DebyeHuckel.h

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/**
 *  @file DebyeHuckel.h
 *    Headers for the DebyeHuckel ThermoPhase object, which models dilute
 *    electrolyte solutions
 *    (see \ref thermoprops and \link Cantera::DebyeHuckel DebyeHuckel \endlink) .
 *
 * Class DebyeHuckel represents a dilute liquid electrolyte phase which
 * obeys the Debye Huckel formulation for nonideality.
 */

/*
 * Copyright (2006) Sandia Corporation. Under the terms of
 * Contract DE-AC04-94AL85000 with Sandia Corporation, the
 * U.S. Government retains certain rights in this software.
 */
#ifndef CT_DEBYEHUCKEL_H
#define CT_DEBYEHUCKEL_H

#include "MolalityVPSSTP.h"
#include "cantera/base/Array.h"

namespace Cantera
{

/*!
 * @name Formats for the Activity Coefficients
 *
 *   These are possible formats for the molality-based activity coefficients.
 */
//@{
#define DHFORM_DILUTE_LIMIT  0
#define DHFORM_BDOT_AK       1
#define DHFORM_BDOT_ACOMMON  2
#define DHFORM_BETAIJ        3
#define DHFORM_PITZER_BETAIJ 4
//@}
/*
 *  @name  Acceptable ways to calculate the value of A_Debye
 */
//@{
#define    A_DEBYE_CONST  0
#define    A_DEBYE_WATER  1
//@}

class WaterProps;
class PDSS_Water;

/**
 * @ingroup thermoprops
 *
 * Class DebyeHuckel represents a dilute liquid electrolyte phase which
 * obeys the Debye Huckel formulation for nonideality.
 *
 * The concentrations of the ionic species are assumed to obey the electroneutrality
 * condition.
 *
 * <HR>
 * <H2> Specification of Species Standard State Properties </H2>
 * <HR>
 *
 * The standard states are on the unit molality basis. Therefore, in the
 * documentation below, the normal \f$ o \f$ superscript is replaced with
 * the \f$ \triangle \f$ symbol. The reference state symbol is now
 *  \f$ \triangle, ref \f$.
 *
 *
 *  It is assumed that the reference state thermodynamics may be
 *  obtained by a pointer to a populated species thermodynamic property
 *  manager class (see ThermoPhase::m_spthermo). How to relate pressure
 *  changes to the reference state thermodynamics is resolved at this level.
 *
 *  For an incompressible,
 * stoichiometric substance, the molar internal energy is
 * independent of pressure. Since the thermodynamic properties
 * are specified by giving the standard-state enthalpy, the
 * term \f$ P_0 \hat v\f$ is subtracted from the specified molar
 * enthalpy to compute the molar internal energy. The entropy is
 * assumed to be independent of the pressure.
 *
 * The enthalpy function is given by the following relation.
 *
 *       \f[
 *         h^\triangle_k(T,P) = h^{\triangle,ref}_k(T)
 *         + \tilde v \left( P - P_{ref} \right)
 *       \f]
 *
 * For an incompressible,
 * stoichiometric substance, the molar internal energy is
 * independent of pressure. Since the thermodynamic properties
 * are specified by giving the standard-state enthalpy, the
 * term \f$ P_{ref} \tilde v\f$ is subtracted from the specified reference molar
 * enthalpy to compute the molar internal energy.
 *
 *       \f[
 *            u^\triangle_k(T,P) = h^{\triangle,ref}_k(T) - P_{ref} \tilde v
 *       \f]
 *
 * The standard state heat capacity and entropy are independent
 * of pressure. The standard state Gibbs free energy is obtained
 * from the enthalpy and entropy functions.
 *
 * The vector Phase::m_speciesSize[] is used to hold the
 * base values of species sizes. These are defined as the
 * molar volumes of species at infinite dilution at 300 K and 1 atm
 * of water. m_speciesSize are calculated during the initialization of the
 * DebyeHuckel object and are then not touched.
 *
 * The current model assumes that an incompressible molar volume for
 * all solutes. The molar volume for the water solvent, however,
 * is obtained from a pure water equation of state, waterSS.
 * Therefore, the water standard state varies with both T and P.
 * It is an error to request standard state water properties  at a T and P
 * where the water phase is not a stable phase, i.e., beyond its
 * spinodal curve.
 *
 * <HR>
 * <H2> Specification of Solution Thermodynamic Properties </H2>
 * <HR>
 *
 * Chemical potentials
 * of the solutes,  \f$ \mu_k \f$, and the solvent, \f$ \mu_o \f$, which are based
 * on the molality form, have the following general format:
 *
 * \f[
 *    \mu_k = \mu^{\triangle}_k(T,P) + R T ln(\gamma_k^{\triangle} \frac{m_k}{m^\triangle})
 * \f]
 * \f[
 *    \mu_o = \mu^o_o(T,P) + RT ln(a_o)
 * \f]
 *
 * where \f$ \gamma_k^{\triangle} \f$ is the molality based activity coefficient for species
 * \f$k\f$.
 *
 * Individual activity coefficients of ions can not be independently measured. Instead,
 * only binary pairs forming electroneutral solutions can be measured.

