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pcsaft_electrolyte.cpp
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#include <vector>
#include <string>
#include <cmath>
#include "math.h"
#include "externals/eigen/Eigen/Dense"
#include "pcsaft_electrolyte.h"
using std::vector;
#if defined(HUGE_VAL) && !defined(_HUGE)
# define _HUGE HUGE_VAL
#else
// GCC Version of huge value macro
#if defined(HUGE) && !defined(_HUGE)
# define _HUGE HUGE
#endif
#endif
vector<double> XA_find(vector<double> XA_guess, vector<double> delta_ij, double den,
vector<double> x) {
/**Iterate over this function in order to solve for XA*/
int num_sites = XA_guess.size();
vector<double> XA = XA_guess;
int idxij = -1; // index for delta_ij
for (int i = 0; i < num_sites; i++) {
double summ = 0.;
for (int j = 0; j < num_sites; j++) {
idxij += 1;
summ += den*x[j]*XA_guess[j]*delta_ij[idxij];
}
XA[i] = 1./(1.+summ);
}
return XA;
}
vector<double> dXAdt_find(vector<double> delta_ij, double den,
vector<double> XA, vector<double> ddelta_dt, vector<double> x) {
/**Solve for the derivative of XA with respect to temperature.*/
int num_sites = XA.size();
Eigen::MatrixXd B = Eigen::MatrixXd::Zero(num_sites, 1);
Eigen::MatrixXd A = Eigen::MatrixXd::Zero(num_sites, num_sites);
double summ;
int ij = 0;
for (int i = 0; i < num_sites; i++) {
summ = 0;
for (int j = 0; j < num_sites; j++) {
B(i) -= x[j]*XA[j]*ddelta_dt[ij];
A(i,j) = x[j]*delta_ij[ij];
summ += x[j]*XA[j]*delta_ij[ij];
ij += 1;
}
A(i,i) = pow(1+den*summ, 2.)/den;
}
Eigen::MatrixXd solution = A.lu().solve(B); //Solves linear system of equations
vector<double> dXA_dt(num_sites);
for (int i = 0; i < num_sites; i++) {
dXA_dt[i] = solution(i);
}
return dXA_dt;
}
vector<double> dXAdx_find(vector<int> assoc_num, vector<double> delta_ij,
double den, vector<double> XA, vector<double> ddelta_dx, vector<double> x) {
/**Solve for the derivative of XA with respect to composition, or actually
rho_i (the molar density of component i, which equals x_i * rho).*/
int num_sites = XA.size();
int ncomp = assoc_num.size();
Eigen::MatrixXd B(num_sites*ncomp, 1);
Eigen::MatrixXd A = Eigen::MatrixXd::Zero(num_sites*ncomp, num_sites*ncomp);
double sum1, sum2;
int idx1 = 0;
int ij = 0;
for (int i = 0; i < ncomp; i++) {
for (int j = 0; j < num_sites; j++) {
sum1 = 0;
for (int k = 0; k < num_sites; k++) {
sum1 = sum1 + den*x[k]*(XA[k]*ddelta_dx[i*num_sites*num_sites + j*num_sites + k]);
A(ij,i*num_sites+k) = XA[j]*XA[j]*den*x[k]*delta_ij[j*num_sites+k];
}
sum2 = 0;
for (int l = 0; l < assoc_num[i]; l++) {
sum2 = sum2 + XA[idx1+l]*delta_ij[idx1*num_sites+l*num_sites+j];
}
A(ij,ij) = A(ij,ij) + 1;
B(ij) = -1*XA[j]*XA[j]*(sum1 + sum2);
ij += 1;
}
idx1 += assoc_num[i];
}
Eigen::MatrixXd solution = A.lu().solve(B); //Solves linear system of equations
vector<double> dXA_dx(num_sites*ncomp);
for (int i = 0; i < num_sites*ncomp; i++) {
dXA_dx[i] = solution(i);
}
return dXA_dx;
}
double pcsaft_Z_cpp(double t, double rho, vector<double> x, add_args &cppargs) {
/**
Calculate the compressibility factor.
*/
int ncomp = x.size(); // number of components
vector<double> d (ncomp);
for (int i = 0; i < ncomp; i++) {
d[i] = cppargs.s[i]*(1-0.12*exp(-3*cppargs.e[i]/t));
}
if (!cppargs.z.empty()) {
for (int i = 0; i < ncomp; i++) {
if (cppargs.z[i] != 0) {
d[i] = cppargs.s[i]*(1-0.12); // for ions the diameter is assumed to be temperature independent (see Held et al. 2014)
}
}
}
double den = rho*N_AV/1.0e30;
vector<double> zeta (4, 0);
double summ;
for (int i = 0; i < 4; i++) {
summ = 0;
for (int j = 0; j < ncomp; j++) {
summ += x[j]*cppargs.m[j]*pow(d[j], i);
}
zeta[i] = PI/6*den*summ;
}
double eta = zeta[3];
double m_avg = 0;
for (int i = 0; i < ncomp; i++) {
m_avg += x[i]*cppargs.m[i];
}
vector<double> ghs (ncomp*ncomp, 0);
vector<double> denghs (ncomp*ncomp, 0);
vector<double> e_ij (ncomp*ncomp, 0);
vector<double> s_ij (ncomp*ncomp, 0);
double m2es3 = 0.;
double m2e2s3 = 0.;
int idx = -1;
for (int i = 0; i < ncomp; i++) {
for (int j = 0; j < ncomp; j++) {
idx += 1;
if (cppargs.l_ij.empty()) {
s_ij[idx] = (cppargs.s[i] + cppargs.s[j])/2.;
}
else {
s_ij[idx] = (cppargs.s[i] + cppargs.s[j])/2.*(1-cppargs.l_ij[idx]);
}
if (!cppargs.z.empty()) {
if (cppargs.z[i]*cppargs.z[j] <= 0) { // for two cations or two anions e_ij is kept at zero to avoid dispersion between like ions (see Held et al. 2014)
if (cppargs.k_ij.empty()) {
e_ij[idx] = sqrt(cppargs.e[i]*cppargs.e[j]);
}
else {
e_ij[idx] = sqrt(cppargs.e[i]*cppargs.e[j])*(1-cppargs.k_ij[idx]);
}
}
} else {
if (cppargs.k_ij.empty()) {
e_ij[idx] = sqrt(cppargs.e[i]*cppargs.e[j]);
}
else {
e_ij[idx] = sqrt(cppargs.e[i]*cppargs.e[j])*(1-cppargs.k_ij[idx]);
}
}
m2es3 = m2es3 + x[i]*x[j]*cppargs.m[i]*cppargs.m[j]*e_ij[idx]/t*pow(s_ij[idx], 3);
m2e2s3 = m2e2s3 + x[i]*x[j]*cppargs.m[i]*cppargs.m[j]*pow(e_ij[idx]/t,2)*pow(s_ij[idx], 3);
ghs[idx] = 1/(1-zeta[3]) + (d[i]*d[j]/(d[i]+d[j]))*3*zeta[2]/(1-zeta[3])/(1-zeta[3]) +
