Mechanistic and Kinetic Studies of the Ring Opening Metathesis Polymerization of Norbornenyl Monomers by a Grubbs Third Generation Catalyst

Article abstract of DOI:10.1021/jacs.9b09752

The mechanism of ring-opening metathesis polymerization (ROMP) for a set of functionalized norbornenyl monomers initiated by a Grubbs third generation precatalyst [(H₂IMes)(pyr)₂(Cl)₂Ru═CHPh] was investigated. Through a series of ¹²C/¹³C and ¹H/²H kinetic isotope effect studies, the rate-determining step for the polymerization was determined to be the formation of the metallacyclobutane ring. This experimental result was further validated through DFT calculations showing that the highest energy transition state is metallacyclobutane formation. The effect of monomer stereochemistry (exo vs endo) of two types of ester substituted monomers was also investigated. Kinetic and spectroscopic evidence supporting the formation of a six-membered chelate through coordination of the proximal polymer ester to the Ru center is presented. This chelation and its impact on the rate of polymerization are shown to vary based on the monomer employed and its stereochemistry. The combination of this knowledge led to the derivation of a generic rate law describing the rate of polymerization of norbornene monomers initiated by a Grubbs third generation catalyst.

Full text of DOI:10.1021/jacs.9b09752

Article
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Cite This: J. Am. Chem. Soc. XXXX, XXX, XXX−XXX
Mechanistic and Kinetic Studies of the Ring Opening Metathesis
Polymerization of Norbornenyl Monomers by a Grubbs Third
Generation Catalyst
†
Michael G. Hyatt,† Dylan J. Walsh,‡ Richard L. Lord,§ Jose G. Andino Martinez,
́
and Damien Guironnet*,‡
†Department of Chemistry, University of Illinois Urbana−Champaign, Urbana, Illinois 61801, United States
‡Department of Chemical and Biomolecular Engineering, University of Illinois Urbana−Champaign, Urbana, Illinois 61801, United
States
§Department of Chemistry, Grand Valley State University, Allendale, Michigan 49401, United States
S
* Supporting Information
ABSTRACT: The mechanism of ring-opening metathesis
polymerization (ROMP) for a set of functionalized
norbornenyl monomers initiated by a Grubbs third generation
precatalyst [(H2IMes)(pyr)2(Cl)2RuCHPh] was investi-
gated. Through a series of 12C/13C and 1H/2H kinetic isotope
effect studies, the rate-determining step for the polymerization
was determined to be the formation of the metallacyclobutane
ring. This experimental result was further validated through
DFT calculations showing that the highest energy transition
state is metallacyclobutane formation. The effect of monomer
stereochemistry (exo vs endo) of two types of ester
substituted monomers was also investigated. Kinetic and spectroscopic evidence supporting the formation of a six-membered
chelate through coordination of the proximal polymer ester to the Ru center is presented. This chelation and its impact on the
rate of polymerization are shown to vary based on the monomer employed and its stereochemistry. The combination of this
knowledge led to the derivation of a generic rate law describing the rate of polymerization of norbornene monomers initiated by
a Grubbs third generation catalyst.
