Interplay of vapor adsorption and liquid imbibition in nanoporous Vycor glass

We have studied the kinetics of spontaneous capillary rise and of the concurrent vapor adsorption in nanoporous, monolithic samples of Vycor glass with a mean pore diameter of $7.5$ nm. As liquids, we have chosen $n$-alcohols ($n=4–10$) whose vapor pressures at room temperature range from $p_0=965$ Pa down to $p_0=0.743$ Pa. Dielectric measurements allow us to achieve spatial selectivity to predefined parts of the porous Vycor glass. In this way, we are able to measure the overall uptake of molecules as well as vapor adsorption from the surroundings in unfilled parts of the pore network, i.e., above the liquid menisci of the rising imbibition front. We show that the latter process is unaltered compared to free adsorption in samples suspended above a liquid reservoir. Only at low vapor pressures, i.e., for long alcohols, vapor adsorption can be neglected and the capillary rise follows the theoretical predictions of the Lucas-Washburn $\sqrt{t}$ law. The more volatile the alcohol, the more important the additional adsorption of molecules becomes. We show that the overall filling process in the pore network is well described by a superposition of the Lucas-Washburn law and the measured vapor adsorption. In addition, the experiments give insight into the vapor diffusion dynamics in the porous matrix.

Figure 1

Sample and pore geometry. (a) Typical nanoporous Vycor sample during an imbibition experiment with imbibition front at position $h\left(t\right)$. Vycor has a spongelike microstructure containing a network of interconnected pores, as illustrated in the circular inset. (b) The individual pores feature a cylindrical cross section. When a capillary flow fills the empty pore space, a parabolic velocity profile is expected. Its width $r_h$ depends on a slip boundary condition (see text). Here, the case of $r_h<r$ and $r=r_L$ ($\theta =0^{\circ }$) is depicted, which was assumed for $n$-alcohols on Vycor glass.

Figure 2

Transport mechanisms and two-dimensional network topologies during vapor adsorption. (a) The transport of molecules inside the pores is largely determined by a Knudsen diffusion in the gaseous phase. The mean free path of vapor particles is typically larger than geometrical constraints. Interactions with the pore walls then govern the molecular movement as opposed to particle-particle interactions in a bulk gas. (b) Adsorbed layers at the pore walls partially retain their mobility. They directly contribute to molecular transport by surface diffusion. (c) Capillary bridges inside the pores provide additional adsorption spots for vapor molecules. Condensation on capillary bridges enables a hydrodynamic advancement of the concave menisci, while disabling Knudsen diffusion along this pore. (d) The network topology gradually changes during the adsorption process. For low pore loadings, adsorption is limited to molecular layers. (e) At higher pore loadings, capillary bridges obstruct the vapor transport inside the pores and parts of the pore volume may have no connection to the bulk vapor.

Figure 3

Schematic of the experimental setup used in this work. It consists primarily of a sealed chamber containing a liquid reservoir and a suspended nanoporous sample. Liquid level and atmosphere can be controlled externally. Both imbibition dynamics and vapor adsorption dynamics can be measured, depending on the amount of liquid inserted. Here, the case of an imbibition experiment is depicted, as the sample touches the liquid surface. The rising liquid inside the porous matrix is detected by means of dielectric spectroscopy.

Figure 4

Dielectric spectra of 1-decanol confined to porous Vycor glass over the course of an experiment: (top) real part and (bottom) imaginary part of permittivity. The dotted lines show the spectra of the empty sample surrounded by air; the continuous lines show the increase in filling ($f=0\%,20\%,40\%,60\%,80\%,100\%$) over time for a sample in contact with the liquid surface.

Figure 5

Evolution of the real part of the total sample capacitance over the course of a typical imbibition experiment, here for 1-decanol. Note the sudden increase at $t=4\mathrm{h}$ due to bulk liquid reaching the stray field when liquid is inserted. The maximum capacity $C_{\text{max}}$ is reached when the sample is entirely filled.

Figure 6

Experimental data on imbibition of $n$-alcohols ($n=4$–10) in Vycor. (a) Filling fractions in imbibition experiments plotted over the square root of time. As viscosity increases for higher chain lengths, imbibition dynamics are slower. (b) Typical imbibition experiment, here for heptanol (C7), showing four characteristic segments: (I) initial phase with systematic deviations from Lucas-Washburn dynamics, (II) linear segment with fit, (III) broadened imbibition front reaches the top of the sample, and (IV) pore space is filled entirely. (c) Visualization of time-dependent imbibition front shape because of adhering bulk liquid menisci. During the first experimental phase (I), an additional liquid influx is expected.

Figure 7

Experimental Lucas-Washburn coefficients for $n$-alcohols, $n=4$–10 ($\circ$) from imbibition experiments compared to theoretical predictions for a no-slip boundary condition (continuous line), with an immobilized molecular layer of water (slip length $b=-0.275\mathrm{nm}$; dotted line) and with an alcohol monolayer oriented in parallel to the pore wall ($b_{\text{Cn}}=-0.4\mathrm{nm}$; dash-dotted line). Note the increasing deviation of $c_h$ for small $n$, i.e., increasing volatility of the liquid. The theoretical values were corrected for vapor adsorption assuming $b=0$ (see Sec. 4c). Then, the resulting corrected Lucas-Washburn prefactors (dashed line) explain the upwards deviations of $c_h$ for small chain lengths. (All lines are linear interpolations of data points calculated for the given molecule lengths.)

Figure 8

Experimental results on vapor adsorption. (a)–(e) Comparison of “free” (dashed line) and “concurrent” (continuous line) adsorption experiments for $n$-alcohols ($n=4$–8). Note that both curves show a good match at small times for all alcohols. (f) For concurrent vapor adsorption, the sample is only partially coated with electrode material and in contact with the liquid, inducing spontaneous imbibition. The measured filling curves then show two different regions: First, adsorbed vapor is detected. (g) After the imbibition front passes the electrode, vapor adsorption and imbibition both contribute to total filling. (h) During free adsorption, the same sample and electrode geometry is subjected to vapor adsorption only, there is no physical contact to the liquid surface (distance not drawn to scale).

Figure 9

Free adsorption dynamics, here for 1-heptanol (C7) [dashed line in Fig. 8], as a function of $t$. Here, three stages with different dynamics can be defined. After an initial step (i), a stage with constant slope (ii) is found that extends over large periods of time. The final stage (iii) with Lucas-Washburn dynamics shows also as a linear segment of the dashed lines in Fig. 8 for large times.

Figure 10

Experimental normalized filling for pentanol ($n=5$) ($\circ$) from imbibition experiments compared to Lucas-Washburn theory (continuous line) and to a superposition of capillary flow and (measured) adsorption from Eq. (13) (dashed line). A linear fit (dotted line) can be applied to the corrected theory in order to obtain corrected Lucas-Washburn coefficients $c_h$.