pow(d[i]*d[j]/(d[i]+d[j]), 2)*2*zeta[2]*zeta[2]/pow(1-zeta[3], 3);
denghs[idx] = zeta[3]/(1-zeta[3])/(1-zeta[3]) +
(d[i]*d[j]/(d[i]+d[j]))*(3*zeta[2]/(1-zeta[3])/(1-zeta[3]) +
6*zeta[2]*zeta[3]/pow(1-zeta[3], 3)) +
pow(d[i]*d[j]/(d[i]+d[j]), 2)*(4*zeta[2]*zeta[2]/pow(1-zeta[3], 3) +
6*zeta[2]*zeta[2]*zeta[3]/pow(1-zeta[3], 4));
}
}
double Zhs = zeta[3]/(1-zeta[3]) + 3.*zeta[1]*zeta[2]/zeta[0]/(1.-zeta[3])/(1.-zeta[3]) +
(3.*pow(zeta[2], 3.) - zeta[3]*pow(zeta[2], 3.))/zeta[0]/pow(1.-zeta[3], 3.);
static double a0[7] = { 0.9105631445, 0.6361281449, 2.6861347891, -26.547362491, 97.759208784, -159.59154087, 91.297774084 };
static double a1[7] = { -0.3084016918, 0.1860531159, -2.5030047259, 21.419793629, -65.255885330, 83.318680481, -33.746922930 };
static double a2[7] = { -0.0906148351, 0.4527842806, 0.5962700728, -1.7241829131, -4.1302112531, 13.776631870, -8.6728470368 };
static double b0[7] = { 0.7240946941, 2.2382791861, -4.0025849485, -21.003576815, 26.855641363, 206.55133841, -355.60235612 };
static double b1[7] = { -0.5755498075, 0.6995095521, 3.8925673390, -17.215471648, 192.67226447, -161.82646165, -165.20769346 };
static double b2[7] = { 0.0976883116, -0.2557574982, -9.1558561530, 20.642075974, -38.804430052, 93.626774077, -29.666905585 };
vector<double> a (7, 0);
vector<double> b (7, 0);
for (int i = 0; i < 7; i++) {
a[i] = a0[i] + (m_avg-1.)/m_avg*a1[i] + (m_avg-1.)/m_avg*(m_avg-2.)/m_avg*a2[i];
b[i] = b0[i] + (m_avg-1.)/m_avg*b1[i] + (m_avg-1.)/m_avg*(m_avg-2.)/m_avg*b2[i];
}
double detI1_det = 0.0;
double detI2_det = 0.0;
double I2 = 0.0;
for (int i = 0; i < 7; i++) {
detI1_det += a[i]*(i+1)*pow(eta, i);
detI2_det += b[i]*(i+1)*pow(eta, i);
I2 += b[i]*pow(eta, i);
}
double C1 = 1./(1. + m_avg*(8*eta-2*eta*eta)/pow(1-eta, 4) + (1-m_avg)*(20*eta-27*eta*eta+12*pow(eta, 3)-2*pow(eta, 4))/pow((1-eta)*(2-eta), 2.0));
double C2 = -1.*C1*C1*(m_avg*(-4*eta*eta+20*eta+8)/pow(1-eta, 5) + (1-m_avg)*(2*pow(eta, 3)+12*eta*eta-48*eta+40)/pow((1-eta)*(2-eta), 3.0));
summ = 0.0;
for (int i = 0; i < ncomp; i++) {
summ += x[i]*(cppargs.m[i]-1)/ghs[i*ncomp+i]*denghs[i*ncomp+i];
}
double Zid = 1.0;
double Zhc = m_avg*Zhs - summ;
double Zdisp = -2*PI*den*detI1_det*m2es3 - PI*den*m_avg*(C1*detI2_det + C2*eta*I2)*m2e2s3;
// Dipole term (Gross and Vrabec term) --------------------------------------
double Zpolar = 0;
if (!cppargs.dipm.empty()) {
double A2 = 0.;
double A3 = 0.;
double dA2_det = 0.;
double dA3_det = 0.;
vector<double> adip (5, 0);
vector<double> bdip (5, 0);
vector<double> cdip (5, 0);
vector<double> dipmSQ (ncomp, 0);
double J2, detJ2_det, J3, detJ3_det;
static double a0dip[5] = { 0.3043504, -0.1358588, 1.4493329, 0.3556977, -2.0653308 };
static double a1dip[5] = { 0.9534641, -1.8396383, 2.0131180, -7.3724958, 8.2374135 };
static double a2dip[5] = { -1.1610080, 4.5258607, 0.9751222, -12.281038, 5.9397575 };
static double b0dip[5] = { 0.2187939, -1.1896431, 1.1626889, 0, 0 };
static double b1dip[5] = { -0.5873164, 1.2489132, -0.5085280, 0, 0 };
static double b2dip[5] = { 3.4869576, -14.915974, 15.372022, 0, 0 };
static double c0dip[5] = { -0.0646774, 0.1975882, -0.8087562, 0.6902849, 0 };
static double c1dip[5] = { -0.9520876, 2.9924258, -2.3802636, -0.2701261, 0 };
static double c2dip[5] = { -0.6260979, 1.2924686, 1.6542783, -3.4396744, 0 };
const static double conv = 7242.702976750923; // conversion factor, see the note below Table 2 in Gross and Vrabec 2006
for (int i = 0; i < ncomp; i++) {
dipmSQ[i] = pow(cppargs.dipm[i], 2.)/(cppargs.m[i]*cppargs.e[i]*pow(cppargs.s[i],3.))*conv;
}
double m_ij;
for (int i = 0; i < ncomp; i++) {
for (int j = 0; j < ncomp; j++) {
m_ij = sqrt(cppargs.m[i]*cppargs.m[j]);
if (m_ij > 2) {
m_ij = 2;
}
J2 = 0.;
detJ2_det = 0.;
for (int l = 0; l < 5; l++) {
adip[l] = a0dip[l] + (m_ij-1)/m_ij*a1dip[l] + (m_ij-1)/m_ij*(m_ij-2)/m_ij*a2dip[l];
bdip[l] = b0dip[l] + (m_ij-1)/m_ij*b1dip[l] + (m_ij-1)/m_ij*(m_ij-2)/m_ij*b2dip[l];
J2 += (adip[l] + bdip[l]*e_ij[i*ncomp+j]/t)*pow(eta, l); // i*ncomp+j needs to be used for e_ij because it is formatted as a 1D vector
detJ2_det += (adip[l] + bdip[l]*e_ij[i*ncomp+j]/t)*(l+1)*pow(eta, l);
}
A2 += x[i]*x[j]*e_ij[i*ncomp+i]/t*e_ij[j*ncomp+j]/t*pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)/
pow(s_ij[i*ncomp+j],3)*cppargs.dip_num[i]*cppargs.dip_num[j]*dipmSQ[i]*dipmSQ[j]*J2;
dA2_det += x[i]*x[j]*e_ij[i*ncomp+i]/t*e_ij[j*ncomp+j]/t*pow(s_ij[i*ncomp+i],3)*
pow(s_ij[j*ncomp+j],3)/pow(s_ij[i*ncomp+j],3)*cppargs.dip_num[i]*cppargs.dip_num[j]*dipmSQ[i]*dipmSQ[j]*detJ2_det;
}
}
double m_ijk;
for (int i = 0; i < ncomp; i++) {
for (int j = 0; j < ncomp; j++) {
for (int k = 0; k < ncomp; k++) {
m_ijk = pow((cppargs.m[i]*cppargs.m[j]*cppargs.m[k]),1/3.);
if (m_ijk > 2) {
m_ijk = 2;
}
J3 = 0.;
detJ3_det = 0.;
for (int l = 0; l < 5; l++) {
cdip[l] = c0dip[l] + (m_ijk-1)/m_ijk*c1dip[l] + (m_ijk-1)/m_ijk*(m_ijk-2)/m_ijk*c2dip[l];
J3 += cdip[l]*pow(eta, l);
