Extraction of Reliable Molecular Information from Diffusion NMR Spectroscopy: Hydrodynamic Volume or Molecular Mass?

Article abstract of DOI:10.1002/chem.201900812

Measuring accurate translational self-diffusion coefficients (D_t) by NMR techniques with modern spectrometers has become rather routine. In contrast, the derivation of reliable molecular information therefrom still remains a nontrivial task. In this paper, two established approaches to estimating molecular size in terms of hydrodynamic volume (V_H) or molecular weight (M) are compared. Ad hoc designed experiments allowed the critical aspects of their application to be explored by translating relatively complex theoretical principles into practical take-home messages. For instance, comparing the D_t values of three isosteric Cp₂MCl₂ complexes (Cp=cyclopentadienyl, M=Ti, Zr, Hf), having significantly different molecular mass, provided an empirical demonstration that V_H is the critical molecular property affecting D_t. This central concept served to clarify the assumptions behind the derivation of D_t=?(M) power laws from the Stokes–Einstein equation. Some pitfalls in establishing log (D_t) versus log (M) linear correlations for a set of species have been highlighted by further investigations of selected examples. The effectiveness of the Stokes–Einstein equation itself in describing the aggregation or polymerization of differently shaped species has been explored by comparing, for example, a ball-shaped silsesquioxane cage with its cigar-like dimeric form, or styrene with polystyrene macromolecules.

Full text of DOI:10.1002/chem.201900812

A Journal of
Accepted Article
Title: Extraction of reliable molecular information from diffusion NMR
spectroscopy: hydrodynamic volume or molecular mass?
Authors: Francesco Zaccaria, Cristiano Zuccaccia, Roberta Cipullo,
and Alceo Macchioni
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To be cited as: Chem. Eur. J. 10.1002/chem.201900812
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Extraction of reliable molecular information from diffusion NMR
spectroscopy: hydrodynamic volume or molecular mass?
Francesco Zaccaria,*[a] Cristiano Zuccaccia,[a] Roberta Cipullo[b] and Alceo Macchioni[a]
Abstract: Measuring accurate translational self-diffusion coefficients
(Dt) by NMR with modern spectrometers has become rather routine.
On the contrary, the derivation of reliable molecular information
become a routine exercise also for non-expert NMR users. On
the other hand, deriving reliable and quantitative molecular
information from Dt is a delicate task, as the correlation between
the latter and molecular size is not always straightforward.[1f]
The master law in this context is the Stokes-Einstein (SE)
equation,[6] which allows the hydrodynamic radius (rH) of the
diffusing species to be derived from its Dt. In its original
formulation, the SE equation is valid only for colloidal particles
moving with uniform velocity in a fluid continuum (i.e. dense
spherical shaped species being significantly larger than the
solvent molecule). As these conditions do not apply to most of
the systems of interest, the SE equation has been modified by
introducing two numerical factors accounting for the shape of the
studied molecule (fs), and the ratio between the rH of the analyte
and that of the solvent (c),[1f] as reported in Equation (1):
therefrom still remains
a non-trivial task. In this paper, two
established approaches to estimate molecular size in terms of
hydrodynamic volume (VH) or molecular weight (MW) are compared.
Ad-hoc designed experiments allow the critical aspects of their
application to be explored, by translating relatively complex
theoretical principles in practical take-home messages. For instance,
comparing the Dt of three isosteric Cp2MCl2 complexes (Cp =
cyclopentadienyl, M = Ti, Zr, Hf), having significantly different
molecular mass, provided an empirical demonstration that VH is the
critical molecular property affecting Dt. This central concept served
to clarify the assumptions behind the derivation of Dt = ƒ(MW) power
laws from the Stokes-Einstein equation. Some pitfalls in establishing
Log(Dt) vs Log(MW) linear correlations for a set of species have
been highlighted by further investigations on selected examples. The
effectiveness of the Stokes-Einstein equation itself in describing
aggregation or polymerization of differently shaped species has
been explored by comparing e.g. a ball-shaped silsesquioxane cage
with its cigar-like dimeric form, or styrene with polystyrene
macromolecules.
