1. Introduction

In our first report^[1] on our new water physics and its model, we showed that the published temperature dependence (0-300 ℃) of the electrical conductivity and mobility of positive and negative electrical charge carriers in pure water behave like trapping-limited drift or transport of electrical charges, with distinct thermal trapping energies (129, 89, 117, 55, 40-37 meV). The first four thermal activation energy, just listed, are especially evident over the standard range of 0-100 ℃, where many experimental data points were obtained at different laboratories and in different times, during the past 50-plus years. In our second reports^[2], for illustration and help to experimentalists as well as a first trial formulation of a quantitative model, we presented the computed theoretical two-terminal characteristics of the electrical charge storage capacitances of pure and impure water, as a function of applied electrical potential, with impure waters containing the univalent acid, HCl, base, NaOH and salt, NaCl. In order to reduce the number of constant parameters and the number of example figures to meet the page count limit, we made two tacit assumptions in our second report^[2]. (ⅰ) The impurities are fully ionized. (ⅱ) The impurity ions are immobilized when trapped by Coulombic attraction to the oppositely charged, immobile, ionized host molecules of the extended water media, (H$_{3}$O)$^{1+}$ and (HO)$^{1-}$. We also assumed, (ⅲ), the immobile neutral water host molecule, (H$_{2}$O)$^{0\pm }$ does not have enough binding force, even from its electrical dipole (denoted by superscript 0$\pm )$, to bind a neutral or even ionized impurity.

To make the description easier comprehensible in our first presentation^[3] and first report^[1], and the algebra easier tractable to approach ''at one glance'' (A1G) physics simplicity in our second report^[2], but still connected to traditional molecular chemistry, we replaced the molecular chemistry symbols of spatially 'isolated' neutral and ionized water molecules in vacuum and gas media, by simplified symbols of 'connected' neutral and ionized water molecules in the extended water media, using the following connection definitions to relate them in vacuum, gas, liquid and solid. The water structure is viewed as an oxygen core surrounded tetrahedrally by four adjacent oxygen cores with two proton trap sites, located approximately at the two trisecting points ($d_{\rm O-H}$ =$1.00 {\mathring{\text{A}}}$ and $d_{\rm O-O}$ =$2.75 {\mathring{\text{A}}}$) on the line joining the two adjacent oxygen cores, O-V-V-O.

(H$_{3}$O)$^{1+}$$\buildrel {\rm def} \over =$V$^{+ }$for the water molecule as the 4-proton trap occupied by one extra proton than its neutral charge or proton-reference state of two trapped protons, or occupied by a total of three protons,

(HO)$^{1-}$$\buildrel {\rm def} \over =$V$^{-}$, occupied by one fewer proton, or a total of one proton; and

(H$_{2}$O)$^{0\pm }$$\buildrel {\rm def} \over =$V$^{0\pm }$, the neutral or reference charge state, occupied by two protons.

This symbol simplification, actually has been traditionally used but not standardized in solid-state physics for the electronic charge at a missing host atom in the extended solid, i.e., an electronic charged atomic or molecular, or even just the electronic vacancy namely, the electronic hole or missing electron. Note: the noun 'vacancy', used in a lattice structure, is precision English, with the lattice points each occupied by a basis (atom or molecule, small or large), when missing then known as vacant and a vacancy particle or quasi-particle. The lattice structure needs not be crystalline or perfectly periodic. It can be slightly imperfect even without a vacancy, arisen from distortion due to random interatomic two-electron or multi-electron bond length and bond angle. Some historical vacancies included: (ⅰ) the $\underline {\textbf{Fermi hole}}$ for the missing one-electron charge distribution in a many-electron and atom crystalline metal, (ⅱ) the $\underline {\textbf{Shockley hole}}$ for a missing valence bond electron in semiconductors, illustrated by his celebrated garage model in his 1950 textbook with the deep-physics title, Electrons and $\underline {\textbf{Holes}}$ in Semiconductors, (ⅲ) the $\underline {\textbf{Dirac hole}}$in vacuum or a missing electron in vacuum known as positron, and (iv) the $\underline {\textbf{anti-particles}}$ or the missing 'elementary' particles in the present definition of 'vacuum' or better, the heretofore not recognized massless space but still containing energy, the electromagnetic and other not yet detectable energy. For the familiar case of semiconductor silicon, a missing host Si, electronically ionized or neutral, has been symboled by (V$_{\rm Si})^{1-}$, (V$_{\rm Si})^{0\pm }$ or (V$_{\rm Si})^{0\mp }$ depending on the choice of the sign of the electronic charge, and also by (V$_{\rm Si})^{2-}$, (V$_{\rm Si})^{3-}$, and (V$_{\rm Si})^{4-}$ for the multiple-electron acceptor charge states, and by (V$_{\rm Si})^{1+}$ for the single-electron donor charge state, which (the many charge states) were supposedly detected in experiments, while the two nearest neighbor Silicon vacancies, the divacancy, symboled by (V$_{\rm Si})_{2}$ in its neutral electrical charge state, seemed relatively more stable, with experimental detections. For a diatomic media, such as the 2-atom-semiconductor GaAs, the vacancy species at the two different host lattice sites in a primitive unit cell, would multiply, to contain (V$_{\rm Ga})^{n+, \, 0\pm, \, \text{or } n-}$ and (V$_{\rm As})^{n+, \, 0\pm, \, \text{or } n-}$.

For the 3-atom-water (H$_{2}$O), there are two vacancy species, the proton vacancy species just described and studied in detail in this series of reports on water, (V$_{\rm P})^{1−}$, which represents a missing proton (not also an electron), or the rather different hydrogen vacancy, V$_{\rm H}$ which is a missing proton with its electron, and the oxygen vacancy species from missing a host oxygen situated a lattice point, V$_{\rm O}$, which is vague and does not distinguish, or does denote both, the vacancy of two very different model states, V$_{\rm O6+}$ and V$_{\rm O8+}$. Strictly speaking, there are two proton vacancy species if the two hydrogen ions (or better, protons) of the water molecule basis are not identical, due to space or/and spin orientations, in which case, we have V$_{\text{H}'}$ and V$_{\text{H}''}$, and if the two protons are identical, then V$_{\text{H}'}$ $=$ V$_{\text{H}''}$ $=$ V$_{\rm H}$. In addition, there are the traditional one-electron energy bands of the valence electrons of the water molecule with the host lattice basis of (H$^{1+}$)$_2$O$^{6+} \equiv$ [(p$^+$)$_2$O$^{6+}$]$^{8+}$, which occur at higher energies than those of protons, due to the lighter electrons and their tighter electron pair bond.

Our endeavor to model water, and liquid state of materials, was triggered by the vast, scientifically taken, experimental data of water and liquids, over more than 200 years, still looking for a satisfactory quantitative theoretical model^{[1, 3]}. In order to take into account of the long-range correlation in the characteristics of the extended water and liquid media, which is lacking in the traditional and recent molecular orbital theory of a small cluster of liquid molecules due to computer limitation, our baseline model was envisioned and formulated from our experiences in our studies of the traditional lattice structure theory, employed by the electronic solid-state physicists to mathematically model the many-electron solids from the derived one-electron energy band and valence-electron bond model, but almost always abbreviated as electronic energy band and valence bond model with the most important terms ''one-electron'' and ''many-electron'' omitted. For our many-particle water and liquid model, the electron particle in the solid-state model is replaced by the proton particle, however, the electron particle is not forgotten, only that its activity range occurs at higher kinetic energies (several electron-volts) or temperature, than that of the active range of the $\sim$2000 (1836. 152) times heavier proton (tens of milli-electron-volts). Thus, the term ''one-electron'' is replaced by ''one-proton'', which is then added to the ''one-electron'', namely, the ''one-electron'' and ''one-proton'' models of liquids, with water, having the H$_{2}$O molecule as the basis, as the simplest and baseline liquid, or at least the most important and abundant liquid on the planet earth, for biological life. And, fortunately, although still obviously model-based, the large mass difference between proton and electron results in the two kinetic energy ranges that can be sufficiently (although not so sharply) separated or ''decoupled in the first order'' to allow for a semi-quantitative understanding of the protonic part of the liquid properties, at proton's lower kinetic energy range (tens of meV) or temperature range (0-100 ℃ and lower), with the electronic part of liquids' properties at electron's higher kinetic energy or energies (a few eV) understood and playing in background, which must be coupled in when dealing with liquid and vapor state phenomena that occur at the higher kinetic energies (both the heat and the electromagnetic energies, traditionally specified by a lattice temperature for heat, $T_{\rm L}$, and an electric (or electromagnetic) temperature for electrons, accelerated in high electric (and also electromagnetic) fields, $T_{\rm E}$, and now also for protons, accelerated in high electric (and also electromagnetic) fields, $T_{\rm P}$. Our initial effort, in the past six months since early summer of 2013, has been on the simplest, which we first envisioned before the generalizations just stated, that is, the one-proton energy band and lattice bond model, with the higher energy one-electron energy band and valence bond model in the background of the picture, such as Fig. 4(c) of our first report^[1], with the two electrons in the electron-pair bond omitted in the figures to simplify the pictures, such as that shown in slides 28 and 29 of our first presentation^[3], which are appended to this report as Figs. A1(a) and A1(b), and the simplification of the mass density or charge density contours shown in Fig. A2. The first encounter of this water physics was traced back to 1959, as shown in Fig. A3 and described in its caption. The similarity between water and silicon or germanium were noticed by chemists at Bell Telephone Laboratory in 1959, which was used to learn new materials properties in 1959 by chemists, such as defect interactions described by chapters in Hannay's monograph cited by Fig. A3. However, the one-electron energy band model was in its infancy and the similarity did not lead to a discovery of the electrical conduction mechanism in pure and impure water in the subsequent half of a century. In the one-proton liquids, quasi-protons, or protonic energy band and bond model, the long-range correlation is the most important feature: the mandatory necessity for a successful model of any extended media, which is missing in the traditional and current chemistry model based on the few-body molecular orbital theory, and in the traditional and current single or few-particle models aimed to unravel the foundation of physics.

This third report extends the examples given in our first two reports^{[1, 2]}. These three reports are based on our first verbal presentation^[3]. In this third report, we extend the theoretical formulation given in our second report^[2], to divalent and multivalent impurities. We also include incomplete ionization or dissociation of the impurity molecules. In addition, we also consider the electrical Coulombic binding of the impurity ions by the oppositely charged, water's four host ion species, the mobile positive and negative quasi-protons, p$^{+}$ and p$^{-}$, and immobile positively and negatively charged proton vacancies, V$^{+}$ and V$^{-}$^[2].

