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THE JOURNAL OF CHEMICAL PHYSICS 134, 044907 (2011)

Precise dipolar coupling constant distribution analysis in protonmultiple-quantum NMR of elastomers

Walter Chass,1,a) Juan Lpez Valentn,2 Geoffrey D. Genesky,3 Claude Cohen,3 andKay Saalwchter1,b)1Institut fr Physik NMR, Martin-Luther-Universitt Halle-Wittenberg, Betty-Heimann-Str. 7, D-06120Halle, Germany2Institute of Polymer Science and Technology (CSIC), C/ Juan de la Cierva 3, 28006 Madrid, Spain3School of Chemical and Biomolecular Engineering, Olin Hall, Cornell University, Ithaca, New York 14853,USA

(Received 7 September 2010; accepted 15 December 2010; published online 27 January 2011)

In this work we present an improved approach for the analysis of 1H double-quantum nuclear mag-netic resonance build-up data, mainly for the determination of residual dipolar coupling constantsand distributions thereof in polymer gels and elastomers, yielding information on crosslink den-sity and potential spatial inhomogeneities. We introduce a new generic build-up function, for useas component fitting function in linear superpositions, or as kernel function in fast Tikhonov reg-ularization (ftikreg). As opposed to the previously used inverted Gaussian build-up function basedon a second-moment approximation, this method yields faithful coupling constant distributions, aslimitations on the fitting limit are now lifted. A robust method for the proper estimation of the errorparameter used for the regularization is established, and the approach is demonstrated for differ-ent inhomogeneous elastomers with coupling constant distributions. 2011 American Institute ofPhysics. [doi:10.1063/1.3534856]

I. INTRODUCTION

The precise analysis of elastomer or gel components andmicrostructure is an important challenge for polymer physics,as this information is key to understanding their intriguingproperties in applications as performance materials, separa-tion membranes, and many more. In the recent years, simpletime-domain 1H solid-state NMR, possibly performed on sim-ple low-field instruments, was demonstrated to be a powerfuland versatile tool to investigate quantitatively the crosslinkdensity of polymer networks.18 The NMR effect is due tothe restrictions to fast segmental fluctuations posed by thecrosslinks and other topological constraints such as entangle-ments, leading to nonisotropic orientation fluctuations. There-fore, intrasegmental dipoledipole couplings are not averagedout completely and residual dipolar couplings (RDC) per-sist, which are directly related to the crosslink density andthus the network structure.911 Residual dipolar couplings aremost simply reflected in the T2 relaxation behavior,9 and de-spite the ambiguities related to the unknown shape of Hahn-echo decay curves,12 attempts have been made to extract evenRDC distributions and, thus, information on crosslink densityinhomogeneities.13

Reliable numerical approaches to the determination ofdipolar coupling constant distributions have for instance beenreported for the case of heteronuclear couplings as obtainedfrom the REDOR (rotational-echo double-resonance) NMRexperiment,14, 15 or for the case of electronelectron dipo-lar couplings from pulsed-electron paramagnetic resonance

a)Electronic mail: walter.chasse@physik.uni-halle.de.b)Electronic mail: kay.saalwaechter@physik.uni-halle.de.

data.16, 17 These situations are favorable in that experimen-tal data are composed of a sum of relaxation-free single-pairresponses, which follow the theoretical prediction more orless exactly. The situation is less favorable for homonucleardipolar couplings in systems with abundant spins, since theresponse of such a multispin system is homogeneous in na-ture and cannot be treated by analytical theory. Our previ-ous work, as reviewed in Ref. 5, has demonstrated that static1H multiple-quantum (MQ) NMR spectroscopy is the tool ofchoice to quantitatively measure homonuclear RDCs and theirdistribution in soft materials.3 Up to now, we have used ageneric yet rather approximate single-RDC Gaussian signalfunction based upon a second-moment approximation. Here,we present a substantially improved approach based on a newgeneric signal function and a reliable protocol for data analy-sis, yielding reliable RDC distributions.

The clear advantage of the MQ NMR experiment arisesfrom the acquisition of two different sets of data, i.e., adouble-quantum (DQ) build-up curve IDQ(DQ) and a refer-ence intensity decay curve Iref(DQ), see Fig. 1. The experi-ment is based upon a pure DQ Hamiltonian which generallyexcites all even quantum orders (thus the nomenclature MQNMR). However, the initial rise of the build-up function isdominated by DQ coherences (thus its nomenclature DQ), yetat longer pulse sequence times DQ, it also comprises higher2n + 2 quantum orders. Similarly, the initial decay of the ref-erence curve is dominated by dipolar-modulated longitudinalmagnetization (quantum order 0), yet contains contributionsfrom higher 2n quantum orders at long times.