detJ3_det += cdip[l]*(l+2)*pow(eta, (l+1));
}
A3 += x[i]*x[j]*x[k]*e_ij[i*ncomp+i]/t*e_ij[j*ncomp+j]/t*e_ij[k*ncomp+k]/t*
pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)*pow(s_ij[k*ncomp+k],3)/s_ij[i*ncomp+j]/s_ij[i*ncomp+k]/
s_ij[j*ncomp+k]*cppargs.dip_num[i]*cppargs.dip_num[j]*cppargs.dip_num[k]*dipmSQ[i]*
dipmSQ[j]*dipmSQ[k]*J3;
dA3_det += x[i]*x[j]*x[k]*e_ij[i*ncomp+i]/t*e_ij[j*ncomp+j]/t*e_ij[k*ncomp+k]/t*
pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)*pow(s_ij[k*ncomp+k],3)/s_ij[i*ncomp+j]/s_ij[i*ncomp+k]/
s_ij[j*ncomp+k]*cppargs.dip_num[i]*cppargs.dip_num[j]*cppargs.dip_num[k]*dipmSQ[i]*
dipmSQ[j]*dipmSQ[k]*detJ3_det;
}
}
}
A2 = -PI*den*A2;
A3 = -4/3.*PI*PI*den*den*A3;
dA2_det = -PI*den/eta*dA2_det;
dA3_det = -4/3.*PI*PI*den/eta*den/eta*dA3_det;
if (A2 != 0) { // when the mole fraction of the polar compounds is 0 then A2 = 0 and division by 0 occurs
Zpolar = eta*((dA2_det*(1-A3/A2)+(dA3_det*A2-A3*dA2_det)/A2)/(1-A3/A2)/(1-A3/A2));
}
}
// Association term -------------------------------------------------------
double Zassoc = 0;
if (!cppargs.e_assoc.empty()) {
int num_sites = 0;
vector<int> iA; //indices of associating compounds
for(std::vector<int>::iterator it = cppargs.assoc_num.begin(); it != cppargs.assoc_num.end(); ++it) {
num_sites += *it;
for (int i = 0; i < *it; i++) {
iA.push_back(it - cppargs.assoc_num.begin());
}
}
vector<double> x_assoc(num_sites); // mole fractions of only the associating compounds
for (int i = 0; i < num_sites; i++) {
x_assoc[i] = x[iA[i]];
}
vector<double> XA (num_sites, 0);
vector<double> delta_ij(num_sites * num_sites, 0);
int idxa = 0;
int idxi = 0; // index for the ii-th compound
int idxj = 0; // index for the jj-th compound
for (int i = 0; i < num_sites; i++) {
idxi = iA[i]*ncomp+iA[i];
for (int j = 0; j < num_sites; j++) {
idxj = iA[j]*ncomp+iA[j];
if (cppargs.assoc_matrix[idxa] != 0) {
double eABij = (cppargs.e_assoc[iA[i]]+cppargs.e_assoc[iA[j]])/2.;
double volABij = _HUGE;
if (cppargs.k_hb.empty()) {
volABij = sqrt(cppargs.vol_a[iA[i]]*cppargs.vol_a[iA[j]])*pow(sqrt(s_ij[idxi]*
s_ij[idxj])/(0.5*(s_ij[idxi]+s_ij[idxj])), 3);
}
else {
volABij = sqrt(cppargs.vol_a[iA[i]]*cppargs.vol_a[iA[j]])*pow(sqrt(s_ij[idxi]*
s_ij[idxj])/(0.5*(s_ij[idxi]+s_ij[idxj])), 3)*(1-cppargs.k_hb[iA[i]*ncomp+iA[j]]);
}
delta_ij[idxa] = ghs[iA[i]*ncomp+iA[j]]*(exp(eABij/t)-1)*pow(s_ij[iA[i]*ncomp+iA[j]], 3)*volABij;
}
idxa += 1;
}
XA[i] = (-1 + sqrt(1+8*den*delta_ij[i*num_sites+i]))/(4*den*delta_ij[i*num_sites+i]);
if (!std::isfinite(XA[i])) {
XA[i] = 0.02;
}
}
vector<double> ddelta_dx(num_sites * num_sites * ncomp, 0);
int idx_ddelta = 0;
for (int k = 0; k < ncomp; k++) {
int idxi = 0; // index for the ii-th compound
int idxj = 0; // index for the jj-th compound
idxa = 0;
for (int i = 0; i < num_sites; i++) {
idxi = iA[i]*ncomp+iA[i];
for (int j = 0; j < num_sites; j++) {
idxj = iA[j]*ncomp+iA[j];
if (cppargs.assoc_matrix[idxa] != 0) {
double eABij = (cppargs.e_assoc[iA[i]]+cppargs.e_assoc[iA[j]])/2.;
double volABij = _HUGE;
if (cppargs.k_hb.empty()) {
volABij = sqrt(cppargs.vol_a[iA[i]]*cppargs.vol_a[iA[j]])*pow(sqrt(s_ij[idxi]*
s_ij[idxj])/(0.5*(s_ij[idxi]+s_ij[idxj])), 3);
}
else {
volABij = sqrt(cppargs.vol_a[iA[i]]*cppargs.vol_a[iA[j]])*pow(sqrt(s_ij[idxi]*
s_ij[idxj])/(0.5*(s_ij[idxi]+s_ij[idxj])), 3)*(1-cppargs.k_hb[iA[i]*ncomp+iA[j]]);
}
double dghsd_dx = PI/6.*cppargs.m[k]*(pow(d[k], 3)/(1-zeta[3])/(1-zeta[3]) + 3*d[iA[i]]*d[iA[j]]/
(d[iA[i]]+d[iA[j]])*(d[k]*d[k]/(1-zeta[3])/(1-zeta[3])+2*pow(d[k], 3)*
zeta[2]/pow(1-zeta[3], 3)) + 2*pow((d[iA[i]]*d[iA[j]]/(d[iA[i]]+d[iA[j]])), 2)*
(2*d[k]*d[k]*zeta[2]/pow(1-zeta[3], 3)+3*(pow(d[k], 3)*zeta[2]*zeta[2]
/pow(1-zeta[3], 4))));
ddelta_dx[idx_ddelta] = dghsd_dx*(exp(eABij/t)-1)*pow(s_ij[iA[i]*ncomp+iA[j]], 3)*volABij;
}
idx_ddelta += 1;
idxa += 1;
}
}
}
int ctr = 0;
double dif = 1000.;
vector<double> XA_old = XA;
while ((ctr < 100) && (dif > 1e-15)) {
ctr += 1;
XA = XA_find(XA_old, delta_ij, den, x_assoc);
dif = 0.;
for (int i = 0; i < num_sites; i++) {
dif += std::abs(XA[i] - XA_old[i]);
}
for (int i = 0; i < num_sites; i++) {
XA_old[i] = (XA[i] + XA_old[i]) / 2.0;
}
}
vector<double> dXA_dx(num_sites*ncomp, 0);
dXA_dx = dXAdx_find(cppargs.assoc_num, delta_ij, den, XA, ddelta_dx, x_assoc);
summ = 0.;
int ij = 0;
for (int i = 0; i < ncomp; i++) {
for (int j = 0; j < num_sites; j++) {
summ += x[i]*den*x[iA[j]]*(1/XA[j]-0.5)*dXA_dx[ij];
ij += 1;
}
}
Zassoc = summ;
}
// Ion term ---------------------------------------------------------------
double Zion = 0;
if (!cppargs.z.empty()) {
vector<double> q(cppargs.z.begin(), cppargs.z.end());
for (int i = 0; i < ncomp; i++) {
q[i] = q[i]*E_CHRG;
}
summ = 0.;
for (int i = 0; i < ncomp; i++) {
summ += cppargs.z[i]*cppargs.z[i]*x[i];
}
double kappa = sqrt(den*E_CHRG*E_CHRG/kb/t/(cppargs.dielc*perm_vac)*summ); // the inverse Debye screening length. Equation 4 in Held et al. 2008.
