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Rheology of Poly(α-olefin) Bottlebrushes: Effect of Self-Dilution by

Alkane Side Chains

Manfred H. Wagner*and Valerian Hirschberg*

Cite This: Macromolecules 2024, 57, 2110−2118

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ABSTRACT: Bottlebrush polymers are combs with extremely

high grafting density along their backbone chain. We consider the

rheology of bottlebrush poly(α-olefins) with side chains ranging

from 6 carbons [poly(1-octene)] to 16 carbons [poly(1-

octadecene)] as investigated by Lopez-Barron et al. (J. Rheol.

2019,63 (6), 917−926). We argue that the backbone chain of

poly(α-olefins) is diluted by the unentangled alkane side chains

and that the rheology of poly(α-olefin) bottlebrushes is equivalent

to that of poly(1-methylethylene), i.e., atactic polypropylene

diluted by a low-molecular-weight solvent. We show that this

approach is in agreement with the decreasing plateau modulus of

the poly(α-olefins) with increasing side chain length, and it allows to replace empirical correlations by a relation based on physical

arguments. The specific strong transient strain hardening in elongational flow shows similar features as observed for entangled

solutions of linear polymers and can be explained by the enhanced relaxation of stretch model if self-dilution of the poly(α-olefins) is

taken into account. Due to the large difference between the disengagement time

and Rouse time

, the orientation and stretch of

backbone chains are well separated, and strain hardening starts after full orientation at Weissenberg numbers above

Wi 0.3

R R

. The amount of strain hardening increases with increasing dilution, and at high

WiR

, all poly(α-olefins) are

expected to reach the same steady-state elongational stress.

1. INTRODUCTION

As summarized in a recent review,

comb and bottlebrush

polymers show a rich variety of rheological and mechanical

properties that can be controlled through their molecular

characteristics, such as the backbone and side chain lengths as

well as the number of branches per molecule or the grafting

density. Of particular interest to industrial applications like film

blowing, fiber spinning, and foaming is the amount of strain

hardening in elongational flow.

In order to induce substantial

strain hardening, at least two branch points with long-chain

branching (LCB) per molecule are needed as shown by McLeish

and Larson.

The side chains length has to be above the

entanglement molecular weight Mein order to pin segments of

the backbone between two branch points to the deforming

matrix and postpone stretch relaxation until the side chains are

withdrawn into the tube of the backbone. Abbasi and co-

workers

investigated the impact of the number of LCB

branches from loosely crafted combs to bottlebrushes and

observed that the strain hardening factor increases significantly

with increasing number of branches. Transient strain hardening

is defined as the relative upward deviation of the elongational

stress growth coefficient from the linear viscoelastic envelope.

The term “bottlebrush” refers to polymer topologies with the

coil size of the side chains being equal to or greater than the

spacing between branch points along the backbone. If the side

chains are shorter and have molecular weights below Me, they

can no longer restrict stretch relaxation of backbone segments,

and there should be no strain hardening at strain rates below the

inverse of the Rouse time. However, recently Lopez-Barron et

al.

2,3

reported that bottlebrush polymers can exhibit substantial

strain hardening even under circumstances where no backbone

segments pinned by side chains exist. They investigated the

elongational rheology of high molecular weight poly(α-olefin)

bottlebrushes ranging from poly(1-octene) to poly(1-octade-

cene) prepared by the polymerization of 1-alkene monomers.