푘
푇
퐵
퐷푡 =
(1)
푐푓 휋휂푟
푠
퐻
where kB is the Boltzmann constant, T the absolute temperature
and η the fluid viscosity. Here, rH represents the radius of a
hypothetical hard sphere that diffuses with the same Dt of the
particle under examination.
An alternative approach to obtain molecular size information
from diffusion NMR is based on the correlation between Dt and
molecular weight (MW) [Equation (2)]:
Introduction
퐷푡 = 퐾 ∙ 푀푊훼
(2)
where K and α are constants typical of the studied molecule at a
given temperature and in a specific solvent. This power law is
actually a derivation of Equation (1), as first proposed by Polson
in 1950.[7] This can be shown by considering the simplest case
of a dense spherical particle; rH can be expressed in terms of
hydrodynamic volume (VH), which in turn is given by the ratio
between MW, the density of the species (ρ) and the Avogadro’s
number (NA) [Equation (3)]:[8]
NMR spectroscopy is the technique of choice for molecular
structural characterization in solution. Along with ‘classical’
multinuclear and multidimensional experiments, diffusion NMR
has emerged as a powerful tool to investigate molecular
interactions in solution and determine molecular size.[1] The
latter subjects are of considerable interest for chemists,
1g,
2]
especially to explore host-guest interactions,[1a,
supramolecular
processes.[3]
aggregations,[1b-e]
and
polymerization
3
푟퐻 = √ 3 푉퐻 =
(3)
(4)
3
√
3
푀푊
Due to the advances in hardware and software development, the
determination of accurate translational self-diffusion coefficients
4휋
4휋 휌푁
퐴
(Dt) by typical pulsed gradient spin echo (PGSE)[2a, 4] or diffusion
푘
푇
푘
푇
퐵
1
퐵
) 푀푊− ⁄
5]
ordered NMR spectroscopy (DOSY)[2b,
experiments has
3
퐷푡 =
= (
3
3
푀푊
3
3
푐푓 휋휂
푐푓 휋휂
√
푠
√
푠
4휋 휌푁
4휋휌푁
퐴
퐴
By substituting the rH expression (3) in the SE Equation (1),
Equation (4) is obtained where all the terms in brackets are
constant for a given particle in a specific solvent and at a
[a]
[b]
Dr. F. Zaccaria, Prof. C. Zuccaccia, Prof. A. Macchioni
Dipartimento di Chimica, Biologia e Biotecnologie and CIRCC
Università di Perugia
Via Elce di Sotto 8, 06123 Perugia, Italy
E-mail: francesco.zaccaria@unipg.it
Prof. R. Cipullo
Dipartimento di Scienze Chimiche,
Università di Napoli Federico II
Via Cintia, 80126 Napoli, Italy
specific temperature, and therefore correspond to the
K
parameter in Equation (2). Analogous correlations between Dt
and MW have been empirically derived for the specific case of
synthetic polymers.[9]
Using Equation (2) in place of (1) has the advantage of providing
molecular size information in terms of MW, which is a generally
more intuitive molecular property than rH. Furthermore, it does
Supporting information for this article is given via a link at the end of
the document.
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not require any explicit assumption on molecular shape, which is
implicitly accounted for in the K and especially α parameter
(extreme cases: α = -0.33 for perfect spheres, α = -1 for linear
rigid rods).[8a, 10]
Equation (2) is typically employed in its logarithmic version
[Equation (5)], providing a linear relationship between Log(Dt)
and Log(MW). The latter has been often exploited to determine
the MW of target species by using internal or external calibration
curves.[5a, 8, 11]
Figure 1. Bis-Cp complexes of the group IV metals.
Table 1. MW, measured Dt and estimated VH of the species shown in
Figures 1-2.
