Karin J Bichler1, Bruno Jakobi2, Victoria García Sakai3, Alice Klapproth4, Richard A Mole4, Gerald J Schneider1,2. 1. Department of Physics &Astronomy, Louisiana State University, Baton Rouge 70803, Louisiana, United States. 2. Department of Chemistry, Louisiana State University, Baton Rouge 70803, Louisiana, United States. 3. ISIS Facility, Rutherford Appleton Laboratory, Harwell Science and Innovation Campus, Chilton, Didcot OX11 0QX, U.K. 4. Australian Nuclear Science and Technology Organisation, New Illawarra Road, Lucas Heights 2234, New South Wales, Australia.
Abstract
We have studied the short-time dynamical behavior of polydimethylsiloxane (PDMS) bottlebrush polymers, PDMS-g-PDMS. The samples have similar backbone lengths but different side-chain lengths, resulting in a shape transition. Quasi-elastic neutron scattering was used to observe the dynamical changes inherent to these structural changes. The combination of data from three spectrometers enabled to follow the dynamics over broad frequency and temperature ranges, which included segmental relaxations and more localized motions. The latter, identified as the methyl group rotation, is described by a threefold jump model and shows higher activation energies compared to linear PDMS. The segmental relaxation times, τs, decrease with increasing molecular weight of the side chains but increase with momentum transfer, Q, following a power law, which suggests a non-Gaussian behavior for bottlebrush polymers.
We have studied the short-time dynamical behavior of polydimethylsiloxane (PDMS) bottlebrush polymers, PDMS-g-PDMS. The samples have similar backbone lengths but different side-chain lengths, resulting in a shape transition. Quasi-elastic neutron scattering was used to observe the dynamical changes inherent to these structural changes. The combination of data from three spectrometers enabled to follow the dynamics over broad frequency and temperature ranges, which included segmental relaxations and more localized motions. The latter, identified as the methyl group rotation, is described by a threefold jump model and shows higher activation energies compared to linear PDMS. The segmental relaxation times, τs, decrease with increasing molecular weight of the side chains but increase with momentum transfer, Q, following a power law, which suggests a non-Gaussian behavior for bottlebrush polymers.
Bottlebrush polymers,
i.e., linear side chains covalently bonded
to a linear backbone, have received great interest in recent times.
These architectural modifications cause shape changes, such as from
spherical into elongated objects, simply by varying the size ratio
of backbone to side-chain length.[1−7] Inherent with the bottlebrush architecture are extraordinary properties,
such as super softness or low viscosity. These properties are desirable
in the development of new materials. Novel composites that take advantage
of shell-only particles or surface modifications can be envisioned.[3,8,9] The resulting macroscopic behavior
is governed by microscopic relaxation processes. However, only little
is known about the dynamics of bottlebrush polymer melts. For polymers
with a linear architecture or very short side chains, the relaxation
spectra are dominated by the backbone.[10] If side chains exceed a certain length, rheology can detect the
emergence of new processes.[11] In bottlebrushes,
the shape transition from elongated into spherical shapes may change
the molecular dynamics of the system. We have investigated the short-time
dynamics of polydimethylsiloxane (PDMS)-based bottlebrush polymers,
PDMS-g-PDMS, for the first time, by quasi-elastic
neutron scattering (QENS). In the current study, three different samples
were considered, with similar backbones but different lengths of the
side chain. The existence of the shape transition in these samples
has been confirmed by small-angle neutron scattering (SANS) experiments.[12]Dynamical processes in polymers evolve
over several orders of magnitude
in time. To cover such a broad time range with QENS, a combination
of three different spectrometers was used in this study, covering
a time range of t = 1 ps to t =
1 ns. By analyzing the dependence of the incoherent intermediate scattering
function, S(Q,t) ≔ Sinc(Q,t), on the time, t, and the momentum transfer, Q, we obtain information about the geometrical confinement
of localized motions as well as the relaxation times of the individual
processes.
