigma^{-3}$.

We now employ again the concept of relative entropy Qian (2001); Altaner
(2017) to estimate the free energy cost arising from the density depletion in
the non-equilibrium density profiles $\rho(r;N)$, which writes for the density
distributions in terms of a nonequilibrium free energy

$\Delta F_{\rm gas}(N)=-T\Delta S_{\rm
gas}=k_{\mathrm{B}}T\int_{0}^{r_{\mathrm{box}}}\mathrm{d}\bm{r}\,\rho(r;N)\ln\left[\frac{\rho(r;N)}{\rho_{0}(N)}\right],$
(12)

where $\rho_{0}(N)$ is the equilibrium reference density for a given remaining
monomer concentration. For the equilibrium systems, $\rho(r;N)=\rho_{0}(N)$
for all distances $r$ is valid and thus $\Delta F_{\rm gas}(N)=0$.

For numerical evaluation of Eq. (12), all non-equilibrium density profiles
were extrapolated for distances between $V^{1/3}/2<r\leq r_{\mathrm{box}}$.
This is because the integration of the density profiles refers to a spherical
volume which fulfills
$M(t)=\int_{0}^{r_{\mathrm{box}}}\mathrm{d}\bm{r}\;\rho(r)$ with
$r_{\mathrm{box}}=(3V/(4\pi))^{1/3}$, while the cubic simulation box with
periodic boundary conditions allows only calculating up to distances
$r=V^{1/3}/2$. Details on the numerical integration and a sensitivity check of
the influence of the nature of the extrapolation can be found in section S3 in
the supporting information.

Figure 6: (a) Nonequilibrium translational free energy $\Delta F_{\rm gas}(N)$
of the free monomer gas around the active center (AC) as a function of chain
length $N$. (b) $\Delta F_{\rm gas}(t)$ as a function of time $t$ for fast
reactions. Evaluated from Eq. (12), see details in the text.

The numerical integration yields the $N$-dependent relative entropy for
growing chains as plotted in Fig. 6(a). We indeed observe substantial
nonequilibrium effects, in particular, the magnitude of $\Delta F_{\rm
gas}(N)$ is generally large for fast reactions, about $3k_{\mathrm{B}}T$,
which is much larger than that for the slow reactions, where only $\Delta
F_{\rm gas}(N)\simeq 0.2k_{\mathrm{B}}T$. Hence, the different density
profiles $\rho(r;N)$ with respect to the corresponding bulk density
$\rho_{0}(N)$ (Fig. 5) result in an appreciable difference in stored free
energy in nonequilibrium.

While in the slow polymerization case, $F_{\rm gas}(N)$ is small and roughly
constant, the fast case exhibits an interesting non-monotonicity in $N$. It
can be better interpreted by inspecting $F_{\rm gas}(t)$ as a function of time
$t$ as plotted in 6(b): For small times, $t\lesssim\tau_{\mathrm{B}}$, first
the depletion hole needs a Brownian (monomer diffusion) time
$t\simeq\tau_{\mathrm{B}}$ to build up. Then, there is a maximum of $F_{\rm
gas}(t)$ where the system for several $\tau_{\mathrm{B}}$ is in a steady-
state. However, as the reaction progresses and more monomers are added to the
growing chain ($t\gtrsim\tau_{\mathrm{B}}$), we report a decrease for $\Delta
F_{\rm gas}$, which is linked to the decreasing bulk monomer density and the
corresponding diminishing entropic cost of forming the low-density depletion
hole. This decrease would not occur in systems with an infinite reservoir of
free monomers where the density around the active center remains constant at
all times, opposite to our simulations with an initially finite number of to-
be-consumed momomers. However, in a typical experiment in a dispersion of many
growing polymer chains also the number of monomers per polymer is finite and
thus will decrease (be consumed) in time during the reaction process. Hence,
such a non-monotonic behavior with a maximum in stored free energy may also
occur experimentally.

###  III.4 Nonequilibrium extra free energy

We are now in the position to plot and discuss the extra free energy in
nonequilibrium, $\Delta F_{\rm neq}=\Delta F_{\rm pol}+\Delta F_{\rm gas}$.
$N$-dependent results are plotted in Fig. 7(a) for slow and fast reaction
speeds. The nonequilibrium contributions to the free energy highly depend on
the reaction speed $\lambda\Delta t$. Importantly, for fast reactions up to
$3~{}k_{\mathrm{B}}T$ are stored in the system. Here, for low $N$, $\Delta
F_{\mathrm{neq}}(N)$ is dominated by the depletion hole in the ideal monomer
gas arising from $\Delta F_{\rm gas}(N)$. Longer chains lead to an increasing
contribution from extended chain conformations $\Delta F_{\rm pol}(N)$.

Figure 7(b) depicts the corresponding time-dependency of the fast
nonequilibrium contribution. After an increase of $\Delta F_{\mathrm{neq}}$ by
around $1.0k_{\mathrm{B}}T$ until $t\approx\tau_{\mathrm{B}}$, the increasing
conformational contribution $\Delta F_{\rm pol}$ and the decreasing density
contribution $\Delta F_{\rm gas}$ balance out and lead to an almost constant
value around $3k_{\mathrm{B}}T$ per chain. Slow reactions exhibit only a very
minor nonequilibrium contribution of the free energy at all times which arises
mostly from the small monomer gas depletion around the AC.