Entry
Name
Cp2TiCl2
Cp2ZrCl2
MW
248.9
292.3
379.6
891.6
1828.1
480.7
945.4
Dt
VH
(
)
( )
퐿표푔(퐷푡) = 훼 ∙ 퐿표푔 푀푊 + 퐿표푔 퐾
(5)
1
2
3
4
5
6
7
11.4±0.6
11.7±0.6
11.5±0.6
5.3±0.3
4.0±0.2
7.0±0.4
4.7±0.2
295±44
280±42
291±43
However, caution is required since this apparently simple and
powerful equation is only valid for a set of molecules under
several assumptions, primarily that they have similar molecular
density.[8b, 10a, 11c, 12]
Cp2HfCl2
POSS(OH)2
POSS-Ti-POSS
MeAl(bht)2
Al(bis-bht)Al
1724±259
3500±525
847±127
2292±344
In this paper, the two approaches based on Equations (1) and
(2) for extracting molecular information from diffusion NMR are
compared. Selected and ad-hoc designed examples allow to
highlight the potential pitfalls behind them, and provide
suggestions for choosing the most appropriate method for
accurately estimating molecular size in solution for each
individual studied molecular system.
Dt measured performed in toluene-d8 at 298K. 5% experimental error
assumed on Dt,[1f, 14] implying 15% uncertainty on VH and calculated MW.
MW in g/mol, Dt values in 1010 m/s2, VH in Å3.
correlate Dt and MW,[8a, 10, 12, 15] which can be extremely useful
and effective if correctly employed within their range of validity.
Strengths and limitations of the modified SE equation
Molecular size information in terms of rH and VH can be derived
from Dt through the SE equation. As they are quite elusive
molecular properties, relative rather than absolute hydrodynamic
radii and volumes are considered in many cases; for instance,
the level of aggregation of a supramolecular system in solution
can be determined by dividing the VH of the aggregate by that of
Results and Discussion
VH and MW: what determines self-diffusion properties?
The first question that might arise from comparing Equation (1)
and (2) concerns the molecular property that actually determines
self-diffusion coefficients: is it VH, MW or both of them? In order
to clarify this point from an experimental point of view, it has
been searched for molecules having the same VH but
significantly different MW. Good candidates have been found in
0
14]
the monomeric species (VH ).[1e,
In this respect, the main
difficulties in the application of Equation (1) are the
determination of the size c and shape fs parameters of both
aggregates and monomers.[16] While the first can be addressed
by using suitable empirical equations,[1f] determining accurate fs
is more problematic, since it requires the shape of the particle to
be accurately known.[17] As these pieces of information are often
unavailable,[18] Equation (1) is frequently employed by simply
assuming a spherical shape, i.e. fs = 1. It has been shown that
this represents a reasonable approximation in many cases.[1f]
For instance, going from a perfectly spherical to a spheroidal
particle having a semi-axis three times longer than the other,
should lead to an increase of fs from 1.0 to only 1.1; furthermore,
the three bis-cyclopentadienyl complexes Cp2MCl2 (Cp
=
cyclopentadienyl, M = Ti, Zr, Hf; Figure 1). Although the atomic
radii of the metal centers are not exactly the same, these three
homologous species are practically isosteric[13] and can be
reasonably considered as having the same VH; conversely, their
MWs are significantly different (~50% increase going from
Cp2TiCl2, 248.9 g/mol to Cp2HfCl2, 379.6 g/mol). The Dt for
these three metallocenes in toluene-d8 solutions at 298 K have
been measured and found to be substantially the same,
considering the 5% experimental uncertainty (Entries 1-3, Table
1).[1f, 14]
0
the fs factor is essentially normalized when considering VH/VH
ratios between species having comparable shapes.[1f]
Some examples can help familiarizing with these considerations.