 *
 *  <H3> Ionic Strength </H3>
 *
 *  Most of the parameterizations within the model use the ionic strength
 *  as a key variable. The ionic strength, \f$ I\f$ is defined as follows
 *
 *  \f[
 *    I = \frac{1}{2} \sum_k{m_k  z_k^2}
 *  \f]
 *
 * \f$ m_k \f$ is the molality of the kth species. \f$ z_k \f$ is the charge
 * of the kth species. Note, the ionic strength is a defined units quantity.
 * The molality has defined units of gmol kg-1, and therefore the ionic
 * strength has units of sqrt( gmol kg-1).
 *
 * In some instances, from some authors, a different
 * formulation is used for the ionic strength in the equations below. The different
 * formulation is due to the possibility of the existence of weak acids and how
 * association wrt to the weak acid equilibrium relation affects the calculation
 * of the activity coefficients via the assumed value of the ionic strength.
 *
 * If we are to assume that the association reaction doesn't have an effect
 * on the ionic strength, then we will want to consider the associated weak
 * acid as in effect being fully dissociated, when we calculate an effective
 * value for the ionic strength. We will call this calculated value, the
 * stoichiometric ionic strength, \f$ I_s \f$, putting a subscript s to denote
 * it from the more straightforward calculation of \f$ I \f$.
 *
 *  \f[
 *    I_s = \frac{1}{2} \sum_k{m_k^s  z_k^2}
 *  \f]
 *
 *  Here, \f$ m_k^s \f$ is the value of the molalities calculated assuming that
 *  all weak acid-base pairs are in their fully dissociated states. This calculation may
 * be simplified by considering that the weakly associated acid may be made up of two
 * charged species, k1 and k2, each with their own charges, obeying the following relationship:
 *
 *   \f[
 *      z_k = z_{k1} +  z_{k2}
 *   \f]
 *  Then, we may only need to specify one charge value, say, \f$  z_{k1}\f$,
 *  the cation charge number,
 *  in order to get both numbers, since we have already specified \f$ z_k \f$
 *  in the definition of original species.
 *  Then, the stoichiometric ionic strength may be calculated via the following formula.
 *
 *  \f[
 *    I_s = \frac{1}{2} \left(\sum_{k,ions}{m_k  z_k^2}+
 *               \sum_{k,weak_assoc}(m_k  z_{k1}^2 + m_k  z_{k2}^2) \right)
 *  \f]
 *
 *  The specification of which species are weakly associated acids is made in the input
 *  file via the
 *  <TT> stoichIsMods </TT> XML block, where the charge for k1 is also specified.
 *  An example is given below:
 *
 * @code
 *          <stoichIsMods>
 *                NaCl(aq):-1.0
 *          </stoichIsMods>
 * @endcode
 *
 *  Because we need the concept of a weakly associated acid in order to calculated
 *  \f$ I_s \f$ we need to
 *  catalog all species in the phase. This is done using the following categories:
 *
 *  -  <B>cEST_solvent</B>                Solvent species (neutral)
 *  -  <B>cEST_chargedSpecies</B>         Charged species (charged)
 *  -  <B>cEST_weakAcidAssociated</B>     Species which can break apart into charged species.
 *                                        It may or may not be charged.  These may or
 *                                        may not be be included in the
 *                                        species solution vector.
 *  -  <B>cEST_strongAcidAssociated</B>   Species which always breaks apart into charged species.
 *                                        It may or may not be charged. Normally, these aren't included
 *                                        in the speciation vector.
 *  -  <B>cEST_polarNeutral </B>          Polar neutral species
 *  -  <B>cEST_nonpolarNeutral</B>        Non polar neutral species
 *
 *  Polar and non-polar neutral species are differentiated, because some additions
 *  to the activity
 *  coefficient expressions distinguish between these two types of solutes. This is the so-called
 *  salt-out effect.
 *
 * The type of species is specified in the <TT>electrolyteSpeciesType</TT> XML block.
 * Note, this is not
 * considered a part of the specification of the standard state for the species,
 * at this time. Therefore,
 * this information is put under the <TT>activityCoefficient</TT> XML block. An example
 * is given below
 *
 * @code
 *         <electrolyteSpeciesType>
 *                H2L(L):solvent
 *                H+:chargedSpecies
 *                NaOH(aq):weakAcidAssociated
 *                NaCl(aq):strongAcidAssociated
 *                NH3(aq):polarNeutral
 *                O2(aq):nonpolarNeutral
 *         </electrolyteSpeciesType>
 * @endcode
 *
 *  Much of the species electrolyte type information is inferred from other information in the
 *  input file. For example, as species which is charged is given the "chargedSpecies" default
 *  category. A neutral solute species is put into the "nonpolarNeutral" category by default.
 *
 * The specification of solute activity coefficients depends on the model
 * assumed for the Debye-Huckel term. The model is set by the
 * internal parameter #m_formDH. We will now describe each category in its own section.
 *
 *
 *  <H3> Debye-Huckel Dilute Limit </H3>
 *
 *  DHFORM_DILUTE_LIMIT = 0
 *
 *  This form assumes a dilute limit to DH, and is mainly for informational purposes:
 *  \f[
 *      \ln(\gamma_k^\triangle) = - z_k^2 A_{Debye} \sqrt{I}
 *  \f]
 *              where \f$ I\f$ is the ionic strength
 *  \f[
 *    I = \frac{1}{2} \sum_k{m_k  z_k^2}
 *  \f]
 *
 *  The activity for the solvent water,\f$ a_o \f$, is not independent and must be
 *  determined from the Gibbs-Duhem relation.
 *
 *  \f[
 *       \ln(a_o) = \frac{X_o - 1.0}{X_o} + \frac{ 2 A_{Debye} \tilde{M}_o}{3} (I)^{3/2}
 *  \f]
 *
 *
 *  <H3> Bdot Formulation </H3>
 *
 *    DHFORM_BDOT_AK       = 1
 *
 *      This form assumes Bethke's format for the Debye Huckel activity coefficient:
 *
 *   \f[
 *      \ln(\gamma_k^\triangle) = -z_k^2 \frac{A_{Debye} \sqrt{I}}{ 1 + B_{Debye}  a_k \sqrt{I}}
 *                        + \log(10) B^{dot}_k  I
 *   \f]
 *
 *  Note, this particular form where \f$ a_k \f$ can differ in multielectrolyte
 *  solutions has problems with respect to a Gibbs-Duhem analysis. However,
 *  we include it here because there is a lot of data fit to it.
 *
 *  The activity for the solvent water,\f$ a_o \f$, is not independent and must be
 *  determined from the Gibbs-Duhem relation. Here, we use:
 *
 *  \f[
 *       \ln(a_o) = \frac{X_o - 1.0}{X_o}
 *        + \frac{ 2 A_{Debye} \tilde{M}_o}{3} (I)^{1/2}
 *                        \left[ \sum_k{\frac{1}{2} m_k z_k^2 \sigma( B_{Debye} a_k \sqrt{I} ) } \right]
 *                        - \frac{\log(10)}{2} \tilde{M}_o I \sum_k{ B^{dot}_k m_k}
 *  \f]
 *    where
 *  \f[
 *     \sigma (y) = \frac{3}{y^3} \left[ (1+y) - 2 \ln(1 + y) - \frac{1}{1+y} \right]
 *  \f]
 *
 * Additionally, Helgeson's formulation for the water activity is offered as an
 * alternative.
 *
 *
 *  <H3> Bdot Formulation with Uniform Size Parameter in the Denominator </H3>
 *
 *  DHFORM_BDOT_AUNIFORM = 2
 *
 *  This form assumes Bethke's format for the Debye-Huckel activity coefficient
 *
 *   \f[
 *    \ln(\gamma_k^\triangle) = -z_k^2 \frac{A_{Debye} \sqrt{I}}{ 1 + B_{Debye}  a \sqrt{I}}
 *                        + \log(10) B^{dot}_k  I
 *   \f]
 *
 *  The value of a is determined at the beginning of the calculation, and not changed.
 *
 *  \f[
 *       \ln(a_o) = \frac{X_o - 1.0}{X_o}
 *        + \frac{ 2 A_{Debye} \tilde{M}_o}{3} (I)^{3/2} \sigma( B_{Debye} a \sqrt{I} )
 *                        - \frac{\log(10)}{2} \tilde{M}_o I \sum_k{ B^{dot}_k m_k}
 *  \f]
 *
 *
 *  <H3> Beta_IJ formulation </H3>
 *
 *  DHFORM_BETAIJ        = 3
 *
 *  This form assumes a linear expansion in a virial coefficient form.
 *  It is used extensively in the book by Newmann, "Electrochemistry Systems",
 *  and is the beginning of more complex treatments for stronger electrolytes,
 *  fom Pitzer and from Harvey, Moller, and Weire.
 *
 *   \f[
 *    \ln(\gamma_k^\triangle) = -z_k^2 \frac{A_{Debye} \sqrt{I}}{ 1 + B_{Debye}  a \sqrt{I}}
 *                         + 2 \sum_j \beta_{j,k} m_j
 *   \f]
 *
 *  In the current treatment the binary interaction coefficients, \f$ \beta_{j,k}\f$, are
 *  independent of temperature and pressure.
 *
 *  \f[
 *       \ln(a_o) = \frac{X_o - 1.0}{X_o}
 *        + \frac{ 2 A_{Debye} \tilde{M}_o}{3} (I)^{3/2} \sigma( B_{Debye} a \sqrt{I} )
 *        -  \tilde{M}_o  \sum_j \sum_k \beta_{j,k} m_j m_k
 *  \f]
 *
 * In this formulation the ionic radius, \f$ a \f$, is a constant. This must be supplied to the
 * model, in an <DFN> ionicRadius </DFN> XML block.
 *
 * The \f$ \beta_{j,k} \f$ parameters are binary interaction parameters. They are supplied to