if (kappa != 0) {
double chi, sigma_k;
summ = 0.;
for (int i = 0; i < ncomp; i++) {
chi = 3/pow(kappa*d[i], 3)*(1.5 + log(1+kappa*d[i]) - 2*(1+kappa*d[i]) +
0.5*pow(1+kappa*d[i], 2));
sigma_k = -2*chi+3/(1+kappa*d[i]);
summ += q[i]*q[i]*x[i]*sigma_k;
}
Zion = -1*kappa/24./PI/kb/t/(cppargs.dielc*perm_vac)*summ;
}
}
double Z = Zid + Zhc + Zdisp + Zpolar + Zassoc + Zion;
return Z;
}
vector<double> pcsaft_lnfug_cpp(double t, double rho, vector<double> x, add_args &cppargs) {
/**
Calculate the natural logarithm of the fugacity coefficients for one phase of the system.
*/
int ncomp = x.size(); // number of components
vector<double> d (ncomp);
for (int i = 0; i < ncomp; i++) {
d[i] = cppargs.s[i]*(1-0.12*exp(-3*cppargs.e[i]/t));
}
if (!cppargs.z.empty()) {
for (int i = 0; i < ncomp; i++) {
if (cppargs.z[i] != 0) {
d[i] = cppargs.s[i]*(1-0.12); // for ions the diameter is assumed to be temperature independent (see Held et al. 2014)
}
}
}
double den = rho*N_AV/1.0e30;
vector<double> zeta (4, 0);
double summ;
for (int i = 0; i < 4; i++) {
summ = 0;
for (int j = 0; j < ncomp; j++) {
summ += x[j]*cppargs.m[j]*pow(d[j], i);
}
zeta[i] = PI/6*den*summ;
}
double eta = zeta[3];
double m_avg = 0;
for (int i = 0; i < ncomp; i++) {
m_avg += x[i]*cppargs.m[i];
}
vector<double> ghs(ncomp*ncomp, 0);
vector<double> denghs(ncomp*ncomp, 0);
vector<double> e_ij(ncomp*ncomp, 0);
vector<double> s_ij(ncomp*ncomp, 0);
double m2es3 = 0.;
double m2e2s3 = 0.;
int idx = -1;
for (int i = 0; i < ncomp; i++) {
for (int j = 0; j < ncomp; j++) {
idx += 1;
if (cppargs.l_ij.empty()) {
s_ij[idx] = (cppargs.s[i] + cppargs.s[j])/2.;
}
else {
s_ij[idx] = (cppargs.s[i] + cppargs.s[j])/2.*(1-cppargs.l_ij[idx]);
}
if (!cppargs.z.empty()) {
if (cppargs.z[i]*cppargs.z[j] <= 0) { // for two cations or two anions e_ij is kept at zero to avoid dispersion between like ions (see Held et al. 2014)
if (cppargs.k_ij.empty()) {
e_ij[idx] = sqrt(cppargs.e[i]*cppargs.e[j]);
}
else {
e_ij[idx] = sqrt(cppargs.e[i]*cppargs.e[j])*(1-cppargs.k_ij[idx]);
}
}
} else {
if (cppargs.k_ij.empty()) {
e_ij[idx] = sqrt(cppargs.e[i]*cppargs.e[j]);
}
else {
e_ij[idx] = sqrt(cppargs.e[i]*cppargs.e[j])*(1-cppargs.k_ij[idx]);
}
}
m2es3 = m2es3 + x[i]*x[j]*cppargs.m[i]*cppargs.m[j]*e_ij[idx]/t*pow(s_ij[idx], 3);
m2e2s3 = m2e2s3 + x[i]*x[j]*cppargs.m[i]*cppargs.m[j]*pow(e_ij[idx]/t,2)*pow(s_ij[idx], 3);
ghs[idx] = 1/(1-zeta[3]) + (d[i]*d[j]/(d[i]+d[j]))*3*zeta[2]/(1-zeta[3])/(1-zeta[3]) +
pow(d[i]*d[j]/(d[i]+d[j]), 2)*2*zeta[2]*zeta[2]/pow(1-zeta[3], 3);
denghs[idx] = zeta[3]/(1-zeta[3])/(1-zeta[3]) +
(d[i]*d[j]/(d[i]+d[j]))*(3*zeta[2]/(1-zeta[3])/(1-zeta[3]) +
6*zeta[2]*zeta[3]/pow(1-zeta[3], 3)) +
pow(d[i]*d[j]/(d[i]+d[j]), 2)*(4*zeta[2]*zeta[2]/pow(1-zeta[3], 3) +
6*zeta[2]*zeta[2]*zeta[3]/pow(1-zeta[3], 4));
}
}
double ares_hs = 1/zeta[0]*(3*zeta[1]*zeta[2]/(1-zeta[3]) + pow(zeta[2], 3.)/(zeta[3]*pow(1-zeta[3],2))
+ (pow(zeta[2], 3.)/pow(zeta[3], 2.) - zeta[0])*log(1-zeta[3]));
double Zhs = zeta[3]/(1-zeta[3]) + 3.*zeta[1]*zeta[2]/zeta[0]/(1.-zeta[3])/(1.-zeta[3]) +
(3.*pow(zeta[2], 3.) - zeta[3]*pow(zeta[2], 3.))/zeta[0]/pow(1.-zeta[3], 3.);
static double a0[7] = { 0.9105631445, 0.6361281449, 2.6861347891, -26.547362491, 97.759208784, -159.59154087, 91.297774084 };
static double a1[7] = { -0.3084016918, 0.1860531159, -2.5030047259, 21.419793629, -65.255885330, 83.318680481, -33.746922930 };
static double a2[7] = { -0.0906148351, 0.4527842806, 0.5962700728, -1.7241829131, -4.1302112531, 13.776631870, -8.6728470368 };
static double b0[7] = { 0.7240946941, 2.2382791861, -4.0025849485, -21.003576815, 26.855641363, 206.55133841, -355.60235612 };
static double b1[7] = { -0.5755498075, 0.6995095521, 3.8925673390, -17.215471648, 192.67226447, -161.82646165, -165.20769346 };
static double b2[7] = { 0.0976883116, -0.2557574982, -9.1558561530, 20.642075974, -38.804430052, 93.626774077, -29.666905585 };
vector<double> a (7, 0);
vector<double> b (7, 0);
for (int i = 0; i < 7; i++) {