The two molecules shown in Figure 2a have been identified as a
suitable introductory case. The silsesquioxane diol POSS(OH)2
has a cage-like structure that can be reasonably considered of
spherical shape. By its reaction with half an equivalent of
Ti(OiPr)4, POSS-Ti-POSS is obtained via iso-propanol
elimination.[19] From a structural point of view, a single Ti atom
bridging two 20-membered Si/O cages does not contribute
significantly to the overall molecular volume. Therefore,
comparing POSS(OH)2 with POSS-Ti-POSS can be seen as a
representative example of a sphere-like particle having radius a
vs a prolate spheroidal-like particle having semi-axis a and b = 2
* a. By applying Equation (1) with fs = 1, the VH of the latter
These results clearly indicate that the hydrodynamic volume, not
the molecular mass, determines self-diffusion rates, as dictated
by the SE equation. This information is crucial to avoid
misconceptions in the manipulation of Dt (vide infra), but it is not
necessarily obvious and sometimes overlooked. In this respect,
it is worth emphasizing that, while the Equation (1) is a
generalized correlation based on first principles, the derivation of
Equation (2) depends on the shape and density of the species
under examination, i.e. it is case-specific. Caution is therefore
required when empirically extrapolating the K and α parameters
for a set of different molecules. These considerations generally
apply also to the other, more elaborated equations proposed to
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(3500 ± 525 Å3) is found to be exactly twice bigger than that of
the former (1724 ± 259 Å3, Entries 4-5, Table 1). This
demonstrates that the modified SE equation can be used within
the spherical approximation to quantitatively evaluate POSS-
dimerization even though it causes an evident change in the
molecular shape. Analogous considerations apply to MeAl(bht)2
and its ‘dimeric’ form Al(bis-bht)Al, shown in Figure 2b. This
couple of molecules represents an even more challenging
example, as their structures deviate even more from a spherical
large non-linear systems. For instance, ion pairs in non-polar
solvents likely form 3D spheroidal aggregates rather than 2D
linear systems; the spherical approximation might still hold in
this case. As a matter of fact, methods based on Equation (1)
1d, 4, 23]
are well-established to study ion pair aggregation.[1c,
Likewise, other common applications concern the study of e.g.
dendrimers,[24] and self-assembled supramolecular systems.[25]
Finally, a peculiar and often tricky case to consider is that of
hollow molecules.[1a-b] Based on the example of Figure 1, it is
reasonable to conclude that species with empty internal cavities
will self-diffuse at the same rate of analogous densely filled
particles having the same dimensions.[26] Therefore, determining
the number of elemental units in e.g. a hollow organometallic
cluster from the VH/VH0 ratio faces an additional hurdle, as not all
the molecular volume (i.e. VH) actually contains an elemental
0
shape. Nevertheless, assuming fs = 1, a VH/VH of ~2 has been
found within experimental uncertainty for Al(bis-bht)Al (2292 ±
344 Å3, Figure 2a) and MeAl(bht)2 (847 ± 127 Å3), as expected.
0
unit (i.e. VH ). This problem has been occasionally addressed by
estimating some empirical correction factors for VH that account
for the internal cavities of the studied molecules, as reported for
instance in the case of methylaluminoxane.[27]
Strengths and limitations of the MW power law
In principle, the application of Equation (2) and (5) for the
determination of MW is rather simple. The Dt of a number of
reference species have to be measured under the same
experimental conditions to build up a Log(Dt) vs Log(MW)
calibration curve; then, the MW of an unknown species can be
estimated from its Dt (or viceversa) by interpolating or
extrapolating it in this calibration curve.
Figure 2. Structures of studied silsesquioxanes and Al-complexes.
While these encouraging examples indicate that some tolerance
is acceptable when dealing with fs, this has not to be considered
as a panacea. In the generality of cases where more or less
flexible and spheroidal molecules are considered, Equation (1)
can be safely used within the spherical approximation to
compare a monomeric species with its dimer, trimer or very short
oligomers,[1e, 14, 20] even in the case of some biologically relevant
peptides,[20d-e] as the associated variations in shape are usually
small.[21] Caution is definitely required when considering longer
polymers or higher aggregates, instead.