 * the object in an <TT> DHBetaMatrix </TT> XML block. There are in principle \f$ N (N-1) /2 \f$
 * different, symmetric interaction parameters, where \f$ N \f$ are the number of solute species in the
 * mechanism.
 * An example is given below.
 *
 * An example <TT> activityCoefficients </TT> XML block for this formulation is supplied below
 *
 * @code
 *  <activityCoefficients model="Beta_ij">
 *         <!-- A_Debye units = sqrt(kg/gmol) -->
 *         <A_Debye> 1.172576 </A_Debye>
 *         <!-- B_Debye units = sqrt(kg/gmol)/m   -->
 *         <B_Debye> 3.28640E9 </B_Debye>
 *         <ionicRadius default="3.042843"  units="Angstroms">
 *         </ionicRadius>
 *         <DHBetaMatrix>
 *               H+:Cl-:0.27
 *               Na+:Cl-:0.15
 *               Na+:OH-:0.06
 *         </DHBetaMatrix>
 *         <stoichIsMods>
 *                NaCl(aq):-1.0
 *         </stoichIsMods>
 *         <electrolyteSpeciesType>
 *                H+:chargedSpecies
 *                NaCl(aq):weakAcidAssociated
 *         </electrolyteSpeciesType>
 *  </activityCoefficients>
 * @endcode
 *
 *  <H3> Pitzer Beta_IJ formulation </H3>
 *
 *  DHFORM_PITZER_BETAIJ  = 4
 *
 *  This form assumes an activity coefficient formulation consistent
 *  with a truncated form of Pitzer's formulation. Pitzer's formulation is equivalent
 *  to the formulations above in the dilute limit, where rigorous theory may be applied.
 *
 *   \f[
 *     \ln(\gamma_k^\triangle) = -z_k^2 \frac{A_{Debye}}{3} \frac{\sqrt{I}}{ 1 + B_{Debye}  a \sqrt{I}}
 *       -2 z_k^2 \frac{A_{Debye}}{3}  \frac{\ln(1 + B_{Debye}  a  \sqrt{I})}{ B_{Debye}  a}
 *                         + 2 \sum_j \beta_{j,k} m_j
 *   \f]
 *
 *
 *  \f[
 *       \ln(a_o) = \frac{X_o - 1.0}{X_o}
 *        + \frac{ 2 A_{Debye} \tilde{M}_o}{3} \frac{(I)^{3/2} }{1 +  B_{Debye}  a \sqrt{I} }
 *        -  \tilde{M}_o  \sum_j \sum_k \beta_{j,k} m_j m_k
 *  \f]
 *
 * <H3> Specification of the Debye Huckel Constants </H3>
 *
 *  In the equations above, the formulas for  \f$  A_{Debye} \f$ and \f$  B_{Debye} \f$
 *  are needed. The DebyeHuckel object uses two methods for specifying these quantities.
 *  The default method is to assume that \f$  A_{Debye} \f$  is a constant, given
 *  in the initialization process, and stored in the
 *  member double, m_A_Debye. Optionally, a full water treatment may be employed that makes
 *  \f$ A_{Debye} \f$ a full function of <I>T</I> and <I>P</I>.
 *
 *   \f[
 *      A_{Debye} = \frac{F e B_{Debye}}{8 \pi \epsilon R T} {\left( C_o \tilde{M}_o \right)}^{1/2}
 *   \f]
 * where
 *
 *  \f[
 *         B_{Debye} = \frac{F} {{(\frac{\epsilon R T}{2})}^{1/2}}
 *  \f]
 *  Therefore:
 * \f[
 *   A_{Debye} = \frac{1}{8 \pi}
 *                 {\left(\frac{2 N_a \rho_o}{1000}\right)}^{1/2}
 *                 {\left(\frac{N_a e^2}{\epsilon R T }\right)}^{3/2}
 * \f]
 *
 * where
 *   - \f$ N_a \f$ is Avogadro's number
 *   - \f$ \rho_w \f$ is the density of water
 *   - \f$ e \f$ is the electronic charge
 *   - \f$ \epsilon = K \epsilon_o \f$ is the permittivity of water
 *   - \f$ K \f$ is the dielectric constant of water
 *   - \f$ \epsilon_o \f$ is the permittivity of free space
 *   - \f$ \rho_o \f$ is the density of the solvent in its standard state.
 *
 * Nominal value at 298 K and 1 atm = 1.172576 (kg/gmol)<SUP>1/2</SUP> based on:
 *   - \f$ \epsilon / \epsilon_0 \f$ = 78.54 (water at 25C)
 *   - T = 298.15 K
 *   - B_Debye = 3.28640E9 (kg/gmol)<SUP>1/2</SUP> m<SUP>-1</SUP>
 *
 * An example of a fixed value implementation is given below.
 * @code
 *   <activityCoefficients model="Beta_ij">
 *         <!-- A_Debye units = sqrt(kg/gmol)  -->
 *         <A_Debye> 1.172576 </A_Debye>
 *         <!-- B_Debye units = sqrt(kg/gmol)/m  -->
 *         <B_Debye> 3.28640E9 </B_Debye>
 *   </activityCoefficients>
 * @endcode
 *
 * An example of a variable value implementation is given below.
 *
 * @code
 *   <activityCoefficients model="Beta_ij">
 *         <A_Debye model="water" />
 *         <!-- B_Debye units = sqrt(kg/gmol)/m  -->
 *         <B_Debye> 3.28640E9 </B_Debye>
 *   </activityCoefficients>
 * @endcode
 *
 *  Currently, \f$  B_{Debye} \f$ is a constant in the model, specified either by a default
 *  water value, or through the input file. This may have to be looked at, in the future.
 *
 * <HR>
 * <H2> %Application within Kinetics Managers </H2>
 * <HR>
 *
 * For the time being, we have set the standard concentration for all species in
 * this phase equal to the default concentration of the solvent at 298 K and 1 atm.
 * This means that the
 * kinetics operator essentially works on an activities basis, with units specified
 * as if it were on a concentration basis.
 *
 * For example, a bulk-phase binary reaction between liquid species j and k, producing
 * a new liquid species l would have the
 * following equation for its rate of progress variable, \f$ R^1 \f$, which has
 * units of kmol m-3 s-1.
 *
 *   \f[
 *    R^1 = k^1 C_j^a C_k^a =  k^1 (C_o a_j) (C_o a_k)
 *   \f]
 * where
 *   \f[
 *      C_j^a = C_o a_j \quad and \quad C_k^a = C_o a_k
 *   \f]
 *
 *  \f$ C_j^a \f$ is the activity concentration of species j, and
 *  \f$ C_k^a \f$ is the activity concentration of species k. \f$ C_o \f$
 *  is the concentration of water at 298 K and 1 atm. \f$ a_j \f$ is
 *  the activity of species j at the current temperature and pressure
 *  and concentration of the liquid phase. \f$k^1 \f$ has units of m3 kmol-1 s-1.
 *
 *  The reverse rate constant can then be obtained from the law of microscopic reversibility
 * and the equilibrium expression for the system.
 *
 *   \f[
 *         \frac{a_j a_k}{ a_l} = K^{o,1} = \exp(\frac{\mu^o_l - \mu^o_j - \mu^o_k}{R T} )
 *   \f]
 *
 *  \f$  K^{o,1} \f$ is the dimensionless form of the equilibrium constant.
 *
 *   \f[
 *    R^{-1} = k^{-1} C_l^a =  k^{-1} (C_o a_l)
 *   \f]
 *
 *  where
 *
 *    \f[
 *       k^{-1} =  k^1 K^{o,1} C_o
 *   \f]
 *
 *  \f$k^{-1} \f$ has units of s-1.
 *
 *  Note, this treatment may be modified in the future, as events dictate.
 *
 * <HR>
 * <H2> Instantiation of the Class </H2>
 * <HR>
 *
 * The constructor for this phase is NOT located in the default ThermoFactory
 * for %Cantera. However, a new DebyeHuckel object may be created by
 * the following code snippets:
 *
 * @code
 *      DebyeHuckel *DH = new DebyeHuckel("DH_NaCl.xml", "NaCl_electrolyte");
 * @endcode
 *
 * or
 *
 * @code
 *    XML_Node *xm = get_XML_NameID("phase", "DH_NaCl.xml#NaCl_electrolyte", 0);
 *    DebyeHuckel *dh = new DebyeHuckel(*xm);
 * @endcode
 *
 * or by the following call to importPhase():
 *
 * @code
 *    XML_Node *xm = get_XML_NameID("phase", "DH_NaCl.xml#NaCl_electrolyte", 0);
 *    DebyeHuckel dhphase;
 *    importPhase(*xm, &dhphase);
 * @endcode
 *
 * <HR>
 * <H2> XML Example </H2>
 * <HR>
 *
 * The phase model name for this is called StoichSubstance. It must be supplied
 * as the model attribute of the thermo XML element entry.
 * Within the phase XML block,
 * the density of the phase must be specified. An example of an XML file
 * this phase is given below.
 *
 * @verbatim
 <phase id="NaCl_electrolyte" dim="3">
  <speciesArray datasrc="#species_waterSolution">
             H2O(L) Na+ Cl- H+ OH- NaCl(aq) NaOH(aq)
  </speciesArray>
  <state>
    <temperature units="K"> 300  </temperature>
    <pressure units="Pa">101325.0</pressure>
    <soluteMolalities>
           Na+:3.0
           Cl-:3.0
           H+:1.0499E-8
           OH-:1.3765E-6
           NaCl(aq):0.98492
           NaOH(aq):3.8836E-6
    </soluteMolalities>
  </state>
  <!-- thermo model identifies the inherited class
       from ThermoPhase that will handle the thermodynamics.
    -->
  <thermo model="DebyeHuckel">
     <standardConc model="solvent_volume" />
     <activityCoefficients model="Beta_ij">
              <!-- A_Debye units = sqrt(kg/gmol)  -->
              <A_Debye> 1.172576 </A_Debye>
              <!-- B_Debye units = sqrt(kg/gmol)/m   -->
              <B_Debye> 3.28640E9 </B_Debye>
              <ionicRadius default="3.042843"  units="Angstroms">
              </ionicRadius>
              <DHBetaMatrix>
                H+:Cl-:0.27
                Na+:Cl-:0.15
                Na+:OH-:0.06
              </DHBetaMatrix>
              <stoichIsMods>
                 NaCl(aq):-1.0
              </stoichIsMods>
              <electrolyteSpeciesType>
                 H+:chargedSpecies
                 NaCl(aq):weakAcidAssociated
              </electrolyteSpeciesType>
     </activityCoefficients>
     <solvent> H2O(L) </solvent>
  </thermo>
  <elementArray datasrc="elements.xml"> O H Na Cl </elementArray>
</phase>
@endverbatim
 *
 *
 */
class DebyeHuckel : public MolalityVPSSTP
{
public:
    //! Default Constructor
    DebyeHuckel();

