**1. Introduction**

Ti3SiC2 belongs to one of the MAX phases (polycrystalline nanolaminates of ternary carbides and nitrides, over 60 + phases) [1]. It possesses the combined characteristics of metal and ceramics, such as being easily machinable, being electrically and thermally conductive, being oxidation resistant, and having a high melting point. Ti3SiC2 can be a potential candidate for structural functional components. Similar to graphite and MoS2, it has a hexagonal structure, endowing it with an exceptional lubricating property. Research on the tribological properties of Ti3SiC2 demonstrated that it is not a special solid lubricant. Contrarily, Ti3SiC2 generally has a high friction coefficient (µ > 0.6) and a high wear rate (WR: in the order of 10−2–10−3 mm3/N m), especially at medium and low temperatures (RT) [2–8]. It has been reported that, in some circumstances (such as with temperature > 600 ◦C, at a speed of >5 m/s, or sliding against some particular tribopair material), polycrystalline Ti3SiC2 displays the relatively low µ and wear rate, which mainly results from the tribo-oxidation transfer films generated on the contact surface [2,6,8–10]. In other situations, the fracture and pulling-out of Ti3SiC2 grains causes three-body abrasive wear, which is the main wear mechanisms for Ti3SiC2.

Subsequently, most of scientific enthusiasts concentrate on the intensifying research of polycrystalline Ti3SiC2 by uniting metal and/or ceramic to enhance its friction and wear behaviors. Ren et al. [11] prepared Ti3SiC2/Cu/Al/SiC composites, and they found that the tribological behaviors of the composite were better than those of polycrystalline Ti3SiC2 at RT and 200 ◦C, while the wear properties of the composite were worse than

**Citation:** Zhang, R.; Zhang, H.; Liu, F. Microstructure and Tribological Properties of Spark-Plasma-Sintered Ti3SiC2-Pb-Ag Composites at Elevated Temperatures. *Materials* **2022**, *15*, 1437. https://doi.org/ 10.3390/ma15041437

Academic Editor: Tomasz Trzepieci ´nski

Received: 29 December 2021 Accepted: 10 February 2022 Published: 15 February 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

that of polycrystalline Ti3SiC2 at elevated temperatures (≥400 ◦C). The incorporation of Al, Cu, SiC and Al2O3 played an important role in reinforcing the bonding of Ti3SiC2 grains and in fixing the soft Ti3SiC2 matrix around them during reciprocate sliding at RT–200 ◦C, where the wear mechanism was abrasive wear. At elevated temperatures (≥400 ◦C), the plastic flow of tribo-oxides layers led to the higher wear rate of the composite than that of Ti3SiC2 where the wear mechanism was adhesive wear. Yang et al. [12] prepared Cu-Ti3SiC2 co-continuous composites. They showed excellent electrical conductivity and wear resistance due to the addition of Cu. Yang et al. [13] explored the tribological behaviors of (TiB2 + TiC)/Ti3SiC2 composites. It was discovered that µ of the composites was bigger compared with that of Ti3SiC2. The addition of TiB2 and TiC could fix the soft matrix around them and scatter the shear stresses. In our earlier work [14], we discussed the tribological behavior of Ti3SiC2/CaF2 composites at elevated temperatures. The prepared Ti3SiC2/CaF2 exhibited better friction and wear property than Ti3SiC2 in a wide temperature range due to the tribo-oxide competition. Islak et al. [15] investigated the effect of reinforcing TiB2 particles (5, 10, and 15 wt.%) on the properties of as-synthesized Ti3SiC2. The addition of TiB2 improved the mechanical properties and thermal diffusivity of the composites. Moreover, the addition of TiB2 significantly increased the wear properties of Ti3SiC2 matrix. Magnus et al. [16] prepared the Ti3SiC2-TiSi2-TiC composites by spark plasma sintering (SPS). They discovered that the tribological property of the composites was attributed to the intrinsic lubricity and the addition of second phase TiC particles. During the sliding, the tribo-oxidative wear was the main wear mechanism; then, it changed to the deformationinduced wear. The transition resulted from the breakdown of the formed tribofilm.

Silver shows some outstanding characteristics (such as good thermal conductivity, favorable ductility, and special mechanical behaviors) and can be widely applied in many fields, such as air-foil bearings, and biomedical and thermal interface materials. It is commonly utilized as a solid lubricant at temperatures below 500 ◦C. Additionally, it has a high diffusion coefficient and easily generates a lower shear stress intersection at the sliding interface [17,18]. Thus, it was significant and practical to incorporate good properties of Ag with Ti3SiC2. The goal of this work was to synthesize Ti3SiC2/Ag composites and to uncover the mechanical and tribological behaviors of the composite at raising temperatures. F. AlAnazi et al. [19] prepared metals-based (Ag and Bi) composites incorporation with 5 vol%, 10 vol%, 20 vol%, and 30 vol% Ti3SiC2 and investigated their tribological properties. They found that the incorporation of Ti3SiC2 improved the hardness and compressive yield strength of the two composites. Additionally, the addition of Ti3SiC2 enhanced the tribological performance of the two composites. Zeng et al. [20] inspected the tribological behaviors of Ti3SiC2/Ag composites with Inconel 718 as the tribopair at RT. At different sliding speeds, the composites showed lower µ and wear rate than that of Ti3SiC2.

Pb is lubricous due to its low shear strength. In our previous work [21,22], we found that the incorporation of PbO (melting point: 888 ◦C) instead of Pb (melting point: 327 ◦C) in Ti3SiC2 can successfully prepare Ti3SiC2/Pb composites and avoid the loss of Pb during the sintering process. The as-prepared composite showed better friction and wear behaviors than Ti3SiC2 at elevated temperatures.

As a result of the above-mentioned facts, it was necessary to suggest some novel composites that can move forward a single step in strengthening the tribological performance of Ti3SiC2 in a wide temperature range. Therefore, in this work, the metal Pb was added to the Ti3SiC2/Ag composite to prepare Ti3SiC2-Pb-Ag composites (TSC-PA). To prevent the loss of Pb during the process of preparing the TSC-PA, PbO was chosen as the source of Pb. The target of this job was to synthesize TSC-PA and to discuss the mechanical and tribological behaviors of the as-synthesized compound at elevated temperatures.


**Potentiodynamic and Galvanodynamic Regimes of Mass Transfer in Flow-Through Electrodialysis Membrane Systems: Numerical Simulation of Electroconvection and Current-Voltage Curve**

#### **Aminat Uzdenova 1,\* and Makhamet Urtenov 2**

Received: 12 February 2020; Accepted: 19 March 2020; Published: 20 March 2020

**Abstract:** Electromembrane devices are usually operated in two electrical regimes: potentiodynamic (PD), when a potential drop in the system is set, and galvanodynamic (GD), when the current density is set. This article theoretically investigates the current-voltage curves (CVCs) of flow-through electrodialysis membrane systems calculated in the PD and GD regimes and compares the parameters of the electroconvective vortex layer for these regimes. The study is based on numerical modelling using a basic model of overlimiting transfer enhanced by electroconvection with a modification of the boundary conditions. The Dankwerts' boundary condition is used for the ion concentration at the inlet boundary of the membrane channel. The Dankwerts' condition allows one to increase the accuracy of the numerical implementation of the boundary condition at the channel inlet. On the CVCs calculated for PD and DG regimes, four main current modes can be distinguished: underlimiting, limiting, overlimiting, and chaotic overlimiting. The effect of the electric field regime is manifested in overlimiting current modes, when a significant electroconvection vortex layer develops in the channel.

**Keywords:** ion-exchange membrane; electrodialysis; current-voltage curve; electroconvection; potentiodynamic regime; galvanodynamic regime; numerical simulation

#### **1. Introduction**

Flow-through electrodialysis (ED) membrane cells are widely used in water purification and the processing of agricultural products (milk, wine, etc.) [1–4]. Electromembrane systems are described by a nonlinear current-voltage curve (CVC), owing largely to the phenomena of concentration polarization, current-induced convection, and water dissociation [5,6]. For dilute electrolyte solutions considered in this article, the main mechanism of overlimiting transfer is electroconvection, as shown by experimental [7–13] and theoretical studies [14–19]. It is customary to distinguish three modes on the CVC of membrane system (Figure 3):

(1) The underlimiting current (ohmic behavior) is the initial linear region of the CVC, which is characterized by a rather high concentration of ions in the region near the membrane. When an electric current flows through the ion-exchange membrane, the ion concentration decreases on one side of the membrane and increases on the other due to the selective transfer of counterions in the membrane (ion concentration polarization). With the increase in the potential drop,

surface [22,23].

almost complete depletion of ions in the region at the membrane surface in the channel of desalination and the transition of the system to the limiting state are observed [20,21]. 1.6 1.8 **Current** Ohmic Limiting Overlimiting

(2) The limiting current is a section of the CVC with a small slope (plateau), which describes the saturation of the current corresponding to the almost complete depletion of ions at the membrane surface [22,23]. 1 1.2 1.4 *i* lim Chaotic

*Membranes* **2020**, *10*, x FOR PEER REVIEW 2 of 18

(2) The limiting current is a section of the CVC with a small slope (plateau), which describes the saturation of the current corresponding to the almost complete depletion of ions at the membrane

(3) The overlimiting current is the region of secondary current growth: with a further increase in the applied potential drop, the current takes on values greater than the limiting. The increase in the electric current essentially indicates an increase in the conductivity of the depleted region. For dilute electrolyte solutions, electroconvection is the main process that partially destroys the depleted region [7–19]. Electroconvection is the entrainment of liquid molecules by ions that form a space charge at the ion-selective surface under the influence of the electric force [24]. The intensity of electroconvection increases significantly with the passage of the overlimiting current when an extended macroscopic space charge region (SCR) is formed at the interface due to the

and the transition of the system to the limiting state are observed [20,21].

