# Shear Thinning from Bond Orientation in Model Unentangled Bottlebrush
Polymer Melts

Utku Gürel  Andrea Giuntoli  University of Groningen, Zernike Institute for
Advanced Materials, Nijenborgh 4, 9747AG Groningen, The Netherlands
<EMAIL_ADDRESS>
###### Abstract

The rheology of molecular brushes remains challenging to control due to the
multiple length scales and relaxation processes involved, and the lack of
direct observation of molecular conformation during flow. We use molecular
dynamics simulations to determine the shear thinning of unentangled
bottlebrush polymers with varying architecture, from linear chains to combs,
to densely grafted bottlebrushes, to star-like and star polymers. We find
shear thinning exponents in line with theoretical and experimental results and
characterise the shape and orientation of bottlebrushes in steady-state flow.
Many shape parameters derived from the gyration tensor show molecular
alignment with the flow for all systems. Yet, the orientation of individual
bonds is what most strongly correlates with the architecture-dependent shear-
thinning exponents. In densely grafted bottlebrushes, the packing of side
chains prevents alignment with the flow causing a reduction in shear thinning.
The molecular insight from our simulations is useful to tune the architecture
of bottlebrushes to control their rheology.

##  1 TOC Graphic

##  2 Introduction

Bottlebrushes are a type of branched polymer composed of a linear backbone and
multiple side chains grafted to it. Their characteristic architecture and the
dense packing of side chains lead to unique properties that have been studied
over the last few decades 1, 2, 3, 4, 5, 6, 7, 8, 9 . The shape and the size
of bottlebrushes are determined by three molecular parameters: the length of
the backbone, the length of the side chains, and the spacing between these
side chains (grafting density). At equilibrium, a bottlebrush is more rigid
and extended than a linear chain with the same backbone length due to the side
chains’ steric repulsion, leading to higher persistence length, stiffness, and
increased packing efficiency 10 . The overall shape of a bottlebrush polymer
is highly dependent on the interplay of architectural parameters. Increasing
grafting density results in more compact and cylindrical molecular shapes. For
a fixed grafting density, increasing side-chain length stretches the backbone
chain and the bottlebrush becomes more rigid 5 . This rich behaviour leads to
highly tunable mechanical and thermal properties of bottlebrushes compared to
linear polymer chains 11 and to softer and more flexible networks 12 . The
control over the architectural composition makes bottlebrushes suitable
candidates for multiple applications in biomedical sciences 13 , soft robotics
14 , pressure-sensitive adhesives 15 , capacitive pressure sensors 16 , and
ultrasoft electronics 17 .

While the configuration of bottlebrushes at equilibrium is somewhat understood
18, 11 , it remains challenging to predict their shape rearrangement under
mechanical deformations and flow. This prediction is necessary to determine
their rheological behaviour and to use them in technological applications such
as 3D printing 19 . It is well-known, for example, that polymer melts exhibit
shear thinning 20 due to the alignment of the polymer chains with the flow
direction 21 . In branched polymers, the shear thinning is governed by
multiple length scales and relaxation times 22, 23 , which gives rise to
unique rheology compared to linear chains 24, 25, 26, 27, 28 that is not
trivial to predict from the molecular architecture. This is complicated by the
challenges in the controlled synthesis and characterisation of branched
polymers 29, 30 .

Several studies have been conducted to investigate the rheological properties
of melts or solutions of bottlebrushes (synthetic or biological) such as their
viscoelasticity 31, 32 , shear-induced crystallinity 33, 34 , and shear-
thinning behaviour in aggrecan solutions 35 . A few experimental studies shed
light on the complex contribution of the bottlebrush architectural parameters
on their melt rheology. Hu et al. showed first the relative importance of the
different relaxations of side chains and backbones 36 . Dalsin et al. reported
an increase in zero-shear viscosity for bottlebrushes of increasing backbone
length and fixed side chain length 37 , with weak molecular weight scaling at
the low molecular weight (corresponding to a star-like regime, where the
backbone and the side chains are of comparable length), and a Rouse-like
scaling for longer backbones, similar to linear unentangled chains. The same
group identifies a star-like to bottlebrush-like transition and recently
reported that this transition also signals the onset of a strain-hardening
regime in extensional flow 38 . López-Barrón et al. found that the complex
viscosity of unentangled bottlebrushes decreases with increasing side-chain
length, due to the decrease of backbone-backbone friction 39 . They recently
attributed a similar strain-hardening of bottlebrushes in extensional flow to
the entanglement of side chains leading to side chain interdigitation and
alignment perpendicular to the flow 27 , though Zografos et al. show that this
effect is present even when side chains are not entangled 38 . Similar
scalings for the linear viscoelasticity of star-like bottlebrushes have also
been reported by Alexandris et al. 40 , linking the rheological behaviour to
the expected relaxation times of bottlebrushes based on their expected
equilibrium configuration.