a[i] = a0[i] + (m_avg-1.)/m_avg*a1[i] + (m_avg-1.)/m_avg*(m_avg-2.)/m_avg*a2[i];
b[i] = b0[i] + (m_avg-1.)/m_avg*b1[i] + (m_avg-1.)/m_avg*(m_avg-2.)/m_avg*b2[i];
}
double detI1_det = 0.0;
double detI2_det = 0.0;
double I1 = 0.0;
double I2 = 0.0;
for (int i = 0; i < 7; i++) {
detI1_det += a[i]*(i+1)*pow(eta, i);
detI2_det += b[i]*(i+1)*pow(eta, i);
I2 += b[i]*pow(eta, i);
I1 += a[i]*pow(eta, i);
}
double C1 = 1./(1. + m_avg*(8*eta-2*eta*eta)/pow(1-eta, 4) + (1-m_avg)*(20*eta-27*eta*eta+12*pow(eta, 3)-2*pow(eta, 4))/pow((1-eta)*(2-eta), 2.0));
double C2 = -1.*C1*C1*(m_avg*(-4*eta*eta+20*eta+8)/pow(1-eta, 5) + (1-m_avg)*(2*pow(eta, 3)+12*eta*eta-48*eta+40)/pow((1-eta)*(2-eta), 3.0));
summ = 0.0;
for (int i = 0; i < ncomp; i++) {
summ += x[i]*(cppargs.m[i]-1)*log(ghs[i*ncomp+i]);
}
double ares_hc = m_avg*ares_hs - summ;
double ares_disp = -2*PI*den*I1*m2es3 - PI*den*m_avg*C1*I2*m2e2s3;
summ = 0.0;
for (int i = 0; i < ncomp; i++) {
summ += x[i]*(cppargs.m[i]-1)/ghs[i*ncomp+i]*denghs[i*ncomp+i];
}
double Zhc = m_avg*Zhs - summ;
double Zdisp = -2*PI*den*detI1_det*m2es3 - PI*den*m_avg*(C1*detI2_det + C2*eta*I2)*m2e2s3;
vector<double> dghsii_dx(ncomp*ncomp, 0);
vector<double> dahs_dx(ncomp, 0);
vector<double> dzeta_dx(4, 0);
idx = -1;
for (int i = 0; i < ncomp; i++) {
for (int l = 0; l < 4; l++) {
dzeta_dx[l] = PI/6.*den*cppargs.m[i]*pow(d[i],l);
}
for (int j = 0; j < ncomp; j++) {
idx += 1;
dghsii_dx[idx] = dzeta_dx[3]/(1-zeta[3])/(1-zeta[3]) + (d[j]*d[j]/(d[j]+d[j]))*
(3*dzeta_dx[2]/(1-zeta[3])/(1-zeta[3]) + 6*zeta[2]*dzeta_dx[3]/pow(1-zeta[3],3))
+ pow(d[j]*d[j]/(d[j]+d[j]),2)*(4*zeta[2]*dzeta_dx[2]/pow(1-zeta[3],3)
+ 6*zeta[2]*zeta[2]*dzeta_dx[3]/pow(1-zeta[3],4));
}
dahs_dx[i] = -dzeta_dx[0]/zeta[0]*ares_hs + 1/zeta[0]*(3*(dzeta_dx[1]*zeta[2]
+ zeta[1]*dzeta_dx[2])/(1-zeta[3]) + 3*zeta[1]*zeta[2]*dzeta_dx[3]
/(1-zeta[3])/(1-zeta[3]) + 3*zeta[2]*zeta[2]*dzeta_dx[2]/zeta[3]/(1-zeta[3])/(1-zeta[3])
+ pow(zeta[2],3)*dzeta_dx[3]*(3*zeta[3]-1)/zeta[3]/zeta[3]/pow(1-zeta[3],3)
+ log(1-zeta[3])*((3*zeta[2]*zeta[2]*dzeta_dx[2]*zeta[3] -
2*pow(zeta[2],3)*dzeta_dx[3])/pow(zeta[3],3) - dzeta_dx[0]) +
(zeta[0]-pow(zeta[2],3)/zeta[3]/zeta[3])*dzeta_dx[3]/(1-zeta[3]));
}
vector<double> dadisp_dx(ncomp, 0);
vector<double> dahc_dx(ncomp, 0);
double dzeta3_dx, daa_dx, db_dx, dI1_dx, dI2_dx, dm2es3_dx, dm2e2s3_dx, dC1_dx;
for (int i = 0; i < ncomp; i++) {
dzeta3_dx = PI/6.*den*cppargs.m[i]*pow(d[i],3);
dI1_dx = 0.0;
dI2_dx = 0.0;
dm2es3_dx = 0.0;
dm2e2s3_dx = 0.0;
for (int l = 0; l < 7; l++) {
daa_dx = cppargs.m[i]/m_avg/m_avg*a1[l] + cppargs.m[i]/m_avg/m_avg*(3-4/m_avg)*a2[l];
db_dx = cppargs.m[i]/m_avg/m_avg*b1[l] + cppargs.m[i]/m_avg/m_avg*(3-4/m_avg)*b2[l];
dI1_dx += a[l]*l*dzeta3_dx*pow(eta,l-1) + daa_dx*pow(eta,l);
dI2_dx += b[l]*l*dzeta3_dx*pow(eta,l-1) + db_dx*pow(eta,l);
}
for (int j = 0; j < ncomp; j++) {
dm2es3_dx += x[j]*cppargs.m[j]*(e_ij[i*ncomp+j]/t)*pow(s_ij[i*ncomp+j],3);
dm2e2s3_dx += x[j]*cppargs.m[j]*pow(e_ij[i*ncomp+j]/t,2)*pow(s_ij[i*ncomp+j],3);
dahc_dx[i] += x[j]*(cppargs.m[j]-1)/ghs[j*ncomp+j]*dghsii_dx[i*ncomp+j];
}
dm2es3_dx = dm2es3_dx*2*cppargs.m[i];
dm2e2s3_dx = dm2e2s3_dx*2*cppargs.m[i];
dahc_dx[i] = cppargs.m[i]*ares_hs + m_avg*dahs_dx[i] - dahc_dx[i] - (cppargs.m[i]-1)*log(ghs[i*ncomp+i]);
dC1_dx = C2*dzeta3_dx - C1*C1*(cppargs.m[i]*(8*eta-2*eta*eta)/pow(1-eta,4) -
cppargs.m[i]*(20*eta-27*eta*eta+12*pow(eta,3)-2*pow(eta,4))/pow((1-eta)*(2-eta),2));
dadisp_dx[i] = -2*PI*den*(dI1_dx*m2es3 + I1*dm2es3_dx) - PI*den
*((cppargs.m[i]*C1*I2 + m_avg*dC1_dx*I2 + m_avg*C1*dI2_dx)*m2e2s3
+ m_avg*C1*I2*dm2e2s3_dx);
}
vector<double> mu_hc(ncomp, 0);
vector<double> mu_disp(ncomp, 0);
for (int i = 0; i < ncomp; i++) {
for (int j = 0; j < ncomp; j++) {
mu_hc[i] += x[j]*dahc_dx[j];
mu_disp[i] += x[j]*dadisp_dx[j];
}
mu_hc[i] = ares_hc + Zhc + dahc_dx[i] - mu_hc[i];
mu_disp[i] = ares_disp + Zdisp + dadisp_dx[i] - mu_disp[i];
}
// Dipole term (Gross and Vrabec term) --------------------------------------
vector<double> mu_polar(ncomp, 0);
