Thermodynamics of imbibition in capillaries of double conical structures-Hourglass, diamond, and sawtooth shaped capillaries-

In this section we reconsider and enlarge our previous studies Iwamatsu (2020, 2022) of the classical capillary model of imbibition (intrusion and extrusion) in a single conical capillary. Though the classical capillary model, which is the simplest case of morphological thermodynamic approach König, Roth, and Mecke (2004); Roth and Kroll (2006), is macroscopic, it is believed to be valid down to the nanoscale Tinti et al. (2023) and would give useful information even to the micro and the nano scale phenomena. In this classical model, the surface free energy $F$ comprises the free energy of the free liquid-vapor surface energy $F_{\rm lv}=\gamma_{\rm lv}S_{\rm lv}$ and that of the liquid-solid surface energy of the capillary wall $F_{\rm sl}=\gamma_{\rm lv}\cos\theta_{\rm Y}S_{\rm sl}$ wetted by the liquid. The total surface free energy is given by

$$F=F_{\rm lv}-F_{\rm sl}=\gamma_{\rm lv}S_{\rm lv}-\gamma_{\rm lv}\cos\theta_{\rm Y}S_{\rm sl},$$

(1)

where $\gamma_{\rm lv}$ and $S_{\rm lv}$ represent the liquid-vapor surface tension and the surface area, respectively, and $S_{\rm sl}$ is the solid-liquid (wet) surface area (see Tab. 1 for the complete list of symbols and their description.). The angle $\theta_{\rm Y}$ is Young’s contact angle defined by Young’s equation, which is expressed as:

$$\cos\theta_{\rm Y}=\frac{\gamma_{\rm sv}-\gamma_{\rm sl}}{\gamma_{\rm lv}},$$

(2)

where $\gamma_{\rm sv}$ and $\gamma_{\rm sl}$ represent the solid-vapor and the solid-liquid surface tensions, respectively. This Young’s contact angle characterizes the wettability of capillary wall. Here, we neglect the effect of gravity since we consider capillaries whose diameters are smaller than capillary length. Also, we neglect the contribution of the line tension, which could play some role in nano scale.

Even though we will not consider the nucleation of liquid droplets or vapor bubbles in the middle of capillary, we will use the terminology "vapor" instead of "gas" throughout this paper. In fact, our model system can be applicable to any binary system of immiscible fluids including non-volatile liquid and gas systems.

We consider an axially symmetric capillary around the $z$ axis whose inlet is at $z=0$. We borrow the concept of transient state of the classical nucleation theory Donne et al. (2022); Tinti et al. (2023), and assume an imbibition pathway along the capillary with a constant Young’s contact angle Iwamatsu (2020, 2022); Tinti et al. (2023). Then, the solid-liquid surface free energy when the liquid-vapor interface reaches $z$ is given by Iwamatsu (2020)

$\displaystyle F_{\rm sl}$

$\displaystyle=$

$\displaystyle 2\pi\gamma_{\rm lv}\cos\theta_{\rm Y}\int_{0}^{z}R(z^{\prime})\sqrt{1+\left(\frac{dR}{dz^{\prime}}\right)^{2}}dz^{\prime},$

(3)

where $R(z)$ is the radius of the capillary at $z$. The liquid-vapor surface free energy is given by

$$F_{\rm lv}=2\pi\gamma_{\rm lv}R(z)^{2}\frac{1-\cos\psi}{\sin^{2}\psi}\simeq\pi\gamma_{\rm lv}R(z)^{2},$$

(4)

where the half-opening angle $\psi$ defined in Fig. 2 is approximated by $\psi=0$ to simplify mathematics. Therefore, a spherical interface is replaced by a flat one Iwamatsu (2020) because inclusion of the spherical interface gives only a small correction which can be included by regarding the capillary radius $R(z)$ as an effective radius Iwamatsu (2022).

The total liquid volume inside the capillary is also approximately given by

$$V(z)=\pi\int_{0}^{z}R_{i}(z^{\prime})^{2}dz^{\prime}+\frac{\pi}{3}R(z)^{3}\nu\left(\psi\right)\simeq\pi\int_{0}^{z}R(z^{\prime})^{2}dz^{\prime}$$

(5)

where a small volume correction of a spherical cap Iwamatsu (2022) (Fig. 2)

$$\nu\left(\psi\right)=\frac{\left(1-\cos\psi\right)^{2}\left(2+\cos\psi\right)}{\sin^{3}\psi}$$

(6)

is neglected. Since the surface free energy $F(z)$ in Eq. (1) is simply given as a function of $z$ by the sum of Eqs. (3) and (4), and the liquid volume $V(z)$ is given by Eq. (5), the liquid pressure $p_{\rm L}(z)$ defined by

$$p_{\rm L}(z)=-\frac{\partial F}{\partial V}=-\frac{1}{dV/dz}\frac{\partial F_{(}z)}{\partial z},$$

(7)

can be analytically calculated as

$$p_{\rm L}(z)=\frac{2\gamma_{\rm lv}}{R(z)}\left(\cos\theta_{\rm Y}\sqrt{1+\left(\frac{dR}{dz}\right)^{2}}-\frac{dR}{dz}\right)$$

(8)

which reduces to the standard Laplace pressure

$$p_{\rm L}(z)=\frac{2\gamma_{\rm lv}\cos\theta_{\rm Y}}{R}$$

(9)

in straight cylinders ($R(z)=R$, $dR/dz=0$).

