1 Introduction

Experimental and theoretical studies of gas adsorption in mesoporous materials have been performed for many decades (Haul et al.1982). The emergence of novel material synthesis routes enabled the fabrication of monodisperse and highly ordered cylindrical pore systems such as MCM-41 (Kresge et al. 1992) or SBA-15 (Zhao et al. 1998) silica materials, allowing for the quantitative validation of theories of adsorption and capillary condensation based on both, macroscopic thermodynamics (Broekhoff 1967) and atomistic descriptions (Ravikovitch et al. 1998). In the early 2000s, a new family of templated mesoporous carbon materials has been developed by using ordered mesoporous silicas as template (Ryoo et al. 2001; Jun et al. 2000). A representative member of this family is CMK-3 carbon derived from negative templating of SBA-15 silica, exhibiting hexagonally ordered carbon nanorods with an intermediate pore space resembling the original silica template. Therefore, the mesopores in CMK-3 are qualitatively different from the cylindrical pores in SBA-15 forming a continuous pore space with the shortest distance between adjacent nanorods. Some consequences of this special pore geometry have been evident already in the very early publications on CMK-3, with the shapes of the nitrogen adsorption isotherm being quite different from those of the original silica templates, and some of them show two step-like ascends at significantly different relative pressures (Jun et al. 2000; Joo et al. 2002). However, the physical origin of these features has not been addressed in detail so far, and gas adsorption in CMK-3 was frequently modeled as if the pores were cylindrical (Jun et al. 2000; Joo et al. 2002). Density functional theory (DFT) kernels developed for calculation of pore size distribution in CMK-3 materials were also based on the cylindrical pore model (Gor et al. 2012). While this model gave quite satisfactory predictions for the mean pore sizes, some of the data showed a “bimodal” size distribution in the mesopore regime, pointing also towards two separate condensation events. Generally, there is a clear discrepancy in the shape of a theoretical isotherm based on the cylindrical pore model with a sharp condensation point, and the much smoother adsorption isotherm measured experimentally on CMK-3 carbons.

To more accurately describe the adsorption isotherms on CMK-3 and alike materials, alternative models were developed, explicitly taking the rod-like geometry into account (Barrera et al. 2013; Jain et al. 2017; Yelpo et al. 2017). These alternative models were mainly based on molecular simulations using Grand Canonical Monte Carlo (GCMC). Barrera et al. (2013) followed the same principal idea as the DFT model by Gor et al. (2012) by representing the pore space of CMK-3 as a collection of cylindrical and slit pores. The interaction between the carbon nanorods and the nitrogen atoms was modeled by a Steele potential for micropores and an integrated, cylindrical Lennard–Jones potential for the mesopores. The resulting kernel was able to reproduce the results from Gor et al. (2012), thus justifying the applicability of the GCMC method. Yelpo et al. (2017) took this approach one step further and determined the carbon potential inside the pore space of CMK-3, creating a more realistic description for the GCMC simulation. Again, kernels were calculated and used to fit experimental data. Furthermore, a TEM image was analyzed and the pore size distribution obtained from TEM was compared to the pore size distribution obtained by their kernel. Jain et al. (2017) took a different route using an all-atom description of several CMK type systems. They represented the carbon rods by an assembly of individual carbon atoms and used a Lennard–Jones-type potential for the individual atoms to describe the interaction between the gas phase and the solid. Interconnections between the individual carbon rods were also introduced, to more accurately describe the CMK materials, which are seemingly necessary to get qualitatively sound results from the simulations. The adsorptive in this study was argon and the adsorption and desorption was modeled with GCMC for CMK-1, CMK-3 and CMK-5. The results from that study gave some insight into the role of imperfections in CMK materials, especially CMK-3.

Although the methods for modeling adsorption based on molecular simulations have become state of the art, there is still a drawback compared to macroscopic theories: generalization of the results for a different adsorbate would require setting up a new time-consuming simulation. In this sense, macroscopic approaches for modeling adsorption in mesoporous materials, such as theories developed by Derjaguin (1992) and Broekhoff and de Boer (1967) (DBdB theory), or by Saam and Cole (1975) are advantageous. While requiring only a few parameters and being not computationally intense, they reliably predict the adsorption isotherms for cylindrical pores not only for simple gases such as nitrogen or argon (Neimark and Ravikovitch 2001), but also for more complex molecules, such as water, methanol, toluene (Lépinay et al. 2015), pentane (Gor et al. 2013) or perfluoropentane (Hofmann et al. 2016).