These charged water-structure or host molecule ions are generated from the neutral quasi-particles in the extended water via capturing a positive or negative quasi-proton. We leveraged the century old chemistry symbols, equations and practice, by compactly expressing the solid-state-physics-based liquid-state-chemistry equations, (H$_{3}$O)$^{1+} \equiv$ V$^{1+} \leftrightarrow$ p$^{+}$ $+$ V$^{0\pm} \equiv {\rm p}^{+}+$ (H$_{2}$O)$^{0\pm }$ and (HO)$^{1-} \equiv$ V$^{1-} \leftrightarrow$ p$^{-}$ $+$ V$^{0\pm} \equiv {\rm p}^{-}+$ (H$_{2}$O)$^{0\pm }$. After several iterations of notations and physics, especially the bond model, the traditional symbol for the missing lattice's host atom, V, was selected and used^[2] for the missing and excess proton particle on the host water-structure's water molecule, with the charge-neutral water molecule as the reference, although only two of its four proton trap sites are occupied, each by one proton. Our model and usage for proton in the water media, is like the century usage by physicists, of the other 'elementary' particle, the electron, in semiconductor and other media, including water and even vacuum. This is in contrast to the traditional chemistry designation of proton, such as the ambiguous hydrogen ion H$^{+}$ or even H$^{1+}$, implying that it is an atom without the valence electron, but which fails to account for the much smaller dimension of the proton (0.5 fm) than the atom (0.5 A or 100 thousand times larger). Nevertheless, we shall still use these two symbols (H$^{+}$ and or H$^{1+})$, leveraging their one-letter simplicity, but, with specific physics-based definitions in 'vacuum' and in a not-vacuum media such as the liquid water or solid water (ice) and in other condensed materials, to be described and defined when first used. Our extended media model of proton brings out the key feature of the proton, namely, as the second and heavier elementary particle, $\sim$2000 (1836.152) times heavier than electron when in vacuum, but actually lighter than the vacuum electron, 50% lighter when the proton is in water, which is ''dressed'' by the adjacent water molecules and its own oxygen atom. It is specified by an effective mass in an extended media (such as water), but still given the full unit of the electron charge, spatially distributed, with an experimentally determined 'horizon' radius in 'vacuum' of about 0.5 fm (10$^{-15}$ meter), in companion with the first and still the lightest ($\sim$2000 lighter or lower in mass) elementary particle, the electron, whose size is much smaller but still unknown today, if ever, way beyond the resolution of the highest energy 'microscope' available today and in the future. (An mass density estimate gives an spherical electron radius of 81.664 am, a $=$ atto $=$ 10$^{-18}$, or 12 (12.245) times smaller than the proton diameter of 1 fm, not like the substantially smaller ($\sim$10$^{-5})$ proton size than the atom size ($\sim$ 0.5 $\times$ 10$^{-8}$ cm $=$ 50 pm $=$ picometer), but still sufficiently smaller to be approximated by a point when modeling electron to work with a proton center.

In our water model, the reference for the number of protons trapped at the multi-proton-vacancy (4) trap sites in water, was shifted by 2 to coincide with the electronic charge number of the water molecule, H$_{2}$O, in the extended host water's lattice structure, in order to relate our new water physics to the traditional empirical molecular orbital water chemistry. In concert with the rigorous quantum statistical mechanics counting, like that of Pauling^[4], based on the Bernal-Fowler water model^[5], $^{ }$that gave the theoretical value of the zero point entropy which was within the experimental error obtained by Giauque and his students^[6], our water molecule, H$_{2}$O, has four protonic trap sites, tetrahedrally surrounding each oxygen core with six positive electron charge, O$^{6+}$, which is surrounded by the eight negatively charged valence electrons (to fill the chemistry's $n$ = 2 or L-shell) two at each of the four tetrahedrally located sites relative to the oxygen core, giving the double negatively charged oxygen core, which in the traditional chemical symbols is written as [O$^{6+}$ $+$ 8e$^{-}$]$^{2-} \equiv$ {(H$^{1+})$$_{0}$[O$^{6+}$ $+$ 8e$^{-}$]$^{2-}$}$^{2-}$ and which is abbreviated by our proton vacancy symbol to give [O$^{6+}$ $+$ 8e$^{-}$]$^{2-} \equiv$ {(H$^{1+})_{0}$[O$^{6+}$ $+$ 8e$^{-}$]$^{2-}$}$^{2-}$ $\buildrel {\rm def} \over =$ O$^{2-}$ $\equiv$ V$^{2-}$. This choice, which we made in our previous reports^{[1, 2]}, was aimed to make the water problem tractable, to attain the level of A1G (At 1 Glance.) in our bilateral one-on-one's between we two authors, even simpler than the back-of-envelop physics, allowing mental picture of the microscopic details of any and all liquid, fluid, and solid phenomena in condensed matters. More explicitly, following the perturbation solution route championed and practiced by Slater^[7] and Shockley^[8] 75 and 65 years ago, and reminded us by Ziman^[9] 40 years later, the starting point is the ideal (meaning pure or no impurity but also perfect or structurally perfect or periodic) water that has the protonic oxygen lattice, with the basis of protonic oxygen core, O$^{2−}$ just defined, acting at the lower energies (tens of milli-electron-volts) and also the electronic oxygen lattice, with the basis that is the electronic oxygen core, O$^{6+}$, and the six valence electrons acting at the higher energies (a few electron volts). Let us now go through a counting exercise as an example. In our protonic oxygen lattice, deduced from that of Bernal-Fowler^[5]-Pauling^[4]-Giauque^[6], each host oxygen has a group of four protonic trapping sites, at the four trisector points, nearest to the oxygen atom, each on the oxygen-oxygen line, which was along the experimentally determined tetrahedral directions. The first positively charged proton can be trapped at one of the four protonic trap sites in one of the two protonic-spin orientations, giving a configuration or space-spin degeneracy of $g_{\rm c}$ $=$ $g_{\rm space}g_{\rm spin}$ $=$ 4 $\times$ 2 $=$ 8, assuming a spherical ground bound state. This leaves three unoccupied protonic traps to trap a second proton which presumably can be trapped in just one proton spin orientation that is the opposite to that of the first trapped proton, on account of the Pauli Exclusion Principle, applied to proton or quasi-proton, therefore, $g_{\rm c}$ $=$ 3 $\times$ 1 $=$ 3. However, the trapping potential well for this second proton is no longer spherical, thus, the assumed $g_{\rm space}$ $=$ 3 must be modified to take into account of this non-spherical symmetry. Then, the third trapped proton by one of the two remaining unoccupied proton traps, could be trapped again in one of the two proton spin orientations, giving $g_{\rm c}$ $=$ 2 $\times$ 2 $=$ 4, again not including non-spherical symmetry. In the Bernal-Fowler water model^[5], employed by Pauling's statistics counting, the one and four trapped-proton states surrounding the oxygen core, are excluded by us due to repulsive electric potential. The spin and charge states, as well as the initial and final sates of the phonons (oxygenic and protonic phonons) that provide the energy conservation for the protonic capture and emission at the proton traps, affect both the protonic equilibrium distributions and the kinetics of the protonic capture and emission transitions at the protonic traps. In our analytical model, we lump these degeneracies and non-spherical symmetries all into an effective ionization or dissociation energy, just like the electron and hole trapping at the imperfection center (physical defects and chemical impurities) in the one-electron theory of the many-body semiconductors. See for examples, our earlier attempts to theorize the single and double donor electron traps in elemental silicon semiconductor in 1971 by Ning and Sah^[10], and in 1974 by Pantelides and Sah^[11]; and our study of the statistical distributions of the electrons at the many-electron and many-hole traps in semiconductors in 1958 by Sah and Shockley^[12] which lumped the excited states and degeneracies into an effective ionization energy, for examples, such as the proton capture transition by a neutral proton trap, converting it to a positively charged proton trap, p$^{+}$ $+$ V$^{0} \to$ V$^{1+}$, and the similar capture of a negatively charged proton-hole, p$^−$ $+$ V$^0$ → V$^−$, and the equivalent (electrical or electromagnetic) circuit representation by Sah in 1967-1971^[13].

Section 2 summarizes and extends our theory of water given in our previous and second report^[2], to include the mobile and immobile impurity ions that are trapped by one or more of the four water-lattice constituent quasi-particles or water's host particles: the mobile positively and negatively charged quasi-protons, p$^{+}$ and p$^{-}$, and the immobile positively and negatively charged water-lattice constituent particles, V$^{+}$ and V$^{-}$. The trapped impurity ions are given by us the name here of mobile and immobile impuri $\underline{\bf ton}$s for the pro$\underline{\bf ton}$ic quasi-particles, reserving the name impuri $\underline{\text{tron}}$s for the elec$\underline{\text{tron}}$ic quasi-particles. The apparent redundancy of p$^{-}$ and V$^{-}$ for the same physical or atomic make-up, was tolerated in the beginning of our study, on account of possible physical defect species in pure water or pure liquids from their flexibilities or the ''softness'' of their lattice or the appreciable or large phonon energies (kinetic) in comparison with that of protons, actually the same origin, that of the protons, in the case of water of the water molecules H$_{2}$O. But in fact, it was soon recognized that p$^{-}$ and V$^{-}$ and also p$^{+}$ and V$^{+}$ are strictly distinguished species in the many particle ensemble picture of water, liquid and any spatially extended materials, which the molecular-bond/molecular-orbital picture fails to distinguish, which was just stated as the pictorially apparent (but false) redundancy of p$^{-}$ and V$^{-}$. The densities of p$^{-}$ and V$^{-}$ (similarly, p$^{+}$ and V$^{+})$ are interdependent, and are connected by the transition energy required for the capture and emission transitions of the proton and proton-hole by the proton trap, including the excited states and space-spin degeneracies.

Section 3 describes the computed electrical conductivity per unit mobility, which is the charged particle concentration, of each charge carrier species in the bulk of water and in water's surface and interfacial channels induced by an applied electric potential or normal electric field, such as surface conduction channel in the field-effect transistor structure, and the bulk conductivity change, in reference to the zero field value, in a diode structure employed in the early days of surface experiments in semiconductors by Brattain and Bardeen in the late 1940's and early 1950's. So, in addition to the transverse electrical conductance, the normal conductance can also be computed and presented as a function of the applied electric potential or field normal to the surface layer. These computed curves also include the concentration of the neutral quasi-particle species, from which the diffusive flux of the neutral species can be obtained when the concentration gradient and the diffusivity are known. The mathematical analysis in this section is based on our earlier (1958) general theoretical frame work of Sah and Shockley^{[12, 13]} for impure semiconductors, which was recently used by us to calculate the electrical charge storage capacitance in impure silicon containing an insulated field-effect electrode, or the MOSC diode (metal oxide silicon-semiconductor capacitor)^[14].

Section 4 gives a short summary.

