Binary fluids in mesoporous materials: Phase separation studied by NMR relaxation and diffusion

Abstract

Relaxation and diffusion measurements were carried out on single and binary liquids filling the pore space of controlled porous glass Vycor with an average pore size of about 4 nm. The dispersion of the longitudinal relaxation time T₁ is discussed as a means to identify liquid-surface interaction based on existing models developed for metal-free glass surfaces. In addition, the change of T₁ and T₂ with respect to their bulk values is discussed, in particular T₂ serves as a probe for the strength of molecular interactions. As the native glass surface is polar and contains a large amount of hydroxyl groups, a pronounced interaction of polar and protic adsorbate liquids is expected; however, the T₁ dispersion, and the corresponding reduction of T₂, are also observed for non-polar liquids such as alkanes and cyclohexane. Deuterated liquids are employed for simplifying data analysis in binary systems, but also for separating the respective contributions of intra- and intermolecular interactions to the overall relaxation rate. Despite the lack of paramagnetic impurities in the glass material, ¹H and ²H relaxation dispersions of equivalent molecules are frequently found to differ from each other, suggesting intermolecular relaxation mechanisms for the ¹H nuclei. The variation of the T₁ dispersion when comparing single and binary systems gives clear evidence for the preferential adsorption of one of the two liquids, suggesting complete phase separation in several cases. Measurement of the apparent tortuosity by self-diffusion experiments supports the concept of a local variation of sample composition within the porespace.

Graphical abstract

1. Introduction

The theoretical description of the T₁ dispersion of liquids in “clean” (i.e., metal-free) porous glass has been established decades ago, distinguishing between the limiting cases of strong and weak adsorption of the molecules on the native glass surface with its large amount of hydroxyl groups [1]. The model predicts vanishing T₁ dispersion in the weak adsorption case, and strongly pronounced dispersion for those molecules that encounter preferential orientation on the surface, allowing for a low-frequency dispersion component by experiencing the surface curvature via anomalous diffusion. Therefore, this process is intramolecular by nature and is expected to lead to equivalent dispersions for protons and deuterons which has indeed been observed for porous glasses [1]. The actual properties of molecular reorientations, which are entering into the relaxation time computation as the spectral density function I(ω) being the Fourier transformation of the autocorrelation function of orientations G(t), have been shown to be related to the geometry of the surface. Neglecting any chemical or physical details for the nature of the interaction, it was shown, based on theoretical predictions by Bychuk and O'Shaughnessy [2], that the surface orientation of the molecules is relevant for the intermediate frequency dependence of the relaxation rate: fast reorientations of molecules travelling through the bulk-like liquid at some distance from the surface do not, despite fast rotation of the molecules, lead to a full isotropization and leave a residual finite value of the autocorrelation function G(t) which is entirely given by the change of orientation at the surface. The latter, for the assumption of one single preferential orientation relative to the surface, can only decay if the molecule moves along a curved surface. Experimental evidence for porous glass with different pore sizes [3] and also for microscopically flat clay platelets [4] has supported the model. The dispersion of the relaxation times, T₁(ω), reflects the surface curvature, which may possess self-similarity within a certain range of length scales. As a consequence of this so-called reorientations mediated by translational displacements (RMTD), non-classical diffusion relative to the surface is observed while molecules still undergo normal Gaussian diffusion during their excursions in the bulk. Removing the bulk excursions by freezing leaves a non-frozen layer close to the surface which must follow classical diffusion and affect the relaxation dispersion T₁(ω); the quantitative demonstration of this change was considered a convincing proof of the RMTD model [5,6].

However, the RMTD approach contains a number of assumptions. First of all, it requires the absence of significant amounts of paramagnetic impurities near the surface which would dominate relaxation if exceeding a critical concentration, leading to a different dispersion which is again modeled by the motion of molecules with respect to the surface. Paramagnetic impurities are typically rare in many types of glasses, but dominate relaxation in natural porous media such as rocks and soils [7,8]. This dominance is observed for spin-1/2 nuclei such as protons, but may also affect, at higher concentration, the T₁ dispersion of nuclei that would normally relax primarily via quadrupolar coupling such as ²H. Even in the absence of unpaired electrons, the limiting cases entering the RMTD approach – either “strong” or “absent” interaction with the surface – often are not met, most frequently in very small pores on a scale of few nanometers. The interaction of liquids with surfaces may then be considered “intermediate”, and the T₁ dispersion will be different for each type of molecule. The actual interaction of molecules with the surface, internal dynamics and intermolecular contributions then need to be considered as non-negligible additional contributions to the RMTD process. In general, only very limited data are available in the literature about T₁(ω) relaxation dispersion in materials of very small pore sizes, as well as deuteron relaxation data that can be employed for the verification of models by comparison with proton relaxation.