    //! Copy constructor
    DebyeHuckel(const DebyeHuckel&);

    //! Assignment operator
    DebyeHuckel& operator=(const DebyeHuckel&);

    //! Full constructor for creating the phase.
    /*!
     *  @param inputFile  File name containing the XML description of the phase
     *  @param id         id attribute containing the name of the phase.
     */
    DebyeHuckel(const std::string& inputFile, const std::string& id = "");

    //! Full constructor for creating the phase.
    /*!
     *  @param phaseRef XML phase node containing the description of the phase
     *  @param id       id attribute containing the name of the phase.
     */
    DebyeHuckel(XML_Node& phaseRef, const std::string& id = "");

    /// Destructor.
    virtual ~DebyeHuckel();

    //! Duplicator from the ThermoPhase parent class
    /*!
     * Given a pointer to a ThermoPhase object, this function will
     * duplicate the ThermoPhase object and all underlying structures.
     * This is basically a wrapper around the copy constructor.
     *
     * @return returns a pointer to a ThermoPhase
     */
    ThermoPhase* duplMyselfAsThermoPhase() const;

    //! @name  Utilities
    //! @{

    /**
     * Equation of state type flag. The base class returns
     * zero. Subclasses should define this to return a unique
     * non-zero value. Constants defined for this purpose are
     * listed in mix_defs.h.
     */
    virtual int eosType() const;

    //! @}
    //! @name  Molar Thermodynamic Properties of the Solution
    //! @{

    /// Molar enthalpy of the solution. Units: J/kmol.
    virtual doublereal enthalpy_mole() const;

    /// Molar entropy. Units: J/kmol/K.
    /**
     * For an ideal, constant partial molar volume solution mixture with
     * pure species phases which exhibit zero volume expansivity:
     * \f[
     * \hat s(T, P, X_k) = \sum_k X_k \hat s^0_k(T)
     *      - \hat R  \sum_k X_k log(X_k)
     * \f]
     * The reference-state pure-species entropies
     * \f$ \hat s^0_k(T,p_{ref}) \f$ are computed by the
     *  species thermodynamic
     * property manager. The pure species entropies are independent of
     * temperature since the volume expansivities are equal to zero.
     * @see SpeciesThermo
     */
    virtual doublereal entropy_mole() const;

