This is the submitted version of a paper published in Journal of magnetic resonance.
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Williamson, N H., Nydén, M., Röding, M. (2016)
The lognormal and gamma distribution models for estimating molecular weight
distributions of polymers using PGSE NMR
Journal of magnetic resonance, 267: 54-62
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The lognormal and gamma distribution models for
estimating molecular weight distributions of polymers
using PGSE NMR
Nathan H. Williamsona,∗, Magnus Nyd´ena,b, Magnus R¨odinga,b,c aFuture Industries Institute, University of South Australia, Mawson Lakes Campus,
Adelaide, SA 5095, Australia.
bSchool of Energy and Resources, UCL Australia, University College London, Torrens
Building, 220 Victoria Square, Adelaide, SA 5000, Australia.
cSP Food and Bioscience, Frans Perssons v¨ag 6, 402 29 G¨oteborg, Sweden.
We present comprehensive derivations for the statistical models and methods for the use of pulsed gradient spin echo (PGSE) NMR to characterize the molecular weight distribution of polymers via the well-known scaling law relating diﬀusion coeﬃcients and molecular weights. We cover the lognormal and gamma distribu-tion models and linear combinadistribu-tions of these distribudistribu-tions. Although the focus is on methodology, we illustrate the use experimentally with three polystyrene samples, comparing the NMR results to gel permeation chromatography (GPC) measurements, test the accuracy and noise-sensitivity on simulated data, and provide code for implementation.
Pulsed gradient spin echo NMR Molecular weight distribution Self-diﬀusion Scaling law Lognormal distribution Gamma distribution Polydispersity 1. Introduction
The physical properties of polymers in solution are determined by the type of polymerization, degree of chain branching, and chain length distribution. An
∗Corresponding author: Nathan H. Williamson, Telephone: +61 (0) 8 8302 3331, Fax: +61 (0) 8 8302 3683, E-mail: firstname.lastname@example.org
Email addresses: email@example.com (Nathan H. Williamson ),
important property is the molecular weight distribution which provides infor-mation on the type of polymerization and the physical properties of polymer mixtures . Self-diﬀusion coeﬃcients correlate with the size distribution of the diﬀusion species and thereby with the molecular weight distribution. For many polymers, self-diﬀusion coeﬃcient D and molecular weight M are related by the scaling law 
D(M ) = KM−ν ⇔ M(D) = K1/νD−1/ν, (1)
where, in the dilute regime, ν∼ 0.5 − 0.8 depending on solvent quality. This comes from the Stokes-Einstein-Sutherland equation [4, 5], in which, through Stokes’ drag acting as a restoring force to the random thermal motion, D is inversely proportional the radius of the Brownian particle. Replacing the radius with a scaling relation between the polymer radius of gyration and molecular weight, Rg= aMν, one obtains Eq. (1). Here, ν is called the Flory exponent 
and describes how the polymer spreads in space. When derived for the polymer chain which takes a random-walk conformation, ν = 1/2, and if excluded-volume eﬀects are taken into account, ν = 3/5 (or, more precisely, 0.588 when calculated by renormalization theory ) .
The scaling law relating D and M opens up the ability to characterize molec-ular weight distributions of polymers using pulsed gradient spin echo (PGSE) NMR , a widely used technique for measuring diﬀusivity of solutes [9, 10], a topic which has attracted some attention since the 1980s. Fleischer  dis-cuss the eﬀects of polydispersity including relaxation-weighting. Chen et al.  determine molecular weight distributions using DOSY (diﬀusion-ordered spec-troscopy) NMR combined with nonparametric inverse Laplace transform (ILT) modeling of diﬀusion coeﬃcient distributions. H˚akansson et al.  discuss the commonly used lognormal distribution model for distributions of diﬀusion co-eﬃcients and molecular weights. Vi´eville et al.  also use DOSY NMR and a nonparametric ILT model. In a series of papers, Gong et al. [15, 16, 17] em-pirically relate a stretched exponential model for the NMR signal attenuation data to the molecular weight distribution. The stretched exponential model can provide some measure of polydispersity, but it is diﬃcult to interpret in terms of a distribution of diﬀusion coeﬃcients [18, 19]. Van Lokeren et al.  discuss determining the molecular weight distribution on binary and ternary polymer mixtures, taking relaxation eﬀects into account. In all, NMR is a promising technique for determining molecular weight distributions of polymers, and an interesting alternative to standard methods such as gel permeation chromatog-raphy (GPC) and dynamic light scattering (DLS). NMR does come with some culprits, though, such as the need for low concentration to avoid microscopic av-eraging, i.e., interactions between polymers with diﬀerent diﬀusion coeﬃcients eﬀectively changing the macroscopic distribution in diﬀusion coeﬃcients. At the microscopic level, the averaging arises from collisions between small polymers with rapid diﬀusive motion and large polymers with slow diﬀusive motion . Additionally, the concentration which is suﬃciently dilute to neglect microscopic averaging eﬀects decreases with increasing molecular weight, which may lead to issues arising from low signal-to-noise ratio . However, these disadvantages
are diﬀerent than those of GPC, for example the need for specialized columns for diﬀerent types of polymers and molecular weight ranges, and of DLS, for example the need for ultra-clean (dust-free) samples. Therefore, there may be situations where it is advantageous to use PGSE NMR methods of determining molecular weight distributions.