Now, we consider two conical capillaries of identical shape with either a converging or a diverging radius (Fig. 3) $R_{\rm C}(z)$ or $R_{\rm D}(z)$ given by:

	$\displaystyle R_{\rm C}(z)=R_{\rm C}(0)-\left(\tan\phi\right)z,\;\;\;\left(0\leq z\leq H\right),$		(10)
	$\displaystyle R_{\rm D}(z)=R_{\rm D}(0)+\left(\tan\phi\right)z,\;\;\;\left(0\leq z\leq H\right),$		(11)

where $\phi(0\leq\phi\leq 90^{\circ})$, and $R_{\rm C}(0)=R_{\rm C}(z=0)$ and $R_{\rm D}(0)=R_{\rm D}(z=0)$, and $H$ represent the tilt angle of the wall, the radius at the inlet ($R_{\rm C}(0)>R_{\rm D}(0)$), and the length of the capillary (Fig. 3). Hence, the capillary parameters in Eqs. (10) and (11) are related by

$$R_{\rm D}(0)=R_{\rm C}(0)-\left(\tan\phi\right)H.$$

(12)

To study the imbibition, we have to specify the geometry of the conical capillaries. We select the tilt angle $\phi$ and the aspect ratio

$$\eta_{\rm C}=\frac{H}{R_{\rm C}(0)}$$

(13)

as the two fundamental parameters to specify the geometry. Therefore, another aspect ratio

$$\eta_{\rm D}=\frac{H}{R_{\rm D}(0)}$$

(14)

is determined from the two fundamental parameters by

$$\frac{1}{\eta_{\rm D}}=\frac{1}{\eta_{\rm C}}-\tan\phi$$

(15)

from Eq. (12). Furthermore, these two fundamental parameters are not independent owing to geometrical constraint $R_{\rm D}(0)\geq 0$ and they satisfy

$$0<\eta_{\rm C}\leq\frac{1}{\tan\phi},$$

(16)

where the equality holds when the capillary is a true cone with $R_{\rm D}(0)=0$. Figure 4 presents the maximum aspect ratio $\eta_{\rm C}=1/\tan\phi$ for given tilt angle $\phi$. A large aspect ratio $\eta_{\rm C}$ is possible only when the tilt angle $\phi$ is low.

The capillary pressure in Eq. (8) becomes a modified Laplace pressure written as

$$p_{{\rm L}(i)}(z)=\frac{2\gamma_{\rm lv}\Pi_{i}\left(\theta_{\rm Y},\phi\right)}{R_{i}(z)},$$

(17)

where the index $i=$ C, D distinguishes the "Converging" and the "Diverging" geometry, and the scaled pressure $\Pi_{i}$ is given by

	$\displaystyle\Pi_{\rm C}\left(\theta_{\rm Y},\phi\right)$	$\displaystyle=$	$\displaystyle\frac{\cos\theta_{\rm Y}}{\cos\phi}+\tan\phi,$		(18)
	$\displaystyle\Pi_{\rm D}\left(\theta_{\rm Y},\phi\right)$	$\displaystyle=$	$\displaystyle\frac{\cos\theta_{\rm Y}}{\cos\phi}-\tan\phi$		(19)

from Eqs. (10) and (11), which determine the sign and the magnitude of the modified Laplace pressure in conical capillaries.

A more accurate pressure formula Iwamatsu (2022), which takes into account the spherical liquid-vapor interface (Fig. 1) was derived in our previous paper Iwamatsu (2022), where the pore radius $R(z)$ in Eq. (17) is replace by an effective radius corrected by the small volume of spherical cap and the scaled pressures in Eqs. (18) and (19) are replaced by Iwamatsu (2022)

	$\displaystyle\Pi_{\rm C}\left(\theta_{\rm Y},\phi\right)$	$\displaystyle=$	$\displaystyle\frac{\cos\theta_{\rm Y}}{\cos\phi}+\frac{2\tan\phi}{1+\sin\left(\theta_{\rm Y}-\phi\right)},$		(20)
	$\displaystyle\Pi_{\rm D}\left(\theta_{\rm Y},\phi\right)$	$\displaystyle=$	$\displaystyle\frac{\cos\theta_{\rm Y}}{\cos\phi}-\frac{2\tan\phi}{1+\sin\left(\theta_{\rm Y}+\phi\right)}.$		(21)

We can recover Eq. (18) and (19) by setting $\psi=0$ (Fig. 1) or $\theta_{\rm Y}-\phi=90^{\circ}$ in Eq. (20) and $\theta_{\rm Y}+\phi=90^{\circ}$ in Eq. (21).

Figure 5 presents the exact (Eqs. (20) and (21)) and the approximate (Eqs. (18) and (19)) scaled pressures $\Pi_{\rm C}\left(\theta_{\rm Y},\phi\right)$ and $\Pi_{\rm D}\left(\theta_{\rm Y},\phi\right)$ as a function of Young’s contact angle $\theta_{\rm Y}$ for a low tilt angle $\phi=10^{\circ}$ and a high tilt angle $\phi=30^{\circ}$. Apparently, they have symmetry $\Pi_{\rm C}\left(\pi-\theta_{\rm Y},\phi\right)=-\Pi_{\rm D}\left(\theta_{\rm Y},\phi\right)$ and $\Pi_{\rm C}\left(\theta_{\rm Y},-\phi\right)=\Pi_{\rm D}\left(\theta_{\rm Y},\phi\right)$. An exact and an approximate curve does not differ appreciably unless the tilt angle $\phi$ and the Young’s contact angle $\theta_{\rm Y}$ are high. In particular, the two curves cross zero exactly at the same critical Young’s angle $\theta_{\rm c(C)}$ and $\theta_{\rm c(D)}$, where the capillary pressure in Eq. (17) vanishes. In order to keep our model as simple as possible, we will continue to use these approximate formulas in Eqs. (4) and (5) which leads to Eqs. (18) and (19).

