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```
# CHNOSZ/wjd.R
# Gibbs energy minimization and supporting functions
# 20111117 jmd
wjd <- function(
# example problem definition
# the formula matrix: composition of the species with elements H, N, O
A = matrix(c(
1,2,2,0,0,1,0,0,0,1,
0,0,0,1,2,1,1,0,0,0,
0,0,1,0,0,0,1,1,2,1),ncol=3,
dimnames=list(NULL,c("H","N","O"))),
# the free energies, G0/RT, at 3500 K
G0.RT = c(
-10.021,-21.096,-37.986,-9.846,-28.653,
-18.918,-28.032,-14.640,-30.594,-26.111),
# initial solution, a positive set of values (numbers of moles)
Y = c(0.1,0.35,0.5,0.1,0.35,0.1,0.1,0.1,0.1,0.1),
# pressure in atmospheres
P = 51,
# number of values of lambda tested at each iteration
nlambda = 101,
# maximum number of iterations
imax = 10,
# the free energy change, as a fraction of the total system energy
# in the current step, below which the iterations will stop
Gfrac = 1e-7
) {
# Gibbs energy minimization
# using steepest descent approach of
# White, Johnson and Dantzig, 1958
# J. Chem. Phys. 28(5), 751-755
# doi:10.1063/1.1744264
# trying to make it faster than version coded 2005-02-10
# notation notes: i for species, j for elements
# use G0 here instead of F0 to stand for standard Gibbs free energy
## most common incorrect call is to have non-positive starting mole numbers
if(any(Y <= 0)) stop("all mole numbers of initial solution (Y) must be positive")
## begin function definitions
# Eq. 18 - coefficients in the system of unknowns
# A: formula matrix
# Y: set of mole numbers
coeff <- function(A,Y) {
# m - number of elements (columns)
m <- ncol(A)
# n - number of species (rows)
n <- nrow(A)
# our matrix of r's is m x m
R <- matrix(0,nrow=m,ncol=m)
for(j in 1:m) {
for(k in j:m) {
# Eq. 17 - r coefficients
r <- sum(A[,j] * A[,k] * Y)
# fill in the matrix
R[j,k] <- r
R[k,j] <- r
}
}
# Eq. 4 - total number of (atomic weights?) of element j
B <- Y %*% A
coeff <- cbind(R,t(B))
B <- c(B,0)
coeff <- rbind(coeff,B)
return(coeff)
}
# Eq. 15 - free energies of the species
f.Y <- function(Y,C) return(Y * (C + log(Y/sum(Y))))
# Eq. 18 - rhs of matrix equation
rhs <- function(A,f.Y) {
# m - number of elements (columns)
m <- ncol(A)
# assemble the sum for each element
rhs <- sapply(1:m, function(j) {
sum(A[,j] * f.Y)
})
# final value is sum over species
rhs <- c(rhs,sum(f.Y))
return(rhs)
}
# Eq. 14 - the resulting mole number vector
X <- function(A,Y,f.Y,mults) {
# m - number of elements (columns)
m <- ncol(A)
# n - number of species (rows)
n <- nrow(A)
# third term: the summation over elements, for each species
X3 <- sapply(1:n, function(i) {
sum(mults[1:m] * A[i,]) * Y[i]
})
# second term: ratio of mole vectors
# (cf. Eq. 19)
X2 <- Y * (tail(mults,1) + 1)
# first term: negative of f.Y
X1 <- -f.Y
# put them together
return(X1 + X2 + X3)
}
## end function definitions
# now set up up the calculation
# keep the initial solution around
Y.0 <- Y
# following Eq. 2
# G0.RT: standard Gibbs free energies
# P: pressure in atmospheres
C <- G0.RT + log(P)
# initialize iteration counter
i <- 0
# initialize system free energy output
F.Y <- numeric()
# initialize fractional distance change output
lambda <- numeric()
# initialize free energy change output
Ffrac <- numeric()
# initialize mass balance results
elements <- t(A) %*% Y
# we iterate
repeat {
# determine f.Y by Eq. 15
f.Y.1 <- f.Y(Y,C)
# compute system free energy
F.Y <- c(F.Y, sum(f.Y.1))
# don't surpass the maximum number of iterations
i <- i + 1
if(i > imax) break
# stop if the last iteration changed the free energy by less than required
if(i > 1) {
d.F.Y <- diff(tail(F.Y,2))
Ffrac <- c(Ffrac, abs(d.F.Y / tail(F.Y,1)))
if(tail(Ffrac,1) < Gfrac) break
}
# set up the system of equations
coeff.1 <- coeff(A,Y)
rhs.1 <- rhs(A,f.Y.1)
# solve the system to get the Lagrange multipliers
mults <- solve(coeff.1,rhs.1)
# get the new (possibly negative) mole values
X.1 <- X(A,Y,f.Y.1,mults)