Figure 7: (a) Nonequilibrium free energy $\Delta F_{\mathrm{neq}}(N)$ of a
growing chain as function of chain length $N$. (b) $\Delta F_{\rm neq}(t)$ as
a function of time $t$ for fast reactions. Defined as in Eq. (3).

We finally discuss briefly the question of what are the possible time scales
of relaxation of the polymer back to equilibrium? In other words, how long
does it take for the polymer to release the stored free energy, if let freely
relaxed after polymerization? We believe there would be a two-step process: 1)
The perturbed nonequilibrium density profile $\rho(r;N)$, which yields $\Delta
F_{\rm gas}(N)$ can be linked to a short-term, almost chain-length independent
energy release within a $\tau_{\mathrm{B}}$-scale. Hence, the major part of
the stored free energy will dissipate relatively quickly back into the system.
2) The second time scale is set by the polymer relaxation. Given the concept
of segmental relaxation of segments $s=N/p$ of mode $p$ of a single chain with
the largest segment $s=N$ for $p=1$ as the Rouse time Rouse (1953) with
$\tau_{\mathrm{R}}\approx N^{2}b_{0}^{2}/(3\pi^{2}D_{0})$, then $\Delta F_{\rm
pol}$ is released gradually back to equilibrium with
$\tau_{\mathrm{R}}\propto\tau_{\mathrm{B}}N^{2}/(3\pi^{2})$, which is rather
slow for long chains. These considerations may change in very dense, many
chain polymer systems, such as melts or glass-like systems. Rubinstein and
Colby (2003)

##  IV Concluding Remarks

In this work, we introduced a framework to analyze data from non-equilibrium
BD chain growth simulations to quantify and characterize the nonequilibrium
free energy stored in a fast growing ideal polymer chain as a function of
chain length $N$ (or time $t$). While we treated the monomer transfer free
energy from bulk into the polymer within equilibrium statistical mechanics,
the non-equilibrium contributions we expressed using the Gibbs-Shannon /
Kullback-Leibler divergence formalisms for relative entropy, applied to the
spatial distributions of monomers (bound and free). Here we assumed, arguably,
that the end-to-end distance is a good coordinate to characterize
conformational distributions. However, this should be checked more carefully
in future work. We also assumed that two contributions, one arising from non-
equilibrium, extended polymer chain conformations and the other from spatio-
temporal correlations of the reaction-perturbed monomer density profiles sum
up to the nonequilibrium free energy.

Within our framework, we demonstrated that a single fast growing chain stores
several $k_{\mathrm{B}}T$ of non-equilibrium free energy for all chain lengths
$N$. In addition, we observed a non-trivial time-dependency of the
nonequilibrium free energy. While the polymer conformational contributions
increase continuously in time which can be rationalized by the competition of
relaxation and observation time scales, Bley, Baul, and Dzubiella (2021) those
of the free monomers were found to be non-monotonous (with a maximum) due to a
short-time built-up of the depletion regions and a long-time decrease of bulk
monomer density. Extrapolating to even longer times, we suspect that even a
minimum can occur again when the polymer conformational contribution dominates
over the free monomer gas contribution. As a consequence, complex kinetics of
free energy release related to the time scales of Brownian diffusion and
segmental chain dynamics can be envisioned for chain relaxation back to
equilibrium after polymerization. We wonder if and how it would be possible in
future to independently validate the calculated nonequilibrium free-energy by
measuring it (or the performed work) in the course of the relaxation towards
equilibrium?

We hope our work serves as a first step towards the characterization of
nonequilibrium thermodynamics in polymer processing. Its theoretical
understanding should be useful to develop novel pathways for targeted energy
storage and release of polymers resulting from non-equilibrium stresses.
Reiter (2020); Ramezani, Baschnagel, and Reiter (2020); Chandran  _et al._
(2019) In other words, if the relaxation back to equilibrium can be
controlled, the extra free energy stored in the system might be convertible
into useful, time-dependent work. How this can be done in detail and how to
define appropriately notions like heat and chemical work Seifert (2012) in
such a process and what are the thermodynamic bounds and fluctuations
relations Evans, Cohen, and Morriss (1993); Jarzynski (1997a, b); Crooks
(1999) will be very interesting challenges for future studies. Furthermore, we
hope that coarse-graining approaches for such nonequilibrium processes will
help reducing the computational complexity of such systems, Schilling (2021)
elucidating at the same time their intricate nonequilibrium thermodynamics.

## Acknowledgements

The authors thank Günter Reiter, Jörg Baschnagel, and Murugappan Muthukumar
for inspiring discussions and useful comments. The authors acknowledge support
by the state of Baden-Württemberg through bwHPC and the German Research
Foundation (DFG) through grant no INST 39/963-1 FUGG (bw - ForCluster NEMO).
This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) via
grant WO 2410/2-1 within the framework of the Research Unit FOR 5099 ”Reducing
complexity of nonequilibrium” (project No. 431945604).

## Data availability statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Supplementary Material

See supplementary material for the derivation of $\Delta F_{\mathrm{eq}}$,
details about the fitting approach for the polymer end-to-end distributions,
and the extrapolation of density profiles of the free monomers around the
active center.

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