2

complete depletion of ions in the region at the membrane surface in the channel of desalination

(3) The overlimiting current is the region of secondary current growth: with a further increase in the applied potential drop, the current takes on values greater than the limiting. The increase in the electric current essentially indicates an increase in the conductivity of the depleted region. For dilute electrolyte solutions, electroconvection is the main process that partially destroys the depleted region [7–19]. Electroconvection is the entrainment of liquid molecules by ions that form a space charge at the ion-selective surface under the influence of the electric force [24]. The intensity of electroconvection increases significantly with the passage of the overlimiting current when an extended macroscopic space charge region (SCR) is formed at the interface due to the polarization of the electric double layer (EDL) (Figures 1 and 2). **Figure 1.** Sketch of a typical current-voltage curve (CVC) of an ion-exchange membrane. The dashed lines *V*cr0, *V*cr1, *V*cr2 indicate changes in the CVC regions: underlimiting current, plateau of the limiting current (*i*lim), overlimiting, overlimiting with chaotic oscillations. 0 0.2 0.4 0.6 0.8 0.0 5.0 10.0 15.0 20.0 25.0 **Voltage** 30.0 35.0 *V*cr0 *V*cr1 *V*cr2

**Figure 2.** Schematic concentration profiles of cations (*с*1, the solid line) and anions (*с*2, the dashed line) in the diffusion layer adjacent to the surface of a cation-exchange membrane (CEM) [25]. The current density *i* is flowing across the system; the electrolyte concentration in the bulk solution, *c*0; the cation concentration at the solution/CEM boundary, *c*1m; different diffusion layer regions are shown: the electroneutral region (**1**), the extended SCR (**2**) and the quasi-equilibrium electric double layer (**3**), **Figure 1.** Schematic concentration profiles of cations (*c*1 , the solid line) and anions (*c*2 , the dashed line) in the diffusion layer adjacent to the surface of a cation-exchange membrane (CEM) [25]. The current density *i* is flowing across the system; the electrolyte concentration in the bulk solution, *c*0 ; the cation concentration at the solution/CEM boundary, *c*1m; different diffusion layer regions are shown: the electroneutral region (**1**), the extended SCR (**2**) and the quasi-equilibrium electric double layer (**3**), respectively. *Membranes* **2020**, *10*, x FOR PEER REVIEW 3 of 18

Cation-exchange membrane

**Figure 3.** Scheme of the flow of an electrolyte solution in the channel between the anion-exchange and cation-exchange membranes, with taking into account the forced flow (shown by arrows) and the development of an electroconvective vortex layer (at the CEM surface). Ion depletion zones are shown **Figure 2.** Scheme of the flow of an electrolyte solution in the channel between the anion-exchange and cation-exchange membranes, with taking into account the forced flow (shown by arrows) and the development of an electroconvective vortex layer (at the CEM surface). Ion depletion zones are shown in blue. Based on [19].

in blue. Based on [19].

Poisson equations for the potential of the electric field [25].

[34], M. Andersen et al. [35], V.A. Kirii et al. [36].

respectively, of Dukhin and Rubinstein [40].

[32].

Later, I. Rubinstein and B. Zaltzman [14] developed a model for describing mass transfer in a diffusion layer at a homogeneous ion-exchange membrane. They found a numerical solution for the Nernst-Planck-Poisson and Navier-Stokes equations under the assumption of local electroneutrality in the solution outside the SCR and using a special condition of electroosmotic slip at the interface of the electroneutral region with the SCR. It was shown that the heterogeneity of the surface is not a necessary condition for the emergence of electroconvection. A characteristic feature of this system is its hydrodynamic instability at sufficiently high potential drop. Several threshold potential drops were established, which separate different phases in the development of electrokinetic instability.

Approaches to electroconvection modelling using the slip condition at the boundary with the SCR were applied by V. Dydek et al. [26], R. Abu-Rjal et al. [27]. Models based on the Nernst-Planck-Poisson and Navier-Stokes equations that directly take into account the formation of the extended SCR were considered in the works of E.A. Demekhin et al. [15,28,29], S.V. Pham et al. [16,30], and K. Druzgalski, E. Karatay et al. [18,31], P. Magnico [32,33]. Numerical studies of electroconvection flows generated at an electrically heterogeneous membrane surface were carried out by S. Davidson et al.

The difference in the CVCs of the ion-exchange membrane without a forced flow during the transition between the limiting and overlimiting current regimes at the increasing and decreasing potential drop was theoretically described in [11,16,28,32]. S.V. Pham et al. examined a wavy membrane and explained hysteretic behavior by the fact that in the decreasing regime the existing depletion zone creates a lateral gradient, which creates a high lateral electric field. Thus, an additional lateral volumetric force is created to maintain the vortex flow. E.A. Demekhin et al. [28] investigated the hysteresis behavior of ideal smooth ion-exchange membranes and showed the dependence of the hysteresis amplitude on the coupling coefficient between the hydrodynamics and the electrostatics. Hysteretic amplitude calculations observed by Demekhin et al. has been confirmed by P. Magnico

Two electroconvection kinds can be distinguished in overlimiting current modes in membrane systems. In the case of a curved or electrically heterogeneous surface, the tangential electric field causes a stable electroosmotic transfer, described in the works of S.S. Dukhin and N.A. Mishchuk [37–39]. In the case of homogeneous membranes in the absence of forced fluid flow, such a kind is not realized: electroconvection appears as a result of hydrodynamic instability, as shown by I. Rubinstein and B. Zaltzman [14]. These two kinds are sometimes termed as electrokinetic modes, surface [22,23].

complete depletion of ions in the region at the membrane surface in the channel of desalination

(2) The limiting current is a section of the CVC with a small slope (plateau), which describes the saturation of the current corresponding to the almost complete depletion of ions at the membrane

(3) The overlimiting current is the region of secondary current growth: with a further increase in the applied potential drop, the current takes on values greater than the limiting. The increase in the electric current essentially indicates an increase in the conductivity of the depleted region. For dilute electrolyte solutions, electroconvection is the main process that partially destroys the depleted region [7–19]. Electroconvection is the entrainment of liquid molecules by ions that form a space charge at the ion-selective surface under the influence of the electric force [24]. The intensity of electroconvection increases significantly with the passage of the overlimiting current

and the transition of the system to the limiting state are observed [20,21].

**Figure 1.** Sketch of a typical current-voltage curve (CVC) of an ion-exchange membrane. The dashed lines *V*cr0, *V*cr1, *V*cr2 indicate changes in the CVC regions: underlimiting current, plateau of the limiting **Figure 3.** Sketch of a typical current-voltage curve (CVC) of an ion-exchange membrane. The dashed lines *V*cr0, *V*cr1, *V*cr2 indicate changes in the CVC regions: underlimiting current, plateau of the limiting current (*i*lim), overlimiting, overlimiting with chaotic oscillations.

current (*i*lim), overlimiting, overlimiting with chaotic oscillations.

The existence of the extended SCR at the electrolyte solution/membrane interface, which is much larger than the region of the equilibrium EDL when a sufficiently high voltage is applied, was first shown by I. Rubinstein and L. Shtilman on the basis of a numerical solution of the Nernst-Planck and Poisson equations for the potential of the electric field [25].

Later, I. Rubinstein and B. Zaltzman [14] developed a model for describing mass transfer in a diffusion layer at a homogeneous ion-exchange membrane. They found a numerical solution for the Nernst-Planck-Poisson and Navier-Stokes equations under the assumption of local electroneutrality in the solution outside the SCR and using a special condition of electroosmotic slip at the interface of the electroneutral region with the SCR. It was shown that the heterogeneity of the surface is not a necessary condition for the emergence of electroconvection. A characteristic feature of this system is its hydrodynamic instability at sufficiently high potential drop. Several threshold potential drops were established, which separate different phases in the development of electrokinetic instability.

**Figure 2.** Schematic concentration profiles of cations (*с*1, the solid line) and anions (*с*2, the dashed line) in the diffusion layer adjacent to the surface of a cation-exchange membrane (CEM) [25]. The current density *i* is flowing across the system; the electrolyte concentration in the bulk solution, *c*0; the cation concentration at the solution/CEM boundary, *c*1m; different diffusion layer regions are shown: the Approaches to electroconvection modelling using the slip condition at the boundary with the SCR were applied by V. Dydek et al. [26], R. Abu-Rjal et al. [27]. Models based on the Nernst-Planck-Poisson and Navier-Stokes equations that directly take into account the formation of the extended SCR were considered in the works of E.A. Demekhin et al. [15,28,29], S.V. Pham et al. [16,30], and K. Druzgalski, E. Karatay et al. [18,31], P. Magnico [32,33]. Numerical studies of electroconvection flows generated at an electrically heterogeneous membrane surface were carried out by S. Davidson et al. [34], M. Andersen et al. [35], V.A. Kirii et al. [36].

electroneutral region (**1**), the extended SCR (**2**) and the quasi-equilibrium electric double layer (**3**), respectively. The difference in the CVCs of the ion-exchange membrane without a forced flow during the transition between the limiting and overlimiting current regimes at the increasing and decreasing potential drop was theoretically described in [11,16,28,32]. S.V. Pham et al. examined a wavy membrane and explained hysteretic behavior by the fact that in the decreasing regime the existing depletion zone creates a lateral gradient, which creates a high lateral electric field. Thus, an additional lateral volumetric force is created to maintain the vortex flow. E.A. Demekhin et al. [28] investigated the hysteresis behavior of ideal smooth ion-exchange membranes and showed the dependence of the hysteresis amplitude on the coupling coefficient between the hydrodynamics and the electrostatics. Hysteretic amplitude calculations observed by Demekhin et al. has been confirmed by P. Magnico [32].

Two electroconvection kinds can be distinguished in overlimiting current modes in membrane systems. In the case of a curved or electrically heterogeneous surface, the tangential electric field causes a stable electroosmotic transfer, described in the works of S.S. Dukhin and N.A. Mishchuk [37–39]. In the case of homogeneous membranes in the absence of forced fluid flow, such a kind is not realized: electroconvection appears as a result of hydrodynamic instability, as shown by I. Rubinstein and B. Zaltzman [14]. These two kinds are sometimes termed as electrokinetic modes, respectively, of Dukhin and Rubinstein [40].

Electroconvection in ED channels with forced fluid flow was investigated by M.Kh. Urtenov et al. [19,41] and R. Kwak et al. [17,42], R. Abu-Rjal et al. [27], P. Magnico [33]. In such channels, the concentration is distributed unevenly along the length of the channel: as the solution moves between the membranes, the electrolyte concentration decreases and the thickness of the diffusion layer increases. In this case, a tangential bulk electric force is formed, that acts on the SCR at the depleted surface of the membrane, even if the membrane is homogeneous. This force causes stationary electroconvection even at underlimiting current densities. According to the terminology of S.S. Dukhin and N.A. Mishchuk [37–39], this type of electroconvection can be considered as an electroosmosis of the first kind. The bulk force is localized at a relatively small distance from the membrane, where viscous forces play an important role due to the adhesion condition. The contribution of electroconvection to the increase of current becomes significant only at the potential drop corresponding to overlimiting currents. In this case, the SCR thickness increases sharply in comparison with the thickness of the equilibrium double layer. At such distances, the role of viscous forces decreases. Therefore, the main contribution to the development of overlimiting transport belongs to the electroosmosis of the second kind. This mode is similar to the Dukhin-Mishchuk mode described above. Nevertheless, it differs in that in the presence of the forced flow, the tangential force necessary for the occurrence of electroconvection arises due to the inhomogeneity of the longitudinal distribution of concentration, and not due to the electrical inhomogeneity of the surface.