Despite the numerous observations and hypothesised behaviour, it remains
challenging to experimentally observe the molecular orientation with the melt
flow of individual bottlebrushes (and branched polymers in general).
Computational studies become then invaluable to precisely determine the
architectural and shape effects of branched polymers in the flow properties of
melts. A few computational papers studied the conformation of nonlinear
polymers such as rings 41, 42 , stars 41, 24 , or comb-like squalene molecules
43 . Qu et al. briefly report a decreasing alignment with the flow direction
related to a decrease in shear thinning for bottlebrushes with increasing
backbone length, but to our knowledge, there are no computational studies of
the effect of chain topology on the conformation under flow and shear thinning
of unentangled bottlebrush polymer melts.

The primary goal of this work is to study the effect of bottlebrush
architecture on the viscosity of the melt for unentangled chains under shear
flow. We use non-equilibrium molecular dynamics (NEMD) simulations to
calculate the flow curve in steady shear for different bottlebrush
architectures at a fixed molecular weight and compare them with the limit
cases of linear chains, comb polymers, and stars. We demonstrate that all
polymer melts exhibit shear thinning, and we also observe a transition from
star-like to bottlebrush-like structures. However, in the investigated
frequency range, we find that the thinning exponent is determined by the
orientation with the flow of individual bonds rather than chain orientation.
Densely grafted bottlebrushes shear-thin the least because the packing of the
side chains prevents their strong alignment with the flow direction, and this
effect decreases with decreasing grafting density from the bottlebrush to the
comb regime. Many other shape parameters that are derived from the principal
components of the gyration tensor also show molecular alignment with the flow
but do not directly correlate with the shear thinning behaviour as a function
of molecular architecture. Our simulations provide important insight into the
bond and chain configuration changes under shear flow that are crucial to
predict and control the rheology of bottlebrush polymer melts in different
architectural regimes.

##  3 Methods

In this work, we use coarse-grained molecular dynamics (CGMD) simulations
under non-equilibrium conditions.

###  3.1 Model

We employ the bead-spring model with reduced Lennard-Jones (LJ) units where
the energy ($\varepsilon$), mass ($m$), and diameter ($\sigma$) of a single
bead are set to $1$ leading to a unity time scale as
$\tau=\sqrt{m\sigma^{2}/\varepsilon}$ 44 . The bonded interactions in the
bead-spring model are given by the finite extensible nonlinear elastic (FENE)
potential

$U_{fene}(r)=-0.5k{R_{0}}^{2}ln\left[1-\left(\frac{r}{R_{0}}\right)^{2}\right]$
(1)

to represent the bonds between connected monomers, where
$k=30\varepsilon/\sigma^{2}$ is the spring constant, $R_{0}=1.5\sigma$ is the
maximum allowed bond length, and $r$ is the distance between two bonded beads.
Non-bonded interactions between the beads are given by the truncated LJ
potential

$U_{LJ}(r)=\left\\{\begin{array}[]{lc}\varepsilon\left[\left(\frac{\sigma^{\text{*}}}{r}\right)^{12}-\left(\frac{\sigma^{\text{*}}}{r}\right)^{6}\right],&r\leq
r_{c}\\\ 0,&r>r_{c}\end{array}\right.$ (2)

to represent the Van der Walls interactions up to a cutoff distance
$r_{c}=2.5\sigma$, where $\sigma^{\text{*}}=2^{1/6}\sigma$ is the length scale
at which the LJ potential attains its minimum value with depth $\varepsilon$.
CGMD simulations are carried out with the LAMMPS (Large-scale Atomic/Molecular
Massively Parallel Simulator) software with periodic boundary conditions in
all three dimensions. (https://www.lammps.org/)45 . Polymer chain snapshots
are rendered with OVITO 46 .

###  3.2 Chain Parameters

We analyse bottlebrush architectures with varying backbone ($N_{BB}$) and side
chain ($N_{SC}$) lengths, specifically,
$\\{(N_{BB},N_{SC},m)\\}=\\{((25,3,1),(20,4,1),(10,9,1),(5,19,1)\\}$, leading
to a total number of beads $N_{BB}(N_{SC}+1)=100$ in a single chain with
$m=1$, $i.e$ one side chain per backbone bead. The total number of chains is
$2,000$ making a total of $200,000$ beads in the melt. Bottlebrushes with
grafting density $m>1$ have the following parameters:
$\\{(N_{BB},N_{SC},m)\\}=\\{(49,4,4),(33,4,2)\\}$. Here, we graft a side-chain
to every $m^{th}$ backbone bead. Both of these systems have a molecular weight
of $M_{w}=101$, and there are $1,980$ chains making the total number of beads
$199,980$ as close as possible to the other systems. Additionally, we compare
bottlebrushes to a system of linear chains with length $L=100$ and a system of
star polymers with $f=9$ arms and $M=11$ beads per arm. Note that these two
additional architectures also satisfy the constraint we set on the molecular
weight $M_{w}=100$ with $2,000$ chains. The chain parameters are summarised in
Table 1.