With the goal of gaining more detailed insight into the relaxation processes of a wider range of liquids, we have therefore carried out (a) ²H relaxation dispersion measurements of perdeuterated liquids and (b) ¹H and ²H studies of binary systems of two fluids in Vycor that are either miscible or immiscible in the bulk. For distinguishing different mechanisms, we have chosen fluids with different polarity as well as protic (water, alcohols) and aprotic liquids (acetone, tetrahydrofuran (THF), cyclohexane, n-alkanes), both as single components and in selected mixtures of equal amounts. In addition to Fast Field Cycling (FFC) profiles, relaxation times and diffusion coefficients were determined at a fixed field of 1.0 T.

2. Theoretical background

The expressions for homonuclear dipolar relaxation or quadrupolar relaxation, respectively, are as follows [9]:

1/T₁ (ω) = const [I(ω) + 4I(2ω)] (1)

1/T₂ (ω) = const/2 [3I(0) + 5I(ω) + 2I(2ω)] (2)

where I(ω) is the spectral density function and the prefactors are given by const_D = (μ₀/4π)² γ⁴ ħ²/5 r⁻⁶ I (I+1) for homonuclear dipolar relaxation, and const_Q = 3/80 (e²qQ/ħ)² (1+ η²/3) for quadrupolar relaxation, respectively; γ is the gyromagnetic ratio, I the spin quantum number, r the internuclear distance, Q the quadrupolar moment, q the electric field gradient and η the asymmetry parameter of the electric field [9]. Note that both expressions differ only by the constant prefactor, i.e., the frequency dependence of T₁ is identical in shape; the prefactor is often assumed to be eliminated by normalizing the measured relaxation times with the bulk liquid value which is a field-independent constant for liquids of low viscosity.

This simple approach, however, is only applicable if the same reorientation process is responsible for the relaxation of ¹H and ²H. In reality, the intermolecular interaction of the dipolar relaxation represents a significant contribution to relaxation, and possesses a different frequency dependence by being governed by the modulation of interspin distances due to relative motion of two molecules. For many bulk liquids, the inter- and intramolecular contributions are comparable in the ¹H relaxation, different from the purely intramolecular ²H quadrupolar relaxation. Since the intermolecular interaction is modulated on a similar timescale as the molecular rotation governing intramolecular relaxation, it likewise leads to a frequency independent constant so that the abovementioned normalization only results in an error in absolute value but not in the shape of the dispersion. For a proper treatment, the experimentally observed relaxation rate must be decomposed into its two components:

1/T_1,2 (ω) = 1/T_1,2,intra (ω) + 1/T_1,2,inter (ω) (3)

One established way of computing both contributions experimentally is by partial deuteration of the liquid which reduces the intermolecular component in proportion to the ¹H spin density.

Note that relaxation via unpaired electrons is equivalent to the case of heteronuclear dipolar coupling and has a formal structure somewhat different from Eqs. (1), (2). In this study of Vycor porous glass, we assume the absence of significant amounts of paramagnetic impurities and will not discuss this situation which has been treated extensively by Korb et al. for other porous materials (see Refs. [7,8] and references therein).

The RMTD model mentioned in the Introduction describes the case of strong interaction, when molecules assume a preferential orientation with respect to the solid surface. This interaction has unspecified duration but ensures that the autocorrelation function of orientation retains a non-zero value at times much exceeding the bulk rotation time. In the other extreme, molecules are considered to have random orientation at the surface, given by weak or absent interaction; they are expected to behave bulk-like. The strong adsorption case has been found to apply for small polar molecules on polar glass surfaces, where hydrogen bonding to the surface hydroxyl groups may be an additional contribution to the preferential molecular reorientation [10]. In particular, RMTD has first been demonstrated to exist for water and D₂O in glass and on silica surfaces, where the agreement of the T₁(ω) dispersion of both ¹H and ²H nuclei in these molecules, as was found in glass of 30 nm pore size, can be seen as proof of the dominating intramolecular nature of the relaxation process [1].

