Article pubs.acs.org/JPCC
On the Cavitation-Like Pore Blocking in Ink-Bottle Pore: Evolution of Hysteresis Loop with Neck Size Phuong T. M. Nguyen, Chunyan Fan, D. D. Do,* and D. Nicholson School of Chemical Engineering, University of Queensland, St. Lucia Qld 4072, Australia ABSTRACT: Studies of adsorption and desorption of argon at 87 K in model inkbottle pores have been carried out using Monte Carlo simulation. We show that the isotherms can be constructed as a composite of isotherms for a set of unit cells with constant pore size. The mechanisms of adsorption and desorption in an ink-bottle pore can be easily understood from the characteristics of these unit cells, providing insight into how the hysteresis loop would evolve in shape and area when the neck size is varied. The key factor controlling the characteristics of the loop is the relative position of the condensation and evaporation pressures of these unit cells. Two features of particular interest are noted: (i) a pore blocking mechanism might be mistaken as a cavitation if cavitation is interpreted as a sudden change in the amount adsorbed along the desorption branch and (ii) the shape of the hysteresis loop switches from type H1 for small neck sizes to type H2 for larger necks but reverts back to type H1 when the neck size approaches the cavity size. function of the properties of the pure fluid but also a function of the cavity size if it is small enough for the adsorbent to make a significant contribution to the overall potential energy. For example, for nitrogen at 77 K, the cavity size is P > PD PD > P > PE PE > P > 0
neck
cavity
remarks
open-end pore open-end pore closed-end pore
closed-end pore closed pore
PA = condensation pressure of the neck PB = condensation pressure of the cavity PC = saturation pressure
closed-end pore open-end pore open-end pore
closed pore
closed pore
closed pore
PD = evaporation pressure of the cavity PE = evaporation pressure of the neck
closed-end pore
mechanisms have been discussed in greater detail elsewhere.16,40 For ink-bottle pores having a neck size smaller than the cavity, there is an interesting evolution of the hysteresis loop. When the neck size is smaller than a critical value WN,c, corresponding to a reduced pressure for evaporation of ∼0.24 (cavitation pressure), the mechanism for evaporation is cavitation. When the neck size is above this critical size, evaporation occurs at a higher pressure than the cavitation 5477
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Figure 3. Adsorption and desorption isotherms for argon at 87 K in closed pores of various dimensions for slit geometry (left) and cylindrical geometry (right).
Figure 4. Adsorption isotherms for argon at 87 K in ink-bottle pores and in the corresponding unit cells when the neck is smaller than the critical neck size (a) for the slit model and (b) for the cylinder model.
pore. Evaporation from the neck then occurs at a lower pressure, corresponding to the evaporation from an openended pore. The behavior of the neck and the cavity along the adsorption and desorption branches is summarized in Table 2. A subtle point about the shape of the hysteresis loops in Group I, which might not have been previously recognized, is
is the same as that of a closed pore having the same dimensions as the cavity. On desorption, the fluid in the cavity is stretched and the evaporation of adsorbate from the cavity occurs at the cavitation pressure, which is greater than the desorption pressure for the neck because of the small neck size. The neck then changes from a closed-end pore to an open-ended 5478
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Table 3. Unit Cells for the Ink-Bottle Pores in Group I curve
slit geometry
cylindrical geometry
1
closed-end pore of 7 nm width and 7 nm length closed pore of 7 nm width and 7 nm length open-end pore of 2.3 nm width and 10 nm length closed-end pore of 2.3 nm width 10 nm length
closed-end pore of 6 nm width and 7 nm length closed pore of 6 nm width and 7 nm length open-end pore of 3.6 nm width and 7 nm length closed-end pore of 3.6 nm width 7 nm length
2 3 4
Table 4. Unit Cells for Ink-Bottle Pores with a Critical Neck Size slit geometry
cylindrical geometry
closed-end pore of 7 nm width and 7 nm length closed pore of 7 nm width and 7 nm length open-end pore of 2.98 nm width and 10 nm length closed-end pore of 2.98 nm width 10 nm length