very recently that the chemical structure of G3 in solution, the
monopyridine complex G3-Mono, was demonstrated to be
different than the isolated solid, the dipyridine complex G3-Di
(Chart 1).33 Acquiring a deeper understanding of the
mechanism of G3 initiated ROMP will guide the development
of new catalysts to address the remaining technical challenges,
such as enhancing the reactivity of low ring-strain monomers,
catalyst stability in solution, and higher endo monomer rates of
polymerization.34−36
INTRODUCTION
■
Ring-opening metathesis polymerization (ROMP) initiated by
ruthenium complexes has evolved as one of the most powerful
controlled polymerization methodologies. The precision and
versatility of ROMP have enabled the synthesis of sequence-
controlled polymers, complex macromolecular architectures,
mechanochemical responsive polymers, and self-healing
polymers.1−8 There have been many improvements to the
reactivity and stability of the multiple generations of Grubbs
catalysts developed over the years, with the third generation
(G3) being the most widely employed for ROMP due to its
high functional group stability, fast polymerization rates, and
ability to initiate living polymerizations.4,9−18 Since its first
report, most advances in ROMP have focused primarily on the
design of substrates (monomer, chain transfer agents,
terminating agents), and in comparison very little work has
been devoted to studying the ruthenium complex it-
self.5,10,19−30 The development of G3 was predicated on the
precise knowledge of the reactivity of the second generation
Grubbs catalyst (G2).31,32 While the general understanding of
the ROMP mechanism is extensive, the rate-determining step
(RDS) of the ruthenium catalyzed polymerization of
norbornenyl monomers remains unknown. In fact it was only
Chart 1. G3 Precatalyst Equilibrium Structures
Received: September 9, 2019
© XXXX American Chemical Society
A
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Beyond the identification of the rate-determining step,
another important aspect of ROMP that is still not fully
understood is the effect of stereochemistry (exo vs endo) and
substituents on the rate of polymerization for norbornenyl
monomers. Achieving a high rate of polymerization is
advantageous for polymer synthesis, as faster rates have been
shown to enable the synthesis of larger polymers with narrower
molecular weight distributions, which has been particularly
important for the synthesis of high density branched polymers
(bottlebrush polymers).37 Exo monomers have long been
known to achieve higher rates of polymerization than their
endo counterparts, thus motivating the isolation of the pure
exo monomer through a tedious and wasteful process.38,39
Beyond stereochemistry, the substituent groups/anchor groups
were also found to impact the rate of polymerization, but this
effect remains poorly understood due to a lack of fundamental
mechanistic understanding.5,37,40 Thus, determining the
molecular features that dictate the rates of polymerization of
ROMP monomers would enable the design of novel ROMP
catalysts and monomers.
This possibility is easily discounted as G3-initiated ROMP is
well-known to be first order in monomer.23,33,37,40 Monomer
coordination, however, could still be the RDS through an
associative or dissociative pathway. We have previously
demonstrated, via a series of kinetic studies, that pyridine is
not coordinated to the metal during the rate determining step
of the polymerization; therefore, this possibility was not
considered.33 The second step of the catalytic cycle is
formation of the metallacyclobutane ring (k2) followed by its
collapse (k3), and ultimately in coordination of pyridine. In
conclusion, the RDS of ROMP can be monomer coordination,
metallacyclobutane formation, or metallacyclobutane collapse.
We chose to investigate the RDS of G3-initiated ROMP using
two common norbornenyl monomers M1 and M2 (Chart 2)
with different ester linkages (2 stereoisomers per monomer;
XX stands for the exo monomer while DD stands for the endo
monomer).
Chart 2. List of Monomers Used
Herein, we report a series of kinetic isotope effect studies to
identify the rate-determining step of G3-initiated ROMP of
norbornenyl monomers. Additionally, IR experiments in
conjunction with kinetic studies provide an explanation for
the differences in reactivity between stereochemistries (exo
and endo) of ester-containing monomers. DFT calculations are
also presented to support our experimental findings and
provide better insight into the interaction of the metal center
and the polymer chain. The cumulative knowledge gained
through this work led to the establishment of a detailed rate
law for the polymerization of norbornenyl-type monomers.
Determination of the RDS was performed through 12C/13C
and 1H/2H kinetic isotope studies focusing on the olefin
double bond of the monomers. The 12C/13C KIE study will
indicate whether the RDS involves the breakage of a carbon−
carbon bond (metallacyclobutane formation or collapse) which
will have a KIE > 1, or if the RDS does not involve the
breaking of a carbon−carbon bond (monomer coordination)
which will have a KIE = 1. The 1H/2H KIE study will
differentiate between metallacyclobutane formation and
collapse. Indeed, the olefin in metallacyclobutane formation
undergoes an sp2 to sp3 hybridization change which will have a
KIE < 1, while metallacyclobutane collapse undergoes an sp3 to
sp2 hybridization change which will have a KIE > 1. The
theoretical KIE values depending on which step is rate
determining are summarized in Table 1.
RESULTS AND DISCUSSION
■
Rate-Determining Step. As the initiators G2 and G3
result in the same active species, we used the previously
established reaction mechanism from G2-initiated ROMP for
the determination of the RDS of G3-initiated ROMP (Scheme
1).31,41 The first step in the mechanism is coordination of the
monomer, which can occur either via an associative pathway
(k1a) or a dissociative pathway (k1b, k1c) involving a pyridine-
free Ru 14-electron complex.42 If pyridine dissociation is the
RDS, then the reaction rate would be zero order in monomer.