While the non-interacting scenario is a limiting case, alkanes in porous glass or aluminum oxide come close to this limit and have shown very weak, although not vanishing, T₁(ω) dispersion [1,11,12]. On the other hand, in the case of an inorganic surface covered with a layer of hydrocarbons (“coke”), the affinity of alkanes generates very similar T₁(ω) dispersion but a flat dispersion with the native surface [13]. In that study, the dispersion of water ¹H nuclei was found to maintain a very pronounced relaxation dispersion for any surface composition, but the surface fractal dimension for different coke content as obtained from the RMTD model changed in full agreement with the surface roughness obtained by independent methods. More recently, similar observations were made in native and oil-treated rocks [14].

In order to make a quantitative prediction of the spectral density function following the RMTD process, the surface roughness needs to be parametrized. For instance, this may involve the assumption of a single radius of curvature, or a fractal distribution of surface modes k following a function S(k) = bk^−χ [6], with 0 < χ < 1. χ = 0 represents equipartition of surface modes; b is a constant being related to the surface roughness. Assuming random walks of the molecules within the bulk phase, it has been suggested that this process follows non-classical Lévy walks so that one obtains [5].

$$I\left(\omega \right)=\frac{b}{4\mathrm{\pi }\sin \left(\frac{\mathrm{\pi }}{2}x\right)}{c}^{-\left(1-x\right)}{\omega }^{-x}$$ (4)

If, on the other hand, “normal” (Gaussian) diffusion is assumed, one finds

$$I\left(\omega \right)=\frac{b}{8\mathrm{\pi }\sin \left(\frac{\mathrm{\pi }}{4}\left(1+\chi \right)\right)}{D}^{-\left(1-\chi \right)/2}{\omega }^{-\left(1+\chi \right)/2}$$ (5)

with the bulk diffusion coefficient D and the adsorption depth h [2] so that c = D/h.

For the most simple assumption of an equipartition of surface modes (χ = 0), this leads to T₁(ω)∼ω^1/2 for normal surface diffusion but to a logarithmic dependence

$$I\left(\omega \right)\sim {\tau }_u\left[2\ln \left(\frac{{\tau }_l}{{\tau }_u}\right)+\ln \left(\frac{1+{\omega }^2{\tau }_u^2}{1+{\omega }^2{\tau }_l^2}\right)\right]$$ (6)

for Lévy walks. In this equation, τ_i = 1/(ck_i) where k_i is the surface mode according to Ref. [5]; k_l and k_u represent the lower and upper cutoff surface modes, and the corresponding cutoff times are given as τ_l = 1/(ck_l) and τ_u = 1/(ck_u), respectively. For times meeting the inequality τ_u « t « τ_l, eq. (6) simplifies to T₁(ω)∼1/ln(ωτ_u) [5]. The temperature dependence of relaxation is governed by the temperature dependence of the self-diffusion coefficient D which, however, vanishes in the limiting case χ = 1. While in porous glasses as well as saponite clays (essentially free of paramagnetic impurities), Lévy walk statistics were found [4,5], motion within gels has been modeled with Gaussian diffusion properties [15]. In fact, the reconstruction of the surface autocorrelation function of motions without the abovementioned assumptions has resulted in functions to decay above a certain cutoff length that correlated with the average pore size for glasses of 4, 30 and 200 nm pore diameter, including Vycor [3,16].

When discussing diffusion measurements, it must be kept in mind that relaxation and diffusion experiments are carried out on entirely different timescales. While relaxation is sensitive to molecular reorientations on the scale of the reciprocal Larmor frequency, i.e., of about 10⁻⁵ s and below, self-diffusion experiments are limited by the experimentally accessible magnetic field gradient strength and typically cover times longer than several 10⁻³ s. In fact, NMR measures the mean-squared displacement in the direction of the applied gradient, not the self-diffusion coefficient itself; the latter is – as in all other techniques – computed from the experimental data based on an assumption. In NMR, this assumption is the validity of the expression

<X²(Δ)> = 2 D Δ (7)

with Δ being the evolution time over which diffusion takes place. If the case of “normal” diffusion, i.e., a Gaussian propagator of displacements X = x(Δ)-x(0) and a proportionality of the mean-squared displacement to time, is not met, this is conventionally expressed by a time-dependence of the self-diffusion coefficient, D(Δ), but it should be considered that this concept may be incompatible with the definition of diffusion coefficients as obtained by other measurement techniques.