Since the modified Laplace pressure $p_{{\rm L}(i)}(z)$ acts as the driving force of liquid intrusion, spontaneous intrusion is possible when the scaled pressure $\Pi_{i}\left(\theta_{\rm Y},\phi\right)$ is positive (see Fig. 5). This occurs when Young’s angle $\theta_{\rm Y}$ is smaller than the critical Yong’s angle $\theta_{{\rm c}(i)}$ ($\theta_{\rm Y}<\theta_{{\rm c}(i)}$). Consequently, the imbibition in a converging capillary is possible but that in a diverging capillary is prohibited when $\theta_{\rm c(D)}<\theta_{\rm Y}<\theta_{\rm c(C)}$. Therefore, a single conical capillary shows diode-like one-way transport: the intrusion into the converging direction will be realized but that into the diverging direction will be prohibited when $\theta_{\rm c(D)}<\theta_{\rm Y}<\theta_{\rm c(C)}$ Iwamatsu (2022).

These two critical Young’s angles $\theta_{{\rm c}(i)}$ are determined from $\Pi_{i}\left(\theta_{{\rm c}(i)},\phi\right)=0$, which gives Iwamatsu (2020, 2022)

	$\displaystyle\theta_{\rm c(C)}$	$\displaystyle=$	$\displaystyle 90^{\circ}+\phi,\;\;\;({\rm Converging}),$		(22)
	$\displaystyle\theta_{\rm c(D)}$	$\displaystyle=$	$\displaystyle 90^{\circ}-\phi,\;\;\;({\rm Diverging}).$		(23)

Geometrically, at this contact angle ($\theta_{\rm Y}=\theta_{{\rm c}(i)}$), the liquid-vapor meniscus becomes flat (see Fig. 2) and the free-energy cost to move the meniscus vanishes because the free energy remain constant irrespective of the location of the meniscus position as $p_{{\rm L}(i)}=\partial F_{i}/\partial V_{i}=0$ at $\theta_{{\rm c}(i)}$. Then, the liquid-vapor interface will be delocalized and able to move freely. The liquid can fill or empty the capillary by the mechanism known as the filling transition of wedge and cone Hauge (1992); Reijmer, Dietrich, and Napiórkowski (1999); Malijevský and Parry (2015) even though the driving force of intrusion is absent ($p_{{\rm L}(i)}=0$).

Figure 6 presents the critical Young’s angles $\theta_{\rm c(C)}$ and $\theta_{\rm c(D)}$ as a function of the tilt angle $\phi$. The critical angle of converging capillary belongs to the "hydrophobic" region $\theta_{\rm c(C)}>90^{\circ}$, whereas that of the diverging one belongs to the "hydrophilic" region $\theta_{\rm c(D)}<90^{\circ}$. In this paper, we will use "hydrophobic" and "hydrophilic" instead of "lyophobic" and "lyophilic" though we will consider general liquid. Therefore, spontaneous imbibition of liquid can occur in converging capillaries even if they are hydrophobic ($90^{\circ}<\theta_{\rm Y}$) as long as $\theta_{\rm Y}<\theta_{\rm c(C)}$.

The two critical angles $\theta_{\rm c(C)}$ in Eq. (22) and $\theta_{\rm c(D)}$ in Eq (23) divide $\left(\phi,\theta_{\rm Y}\right)$ space in Fig. 6 into three regions I, II and III. In the region I, the spontaneous liquid intrusion occurs both in converging capillaries and in diverging capillaries because $\theta_{\rm Y}<\theta_{\rm c(D)}<\theta_{\rm c(C)}$. In the region II, the spontaneous intrusion only in converging capillaries is possible and that in diverging capillaries is prohibited because $\theta_{\rm c(D)}<\theta_{\rm Y}<\theta_{\rm c(C)}$. Therefore, in this region II, a single conical capillary functions as a liquid diode Singh, Kumar, and Khan (2020); Iwamatsu (2022). A larger tilt angle $\phi$ is advantageous from Fig. 6 to expand the region II. However, it requires a short capillary length $H$ with a low aspect ratio $\eta_{\rm C}$ from Fig. 4. In converging conical capillaries, furthermore, the spontaneous liquid intrusion can occur even if the wall is hydrophobic as long as $90^{\circ}<\theta_{\rm Y}<\theta_{\rm c(C)}$ or $90^{\circ}<\theta_{\rm Y}<90^{\circ}+\phi$ from Eq. (22). In fact, the spontaneous intrusion (infiltration) in hydrophobic and converging conical capillaries has been observed by the molecular dynamic simulation Liu et al. (2009).

In the region III, the spontaneous liquid intrusion is prohibited both in converging capillaries and in diverging capillaries because $\theta_{\rm c(D)}<\theta_{\rm c(C)}<\theta_{\rm Y}$. Only the forced imbibition, which is realized by applying the external (infiltration) pressure (Fig. 3) to liquid or vapor, is possible. Further investigation of the free energy landscape Iwamatsu (2020) is necessary to understand the details of the forced imbibition process.

When the modified Laplace pressure in Eq. (17) is negative, i.e., $p_{{\rm L}(i)}(z)<0$ or $\Pi_{i}\left(\theta_{\rm Y},\phi\right)<0$, the spontaneous liquid intrusion is prohibited. It is necessary to apply a positive external pressure $p_{\rm ext}=p_{\rm l}-p_{\rm v}>0$ (Fig. 3) to cancel this negative Laplace pressure to force the intrusion of liquid. On the other hand, when the capillary is completely filled by a positive Laplace pressure $p_{{\rm L}(i)}(z)>0$, it is necessary to apply a negative external pressure $p_{\rm ext}<0$ to cancel this positive Laplace pressure to force extrusion of liquid from the capillary. To determine the magnitude of the applied pressure $p_{\rm ext}$, we have to understand the free energy landscape of imbibition.