Table 1:  Architectural parameters for bottlebrushes with backbone length $N_{BB}$ and side chain length $N_{SC}$, linear polymers of length $L$, star polymers of $f$ arms with arm length $M$. The molecular weight is given by $N_{BB}(1+N_{SC}/m)$ for bottlebrushes and by $fM+1$ for stars. Topology | $N_{BB}$ | $N_{SC}$ | $m$
---|---|---|---
Bottlebrush | 25 | 3 | 1
20 | 4 | 1
10 | 9 | 1
5 | 19 | 1
33 | 4 | 2
49 | 4 | 4
Linear | L=100
Star | f=9, M=11

###  3.3 Equilibration protocol

To prepare the polymer melt, we first randomly pack the desired number of
polymer chains in the simulation box followed by an energy minimisation step.
Then we equilibrate the system in the NPT ensemble (constant number of
particles, pressure, and temperature) using a Nose-Hoover thermostat at
temperature $<T>=1.0$ and pressure $<P>=0.1$. We choose this pressure value
since it is close to atmospheric pressure 47 . We run an initial equilibration
for $5\times 10^{7}$ simulation steps and additional $5\times 10^{6}$
production steps with an integration step of $0.005\tau$ ensuring that the
initial end-to-end vector correlations of polymer chains are lost (see Fig.
S1). We consider the longer subchains of bottlebrushes (either the backbone or
the side chain) in the correlation calculations. For star polymers, we
consider the chain segment from the core to a terminal arm bead. The
correlations are averaged over all subchains and for all chains in the melt.
We also have a final average over 3 independent simulation runs each of which
starts with a different initial configuration of polymer chains at different
random initial velocities.

###  3.4 Deformation protocol

After the creation of the polymer melt, we apply a simple shear with a fixed
shear rate on the $xy-$plane where the directions of shear, gradient, and
vorticity are $\hat{x}$, $\hat{y}$, and $\hat{z}$ respectively. The SLLOD
equations of motion are integrated with Lees-Edwards boundary conditions at
constant volume. We measure the average stress values $\sigma_{xy}$ of all
beads in the system which is a function of the total strain
$\gamma=\dot{\gamma}t$. We then calculate the viscosity
$\eta=\sigma_{xy}/\dot{\gamma}$ of each system after deforming them up to a
total strain of $\gamma=100$. The overall shearing procedure is the same as in
Ref. 48 . We explore a range of shear rates from $\dot{\gamma}=10^{-3}$ to
$\dot{\gamma}=0.2$ similar to Ref. 49 observing unstable viscosity data for
shear rates lower than $10^{-3}$ as they also report in their work. We observe
an unexpected drop in the pressure of the system when we try to simulate at
lower shear rates. Also, at shear rates larger than $0.2$, the bonds in the
system become unrealistically large and cause the simulations to fail. To
ensure the stability of our simulations at high shear rates, we reduce the
integration time step to $0.001\tau$ and check that the temperature fluctuates
stably around the initially set value.

###  3.5 Definition of architectural parameters

We computed the following architectural parameters, most of which are a
function of gyration tensor eigenvalues. The correlation between these
parameters is shown in Fig. 5. We plot them as a function of the shear rate in
Fig. S3 along with the corresponding eigenvalues in Fig. S4.

  * •

Orientation resistance $\tan 2\phi=\frac{<G_{xy}>}{<G_{xx}>-<G_{yy}>}$

  * •

Asphericity
$b=\lambda^{2}_{max}-\frac{1}{2}\left(\lambda^{2}_{min}+\lambda^{2}_{mid}\right)$

  * •

Acylindricity $c=\lambda^{2}_{mid}-\lambda^{2}_{min}$

  * •

Prolateness
$p=\frac{\left(2\lambda_{min}-\lambda_{mid}-\lambda_{max}\right)\left(2\lambda_{mid}-\lambda_{min}-\lambda_{max}\right)\left(2\lambda_{max}-\lambda_{min}-\lambda_{mid}\right)}{2\left(R^{2}_{g}-\lambda_{min}\lambda_{mid}-\lambda_{mid}\lambda_{max}-\lambda_{min}\lambda_{max}\right)}$