In the same context, different concepts of the term “tortuosity” exist, where the common idea is that molecules in a porous medium move “slower” as compared to bulk since they need to traverse longer paths to reach a particular distance in 3d topological space; this may be a consequence of pure geometric restriction, but also of extended residence times of a molecule on the surface due to interactions. The behavior of chain molecules in size-exclusion chromatography, often using functionalized pore surfaces, is an extreme example of this effect. In this study of small molecules, we assume the phenomenological definition of considering the tortuosity τ as the ratio of the self-diffusion coefficient in bulk and in the porous medium, respectively; for vanishing time dependence this is identical to the ratio of the respective mean-squared displacements. In porous media, self-diffusion is indeed dependent on the choice of the diffusion period Δ, in fact this time dependence can be exactly computed for defined geometries and becomes pronounced if the root mean square (rms) displacement is on the order of the structural size of the system, most often the pore size. Time-dependent self-diffusion measurements have therefore been employed frequently to determine the pore size distribution of porous media [[17], [18], [19]]. Under the assumption of purely geometric effects, the self-diffusion coefficient decays from its bulk value at very small displacements to a limiting value at displacements far exceeding the largest structural size of the system. For all liquids studied here, a room temperature bulk self-diffusion coefficient on the order 10⁻⁹ m² s⁻¹ and diffusion times exceeding 10 ms result in rms displacements of at least several micrometers, about three orders of magnitude larger than the average pore size of Vycor. In earlier works [20,21] it was shown that D is indeed time-independent in the experimentally accessible range, thus excluding the presence of voids or structural inhomogeneities on a scale of μm or larger. It is henceforth assumed that D is time-independent and that the reduction of D inside the porous glass is entirely due to geometrical hindrance, which results in identical values of tortuosity for all liquids. Deviations from this simplified scenario will be discussed.

It must finally be noted that, while self-diffusion in a single-component fluid is a well-defined quantity under conditions accessible to the NMR experiment, the relative mobility of molecules in binary mixtures is less straightforward to interpret. Even in the absence of a concentration gradient, the so-called intradiffusion coefficient in a mixture of two liquids A and B is dependent on composition and may follow strongly non-linear behavior as a result of the interactions between molecules of type A and B. The diffusivity of a mixture A + B can assume values that are either larger or smaller than the respective values of both liquids A and B separately. A large body of literature deals with the proper determination and prediction of the concentration dependence of diffusive properties on systems of two and more components and will not be discussed here. However, it is important to note that the interpretation of experimental results by the assumption of a geometrical tortuosity may fail if a binary system of either miscible or immiscible liquids is studied in a mesoporous material, as will be addressed in the Results section.

3. Experimental

Vycor porous glass (Corning 7930) with a nominal pore size of 4 nm was purchased in rods of approx. 3 mm diameter and was cut into pieces of 20–30 mm length. Following the recommended procedure, all samples were first boiled for 60 min in a 30% H₂O₂ solution and subsequently dried in vacuum for around 12 h to remove organic contaminants which are adsorbed from the atmosphere. This process was carried out twice and the cleaned samples were then placed in vials filled with the appropriate liquid for 24 h. The following liquids were employed: water, acetone, ethanol, hexanol, n-hexane, n-tetradecane, cyclohexane, tetrahydrofuran; as well as the deuterated compounds D₂O, acetone-d6, nonane-d20, cyclohexane-d12, tetrahydrofuran-d8. Glass rods were taken out of these vials and wiped superficially to remove bulk liquid, they were then placed in either 5 mm OD (one piece) or 10 mm OD (three pieces) NMR tubes and flame-sealed after the free volume was filled by a glass rod. Mixtures of 50:50 vol-% of miscible liquids were prepared the same way. In this work, results for mixtures of THF and cyclohexane (one deuterated, one undeuterated) are shown. All undeuterated liquids as well as D₂O were purchased from Sigma-Aldrich, Taufkirchen, Germany; deuterated THF, acetone and cyclohexane were obtained from Deutero GmbH, Kastellaun, Germany. Nonane-d20 was obtained from Akademie der Wissenschaften der DDR, Zentralinstitut für Isotopen-und Strahlenforschung.