Experimental Section
QENS measures
the dynamic structure factor which comprises coherent
and incoherent parts resulting from the scattering event such that S(Q,ω) = Scoh(Q,ω) + Sinc(Q,ω). The samples studied here are protonated PDMS-g-PDMS bottlebrush polymers, and as such, because of the
large incoherent scattering cross section of hydrogen, σinc = 80.26 × 10–24 cm2,
the measured dynamical structure factor is dominated by the incoherent
part. Therefore, in our case, S(Q,ω) = Sinc(Q,ω).The measured intensity, Sexp(Q,ω), is a convolution of the pure sample scattering, Sinc(Q,ω), with the instrument-specific
resolution, R(Q,ω), i.e., Sexp(Q,ω) = Sinc(Q,ω) ⊗ R(Q,ω). The resolution, which represents the
longest timescale that can be probed by the instrument, is a fundamental
property of the spectrometer used to measure the signal. As the temperature
is changed, it is not uncommon for a signal to shift from the dynamic
range of one instrument to another. In reciprocal and energy space,
the resolution is convoluted with the sample scattering, while if
one performs the time Fourier transformation of the data to get the
intermediate scattering function, Sinc(Q,t), the convolution is a simple
multiplication.Doing this implies that it becomes easier to
account for the resolution
and thus compare the dynamic structure factor across a number of instruments.In order to cover a broad dynamical range, we have used a
combination
of three different QENS instruments, the time-of-flight spectrometer
Pelican[13] at ANSTO, the time-of-flight
backscattering instrument IRIS[14] at ISIS,
and the backscattering instrument EMU[15] at ANSTO. Each of them has a different energy transfer and energy
resolution, resulting in different time ranges. The technical specifications
of the instruments are summarized in Table .
Table 1
Specifications of
the Different QENS
Instruments Used for Our Experiment
location
instrument
energy resolution
(μeV)
energy range (meV)
Q-range (Å–1)
ANSTO
EMU
1.1
–0.028 to 0.028
0.4–1.8
ISIS
IRIS
17.5
–0.55 to 0.57
0.6–1.6
ANSTO
Pelican
65
–19.99 to 2.59
0.2–1.8
All measurements were
conducted in a temperature range of T = 75 K to T = 300 K, accompanied with
a measurement of the instrument resolution at low temperature, specifically
at T = 1.5 K on EMU and Pelican and at T < 10 K on IRIS. For the temperature control, a closed-cycle cryostat
with a secondary helium circulation loop, controlled by a needle valve
(EMU and Pelican), and a closed-cycle refrigerator (IRIS) were used.
This allowed a base temperature of T < 10 K.For the normalization against detector efficiency, standard vanadium
calibration data were used for each instrument. The samples were loaded
into aluminum cans with aluminum screws provided by ANSTO, and all
measurements were carried out with these same cans, to ensure consistency
over the three instruments. The sample gap was chosen to be 0.1 mm,
which ensures 90% transmission and minimizes multiple scattering.
For all experiments, the program MANTID[16] was used for data reduction, employing existing routines specific
for each of the three spectrometers.
Backscattering Instrument—EMU
EMU is a neutron
backscattering instrument located at ANSTO, Sydney, Australia.[15] Neutrons with a wavelength of λ = 6.28
Å are selected via a pyrolytic graphite crystal
premonochromator via their (002) Bragg reflection.
The continuous beam is split by a background chopper into several
bunches, i.e., the neutron beam is pulsed to reduce background scattering.
The following graphite chopper, that is, a disc chopper with graphite
elements located at the periphery, deflects the neutrons, in its closed
position, to the Doppler drive. The Doppler drive oscillates with
a maximum velocity of vmax = 4.7 m/s and
shifts the neutrons in energy, creating an energy distribution of
(Ei ± δE)(t). The Doppler-shifted neutrons pass the graphite chopper
in the open position and scatter at the sample. A small number of
scattered neutrons, fulfilling the Bragg condition, are reflected
by the analyzer array crystals, Si(111), and pass the sample a second
time. The backscattered neutrons are collected at the detectors. The
time dependence of the Doppler drive and therefore of δE(t) is known. This allows to obtain the
energy transfer ΔE of the sample.