Among others,[5a, 8, 10a, 11a-c, 15a, 25b, 28] a typical, successful case of
application is the determination of the average MW of linear
synthetic polymers, for which a plethora of examples is available
9a, 15b, 29]
in the literature.[3a,
This approach relies on the
assumption of
a
monomodal MW distribution,[30] although
advanced methods to determine polydispersity by diffusion NMR
have been proposed.[9a, 31] The application of Equation (2) or (5)
is especially advantageous in this case. The calibration curve is
usually built by using known samples of the same kind of
polymer of the analyte, having different and independently
determined MWs (e.g. by gel permeation chromatography or
mass spectrometry). Within a certain range of Pn,[32] all the
contributions to Dt that are especially difficult to estimate for
polymers (i.e. in terms of shape, density, intramolecular
interactions affecting the friction factor) can be assumed to be
constant for all these species, and are empirically accounted for
in the fitted K and α parameters. Thus, as the MW of the analyte
is determined with respect to its shorter or longer homologues
(not to the monomer), the aforementioned problems related to
size estimation of polymers are circumvented, as all the critical
factors are normalized.
The choice of the reference species is indeed crucial to obtain a
reliable MW estimation; the calibration set has to be sufficiently
large and compatible with the analyte in terms of molecular
shape and composition (and therefore molecular density).[10a, 11c]
Concerning the latter, the contribution of ‘heavy’ atoms like
metals or halides, as well as of molecular cavities, has to be
carefully accounted for, as it will likely impact molecular density
significantly. Likewise, also comparing hydrogenated and
deuterated species can be problematic.[11c] In this respect, it is
The case of linear polymers is particularly problematic. For
instance, the Dt of styrene (19.8 * 10-10 m/s2) and of a
commercial monodisperse 10 kDa polystyrene sample (PS; 1.43
* 10-10 m/s2) have been measured under analogous conditions.
keeping into account their different c parameter. Nevertheless,
0
the estimated VH/VH suggests a polymerization degree (Pn) of
~640, clearly overestimating the independently determined value
of ~97. This discrepancy is likely due to the fact that the fs of a
single styrene unit is significantly different from that of a linear
polystyrene macromolecule and, in particular, the latter deviates
noticeably from the spherical approximation. Furthermore, it is
known that the self-diffusion of polymers in dilute solutions
exhibits some peculiarities with respect to typical small
compounds,
as
occasional
interactions
between
macromolecules lead to an increase of their frictional coefficient
(i.e. a decrease of Dt) with respect to the solvent-dominated
diffusion limit.[9c, 9d, 22] This makes the physical definition of VH
even more complex to grasp in the case of polymers.
The above considerations do not necessarily imply that only
limited ranges of polymerization/aggregation can be analysed,
as the modified SE equation can still work when considering
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Figure 3. Set of molecules used to build up the calibration curve of Figure 4a (a), and species used as stress test (b). R = iso-butyl.
Figure 4. Log(Dt) vs Log(MW) plot obtained for the set of molecules of Figure 3a (a), and comparison with analogous plots obtained using the stress test
molecules of Figure 3b (b). □ = molecules in the calibration set (Figure 3a); Δ = molecules used as stress test (see Figure 3b and main text). Fitted equations for
the points in red and blue reported for semi-quantitative comparison.
Table 2. MW and experimental Dt of the species in Figure 3, and calculated MW using the calibration curve of Figure 4a.