In this work, we provide the methods and derivations for the use of PGSE NMR to estimate the distribution of diﬀusion coeﬃcients and the correspond-ing molecular weight distribution of polymer samples. Though pulsed gradient stimulated echo (PGSTE) experiment are used, the methods can in principle be applied to the analysis of any diﬀusion measurement within the greater class of single and double PGSE methods. We cover the lognormal and gamma distri-bution models and linear combinations of these distridistri-butions. Both models have been used previously for modeling NMR signal attenuation data. However, we provide comprehensive derivations of the relevant number-weighted and mass-weighted distributions, which are of interest for polymer characterization, and provide Matlab source code for easy implementation (see Supplementary ma-terial). This research is aimed at those who wish to implement PGSE NMR methods of determining polymer molecular weight distributions based on log-normal and gamma distribution models. Although the focus is on methodology, we illustrate the use experimentally with three polystyrene samples, comparing the NMR results to gel permeation chromatography (GPC) experiments, and we perform two simulation studies, testing the accuracy and sensitivity of the models on data sets simulated from known polymer molecular weight distribu-tions.
2.1. General theory
For a monodisperse solute, i.e., for a single self-diﬀusion coeﬃcient, the signal attenuation in a PGSE NMR experiment is given by the standard Stejskal-Tanner equation
I(b) = I0exp (−bD) (2)
with self-diﬀusion coeﬃcient D, initial signal intensity I0= I(0), and the
inde-pendent variable b, which controls the sensitivity of the signal I(b) to diﬀusion [22, 23]. For a polydisperse solute, analysing the initial attenuation using Eq. (2) gives the mean diﬀusivity of the solute,⟨D⟩ . However, as attenuation is increased, Eq. (2) becomes invalid as the signal attenuation is not a single exponential. A common way to model the polydispersity is to use the stretched exponential model, I(b) = I0exp ( −(bD0)β ) , (3)
which is able to express polydispersity to some extent through the ’stretch’ parameter β [18, 19]. However, as has been stressed before, the relation between spread and the value of the β parameter is complicated and the model is diﬃcult to interpret in terms of a distribution of diﬀusion coeﬃcients. There is, for
example, no simple way of extracting a mean diﬀusion coeﬃcient ⟨D⟩, but it can be shown that
⟨1 D⟩ = 1 βΓ ( 1 β ) D0 (4) where Γ is the gamma function (P.T. Callaghan, personal communication). However, because ⟨1/D⟩−1 ̸= ⟨D⟩, the estimate of ⟨D⟩ will be weighted to-ward the slowly diﬀusing molecules. This approach has been shown to give a 20% underestimate of ⟨D⟩ when applied to data simulated for a lognormal molecular weight distribution with polydispersity index, PDI, = 2 and ν = 0.6, and the disparity increases with polydispersity . Hence, a more physically plausible model is to explicitly model a superposition of exponential decays,
I(b) = I0
wD(D) exp (−bD)dD, (5)
choosing a functional form for the (probability) distribution of self-diﬀusion coeﬃcients, wD(D). In general, the true functional form of the distribution is
unknown . However, a couple of common parametric models for distributions of polymer self-diﬀusion coeﬃcients, wD(D), are the lognormal [11, 21, 13] and
gamma distributions [25, 26].
Because the NMR signal is weighted by the number of protons, which scales with molecular weight, and not by the number of molecules, wD(D) is a
mass-weighted distribution. From wD(D) and assuming Eq. (1), we shall see that
obtaining the mass-weighted molecular weight distribution, wM(M ), and
there-fore the mass-weighted average molecular weight, Mw =⟨M⟩w, is
straightfor-ward. Though GPC is also mass-weighted, other experimental techniques may be number-weighted such that comparisons require a conversion between the mass weighted distribution and the number weighted distribution. In addi-tion to Mw, molecular weight distributions are typically characterized by the
average of the number-weighted distribution, Mn=⟨M⟩n and polydispersity
in-dex, PDI = Mw/Mn . The peak molecular weight, number-weighted, Mp,n,
and mass-weighted, Mp,w, are sometimes reported, mostly for narrow molecular
weight distributions such as those of reference and calibration standards. We will proceed by presenting our solutions of these for the lognormal and gamma distribution models. We present general polymer distribution equations used in deriving the distribution parameters, as well as equations for calculating the number-weighted distributions and peak molecular weights in the Appendices.
