Vij D.R. Handbook of Applied Solid State Spectroscopy

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1. Nuclear Magnetic Resonance Spectroscopy

An example of motion that produces changes in the isotropic chemical

shifts is the ring S-ringflips in poly(p-vinylene phenylene) (PPV). Figure

1.17b shows the 2D-MAS exchange of PPV acquired at 258 K with a

spinning frequency of 7.5 kHz. The presence of hetero-cross-peaks linking the

2,5 and the 3,6 lines is remarkable, indicating the occurrence of an exchange

process where the isotropic chemical shifts of the 2,5 and 3,6 carbons are

interchanged. Since the isotropic chemical shift of the 2,5 and 3,6 carbons

depends on the conformation of these segments relative to the vinylene

carbon, the interchange in the isotropic chemical shift can be attributed to a

180q rotation of the phenylene rings around the 1,4 axis. In the particular case

of PPV, slow conformational dynamics are directly associated with chain

disorder, which affects its photoluminescence behavior [109]. Some other

examples of applications of rotor synchronized 2D MAS exchange NMR are

the study of conformational dynamics in amorphous and semicrystalline

polymers, liquid crystals and the study of tautomerism, and molecular

dynamics in molecular crystals [110–112].

2D exchange NMR experiments have also been used for studying local

structure of amorphous solids. This is possible because for dipolar coupled

spins 2D exchange NMR is capable of probing the relative orientation of the

principal axis of the individual chemical shift tensors, which can be translated

in terms of molecular torsion angles. In these cases the exchange process is

not governed by molecular motions, but by local magnetization exchange,

known as spin-diffusion, between directly bonded nuclei [113]. In fact, in

studies of molecular reorientation, spin-diffusion is a limiting factor, because

its effects can be confused with the exchange due to molecular motion. For

this reason, low abundant and low J nuclei such as

C or

H are preferable for

probing molecular rotations, but the mixing times have to be kept short

(usually smaller than 1 s) to avoid spin-diffusion effects [114]. However, in

systems of highly abundant nuclei the dominant exchange process will be

spin-diffusion. Since the NMR frequencies are orientation-dependent, if a

nucleus in a given molecular segment exchanges its magnetization with

another nucleus in a neighbor site during t

, it will experience different

frequencies before and after t

. The shape of the resulting 2D pattern will

depend on the relative orientation of the two molecular segments. However, it

is very important to guarantee that spin-diffusion will be the only exchange

mechanism, i.e., that no slow motion takes place during t

. 2D exchange has

been used for determining relative orientations between different carbon sites

in many organic materials [115–118]. For that, the use of a

C-

C or

N-

pair is necessary, which has been provided by specific

C or

N isotopic

labeling at the sites of interest. Other applications of the method not requiring

isotopic labeling are those where

P nuclei are used to probe relative

molecular orientations.

P is a spin 1/2, 100% abundant nucleus, found in

many organic and inorganic systems. An example of these studies is the

measurement of relative tensor orientations of different phosphate units (Q

1.7 Molecular Dynamics and Local Molecular Conformation in Solid Materials

and Q

units in NMR terminology [9]); see molecular structure in Figure 1.18

in crystalline diphosphate Ca

and in binary sodium phosphate glasses

[119].

Figure 1.18 a) 2D exchange spectra using slow MAS for measuring the relative tensor

orientations for a diphosphate Ca

. b) Same as a) for a binary sodium phosphate glass with

sodium to phosphorous ratio of 1:4. Molecular representation of the relative orientations of Q

and Q

-Q

connectivities and their anisotropic powder patterns are also shown. (Adapted

In this case it is advantageous to use a slightly different version of the 2D

exchange NMR, where slow MAS is applied together with radio frequency

pulses in such a way that the local transfer of magnetization is enhanced

during the mixing time (radio frequency–driven spin-diffusion — RFDR)

[113]. For crystalline diphosphate Ca

, which is composed of only Q

units, a single exchange pattern is obtained, Figure 1.18a. From this spectrum

the relative orientation of two neighbor phosphate units can be estimated. In

contrast, for the binary phosphate glass, where Q

and Q

units are present, the

2D exchange spectrum is a superposition of the individual spectra for Q

- Q

, Q

- Q

, according to their statistical weights; see Figure 1.18b. In this

case, because of the spectral superposition, it is difficult to determine

unambiguously the relative tensor orientation for each connectivity, but the

spectrum clearly reveals the predominance of Q

-Q

connectivities in the

glass.