From a molecular point of view, these are the simplest

bottlebrush polymers as both the backbone and the side chains

consist of linear alkanes. The lower plateau moduli of the

poly(α-olefins) were explained by steric repulsion between the

side chains, with longer side chains leading to larger persistence

lengths and resulting in higher entanglement molecular weights

Received: November 26, 2023

Revised: January 28, 2024

Accepted: February 1, 2024

Published: February 21, 2024

Article

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https://doi.org/10.1021/acs.macromol.3c02430

Macromolecules 2024, 57, 2110−2118

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Me. Lopez-Barron and co-workers

2,3

observed substantial

transient strain hardening of the elongational stress growth

coefficient of the poly(α-olefins) investigated and attributed this

to an increase in side chain interdigitation as soon as the

polymers align in the flow direction. This was postulated to

increase intermolecular friction, leading to enhanced transient

strain hardening. However, to date, no constitutive model has

been advanced to predict the elongational rheology of these

bottlebrush polymers with entangled backbone and unentangled

side chains and explain quantitatively why they are capable of

strain hardening.

According to a recent review by Matsumiya and Watanabe,

the change of monomeric friction with segmental orientation is

the key in understanding the elongational rheology of linear

polymer melts and concentrated polymer solutions. While the

universality of the linear viscoelastic behavior of well-entangled

monodisperse linear polymer melts and solutions is well

established based on only three material parameters (plateau

modulus, characteristic time, and number of entanglements),

this universality is lost in the nonlinear viscoelastic regime as

especially apparent in elongational flow, where differences in the

rheology of melts and solutions become apparent. While the

elongational viscosity of linear polymer melts shows monoto-

nous strain-rate thinning, solutions show strain-rate thinning,

followed by strain-rate hardening, see, e.g., ref 9. According to

Ianniruberto et al.,

31,32

strain-rate thinning is caused by the

reduction of segmental friction of highly oriented/stretched

polymer chains in linear melts under fast extensional flows. The

magnitude of this friction reduction diminishes with decreasing

polymer concentration in solution, resulting in the nonun-

iversality of the elongational rheology of polymer melts and

solutions. However, so far the effect of friction reduction has

been modeled by the use of empirical functions related to

segmental orientation or elongational stress with parameters

fitted to experimental data of elongational viscosities. A

perspective on recent findings in extensional rheology for

polymer melts and solutions with different macromolecular

architectures was presented by Huang.

In the context of tube models with varying tube diameter,

Wagner and Narimissa

proposed the enhanced relaxation of

stretch (ERS) model and showed that the stretch evolution

equation of the ERS model can be expressed in terms of

monomeric friction reduction. Instead of empirical correlations

between friction coefficient and segmental orientation, the ERS

model provides an analytical and parameter-free relation of

friction reduction as a function of chain stretch. By primitive

chain network simulations using the parameter-free universal

relation of monomeric friction reduction derived from the ERS

model, Wagner et al.

demonstrated the equivalence with

empirical friction reduction models for three poly(propylene

carbonate) melts and a polystyrene (PS) melt.

The objective of this contribution is to show that the

elongational rheology of the poly(α-olefin) bottlebrushes

investigated by Lopez-Barron and co-workers

can be explained

by the ERS model, if dilution of the backbone by the side chains

is taken into account. We first summarize the molecular and

linear viscoelastic characterization of the poly(α-olefins) and

show that the decreasing plateau modulus with increasing side

chain length is a consequence of self-dilution of the backbone by

the alkane side chains. We then give a short summary of the ERS

model, followed by a comparison of experimental data and

model predictions. The modeling is based exclusively on the

molecular and linear viscoelastic characterization of the poly(α-

olefins) and without the use of any free parameter.

2. MOLECULAR AND LINEAR VISCOELASTIC

CHARACTERIZATION OF POLY(α-OLEFIN)

BOTTLEBRUSHES

The molecular properties of the bottlebrush poly(α-olefins)

with side chains from 6 carbons (poly(1-octene)) to 16 carbons

(poly(1-octadecene)) are taken from Lopez-Barron et al.