The thermodynamics of forced imbibition process is described by the free energy König, Roth, and Mecke (2004); Roth and Kroll (2006); Iwamatsu (2020, 2022); Donne et al. (2022); Alzaidi et al. (2022)

$$\Omega_{i}=F_{i}-p_{\rm ext}V_{i},$$

(24)

where $F_{i}$ is the surface free energy in Eq. (1) and $V_{i}$ is the liquid volume inside the capillary given by Eq. (5). Then, the driving capillary pressure becomes

$$p_{i}(z)=-\frac{\partial\Omega_{i}}{\partial V_{i}}=p_{\rm ext}+p_{{\rm L}(i)}(z),$$

(25)

where $p_{{\rm L}(i)}(z)$ is the modified Laplace pressure in Eq. (17). If the driving pressure $p_{i}(z)$ is always positive within the capillary ($0\leq z\leq H$), the intrusion of liquid into the whole capillary is realized. If the driving pressure is always negative within the capillary, extrusion of liquid from the whole capillary is achieved, and the capillary will be empty and filled by vapor.

The liquid intrusion starts at the inlet ($z=0$) when $p_{i}(z=0)\geq 0$ in Eq. (25). The critical external pressure $p_{\rm ext}=p_{c(i)}$ is given by the condition $p_{i}(z)=0$ at $z=0$, which leads to

$$p_{\rm{c}(i)}=-p_{{\rm L}(i)}(0)=-\frac{2\gamma_{\rm lv}\Pi_{i}\left(\theta_{\rm Y},\phi\right)}{R_{i}(0)}.$$

(26)

To visualize the pressure and the free energy landscape, we introduce the non-dimensional pressure $\tilde{p}$ and the free energy $\omega_{i}$ through Iwamatsu (2020, 2022)

$$\Omega_{i}=\gamma_{\rm lv}\pi R_{i}^{2}(0)\omega_{i}(\tilde{z}),$$

(27)

and the non-dimensional quantities

$$\tilde{z}=\frac{z}{H},\;\;\;(0\leq\tilde{z}\leq 1)$$

(28)

$$\tilde{p}=\frac{Hp_{\rm ext}}{\gamma_{\rm lv}},\;\;\;\alpha_{i}=\frac{H\tan\phi}{R_{i}(0)}=\eta_{i}\tan\phi.$$

(29)

Therefore, the non-dimensional critical pressures are given by

	$\displaystyle\tilde{p}_{\rm c(C)}$	$\displaystyle=$	$\displaystyle\frac{Hp_{\rm c(C)}}{\gamma_{\rm lv}}=-2\eta_{\rm C}\Pi_{\rm C}\left(\theta_{\rm Y},\phi\right)$		(30)
	$\displaystyle\tilde{p}_{\rm c(D)}$	$\displaystyle=$	$\displaystyle\frac{Hp_{\rm c(D)}}{\gamma_{\rm lv}}=-2\eta_{\rm D}\Pi_{\rm D}\left(\theta_{\rm Y},\phi\right)$		(31)

from Eq. (26).

Figure 7 presents the non-dimensional critical pressure $\tilde{p}_{{\rm c}(i)}$ which corresponds to $p_{\rm{c}(i)}$ as a function of Young’s angle $\theta_{\rm Y}$ when $\phi=10^{\circ}$ and $\eta_{\rm C}=4.0$. It also shows the two other characteristic pressures $\tilde{p}_{{\rm e}(i)}$ and $\tilde{p}_{{\rm s}(i)}$, whose meaning will be apparent soon. Note that this critical pressure $p_{c(i)}$ does not represent the entrance barrier pressure due to the potential barrier from atomic interactions Mo et al. (2015).

The free energy landscape $\Omega_{i}$ in Eq. (24) can be analytically calculated from Eqs. (3)-(5) and the non-dimensional free energy $\omega_{i}$ is given by cubic polynomials of $\tilde{z}$ Iwamatsu (2020, 2022):

	$\displaystyle\omega_{\rm C}\left(\tilde{z}\right)$	$\displaystyle=$	$\displaystyle\left(\tilde{p}_{\rm c({\rm C})}-\tilde{p}\right)\tilde{z}-\alpha_{\rm C}\left(\frac{\tilde{p}_{\rm c(C)}}{2}-\tilde{p}\right)\tilde{z}^{2}-\frac{1}{3}\alpha_{\rm C}^{2}\tilde{p}\tilde{z}^{3},$
	$\displaystyle\omega_{\rm D}\left(\tilde{z}\right)$	$\displaystyle=$	$\displaystyle\left(\tilde{p}_{\rm c(D)}-\tilde{p}\right)\tilde{z}+\alpha_{\rm D}\left(\frac{\tilde{p}_{\rm c(D)}}{2}-\tilde{p}\right)\tilde{z}^{2}-\frac{1}{3}\alpha_{\rm D}^{2}\tilde{p}\tilde{z}^{3}$

for the converging and the diverging capillary, where we have dropped the constant free energy from the liquid vapor surface tension when the meniscus is located at the inlet ($\tilde{z}=0$). Therefore, the origin of the free energy is always zero at $\tilde{z}=0$ Iwamatsu (2020).

Figure 8 presents the free energy landscape of imbibition along the pathway from $\tilde{z}=0$ (inlet) to $\tilde{z}=1$ (outlet), where the free energies are further scaled as

	$\displaystyle\tilde{\omega}_{\rm C}\left(\tilde{z}\right)$	$\displaystyle=$	$\displaystyle\omega_{\rm C}\left(\tilde{z}\right)$		(34)
	$\displaystyle\tilde{\omega}_{\rm D}\left(\tilde{z}\right)$	$\displaystyle=$	$\displaystyle\frac{R_{\rm D}^{2}(0)}{R_{\rm C}^{2}(0)}\omega_{\rm D}\left(\tilde{z}\right)=\left(1-\eta_{\rm C}\tan\phi\right)^{2}\omega_{\rm D}\left(\tilde{z}\right)$		(35)

to make the scale of vertical axis (energy) common to both the converging capillary and the diverging capillary (see Eq. (27)) since

$$\frac{R_{\rm D}^{2}(0)}{R_{\rm C}^{2}(0)}=\left(1-\eta_{\rm C}\tan\phi\right)^{2}$$

(36)

from Eq. (12).