Attempts to describe the adsorption and capillary condensation for geometries other than simple slit- or cylindrical pores were carried out by Philip (1977b) already in the 1970s for two parallel cylinders. This idea was further developed by Morishige and Nakahara (2008) into a comprehensive theoretical framework for the transition from a liquid film phase to a “bridged” phase, effectively spanning the void space between the two adjacent cylinders. Dobbs and Yeomans (1993) extended the approach of Philip by numerically minimizing the grand potential of different configurations of liquid in the open pore space between cylindrical rods located on a square lattice. However, that paper was published several years before the emergence of CMK-3, and to the best of our knowledge its predictions were never adapted to the hexagonal geometry and neither have they been compared to real experimental data.

Here we solve the problem of the hexagonal geometry of CMK-3 adapting the thermodynamic model from Ref. (Dobbs and Yeomans 1993). We calculate explicit solutions of the corresponding non-linear differential equations for two different adsorbates, namely nitrogen and n-pentane, by using appropriate reference isotherms to model the solid–fluid interaction (Gor and Neimark 2010, 2011). Equilibrium phase diagrams separating a “film phase” (liquid film on the carbon nanorods), a “bridged phase” (liquid bridges between adjacent nanorods) and a “filled” phase (entire liquid filled pore space) are obtained. Calculations with a simplified, analytical model of the “bridged phase” are performed to elucidate numerical results and provide comparison to traditional characterization approaches such as the Kelvin-Cohan (Neimark et al. 2003) equation. Predictions from the numerical results are compared with experimentally measured adsorption isotherms from CMK-3-like carbon materials (Koczwara et al. 2017) using nitrogen at 77 K and n-pentane at 290 K.

2 Theoretical model

For the thermodynamic description of adsorption in CMK-3-like materials we employ the first variation of the grand potential of three competing unique distributions of liquid-like phases in the open mesopore space (see Fig. 1).

  • “Separated” phase: an adsorbed layer (liquid-like film) is present on each individual rod, but the films are not connected with each other.

  • “Bridged” phase: a liquid bridge exists between neighboring rods, with a void space remaining between the three rods.

  • “Filled” phase: The entire space between the carbon nanorods is filled with liquid.

Fig. 1
figure 1

Sketch of the top-view pore-space geometry of the hexagonally arranged cylindrical carbon nanorods (grey) with radius r and distance D. Dark blue areas indicate the “separated” phase (liquid adsorbed film around each cylinder), while liquid bridges between the rods are labeled by light blue. The image at the bottom shows an enlarged detail including the two cylindrical coordinate systems used. For the “separated” phase the origin is set into the center of the rod (black system) and is denoted with the subscript 1, while for the “bridged phase” the origin is set into the center of the triangle (red) with the subscript 2 in the respective equations (Color figure online)

All derivations are based on the assumption that the aspect ratio between the diameter of the rods and their length is small, meaning that we can restrict the description to the plane perpendicular to the rod axis. Because of the radial symmetry of the rods we use cylindrical coordinates. To simplify calculations, we consider two different coordinate systems, with the “separated” phase having its origin in the center of the rods, while for the “bridged” phase the origin is located in the center of the unit cell set up by the three rods (see Fig. 1). To model fluid adsorption in this system, the grand potential per unit length of the rod \(\Omega \) is treated as a functional \(\Omega \left(l\right)\) of the liquid profile, where l is the radial coordinate. We assume uniform density of the liquid, meaning that \(\Omega \) will depend on geometry only. This results in

$$ \Omega = \int_{{\theta _{a} }}^{{\theta _{b} }} {f\left( {\theta ,l,l_{\theta } } \right)d\theta } $$

with θa and θb being the limits of the angular interval, l being the distance of the vapor–liquid interface from the origin as a function of θ, and lθ being the derivative of l with respect to θ.

2.1 Governing equations for the phases

Adapting the equation for the grand potential of the “separated” phase from (Dobbs and Yeomans 1993) to the hexagonal geometry, we have

$$ \Omega \left( {l_{1} } \right) = 6\int_{0}^{{\frac{\pi }{6}}} {d\theta _{1} \left( {\gamma \left( {l_{1}^{2} + l_{{\theta _{1} }}^{2} } \right)^{{\frac{1}{2}}} } \right)} + \Delta \tilde{\mu }\left( {6\int_{0}^{{\frac{\pi }{6}}} {d\theta _{1} \frac{1}{2}l_{1} \left( {\theta _{1} } \right)^{2} - \frac{\pi }{2}r^{2} } } \right) + 6\int_{0}^{{\frac{\pi }{6}}} {d\theta _{1} V\left( {l_{1} ,\theta _{1} } \right)}. $$