For systems with forced flow at the threshold potential drop, *V*cr1 (Figure 3), single electroconvective vortices rotating in the same direction are formed in the region near the membrane surface (Figure 2) [17,19]. Vortices mix the electrolyte solution in the area near the membrane, which partially destroys the depletion layer and provides the regime of overlimiting current. Due to the forced flow, the vortices move along the solution/membrane interface towards the channel outlet. This movement of the vortices causes current density fluctuations on the CVC [19]. As the potential drop increases, the size of the vortices increases; at a certain potential drop (*V*cr2 on Figure 3) single vortices transform into large vortex complexes consisting of several vortices rotating in opposite directions [19,33]. As a result, the amplitude of the current density oscillations (or potential drop) increases and oscillations become chaotic [19]. P. Magnico investigated the role of electroconvective vortices in the fluid motion using the Lagrangian approach [33]. In this way, trajectories were constructed that reflect the ejection from the mixing layer, trapping by a growing vortex or merging vortices.

The electrical regime in membrane devices as a rule is determined in two ways: potentiodynamic (PD), when a potential drop in the system is set (constant, linearly increasing, periodically changing in time, etc.), and galvanodynamic (GD) when the current density is set (constant, linearly increasing, periodically changing in time, etc.).

Theoretical studies of transport processes taking into account the formation of the extended SCR and the development of electroconvection in membrane systems were mainly carried out for the PD regime using the equations of Navier-Stokes, Nernst-Planck and Poisson for the electric field potential [14–19,28,32,33]. The description of the GD regime caused difficulties associated with the absence of a differential equation for the current density. One approach to describing the ion transport in the membrane system in the GD regime is the decomposition of the system of Nernst-Planck and Poisson equations based on the assumption of local electroneutrality of the electrolyte solution [43,44]. In this approach, the distribution of a current density in the system is obtained using the electric current stream function. However, approaches based on the local electroneutrality assumption do not allow taking explicitly into account the effect of the SCR, which is formed at the solution/membrane boundary. Recently, these difficulties have been overcome using an approach involving the solution of the Poisson equation with a boundary condition determining the potential gradient through the current density [45,46].

This article presents numerical calculations of the CVCs and the hydrodynamic response of the electrolyte solution in flow-through membrane systems in the PD and GD regimes of the electric field. The structure of the electroconvective vortex layer is compared for these modes. For the first time, the hysteresis amplitude is calculated for flow-through systems in the PD and GD regimes. *Membranes* **2020**, *10*, x FOR PEER REVIEW 5 of 18

#### **2. Mathematical Models**

CVCs were calculated: **2. Mathematical Models**

value, then to decrease from this value to 0:

*2.1. Formulation for the PD Regime*

19]:

(1) for the PD regime, when the potential drop, ∆ϕ, is set to increase from 0 to a certain value, then to decrease from this value to 0: CVCs were calculated: (1) for the PD regime, when the potential drop, , is set to increase from 0 to a certain value,

$$
\Delta\varphi = \begin{cases}
 at\_\prime & t \le t\_{1\prime} \\
 2at\_1 - at\_\prime & t > t\_{1\prime}
\end{cases}
\tag{1}
$$

,

1

where α > 0 is the potential sweep speed, *t*1 is the point in time at which the regime of the increasing potential drop is replaced by decreasing regime. 1 where α > 0 is the potential sweep speed, *t*1 is the point in time at which the regime of the increasing

,

(2) for the GD regime, when the average current density, *iav*, is set to increases from 0 to a certain value, then to decrease from this value to 0: potential drop is replaced by decreasing regime. (2) for the GD regime, when the average current density, *av i*, is set to increases from 0 to a certain

$$\mathbf{i}\_{av} = \begin{cases} \beta t\_{\prime} & t \le t\_{2\prime} \\ 2\beta t\_{2} - \beta t\_{\prime} & t > t\_{2\prime} \end{cases} \tag{2}$$

where β > 0 is the sweep speed of the current density, *t*2 is the point in time at which the regime of increasing current density is replaced by decreasing. where > 0 is the sweep speed of the current density, *t*2 is the point in time at which the regime of increasing current density is replaced by decreasing.

The calculations are based on the 2D mathematical models of the overlimiting transfer enhanced by electroconvection in a flow-through ED cell for the PD [19,47] and GD [46] regimes. To simplify the numerical solution, we consider the processes in half of the ED channel at the surface of the cation-exchange membrane (CEM), Figure 4. Let *x* and *y* be the transverse and longitudinal coordinates, respectively; *x* = 0 relates to the middle of the ED channel, *x* = *h* is the electrolyte solution/CEM interface; *y* = 0 corresponds to the inlet and *y* = *l* to the outlet of the channel. The calculations are based on the 2D mathematical models of the overlimiting transfer enhanced by electroconvection in a flow-through ED cell for the PD [19,47] and GD [46] regimes. To simplify the numerical solution, we consider the processes in half of the ED channel at the surface of the cationexchange membrane (CEM), Figure 4. Let *x* and *y* be the transverse and longitudinal coordinates, respectively; *x* = 0 relates to the middle of the ED channel, *x* = *h* is the electrolyte solution/CEM interface; *y* = 0 corresponds to the inlet and *y = l* to the outlet of the channel.

**Figure 4.** Scheme of the system under consideration: half of the desalination electrodialysis (ED) cell adjacent to CEM. Schematic concentration profiles of cations (*c*1, solid line) and anions (*c*2, dashed line), direction of the electric current *i* , forced flow velocity *V* are shown. **Figure 4.** Scheme of the system under consideration: half of the desalination electrodialysis (ED) cell adjacent to CEM. Schematic concentration profiles of cations (*c*1 , solid line) and anions (*c*2 , dashed line), direction of the electric current → *i* , forced flow velocity → *V* are shown.

Re

*p*

*V V*

( )

*t*

*V*

The non-stationary process of transfer of binary electrolyte ions in membrane systems in the

*V Kel*

,

 div*V* 0

(3)

#### *2.1. Formulation for the PD Regime*

The non-stationary process of transfer of binary electrolyte ions in membrane systems in the absence of chemical reactions, with taking into account electroconvection, is written as follows [14–19]:

$$
\frac{\overrightarrow{\partial V}}{\partial t} + (\overrightarrow{V}\nabla)\overrightarrow{V} = -\nabla p + \frac{1}{\text{Re}}\Delta \overrightarrow{V} + K\_{\text{el}}\Delta\varphi\nabla\varphi,\text{ div}\overrightarrow{V} = 0\tag{3}
$$

$$
\stackrel{\rightarrow}{j}\_{i} = -z\_{i}D\_{i}c\_{i}\nabla\varphi - D\_{i}\nabla c\_{i} + \text{Pec}\_{i}\stackrel{\rightarrow}{V}\_{\prime} \text{ i = 1, 2} \tag{4}
$$

$$\frac{\partial c\_i}{\partial t} = -\frac{1}{Pe} \text{ div } \stackrel{\rightarrow}{j}\_{i'} \text{ : } i = 1, 2 \tag{5}$$

$$-\varepsilon\Delta\varphi = z\_1\mathbf{c}\_1 + z\_2\mathbf{c}\_2.\tag{6}$$

$$\stackrel{\rightharpoonup}{i} = z\_1 \stackrel{\rightharpoonup}{j}\_1 + z\_2 \stackrel{\rightharpoonup}{j}\_2 - \varepsilon \operatorname{ Pe} \frac{\partial}{\partial t} (\operatorname{V} \phi) \tag{7}$$

Equations (3)–(7) are given in dimensionless form. We scale time, *t*, by the value *h*/*V*0; spatial coordinates, *x* and *y*, by the thickness of the considered region *h* (half of the ED channel thickness); velocity, → *V*, by the average velocity of the forced flow *V*0; pressure, *p*, by the value ρ*V* 2 0 ; concentration of the i-th ion, *ci* , by the electrolyte concentration in the bulk solution *c*0; electric potential,ϕ, by the value *RT*/*F*; individual ion diffusion coefficients, *D*1 and *D*2, by the electrolyte diffusion coefficient *D* = *D*1*D*2(*z*1 − *z*2)/(*D*1*z*1 − *D*2*z*2); current density, → *i* , by the value *Dc*0*F*/*h*; ion flux → *j i* by the value *Dc*0/*h*. Here *Re* = *V*0*h*/ν is the Reynolds number, *Pe* = *V*0*h*/*D* is the Peclet number, ε = *RT*ε0ε*r*/(*c*0*F* 2*h* 2 ) = 2(*LD*/*h*) 2 and *Kel* = ε0ε*rR* 2*T* 2/(ρ0*V* 2 0 *F* 2*h* 2 ) are the dimensionless parameters; *zi* is the charge number of the i-th ion; *F* is the Faraday constant; *R* is the gas constant; *T* is the absolute temperature; ε0 is the dielectric permittivity of vacuum; ε*r* is the solution relative permittivity (assumed constant); ρ0 is the solution density (assumed constant), ν is the kinematic viscosity.

→ *V*, *p*, → *j* 1 , → *j* 2 , *c*1, *c*2, ϕ, *i*x, *i*y are unknown function of *t*, *x* and *y*. The Navier-Stokes equations, Equations (3), describe the velocity field under the action of the forced flow and the electric body force. The equations of Nernst-Planck, Equations (4), material balance, Equations (5), and Poisson, Equation (6), describe the ion concentration and potential fields. Equation (7) is a formula for the total current density, including the conduction current, → *i c* = *z*1 → *j* 1 + *z*2 → *j* 2 , and displacement current, → *i d* = −ε *Pe* ∂ ∂*t* (∇ϕ). For the calculations of this article, the displacement current, *id,* is negligible (less than 10−7 ).

The system of Equations (3)–(7) is supplemented by the boundary conditions [19,47]. At the channel inlet (*x* ∈ [0, *h*], *y* = 0), the velocity profile is parabolic and satisfies Poiseuille's law (expressions for half of the ED channel):

$$V\_{\mathbf{x}}(\mathbf{x},0,t) = 0, \ V\_{\mathbf{y}}(\mathbf{x},0,t) = \mathbf{1}.5(1-\mathbf{x}^2). \tag{8}$$

In model from [19], the condition of uniform distribution along *x* for ion concentration at the channel inlet is accepted:

$$c\_i(\mathbf{x}, \mathbf{0}, t) = 1, \ i = 1, \ 2. \tag{9}$$

In this paper, instead of condition (9), the Danckwerts' boundary condition is used, which determines that arrival rate of ions into the channel is equal to the rate with which they cross the plane *y* = 0 by the combination of flow, electromigration, and diffusion [48]:

$$\left(-z\_i D\_i c\_i \nabla \varphi - D\_i \nabla c\_i + Pe\_i \overrightarrow{V}\right)(x, 0, t) = Pe\_i \sigma \stackrel{\rightarrow}{V} \dot{\iota} = 1 \text{ 2},\tag{10}$$

where *c*0 = 1 is the input electrolyte concentration. The advantage of this condition in comparison with condition (9) is the absence of a special feature of the distribution of ion concentration near the point (*h*, 0). At the numerical implementation with condition (10), the accuracy of fulfilling the condition that

the tangential current density through the inlet vanishes, *iy*(*x*,0,*t*) = 0, is higher than with condition (9) (Appendix A).