closed-end pore of 6 nm width and 7 nm length closed pore of 6 nm width and 7 nm length open-end pore of 4 nm width and 7 nm length closed-end pore of 4 nm width 7 nm length
that the isotherm of the ink-bottle pore is the composite of these unit cells. The isotherms of the unit cells are shown in the middle and bottom panels in Figure 4. The middle panels show the isotherms for the unit cells having the same dimensions as the cavity, one of which is a closed-end pore (Curve 1) and the other is a closed pore (Curve 2). The bottom panels show isotherms for the unit cells having the same dimensions as the neck: one is an open-end pore (Curve 3) and the other is a closed-end pore (Curve 4). The condensation in the ink-bottle pore at A is associated with the neck because the pressure at this point is the same as the condensation pressure for the open-end pore with the same width WN (Curve 3). For pressures greater than PA (pressure at the point A), the cavity behaves as a closed pore, and as pressure is increased, condensation occurs at B, associating with the cavity because PB is the same as the condensation pressure in the closed pore with the same width, WC (Curve 2). For desorption from C (filled pore), the cavity behaves like a closed pore and the neck behaves like a closed-end pore. When the pressure reaches point D, an abrupt evaporation occurs, which is associated with the cavitation of the fluid in the cavity because PD is the same as the evaporation pressure of the closed pore of width WC. At this point, the cavity continues to behave as a closed pore, whereas the neck changes from a closed-end pore to an open-ended pore. As the pressure is further decreased, a second evaporation from the neck occurs at E because PE is the same as the evaporation pressure of the openended pore of width WN. The mesoscopic configurations of adsorption and desorption in Group I ink-bottle pore for points A−E are presented in Figure 5. It may be concluded that it is possible to construct the isotherm for an ink-bottle pore by summing the isotherms of
Figure 5. Typical adsorption−desorption isotherms in the ink-bottle pore in group I.
that the loops in the top four isotherms in Figure 2, might be classified as Type H1 because they appear to be associated with a vertical condensation and evaporation in the cavity. However, if the cavity size is made larger, then the portion of the isotherm before condensation occurs in the cavity, which is associated with the amount of adsorbate in the adsorbed film on the cavity walls, is much greater than the amount adsorbed at condensation; as a result the hysteresis loop is more Type H2 than H1. To substantiate this, we present in Figure 3 the adsorption isotherms of closed pores having various dimensions. To examine the mechanisms of adsorption and desorption in Group I in more detail, we consider a neck size of 2.3 nm for the slit model and 3.6 nm for the cylinder. Their isotherms exhibit a double hysteresis loop, as shown in the top figures in Figure 4a (for the slit) and Figure 4b (for the cylinder). We define unit cells of uniform pore size in Table 3 and then show
Figure 6. Composite isotherms of an ink-bottle pore in group I (a) with neck size of 2.3m for slit and (b) with neck size of 4 nm for cylinder. 5479
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Figure 7. Adsorption isotherms for argon at 87 K in ink-bottle pores and in the corresponding unit cells when the neck is equal to the critical neck size (a) for the slit and (b) for the cylinder.
Figure 8. Schematic diagram for desorption in an ink-bottle with the neck is equal to critical neck size.
Figure 9. Schematic diagram of adsorption in an ink-bottle pore with neck smaller than WN*.
the relevant unit cells, weighted with respect to their accessible volumes for these small neck sizes. The composite isotherm constructed from the isotherms of the unit cells is shown in Figure 6, together with the directly simulated isotherm for the
ink-bottle pore. The constructed composite isotherm corresponds closely to the isotherm from the direct simulation. The small differences are due to adsorption in the small conical 5480
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Figure 10. Adsorption isotherms for ink-bottle pores and their unit cells when the neck is smaller than WN* (a) slit pores and (b) cylindrical pores.
