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Chemistry
1-1-2006
Molecular Mass and Dynamics of Poly(Methyl
Acrylate) in the Glass Transition Region
Burak Metin
Frank D. Blum
Missouri University of Science and Technology
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Recommended Citation
B. Metin and F. D. Blum, "Molecular Mass and Dynamics of Poly(Methyl Acrylate) in the Glass Transition Region," Journal of Chemical
Physics, American Institute of Physics (AIP), Jan 2006.
The definitive version is available at http://dx.doi.org/10.1063/1.2162879
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THE JOURNAL OF CHEMICAL PHYSICS 124, 054908 共2006兲
Molecular mass and dynamics of poly„methyl acrylate…
in the glass-transition region
Burak Metin and Frank D. Bluma兲
Department of Chemistry and Materials Research Center, University of Missouri-Rolla, Rolla,
Missouri 65409-0010
共Received 12 October 2005; accepted 5 December 2005; published online 3 February 2006兲
The segmental dynamics of bulk poly共methyl acrylate兲 共PMA兲 were studied as a function of
molecular mass in the glass-transition region using 2H NMR and modulated differential scanning
calorimetry 共MDSC兲. Quadrupole-echo 2H NMR spectra were obtained for four samples of
methyl-deuterated PMA-d3 with different molecular masses. The resulting spectra were fit using
superpositions of simulated spectra generated from the MXQET simulation program, based on a
model incorporating nearest-neighbor jumps from positions on the vertices of a truncated
icosahedron 共soccer-ball shape兲. The lower molecular-mass samples, influenced by the presence of
more chain ends, showed more heterogeneity 共broader distribution兲 and lower glass transitions than
the higher molecular-mass samples. The MDSC experiments on both protonated and deuterated
samples showed behavior consistent with the NMR results, but temperature shifted due to the
different frequency range of the measurements in terms of both the position and breadth of the glass
transition as a function of molecular mass. © 2006 American Institute of Physics.
关DOI: 10.1063/1.2162879兴
INTRODUCTION
Polymeric and other glass formers have been extensively
studied to give a clearer understanding of the dynamics
around the glass transition.1 Several theoretical approaches,
thermodynamic,3
and
including
free
volume,2
4
mode-coupling theories, have been used to describe the phenomenon behind the glass transition. However, the segmental
dynamics of polymers through the glass transition, by which
a glassy polymer becomes rubbery, are still not well understood. Different experimental techniques, such as NMR,
thermomechanical analysis 共TMA兲, dynamic mechanical
analysis 共DMA兲, differential scanning calorimetry 共DSC兲,
and dilatometry, have all been used to probe this important
phenomenon.5,6
The concept of motional heterogeneity in the glasstransition region of polymers was probed over 20 years ago.7
More recently, dynamical heterogeneity was defined as the
experimental selection of a dynamically distinguishable subensemble in a material around the glass-transition temperature 共Tg兲 and the observation of the subensemble’s return to
its original 共equilibrium兲 state.8 The heterogeneous scenario
describes the nonexponential decay of relaxation functions as
a distribution of the relaxation times present in the system
around the glass-transition region.9 A proper understanding
of the dynamics of glass formers requires sophisticated experimental techniques and theoretical work.
A number of experimental techniques have been applied
to study the dynamics of glass formers, including
one-dimensional8,10–17 共1D兲 and two-dimensional18–21
NMRs, and calorimetry.22–24 These techniques have provided
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-9606/2006/124共5兲/054908/10/$23.00
both quantitative and qualitative bases for some understanding of dynamics in glass formers. Relaxation-time measurements have been particularly useful. Nonexponential structural relaxation, a consequence of a distribution of relaxation
times, has been seen close to Tg in both relaxation-time and
calorimetric measurements for organic glass formers8,10,22
and amorphous polymers.23,24
Deuterium NMR line shapes of deuterated polymers are
usually dominated by the quadrupole interaction, as it is significantly larger than other effects, such as dipolar couplings.
Segmental motions may average the quadrupole couplings
and, consequently, affect the line shapes. For deuterons, in
our case those in C–D bonds, each different orientation of
the electric-field gradient in the principle axis system 共with
respect to the magnetic-field axis兲 gives rise to two transitions, producing a series of doublets. The splitting between
doublets is given as5
⌬␯Q = 23 关e2qQ/h兴关 21 共3 cos2 ␪共t兲 − 1兲
+ 21 共␩ sin2 ␪共t兲cos 2␾共t兲兲兴 ,
共1兲
where e2qQ / h is the quadrupole-coupling constant 共typically
167 kHz for aliphatic C–D’s兲, signifying the strength of the
coupling between the nuclear quadrupole moment and the
electric-field gradient 共EFG兲; ␪ and ␾ are the Euler angles
used to define the relative orientation of the principle axis
system of the EFG 共C–D bond direction兲 with respect to the
external magnetic field; and ␩ is the asymmetry parameter
that can be taken in our experiments as 0 for our approximately axially symmetric C–D bonds. The 2H NMR spectra
for unoriented solidlike polymers consist of powder patterns
from randomly oriented C–D bonds. Motions of the C–D
bond vectors lead to changes in line shapes, including narrowing of the powder patterns. Collapse to a single narrow
124, 054908-1
© 2006 American Institute of Physics
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054908-2
J. Chem. Phys. 124, 054908 共2006兲
B. Metin and F. D. Blum
resonance occurs with fast isotropic motions. Modeling of
the line shapes, based on specific mechanisms and rates of
motion, can provide insight into the polymer dynamics.