The condition for the electric potential is obtained from Equations (4) and (7) considering the zero tangential current density through the inlet, *iy*(*x*,0,*t*) = 0, (the tangential component of the displacement current, *id y*, is negligible):

$$\frac{\partial \varPhi}{\partial y}(x,0,t) = -\frac{1}{z\_1^2 D\_1 + z\_2^2 D\_2} \left( z\_1 D\_1 \frac{\partial c\_1}{\partial y} + z\_2 D\_2 \frac{\partial c\_2}{\partial y} \right) \tag{11}$$

At the channel outlet (*x* ∈ [0, *h*], *y* = *l*) the velocity profile is again parabolic; the sum of diffusion and migration tangential components of the cation (*i* = 1) and anion (*i* = 2) fluxes is zero; the tangential derivative of the potential is set to be zero:

$$V\_{\mathbf{x}}(\mathbf{x}, l, t) = 0, \ V\_{\mathbf{y}}(\mathbf{x}, l, t) = 1.5(1 - \mathbf{x}^2) \tag{12}$$

$$\left(-\frac{\partial c\_i}{\partial y} - z\_i c\_i \frac{\partial \varphi}{\partial y}\right)(x, l, t) = 0, \; i = 1, 2\tag{13}$$

$$\frac{\partial \Phi}{\partial y}(\mathbf{x}, l, t) = 0.\tag{14}$$

At *x* = 0, *y* ∈ [0, *l*] (middle of the ED channel) the following conditions are applied:

$$V\_{\mathbf{x}}(0, y, t) = 0, \ V\_{\mathbf{y}}(0, y, t) = 1.5 \tag{15}$$

$$c\_i(0, y, t) = 1, \ i = 1, \ 2 \tag{16}$$

$$
\varphi(0, y, t) = 0.\tag{17}
$$

At *x* = 1, *y* ∈ [0, *l*] (the solution/membrane interface), the no-slip condition (18) is applied; the counterion concentration, *c*1, is set as a constant value *Nc* greater than the bulk solution concentration, Equation (19), [25]; continuous flow of co-ions, Equation (20); the potential drop is set, Equation (21):

$$V\_{\mathbf{x}}(\mathbf{1}, \mathbf{y}, t) = \mathbf{0}, \ V\_{\mathbf{y}}(\mathbf{1}, \mathbf{y}, t) = \mathbf{0} \tag{18}$$

$$
\sigma\_1(1, y, t) = N\_\circ \tag{19}
$$

$$\left(-D\_{2}\frac{\partial c\_{2}}{\partial \mathbf{x}} - z\_{2}D\_{2}c\_{2}\frac{\partial \varphi}{\partial \mathbf{x}}\right)(\mathbf{1},y,t) = \frac{(1-T\_{1})}{z\_{2}}\dot{i}\_{\mathbf{x}}(\mathbf{1},y,t)\tag{20}$$

$$
\varphi(1, y, t) = \Delta \varphi \tag{21}
$$

The potential drop, ∆ϕ, is given by Equation (1).

Thus, the formulation of the model for the PD regime includes the system of Equations (3)–(7) and boundary conditions (8), (10)–(21). The average over the channel length current density is calculated as [46]:

$$i\_{av} = \frac{1}{l} \int\_0^l \int\_0^1 i\_x dx dy \tag{22}$$

#### *2.2. Formulation for the GD Regime*

To describe the GD regime, Equations (3)–(6) and boundary conditions (8), (10)–(20) are used similarly to PD regime, but there are two differences. First, at the boundary *x* = 1, *y* ∈ [0, *l*] (solution/membrane interface), instead of condition (21), normal to the membrane surface component of the electric field strength is specified as function of the electric current density [46]:

$$\frac{\partial\varphi}{\partial\mathbf{x}}(\mathbf{1},y,t) = -\left(\frac{\left(\mathbf{i}\_{\mathbf{x}} + \varepsilon \operatorname{Pe}\frac{\partial^2\varphi}{\partial\mathbf{x}\partial t} + z\_1\mathcal{D}\_1\frac{\partial\mathbf{c}\_1}{\partial\mathbf{x}} + z\_2\mathcal{D}\_2\frac{\partial\mathbf{c}\_2}{\partial\mathbf{x}}\right)}{z\_1^2\mathcal{D}\_1\mathbf{c}\_1 + z\_2^2\mathcal{D}\_2\mathbf{c}\_2}\right)(\mathbf{1},y,t) \tag{23}$$

Condition (23) was obtained from Equations (4) and (7) [46,49].

Secondly, an additional equation is introduced to determine the distribution of current density, which is required by the boundary condition (23). For this purpose, the method of electric current flow function is used [43–46]. According to this method, the electric current stream function, η, is determined:

$$\dot{i}\_x = \frac{\partial \eta}{\partial y'} \; \dot{i}\_y = -\frac{\partial \eta}{\partial x} \tag{24}$$

Then the equation and boundary conditions for η are introduced to the mathematical formulation of the model [45,46]:

$$\begin{split} \Delta \eta &= -\left( \left( z\_1^2 D\_1 \frac{\partial c\_1}{\partial y} + z\_2^2 D\_2 \frac{\partial c\_2}{\partial y} \right) \frac{\partial \psi}{\partial x} - \left( z\_1^2 D\_1 \frac{\partial c\_1}{\partial x} + z\_2^2 D\_2 \frac{\partial c\_2}{\partial x} \right) \frac{\partial \psi}{\partial y} \right) + \\ &+ \text{Pe}\left( z\_1 \frac{\partial c\_1}{\partial y} + z\_2 \frac{\partial c\_2}{\partial y} \right) V\_x - \text{Pe}\left( z\_1 \frac{\partial c\_1}{\partial x} + z\_2 \frac{\partial c\_2}{\partial x} \right) V\_y + \text{Pe}\left( z\_1 c\_1 + z\_2 c\_2 \right) \left( \frac{\partial V\_x}{\partial y} - \frac{\partial V\_y}{\partial x} \right) . \end{split} \tag{25}$$

$$\frac{\partial \eta}{\partial \mathbf{x}}(0, y, t) = 0, \,\frac{\partial \eta}{\partial \mathbf{x}}(1, y, t) = 0, \,\eta(\mathbf{x}, 0, t) = 0, \,\eta(\mathbf{x}, l, t) = i\_{\text{av}} l \tag{26}$$

The boundary conditions (26) were derived under the simplifying assumption that the current through the channel outlet *iy*(*x*,*l*,*t*) ≈ 0 (due to its smallness, Figure A2). Therefore, average current density, *i*av, can be used as a parameter determining the electrical regime in the system:

$$i\_{av} = \frac{1}{l} \int\_0^l i\_{\mathbf{x}}(0, y, t) dy = \frac{1}{l} \int\_0^l i\_{\mathbf{x}}(1, y, t) dy \tag{27}$$

Thus, current density *ix* in boundary condition (23) is determined by Formula (24).

Thus, the formulation of the model for the GD regime includes the system of Equations (3)–(6), (25) and boundary conditions (8), (10)–(20), (23) and (26).

#### *2.3. Numerucal Implementation*

Numerical solutions were found by the finite element method using Comsol Multiphysics 5.1 software package. The results presented below were obtained using a non-uniform unstructured triangular computational grid consisting of about 55,000 elements. The density of the mesh elements was increased near the solution/membrane boundary: 1000 elements were set using the "Distribution" node. The influence of the quality of the computational mesh was tested by comparing solutions for two meshes consisting of about 41,000 elements (when "Distribution" node set 700 elements on the solution/membrane boundary) and 55,000 elements (with 1000 elements on the boundary). The difference in the values of the threshold potential drop of the transition to the overlimiting current mode (both in increasing and decreasing regimes) did not exceed 2%.

The following modules are used to implement the model for the GD regime: "Laminar flow" for the Navier-Stokes Equation (3); "Transport of Diluted Species" for the anions and cations concentrations fields, Equations (4) and (5); "Poisson's equation" for the electric potential fields, Equation (6); "General form PDE" for the electric current stream function, Equation (25). For spatial discretization of the concentration, potential, and the electric current stream function fields, the quadratic Lagrange interpolation functions are used. The "Laminar flow" module has the "P2 + P1" discretization that means second order elements for the velocity components and linear elements for the pressure field [50].

For time-depended calculations a segregated node with implicit time-stepping method BDF (backward differentiation formulas) is used [50]. One segregated iteration consists of executing two segregated step: in the first step, concentration, potential and electric current stream function are calculated; on the second, speed and pressure are calculated. At each step, the multifrontal massively parallel sparse direct solver (MUMPS) method [50] is used.

The time step is automatically determined by the solver so that the requirement for the relative tolerance is met (its value was set equal to 10−8 ). With a decrease in the relative tolerance by a factor of 10, the change in the threshold potential drop of the transition to the overlimiting mode did not exceed 1%.

The implementation of the PD regime is similar to the described for the GD regime with the difference that the equation for the electric current stream function (25) is excluded from the calculation process and the boundary condition for the potential (23) changes to (21).

#### **3. Results**

#### *3.1. Parameters Used in Computations*

The results of simulation presented here are obtained for a flow-through ED cell (Figure 4) in the case of dilute NaCl solutions. The dimensionless parameters ε = 3.05 × 10−8 , *Pe* = 589, *Re* = 1.07, *K*el = 5.23 × 10−4 , which correspond to the following system parameters: the thickness of the considered region *h* = 0.5*H*, where *H* = 0.5 × 10−3 m is the intermembrane distance; the channel length *l* = 10−3 m; the average velocity of forced flow *V0* = 3.8 × 10−3 m/s; the electrolyte solution density ρ*0* = 1002 kg/m3 ; the kinematic viscosity ν = 0.89 × 10−6 m2 /s; the input concentration of the electrolyte solution of NaCl *c*0 = 0.1 mol/m3 ; the temperature *T* = 298 K; the diffusion coefficients of cations *D*1 = 1.33 × 10−9 m2 /s and anions *D*2 = 2.05 × 10−9 m2 /s; the cation transport number in the membrane *T*1 = 0.972 and that in the solution *t*1 = 0.395; the ion charge numbers *z*1 = 1, *z*2 = −1. To simplify the numerical solution, the ratio of the counterion concentration at the solution/CEM boundary to its value in the bulk solution *N*c was taken as *N*c = 1. This value is less than in real systems, however, as Urtenov et al. [51] have shown, when *N*c ≥ 1, the value *N*c does not essentially affect the distribution of concentrations and potential in the extended SCR.

The sweep speeds of the potential drop (α = 0.0064) and average current density (β = 0.0003) are chosen sufficiently small and the solution can be considered quasi-stationary, that is, their values do not affect the CVCs trend.