1−3

They also considered an atactic poly(1-methylethylene), i.e.,

atactic polypropylene (aPP) at two temperatures to compare the

rheological response of the bottlebrush polymers with that of a

linear polyolefin. The entanglement molecular weight was

calculated by

MRT

(1)

is the density measured at 25 °C for aPP, PO, PD, and PDD

and at 70 °C for PTD and POD, Rthe gas constant, Tthe

absolute temperature, and

the plateau modulus of the

poly(α-olefins). In addition, Table 1 summarizes the number Z

= Mw/Meof entanglements, the glass transition temperature Tg,

and the melting temperature Tm(measured as the second (high

temperature) peak in the DSC traces

). All bottlebrush

polymers have well-entangled backbones.

From the mastercurves of G′and G″, parsimonious relaxation

spectra were obtained

G t g t( ) exp( / )

(2)

for characterization of the linear viscoelasticity all polymer

systems considered here. The partial moduli giand relaxation

times

as determined by the IRIS software

9,10

resulted in

excellent agreement with the linear viscoelastic data of G′and

G″, see Figure 1 and the Support Information.

We calculated

the zero-shear viscosity

and the mean quadratic average of the

relaxation times taken as disengagement time

from the

discrete relaxation spectra:

(3)

and

Table 1. Molecular Characterization of Bottlebrush

polymer Mw[kg/mol] Mw/Mn[-] Me[kg/mol]

[kPa] Z[-] Tg[°C] Tm[°C] ρ[kg/m3]

poly(1-methylethylene), aPP 233 1.82 3.88 590 60 864

poly(1-octene), PO 3030 2.41 27.1 72.5 112 −63.5 792

poly(1-decene), PD 3160 2.43 38.0 51.1 83.2 −68.0 −0.65 784

poly(1-dodecene), PDD 3960 2.48 55.3 35.9 71.6 −66.5 25.9 801

poly(1-tetradecene), PTD 4820 2.12 74.2 28.2 65.0 43.1 803

poly(1-octadecene), POD 5870 2.50 109 20.1 53.9 54.1 762

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iii

(4)

We use Osaki’s approach

for the quantification of the Rouse

stretch relaxation time

, which extrapolates the Rouse time of

unentangled polymer systems to the Rouse time of the melt and

takes into account the power of 3.4 scaling of the zero-shear

viscosity with molecular weight M

2.4

{

(5)

As the bottlebrush melts considered are polydisperse with

polydispersities Mw/Mnof 1.8−2.5 and the Rouse time will be

dominated rather by the longer polymer chains, we identify M

here with the z-average Mzof the molecular weight and assume a

log-normal molecular weight distribution, i.e.,

M M M M/

(6)

The critical molecular weight Mc, which is the molecular

weight when the entanglement effect becomes apparent by a

change of the power-law exponent of the zero-shear viscosity

scaling from 1 to 3.4, is typically related to Meby a factor of 2 to

3. We take the relation

M M2.5

c e

(7)

in the following. As seen from Table 2,

and

are separated by

2−3 orders of magnitude. Due to the different molecular weights

and the different measurement temperatures T,

and

of the

poly(α-olefins) cannot be compared directly between the

samples.

As seen in Table 1, the plateau modulus

of the bottlebrush

poly(α-olefins) decreases with increasing number

Nsc

of side

chain bonds or increasing molecular weight

per backbone

bond. Fetters et al.

reported the following empirical

correlation for poly(α-olefins),

G m41.84

N b

1.58

(8)

with the prefactor in units of MPa and based on the range of

m35 g/mol 56 g/mol

. Fetters and co-workers noted that

this relation is strictly empirical, and there is no justification for

this correlation, except that it fits the data. Lopez-Barron and co-

workers

expressed the dependence of

Nsc

by a power law

decay function of the form

G N

1.05 1.05

N sc

1.47 b

1.47

{

(9)

with the prefactor again in units of MPa. As shown in Figure 1

(dashed lines), both empirical correlations result in good fits to

the

data of the poly(α-olefins). The additional data point for

poly(1-hexene) is taken from ref 1.

