The basic assumptions include electroneutrality, negligible fluid momentum, non-deformable pores, and the transport equations of only one species ion concentration in two phases solid and liquid electrolyte. For the liquid electrolyte phase, we can refer to Eqs.
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The source term in both these equations is proportional to the nonlinear Butler—Volmer equation that here represents the transfer of charges between the phases and depends on ion concentration fields and potential in both phases. Although this equation is treated as a macroscopic balance developed as a way to include surface DLVO effects and electrochemistry for the local current at the electrode surface, its derivation does not usually incorporate any correction or upscaling for a porous medium.
This assumption is appropriate, provided a slow reaction timescale compared to the electrolyte diffusion, and we can therefore approximate the total source term locally with the Butler—Volmer equation, simply scaled by the specific surface. Unfortunately, these approximations are not valid for the solid phase in which the ion diffusivity is typically extremely slow usually of at least four orders of magnitude. This is equivalent to the well-known problem of upscaling high-contrast media.
Therefore, in principle, the full microscale solution would be needed at each point. However, assuming the solid phase is made of non-connected spherical particles, Newman and Tiedemann proposed a hybrid micro—macromodel where, at each macroscopic point, a diffusion equation is solved for the ion concentration in a microscopic radial coordinate only.
Here, the phase exchange term the Butler—Volmer kinetics is applied as a flux at the external boundary. Several improvements polydispersed particles or further approximations e. The potential of the solid is instead averaged or homogenized over the solid electrode porous structure partially in contradiction with the assumption of non-connected spherical particles and represented as a macroscopic Poisson equation with effective solid conductivity. While this model has clearly several limitations, it still represents a milestone in understanding the dynamic behavior of lithium-ion batteries, as a compromise between more complex electrochemical molecular models and system-scale equivalent circuit models for control and real-time applications.
There are multiple means of introducing drugs into the body, e. The hydrogel is carefully laced with the drug of interest e. The polymer can be cross-linked , which keeps the polymer from dispersing completely, or not.
The physical process of the liquid entering into a hydrogel is due to osmotic pressure and is a consequence of the liquid having a preference to being next to the solid polymer instead of being in a bulk phase. At the microscale, these mechanisms are a consequence of the solid polymer having a slightly charged surface, resulting in hydration by water, or in the case of the on-demand release mechanism, an electric field that affects the deformation of the polymer.
A human body is a structure where the electrical double layers are ubiquitous. While human tissue is mainly composed of water, the body appears without a doubt as a solid structure. The high water content is vital as renewal of tissue is constantly going on, and supply and drainage of material have to happen from capillaries to cells through diffusion. The reason why, despite the high water content, human tissues appears solid is because most of the water is bound in electrical double layers in and outside the living cells. The cytoplasm is understood as nanoporous network of biological polymers, named cytoskeleton.
The cytoskeleton is charged in many living cells. The extracellular space of cartilage, intervertebral disk, and bone have fixed charges very much like clay minerals has. One of the most salient features of the remodeling process of a human tissue is that the tissue itself organizes its renewal so as to resist existing mechanical load better. The ability of biological tissue to sense its mechanical load is known as mechanotransduction. There is a suspicion that electrical double layers may play an important role in this mechanotransduction.
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Large deformations usually play an important role in applications of biological porous media Huyghe and Janssen The osmotic pressure in blood is tightly regulated by the kidneys, indicating that osmotic pressure is a vital aspect of physiology. Small departures from physiological salt concentrations quickly result in swelling and disease, and it was known even thousands of years ago ayurvedic scriptures that swelling was a key symptom of disease. The osmotic pressure defined in Eq.
The change of chemical potential drives osmosis, ensuring that the majority of water molecules in the human body are bound to the ionized solid. Differences in charge density between the intracellular and extracellular environment cause differences between intracellular osmotic pressure and extracellular osmotic pressure. This difference in osmotic pressure across a permeable cell membrane results in a jump in mechanical pressure across the membrane.
The smooth round shape of cell membranes is a direct consequence of this pressure jump. Ions are sensitive to the difference in charge density as well. Intracellular ionic concentrations differ significantly between intracellular space and extracellular space Guyton and Hall Part of these differences can be explained by the hybrid mixture theory mentioned in the previous sections.