Time-of-Flight
Backscattering Instrument—IRIS
IRIS is an indirect
geometry time-of-flight spectrometer located
at the pulsed neutron source ISIS, Didcot, United Kingdom.[14] The incident neutron beam from the spallation
source is wavelength-selected by a chopper, with a frequency of f = 50 Hz, resulting in a bandwidth of neutrons centered
at λ = 6.66 Å. These neutrons scatter upon interaction
with the sample in all directions and hit an analyzer array of pyrolytic
graphite. Similar to EMU, those neutrons, fulfilling the Bragg condition,
in this case, the (002) reflection, are reflected back from the analyzers
and collected at the detectors. By knowing the total time-of-flight
time, t, of the neutrons together with the distances L1, source–sample, and L2, sample–analyzer–detectors, the energy
exchange within the sample can be determined.
Time-of-Flight Spectrometer—Pelican
Pelican
is a direct time-of-flight geometry cold-neutron spectrometer at ANSTO,
Sydney, Australia.[13] The incoming neutrons
are monochromatized and wavelength-selected by a focusing crystal
and pulsed by two Fermi choppers, which are phased in order to remove
λ/2 from the neutron beam. These pulsed neutrons of λ
= 6 Å scatter at the sample in all directions to be collected
at the detectors. The time, t0, needed
for elastic scattering without energy transfer can be determined based
on the initial energy, Ei, and the pathlength,
which the neutrons need to travel until the detector is reached. The
energy transfer can be determined by taking the difference in the
“real” flight time, ttof, which was needed including energy exchange and the elastic time, t0. This results in the scattering intensity
depending on the scattering angle and time. This instrument is able
to detect larger energy transfers compared to EMU and IRIS. Thus,
the simple approximation for the momentum transfer based on elastic
scattering is not valid, and the law of cosines is applied.
Samples
The samples used in this study are bottlebrush
polymer melts based on PDMS, i.e., PDMS-g-PDMS bottlebrushes.
Details of the synthesis route have been previously described in a
related publication.[12] These samples have
been analyzed prior by dielectric spectroscopy and SANS.[12] For completeness, the resulting molecular weights,
together with the polydispersity index (PDI) and the grafting density,
are summarized in Table . A pictorial representation of the resulting shapes of the different
bottlebrush polymers is illustrated in Figure .
Table 2
Number-Average
Molecular Weight, Mn, PDI, and Grafting
Density, ρ*, of the
Three Bottlebrush Samples and Their Single Component
name
single chain Mn (g/mol) (PDI)
bottlebrush Mn (kg/mol) (PDI) [ρ*]
side chain short
298 (1.0)
95 (1.3) [89]
side chain middle
1800 (1.2)
157 (1.1) [30]
side chain long
11,500 (1.1)
1106 (1.2)
[41]
backbone short
16,500 (1.1)
backbone middle/long
13,500 (1.2)
Figure 1
Simplified illustration of the shape of the
PDMS-g-PDMS bottlebrush polymers with (a) Mnside chain =
298 g/mol, (b) Mnside chain = 1800 g/mol, and (c) Mnside chain = 11,500 g/mol.
Simplified illustration of the shape of the
PDMS-g-PDMS bottlebrush polymers with (a) Mnside chain =
298 g/mol, (b) Mnside chain = 1800 g/mol, and (c) Mnside chain = 11,500 g/mol.
Results
At the lowest temperatures,
a weak decay with almost Q-independent relaxation
times is observed in the accessible time
range, as seen in Figure a,b. This indicates a localized motion, which
at such temperatures, where the segmental dynamics are assumed to
be frozen or too slow, can be attributed to the rotation of the methyl
groups because all protons of the bottlebrush polymer are parts of
a methyl group (Figure ).[17]
Figure 3
Intermediate scattering function, S(Q,t), vs time, t, for the sample
with Mnside chain = 11,500 g/mol for four different temperatures:
(a) T = 75 K, (b) T = 100 K, (c) T = 150 K, and (d) T = 175 K. Solid lines
represent the data descriptions with the model function for low temperatures, eq .