While these are empirical correlations, we argue that the

backbone of the bottlebrush poly(α-olefin) is equivalent to

diluted aPP, i.e., the side chains of the poly(α-olefin) polymers

dilute their backbone chain. We assume that the effect of the side

chains on the rheology is equivalent to that of a solvent

consisting of low-molecular-weight oligomers, except that the

oligomers are attached to the backbone. We recall that the

plateau modulus of, e.g., PS dissolved in oligomers of the same

chemistry decreases with the polymer fraction

according to

, see, e.g.

9,14,29

We take the dilution exponent as equal to

1, while larger values of the dilution exponent (1 to 1.3) found

experimentally can be attributed to the enhanced relaxation of

chain ends as reported by Shahid et al.

We also note that Liu et

al.

investigated the solvent effect of the short arm of entangled

asymmetric PS star polymers. They demonstrated that the effect

of the short arm on the plateau modulus is equivalent to the

effect of short chains in binary blends consisting of linear long

and short chains with equivalent weight fractions as in the

asymmetric stars.

We therefore expect that the plateau modulus

of the

poly(α-olefins) decreases with the virtual fraction

of aPP

contained in the poly(α-olefins) according to

G G

N PP m

(10)

GPP

is the plateau modulus of aPP. The ratio

m m N/ 3/( 2)

mPP b sc

= = +

of the molecular weight

m21 g/mol

PP =

per backbone bond of aPP to the respective

molecular weight

of the α-olefins is given in Table 2.

Equation 10 is shown in Figure 1 by the continuous red line and

results in description of the

data based on reasonable

assumptions without the need for any fit parameter. We can also

Figure 1. Plateau modulus

as a function of the molecular weight

per backbone bond of α-olefins. Data (symbols) from Lopez-Barron

and co-workers

and Fetters et al.

Table 2. Linear Viscoelastic Characterization of the Poly(α-olefin) Polymers at the Measurement Temperatures T

polymer T[°C] τR[s] τd[s] η0[Pa s] ϕm[-] ϕ[-] U/(3kT) [-]

aPP 25 °C 25 79.7 6.23 ×1043.43 ×1091 1 47

aPP 46 °C 46 2.21 1.89 ×1031.02 ×1081 1 44

PO 25 13.5 6.12 ×1042.32 ×1080.375 0.35 47

PD 7.6 24.8 5.32 ×1041.83 ×1080.300 0.29 50

PDD 3.3 31.7 1.92 ×1051.36 ×1080.250 0.25 51

PTD 60 0.56 7.57 ×1021.52 ×1060.214 0.22 42

POD 70 1.93 1.86 ×1033.37 ×1060.167 0.18 41

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derive the fraction of aPP backbone contained in the poly(α-

olefins) directly by the ratio of

GPP

G G/

N PP

(11)

As seen from Table 2, there is excellent agreement of with

, which confirms the assumption of backbone dilution. Thus,

as far as the rheological behavior is concerned, we may consider

the bottlebrush poly(α-olefins) as self-diluted aPP, i.e., as

diluted aPP with the solvent molecules attached to the

backbone. The effective fraction of the aPP backbone

decreases with increasing side chain length from 1 for aPP to

0.35 for PO and to 0.18 for POD. The question may arise, why

aPP is the correct reference for self-dilution of the poly(α-

olefins) and why it is not polyethylene (PE). As also shown in

Figure 1 (dotted red line), the relation

G G m m( / )

N PE PE b

with

G2500 kPa

PE =

and

m14 g/mol

PE =

does not fit the

plateau moduli of the poly(α-olefins). The steric effect of the

methyl group in aPP reduces chain flexibility in comparison to

PE as expressed by the decreased plateau modulus or the

increased entanglement molecular weight of aPP. The same

steric effect on the backbone flexibility is caused by the first

methylene group (c1) of the alkane side chains of the poly(α-

olefins), while the effect of c2 and higher, i.e., the effect of the

number

of side chain bonds on the plateau modulus can

be expressed by eq10 with the aPP fraction

m m N/ 3/(3 1)

mPP b sc

= = +

Table 2 summaizes the linear viscoelastic characterization of

the poly(α-olefin) polymers at the measurement temperature T.

