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A methodology of hydrodynamic complexity in topologically hyper-branched polymers undergoing hierarchical multiple relaxations

01 Jul 2020-Macromolecular Chemistry and Physics (Wiley)-Vol. 221, Iss: 13, pp 2000052

About: This article is published in Macromolecular Chemistry and Physics.The article was published on 2020-07-01 and is currently open access. It has received 2 citation(s) till now.

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DOI: 10.1002/macp.202000052 ((please add manuscript number))
Article type: Full Paper
A methodology of hydrodynamic complexity in topologically hyper-branched polymers
undergoing hierarchical multiple relaxations
Haibao Lu
*, Xiaodong Wang
, Mokarram Hossain
and Yong Qing Fu
Prof. H. B. Lu and X. D. Wang
Science and Technology on Advanced Composites in Special Environments Laboratory,
Harbin Institute of Technology, Harbin 150080, China
Dr. M. Hossain
Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea
University, Swansea, UK
Prof. Y. Q. Fu
Faculty of Engineering and Environment, University of Northumbria, Newcastle upon Tyne,
Keywords: hyperbranched, thermodynamics, relaxation
A hydrodynamic model was proposed to describe conformational relaxation of molecules,
viscoelasticity of arms and hierarchical multiple-shape memory effect (multi-SME) of hyper-
branched polymer. Fox-Flory and Boltzmann’s principles were employed to characterize and
predict the hierarchical relaxations and their multi-SMEs in hyper-branched polymers. A
constitutive relationship among relaxation time, molecular weight, glass transition
temperature and viscoelastic modulus was then formulated. Results revealed that molecular
weight and number of arms of the topologically hyper-branched polymers significantly
influence their hydrodynamic relaxations and shape memory behaviors. The effectiveness of
model has been demonstrated by applying it to predict mechanical and shape recovery
behaviors of hyper-branched polymers, and the theoretical results show good agreements with
the experimental ones. We expect this study provides an effective guidance on designing
multi-SME in topologically hyper-branched polymers.

1. Introduction
Shape memory polymers (SMPs) are featured with shape memory effect (SME) which
enables them being smart and responsive materials through shape deformation/recovery in the
presence of external stimuli, such as thermal heating,
or electrical
SME is induced by partially segmental relaxation of macromolecular chains, where
the hard segments are kept frozen to memorize the permanent conformation and the soft ones
are kept active to undergo entropic conformation relaxation.
Multi-SMEs can be realized in
the SMPs, in which the multi-step relaxation is introduced for the soft segments in the SMP
In practice, the first strategy for generating multi-SME is to generate the
multiple segments by incorporating several discrete thermal transition components into
polymer macromolecules.
The second one is to generate soft segments with a broad
thermal transition in the polymer macromolecules.
Multi-SME in the SMPs has attracted
extensive attention due to their capabilities of shape recovery step by step,
which creates
great potentials for the practical applications in smart textiles, artificial intelligence robots,
bio-medical engineering.
Hyper-branched polymer is a popular functional material and consisted of multiple arms
connected to a central core.
The core can be an atom, a molecule, or a macromolecule,
while the arms are consisted of homogeneous or heterogeneous macromolecular chains.
Generally, the homogeneous arms have equal lengths and uniform structures, whereas the
heterogeneous ones have varied lengths and structures.
Recently, multi-SME has been
discovered and explored in the homogeneous hyper-branched polymers. Due to their discrete
thermal transitions of core and arms,
their shape recovery strength has been significantly
As it is well known, modelling of the multi-SME in hyper-branched polymers
is critical to explore their working mechanism and hydrodynamics.
However, this is
difficult because their branch-to-branch topology structures and hydrodynamics are quite

different from those of their linear counterparts.
Till now, few theoretical studies have been
carried out to study their hydrodynamic behaviors.
In this study, Fox-Flory equation
is initially employed to characterize the glass transition
temperature (T
) and relaxation behaviors of topologically hyper-branched polymers.
According to the Boltzmann’s hydrodynamic principle,
a constitutive relationship is
formulated with the consideration of molecular weight, number of arms, T
and relaxation
time. The proposed model is then employed to characterize and predict hierarchical and
multiple relaxations (e.g., relaxations of the core and arms, respectively) and their multi-
SMEs in hyper-branched polymers. Furthermore, effects of molecular weight and number of
arms on hydrodynamic relaxations and shape memory behaviors of topologically hyper-
branched polymers have been explored. Finally, the working principle of the complex multi-
arms undergoing molecular-level hierarchical multiple relaxations is explored and discussed
in order to link with the multi-SME in topologically hyper-branched polymers. Theoretically
obtained results are compared with the experimental data reported in literature in order to
verify the accuracy of proposed model. This study is expected to provide an effective strategy
to explore the hydrodynamic principle of multi-SME in the topologically hyper-branched
2. Theoretical framework
The SME in amorphous SMPs occurs within the glass transition zone.
is an
essential parameter for the hydrodynamics in the hyper-branched polymers.
of the
hyper-branched polymer is determined by the number of arms (
) and molecular weight
) based on the following equation:
M M fM

is the
is a polymer-specific constant (the constant is
proportional to the number of ends in the molecule),
are the molecular weights
of the arms and core of the hyper-branched polymer, respectively.
Figure 1 plots the calculated results of
based on equation (1) as functions of molecular
weight (
) and number of arms (
). As revealed from Figure 1(a), the molecular weight
) has a significant influence on the
, which is increased from 320 K to 360 K with an
values from 0.5B to 1.0B at a given
=400 K. As reported in literature,
the molecular weight has a significant influence on the
of the macromolecule chains,
due to their constitutive relationships. Therefore, the
is gradually increased with an
increase in molecular weight (
Meanwhile, effect of number of arms (
) on
has been investigated and the results are
shown in Figure 1(b). The analytical results reveal that the
is gradually decreased from
390 K to 370 K with an increase in the number of arms from
=2 to
=6, at a given
molecular weight of
. While the
is gradually increased with an increase in the
molecular weight ratio (
0 a
), mainly due to the increase in the molecular weight of core
). It is found that there is a distinct difference in the effect of molecular weight on the
between the hyper-branched polymers and the conventional polymers with a sharp transition.
of topologically hyper-branched polymer is not proportional to molecular weight, but
determined by the number of arms. That is to say, the
will be increased with an increase in
the molecular weight ratio of core and arms. However, it is then decreased with an increase in
the number of arms at a given constant weight ratio.
[Figure 1]

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