1.7.3 One-Dimensional Exchange NMR Experiments

Beyond the 2D exchange NMR methods, there is a class of one-dimensional

(1D) techniques that have been used for extracting two- and multiple-time

correlation functions of rotational molecular motions occurring on the

millisecond-to-second time-scale [39, 75, 77, 80, 81, 102, 120, 121]. In

particular, the stimulated-echo experiment (STE) [35, 39, 77] has been widely

employed for studying time scale, amplitude, and heterogeneity of slow

motions in organic and inorganic solids. The pulse sequence for the basic

stimulated-echo experiment is shown in Figure 1.19a. After the excitation

pulse, the quantum coherences (or magnetization components) evolve during

the evolution period W under the action of an anisotropic NMR interaction

(usually chemical shift or quadrupolar). A store pulse is then applied with

1. Nuclear Magnetic Resonance Spectroscopy

adequate phase and amplitude in order to convert a selected single-quantum

coherence that evolved during W into a term that does not evolve during the

long mixing time, because it commutes with the spin Hamiltonian (usually it

is represented by T

— spin-alignment , or I

— Zeeman order). Finally, a read-

out pulse is applied to the system to reconvert the stored coherence into an

observable magnetization. Using the pulse sequence shown in Figure 1.19a

and disregarding relaxation effects, the NMR signal can be written as















Ĳ,, sinȦ 0 Ĳ sin ȦStt A tt ,

(1.87)

where



Ȧ 0 and





Ȧ t are the NMR frequencies before and after t

. The

amplitude A depends on the flip angles and phases of the store and read-out

pulses, which are chosen according to the nuclear spin interaction used for

probing the molecular motion [77]. The maximum value of





Ĳ,,Stt occurs

at t = W (echo maximum) and depends on the relation between



Ȧ 0 and





Ȧ t . The decrease of the echo amplitude as a function of t

provides the

two-time correlation function of the exchange process. The number of sites

involved in the exchange process also affects the stimulated-echo, in

particular its intensity at long mixing times. The longitudinal relaxation, T

also produces echo decay as a function of the mixing time, which may happen

in the same time scale of the decay due to molecular motion. Thus, to

guarantee that only motion effects are encoded in the stimulated-echoes, the

data must be normalized by the T

decay, which can be obtained by setting the

evolution time W = 0 and acquiring the echo intensities as a function of t

Corrections for T

relaxation during the W period are performed by

normalizing the echo intensity by its value at short mixing times. STE

experiments were extensively used to study molecular dynamics in organic

systems. Most of these applications used

H STE, where the anisotropic

quadrupolar interaction is employed to probe the molecular motion. In this

case spin-alignment echoes (the T

term is stored during t

) are produced and

the expression for the STE signal is



















em 0 1 2 Q e Q m

,sin2Ȥ sin 2Ȥ sin Ȧ 0sinȦStt t M t t t ,

(1.88)

where it becomes clear that the pulse flip angles

Ȥ and

Ȥ that maximize the

signal are each equal to 55q.

H spin-alignment echoes have been used for many years in the study of D-

and E-relaxation in supercooled liquids [35, 97, 122, 123]. Important informa-

tion about the motional rates, motional geometry, and distribution of

correlation times have been obtained using this technique. New experiments

obtained by adding extra mixing times and evolution periods in the

stimulated-echo pulse sequence have also made it possible to obtain multiple

time correlation functions. These techniques were used to elucidate the origin

of the non-exponential behavior of the correlation functions that characterize

the molecular motions responsible for the glass transition of supercooled

systems [73, 74, 78, 95, 123, 124]. The results revealed that non-exponen-

1.7 Molecular Dynamics and Local Molecular Conformation in Solid Materials

tiality in the correlation functions are associated with dynamic hetero-

geneities with lifetimes comparable to that of the D-relaxation, and their sizes

were estimated [74, 124]. More recently, STE experiments were also applied

to study the slow exchange of the

Li and

Be cations in inorganic solid

electrolytes [39, 78, 125]. In these cases the experiment was adapted to spin

3/2 systems. With the appropriate phase cycling [125], the echo intensity can

be written as













m0Q Qm

Ĳ,, sinȦ 0 Ĳ sin ȦStt M tt

(1.89)