pressure is greater than PD (or PE), the neck behaves like a closed-end pore and desorption starts as the meniscus recedes from the pore mouth. When the meniscus reaches the junction between the neck and the cavity, that is, when the fluid in the neck has evaporated and the pressure reaches the cavitation pressure, there is an instant evaporation from both the neck and cavity (D→E in the schematic in Figure 8). This mechanism explains why there is a critical neck size, which is the size at which the pressure for evaporation from the closed-end unit cell of this size (Curve 4) is the same as the cavitation pressure or the pressure at which there is evaporation from a closed unit cell of width WC (Curve 2). This type of hysteresis loop has been observed by Casanova et al.41,42 for nanoporous silicone. By optical interferometry, the same conclusion has also made: the mechanisms of filling and emptying processes for the narrow neck are the same as those of corresponding individual pore, but those mechanisms for the cavity are different with the corresponding closed-end pore because its does not have a direct access to the gas reservoir. 4.3. Group II: the Neck Size Is Greater than the Critical Size. In this group, the mechanism for evaporation from the cavity changes from cavitation to pore blocking. This can be recognized because the evaporation pressure is greater than the cavitation pressure. Moreover, when the neck is larger than the
Table 5. Unit Cells for the Ink-Bottle Pores with Neck Size Smaller than WN* slit geometry
cylindrical geometry
closed-end pore of 7 nm width and 7 nm length closed pore of 7 nm width and 7 nm length open-end pore of 3.65 nm width and 10 nm length closed-end pore of 3.65 nm width 10 nm length
closed-end pore of 6 nm width and 7 nm length closed pore of 6 nm width and 7 nm length open-end pore of 4.4 nm width and 7 nm length closed-end pore of 4.4 nm width 7 nm length
section of the ink-bottle pore, which has not been accounted for in the composite isotherm. 4.2. Neck Size Is Equal to the Critical Neck Size. The isotherms for the ink-bottle pore with a critical neck size for the corresponding unit cells (defined in Table 4) are presented in Figure 7. The mechanisms of adsorption and desorption in this pore are very similar to that of the Group I. The only difference is that the pressure for evaporation from the cavity via a cavitation process is now exactly the same as the pressure for evaporation from the neck, that is, PD = PE. It has been established that the evaporation from a cavity is due to cavitation; when the 5481
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Figure 11. Adsorption isotherms for ink-bottle pores and their unit cells when the neck is larger than WN* (a) slit pores and (b) cylindrical pores.
Table 6. Unit Cells for the Ink-Bottle Pores with Neck Size Larger than WN* slit geometry
cylindrical geometry
closed-end pore of 7 nm width and 7 nm length closed pore of 7 nm width and 7 nm length open-end pore of 4.99 nm width and 10 nm length closed-end pore of 4.99 nm width 10 nm length
closed-end pore of 6 nm width and 7 nm length closed pore of 6 nm width and 7 nm length open-end pore of 5.6 nm width and 7 nm length closed-end pore of 5.6 nm width 7 nm length
critical size but smaller than a threshold value WN* (WN* = 5 nm for both slit and cylinder in this system) the hysteresis loop is type H2 with a two-step condensation (associated with the neck and the cavity) and a steep evaporation, which is usually interpreted as the fingerprint of a cavitation process, but here the steep evaporation is due to emptying of the fluid from the cavity, induced by the evaporation in the neck; we refer to this as “advanced evaporation from the cavity”. When the neck is greater than WN*, the hysteresis loop is of Type H1. Condensation occurs in just one step, which is due to the “advanced condensation in the neck”, facilitating condensation in the cavity. To elaborate this, we analyze the adsorption isotherm for ink-bottle pores and the corresponding unit cells for these cases.
Figure 12. Schematic diagram of adsorption in the ink-bottle pore when the neck is larger than WN*.