Relaxation dispersion measurements, i.e., frequency-dependent longitudinal relaxation times T₁(ω), were obtained on a Stelar Fast Field Cycling (FFC) relaxometer (Stelar, Mede, Italy) at magnetic field strengths between 0.02 mT and 0.7 T. For detection, the probes were tuned to 11 MHz (¹H) and 3 MHz (²H), respectively, with the detection field set at the corresponding strength of 0.26 T or 0.46 T. Signal acquisition was realized with a CPMG pulse sequence. All T₁ relaxation curves were found to follow a monoexponential function. All experiments were carried out at room temperature (293 K). Typically 4 accumulations, respectively, were carried out for ¹H nuclei, and 16–32 accumulations for ²H.

Additional relaxation measurements of T₁ and T₂ at constant field strength were carried out at 1.0 T (SpinSolve, ¹H only) and 7.05 T (Bruker Avance III, ¹H and ²H). All relaxation functions were found to be monoexponential within experimentally given error limits.

¹H self-diffusion measurements on selected samples were carried out on the abovementioned Spinsolve spectrometer employing pulsed gradient stimulated echo (PGSTE) sequences with gradient pulse durations δ of 5 ms and pulse separations Δ of 200 ms.

4. Results and discussion

4.1. Relaxation of single-component liquids

The frequency dependence (i.e., dispersion) of the longitudinal relaxation time, T₁(ν), of the ¹H nuclei of several pore-filling liquids is presented in Fig. 1 [21], where the conventional practice has been followed to indicate ν = ω/2π instead of the Larmor frequency ω proper. As in all other experiments, ¹H relaxation was found to be monoexponential within at least one order of magnitude. The results were normalized by the respective bulk value to facilitate comparison. The findings generally follow the tendency described in early studies [1] carried out in glass with a larger pore size (30 nm) where adsorbates clearly fall into two categories, polar molecules (water, acetone, alcohols) with a strong dispersion and non-polar ones (alkanes, cyclohexane) with a small but not vanishing dispersion. In Vycor, however, this grouping is less obvious, and one observes a rather gradual transition from weak dispersion (long alkanes) to strong dispersion (acetone, water). The “strength” of the dispersion may be approximated either by an apparent slope in the double-logarithmic representation, indicating a power-law T₁ ∼ ω^γ, or by the ratio of the high-frequency and low-frequency limits of T₁. The latter is not consistently available since the observed values show frequency dependence even at the upper and lower limits that were accessible experimentally. A crude estimate for the degree of dispersion, which is considered as a measure of interaction strength of the molecules with the surface, is therefore given by the ratio T₁/T₂ at a given magnetic field strength. This ratio has been established as a suitable quantity in other porous media studies, foremost in oil well logging, and will be discussed subsequently.

Fig. 1.

Relaxation dispersion T₁(ν) of the ¹H nuclei of liquids in Vycor porous glass at pore-filling conditions. Experimentally determined values have been normalized by the respective bulk values in order to facilitate comparison (modified from Ref. [21]).

Assuming the validity of the RMTD model which does not specify interactions or molecular geometry, a difference in dispersion must be assigned to a difference in molecular reorientations on the surface. At the same time, one proof for the RMTD model would be the equivalence of the T₁ dispersion for ¹H and ²H nuclei as has been reported for the case of water before [1].

Fig. 2a summarizes the results for four deuterated liquids that have been prepared in Vycor under identical conditions as the liquids presented in Fig. 1; nonane-d20 is considered in comparison to n-hexane and n-tetradecane which both show significant but rather weak dispersions. Unlike for ¹H, a clear distinction between polar and non-polar liquids is not found any more for the ²H relaxation; while the dispersion of D₂O appears to be weaker than for water H₂O, it is stronger for the remaining three deuterated liquids acetone, cyclohexane and nonane. The relaxation rate induced by the presence of the pore surface becomes even more similar when plotted as a difference to the bulk values (Fig. 2b). In general, the qualitative difference between ¹H and ²H relaxation dispersion of the same molecules suggests that the RMTD model is insufficient to describe the experimental results, it is rather possible that an additional, intermolecular relaxation contribution becomes relevant. This also supports the gradual change of relaxation properties for the liquids of different polarity and size. For the particularly strong response of acetone, a surface reaction as has been observed on alumina surfaces [10] is unlikely but cannot be ruled out.