Although a wide variety of NMR techniques are known,
ID 2H quadrupole-echo experiments have continued to be
used for characterizing segmental dynamics in various
systems.12–21,25–28 Deuterium NMR line shapes provide information about segmental dynamics through the glasstransition region due to the coupling of the C–D bond vectors with the cooperative motions of the polymer chains.19
Lin and Blum26 reported the effect of molecular mass on
segmental dynamics through the glass-transition region for
poly共methyl acrylate兲-d3 共PMA-d3兲. Analysis of the 2H NMR
spectra through the glass-transition region indicated the presence of at least two motionally different components in a low
molecular-mass 共LMM兲 sample. These components were labeled as more-mobile and less-mobile fractions. In contrast, a
single component in a very high molecular-mass 共HMM兲
sample was observed. Based on these results, the segmental
dynamics of the LMM PMA samples were considered to be
heterogeneous while that of the HMM sample was considered to be homogeneous. However, the PMA samples were
fairly polydisperse and only two samples were studied. A
better understanding of the polymeric systems may be obtained by using more samples of lower polydispersity.
In this paper, we report studies of the segmental dynamics of four different relatively monodisperse PMA-d3
samples, through the glass-transition region, using 2H
quadrupole-echo NMR and modulated differential scanning
calorimetry 共MDSC兲. A jump model, based on the geometry
of a truncated icosahedron 共soccer ball兲, was used to simulate
the experimental 2H NMR spectra and characterize the effect
of molecular mass on the segmental dynamics of PMA. The
effects of both molecular mass and polydispersity on segmental dynamics were apparent in both NMR and MDSC
measurements.
EXPERIMENT
Methyl acrylate-d3 was synthesized using methanol-d4
共CIL兲, acryloyl chloride 共Aldrich, 96%兲, triethylamine 共Alfa
Aesar, 99.9%兲, and toluene. Acryloyl chloride was purified
by fractional distillation and other chemicals were used without further purification. Methanol-d4 共11.23 ml, 0.28 mols兲
was mixed with triethylamine 共34.8 ml, 0.25 mols兲 in a
round-bottomed flask containing 25 ml toluene. A mixture of
acryloyl chloride 共20.5 ml, 0.25 mols兲 and 15 ml toluene
was added dropwise into the mixture prepared at 0 ° C over
30– 45 min. The mixture was stirred at 0 ° C for 3 – 4 h and
then stirred at room temperature for 20– 24 h. The reaction
was stopped with the addition of a concentrated aqueous
NaHCO3 solution. The product was washed twice with a
concentrated NaHCO3 solution and three times with distilled
water. The toluene and methyl acrylate-d3 mixture was isolated and the product was dried over CaH2 for 18 h. After
vacuum distillation, a 70%–75% yield was determined.
Atom transfer radical polymerization 共ATRP兲 was used
to synthesize poly共methyl acrylate兲-d3 with narrow
polydispersities.29 Methyl acrylate-d3, N , N , N⬘ , N⬘ , N⬙-
TABLE I. Molecular mass of the poly共methyl acrylate兲-d3 samples.
Name
PMA-5 K
PMA-38 K
PMA-77 K
PMA-165 K
M w 共g/mol兲
M n 共g/mol兲
Polydispersity
5 600
38 000
77 000
165 000
5 500
33 500
61 000
116 000
1.02
1.15
1.26
1.42
pentamethyldiethylenetriamine 共PMDETA, Aldrich, 99%兲,
and ethyl 2-bromopropionate 共2-EBP, Aldrich, 99%兲 were
used as received without further purification. CuBr 共Aldrich,
98%兲 was added to a 50 ml round-bottomed flask, and that
was sealed with a rubber septum, and then purged with nitrogen to remove oxygen. Degassed toluene, monomer, and
amine ligand 共PMDETA兲 were added with nitrogen-purged
syringes. The mixture was purged with nitrogen for 10 min.
The initiator 共2-EBP兲 was added with nitrogen-purged syringes and the flask was immersed in an oil bath at 90 ° C for
8 – 24 h. Depending on the molecular mass targeted, the ratio
of 关monomer兴/关initiator兴 was changed for each reaction. The
protonated PMA samples, used in the MDSC experiments,
were also synthesized using the ATRP technique.