2. Physics and Theory of the Amphoteric Pure and Impure Protonic Water

In this third report, we extend the physics and theory described in our second report^[2] for the extended water media, which we modeled by our lattice structured water, [(H$_{2}$O)$_{N\to \infty}$]$^{0}$, where $N$ is the number of water molecule in the extended volume. For the cubic ice and water with two water molecules in one primitive or 'smallest' unit cell which would fill up the entire extended (infinite) volume of the water if it were undistorted and truly periodic (then primitive), the number of the primitive or smallest unit cell is then $N$/2. In the usual water, it is no longer the smallest cell due to liquids' geometric distortion, which if not completely random, could result in some low-density trap states from some correlated distortions, that could be extended traps for the mobile protons and distinguish our p$^{-}$ and V$^{-}$ quasi-particles. For the hexagonal ice and water, which is 70 some percent in natural abundance, there are four water molecules in the primitive or smallest unit cell, therefore, the number or primitive or smallest unit cell is $N$/4, with the same concerns on distortion and correlated distortion. The consideration just given is from an anticipated development of the theory of water as a perturbation, however large, of the well-developed theory of the crystalline or perfect and infinitely large matter in solid-state and semiconductor physics introduced by Slater^[7] and Shockley^[8] and further elaborated and advocated by Ziman^[9]. However, without any distortion and the same four-proton trap geometry of tetrahedral trisector (need not exactly be $d_{\rm O-O}$/3) site, the distinction between p$^{-}$ and V$^{-}$ and between p$^{+}$ and V$^{+}$ is fundamental statistical mechanics, seen by the simple picture that their interparticle distances is large (several thousand or more) compared with the intertrap or interhost-core distance, $d_{1-2}$ $\gg$ $d_{\rm O-O}$ where 1 and 2 are any two of the four particles including itself, such as the four 4-distance groups (inter-different-particle distance repeated, 4 $+$ 3 $\times$ 4/2 $=$ 10 unique pairs): $d_{1-2}= d_{\rm p+-p+}, d_{\rm p+-p-}, d_{\rm p+-V+}, d_{\rm p+-V-}; d_{\rm p--p-}, d_{\rm p--p+}, d_{\rm p--V+},$ $d_{\rm p--V-}; d_{\rm V+-V+}, d_{\rm V+-V-}, d_{\rm V+-p+ }, d_{\rm V+-p-}$; and $d_{\rm V--V-}$, $d_{\rm V--V+}, d_{\rm V--p+}, d_{\rm V--p-}$. And the interparticle distances can be estimated by (particle density)$^{-1/3}$, for example in the pure water at the standard condition of 25 ℃, one atmospheric pressure and 1 gm/cm$^{3}$ density, we have $N_{{\rm H}_2{\rm O}}= 3.3429 \times 10^{22}$ molecule/cm$^{3}$ or an inter-H$_{2}$O distance or inter-oxygen-core (O$^{2-}$ for the one-proton protonic energy bands, or O$^{6+}$ for the one-electron electronic plus the one-proton protonic energy bands) of $d_{\text{H}_2\text{O-H}_2\text{O}} =d_{\rm O-O}= (N_{{\rm H}_2{\rm O}})^{-1/3}= (3.3429 \times 10^{22})^{-1/3}$ cm =$3.1043 {\mathring{\text{A}}}$ (or slightly more accurate using the two water molecules in a cubic ice-water primitive unit cell, or the four water molecules in a hexagonal ice-water primitive unit cell) while the inter-proton and inter-protonhole or interprotole distance is $d_{\rm p+/p+}=d_{\rm p-/p-}=d_{\rm pi/pi}=d_{\rm p+/p−}(2)^{1/3}$ $=$ ($p_{\rm i})^{-1/3}$ $=$ (7.334 $\times$ 10$^{13})$$^{-1/3}$ cm $=$ 2.389 $\times$ 10$^{-5}$ cm =$2389 {\mathring{\text{A}}}$, or there is one proton or proton-hole among (3.3429 $\times$ 10$^{22}$/7.334 $\times$ 10$^{13})$ $=$ 4.558 $\times$ 10$^{8}$ water molecules, hence Henry's Law of dilute solution and the Boltzmann statistical distribution of the particle number or particle volume density (cm$^{-3})$ as a function of the particle kinetic energy, applies well.

In this present extension and generalization, we include all conceivable quasi-particles (electrons, protons, atoms and molecules and their ions) in the to-be-distorted ''perfectly periodic'' water media lattice structure, the $\underline {\text{host lattice}}$, as well as the correlated geometrical distortions of the 'soft' host lattice. We take into account of the generation-recombination-trapping or ionization-deionization $=$ dissociation-association kinetics of the host particles, as well as their transports by diffusion in the presence of a concentration gradient, and by drift if ionized or charged, in the presence of an electric field.

For the impure water, we extend our model to include the ionization and deionization of the impurity molecules and association-dissociation of the impurity and host ion pairs, not excluding even the neutral host molecule, the neutral water molecule, V$^{0}\equiv V^{0\pm } \buildrel {\rm def}\ \over =\ $ [H$_{2}$O]$^{0\pm }$, allowing its charge-neutral square-well (or dipolar-quadripolar-multipolar) potential, to possibly be sufficient to bind an impurity ions. These are coined by us as the impuritons. For the simplest impurities, namely the univalent acidic and basic impurities, and salts, respectively H$^{1+}$A$^{1-}$ and B$^{1+}$(OH)$^{1-}$, and B$^{1+}$A$^{1-}$, the bond strength and bond length between the oppositely charged particles, giving the electrically neutral impuritons, the immobile [B$^{1+}$V$^{-}$]$^{0}$ and [V$^{+}$A$^{1-}$]$^{0}$, and the mobile [p$^{+}$A$^{1-}$]$^{0}$ and [B$^{1+}$p$^{-}$]$^{0}$, can be estimated by the Bohr model with the Coulomb attractive potential well, screened by the dipolar water molecules ($\varepsilon _{\text{H}_2\text{O}}\varepsilon _{\rm o}= 80 \varepsilon_{\rm o}$ static) if there is a sufficient number of water molecules within the bound-orbit sphere, using the density of state effective masses of the host particles from the pair generation-recombination equilibrium constant or intrinsic carrier density, assuming symmetry or some specific asymmetry between the proton and proton-hole (protole) in the one-proton energy bands of the water host. For the electrically neutral trap of proton, from the neutral but polar host water, [H$_{2}$O]$^{0\pm }$, and also from the electrically neutral impuritons, [p$^{+}$A$^{1-}$]$^{0\pm }$ and [B$^{1+}$p$^{-}$]$^{0\pm }$, their binding strength (or proton positivity) and bond length could also be estimated by the square-well potential (or the r$^{-6}$ Morse Potential), or the single charged Coulomb potential for two oppositely charged protonic particles or host vacancies via CMS if desired.

In this analysis, instead of starting from the four empty proton traps around each oxygen core that contains also the eight valence electron (H$_{0}$O$^{2-})$$\equiv$O$^{2-}$, we shift the reference by two to coincide with the charge number zero, using the Vacancy symbol V$^{0}$ with its concentration abbreviated by its symbol, that is, [(H$_{2}$O)$^{0}$] $\equiv$ $V^{0}$. Similarly, (H$_{3}$O)$^{1+} \equiv$ $V^{+}$ and (HO)$^{1-}\equiv$ $V^{-}$. Thus, for pure water, we have the five host lattice quasi-particles, $p^{+}$, $p^{-}$, $V^{+}$, $V^{0}$ and $V^{-}$ where the number conservation condition for the water molecule is $V^{\rm T}=V^{\rm Total}=V^{+}+ V^{0} + V^{-} = N_{\text{H}_2\text{O}}$ $=$ $N(x_{0}$, $y_{0}$, $z_{0}$, $t_{0})$ for some imposed initial and boundary conditions which could even vary during the experiment or the relaxation of the system at any space location or point in the system ($x_{0}$, $y_{0}$, $z_{0})$ and certainly during the initial transients before a steady-state is reached. For the usual, preferred situation, to provide easy interpretation of the experiment, the experimentalist would set $N(x_{0}$, $y_{0}$, $z_{0}$, $t_{0})$ $=$ constant or zero. However, for sensitivity enhanced measurements of the fundamental parameters, such as the three binding energies of the two mobile, electrical charge carrying host particles, the proton and the proton-hole or protole, and their capture and emission rate coefficients at the host lattice's electrically neutral proton trap, V$^{0}$, an experiment could be a tiny impurity droplet, dropped into a beaker of pure water with a volume much larger than the tiny droplet, to illustrate how this foreign droplet spreads out in the water, which would also give the diffusion coefficient and the drift mobility. For symbol simplicity and illustration ease, we shall include one impurity species in this water physics experiment and theory, to be denoted by BA where B also stands for Base and A, Acid. The simple acid is HA where H is the hydrogen or the acid part of the water molecule, H$_{2}$O or ${\bf H}$OH, and A is the acidic univalent impurity atom, such as F, Cl, Br and I. Similarly, the simple base is BOH where OH is the base part of the water molecule, H$_{2}$O or H${\bf OH}$, and B is the basic univalent impurity atom, such as Li, Na, K, Rb and Cs. Furthermore, BA can be a salt added to the pure water host. It can be ionized into two ionic atomic-molecular impurity species of one or more ($n$) electron charge on each, BA $\leftrightarrow$ B$^{n+}$ $+$ A$^{n-}$, $n =$ 1 for monovalent, $n =$ 2 for divalent, $n=$ 3 for trivalent, $\cdots$. This generalization can be readily expanded to many ionic impurity species of single-atom with multi-electron charges and multi-atom molecular species with multi-electron charges, such as inorganic, organic, and biological molecules.

The partial differential equations of the macroscopic water density, for protonic electrical conduction by positive and negative quasi-protons (proton and proton-hole), are similar to the Shockley Equations for electrons and holes (or negative and positive electrons) in solid-state semiconductors. These have been given the abbreviated symbols of (e$^{-})_{\rm i }\equiv n$ and (h$^{+})_{\rm b}\equiv p$, where i $=$ interstitial and b $=$ bond. And e $=$ electron with a mass, while h $=$ hole $=$ missing; which normally does not distinguish the silicon-core host with its four valence bond electrons as an amphoteric electron traps with V$^{0 }\equiv$ {Si$^{4+}$[(e$^{-})_{\rm b}$]$_{4}$}$^{0}$, V$^{+ }\equiv$ {Si$^{4+}$[(e$^{-})_{\rm b}$]$_{3}$}$^{+ }\equiv$ (h$^{+})_{\rm b}\equiv$ p and V$^{- }\equiv$ {Si$^{4+}$[(e$^{-})_{\rm b}$]$_{4}$[(e$^{-})_{\rm i}$]$_{1}$}$^{-}\equiv$ [V$^{0}$(e$^{-})_{\rm i}$]$^{-}$, in which V$^{+}$ and p have the same physical composite or V$^{+}$ is the hole or the negative quasi electron, while V$^{-}$ is an electron bound to silicon lattice, or the negatively charged quasi-electron. In the semiconductor case, conventionally, however, V is used to indicate a missing host atom or a host vacancy, V$_{\rm Si}$ with the silicon core, Si$^{4+}$, missing, hence (V$_{\rm Si}$$^{4+})$$^{4-}$ which can give off the electron on one to four of the pair electron bonds, and also one of the four remaining bond electrons, so shifting the reference to coincide with the neutral configuration, then we have, V$_{\rm Si}^{0 } \equiv$ (Si$^{0}$)$_{0}$}$^{0} \equiv$ {(Si$^{4+})_{0}$ ([(e$^{-})_{\rm b}$]$_{4}$)$_{0}$}$^{0} \equiv$ V$^{0}$, V$_{\rm Si}^{+} \equiv$ {(Si$^{+})_{0}$}$^{+} \equiv$ {(Si$^{4+})_{0}$([(e$^{-})_{\rm b}$]$_{5})_{0}$}$^{+ }\equiv$ V$^{+ }$ and V$_{\rm Si}^{- }\equiv$ {(Si$^{-})_{0}$}$^{-} \equiv$ {(Si$^{4+})_{0}$([(e$^{-})_{\rm b}$]$_{3})_{0}$}$^{-} \equiv$ V$^{-}$.

In our previous, second, report^[2], we listed the proposed Water Equations for the diffusion, drift and generation-recombination-trapping (DDGRT) for each quasi-particle in pure and impure water. These are now extended to include the impurity-host pairs, which we shall call the impuritons. The symbols and terms are defined in Appendix A when not self-evident.

In order to provide a simple mental picture to think about and to mathematically describe the various quasi-particles' DDGRT events that are continuously occurring, randomly, over the entire volume of the water media in the sense of homogenously disordered media, such as that advocated by Ziman^[9], we divide the physical space of the water media into two regions, the lattice (bond) and the tunnel (interstitial) regions. Their correspondence to the interbond (lattice) and interstitial (tunnel) proton bond model and to the low (lattice) and high (tunnel) potential energy regions of the one-proton potential energy contour enables this division. The latter, the tunnels, are the hexagonal tunnels exiting in both the hexagonal ice-water and the cubic ice-water, with sufficiently large dimension (or hexagonal diameter) that the larger impurity ions and molecules can fit into and transport through (diffusion and drift, although rather slowly, even limited by trapping) but not the lattice. Indeed, these larger impurity ions could still be trapped on the 'interface' between these two regions and lodged inside the lattice region from the softness of the water lattice, becoming immobilized to some extent, which is mathematically represented by the impuritons just described, from which the viscosity of the fluid can be modeled microscopically.