Entry
Name
Al2Me6
BHT
MW
Dt
Log(MW)
2.06
2.34
2.39
2.44
2.63
2.68
2.84
2.95
2.98
4.00
4.46
Log(Dt)
-8.82
-8.98
-9.00
-9.04
-9.12
-9.15
-9.21
-9.28
-9.33
-9.85
-10.08
calc. MW
112±16
227±33
243±36
290±43
420±62
488±72
615±91
847±125
1047±154
10300±1515
28593±4027
1
2
3
4
5
6
7
8
9
114.2
220.4
244.3
276.4
424.7
480.7
700.2
891.6
945.4
10110
28770
15.1±0.8
10.5±0.5
10.1±0.5
9.2±0.5
7.6±0.4
7.0±0.4
6.21±0.4
5.3±0.3
4.7±0.2
1.4±0.1
0.84±0.04
Ph3CH
Me2Al(bht)
bis-BHT
MeAl(bht)2
(tBuPh)Ind2ZrMe2
POSS
Al(bis-bht)Al
PS 1
10
11
PS 2
12
13
14
Cp2ZrMe2
(Me)Ind2ZrMe2
POSS-Ti-POSS
251.5
463.3
1828.1
12.5±0.4
8.3±0.5
4.0±0.2
2.40
2.66
3.26
-8.90
-9.08
-9.40
161±24
349±51
1415±208
Data ordered according to increasing MW. Dt measured in toluene-d8 at 298K. 5% experimental error assumed on Dt,[1f, 14]
implying 15% uncertainty on VH and calculated MW. Dt values in 1010 m/s2, MW in g/mol.
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instructive to recall the example reported in Figure 1 (see also
Entries 1-3, Table 1). A single atom is replaced going from
Cp2TiCl2 to Cp2ZrCl2 and Cp2HfCl2, and this results in a
significant variation of molecular density. This implies that, in
order to obtain reliable molecular information, the MWs of these
three complexes should be estimated separately by using three
specific calibration curves for titanocenes, zirconocenes and
hafnocenes, respectively.
Conclusions
Although diffusion NMR techniques are powerful tools to
investigate molecular size in solution, some caution is required
when quantitative information is desired. In this paper, the
performances of two established protocols to estimate VH or MW
of target molecules from their Dt have been compared,
highlighting their strengths and limitations. Practical examples
have allowed to recall the basic principles describing
translational self-diffusion of molecules in solution, the most
important being that particles with same dimensions diffuse at
the same rate independently from their mass. Starting from this
crucial observation, the analysis of ad hoc selected case studies
has allowed to translate more elaborate theoretical concepts in
simpler prescriptions for routine applications.
Obviously, some flexibility can be acceptable in the selection of
the reference species, depending on the type of problem that
has to be addressed. In some case, it has been shown that
chemically useful information can be obtained also from
correlations established among
a quite diversified set of
molecules,[8a, 11a, 12, 15a] but to what extent can this be done? The
answer to this question largely depends on the specificity of the
case study. However, a general, important point to make is that,
even if a good linear correlation is obtained for certain group of
molecules, it does not imply that the resulting plot can be used
as a reliable calibration curve.
The take-home messages can be summarized saying that
deriving rH and VH, by using the SE equation, is preferable when
molecules with non-uniform molecular density are investigated.
In many cases, this approach can be highly simplified by
assuming a spherical shape of the studied species; the resulting
For instance, the species reported in Figure 3a have been
selected to construct the Log(Dt) vs Log(MW) curve shown in
Figure 4a. An excellent linear fitting has been obtained (R2
0
VH/VH ratios are generally reliable parameters to quantitatively
>
estimate dimerization or trimerization processes, and higher
aggregation or polymerization degrees provided that the
molecular shape of the monomer is reasonably retained. The
determination of MW from Dt using Equations (2) or (5) is
instead preferable when known homologues of the analyte can
be used as reference compounds, and it is especially convenient
for species deviating significantly from the spherical
approximation and/or having peculiar factors determining Dt (e.g.
linear macromolecules).
0.998), despite the various nature of the reference molecules
spanning a very wide range of MW (Entries 1-11, Table 2).
Within a reasonable degree of approximation, only the subgroup
including the small organic molecules (e.g. Ph3CH, BHT, bis-
BHT) and the Al-complexes (e.g. Me2Al(bht), MeAl(bht)2,
Al(bis-bht)Al) can be assumed to have comparable shape and
density, since variations in molecular shape are not dramatic
(vide supra) and ρ should not be affected considerably by the
presence of ‘light’ metal centers like the aluminium ones.