The subscript 1 denotes the “separated” phase with l1 ≡ l(θ1) and \({l}_{{\theta }_{1}}\)≡ dl1/dθ1. The first part of the equation describes the contribution of the liquid–vapor interfacial energy γ, the second term describes the influence of the chemical potential \(\Delta \tilde{\mu }\) of the liquid, and the last term describes the energy due to the solid–liquid film potential \(V\left(l,\theta \right)\). The chemical potential per unit volume \(\Delta \tilde{\mu }\) is:

$$ {\Delta }\tilde{\mu } = \frac{{R_{g} T}}{{v_{l} }}\ln \left( {\frac{p}{{p_{0} }}} \right) $$

where vl is the molar volume of the liquid, Rg is the universal gas constant, T is the absolute temperature and \(p/{p}_{0}\) is the relative pressure. The last term in Eq. 2 represents the integrated effect of the solid–fluid interactions, which can be related to Derjaguin’s disjoining pressure Π

$$ V\left( {l,\theta } \right) = \int\limits_{{l\left( \theta \right)}}^{{l_{{max}} }} {\Pi \left( {l,\theta } \right)l dl} $$

where lmax is the maximum value of the profile to be considered, which helps keeping the integration limited to a unit cell in ordered systems. The detailed discussion of Π(l) is given in Sect. 2.2.

Application of the Euler–Lagrange equation to Eq. 2 leads to a second order, non-linear, ordinary differential equation, which minimized the grand potential in any radially symmetric case:

$$ \gamma \frac{d}{{d\theta _{1} }}\left( {\frac{{l_{{\theta _{1} }} }}{{\left( {l_{1}^{2} + l_{{\theta _{1} }}^{2} } \right)^{{\frac{1}{2}}} }}} \right) - \gamma \frac{{l_{1} }}{{\left( {l_{1}^{2} + l_{{\theta _{1} }}^{2} } \right)^{{\frac{1}{2}}} }} + l_{1} \left( {\Delta \tilde{\mu } - \Pi \left( {\theta _{1} ,l_{1} } \right)} \right) = 0. $$

Noteworthy, for cylindrical pores Eq. 5 reduces to Derjaguin’s equation (Broekhoff 1967; Derjaguin 1992).

For the “bridged” phase we set the origin of the coordinate system in the center of the rod (Philip 1977b; Dobbs and Yeomans 1993; Gatica et al. 2002) as depicted in Fig. 1 with subscript 2 (l2 ≡ l(θ2)) and Eq. 2 becomes

$$ \Omega \left( {l_{2} } \right) = 6\int\limits_{0}^{{\frac{\pi }{3}}} {d\theta _{2} \left( {\gamma \left( {l_{2}^{2} + l_{{\theta _{2} }}^{2} } \right)^{{\frac{1}{2}}} } \right)} + \Delta \tilde{\mu }\left( {{\sqrt 3 }R^{2} - \frac{\pi }{2}r^{2} - 6\int\limits_{0}^{{\frac{\pi }{3}}} {d\theta _{2} \frac{1}{2}l_{2} \left( {\theta _{2} } \right)^{2} } } \right) + 6\int\limits_{0}^{{\frac{\pi }{3}}} {d\theta _{2} V\left( {l_{2} ,\theta _{2} } \right)}. $$

Here, R = D/2 is the half-distance between the centers of the adjacent rods and r is the radius of the rods (see Fig. 1), and we change the integration limit from \(\frac{\pi }{6}\) to \(\frac{\pi }{3}\) because of the change in origin from the center of a rod to the center of the interstitial void space.

Minimizing the functional we obtain again a second order, non-linear ordinary differential equation

$$ \gamma \frac{d}{{d\theta _{2} }}\left( {\frac{{l_{{\theta _{2} }} }}{{\left( {l_{2}^{2} + l_{{\theta _{2} }}^{2} } \right)^{{\frac{1}{2}}} }}} \right) - \gamma \frac{{l_{2} }}{{\left( {l_{2}^{2} + l_{{\theta _{2} }}^{2} } \right)^{{\frac{1}{2}}} }} - l_{2} \left( {\Delta \tilde{\mu } - \Pi \left( {\theta _{2} ,l_{2} } \right)} \right) = 0. $$

In the case of the completely filled pore space (“filled” phase) the sole contribution to the grand potential is the liquid inside the filled pore space

$$ {\Omega } = \Delta\tilde{\mu } \left( {\sqrt 3 R^{2} - \frac{\pi }{2}r^{2} } \right). $$

We note that the values of the grand potential obtained in this study are not absolute but are offset by a constant contribution (containing the surface energy of the solid). The discussion of these terms is given elsewhere (Gor and Neimark 2010). While these terms do not affect the transition points, they contribute to the solvation pressure in the pore and affect the thermodynamic properties of the fluid (Hill 1952).

