Introduction

Due to the growing interest in porous gel materials for applications in industry and medicine, the need to control their properties becomes a cause for intensive investigations. Despite the extensive research material in the field of gels, investigation of many aspects of their chemical and physical properties is still unresolved. After two decades of research, we still do not have an answer to the fundamental question: what are the structural requirements for a molecule to have the ability of the liquid [5,6,7,8,9]. Gelating materials that form two-phase, stable in definite ranges of temperature, but thermally reversible physical gels represent a basis for manufacturing of foodstuffs, cosmetics and pharmaceutical agents. By a proper choice of the solvent and gelator, one can influence on the geometry of the rigid network that is formed in the gel. The rigid network extracted from the gel (so-called xerogel or aerogel) can represent an ideal matrix, enabling one to uniformly distribute an applied drug delivered in the form of an ointment or cream. Nevertheless, getting thorough knowledge and description of these materials still remains a great challenge. Physical gel is formed by mixing a specific liquid with a gelating material, at proper thermodynamic conditions, which leads to forming a ‘new’ material, with viscoelastic properties different from either solvent or the gelating material. A rigid network structure forms due to a self-organization of gelator molecules by various physical interactions: van der Waals, hydrogen bonds, π-bonds, etc. [10]. However, in such system no irreversible (chemical bounds) interactions between the solvent and the gelating material occur; hence, the separation of the liquid phase and the rigid gel formed becomes possible, opposite than in the case of chemical gels. In general, a physical gel is characterized by the fact that the sample volume is occupied by two separate phases: solvent and gelator. In this way, one obtains a new material of porous structure, which can be described by such characteristic parameters as tortuosity, porosity and the size of pores and barriers. The gelator Gluco-NO2 used to form the gel studied is a unique sugar-based gelator, as it can gelatinize not only organic solvents but also water. Such examples of ‘bifunctional’ gelators are sparse in the literature [11]. This organogel belongs to a class of gelling materials with a potentially large spectrum of applications. This is a non-toxic derivative of sugar, which forms stabile gels with several polar and non-polar solvents. However, the application of this compound requires a detailed knowledge of its physical and chemical properties. Recently, we studied the thermal properties and the microstructure of the hydrogel of Gluco-NO2 and the molecular dynamics of water in the gel network structure [12]. In the current paper, the measurements of the diffusion coefficient of toluene (C6H5CH3, methylbenzene) confined in the methyl-4,6-O-(p-nitrobenzylidene)-α-d-glucopyranoside-based gel and structural parameters of rigid matrix of gel are presented.

Method

The pulsed gradient spin echo (PGSE) is a very important method of nuclear magnetic resonance (NMR) for measuring the self-diffusion coefficient of small molecules in the presence of gelator network, and therefore, it was employed to obtain diffusion behaviour of toluene in the organogel matrix of Gluco-NO2 [13,14,15]. The time-dependent effective diffusion coefficient D eff(∆) measurements of toluene in the Gluco-NO2 organogels network were taken with Bruker Avance pulse spectrometer operating at 300 MHz (7,14 T) and equipped with magnetic field gradients. The diffusion coefficients were measured using the pulsed gradient spin echo (PGSE) pulse sequence introduced by Stejskal and Tanner [16]. The D eff(∆) values were determined by using the relationship between the echo signal and the field gradient parameter:

$$ A\left( {g, \,\Delta } \right) = A\left( 0 \right) \times \exp \left[ { - \gamma^{2} g^{2} D_{\text{eff}} \left( \Delta \right)\delta^{2} \left( {\Delta - \frac{\delta }{3}} \right)} \right] $$
(1)

where A(g, ∆) and A(0) are the echo signal intensities at t = TE (echo time) with and without the field gradient pulse of strength g, respectively. Here, γ is the magnetogyric ratio of the proton, g is the field strength, δ is the duration of the gradient pulses and Δ is the gradient pulse interval, termed the diffusion time. The field strength g values employed in our experiment are 0–1 T m−1, in equal intervals of 60 mT m−1. The diffusion coefficients D eff were studied as a function of the gradient pulse interval Δ in the range from 10 to 300 ms. The duration of the gradient δ in all experiments was 2 ms. The measurements were taken along the z direction of the cylindrical gel samples and at 20 °C. Equation 1 can be rewritten as A(g, Δ) = A(0) × exp(−b i D eff(Δ)) with b i values (γ 2 g 2 δ 2(Δ−δ/3)) range of 0–8 × 1010 s m−2 for highest diffusion time Δ and gradient strength g.

