Diffusive Transport in Pores. Tortuosity and Molecular Interaction with

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Diffusive Transport in Pores. Tortuosity and Molecular Interaction with the Pore Wall Fredrik Elwinger, Payam Pourmand, and Istvan Furo J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b03885 • Publication Date (Web): 15 Jun 2017 Downloaded from http://pubs.acs.org on June 20, 2017

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The Journal of Physical Chemistry

Diffusive Transport in Pores. Tortuosity and Molecular Interaction with the Pore Wall

Fredrik Elwinger,a,b Payam Pourmanda and István Furóa* a

Division of Applied Physical Chemistry, Department of Chemistry, KTH Royal Institute of Technology, SE-10044 Stockholm, Sweden b

GE Healthcare Bio-Sciences AB, Björkgatan 31, SE-75184 Uppsala, Sweden *Corresponding author, E-mail: [email protected]

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Abstract The self-diffusion of neat water, dimethylsulfoxide (DMSO), octanol and the molecular components in a water-DMSO solution were measured by 1H and 2H NMR diffusion experiments for those fluids imbibed into Controlled Pore Glasses (CPG). Their highly interconnected structure is scaled by pore size and shows invariant pore topology independently of the size. The nominal pore diameter of the explored CPGs varied from 7.5 nm to 72.9 nm. Hence, the ∼μm mean-square diffusional displacement during the explored diffusion times was much larger than the individual pore size and the experiment yielded the average diffusion coefficient. Great care was taken to establish the actual pore volumes of the CPGs. Transverse relaxation experiments processed by Inverse Laplace Transformation were performed to verify that the liquids explored filled exactly the available pore volume. Relative to the respective diffusion coefficients obtained in bulk phases, we observe a reduction in the diffusion coefficient that is independent of pore size for the larger pores and becomes stronger towards the smaller pores. Geometric tortuosity governs the behavior at larger pore sizes while the interaction with pore walls becomes the dominant factor at our smallest pore diameter. Deviation from the trends predicted by the Renkin equation indicates that the interaction with the pore wall is not just simple steric one but is in part dependent on the specific features of the molecules explored here.


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Introduction Molecular diffusion in (broadly defined) pores is the process that defines the outcome in a number of natural and technical phenomena.1-3 Specifically, it is a decisive feature4 when it comes to chromatography, the method by which current biotechnology achieves pure solutions of selected biomolecules. There are several factors that influence diffusion in pores.5 One is the pore structure, often described in terms of pore shape and interconnectivity. In random porous materials, pore structure is not easy to characterize. There are several methods including gas sorption, mercury intrusion and cryo-porometries6-8 all of which can provide some integral measures relevant for the porous structure. Yet, all those methods fail, sometimes miserably, when it comes to irregularities in the structure such as “ink-bottle” shapes where the word “ink-bottle” wishes to indicate strong variation of pore cross section (size and shape) along passages. True imaging methods like micro-CT9 may yet provide solution to this problem but, as of today, neither resolution, contrast, nor statistics seems satisfying when it comes to pores below the 100 nm size range. Depending on interconnectivity and topology, even infrequent pore ends may prolong diffusional pathways and thereby limit diffusion over length scales larger than the pore sizes. A particular issue with pore structure determination is that, in many systems, pore structure changes when the solvent filling the pores is removed. Yet, many of the methods do not work with solvent-filled pores (exceptions are cryoporometry8 and micro-CT9). Hence, with pore structure typically being defined on a sparse manner, it is not surprising that the complex influence of that on diffusive transport is typically and conveniently hidden behind the seemingly simple concept of “tortuosity”.1, 2, 10, 11 The name expresses

the

fact

that

the

diffusion

pathways

in

a

porous

network

are

twisted/winding/convoluted and are altogether longer as compared to that between two points for the same material in its bulk phase. If dynamics on the molecular scale is unchanged, then longer pathways lead to smaller linear displacements and thereby an apparent diffusion coefficient Deff that is smaller than its corresponding bulk value D0. Often, tortuosity τ is defined by the ratio of the path length over linear displacement (mathematically, the arc-chord ratio). That also yields, via the well-known linear dependence of the mean-square diffusional displacement on the diffusion coefficient, that τ = (D0/Deff)1/2. Regarding definition via diffusion coefficients, this is not the only or dominant formula10 and τ = (D0/Deff) has a similarly wide circulation (and there are still other available expressions). Nevertheless and irrespective of the particular form of the tortuosity expression, an apparently simple recipe for 3 ACS Paragon Plus Environment