To solve the differential equations Eqs. 5 and 7, two values of the film thickness or the slope at the boundaries of the unit cell have to be known. The boundary conditions for the “separated” phase, with the origin fixed at the center of the rod, are given by:

$$ \left. {l_{{\theta_{1} }} } \right|_{{\theta_{1} = 0}} = 0 \,and\,\left. {l_{{\theta_{1} }} } \right|_{{\theta_{1} = \frac{\pi }{6}}} = 0 $$

Similarly, for the “bridged” phase the conditions read:

$$ \left. {l_{{\theta_{2} }} } \right|_{{\theta_{2} = 0}} = 0 \, and \,\left. {l_{{\theta_{2} }} } \right|_{{\theta_{2} = \frac{\pi }{3}}} = 0 $$

Using these boundary conditions, the two-point Neumann boundary value problem was solved numerically, as outlined in Appendix A.

2.2 Determination of the solid–fluid interaction potential

The key term in the macroscopic theories of adsorption and capillary condensation is the term related to the solid–fluid interaction potential. Derjaguin used the concept of disjoining pressure Π(h) to represent it, where h is the film thickness, which can more generally be defined as the shortest distance between the substrate surface and the liquid–vapor interface. Disjoining pressure isotherms for adsorption of fluids on a flat surface are often modeled via the Frenkel-Halsey-Hill equation (Halsey 1948; Hill 1952):

$$ {\Pi }\left( h \right) = - \frac{{R_{g} T}}{{v_{l} }}\frac{k}{{\left( {\frac{h}{{h_{0} }}} \right)^{m} }} $$

where h0 = 0.1 nm and k and m are the two free parameters of the model.

While the DBdB theory (Broekhoff 1967; Derjaguin 1992) typically neglects the curvature of the pores and uses Eq. 11 directly to represent the solid–fluid interactions in the cylindrical pore, we take here the curvature of the carbon nanorods into account. We use the integrated solid–fluid potential for an infinite rod derived for arbitrary inverse power-law potentials (Philip 1977a):

$$ {\Pi }\left( h \right) = - \frac{{\pi^{\frac{3}{2}} {\Gamma }\left( {\frac{ \epsilon - 1}{2}} \right)}}{{{\Gamma }\left( {\frac{ \epsilon }{2}} \right)}} \alpha r^{2} \left( {h + r} \right)^{1 - \epsilon} F_{2;1} \left( {\frac{ \epsilon - 1}{2};\frac{ \epsilon - 1}{2};2;\left( {\frac{r}{h + r}} \right)^{2} } \right) $$

where Π is the film potential, α is the interaction parameter, ε is the exponent of the inverse-power-law, \(\Gamma \) denotes the Eulerian gamma function, and F2;1 is the generalized hypergeometric function. The parameters α and ε are determined by fitting from reference isotherms, assuming infinite, planar substrates in any case. Consequently, one can think of this as an empirical function, with the units of α depending on the value of ε. These parameters can be readily related to the parameters of the Frenkel-Halsey-Hill equation (Eq. 11):

$$ m = \epsilon{-}3 $$
$$ R_{g} Tkh_{0}^{m} = \frac{2\pi \alpha }{{\left( { \epsilon - 3} \right)\left( { \epsilon - 2} \right)}} $$

Finally, we take into account that the adsorbing fluid interacts not with a single rod, but with the three rods of the unit cell and sum up the potentials at each single point of consideration, similar to the quadratic lattice considered in (Dobbs and Yeomans 1993).

3 Computational results

3.1 Reference isotherms and disjoining pressure

Theoretical predictions of adsorption isotherms require the knowledge of the disjoining pressure isotherm Π(h). In order to be able to compare our numerical results with experimental data, we used available experimental reference isotherms from nitrogen and n-pentane adsorption on carbon (Fig. 2). For nitrogen adsorption at 77 K we used the literature data of activated carbon annealed at high temperature (2000 °C) for 2.5 h (Silvestre-Albero et al. 2014). For n-pentane, no literature data were available. Therefore, we employed own data from n-pentane adsorption on a carbon xerogel thermally annealed at 1800 °C for 50 min. This sample contained a negligibly small amount of micropores, and mesopores of some tens of nanometers in size (Balzer 2018). Ideally, the reference isotherm should have been measured on a non-porous or at least a macroporous only sample, but no such sample was available to perform n-pentane adsorption measurements. The presence of large mesopores is probably the reason why the fit in Fig. 2b deviates from the data at relative pressures above 0.6. Nevertheless, the first layers are properly described by the Frenkel-Halsey-Hill (FHH) isotherm and should therefore approximate the interaction of the fluid molecules with the carbon substrate with sufficient accuracy. The interaction parameters derived from the fits of Eqs. 11, 13 and 14 to the corresponding reference isotherms are presented in Table 1.

