The interlayer pore width of the osmotic swelling is calculated from a regression curve (Equations (11) and (12)) in Figure 2 plotting the relation between the effective montmorillonite density and basal spacing. The basal spacing of the mixed state of the osmotic and crystalline swelling was at 700 kg/m3ρem < 1,000 kg/m3 for Na-montmorillonite. As the abundance ratio of the number of stacks attributed to the osmotic or crystalline swelling was difficult to evaluate precisely, we assumed that the osmotic swelling stacks had an abundance ratio of 100% at ρem = 700 kg/m3 and 0% at ρem = 1,000 kg/m3, and vice versa for the crystalline swelling. The abundance ratio of the stacks between these densities was calculated from linear interpolation. The basal spacing of the mixed state consisting of the 3WH and 2WH states was estimated at 1,300 kg/m3 < ρem < 1,600 kg/m3 for Na-montmorillonite and Ca-montmorillonite. We assumed that the stacks had an abundance ratio of 100% at ρem = 1,300 kg/m3 and 0% at ρem = 1,600 kg/m3 for the 3WH state, and vice versa for the 2WH state. Between these densities, the abundance ratio of the stacks was calculated in the same manner used for Na-montmorillonite at 700 kg/m3ρem < 1,000 kg/m3. The external pore width was calculated by Equations (1), (11), (12), and (20) using the calculation parameters as shown in Table 1. The external pore width decreases as the effective montmorillonite density increases, and also as the number of laminations in the stack (n) decreases (Figure 5). If the number of laminations exceeds 4 for Na-montmorillonite, the external pore width at ρem = 1,000 kg/m3 is larger than that at ρem = 700 kg/m3, based on the assumed ratio of the osmotic swelling to the 3WH states. Having a large external pore width implies that the specimen has enough space for montmorillonite swelling. Since only crystalline swelling was observed at ρem = 1,000 kg/m3 (Kozaki et al. 1998), it would be unreasonable to assume that the external pore width at ρem = 1,000 kg/m3 is larger than that at ρem = 700 kg/m3. In contrast to the unreasonable behavior observed for Na-montmorillonite with an n over 3, the external pore width in the case of Ca-montmorillonite decreases monotonically as the effective montmorillonite density increases in every lamination number. This difference between Na-montmorillonite and Ca-montmorillonite was due to the difference in their swelling behavior. Although Na-montmorillonite could form the osmotic state (the d-value was over 3.2 nm) under any effective montmorillonite density condition other than 1,000 kg/m3ρem, Ca-montmorillonite could swell only up to the 3WH state (the d-value was 1.88 nm) under the density condition of ρem < 1,000 kg/m3. Therefore, we determined that the number of laminations for Na-montmorillonite was 2 and 3, while that for Ca-montmorillonite was an arbitrary number above 2.
Table 1

Properties of montmorillonite particle

 Diameter of montmorillonite particle (2R) 3.91 × 10−7 [m] (Suzuki et al. 2012) Thickness of montmorillonite particle (t) 9.55 × 10−10 [m] (Tournassat et al. 2003) Particle density of montmorillonite (ρm) 2,770 (kg/m3) (Komine 2008)
 Diameter of montmorillonite particle (2R) 3.91 × 10−7 [m] (Suzuki et al. 2012) Thickness of montmorillonite particle (t) 9.55 × 10−10 [m] (Tournassat et al. 2003) Particle density of montmorillonite (ρm) 2,770 (kg/m3) (Komine 2008)
Figure 5

Relation between effective montmorillonite density and external pore width (n represents the number of laminations in a single stack).

Figure 5

Relation between effective montmorillonite density and external pore width (n represents the number of laminations in a single stack).

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