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determining tortuosity is to compare diffusion coefficients in bulk liquid and in the same bulk liquid imbibed in the investigated porous material. Yet, this recipe fails because pore walls are not simple reflective obstacles. Instead, they may interact with the molecules of the pore-filling liquid and, depending on the nature of that interaction, change their translational dynamics on the molecular scale. This latter feature is often represented by a spatially dependent diffusion coefficient that is, typically, lower adjacent to the pore walls. Hence, over the length scale of individual pores the diffusion coefficient Dpore becomes a population average of the locally different diffusion coefficients within the pore volume. These two effects, tortuosity and wall interactions, are not easy to separate. In this work, we take yet another step towards characterizing them distinctly. Here, we use diffusion NMR to characterize the self-diffusion of various molecules such as water, dimethylsulfoxide (DMSO), and 1-octanol in pure phases and some of their mixtures. Comparisons are then made between the self-diffusion coefficients obtained, on one hand, in bulk and, on the other hand, for liquids imbibed in Controlled Pore Glasses (CPG).12, 13 As another element, we perform these experiments in a series of CPGs whose pore size (nominal pore diameter) varies from 7.5 to 72.9 nm. CPGs are assumed to have a scaled pore structure: that is, their topology is supposed to be the same, with the size of all features provided by a single multiplying factor.14-16 The outcome is a quantitative assessment of the interaction of the various solvents with the pore wall. The qualitative outcome is an indication of the size range where wall interaction has a significant effect on the apparent tortuosity (that is, (D0/Deff)α with α seemingly being the matter of taste).

Materials and methods

Materials and samples The CPGs (see Table 1 for selected properties) were from Millipore and supplied as 120-150 µm diameter porous particles. The pore-filling solvents were D2O (99.9% D, Cambridge Isotope Laboratories), dimethylsulfoxide (DMSO, purity ≥99.9%, Merck), deuterated DMSO (DMSO-D6, D 99.9%, Cambridge Isotope Laboratories), 1-octanol (purity ≥99.9%, SigmaAldrich). For determination of interparticle liquid volume (see below), native dextran polymer


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produced in-house (GE Healthcare) by the bacterium Leuconostoc mesenteroides BF-512 was used. The average molecular weight of the latter was ∼30×106 g/mol. The CPG samples were first dried at 105○C in an oven for at least 24 hours and then kept in dessicator until used. For NMR diffusion and relaxation experiments, dry CPG particles were weighed into a 5 mm NMR tube up to height of approx. 13 mm (corresponding to a weight in the order of 100 mg). A particular solvent was then weighed into the NMR tube using a micropipette. For diffusion measurements, the amount of added liquid corresponded to complete pore filling (see below) inside the CPG particles. For the transverse relaxation experiments aimed at verifying that liquid added in a volume equivalent to the determined pore volume got completely imbibed in the pores, liquid was added successively corresponding to 50, 70, 90, 95, 100, 105, 110, and 130% of the pore volume in CPG. After having added any liquid to a CPG sample in an NMR tube, the tube was first centrifuged at 1000G for 1 min to force all liquid into the bed of CPG. Then vigorous vortex mixing was applied for 2-5 min until the samples looked homogeneous. Finally, the samples were centrifuged again at 1000G for 1 min to force any excess liquid down (ideally there should be no excess liquid outside the CPG particles). Diffusion (at 100% pore filling) and relaxation (at the pore filling steps specified above) experiments were performed after having treated the particular sample on this manner. Separate samples were prepared for determining the pore volume in the different CPG materials. An empty PD-10 column (in effect, a 10 ml sample tube equipped with a filter in the bottom, GE Healthcare) with plugs both at the top and at the bottom (below the filter) was weighed. About 350 mg dry CPG was weighed into the column. The bottom and top plugs on the column were removed and the CPG sample inside the column was washed carefully with distilled water. The interparticular water (or rather, most of it) was then removed by centrifugation (1000G, 1 min), the plugs were put back and the column was weighed again. 0.5 ml probe solution with 0.5 wt% native dextran in water with 5 vol% D2O was weighed into the column using a micropipette and the sample was mixed vigorously on a vortex mixer for 1 min and then put on shaking-table (1100 rpm, 1 hour) to allow the solution to equilibrate with CPG. After the equilibration the column plugs were removed and the sample was evacuated again by centrifugation (1000G, 1 min) and 400 µl of the filtrate was transferred with a micropipette to a 5 mm NMR tube.