Fig. 2
figure 2

Plot of the film thickness h of the reference isotherms for nitrogen on annealed activated carbon [data from (Silvestre-Albero et al. 2014)] (a), and n-pentane on a carbon xerogel (b). The experimental reference isotherms are shown by (black) squares, the FHH-fits with solid (red) lines (Color figure online)

Table 1 Fitting parameters of the FHH isotherm (see Eq. 11) to the reference isotherms

We note the significance of the Cole-Saam approach (Saam and Cole 1975) for our work, which takes the curvature of the solid surface explicitly into account. In cylindrical mesopores, where capillary condensation usually happens at relatively low film thickness (only few monolayers of adsorbate), the effect of curvature is small. In our model, however, we need to consider the film potential at distances of several nanometers when solving the equations for the “bridged” phase. Figure 3 compares the disjoining pressure as a function of film thickness h for a flat surface and a single carbon cylinder with a radius of 3.7 nm. At low film thicknesses the overall difference between both potentials is small, but at distances above 1 nm, in case of the flat surface the overall potential is overestimated by 20 to 50 percent.

Fig. 3
figure 3

Disjoining pressure calculated for a flat surface and for the surface of a cylinder with radius r = 3.7 nm for n-pentane on carbon (Balzer 2018). The inset shows the relative difference of the data for the flat surface in regard to the cylindrical potential as a function of the distance to the substrate

3.2 Calculated isotherms and phase diagrams

To predict adsorption isotherms for the geometry outlined in Fig. 1, the liquid–gas interface profiles for the “separated” and “bridged” phases were calculated by solving Eqs. 5 and 7, respectively. According to the available experimental data (see Sect. 4 and Appendix B), the rod radius r was varied between 3.3 and 4.4 nm in 0.05 nm steps, while the rod distance D was fixed at a value of 10.1 nm. Temperatures used in the simulations were 77.4 K and 290 K for nitrogen and n-pentane respectively, according to the temperatures at which the adsorption measurements were carried out. Experimental details are outlined in Appendix B. The differential equations were solved numerically by applying a finite difference scheme using a custom written code, outlined in some detail in Appendix A. Solutions were obtained for 20 equidistant relative pressure values ranging from 0.01 to 0.95 for both sets of interaction parameters.

From the interface profiles l(θ) the grand potentials were calculated using Eqs. 2 and 6 for the “separated” and “bridged” phases, respectively, and Eq. 8 was employed for the “filled” phase. The thermodynamically stable phase at a chosen pressure is now simply given by the lowest value of Ω, with the exact relative pressure values at which transitions between the phases occur obtained via interpolation and root finding. A selection of profiles of the “bridged” phase for \(D/r =2.7\) are shown in Fig. 4a. With these profiles, the grand potential Ω of the “bridged” phase can be calculated (Eq. 6) and compared to the grand potentials of the “separated” and “filled” phases as shown in Fig. 4b. As can be seen in Fig. 4b, the “separated” phase is stable for low relative pressures, followed by the “bridged” phase and finally the “filled” phase. Figure 4c displays the corresponding nitrogen adsorption isotherm for a ratio D/r = 2.7, calculated from the “separated” and “bridged” profiles. The circles in Fig. 4c correspond to the film profiles for the “bridged” phase shown in Fig. 4a. With increasing relative pressure, the void space shrinks and changes its shape from rather triangular towards more circular, upon which the “bridged-to-filled” transition happens.

Fig. 4
figure 4

a 2D interface profiles for the “bridged” phase. With increasing relative pressure, the overall size of the void space decreases, while the shape of the void space changes from more triangular towards more circular. The four relative pressures at which the profiles in (a) are shown are indicated by blue circles in panel (c). b The grand potential as a function of relative pressure for the three different phases (blue: “separated” phase; green: “bridged” phase; red: “filled” phase) for nitrogen on carbon and D/r = 2.7. c Corresponding adsorption isotherm (Color figure online)

From the isotherms determined for different D/r ratios we can extract phase diagrams, showing the stability range of the respective phases as a function of relative pressure and D/r ratio. This ratio can be related to the maximum inscribed radius \({r}_{u}^{*}\) between the three cylindrical rods

$$ r_{u}^{*} = \frac{\sqrt 3 }{3} D - r $$

or in a dimensionless representation

$$ r^{\prime}_{u} = \frac{{r_{u}^{*} }}{r} = \frac{\sqrt 3 }{3}\frac{D}{r} - 1 $$

linking the reduced sizes to the pore geometry considered in earlier work (Ryoo et al. 2001).