The polymer properties were characterized by light scattering, 1H NMR, and MDSC. Molecular-mass measurements
of the PMA samples were made using an OPTILAB DSP
interferometer refractometer and a DAWN EOS lightscattering instrument 共Wyatt Technology, Santa Barbara,
CA兲. The specific refractive index increment 共dn / dc兲 values
for PMA and PMA-d3 were both determined to be
0.063 ml/ g at 690 nm in tetrahydrofuran 共THF兲, using the
OPTILAB DSP. Molecular-mass information and sample
designations of the polymers are shown in Table I. The stereoregularities of PMA samples were determined with 1H
NMR. Analyses of the intensities of the backbone methylene
resonances of the PMA samples yielded isotactic 共m兲 diad
fractions of 0.48 for PMA-5 K, 0.51 for PMA-38 K, 0.52 for
PMA-77 K, and 0.57 for PMA-165 K. Thermal characterization was made using a MDSC 共TA Instruments, New Castle,
DE兲, with a modulation rate of ±0.5 ° C per 40 s, and a
heating rate of 3 ° C / min. UNIVERSAL ANALYSIS software
共TA Instruments兲 was used to process the MDSC data.
Deuterium NMR spectra of the samples were obtained
using a VARIAN VXR-400/S spectrometer with a 2H frequency of 61.39 MHz. The quadrupole-echo pulse sequence
共delay− 90y − ␶ − 90x − ␶ − acquisition兲 was used in the experiments, and the 90° pulse width was 2.8 ␮s with an echo time
共␶兲 of 27 ␮s. Typically, 256 scans were taken for each spectrum. The intensities were measured for each spectrum and
the intensities reported were for the Boltzmann factors. The
probe response was checked at various temperatures and was
found to be consistent over the range studied. The exchange
during pulses was neglected as the echo delay was about ten
times longer than the pulse width.
A FORTRAN program, MXQET,30,31 which includes the effect of finite pulse width and virtual free induction decay
共FID兲 on the 2H NMR line shapes, was used to simulate the
experimental NMR line shapes. Jump models, with a tetrahedral symmetry, have been previously used to model relax-
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054908-3
J. Chem. Phys. 124, 054908 共2006兲
Molecular mass and dynamics of poly共methyl acrylate兲
艋 1.0⫻ 1011 Hz. The weighting factor, w共ki兲, for each simulated line shape with different jump rates, S共ki兲, was found
by solving a system of linear equations:
X
E共T兲 = 兺 w共ki兲S共ki兲.
共2兲
i=1
The discrete jump model of Torchia and Szabo34 was adapted
to determine the correlation functions at each ki. A correlation function, Cm共t兲, of the form
FIG. 1. Truncated icosahedron 共soccer ball兲 geometry used for the simulations. A jump occurs from site 0 to any of the neighboring sites 1, 2, and 3
with equal probabilities and a jump rate 共k兲. New jumps occur from the new
site and jump backs may also occur. Eventually, all of the sites may be
sampled.
ation times32 and 2H 2D-exchange33 experiments in polymeric systems. In this study, the simulation program was
adopted based on a truncated icosahedron 共soccer ball兲. The
vertices were defined as the jump sites 共60 sites兲 for the
symmetry axis of the methyl group 共effective C–D bond vector兲. Several other geometric models, tetrahedron 共4 sites兲,
octahedron 共6 sites兲, dodecahedron 共20 sites兲, and icosahedron 共12 sites兲, were also studied with the simulation program. Of these models, the soccer-ball model provided a set
of line shapes that was closest to the experimental ones,
probably because this model came closest to a small jump
model. The program provides the flexibility of defining different jump mechanisms, other than the already available
random and nearest-neighbor jumps. To simulate the diffusive reorientational motion with small jumps for an amorphous polymeric system, a 60⫻ 60 exchange matrix was designed to allow jumps to occur from a given site to one of its
three nearest neighbors. A schematic for the soccer ball and
the jump mechanism is shown in Fig. 1. Jumps from site 0 to
sites at 1, 2, or 3 were allowed to occur with equal probability. The jump mechanism was repeated at the new position,
following the completion of a jump. Eventually, all of the
jump sites 共vertices of the soccer ball兲 were sampled. A reduced quadrupole-coupling constant of 50 kHz was used in
the simulations to account for the fast methyl group rotation
around its symmetry axis. A 27 ␮s pulse spacing and a
2.8 ␮s pulse width were used for the quadrupole-echo pulse
sequence in the simulations.
A series of simulated line shapes with different jump
rates was produced. A mathematical routine was applied to
fit the experimental line shapes to a superposition of simulated spectra. The weighting factors of each of the simulated
spectra were found by solving a system of linear equations
with MATLAB 共The Mathworks, Inc., Natick, MA兲. A constrained least-square fit was applied to find the positive
weighting factors of a series of spectra.