Conversely, the same considerations hardly apply to the other
species in the calibration set.
Experimental Section
To gain further insight in this correlation, zirconocenes Cp2ZrMe2
(251.5 g/mol) and (Me)Ind2ZrMe2 (463.3 g/mol), and POSS-Ti-
POSS (1828.1 g/mol) in Figure 3b can be considered as suitable
stress tests for this calibration curve, as they are analogous to
the reference species (tBuPh)Ind2ZrMe2 and POSS(OH)2.[33]
The estimated MW for these three analytes are found to be 161
± 24, 349 ± 51 and 1415 ± 208 g/mol, respectively, differing
significantly from the expected values (Entries 12-14, Table 2). A
graphical analysis of results can help rationalizing these
observations. Figure 4b compares the calibration curve of Figure
4a (squared points) with the Log(Dt) vs Log(MW) plots obtained
for the aforementioned three dimethyl zirconocenes (red points)
and two silsesquioxanes (blue). As they are established among
homologous species, the latter two linear correlations are
definitely more reliable than the former, although the limited
number of experimental points provide only semi-quantitative
information. By analysing the graph in Figure 4b, it can be
concluded that (tBuPh)Ind2ZrMe2 and POSS(OH)2 are found to
correlate with the other species in Figure 3a just accidentally,
because it happens that they lie at the intersections between the
curve of Figure 4a and those specific for zirconocenes and
silsesquioxanes, respectively. Analogous considerations likely
apply to the two PS polymers, the Dt of which cannot be directly
compared to those of small molecules, as discussed in the
previous paragraphs.
Manipulations of air-sensitive compounds were performed under rigorous
exclusion of oxygen and moisture in flame-dried Schlenk-type glassware
interfaced to a high-vacuum line (< 10-5 Torr), or in a nitrogen-filled
MBraun glovebox (<0.5 ppm O2). Molecular sieves (4Å, MS) were
activated for 24 h at ca. 200-230 °C under dynamic vacuum. Toluene-d8
was freeze-pump-thaw degassed on a high-vacuum line, dried over Na/K
alloy, vacuum-transferred to a dry storage-tube with a PTFE valve and
stored over activated MS in the glovebox. Bis-cyclopentadienyl
complexes, triphenylmethane, POSS, trimethylaluminum, styrene, BHT
and bis-BHT were purchased from Sigma-Aldrich, while Ti(OiPr)4 from
Abcr and PS1-PS2 from Agilent Technologies, and used without further
purifications.
Me2Al(bht),[34]
MeAl(bht)2,[34]
(Me)Ind2ZrMe2,[35]
(tBuPh)Ind2ZrMe2[36] and POSS-Ti-POSS[19] were synthesized according
to previously established procedures.
NMR experiments were performed using a Bruker Avance III HD 400
instrument equipped with a smartprobe (400 MHz for 1H) with a z
gradient coil. 1H NMR spectra were referenced to the residual protons of
the deuterated solvent used; 13C NMR spectra were referenced internally
to the D-coupled 13C resonances of the NMR solvent. To describe the
multiplicity of the signals, the following abbreviations are used: s, singlet.
DOSY and PGSE NMR measurements were performed at 298K in
toluene-d8 by using the standard double-stimulated echo pulse sequence
without spinning. The shape of the gradients was rectangular, their
duration (δ) was 4 ms, and their strength (G) was varied during the
experiments. All the spectra were acquired using 32K points, between 16
and 128 scans, a spectral width of 6400 Hz, an acquisition time of 0.5 s
and a relaxation delay of 1 s per transient. The DOSY spectra were
acquired using 128 or 256 increments and processed by means of the
Bruker Dynamics Center software package (version 2.5.5), using a line
broadening of 1.0 Hz in the direct dimension. The DOSY maps were
obtained by using the Inverse Laplace Transform routine and choosing
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10.1002/chem.201900812
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128 or 256 points in the vertical dimension. Toluene-d8 was used as
internal standard to account for temperature and viscosity fluctuations; its
Dt was calibrated using an external sample of HDO in D2O[37] under the
same exact conditions. Dt data were treated as described in the literature
in order to derive the hydrodynamic dimensions.[1f, 14] In the cases where
the aliphatic resonance of some species overlap with that of toluene-d8,
self-diffusion coefficients were calculated with respect to that of Ph3CH.