In Figs. 5a and 6a we show the resulting phase diagrams for nitrogen and n-pentane, respectively. For all calculations we used a fixed nanorod distance D = 10.1 nm, which is the mean value determined experimentally for the sample discussed (see Appendix B). Variation of D in the calculations was also considered, but its influence was found to be minor as compared to the impact of the nanorod radius r. The phase diagrams for the two fluids show a similar overall trend. Both adsorbates show a bridging transition at very low relative pressures for the smallest D/r ratio, which corresponds to a small distance of the rods of 1.3 nm only. With increasing D/r, the “separated” to “bridged” phase transition appears at increasingly larger relative pressures. While the “bridging” transition is strongly dependent on the nanorod radius, the “bridged” to “filled” transition appears at very similar relative pressures for all investigated radii (i.e. the corresponding phase boundary is almost vertical). The location of the phase boundaries are clearly different for the two investigated fluids, reflecting the different fluid–solid interactions. In particular, for a D/r ratio > 3.1 (corresponding to a mesoporosity > 62%), there is no more “bridged” phase for nitrogen, while for n-pentane there is still a quite broad stability range for the “bridged” phase.

Fig. 5
figure 5

a Calculated phase diagram for nitrogen in CMK-3-like carbon at 77 K showing the “separated” phase (green), the “bridged” phase (white), and the “filled” phase (red). b Nitrogen (77 K) adsorption isotherm of hierarchically porous CMK-3-like carbon (black), and the derivative of the isotherm (grey). Vertical lines are drawn at the relative pressure of the two local maxima of the derivative indicating the pressure of the “separated-to-bridged” and the “bridged-to-filled” phase transitions in this sample. The grey vertical regions represent the uncertainty of the maximum derived from the experimental isotherm. The horizontal line in panel a indicates the ratio D/r of 2.6 measured with SAXS for the present sample, with the grey horizontal region representing the uncertainty of the experimentally determined D/r ratio (Color figure online)

Fig. 6
figure 6

a Calculated phase diagram for n-pentane in CMK-3-like carbon at 290 K showing the “separated" phase (green), the “bridged” phase (white), and the”filled” phase (red). b n-pentane (290 K) adsorption isotherm in hierarchically porous CMK-3-like carbon (black), and the derivative of this curve (grey). Vertical lines are drawn at the relative pressure of two maxima of the derivative, indicating the pressure of the “separated-to-bridged” phase transition and the “bridged-to-filled” phase transition in this sample. The grey region represents the uncertainty of the maximum derived from the experimental isotherm. The horizontal line in a indicates the ratio D/r of 2.6 measured with SAXS for the present sample, with the grey horizontal region representing the uncertainty in the experimentally determined D/r ratio (Color figure online)

4 Comparison with experiment

Despite of some experimental hints towards a double condensation transition already in the very early papers on CMK-3 carbons (Jun et al. 2000; Joo et al. 2002), and the corresponding appearance of a “bimodal” mesopore diameter distribution (Gor et al. 2012), the idea of a “film-to-bridged” phase transition without accompanying complete pore filling was not confirmed experimentally so far. The reason might be that the two steps in the adsorption isotherms are not, or at least not unambiguously, seen in many experimental data sets of CMK-3. For instance, in ref. (Gor et al. 2012), one sample showed only a very slight indication of a second step, while for the other sample this second step was clearly visible. This is probably a consequence of a quite large amount of disorder in the arrangement and a large surface roughness of the nanorods, which may smear out such transitions. Here we provide some experimental evidence that the experimentally observed double steps may indeed be related to the two transitions predicted by the thermodynamic model in the previous section. We have chosen a carbon sample with hierarchical porosity, synthesized via nanocasting into a hierarchical silica sample with SBA-15 type cylindrical mesopores. The resulting carbon sample exhibits a micro-/meso-/macroporous structure with the hexagonally ordered cylindrical carbon nanorods forming a CMK-3-like pore geometry (Koczwara et al. 2017). The structural parameters characterizing the mesopore space, D = 10.1 nm and r = 3.9 nm obtained from SAXS (see Appendix B), correspond to a mesoporosity of 46% and a ratio D/r = 2.6. Figures 5b and 6b show the experimentally determined adsorption isotherms of the sample for nitrogen at 77 K and n-pentane at 290 K, respectively (see Appendix B for experimental details). The isotherms show a rapid increase at very low pressures, which is attributed to the filling of micropores within the carbon nanorods. Besides this micropore filling, two shoulders with inflection points at p/p0 ≈ 0.35 and p/p0 ≈ 0.7 for nitrogen, and at p/p0 ≈ 0.20 and p/p0 ≈ 0.6 for n-pentane are clearly recognized. This non-monotonic behavior becomes even clearer when considering the first derivative of the adsorption isotherm (grey curve in Figs. 5b and 6b). Thus, the hierarchical porous carbon material seemingly exhibits two distinct condensation events in adsorption for both nitrogen and n-pentane, but only a single one in desorption (shown in Appendix B). This is consistent with the study by Gor et al. (2012), where high resolution nitrogen adsorption isotherms recorded for CMK-3 samples showed two step-like features in adsorption, but only a single evaporation event for desorption. The data are also qualitatively in agreement with the very early papers on CMK-3 (Jun et al. 2000; Joo et al. 2002), but unfortunately all those data sets did not provide values for the nanorod radius nor their distance, which prevented to include them into our analysis. We mention that although the hierarchical sample investigated here exhibits also macropores, their size range (micrometers) is not compatible with a condensation event at p/p0 ≈ 0.7 for nitrogen.