Scheme 1. Proposed Mechanism for ROMP of Norbornene
(Ligands Omitted for Clarity)
Table 1. Predicted KIE Values for Each Proposed RDS
Proposed RDS
12C/13C
1H/2H
monomer coordination
metallacyclobutane formation
metallacyclobutane collapse
1
>1
>1
1
<1
>1
The 12C/13C KIE was determined using the natural
abundance enrichment KIE technique developed by Singleton
and Thomas, as it allows for the determination of small KIEs
without the need to isotopically label the starting material.43
This technique consists of performing the reaction to high
conversion and isolating the residual substrate for isotope
analysis. This operation requires the use of large amounts of
substrate in order to isolate sufficient quantities of unreacted
B
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reagent for analysis. Using M2-DD, we performed a 5-g scale
polymerization (18 mg G3-Di) to 98.5% conversion. The
unreacted monomer was isolated by precipitating the polymer
and was purified via preparative thin layer chromatography. As
the enrichment of the 13C content is expected to be small we
used three independent quantification methods (13C NMR, 1H
NMR and isotope mass spectrometry) to determine the
change in the 13C content. From 13C NMR we measured a
3.3% enrichment in the 13C content at the CC position
which corresponds to a 12C/13C KIE of 1.0077 ( 0.001). From
1H NMR, we measured a 3.5% enrichment of the 13C content,
by comparing the olefin peak and its 13C satellites, which
corresponds to a 12C/13C KIE of 1.008 ( 0.004). Then, we
used isotope mass spectrometry to burn the sample and
measure the relative amounts of 13CO2/12CO2 in the
combustion product to show that there is a 4.6% enrichment
of the 13C content at the olefin position, which corresponds to
using the ORCA 4.0.1 package (see Supporting Information
for details). Our initial DFT calculations looked at the
thermodynamics of pyridine dissociation from G3-Di to
form G3-Mono, which served as an experimental benchmark
for our calculations. A free energy of −2 kcal/mol was
calculated when considering the effects of methylene chloride
as a solvent using the conductor-like polarizable continuum
model, CPCM. Although this moderate preference for G3-
Mono in solution deviates slightly from the experimentally
determined value of ΔG (+1 kcal/mol),33 the solution phase
enthalpy calculated using the electronic energy derived from
the solvation free-energy calculation and the zero-point energy,
is +12 kcal/mol. This value compares reasonably well with the
experimentally determined ΔH of +10 kcal/mol. Furthermore,
the ring opening of the three norbornenyl monomers were
calculated to be exergonic, which is consistent with the release
of the ring strain of the monomers.45
The transition state energies for the formation of the
metallacyclobutane using norbornene, M1-XX and M1-DD
(19.8, 19.7 and 22.3 kcal/mol, respectively), are the highest
energies of the reaction profile, which is consistent with our
experimental results that formation of the metallacyclobutane
ring is the RDS. It is worth noting the higher transition state
energies for the endo monomer (M1-DD, TS1b) over the exo
monomer (M1-XX, TS1c). This difference is consistent with
the experimental observation that the endo monomer
polymerizes at a slower rate than the exo monomer. In
contrast, the collapse of the metallacyclobutane ring is nearly
barrierless. Such a low energy transition state is expected due
to the strain present in the four-membered ring of the
metallacyclobutane. This result was confirmed for norbornene
and M1-DD, and even though we were unable to locate a TS
for M1-XX, there is a consistent trend among the three profiles
as can be seen in Figure 2. Besides validating our experimental
evidence obtained through kinetic studies, the determination
of the transition state energy via DFT enables us to rapidly
probe whether the RDS of the reaction remains the same using
other norbornene derivatives; this is important, as a change in
the RDS leads to a change in the rate law for the
polymerization.
a
12C/13C KIE of 1.01097 ( 0.00013). The 12C/13C KIE data
from each measurement technique agree that ROMP has a
12C/13C KIE for the olefinic carbon greater than 1, which rules
out monomer coordination as the RDS. We repeated the
12C/13C KIE study with another monomer, M1-DD, and
measured an enrichment in 13C of 3.9% via 13C NMR which
corresponds to a 12C/13C KIE of 1.012 ( 0.015). These results
suggest that the olefinic carbon−carbon bond is breaking
during the rate-determining step (metallacyclobutane forma-
tion or metallacyclobutane collapse).