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A key feature of cell physiology is the jump in electric potential across the cell membranes. This jump amounts to several tens of mV. Equating the electrochemical potential, Eq. Nevertheless, some caveat should be mentioned in the context of a multiple-ion system like a human cell. In most ionic theories describing electrical double layers, ions are described as point charges. More advanced theories are needed to deal with heteroionic systems and the interaction between ions of same charge. Substantial effort has been invested in the understanding of why potassium concentration inside cells is higher than outside, and why sodium concentration inside cells is lower than outside.
Such a dichotomy cannot be explained by most existing ionic theories. This dependence is the pivotal property allowing the electrical double layers to sense strain. Although it is too simplistic to present the response of a complex cell such as the endothelial cell as a purely electrochemical response, the very principle of mechanotransduction through ionic flux is demonstrated by this example. In this paper, the fundamental concepts used to model flow and transport in charged porous media have been introduced. While the microscopic description of physical and chemical processes in charged porous media have been explained, Darcy-scale models are required to simulate the behavior of these systems at larger physical scales.
We first note that classical microscale equations e.
This means to solve the ions transport in electrically charged systems, the electric field needs to be coupled with the transport equation referred to as Poisson—Nernst—Planck Eqs. As a result, the expression for the disjoining pressure based on the equilibrium Poisson—Boltzmann relation is not valid.
Similar to the preliminary work of Joekar-Niasar and Mahani , non-equilibrium disjoining pressure and non-equilibrium osmotic pressure should have constitutive expressions that are valid at non-equilibrium. It is important to note that there are several underlying assumptions in Poisson—Boltzmann and Poisson—Nernst—Planck equations that limit their validity under some conditions such as high concentrations. For example, in the original form of these theories, ions are considered as point charges no size assigned and consequently the concentration can theoretically increase to infinity.
Although this assumption might be valid at low concentrations, at high concentrations the size of ions is important and leads to significant deviation of observations from theory. Electroneutrality is another important condition that should be carefully considered in computational simulation of electrokinetics in porous media.
The condition applied to enforce electroneutrality is highly scale dependent. For example, electroneutrality should only be enforced within the electrical diffuse layer if the surface charge density is incorporated as well, while in the bulk concentration, where the electrical potential is negligible, the net charge concentration should be zero. While this assumption might be valid for porous media with large pores compared to the Debye length , in very tight systems such as clays, shale, as well as in thin films the cross-coupling diffusion coefficient of ions should be derived including the contribution of the diffuse layer.
In very tight systems, the pores can function as selective membranes which are partially permeable to some ions. Although numerical simulation of flow and transport in microscopic charged systems such as microfluidics is feasible, solving the electrokinetics coupled with flow and transport at macroscopic systems such as soils, batteries, and rocks is very challenging, if not infeasible, due to huge discrepancy between the physical scale of diffuse layers versus that of macroscopic porous media, complex nonlinear physical and chemical phenomena, limited computational.
Therefore, theoretical upscaling of microscopic phenomena is essential to obtain meaningful macroscopic theories that can be used to model physical problems. However, due to the very complex and highly nonlinear nature of processes in charged porous media, the macroscopic systems at the continuum-scale are only partially understood and our capability to rigorously describe such complex processes is still limited. In Sect. Upscaling via homogenization is difficult due to the highly varying dependent variables, the nonlinear nature of the microscopic equations, and the uncertainty of capturing all the necessary microscopic physics, but homogenization approaches are becoming more sophisticated and improving.
Upscaling via hybrid mixture theory avoids the difficulty of the microscale nonlinear equations, but looses all the microscopic information such as geometry, and it is unclear how macroscopic quantities are related to pore structure. Overall, the difficulty in modeling such complex media is not only in determining what aspects are crucial, but it is also clear that we do not have a clear understanding of all the physics. Examples provided here include diffusion in bentonite clay and lithium-ion batteries and ionic concentration differences in biological cell membranes.
Skip to main content Skip to sections. Advertisement Hide. Download PDF. Transport in Porous Media pp 1—32 Cite as. Open Access. First Online: 12 March The distribution of ions in the vicinity of the surface is theoretically defined using the Boltzmann distribution normal to the surface. The Boltzmann distribution provides an equilibrium ionic concentration as a function of the electric potential.