2.2. Lognormal distribution model
We consider a model where wD(D) in Eq. (5) is a lognormal distribution, wD(D) = 1 DσD √ 2πexp ( −(log D− µD)2 2σ2 D ) . (6)
A well-known property of the lognormal distribution family is that it is closed under powers and multiplication. This implies that wM(M ) is also lognormal,
with parameters µM = log K− µD ν (7) σM = σD ν . (8)
Further, it holds that
Mn= exp ( µM− σ2M/2 ) , (9) Mw= exp ( µM + σM2 /2 ) , (10)
where Mw is the mean of the lognormal distribution and Mn is the mean of nm(M ) (see Appendices), and therefore
PDI = exp(σM2 ). (11)
For a lognormal mixture, i.e., a diﬀusion coeﬃcient distribution modeled by a linear combination of two or more lognormals, obtaining wM(M ) from wD(D)
is still straightforward, because by linearity it too is a mixture of lognormals; hence, we can use Eqs. (7) and (8) for each component in the mixture. However, the calculations of the number-weighted and mass-weighted average molecular weights involve integrals over the entire distribution. From this, we obtain
Mn= ( n ∑ i=1 θiexp ( −µM,i+ σ2M,i/2 ))−1 , (12) and Mw= n ∑ i=1 θiexp ( µM,i+ σM,i2 /2 ) (13) where θi is the weight of the ith lognormal component. The polydispersity
index, PDI, is just Mw/Mn. It should be pointed out that, in the case of several
components, it may make sense to compute the Mn, Mw, and PDI for each
component individually rather than for the combined distribution.
2.3. Gamma distribution model
We consider a model where wD(D) is a gamma distribution, wD(D) =
α−1exp (−βD). (14)
This yields that the decay model in Eq. (5) has the closed-form expression
I(k) = I0 ( β β + k )α , (15)
so the main advantage of the gamma distribution model is fast computation (see  for details). Under a change-of-variables, using Eq. (1), it holds that
wM(M ) becomes wM(M ) = βαKα Γ (α) νM −να−1exp(−βK Mν ) . (16)
We notice that for ν = 1, this is an inverse gamma distribution. Altogether, though, it is a family of distributions that we would call generalized inverse gamma distributions (analogous to the generalized gamma distributions ). Further, it holds that
Mn= β1/νK1/νΓ (α) Γ (α + 1/ν) , (17) Mw= β1/νK1/νΓ (α− 1/ν) Γ (α) , (18) and PDI = Γ (α + 1/ν) Γ (α− 1/ν) Γ (α)2 . (19)
For a gamma mixture, wM(M ) is, by linearity, a sum of components with
the form of Eq. (16). Furthermore,
Mn= ( n ∑ i=1 θi βi−1/νK−1/νΓ (αi+ 1/ν) Γ (αi) )−1 , (20) Mw= n ∑ i=1 θi β1/νi K1/νΓ (αi− 1/ν) Γ (αi) , (21)
where θi is the weight of the ith gamma component, and PDI = Mw/Mn.
3. Materials and methods
For calibration of the method, i.e., to find K and ν of Eq. (1) for dilute polystyrene (PS) in deuterated chloroform (CDCl3), PS standards (2,430 Mw,
3,680 Mw, 13,700 Mw, 18,700 Mw, 29,300 Mw, 44,000 Mw, and 212,400 Mw,
Sigma-Aldrich) were each mixed to 0.1 % w/w in CDCl3 (99.8 atom %
deu-terium, Sigma-Aldrich). These concentrations were chosen to be dilute enough to avoid microscopic averaging eﬀects [13, 21] but still give suﬃcient signal. The solutions were pipetted into standard 5 mm NMR tubes and flame-sealed to prevent convection caused by solvent evaporation. 1H pulsed gradient
stim-ulated echo (PGSTE) NMR experiments were performed at ambient conditions on a 600 MHz NMR spectrometer (Bruker, Germany) with a micro5 probe and Diﬀ30 gradient set. Relaxation weighting is best avoided with the stimulated echo rather than the traditional spin echo because T1is typically longer than T2
for polymers . The T2 and T1 were much longer than τ (time between first
and second 90◦) and the storage time (time between the second and third 90◦), respectively, and therefore there were no relaxation-weighting eﬀects. Tempera-ture control was not used because this caused convection in the sample, a prob-lem which chloroform is especially prone to . In order to keep temperature control, others  have used a convection-compensated double stimulated-echo pulse sequence . The sinusoidal gradient pulse yields that
b = (γgδ)2 4 π2 ( ∆−δ 4 ) , (22)
with proton gyromagnetic ratio γ, time lapse ∆ between the leading edges of the gradient pulses, gradient pulse duration δ, and gradient strengths g. The measurements used ∆ = 50 ms, δ = 1.58 ms, repetition time 10 s, 32 scans, and 16 gradient points with the maximum g chosen such that the attenuation was observed until 30 % of the original signal intensity remained. The reason for disregarding the remaining decay was that it allowed for obtaining the mean diﬀusivity by fitting a single exponential decay to the data, and also that gel per-meation chromatography (GPC) provided PDI values between 1.04−1.08 which is near-monodisperse. (Additionally, fits of a lognormal model to the standards were performed, in order to verify the robustness of this approach. It was found that, first, the estimated mean diﬀusion coeﬃcient was virtually independent of model choice, which is expected for a near-monodisperse distribution, and second, that the estimated mean diﬀusion coeﬃcient was also virtually inde-pendent of the degree of attenuation captured in the measurements. Hence, we regard the single exponential model as a robust and parsimonious choice for analysing the calibration measurements).