for best pulse flip angles F

and F

equal to 45q. The phase cycling employed

in reference [78] was used to provide pure quadrupolar order, stimulated-

echoes (spin-alignment echoes) as in the case of

H. However, homonuclear

dipolar coupling between the cations also contributes to the echo intensity.

This is shown in Figure 1.19b, where the regular solid-echo spectra of the

ionic conductor Li

(PO

)

are compared with the spin-alignment spectra

(Fourier transform of the STE signal) recorded with t

= 50 ms. It is evident

that the satellite contributions in the spin-alignment spectra are much more

prominent than in the solid-echo spectra. However, the presence of significant

central lines indicates that dipolar order is also produced.

Figure 1.19 a) Pulse sequence for stimulated-echo experiments. b) Solid-echo spectra of

(PO

)

(left) as compared with

Li spin-alignment echo spectra (right) at two

temperatures. c) t

dependence of the

Li spin-alignment echoes amplitudes in the Li

(PO

)

ionic conductor. The activation energy barrier for the Li hopping was estimated as 11,300 K/k

Fortunately, at short evolution times W the echo intensity is dominated by

quadrupolar order, making it possible to select pure (or mostly pure)

quadrupolar order echoes [78]. Because the quadrupolar frequency depends

on the strength and orientation of the electric field gradient at the cation site,

any modification in these parameters’ changes Z

. As a consequence, jumps

of the ions between electrically non-equivalent sites change the quadrupolar

frequency, making it possible to detect the cation motion in STE experiments.

Again, the measurement of the stimulated-echo amplitude as a function of the

mixing time provides the two-time correlation function of the ionic motions,

whose decay time is the average ion-hopping time. Figure 1.19c shows the

mixing time dependence of the spin-alignment echo amplitudes for different

1. Nuclear Magnetic Resonance Spectroscopy

temperatures in the Li

(PO

)

ionic conductor. From these measurements

the mean correlation times (decay times) as a function of the temperature and

the activation energy of the ionic motion were obtained [78]. Other STE

studies on ionic conductors have been carried out using

109

Ag (spin ½

nucleus) NMR. In this case two-time and three-time correlation functions

were obtained, which allowed one to conclude that the non-exponentiality

observed in the correlation function is due to a correlation time distribution,

i.e., dynamic heterogeneities, rather than to an intrinsic non-exponentiality

[126].

Concerning

C stimulated-echo experiments, these have been widely

employed for studying not only the time scale, but also the amplitude of slow

motions in organic solids. However, the use of this technique for studying

complex motions suffers from the same problem as the regular 2D exchange

experiment, i.e., there is no specific site resolution, and it is very difficult to

characterize the exchange intensity if only a small fraction of the molecular

segments takes part in the molecular motion. To overcome one of these

problems a 1D pure exchange version of the stimulated-echo technique has

also been developed. This technique, named 1D pure exchange NMR (1D-

PUREX), suppresses the signals of immobile molecules and thus provides

spectra selectively from segments that reorient during the mixing time. Thus,

the amplitude and time scale of slow motions can be determined even for a

small fraction of mobile segments [101, 102]. The pulse sequence for the 1D-

PUREX NMR experiment is obtained from the 2D-PUREX pulse sequence,

Figure 1.16, simply by removing the t

evolution and the first z-period (the

two 90q pulses flanking the t

period). Hence, after the excitation the spins

immediately start their evolution under the chemical shift anisotropy (CSA)