The distributions of cations and anions of a symmetric z-valent electrolyte e. Open image in new window. To investigate the dynamics electrical double layer and the ionic transport, the coupling between electric field Poisson equation, which is valid under equilibrium and non-equilibrium conditions and transport of the ions are combined. The transport of ions within the diffuse layer is governed by the Nernst—Planck equation , derived from the Smoluchowski diffusion equation Israelachvili and Pashley To obtain the bulk-phase counterpart, Eq.
There are two forces commonly referred to that appear in the conservation of linear momentum. And finally, we provide the resulting constitutive equation for the rate of exchange of mass for component j , see Eq. At the microscale, we begin with the Poisson—Nernst—Planck equations for ion transport in a fluid, in Eqs. A periodic structure consists of spherical solids with a solid phase that has a surface charge density. An example of a homogenization approach that alleviates many of these shortcomings is that of Moyne and Murad Because Moyne and Murad is interested in modeling montmorillonite clay, the geometry of the periodic structure is taken to be two parallel solid plates with a fluid consisting of water with two charged species, an anion and a cation.
Because the fictitious fluid is not in contact with the charged solid phase, it is electrically neutral so that the concentration of anions and cations are equal.
We note that even though the hybrid mixture theory, HMT, and homogenization approaches are quite different approaches, the resulting macroscopic equations have quite a bit in common. Comparing the convection—diffusion equations among the three approaches requires doing a similar manipulation for The crystal units of clay minerals carry an unbalanced electrical charge on the surfaces and edges. Charge imbalance resulting from the isomorphous substitution is balanced by the exchangeable cations that are located between the unit layers and on the surfaces of particles. The arrangement of unit layers for three common clay minerals including kaolinite, illite, and smectite is shown in Fig.
The pore system of clays is a distribution of pore sizes in which mass transport is heavily affected by the surface interactions and charge imbalance. The interlayer pores only contain water and exchangeable ions. Ions form a diffuse double-layer system around the particles that normally exist to balance the charge of the clay surface. The diffuse double layer that is formed at the meso- and macroscale contains both cations and anions as shown in Fig. The vicinal fluid consists of the fluid within the interlayer and a small portion of water in the micropores that is close to the particle surface and is typically considered relatively immobile compared to water in the macropores, and some authors treat this portion of water as part of the solid phase Pusch ; Hueckel Practically, the fluid in the interlayer pores contributes very little to fluid migration, whereas the micro- and macropores are likely to act as major fluid pathways Pusch ; Hueckel The above-mentioned findings indicate that in general larger values for the effective diffusion coefficients of cations and smaller values for anions than those for water tracers neutral species have been found commonly in compacted clays Appelo and Wersin Appelo, C.
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Coupled Processes in Charged Porous Media: From Theory to Applications | SpringerLink
Nature , — CrossRef Google Scholar. Bartels, W. Fuel , — Basu, S. Colloid Interface Sci. Ben-Yaakov, S. Acta 36 12 , — CrossRef Google Scholar.
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The de-embedding and the extraction method were first tested for the quartz substrate fused silica , which is known to have a constant dielectric permittivity of 3. The extraction method is described in detail in [ 13 ]. We can see that the curves show continuity between the two frequency ranges and the extracted values of the permittivity are 3.
These results are very close to the literature value of quartz permittivity 3. They were thus used to characterize the porous Si layer in the above frequency ranges. This is applied to structures where the filling fraction is comparable to the porosity [ 23 ]. It introduces the spectral density function g n,P to take into account the nanotopology of the material. Figure 4 Dielectric permittivity of porous Si as a function of porosity.
Similarly, the value of the loss tangent is between 0. In the 1-toGHz range, the results are not reliable due to the contact resistance between the RF probes and the pads and were omitted. This effect becomes negligible at higher frequencies. In order to demonstrate the high performance of porous Si for use as a substrate for RF and millimeter-wave devices, a comparison was made between this substrate and three other substrates used in the same respect.
Identical CPW TLines were integrated on the four different substrates, their S-parameters were measured, and the propagation constant for each line was extracted. More specifically, trap-rich HR-Si reduces losses from 4. Both the above substrates show similar performance with quartz, which is a non-Si, off-chip substrate. J Appl Phys , In Top.
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