The method was then tested on three types of PS; first, a monodisperse PS standard (Mw = 44,000 g mol−1, Sigma-Aldrich) which was also used for
cali-bration, second, a monomodal PS (Mw= 190,000 g mol−1, Sigma-Aldrich), and
third, a bimodal PS (Mw = 35,000 g mol−1, Sigma-Aldrich). The monomodal
and bimodal PS were also dissolved in CDCl3 yielding 0.02 % (w/w) polymer
and 0.1 % (w/w) polymer, respectively. For the measurement of the monodis-perse sample, g was varied linearly to 5.5 T/m over 32 gradient points, for the monomodal sample, g was varied linearly to 10 T/m over 32 gradient points, and for the bimodal sample, g was varied linearly to 10 T/m over 128 gradi-ent points. Experimgradi-ents were performed in triplicate and the number of scans were 32 for the monodisperse, 128 for the monomodal, and 32 for the bimodal sample. Otherwise, the experimental settings were as stated for the calibration measurements.
Molecular weight distribution measurements of the PS samples were per-formed using gel permeation chromatography (GPC). The samples were dis-solved for 1 h at ambient conditions, and characterized using an Agilent PL-GPC 220 system flowing 1,2,4-trichlorobenzene at 1 ml/min through three PLgel 10
µm mixed-B columns at 150◦C using a 200 µl injection volume and either 1.5
were used for column calibration and the molecular weights were calculated using relative calibration.
Data analysis was performed in Matlab R2015a (Mathworks, Natick, MA, US). The Supplementary material contains Matlab source code and raw data for the molecular weight determination of the monomodal and bimodal PS samples, including calibration and Monte Carlo error analysis.
In order to assess the accuracy of the model with regards to signal-to-noise ratio (SNR = I0/σnoise), PGSE data sets from known polymer molecular weight
distributions were simulated, adding gaussian noise with varying SNRs. At each SNR, 50 data sets were simulated in order to measure the noise sensitivity of the fit as seen from the spread in the 50 resulting Mw and PDI values. Two
simulation studies were performed in which the parameters were chosen to em-ulate the experiments performed on the monomodal and bimodal PS samples. The first and second simulation study used the same parameters as in the ex-periments performed on the monomodal PS sample and bimodal PS sample, respectively, and K and ν were set equal to the values found in calibration. In the first simulation study, the molecular weight distribution took the form of a single-component gamma distribution with Mw= 200, 000 and PDI=2. In the
second simulation study, the molecular weight distribution took the form of a two-component lognormal distribution with Mw= 1, 000, PDI=2, and θ = 0.5
for component 1, and Mw= 100, 000 and PDI = 2 for component 2.
4. Results and discussion
4.1. Molecular weight distributions of polystyrene samples
The scaling parameters were estimated by a least squares fit of Eq. (1) to the found⟨D⟩ and reported Mwfor the monodisperse standards. Error analysis
was performed using a two-step Monte Carlo procedure in the spirit of Alper and Gelb . In the first step, errors of the estimated diﬀusion coeﬃcients were assessed by adding Gaussian errors with the same standard deviation as the residuals of the original fits, creating a set of nmc = 103 artificial data sets
and as many corresponding Monte Carlo replicates of the diﬀusion coeﬃcients. In the second step, the same procedure is applied to the residuals of the linear fit to the scaling law, taking also the lack of fit to a straight line in log-log space into account. Fig. 1 shows the calibration with error bounds. The obtained values (with 95 % confidence intervals) were K = 2.44± 0.96 × 10−8 and ν = 0.539±0.039. The value of ν is within the range of values presented in literature for polystyrene in chloroform (∼ 0.5 − 0.6) [2, 34, 31].