Theoretical background

Porosity

Porosity ϕ is a quantity characterizing the porous material with regard to the amount of free space in the whole volume of the sample. It is expressed as a ratio of the volume of free space within the pores V pore to the total volume of the sample V sample, Φ = V pore /V sample. The volume fraction of the rigid matrix \(\upvarphi=V_{\text{gelator}}/V_{\text{sample}}\) of the porous material is connected with porosity ϕ by relation 1−\(\upvarphi\) = ϕ [17, 18].

Tortuosity

Geometrical tortuosity τ of a system composed of solvent molecules and molecules representing obstacles for the diffusing liquid is defined as a ratio of the distance to be followed by a diffusing particle and the effective path length L e, corresponding to the geometric size L of the sample. The shortest path to be followed by a particle diffusing in the liquid (in the ‘bulk’ state) to go as far as the sample size is exactly equal to that size. The tortuosity of this system can be expressed as τ = L e /L [1, 19,20,21]. The coefficient τ = 0 describes the geometry of the porous materials, in which all the pores are separated from each other. The solvent molecule trapped in a pore cannot diffuse to another one. The tortuosity near to 1 is characteristic for porous materials in which solvent molecules can diffuse by channels between pores without significant resistance from the matrix. Such situation occurs when the pore size is comparable to the gel matrix backbone size arising by diffusing molecules. Generally, in the case of open geometry where the pores are well connected by the channels, with an increase in tortuosity arising when the number and size of barriers that must be overcome by diffusible molecule increases [22]. Many proposals can be encountered in the literature of the tortuosity–porosity relation for the porous systems under consideration [23]. Depending on the shape of the elements constituting the rigid network of the porous material, these relations have an exponential or linear character. The tortuosity for mica, textile fibres, kaolinite or bituminous soil with a high porosity range (~60–90%) exponentially increases with porosity increase but for medium range of porosity (~34–45%) of packed beds of spheres the tortuosity linearly decreases with porosity increase [18]. In general, on the assumption that the pores are open, the value of tortuosity of the system decreases with increasing porosity and a diffusing particle can easily move within the rigid network of the porous material.

Surface-to-volume ratio

The next parameter characterizing the porous systems is the surface-to-volume ratio (S/V p, [m−1]) of the pores. This quantity can be determined on the basis of the investigation of the diffusion coefficient of the liquid in the pores. The ratio S/V p provides information on the pore size, but not on its shape; the latter can be varied arbitrarily on condition of conserving the surface-to-volume ratio value [2, 4, 24, 25].

Sample preparation

The gels of Gluco-NO2 (the gelator obtained from non-commercial synthesis) with toluene (the solvent with purity of 99.8% were obtained commercially from the Sigma-Aldrich and used as supply) were synthesized according to the method described elsewhere [26]. Concentrations of 2.0, 3.0, 4.0 and 5.0 wt% of Gluco-NO2 were chosen to form the gels with toluene. The gels are prepared by mixing the appropriate amounts of the gelator and solvent in a closed capped tube and by subsequent heating the mixture to the temperature near 373 K (lower than the boiling point of toluene—384 K) until the solid is dissolved. Next, cooling the solution below the characteristic gelation temperature T gs (315 K–335 K for 2.0–5.0 wt% range of Gluco-NO2 in toluene) brings out the transition to the gel phase. As a result, thermoreversible, slightly opaque gels were obtained for each concentration.