Fig. 2.

(a) Relaxation dispersion T₁(ν) of the ²H nuclei of liquids in Vycor porous glass at pore-filling conditions. Experimentally determined values have been normalized by the respective bulk values in order to facilitate comparison. (b) Plots of the same data as the difference of the relaxation rates, R₁(ν), and the corresponding bulk values.

4.2. Relaxation of two-component liquids

Mixtures of cyclohexane and THF, as well as methanol and THF, had been investigated by NMR relaxometry before as examples for different adsorption properties of fluids [22]; in that work the substrate had been γ-alumina. THF was identified as the preferentially adsorbed species compared to cyclohexane, but less preferentially adsorbed compared to methanol; accordingly, the relaxation rate of THF was found to increase and decrease, respectively, upon increasing THF concentration. The actual shape of the dispersion, however, remained virtually unchanged for cyclohexane/THF, with a slightly visible effect for the methanol/THF mixtures.

In this study, we have compared 50:50 (by volume) liquid mixtures with the corresponding pure liquids, with either both components containing ¹H nuclei or one component being perdeuterated. Fig. 3 shows the data for the system cyclohexane/THF with one protonated and one deuterated compound each, and the results for the protonated THF for comparison. Apparently, the dispersion of cyclohexane is practically absent except of a high-frequency variation above several MHz of Larmor frequency; this compares with a T₁ of 3.65 s (frequency independent) in the bulk mixture so that a reduction of T₁ by a factor of 2–3 is indeed observed. The observation is at variance with the consistent finding of a significant dispersion of approximately T₁∼ω^0.2 to ω^0.25 for cyclohexane in Vycor at saturation conditions (see Fig. 1 and [11]). This lack of dispersion was also found for similar systems in Vycor such as cyclohexane/acetone and immiscible binary liquids employing n-alkanes (data not shown). For the THF dispersion, a mere vertical shift with a virtually unchanged slope of T₁∼ω^0.6 is observed. The dispersion possibly extends towards much higher Larmor frequency since the T₁ value in the bulk mixture of 4.37 s is not reached by far. For both liquids, the dispersion is stronger in Vycor than in alumina of similar pore size [12], whereas the tendency is the same if even more pronounced.

Fig. 3.

Relaxation dispersion T₁(ν) of the ¹H nuclei of liquids in Vycor porous glass at pore-filling conditions for mixtures of tetrahydrofuran (THF) and cyclohexane at 50:50% volume fraction compared with the dispersion of pure THF in Vycor. In the mixtures, one component was deuterated and the ¹H signal of the remaining component (marked in boldface) was measured.

As the relative shift of the absolute values of the THF relaxation times is concerned, we would only like to enumerate possible reasons without providing quantitative evidence at this stage: First, mixing with a perdeuterated substance generally reduces spin density and therefore the intermolecular contribution to relaxation, leading to longer relaxation times. This effect can be verified by the available bulk liquid results for ¹H T₂ and T₁, respectively. Second, since the viscosity of a mixture can be either significantly larger or smaller than that of the individual components, a corresponding shift of the bulk relaxation times must follow as they are dominated by the molecular rotation and translation dynamics; the surface dynamics, assumed to be responsible for the T₁ dispersion, may be directly proportional to this timescale. Third, the possibility of a partial or complete phase separation inside the pore space must be taken into account; in fact, the absence of T₁ dispersion of cyclohexane in the presented example suggests that cyclohexane molecules either do not have contact with the glass surface at all, or have insufficient opportunities to interact with this surface in order to generate detectable relaxation. Such a tendency towards phase separation has already been suggested in Ref. [11] and is the topic of research employing Molecular Dynamics simulations. Fourth, it has been suggested that the liquid density in nanometer-sized pores can vary by large amounts [23,24] — a lower molecular density will again reduce the intermolecular contribution to relaxation, while it will also facilitate diffusion and therefore affect the molecular reorientation dynamics. At this stage, it must be beyond the task of this work to separate and quantify these different contributions, although a change in the shape of the relaxation dispersion is expected to be the consequence of the adsorption probability alone, and therefore of the spatial distribution of molecules inside the pores.