3. ERS MODEL WITH DILUTION

The ERS model is a special form of the molecular stress function

(MSF) model, which is a generalized tube segment model with

strain-dependent tube diameter.

15−17

The extra stress tensor

t( )

of the MSF model is given by a history integral of the form

tG t t

tf t t t t tS( ) (’)

’( , ’) ( , ’)d ’

(12)

G(t) is the relaxation modulus, tis the time of observation

when the stress is measured, and t’ indicates the time when a

tube segment was created by reptation. The strain measure

SDE

represents the contribution to the extra stress tensor originating

from the affine rotation of the tube segments according to the

“Independent Alignment (IA)” assumption of Doi and

Edwards

18,19

and is given by

t t

t tSu u S( , ’) 5 ’ ’

’5 ( , ’)

(13)

with

t tS( , )

being the relative second-order orientation tensor.

u u

is the dyad of a deformed unit vector

t tu u ( , )=

u F u’t

= ·

(14)

t tF F ( , )

t t

1 1

is the relative deformation gradient tensor, and

u’ is the length of

. The orientation average is indicated by

<...>0,

i.e., an average over an isotropic distribution of unit vectors

f f t t( , )=

represents the inverse of the relative tube

diameter a/a0, and at the same time, the relative length l(t,t’)

of a deformed tube segment

f t t a

a t t

l t t

( , ’) ( , ’)

( , ’)

= =

(16)

At time t = t’, the tube segment was created with equilibrium

tube diameter a0and equilibrium length l0. Equation 16 is a

direct consequence of equation (A9) of Doi and Edwards,

who

showed that the line density n/l, i.e., the number nof Kuhn

segments of length bthat are found per length lof the tube, is a

well-defined thermodynamic quantity and defines the tube

diameter aby the relation n/l = a/b2. For

, eq 12 reduces to

the original Doi−Edwards IA model.

SDE

is determined directly by the deformation history

according to eq14, while the stretch fis found as solution of

an evolution equation considering affine tube segment

deformation balanced by enhanced Rouse relaxation.

With

increasing stretch and decreasing tube diameter a, the number of

monomers in a control volume of length and diameter awill

decrease with the consequence of ERS in this control volume.

The incremental increase of the relaxation rate with stretch is

proportional to the fourth power of the stretch, and the

relaxation rate therefore is proportional to the fifth power of the

stretch. The evolution equation of the stretch f(t,t’) is obtained

as a balance of extension rate versus relaxation rate

tff f

K S( : ) 1(1 ) ( 1)

4 5

(17)

with initial condition

f t t( , 0) 1= =

. The first term of the

right-hand side of eq 17 expresses affine deformation with K

being the deformation rate gradient, the second term Rouse

stretch relaxation with Rouse time

, and the third term takes

into account ERS on smaller length scales. is the volume

fraction of the polymer in solution. Note that for small stretches

with

f1 1< <

, eq17 reduces to the classical relation

tf fK S( : ) 1( 1)

(18)

i.e., the evolution of stretch depends only on the deformation

rate and the Rouse time. As shown by ref 10, the evolution eq17

can alternatively be expressed as a reduction of the monomeric

friction coefficient by

f f f f

5(1 ) ( 1)

04 4 4 3 2

+ + + + +

(19)

with

being the equilibrium friction coefficient. Thus, in this

interpretation of the ERS model, friction decreases with

increasing stretch, but the reduction of friction is delayed and

reduced at lower polymer concentration.

According to the entropic fracture hypothesis,

brittle or

elastic fracture will occur when the strain energy of a chain

segment reaches the bond energy Uof a carbon−carbon bond.

Due to thermal fluctuations, the total strain energy of the chain

segment will be concentrated on one C−C bond by thermal

fluctuations, and the bond then ruptures. This leads to crack

initiation and within a few milliseconds to macroscopic fracture.