  * •

Shape anisotropy $\kappa=\frac{b^{2}+\frac{3}{4}c^{2}}{R^{4}_{g}}$

##  4 Results and Discussion

Figure 1 shows the architectures studied. We change chain branching to obtain
star, comb and bottlebrush polymers to study the topology effect. To avoid
molecular weight contributions, all architectures have the same number of
beads ($M_{w}=100\pm 1$), which also constrains our choice for the limit
architectures of linear chains and star polymers. Bottlebrush and comb
architectures are defined by three parameters: (i) the number of backbone
beads $N_{BB}$, (ii) the number of side chain beads $N_{SC}$, and (iii) the
spacing between grafted side chains ($m$). Hence, the set
$\\{(N_{BB},N_{SC},m)\\}$ encodes all architectural information. Densely
grafted bottlebrush regime corresponds to $m=1$ and can be classified into
three regions depending on the ratio $N_{BB}/N_{SC}$ : Bottlebrush-like,
sphere-like, and star-like 18 . Bottlebrush-like polymers have significantly
larger backbone lengths compared to their side chains ($N_{BB}>>N_{SC}$) which
allow us to consider the backbone as a diluted linear chain. Sphere-like
polymers have comparable backbone and side chain lengths ($N_{BB}\sim N_{SC}$)
and the backbones of star-like polymers are much shorter than their side
chains ($N_{BB}<N_{SC}$). The architectures with $m\geq N_{SC}$ correspond to
loosely grafted combs, and $m\approx N_{SC}$ marks the transition region
between loosely grafted combs to densely grafted combs 11 .

Figure 1: Simulated polymer architectures at fixed molecular weight
$M_{w}=100$. (a) Linear chain of length 100. (b) Sphere-like polymer with
$N_{BB}=10,N_{SC}=9,m=1$. (c) Star-like polymer with $N_{BB}=5,N_{SC}=19,m=1$.
(d) Star polymer with $f=9,M=11$. (e) Bottlebrush with
$N_{BB}=49,N_{SC}=4,m=4$. (f) Bottlebrush with $N_{BB}=33,N_{SC}=4,m=2$. (g)
Bottlebrush with $N_{BB}=20,N_{SC}=4,m=1$. (h) Bottlebrush with
$N_{BB}=25,N_{SC}=3,m=1$. We focus on capturing the linear-to-star transition
by increasing $N_{SC}/N_{BB}$ ratio (left column) and the linear-to-
bottlebrush transition (right column) by decreasing m.

###  4.1 Shear thinning of polymer melts

Here we show the shear thinning behaviour of polymer melts with different
molecular architectures. We calculate the viscosity as

$\eta(\dot{\gamma})=\frac{\sigma_{xy}}{\dot{\gamma}},$ (3)

where $\sigma_{xy}$ is the shear stress at steady flow given by the average
per-monomer stress, and $\dot{\gamma}$ is the shear rate. The viscosity flow
curves of polymer melts with different architectures are reported in Fig. 2
and show a clear shear thinning trend. We fit the data with a power-law model
$\eta=K\dot{\gamma}^{n-1}$ where $n$ is the shear thinning exponent and $K$ is
the flow consistency index. Shear thinning exponents for all systems are
reported in Table 2. The value of $n$ ranges between 0 and 1 and smaller
values of $n$ indicate stronger shear thinning. The prediction of this
exponent for unentangled linear chains is $n=0.5$ which is known from the
theory of polymer physics and confirmed by experiments 50, 51 . We can confirm
that the linear chains in our system are not entangled by the measurement of
the scaling exponent $n$. Since the bottlebrushes in our systems are
unentangled, we also choose unentangled linear chains to make a better
comparison between different architectures. Note that for entangled systems,
the initial drop in the viscosity during the start-up shear is attributed to
the chain disentanglement 52 . Hence, we exclude this mechanism by choosing
unentangled linear chains, as later proven by their shear-thinning behaviour
(see Table 2 ). We would expect a kink in the viscosity curve around
$\dot{\gamma}\sim 10^{-4}$ based on the relaxation times $\tau_{R}$ of the
longest subchain of our polymers (see Fig. S1). For shear rates lower than
$\tau^{-1}_{R}$, we would reach the Newtonian plateau. Although we try to
reach this plateau, the viscosity data fluctuate largely for
$\dot{\gamma}<10^{-3}$, and the simulations become unstable. Nevertheless,
based on the relaxation times $\tau_{R}$ in Fig. S1 and Rouse model arguments
for unentangled polymers, we might expect a higher viscosity at lower shear
rates for linear chains and other elongated architectures, given that the
total molecular weight is fixed 50 . A crossover point is then expected at
intermediate shear rates around $\dot{\gamma}\sim 1/\tau_{R}$, which is
roughly what we observe in Fig. 2.