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Specific pore volume of the CPG materials determined by NMR intensity measurements The experiments were performed on a Bruker 300 MHz Avance III-HD spectrometer. Spectral intensities were measured, on one hand, in the filtrate samples obtained as described above and, on the other hand, in the probe solution that was used in the initial step of sample preparation yielding the filtrate at the end. The goal was to determine two experimental parameters: Vintra, the total volume of the porous network within the CPG particles and Vinter, the volume of interparticle liquid that was left in the particle pack after the centrifugation step above. The material parameter of interest was Vintra while Vinter had to be measured to avoid experimental error in determining Vintra. The experimental principle was that D2O present in the probe solution is diluted by the normal water present in the water volume (Vintra + Vinter) of the centrifuged particle pack. On the other hand, native dextran with its 30×106 g/mol molecular size did not access the pores, only Vinter. Hence,

⎛ I dextran ⎞ Vinter = Vsol ⎜ initial −1 ⎜ I dextran ⎟⎟ eq ⎝ ⎠

dextran where I initial

(1a)

⎛ I D2O ⎞ (1b) Vintra + Vinter = Vsol ⎜ initial − 1 ⎟⎟ D O ⎜I 2 eq ⎝ ⎠ 1 dextran and I eq are the H NMR intensities of the dextran signal (see 1H spectrum in

SI) of the initial test solution and the filtrate solution obtained by equilibration by the particle D2O pack, respectively, while I initial and I eqD2O are the corresponding 2H NMR intensities and Vsol is

the volume of the added probe solution. A single-pulse 2H experiment performed on the lock-channel of a conventional highresolution probe was used to measure 2H integral intensities arising from D2O in the samples. The 90° pulse length was calibrated to 230 µs at 3.5 W rf power. The signal was acquired with eight scans with the total recycle delay set to >5 s. Prior to Fourier transformation, exponential apodization by 0.3 Hz line broadening was applied. The obtained spectra were subject of automatic (5th order polynomial) baseline correction before integration. Detecting the 1H NMR signal for large dextran not able to access pores was complicated by two factors. First, the dominant 1H signal from H2O had to be suppressed and, secondly, the signal from small-molecular impurities in the native dextran (mostly fructose and glucose, 6 ACS Paragon Plus Environment

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a few %) had to be cancelled, too. Both of these were achieved by having recorded the pulsedfield-gradient stimulated echo signal with gradient amplitude and timing parameters (i.e., 100 ms diffusion time, gradient pulse length of 3 ms and gradient strength 0.56 T/m) so that the diffusion weighting factor b (defined in Eq. 2 below) was sufficiently high to completely eliminate the signals from small and therefore rapidly diffusing molecules. The signal from the dextran (see 1H spectrum in SI) was only slightly attenuated. The 90° pulse length was calibrated to 15.5 µs at 7.0 W rf power. 32 scans were acquired with a recycle delay that was more than five times T1 (measured by inversion recovery) for the slowest-relaxing dextran signal. Prior to Fourier transformation, exponential apodization by 0.3 Hz line broadening was applied. The obtained spectra were subject of automatic (5th order polynomial) baseline correction before integration. Here we note that, despite this complication, dextran was selected here as a non-pore marker because other tested polymers of large molecular size, for example, 4×106 g/mol poly(ethylene oxide), became immobilized (as inferred from the huge line broadening it exhibited) by adsorption to the CPG (external and/or internal) surfaces. The results obtained are presented below in Table 1. It is noteworthy that the specific pore volumes and thereby porosities obtained by the manufacturer seem to be slightly lower than that obtained by us. The difference is particularly large for the smallest pores. We ascribe this difference to pore blocking in the mercury porosimetry experiments used by the manufacturer. In contrast to that, we explored the pore space by a wetting liquid that must have exhibited better pore penetration.