In Fig. 8, we present the free energy landscapes at the selected pressures $\tilde{p}=0$ (spontaneous imbibition), $\tilde{p}_{{\rm c}(i)}$, $\tilde{p}_{{\rm e}(i)}$ and $\tilde{p}_{{\rm s}(i)}$. The first characteristic pressure $\tilde{p}_{{\rm c}(i)}$ given by Eqs. (30) and (31) is the critical pressure which characterizes the onset of imbibition at the inlet ($d\tilde{\omega}/d\tilde{z}|_{\tilde{z}=0}=0$ at $z=0$). The second characteristic pressure $\tilde{p}_{{\rm e}(i)}$ are given by Iwamatsu (2020, 2022)

	$\displaystyle\tilde{p}_{\rm e(C)}$	$\displaystyle=$	$\displaystyle\frac{3\left(2-\alpha_{\rm C}\right)}{2\left(3-3\alpha_{\rm C}+\alpha_{\rm C}^{2}\right)}\tilde{p}_{\rm c(C)}.$		(37)
	$\displaystyle\tilde{p}_{\rm e(D)}$	$\displaystyle=$	$\displaystyle\frac{3\left(2+\alpha_{\rm D}\right)}{2\left(3+3\alpha_{\rm D}+\alpha_{\rm D}^{2}\right)}\tilde{p}_{\rm c(D)}.$		(38)

where the free energy of the completely empty state and that of the completely filled state becomes equal $\tilde{\omega}\left(\tilde{z}=0\right)=\tilde{\omega}\left(\tilde{z}=1\right)=0$ (see Fig. 8). This condition is similar to the two-phase coexistence of first-order phase transition.

The third characteristic pressure $\tilde{p}_{{\rm s}(i)}$ given by Iwamatsu (2020, 2022)

	$\displaystyle\tilde{p}_{\rm s(C)}$	$\displaystyle=$	$\displaystyle\frac{\tilde{p}_{\rm c(C)}}{1-\alpha_{\rm C}},$		(39)
	$\displaystyle\tilde{p}_{\rm s(D)}$	$\displaystyle=$	$\displaystyle\frac{\tilde{p}_{\rm c(D)}}{1+\alpha_{\rm D}},$		(40)

characterize the stability limit of intruded liquid at the outlet ($d\tilde{\omega}/d\tilde{z}|_{\tilde{z}=1}=0$ at $z=H$) when the liquid starts to flow out from the outlet. In fact, they correspond simply to the modified Laplace pressure at the outlet ($z=H$):

	$\displaystyle p_{\rm s(C)}$	$\displaystyle=$	$\displaystyle-p_{\rm L(C)}(H)=-\frac{2\gamma_{\rm lv}\Pi_{\rm C}\left(\theta_{\rm Y},\phi\right)}{R_{\rm C}(H)},$		(41)
	$\displaystyle p_{\rm s(D)}$	$\displaystyle=$	$\displaystyle-p_{\rm L(D)}(H)=-\frac{2\gamma_{\rm lv}\Pi_{\rm D}\left(\theta_{\rm Y},\phi\right)}{R_{\rm D}(H)},$		(42)

in the original unit from Eqs, (10), (11) and (17). Therefore, the characteristic pressures $\tilde{p}_{{\rm c}(i)}$ and $\tilde{p}_{{\rm s}(i)}$ of the converging and the diverging capillary are related by

	$\displaystyle p_{\rm s(C)}$	$\displaystyle=$	$\displaystyle\frac{\Pi_{\rm C}\left(\theta_{\rm Y},\phi\right)}{\Pi_{\rm D}\left(\theta_{\rm Y},\phi\right)}p_{\rm c(D)}.$		(43)
	$\displaystyle p_{\rm s(D)}$	$\displaystyle=$	$\displaystyle\frac{\Pi_{\rm D}\left(\theta_{\rm Y},\phi\right)}{\Pi_{\rm C}\left(\theta_{\rm Y},\phi\right)}p_{\rm c(C)}.$		(44)

because $R_{\rm C}(H)=R_{\rm D}(0)$ and $R_{\rm D}(H)=R_{\rm C}(0)$ so that $\tilde{p}_{\rm s(C)}$ and $\tilde{p}_{\rm c(D)}$, and $\tilde{p}_{\rm s(D)}$ and $\tilde{p}_{\rm c(C)}$ run almost in parallel in Fig. 7.

When the external pressure $\tilde{p}$ is between $\tilde{p}_{{\rm c}(i)}$ and $\tilde{p}_{{\rm s}(i)}$ (e.g. Fig. 8 for $\tilde{p}=\tilde{p}_{{\rm e}(i)}$), the free energy landscape exhibits an extremum at $\tilde{z}_{{\rm ex}(i)}$ given by Iwamatsu (2020)

	$\displaystyle\tilde{z}_{\rm ex(C)}$	$\displaystyle=$	$\displaystyle\frac{1}{\alpha_{\rm C}}\left(1-\frac{\tilde{p}_{\rm c(C)}}{\tilde{p}}\right),$		(45)
	$\displaystyle\tilde{z}_{\rm ex(D)}$	$\displaystyle=$	$\displaystyle-\frac{1}{\alpha_{\rm D}}\left(1-\frac{\tilde{p}_{\rm c(D)}}{\tilde{p}}\right),$		(46)

which move from $\tilde{z}_{{\rm ex}(i)}=0$ at $\tilde{p}=\tilde{p}_{{\rm c}(i)}$ to $\tilde{z}_{{\rm ex}(i)}=1$ at $\tilde{p}=\tilde{p}_{{\rm s}(i)}$ from Eqs. (39) and (40), and their free energies become Iwamatsu (2020)