Figure 2: Shear viscosity of different architectures as a function of shear
rate. Data are shown by solid markers, and the lines are exponential fit to
the data given by the expression $\eta=K\dot{\gamma}^{n-1}$. The drop in
viscosity with increasing shear rate indicates shear thinning for all systems.
We observe an architectural difference between systems with the same molecular
weight, with the strongest and weakest shear thinning for linear chains and
dense bottlebrushes, respectively. The snapshots show a typical configuration
of a single bottlebrush chain of $N_{BB}=25$, $N_{SC}=3,m=1$ and a linear
chain at the lowest (left) and the highest (right) shear rates. The simulated
architectures are elongated and aligned with the flow at high shear rates, and
they have a more globular shape at low shear rates.

One would expect stronger shear thinning for architectures that better align
with the flow at an increasing shear rate. Yet, for complex branched
architectures the interpretation of what ”alignment with the flow” means is
not trivial. In the following sections we thoroughly analyse the dependence on
the shear rate of multiple shape parameters and their relation with the melt
shear thinning behaviour, but we can already make some observations based on
the viscosity data. Linear chains easily elongate in the flow direction and
show the strongest shear thinning. Densely grafted architectures (a-b-c-d-g-h
in Fig. 1) do not show a clear trend with varying $N_{BB}/N_{SC}$ ratio from
the limits of linear chains to star polymers. Instead, bottlebrush-like
systems (architectures g and h) show the weakest shear thinning behaviour.
Another observation we can make is the effect of grafting density on shear
thinning. Comparing the viscosity data in Fig. 2 for the topologies with the
same side-chain length ($N_{SC}=4$), the loosely grafted architecture with
$m=4$ has the lowest viscosity at the high shear rate, followed by more
densely-grafted architectures with $m=2$ and $m=1$ respectively. Therefore,
the increasing spacing between grafted side chains enhances the shear
thinning. We discuss the molecular origin of these trends in the following
sections.

Table 2: Shear thinning exponent n for different architectures Architecture | $n$
---|---
$Linear$ | 0.507
$N_{BB}=5$, $N_{SC}=19,m=1$ | 0.574
$N_{BB}=49$, $N_{SC}=4,m=4$ | 0.629
$N_{BB}=10$, $N_{SC}=9,m=1$ | 0.635
$Star$ | 0.642
$N_{BB}=33$, $N_{SC}=4,m=2$ | 0.671
$N_{BB}=20$, $N_{SC}=4,m=1$ | 0.696
$N_{BB}=25$, $N_{SC}=3,m=1$ | 0.704

###  4.2 Bond orientation parameter

Orientational order is the driving mechanism in the shear thinning of complex
fluids 21 , but its definition for branched polymers is not trivial. In fact,
one would expect the shear thinning to be correlated to some metric derived
from the gyration tensor, but we report later in detail that this is not the
case for our system. Instead, we find a strong correlation between shear
thinning and the orientation of the bonds. To quantify it, we measure the bond
orientation parameter $<P_{2}>$ which is defined as

$<P_{2}(\cos\theta)>=\frac{3\left<\cos^{2}\theta\right>-1}{2}$ (4)

where $P_{2}$ is the second order Legendre polynomial, $\theta$ is the angle
between the bond vector and the unit vector $\hat{x}$. The average $<.>$ is
taken over all bonds in a single chain and over all chains in the system. We
also calculate $P_{2}$ for backbones and side chains separately (see Fig. S2).
$<P_{2}>$ takes values between $-0.5$ and $1$ where $<P_{2}>=-0.5$ indicates a
perpendicular alignment to the flow direction, and $<P_{2}>=1$ a perfect
parallel alignment. Under equilibrium conditions, $<P_{2}>$ becomes $0$
indicating a random orientation without any directional preference. We also
measure and report $<P_{2}>$ at equilibrium in Table 3 confirming that the
orientation under equilibrium conditions is $0$ as expected.