during the period W. Then, the magnetization is stored along the z-direction, so

that it does not precess or dephase during a long mixing time, t

. In the

absence of slow motions during t

, the CSA is refocused after a read-out

pulse and another evolution period W (total evolution time of W

CSA

= 2W). At

this point, the magnetization is stored again along the z direction during a

short period t

(<< t

) and then flipped back to the transverse plane for

detection. If segmental reorientations occur during t

, the orientation-

dependent frequency changes, and the CSA is not completely refocused. The

resulting dephasing of the stimulated-echo is observed as a decrease in the

detected spectral intensity. To remove effects of T

and T

relaxations during

and W

CSA

, respectively, a reference spectrum S

= S(t

, GW

CSA

)—which has

the same relaxation factors, but no motion effects during t

—is measured. S

is obtained by simply swapping t

and t

. To remove the contribution of

segments that do not move on the milliseconds time scale, the 1D-PUREX

intensity S = S(t

, GW

CSA

) is subtracted from the reference spectrum S

Dividing this difference by S

, one can obtain a normalized pure-exchange

intensity E(t

, GW

CSA

) = 'S/S

, which does not contain relaxation effects and

just accounts for the signals from the reorienting segments. For discrete jumps

1.7 Molecular Dynamics and Local Molecular Conformation in Solid Materials

with correlation times W

in the range of milliseconds to seconds (slow-motion

regime), the normalized pure-exchange intensity E(t

, GW

CSA

) obtained by

integrating the spectra yields the same information as the standard stimulated-

echo experiment. In this motional regime, E(t

, GW

CSA

) can be approximated

by the following expression:











CSA

mCSA m 1 2

CSA

m12

,įĲ 12sin

12sin ,

Et t



§·

ªº

I ::

¨¸

¬¼

©¹

ªº

§·

ªº

I ::

«»

¨¸

¬¼

©¹

¬¼

(1.90)

where f

is the fraction of slow mobile segments (mobile fraction), M is the

number of magnetically non-equivalent sites accessible to the moving

segments, I(t

) is the correlation function of the molecular motion, and :

–

is the difference between the NMR frequencies before and after t

Therefore, the t

-dependence of E( t

CSA

) yields the time scale of the

motion, while its W

CSA

-dependence provides information on the amplitude of

the slow molecular reorientations that occurred during t

Figures 1.20a–b show, respectively, the t

- and the W

CSA

-dependence of

E(t

, GW

CSA

) curves for DMS at 293 K. A correlation time of 15 ms was

obtained from the t

-dependence of the E(t

, GW

CSA

) curves. Note that the final

intensity in both curves is 0.5, which is exactly what is expected for the DMS

two-site jump motion. Besides, as shown in the simulation for motions with

distinct reorientation angles, the 1D-PUREX experiment is very sensitive to

small-angle rotations. The best fit to the DMS experimental data occurs for a

reorientation angle of 70q. This is equivalent to 110q due to the symmetry

property of the axially symmetric CSA tensor of DMS that makes

reorientation by E

and 180q–E

equivalent. Figure 1.20c shows the reference

, the 1D-PUREX S, and the difference S

– S spectra obtained for non-

labeled PMMA at T = 295 K. The S

– S spectrum shows significant intensity

for all carbons confirming that the E-relaxation of PMMA involves both the

side group and main chain motions. The W

CSA

-dependence of E(t

, GW

CSA

) for

the carbonyl group in PMMA is shown in Figure 1.20d. The bimodal

character of the curve is promptly recognized, showing the presence of both

small (slow build-up) and high (fast build-up) angle motions. The best-fit

simulation of the E(t

, GW

CSA

) 1D-PUREX intensities as a function of W

CSA

also shown. In the simulations, the three-component model discussed above in

the context of Figure 1.13 was used. All small-angle rotations were simulated

using Gaussian distributions of reorientation angles centered at 0q with root

mean square widths V. The E(t

, GW

CSA

) vs. W

CSA

curves for each component

and the corresponding distribution of rotation angles for the small-angle

motion are shown in Figure 1.20d.