Vertical dashed lines in Figs. 5 and 6 denote the experimental transition pressures given by the maxima of the derivative, and grey intervals visualize the uncertainty range. It is seen that the second transition (“bridged-to-filled”) agrees with the calculated phase transition pressure for both fluids within the experimental error. In fact, this transition is not very sensitive to the D/r ratio, as the phase boundary is almost vertical in Figs. 5a and 6a. We note that this transition around p/p0 ≈ 0.7 also agrees quite well with the second hump in the isotherm shown in Ref. (Gor et al. 2012). For the “separated-to-bridged” phase, the experimental transition lines cross the phase boundary at D/r ≈ 2.65 for nitrogen and at D/r ≈ 2.7 for n-pentane, respectively. The horizontal line drawn in Figs. 5a and 6a indicates the D/r value of 2.6 measured experimentally with small-angle X-ray scattering (SAXS). The agreement between calculated phase diagram and the experimental result for this given D/r ratio appears to be excellent within the experimental errors.

5 Discussion

The thermodynamic model of a bridging transition between two cylindrical rods was developed by Philip (1977b) and extended to a quadratic array of four cylindrical rods by Dobbs and Yeomans (1993). The latter work allowed predicting three different thermodynamic phases during physical adsorption of fluids in such a system, i.e., a “separated” phase (liquid-like adsorbed film on the cylinders), a “bridged” phase (liquid bridges between the cylinders), and a “filled phase”, where the entire space between the cylinders is filled with liquid. The transitions between these phases are first-order and should be observable in experiments by discontinuous, step-like events in the adsorption isotherms at specific relative vapor pressures. Unfortunately, the geometry of four cylinders on a square lattice proposed in (Dobbs and Yeomans 1993) is not realized experimentally for mesoporous materials, and also the hexagonal rod arrangement realized experimentally with CMK-3 came up only several years after Ref. (Dobbs and Yeomans 1993) was published. This is probably the reason why this elegant thermodynamic treatment of a complex, but still solvable pore space geometry has not found further attention so far. On a side note we mention that although such continuum approaches often exhibit deviations from microscopic simulations particularly at small pore sizes below 5 nm, general trends and at least qualitative agreement is still to be expected (Ravikovitch and Neimark 2000).

In the present work we have reformulated the theory of Dobbs and Yeomans for a hexagonal lattice, and we have calculated theoretical phase diagrams for the adsorption of nitrogen and n-pentane on CMK-3 carbon by deriving the fluid–solid interactions from respective reference isotherms. The relative pressures of the predicted phase transitions are qualitatively consistent with condensation events between two neighboring rods and in the space between three hexagonally arranged rods, respectively, employing the Kelvin-Cohan equation (Appendix C). The predictions of the model were then compared with experimental data from a CMK-3 type sample. Unfortunately, the experimental adsorption isotherms did not unambiguously show two step-like features related to the two transitions. Yet, the shape of the isotherms clearly reveals two “discontinuities” at relative pressures consistent with predictions from the model (Figs. 5 and 6). This agrees with earlier work from other authors on CMK-3 (Gor et al. 2012; Jun et al. 2000; Ryoo et al. 2001), where a second “shoulder” was clearly observed, although this feature was not discussed in these papers. Gor et al. (2012) used N2 isotherms on CMK-3 to derive a pore size distribution from QSDFT, and observed a second class of mesopore sizes which would be consistent with a second condensation event. However, the experimentally observed step height of the transitions does not agree with the theoretical predictions. The experimentally observed second step of the “bridged-to-filled” transition (Fig. 5b) is much smaller than the one predicted (Fig. 4b). There are two possible reasons for this deviation: First, the mean-field theory used here predicts a more abrupt transition than a theory which would consider the density variation in the condensed phase, such as based on molecular simulations. Second, we may attribute this deviation also to the disorder in the system (see Fig. 7a). There is indeed strong evidence in literature that the nanorods exhibit a strongly corrugated surface with carbon cross-bridges between them (Solovyov et al. 2002). As sketched in Fig. 7b, such cross bridges might be local condensation points, and the geometry of the “bridged” phase might be realized only locally over a restricted volume (Fig. 7c). Due to the disorder of the carbon nanowires, specific locations in the sample may exhibit small interstitial spaces in-between three 2D-hexagonally ordered carbon nanowires, which fill already at lower pressures, thus the “separated-to-bridged” transition and the “bridged-to-filled” transition not being separately resolved, as sketched in Fig. 7d, I and II. This would naturally lead to larger filling fractions for the “separated-to-bridged” transition, with a broad transition-pressure regime, as observed experimentally. As a consequence, only a considerably smaller pore volume fraction than predicted by our model (Fig. 4c) would contribute to the “bridged-to-filled” transition. We note that the structural parameters (\(D\) and \(r\)) from SAXS (see Fig. 8) are somewhat ambiguous, as they are related to the highly ordered part of the pore space. It has been shown already for SBA-15 silica (Jähnert et al. 2009; Findenegg et al. 2010) that there may be a considerable amount of “disordered porosity”, which we expect to be even higher for CMK-3 due to the additional synthesis step using SBA-15 as a template. We speculate that as long as the distance of mutual contact points between neighboring cylinders along the cylinder axis is clearly larger than the distance between the cylindrical rods, the geometry sketched in Fig. 1 would exist at least locally (see Fig. 7d, I), enabling in principle the proposed transitions. We are however fully aware of the fact that our experimental model system is much more complicated than the theoretical model. Consequently, the observed agreement between the calculated and the measured phase transition pressures is a strong indication, but no final proof for the existence of such capillary bridges between nanorods in CMK-3. A final proof would require the availability of samples with much higher structural order, which to our knowledge are not available so far.