Results and discussion

The porosity of gels ϕ = V pore /V sample was calculated using density of toluene (0.8623 g cm−3) and Gluco-NO2 (0.533 g cm−3) as 0.968, 0.952, 0.937 and 0.921 for 2.0, 3.0, 4.0 and 5.0 wt% gel, respectively. Due to the lack of literature data, the gelator density was estimated experimentally as 0.533 g cm−3 by weighed maximal packed under pressure of 5 MPa Gluco-NO2 powder in a known volume. In the relation of ϕ = V pore /V sample, V pore and V sample correlate with toluene and dissolved Gluco-NO2 in toluene volume, respectively. Figure 1 presents the time-dependent diffusion coefficients D eff(∆) of toluene for 2.0, 3.0, 4.0, and 5.0 wt% gel as a function of the gradient pulse interval Δ. The data were scaled to the diffusion coefficient of bulk toluene D 0  = 2.28 × 10−9 m2 s−1. All measurements were taken at 293 K. The NMR data of A(g, Δ) were one-exponential over the full range of wt% of samples up to Δ = 200 ms. Above this diffusion time, a little deviation from exponential behaviour in gradient value higher than 0.5 T m−1 was observed to give in fitting procedure of Eq. 1, the R 2 parameter in the range of 0.8–0.9. Under Δ = 200 ms of diffusion time, the R 2 was always bigger than 0.9.

Figure 1
figure 1

The time-dependent diffusion coefficient D eff(∆) of toluene to its bulk diffusion coefficient D 0 ratio in the gel matrix of gelator Gluco-NO2 studied as a function of gelator concentration

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The organogel studied consists of a very large amount of toluene (the range from 95 to 98 wt%) confined in a gel network composed of the gelator aggregates. The experimental results clearly show that for low Δ values (below 100 ms), the diffusion coefficient values nearly coincide despite the different gel concentrations. The apparent differences occur for longer diffusion times (above 100 ms). Therefore, one can distinguish two time regimes. Generally, in the function of diffusion time ∆, D eff(∆)/D 0 ratio decreases with increasing gelator concentration in the sample. The change in behaviour of D eff(∆)/D 0 for 3 wt% gel at the time limit of ~0.1 s is clearly visible in Fig. 2. In a short-time regime, only those molecules that are close to the solid matrix experience the effect arising from the barrier, and as long as most molecules do not experience multiple reflections from the walls of the matrix, the behaviour of the diffusion coefficient will be the result of pore size by S/V p ratio (ratio of pore surface to its volume with dimension of m−1). In a long-time regime, the multiple reflections of the solvent molecules at the solid matrix occur. The pore space no longer matters and the behaviour of the diffusion coefficient varies with a change in tortuosity of a porous matrix [27]. These two time regimes are separated by a characteristic, so-called threshold time ∆th, which sets the time to change the slope of the measured diffusion values. Based on the analysis of the behaviour of time-dependent diffusion coefficient in all samples, the threshold time was designated as 89, 65, 58 and 40 ms for 2.0, 3.0, 4.0 and 5.0 wt% gel, respectively.

Figure 2
figure 2

The time-dependent diffusion coefficient D eff(∆)/D 0 ratio of toluene in the 3.0wt% gelator Gluco-NO2 concentration in gel. The data shown in logarithmic scale of time

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In the short-time regime 0–0.1 s, D eff(∆)/D 0 versus Δ dependence is a function of surface to volume of the pores S/V p in the system according to equation proposed by Mitra et al. [3]:

$$ \frac{D\left( \Delta \right)}{{D_{0} }} \approx 1 - \frac{4}{9\sqrt \pi }\frac{S}{{V_{\text{P}} }}\sqrt {D_{0} \Delta } $$
(2)

On the other hand, in the long-time regime 0.1–0.3 s, the knowledge of the D eff(∆)/D 0 versus Δ dependence enables one to find with a very good approximation the tortuosity of the porous system according to equation reported in the literature [1]:

$$ \frac{D\left( \Delta \right)}{{D_{0} }} \approx \frac{1}{\tau } $$
(3)