    /// Molar Gibbs function. Units: J/kmol.
    virtual doublereal gibbs_mole() const;

    /// Molar heat capacity at constant pressure. Units: J/kmol/K.
    virtual doublereal cp_mole() const;

    //! Molar heat capacity at constant volume. Units: J/kmol/K.
    /*
     *      (HKM -> Bump up to Parent object)
     */
    virtual doublereal cv_mole() const;

    //@}
    /** @name Mechanical Equation of State Properties
     //@{
     *
     *   In this equation of state implementation, the density is a
     *   function only of the mole fractions. Therefore, it can't be
     *   an independent variable. Instead, the pressure is used as the
     *   independent variable. Functions which try to set the thermodynamic
     *   state by calling setDensity() may cause an exception to be
     *   thrown.
     */

    //! Return the thermodynamic pressure (Pa).
    /*!
     * For this incompressible system, we return the internally stored
     * independent value of the pressure.
     */
    virtual doublereal pressure() const;

    //! Set the internally stored pressure (Pa) at constant
    //! temperature and composition
    /*!
     *  This method sets the pressure within the object.
     *  The water model is a completely compressible model.
     *  Also, the dielectric constant is pressure dependent.
     *
     *  @param p input Pressure (Pa)
     *
     * @todo Implement a variable pressure capability
     */
    virtual void setPressure(doublereal p);

protected:
    //! Calculate the density of the mixture using the partial
    //! molar volumes and mole fractions as input
    /*!
     * The formula for this is
     *
     * \f[
     * \rho = \frac{\sum_k{X_k W_k}}{\sum_k{X_k V_k}}
     * \f]
     *
     * where \f$X_k\f$ are the mole fractions, \f$W_k\f$ are
     * the molecular weights, and \f$V_k\f$ are the pure species
     * molar volumes.
     *
     * Note, the basis behind this formula is that in an ideal
     * solution the partial molar volumes are equal to the pure
     * species molar volumes. We have additionally specified
     * in this class that the pure species molar volumes are
     * independent of temperature and pressure.
     */
    virtual void calcDensity();

public:
    //! Set the internally stored density (gm/m^3) of the phase.
    /*!
     * Overwritten setDensity() function is necessary because the
     * density is not an independent variable.
     *
     * This function will now throw an error condition
     *
     * @internal May have to adjust the strategy here to make
     * the eos for these materials slightly compressible, in order
     * to create a condition where the density is a function of
     * the pressure.
     *
     * This function will now throw an error condition if the
     * input isn't exactly equal to the current density.
     *
     * @todo Now have a compressible ss equation for liquid water.
     *       Therefore, this phase is compressible. May still
     *       want to change the independent variable however.
     *
     * @param rho Input density (kg/m^3).
     */
    void setDensity(const doublereal rho);

    //! Set the internally stored molar density (kmol/m^3) of the phase.
    /**
     * Overwritten setMolarDensity() function is necessary because the
     * density is not an independent variable.
     *
     * This function will now throw an error condition if the input
     * isn't exactly equal to the current molar density.
     *
     * @param conc   Input molar density (kmol/m^3).
     */
    virtual void setMolarDensity(const doublereal conc);

    //! Set the temperature (K)
    /*!
     * This function sets the temperature, and makes sure that
     * the value propagates to underlying objects, such as
     * the water standard state model.
     *
     * @param temp Temperature in kelvin
     */
    virtual void setTemperature(const doublereal temp);

    //! Set the temperature (K) and pressure (Pa)
    /*!
     *  Set the temperature and pressure.
     *
     * @param t    Temperature (K)
     * @param p    Pressure (Pa)
     */
    virtual void setState_TP(doublereal t, doublereal p);

    /**
     * @}
     * @name Activities, Standard States,  and Activity Concentrations
     *
     * The activity \f$a_k\f$ of a species in solution is
     * related to the chemical potential by \f[ \mu_k = \mu_k^0(T)
     * + \hat R T \log a_k. \f] The quantity \f$\mu_k^0(T,P)\f$ is
     * the chemical potential at unit activity, which depends only
     * on temperature and the pressure.
     * Activity is assumed to be molality-based here.
     * @{
     */

    //! This method returns an array of generalized concentrations
    /*!
     * \f$ C_k\f$ that are defined such that
     * \f$ a_k = C_k / C^0_k, \f$ where \f$ C^0_k \f$
     * is a standard concentration
     * defined below.  These generalized concentrations are used
     * by kinetics manager classes to compute the forward and
     * reverse rates of elementary reactions.
     *
     * @param c Array of generalized concentrations. The
     *          units depend upon the implementation of the
     *          reaction rate expressions within the phase.
     */
    virtual void getActivityConcentrations(doublereal* c) const;

    //! Return the standard concentration for the kth species
    /*!
     * The standard concentration \f$ C^0_k \f$ used to normalize
     * the activity (i.e., generalized) concentration in
     * kinetics calculations.
     *
     * For the time being, we will use the concentration of pure
     * solvent for the the standard concentration of all species.
     * This has the effect of making reaction rates
     * based on the molality of species proportional to the
     * molality of the species.
     *
     * @param k Optional parameter indicating the species. The default
     *         is to assume this refers to species 0.
     * @return
     *   Returns the standard Concentration in units of
     *   m<SUP>3</SUP> kmol<SUP>-1</SUP>.
     */
    virtual doublereal standardConcentration(size_t k=0) const;

    //! Returns the units of the standard and generalized concentrations.
    /*!
     * Note they have the same units, as their
     * ratio is defined to be equal to the activity of the kth
     * species in the solution, which is unitless.
     *
     * This routine is used in print out applications where the
     * units are needed. Usually, MKS units are assumed throughout
     * the program and in the XML input files.
     *
     * The base ThermoPhase class assigns the default quantities
     * of (kmol/m3) for all species.
     * Inherited classes are responsible for overriding the default
     * values if necessary.
     *
     * On return uA contains the powers of the units (MKS assumed)
     * of the standard concentrations and generalized concentrations
     * for the kth species.
     *
     * @param uA Output vector containing the units
     *  uA[0] = kmol units - default  = 1
     *  uA[1] = m    units - default  = -nDim(), the number of spatial
     *                                dimensions in the Phase class.
     *  uA[2] = kg   units - default  = 0;
     *  uA[3] = Pa(pressure) units - default = 0;
     *  uA[4] = Temperature units - default = 0;
     *  uA[5] = time units - default = 0
     * @param k species index. Defaults to 0.
     * @param sizeUA output int containing the size of the vector.
     *        Currently, this is equal to 6.
     * @deprecated To be removed after Cantera 2.2.
     */
    virtual void getUnitsStandardConc(double* uA, int k = 0,
                                      int sizeUA = 6) const;

    //! Get the array of non-dimensional activities at
    //! the current solution temperature, pressure, and solution concentration.
    /*!
     *
     * We resolve this function at this level by calling
     * on the activityConcentration function. However,
     * derived classes may want to override this default
     * implementation.
     *
     * (note solvent activity coefficient is on molar scale).
     *
     * @param ac  Output vector of activities. Length: m_kk.
     */
    virtual void getActivities(doublereal* ac) const;