if (!cppargs.dipm.empty()) {
double A2 = 0.;
double A3 = 0.;
double dA2_det = 0.;
double dA3_det = 0.;
vector<double> dA2_dx(ncomp, 0);
vector<double> dA3_dx(ncomp, 0);
static double a0dip[5] = { 0.3043504, -0.1358588, 1.4493329, 0.3556977, -2.0653308 };
static double a1dip[5] = { 0.9534641, -1.8396383, 2.0131180, -7.3724958, 8.2374135 };
static double a2dip[5] = { -1.1610080, 4.5258607, 0.9751222, -12.281038, 5.9397575 };
static double b0dip[5] = { 0.2187939, -1.1896431, 1.1626889, 0, 0 };
static double b1dip[5] = { -0.5873164, 1.2489132, -0.5085280, 0, 0 };
static double b2dip[5] = { 3.4869576, -14.915974, 15.372022, 0, 0 };
static double c0dip[5] = { -0.0646774, 0.1975882, -0.8087562, 0.6902849, 0 };
static double c1dip[5] = { -0.9520876, 2.9924258, -2.3802636, -0.2701261, 0 };
static double c2dip[5] = { -0.6260979, 1.2924686, 1.6542783, -3.4396744, 0 };
const static double conv = 7242.702976750923; // conversion factor, see the note below Table 2 in Gross and Vrabec 2006
vector<double> dipmSQ (ncomp, 0);
for (int i = 0; i < ncomp; i++) {
dipmSQ[i] = pow(cppargs.dipm[i], 2.)/(cppargs.m[i]*cppargs.e[i]*pow(cppargs.s[i],3.))*conv;
}
vector<double> adip (5, 0);
vector<double> bdip (5, 0);
vector<double> cdip (5, 0);
double J2, dJ2_det, detJ2_det, J3, dJ3_det, detJ3_det;
double m_ij;
double m_ijk;
for (int i = 0; i < ncomp; i++) {
for (int j = 0; j < ncomp; j++) {
m_ij = sqrt(cppargs.m[i]*cppargs.m[j]);
if (m_ij > 2) {
m_ij = 2;
}
J2 = 0.;
dJ2_det = 0.;
detJ2_det = 0;
for (int l = 0; l < 5; l++) {
adip[l] = a0dip[l] + (m_ij-1)/m_ij*a1dip[l] + (m_ij-1)/m_ij*(m_ij-2)/m_ij*a2dip[l];
bdip[l] = b0dip[l] + (m_ij-1)/m_ij*b1dip[l] + (m_ij-1)/m_ij*(m_ij-2)/m_ij*b2dip[l];
J2 += (adip[l] + bdip[l]*e_ij[i*ncomp+j]/t)*pow(eta, l); // i*ncomp+j needs to be used for e_ij because it is formatted as a 1D vector
dJ2_det += (adip[l] + bdip[l]*e_ij[i*ncomp+j]/t)*l*pow(eta, l-1);
detJ2_det += (adip[l] + bdip[l]*e_ij[i*ncomp+j]/t)*(l+1)*pow(eta, l);
}
A2 += x[i]*x[j]*e_ij[i*ncomp+i]/t*e_ij[j*ncomp+j]/t*pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)/
pow(s_ij[i*ncomp+j],3)*cppargs.dip_num[i]*cppargs.dip_num[j]*dipmSQ[i]*dipmSQ[j]*J2;
dA2_det += x[i]*x[j]*e_ij[i*ncomp+i]/t*e_ij[j*ncomp+j]/t*pow(s_ij[i*ncomp+i],3)*
pow(s_ij[j*ncomp+j],3)/pow(s_ij[i*ncomp+j],3)*cppargs.dip_num[i]*cppargs.dip_num[j]*dipmSQ[i]*dipmSQ[j]*detJ2_det;
if (i == j) {
dA2_dx[i] += e_ij[i*ncomp+i]/t*e_ij[j*ncomp+j]/t*pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)
/pow(s_ij[i*ncomp+j],3)*cppargs.dip_num[i]*cppargs.dip_num[j]*dipmSQ[i]*dipmSQ[j]*
(x[i]*x[j]*dJ2_det*PI/6.*den*cppargs.m[i]*pow(d[i],3) + 2*x[j]*J2);
}
else {
dA2_dx[i] += e_ij[i*ncomp+i]/t*e_ij[j*ncomp+j]/t*pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)
/pow(s_ij[i*ncomp+j],3)*cppargs.dip_num[i]*cppargs.dip_num[j]*dipmSQ[i]*dipmSQ[j]*
(x[i]*x[j]*dJ2_det*PI/6.*den*cppargs.m[i]*pow(d[i],3) + x[j]*J2);
}
for (int k = 0; k < ncomp; k++) {
m_ijk = pow((cppargs.m[i]*cppargs.m[j]*cppargs.m[k]),1/3.);
if (m_ijk > 2) {
m_ijk = 2;
}
J3 = 0.;
dJ3_det = 0.;
detJ3_det = 0.;
for (int l = 0; l < 5; l++) {
cdip[l] = c0dip[l] + (m_ijk-1)/m_ijk*c1dip[l] + (m_ijk-1)/m_ijk*(m_ijk-2)/m_ijk*c2dip[l];
J3 += cdip[l]*pow(eta, l);
dJ3_det += cdip[l]*l*pow(eta, (l-1));
detJ3_det += cdip[l]*(l+2)*pow(eta, (l+1));
}
A3 += x[i]*x[j]*x[k]*e_ij[i*ncomp+i]/t*e_ij[j*ncomp+j]/t*e_ij[k*ncomp+k]/t*
pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)*pow(s_ij[k*ncomp+k],3)/s_ij[i*ncomp+j]/s_ij[i*ncomp+k]/
s_ij[j*ncomp+k]*cppargs.dip_num[i]*cppargs.dip_num[j]*cppargs.dip_num[k]*dipmSQ[i]*
dipmSQ[j]*dipmSQ[k]*J3;
dA3_det += x[i]*x[j]*x[k]*e_ij[i*ncomp+i]/t*e_ij[j*ncomp+j]/t*e_ij[k*ncomp+k]/t*
pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)*pow(s_ij[k*ncomp+k],3)/s_ij[i*ncomp+j]/s_ij[i*ncomp+k]/
s_ij[j*ncomp+k]*cppargs.dip_num[i]*cppargs.dip_num[j]*cppargs.dip_num[k]*dipmSQ[i]*
dipmSQ[j]*dipmSQ[k]*detJ3_det;
if ((i == j) && (i == k)) {
dA3_dx[i] += e_ij[i*ncomp+i]/t*e_ij[j*ncomp+j]/t*e_ij[k*ncomp+k]/t*pow(s_ij[i*ncomp+i],3)