Figure 2
Chemical structure of the PDMS-g-PDMS bottlebrush
polymer with all protons located in the methyl groups.
Chemical structure of the PDMS-g-PDMS bottlebrush
polymer with all protons located in the methyl groups.Intermediate scattering function, S(Q,t), vs time, t, for the sample
with Mnside chain = 11,500 g/mol for four different temperatures:
(a) T = 75 K, (b) T = 100 K, (c) T = 150 K, and (d) T = 175 K. Solid lines
represent the data descriptions with the model function for low temperatures, eq .By comparing the glass-transition temperature, determined by dielectric
spectroscopy by Jakobi et al.(12) (Table ), with the measurement temperature of Figure a,b, it reveals that these two temperatures
are below Tg, thus the segmental dynamics
are frozen.
Table 3
Glass-Transition Temperature, Tg, of the Three Samples, Taken from the Publication
of Jakobi et al.(12)
Mnside chain
Tg (K)
298
140.4 ± 0.2
1800
146.2 ± 0.5
11,500
144.6 ± 0.7
With increasing temperature, the amplitude decreases
slightly and
the decay is still almost independent of the momentum transfer, which
adds further support for the localized motions. In the long-time region,
a plateau, intensifying with increasing temperature, emerges (Figure c,d). Similar behavior
is seen for all three samples. The data of Figure c,d are measured above Tg; however, based on dielectric spectroscopy data,[12] in this temperature range, the segmental dynamics
are too slow to be in the available time window for QENS.Further
increasing the measurement temperature shifts the next
slower dynamical process, i.e., segmental relaxation, into the available
time window of QENS. Here, the methyl group dynamics are too fast
to show any contributions to the intermediate scattering function, Sinc(Q,t).
In this temperature region, a full decay to 0 is visible for all Q-values in the accessible time range (Figure ). Similar behavior is observed
in all three samples (Figures S1–S4 in the Supporting Information).
Figure 4
: Intermediate scattering function, S(Q,t), vs time, t, for
the sample with Mnside chain = 11,500 g/mol for two temperatures,
(a) T = 250 K and (b) T = 300 K.
Solid lines represent the data descriptions with the model function
for high temperatures, eq .
: Intermediate scattering function, S(Q,t), vs time, t, for
the sample with Mnside chain = 11,500 g/mol for two temperatures,
(a) T = 250 K and (b) T = 300 K.
Solid lines represent the data descriptions with the model function
for high temperatures, eq .
Data Analysis
The experimental data
suggest two different
processes in the available time range, separated by temperature. Based
on the results, the first process at low temperatures is identified
as the methyl group rotation, whereas the second process is dominated
by the segmental relaxation.Methyl group rotations are localized
and confined by their geometry, i.e., by the positions the hydrogen
atoms can occupy. The mathematical framework of the so-called elastic
incoherent structure factor (EISF) is very useful for describing localized
motions (represented as index ), which
leads towith the Debye–Waller factor
and a
prefactor A. Because polymers show a distribution
of different relaxation times, the Kohlrausch–Williams–Watts
(KWW) function, , is an appropriate way
to describe the
relaxation times of the involved processes.[17−20] In this context, describes
the relaxation time of the methyl
groups and is
the stretching exponent, resembling
the width of the relaxation time distribution.Typical for the
EISF of the methyl group rotation is the so-called
threefold jumpwith j0 being
the zeroth-order spherical Bessel function and R being
the circle radius.[20−22] Here, the hydrogen atoms can jump between three equidistant
positions on a circle with radius R (Figure a). The threefold jump, represented
by the EISF, is an oscillation around EISF = 1/3 for large Q-values (Figure b). This assumption is also valid for our sample systems,
as seen later in the data discussion.
Figure 5
(a) Three-dimensional illustration of
a methyl group, attached
to a silicon atom, including the radius, R, of the
circle spanned by the position of the hydrogen atoms. (b) EISF, vs QR of the methyl group rotation, that is, the motions of
the H atoms, confined to a potential of a threefold jump.