The critical stretch at fracture,

, is given by

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kT3

(20)

with kbeing the Boltzmann constant and Tthe absolute

temperature. This fracture criterion has been shown to be in

agreement with experimental evidence of polymer melts and

solutions, see, e.g., refs 21−24.

4. COMPARISON OF ERS MODEL PREDICTIONS AND

ELONGATIONAL VISCOSITY DATA OF

POLY(α-OLEFIN) POLYMERS

Predictions of the ERS model with stress tensor eq12 and

evolution of stretch eq17, and with the parameters

and

from Table 2, are compared in Figures 2 and 3to the data

(symbols) of the elongational stress growth coefficient

t( )

reported by Lopez-Barron and co-workers.

The atactic PP

(Figure 2) at 25 and 46 °C shows the typical elongational

behavior of a linear polymer with transient strain hardening

starting at elongations rates

1/ R

with

calculated by eq5,

followed by monotonous increasing transient strain-hardening

with increasing strain rate. Calculated results (lines) of the ERS

model are in line with this strain hardening behavior. The data of

the elongational stress growth coefficient of aPP at 46 °C show

some indication of transition to a steady-state elongational

viscosity, in qualitative agreement with model predictions. At 25

°C and at small strain rates,

t( )

shows a maximum which is

indicative of failure by inhomogeneous deformation, while at

higher strain rate, sample fracture is observed before a steady

state is reached. The maximal elongational viscosity reached

experimentally is somewhat below the steady-state elongational

viscosity predicted by the model, and according to the fracture

criterion of eq 20, fracture would only be expected at the highest

strain rate of

10 s 1

as indicated by the abrupt end of the

predicted line before reaching the steady state.

For all the poly(α-olefin) bottlebrushes (Figure 3), the onset

of strain hardening is in agreement with model predictions based

on the Rouse times calculated by eq5. Also, the start-up of the

elongational viscosity is well described by the ERS model for all

elongation rates investigated. The samples were allowed to

elongate in the Sentmanat extensional rheometer (SER) beyond

the recommended maximum Hencky strain of

, above

which the sample starts to wind on itself. All the samples broke

before reaching a Hencky strain of 4, except PTD (Figure 3d)

and POD (Figure 3e) at strain rates above 1 s−1. For PO (Figure

3a) and POD (Figure 3e), the maximal values of

t( )

are

mostly in agreement with predictions, although the transition to

a steady-state elongational viscosity is only seen at the lowest

strain rates. Fracture according to the fracture criterion of eq20

is only expected at the highest strain rates and is indicated again

by the abrupt end of the predicted lines. The samples of PD

(Figure 3b) and PDD (Figure 3c) fractured already at much

smaller strains than expected. We note that PD and PDD were

measured at rather low temperatures, PD at only 7K above the

melting temperature, and PDD above the glass transition

temperature but below the melting temperature. While the ERS

model gives a consistent description of the elongational stress

growth of aPP and the poly(α-olefin) bottlebrushes, the cause of

the premature fracture particularly of the samples of PD and

PDD remains unclear.

Due to the different measurement temperatures and the

different molecular weights, a direct comparison of the time-

dependent elongational stress growth coefficients

t( )

and the

strain hardening of the poly(α-olefin) bottlebrushes is not

possible. However, from Figures 2 and 3and except for PD and

PDD, we may conclude the steady-state elongational viscosity

predictions of the ERS model are in general agreement with the

maximal elongational viscosity data measured and can therefore

be used for a comparison of the elongational behavior of the

poly(α-olefin) bottlebrushes. The normalized steady-state

elongational viscosity

E 0

or its maximal value in the case of

fracture predicted by the ERS model is plotted in Figure 4a as a

function of Weissenberg number

WiR R

. Lines are

calculated by the ERS model and symbols indicate the calculated

values at the experimental strain rates. Note that this is a

temperature-invariant representation. aPP shows a monoto-

nously decreasing elongational viscosity with a change of slope

when strain hardening begins. The slope after the kink is

approximately −0.4 as indicated by the red straight line. This is

similar to the slope of

log E

as a function of

log

observed for

other linear polymers, and it is higher than the slope of −1/2

expected from a one-mode ERS model. The higher slope is

caused by the width of the relaxation spectrum, as already

Figure 2. Comparison of the elongational stress growth coefficient

t( )

data (symbols) to the calculated results of the ERS model (lines) for aPP.

