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#' @title Flory-Huggins Isotherm Non-Linear Analysis
#' @name floryhugginsanalysis
#' @description Flory-Huggins isotherm model describes the degree of surface coverage
#' characteristics of the adsorbate on the adsorbent. It describes the nature of the
#' adsorption process regarding the feasibility and spontaneity of the process. The theory
#' of the Flory-Huggins provides the mathematical model for the polymer blends'
#' thermodynamics.
#' @param Ce is equal to Co which is the numeric value for the initial concentration
#' @param theta is the fractional surface coverage
#' @import nls2
#' @import Metrics
#' @import stats
#' @import ggplot2
#' @return the nonlinear regression, parameters for Flory-Huggins isotherm, and model
#' error analysis
#' @examples theta <- c(0.19729, 0.34870, 0.61475, 0.74324, 0.88544, 0.89007, 0.91067, 0.91067, 0.96114)
#' @examples Ce <- c(0.01353, 0.04648, 0.13239, 0.27714, 0.41600, 0.63607, 0.80435, 1.10327, 1.58223)
#' @examples floryhugginsanalysis(Ce, theta)
#' @author Jemimah Christine L. Mesias
#' @author Chester C. Deocaris
#' @references Flory, P. J. (1971). Principles of polymer chemistry. Cornell Univ.Pr.
#' @references Foo, K. Y., and Hameed, B. H. (2009, September 13).
#' <doi:10.1016/j.cej.2009.09.013> Insights into the modeling of adsorption isotherm
#' systems. Chemical Engineering Journal.
#' @export
#'
# Building the Flory-Huggins isotherm nonlinear form
floryhugginsanalysis <- function(Ce, theta){
x <- theta
y <- Ce
data <- data.frame(x, y)
# Flory-Huggins isotherm nonlinear equation
fit1 <- y ~ x/(KFH*(1-x)^nFH)
# Setting of starting values
start1 <- list(KFH = 100 , nFH = 1)
# Fitting of the Flory-Huggins isotherm via nls2
fit2 <- nls2::nls2(fit1, start = start1, data=data,
control= nls.control(maxiter= 100, warnOnly=TRUE),
algorithm= "port")
print("Flory HUggins Parameters")
print(summary(fit2))
print("Akaike Information Criterion")
print(AIC(fit2))
print("Bayesian Information Criterion")
print(BIC(fit2))
# Error analysis of the Flory-Huggins isotherm model
errors <- function(y) {
rmse <- Metrics::rmse(y, predict(fit2))
mae <- Metrics::mae(y, predict(fit2))
mse <- Metrics::mse(y, predict(fit2))
rae <- Metrics::rae(y, predict(fit2))
N <- nrow(na.omit(data))
SE <- sqrt((sum(y-predict(fit2))^2)/(N-2))
colnames(y) <- rownames(y) <- colnames(y)
list("Relative Mean Squared Error" = rmse,
"Mean Absolute Error" = mae,
"Mean Squared Error" = mse,
"Relative Absolute Error" = rae,
"Standard Error for the Regression S" = SE)
}
a <- errors(y)
print(a)
rsqq <- lm(theta~predict(fit2))
print(summary(rsqq))
# Graphical representation of the Flory-Huggins isotherm model
### Predicted parameter values
parsFloryHuggins <- as.vector(coefficients(fit2))
pars_KFH <- parsFloryHuggins[1L];
pars_nFH <- parsFloryHuggins[2L];
rhs <- function(x) (x/(pars_KFH*(1-x)^pars_nFH))
#### Plot details
ggplot2::theme_set(ggplot2::theme_bw(10))
ggplot2::ggplot(data, ggplot2::aes(x = x, y = y)) + ggplot2::geom_point(color ="#3498DB" ) +
ggplot2::geom_function(color = "#D35400", fun = rhs ) +
ggplot2::labs(x = expression(paste(theta)),
y = "Ce",
title = "Flory-Huggins Isotherm Nonlinear Model",
caption = "PUPAIM") +
ggplot2::theme(plot.title=ggplot2::element_text(hjust = 0.5)) + ggplot2::coord_flip()
}
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