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Quantifying size exclusion by diffusion NMR: A versatile method to measure pore access and pore size Fredrik Elwinger, Jonny Wernersson, and Istvan Furo Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.8b02474 • Publication Date (Web): 27 Aug 2018 Downloaded from http://pubs.acs.org on August 28, 2018
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Analytical Chemistry
Quantifying size exclusion by diffusion NMR: A versatile method to measure pore access and pore size Fredrik Elwingera,b, Jonny Wernerssonb and István Furóa,b,* a
Division of Applied Physical Chemistry, KTH Royal Institute of Technology, Teknikringen 36, SE-10044 Stockholm, b Sweden. GE Healthcare Bio-Sciences AB, Björkgatan 31, SE-75184 Uppsala, Sweden.
ABSTRACT: Size-exclusion quantification NMR spectroscopy (SEQ-NMR) is introduced for measuring equilibrium distribution coefficients, Keq, in porous media. The porous medium is equilibrated with a polydisperse polymer solution. The original bulk polymer solution and the polymer solution after equilibration but in the absence of the porous medium are analyzed by NMR diffusion experiments. The joint evaluation of the two diffusion attenuation curves under suitable constraints provides the extent by which polymer fractions of particular size were depleted from the solution by pore access. This procedure yields Keq versus polymer probe size, the selectivity curve that in turn can provide the pore size and its distribution. Simulations probe the performance of the method that is demonstrated experimentally in chromatographic media using dextran polymers. SEQ-NMR and inverse size-exclusion chromatography (ISEC) yield selectivity curves that virtually coincide. Crucial advantages with SEQ-NMR, such as versatility with regard to both the polymer used and porous system explored, high speed, potential for automation, and small required sample volume, are discussed.
INTRODUCTION Natural or human-made porous materials are at the heart of many phenomena, both natural and technological. As one example, chromatography is an essential method of modern biotechnology and chemistry, both for purification and characterization. The performance of many chromatographic separations is determined to a large extent by partitioning of the target molecules between the porous network within the particles/beads of chromatographic medium (the stationary phase) and the liquid (mobile) phase. This partitioning, that may be dependent on either size, shape or other molecular properties, is described in terms of a distribution coefficient, Keq, defined here as the ratio of pore volume available for a certain molecule over the complete pore volume. While in pores, molecular displacement is typically reduced. This is exploited in, for example, size-exclusion chromatography (SEC), where molecules are separated due to their difference in size and shape, which in turn determines the fraction of pores accessible to them. Hence, Keq has direct influence on the chromatographic resolution.1-2 The variation of Keq with polymer probe size, also called the selectivity curve, quantifies the fraction of pores that may be accessed by polymers of a given size and thereby the curve contains information about the pore size and its distribution. Keq can be measured with inverse size-exclusion chromatography (ISEC),3-10 where solutions with probe molecules (most often polymers) of different and well-defined sizes are allowed to flow through a column packed with porous particles/beads. Suitably small probe molecules equilibrate over the solvent volume constituted
by both the inter-particle bulk and intra-particular pore space. In the absence of any (unwanted) adsorption onto matrix material, the selectivity curve depends on the pore size and its distribution is captured by suitable models.3, 68, 10
Structural properties in porous materials such as pore size distributions represent a class of properties that are difficult to access,11-12 particularly in systems that are saturated by liquids. As one noteworthy class, the porous structure of many soft hydrogels collapses upon drying and can thus not be analyzed at all with conventional methods like mercury intrusion or nitrogen sorption porosimetries. Potential alternative techniques exist, including thermoporometry and NMR-cryoporometry.13-14 The latter has proven to be an accurate tool for determination of pore size distributions in various organic hydrogels,15-16 including chromatography media.17-18 Yet, alternative methods for