The subscript energies in meV for $\underline{\text{the grt transitions}}$, sometimes indicated to delineate the phonon partners of the specific grt transitions, are those zone-center phonons from the vibrational motion of the protons on the water lattice, designated here by us as the hydrogen-phonon (H$^{1+}$-phonons) or $\underline{\bf \text{proton-phonons}}$. The lower energy $\underline{\text{scattering transitions}}$ of the protons, that limit the proton diffusivity and mobility, analogous to those in electronic semiconductors, are caused by the lowest energy oxgen (O$^{6+})$ phonons, which are elastic scatterings at and near $E(q=0)=0$, with only velocity direction randomization by the longitudinal acoustical oxygen phonons, and by the inelastic higher energy oxgen (O$^{6+})$ interband and intervalley optical phonons. The phonon energy band or frequency-wave-number diagram, $E(\boldsymbol{q}) \equiv h\nu (\boldsymbol{q}) = h\omega (\boldsymbol{q})$, contains $4\times3\times3=36$ branches from the 4 H$_{2}$O molecules in the primitive unit cell of the hexagonal ice and water. They were described by us in our first two reports^{[1, 2]}. Of the 36 branches, 12 ($=1\times4\times3$) are the low-frequency branches of the acoustical and optical phonons from the vibrations of the oxygen atoms on the oxygen lattice, which were abbreviated by us as the $\underline {\textbf{Oxygen-Phonons or Atomic-Phonons}}$. The remaining 2 $\times$ 4 $\times$ 3 $=$ 24 branches are the high-frequency branches from the vibrations of the two protons of each water molecules on the water lattice, abbreviated by us as the hydrogen vibration phonons or proton vibration phonons, or just $\underline {\textbf{Hydrogen-Phonons}}$ or $\underline {\textbf{Protonic-Phonons}}$. Thus, the total of 36 branches can be called $\underline {\textbf{Molecular-Phonons}}$ or $\underline {\textbf{Water-Phonons}}$ or explicitly H$_{2}$O-Phonons; and if 100% Deuteron replacement of Hydrogen, then the Heavy-Water-Phonons or D$_{2}$O-Phonons, and if only a low concentration of Deuterons, then the (DHO)$_{m\sim \text{10-say}\ll n}$(H$_{2}$O)$_{n\to \infty }$ $\textbf{Deuteron-Local-Mode}$ or $\underline {\textbf{Deuteron-Bound-Phonon}}$ $\underline {\text{in the water-phonon bands}}$. Their frequencies or energies can be estimated as follows, using the simple Mass model, not taking account of the interatomic force difference, which were given in our first two reports^{[1, 2]}. Thus, the 12 low-frequency branches, namely, the oxygen phonons, come approximately from the vibrations of the four oxygen 'core' O$^{6+}$ (4 atoms and 3 degree of freedom^[7] in the primitive hexagonal unit cell, disregarding the two protons attached to each oxygen which only increase the mass $\underline {\textbf{from}}$ the 16 (proton-mass-unit) approximate-spherical oxygen-mass-charge, $M_{\rm O6+}$($x, y, z$), in a size of approximately $r_{\rm O6+}= (0.430 + 0.413)/2 = 0.422$ A (or an oxygen nucleus point, $M_{\rm O8+}$, of radius $\sim$0.5 fm $=$ $5\times10^{-6}$ A) $\underline {\textbf{to}}$ a nonspherical or half tetrahedral distribution of mass-charge of a size of about ($r_{\rm O6+}+d_{\rm O-H}/2)= (0.422+0.5) = 0.9$ A. This lowers the vibrating frequency by only ($M_{\rm H2O}/M_{\rm O})^{1/2}$ $=$ (18/16)$^{1/2 }$ $=$ 1.06066, giving about 6% ($\sim$6.066) lower in phonon frequency or energy, but the vibrating mass is still small in size ($r_{\rm O6+}+d_{\rm O-H}/2\sim 0.9$ A) in comparison with the inter-oxygen distance ($d_{\rm O-O}\sim 2.75$ A), so the point mass model with the force change neglected is not too bad. CMC gives similar estimate. The $2\times4\times3 = 24$ high-frequency branches of the phonon energy band diagram from the two hydrogens in each water molecule, are approximately ($M_{\rm O}/2M_{\rm H})^{1/2}$ $=$ (16/2)$^{1/2}$ $=$ 2.8 times and ($M_{\rm O}/M_{\rm H})^{1/2 }$ $=$ (16/1)$^{1/2}$ $=$ 4 times higher in frequencies or vibrational energies than those of the lower branches from the oxygen^{[1, 2]}. These protonic-phonons are the ones which assist in the generation-recombination-trapping of the protons on the water lattice, while the oxgen-phonons are those which scatter the protons and proton-holes and limit their diffusivity and drift mobility.

3. Electrical Conductance of Water, Pure and Impure

The electrical conductance consists of the sum of that from the positive and negative mobile species of the electrically charged particles. In pure water, they are the positive proton and the negative protons or proton-holes (protoles). In impure water, in addition to the positive and negative protons, they include also the charged mobile impurity ions and the mobile species of the impuritons which are bound pairs formed by the mobile impurity ions and the mobile positive proton and negative proton-holes (protoles).

The electrical conductance is the product of the mobility and the volume density or concentration of the mobile charge carriers. Thus, the total conductance of an impure water, containing the univalent impurity ions A$^{-}$ and B$^{+}$, is given by the sum $q(\mu_{+}p^{+}+\mu _{-}p^{-}+\mu _{\rm A-}A^{-}+\mu _{\rm B+}B^{+})$. In our model, only the electrically neutral mobile impuritons are considered, not the charged mobile impuritons, which would have given some electrical conductivity too, if existed. Let's call the charged impuritons, the $\textbf{affinitons}$, mobile or immobile, again leaving affinit$\textbf{r}$ons to electrons. They may not exist or may not be stable under the standard condition, because the proton affinity to a neutral impuriton is likely very weak, hence its affiniton is easily dissociated to stay at their original neutral charge state, for examples, the protonic analogy of the well-known electron affinity are: the acid affinitons [$p^{+}(p^{+}A^{-})^{0\pm }]^{1+} \equiv$ [($p^{+})_{2}A^{-}]^{1+ }$ $\to$ $p^{+} + (p^{+}A^{-})^{0\pm }$ and the base affinitons [$p^{-}(p^{-}B^{1+})^{0\pm }]^{1-}\equiv$ [($p^{-})_{2}B^{1+}]^{1- }$ $\to$ $p^{-} + (p^{-}B^{1+})^{0\pm }$. But if they exist under certain conditions, such as very low temperatures and very high pressures, the preceding chemical equations can easily be applied, with $\to$ replaced by $\leftrightarrow$.

In this report, the mobility formula of each charged mobile particle species in water is not derived and not calculated, other than that the existing experimental mobility data of the positive charge and negative charge carriers suggested that the mobility and the drift current are trapping limited and are inversely proportional to the concentration of the ionized traps. Therefore, the variations of the concentrations of the particles in water give the characterization of the electrical properties of water. The formulas used in the numerical computation of the curves presented in the illustration figures are given in Appendix A for pure and impure waters. These illustration curves include the variation of the concentrations with temperature and electric potential at a contact surface, aimed to guide experimental determination of the fundamental parameters that characterize pure and impure water. The figure captions give the numerical values of the parameters used in computing the curves shown in each figure. The special characteristics and unique features of the results shown by each figure are described and explained in the following paragraphs.

Figure 1 gives in six parts the properties of the five pure-water host particles, p$^{+}$, p$^{-}$, V$^{+}$, V$^{0}$, and V$^{-}$. Part (a) gives their concentrations and (b), the normalized concentration as a function of the water temperature (0 to 100 ℃) as 1000/$T$(K). Part (c) gives their excess (deficient is negative) areal charge density (q/cm$^{2})$ referred to flat-band, integrated through the thickness of the water layer, defined by $\Delta Q_{\rm p+}(V_{\rm S}, T=T_{1}) \buildrel {\rm def} \over = Q_{\rm p+}(V_{\rm S, }T=T_{1})- Q_{\rm p+}(V_{\rm S}=0, T=T_{1})$, as a function of applied surface potential ($V_{\rm S }=-600$ to $+600$ mV) for the five temperatures, $T_{1}=0$, 25, 50, 70, 100 ℃. Part (d) gives the slope (d/d$V_{\rm S})\Delta Q_{\rm p+}(V_{\rm S}$, $T=T_{1})$ vs $V_{\rm S}$ at the five temperatures to illustrate the peaks, valleys and inflection points that could be helpful in the experimental measurements of the properties of the pure water. Parts (e) and (f) are similar to (c) and (d) but with the surface potential replaced by the voltage ($-5$ V to $+5$ V) applied to an insulated gate electrode relative to a ohmic electrode, in analogy to the silicon metal-oxide-semiconductor capacitor. The input data used to compute these curves are given in the Figure caption, including: (ⅰ) the experimental energy gap as a function of temperature, $E_{\rm p+ }- E_{\rm p-} = E_{\rm p+p-}(T)$, computed from the industrial consensus LSF formula fitted to the experimental data which was cited and given in our second report^[2], and (ⅱ) the experimental trapping energies, ($E_{\rm p+} - E_{1, 0})$ for protons and ($E_{0, -1}- E_{\rm p-})$ for proton-holes (protoles) which we reported^[1-3] hidden in the industrial consensus experimental data, in the three temperature ranges (meV, meV): 0-25 ℃ (128, 128), 25-70 ℃ (88, 88) and 70-100 ℃ (55, 117). The smaller proton trapping energy (55 meV) than that of proton-hole (117 meV) in the 70-100 ℃ range gives a large ratio of p$^{+}$/p$^{-}$, making pure water acidic.