The monodisperse PS standard was analyzed assuming a (one-component) lognormal and a (one-component) gamma distribution of diﬀusion coeﬃcients. Error analysis was based on the calibration error as well as on the residuals of the fit to the signal attenuation in order to obtain the full accumulated error. Fig. 2 shows the signal attenuation and the fits together with the estimated diﬀusion
Molecular weight (g/mol) 103 104 105 106 10-11 10-10 10-9 Diffusion coefficient (m 2/s)
Figure 1: (Single column figure) Calibration of the scaling law parameters using seven PS standards (black dots), showing the estimated scaling law (solid black line) with a 95 % point-wise confidence interval (dashed black lines).
coeﬃcient and molecular weight distributions. Tab. 1 shows the estimated values (with 95 % confidence intervals) of Mn, Mw, and PDI from the two models as
well as GPC for comparison. The estimated gamma and lognormal distributions are indistinguishable and closely resemble the results from GPC.
NMR (Lognormal) NMR (Gamma) GPC
Mn(g/mol) 46, 600± 6, 000 46, 600± 5, 700 42, 700± 3, 700
Mw(g/mol) 48, 200± 6, 400 48, 200± 6, 000 45, 200± 1, 800
PDI 1.03± 0.04 1.03± 0.04 1.06± 0.05
Table 1: (Single column table) Results of molecular weight distribution parameters for the monodisperse PS standard, showing estimated values for the lognormal model, for the gamma model, and for GPC as a comparison, with 95 % confidence intervals. The GPC confidence intervals are based on the 2 replicates.
The monomodal PS sample was analyzed in the same way as the monodis-perse PS standard. Fig. 3 shows the signal attenuation and the fits together with the estimated diﬀusion coeﬃcient and molecular weight distributions. Tab. 2 shows the estimated values (with 95 % confidence intervals) of Mn, Mw, and
b (× 1011 s/m2) 0 0.2 0.4 0.6 0.8 1 1.2 10-2 100 Intensity D (m2/s) 10-11 10-10 10-9 Probability M (g/mol) 104 105 106 Probability a b c
Figure 2: (Single column figure) Results for the monodisperse PS standard for the lognormal (red) and gamma (blue) models, showing (a) the experimental signal attenuation (black dots) and the model fits, (b) the estimated distributions of diﬀusion coeﬃcients, and (c) the esti-mated distributions of molecular weights along with the GPC reference measurement (black line).
In , the gamma and lognormal model fits were tested on simulated data of both single-component lognormal and gamma diﬀusion coeﬃcient distribu-tions with SNR = 1000. From this study, it was found that the models could recover both the mean and the coeﬃcient of variation (CV = mean/std. devia-tion) of the diﬀusion coeﬃcient distribution for a CV range of 0 to 50 %, which corresponds to a PDI range of 1 to 2 using the K and ν values found for PS in chloroform. Therefore, the gamma and lognormal distribution models are able to accurately estimate the parameters of the molecular weight distributions for monomodal distributions which appear approximately lognormal and with PDIs between 1 and 2. These observations are aﬃrmed by the agreement which we have found between the molecular weight distributions from analysis of the PG-STE NMR experiments and GPC experiments performed on the monodisperse
b (× 1011 s/m2) 0 0.5 1 1.5 2 2.5 3 3.5 4 10-4 10-2 100 Intensity D (m2/s) 10-12 10-11 10-10 10-9 Probability M (g/mol) 104 105 106 Probability a b c
Figure 3: (Single column figure) Results for the monomodal PS data set for the lognormal (red) and gamma (blue) models, showing (a) the experimental signal attenuation (black dots) and the model fits, (b) the estimated distributions of diﬀusion coeﬃcients, and (c) the estimated distributions of molecular weights along with the GPC reference measurement (black line).
and monomodal PS samples.
The bimodal PS sample was analyzed assuming a two-component lognormal and a two-component gamma distribution of diﬀusion coeﬃcients. Fig. 4 shows the signal attenuation and the fits, together with the estimated diﬀusion coef-ficient and molecular weight distributions. Tab. 3 shows the estimated values (with 95 % confidence intervals) of Mn, Mw, and PDI for the two models as
well as GPC. Compared to the distribution estimated by GPC, the NMR dis-tribution models gave similar estimates for the component of larger molecular weight, component 2, but over-estimated Mn, Mw, and PDI of the component
of smaller molecular weight, component 1.