The Dt of Ph3CH in toluene-d8 was separately calibrated using the
diffusion coefficient of residual solvent resonance in a dilute solution (0.1
mM). The obtained Stejskal-Tanner[38] plots are reported in the
Supporting Information.
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Synthesis of Al(bis-bht)Al. TMA (3 µL, 20 µmol) was added to a
solution of BHT (4.4 mg, 20 µmol) in toluene-d8 (1.3 mL). This reaction
proceeds rapidly giving Me2Al(bht), along with minor amounts of
MeAl(bht)2 and residual TMA.[34] Upon addition of bis-BHT (6.4 mg, 15
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while Me2Al(bht) remains unchanged and TMA forms small amounts of
oligomeric Al-phenolate complexes. The minor side products do not
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represent
a
significant hurdle for accurate and complete NMR
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characterization of Al(bis-bht)Al , as well as for its analysis by DOSY
NMR (accomplished by monitoring diagnostic signals being well
separated from those of other species, e.g. CH2 protons).
1H NMR (400 MHz, toluene-d8):
7.32 (s, H3), 7.15 (s, H3‘), 4.06
(s, H1), 2.32 (s, H1‘), 1.60 (s, H6),
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1.59 (s, H6‘), -0.24 (s, H8) ppm.
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13C NMR (100 MHz, toluene-d8):
152.2 (C7 and C7‘), 138.1 (C4
and C4‘), 131.9 (C2), 127.6 (C2‘),
125.8 (C3), 125.7 (C3‘), 41.7 (C1), 34.8 (C5 and C5‘), 31.6 (C6 and C6‘),
21.0 (C1‘), -9.5 (C8) ppm.
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Acknowledgements
[16] It is worth emphasizing here that, even though Dt
∝
VH,
a
direct
comparison of Dt vs Dt0 is not necessarily equivalent to consider VH/VH
ratios, as significant variations in fs and especially c parameters going
from the monomer to the polymer/aggregate would be not accounted for
properly in the former case.
0
This work has been financially supported by PRIN 2015
(20154X9ATP_004), University of Perugia and MIUR (AMIS,
“Dipartimenti di Eccellenza - 2018-2022” program). The authors
thank the group of Prof. A. Z. Voskoboynikov and Dr. D. V.
Uborsky from the Lomonosov Moscow State University, Russia,
for kindly providing bis-indenyl metallocene samples. F.Z. thanks
INSTM and CIRCC for a post-doc grant.
[17] a) For instance, the Perrin’s equation (ref. 17b) allows to derive accurate
rH for prolate (cigar-like) and oblate (disk-like) spheroidal particles,
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elaborated approaches are necessary for species having three different
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T.; Wang, X.-Q.; Cai, J.; Xu, B.; Wang, M.; Liu, C.; Yang, H.-B.; Li, X., J.
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Keywords: DOSY, PGSE, hydrodynamic volume, molecular
weight, Stokes Einstein
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Francesco Zaccaria,* Cristiano
Zuccaccia, Roberta Cipullo and Alceo
Macchioni
Page No. – Page No.
Extraction of reliable molecular
information from diffusion NMR
spectroscopy: hydrodynamic volume
or molecular mass?
Diffusion NMR techniques can be used to determine the size of molecules in
solution. Selected examples allow the basic principles underlying translational self-
diffusion to be recalled and derive practical lessons for the accurate manipulation of
self-diffusion coefficients (Dt).
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