With the two signal functions, it is possible to ob-tain structure information about the polymer network in-dependently of relaxation effects by normalizing the DQ

0021-9606/2011/134(4)/044907/10/$30.00 2011 American Institute of Physics134, 044907-1

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

http://dx.doi.org/10.1063/1.3534856http://dx.doi.org/10.1063/1.3534856http://dx.doi.org/10.1063/1.3534856mailto: walter.chasse@physik.uni-halle.demailto: kay.saalwaechter@physik.uni-halle.de

044907-2 Chass et al. J. Chem. Phys. 134, 044907 (2011)

FIG. 1. Experimental double-quantum (IDQ), reference (Iref), and normal-ized DQ (InDQ) intensities for the natural rubber sample NR-C8, as well asthe result of a fit of the latter with Eq. (3) yielding DG/2 = 595 Hz andG/2 = 166 Hz. The dashed lines indicate the fitting limit. Note that theuncertainty for InDQ increases at long times due to the division of increas-ingly small quantities.

buildup through point by point division by the sum relaxationfunction IMQ = IDQ + Iref, possibly after subtraction of(usually exponential) long-time signal tails related to networkdefects or solvent: InDQ = IDQ/(IMQtails). The resultingnormalized DQ build-up function must reach a relative am-plitude of 0.5 in the long-time limit for theoretical reasons(equal partition among all excited even quantum orders) andis to a good approximation independent of temperature. Thestructural information in the form of RDCs is obtained by fit-ting such data to appropriate functions.

As mentioned, up to now we have used an approxi-mated build-up function based on a static second-momentapproximation:3, 11, 18

InDQ(DQ, Dres) = 0.5(1 exp { 25 D2res 2DQ}) , (1)

where Dres is an apparent RDC characteristic of the wholemonomer unit representing of course the averaged action ofmany pair couplings. This build-up function was shown to fitvery well both simulated data of multispin systems as wellas data measured on homogeneous single-component elas-tomers. In the latter, the segmental fluctuations are all subjectto the same crosslink induced anisotropy, and a discussion ofthis a priori unexpected finding and the polymer-physical im-plications is found in Ref. 19.

In case of RDC distributions, i.e., in inhomogeneouspolymer networks with broad or even multimodal distributionof RDCs, such as in swollen polymer networks or networkswith spatially separated bimodal or multimodal chain lengthdistributions, the above function does not give a proper fit,because it only considers a single RDC constant Dres. Theresponse is then generally given by a Fredholm integral equa-tion (a distribution integral):

g(DQ) =

0K [Dres, DQ] f (Dres)dDres, (2)

where the function g(DQ) represents the measured data andf (Dres) is the RDC distribution. Taking Eq. (1) as (approxi-mate) kernel function K [Dres, DQ] and assuming that the dis-tribution is Gaussian with the average RDC denoted as DGand the standard deviation G, one can obtain an analyticalfitting function:3

InDQ(DQ, DG, G) = 12

1

exp

{

25 D

2G

2DQ

1+ 45 2G 2DQ

}

1 + 45 2G 2DQ

, (3)

which applies for moderate distributions with G < DG. Notethat from now on, Eq. (1) is referred to as Gaussian fittingfunction, while Eq. (3) is the Gaussian-distributed fittingfunction.

More generally, one may try to fit linear superpositionsof Eqs. (1) or (3) in order to model a multimodal distributionor resort to a numerical inversion procedure. The latter meansthat Eq. (2) has to be inverted to obtain f (Dres), which is anill-posed problem, related to the well-known case of the in-verse Laplace transformation. The program ftikreg,20, 21 basedon the fast Tikhonov regularization algorithm, combined withthe Gaussian build-up function [Eq. (1)] as kernel function,was until now our method of choice to obtain an estimate ofthe distribution function.

However, one needs to keep in mind that Eq. (1) ap-proximates simulated as well as true data only for nDQ in-tensities up to 0.45, corresponding to a time limit of maxDQ= 2.4/Dres, see Fig. 1, resulting in systematic errors in thecomponent fractions or the distribution shape obtained byftikreg, in particular for networks with a dominating high-Drescomponent. In addition, the regularization result is sensitivelydependent on a user-defined error parameter , which shouldideally reflect the uncertainty for each data point. This un-certainty, however, is not constant for normalized InDQ data,as the noise-related error in the two experimental functionsis constant, but its relative importance increases at long timesdue to division of small quantities. Due to these limitations tonumerical regularization, in most of our previous papers weus