To differentiate between metallacyclobutane formation and
1
metallacyclobutane collapse, we performed a H/2H kinetic
isotope effect study. We synthesized a partially deuterated M2-
DD that is 75% deuterated at each of the four positions
indicated in Figure 1 and performed a 3-g scale polymerization
Having determined the RDS, we derived the rate law for the
polymerization of norbornenyl monomers initiated by G3 (eq
1) (Keq 1 is shown in Scheme 2). This rate law predicts non-
first-order behavior for [monomer], but experiments show that
it is first order. This is explained by pyridine coordination
Figure 1. Partially deuterated M2-DD.
(11.3 mg of G3-Di) to 98.3% conversion. The residual
monomer was isolated, purified by preparative thin layer
chromatography, and analyzed by 1H NMR. An enrichment in
1H of 15.6% was measured at the olefinic protons which
being much stronger than the monomer coordination
[Pyr]
Keq1[monomer]
(
≫ 1) which makes the ‘+1’ in the denominator
1
corresponds to a H/2H KIE of 0.965 ( 0.001). A KIE of
insignificant. This simplified rate law is also consistent with
0.9987 ( 0.0005) was observed at the bridgehead (position b)
which is consistent with there being no reaction at that
previously reported data.33
Theoretical Rate Law with Step 2 as RDS:
position. Since the olefinic position has a H/2H KIE that is
1
less than 1 and a 12C/13C KIE that is greater than 1, that means
that only metallacyclobutane formation (step 2) is consistent
with being the RDS (Scheme 1).
k2Keq1[Ru]o[monomer]
[Pyr]
k2[Ru]o
Rate =
≈
i
j
y
z
[Pyr]
j1 +
z
z
j
Keq1[monomer]
1
We also observed a H/2H KIE of 0.953 ( 0.002) at the
k
{
methylene bridge at “position d2” and a H/2H KIE of 0.965
1
(1)
( 0.001) at “position d1”. Since these two positions are not
undergoing an sp2 to sp3 hybridization change, we did not
initially expect an inverse KIE for these positions. We suspect
that the inverse KIE at “positions d1 and d2” is due to some
type of γ hyperconjugation effect.44
The conclusions of the KIE experiments were further
validated by analyzing the reaction through DFT calculations
Exo-Endo Rate Study and Chelation Effects. Endo-
substituted norbornenes are known to undergo ROMP
significantly slower than their exo analogs. This difference in
reactivity is presumably caused by a combination of the
sterics/electronics of the monomer (referred to as monomer
control) and the sterics/electronics of the growing polymer on
the Ru center (referred to as polymer control). The origin of
C
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Figure 2. Free energy profile for the initiation of G3 with norbornene, M1-DD, and M1-XX.
the exo/endo rate difference has been studied before with
dicyclopentadiene, where kinetic evidence suggests that the
steric encumbrance of the substituents in the endo monomer
slows down the polymerization of this isomer (monomer
control).38,39
Scheme 2. Thermodynamic Equilibria between Ester,
Monomer, and Pyridine Coordination to the Ruthenium
Center
For ester-containing monomers, it has been suggested that
the ester from the proximal repeating unit can coordinate to
the Ru center and slow down the polymerization (Keq 2
,
Scheme 2). This ester coordination was postulated to be the
main cause for the rate differences between M2 and an M1-like
monomer.21 However, DFT calculations and kinetic measure-
ments in the presence of chelate-opening agents did not
validate this hypothesis for structurally similar monomers.37 In
another report, DFT calculations suggest that within one
monomer set (M1) the endo polymer ester coordinates more
Figure 3. Overlaid IR spectrum for the monomer (blue), polymer (red), and chelate (green) for M1-DD (left) and M1-XX (right).
D
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Figure 4. (a) M2-DD plot of ln (kobs) versus ln([pyridine]). (b) M1-DD plot of ln(kobs) versus ln([pyridine]) at constant Ru loading. (c) M1-DD
plot of ln(kobs) versus ln([Ru]) without added pyridine.