1. Nuclear Magnetic Resonance Spectroscopy

Figure 1.20 a) E(t

, GW

CSA

) vs. t

curve of DMS at 293 K. b) E(t

, GW

CSA

) vs. W

CSA

curve of

DMS at 293 K. c) Reference S

, exchange S, and difference S

– S 1D-PUREX spectra of

PMMA. d) E(t

, GW

CSA

) vs. W

CSA

build-up curve for the carbonyl groups of PMMA together

with simulated curves based on the model discussed in the text (components I, II, III). (Adapted

Following the idea of the rotor synchronized 2D MAS exchange

experiments, there is also a series of 1D MAS experiments especially

designed for studying slow molecular reorientations using spinning sidebands

[80, 81]. Named one-dimensional exchange spectroscopy by sideband

alternation (ODESSA), this is an alternative version of the 2D MAS

exchange. In this method, the first evolution period in Figure 1.11c is set to

half of the rotor period, t

= t

/2. Then, after the rotor synchronized mixing

period, the signal is detected during the period t = t

. The magnetization stored

along the z direction during the mixing periods corresponds to distinct

polarizations of the spinning sidebands. During the mixing period, the

molecular reorientations redistribute the polarization among the various

spinning sidebands, which makes their intensities dependent on the molecular

motion that occurred during t

. From a series of spectra acquired with

different mixing periods, dynamic parameters such as the correlation time

and, in favorable cases, the amplitude and number of accessible sites for the

motion can be obtained. Other variants of these techniques were proposed in

order to obtain pure absorptive MAS spectra for systems with many non-

equivalent sites (time-reverse ODESSA) and to separate the sideband

intensities of different chemical groups (PATROS) [80, 81, 83]. In the

particular case of tr-ODESSA the correlation time of the motion can be

obtained either by the increase of the spinning sidebands intensities or the

decrease of the centerband intensity as a function of the mixing time.

1.7 Molecular Dynamics and Local Molecular Conformation in Solid Materials

Figure 1.21 shows an example of the use of the tr-ODESSA pulse

sequence to study molecular motions responsible for the D- and E-relaxation

of the glassy polymer poly(n-hexyl methacrylate) (PHMA). Using tr-

ODESSA, the correlation times and their distribution were obtained as a

function of temperature for the side group and main chain motion [92]. This

made it possible to conclude that the alkyl side group contributes in the

dynamic window of the NMR exchange experiments mainly to the E-

relaxation, while the main chain and carboxyl groups contribute to both D-

and E-relaxations.

Figure 1.21 a)

C CP-MAS spectrum of poly(n-hexyl methacrylate) (PHMA) at Q

= 3 kHz.

COO spinning sidebands are indicated with asterisks. b) t

-dependence of tr-ODESSA

American Chemical Society)

The main disadvantages of the above discussed MAS exchange methods is

the necessity of observing sidebands in the spectrum. This is particularly

problematic for

C studies in chemically complex systems where the overlap

between the sidebands of different chemical groups may preclude the accurate

intensity measurement. This was overcome by a class of methods called

centerband-only detection of exchange (CODEX), where slow molecular

motions are quantified on the basis of the line intensities of a high resolution

MAS spectrum, without the necessity of observing MAS sidebands. The

CODEX experiment is essentially the MAS version of 1D-PUREX NMR [75,

121]. The averaging of the chemical shift anisotropy under MAS provides

higher site resolution and sensitivity. However, MAS would make the pulse

sequence insensitive to segmental orientations and reorientations. Then, it is

necessary to reintroduce (recouple) the chemical shift anisotropy during the

periods used to codify the exchange process, i.e., the evolution periods, Nt

This is performed by using a train of rotor synchronized 180q (S) pulses

capable of reintroducing the chemical shift anisotropy even under high MAS

spinning frequencies. This scheme, known as recoupling of the CSA, was

originally developed for using in the so-called rotational echo double

resonance (REDOR) experiments, which are a very useful class of NMR

methods for measuring internuclear distances [47]. CSA recoupling during the

1. Nuclear Magnetic Resonance Spectroscopy

evolution periods is achieved by replacing each evolution period W in the 1D-

PUREX pulse sequence by a train of N/2 180q pulses spaced by t

/2, where t

is the sample rotation period; see Figure 1.22a. This leads to a total evolution-

and-refocusing period of Nt

for CODEX. With the CSA recoupling, the phase

accumulated during the first and second evolution periods before ()

) and

after ()

) the mixing time are given by

rr r

/2 /2

1 iso aniso,1 iso aniso,1 aniso,1

0/2 0

ȦȦ() ȦȦ() Ȧ ()

tt t

tdt tdt N tdt

§·

ªºªº

)     