We have tested the ability of the model to estimate real polymer molecular weight distributions as compared to GPC. We note that the shape of the
distri-NMR (Lognormal) NMR (Gamma) GPC
Mn(g/mol) 82, 500± 10, 800 90, 600± 12, 900 86, 900± 15, 800
Mw(g/mol) 201, 000± 48, 800 220, 200± 58, 100 193, 300± 8, 800
PDI 2.44± 0.37 2.43± 0.34 2.24± 0.35
Table 2: (Single column table) Results of molecular weight distribution parameters for the monomodal PS data set, showing estimated values for the lognormal model, for the gamma model, and for GPC as a comparison, with 95 % confidence intervals. The GPC confidence intervals are based on the 3 replicates.
NMR (Lognormal) NMR (Gamma) GPC Mn(g/mol) 3, 200± 800 3, 300± 700 1, 300± 100 Component 1 1, 400± 300 1, 500± 300 700± 200 Component 2 51, 800± 14, 500 64, 400± 10, 100 40, 700± 2, 600 Mw(g/mol) 69, 000± 13, 500 70, 800± 14, 100 50, 800± 3, 900 Component 1 2, 500± 1, 100 3, 600± 900 1, 100± 700 Component 2 115, 000± 23, 100 124, 300± 24, 600 103, 500± 4, 300 PDI 21.40± 8.43 21.55± 8.27 38.48± 2.41 Component 1 1.85± 0.87 2.42± 0.56 1.55± 0.09 Component 2 2.22± 0.56 1.93± 0.31 2.54± 0.19
Table 3: (Single column table) Molecular weight distribution parameter results for the bimodal PS data set, showing estimated values for the lognormal model, for the gamma model, and for GPC as a comparison, with 95 % confidence intervals. The GPC confidence intervals are based on the 3 replicates. Values for the whole distribution as well as for the two components separately are reported.
butions found by GPC were approximately lognormal, providing more evidence that lognormal and gamma distributions are good models for polymer samples. In the monodisperse and monomodal cases, both models estimated distribu-tions which were not statistically diﬀerent from that of GPC. However, in the bimodal case, we saw inconsistency between NMR and GPC in their estimates of the distribution for component 1. Though 128 gradient points were used to better sample the attenuation caused by both components, given that the number of fit parameters increased between these two cases, an obvious culprit could have been noisy data. However, there is nothing to say that GPC gave the correct molecular weight distributions. The lower limit of the defined molecular weight operating range for the GPC column used was 500, and a significant por-tion of component 1 was measured to be below this molecular weight (see Fig. 4). This could have impacted the accuracy of the GPC estimates of component 1. To assess whether noise could have impacted the results in the bimodal case, we simulated noisy data from known molecular weight distributions, first for a
b (× 1011 s/m2) 0 0.5 1 1.5 2 2.5 3 3.5 4 10-4 10-2 100 Intensity D (m2/s) 10-11 10-10 10-9 Probability M (g/mol) 102 103 104 105 106 Probability a b c
Figure 4: (Single column figure) Results for the bimodal PS data set for the lognormal (red) and gamma (blue) models, showing (a) the experimental signal attenuation (black dots) and the model fits, (b) the estimated distributions of diﬀusion coeﬃcients, and (c) the estimated distributions of molecular weights along with the GPC reference measurement (black line).
monomodal distribution and second for a bimodal distribution.
In the first simulation study, single-component gamma and lognormal mod-els were fit to simulated PGSE data from a known single-component gamma molecular weight distribution. Randomly generated Gaussian noise was added to the signal attenuation data to make 50 distinct data sets at each signal-to-noise ratio. It was found that both models were able to accurately estimate Mw
and PDI for SNR > 50, and the spread of these estimates from the 50 fits at each noise level became relatively insensitive to noise for SNR > 100. This is indicated by the mean values and the error bars, showing the 2.5 % and 97.5 % percentiles of the obtained estimates, of Mwand PDI at varying SNRs, presented
data estimated to be greater than 1,000, this aﬃrms that noise negligibly im-pacted the results of these measurements and the slight inconsistency between GPC and NMR was due to other reasons.
101 102 103 104 ×105 0 2 4 6 Mw (g/mol) SNR 101 102 103 104 2 4 6 8 PDI a b
Figure 5: (Single column figure) Results of the mean values and the 2.5 % and 97.5 % per-centiles of (a) Mwand (b) PDI estimated from fits of the lognormal (red) and gamma (blue)
models to attenuation data simulated with varying SNRs assuming a gamma polymer distri-bution with Mw= 200, 000 g/mol and PDI = 2.
As a guide for experimental PGSE methods for estimating monomodal poly-mer molecular weight distributions, these simulations indicate that suﬃciently accurate (seen by the mean of the estimates) and precise (seen by the error bars) distribution estimates can be obtained at SNRs as low as 100, and additional experimental scans (averages) may have negligible impact on results. Both the lognormal and gamma models can be regarded as good model choices for accu-rate estimation of Mn, Mw, and PDI (also shown in ). Given that a model
is a suitable choice to begin with, obtaining a very high SNR is not a major concern to ensure suﬃcient accuracy. The precision of estimated values can be measured by the confidence intervals obtained from Monte Carlo error analysis as well as by the variance obtained from performing multiple experiments.