(a) Three-dimensional illustration of
a methyl group, attached
to a silicon atom, including the radius, R, of the
circle spanned by the position of the hydrogen atoms. (b) EISF, vs QR of the methyl group rotation, that is, the motions of
the H atoms, confined to a potential of a threefold jump.At higher temperatures, the segmental relaxation prevails
with
some contributions of the Debye–Waller factor. There are no
geometrical constraints imposed on the segmental relaxation; therefore,
an additional EISF is not needed for the data description, and the
model function is reduced towith τs being the segmental
relaxation time and βs being the associated stretching
parameter of the KWW function.Both model functions have a large
number of adjustable parameters.
Therefore, to improve the stability of the fitting function, reasonable
assumptions are needed. For the low-temperature function (eq ), the atomistic mean-square
displacement u2 as well as the relaxation
time of the localized motion and the associated stretching parameter are
assumed to be independent of Q. In the case of the
Debye–Waller factor, the Q-dependence is already
included as the Q2 factor within the exponential
function, and therefore, u2 only depends
on temperature. The relaxation
time of the localized motion occurs naturally on a well-restricted
geometry, and therefore, no significant Q-dependence
is detectable.[22] Furthermore, QENS experiments
on similar sample systems have not shown any substantial dependence
on Q within the experimental error for the stretching
parameter, .[20,23] The EISF shows an explicit Q-dependence, which
is determined separately.In the case of the segmental relaxation,
described with the model
function for high temperatures (eq ), the relaxation times show a pronounced Q-dependence.[17,24,25] Therefore, τs needs to be determined for each Q-value independently. The stretching parameter, βs, verified by single fits, and u2 are treated to be independent of the momentum transfer.All Q-values of one temperature are fitted with
a so-called “Global Fit”. By means of this, all Q-values are simultaneously described with the respective
model function, whereby the Q-independent parameters
are shared over all Q-values. Shared implies that
all values do not depend on Q and should therefore
be the same for all momentum transfers.
Discussion
The
plateau in the long-time region at the low temperatures raises
the question whether the geometrical constraint of the methyl groups
causes this effect. Adjusting the intermediate scattering function
for the EISF and the Debye–Waller factor according to eq results in a full
decay to 0 for all Q-values, as seen in Figure . This supports the
assumption that the plateau is
caused by the constrained dynamics of the methyl group rotation. Similar
results have been reported for poly(vinyl acetate)[26] and poly(methyl methacrylate).[20]
Figure 6
Adjusted
intermediate scattering function, S′(Q), vs time, t, for the sample with Mnside chain = 11,500 g/mol for all Q-values at temperature T = 175 K, cf. text.
Adjusted
intermediate scattering function, S′(Q), vs time, t, for the sample with Mnside chain = 11,500 g/mol for all Q-values at temperature T = 175 K, cf. text.The resulting EISF values show a particular dependence on the temperature
and the momentum transfer (Figure S5).
Describing these dependencies with the classical threefold jump (eq ), known for methyl group
rotations, implies a temperature-dependent radius of the methyl group
rotation, which increases by ∼28% within ΔT = 100 K and exceeds the maximum radius, that is possible based on
the bond length. Therefore, a modified threefold jump model has been
used including the fraction, m, of participating
protons on the signal in the available time window.[20,27,28]Here, a temperature-independent
radius, R, of
the methyl group rotation has been calculated based on the H–H
distance of 1.79 Å,[29] resulting in R = 1.033 Å. As seen in the inset of Figure , the number of protons contributing
to the measurement signal increases with increasing temperature and
reaches 100% for the highest temperatures. This approach has previously
also been applied to poly(methyl methacrylate) studies.[30] Interestingly, a nonsystematic change of the
participating fraction depending on the molecular weight of the side
chains is visible, especially at low temperatures. This could be due
to the differences in the grafting density; however, a more detailed
study would be needed to verify this in detail.
Figure 7
EISF, adjusted for the
participation protons vs QR for the sample with Mnside chain = 11,500 g/mol at different
temperatures. The solid black line represents the motion of a hydrogen
atom, confined to the potential of a threefold jump. Inset: Fraction
of participating protons, m, vs temperature, T, for all three samples.