	$\displaystyle\omega_{\rm ex(C)}$	$\displaystyle=$	$\displaystyle-\frac{\left(\tilde{p}-\tilde{p}_{\rm c(C)}\right)^{2}\left(2\tilde{p}+\tilde{p}_{\rm c(C)}\right)}{6\alpha_{\rm C}\tilde{p}^{2}},$		(47)
	$\displaystyle\omega_{\rm ex(D)}$	$\displaystyle=$	$\displaystyle\frac{\left(\tilde{p}-\tilde{p}_{\rm c(D)}\right)^{2}\left(2\tilde{p}+\tilde{p}_{\rm c(D)}\right)}{6\alpha_{\rm D}\tilde{p}^{2}},$		(48)

from Eqs. (LABEL:eq:D32) and (LABEL:eq:D33), which correspond to a minimum when $\omega_{{\rm ex}(i)}<0$ (Figs. 8(a) and (d)) and a maximum when $\omega_{{\rm ex}(i)}>0$ (Figs. 8(b) and (c)). From Eqs. (27), (47) and (48), and noting $\tilde{p}\sim O(1)$ from Tab. 2, we have roughly

$$\left|\Omega_{{\rm ex}(i)}\right|=\gamma_{\rm lv}\pi R_{i}^{2}(0)\left|\omega_{{\rm ex}(i)}\right|\simeq\frac{\gamma_{\rm lv}\pi R_{i}^{2}(0)}{\alpha_{i}}=\frac{\gamma_{\rm lv}\pi R_{i}^{3}(0)}{H\tan\phi},$$

(49)

which gives, for example, $\Omega_{{\rm ex}(i)}\simeq 1.3\times 10^{-15}$ J $\gg kT\sim 4.0\times 10^{-21}$ J for water ($\gamma_{\rm lv}=0.072$ J/m²) in $R_{i}(0)=0.1$ $\mu$m, $H=1.0$ $\mu$m, and $\phi=10^{\circ}$ capillary. Of course, the thermal fluctuation $kT$ may not be negligible when the size of capillary is reduced to 1/100 ($R(0)=1$ nm and $H=10$ nm).

In Fig. 8, the free energy landscapes of a converging capillary with $\phi=10^{\circ}$ and $\eta_{\rm C}=4.0$ are presented in Figs. 8(a) and (c), and those of a diverging capillary are presented in Figs. 8(b) and (d). The numerical values of $\tilde{p}_{{\rm c}(i)}$, $\tilde{p}_{{\rm e}(i)}$ and $\tilde{p}_{{\rm s}(i)}$ used are tabulated in Tab. 2, and $\alpha_{\rm C}\simeq 0.705$ and $\alpha_{\rm D}\simeq 2.393$ from Eq. (29). In original unit, $p_{\rm ext}=\gamma_{\rm lv}\tilde{p}/H$ from Eq. (29). So, for example, $\tilde{p}=1.0$ corresponds to $p_{\rm ext}=7.2\times 10^{4}$ Pa for water in $H=1.0$ $\mu$m capillary.

The wettability of capillary in Figs. 8(a) and (b) is hydrophobic with $\theta_{\rm Y}=120^{\circ}$ so that the spontaneous liquid intrusion is prohibited. Hence, the landscapes presented in Figs. 8(a) and (b) represent the pathway of liquid intrusion by positive applied pressures which are termed intrusion or infiltration pressure Liu et al. (2009); Goldsmith and Martens (2009); Mo et al. (2015). The wettability of capillary in Figs. 8(c) and (d) is hydrophilic with $\theta_{\rm Y}=60^{\circ}$ so that the capillary is filled by spontaneously intruded liquid. Hence, the landscapes presented in Figs. 8(c) and (d) represent the pathway of the liquid extrusion or the vapor intrusion when the applied pressure is negative Iwamatsu (2020). That is when the liquid pressure is lower than the vapor pressure. The free energy landscape of the initial state at $\tilde{p}=0$ is that of the spontaneous imbibition and is monotonically increasing (Figs. 8(a) and (b)) or decreasing (Figs. 8(c) and (d)) function indicating the complete liquid extrusion or intrusion.

This free energy landscape of the intrusion and the extrusion in Fig. 8 can be interpreted as that of the wetting and the drying transition Dietrich (1988); Rauscher and Dietrich (2008) of complete wetting and drying in conical capillaries induced by the external applied pressure. Figure 9 presents the phase diagrams of a converging and a diverging conical capillary. The complete intrusion corresponds to the complete wetting and the complete extrusion corresponds to the complete drying. Therefore, the characteristic pressure $\tilde{p}_{{\rm e}(i)}$ corresponds to the "binodal" and the two critical pressures $\tilde{p}_{{\rm c}(i)}$ and $\tilde{p}_{{\rm s}(i)}$ correspond to the upper (lower) and the lower (upper) "spinodals" in the language of wetting transition. The free energy landscape with a maximum, which acts as a free energy barrier, indicates the first order-like wetting and drying transition with pressure hysteresis and meniscus jumps, while that with a minimum indicates the second order-like transition with continuous change of the meniscus position. In the former case, the meniscus trapped at the inlet (complete drying state) or at the outlet (complete wetting state) separated by the free energy barrier can be in the "metastable" state so that it can be destroyed and transition (intrusion or extrusion) can be induced by any perturbation.