Fig. 7
figure 7

Sketch of a “realistic” 3D model of the carbon nanorods (a), their mean distance and radius corresponding to the values obtained from SAXS. b and c show sketches of vertical cuts for pressures below (b), and above (c) the “separated-to-bridged” transition, with the liquid adsorbate shown in opaque green. The liquid-like film covering the nanorods is omitted in c for better visualization, and only condensed regions in small “constrictions” are shown, which will act as nucleation sites for the “bridged” phase. In d, two top-view sketches at cuts through the positions I and II in c are shown, demonstrating the local existence of the “bridged” and the “filled” phases in different regions along the rod axis

Although kernels of isotherms from molecular simulations provide satisfactory fits to experimental data from CMK-3 carbons, none of them discusses the existence of liquid bridges spanning the shortest distance between neighboring nanorods (Yelpo et al. 2017; Barrera et al. 2013; Jain et al. 2017). The adsorption isotherms in these models are usually derived assuming the existence of a spinodal, which is not necessarily the case in highly conjugated pore spaces (Gommes and Roberts 2018). Our purely thermodynamic equilibrium approach was able to quite accurately predict the experimentally observed transition pressures. This implies that nucleation events in the void space between the nanorods must help overcoming the activation barrier right at the pressure of equilibrium, which consequently means that a spinodal transition is not present. Jain et al. (2017) performed simulations of argon adsorption on CMK-3 including interconnections between the individual nanorods. They could clearly show the influence of such irregularities on the general shape of the adsorption isotherm. The deliberately introduced interconnections basically shifted the relative pressure for condensation to significantly lower values, meaning that smaller structures within the void space could very well serve as nucleation sites. Hence, the presence of disorder and possible carbon interconnects in CMK-3 might be even the key to the formation of liquid bridges between neighboring rods by providing the nucleation sites for the “bridged” phase. The fact that no spinodal transition is needed to determine the relative pressures of condensation sheds light on the underlying physical processes and their relation to the actual structures present in the material.

6 Conclusion

In conclusion, we presented a comparison between computational and experimental results on the structural characterization of a monolithic CMK-3-like material using nitrogen and n-pentane adsorption and small angle X-ray scattering. Following an earlier theoretical approach (Dobbs and Yeomans 1993), three different phases of the fluid in the pore space (“separated”, “bridged” and “filled” phases) are proposed, and their grand potentials are minimized by the geometric arrangement of liquid inside the open pore space in the monolithic CMK-3-like material. The theoretical predictions for the adsorption isotherms of nitrogen and n-pentane on the carbon material for varying nanorod radii but constant nanorod distances were used to construct phase diagrams, linking the “separated-to-bridged” and “bridged-to-filled” phase transitions to experimental adsorption data and structural data from small angle X-ray scattering. For both, nitrogen and n-pentane adsorption, fair agreement between the theoretical predictions and experimental results is found, indicating that this model is able to qualitatively describe the physical processes governing adsorption in the open pore space of CMK-3 like materials. The resulting mean pore size is in good agreement with earlier work using state-of-the-art methods (Gor et al. 2012).