THEORY
The experimental line shapes at different temperatures, E共T兲, were fitted with the addition of a number of
simulated line shapes based on different jump rates 共ki,
i = 1 , 2 , 3 , . . . , X兲. We found that adequate fits could be obtained with X = 94, spanning a range of 1.0⫻ 102 Hz艋 ki
2
Cm共t,ki兲 =
2
兺 兺
a=−2
共2兲
dma
共␪兲dma⬘共␪兲ei共a−a⬘兲␾Caa⬘共t,ki兲, 共3兲
共2兲
a⬘=−2
where
N
N
N
共n兲 共0兲 共n兲 共2兲
Caa⬘共t,ki兲 = 兺 兺 兺 e␭ntK共0兲
x Kx K y K y d0a 共⌰x兲d0a⬘
共2兲
x=1 y=1 n=2
⫻共⌰y兲ei共a␾x−a⬘␾y兲
共4兲
was used for discrete jumps.31 In Eqs. 共3兲 and 共4兲, the ␭n’s
are eigenvalues and columns of K共n兲 are the corresponding
eigenvectors of a symmetric 60⫻ 60 exchange matrix A.
This was used in the simulations so that the jumps would
occur with a ki to one of the three nearest neighbors in the
crystal-fixed axis frame,
Axy = 共AxyAyx兲1/2
Axy = −
兺 Ayx
for x ⫽ y,
for x = y.
共5a兲
共5b兲
x⫽y
共2兲
共2兲
共⌰x兲 and d0a 共⌰y兲 are the reduced second-rank Wigner
d0a
⬘
rotation elements, ␪ and ␾ are the spherical polar angles that
define the orientation of the magnetic field in the crystalfixed axis frame, and ⌰ and ␾ are angles used to define jump
sites in the crystal-fixed axis frame.
A Kohlrausch-Williams-Watts 共KWW兲 function of the
form
关Cm共t兲 − Cm共⬁兲兴
␤
= e−共t/␶兲
关Cm共0兲 − Cm共⬁兲兴
共6兲
provided suitable fits to the correlation functions at each
jump rate where ␶ is the effective correlation time and ␤ the
width parameter. The necessity of a width parameter in Eq.
共6兲 to fit the correlation functions at each jump rate, can be
understood from Eq. 共4兲 as it is basically the sum of exponentials.
The correlation function for an experimental spectrum at
a temperature f共T兲 was determined by taking the weight average of the correlation functions of each jump rate that contributed to the fitting of the experimental NMR line shape.
X
f共T兲 = 兺 w共ki兲exp − 共t/␶ki兲␤ki .
共7兲
i=1
An average correlation time, 具␶典, for each spectrum, was then
estimated from the fitted parameters to the correlation functions, as shown below.
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054908-4
J. Chem. Phys. 124, 054908 共2006兲
B. Metin and F. D. Blum
FIG. 2. Experimental 共−兲 and simulated 共. . .兲 2H NMR spectra for bulk PMA-5 K as a function of temperature.
具␶典 =
冕兺
⬁ X
0 i=1
w共ki兲exp − 共t/␶ki兲␤kidt.
共8兲
RESULTS
The experimental and simulated 2H NMR spectra for the
bulk PMA-5 K sample are shown in Fig. 2. The spectra have
all been normalized to a constant height. The fits are almost
indistinguishable from the experimental spectra. The experimental line shapes of the PMA-5 K sample indicate the presence of multiple motional components in some of the spec-
tra. At 20 ° C, the spectrum consists of a Pake pattern. At
25 ° C, a small number of segments with faster motions
cause a middle peak to appear. The intensity of this motionally narrowed resonance increases through the “glasstransition region” until the powder partem completely collapses into a single resonance. At 40 ° C and above, there is
no evidence of any residual powder pattern. The change from
a Pake pattern to a single resonance occurs in the 25– 40 ° C
range.
2
H NMR spectra for the PMA-38 K sample at different
temperatures are shown in Fig. 3. The spectra are similar to
those for the PMA-5 K sample except that they are shifted to
FIG. 3. Experimental 共−兲 and simulated 共. . .兲 2H NMR spectra for bulk PMA-38 K as a function of temperature.
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054908-5
Molecular mass and dynamics of poly共methyl acrylate兲
J. Chem. Phys. 124, 054908 共2006兲
FIG. 4. Experimental 共−兲 and simulated 共. . .兲 2H NMR spectra for bulk PMA-77 K as a function of temperature.
higher temperatures and the intensities of the narrow components in the intermediate regime are reduced. At 20 and
25 ° C, the line shape is that of a solid powder pattern. At
35 ° C, part of the powder pattern collapses into a central
resonance. At higher temperatures, the powder pattern collapses to multicomponent spectra through the glass-transition
region.
For the PMA-77 K sample, the 2H NMR line shapes are
shown in Fig. 4. From 25 to 40 ° C, there is little change in
the powder pattern, indicating that the mobility of the polymer chains has not changed much on the 2H NMR time
scale. Distinct evidence of a small motionally narrowed com-
ponent does not appear until 50 ° C. The spectra collapse to a
broad compound resonance at 55 ° C due to higher segmental
mobility. The resonance becomes narrower at higher temperatures following the collapse of the powder pattern.