A horizontal dashed line in Fig. 2 defines as inverse of tortuosity and so-called tortuosity limit. The tortuosity limit 1/τ calculated as a fitting parameter of long-time diffusion regime by Eq. 3 gives 0.67, 0.61, 0.56 and 0.41 for 2.0, 3.0, 4.0 and 5.0 wt% gel, respectively. This means that in the investigated range of gelator concentration in gel phase, the rigid matrix is composed of well-connected pores. The effective path length L e in comparison with geometrical size of the sample L for diffused molecules increases by a value of tortuosity 1.48, 1.63, 1.79 and 2.24 with increasing gelator concentration from 2.0 to 5.0 wt%. The S/V p ratio was obtained as fitting parameter of Eq. 2 to short-time diffusion regime. The S/V p parameter increases with increasing gelator concentration in the gel phase. The value of 0.19, 0.26, 0.32 and 0.38 μm−1 for 2.0, 3.0, 4.0 and 5.0 wt% gel, respectively, was obtained. On the basis of S/V p parameter, spherical pore diameter d can be calculated by relations:

$$ \frac{S}{{V_{\text{p}} }} = \frac{6}{d} $$
(4)

For comparison, the size of pores also was calculated using the time threshold by the mean-squared displacement r 2 relationship [14, 28, 29]:

$$ \langle{r^{2}}\rangle = \left( {2\Delta_{\text{th}} D_{0} } \right)^{{\frac{1}{2}}} $$
(5)

The value of the mean square displacement 〈r 2〉 of freely diffusing particles in a given time reflects the size of the space where diffusion occurs. The results are shown in Fig. 3.

Figure 3
figure 3

The pore diameter as a function of gelator concentration calculated on the basis of S/V p (open circles) and ∆th (open squares) values. The diamonds represent average value of calculated by Eqs. 4 and  5 pore diameters

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The differences in pores size between that calculated on the basis of S/V p and ∆th values are due to the fact of difficulty of accurately indicating the ∆th on the timescale for low gelator concentration in the gel. With the increasing Gluco-NO2 content in the system, the boundary between short and long diffusion regimes is better visible, resulting in a much higher accuracy of the pore size calculation. The parameters obtained by fitting the results describing the real structure of the gel, wherein the shape of the pores is unknown, have been adapted to the spherical shape only. However, we can assume that the real pore size will not be much different from the calculated values, because in the case of the 2 wt% of Gluco-NO2 gel in toluene at 300 K the diffraction pattern was observed [30]. This means that internal rigid structure at least locally is ordered and relatively monodisperse. This ordering may be disrupted with increasing temperature. In the publication of [30], the pore size using pore-hopping model was calculated as 64 μm for 2 wt% gel at 300 K, but this value strongly depends on the rate of temperature change during gelation process and even on the size of the tubes in which the process takes place. The gel structure changes with increasing temperature up to a limit temperature T gs above which the gel starts to flow. The T gs temperature was determined by visual observation of the toluene gel as 315–335 K in the range of gelator concentration from 2.0 to 5.0 wt%. So, it can be expected that the pore size in gel calculated at 293 K will vary with that obtained at 300 K. On the basis of the obtained results, we can conclude that in the investigated gels the average pore sizes range from several to tens of micrometres. With a good approximation to a circular cross-sectional shape, the diameter of the barrier (distance between the matrix backbone) for diffused molecules in the gel can be calculated as [31]:

$$ b = 6\left( {\phi^{ - 1} - 1} \right)\frac{{V_{\text{p}} }}{S} $$
(6)