# what are the directional changes
D <- X.1 - Y
# lambda is the fractional amount we go along that direction;
# WJD58 give two constraints but no specific exploration procedure.
# first constraint is that all mole numbers are positive.
if(any(X.1 < 0)) {
# find the lowest value of lambda where any mole
# number becomes zero (the species is zapped, not allowed!)
lam <- -Y/D
lam[lam < 0] <- 1
lamzap <- min(lam)
lastlam <- nlambda - 1
} else {
# all mole numbers are positive; we can take it all the way
lamzap <- 1
lastlam <- nlambda
}
# let's explore lambda between 0 and lamzap
# (including lamzap if it's 1)
lams <- seq(0,lamzap,length.out=nlambda)[1:lastlam]
# second constraint is that the derivative of free energy
# doesn't go positive ... Eq. 20
d.f.Y <- function(lambda,Y,D,C) {
d.f.Y <- sum(D * (C + log( (Y + lambda * D) / (sum(Y) + lambda * sum(D)) )))
return(d.f.Y)
}
# what are the free energy derivatives
d.f.Y.1 <- sapply(lams,d.f.Y,Y=Y,D=D,C=C)
# if any are positive, exclude those lambdas
lams[d.f.Y.1 > 0] <- 0
# take the highest lambda
lambda <- c(lambda,max(lams))
# we now have lambda, so we can calculate a new value for Y
Y <- Y + tail(lambda,1) * D
# it might be wise to check the mass balance
elements <- cbind(elements,t(A) %*% Y)
# next iteration
}
# the result is in 'X' to be consistent with notation of WJD58
X <- Y
# the initial solution is in 'Y'
Y <- Y.0
# return problem definition and results
return(list(A=A,G0.RT=G0.RT,Y=Y,P=P,X=X,F.Y=F.Y,lambda=lambda,Ffrac=Ffrac,elements=elements))
}
element.potentials <- function(w, plot.it=FALSE, iplot=1:ncol(w$A)) {
# calculate the chemical potentials of the elements
# from the output of wjd(), using all or some of the combinations
# of species that are compositionally independent
# 20111126 jmd
# put the species in abundance order
oX <- order(w$X)
# the mole fractions, formulas, and energies of the species in this order
X <- w$X[oX]
A <- w$A[oX,]
G0.RT <- w$G0.RT[oX] + log(w$P)
# get the combinations of species that are compositionally independent
ic <- invertible.combs(A)
# a function to calculate chemical potentials of the elements for the ith combination of species
mu <- function(i) {
myA <- A[ic[i,],]
# chemical potentials (/RT) of the species: G0/RT + ln(mole fraction)
myB <- (G0.RT + log(X/sum(X)))[ic[i,]]
# chemical potentials of the elements
myX <- solve(myA,myB)
}
# run the calculation over all combinations
ep <- t(sapply(1:nrow(ic),mu))
# keep names of the elements
colnames(ep) <- colnames(w$A)
# to make a plot
if(plot.it) {
par(mfrow=c(length(iplot),1))
for(i in iplot) {
ylab <- as.expression(substitute(mu[x]/RT,list(x=colnames(ep)[i])))
plot(ep[,i],xlab="species combination",ylab=ylab, pch=19)
title(main=paste("max difference (range) =",format(diff(range(ep[,i])),digits=2)))
}
}
return(ep)
}
is.near.equil <- function(w, tol=0.01, quiet=FALSE) {
# given the output of wjd(), make a simple test for equilibrium
# that is, that the chemical potentials of the elements are nearly
# the same when calculated using different sets of species in the system
ep <- element.potentials(w)
# stop if we don't have at least two combinations
if(nrow(ep) < 2) stop("can not test for equilibrium because species abundances are determined")
# equilibrium unless proven guilty
ine <- TRUE
for(i in 1:ncol(ep)) if(diff(range(ep[,i])) > tol) ine <- FALSE
if(!ine & !quiet) {
# talk about the differences in chemical potentials
epdiff <- abs(apply(apply(ep, 2, range), 2, diff))