characterizing pore size distribution are sought after. Regarding ISEC, its advantages are experimental and instrumental simplicity. It has also significant drawbacks. First, it requires a time- and material-consuming preparation – packing a column with the porous medium. For soft gels, packing can be tedious and prone to faults such as in bed integrity that limit reproducibility and, thereby, accuracy. Column packing is difficult to automate and therefore ISEC is not well suited for high-throughput analysis. Size resolution is largely determined by bed height and high resolution demands a lot of material that is a limitation when exploring novel media. Long columns of soft gels also suffer from high back-pressures.3 Secondly, one must have access to a suitable series of probe polymers.3 Such a series
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must consist of polymer fractions of low polydispersity that makes them expensive (and, for many polymers, not readily available). Some remedies exist. Hence, size-exclusion can also be performed in batch mode4, 19-22 where the porous medium is equilibrated with a solution of a probe molecule, again of low polydispersity. The pore volume accessible for the probe is determined from its relative concentrations before and after equilibration. As no column is required, one saves both time and cost and may both parallelize and automate, such as with multiwell plates, to permit high-throughput analysis. The batch mode has moreover proved to be better than ISEC in determining low Keq (near Keq=0).19, 22 The direct precursor of the experimental principle we present here is the “mixed solute exclusion” batch method.22 In that type of experiment, a test solution that contains a mixture of suitable polymers and oligomers over a wide size range is equilibrated with the porous medium. The result is a differential dilution as a function of molecular size that can then be detected by gel permeation chromatography (GPC). Drawbacks are that multiple GPC columns matching the pore size range of the investigated medium are required and those columns employed must provide very high accuracy and resolution. Here we present a new technique, termed size-exclusion quantification NMR (SEQ-NMR), where a diffusion NMR experiment is employed, essentially as a replacement for the set of GPC columns for measuring differential dilution of a mixed probe solution. As is shown below, one can obtain the difference between size distributions of probe molecules before and after equilibration with the porous medium by analyzing the experimental diffusional decays with a numerical procedure that is akin to numerical implementations of inverse Laplace transformation. This difference yields the selectivity curve. We shall first discuss the theoretical background and then the simulated performance of the method. Finally, results of proof-of-concept experiments with dextran probes and an agarose-based chromatographic medium are presented. As discussed at the end, there are plenty of advantages with the proposed method, the most important of which is universality both with regard to the probe species and the porous systems. THEORETICAL BACKGROUND In typical NMR diffusion experiments (pulsed-field-gradient spin-echo NMR, also known by acronyms like PGSEor PFG-NMR and lately as DOSY-NMR) magnetic field gradient pulses with length δ and strength g are employed to make the NMR frequency spatially dependent. When used in suitable combinations, typically in pair separated by time Δ, the encoded quantity becomes the displacement of the spin-bearing particles over the Δ period (also called the diffusion time). If the displacement is diffusive, it manifests itself as attenuation of NMR peak intensities I upon increasing g. For a single diffusing species in a homogeneous material characterized by a single self-diffusion coefficient D, the attenuation E becomes:23-24
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I (b) = exp(-bD) I (0)
(1a)
2 b = γ 2 g eff δ 2 (Δ − δ / 4)
(1b)
E=
with
where g is the gyromagnetic ratio of the explored NMR nucleus. Eq 1b is specifically valid for the half-sine shaped pulses used in this work, with geff = 2·gmax/π where gmax is the maximum value of the gradient within the pulse. For several species contributing to the same NMR signal and diffusing with their own characteristic diffusion coefficient Di, the decay becomes the sum of individual components: n
E = å Pi exp(-bDi )
(2)
i =1
where Pi is the relative population of the spin-bearing nun
clei within species i, i.e.