Table 1. Material parameters of the explored Controlled Pore Glasses, both as supplied by the manufacturer and as measured here. d (nm)a

Δd (%)b

V (mL/g)c

V (mL/g)d

S (m2/g)e

𝜑f

CPG75

7.5

6

0.47

0.55

140

0.59

CPG115

11.5

7

0.49

0.61

120

0.62

CPG156

15.6

6

0.81

0.82

91

0.68

CPG237

23.7

4

0.95

1.00

78.8

0.73

CPG313

31.3

6

1.00

1.07

66.7

0.74

CPG507

50.7

4

1.00

0.97

43.5

0.72

CPG729

72.9

6

0.75

0.76

24.9

0.67

Label

a

Nominal pore diameter, determined by mercury porosimetry, supplied by the manufacturer.



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b c

80% of pores within the given range, as supplied by the manufacturer.

Specific pore volume supplied by the manufacturer. dSpecific pore volume obtained here.

e

Specific pore surface supplied by the manufacturer. Slightly different values were obtained by independent gas sorption experiments.17 f

Pore volume fraction derived from the specific pore volume data V obtained here, with the assumption of 2.65 g/cm3 density for the solid matrix.

Transverse relaxation experiment to test pore filling by trial liquids As mentioned in the previous section, pore filling is easier to achieve by liquids that wet the pore surface. In this work, we explored several different liquids with different surface tensions. While all were wetting silica, we wanted to make certain that, when added to the dry CPG in an amount that was set to provide accurate filling of the pore space and by the procedure outlined above, all liquid imbibed equally well into the porous network. To test that this was the case, we performed transverse relaxation measurements using the CPMG experiment18 on a Bruker 300 MHz Avance III-HD spectrometer. Performed with the pulse sequence 90°-(τ180°-τ)n , where τ is the interpulse delay and n is the number of spin-echo-blocks, the decay of the single-point intensities for odd echoes at the midpoint between the 180° pulses has been recorded. Both 1H and 2H experiments were performed with rf pulse parameters as provided above. For both 1H and 2H experiments, τ was set to 100 µs and n to a high enough value to let the signal reach the noise level. The recycle delay was set long enough to allow for full thermal relaxation (tested by inversion recovery for the selected liquids). Experiments were performed in samples that were filled by liquids up to particular fractions of the pore volume determined by us, see above. Relaxation data was converted to T2-distributions using the inverse Laplace transform algorithm UPEN (Uniform-Penalty, version UPEN3D6).19, 20 The number of T2-relaxation output points in the distribution was set to 200. Otherwise, the default settings in UPEN were used. Since this test method is a novel experiment, we present its outcome in the Results section below.


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Diffusion measurements Diffusion NMR experiments21 were performed in samples where a particular liquid was added in a volume that corresponded to complete filling of the pore volume in the selected CPG. For each CPG, the diffusion coefficients of the compounds in the following solutions (numbered as I-V) were measured; water (solution I: 10% H2O in D2O, 1H detection), DMSO (solution II, 1