    //! Get the array of non-dimensional molality-based
    //! activity coefficients at
    //! the current solution temperature, pressure, and solution concentration.
    /*!
     *  note solvent is on molar scale. The solvent molar
     *  based activity coefficient is returned.
     *
     *  Note, most of the work is done in an internal private routine
     *
     * @param acMolality Vector of Molality-based activity coefficients
     *                   Length: m_kk
     */
    virtual void
    getMolalityActivityCoefficients(doublereal* acMolality) const;

    //@}
    /// @name  Partial Molar Properties of the Solution
    //@{

    //! Get the species chemical potentials. Units: J/kmol.
    /*!
     *
     * This function returns a vector of chemical potentials of the
     * species in solution.
     *
     * \f[
     *    \mu_k = \mu^{\triangle}_k(T,P) + R T ln(\gamma_k^{\triangle} m_k)
     * \f]
     *
     * @param mu  Output vector of species chemical
     *            potentials. Length: m_kk. Units: J/kmol
     */
    virtual void getChemPotentials(doublereal* mu) const;

    //! Returns an array of partial molar enthalpies for the species
    //! in the mixture. Units (J/kmol)
    /*!
     * For this phase, the partial molar enthalpies are equal to the
     * standard state enthalpies modified by the derivative of the
     * molality-based activity coefficient wrt temperature
     *
     *  \f[
     * \bar h_k(T,P) = h^{\triangle}_k(T,P) - R T^2 \frac{d \ln(\gamma_k^\triangle)}{dT}
     * \f]
     * The solvent partial molar enthalpy is equal to
     *  \f[
     * \bar h_o(T,P) = h^{o}_o(T,P) - R T^2 \frac{d \ln(a_o}{dT}
     * \f]
     *
     * The temperature dependence of the activity coefficients currently
     * only occurs through the temperature dependence of the Debye constant.
     *
     * @param hbar    Output vector of species partial molar enthalpies.
     *                Length: m_kk. units are J/kmol.
     */
    virtual void getPartialMolarEnthalpies(doublereal* hbar) const;

    //! Returns an array of partial molar entropies of the species in the
    //! solution. Units: J/kmol/K.
    /**
     * Maxwell's equations provide an insight in how to calculate this
     *   (p.215 Smith and Van Ness)
     *   \f[
     *      \frac{d\mu_i}{dT} = -\bar{s}_i
     *   \f]
     *
     * For this phase, the partial molar entropies are equal to the
     * SS species entropies plus the ideal solution contribution.following
     * contribution:
     *  \f[
     *     \bar s_k(T,P) =  \hat s^0_k(T) - R log(M0 * molality[k])
     * \f]
     * \f[
     *      \bar s_{solvent}(T,P) =  \hat s^0_{solvent}(T)
     *                  - R ((xmolSolvent - 1.0) / xmolSolvent)
     * \f]
     *
     * The reference-state pure-species entropies,\f$ \hat s^0_k(T) \f$,
     * at the reference pressure, \f$ P_{ref} \f$,  are computed by the
     * species thermodynamic
     * property manager. They are polynomial functions of temperature.
     * @see SpeciesThermo
     *
     *  @param sbar    Output vector of species partial molar entropies.
     *                 Length = m_kk. units are J/kmol/K.
     */
    virtual void getPartialMolarEntropies(doublereal* sbar) const;

    //! Return an array of partial molar heat capacities for the
    //! species in the mixture.  Units: J/kmol/K
    /*!
     * @param cpbar   Output vector of species partial molar heat
     *                capacities at constant pressure.
     *                Length = m_kk. units are J/kmol/K.
     */
    virtual void getPartialMolarCp(doublereal* cpbar) const;

    //! Return an array of partial molar volumes for the
    //! species in the mixture. Units: m^3/kmol.
    /*!
     * For this solution, the partial molar volumes are normally
     *  equal to theconstant species molar volumes, except
     * when the activity coefficients depend on pressure.
     *
     * The general relation is
     *
     *       vbar_i = d(chemPot_i)/dP at const T, n
     *              = V0_i + d(Gex)/dP)_T,M
     *              = V0_i + RT d(lnActCoeffi)dP _T,M
     *
     *  @param vbar   Output vector of species partial molar volumes.
     *                Length = m_kk. units are m^3/kmol.
     */
    virtual void getPartialMolarVolumes(doublereal* vbar) const;

    //@}

    /**
     * @name Chemical Equilibrium
     * @{
     */

    //!This method is used by the ChemEquil equilibrium solver.
    /*!
     * It sets the state such that the chemical potentials satisfy
     * \f[ \frac{\mu_k}{\hat R T} = \sum_m A_{k,m}
     * \left(\frac{\lambda_m} {\hat R T}\right) \f] where
     * \f$ \lambda_m \f$ is the element potential of element m. The
     * temperature is unchanged.  Any phase (ideal or not) that
     * implements this method can be equilibrated by ChemEquil.
     *
     * @param lambda_RT Input vector of dimensionless element potentials
     *                  The length is equal to nElements().
     */
    virtual void setToEquilState(const doublereal* lambda_RT) {
        throw NotImplementedError("DebyeHuckel::setToEquilState");
    }

    //@}

    /*
     *  -------------- Utilities -------------------------------
     */

    //! Initialize the object's internal lengths after species are set
    /**
     * @internal Initialize. This method is provided to allow
     * subclasses to perform any initialization required after all
     * species have been added. For example, it might be used to
     * resize internal work arrays that must have an entry for
     * each species.  The base class implementation does nothing,
     * and subclasses that do not require initialization do not
     * need to overload this method.  When importing a CTML phase
     * description, this method is called just prior to returning
     * from function importPhase().
     *
     * Cascading call sequence downwards starting with Parent.
     *
     * @internal
     */
    virtual void initThermo();

    //! Process the XML file after species are set up.
    /*!
     *  This gets called from importPhase(). It processes the XML file
     *  after the species are set up. This is the main routine for
     *  reading in activity coefficient parameters.
     *
     * @param phaseNode This object must be the phase node of a
     *             complete XML tree
     *             description of the phase, including all of the
     *             species data. In other words while "phase" must
     *             point to an XML phase object, it must have
     *             sibling nodes "speciesData" that describe
     *             the species in the phase.
     * @param id   ID of the phase. If nonnull, a check is done
     *             to see if phaseNode is pointing to the phase
     *             with the correct id.
     */
    virtual void  initThermoXML(XML_Node& phaseNode, const std::string& id);