We show the values of $<P_{2}>$ at all shear rates in Fig. 3 with solid
markers. $<P_{2}>$ converges to $0$ at low shear rates, recovering the
equilibrium value, whereas it increases with increasing shear rate as a result
of an increased bond orientation. We expect $P_{2}$ to reach a plateau at the
infinite shear rate limit imposed by both topological constraints and the
chain stiffness 53 . Although the perfect alignment with flow indicates a
value of 1 for $P_{2}$, this limit is never reached because of the entropic
contributions on the order of $\sim k_{B}T$. The interactions between
different chains prevent them from moving on a straight line on the x-axis and
hence aligning perfectly. Plateau values of $P_{2}$ have been previously
reported up to $0.5$ for squalane, a comb-like unentangled polymer, at
different thermodynamic conditions 54 and up to $0.8$ for rigid colloidal rods
53, 55 . Note that reaching this plateau experimentally is not possible due to
the high shear rates required, and even our simulations become numerically
unstable beyond the data points presented here. Nevertheless, we can fit our
data with the generalised logistic function (solid lines) that is given by the
expression
$P_{2}(\dot{\gamma})=a+\frac{-a}{1+\left(\frac{\dot{\gamma}}{c}\right)^{0.82}}$.
The fit parameter $a$ describes the behaviour at the infinite shear rate
regime when the bond orientation reaches a plateau, and $c$ is a
characteristic shear rate at which bonds transition from their equilibrium
configuration to an alignment with the flow. $1/c$ is roughly related to the
inverse of the bond relaxation time, on the order of $~{}10^{2}\tau$, as we
discuss in Fig. S1(c). The exponent of $\frac{\dot{\gamma}}{c}$ determines how
sharp the transition is from $0$ to $a$. The value $0.82$ is the best fit to
data and is fixed for all systems. The fact that a single exponent is
sufficient for all systems indicates that it does not strongly depend on the
architecture, though we don’t have a particular insight into this specific
parameter.

Figure 3: Bond orientation parameter of different architectures as a function
of shear rate. Data are shown by solid markers and the lines are generalised
logistic function fits. Individual bonds in polymer chains align better with
the flow at high shear rates, leading to an increase in shear thinning. Inset:
Predicted value $a$ of bond orientation at high shear rate regime as a
function of shear thinning exponent $n$. Fit parameters are shown with solid
markers, and the error in the fit is given by twice the standard deviation.
Inverse linear relation indicates that the shear thinning increases when the
bonds orient more with the flow.

In the range of shear rates studied, the progressive bond orientation with the
flow governs the shear-thinning behaviour of all architectures: the higher the
bond orientation, the stronger the shear thinning, see the inset of Fig. 3. We
observe that bottlebrush orientation increases with increasing $m$ for the
same side-chain length ($N_{SC}=4$). The loosely-grafted bottlebrushes with
$m=4$ orient better than more densely-grafted bottlebrushes with $m=2$ and
$m=1$. This trend is the same as their high shear rate viscosity values. The
increase in the spacing between the grafted side chains allows the bonds in
the system to orient better with the flow; thus, leading to a stronger shear
thinning behaviour in Fig. 2. This is due to the local frustration of the
bonds in densely-grafted bottlebrushes imposed by the existence of side
chains, see Fig. S2. When the crowding around the backbone increases, the
bonds become more resistant to orientation. As a result, bottlebrushes with
$m>1$ exhibit higher degrees of shear thinning when $m$ is gradually
increased. Star polymers and star-like bottlebrushes have fewer bonds around
the frustrated region near the backbone and the average bond orientation is
stronger leading to a stronger shear thinning.

###  4.3 Shape Parameters

We expect that the orientation of the molecules with the flow plays an
important role in determining the system rheology, and here we calculate
different shape parameters by computing the radius of gyration tensor that is
commonly used to determine the shape, size, and structure of polymer chains 56
. The components of the gyration tensor are given by

$G_{ij}=\frac{1}{N}\sum_{k=1}^{N}\left(r_{ik}-r_{i}^{(CM)}\right)\left(r_{jk}-r_{j}^{(CM)}\right)$
(5)

where $N$ is the total number of beads in a single chain, $r_{ik}$ is the
Cartesian components ($x,y,z$) of the $k^{th}$ monomer and $r_{i}^{(CM)}$ is
the $i^{th}$ component of the centre of mass (CM) of the chain. We diagonalise
the gyration tensor to obtain its eigenvalues
$\lambda^{2}_{max}>\lambda^{2}_{mid}>\lambda^{2}_{min}$ in descending order.
The trace of this tensor gives the radius of gyration $R_{g}$ for a polymer
chain as $R_{g}^{2}=\lambda^{2}_{max}+\lambda^{2}_{mid}+\lambda^{2}_{min}$. We
show the radius of gyration $R_{g}^{2}$ and the diagonal components $G_{ii}$
in Fig. 4 which are normalised by their respective $R^{2}_{g0}$ values at
equilibrium to identify the changes under shear with respect to equilibrium
conditions. The equilibrium values of $R_{g0}^{2}$ are given in Table 3 along
with the average bond orientations of different architectures. Note that the
curves in Fig. 4 all start from $1$ at equilibrium.

