TTS {TTS}

Time-Temperature Superposition (TTS) analysis

Description

The Master Curve at a specific temperature is estimated using Time-Temperature Superposition (TTS) procedures. The Master Curve means the variation of a specific viscoelastic property of the selected material depending on time or frequency. TTS procedures provide the viscoelastic property variation at the selected temperature in a wider interval of time or frequency than in the experimental case. The Master Curve is estimated for each selected reference temperaure using TTS procedures. Three TTS methodologies are implemented in this package: the two wider used methods, Arrhenius based methods and WLF, and the newer methodology based on derivatives procedure. The Master Curve is smoothed by B-splines basis.

Usage

TTS(
  x,
  reference.temperature = 150,
  n = 100,
  nB = 100,
  method = c("derived", "WLF", "Arrhenius")
)

Arguments

x

Matrix or data frame composed of three columns: a numeric column vector with the experimental frequencies (in logarithmic scale, base-ten), the modulus (E' or G') base-ten logarithm vector and, finally the corresponding temperatures vector.

reference.temperature

Value of the selected reference temperatura at which the Master Curve of modulus will be obtained, the default value of temperature is 150.

n

Number of partitions in the frequency domain in order to fit the B-spline basis. The default value is 100.

nB

Number of bootstrap replicates to estimate confidence intervals of master curve fitting. The default is 100.

method

A string vector composed of one of the following options: "derived" (by default), "WLF" and "Arrhenius".

Details

El New method for estimating shift factors in time-temperatura superposition models (Naya et al., 2013) opens the possibility to perform the TTS function. The horizontal and vertical shifts are calculated. Namely, the different methods are differenciated due to the expression for estimating the horizontal shifts, aT. The "derivated" method is based on the application of horizontal shifts to the moduli derivatives (depending on the frequency) and thus obtaining the Master Curve at the selected temperature:

(dE')/dx(x+aT) -> (dE')/dx(x)

WLF method is defined by the parametric expression:

Log10(aT)=-C1*(T-To)/(C2+(T-To))

Where C1 and C2 are constants to be estimated, T is the temperature and To the reference temperature.

Arrhenius method is defined by the parametric expression:

Log10(aT)=Ea*((1/T)-(1/To))*log10(2.718282)/R

Where Ea is the activation energy, R = 8.314 J/mol is the universal gas constant, T is the absolute temperature (Kelvin degrees), and To the reference temperature (Celsius degrees).

The vertical shifts, bT, are calculated taking into account the vertical distance between the moduli curves.

Value

The function returns a list composed of the following outputs:

data

Input experimental data.

aT

Numerical vector of horizontal shifts between the modulus curves.

bT

Numerical vector of vertical shifts between the modulus curves.

TTS.data

Master Curve Data frame defined by three columns: log10frequency, log10module and temperature.

ref.temp

Reference temperature value.

TTS.gam

Data frame of the Generalized Additive Model with B-splines (GAM) estimate of the Master Curve. It contains two columns: frequency and Prediction.

I.lower

Lower limit of bootstrap confidence interval corresponding to the estimated B-splines Master Curve.

I.upper

Upper limit of bootstrap confidence interval corresponding to the estimated B-splines Master Curve.

residuals

Residuals corresponding to the GAM with B-splines Master Curve fitting.

Author(s)

Antonio Meneses [e-mail], Salvador Naya [e-mail] and Javier Tarrio-Saavedra [e-mail]

References

Naya, S., Meneses A., Tarrio-Saavedra, J., Artiaga R., Lopez-Beceiro, J. and Gracia-Fernandez C. (2013) New method for estimating shift factors in time-temperatura superposition models. Journal of Thermal Analysis and Calorimetry. ISSN 1388-6150. DOI 10.1007/s10973-013-3193-1.

Williams, M. L. (1964) Structural analysis of Viscoelastic materials. AIAA Journal, 785-808.

Artiaga R., Garcia A. Fundamentals of DMA. In: 'Thermal analysis. Fundamentals and applications to material characterization' (ed.: Artiaga R.) Publicaciones de la Universidade da Coruna, A Coruna, Spain, 183-206 (2005).

Examples

## Polycarbonate dataset
data(PC)
x=PC
## TTS function applied to polycarbonate.
derive=TTS(x,reference.temperature=150, method=c("derived","WLF","Arrhenius"))
names(derive)
##[1] "data"      "aT"        "bT"        "TTS.data"  "ref.temp"  "TTS.gam"
##[7] "I.lower"   "I.upper"   "residuals"
## Horizontal shifts vector of modulus versus frequency curves.
derive$aT
## Reference temperature
derive$ref.temp