4.3.1. Advanced Evaporation or Cavitation-Like Pore Blocking. We show in Figures 9 and 10 the examples for inkbottle pores with neck size smaller than WN*. The dimensions of the corresponding unit cells are given in Table 5. It is seen that the mechanism of adsorption is similar to that for Group I. The adsorption branch in the ink-bottle pore proceeds by a two-step condensation: the first step (at A) corresponds to condensation in the neck, which occurs at the same pressure as the corresponding open-ended pore of width WN (Curve 3 in 5482
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cylindrical cross sections, as a function of neck size. We find that the shape and area of the hysteresis loop changes significantly with the neck size. When the neck is smaller than a critical neck size, WN,c, evaporation is via a cavitation mechanism and the hysteresis loop is type H1 in the IUPAC classification, with a steep condensation and a steep evaporation at the cavitation pressure, which is governed only by the fluid properties. When the neck is larger, but smaller than a critical width WN*, the hysteresis loop is type H2, with a steeply sloping desorption branch. However, the mechanism of evaporation does not involve bubble formation in the pore interior, but an advancing evaporation from the cavity when the meniscus in the neck reaches the junction. We have termed this mechanism “cavitation-like pore blocking”. It can be distinguished from cavitation by a knee shape in the desorption branch, which is seen before the evaporation pressure, which is higher than the cavitation pressure. When the neck is larger than WN*, the hysteresis loop is type H1, which is attributed to an advanced condensation in the neck, in addition to the advanced evaporation from the cavity. The former phenomenon occurs when the neck is instantly filled as it changes from an open-ended pore to a closed pore when the meniscus in the cavity has reached the junction. The isotherm for an ink-bottle pore can be constructed as the composite of isotherms for a set of unit cells with constant sizes comprising: (1) a closed pore of size WC, (2) a closed-end pore of size WC, (3) a closed-end pore of size WN, and (4) an openend pore of size WN.
Figure 10), and the second step (at B) corresponds to condensation in the cavity that occurs at the same condensation pressure as the closed pore of width WC (Curve 2 in Figure 10). Upon desorption from a filled pore (at C), a meniscus develops at the mouth of the neck and recedes more readily than in Group I because of the larger neck size. This results in a knee in the desorption branch before point D, which was not observed for the Group I models. At D, the meniscus has reached the junction between the neck and the cavity, and the cavity now becomes a closed-end pore with one end exposed to the surroundings. Because the pressure at this point is below the evaporation pressure of the corresponding closed-end pore (Curve 1 in Figure 10), the cavity cannot sustain a condensate and the fluid evaporates at point D. This cannot be regarded as a cavitation mechanism because there is no gas-like core surrounded by adsorbed film, as would be the case according to the strict definition of cavitation. We have termed this process “cavitation-like pore blocking” because the hysteresis loop has a similar shape to that found for a true cavitation, but here the cavity is emptied at a pressure higher than the cavitation pressure. This process of advanced evaporation from the cavity, due to the evaporation from a neck larger than the critical size, has not been previously recognized in the literature. 4.3.2. Advanced Condensation in the Neck and Advanced Evaporation from the Cavity. When the neck is larger than WN*, we observe a phenomenon that we have called “advanced condensation in the neck”, in addition to “advanced evaporation from the cavity” described in Section 4.3.1. Figure 11 shows the adsorption isotherms for an ink-bottle pore with neck size larger than WN* and for the corresponding unit cells whose dimensions are given in Table 6. The mechanisms of adsorption and desorption of the ink-bottle pore are presented in Figure 12. Adsorption follows a molecular layering process for pressures less than PA: the cavity behaves as a closed-end pore and the neck behaves as an open-end pore. In contrast with the previous cases for which the neck is filled first, the adsorption proceeds by filling the cavity via the advance of a meniscus from the closed-end. This is because the neck is wide enough for the condensation pressure of the open-ended pore of width WN (curve 3) to be higher than the pressure at which the closedend pore of width WC (curve 1) is filled. (See Figure 11.) At A, the cavity is completely filled and the meniscus in the cavity has reached the junction; the neck then changes from an openended pore (unit cell of Curve 3) to a closed pore (Curve 4). Because the closed-end pore of width WN must be filled at this pressure according to Curve 4, it cannot sustain a gas phase in the pore interior, and as a result, a adsorption increases steeply at PB, corresponding to the complete filling of the remaining parts of the cavity and the neck. We refer to this as “advanced condensation” in the neck, which occurs when the size of the neck is close to that of the cavity. Desorption from a filled pore is similar to the case when the neck is smaller than WN* and there is “advance evaporation from the cavity” at a pressure PD, higher than the cavitation pressure. This occurs when the cavity changes from a closed pore to a pore with a closed end and the meniscus has reached the junction. This phenomenon is called “cavitation-like pore blocking”.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This project is supported by the Australian Research Council. REFERENCES
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5. CONCLUSIONS We have presented a simulation study of adsorption and desorption of argon at 87 K in ink-bottle pores with slit and 5483
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