H detection), water with 10vol% DMSO added (solution III: 10vol% DMSO in D2O, 2H

detection), 10vol% DMSO in water (solution IV: 10vol% DMSO-D6 in H2O with 2H detection), and 1-octanol (solution V: 1H detection). The 1H experiments were performed using conventional high-resolution probes with approx. 0.56 T/m maximum gradient strength either on a Bruker 500 MHz Avance III-HD spectrometer (solution I) or on Bruker 300 MHz Avance III-HD spectrometer (solution II and V). The 2H measurements in solutions III and IV demanded stronger gradients and were performed on a Bruker 500 MHz Avance III-HD spectrometer equipped with a DIFF30 probe (used up to 6 T/m magnetic field gradient strength, though capable of delivering maximum 18.3 T/m). All diffusion measurements were performed at 25.0°C with recycle delays set to at least five T1. To suppress the effect of internal magnetic field gradients on the measured diffusion coefficient, the Cotts 13-interval stimulated echo pulse sequence incorporating bipolar gradient pairs separated by 180° pulses was used.22 Rectangularly shaped gradient pulses (with rise and fall times of approx. 100 µs and with stabilization times set to exceed those so that stabilization tests showed no detectable disturbance) were used in all cases and the gradient strength g was incremented to large enough values to suppress the signal by approximately one order of magnitude. The length of each gradient pulse δ in the bipolar pulse pair was 0.85 – 2.5 ms, depending on sample and nucleus. In 1H experiments, the first rf pulse in the sequence was, if needed, set to short tip angles (30°) to avoid radiation damping effects. The diffusion coefficients were extracted by fitting the suitably modified StejskalTanner equation22 to the diffusional decays with E (b) ≡

S (b) = exp(−bD ) S (0)

(2a)

and b = γ 2 g 2δ 2 (6τ1 + 4τ 2 − 2δ / 3)

(2b)



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where 2τ1 corresponds to the period between the first two 90° pulses in the pulse sequence and τ2 is the time between the second and third 90° pulses.22 The diffusion time, defined as the time between the onset of the encoding and decoding bipolar gradient pulse pairs, is thus given by Δ = 2τ 1 + τ 2 .

(2c)

The gradient strength was suitably calibrated23, 24 for separate spectrometers/probes.

Figure 1. Example of the diffusional attenuation curve obtained with Δ= 20 ms and the fit of eq (2) to it for a liquid (DMSO) diffusing in CPG237. Factor b is as defined in eq (2b).

To be able to leave an imprint in simple stimulated-echo NMR diffusion experiments, the internal gradients must have varied over the length scale comparable to the mean diffusional displacement. The latter being in the order of micrometers, it is clear that it is not the magnetic field inhomogeneity within the individual pores that was relevant for our experiments. Rather, the variation over the whole porous grain is the important factor, and that has probably originated, for our roughly spherical grains, from the diamagnetic magnetization of the neighboring particles. While there are more advanced methods for suppressing the effect of internal gradients,25 we found that the Cotts 13-interval stimulated echo pulse sequence22 sufficed to provide accurate diffusion coefficients Deff within the particles. As proof for that, there was no significant difference for the initial (by one order of magnitude) decays of the diffusional attenuation curves obtained with the diffusion time set to Δ = 20, 50 or 100 ms (see SI, the tail of the decays became pronouncedly non-exponential for long Δ, as expected 10 ACS Paragon Plus Environment

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for internal gradients over the length scale of the grains). This latter observation (that is, the lack of Δ-dependence of the initial decay) was valid for any of the liquids explored and thereby the effect of the internal gradients was negligible irrespective of the magnetic susceptibility of the pore filling liquid. Hence, Deff was obtained from the initial decay as shown by Fig. 1. Negligible effect from internal gradients is also shown from having obtained (well within the estimated +/- 1% experimental error) identical Deff from experiments in identical samples at 300 MHz and 500 MHz resonance frequencies.

Results and discussion

Pore filling As discussed above, the filling of pores has been investigated by analyzing the results of transverse relaxation experiments. The experimental principle was as follows. Filling the pores with a small amount of wetting liquid (that is, below pore saturation) sets a large fraction of the liquid molecules in contact with the pore wall. Those molecules experience slower motional correlation times than that in bulk and therefore exhibit faster transverse relaxation. In addition, molecular displacements within the pores may also be slowed down, leading a slower averaging of, for example, internal magnetic field gradients. That effect also leads to faster transverse relaxation at low pore filling. These effects were demonstrated earlier.26, 27 Moreover, partial pore filling creates additional vapor/liquid interfaces within the pores and that increases further the magnetic field gradients and, thereby, the transverse relaxation rate. In summary, upon increasing pore filling the transverse relaxation times are supposed to increase.