    //! Return  the Debye Huckel constant as a function of temperature
    //! and pressure (Units = sqrt(kg/gmol))
    /*!
     *  The default is to assume that it is constant, given
     *  in the initialization process, and stored in the
     *  member double, m_A_Debye. Optionally, a full water treatment may be employed that makes
     *  \f$ A_{Debye} \f$ a full function of T and P.
     *
     *   \f[
     *      A_{Debye} = \frac{F e B_{Debye}}{8 \pi \epsilon R T} {\left( C_o \tilde{M}_o \right)}^{1/2}
     *   \f]
     *  where
     *  \f[
     *         B_{Debye} = \frac{F} {{(\frac{\epsilon R T}{2})}^{1/2}}
     *  \f]
     *  Therefore:
     *  \f[
     *   A_{Debye} = \frac{1}{8 \pi}
     *                 {\left(\frac{2 N_a \rho_o}{1000}\right)}^{1/2}
     *                 {\left(\frac{N_a e^2}{\epsilon R T }\right)}^{3/2}
     *  \f]
     *
     *  where
     *  - Units = sqrt(kg/gmol)
     *  - \f$ N_a \f$ is Avogadro's number
     *  - \f$ \rho_w \f$ is the density of water
     *  - \f$ e \f$ is the electronic charge
     *  - \f$ \epsilon = K \epsilon_o \f$ is the permittivity of water
     *  - \f$ K \f$ is the dielectric constant of water,
     *  - \f$ \epsilon_o \f$ is the permittivity of free space.
     *  - \f$ \rho_o \f$ is the density of the solvent in its standard state.
     *
     *  Nominal value at 298 K and 1 atm = 1.172576 (kg/gmol)<SUP>1/2</SUP>
     *  based on:
     *    - \f$ \epsilon / \epsilon_0 \f$ = 78.54 (water at 25C)
     *    - T = 298.15 K
     *    - B_Debye = 3.28640E9 (kg/gmol)<SUP>1/2</SUP> m<SUP>-1</SUP>
     *
     * @param temperature Temperature in kelvin. Defaults to -1, in which
     *                    case the   temperature of the phase is assumed.
     *
     * @param pressure Pressure (Pa). Defaults to -1, in which
     *                    case the pressure of the phase is assumed.
     */
    virtual double A_Debye_TP(double temperature = -1.0,
                              double pressure = -1.0) const;

    //! Value of the derivative of the Debye Huckel constant with
    //! respect to temperature.
    /*!
     * This is a function of temperature and pressure. See A_Debye_TP() for
     * a definition of \f$ A_{Debye} \f$.
     *
     * Units = sqrt(kg/gmol) K-1
     *
     * @param temperature Temperature in kelvin. Defaults to -1, in which
     *                    case the   temperature of the phase is assumed.
     *
     * @param pressure Pressure (Pa). Defaults to -1, in which
     *                    case the pressure of the phase is assumed.
     */
    virtual double dA_DebyedT_TP(double temperature = -1.0,
                                 double pressure = -1.0) const;

    //! Value of the 2nd derivative of the Debye Huckel constant with
    //! respect to temperature as a function of temperature and pressure.
    /*!
     * This is a function of temperature and pressure. See A_Debye_TP() for
     * a definition of \f$ A_{Debye} \f$.
     *
     * Units = sqrt(kg/gmol) K-2
     *
     * @param temperature Temperature in kelvin. Defaults to -1, in which
     *                    case the   temperature of the phase is assumed.
     *
     * @param pressure Pressure (Pa). Defaults to -1, in which
     *                    case the pressure of the phase is assumed.
     */
    virtual double d2A_DebyedT2_TP(double temperature = -1.0,
                                   double pressure = -1.0) const;

    //! Value of the derivative of the Debye Huckel constant with
    //! respect to pressure, as a function of temperature and pressure.
    /*!
     * This is a function of temperature and pressure. See A_Debye_TP() for
     * a definition of \f$ A_{Debye} \f$.
     *
     * Units = sqrt(kg/gmol) Pa-1
     *
     * @param temperature Temperature in kelvin. Defaults to -1, in which
     *                    case the   temperature of the phase is assumed.
     *
     * @param pressure Pressure (Pa). Defaults to -1, in which
     *                    case the pressure of the phase is assumed.
     */
    virtual double dA_DebyedP_TP(double temperature = -1.0,
                                 double pressure = -1.0) const;

    //!Reports the ionic radius of the kth species
    /*!
     * @param k  species index.
     */
    double AionicRadius(int k = 0) const;

    //! Returns the form of the Debye-Huckel parameterization used
    int formDH() const {
        return m_formDH;
    }

    //! Returns a reference to M_Beta_ij
    Array2D& get_Beta_ij() {
        return m_Beta_ij;
    }

private:
    //!  Static function that implements the non-polar species
    //!   salt-out modifications.
    /*!
     *   Returns the calculated activity coefficients.
     *
     * @param IionicMolality Value of the ionic molality (sqrt(gmol/kg))
     */
    double _nonpolarActCoeff(double IionicMolality) const;

    //!      Formula for the osmotic coefficient that occurs in the GWB.
    /*!
     * It is originally from Helgeson for a variable
     *      NaCl brine. It's to be used with extreme caution.
     */
    double _osmoticCoeffHelgesonFixedForm() const;

    //!      Formula for the log of the water activity that occurs in the GWB.
    /*!
     * It is originally from Helgeson for a variable
     *      NaCl brine. It's to be used with extreme caution.
     */
    double _lnactivityWaterHelgesonFixedForm() const;
    //@}

protected:
    //! form of the Debye-Huckel parameterization  used in the model.
    /*!
     * The options are described at the top of this document,
     * and in the general documentation.
     * The list is repeated here:
     *
     * DHFORM_DILUTE_LIMIT  = 0       (default)
     * DHFORM_BDOT_AK       = 1
     * DHFORM_BDOT_AUNIFORM = 2
     * DHFORM_BETAIJ        = 3
     * DHFORM_PITZER_BETAIJ = 4
     */
    int m_formDH;

    /**
     * Format for the generalized concentration:
     *
     *  0 = unity
     *  1 = molar_volume
     *  2 = solvent_volume    (default)
     *
     * The generalized concentrations can have three different forms
     * depending on the value of the member attribute m_formGC, which
     * is supplied in the constructor.
     *                          <TABLE>
     *  <TR><TD> m_formGC </TD><TD> GeneralizedConc </TD><TD> StandardConc </TD></TR>
     *  <TR><TD> 0        </TD><TD> X_k             </TD><TD> 1.0          </TD></TR>
     *  <TR><TD> 1        </TD><TD> X_k / V_k       </TD><TD> 1.0 / V_k    </TD></TR>
     *  <TR><TD> 2        </TD><TD> X_k / V_N       </TD><TD> 1.0 / V_N    </TD></TR>
     *                         </TABLE>
     *
     * The value and form of the generalized concentration will affect
     * reaction rate constants involving species in this phase.
     *
     * (HKM Note: Using option #1 may lead to spurious results and
     *  has been included only with warnings. The reason is that it
     *  molar volumes of electrolytes may often be negative. The
     *  molar volume of H+ is defined to be zero too. Either options
     *  0 or 2 are the appropriate choice. Option 0 leads to
     *  bulk reaction rate constants which have units of s-1.
     *  Option 2 leads to bulk reaction rate constants for
     *  bimolecular rxns which have units of m-3 kmol-1 s-1.)
     */
    int m_formGC;