To test the correctness of our new water physics and model and its computed numerical results, since the model is not the conventional whose solutions are familiar from solid semiconductor such as silicon, we use the hypothetical model of complete symmetry between the positive and negative protons, both in their energy bands (E-k or effective masses) and in their traps ($E_{\text{trap-binding-energy}})$. The results are given by the $\underline {\textbf{black}}$ $\underline { \text{continuous}}$ (positive values) and $\underline {\text{brokens}}$ (negative values) $\underline {\text{lines}}$ in all six parts of Fig. 1 for the hypothetical identical, symmetrical, and water temperature independent (0-100 ℃) proton trap (V$^{+})$ and proton hole trap (V$^{-})$, having the same transition energy for proton and proton-hole capture by the neutral amphoteric proton trap, V$^{0}$, with emitted phonon energy of $\hbar\omega _{\rm phonon}=$128 meV, namely, V$^{0}$ $+$ p$^{+} \leftrightarrow$ V$^{+}_{ }$ $+$ $\hbar\omega _{\text{128 meV}}$, and V$^{0}$ $+$ p$^{- }\leftrightarrow$ V$^{-}$ $+$ $\hbar\omega _{\text{128 meV}}$. Detailed molecular-orbital picture (assuming valid in this sufficiently localized phenomena) and general proton energy band model (presumably valid but rather elusive and doubtful generality) do not assure such general symmetry, even simply just from the differences between the corresponding two initial states and corresponding two final states and their degeneracies, and also from the differences in the hydrogen-or-protonic phonon local modes containing one, two and three protons, not the same number of protons. Nevertheless, the complete symmetry example does provide us the baseline to illustrate the deviations-variations due to asymmetry between the proton and proton-hole trapping transitions. Figures 1(a) and 1(b) confirm the expected results: $p^{+} = p^{-} = p_{\rm i }$ and $V^{+ }= V^{-} \ll V^{0} \cong P_{+ }= P_{- }=$ Effective Density of States (DOS$_{\rm eff}) = 5.3796\times10^{18}$ cm$^{-3}$ $\ll N_{\text{H}_2\text{O}} = 3.3429\times10^{22}$ H$_{2}$O/cm$^{3}$ (1 gram/cc), the latter, DOS$_{\rm eff} \ll N_{\text{H}_2\text{O}}$, is reminiscent of that of solid-state crystalline semiconductors, such as Si and Ge. $\underline {\text{The second expected results}}$ is the dependence on the trapping energy of the completely symmetrical proton and proton hole trapping in the temperature ranges of $T=$ 0-25 ℃ and 25-70 ℃. This is shown by the two groups of lines for the two trapping energies, 128 meV and 88 meV, confirming that more trapping or less detrapping-emission of proton and proton-hole when the traps are shallower (88 meV vs 128 meV) and when the temperature is higher. $\underline {\text{The third expected result}}$, perhaps the most important, comes from the asymmetry of the proton and proton-hole trapping, expressed by the difference in trapping energy of proton (55 meV) and proton hole (117 meV) given by the experimental data in the 70-100 ℃ range, with the computed theoretical result, shown in Fig. 1(b), of p$^{+}$/p$^{-} > \sim$8 (2.85289/0.35025 to 2.62909/0.38036 $=$ 8.13902 to 6.914239 $\sim$ 8 to 7), namely, pure water should be rather ''acidic'' in its upper liquid temperature range. This last result also suggests that the traditional method of obtaining the mobility of the two electrical charge carriers in pure water may need modifications because the traditional method depends on a theoretical extrapolation of the electrical conductivity data of acidic and basic impure waters to zero acid impurity and base impurity concentrations. However, the traditional theory does not include the trapping limitation of the two charge carrier species in pure water. Such more accurate evaluation of the experimental results could also provide the microscopic details and fundamental properties of the proton energy band model of water. These include the asymmetry of (ⅰ) the density of states of proton and the proton-hole and (ⅱ) the trapping of the proton and the proton hole by the neutral water molecule. The latter could be expected even just on theoretical many-body physics grounds, namely, from the space-spin degeneracy of the ground and excited states of the two proton traps, V$^{+ }$ and V$^{-}$, as well as their extended, multi-particle, and not-spherical and not-point or extended charge-and-$\underline {\text{mass}}$ distribution in space or in the primitive unit cell, which determine the density of their initial and final states, shown explicitly by our modified chemical formulas: V$^{0}$ $\buildrel {\rm def} \over =$ [2p$^{+}$ $+$ O$^{6+}$ $+$ 8e$^{-}$]$^{0\pm } \equiv$ {(p$^{+})_{2}$[O$^{6+}$ $+$ 8e$^{-}$]$^{2-}$}$^{0\pm }$, V$^{+}$ $\buildrel {\rm def} \over =$ [3p$^{+}$ $+$ O$^{6+}$ $+$ 8e$^{-}$]$^{0\pm } \equiv$ {(p$^{+})_{3}$[O$^{6+}$ $+$ 8e$^{-}$]$^{2-}$}$^{1+}$, and V$^{-}$ $\buildrel {\rm def} \over =$ [1p$^{+}$ $+$ O$^{6+}$ $+$ 8e$^{-}$]$^{1-} \equiv$ {(p$^{+})_{1}$[O$^{6+}$ $+$ 8e$^{-}$]$^{2-}$}$^{1-}$. In addition, the protonic-phonons at high densities, and the protonic-local-modes at low densities, $\underline {\text{not only}}$ are the very protons of these extended host vacancies and proton traps, $\underline {\text{but also}}$ are those responsible for the proton-local-modes, that determine the magnitude of coupling between the phonon-local-modes and the protons and the proton-holes. Nevertheless, our macroscopic statistics models and quasi-particles are still adequate models and physics-based representations of the bulk water, containing many water molecules and, most important, having reached the thermodynamic steady-$\underline {\text{macroscopic}}$-state, that is what experiments observe. The theoretical derivative characteristics in Figs. 1(d) and 1(f) show peaks, valleys and inflections, which could be signatures and landmarks for the experimental measurements of these pure water properties.

Figure 2 is given to test the use of the experimental effective density of states of proton and proton-hole in the energy band model for the total proton trap density from pure water measurement of the charge carrier concentration, $p_{\rm i}$ and its temperature dependence, $V_{\rm T} = V^+ + V^0 + V^− = P^+ = P^− = 5.38\times10^{18}$ cm$^{−3}$. The completely symmetrical water is used to illustrate in Fig. 1. We recomputed, in Fig. 2, the six parts of Fig. 1 by increasing it nearly four orders of magnitude to the molecular value, $V^{\rm T} = V^+ + V^0 + V^− = N_{\text{H}_2\text{O}} = 3.34\times10^{22}$ cm$^{−3}$. Comparing the corresponding parts of Figs. 1 and 2, one can see that the general characteristics are similar, only occurring at higher values in Fig. 2, offering the delineation via experiments.

The effects of impurity in water are illustrated by the six parts of Fig. 3 for the acidic impure water with the univalent acidic impurity, HA. Due to page limitation, the similar results for the basic and salty impurity waters are deferred to a future report, which illustrates the effects of impuritons and salts on the acidity and basicity of impure water. A most important result from impurity is its concentration threshold, shown in both of the two parts of Fig. 3(a). The left part of Fig. 3(a) shows that the concentration of the proton, $p^+$, starts to rise, approaching proportionality, when the acid impurity concentration HA rises above the total trap concentration, $V^{\rm T}$, in this case, the density of states value, $V^{\rm T}= P^+$, from the pure water data of carrier concentration, $p_{\rm i}$, and its ionization energy, or the protonic energy gap, $E_{\rm p+p−}$. The right part of Fig. 3(a) shows the increase of this threshold to the molecular value from the higher $V^{\rm T} = N_{\text{H}_2\text{O}}$ used. This difference provides the experimental determination of the effective density of states of protons using acid water and of proton-holes using base water. Figure 3(b) shows the relatively small dependence on temperature, tested at two temperatures, 25 ℃ and 70 ℃. Figures 3(c) to 3(f) give the field-effect concentrations and its surface-potential derivative as a function of the surface potential and applied gate voltage with the $N_{\rm HA}/N_{\text{H}_2\text{O}}$ increased from $10^{−5}$ to $10^{−1}$. They illustrate the shift of the derivative peak from the trapped charge ($V^+$ for the acid impurity) to higher positive surface potentials or more accumulation of the majority carriers, as the acid impurity increases, offering further experimental determination of the proton energy band and proton trap parameters.

4. Summary

In this third report, we have presented the theoretical electrical conductivity characteristics of pure and impure water, at thermodynamic equilibrium. Our Water Physics Equations are extended from those given in our second and previous report^[2] by the addition of mobile and immobile neutral impuritons, which are respectively composed of the impurity ion that is quantum mechanically bound to the oppositely charged proton and proton trap. Computed pure water characteristics illustrate the effects of density of protonic traps from the effective density of states of the energy band model and from water molecule concentration of the molecular bond model. The impurity concentration threshold of charge carrier density with increasing impurity concentration is also illustrated which would be further modified by the presence of impuritons.

Acknowledgements

This study is supported by the Xiamen University, China. We thank XMU President Zhu Chongshi (朱崇实) and Physics Dean Wu Chenxu (吴晨旭) for freedom to study subjects of our interest. BJ is also supported by the CTSAH Associates of Florida, USA, which was founded by the late Linda Su-nan Chang Sah. TS thanks Robert and Dinah Sah for recent informational discussions which reignited TS' chemistry interest that was started during 1956-1958 while working with several Chemists (Gordon Moore, Jean Hoerni, Doug Tremere and Harry Sello) and Physicist Jay Last, under Shockley's direction at Shockley Semiconductors Laboratory of Beckman Instruments. This continued during 1959-1964 at Fairchild Semiconductor with the addition of Chemists Andy Grove and Bruce Deal plus one third of TS's 64-member Fairchild team, who were chemists and chemical engineers, on the development of the first generation silicon transistor and integrated circuit technology. TS' chemistry interest was further intensified at the home front by Chemists Linda Chang (Tsang) during 1958-2003 and Floris Tsang (Chang) in 1961, and the life-science biologist and bioengineer, Dinah and Robert, starting in 1967-1968 when they first learned chemistry at the University High School in Urbana, Illinois, then at MIT and Harvard-Med School. For this intrusion into water chemistry and physics, the authors are benefitted by the pioneering studies and explanations made by numerous authors of journal articles in the last 80+ years^[4-6]. [See also those cited in our first two reports^{[1, 2]} and their references.] The authors are especially influenced by the textbooks written by the late expert teachers on this subject, started by Slater's 1939 $\mathit{\boldsymbol{Introduction\;to\; Chemical\;Physics}}$^[7], Shockley's 1950 $\mathit{\boldsymbol{Electrons \;and\;Holes\;in\;Semiconductors}}$^[8], and Ziman's 1979 $\mathit{\boldsymbol{Models\;of\;Disorder\;}}$-$\mathit{\boldsymbol{The\;theoretical\;physics\;of\;homogenously\;disordered \;systems}}$^[9].

Appendix A: Graphic Symbol and Water Equations

This appendix gives the equations of the concentrations of the particle species. We start with the abbreviated graphic symbol for the water molecule of nuclear protons, atomic electrons, atomic and molecule protons, and atomic oxygen, which were defined and described in slides 27 and 28 of our first presentation^[3], copied to Figs. A1(a) and A1(b) in this appendix, which is a simplification of the constant mass or charge density contour map, traditionally employed in chemistry's molecular orbital theory, such as the contours shown in Fig. A2 in this appendix, for the hydrogenation of the group Ⅲ acceptors (B, Al, Ga and In) in the solid-state semiconductor silicon, which TS sketched, based on the experimental high-frequency capacitance-voltage ($C$-$V$) data taken and reported by Sun, Tzou, Pan and Hsu in 1983 on aluminum-gate and silicon-gate Metal-Oxide-Silicon capacitors (MOSC's) on p-type and n-type silicon substrates, using the first silicon wafers from the silicon gate technology, pure and impurity doped silicon, provided by Yu Hwa-Nien of the IBM T. J. Watson Research Center at Yorktown Height, NY.

This appendix gives the equations of the concentrations of the particle species. We start with the abbreviated graphic symbol for the water molecule of nuclear protons, atomic electrons, atomic and molecule protons, and atomic oxygen, which were defined and described in slides 27 and 28 of our first presentation^[3], copied to Figs. A1(a) and A1(b) in this appendix, which is a simplification of the constant mass or charge density contour map, traditionally employed in chemistry's molecular orbital theory, such as the contours shown in Fig. A2 in this appendix, for the hydrogenation of the group Ⅲ acceptors (B, Al, Ga and In) in the solid-state semiconductor silicon, which TS sketched, based on the experimental high-frequency capacitance-voltage ($C$-$V$) data taken and reported by Sun, Tzou, Pan and Hsu in 1983 on aluminum-gate and silicon-gate Metal-Oxide-Silicon capacitors (MOSC's) on p-type and n-type silicon substrates, using the first silicon wafers from the silicon gate technology, pure and impurity doped silicon, provided by Yu Hwa-Nien of the IBM T. J. Watson Research Center at Yorktown Height, NY.