In the second simulation study, two-component gamma and lognormal mod-els were fit to simulated PGSE data from a known two-component lognormal molecular weight distribution. The mean values and the 2.5 % and 97.5 % percentiles of the estimates of Mw and PDI are shown in Fig. 6. The values
estimated for SNR < 40 were unstable and are not presented. The mean values of Mw and PDI for the component of larger molecular weight (component 2)
are consistent with the known values at all SNRs presented. For component 2, the spread of the estimates of Mw and PDI, as seen by the error bars, are
spread of the estimates of Mwand PDI for the component of smaller molecular
weight (component 1) decrease to this level only after roughly SNR > 3, 000. Additionally, the mean values of Mwand PDI from the two-component gamma,
the incorrect model for the data, are noticeably larger than the known values. It appears that parameter estimates are sensitive to noise and model choice for the two-component case. Interestingly, at SNR < 200 the spread of the parameter estimates from the two-component lognormal distribution model, which is the correct model, are larger than those for the two-component gamma; in this case, the correct model has even more sensitivity to noise than the incorrect model.
As a guide for experimental PGSE methods for bimodal polymers, the sim-ulations indicate that to obtain the same level of accuracy and precision for a bimodal distribution which can be obtained for a monomodal distribution at SNR = 100 requires SNR = 3, 000 and may not be possible due to incorrect model choice. Such high SNR is diﬃcult to obtain for dilute polymer solutions, and it is generally not possible to know the correct model a priori. Therefore, settling for less than optimal SNR and accepting the limitations of the models may be necessary. However, the simulations also indicate that one can antici-pate more accurate estimates of component 2 than component 1. This is due to the order of attenuation of the components and the information contained in the PGSE experiment. The signal from component 2 attenuates over larger b values (Eq. (2)) than the signal from component 1 such that there exists a b value after which the attenuation contains negligible information about component 1. This attenuation region where information about component 2 is separated from in-formation about component 1 makes component 2 easier to fit. Furthermore, no attenuation reign exists where the information about component 1 is sepa-rated from the information about component 2, therefore, potentially limiting the accuracy of the estimate of component 1.
Results of the second simulation study can be used to inform the experimen-tal results of the bimodal PS sample. The SNRs of the experiments performed on the bimodal PS sample were roughly 500, as estimated by the residuals of the fits. The PDI and Mwof component 1 were over-estimated when compared
to GPC (see Tab. 3). Insuﬃcient SNR and the use of incorrect models were seen to each, independently, lead to such an over-estimate in the second simulation study. However, comparing the over-estimates of Mw of component 1 between
the experimental results, seen in Tab. 3, and the simulation study, seen in Fig. 6, the diﬀerences between the NMR and GPC estimates are significantly greater than the diﬀerence between the mean of the estimates and the known value; the diﬀerences are too great to attribute noise alone. We can extrapolate that the diﬀerence between the mean and actual values of parameter estimates for com-ponent 1 will increase as the model choice becomes less correct. Though both components of the distribution estimated by GPC appear roughly lognormal in shape in Fig. 4, perhaps they are not suﬃciently so. Therefore, the cause of the over-estimates in the experimental results of PDI and Mw of component
1 for the bimodal PS sample may be the use of incorrect distribution models combined with noise, however, inaccurate GPC estimates still can not be ruled out.
We provide comprehensive derivations of statistical models and methods for the use of PGSE NMR measurements to characterize the molecular weight distri-bution of polymers via the well-known scaling law relating diﬀusion coeﬃcients and molecular weights, covering the lognormal and gamma distribution models. We illustrate its use on three polystyrene samples, test its accuracy and sensi-tivity to noise on simulated data, and provide a Matlab R2015a (Mathworks, Natick, MA, US) implementation (see Supplementary material). We conclude that although NMR has its drawbacks for molecular weight distribution estima-tion, it is complementary to, for example, gel permeation chromatography and is a useful part of the diﬀusion NMR analysis toolbox.
This research was funded by the South Australian Government Premier’s Re-search and Industry Grant project ’A Systems Approach to Surface Science’, as well as the Australian Government International Presidents Scholarship (IPS).
6. Supplementary material
A Matlab R2015a (Mathworks, Natick, MA, US) implementation of the mod-els, that also reads raw Bruker data files, is provided as supplementary material. The implementation includes support for Monte Carlo error analysis. It also of-fers the option of adding an extra single exponential component to account for solvent signal (which did not have to be used in the current experimental study).