    //! Vector containing the electrolyte species type
    /*!
     * The possible types are:
     *  - solvent
     *  - Charged Species
     *  - weakAcidAssociated
     *  - strongAcidAssociated
     *  - polarNeutral
     *  - nonpolarNeutral
     *  .
     */
    vector_int  m_electrolyteSpeciesType;

    /**
     *  a_k = Size of the ionic species in the DH formulation
     *        units = meters
     */
    vector_fp m_Aionic;

    //! Current value of the ionic strength on the molality scale
    mutable double m_IionicMolality;

    /**
     * Maximum value of the ionic strength allowed in the
     * calculation of the activity coefficients.
     */
    double m_maxIionicStrength;

public:

    /**
     * If true, then the fixed for of Helgeson's activity
     * for water is used instead of the rigorous form
     * obtained from Gibbs-Duhem relation. This should be
     * used with caution, and is really only included as a
     * validation exercise.
     */
    bool m_useHelgesonFixedForm;
protected:

    //! Stoichiometric ionic strength on the molality scale
    mutable double m_IionicMolalityStoich;

public:
    /**
     * Form of the constant outside the Debye-Huckel term
     * called A. It's normally a function of temperature
     * and pressure. However, it can be set from the
     * input file in order to aid in numerical comparisons.
     * Acceptable forms:
     *
     *       A_DEBYE_CONST  0
     *       A_DEBYE_WATER  1
     *
     * The A_DEBYE_WATER form may be used for water solvents
     * with needs to cover varying temperatures and pressures.
     * Note, the dielectric constant of water is a relatively
     * strong function of T, and its variability must be
     * accounted for,
     */
    int m_form_A_Debye;

protected:
    //! Current value of the Debye Constant, A_Debye
    /**
     * A_Debye -> this expression appears on the top of the
     *            ln actCoeff term in the general Debye-Huckel
     *            expression
     *            It depends on temperature and pressure.
     *
     *            A_Debye = (F e B_Debye) / (8 Pi epsilon R T)
     *
     *            Units = sqrt(kg/gmol)
     *
     *            Nominal value(298K, atm) = 1.172576 sqrt(kg/gmol)
     *                  based on:
     *                    epsilon/epsilon_0 = 78.54
     *                           (water at 25C)
     *                    T = 298.15 K
     *                    B_Debye = 3.28640E9 sqrt(kg/gmol)/m
     *
     *            note in Pitzer's nomenclature, A_phi = A_Debye/3.0
     */
    mutable double m_A_Debye;

    //! Current value of the constant that appears in the denominator
    /**
     * B_Debye -> this expression appears on the bottom of the
     *            ln actCoeff term in the general Debye-Huckel
     *            expression
     *            It depends on temperature
     *
     *            B_Bebye = F / sqrt( epsilon R T / 2 )
     *
     *            Units = sqrt(kg/gmol) / m
     *
     *            Nominal value = 3.28640E9 sqrt(kg/gmol) / m
     *                  based on:
     *                    epsilon/epsilon_0 = 78.54
     *                           (water at 25C)
     *                    T = 298.15 K
     */
    double m_B_Debye;

    //! Array of B_Dot values
    /**
     *  This expression is an extension of the Debye-Huckel expression used
     *  in some formulations to extend DH to higher molalities. B_dot is
     *  specific to the major ionic pair.
     */
    vector_fp  m_B_Dot;

    /**
     *  These are coefficients to describe the increase in activity coeff for
     *  non-polar molecules due to the electrolyte becoming stronger (the
     *  so-called salt-out effect)
     */
    vector_fp m_npActCoeff;


    //! Pointer to the  Water standard state object
    /*!
     *  derived from the equation of state for water.
     */
    PDSS_Water* m_waterSS;

    //! Storage for the density of water's standard state
    /*!
     * Density depends on temperature and pressure.
     */
    double m_densWaterSS;

    //! Pointer to the water property calculator
    WaterProps* m_waterProps;

    //! Temporary array used in equilibrium calculations
    mutable vector_fp      m_pp;

    //! vector of size m_kk, used as a temporary holding area.
    mutable vector_fp      m_tmpV;

    /**
     * Stoichiometric species charge -> This is for calculations
     * of the ionic strength which ignore ion-ion pairing into
     * neutral molecules. The Stoichiometric species charge is the
     * charge of one of the ion that would occur if the species broke
     * into two charged ion pairs.
     *  NaCl ->   m_speciesCharge_Stoich = -1;
     *  HSO4- -> H+ + SO42-              = -2
     *      -> The other charge is calculated.
     * For species that aren't ion pairs, it's equal to the
     * m_speciesCharge[] value.
     */
    vector_fp  m_speciesCharge_Stoich;

    /**
     *  Array of 2D data used in the DHFORM_BETAIJ formulation
     *  Beta_ij.value(i,j) is the coefficient of the jth species
     *  for the specification of the chemical potential of the ith
     *  species.
     */
    Array2D m_Beta_ij;

    //!  Logarithm of the activity coefficients on the molality scale.
    /*!
     *       mutable because we change this if the composition
     *       or temperature or pressure changes.
     */
    mutable vector_fp m_lnActCoeffMolal;

    //! Derivative of log act coeff wrt T
    mutable vector_fp m_dlnActCoeffMolaldT;

    //! 2nd Derivative of log act coeff wrt T
    mutable vector_fp m_d2lnActCoeffMolaldT2;

    //! Derivative of log act coeff wrt P
    mutable vector_fp m_dlnActCoeffMolaldP;

private:
    //! Initialize the internal lengths.
    /*!
     * This internal function adjusts the lengths of arrays based on
     * the number of species.
     */
    void initLengths();

    //! Calculate the log activity coefficients
    /*!
     * This function updates the internally stored natural logarithm of the
     * molality activity coefficients. This is the main routine for
     * implementing the activity coefficient formulation.
     */
    void s_update_lnMolalityActCoeff() const;

    //! Calculation of temperature derivative of activity coefficient
    /*!
     *   Using internally stored values, this function calculates
     *   the temperature derivative of the logarithm of the
     *   activity coefficient for all species in the mechanism.
     *
     *   We assume that the activity coefficients are current in this routine
     *
     *   The solvent activity coefficient is on the molality scale. Its derivative is too.
     */
    void s_update_dlnMolalityActCoeff_dT() const;

    //! Calculate the temperature 2nd derivative of the activity coefficient
    /*!
     *   Using internally stored values, this function calculates
     *   the temperature 2nd derivative of the logarithm of the
     *   activity coefficient for all species in the mechanism.
     *
     *   We assume that the activity coefficients are current in this routine
     *
     *   solvent activity coefficient is on the molality
     *   scale. Its derivatives are too.
     */
    void s_update_d2lnMolalityActCoeff_dT2() const;

    //! Calculate the pressure derivative of the activity coefficient
    /*!
     *   Using internally stored values, this function calculates
     *   the pressure derivative of the logarithm of the
     *   activity coefficient for all species in the mechanism.
     *
     *   We assume that the activity coefficients, molalities,
     *   and A_Debye are current.
     *
     *   solvent activity coefficient is on the molality
     *   scale. Its derivatives are too.
     */
    void s_update_dlnMolalityActCoeff_dP() const;
};

}

#endif