å P = 1. For homopolymers, P i =1
i
i
approximates well the relative population in molar monomeric units. In the continuous limit (such as for polydisperse polymers), one obtains: ¥
E (b) = ò P( D)exp(-bD)d D
(3)
0
where P(D) is the distribution of diffusion coefficients. In case of polymers, this can – if so wished – be turned into a size (weight) distribution via a previously established scaling relationship between hydrodynamic radius, rH, and molecular weight, M
rH = aM a
(4)
in combination with the Stokes-Einstein equation8, 24-28 D=
kT 6ph rH
(5)
where a and α (the latter often called the Mark-Houwink exponent) are material-dependent constants, k is the Boltzmann factor, T is the temperature, and h is the dynamic viscosity of the solution. Generally speaking, the multiexponential attenuation curve E(b) is the experimental observation while P(D) is often the sought distribution of molecular properties, connected to each other by the Laplace integral in eq 3. In principle (see some considerations in SI), one could obtain P(D) via inverse Laplace transformation (ILT). However, ILT is an ill-posed mathematical problem with a very strong sensitivity to experimental noise. In addition, its result is not unique and does not depend continuously on the data.29-34 A practical help is to add prior information to the inversion routine (that typically involves least-squares fitting), often in the form of some regularization or constraints.29-39 Apparently successful examples include the
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Analytical Chemistry
measurement of relaxation time distributions in porous media35-36 or assessing the size distribution for polymers by NMR diffusion experiments.37, 39 In the SEQ-NMR method proposed here, the pore size in a water-saturated porous material, here a pack of chromatographic beads, is investigated. In the actual experiment, the water-saturated porous material is equilibrated with a solution of polymeric probes whose size distribution is such that it covers the pore size distribution. A polymer fraction indexed by i is depleted from the solution outside the pores because it also accesses that portion Vi of the intra-bead pores that are sufficiently large to permit penetration. Note that i indexes the fractions in ascending order regarding pore access, hence i = 1 belongs to the polymers with least pore penetration. In this discrete model, the equilibrium distribution coefficient of the particular polymer fraction i, Keqi is set as
Keqi =
Vi Vintra
,
(6)
where the complete liquid volume in the pores is denoted Vintra. One sample to be analyzed is a bulk solution of a polydisperse polymer whose diffusional attenuation is described by eq 2. The other sample is the solution obtained after having equilibrated the initial polymer solution of volume Vsol with the bead pack containing liquid-saturated pores. The total solvent volume within the bead pack before equilibration is Vtotal = Vintra + Vinter, where Vinter represents the liquid between the beads but not within the pores (in practice, this latter liquid fraction is difficult to avoid, see Sample preparations). As a result of the equilibration with the pores, the diffusional attenuation of the polymer signal of the probe solution outside the bead pack changes from eq 2 to N
Vsol Pi exp(-bDi ) . i =1 Vsol + Vinter + Vi
Eeq (b) = å
(7)
We stress that this attenuation is experimentally recorded in a bulk solution that has been removed from the bead pack after the equilibration step, see Sample preparations. The essence of the method we describe herein is that the diffusional attenuation curves acquired before and after equilibration with a water-saturated porous material are least-squares fitted simultaneously with eqs 2 and 7 yielding (from the extracted Vi values) Keqi as output via eq 6. The sole constraints are that (i) the larger the probe molecule the less is the accessed pore volume, and (ii) the largest probe molecule cannot be excluded from a volume larger than the complete pore volume, and (iii) the smallest probe molecule cannot access more volume than the complete pore volume. They are summarized, respectively, as
Vi +1 ³ Vi
(8a)
V1 ³ 0
(8b)
VN £ Vintra
(8c)
where N is the number of discrete polymer fractions by which we choose to represent the stock solution. Input parameters Vintra and Vinter for the fit are obtained by separate straightforward experiments, see Experimental results and discussion. We stress here that, with this procedure, we do not directly seek any information about P(D). Hence, the method outlined here does not conform to any ILT procedure and does not attempt to extract information about the size-dependent depletion of polymers from the test solution in form of a difference between two separate distribution curves. We re-visit this crucial point below. SIMULATIONS The simulations performed using Matlab consisted of two steps. First, within the framework of a suitable polymer, pore and distribution coefficient model, we calculated diffusional attenuations (corresponding to eqs 2 and 7). After having had added noise to those, we then simulated how well we could re-capture the features of the porous system by analyzing the noise-colored data by the procedure presented in the previous section. In our test experiments