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Figure 3. Comparison of the elongational stress growth coefficient

t( )

data (symbols) to the predictions of the ERS model (lines) for (a) poly(1-

octene), (b) poly(1-decene), (c) poly(1-dodecene), (d) poly(1-tetradecene), and (e) poly(1-octadecene).

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documented earlier for monodisperse PS melts.

For the

poly(α-olefin) bottlebrushes, the elongational viscosity first

decreases with increasing

WiR

, then increases and decreases

again after a maximum in

.Figure 4a reveals a clear pattern of

the strain hardening. The maximum in the elongational viscosity

is the larger, the smaller the fraction of the backbone chain. The

drop of

at high

WiR

is due to elastic fracture. We note that the

elongational behavior of the poly(α-olefin) polymers is

strikingly similar to that of entangled solutions of 3900 kg/

mol polystyrene PS3900 in diethyl phthalate (DEP) investigated

by Bhattacharjee and co-workers

and Acharya and co-

workers.

This is shown in Figure 4b for solutions of

PS3900/DEP with 17/15/10 weight% of PS and compared to

POD with aPP backbone fraction of 18%. Due to the large

difference between disengagement time

and Rouse time

the elongational viscosity of the poly(α-olefin) bottlebrushes

and the PS3900/DEP solutions is first reduced by increasing

orientation, followed by strong strain-rate hardening when the

Weissenberg number is approaching

Wi 1

. Lines are

predictions of the ERS model. Similar trends were also reported

for other high molecular weight PS samples dissolved in

oligomeric styrene.

14,28,29

5. DISCUSSION AND CONCLUSIONS

The rheology of bottlebrush poly(α-olefin) polymers can be

explained by considering that the alkane side chains with 6−16

carbons dilute the backbone chain. As far as the rheology is

concerned, poly(α-olefin) can be considered as diluted aPP with

the solvent molecules consisting of alkane oligomers attached to

the backbone. The plateau modulus of the poly(α-olefins)

decreases with increasing length of the side chain according to

G G

aPP

0 2

, and the diluted fraction of the backbone derived

from this relation agrees with the ratio of molecular weight per

backbone bond of aPP to that of the poly(α-olefins). The Rouse

time

as obtained from Osaki’s relation

is in good concord

with the onset of transient strain hardening oberseved

experimentally at elongation rates

1/ R

, if we consider

that the effective

in the case of the polydisperse melts is

determined by the longer molecules. Overall, the ERS model

and the fracture criterion of eq 20 provide a consistent

description of the elongational rheology of the bottlebrush

poly(α-olefins), even with the early fracture especially of the PD

and PDD samples remaining unexplained.