Water Equations -The Xiada Water Equations.

Here are the symbols and formulas used in the computations of the curves given in the figures.

A1. DDGRT Water Equations for the positive quasi-proton p$^{+}$

(A1) $q\partial p^{+}/\partial t = -\boldsymbol{\nabla}\cdot \boldsymbol{j}_{\rm p+}+ q(g_{\rm p+p-} - r_{\rm p+p-})_{\rm \sim 600meV} + q(g_{\rm p+V0} -r_{\rm p+V0})_{\rm \sim 128, 88, 55meV}$+ $q(g_{\rm p+A-} - r_{\rm p+A-})_{\text{undecided-if-in-data}} + q\text{(n-valence-Acid)}_{\rm undecided}$,

(A1A) $\boldsymbol{j}_{\rm p+} = + q \boldsymbol{\mu}_{\rm p+}\boldsymbol{E}p^{+ }- q\boldsymbol{D}_{\rm p+} \nabla p^{+}$ $=-q \boldsymbol{\mu} _{\rm p+}p^{+} \nabla V_{\rm p+},$

(A1B1) $g_{\rm p+p-} - r_{\rm p+p-}= g_{\rm p+p-0}V^{0}V^{0} - r_{\rm p+p-0}p^{+}p^{- } \quad\text{(lattice-lattice, DDmobile)}$,

(A1B2) $g_{\rm p+V0} - r_{\rm p+V0}$= $e_{\rm p+V0}V^{+ }- c_{\rm p+V0} p^{+}V^{0 }$ $\quad\text{(lattice-lattice, immobile)}$,

(A1C1) $g_{\rm p+A-} - r_{\rm p+A- }$= $e_{\rm p+A-}[p^{+}A^{-}]- c_{p+A-}p^{+}A^{-}\quad$ $\text{(lattice-tunnel, DDmobile)},$

(A1D201) $\text{divalent-acid}_{n=2, p+=0+1}$ = $e_{p+201}[(p^{+})^{1}A^{2-}]^{1-}- c_{p+201}(p^{+})^{1}A^{2-} \quad$ $\text{(lattice-tunnel, DDmobile)},$

(A1D202) $\text{divalent-acid}_{n=2, p+=0+2}$ = $e_{p+202}[(p^{+})^{2}A^{2-}]^{0\pm }- c_{p+202}(p^{+})^{2}A^{2- } \quad\text{(lattice-tunnel, Dmobile)}$,

(A1D211) $\text{divalent-acid}_{n=2, p+=1+1 }$=$e_{p+211}[(p^{+})^{2}A^{2-}]^{0\pm }$-$c_{p+211}(p^{+})^{1}[(p^{+})^{1}A^{2-}]^{1-} \quad\text{(lattice-tunnel, Dmobile)}$.

A2. DDGRT Water Equations for the negative quasi-proton p$^{-}$ (or proton-hole $=$ protole)

(A2) $q\partial p^{-}/\partial t = + \boldsymbol{\nabla}\cdot \boldsymbol{j}_{\rm p-}+ q(g_{\rm p+p-}- r_{\rm p+p-})_{\rm \sim 600meV} + q(g_{\rm p-V0} - r_{\rm p-V0})_{\rm \sim 128, 88, 55, 117meV}$ +$q(g_{\rm p-B+}- r_{\rm p-B+})_{\text{undecided-if-in-data}} + q({\text{n-valence-Base}}),$

(A2A) $\boldsymbol{j}_{\rm p-} = + q \boldsymbol{\mu}_{\rm p-}\boldsymbol{E}p^{-} + q\boldsymbol{D}_{\rm p-}\nabla p^{-}$=$- q \boldsymbol{\mu} _{\rm p-}p^{- } \nabla V_{\rm p-}$,

(A2B1) $g_{\rm p+p- }- r_{\rm p+p-}$ = $g_{\rm p+p-0} V^{0}V^{0}-r_{\rm p+p-0}p^{+}p^{-} \quad\text{[Same as (1B1).]}$,

(A2B2) $g_{\rm p-V0} - r_{\rm p-V0}$=$e_{\rm p-V0}V^{- }-c_{\rm p-V0}p^{-}V^{0} \quad\text{[Negative or 'anti' (1B2).]}$,

(A2C1) $g_{\rm p-B+} -r_{\rm p-B+}$ = $e_{\rm p-B+}(p^{-}B^{+})^{0} - c_{\rm p-B+}p^{-}B^{+} \quad\text{[Negative or 'anti' (1C1).]}$,

(A2D201) $\text{divalent-base}_{n=2, p-=0+1}$= $e_{\rm p-201}[(p^{-})_{1}B^{2+}]^{1- }- c_{\rm p-201}(p^{-})^{1}B^{2+} \quad$ $\text{(lattice-tunnel, DDmobile)}$,

(A2D202) $\text{divalent-base}_{n=2, p-=0+2}$= $e_{\rm p-202}[(p^{-})_{2}B^{2+}]^{0\pm}- c_{\rm p-202}(p^{-})^{2}B^{2+} \quad$ $\text{(lattice-tunnel, Dmobile)}$,

(A2D211) $\text{divalent-base}_{n=2, p-=1+1 }$= $e_{\rm p-211}[(p^{-})_{1}(p^{-})_{1}B^{2+}]^{0\pm}-c_{\rm p-211}(p^{-})_{1}[(p^{-})_{1}B^{2+}]^{1-} \quad$ $\text{(lattice-tunnel, Dmo).}$

A3. DDGRT Water Equations for the univalent negative acidic ion A$^{1-}$ $\equiv$A$^{-}$

(A3) $q\partial A^{-}/\partial t = +\boldsymbol{\nabla}\cdot \boldsymbol{j}_{\rm A-} + q(g_{\rm H+A-} - r_{\rm H+A-}) + q(g_{\rm V+A-} - r_{\rm V+A-})$+ $q(g_{\rm p+A-} - r_{\rm p+A-}) + q(\text{n-valence-Acid})$,

(A3A) $\boldsymbol{j}_{\rm A-}$ = $+ q \boldsymbol{\mu}_{\rm A-} \boldsymbol{E}A^{-} + q\boldsymbol{D}_{\rm A-} \nabla A^{-}$ = $- q \boldsymbol{\mu}_{\rm A-}A^{-} \nabla V_{\rm A-}$,

(A3B1) $g_{\rm H+A-} - r_{\rm H+A- }$= $g_{\rm H+A-0}(H^{+}A^{-})^{0}- r_{\rm H+A-0}H^{+}A^{-} \quad$ $(\text{tunnel-tunnel, Dmobile), }$

(A3B2) $g_{\rm V+A-} - r_{\rm V+A-}$= $e_{\rm V+A-}(V^{+}A^{-})^{0}-c_{\rm V+A-}V^{+}A^{- } \quad$ $\text{(lattice-tunnel, immobile), }$

(A3C1) $g_{\rm p+A-} - r_{\rm p+A- }$= $e_{\rm p+A-}(p^{+}A^{-})^{0}- c_{\rm p+A-} p^{+}A^{-} \quad$ $\text{(lattice-tunnel, Dmobile), }$

(A3Dp$^{+}$201) $\text{divalent-acid}_{n=2, p+=0+1}$ = $e_{\rm p+201}[(p^{+})_{1}A^{2-}]^{1- }- c_{\rm p+201}(p^{+})^{1}A^{2-} \quad$ $\text{(lattice-tunnel, DDmobile), }$

(A3Dp$^{+}$202) $\text{divalent-acid}_{n=2, p+=0+2}$= $e_{\rm p+202}[(p^{+})_{2}A^{2-}]^{0\pm}- c_{\rm p+202}(p^{+})^{2}A^{2-}\quad$ $\text{(lattice-tunnel, Dmobile), }$

(A3Dp$^{+}$211) $\text{divalent-acid}_{n=2, p+=1+1}$ = $e_{\rm p+211}[(p^{+})_{2}A^{2-}]^{0\pm }-c_{\rm p+211}(p^{+})^{1}[(p^{+})_{1}A^{2-}]^{1-} \quad$ $\text{(lattice-tunnel, Dmob), }$

(A3DV$^{+}$201) $\text{divalent-acid}_{n=2, V+=0+1}$= $e_{\rm V+201}[(V^{+})_{1}A^{2-})^{1-}- c_{\rm V+201}(V^{+})^{1}A^{2-} \quad$ $\text{(lattice-tunnel, immobile), }$

(A3DV$^{+}$202) $\text{divalent-acid}_{n=2, V+=0+2}$= $e_{\rm V+202}[(V^{+})_{2}A^{2-}]^{0\pm}- c_{\rm V+202}(V^{+})^{2}A^{2-}\quad$ $\text{(lattice-tunnel, immobile), }$

(A3DV$^{+}$211) $\text{divalent-acid}_{n=2, V+=1+1}$=$e_{\rm V+211}[(V^{+})_{2}A^{2-}]^{0\pm} -c_{\rm V+211}(V^{+})^{1}[(V^{+})_{1}A^{2-}]^{1-} \quad$ $\text{(lattice-tunnel, immo), }$

(A3D1C) $q\partial A^{2-}/\partial t$= $+ \boldsymbol{\nabla}\cdot \boldsymbol{j}_{\rm A2-} + q(3Dp^{+}201)$ +$q(3Dp^{+}202) + q(3DV^{+}201) + q(3DV^{+}202)$

(A3D1F) $\boldsymbol{j}_{\rm A2-}$ =$+ q \boldsymbol{\mu}_{\rm A2-}\boldsymbol{E}A^{2-}+ q\boldsymbol{D}_{\rm A2-} \nabla A^{2-}$ =$- q\boldsymbol{\mu}_{\rm A2-}A^{2-} \nabla V_{\rm A2-}$,

(A3D2C) $q\partial [(p^{+})_{1}A^{2-}]^{1-}/\partial t$=$+\boldsymbol{\nabla}\cdot \boldsymbol{j}_{\rm p+A2-}- q(3Dp^{+}201) + q(3Dp^{+}211)$,

(A3D2F) $\boldsymbol{j}_{\rm p+A2-}$= $+ q \boldsymbol{\mu}_{\rm p+A2-}\boldsymbol{E}[(p^{+})_{1}A^{2-}]^{1-}$+ $q\boldsymbol{D}_{\rm p+A2-} \nabla [(p^{+})_{1}A^{2-}]^{1-}$=$- q \boldsymbol{\mu}_{\rm p+A2-}[(p^{+})_{1}A^{2-}]^{1-} \nabla V_{\rm p+A2-},$

(A3D3C) $\partial [(p^{+})_{2}A^{2-}]^{0\pm}/\partial t$ = $-\boldsymbol{\nabla}\cdot \boldsymbol{f}_{\rm p+2A2-}- (3Dp^{+}202)- (3Dp^{+}211)$,

(A3D3F) $\boldsymbol{f}_{\rm p+2A2-}$= -$\boldsymbol{D}_{\rm p+2A2-} \nabla [(p^{+})_{2}A^{2-}]^{0\pm}$=-$(\boldsymbol{D}_{\rm p+2A2-}/k_{\rm B}T)[(p^{+})_{2}A^{2-}]^{0\pm} \nabla V_{\rm p+2A2-}$.