It is relatively easy to imagine the scenario of intrusion and extrusion from the phase diagram in Fig. 9. However, to understand more complex scenarios in double-conical capillaries in the next section, it is helpful to look into the details of the intrusion and the extrusion processes from the free energy landscapes in Fig. 8(a)-(d), which are summarize in Tab. 3(a)-(d) and are as follows:

1. (1)

   Figure 8(a) presents the free energy landscape of the liquid intrusion or infiltration Liu et al. (2009); Mo et al. (2015) in a hydrophobic ($\theta_{\rm Y}>\theta_{\rm c(C)}$) converging capillary. By increasing the (positive) applied pressure from $\tilde{p}=0$ (a long thin solid down arrow in Fig. 8(a)), the intrusion starts at $\tilde{p}_{\rm c(C)}$ (Eq. (30)) and is completed at $\tilde{p}_{\rm s(C)}$ (Eq. (39)). When $\tilde{p}_{\rm c(C)}\leq\tilde{p}\leq\tilde{p}_{\rm s(C)}$ (Fig. 9(a)), the landscape shows a minimum (MIN), which acts as the trap for the liquid-vapor meniscus. The location of MIN moves continuously from the inlet ($\tilde{z}=0$) toward the outlet ($\tilde{z}=1$). Therefore, the intrusion occurs gradually (a short thick solid right arrow in Fig. 8(a)). In fact, such a gradual motion of meniscus is observed in the molecular dynamic simulation Liu et al. (2009). In addition, the reverse process of liquid extrusion (a long thin broken up arrow and a short thick broken left arrow in Fig. 8(a)) is also gradual and completed at $\tilde{p}_{\rm c(C)}$ (see Tab. 3(a)).

2. (2)

   Figure 8(b) presents the landscape of the liquid intrusion in a hydrophobic ($\theta_{\rm Y}>\theta_{\rm c(D)}$) diverging capillary (Fig. 9(b)). The free energy landscape shows a maximum (MAX), which acts as the barrier. As the pressure is increased from $\tilde{p}_{\rm s(D)}$ towards $\tilde{p}_{\rm c(D)}$ (a long thin solid down arrow in Fig. 8(b)), the meniscus is trapped at the inlet and is in metastable state, while the location of MAX moves from the outlet ($\tilde{z}=1$) toward the inlet ($\tilde{z}=0$). Therefore, the intrusion occurs abruptly at $\tilde{p}_{\rm c(D)}$ when the barrier reaches the inlet and disappears. The meniscus jumps from the inlet to the outlet at $\tilde{p}_{\rm c(D)}$ (a short thick solid right arrow in Fig. 8(b)). Hence, this hydrophobic diverging capillary could act as a pneumatic switch Preston et al. (2019). Of course, any perturbations, such as mechanical vibration, thermal fluctuation may assist the meniscus to overcome the barrier estimated in Eq. (49). In the reverse process of depressing $\tilde{p}$ from $\tilde{p}_{\rm c(D)}$ to $\tilde{p}_{\rm s(D)}$ (a long thin broken up arrow in Fig. 8(b)), the meniscus jumps from the outlet to the inlet at $\tilde{p}_{\rm s(D)}$ (a short thick broken left arrow in Fig. 8(b)). Therefore, we will observe a pressure hysteresis between $\tilde{p}_{\rm c(D)}$ and $\tilde{p}_{\rm s(D)}$ (Tab. 3(b)).

3. (3)

   Figure 8(c) presents the landscape of the liquid extrusion or the vapor intrusion in a hydrophilic ($\theta_{\rm Y}<\theta_{\rm c(C)}$) converging capillary. By increasing the absolute magnitude of negative applied pressure from $\tilde{p}=0$ (a long thin solid up arrow in Fig. 8(c)), the complete extrusion occurs at $\tilde{p}_{\rm s(C)}$ (<0, Tab. 2, Fig. 9(a)). The free energy landscape shows a maximum (MAX). Therefore, the extrusion occurs abruptly. Since the extrusion of liquid from a converging capillary is the intrusion of vapor into a diverging capillary, this process in Fig. 8(c) is similar to that in Fig. 8(b). The reverse process of the complete liquid intrusion by depression (a long thin broken down arrow in Fig. 8(c)) occurs also abruptly (a short thick broken right arrow in Fig. 8(c)) at $\tilde{p}_{\rm c(C)}$ (<0). Again, we will observe a pressure hysteresis between $\tilde{p}_{\rm c(C)}$ and $\tilde{p}_{\rm s(C)}$ (Tab. 3(c)) .

4. (4)

   Figure 8(d) presents the landscape of the liquid extrusion in a hydrophilic ($\theta_{\rm Y}<\theta_{\rm c(D)}$) diverging capillary. The extrusion occurs gradually as the landscape shows a minimum (MIN), and is completed at $\tilde{p}_{\rm c(D)}$. This process in Fig. 8(d) is similar to that in Fig. 8(a). The reverse process of the complete liquid intrusion is also gradual and is completed at $\tilde{p}_{\rm s(D)}$ (Tab. 3(d)).

The external pressure necessary to complete the intrusion and the extrusion can be predicted simply from the highest Laplace pressure, which is achieved either at the inlet or the outlet where the cross section is narrowest in conical capillaries. However, the intrusion and the extrusion process can be either gradual or abrupt depending on the shape of the free energy landscape. They occur gradually if the free energy landscape has a minimum (Figs. 8(a) and (d)), while they occur abruptly if the landscape has a maximum (Figs. 8(b) and (c)). We note again that this free energy maximum is not the nucleation barrier of critical droplet or bubble nucleated in the middle of capillaries Lefevre et al. (2004); Remsing et al. (2015); Tinti et al. (2017, 2023).

The free energy maximum, which occurs in a hydrophobic diverging capillary (Fig. 8(b)) and a hydrophilic converging capillary (Fig. 8(c)), always accompanies a pressure hysteresis:

	$\displaystyle\Delta p_{\rm hyst}$	$\displaystyle=$	$\displaystyle p_{\rm c(C)}-p_{\rm s(C)},\;\;\;({\rm converging}),$		(50)
		$\displaystyle=$	$\displaystyle p_{\rm c(D)}-p_{\rm s(D)},\;\;\;({\rm diverging}),$		(51)

or

$$\Delta p_{\rm hyst}=2\gamma_{\rm lv}\left|\Pi_{\rm i}\left(\theta_{\rm Y},\phi\right)\right|\left(\frac{1}{R_{\rm C}(0)}-\frac{1}{R_{\rm D}(0)}\right)$$

(52)

from Eqs. (26), (41) and (42). Therefore, the pressure hysteresis is simply the difference between the modified Laplace pressure at the inlet and that at the outlet, or between the highest and the lowest modified Laplace pressure. Based on these scenarios in Fig. 8 and Tab. 3, we will discuss the intrusion and the extrusion in double conical capillaries in the next section.