Figure 5. Plot of 1/kobs versus [pyridine] for M1-DD (a) and for M1-XX (b).
strongly than the exo polymer ester and, thus, is responsible in
part for the exo/endo rate difference.5 We initiated an
experimental kinetic and spectroscopic investigation to further
understand whether ester coordination causes the exo/endo
rate difference, and how ester coordination changes between
monomer types.
Evidence supporting the chelate formation led us to
reconsider the rate law for the polymerization, as ester
coordination should compete with monomer (and pyridine)
coordination and thus slow down the rate (Scheme 2). Taking
that into consideration, we derived a new rate law under pre-
equilibrium conditions (eq 2).
Theoretical Rate Law for Scheme 2 with k2 as the RDS:
We first investigated the formation of the chelate
spectroscopically. In CH2Cl2 2 equiv of M2-DD was reacted
with 1 equiv of G3-Mono at room temperature, and the crude
mixture was analyzed by FTIR. No new carbonyl peak in the
IR consistent with a chelated structure was observed with M2-
DD. The same experiment with M1-DD, however, resulted in
a new lower energy IR band at 1691 cm−1 compared to that of
the polymer at 1691 cm−1, which is consistent with the
electron donation from the Ru into the chelated carbonyl
weakening the bond (Figure 3).46 This chelate was also
observed for the other isomer M1-XX. The intensity of the
lower energy carbonyl signals could be diminished upon
addition of excess pyridine or completely suppressed upon
addition of a stronger ligand (4-(dimethylamino)pyridine),
which supports the conclusion that the new signal at 1733
cm−1 is the chelated complex (see Supporting Information
Figure S5). While M1 and M2 are structurally very similar,
only M1 resulted in a coordinated ester; we postulate that the
reason for this difference is due to the more stable ring size of
the chelate for M1 (six member for M1 and eight member for
M2). While no other chelating structures have been observed
previously with G2 or G3 initiated polymerizations, these
spectroscopic observations are consistent with a previous
report using G1 that observed a new alkylidene signal in the
1H NMR consistent with a coordinated ester using an M1 type
monomer.47,48
k2[Ru]o
Rate =
i
y
[Pyr]
1
j
z
j1 +
+
z
z
j
Keq1[monomer]
Keq2[monomer]
k
{
k2[Ru]o[monomer]
≈
i
y
[Pyr]
Keq1
1
Keq2
j
z
j
j
+
z
z
(2)
k
{
In eq 2, the ester coordination term is additive to that of
pyridine coordination, meaning that if ester coordination is
non-negligible the observed pyridine order of the reaction
should be less than 1. M2-DD was determined experimentally
to be inverse first order in pyridine (Figure 4a), suggesting that
any ester coordination is negligible under pre-equilibrium
conditions. This is consistent with the fact that no chelate was
detected by IR spectroscopy for this monomer. For M1-DD,
we measured that the polymerization is inverse 0.29 order in
pyridine (Figure 4b) suggesting that ester chelation competes
with monomer and pyridine for the coordination site which
again is consistent with the chelate observed by IR spectros-
copy.
In the absence of extra pyridine, the polymerization of M1-
DD was measured to be 0.66 order in ruthenium (Figure 4c),
which is consistent with G3-Di releasing 1 equiv of pyridine
into solution and ester coordination. The polymerization
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Scheme 3. Representation of How Ester Chelation Affects the Rate of Polymerization, But Not Catalyst Initiation
becomes first order in ruthenium when excess pyridine is
added to the reaction mixture (see Supporting Information
Figure S7). The relative binding strength of the chelating ester
for a given monomer relative to pyridine can be determined by
studying the rate of polymerization as a function of pyridine
concentration. By rearranging eq 2 we obtain a linear
relationship between 1/kobs and [pyridine] (eq 3); the ratio
between the slope and y-intercept of this linear equation
corresponds to the relative binding strength (eq 4). For M1-
DD we measured pyridine coordination to be 527 times
stronger than ester coordination (Figure 5a), but because of
the low concentration of pyridine in solution the majority of
the catalyst shifts to the ester bound state. In comparison, for
M1-XX, pyridine coordination was measured to be 1326 times
stronger than ester coordination (Figure 5b). Overall this
means that the endo polymer ester chelates ca. 2.5 times more
strongly than the exo isomer, which establishes that ester
coordination contributes to the differences in reactivity
between the endo and exo isomers for M1.