A higher molecular-mass sample, PMA-165 K, was also
studied. The 2H NMR spectra for this sample are shown in
Fig. 5. At 50 ° C, the spectrum of the PMA-165 K sample
appears almost as a rectangular pattern due to the superposition of a residual powder pattern and a motionally narrowed
共mobile兲 middle. Only a very small amount of the mobile
component occurs with a residual powder pattern. Below
50 ° C, there is no obvious indication of a mobile component
FIG. 5. Experimental 共−兲 and simulated 共. . .兲 2H NMR spectra for bulk PMA-165 K as a function of temperature.
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054908-6
J. Chem. Phys. 124, 054908 共2006兲
B. Metin and F. D. Blum
The simulations of the experimental 2H NMR line
shapes are shown as the dotted lines in Figs. 2–5. In most
cases, the fits were excellent and, consequently, difficult to
distinguish for the experimental spectra in the figures. A series of jump rates, from 1.0⫻ 102 to 1.0⫻ 1011 Hz, was used
to create a series of simulated line shapes. The jump rates
were selected so that as many spectra 共with differing jump
rates兲 as possible could be used to fit the experimental spectra in a reasonable amount of time. The best fits from the
MATLAB program, to a system of linear equations, provided
the contributions from each simulated line shape to the final
fit. The jump rates were grouped into three different regimes:
slow, intermediate, and fast with respect to their powderpattern intensities. The slow regime 共102 – 104 Hz兲 was identified as the group of exchange rates that did not result in
significant intensity loss during the quadrupole echo. The
intermediate regime 共104 – 106 Hz兲 was classified as the region where the exchange rates resulted in a significant intensity loss due to exchange-induced echo dephasing. Finally,
the fast regime 共106 – 1011 Hz, on the fast side of the intensity
versus jump rate curve兲 was the region where jump rates also
resulted in little loss of intensity. For simplicity, the resulting
fractions of each motional component in the different motional regimes, used in the fittings at different temperatures,
are given in Table II.
We were able to effectively recreate all of the experimental spectra with combinations of simulated spectra. The
details of the distributions of these jump rates are shown in
Fig. 7. The bar heights indicate the weighting factors of the
simulated spectra at the different jump rates used to fit the
experimental spectra. Several spectra that were included in
the fit but had very low intensities are not observable due to
the scale of the figure. Most of the simulated spectra did not
contribute to the fits 共i.e., they had zero intensity兲. In the
figures for each sample, it is easy to observe the most intense
contributions move from the slow to the intermediate time
regime as the temperature is increased. At the highest temperature, some moderate contributions from the fast regime
are noted, especially for the lowest molecular-mass sample.
Some rather narrow features in the spectra, such as those in
FIG. 6. Cumulative molar mass distributions of the PMA-d3 samples from
light scattering.
superimposed on the powder pattern. The similarity of spectral features of the PMA-165 K and PMA-77 K samples was
also evident at 55 ° C and above where, basically, a single
broad resonance with almost no other distinguishable components was observed.
The small intensity of the motionally narrowed component in the PMA-165 K spectrum at 50 ° C 共Fig. 5兲 might be
attributed to the broader polydispersity of the sample. Cumulative molar mass distributions from light-scattering data for
the PMA-38 K, PMA-77 K, and PMA-165 K samples are
shown in Fig. 6. These distributions show the mass fraction
of a material having molar mass less than the given molar
mass interval. The distributions approach zero at low molar
masses and unity at high molar masses. The data for PMA5 K are not shown in the figure because that polymer does
not scatter enough light. A molecular-mass region, marked
by the vertical line at 5 ⫻ 10 g / mol, represents the rough
limit between the low and high molecular-mass regimes
共vide infra兲. Based on this distinction, PMA-5 K and PMA38 K would be labeled primarily as LMM samples, while
PMA-77 K and PMA-165 K would be considered to be
HMM samples. It can be observed that, even for PMA165 K, a small amount of polymer exists in the LMM region.
We believe that this small component accounts for the weak
narrow resonance in that sample 共i.e., Fig. 5 at 50 ° C兲.
TABLE II. Motional components used to simulate experimental line shapes. The components are slow 共k 艋 1.0⫻ 104 Hz兲, intermediate 共1.0⫻ 104 ⬍ k 艋 1.0
⫻ 106 Hz兲, and fast 共1.0⫻ 106 ⬍ k 艋 1.0⫻ 1011 Hz兲 where k is the jump rate. Given as the percentages of each component in the simulated spectra.
T
共°C兲
20
25
30
35
40
45
50
55
60
65
70
75
PMA-5 K
PMA-38 K
PMA-77 K
PMA-165 K
Slow
Interm.
Fast
Slow
Interm.
Fast
Slow
Interm.
Fast
Slow
Interm.