*pow(s_ij[j*ncomp+j],3)*pow(s_ij[k*ncomp+k],3)/s_ij[i*ncomp+j]/s_ij[i*ncomp+k]/s_ij[j*ncomp+k]
*cppargs.dip_num[i]*cppargs.dip_num[j]*cppargs.dip_num[k]*dipmSQ[i]*dipmSQ[j]
*dipmSQ[k]*(x[i]*x[j]*x[k]*dJ3_det*PI/6.*den*cppargs.m[i]*pow(d[i],3)
+ 3*x[j]*x[k]*J3);
}
else if ((i == j) || (i == k)) {
dA3_dx[i] += e_ij[i*ncomp+i]/t*e_ij[j*ncomp+j]/t*e_ij[k*ncomp+k]/t*pow(s_ij[i*ncomp+i],3)
*pow(s_ij[j*ncomp+j],3)*pow(s_ij[k*ncomp+k],3)/s_ij[i*ncomp+j]/s_ij[i*ncomp+k]/s_ij[j*ncomp+k]
*cppargs.dip_num[i]*cppargs.dip_num[j]*cppargs.dip_num[k]*dipmSQ[i]*dipmSQ[j]
*dipmSQ[k]*(x[i]*x[j]*x[k]*dJ3_det*PI/6.*den*cppargs.m[i]*pow(d[i],3)
+ 2*x[j]*x[k]*J3);
}
else {
dA3_dx[i] += e_ij[i*ncomp+i]/t*e_ij[j*ncomp+j]/t*e_ij[k*ncomp+k]/t*pow(s_ij[i*ncomp+i],3)
*pow(s_ij[j*ncomp+j],3)*pow(s_ij[k*ncomp+k],3)/s_ij[i*ncomp+j]/s_ij[i*ncomp+k]/s_ij[j*ncomp+k]
*cppargs.dip_num[i]*cppargs.dip_num[j]*cppargs.dip_num[k]*dipmSQ[i]*dipmSQ[j]
*dipmSQ[k]*(x[i]*x[j]*x[k]*dJ3_det*PI/6.*den*cppargs.m[i]*pow(d[i],3)
+ x[j]*x[k]*J3);
}
}
}
}
A2 = -PI*den*A2;
A3 = -4/3.*PI*PI*den*den*A3;
dA2_det = -PI*den/eta*dA2_det;
dA3_det = -4/3.*PI*PI*den/eta*den/eta*dA3_det;
for (int i = 0; i < ncomp; i++) {
dA2_dx[i] = -PI*den*dA2_dx[i];
dA3_dx[i] = -4/3.*PI*PI*den*den*dA3_dx[i];
}
vector<double> dapolar_dx(ncomp);
for (int i = 0; i < ncomp; i++) {
dapolar_dx[i] = (dA2_dx[i]*(1-A3/A2) + (dA3_dx[i]*A2 - A3*dA2_dx[i])/A2)/pow(1-A3/A2,2);
}
if (A2 != 0) { // when the mole fraction of the polar compounds is 0 then A2 = 0 and division by 0 occurs
double ares_polar = A2/(1-A3/A2);
double Zpolar = eta*((dA2_det*(1-A3/A2)+(dA3_det*A2-A3*dA2_det)/A2)/(1-A3/A2)/(1-A3/A2));
for (int i = 0; i < ncomp; i++) {
for (int j = 0; j < ncomp; j++) {
mu_polar[i] += x[j]*dapolar_dx[j];
}
mu_polar[i] = ares_polar + Zpolar + dapolar_dx[i] - mu_polar[i];
}
}
}
// Association term -------------------------------------------------------
vector<double> mu_assoc(ncomp, 0);
if (!cppargs.e_assoc.empty()) {
int num_sites = 0;
vector<int> iA; //indices of associating compounds
for(std::vector<int>::iterator it = cppargs.assoc_num.begin(); it != cppargs.assoc_num.end(); ++it) {
num_sites += *it;
for (int i = 0; i < *it; i++) {
iA.push_back(it - cppargs.assoc_num.begin());
}
}
vector<double> x_assoc(num_sites); // mole fractions of only the associating compounds
for (int i = 0; i < num_sites; i++) {
x_assoc[i] = x[iA[i]];
}
vector<double> XA (num_sites, 0);
vector<double> delta_ij(num_sites * num_sites, 0);
int idxa = 0;
int idxi = 0; // index for the ii-th compound
int idxj = 0; // index for the jj-th compound
for (int i = 0; i < num_sites; i++) {
idxi = iA[i]*ncomp+iA[i];
for (int j = 0; j < num_sites; j++) {
idxj = iA[j]*ncomp+iA[j];
if (cppargs.assoc_matrix[idxa] != 0) {
double eABij = (cppargs.e_assoc[iA[i]]+cppargs.e_assoc[iA[j]])/2.;
double volABij = _HUGE;
if (cppargs.k_hb.empty()) {
volABij = sqrt(cppargs.vol_a[iA[i]]*cppargs.vol_a[iA[j]])*pow(sqrt(s_ij[idxi]*
s_ij[idxj])/(0.5*(s_ij[idxi]+s_ij[idxj])), 3);
}
else {
volABij = sqrt(cppargs.vol_a[iA[i]]*cppargs.vol_a[iA[j]])*pow(sqrt(s_ij[idxi]*
s_ij[idxj])/(0.5*(s_ij[idxi]+s_ij[idxj])), 3)*(1-cppargs.k_hb[iA[i]*ncomp+iA[j]]);
}
delta_ij[idxa] = ghs[iA[i]*ncomp+iA[j]]*(exp(eABij/t)-1)*pow(s_ij[iA[i]*ncomp+iA[j]], 3)*volABij;
}
idxa += 1;
}
XA[i] = (-1 + sqrt(1+8*den*delta_ij[i*num_sites+i]))/(4*den*delta_ij[i*num_sites+i]);
if (!std::isfinite(XA[i])) {
XA[i] = 0.02;
}
}
vector<double> ddelta_dx(num_sites * num_sites * ncomp, 0);
int idx_ddelta = 0;
for (int k = 0; k < ncomp; k++) {
int idxi = 0; // index for the ii-th compound
int idxj = 0; // index for the jj-th compound
idxa = 0;
for (int i = 0; i < num_sites; i++) {
idxi = iA[i]*ncomp+iA[i];
for (int j = 0; j < num_sites; j++) {
idxj = iA[j]*ncomp+iA[j];
if (cppargs.assoc_matrix[idxa] != 0) {
double eABij = (cppargs.e_assoc[iA[i]]+cppargs.e_assoc[iA[j]])/2.;
double volABij = _HUGE;
if (cppargs.k_hb.empty()) {
volABij = sqrt(cppargs.vol_a[iA[i]]*cppargs.vol_a[iA[j]])*pow(sqrt(s_ij[idxi]*