Table 2. Relative Rates of Initiation and Polymerization for
ROMP of M1 and M2
Monomer
Initiation
Polymerization
M1-XX vs M1-DD
M2-XX vs M2-DD
M1-XX vs M2-XX
M1-DD vs M2-DD
4.1
4.7
1.6
1.9
10.2
5.2
NA
NA
difference. M2-XX was then measured to polymerize 5.2
times faster than M2-DD; the agreement between the
initiation and polymerization values suggests that the differ-
ences in rate for polymerization between exo and endo for M2
are primarily due to the incoming monomer (monomer
control), which is consistent with the absence of chelate
formation.
M1-XX was determined to initiate 4.1 times faster than M1-
DD, while it polymerizes 10.2 times faster. This is consistent
with both monomer and polymer control contributing to the
rate difference between exo and endo for M1, which again is
consistent with the formation of an ester chelate. Comparing
between monomer types we see that both M1-XX and M1-DD
initiate faster than their M2 counterparts, which suggests that
the substituent groups coming off the norbornene also affect
the reactivity of the monomer.
Rearrangement of eq 2:
k2[Ru]o
kobs
≈
i
y
[Pyr]
1
Keq2
j
z
j
j
+
z
z
Keq1
k
{
[Pyr]
1
kobs
1
≈
+
Keq1k2[Ru]o
Keq2k2[Ru]o
(3)
(4)
CONCLUSIONS
Relative Binding Strength of Pyridine and Chelate:
■
The rate-determining step for ROMP of a norbornene
monomer using G3 was determined to be formation of the
metallacyclobutane ring using 12C/13C and 1H/2H kinetic
isotope effect studies. This result was further validated by DFT
calculations showing that the metallacyclobutane formation
presented the higher transition state energy. The knowledge of
the rate-determining step led to the derivation of a rate law
describing the kinetics of ROMP of norbornene monomers.
The origin of the exo/endo rate of polymerization was
investigated through a series of kinetic experiments. The
difference in rate appears to be primarily due to the
stereochemistry of the incoming monomer for M2, while
chelation (through ester coordination) and the incoming
monomer contribute to M1. The ability of the M1 polymer
ester to coordinate to the Ru center is exemplified by the
inverse 0.29 pyridine order in the rate law and the chelated
species being observable in the IR. This in-depth mechanistic
study provides the fundamental understanding needed to
design next generation ROMP catalysts and monomers.
Keq2
Slope
Intercept
≈
Keq1
We also probed the coordination of the ester via DFT and
confirmed that the carbonyl oxygen from the proximal ester
group from M1-XX forms a stable six membered chelate
(Figure 2, Structure 7c). Going from the ester complex to the
pyridine complex, the ΔG was determined to be −3.4 kcal/mol
from DFT, which is in good agreement with the experimentally
determined value of −4.3 kcal/mol (based on 1326
equilibrium constant at 24.5 °C).
Having established that ester coordination contributes to the
differences in rates of polymerization between the exo and
endo monomers for M1, we went on to probe whether the
incoming monomer (monomer control) contributes as well.
The extent of monomer control can be determined
experimentally by studying the difference in rate of initiation
of G3 with the pure exo and endo isomers, as initiation has no
polymer control contribution (Scheme 3). By comparing the
rates of polymerization for each pure monomer to the rate of
initiation, we can determine to what extent polymer and
monomer control contribute to the exo/endo rate difference.
As can be seen in Table 2, the rate of initiation using the
M2-XX monomer is 4.7 times faster than that with M2-DD,
which shows that the stereochemistry of the incoming
monomer is a major contributor to the exo/endo rate
ASSOCIATED CONTENT
* Supporting Information
The Supporting Information is available free of charge on the
ACS Publications website at DOI: 10.1021/jacs.9b09752.
■
S
All experimental methods, materials and characterization
(PDF)
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AUTHOR INFORMATION
■
Corresponding Author
*guironne@illinois.edu
ORCID
Richard L. Lord: 0000-0001-6692-0369
Damien Guironnet: 0000-0002-0356-6697
Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS
■
Umicore is acknowledged for the generous donation of
metathesis catalyst. Major funding for the 500 MHz Bruker
CryoProbe was provided by the Roy J. Carver Charitable Trust
to the School of Chemical Sciences NMR Lab. This work was
funded by NSF CHE 18-00068 and NSF DMR 17-27605.
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