Fast
35.2
6.3
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
64.8
93.7
99.97
98.9
96.8
95.9
81.9
60.7
44.9
33.7
28.8
¯
¯
¯
0.03
1.1
3.2
4.1
18.1
39.3
55.1
66.3
71.2
100
70.2
38.8
3.3
2.9
¯
¯
¯
¯
¯
¯
29.8
61.2
96.7
97.1
99.9
99.9
97.9
91.5
80.9
58.2
50.3
40.2
¯
¯
¯
¯
0.1
0.1
2.1
8.5
20.1
41.8
49.7
59.8
78.5
54.2
46.8
40.3
12.5
13.1
0.1
¯
¯
¯
¯
¯
21.5
45.8
53.2
59.2
87.5
86.9
99.9
100
100
98.5
88.9
84.7
¯
¯
¯
¯
¯
¯
¯
¯
¯
1.5
11.1
15.3
100
59.9
52.6
37.4
12.5
11.8
0.2
¯
¯
¯
¯
¯
¯
40.1
47.4
62.6
87.5
88.2
99.8
100
99.7
99.6
91.9
98.7
¯
¯
¯
¯
¯
¯
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0.3
0.4
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2.3
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054908-7
Molecular mass and dynamics of poly共methyl acrylate兲
J. Chem. Phys. 124, 054908 共2006兲
FIG. 7. Distributions of jump rates from the simulation of each sample at different temperatures.
the middle of the powder patterns or on top of an already
narrowed spectrum require very small amounts of fast components for a good fit. Components with as little as 0.01% of
the spectral intensity were required to fit some of these sharp
spectral features.
A few selected correlation functions and their nonexponential fits are shown in Fig. 8 to illustrate their nature. The
wide range of decays made it necessary to display some of
the correlation functions with a different time scale on the
inset. The best fits for the correlation functions seemed to be
obtained with a KWW-type function. Fittings were done for
different ki values that ranged from 102 to 1011 Hz. A ␤
parameter of 0.63± 0.02 was sufficient to fit all of the correlation functions tested.
Average correlation times, 具␶典, for each of the samples,
as a function of temperature, were estimated using the fitting
parameters of Eq. 共8兲. Plots of log共具␶典兲 versus temperature
for each sample are shown in Fig. 9. As expected, based on
their spectra, the behaviors of 具␶典 for the PMA-77 K and
PMA-165 K samples were very similar. The behavior for the
PMA-38 K sample was faster than the other two, especially
at the lower temperatures. The PMA-5 K sample had even
faster correlation times, especially at higher temperatures
when its dynamics were much faster than the other samples.
Thermal analysis experiments, using MDSC, were performed on both protonated and deuterated PMA samples.
Glass-transition curves for the deuterated samples are shown
in Fig. 10 as the derivative of the reversible heat flow as a
function of temperature. The peak maxima of the derivative
of reversible heat flow curves were chosen as the Tg of the
samples. These points roughly correspond to the middle of
the endotherm of the traditional DSC curves. It is observed
that the LMM samples have broader glass-transition widths
and lower Tg’s than the HMM samples.
The results of MDSC experiments performed on protonated PMA samples are shown in Fig. 11, along with the
FIG. 8. The nonexponential fits for representative correlation functions at
different jump rates. The inset shows the fits for the faster decay functions.
FIG. 9. Average correlation times, 具␶典, obtained from fitting the correlation
functions of each experimental spectra through line-shape simulations.
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054908-8
J. Chem. Phys. 124, 054908 共2006兲
B. Metin and F. D. Blum
FIG. 10. MDSC derivative curves for the PMA-d3 samples. The peak is
taken as the Tg reported and the breadth as the width at half height.
results for the deuterated samples. The values of the Tg and
the widths 共shown as the error bars兲 are plotted as a function
of molecular mass. Transition widths were calculated using
the derivative of reversible heat versus temperature curve
and estimated as the width at half height of the glasstransition peak. For each sample, the HMM samples have
narrower glass-transition widths than the LMM samples do.
While transition widths at half height were 7.5– 9.3 ° C for
LMM samples, they were 5.6– 7.1 ° C for the HMM samples.
The solid curve is the fit to the Fox-Flory model 共vide infra兲.
The results of the Tg’s for both NMR and MDSC experiments for deuterated PMA samples are compared in Fig. 12.
The NMR Tg’s were estimated as the temperatures at which
the intensity loss was greatest in the Boltzmann-corrected
intensity versus temperature curve 共Fig. 13兲. The MDSC
curve shown is that derived from the Fox-Flory fits 共Fig. 11兲.
The curve shown for the NMR data is the same as that for
the MDSC curve, except that it is shifted by 37 K 共vide
infra兲. The NMR intensity minima, shown in Fig. 13, occurred at temperatures where the powder pattern collapsed to
a single broad resonance. The collapse of the powder pattern
and the maximum intensity loss occurred at 30 ° C for PMA5 K, 40 ° C for PMA-38 K, and 55 ° C for PMA-77 K and
PMA-165 K. The NMR glass-transition widths were esti-
FIG. 11. Glass-transition temperatures 共Tg’s兲 and widths of PMA and PMAd3 samples measured with MDSC. The bars represent the widths 共half
height兲 of the transition as measured from the derivative curves. The solid
curve is the result of the fit to the Fox-Flory model.