¨¸

¬¼¬¼

©¹

³³ ³

rr r

/2 /2

2 iso aniso,2 iso aniso,2 aniso,2

0/2 0

ȦȦ () ȦȦ () Ȧ ()

tt t

tdt tdt N tdt

§·

ªºªº

)   

¨¸

¬¼¬¼

©¹

³³ ³

(1.91)

where

aniso,1

()tZ and

aniso,2

()tZ are the NMR frequencies due to the anisotropic

part of the chemical shift under MAS, equation (1.71), before and after t

respectively. Note that equation (1.73) was used to derive the expressions for

the phases. If Z

(t) = Z

(t), i.e., if no exchange occurs, equation (1.91) leads to

a total phase accumulation of )

+ )

= |)

| – |)

| = 0; in other words, the

magnetization is refocused in a stimulated-echo along its original direction.

Figure 1.22 a) Pulse sequence for CODEX experiments. b) CODEX S, reference S

, and the

difference S

– S, spectra obtained for the molecular crystal methylmalonic acid (left) and

difference S

– S, spectrum as a function of t

for PMMA (right) (see chemical structures in the

insets). c) t

-dependence of the pure-exchange CODEX intensity for PMMA, methylmalonic

acid, and DMS. d) Nt

-dependence of the E(t

, GNt

) intensity obtained for DMS using

1.7 Molecular Dynamics and Local Molecular Conformation in Solid Materials

Molecular motions during t

reduce the stimulated-echo and, as a result,

the observed line intensities. Therefore, the analysis of the motional amplitude

and time scale can be performed in the same way as for the 1D-PUREX

measurements. The CODEX intensity S = S(t

, GNt

) and the reference

intensity with no motion effects, S

= S(t

, GNt

), also obtained after swapping

and t

, are combined to produce the normalized pure exchange intensity,

E(t

, GNt

). E(t

, GNt

) plotted as a function of t

or Nt

provides information

on the motional rate and amplitude, respectively. The main advantage of

CODEX is the possibility of studying the molecular motion of each chemical

site independently, since a high resolution MAS spectrum is acquired. This

makes possible the probing of local motions at each individual site of

chemically complex systems. Figure 1.22b shows the CODEX S, the reference

, and the difference S

– S spectra obtained for the molecular crystal

methylmalonic acid and for the glassy polymer PMMA (see molecular

structures in the insets).

The absence of lines in the difference spectrum S

– S, of methylmalonic

acid indicates that the molecular segments of methylmalonic acid do not

undergo slow rotations. In contrast, molecular groups in both side groups and

main chain of PMMA execute molecular motions in the millisecond-to-second

time scale. Figure 1.22c shows the t

-dependence of the pure-exchange

CODEX intensity for PMMA, methylmalonic acid, and DMS. The resulting

curves are the correlation functions of these dynamics on the millisecond time

scale. For DMS, a single exponential curve is obtained, while PMMA shows

some non-exponentiality. Methylmalonic acid shows no exchange except for

C spin-diffusion (~5%). E(t

, GNt

) plotted as a function of Nt

for DMS is

shown in Figure 1.22d. Simulations with various reorientation angles are also

shown. As in the case of 1D-PUREX experiments, high sensitivity to small-

angle reorientations can be observed in these curves. As expected, the best fit

to the DMS experimental data is shown for a reorientation of 70q, i.e., 110q, as

explained before.

Many extensions of the CODEX technique have appeared, including an

extension to a four-time pulse sequence used for studying motional memory

[75, 127]; a triple-resonance CODEX applied to study protein dynamics

[128]; scaled-CODEX, used for measurements at low spinning rates [120];

multiple-quantum CODEX used for torsion angle determination in solids

[129]; and

F CODEX, was used to study the oligomerization in lipid bilayer

proteins [130]; and a CODEX variant to measure internuclear distances

between a spin ½ and a quadrupolar nucleus [131].

1.7.4

Conformation-NMR

The experimental elucidation of molecular conformations in non-

oriented solids, such as amorphous polymers, unoriented polypeptides, or

polycrystalline proteins, is a great challenge. Solidstate NMR provides a set of