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Figure 2. The distribution of the transverse relation time T2 obtained by Inverse Laplace Transformation of the transverse relaxation decay from a 1H CPMG experiment on water in CPG507. The water added to the dry CPG corresponded to 130% of the experimentally measured pore volume. The full width at half-maximum (FWHM) of the distribution is marked with the arrow.

Furthermore, pore structure is, even in the rather regular CPGs, random and therefore is bound to exhibit irregularities. In other words, one can plausibly expect that the increase of the transverse relaxation rate in a partially filled porous structure exhibits local variation. If so, one also expects that the distribution of the transverse relaxation times T2, obtained on a manner discussed in the previous section and illustrated in Fig. 2, becomes broader if liquids can only partially fill the pore space. One can expect extra broadening also if there is superfluous liquid in the system since in that case a fraction of the liquid remains outside the pores and, in those liquid domains, the transverse relaxation attains a value that is assumedly similar to that exhibited in bulk liquid. In summary, one expects that the distribution of transverse relaxation rates becomes narrowest at complete pore filling.


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Figure 3. T2 distribution width versus the degree of pore filling (% of total pore volume, as obtained from the specific volume determined by us) for CPGs of different pore diameters as obtained by (a) 1H and (b) 2H as probe nuclei. The dashed lines mark pore volume as specified by the supplier based on data from mercury intrusion porosimetry, for CPG 115 (pore diameter 115Å, red) and CPG 237 (pore diameter 237Å, blue).

Indeed, as illustrated by some of the distribution width data in Fig. 3, the experimental behavior seems to follow these expectations. Since different nuclei exhibit different sensitivity to different relaxation mechanisms, it is not expected that they show the same quantitative behavior. Yet, for both 1H and 2H data and for all investigated liquids, we found that the T2 distribution width takes its minimum value in the 90-100% pore filling range even though there are clear differences between the different liquids and CPGs as concerning quantitative trends. This latter discrepancy we ascribe in part to the known sensitivity of ILT results to experimental noise. Yet, the data also clearly indicate that - upon having supplied initially dry 13 ACS Paragon Plus Environment


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CPGs with either water, DMSO, or mixtures of those or octanol in a volume that corresponds to the pore volume in that given CPG - one obtains complete pore filling. Hence, the diffusion experiments in such systems report on translational dynamics in continuous liquid-filled regions.

The pore structure and its scaling properties in CPG materials CPGs are prepared by quenching a molten glass mixture below its critical point and then letting it phase separate for a set time. After that and at a yet lower temperature, one phase consisting of B2O3 is leached out. After some additional etching of the remaining SiO2 matrix by NaOH, the procedure ultimately yields a porous structure that has a typical porosity in the range of 50-75%. The initial spinodal phase separation step defines the pore shape that is best characterized as a multiconnected random tubular network that seems bicontinuous both as regarding the pore space and the matrix.13, 14 The process seems to follow a simple scaling function of time permitted for the phase separation to proceed so that the pore structure retains its general shape and topology but its characteristic size increases. Hence, larger average pore sizes are created by setting longer phase separation times.15, 16 Experiments have also shown that the pore structure is homogeneous on length scales larger than the average pore size.28 These features make CPGs a good model material for investigating the effect of wall interaction with the solvents on the transport properties. Namely, the similarity of the structures, apart the linear scale, makes that one expects the same average tortuosity on length scales that are far above the pore size.


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Figure 4. Self-diffusion coefficient versus CPG pore diameter. (a) Self-diffusion coefficients normalized to the bulk diffusion coefficient D0 of the respective liquid. (b) Self-diffusion coefficients normalized to the average diffusion coefficient for the three CPGs with largest pores. (c) The same data as in (b) compared to predictions based on the Renkin equation (see SI) for DMSO (blue), 1-octanol (green), and water (red). The symbols represent DMSO (▲), 15 ACS Paragon Plus Environment


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DMSO in (10vol% DMSO : 90vol% water) ( ), 1-octanol (■), water in (10vol% DMSO : 90vol% water) (○), and water (●).