The increasing $G_{xx}$ and decreasing $G_{yy}$ (Fig. 4) indicate that all
architectures extend in the flow direction and compress in the gradient
direction leading to the orientation of the whole molecule in the
$\hat{x}$-direction. Note that linear chains have the highest increase in
$R_{g}^{2}$ and $G_{xx}$ , while the lowest increase is observed for star
polymers. Moreover, we infer a compression along the vorticity direction from
the decrease in $G_{zz}$ as expected from shear-thinning polymer melts 57 . We
agree with Ref. 57 that increasing branching makes $G_{zz}$ insensitive to
shear. This compression is more prominent for linear chains as they have more
space to compress in the $z$-direction than other architectures. The star-like
(5,19,1) and the sphere-like (10,9,1) bottlebrushes also show a decrease in
$G_{zz}$ at low shear rates. For the remaining systems, $G_{zz}$ either stays
constant or slightly increases with the increasing shear rate. This indicates
that the chains resist the orientation along the flow direction and, as a
result, expand in the vorticity direction. Note that $G_{xx}$ does not
correlate with the viscosity directly. Some architectures orient better in the
flow direction, yet they do not show a stronger shear thinning. What we infer
from the analysis of the conformational properties is that the molecules can
orient with the flow as a whole; however, shape parameters based on the
gyration tensor are not the key predictors of the architecture dependence on
shear thinning.

Table 3:  Equilibrium properties Architecture | $R_{g0}^{2}$ | $P_{2}$
---|---|---
$Linear$ | 27.00 | -0.00081
$N_{BB}=49$, $N_{SC}=4$, $m=4$ | 16.45 | 0.00004
$N_{BB}=5$, $N_{SC}=19$, $m=1$ | 14.85 | 0.00043
$N_{BB}=33$, $N_{SC}=4$, $m=2$ | 12.84 | 0.00052
$N_{BB}=25$, $N_{SC}=3$, $m=1$ | 10.84 | 0.00088
$N_{BB}=20$, $N_{SC}=4$, $m=1$ | 9.86 | -0.00017
$N_{BB}=10$, $N_{SC}=9$, $m=1$ | 9.86 | -0.00030
$Star$ | 9.58 | -0.00078
Figure 4: Radius of gyration and the diagonal components of the gyration
tensor. (a): Radius of gyration squared calculated as the trace of the
gyration tensor. Increasing $R_{g}^{2}$ indicates an extension and stretching
of polymer chains under shear. (b): Increasing $x$-component of the gyration
tensor represents the extension along the shear direction with increasing
shear rate. (c-d): Decreasing $y$ and $z$-components of the gyration tensor
indicates compression with increasing shear rate on the gradient and the
vorticity directions respectively.

###  4.4 Correlations of architectural parameters with viscosity

One could think that it is not $G_{ii}$, but some geometrical shape
combination of them, e.g. asphericity, which correlates with the shear
thinning. We test various architectural parameters which are a function of
gyration tensor components with different weights and combinations. Indeed,
all shape parameters investigated behave as expected under flow. For instance,
the asphericity increases and this increase is more prominent for stars and
star-like systems than bottlebrush-like systems. However, no other shape
parameter correlates as well with shear thinning as $P_{2}$, across different
architectures. We quantify this by the Spearman coefficient given in Fig. 5,
and the full data sets of these parameters as a function of shear rate for all
architectures are reported in Fig. S3-S4. We consider these parameters as a
function of shear rate, which is an independent variable, and compute the
correlation between two data sets by sorting them from low to high shear rate.
The correlations between these parameters and the viscosity are reported in
Table 4 in descending order of the absolute value. Although $G_{yy}$
correlates with viscosity as much as $P_{2}$ does, it cannot explain the
ranking of viscosities for different architectures at high shear rates. On the
other hand, $P_{2}$ is able to describe this trend for all architectures. A
natural parameter for this study would be $G_{xx}$ as we apply the shear in
the $x$-direction; however, it does not correlate with viscosity. Hence, the
reason for high correlations in $G_{yy}$ is a secondary effect as the chains
have to compress in the gradient direction when the shear rate is increased.
There is no strong correlation between the viscosity and all the other
quantities reported in Table 4.

Figure 5: Spearman correlation coefficient between different parameters. We
report the correlations between shape parameters and the viscosity that are
given in Table 4 sorted by decreasing absolute value. We also see the
correlations between different shape parameters. The red block in the middle
of the map corresponds to high correlations of shape parameters expressed in
terms of gyration tensor eigenvalues. Hence, we expect them to be somewhat
correlated. Smaller values of the relative shape anisotropy ($\kappa$)
determine how isotropic a configuration is. Similarly for asphericity ($b$)
and acylindricity ($c$). Detailed definitions of these parameters are given in
the Methods.