FIG. 12. NMR and MDSC glass-transition temperatures for PMA-d3. The
error bars represent the width of the transitions. Both solid curves are from
the Fox-Flory fit in Fig. 11 with the NMR curve shifted vertically based on
the WLF shift factors.
mated as the width at half height in Fig. 13. Those widths
were 37.5 ° C for PMA-5 K, 34.6 ° C for PMA-38 K,
34.0 ° C for PMA-77 K, and 33.9 ° C for the PMA-165 K
sample. The Tg widths from the intensity losses did not vary
as much as those from the MDSC experiments.
DISCUSSION
The Pake powder patterns 共slow regime兲 for PMA-d3
samples collapsed to motionally narrowed resonances 共fast
regime兲 with increasing temperature due to changes in the
segmental mobility of the polymer. In the intermediate regime, the spectra consisted of superpositions of both. Motional narrowing with temperature was initially observed as a
broadening of the powder pattern and then the appearance of
a central resonance in the spectra. At a given temperature, the
relative intensity of the central 共motionally narrowed兲 components decreased from PMA-5 K ⬎ PMA-38 K ⬎ PMA77 K ⬎ PMA-165 K. Further increases in temperature resulted in the complete collapse of the powder patterns.
The initial appearance of the mobile components in the
2
H spectrum can be attributed to chain ends,26 since they are
the most mobile portions of the polymers. The intensity of
the middle component as a function of molecular mass, at a
given temperature, should decrease with the molecular mass
of the polymer. Changes in the glass-transition temperatures
with the concentration of chain ends have been shown pre-
FIG. 13. Temperature dependence of the Boltzmann-corrected intensity loss
due to exchange-induced dephasing during the echo-delay time for PMA-d3.
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054908-9
J. Chem. Phys. 124, 054908 共2006兲
Molecular mass and dynamics of poly共methyl acrylate兲
viously for linear35,36 and star-shaped37 polystyrene 共PS兲. In
addition to the number of chain ends, their chemical
composition38 also affects the glass transition, as shown for
dendritic polyethers and polyesters. Specifically labeled end
groups coupled with deuterium 2H NMR,39 specular neutron
reflectivity,40 and electron-spin resonance41 experiments
have all directly confirmed the additional mobility of the
chain ends relative to the middles of the chains.
A trend towards a more homogeneous dynamics in the
glass-transition region was also observed in the 2H NMR
spectra with increasing molecular mass. The superposition of
more-mobile and less-mobile components in the PMA-5 K
and PMA-38 K samples clearly indicated the heterogeneity
of the segmental dynamics through the glass-transition region for those samples. The apparent heterogeneity in the
line shapes of the PMA-77 K and PMA-165 K spectra was
much less, although there was still an obvious middle component at 50– 55 ° C, due to the chain ends. Lin and Blum26
also found similar results with respect to the homogeneity of
segmental dynamics for low 共to medium兲 and high
molecular-mass samples. Unfortunately, in that study, some
of the details of the molecular-mass effects were not obvious
due to the polydispersity of those samples.
The presence of only a very small amount of spectral
heterogeneity through the glass-transition region for the
PMA-77 K suggests that this sample should be considered as
being in the HMM regime. In contrast, there was strong heterogeneity in the spectra of the LMM PMA-5 K and PMA38 K samples. Based on these results and the temperature
dependence of the Tg shown in Figs. 11 and 12, we estimate
the rough division between HMM and LMM at about
50 kg/ mol. This is the value for the guideline shown in Fig.
6 which clearly shows the presence of small amounts of
LMM material in the HMM samples. These LMM components are likely responsible for the presence of the middle
component in the HMM sample spectra.
The detailed picture of the distribution of jump rates
used to fit the experimental spectra at different temperatures
共Fig. 7兲, was indicative of the breadth of segmental motions
that occur in the polymer. At higher temperatures, the major
components were in the intermediate regime with only small
contributions from the fast regime. Small contributions from
fast jump rates were always present for each sample above
60 ° C. However, these contributions were only noticeable
for the PMA-5 K and PMA-38 K samples. The homogeneity
of the segmental dynamics with increasing molecular mass
was also seen in the distributions of jump rates. The distribution of jump rates appeared to be narrower for the PMA77 K and PMA-165 K samples. This was especially obvious
at higher temperatures.
We note that it was also possible to fit our line shapes
with random jumps on the soccer-ball shapes. This approach
was considered less physical than the one taken because it
was inconsistent with the small jump model validated for
amorphous polymers.19,42–44 Nevertheless, it is interesting to
compare the results between the two approaches, i.e., small
and random jumps. For random jumps, the individual correlation functions were exponential, or
关Cm共t兲 − Cm共⬁兲兴
= e−1/␶ ,
关Cm共0兲 − Cm共⬁兲兴
共9兲
where the correlation time ␶ = 共kN兲−1, where N is the number
of jump sites and k is the jump rate. The spectra fit in this
way also have significantly slower correlation times, as one
might expect because random jumps would cause the correlation function to decay much faster than nearest-neighbor
jumps.