Figure 4 shows the graph of structural parameters (porosity ϕ, tortuosity τ and pore d and barrier b diameters of the gels) as a function of gelator concentration in the samples. In the range from 2.0 to 5.0 wt% of gelator concentration in the samples, the pore diameters d and porosity ϕ are changing about 5%. The increasing diameters of the barrier in the same range of gelator concentration in the samples are about 10%. The porosity of the samples decreases slowly as a function of gelator concentration with increasing diameter of the element of rigid structure b. Such situation is possible in systems in which the size of the aggregates forming the rigid matrix is at least several times smaller than the pore size. The variation of about 50% of tortuosity parameter in the range of 2.0–5.0 wt% of gelator concentration in the samples indicates that the expansion of the real rigid matrix in the system in 3D space is a directional not homogenous process. The tortuosity parameter suggests that the large aggregates formed from bundles of thin fibres in cross section have the irregular flattened shape similar to multiple star whose arms have different lengths. On the basis of the structural parameters obtained from diffusion measurements performed for investigated materials, we can suggest that the solid matrix of gel is composed of the fibres of a few micrometers in diameter with irregular cross-section shape. The investigated gelator belongs to the so-called bifunctional sugar-based gelators, which in contact with polar or non-polar solvents forms in the gel phase different solid structures due to the existence of the solvent–gelator molecules dipole–dipole interaction [26]. Generally, for this class of gelator the shape of aggregates changes from short and thick for less polar solvent to very long and thin for higher polarity of solvent [32]. In water, 1-propanediol or 1-butanol, the gelator Gluco-NO2 forms about 1–5-μm-thin fibres with a length of 100–300 μm [12, 33]. In comparison with these solvents, the toluene molecules have a small dipole moment equal to 0.36 D. Knowing the diameter of the aggregates (calculated from diffusion measurements), we can conclude that in the case of toluene the length of aggregates should contain from a few to several times more than thickness. In water with dipole moment of 1.85 D, Gluco-NO2 forms the fibres with ~10 and ~300 μm thin and length, respectively [12]. On the basis of the time-dependent diffusion coefficient of toluene in Gluco-NO2 gels, the 1–2-μm-thin and 10–20-μm-length fibres can be concluded by tortuosity analysis.

Figure 4
figure 4

Graph of the characteristic gel parameters (porosity, tortuosity and pore and barrier diameters) describing the internal structure of gels as a function of gelator concentration

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Conclusion

The paper reports the theoretical analysis based on diffusometry method of internal structure of methyl-4,6-O-(p-nitrobenzylidene)-α-d-glucopyranoside (Gluco-NO2) gel with toluene. With the slow decreasing porosity ϕ in the range of 0.968–0.921 for 2.0 and 5.0 wt% of Gluco-NO2 in toluene, respectively, the tortuosity τ parameter increases from 1.48 to 2.24 value (Fig. 4—open diamonds) and pore size d is reduced by a half (from 31 to 15 μm—Fig. 4—open circles). In the publication of [30], the pore size using pore-hopping model [34,35,36] was calculated as 64 μm for 2 wt% gel with toluene at 300 K, but this value strongly depends on the rate of temperature change during gelation process and even on the size of the tubes in which the process takes place. The results obtained in this work (the thick and length of rigid fibres of few and a dozen of μm, respectively) very well correspond with simulation of porosity–tortuosity relation which suggests small parallelepipeds shape of solid fibres [37]. The effective path length L e for diffusing molecule increases twice for the 5 wt% gel sample compared to 2 wt% gel (L e = τ × L). These results suggest that the solid matrix backbone size varies slightly in the range of 2.0–5.0 wt% of Gluco-NO2 in toluene. As shown by the time-dependent effective diffusion coefficient analysis, these dimensions vary from 1 up to 1.3 μm (Fig. 4—closed circles). The results correspond very well with the optical microscopy image of 2.0 wt% Gluco-NO2 in toluene published by Bielejewski et al. where the 1 μm in thickness (analogous to a 1-μm dimension calculated at this work for 2 wt% gel) and 5–10 μm in length fibres are clearly visible (see [32]—Fig. 6a). The porosity–tortuosity theory, based on the time-dependent diffusion coefficient of liquid phase, is usually used to analyse rigid porous materials such as zirconia, zeolites, kaolinite and pumice stones [38, 39]. This work shows that it can also be successfully used in the analysis of soft porous materials such as gels.