7.1. General equations
A mass-weighted distribution, wD(D), is related to a number-weighted
dis-tribution, nD, by wD(D) = M (D)nD(D) ∫∞ 0 M (D)nD(D)dD (23) and nD(D) = wD(D)/M (D) ∫∞ 0 wD(D)/M (D)dD . (24)
where the molecular weight as a function of diﬀusion coeﬃcient, M (D), is given by Eq. (1). Equations (23) and (24) apply equally for relating molecular weight distributions, wM(M ) and nM(M ), if all instances of D are replaced with M .
Given a function for wM(M ) or nM(M ), one can derive the number-weighted
average molecular weight , Mn=⟨M⟩n= ∫ ∞ 0 M nM(M )dM = 1/ ∫ ∞ 0 wM(M )/M dM, (25)
the mass-weighted average molecular weight, Mw=⟨M⟩w= ∫ ∞ 0 M wM(M )dM = ... ∫ ∞ 0 M2nM(M )dM/ ∫ ∞ 0 M nM(M )dM, (26)
and the polydispersity index,
PDI = Mw Mn = ⟨M 2⟩ n ⟨M⟩2 n . (27)
7.2. Lognormal distribution model
Using Eq. (24), the number-weighted distributions corresponding to the log-normal distributions of wD(D) and wM(M ) are,
nD(D) = D1/ν−1 σD √ 2πexp ( −µD/ν− σ2D/(2ν 2 ))exp ( −(log D− µD)2 2σ2 D ) (28) and nM(M ) = 1 M2σ M √ 2πexp ( µM − σM2 /2 ) exp ( −(log M− µM)2 2σ2 M ) . (29)
The peak molecular weights are Mp,n= exp ( µM− 2σM2 ) (30) and Mp,w= exp ( µM − σ2M ) . (31)
In the multi-component lognormal case, calculating nD becomes somewhat
more complicated than in the previous one-component case, because the re-weighting step must now be performed over the entire mixture, so that
nD(D) = 1 Z n ∑ i=1 θi 1 D1+1/νσ D,i √ 2πexp ( −(log D− µD,i)2 2σ2 D,i ) (32) where Z = n ∑ i=1 θiexp ( µD,i/ν + 2 (σD,i/ν) 2) . (33)
Obtaining nM from wM(M ) once again requires some more work, and nM(M ) = 1 Z n ∑ i=1 θi 1 M2σ M,i √ 2πexp ( −(log M− µM,i)2 2σ2 M,i ) (34) where Z = n ∑ i=1 θiexp ( −µM,i+ σM,i2 /2 ) . (35)
7.3. Gamma distribution model
When wD(D) is a gamma distribution, interestingly, nD(D) is also a gamma
distribution, with a slightly modified parameter,
Γ (α + 1/ν)D
α+1/ν−1exp (−βD). (36)
Hence, nM(M ) is also a generalized inverse gamma distribution, slightly
modi-fied from wM(M ), nM(M ) = βα+1/νKα+1/ν Γ (α + 1/ν) νM −ν(α+1/ν)−1exp(−βK Mν ) . (37)
The peak molecular weights are Mp,n= ( να + 2 νβK )−1/ν (38) and Mp,w= ( να + 1 νβK )−1/ν . (39)
Here, it must hold that α > 1/ν. For a gamma mixture,
wD(D) = n ∑ i=1 θi βαi i Γ (αi) Dαi−1exp (−β iD), (40) nD(D) = 1 Z n ∑ i=1 θi βαi i Γ (αi) Dαi+1/ν−1exp (−β iD), (41) where Z = n ∑ i=1 θiβi−1/ν Γ (αi+ 1/ν) Γ (αi) (42) Obtaining nM is nM(M ) = 1 Z n ∑ i=1 θi βαi i K αi Γ (αi) νM−ν(αi+1/ν)−1exp ( −βiK Mν ) , (43) where Z = n ∑ i=1 θi βi−1/νK−1/νΓ (αi+ 1/ν) Γ (αi) . (44)
101 102 103 104 0 2000 4000 6000 Mw (g/mol) 101 102 103 104 2 4 6 8 PDI 101 102 103 104 ×105 0 2 4 6 Mw (g/mol) SNR 101 102 103 104 2 4 6 8 PDI c a b d
Figure 6: (Single column figure) Results of the mean values and the 2.5 % and 97.5 % per-centiles of (a) Mw and (b) PDI of component 1 and (c) Mw and (d) PDI of component 2
estimated from fits of the lognormal (red) and gamma (blue) models to attenuation data sim-ulated with varying SNRs assuming a two-component lognormal polymer distribution with
Mw = 1, 000 g/mol and PDI = 2 for component 1 Mw = 100, 000 g/mol and PDI = 2 for component 2.
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