imax <- which.max(epdiff)
message("is.near.equil: solution has variation of ", epdiff[imax], " in mu/RT of ", names(epdiff)[imax])
}
return(ine)
}
guess <- function(
A = matrix(c(
1,2,2,0,0,1,0,0,0,1,
0,0,0,1,2,1,1,0,0,0,
0,0,1,0,0,0,1,1,2,1),ncol=3,
dimnames=list(NULL,c("H","N","O"))),
B = c(2,1,1), method="stoich", minX=0.001, iguess=1, ic=NULL
){
# given the elemental stoichiometries of a set of species (A)
# and the number of moles of elements (B)
# find moles of species that satisfy mass balance and are all positive
# generally this will be one of the solutions of an underdetermined system
# first of all, we can't do anything if all there are no elements
if(all(B==0)) stop("there are zero moles of all elements")
# if method="central" get central solution using limSolve package 20120919
if(identical(method, "central")) {
if(!"limSolve" %in% row.names(installed.packages())) {
message("guess: skipping 'central' method as limSolve package is not available")
} else {
# the inequality constraints for moles of species
G <- diag(nrow(A))
# minX is the minimum mole number we will accept
H <- rep(minX, nrow(A))
# get a solution
X <- limSolve::xranges(E=t(A), F=B, G=G, H=H, central=TRUE, full=TRUE)[, "central"]
return(X)
}
}
if(identical(method, "stoich")) {
# if method="stoich" use a stoichiometric approach: 20111231 jmd
# - select one of the (many) species combinations (ic) that
# make a square, invertible stoichiometric matrix (the "variable" species)
# - assign equal mole numbers to all the "other" species (Xother),
# such that any element has at most max.frac fraction of the desired amount (B)
# (max.frac is scanned from 0.01 to 0.99)
# - calculate the mole numbers of the stoichiometry-setting species
# that give the desired elemental composition; accept the provisional
# solution if all numbers are positive
# arguments:
# A - the stoichiometric matrix
# B - the moles of elements
# iguess - which provisional guess to return (NULL for all)
# ic - which specific combination of species to test (NULL for all)
# get the various combinations of species that are
# stoichiometrically independent
combs <- invertible.combs(A)
# we will potentially process all of them unless a specific one is identified
if(is.null(ic)) ic <- 1:nrow(combs)
# a counter to keep track of the provisional guesses
iprov <- 0
# where to store the output if we want all guesses
out <- list()
for(i in ic) {
# which species are the variable ones
ivar <- combs[i,]
# moles of elements for one mole of all of the other species
Bother.1 <- colSums(A[-ivar, , drop=FALSE])
# which element is proportionally most highly represented w.r.t. the desired composition
imax <- which.max(Bother.1/B)
# max.frac - the highest fraction of contribution to moles of elements by the "other" species
for(max.frac in c(0.01, 0.05, 0.1, 0.25, 0.5, 0.75, 0.9, 0.95, 0.99)) {
# the number of moles of all "other" species that give max.frac of any element
Xother <- max.frac/(Bother.1/B)[imax]
# moles of elements for this number of moles of all the other species
Bother <- Bother.1 * Xother
# moles of elements that are left for the variable species
Bvar <- B - Bother
# now solve for the number of moles of the variable species
Xvar <- solve(t(A[ivar,]),Bvar)
# stop the search if we found a positive solution
if(all(Xvar > 0)) break
}
# put them together
X <- numeric(nrow(A))
X[-ivar] <- Xother
X[ivar] <- Xvar
# add names
names(X) <- rownames(A)