#### *3.2. Current-Voltage Curves*

Figure 5 shows the CVCs calculated for the GD and PD regimes. All CVCs have a linear initial part (denoted by 1 in Figure 5a), a sloping plateau (2 in Figure 5a), and an overlimiting current (3,4 in Figure 5a), which qualitatively corresponds to the existing experimental [5,7,9,13] and theoretical [16,19,31] studies about the CVCs of membrane systems. Note that the limiting current density of the calculated CVCs, determined by the point of intersection of the tangents drawn to the initial part and to the sloping plateau of the curve is close to *i*lim, calculated using Leveque's Equation (28) (values differ by less than 2%) [47]:

$$i\_{\rm lim} = \frac{1}{T\_1 - t\_1} \left( 1.47 \left( \frac{4h^2 V\_0}{lD} \right)^{1/3} - 0.2 \right) \tag{28}$$

At the underlimiting and limiting current modes (regions *1* and *2* on Figure 5a, respectively) of the CVCs calculated for PD and GD regimes coincide with high accuracy (the difference is less than 0.01*i*lim). In these modes (at current densities *i*av/*i*lim ≤ 1 or potential drop ∆ϕ < *V*cr1), electroconvective vortices are not observed in the fluid flow (Figure *Membranes* **2020**, *10*, x FOR PEER REVIEW 6a). 10 of 18

**Figure 5.** (**a**) CVCs calculated for the potentiodynamic (PD) (increasing —purple line, decreasing —green line) and galvanodynamic (GD) (increasing *i*av—red line, decreasing *i*av—blue line) regimes. The dotted line shows the limiting current density, *i*lim, calculated using Leveque's Equation (28). The dashed lines *V*cr0, *V*cr1, *V*cr2 indicate changes in the CVC regions: underlimiting current **1**, plateau of the limiting current **2**, overlimiting **3**, overlimiting with chaotic oscillations **4**. (**b**) enlarged **Figure 5.** (**a**) CVCs calculated for the potentiodynamic (PD) (increasing ∆ϕ—purple line, decreasing ∆ϕ—green line) and galvanodynamic (GD) (increasing *i*av—red line, decreasing *i*av—blue line) regimes. The dotted line shows the limiting current density, *i*lim, calculated using Leveque's Equation (28). The dashed lines *V*cr0, *V*cr1, *V*cr2 indicate changes in the CVC regions: underlimiting current **1**, plateau of the limiting current **2**, overlimiting **3**, overlimiting with chaotic oscillations **4**. (**b**) enlarged fragment of (**a**).

fragment of (**a**). At the underlimiting and limiting current modes (regions *1* and *2* on Figure 5a, respectively) of the CVCs calculated for PD and GD regimes coincide with high accuracy (the difference is less than 0.01*i*lim). In these modes (at current densities *i*av/*i*lim ≤ 1 or potential drop < *V*cr1), electroconvective vortices are not observed in the fluid flow (Figure 6a). At the overlimiting current modes of the CVCs calculated for the both regimes single electroconvective vortices rotating in the same direction (for region *3* on Figure 5a; Figure 6b, c) and At the overlimiting current modes of the CVCs calculated for the both regimes single electroconvective vortices rotating in the same direction (for region *3* on Figure 5a; Figure 6b,c) and large vortex complexes consisting of several vortices rotating in opposite directions (for region *4* on Figure 5a; Figure 6d, e) are formed in the region near the membrane surface. Movement of the vortices causes current density fluctuations in the PD regime and potential drop fluctuations in the GD regime (regions *3* and *4* on Figure 5a). At the same time, the trends of the overlimiting current regions of the CVCs in both regimes approximately coincide (Figure 5b).

large vortex complexes consisting of several vortices rotating in opposite directions (for region *4* on Figure 5a; Figure 6d, e) are formed in the region near the membrane surface. Movement of the vortices causes current density fluctuations in the PD regime and potential drop fluctuations in the GD regime (regions *3* and *4* on Figure 5a). At the same time, the trends of the overlimiting current regions of the

To quantitatively describe the electroconvection vortex layer, parameters such as the thickness, *hec*, and length, *lec*, and density of vortices, *dec*, of this layer were determined. For systems with forced flow, the vortex sizes are not stable and depend on its position in the channel [17,19,47]; therefore, at a given point in time the thickness, *hec*, was determined as the distance from the membrane surface to the farthest edge of the closed streamline forming the biggest vortex [47] (Figure 6b). At each moment of time, the electroconvective vortex layer is a set of successive vortices and vortex structures. Wherein, this layer appears in the region at the channel outlet. Therefore, length, *lec*, was defined as the distance from the outlet to the farthest edge of the closed streamline forming the first (from the inlet) vortex (Figure 6b). Thus, *hec* and *lec* characterize the dimensions of the electroconvective vortex layer, that is the maximum transverse dimension of the biggest vortex and the length of the entire

CVCs in both regimes approximately coincide (Figure 5b).

*3.3. Electroconvective Vortex Layer*

*Membranes* **2020**, *10*, x FOR PEER REVIEW 11 of 18

**Figure 6.** Distribution of cation concentration (the magnitude is shown by different colors), solution streamlines (white lines) in the area at the membrane surface. Calculation for the increasing GD **Figure 6.** Distribution of cation concentration (the magnitude is shown by different colors), solution streamlines (white lines) in the area at the membrane surface. Calculation for the increasing GD regime at *i*av/*i*lim = 1 (**a**), 1.1 (**b**), 1.25 (**c**), 1.3 (**d**), 1.35 (**e**). To improve the visibility of the electroconvective vortex layer, the scale along the *x* axis is set larger than the *y* axis, thus the shape of the vortices is deformed.

#### *3.3. Electroconvective Vortex Layer*

To quantitatively describe the electroconvection vortex layer, parameters such as the thickness, *hec*, and length, *lec*, and density of vortices, *dec*, of this layer were determined. For systems with forced flow, the vortex sizes are not stable and depend on its position in the channel [17,19,47]; therefore, at a given point in time the thickness, *hec*, was determined as the distance from the membrane surface to the farthest edge of the closed streamline forming the biggest vortex [47] (Figure 6b). At each moment of time, the electroconvective vortex layer is a set of successive vortices and vortex structures. Wherein, this layer appears in the region at the channel outlet. Therefore, length, *lec*, was defined as the distance from the outlet to the farthest edge of the closed streamline forming the first (from the inlet) vortex (Figure 6b). Thus, *hec* and *lec* characterize the dimensions of the electroconvective vortex layer, that is the maximum transverse dimension of the biggest vortex and the length of the entire layer at a moment in time. Another important parameter characterizing the electroconvective vortex layer is the density of vortices, *dec*; that is, the number of vortices per unit of length.

The values of *hec*, *lec*, *dec*, were calculated for the PD regime for the potential drop ∆ϕ = 23.4, 23.6, . . . , 31 (the results are indicated by crosses and trend lines in Figure 7a,c,e). Figure 7b,d,f show values of *hec*, *lec*, *dec*, calculated for the GD regime at the current density *i*av/*i*lim = 1, 1.01, . . . , 1.4. The increase in the length of the electroconvective vortex layer, *lec*, is limited by the moment (∆ϕ ≈ 28 or *i*av/*i*lim ≈ 1.24), when this layer occupies almost the entire length of the channel (*l* = 4), Figure 7a,b. The thickness of the electroconvective vortex layer, *hec*, increases approximately linearly with increasing potential drop (or current density) everywhere in the considered range of ∆ϕ (or *i*av) values, except for the initial region of rapid growth at ∆ϕ ≈ *V*cr1 (or *i*av/*i*lim ≈ 1), Figure 7c,d. Saturation of the thickness, *hec*, is not observed.

Figure 7e,f show a decrease in the vortices density, *d*ec, in the range of the potential drop (or current density) corresponding to a rapid increase in the size of the electroconvective vortex layer; and the increase density, *d*ec, in the range of the development of the large vortex complexes consisting of several vortices rotating in opposite directions (Figure 6d,e).

The described behavior of the system is characteristic of both the PD and GD regimes, both for increasing and decreasing cases. To compare the parameters of the electroconvective vortex layer in the GD and PD regimes, the average current densities, *iav* corresponding to ∆ϕ= 23.4, 23.6, . . . , 31 for the PD regime and the average values of the potential drop, ∆ϕ, corresponding to *i*av/*i*lim = 1, 1.01, . . . 1.4 were calculated for GD regime. Figure 7 also show the dependences *hec*, *lec*, *dec*, on ∆ϕ, calculated for the GD regime and on *iav*, calculated for the PD regime. Figure 7 shows that the dependences of the parameters of the electroconvective vortex layer on the potential drop and current density in the PD and GD regimes are approximately the same.

observed.

the vortices is deformed.

regime at *i*av/*i*lim = 1 (**a**), 1.1 (**b**), 1.25 (**с**), 1.3 (**d**), 1.35 (**e**). To improve the visibility of the electroconvective vortex layer, the scale along the *x* axis is set larger than the *y* axis, thus the shape of

..., 31 (the results are indicated by crosses and trend lines in Figures 7a,c,e). Figures 7b,d,f show values of *hec*, *lec*, *dec*, calculated for the GD regime at the current density *i*av/*i*lim = 1, 1.01, …, 1.4. The increase in

when this layer occupies almost the entire length of the channel (*l* = 4), Figures 7a,b. The thickness of the electroconvective vortex layer, *hec*, increases approximately linearly with increasing potential drop

≈ *V*cr1 (or *i*av/*i*lim ≈ 1), Figures 7c,d. Saturation of the thickness, *hec*, is not

(or *i*av) values, except for the initial

= 23.4, 23.6,

≈ 28 or *i*av/*i*lim ≈ 1.24),

The values of *hec*, *lec*, *dec*, were calculated for the PD regime for the potential drop

the length of the electroconvective vortex layer, *lec*, is limited by the moment (

(or current density) everywhere in the considered range of

**Figure 7.** Dependences of the length, *lec*, thickness, *hec*, and vortex density, *dec*, of the electroconvective vortex layer on the potential drop, respectively **(a), (c), (e**), and current density **(b), (d), (f)**. **Figure 7.** Dependences of the length, *lec*, thickness, *hec*, and vortex density, *dec*, of the electroconvective vortex layer on the potential drop, respectively (**a**), (**c**), (**e**), and current density (**b**), (**d**), (**f**). Calculations for the PD regime are indicated by crosses and for the GD regime by rhombuses. The solid lines (Pol. appr.) indicate polynomial approximation of the corresponding data.

#### *3.4. Comparison of Increasing and Decreasing Regimes (Hysteretic Behavior)*

The differences in the CVCs calculated with increasing and decreasing potential drop (or average current density) are manifested in the overlimiting current mode both in the PD and GD cases.

The critical potential drop of the transition to the overlimiting current mode in the increasing regime, *V*cr1Inc, is larger than the corresponding value in the decreasing regime, *V*cr1Dec (Figure 5b): Vcr1Inc ≈ 24.52 and Vcr1Dec ≈ 23.92 (these values approximately coincide for the PD and DG regimes). The hysteresis amplitude (∆*V*cr1 = *V*cr1Inc − Vcr1Dec ≈ 0.6) is less than the difference in the potential drops correspond for the appearance and disappearance of vortices (determined by the values of *hec*, *lec*), which is about 1.2. This is due to the fact that the transition between the limiting and overlimiting

modes on CVC appears only when the thickness *hec* exceeds approximately 0.02. The electroconvective vortex layer of the smaller thickness only causes fluctuations of small amplitude in the CVC.