Our findings align with recent experimental results and provide a precise
picture of the molecular rearrangements giving rise to the observed
rheological behaviour of our melts. Like Zografos et al. 38 and Lopez et al.
27 , for densely grafted bottlebrushes we identify a star-like to bottlebrush-
like transition with power law exponents for the flow curves progressively
decreasing and converging to a Zimm scaling of $\sim-0.33$ with increasing
$N_{BB}$ 58 , even if in our case the total molecular weight is constant
instead of progressively increasing $N_{BB}$ at fixed side chain length. We
attribute the shear thinning behaviour to the orientation of the individual
bonds rather than the backbone relaxation, in line with the computational
findings of Qu et al. 59 . This idea would also be in line with the recent
findings that a monomeric friction coefficient in Brownian dynamics
simulations of Fraenkel chains is necessary to correctly capture experimental
viscosity data for unentangled linear polymers under flow 60, 61, 62 . Yet,
this might be caused by the high shear rates accessible to our simulations,
close to the inverse of the bond relaxation times $\tau_{b}$, while different
architectural effects might play a more important role at lower shear rates
that we could not access. Nevertheless, we agree with the idea that for
densely grafted bottlebrushes the side chains are not able to align well in
the flow direction, which might lead to strain-hardening in extensional flow
38, 27 . Still, we attribute this to the steric repulsion of side chains close
to the backbones, rather than to interdigitation. In fact, both in densely and
loosely grafted bottlebrushes the side chains still are somewhat aligned with
the flow ($P_{2}>0$, see Fig. S2) and not really interlocked in the direction
perpendicular to the flow.

Table 4:  Spearman correlation coefficient between different parameters and viscosity Parameter | Spearman Coefficient
---|---
$P_{2}$ | -0.96
$G_{yy}$ | 0.95
$\lambda_{min}$ | 0.89
$\tan 2\phi$ | -0.81
$G_{xx}$ | -0.51
Shape Anisotropy ($\kappa$) | -0.50
Asphericity ($b$) | -0.48
$\lambda_{max}$ | -0.47
$R_{g}^{2}$ | -0.46
Prolateness ($p$) | -0.43
$G_{xy}$ | -0.30
$G_{xz}$ | 0.23
$G_{zz}$ | 0.23
Acylindricity ($c$) | -0.23
$\lambda_{mid}$ | 0.18
$G_{yz}$ | 0.05

##  5 Conclusions

We have studied the shear thinning of bottlebrush polymer melts under shear
flow using non-equilibrium molecular dynamics simulations. We explore a wide
range of architectures at fixed molecular weight, transitioning from linear
chains to combs, to densely grafted bottlebrushes, to star-like and star
polymers. We report a range of shear thinning exponents from a Rouse regime
for linear and star-like systems to a Zimm regime for cylindrical
bottlebrushes, observing a star-like to bottlebrush-like transition in line
with recent experiments38 . We observe directly the alignment of all molecules
with the flow direction at an increasing shear rate, which is more pronounced
for linear chains and for bottlebrush-like polymers than for stars due to
their anisotropic shape. We quantify molecular alignment in the form of
several parameters extracted from the gyration tensor such as the principal
components along and perpendicular to the flow, and shape parameters such as
asphericity or prolateness. Yet, in the range of shear rates considered, the
alignment of molecular bonds explains the architecture effect on shear
thinning, rather than the molecular alignment. Linear chains show the
strongest shear thinning because all bonds can easily align with the flow,
while this is hindered for the side chains of cylinder-like bottlebrushes,
which then shear thin the least. Lopez et al. 27 and Zografos et al. 38
attribute the strain hardening observed in extensional flow in the cylindrical
bottlebrush regime to this bottlebrush backbone alignment and increased
friction due to side chain interdigitation; we see a similar alignment, but
our results suggest that the effect on rheology is rather caused by the
orientation of the backbone and side chain bonds. Overall, our findings line
up with the intuition obtained from experimental results on the rheology of
bottlebrushes in different regimes but offer important insight into the
molecular configurations of different architectures under flow that is crucial
to determine and control the flow properties of bottlebrush melts by tuning
their architecture. It will be important in future work to extend these
findings to a wider range of molecular weights and shear rates, and to
eventually quantify analytically the architecture-structure-property
relationships under flow for this class of branched polymers.

##  6 Code availability

In an effort to promote Open Science practices, the LAMMPS code used to
perform our simulations is available on the GitHub page of our group
https://github.com/giuntoli-group/bottlebrush-shear-thinning

We thank the Center for Information Technology of the University of Groningen
for their support and for providing access to the Hábrók high performance
computing cluster. This work made use of the Dutch national e-infrastructure
with the support of the SURF Cooperative using grant no. EINF-2648. We thank
Daniele Parisi for the useful discussions and Yağmur Öztürk for proofreading.

This material is available for download from http://pubs.acs.org.

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