EISF, adjusted for the
participation protons vs QR for the sample with Mnside chain = 11,500 g/mol at different
temperatures. The solid black line represents the motion of a hydrogen
atom, confined to the potential of a threefold jump. Inset: Fraction
of participating protons, m, vs temperature, T, for all three samples.After adjusting the EISF for the temperature dependence of the
participating proton fraction, m, all values obtained
for the different temperatures superimpose, as seen in Figure . Here, the data can be well
described with the threefold jump model, which agrees with the results
in the literature of methyl group dynamics.[21,31]The resulting stretching parameter, , for
all three samples, which is associated
with the relaxation time distribution, shows a weak temperature dependence,
as illustrated in Figure S6 in the Supporting Information. With increasing temperature, increases
and tends to saturate at high
temperatures,[20] for the two bottlebrush
polymers having Mnside chain = 1800 g/mol and Mnside chain = 11,500 g/mol. A potential consequence could be narrowing of the
relaxation time distribution as the temperature increases. It could
be connected to the fraction of protons contributing to the signal.
In the case of the Mnside chain = 298 g/mol sample, the parameter
peaks and does not show a clear
connection with the fraction of participating protons. At high temperatures,
the motions of almost all protons, which are mainly located in the
methyl groups, are within the available time window. Therefore, the
relaxation times are more uniform and less distributed, resulting
in higher values.
Because of the different final
values, which of
these two samples approximates, the
grafting density could have some influence on the final distribution
width in the high-temperature region. However, at this point, a more
detailed statement is not possible.Focusing on the temperature
dependence of the relaxation times
for the localized motions, , a similar behavior is seen for all three
samples. With increasing temperature, the relaxation time decreases,
i.e., the relaxation speeds up (Figure ).
Figure 8
Relaxation times, τ, vs 1000/T of
all three
samples for the two relaxation processes. Round symbols refer to the
methyl group dynamics measured by QENS, described by the Arrhenius
law, and squared symbols refer to the segmental dynamics obtained
by dielectric spectroscopy and described by the Vogel–Fulcher–Tammann
(VFT) law.[12]
Relaxation times, τ, vs 1000/T of
all three
samples for the two relaxation processes. Round symbols refer to the
methyl group dynamics measured by QENS, described by the Arrhenius
law, and squared symbols refer to the segmental dynamics obtained
by dielectric spectroscopy and described by the Vogel–Fulcher–Tammann
(VFT) law.[12]This behavior follows an Arrhenius law, which implies a thermally
activated process with a characteristic activation energy, EA, describing the energy barrier the methyl
group needs to overcome to perform the threefold jump, that is, a
rotation around the Si–C bond to the next possible hydrogen
position. The resulting fit parameters of the Arrhenius law are summarized
in Table . For all
three samples, the values are on the same orders of magnitude. Furthermore,
literature values of QENS experiments on linear PDMS of EA = (4.5 ± 0.5) kJ/mol[18,19] or EA = 5.2 kJ/mol[32] imply
that the methyl group dynamics only slightly depend on the molecular
weight. However, comparing the linear and bottlebrush architectures
shows significantly different activation energies, as seen in Table . As observed for
numerous polymers, CH3 rotations depend on the local environment.[20,22,33] According to Jakobi et
al.,[12] the monomer density is
higher closer to the backbone based on the radial monomer density
distribution function deducted from the form factor description of
the SANS data. Therefore, the environment along the side chains could
be assumed to change. This leads to more confined CH3 groups
closer to the backbone and CH3 groups more equivalent to
linear PDMS at the outer positions of the side chains, resulting in
higher activation energies on average.