Lopez-Barron and co-workers

attributed the strain hardening

of bottlebrush poly(α-olefins) to an increase in side chain

interdigitation as soon as the polymers align in the flow

direction. They postulated an increase in intermolecular friction

by side chain interaction. Based on friction reduction

interpretation of the ERS model according to eq19, we can

now test the validity of this conjecture. Figure 5 presents the

normalized elongational viscosity

E 0

as a function of the

tensile stress

, and the tensile stress is normalized by the

plateau modulus

. All the curves coincide initially until

reaching the vertical red line at

E N

, which signifies full

orientation according to the Doi−Edwards IA model. We

conclude that the first part of the conjecture is confirmed. Strain

hardening starts after full orientation of the polymer chains, and

the distinct separation of orientation and stretch is due to the

large difference between disengagement time

and Rouse time

of the polymers considered. However, concerning the strain

hardening part of the conjecture in terms of intermolecular

friction, instead of an increase of friction by interdigitation,

strain hardening is rather caused by delayed and reduced friction

reduction with a decreasing fraction of the diluted aPP backbone

as seen from eq19. Alternatively, in the tube model

interpretation of the ERS model, the tube diameter at

equilibrium increases from the tube diameter

aPP

of aPP to

Figure 4. Normalized elongational viscosity

E 0

as a function of Weissenberg number

WiR

for (a) of poly(α-olefin) bottlebrushes, and (b) for 17/

15/10 weight% of PS3900 dissolved in DEP

26,27

and for POD. Straight red line in (a) indicates a slope of −0.4.

Figure 5. Normalized elongational viscosity

( )/

EE0

as a function of

elongational stress

normalized by the plateau modulus

. Vertical

red line indicates

E N

Macromolecules pubs.acs.org/Macromolecules Article

https://doi.org/10.1021/acs.macromol.3c02430

Macromolecules 2024, 57, 2110−2118

2116

the tube diameter

a a

0 0

aPP 1/2

of the poly(α-olefins) due to

self-dilution in agreement with their smaller plateau modulus. A

larger tube diameter allows for a larger stretch. This is reflected

in the relaxation term of the stretch evolution eq17 by the factor

, which leads to high strain rates to a steady-state stretch

fss

ref 10

f Wi5 5

= =

(21)

The limiting stretch

fss

increases with increasing dilution and

decreasing polymer fraction; i.e., more diluted polymer solutions

show more strain hardening than less diluted ones, see, e.g., refs

9,28,29. This results in a limiting elongational stress of

G f G Wi G Wi5 5 5 5 5

E N ss

R N

aPP

= = =

(22)

The universal relation 22 is shown in Figure 6 by the straight

red line with a slope of 1/2. All the poly(α-olefins) reach the

same elongational stress as given by eq22 at high

WiR

. At even

higher

WiR

, the stress is expected to level off due to sample

fracture. We can also see from Figure 6 that significant strain

hardening starts already at Weissenberg numbers above

Wi 0.3

In conclusion, the specific elongational rheology of poly(α-

olefin) bottlebrushes with entangled backbone and unentangled

side chains can be explained by self-dilution of the backbone by

the side chains and can be described by the ERS model. This is

expected to be true for other bottlebrush polymer systems with

entangled backbones as well, as long as the side chains are not

entangled, such as the model poly(n-alkyl methacrylate)s

investigated by Wu et al.

Based on the data of Abbasi et al.,

Hirschberg et al.

have shown that from bottlebrushes to

loosely grafted combs, the diluted plateau modulus of the

backbone decreases with the square of the backbone fraction

. Therefore, we expect that our modeling will still be valid if the

graft density decreases from bottlebrushes to loosely crafted

combs to linear aPP (as shown here in Figure 2), as long as the

side chains are not entangled. However, a qualitative change of

the elongational behavior occurs,

7,35

if the length of the side

chains approaches the entanglement limit, and/or the backbone

after dilution will no longer be entangled.

■AUTHOR INFORMATION

Corresponding Authors

Manfred H. Wagner −Polymer Engineering/Polymer Physics,

Berlin Institute of Technology (TU Berlin), 10587 Berlin,

Germany; orcid.org/0000-0002-1815-7060;

Email: [email protected]

Valerian Hirschberg −Institute of Chemical Technology and

Polymer Chemistry (ITCP), Karlsruhe Institute of Technology

(KIT), 76131 Karlsruhe, Germany; orcid.org/0000-0001-

8752-930X; Email: [email protected]

Complete contact information is available at:

https://pubs.acs.org/10.1021/acs.macromol.3c02430

Funding

No funding was received to assist with the preparation of this

manuscript.

Notes

The authors declare no competing financial interest.

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