A4.DDGRT Equations for the univalent positive basic ion $\boldsymbol{B}^{1+}{\equiv \boldsymbol{B}}^{+}$

(A4) $q \partial B^{+}/\partial t$=$- \boldsymbol{\nabla}\cdot \boldsymbol{j}_{\rm B+}$+ $q(g_{\rm B+OH-}- r_{\rm B+OH-})$ + $q(g_{\rm B+V-}$-$r_{\rm B+V-})\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+q(g_{\rm B+p-} - r_{\rm B+p-}) + q (\text{n-valence-Base})$,

(A4A) $\boldsymbol{j}_{\rm B+}$= $+ q \boldsymbol{\mu} _{\rm B+} \boldsymbol{E}B^{+} + q\boldsymbol{D}_{\rm B+} \nabla B^{+}$ = $-q \boldsymbol{\mu} _{\rm B+}B^{+ } \nabla V_{\rm B+}$,

(A4B1) $g_{\rm B+OH- }- r_{\rm B+OH-}$ = $g_{\rm B+OH-0}[B^{+}(OH)^{-}]^{0\pm} -r_{\rm B+OH-0}B^{+}(OH)^{1-} \quad$ $\text{(tunnel-tunnel, Dmobile), }$

(A4B2) $g_{\rm B+V-}- r_{\rm B+V- }$= $e_{\rm B+V-}(B^{+}V^{-})^{0\pm }- c_{\rm B+V- }B^{+}V^{- } \quad$ $\text{(tunnel-lattice, immobile), }$

(A4C1) $g_{\rm B+p-} - r_{\rm B+p-}$= $e_{\rm B+p-}(B^{+}p^{-})^{0\pm}- c_{\rm B+p-}B^{+}p^{- } \quad$ $\text{(tunnel-lattice, Dmobile), }$

(A4Dp$^{-}$201) $\text{divalent-base}_{n=2, p-=01}$ = $e_{\rm p-201}[B^{2+}(p^{-})_{1}]^{1- }- c_{\rm p-201}B^{2+}(p^{-})^{1}\quad$ $\text{(lattice-tunnel, DDmobile)},$

(A4Dp$^{-}$202) $\text{divalent-base}_{n=2, p-=02}$= $e_{\rm p-202}[B^{2+}(p^{-})_{2}]^{0\pm}- c_{\rm p-202}B^{2+}(p^{-})^{2} \quad$ $\text{(lattice-tunnel, Dmobile), }$

(A4Dp$^{-}$211) $\text{divalent-base}_{n=2, p-=1+1}$= $e_{\rm p-211}[B^{2+}(p^{-})_{2}]^{0\pm }-c_{\rm p-211}[B^{2+}(p^{-})_{1}]^{1-}(p^{-})^{1} \quad$ $\text{(lattice-tunnel, Dmob), }$

(A4DV$^{-}$201) $\text{divalent-base}_{n=2, V-=0+1}$= $e_{\rm V-201}[B^{2+}(V^{-})_{1}]^{1- }-c_{\rm V-201}B^{2+}(V^{-})^{1} \quad$ $\text{(lattice-tunnel, immobile)},$

(A4DV$^{-}$202) $\text{divalent-base}_{n=2, V-=0+2}$ = $e_{\rm V-202}[B^{2+}(V^{-})_{2}]^{0\pm }-c_{\rm V-202}B^{2+}(V^{-})^{2} \quad$ $\text{(lattice-tunnel, immobile), }$

(A4DV$^{-}$211) $\text{divalent-base}_{n=2, V-=1+1 }$=$e_{\rm V-211}[B^{2+}(V^{-})_{2}]^{0\pm }-c_{\rm V-211}[B^{2+}(V^{-})_{1}]^{1+}(V^{-})^{1} \quad$ $\text{(lattice-tunnel, immo)},$

(A4D1C) $q \partial B^{2+}/\partial t$ = $- \boldsymbol{\nabla}\cdot \boldsymbol{j}_{\rm B2+}$+ $q(4Dp^{-}201)$ + $q(4Dp^{-}202) + q(4DV^{-}201) + q(4DV^{-}202),$

(A4D1F) $\boldsymbol{j}_{\rm B2+}$ =$+ q \boldsymbol{\mu} _{\rm B2+}\boldsymbol{E}B^{2+}- q\boldsymbol{D}_{\rm B2+}\nabla B^{2+}$=$- q \boldsymbol{\mu}_{\rm B2+}B^{2+} \nabla V_{\rm B2+},$

(A4D2C) $q\partial [\boldsymbol{B}^{2+}(p^{-})_{1}]^{1+}/\partial t$ =$-\boldsymbol{\nabla}\cdot \boldsymbol{j}_{\rm B2+p-}- q(4Dp^{-}201) + q(4Dp^{-}211),$

(A4D2F) $\boldsymbol{j}_{\rm B2+p-}$ = $+ q \boldsymbol{\mu} _{\rm B2+p-}\boldsymbol{E}[B^{2+}(p^{-})_{1}]^{1+ }-q\boldsymbol{D}_{\rm B2+p-}\nabla [B^{2+}(p^{-})_{1}]^{1+}$=$-q\boldsymbol{\mu}_{\rm B2+p-}[B^{2+}(p^{-})_{1}]^{1+} \nabla V_{\rm B2+p-},$

(A4D3C) $\partial [B^{2+}(p^{-})_{2}]^{0\pm}/\partial t$ = $- \boldsymbol{\nabla}\cdot \boldsymbol{f}_{\rm B2+p-2}- (4Dp^{-}202)-(4Dp^{-}211)$,

(A4D3F) $\boldsymbol{f}_{\rm B2+p-2}$= $- \boldsymbol{D}_{\rm B2+p-2} \nabla [B^{2+}(p^{-})_{2}]^{0\pm}$=$- (\boldsymbol{D}_{\rm B2+p-2}/k_{\rm B}T)[B^{2+}(p^{-})_{2}]^{0\pm}\nabla V_{\rm B2+p-2}$.

A5. DDGRT Equations for the univalent neutral salt molecule $(\boldsymbol{BA})^{0 }\equiv (\boldsymbol{B}^+\boldsymbol{A}^{-})^{0} \Leftrightarrow \boldsymbol{B}^{+}{ + \boldsymbol{A}}^{-}$

(A5) $\partial (BA)^{0}/\partial t$ =$- \boldsymbol{\nabla}\cdot \boldsymbol{f}_{\rm BA0}+ (g_{\rm BA}- r_{\rm BA}),$

(A5A) $\boldsymbol{f}_{\rm AB0} = -\boldsymbol{D}_{\rm BA} \nabla (BA)^{0}$ =$-(D_{\rm BA0}/k_{\rm B}T)(BA)^{0}\nabla V_{\rm BA0},$

(A5B1)$g_{\rm BA}- r_{\rm BA}$= $- [g_{\rm B+A-}(BA)^{0}-r_{\rm B+A-}B^{1+}A^{1-}]$.

A6. Poisson Equation, Number Conservation, Current Continuity-Kirchoff's Law

(A6) $\boldsymbol{\nabla}\cdot \varepsilon \boldsymbol{E}$ = $q ( p^{+}- p^{- }+ V^{+}- V^{- }+ B^{+} - A^{-}$ + $\text{charged-n_valent-impuritons), }$

(A7)$N_{\rm H2O}$ = $V^{+}(x, y, z, t) + V^{0}(x, y, z, t) + V^{-}(x, y, z, t)$+ $\text{ All Immobile Impuritons, }$

(A8A) $N_{\rm HA} = A^{-}(x, y, z, t) + A^{0}(x, y, z, t)$+ $\text{Mobile and Immobile Acidic Impuritons, }$

(A8B) $N_{\rm BOH}$ = $B^{+}(x, y, z, t) + B^{0}(x, y, z, t)$+ $\text{Mobile and Immobile Basic Impuritons, }$

(A8C) $N_{\rm BA}$ = $B^{+}(x, y, z, t) + (BA)^{0}(x, y, z, t) + A^{-}(x, y, z, t)$+ $\text{Mobile and Immobile Impuritons, }$

(A9) $\boldsymbol{\nabla} [(\partial /\partial t)(\varepsilon \boldsymbol{E})$ + $\boldsymbol{j}_{\rm P+} +\boldsymbol{j}_{\rm P-}$+ $\boldsymbol{j}_{\rm B+} +\boldsymbol{j}_{\rm A-}$+ $\text{charged impuritons}] = 0 = \textbf{Kirchoff's Current Law}$.

Water Equations at Thermodynamic Equilibrium
-The Xiada Water Equations at Thermodynamic Equilibrium.

A10. Water Equations for Pure Water at Thermodynamic Equilibrium

(A10) $q\partial p^{+}/\partial t = 0$ = $-\boldsymbol{\nabla}\cdot\boldsymbol{j}_{\rm p+}+ q(g_{\rm p+p-} - r_{\rm p+p-})_{\rm \sim 600meV}$+ $q(g_{\rm p+V0} - r_{\rm p+V0})_{\rm \sim 128, 88, 55meV} = 0 + 0+ 0.$

(A11A0) $\boldsymbol{j}_{\rm p+}$=$+ q\boldsymbol{\mu} _{\rm p+}\boldsymbol{E}p^{+ }-q\boldsymbol{D}_{\rm p+} \nabla p^{+} =-q \boldsymbol{\mu}_{\rm p+}p^{+}\nabla V_{\rm p+}= 0$,

(A11A1) $\therefore V_{\rm P+} = V_{\rm F} = E_{\rm F}/q$= $\text{Fermi Potential for all species = space-time constant, }$

(A11B1a) $g_{\rm p+p-}- r_{\rm p+p-}$= $g_{\rm p+p-0}V^{0}V^{0}- r_{\rm p+p-0}p^{+}p^{-} = 0,$

(A11B1b) $\therefore (g_{\rm p+p-0}{ V}^0{ V}^0-r_{\rm p+p-0}p^{+}p^{-}) = 0 \text{ or},$

(A11B1c) $p^{+}p^{-}$= $g_{\rm p+p-0}/r_{\rm p+p-0} { V}^0{ V}^0\\ = p_{\rm i}^{2}\ (\because { V}^0 \text{ is nearly constant}\approx N_{\text{H}_2\text{O}}\gg { V}^++{ V}^-.)\\$ $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$=$\text{Equilibrium Constant } K_{\rm p} = p_{\rm i0}^{2} \exp(-E_{\rm p+p-}/k_{\rm B}T)$,

(A11B1d) $p^{+} = P_{+} \exp[-(E_{\rm p+}-E_{\rm F})/k_{\rm B}T]$= $p_{\rm i} \exp[+(E_{\rm F}-E_{\rm I})/k_{\rm B}T]$,

(A11B1e) $p^{-} = P_{-} \exp[+(E_{\rm p-}-E_{\rm F})/k_{B}T]$= $p_{\rm i} \exp[+(E_{\rm I}-E_{\rm F})/k_{\rm B}T],$

(A11B2a) $g_{\rm p+V0}- r_{\rm p+V0}$= $e_{\rm p+V0}V^{+}- c_{\rm p+V0}p^{+}V^{0}= 0,$

(A11B2b) $\therefore e_{\rm p+V0}/c_{\rm p+V0}$ = $p^{+}(V^{+}/V^{0})\equiv p^{+} \exp[(E_{\rm F}-E_{1, 0})/k_{\rm B}T]\\$ $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$= $P_{+} \exp[-(E_{\rm p+ }-E_{1, 0})/k_{\rm B}T]$ =$p_{\rm i} \exp[(E_{1, 0}-E_{\rm I})/k_{\rm B}T],$

where $E_{\rm C}-E_{1, 0} =$ effective ionization energy of $V^{0}$ via capture of a $p^{+}$ converting it to $V^{+}$, described by the chemical reaction equation $V^{0 }+ p^{+}\leftrightarrow V^{+}+\hbar\omega_{\rm phonon}$.