Table 4
Parameters
Obtained from Fitting the
Temperature Dependence of the Relaxation Times of the Methyl Groups
in Figure by the
Arrhenius Law, Including the Molecular Weights of the Side Chains, Mnside chain, the Relaxation Time for Infinitely High Temperatures, τ∞, and the Activation Energy, EA
Mnside chain (g/mol)
τ∞ (s)
EA (kJ/mol)
298
(2.04 ± 0.72) × 10–13
6.0 ± 0.3
1800
(1.67 ± 0.11) × 10–13
6.5 ± 0.1
11,500
(3.21 ± 0.22) × 10–13
5.8 ± 0.1
Concentrating
on the different side-chain lengths, slight differences
occur, which could be attributed to the different grafting densities,
influencing the number of different methyl groups.The obtained
segmental relaxation times, τs, from
the high temperatures, show a particular Q-dependence,
which follows a power law, with the slope parameter s, as seen in Figure .
Figure 9
Segmental relaxation
time, τs, vs momentum transfer, Q, of all three samples for two temperatures, (a) T = 250 K and (b) T = 300 K. Solid lines
are the power law description with eq .
Segmental relaxation
time, τs, vs momentum transfer, Q, of all three samples for two temperatures, (a) T = 250 K and (b) T = 300 K. Solid lines
are the power law description with eq .While the relaxation
becomes faster with increasing momentum transfer,
increasing the side-chain length slows down the segmental relaxation
times. Also, a temperature effect is visible. Increasing the temperature,
by ΔT = 50 K, accelerates the relaxation by
around half a decade. This behavior exists for each sample. The resulting
power laws associated with each temperature and sample are summarized
in Table .
Table 5
Parameters Obtained from the Power
Law Description of the Segmental Relaxation Times, τs, as a Function of Momentum Transfer, Q, in Figure , Including the Number-Average
Molecular Weight of the Side Chains, Mnside chain,
and the Slope Parameters, s, for the Two Temperatures, T = 250 K and T = 300 K
Mnside chain
s (T = 250 K)
s (T = 250 K)
298
(8.52 ± 0.02)
(7.24 ± 0.09)
1800
(6.35 ± 0.13)
(4.96 ± 0.12)
11,500
(5.31 ± 0.12)
(4.24 ± 0.10)
For both temperatures,
a pronounced decrease of the slope with
increasing side-chain length is observed, as indicated by the s parameter. Here, a stronger dependence is seen for the
shortest side chains at both temperatures. With increasing temperatures, s reduces, thus the Q-dependence of the
relaxation time on the momentum transfer is reduced.The stretching
parameter, βs, associated with
the segmental dynamics increases slightly with temperature and molecular
weight of the side chains, as seen in the inset of Figure . Here, the highest values
are obtained for the bottlebrush polymer with Mnside chain =
11,500 g/mol. For linear polymers, βs = 0.4–0.6
is reported.[17,34] Our values are continually below
βs = 0.4 for all side-chain lengths, i.e., our relaxation
time distribution is broader compared to that of linear polymers.
Because with increasing Mnside chain, the values increase,
it could be assumed that the low βs values are routed
in the bottlebrush architecture and may reach values known for linear
polymers with sufficiently long side chains. This is in agreement
with recent findings of dielectric spectroscopy.[12] Here, a reduction of segmental relaxation times was found,
by comparing those from linear side chains to the grafted side chains.
The strongest reduction was reported for the shortest side chains
and almost identical relaxation times for the longest side-chain length.
This could suggest different relaxation times within the side chains
of the bottlebrush polymers, i.e., a distribution of relaxation times.
Furthermore, it could be interpreted in a way that the inner segments
are influenced the most, and with increasing side-chain length, this
effect saturates and finally, relaxation times approximate the relaxation
times of a linear polymer. Thus, the distribution narrows with increasing
side-chain length. This is also reflected in the stretching parameter
β. The longer the side chain length,
the larger and more similar the βs values are to
those of linear polymers. Therefore, the relaxation times show a narrower
distribution for longer side-chain lengths.
Figure 10
Slope parameter, s, vs stretching parameter, βs, for all
three PDMS-g-PDMS bottlebrush polymers.
Inset: stretching parameter, βs, vs temperature, T, for all three samples.
Slope parameter, s, vs stretching parameter, βs, for all
three PDMS-g-PDMS bottlebrush polymers.