It should be mentioned that the ratio of relaxation times in confinement and in bulk or, as introduced above, the ratio T₁/T₂ in confinement (being identical to unity in bulk by theoretical assumptions of rotation of translation of molecules without internal dynamics) can be understood as measures of the relative interaction strength of molecules. The ratio T₁/T₂ is dependent on Larmor frequency and may not constitute a “good” parameter in this study where the experimental range of magnetic fields does not cover the whole dispersion region. On the other hand, the ratio T₂^bulk/T₂^Vycor at a constant magnetic field strength gives a good estimate of the molecular interaction with the porous medium.

This ratio T₂^bulk/T₂^Vycor approaches or exceeds 1000 for strongly polar liquids such as water and acetone in Vycor; the same order of magnitude is found for both ¹H and ²H relaxation. For cyclohexane in mixtures with THF or acetone, these ratios were found to be only 2.3 and 1.3 for ¹H and 1.8 and 1.4 for ²H, respectively, at similar Larmor frequencies of 43 MHz and 46 MHz. This compares to a ratio T₁/T₂ of 7.4 for cyclohexane as a single liquid filling Vycor. Therefore the dynamics of cyclohexane in the mixture is barely hindered by the presence of the glass surface at all, which is indeed remarkable considering that the studied liquids are fully miscible in bulk so that the occurrence of stable phase boundaries inside the porespace cannot be assumed a priori.

4.3. Diffusion measurements

If the concept of phase separation, partial or complete, were indeed correct, one would expect the overall diffusivity of molecules in mixtures to be affected. For instance, if one assumes the picture of one component completely covering the surface, the available diffusion trajectories for its molecules would be even more reduced compared to the liquid-filled porous medium, and a lower diffusion coefficient would be expected. In the case of isolated pockets of one component, the mean-squared displacement will be limited to a finite value and the computed diffusion coefficient will approach zero for long observation times.

The apparent tortuosity, i.e., the ratio of the bulk and the confined self-diffusion coefficient, had been determined to be in the range of 5–7 for simple liquids in Vycor [20,25], with the possible exception of larger molecules that might encounter increased hindrances to their mobility. These findings could be qualitatively confirmed for ²H measurements of several liquids in this study (data not shown).

For all mixtures studied so far (water with THF or acetone; cyclohexane with THF or acetone; THF and acetone, generally with one component in its deuterated form and the proton-containing molecules measured), the reduction of the self-diffusion coefficient with respect to the bulk value ranged between 2.7 and 12.0. In most cases, a large apparent tortuosity for the one component agrees with a small value for the other. This appears indicative of a concentration gradient, or partial phase separation, within the pore space. When considering that the self-diffusion coefficients, just as the viscosity, is strongly concentration-dependent in mixtures, a variation of local concentration can explain the observed change in self-diffusion coefficient and the comparison with the corresponding bulk values is rather meaningless, since the reference at a given concentration (50:50) is not reflected in the actual liquid distribution within the pore space, although the total amount of liquid inside Vycor is equivalent to that value and has been confirmed by quantitative analysis of the ¹H spectra of the corresponding mixtures of undeuterated liquids.

For the example discussed earlier, cyclohexane and THF, self-diffusion coefficients are reduced by factors of 9.9 (cyclohexane) and 12.0 (THF), respectively, assuming almost identical values of 2.24 × 10⁻¹⁰ m²/s in Vycor at T = 293 K (see Fig. 4), which is certainly a coincidence. For comparison, the corresponding ratios in the Vycor sample filled with cyclohexane and acetone are 4.3 (cyclohexane) and 7.6 (acetone) while the measured values were 6.4 × 10⁻¹⁰ m²/s and 5.2 × 10⁻¹⁰ m²/s, respectively. Cyclohexane in mixtures with either THF or acetone features a virtual absence of T₁ dispersion. If the assumption of a complete separation were correct, cyclohexane would be found in the center of the porespace, without contact to the interface, and the corresponding self-diffusion coefficients of the cyclohexane phase itself would be expected to be identical, equal volume fractions of liquids assumed. However, the self-diffusion coefficients of cyclohexane inside Vycor differ by a factor of 3. Interestingly, cyclohexane completely filling the porespace of Vycor was found to possess a value of 2.2 × 10⁻¹⁰ m²/s at 294 K – its intraporous self-diffusion coefficient in a mixture with THF therefore is similar to the pore-filling single phase, while in the mixture with acetone, it diffuses three times faster, compared to twice faster in the bulk mixture. These findings combined suggest that a full phase separation of both liquids inside Vycor cannot be assumed, despite the conspicuous influence onto the cyclohexane ¹H relaxation dispersion.