Based on the knowledge of the imbibition in a single conical capillary, we consider the intrusion and the extrusion in double conical capillaries. Specifically, we consider four capillaries: hourglass, diamond, sawtooth-1, and sawtooth-2 shaped capillaries illustrated in Fig. 1.

Figure 10 summarizes the scenarios of spontaneous intrusion in double conical capillaries. The intrusion occurs from the left liquid reservoir to the right vapor reservoir. Since the spontaneous intrusion can occur only in converging capillaries and not in diverging capillaries when $\theta_{\rm c(C)}>\theta_{\rm Y}>\theta_{\rm c(D)}$, there are three scenarios: complete filling, half-filling, and complete empty. As we consider the quasi-static thermodynamic transient state, the dynamical effects, such as viscous resistance, pinning, vortex etc. at the junction and the entrance are neglected.

Figure 10(a) illustrates the three scenarios in hourglass shaped capillaries. The capillary is completely empty when $\theta_{\rm Y}>\theta_{\rm c(D)}$. It is half-filled when $\theta_{\rm c(C)}>\theta_{\rm Y}>\theta_{\rm c(D)}$, and it is completely filled when $\theta_{\rm c(D)}>\theta_{\rm Y}$. These three scenarios can be confirmed from the free energy landscape in the next subsection. Figure 10(b) illustrates the two scenarios in diamond shaped capillaries: the capillary is completely empty when $\theta_{\rm Y}>\theta_{\rm c(D)}$, and it is completely filled when $\theta_{\rm c(D)}>\theta_{\rm Y}$.

In Figs. 10(c) and (d), we illustrate the scenarios in two sawtooth shaped capillaries. In these cases, a vertical wall at the junction (a shaded pierced-coin shaped vertical wall in Figs. 1(c) and (d)) will affect hydrodynamics of flow. Here, we neglect various hydrodynamic effects and concentrate on the results obtained purely from the thermodynamic free energy consideration.

Figure 10(c) illustrates the three scenarios in sawtooth-1 shaped capillaries (Fig. 1), where the capillary is completely empty when $\theta_{\rm Y}>\theta_{\rm c(C)}$, half-filled when $\theta_{\rm c(C)}>\theta_{\rm Y}>90^{\circ}$, and completely filled when $\theta_{\rm c(D)}>\theta_{\rm Y}$. The half-filling arises because the vertical wall at the junction acts as a free energy barrier or a hydrophobic gate for liquid intrusion when the vertical wall is hydrophobic ($\theta_{\rm Y}>90^{\circ}$), which will be discussed more quantitatively in the next subsection using the free energy landscape. Figure 10(d) illustrates the two scenarios in sawtooth-2 shaped capillary (Fig. 1), where the capillary is completely empty when $\theta_{\rm Y}>\theta_{\rm c(D)}$ and is completely filled when $\theta_{\rm c(D)}>\theta_{\rm Y}$. In contrast to the sawtooth-1 shaped capillary, the vertical wall at the junction does not act as the free energy barrier when $\theta_{\rm c(D)}>\theta_{\rm Y}$ because the wall is hydrophilic as $90^{\circ}>\theta_{\rm c(D)}$. Those scenarios will be the initial state of the forced imbibition which is the subject of the next subsection.

The scenarios illustrated in Figs. 10(c) and (d) are thermodynamic equilibrium states purely judged from the free energy minimum. There are also a completely filled and a half-filled metastable state, which has the free energy higher than the equilibrium stable states. These metastable states can exist because they are separated from the equilibrium state by the free energy barrier at the junction. This can be clearly seen in the free energy landscape, which is the subject of the next subsection.

If a straight cylindrical capillary could be mechanically deformed He et al. (2014); Cao (2019) into a converging-diverging hourglass or a diverging-converging diamond shaped capillary, we can imagine a mechanical switch illustrated in Fig. 11 when both ends (inlet and outlet) of the capillary are immersed in the liquid reservoir. When a hydrophobic straight cylinder is deformed into an hourglass shaped capillary and Young’s contact angle satisfies $\theta_{\rm c(C)}>\theta_{\rm Y}>90^{\circ}$, the liquid will intrude from both ends and fill the hourglass shaped capillary (Fig 11(a)). When a hydrophilic straight cylinder completely filled by liquid is deformed into a diamond shaped capillary and Young’s contact angle satisfies $\theta_{\rm c(D)}<\theta_{\rm Y}<90^{\circ}$, the liquid will extrude from the capillary and the capillary will be completely empty (Fig 11(b)). Of course, the effect of vapor should be negligible or dissolved into or released from the liquid inside the capillary

To study the forced imbibition in double conical capillaries, we have to combine the free energy landscape of a single conical capillary considered in section II.3. The free energy landscape of a double conical capillary is simply a combination of that of a single converging capillary $\tilde{\omega}_{\rm C}(\tilde{z})$ and a diverging capillary $\tilde{\omega}_{\rm D}(\tilde{z})$. We present the free energy landscapes in Figs. 12 to 15. The parameters $\phi=10^{\circ}$ and $\eta_{\rm C}=4.0$ which characterize the converging and the diverging capillary are the same as those used in section II so that the critical Young’s angles are $\theta_{\rm c(C)}=90+10=100^{\circ}$ and $\theta_{\rm c(D)}=90-10=80^{\circ}$, and the non-dimensional characteristic pressures are given in Tab. 2. The double conical capillaries are twice as long as a single conical capillary considered in section II. So, we consider the imbibition pathway along $0\leq z\leq 2H$ or $0\leq\tilde{z}\leq 2$ in the non-dimensional unit.