The calculations in this article confirm the existence of the hysteretic behavior for flow-through channels in both PD and GD regimes. In the transition region between the limiting and overlimiting modes at the fixed potential drop in the decreasing regime, the length, *lec*, and thickness, *hec*, of the electroconvective vortex layer are greater than in the increasing regime (Figure 7a,c). At the fixed average current density, the length, *lec*, and thickness, *hec*, of the electroconvective vortex layer in the decreasing and increasing regimes approximately coincide (Figure 7b,d). In this region (∆ϕ ≈ *V*cr1, *i*av/*i*lim ≈ 1), the density of vortices, *dec*, is higher in the increasing regime compared to the decreasing (Figure 7e,f).

In addition, the critical potential drop of the transition to chaotic oscillations in the increasing regime, *V*cr2Inc, is also larger than the corresponding value in the decreasing regime, *V*cr2Dec (Figure 5b): *V*cr2Inc ≈ 28.88 and *V*cr2Dec ≈ 27.44 for GD case; *V*cr2Inc ≈ 30.36 and *V*cr2Dec ≈ 28.61 for PD case. In the region of chaotic oscillations of the CVCs, the length and thickness of the electroconvective vortex layer oscillate in the same range for the increasing and decreasing regimes, but the density of vortices in the decreasing regime is higher. This is due to the fact that vortex complexes consisting of many vortices are maintained at the lower potential drop in the decreasing regime compared to increasing one.

#### **4. Conclusions**

On the CVCs calculated for the PD and DG regimes, four main current modes can be distinguished: underlimiting, limiting, overlimiting, and chaotic overlimiting. The influence of the electric field regime is manifested in the overlimiting current modes when a significant electroconvection vortex layer develops in the channel. The slipping of vortices along the membrane surface under the action of the forced flow leads to fluctuations in the current density at the PD regime and oscillations in the potential drop at the GD regime. The trend lines of the overlimiting sections of the CVCs for the PD and GD regimes are approximately the same, since the values of the parameters of the electroconvective vortex layer at the same values of the potential drop (or current density) in these modes are quite close.

At the fixed potential drop, the length and thickness of the electroconvective vortex layer in the decreasing regime (PD or GD) is greater than in an increasing one. This leads to the formation of a hysteresis loop in the transition region between the limiting and overlimiting regions of the CVCs. There is also a difference in the critical potential drop of the transition to the chaotic oscillations mode in the increasing and decreasing regimes.

Thus, the development of electroconvection determines the influence of the electric field regime on the processes of ion transfer in membrane systems.

**Author Contributions:** Formal Analysis, Funding Akquisition, Investigation, Visualization, Writing—original draft, A.U.; Methodology, A.U. and M.U.; Writing—review & editing, A.U. and M.U. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Russian Foundation for Basic Research grant number 18-38-00572.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **Appendix A**

Figure A1 shows the concentration profiles for the channel inlet, *y* = 0, and at a short distance from it, *y* = 0.001, calculated with the condition of the uniform ion distribution (9) and with Danckwerts' condition (10) at ∆ϕ = 19.5. In the first case, the concentration profiles vary significantly in the longitudinal direction: for the section *y* = 0.001, they are lower than for *y* = 0; in the second case, concentration profiles practically coincide (maximal difference less than 0.01). As a result, in the calculation with condition (9), a stationary vortex is formed at the inlet at ∆ϕ < *V*cr1, that is, earlier than that for the rest of the channel (Figure A2). When condition (10) is used, vortices at the inlet

c1

**Appendix A**

density) in these modes are quite close.

in the increasing and decreasing regimes.

on the processes of ion transfer in membrane systems.

**Conflicts of Interest:** The author declares no conflict of interest.

draft, A.U.; Methodology, A.U. and M.U.; Writing - review & editing, A.U. and M.U.

appear at ∆ϕ > *V*cr1 during the growth of the electroconvective vortex layer, the formation of which begins at the outlet (Figure 6e). than that for the rest of the channel (Figure A2). When condition (10) is used, vortices at the inlet appear at ∆φ > *V*cr1 during the growth of the electroconvective vortex layer, the formation of which begins at the outlet (Figure 6e).

calculation with condition (9), a stationary vortex is formed at the inlet at ∆φ < *V*cr1, that is, earlier

*Membranes* **2020**, *10*, x FOR PEER REVIEW 14 of 18

surface under the action of the forced flow leads to fluctuations in the current density at the PD regime and oscillations in the potential drop at the GD regime. The trend lines of the overlimiting sections of the CVCs for the PD and GD regimes are approximately the same, since the values of the parameters of the electroconvective vortex layer at the same values of the potential drop (or current

At the fixed potential drop, the length and thickness of the electroconvective vortex layer in the decreasing regime (PD or GD) is greater than in an increasing one. This leads to the formation of a hysteresis loop in the transition region between the limiting and overlimiting regions of the CVCs. There is also a difference in the critical potential drop of the transition to the chaotic oscillations mode

Thus, the development of electroconvection determines the influence of the electric field regime

Figure A1 shows the concentration profiles for the channel inlet, *y* = 0, and at a short distance from it, *y* = 0.001, calculated with the condition of the uniform ion distribution (9) and with Danckwerts' condition (10) at ∆φ = 19.5. In the first case, the concentration profiles vary significantly in the longitudinal direction: for the section *y* = 0.001, they are lower than for *y* = 0; in the second case,

**Author Contributions:** Formal Analysis, Funding Akquisition, Investigation, Visualization, Writing - original

**Funding:** This research was funded by Russian Foundation for Basic Research grant number 18-38-00572.

**Figure A1.** (**a**) Concentration profiles of cations (*c*1, solid lines) and anions (*c*2, dashed lines) in sections *y*= 0 and *y* = 0.001. Calculation for the PD regime at ∆φ = 19.5 with condition (9) (red lines) and with Danckwerts' condition (10) (blue lines). (**b**) enlarged fragment of (**a**). **Figure A1.** (**a**) Concentration profiles of cations (*c*1 , solid lines) and anions (*c*2 , dashed lines) in sections *y* = 0 and *y* = 0.001. Calculation for the PD regime at ∆ϕ = 19.5 with condition (9) (red lines) and with Danckwerts' condition (10) (blue lines). (**b**) enlarged fragment of (**a**). *Membranes* **2020**, *10*, x FOR PEER REVIEW 15 of 18

**Figure A2.** Distribution of cation concentration (the magnitude is shown by different colors), solution streamlines (white lines) in the area at the membrane surface. Calculation for the PD regime at ∆φ = 19.5 with condition (9) (**a**) and with Danckwerts' condition (10) (**b**). To improve the visibility of the electroconvective vortex layer, the scale along the *x* axis is set larger than the *y* axis, thus the shape of **Figure A2.** Distribution of cation concentration (the magnitude is shown by different colors), solution streamlines (white lines) in the area at the membrane surface. Calculation for the PD regime at ∆ϕ = 19.5 with condition (9) (**a**) and with Danckwerts' condition (10) (**b**). To improve the visibility of the electroconvective vortex layer, the scale along the *x* axis is set larger than the *y* axis, thus the shape of the vortices is deformed.

inlet, *i*y av(*x*,0*,t*), and outlet, *i*y av(*x*,*l,t*), boundaries on time are shown. In the considered range of the potential drop, the current *i*y av(*x*,0*,t*), calculated with condition (10), does not exceed 5 × 10−5

condition (9), this value reaches 0.09 *i*lim. As a result, at the calculations with condition (10), the plateau angle of the limiting current decreases; the overlimiting region of the CVC lies lower (Figure A3a).

(**a**) (**b**)

0.00005

*i*y av(x,*l,*t)/*i*lim

0.0001

0

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0

∆φ

Eq. (10) Eq. (9)

*i*lim; for

the vortices is deformed.

*i*y av(x,0,t)/*i*lim

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0

Eq. (9) Eq. (10)

∆φ

0.7

0.8

*x*

**(a)**

0.9

1.0

0.7

**(b)**

0.8

*x*

0.9

1.0 0

the vortices is deformed.

In the Figure A3a,b, the dependences of the average tangential current density through the inlet, *i*y av(*x*,0*,t*), and outlet, *i*y av(*x*,*l,t*), boundaries on time are shown. In the considered range of the potential drop, the current *i*y av(*x*,0*,t*), calculated with condition (10), does not exceed 5 × 10−5 *i*lim; for condition (9), this value reaches 0.09 *i*lim. As a result, at the calculations with condition (10), the plateau angle of the limiting current decreases; the overlimiting region of the CVC lies lower (Figure A3a). In the Figures A3a,b, the dependences of the average tangential current density through the inlet, *i*y av(*x*,0*,t*), and outlet, *i*y av(*x*,*l,t*), boundaries on time are shown. In the considered range of the potential drop, the current *i*y av(*x*,0*,t*), calculated with condition (10), does not exceed 5 × 10−5 *i*lim; for condition (9), this value reaches 0.09 *i*lim. As a result, at the calculations with condition (10), the plateau angle of the limiting current decreases; the overlimiting region of the CVC lies lower (Figure A3a).

**Figure A2.** Distribution of cation concentration (the magnitude is shown by different colors), solution streamlines (white lines) in the area at the membrane surface. Calculation for the PD regime at ∆φ =

0 0.2 0.4 0.6 0.8 1

0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0

*y*

*Membranes* **2020**, *10*, x FOR PEER REVIEW 15 of 18

0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0

*y*

**Figure A3.** Average tangential current density through the inlet (**a**) and outlet (**b**); CVCs (**c**). The calculation results with condition (9) (red lines) and Danckwerts' condition (10) (blue lines) for the PD regime at = 0.025, other parameters as in p. 3.1. **Figure A3.** Average tangential current density through the inlet (**a**) and outlet (**b**); CVCs (**c**). The calculation results with condition (9) (red lines) and Danckwerts' condition (10) (blue lines) for the PD regime at α = 0.025, other parameters as in p. 3.1.

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10. Pismenskaya, N.D.; Nikonenko, V.V.; Belova, E.I.; Lopatkova, G.Y.; Sistat, P.; Pourcelly, G.; Larchet, C. Coupled convection of solution near the surface of ion-exchange membranes in intensive current regimes.

11. Rubinstein, S.M.; Manukyan, G.; Staicu, A.; Rubinstein, I.; Zaltzman, B.; Lammertink, R.G.H. Direct Observation of a Nonequilibrium Electro-Osmotic Instability. *Phys. Rev. Lett.* **2008**, *101*, 236101.