Fig. 4.

Diffusion decays measured by a pulsed field gradient (PFG) sequence of mixtures of tetrahydrofuran (THF) and cyclohexane at 50:50% volume fraction (same samples as in Fig. 3). Each dataset corresponds to the proton-containing substance marked in boldface. Signal is normalized to 1 in the absence of gradients (g = 0).

5. Conclusion

NMR relaxation of proton-containing liquids and their deuterated homologues were found to possess different dispersion properties. While the general trend of enhanced dispersion, and larger reduction of T₂ and also T₁ with respect to their bulk values, of polar liquids compared to non-polar liquids remains present, even non-polar liquids show a strong T₁ dispersion in Vycor that cannot be fully explained by the “weak interaction” case of the RMTD model. One remarkable finding of the experiments of binary systems in Vycor is the increase of T₂ and the essential absence of any T₁ dispersion for alkanes and cyclohexane; in the latter case, cyclohexane is introduced into the pore space as a miscible binary system but the absence of dispersion suggests a phase separation which either isolates the cyclohexane molecules from the surface, or does not allow them to spend sufficient time to undergo effective relaxation either by RMTD or by direct interaction with the surface nuclei. This finding is corroborated by published data for cyclohexane in aluminum oxide porous media. For mixtures of polar liquids, the change is less dramatic but the degree of dispersion correlates with the affinity of the molecule to the surface. Two liquids filling the porespace of Vycor at equal amounts indeed can show entirely different relaxation properties, suggesting different dynamics even within these rather small pores. The difference of dispersion of ¹H and ²H nuclei suggests a contribution of intermolecular processes for the former; modelling the exact mechanisms for both ¹H and ²H relaxation properties in these systems will be the task of a future work.

Further evidence for a partial or complete phase separation inside the pores comes from the change of the ratios of the self- or intra-diffusion coefficient between two liquids, and between the confined system and the bulk mixtures. The locally different concentrations inside the pore space, and the apparently affected tortuosity of molecules moving within an immiscible binary system, are two possible contributions for explaining the observed diffusion properties on length scales of tens of micrometers, but a consistent picture is not yet found except that a complete phase separation between miscible fluids is inconsistent with diffusion results. Much as PFG NMR has helped discuss the distribution and flow properties of binary fluids in macroscopically porous media by comparison with computational fluid dynamics, NMR relaxation dispersion combined with diffusion and other data such as two-dimensional spectroscopy techniques and multiple quantum coherence evolution represents powerful tools for a non-invasive study of molecular dynamics on the nanometer scale.

CRediT authorship contribution statement

Siegfried Stapf: Conceptualization, Investigation, Supervision, Project administration, Writing – original draft, Writing – review & editing. Niklas Siebert: Investigation, Validation. Timo Spalek: Investigation, Validation. Vincent Hartmann: Investigation, Validation. Bulat Gizatullin: Formal analysis, Supervision, Writing – review & editing. Carlos Mattea: Investigation, Validation, Formal analysis, Supervision, Writing – review & editing.

Declaration of competing interest

Siegfried Stapf is a Guest Editor for this issue of MRL and was not involved in the editorial review or the decision to publish this article. All authors declare that there are no competing interests.

Biography

Siegfried Stapf obtained his PhD at Ulm University in the group of Rainer Kimmich. Following a Postdoctoral position at University of Nottingham he obtained the habilitation at RWTH Aachen with Bernhard Blümich. He holds a professor position at University of Technology at Ilmenau since 2007. His main research activities and interests cover all aspects of NMR relaxation and diffusion in porous media, as well as NMR applications to polymers and other complex fluids.