below, we used a mixed stock solution of dextran standards and glucose to probe the pore space. The molecular weight distributions of the individual dextran standards were obtained experimentally using lognormal models applied to SEC triple detector system (TDS) data (see SI). The molecular weight distribution of the resulting stock solution was assumed to be the simple population average of those individual distributions. That molecular weight distribution was discretized in steps of 180 g/mol (the molecular weight of glucose, one monomer of dextran) from 180 g/mol (corresponding to j = 8566, where j is the index in eq 9 below) up to 10Mw (corresponding to j = 1) where Mw = 154000 is the average molecular weight for the largest dextran, see SI. The molecular-weight scale was transformed, with a and α (see SI), to a diffusion/hydrodynamic radius scale via eqs 4 and 5. The diffusional attenuation curve characteristic of the simulated stock solution (see Figure 1) was then obtained by eq 2, using the same b factors as in the experiments (see Experimental section). As next step, a distribution coefficient for each discrete polymer fraction j with hydrodynamic radius rH,j was calculated by adopting a cylindrical pore model6, 40 ¥
2
æ rH , j ö Keq j = ò f (r ) ç1 ÷ dr r ø è rH , j
(9)
where f(r) is the distribution of pore volume comprising cylindrical pores of radius r. The performance of SEQ-NMR was tested by simulating simple cases of a one- and a twopore model, i.e. with f(r) = d(r-rp) and f(r) = PAd(r-rpA) + (1PA)d(r-rpB), respectively, where d is the delta-function, rp, rpA and rpB are the pore radii, and PA is the volume fraction of pores with radius rpA. In the simulations rp was set to 6 nm, and rpA and rpB to 2 and 6 nm, respectively, and PA to 0.5. Volumes Vinter, Vintra and Vsol were set to have the same
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relative proportions as those in our experimental system (see below). From the Keqj values in eq 9, we obtained the attenuation curve simulated for an equilibrated system via eqs 6 and 7.
Figure 1. Simulated NMR diffusional attenuation curves for a probe solution containing polymers in a broad size range before (open symbols) and after (full symbols) equilibration with a porous medium. The inset shows the magnified data for the decay tails. The simulation was performed with SNR=1500, see text for details. The continuous lines represent results of simultaneous fits with the fmincon routine (see text).
The trends in Figure 1 illustrate well the effect of equilibration on the probe solution: a large fraction of small molecules, dominantly contributing to the initial parts of the decays, is lost from the probe solution by penetration to the pores while the concentration of larger molecules, that dominate the tail of the decays, remains roughly the same. To the simulated diffusional attenuation curves (stock solution and equilibrated solution), zero-mean normaldistributed noise was added at each point, with standard deviation set to 1/SNR (relative to the first point in the attenuation curve of the stock solution) and with the signalto-noise ratio SNR ranging 100 < SNR < 10000. The curve pairs, like that in Figure 1, were then simultaneously fitted with eqs 2 and 7, with constraints as in eq. 8. The fitting was performed with the ‘fmincon’ (minimization of constrained nonlinear multivariate function) routine in Matlab. The independent variables were the values of hydrodynamic radius rH,i distributed reciprocally in the range rH,N < rH,i < rH,1 and the results were the Vi (i = 1, …,N) volumes quantifying the pore volumes accessible for a probe molecule fraction with the given size rH,i and initial population Pi. Finally, Keqi values, see examples in Figure 2, were derived from Vi via eq 6. The upper and lower limits of rH must be set to include the full range of polymer standards. As lower limit, the size of glucose for rH,N suited well. As long as rH,1 was set high enough (see SI, roughly twice the average rH of our largest stock component), the outcome is rather insensitive to its exact position, as is illustrated by the data in SI. The sole effect of setting the rH,1 unnecessarily high is that the derived Keq points map the actual curve more sparsely.
Figure 2. SEQ-NMR results obtained from simulated data for a porous medium with cylindrical pores of (a) 6 nm radius or (b) 2 nm and 6 nm radii with equal volume fractions. The simulations were performed with SNR = 1500 and with N = 9 hydrodynamic radii distributed reciprocally in the indicated range whose upper limit rH,1 was set to 1.7 times the average rH value for the largest dextran (see text). The error bars represent ±σ confidence intervals as given by the scatter over 100 simulated data sets. The continuous lines represent the exact model values as provided by eq 9.
The number N of the assumed distinct polymer fractions representing the system is intimately connected to the persistent problem with uniqueness and noise sensitivity of ILT29-34 and/or deconvoluting multiexponential decays, both generally29, 41-43 and specifically in diffusion NMR.44 As far as we know, there is no generally accepted and quantitative answer to this problem. Here we simply note that setting N too low (in our and similar cases, to about