12. Kwak, R.; Guan, G.; Peng, W.K.; Han, J. Microscale electrodialysis: Concentration profiling and vortex

conditions of free convection. *J. Membr. Sci.* **1992**, *71*, 105–116. doi:10.1016/0376-7388(92)85010-G. 8. Zabolotsky, V.I.; Nikonenko, V.V.; Pismenskaya, N.D.; Laktionov, E.V.; Urtenov, M.K.; Strathmann, H.; Wessling, M.; Koops, G.H. Coupled transport phenomena in overlimiting current electrodialysis. *Sep. Purif.*

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© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

*Article*


*2.6. Statistical Analysis*

Data were analyzed by one-way ANOVA followed by Tukey's test or paired *t*-test using GraphPad Prism version 6.0 (GraphPad Software Inc., San Diego, CA, USA. Differences were considered significant at *p* < 0.05.

## **3. Results**


**2. Multiple mechanisms of vitamin K function**

The classical function of vitamin K is a cofactor of GGCX which was clarified in the 1970s [1, 2]. GGCX catalyzes the addition of a carboxyl group to glutamate residues in the substrate proteins, which is coupled by oxidization of vitamin K hydroquinone to vitamin K epoxide. Vitamin K-dependent coagulation factors (II, VII, IX, and X) are well known substrates for GGCX. They become active when several glutamate residues are γ-carboxylated. So far, 18 human proteins are reported to be γ-carboxylated and their functions are regulated by γcarboxylation status in most of them. It is known that cyclic use of vitamin K is necessary for its function as a cofactor for GGCX [4]. To be recycled, vitamin K epoxide should be reduced by an enzyme called vitamin K epoxide reductase (VKOR). Warfarin, which has an anticoagulant activity, inhibits VKOR, causing a subsequent decrease in GGCX activity (**Figure 1**).

**Figure 1.** Multiple mechanisms of vitamin K actions. (1) GGCX catalyzes conversion of glutamate residues into Gla residues by incorporating an additional carboxyl group to glutamate. This reaction requires cyclic use of vitamin K. Vitamin K epoxide reductase (VKOR) is required for recycling vitamin K which is oxidized during γ-glutamyl carboxylation. Warfarin inhibits VKOR and vitamin K recycling, thereby suppressing GGCX activity. Covalent binding of vitamin K epoxide and a target protein is also proposed as a novel mode of vitamin K action which is dependent on GGCX activity. (2) Vitamin K also functions as a ligand of steroid and xenobiotic receptor (SXR) and its murine homolog, pregnane X receptor (PXR). On vitamin K binding, SXR/PXR forms heterodimers with 9-cis-retinoid acid receptor (RXR), and this complex binds to SXR-responsive elements (SXRE) within the promoter or enhancer regions of target genes. (3) Vitamin K also activates protein kinase A (PKA) with unknown mechanism. This action was suppressed by PKA inhibitor but not affected by stimulation with SXR agonist or knocking down of GGCX. Typical substrate of PKA is CREB (cyclic AMP-responsive element binding protein) and it binds to CRE (cyclic AMP-responsive element) within

Vitamin K, SXR, and GGCX http://dx.doi.org/10.5772/63983 23

the promoter or enhancer regions of target genes when CREB is phosphorylated.

German; thus it is called vitamin K in English named after the initial letter of its German word. He shared the Nobel Prize in Physiology or Medicine in 1943 with an American biochemist Dr. Edward A. Doisy who later identified the structure of vitamin K. During the 1970s, the mechanism of vitamin K began to be revealed with the discovery, namely, vitamin K was necessary for γ-carboxylation of some coagulation factors which is catalyzed by an enzyme called γ-glutamyl carboxylase (GGCX) [1, 2]. Interestingly, warfarin, which inhibits vitamin K function, was in medical use since 1954, and vitamin K administration to newborn babies for preventing intracranial hemorrhage started in many countries in the 1960s before

Recently, epidemiological and clinical studies suggested that vitamin K is related to various physiological and pathological processes besides coagulation. Based on these studies, vitamin K was approved to be used as a drug preventing osteoporotic fracture in several Asian countries. Moreover, for these two decades, another mode of vitamin K action has been elucidated. We discovered vitamin K functions as a ligand for a nuclear receptor, SXR (steroid and xenobiotic receptor), and its murine ortholog, PXR (pregnane X receptor) [3], which have physiological or pathological significance. Summing up, vitamin K plays important roles in

In this chapter we are going to introduce novel mechanism of vitamin K action mediated by SXR/PXR as well as recent findings concerning classical vitamin K action mediated by GGCX. Then we would like to discuss the functions of vitamin K in some aging-related diseases based

The classical function of vitamin K is a cofactor of GGCX which was clarified in the 1970s [1, 2]. GGCX catalyzes the addition of a carboxyl group to glutamate residues in the substrate proteins, which is coupled by oxidization of vitamin K hydroquinone to vitamin K epoxide. Vitamin K-dependent coagulation factors (II, VII, IX, and X) are well known substrates for GGCX. They become active when several glutamate residues are γ-carboxylated. So far, 18 human proteins are reported to be γ-carboxylated and their functions are regulated by γcarboxylation status in most of them. It is known that cyclic use of vitamin K is necessary for its function as a cofactor for GGCX [4]. To be recycled, vitamin K epoxide should be reduced by an enzyme called vitamin K epoxide reductase (VKOR). Warfarin, which has an anticoagulant activity, inhibits VKOR, causing a subsequent decrease in GGCX activity

the enzymatic mechanisms of vitamin K function had been clarified.

wide variety of biological process in various modes of actions.

**2. Multiple mechanisms of vitamin K function**

on recent discoveries.

22 Vitamin K2 - Vital for Health and Wellbeing

(**Figure 1**).

**Figure 1.** Multiple mechanisms of vitamin K actions. (1) GGCX catalyzes conversion of glutamate residues into Gla residues by incorporating an additional carboxyl group to glutamate. This reaction requires cyclic use of vitamin K. Vitamin K epoxide reductase (VKOR) is required for recycling vitamin K which is oxidized during γ-glutamyl carboxylation. Warfarin inhibits VKOR and vitamin K recycling, thereby suppressing GGCX activity. Covalent binding of vitamin K epoxide and a target protein is also proposed as a novel mode of vitamin K action which is dependent on GGCX activity. (2) Vitamin K also functions as a ligand of steroid and xenobiotic receptor (SXR) and its murine homolog, pregnane X receptor (PXR). On vitamin K binding, SXR/PXR forms heterodimers with 9-cis-retinoid acid receptor (RXR), and this complex binds to SXR-responsive elements (SXRE) within the promoter or enhancer regions of target genes. (3) Vitamin K also activates protein kinase A (PKA) with unknown mechanism. This action was suppressed by PKA inhibitor but not affected by stimulation with SXR agonist or knocking down of GGCX. Typical substrate of PKA is CREB (cyclic AMP-responsive element binding protein) and it binds to CRE (cyclic AMP-responsive element) within the promoter or enhancer regions of target genes when CREB is phosphorylated.

Recently, another mode of GGCX-dependent vitamin K function was reported in the study of proapoptotic effect of vitamin K. Handa et al. found proapoptotic protein Bak was covalently modified by vitamin K epoxide and regulated by its modification [5]. This function is dependent on GGCX-mediated vitamin K function since GGCX activity is required to generate vitamin K epoxide (**Figure 1**).

Later, among several nutrients including vitamin D and calcium, vitamin K was shown to be the only nutrient that is significantly correlated with hip fracture incidence in Japanese population [14]. Furthermore, the fracture-preventing effect of vitamin K was observed in several clinical studies in Japan, which was confirmed by meta-analysis [15]. Based on these results, vitamin K2 is used for treatment of osteoporosis in several Asian countries. We previously reported a functional single nucleotide polymorphism (SNP) in *GGCX* that causes higher enzymatic activity correlated with higher bone mineral density in elderly Japanese women [16], suggesting bone-protective function of vitamin K is related to GGCX activity. Osteocalcin, one of the substrates of GGCX, is specifically expressed in osteoblastic lineage. The concentration of undercarboxylated form of osteocalcin (ucOC) in serum was reported to be positively correlated with fracture risk [17]. Measurement of ucOC has been clinically used to decide the indication of vitamin K for treatment of osteoporosis in Japan. These support the

Vitamin K, SXR, and GGCX http://dx.doi.org/10.5772/63983 25

Vitamin K also has some epidemiological evidences in relationship with another skeletal disease, osteoarthritis. Low vitamin K intake was correlated to the prevalence of osteoarthritis both in North America and in Japan [18–20]. Unfortunately, therapeutic effect of vitamin K for established osteoarthritis was not proven by a trial [21], suggesting that the study period was

**4. Paradoxical GGCX-mediated vitamin K functions on bone metabolism**

It is difficult to evaluate vitamin K function on bone tissue mediated by GGCX *in vivo* due to its dominant effect on coagulation activity. For example, it is impossible to measure bone mineral density of adult mice systemically lacking GGCX because *Ggcx*-knockout mice die before birth or on the day of birth with massive bleeding [22]. To overcome this obstacle, we utilized Cre/loxP system which enables tissue-/organ-specific knockout of GGCX dependent on promoter activity [23] and generated osteoblast-specific GGCX-deficient mice by crossing with *Col1a1*-Cre mice [24]. Since osteoblasts express several substrates of GGCX including osteocalcin, we assumed bone-protective effect of vitamin K is mediated by GGCX activity in osteoblasts. Surprisingly, the bone mineral density was increased in osteoblast-specific GGCXdeficient mice and aberrant mineralization was observed in these mice by ultrastructural analysis. This result indicates that GGCX in osteoblast may not contribute to bone-protective effect of vitamin K. Moreover, it is contradicting to the clinical studies on *GGCX* SNPs or ucOC described above. We speculate that GGCX activity in other tissue is responsible for boneprotective effect of vitamin K and/or vitamin K function mediated by SXR/PXR that would be more important in the bone tissue. Further studies are necessary to clarify this enigma. It is noteworthy that osteocalcin-deficient mice have been shown to have mechanically stronger bone than wild-type mice [25], suggesting that the decrease of carboxylated osteocalcin, rather

contribution of GGCX activity to bone-protective effect.

too short or vitamin K has only preventive effect.

than increase of ucOC, has "bone strengthening effect."

On the other hand, we discovered GGCX-independent mode of vitamin K function mediated by transcriptional regulation [3] as compared to posttranscriptional modifications explained above. Vitamin K was found to be one of the ligands of the nuclear receptor, SXR, and its murine ortholog, PXR. This receptor is also called NR1I2 according to standardized nomenclature designated by the nuclear receptor committee. In 1998, SXR/PXR was cloned as a novel nuclear receptor that is mainly expressed in the liver and intestine [6]. At first, its functions were characterized as a ligand-dependent transcription factor which is activated by various pharmaceutical agents and xenobiotic compounds [7]. It was originally classified as an orphan receptor since the endogenous ligand was not known when it was cloned. It was later shown that some kinds of secondary bile acids (such as lithocholic acid) could be endogenous ligands for this receptor [8, 9]. It forms a heterodimer with 9-cis-retinoid acid receptor (RXR) on ligand stimulation. This complex then binds to SXR-responsive elements (SXRE) in the promoter or enhancer regions of target genes (**Figure 1**). Some of its target genes are the drug-metabolizing enzyme, such as *CYP3A4*, and the ABC (ATP-binding cassette) family transporter, *MDR1*. Because of that, a function of SXR/PXR is considered as a xenobiotic sensor-inducing genes involved in detoxification and drug excretion [10] and named as such. The discovery of novel vitamin K function as a ligand for SXR/PXR indicated that physiological and pathological processes mediated by PXR/SXR would be affected by vitamin K.