The relationship between Tg’s and molecular mass is not
new. Our data confirm this relationship for PMA and also
extends it to the Tg’s measured with NMR. It has been shown
that, after a certain molecular mass is attained, the Tg does
not change significantly with molecular mass.23,35,36,45,46 This
is observed in the similarity of the spectra for PMA-77 K
共Fig. 4兲 and PMA-165 K 共Fig. 5兲, where the NMR glasstransition region does not change much with molecular mass.
The MDSC results also confirmed the molecular-mass dependence of Tg. The molecular-mass dependence of Tg was
characterized with the well-known Fox-Flory model35,36 for
the calorimetric results in Fig. 11.
Tg = Tg共⬁兲 − K/M .
共10兲
The data were analyzed for M ⬎ 2 ⫻ 104 and the FoxFlory constant, K, was found to be equal to ⬃2.5⫻ 105.
Boyer46 showed a relationship between K and Tg共⬁兲 for different polymers. The variation in K with the polymer type
was shown clearly for polymers with M ⬎ 104. The K value
for PMA estimated in our study was higher than the one
expected 共5 ⫻ 104兲 based on the Tg curve for similar polymers by Boyer.46 However, our value is acceptable given that
monodisperse PMA samples have not been available until
recently and the variation of K’s for poly共vinyl acetate兲
共PVAc兲, poly共vinyl chloride兲 共PVC兲, and s-PMMA in Boyer’s work is moderately large. The very low molecular-mass
sample 共M ⬍ 2 ⫻ 104兲 was not included in this analysis because it fell far from the line for the other samples due to its
very low molecular mass.
NMR Tg’s and their widths were estimated using the
intensity curves shown in Fig. 13. The greatest intensity loss
occurred when the correlation time was on the order of the
reciprocal of the echo-delay time. An echo-delay time of
27 ␮s, which is reciprocal of a quadrupole splitting of
37 kHz for PMA-d3 samples, was used in the experiments.
Thus, the Tg’s estimated were those that should have corresponded to a splitting of about 40 kHz, which is the time
scale for these measurements. The NMR glass-transition
widths, taken as the width of the intensity loss curves at half
height 共Fig. 13兲, can be compared with those from the
MDSC derivative curves in Fig. 12. The NMR transition
widths decreased slightly with increased molecular mass and
are much broader than those from MDSC. The molecularmass dependence is consistent with the LMM samples having larger motional heterogeneity.
Our NMR Tg measurements were around 40 K higher
than those from the MDSC experiments. A very similar difference was found from the collapse of the 2H powder pattern and DSC measurements of Tg for methyl-labeled poly共methyl methacrylate兲 共Ref. 47兲 and poly共vinyl acetate兲-d3.28
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054908-10
B. Metin and F. D. Blum
Publications by McCall48 and Chartoff et al.49 demonstrated
how different experimental techniques yielded frequencydependent Tg values. The changes in Tg with frequency can
be estimated using the Williams, Landel, and Ferry 共WLF兲
model based on the universal constants of C1 = 17.44 and
C2 = 51.6.50 The correlation times used in the WLF equation
were 100 s 共DSC兲 共Refs. 19 and 50兲 and the 具␶典 associated
with the minimum in the intensity loss curves 共NMR兲. The
WLF model predicts that the NMR Tg should be 37 K higher
than the MDSC Tg. This value 共37 K兲 was used for the displacement of the Tg 共NMR兲 curve in Fig. 12. Excellent
agreement 共within a couple of degrees兲 with the experiments
was found. Therefore, the NMR and MDSC experiments are
both very consistent in terms of their picture of the dynamics
of the polymer through the glass-transition region.
CONCLUSIONS
Deuterium NMR line shapes were quite sensitive to the
dynamics in the glass-transition region. The line shapes were
simulated using the MXQET program based on a model of
nearest-neighbor jumps on a truncated icosahedron 共soccer
ball兲. The line shapes and their simulations provide a means
to identify distributions of motional rates in polymers. Lower
molecular-mass samples showed significant heterogeneity in
their dynamics, while higher molecular-mass samples were
more homogenous through the glass-transition region. This
effect was clearly due to the presence and number of chain
ends which were identified as responsible for the more
highly mobile narrowed resonances. The extent of heterogeneous dynamics was clearly evident when a superposition of
a Pake powder pattern and the motionally narrowed resonance were observed.
The MDSC studies were consistent with the NMR results. Selective deuteration did not affect the calorimetric Tg
of the polymers and the molecular-mass dependence was
consistent with the Flory-Fox model. The homogeneity of
the dynamics manifested itself in the narrower glasstransition widths for HMM samples in the MDSC experiments. The polydispersity of the polymer samples was also
seen to have an effect on the segmental dynamics.
ACKNOWLEDGMENTS
The authors acknowledge the National Science Foundation 共NSF兲 Grant 共DMR-0412320兲 and the University of
Missouri-Rolla for financial support of this research. The authors also acknowledge Professor R. L. Vold for providing
and assisting with the simulation program.
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