pyvisco.shift

Collection of functions to apply the time-temperature superposition principle to create a master curve from measurements performed at different temperatures.

Functions

WLF(Temp, RefT, WLF_C1, WLF_C2)	Calculate the Williams-Landel-Ferry (WLF) equation [1].
fit_WLF(RefT, df_aT)	Fit the Williams-Landel-Ferry (WLF) equation [1] to a set of shift factors.
fit_poly(df_aT)	Fit polynomial functions of degree 1 to 4 to a set of shift factors.
plot(df_aT, df_WLF, df_C)	Plot shift factors and shift functions.
poly1(x, C0, C1)	Calculate a polynomial function of degree 1 with a single variable x.
poly2(x, C0, C1, C2)	Calculate a polynomial function of degree 2 with a single variable x.
poly3(x, C0, C1, C2, C3)	Calculate a polynomial function of degree 3 with a single variable x.
poly4(x, C0, C1, C2, C3, C4)	Calculate a polynomial function of degree 4 with a single variable x.

pyvisco.shift.WLF(Temp, RefT, WLF_C1, WLF_C2)

Calculate the Williams-Landel-Ferry (WLF) equation [1].

Parameters:

• Temp (numeric) – Evaluation temperature of the shift factor.

• RefT (numeric) – Reference temperature chosen to construct the master curve.

• WLF_C1 (numeric) – Empirical constants. (Obtained from fitting the shift factor a_T)

• WLF_C2 (numeric) – Empirical constants. (Obtained from fitting the shift factor a_T)

Returns:

log_aT – The decadic logarithm of the WLF shift factor.

Return type:

numeric

References

[1] Williams, Malcolm L.; Landel, Robert F.; Ferry, John D. (1955). “The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-forming Liquids”. J. Amer. Chem. Soc. 77 (14): 3701-3707. doi:10.1021/ja01619a008

pyvisco.shift.poly1(x, C0, C1)

Calculate a polynomial function of degree 1 with a single variable x.

Parameters:

• x (numeric) – Input variable.

• C0 (numeric) – Polynomial coefficients

• C1 (numeric) – Polynomial coefficients

Returns:

Result of the polynomial function.

Return type:

numeric

pyvisco.shift.poly2(x, C0, C1, C2)

Calculate a polynomial function of degree 2 with a single variable x.

Parameters:

• x (numeric) – Input variable.

• C0 (numeric) – Polynomial coefficients

• C1 (numeric) – Polynomial coefficients

• C2 (numeric) – Polynomial coefficients

Returns:

Result of the polynomial function.

Return type:

numeric

pyvisco.shift.poly3(x, C0, C1, C2, C3)

Calculate a polynomial function of degree 3 with a single variable x.

Parameters:

• x (numeric) – Input variable.

• C0 (numeric) – Polynomial coefficients

• C1 (numeric) – Polynomial coefficients

• C2 (numeric) – Polynomial coefficients

• C3 (numeric) – Polynomial coefficients

Returns:

Result of the polynomial function.

Return type:

numeric

pyvisco.shift.poly4(x, C0, C1, C2, C3, C4)

Calculate a polynomial function of degree 4 with a single variable x.

Parameters:

• x (numeric) – Input variable.

• C0 (numeric) – Polynomial coefficients

• C1 (numeric) – Polynomial coefficients

• C2 (numeric) – Polynomial coefficients

• C3 (numeric) – Polynomial coefficients

• C4 (numeric) – Polynomial coefficients

Returns:

Result of the polynomial function.

Return type:

numeric

pyvisco.shift.fit_WLF(RefT, df_aT)

Fit the Williams-Landel-Ferry (WLF) equation [1] to a set of shift factors.

Parameters:

• RefT (numeric) – Reference temperature chosen to construct the master curve.

• df_aT (pandas.DataFrame) – Contains the decadic logarithm of the shift factors log_aT and the corresponding temperature values T in degree Celsius.

Returns:

df – Contains the necessary parameters to calculate the WLF equation (RefT, WLF_C1, WLF_C2).

Return type:

pandas.DataFrame

See also

shift.WLF

Calculates the WLF equation.

Note, -----, Too, a, an, empirical, only, is

References

[1] Williams, Malcolm L.; Landel, Robert F.; Ferry, John D. (1955). “The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-forming Liquids”. J. Amer. Chem. Soc. 77 (14): 3701-3707. doi:10.1021/ja01619a008

pyvisco.shift.fit_poly(df_aT)

Fit polynomial functions of degree 1 to 4 to a set of shift factors.

Parameters:

df_aT (pandas.DataFrame) – Contains the decadic logarithm of the shift factors log_aT and the corresponding temperature values T in degree Celsius.

Returns:

• df_C (pandas.DataFrame) – Contains the coefficients to calculate the polynomial shift functions of degree 1 to 4 for temperatures in degree Celsius.

• df_K (pandas.DataFrame) – Contains the coefficients to calculate the polynomial shift functions of degree 1 to 4 for temperatures in Kelvin.

• Note

• —–

• The coefficients of the polynomial shift funtions are dependent on the

• Temperature unit. Hence, two different dataframes are provided for

• temperatures described in Celsius and Kelvin. For temperatures in Kelvin,

• at least 5 significant figures should be used for the polynomial

• coefficients to obtain accurate results for the polynomial shift functions.

• The interconversion from degree Celsius (T_C) to Kelvin (T_K) is performed

• as (T_K = T_C + 273.15.)

pyvisco.shift.plot(df_aT, df_WLF, df_C)

Plot shift factors and shift functions.

Parameters:

• df_aT (pandas.DataFrame) – Contains the decadic logarithm of the shift factors log_aT and the corresponding temperature values T in degree Celsius.

• df_WLF (pandas.DataFrame) – Contains the necessary parameters to calculate the WLF equation (RefT, WLF_C1, WLF_C2) in degree Celsius.

• df_C (pandas.DataFrame) – Contains the coefficients to calculate the polynomial shift functions of degree 1 to 4 for temperatures in degree Celsius.

Returns:

• fig (matplotlib.pyplot.figure) – Matplotlib figure instance.

• df_shift (pandasDataFrame) – Contains the data used to create the plot.