(A11B2c) $E_{S+1, S} \buildrel {\rm def} \over = E_{\rm p+} + k_{\rm B}T {\rm log}_{\rm e}(Z_{\rm S}/Z_{\rm S+1}) \\$= $\text{Effective Ionization Energy for the transition from charge State } S \text{ to } S+1^{[12, \, 13]}, \\$

(A11B2d) $Z_{\rm S} = \text{States Sum (Zustandssumme)}^{[8, \, 12, \, 13] }=\sum _{i }g_{{\rm S}, i} \exp[-(E_{s.i}-E_{\rm p+})/k_{\rm B}T], \\$ $\;\;\;\;\;\;\;\;i = i\text{-th excited state}, g_{{\rm S}, i } = \text{space-spin degeneracy of } i\text{-th excited state. }$

(A11B2e) ${ V}^+:{ V}^0:{ V}^-$=$\exp[(E_{\rm V+}-E_{\rm F}/k_{\rm B}T]:1:\exp[(E_{\rm F}-E_{\rm V-}/k_{\rm B}T].$

(A20) $q\partial p^{-}/\partial t$ = $+ \boldsymbol{\nabla}\cdot \boldsymbol{j}_{p-}$+ $q(g_{\rm p+p-}- r_{\rm p+p-})_{\rm \sim 600meV}$ + $q(g_{\rm p-V0} - r_{\rm p-V0})_{\rm \sim 128, 88, 55, 117meV} = 0 + 0 + 0$.

(A22A0) $\boldsymbol{j}_{\rm p-}$= +$q \boldsymbol{\mu} _{\rm p-}\boldsymbol{E}p^{-}$ + $q\boldsymbol{D}_{\rm p-}\nabla p^{-} = -q\boldsymbol{\mu} _{\rm p-}p^{- } \nabla V_{\rm p-} = 0$,

(A22A1) $\therefore V_{\rm P-} = V_{\rm F} = E_{\rm F}/q$ = $\text{Fermi Potential for all species} = \text{space-time constant}$

(A22B1a) $g_{\rm p+p-}- r_{\rm p+p-}$ =$g_{\rm p+p-0} V^{0}V^{0}-r_{\rm p+p-}p^{+}p^{-} = 0 \quad\text{ Same as (11B1a).}$

(A22B2a) $g_{\rm p-V0} - r_{\rm p-V0 }$= $e_{\rm p-V0}V^{- }- c_{\rm p-V0}p^{-}V^{0} = 0 \quad$ $\text{Negative or 'anti' (11B2a). Therefore, }$

(A22B2b) $\therefore e_{\rm p-V0}/c_{\rm p-V0} = p^{-}(V^{-}/V^{0})\equiv p^{-} \exp[(E_{\rm F}-E_{0, -1})/k_{\rm B}T]\\$ $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= P_{-} \exp[+(E_{\rm p-}-E_{0, -1})/k_{\rm B}T]$ = $p_{\rm i} \exp[-(E_{0, -1}-E_{\rm I})/k_{\rm B}T]$.

(A26) $\boldsymbol{\nabla }\cdot\varepsilon \boldsymbol{E }= \rho$ = $q ( p^{+} - p^{- }+ V^{+} - V^{-}) \ne 0 \text{ (Any Band-Bending at Equilibrium, zero current.).}$

(A26A) $\therefore p^{+}(x, y, z)$ -$p^{-}(x, y, z) + V^{+}(x, y, z) - V^{-}(x, y, z)$ =$0 \text{ (Flatband, } \boldsymbol{E} =0, V_{\rm S}=0.).$

(A27) $N_{\text{H}_2\text{O}}(x, y, z, t) = V^{+}(x, y, z, t) + V^{0}(x, y, z, t) + V^{-}(x, y, z, t)$

(A27a) $= V^{+}(x, y, z) + V^{0}(x, y, z) + V^{-}(x, y, z) = \text{ constant (at any Band-Bending)}\\$ $\because V^{+}, V^{0} \text{ and } V^{-} \text{ or } O^{2-} \text{are immobile. Only } p^{+} \text{and } p^{-} \text{are mobile which are captured}\\$ $\text{ by } V^{0} \text{ to give } V^{+} \text{and } V^{-} \text{ to change the local concentration of } V^{0}, V^{+} \text{ and } V^{-}\\[1mm] \text{ but } V^{+} + V^{0} + V^{-} = N_{\text{H}_2\text{O}} = \text{constant}.$

(A27b) $= V^{+}+ V^{0} + V^{-} = \text{constant} + \text{constant} + \text{constant (at Flatband).}$

(A27c) ${\displaystyle ∰}$ $N_{\text{H}_2\text{O}}(x, y, z, t)\text{d}x\text{d}y\text{d}z= N_{\text{H}_2\text{O} \text{ in a Volume} }$=$\text{constant (Any Band-Bending). }$

A30. Additional Equations for Water with Univalent Acid Impurity HA at Equilibrium

(A30) $q\partial A^{-}/\partial t$= $+\boldsymbol{\nabla}\cdot \boldsymbol{j}_{\rm A-}$+ $q(g_{\rm H+A-}$ -$r_{\rm H+A-})+ q(g_{\rm V+A-}$ -$r_{\rm V+A-}) + q(g_{\rm p+A-} - r_{\rm p+A-}) = 0+0+0+0.$

(A30A0) $\boldsymbol{j}_{\rm A-}$= $+ q\boldsymbol{\mu }_{\rm A-}\boldsymbol{E}A^{- }\boldsymbol{+ }q\boldsymbol{D}_{\rm A-}\nabla A^{-}$=$\boldsymbol{- }q\boldsymbol{\mu }_{\rm A-}A^{-}\nabla V_{\rm A-}= 0.$

(A30A1) $\therefore V_{\rm A-} = V_{\rm F} = E_{\rm F}/q$=$\text{Fermi Potential for all species} = \text{space-time constant}.$

(A30B1a) $g_{\rm H+A-} - r_{\rm H+A- }$=$g_{\rm H+A-0}(H^{+}A^{-})^{0} -r_{\rm H+A-0}H^{+}A^{- }= 0$.

(A30B1b) $g_{\rm H+A-0}\boldsymbol{/}r_{\rm H+A-0} = H^{+}A^{-}/(H^{+}A^{-})^{0}$= $H^{+}\exp[-(E_{\rm F}-E_{\rm H+A-})/k_{\rm B}T] .$

(A30B1c) $\sim p^{+}\exp[-(E_{\rm F}-E_{\rm H+A-})/k_{\rm B}T]=$ $P_{+}\exp[-(E_{\rm p+}-E_{\rm H+A-})/k_{\rm B}T]\\$ $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \to 1 \text{ (completely ionized and absorbed into water lattice)}.$

(A30B2a) $g_{\rm V+A-} - r_{\rm V+A- }= e_{\rm V+A-}(V^{+}A^{-})^{0 }\boldsymbol{- }c_{\rm V+A-}V^{+}A^{- } = 0.$

(A30B2b) $\therefore e_{\rm V+A-}\boldsymbol{/}c_{\rm V+A-} = V^{+}A^{-}/(V^{+}A^{-})^{0}$= $V^{+}\exp[(E_{\rm F}-E_{\rm V+A-})/k_{\rm B}T]\\$ $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$=$V^{0}\exp[-(E_{\rm V+A-}E_{\rm V+})/k_{\rm B}T].$

(A30C1a) $g_{\rm p+A-} - r_{\rm p+A- }= e_{\rm p+A-}(p^{+}A^{-})^{0 }\boldsymbol{-}c_{\rm p+A-}p^{+}A^{- } = 0.$

(A30B1b) $\therefore e_{\rm p+A-}\boldsymbol{/}c_{\rm p+A-} = p^{+}A^{-}/(p^{+}A^{-})^{0} \equiv p^{+}\exp[-(E_{\rm F}-E_{\rm p+A-})/k_{\rm B}T]\\$ $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=P_{+}\exp[-(E_{\rm p+}- E_{\rm p+A-})/k_{\rm B}T) = p_{\rm i}\exp[(E_{\rm p+A-}-E_{I})/k_{\rm B}T].$

(A36) $\boldsymbol{ \nabla }\cdot\varepsilon \boldsymbol{E }$= $q ( p^{+} - p^{- }+ V^{+} - V^{-} - A^{-}) \ne 0$ $\text{ (Any Band-Bending at Equilibrium, zero current.)}.$

(A36A) $\therefore p^{+}(x, y, z) - p^{-}(x, y, z) + V^{+}(x, y, z) - V^{-}(x, y, z) - A^{-}(x, y, z)$ = $0\ (\text{Flatband}, \boldsymbol{E} =0, V_{\rm S}=0.)$

(A37) $N_{\text{H}_2\text{O}}(x, y, z, t) = V^{+}(x, y, z, t) + V^{0}(x, y, z, t) + V^{-}(x, y, z, t) + V^{+}A^{-}(x, y, z, t)$

(A37a) $=V^{+}(x, y, z) + V^{0}(x, y, z) + V^{-}(x, y, z) + V^{+}A^{-}\text{(flatband)}\\$

         $= \text{constant (Any Band-Bending.)}\\$

          $\because V^{+}, V^{0} \text{ and } V^{-} \text{ or } O^{2-} \text{are immobile. Only } p^{+} \text{and } p^{-} \text{are mobile which are}\\$

         $\text{captured by } V^{0} \text{ to give } V^{+} \text{ and } V^{-} \text{ to change the local concentration of } V^{0}, V^{+} \text{ and } V^{-} \text{ but}\\$

         $V^{+} + V^{0} + V^{-} = N_{\text{H}_2\text{O}} =\text{ constant.}$

(A37b)         $= V^{+}+ V^{0} + V^{-} = \text{constant } + \text{ constant } + \text{ constant (Flatband).}$

(A37c) ${\displaystyle ∰}$$N_{\text{H}_2\text{O}}(x, y, z, t)\text{d}x\text{d}y\text{d}z = N_{\text{H}_2\text{O}\text{-Total-Volume} }= \text{constant (Any Band-Bending).}$

(A38) $N_{\rm HA }(x, y, z, t) = A^{-}(x, y, z, t) + [H^{+}A^{-}]^{0}(x, y, z, t) + [p^{+}A^{-}]^{0}(x, y, z, t) + [V^{+}A^{-}]^{0}(x, y, z, t)$

(A38a)            $= A^{-}(x, y, z) + [H^{+}A^{-}]^{0}\text{ (flatband) }+ [p^{+}A^{-}]^{0}\text{ (flatband) }+ [V^{+}A^{-}]^{0}\text{ (flatband)}\\ $

            $\ne \text{constant at any Band-Bending because } A^{-} \text{ is charged by capturing } p^{+} \text{and } p^{-}$

            $\text{ which are mobile.}$

(A38b)        $= \text{constant} + \text{constant} + \text{constant} + \text{constant (at Flatband)}.$

${\displaystyle ∰}$$N_{\rm HA}(x, y, z, t){\rm{d}}x{\rm{d}}y{\rm{d}}z = N_{\textbf{HA in Total-Volume}}= \text{constant (Any Band-Bending).}$

A40. Additional Equations for Water with Univalent Base Impurity BOH at Equilibrium

Equations are similar to those given by equations (A3nx) for Univalent Acid Impurity HA.