# if all the moles are positive, this is a provisional
# guess, otherwise make the result NA
if(any(Xvar <= 0)) X <- NA
else iprov <- iprov + 1
# return the result if we're at the correct guess number
if(is.null(iguess)) out <- c(out,list(X))
else if(iprov==iguess) return(X)
}
# if we're here, we should return all guesses, or
# make an error (the requested guess number doesn't exist)
if(is.null(iguess) & iprov > 0) return(out)
else {
if(is.null(iguess)) iguess <- "[ALL]"
stop(paste("you asked for guess number ",iguess,
" but there are only ",iprov,
" that satisfy all stoichiometric constraints",sep=""))
}
}
# if we're here we didn't find a guessing method
stop("no method found")
}
run.wjd <- function(ispecies, B=NULL, method="stoich", Y=run.guess(ispecies, B, method),
P=1, T=25, nlambda=101, imax=10, Gfrac=1e-7, tol=0.01) {
### set up a Gibbs energy minimization
### using compositions and standard Gibbs energies of species
### from database in CHNOSZ 20120101 jmd
## get the stoichiometric matrix for the species
A <- i2A(ispecies)
## assemble the standard molal Gibbs energies of the species
s <- subcrt(ispecies, P=P, T=T, property="G", exceed.Ttr=TRUE)
G0 <- sapply(1:length(s$out), function(i) s$out[[i]]$G)
R <- 1.9872 # gas constant, cal K^-1 mol^-1
G0.RT <- G0/R/convert(T, "K")
## if Y is provided use that as initial guess
if(!missing(Y)) {
# giving both Y and B is not allowed
if(!is.null(B)) stop("Y and B can not both be provided")
# the length of Y must be equal to number of species
if(length(Y) != nrow(A)) stop("Y must have same length as number of species")
# a single guess
w <- wjd(A, G0.RT, Y, P=P, nlambda=nlambda, imax=imax, Gfrac=Gfrac)
} else {
# if we're using method "central" there is only one guess
if(method=="central") {
w <- wjd(A, G0.RT, Y, P=P, nlambda=nlambda, imax=imax, Gfrac=Gfrac)
} else {
# for method "stoich" loop over all the guesses created by run.guess
Y <- Y[!is.na(Y)]
for(i in 1:length(Y)) {
w <- wjd(A, G0.RT, Y[[i]], P=P, nlambda=nlambda, imax=imax, Gfrac=Gfrac)
if(is.near.equil(w, tol=tol)) {
message("run.wjd: got within tolerance on initial solution ", i, " of ", length(Y))
break
}
if(i==length(Y)) message("run.wjd: tested ", length(Y), " initial solutions")
}
}
# only return a near equilibrium solution
if(!is.near.equil(w, tol=tol)) {
stop(paste("couldn't find a solution within mu/RT tolerance of", tol))
}
}
return(w)
}
run.guess <- function(ispecies, B=NULL, method="stoich", iguess=NULL) {
## run guess() using species from database 20120612
# get the stoichiometric matrix for the species
A <- i2A(ispecies)
# we need B
if(is.null(B)) stop(paste("please provide B (numbers of moles of ", paste(colnames(A), collapse=", ") , ")", sep=""))
# if B is a formula, turn it into a vector
if(is.character(B)) {
# zero formula with elements in same order as A
zero <- paste(colnames(A), "0", collapse="", sep="")
# sum of zero and B
mB <- makeup(c(zero, B), sum=TRUE)
# then turn it into a vector
B <- as.numeric(unlist(mB))
} else B <- as.vector(B)
# setup initial guess
Y <- guess(A, B, method=method, iguess=iguess)
# take away NA guesses
Y <- Y[!is.na(Y)]
return(Y)
}
equil.potentials <- function(w, tol=0.01, T=25) {
## return the average of the element.potentials, only if w is.near.equil 20120613
R <- 1.9872 # gas constant, cal K^-1 mol^-1
if(!is.near.equil(w, tol=tol)) return(NULL)
else return(colMeans(element.potentials(w)) * R * convert(T, "K"))
}
```

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