There is another mode of vitamin K function which modulates activation of signal transduction pathway. This is inferred by existence of some genes induced by vitamin K, not by SXR agonist, rifampicin [11]. This induction was not affected by knocking down of GGCX suggesting that this is γ-carboxylation-independent pathway. Expression of those genes was suppressed by protein kinase A (PKA) inhibitor, showing the novel vitamin K function as a modulator of PKA activity (**Figure 1**).

Inhibition of another protein kinase, protein kinase C (PKC) α and ε, by vitamin K was also reported [12]. Inhibition of IKK (inhibitor of nuclear factor kappa B kinase) and subsequent inhibition of NFkB (nuclear factor kappa B) were observed. Whether this function of vitamin K is independent of mechanisms described above remains to be elucidated.


During the project, the survey team received strong support and help from many leaders and experts from the Nanjing Center, China Geological Survey and the Zhejiang Institute of Geological Survey. Prof. Rixiang Gong, Kongzhong Wang, and Zhongda Chen have guided the project team in the fieldwork for many times. In order to compile the report, the experts including Prof. Yanjie Zhang from the Nanjing Center, China Geological Survey, Prof. Yida Luo and Xiaoyong Wu from the Zhejiang Institute of Geological Survey put forward many valuable suggestions and opinions. The survey team also received thoughtful guidance and help in investigating and studying stratigraphic paleontology in the area from Academician Xu Chen, Prof. Yuandong Zhang, Prof. Peng Tang, Dr. Xiang Fang and Dr. Xuan Ma, of the Nanjing Institute of Geology and Paleontology, Chinese Academy of Sciences, Prof. Zhiqiang Zhou of the Xi'an Center, China Geological Survey, Joe Botting and Lucy Muir of the National Museum of Wales, as well as Prof. Guohua Yu of Zhejiang Institute of Geological Survey. In the effort, Academician Chen Xu, Prof. Yuandong Zhang, and Dr. Xuan Ma assisted in identifying the graptoliths, Prof. Peng Tang helped in identifying the chitin fossils, and Prof. Zhiqiang Zhou lent a support in confirming the trilobite fossils.

In addition, we extend gratitude to some colleagues for their participation in the project and generous guidance for this work, in no special order they include Jian Liu, Fenglong Liu, Zhen Wang, Ming Wu, Shengqiang Yu, Jinhua Chen, Shuanghui Xu, Zengcai Tang, Wenjie Hu, and Huaisheng Xie.

The authors sincerely appreciate the support and help of all the leaders, experts, and colleagues.

Due to the limited knowledge and time, there are inevitably some problems and deficiencies in this book, which we sincerely wish readers to criticize and correct.

Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.


**Abstract**

The unique clinical and surgical experience with cardiac valve allografts is presented in the study. Cutting-edge approach with regard to clinical course of the disease, particular diagnostic findings, patient's preference, and allograft accessibility is highlighted in case series. State-of-the-art techniques, initial and mid-term results are summarized with a specific focus on allograft tissue application in cardiovascular surgery. Four patients underwent surgery during the period between February 2020 and February 2023 with inferior vena cava tumor involvement, destructive double valve endocarditis, rheumatic aortic and mitral insufficiency, and severe tricuspid regurgitation in patients with large ostium secundum and atrial fibrillation. All patients demonstrated an uneventful postoperative course.

**Keywords:** allograft, inferior vena cava, endocarditis, rheumatic heart disease, tricuspid replacement


*3.3. Main Findings*

Thirteen articles were selected and subjected to thematic analysis, showing different themes. The discussion used seven themes based on the information retrieved from them. Those seven themes fall into the broad categories of stressors and coping strategies. The first four themes are related to the nursing students' stressors, while the last three are related to the coping strategies employed by those students.

#### 3.3.1. Theme 1: Nursing Students' Stressors

Nine articles were concerned with studies on COVID-19-related stressors, from which four subthemes emerged. The first subtheme is "stress from distance learning", in which issues such as remote learning's psychological impacts are covered. The second subtheme is "stress from assignments and workload". The third subtheme is "stress from clinical training". The fourth subtheme is "fear of infection", and it covers issues such as feeling isolated and worrying about getting infected.

#### Stress from Distance Learning

Eight articles provided a link between the COVID-19 pandemic and nursing students' stress, with a focus on various stressors, including distance learning. During the COVID-19 pandemic, distance learning is a significant source of stress for nursing students. A cross-sectional study was conducted in 13 different nursing colleges in Nepal on 1116 participants [18]. The study aimed to assess the impact of E-learning during the COVID-19 pandemic among nursing students. The study found that many students suffered from the disruption of online classes related to the technological issues that occurred since higher learning institutions had moved to online classes. More than 63.2% of nursing students suffered from electricity problems, while 63.6% of nursing students suffered from internet problems, and only 64.4% of nursing students had internet access in the home for their online classes. Another cross-sectional study by [7] in Israel on 244 nursing students at Ashkelon Academic College assessed the levels of anxiety and coping strategies among nursing students during the COVID-19 pandemic. The study discovered that nursing students suffered significantly high levels of anxiety due to challenges presented by distance learning.

A qualitative study by [16] on 33 nursing students in Croatia explored how nursing students perceived the COVID-19 pandemic and their studying experience during the period. Distance learning presented many challenges to nursing students, including difficulties concentrating, as opposed to what they would do in a typical lecture room or face-to-face environment. The study also noted that nursing students found it difficult to remember and develop the motivation to undertake distance learning.

An observational and prospective study by [17] was on 142 nursing students in their second year in Murcia, Spain. The study's purpose was to assess the levels of stress among nursing students before and during the COVID-19 lockdown and its influence on taking online exams. The study established that levels of stress significantly increased among nursing students after lockdown. In addition, the study noted that the students who failed the online exam had higher levels of stress compared to those who passed.

The perceived stress levels and poor concentration emerged in another cross-sectional study by [19]. The study on 662 nursing students in Turkey evaluated nursing students' views on the COVID-19 pandemic and their perceived stress levels. The study proved that the nursing students suffered from moderate stress levels, but they had higher levels of stress than students assessed in the previous year. Nursing students expressed concerns about their clinical practice and inadequate clinical skills related to the interruption of education and moving to online learning during the pandemic.

Another cross-sectional descriptive study by [10] recruited 184 nursing students from universities in Nepal. The study assessed the factors associated with perceived stress, anxiety, and insomnia during the COVID-19 pandemic among nursing students. The study showed that 29.9% of the nursing students were afraid of delayed graduation, 36.4% suffered from costly mobile data and the necessity of spent money on mobile charging devices for their online classes, 17.4% had difficulties attending online access, 29.3% had difficulties concentrating, and 15.2% they were afraid of failure because they were unable to understand the online classes. The findings were confirmed in another study by [11], where they undertook a cross-sectional descriptive study on 244 nursing students in India. The authors assessed the perceived stress among nursing students during the COVID-19 lockdown. The study established that nursing students had moderate levels of stress related to a lack of resources and distance learning challenges.

Moreover, in a mixed-methods study by [12] that was conducted in Jordan on 335 nursing students, they analyzed the stress levels and sources of stressors related to distance learning and experienced by nursing students during the COVID-19 pandemic. It became evident that overall stress levels were higher among nursing students with low family income; 84.2% of the participants had a financial burden from paying for internet services. Furthermore, the Jordanian study found that distance learning has presented many stressors to nursing students, including difficulties concentrating because of distracting environments and no private areas for studying, limited resources, unorganized workloads, and a lack of strategies for standardized distance learning.

#### Stress from Assignments and Workload

Three articles reported that during the COVID-19 pandemic, assignments and workloads were a vital stress source for nursing students. One of these was the study conducted by [10], which reported that the global pandemic has affected university students in many ways. For nursing students working in a hospital, 44.4% suffered from long hours of duty, and 16.7% experienced increased workloads related to increased numbers of patients infected with the COVID-19 virus. Correspondingly, a study conducted by [7] reported that, in relation to the increase in the number of cases of the COVID-19 virus, there was a need to hire nursing students due to labor shortages in hospitals and in the community during the pandemic. Approximately 69% of the nursing students employed by hospitals had increased levels of anxiety. According to a descriptive study by [5], conducted in the United States among nursing students, learners showed difficulties in handling assignments, too. The study found that of the 84% of nursing students feeling anxious and overwhelmed, 62% had difficulty handling the academic workload, while 20% of nursing students had stress and difficulty writing assignments.


**2.2 How farmers plant soybeans in paddy fields**

The remaining straw in paddy fields is spread over the soil surface (**Figure 3b**). This method can inhibit the growth of weeds and reduce the evaporation of groundwater and prevent the attack of fly pests on peanut seeds (**Figure 4**).


2.6.1. Basic Model: Definition and Test

Recalling the information provided in the Introduction, INFONAVIT is the organization that manages social housing for workers. As portrayed in its official housing catalogue, they used to build the same housing model in different cities and states due to having the most economic design and construction [20]. The principal aim of our project was applying the Mexican standard to social housing, so the first step in the testing stage is the definition of a universal basic model, using the analogous example from the INFONAVIT housing catalogue. This model is the most replicated and affordable house in the catalogue.

The construction system used for the basic model was a reinforced concrete structure, with brick walls with plaster (17 cm wide). The floor was a 30 cm foundation bed/slab, and the ceiling was a joist and block concrete slab. After designating the tested climatic cities for the testing exercise, the base model was simulated for each of the EPW weather files (corresponding to each city). The add-on for Open Studio was run, and the model compliance with NOM-020-ENER-2011 was tested. None of the climatic zone models complied with the standard because all resulted in a negative energy savings of –151%. In other words, if the reference model had an allowed heat gain limit of 1519 Watts for thermal envelope, the base model had a calculated heat gain of 3816 Watts for thermal envelope. It exceeds the limit almost 150%. These results showed the unsuitability of using the same housing model for all cities without considering the climatic characteristics.


**Author details**

Carmen Purdel1 , Mihaela Ilie1 and Denisa Margina2\*

\*Address all correspondence to<EMAIL_ADDRESS>
1 Carol Davila University of Medicine and Pharmacy, Faculty of Pharmacy, Department of Toxicology, Bucharest, Romania

2 Carol Davila University of Medicine and Pharmacy, Faculty of Pharmacy, Department of Biochemistry, Bucharest, Romania