The CPGs investigated here have an average nominal pore size in the 7.5 – 72.9 nm range. On the other hand, the self-diffusion coefficients we obtained for the various probe liquids was in the order of 10-10 – 10-9 m2/s. Even at the minimum diffusion time of Δ = 20 ms explored here, the mean-square displacement becomes >1 µm, that is far above the pore size. The pore network was stated to be multiconnected13, 14 and the lack of Δ–dependence in the initial diffusional decay (as that in Fig.1) also indicates that there is no significant obstruction21 (that is, no displacement out of given geometric enclosure). Regarding the invariance of pore shape and topology of the CPG structures, we have two results in slight apparent contradiction with each other. On one hand, Table 1 shows that porosity varies a bit along the CPG series, from 𝜑 = 0.59 at lowest pore size (7.5 nm) to 𝜑 = 0.74 at the middle of the range (31.3 nm). Invariant pore topology should provide invariant porosity. On the other hand, Fig. 4 shows that for the three largest pore sizes the diffusion coefficients are virtually the same irrespective of the pore liquid, even though there is a sizeable variation (from 0.67 to 0.74) in the porosity. This latter observation shows that tortuosity expressed as τ = (D0/Deff) should vary only weakly upon varying 𝜑. In the rich flora of available (and often phenomenological or semi-phenomenological) theories,10 there are several choices consistent with this observation. Hence, to summarize, even though porosity varies slightly over the selected CPGs, the purely geometric tortuosity factor (that is, tortuosity that solely depends on excluding diffusive pathways from the region occupied by the matrix) should remain close to invariant over the range of glasses. Contrary to this expectation, the ratio D0/Deff exhibits a strong variation towards the smallest pore sizes.

The effect of solvent-wall interaction on diffusion There are two systematic attempts29, 30 known to us to characterize the pore size dependence of diffusion in liquids with results that are partly contradictory to each other. Kärger et al investigated the diffusion of water, methanol, dodecane and dodecene in porous glasses with average pore diameters 1.1-1.4, 5.8-7 and 45 nm.30 As one shortcoming relative to the data here, their glasses into which the solvents were imbibed may have exhibited different 16 ACS Paragon Plus Environment

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structures/topologies (that is, were not scalable to each other). Thanks to the development since in the area of porous-glass manufacturing, our sample series is far better suited to investigate the onset of the wall interaction on transport, both because of the scalable structure but also because of the more even spacing of available pore sizes. Yet, in their smallest pores Kärger et al have clearly observed that the interaction of the liquid with the pore wall is the factor that dominates over geometric tortuosity in defining diffusive transport. A few years later, Fukuda et al investigated the diffusion of water molecules in porous silica glasses.29 Their home-made glass samples31 were prepared in a manner similar to that used for preparing CPGs and, assumedly, they thereby obtained scalable structures, too, with narrow pore size distributions akin to that in current commercial CPGs. Yet, with only three investigated average pore diameters of 3.8, 16, and 45 nm they failed to recognize that their data (if re-plotted similarly to Fig. 4) clearly indicates the presence of interactions with the wall (specific and/or hydrodynamic, see below). Hence, their erroneous conclusion was that diffusion in the 3.8 - 45 nm pore size range is solely dependent on the geometric tortuosity. Kimmich et al arrived at the same conclusion, yet on the basis of data recorded at even fewer pore sizes (4 nm and 30 nm) and those with possibly different pore topologies.32 Our data presented in Fig. 4 indicate that geometric tortuosity fully defines diffusive transport for pore diameters above 30 nm. The reason for the small difference in Fig. 4a between, on one hand, water (and water with 10vol% DMSO), on the other hand, DMSO, octanol, and DMSO dissolved in water is unclear. The feature is reproducible and it seems not to be connected with differences in pore filling. One could, of course, assume that – within the limits as defined by Fig. 3 – a few % of water remained outside of the pore network and that leads to its faster average diffusion while all DMSO got imbibed. Yet, different molecular components, water and DMSO, in a solution behave also differently and, therefore, this feature cannot be connected to difference in pore filling. Our hypothesis is that this difference between the solvents is caused by a small fraction of micropores with ~nm pore sizes present in the system in which molecules may become trapped for times long enough (>>ns) to limit their diffusion but short enough (

Diffusive Transport in Pores. Tortuosity and Molecular Interaction with

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