s_ij[idxj])/(0.5*(s_ij[idxi]+s_ij[idxj])), 3);
}
else {
volABij = sqrt(cppargs.vol_a[iA[i]]*cppargs.vol_a[iA[j]])*pow(sqrt(s_ij[idxi]*
s_ij[idxj])/(0.5*(s_ij[idxi]+s_ij[idxj])), 3)*(1-cppargs.k_hb[iA[i]*ncomp+iA[j]]);
}
double dghsd_dx = PI/6.*cppargs.m[k]*(pow(d[k], 3)/(1-zeta[3])/(1-zeta[3]) + 3*d[iA[i]]*d[iA[j]]/
(d[iA[i]]+d[iA[j]])*(d[k]*d[k]/(1-zeta[3])/(1-zeta[3])+2*pow(d[k], 3)*
zeta[2]/pow(1-zeta[3], 3)) + 2*pow((d[iA[i]]*d[iA[j]]/(d[iA[i]]+d[iA[j]])), 2)*
(2*d[k]*d[k]*zeta[2]/pow(1-zeta[3], 3)+3*(pow(d[k], 3)*zeta[2]*zeta[2]
/pow(1-zeta[3], 4))));
ddelta_dx[idx_ddelta] = dghsd_dx*(exp(eABij/t)-1)*pow(s_ij[iA[i]*ncomp+iA[j]], 3)*volABij;
}
idx_ddelta += 1;
idxa += 1;
}
}
}
int ctr = 0;
double dif = 1000.;
vector<double> XA_old = XA;
while ((ctr < 100) && (dif > 1e-15)) {
ctr += 1;
XA = XA_find(XA_old, delta_ij, den, x_assoc);
dif = 0.;
for (int i = 0; i < num_sites; i++) {
dif += std::abs(XA[i] - XA_old[i]);
}
for (int i = 0; i < num_sites; i++) {
XA_old[i] = (XA[i] + XA_old[i]) / 2.0;
}
}
vector<double> dXA_dx(num_sites*ncomp, 0);
dXA_dx = dXAdx_find(cppargs.assoc_num, delta_ij, den, XA, ddelta_dx, x_assoc);
int ij = 0;
for (int i = 0; i < ncomp; i++) {
for (int j = 0; j < num_sites; j++) {
mu_assoc[i] += x[iA[j]]*den*dXA_dx[ij]*(1/XA[j]-0.5);
ij += 1;
}
}
for (int i = 0; i < num_sites; i++) {
mu_assoc[iA[i]] += log(XA[i]) - 0.5*XA[i] + 0.5;
}
}
// Ion term ---------------------------------------------------------------
vector<double> mu_ion(ncomp, 0);
if (!cppargs.z.empty()) {
vector<double> q(cppargs.z.begin(), cppargs.z.end());
for (int i = 0; i < ncomp; i++) {
q[i] = q[i]*E_CHRG;
}
summ = 0.;
for (int i = 0; i < ncomp; i++) {
summ += cppargs.z[i]*cppargs.z[i]*x[i];
}
double kappa = sqrt(den*E_CHRG*E_CHRG/kb/t/(cppargs.dielc*perm_vac)*summ); // the inverse Debye screening length. Equation 4 in Held et al. 2008.
if (kappa != 0) {
vector<double> chi(ncomp);
vector<double> sigma_k(ncomp);
double summ1 = 0.;
double summ2 = 0.;
for (int i = 0; i < ncomp; i++) {
chi[i] = 3/pow(kappa*d[i], 3)*(1.5 + log(1+kappa*d[i]) - 2*(1+kappa*d[i]) +
0.5*pow(1+kappa*d[i], 2));
sigma_k[i] = -2*chi[i]+3/(1+kappa*d[i]);
summ1 += q[i]*q[i]*x[i]*sigma_k[i];
summ2 += x[i]*q[i]*q[i];
}
for (int i = 0; i < ncomp; i++) {
mu_ion[i] = -q[i]*q[i]*kappa/24./PI/kb/t/(cppargs.dielc*perm_vac)*
(2*chi[i] + summ1/summ2);
}
}
}
double Z = pcsaft_Z_cpp(t, rho, x, cppargs);
vector<double> mu(ncomp, 0);
vector<double> lnfugcoef(ncomp, 0);
for (int i = 0; i < ncomp; i++) {
mu[i] = mu_hc[i] + mu_disp[i] + mu_polar[i] + mu_assoc[i] + mu_ion[i];
lnfugcoef[i] = mu[i] - log(Z); // the natural logarithm of the fugacity coefficient
}
return lnfugcoef;
}
vector<double> pcsaft_fugcoef_cpp(double t, double rho, vector<double> x, add_args &cppargs) {
/**
Calculate the fugacity coefficients for one phase of the system.
*/
int ncomp = x.size(); // number of components
vector<double> lnfug = pcsaft_lnfug_cpp(t, rho, x, cppargs);
vector<double> fugcoef(ncomp, 0);
for (int i = 0; i < ncomp; i++) {
fugcoef[i] = exp(lnfug[i]); // the fugacity coefficients
}
return fugcoef;
}
double pcsaft_p_cpp(double t, double rho, vector<double> x, add_args &cppargs) {
/**
Calculate pressure
*/
double den = rho*N_AV/1.0e30;
double Z = pcsaft_Z_cpp(t, rho, x, cppargs);
double P = Z*kb*t*den*1.0e30; // Pa
return P;
}
double pcsaft_ares_cpp(double t, double rho, vector<double> x, add_args &cppargs) {
/**
Calculate the residual Helmholtz energy
*/
int ncomp = x.size(); // number of components
vector<double> d (ncomp);
for (int i = 0; i < ncomp; i++) {
d[i] = cppargs.s[i]*(1-0.12*exp(-3*cppargs.e[i]/t));
}
if (!cppargs.z.empty()) {
for (int i = 0; i < ncomp; i++) {
if (cppargs.z[i] != 0) {
d[i] = cppargs.s[i]*(1-0.12); // for ions the diameter is assumed to be temperature independent (see Held et al. 2014)
}
}
}
double den = rho*N_AV/1.0e30;
vector<double> zeta (4, 0);
double summ;
for (int i = 0; i < 4; i++) {
summ = 0;
for (int j = 0; j < ncomp; j++) {