Inset: stretching parameter, βs, vs temperature, T, for all three samples.In general, the shape parameter, s, can be related
to the stretching parameter, β,
which allows to clarify the underlying behavior of the system. For
systems showing a Gaussian behavior, this relationship can be described
by s = 2/βs.[17,35] Using this relationship results in the solid green line in Figure . The fit values,
βs (filled symbols), for our PDMS-g-PDMS bottlebrush polymers are different, hence not confirming Gaussian
behavior. However, the relationship of s and βs can be well described with s = 1.3/βs, as shown with the solid purple line in Figure . Based on this, we suggest
a non-Gaussian behavior for all our three samples in the available Q-range, which is equivalent to a heterogeneous system.
Similar descriptions have been reported for polybutadiene, s = 1.4/βs, and for polyisoprene, s = 1.3/βs.[31,36]To verify
that the segmental relaxation, obtained by QENS, belongs
to the α-relaxation, as determined by dielectric spectroscopy,
a comparison of the relaxation times obtained from both techniques
is very common.[17,37,38] The relaxation times obtained by dielectric spectroscopy together
with τs for every Q-value are shown
in Figure for all
three samples. Here, the same behavior as reported by dielectric spectroscopy
is seen in the data obtained by QENS. With increasing temperature,
the relaxation times decreases.
Figure 11
Segmental relaxation times, τs, vs 1000/T of the three samples with different
side-chain lengths,
obtained by dielectric spectroscopy and QENS. (a) Mnside chain = 298 g/mol, (b) Mnside chain = 1800 g/mol, and (c) Mnside chain = 11,500 g/mol. Solid lines in the low-temperature region are data
descriptions of the dielectric spectroscopy data by the VFT equation.
Segmental relaxation times, τs, vs 1000/T of the three samples with different
side-chain lengths,
obtained by dielectric spectroscopy and QENS. (a) Mnside chain = 298 g/mol, (b) Mnside chain = 1800 g/mol, and (c) Mnside chain = 11,500 g/mol. Solid lines in the low-temperature region are data
descriptions of the dielectric spectroscopy data by the VFT equation.
Summary and Conclusions
We have
used QENS to follow the short-time dynamics on PDMS-g-PDMS bottlebrush polymers. Three different samples have
been used, having different side-chain lengths, while the backbone
length is similar. SANS revealed a shape transition from ellipsoidal
toward an elongated shape, with decreasing side-chain length.[12] In order to cover a broad time range for dynamical
studies, a combination of three different neutron spectrometer instruments
was used, resulting in a time range from 1 ps to 1 ns. This time window
includes two processes: localized motions at low temperatures and
segmental dynamics at higher temperatures. The localized motions have
been identified as methyl group rotations, which are confined by their
geometry and can be described by a threefold jump model considering
the participating fraction of the protons in the available time window.
While the relaxation time of this dynamical process is independent
of the side-chain length and the momentum transfer, the segmental
relaxation times show a well-pronounced dependence on both the molecular
weight of the side chains and the momentum transfer. Here, the relaxation
time decreases with increasing momentum transfer and decreasing side-chain
length. The Q-dependence of the segmental relaxation
times follows a power law, resulting in the slope parameter s. Based on this parameter, a relationship to the stretching
parameter, βs, can be established, suggesting a non-Gaussian
behavior for our PDMS-g-PDMS bottlebrush polymers.
Authors: Sarit Dutta; Matthew A Wade; Dylan J Walsh; Damien Guironnet; Simon A Rogers; Charles E Sing Journal: Soft Matter Date: 2019-04-03 Impact factor: 3.679
Authors: William F M Daniel; Joanna Burdyńska; Mohammad Vatankhah-Varnoosfaderani; Krzysztof Matyjaszewski; Jarosław Paturej; Michael Rubinstein; Andrey V Dobrynin; Sergei S Sheiko Journal: Nat Mater Date: 2015-11-30 Impact factor: 43.841
Authors: Vahid Asadi; Xuecong Li; Francesco Simone Ruggeri; Han Zuilhof; Jasper van der Gucht; Thomas E Kodger Journal: Polym Chem Date: 2022-07-20 Impact factor: 5.364