Kinetic Coefficients¶

Overview¶

The polykin.kinetics module offers a number of classes to deal with the types of kinetic coefficients commonly encountered in polymerization kinetics.

Class	Required arguments	Optional arguments
Arrhenius	k0, EaR	T0, Tmin, Tmax, unit, symbol, name
Eyring	DSa, DHa	kappa, Tmin, Tmax, symbol, name
PropagationHalfLength	kp	C, ihalf, name
TerminationCompositeModel	k11, icrit	aS, aL, name

The meanings of all arguments, along with their corresponding units, are documented in the respective docstrings.

Arrhenius and Eyring¶

In [1]:

# %pip install polykin
from polykin.kinetics import Arrhenius, Eyring

# %pip install polykin from polykin.kinetics import Arrhenius, Eyring

To instantiate a kinetic coefficient, we call the respective class constructor with the desired argument values. Here are some examples of Arrhenius coefficients.

In [2]:

# Arrhenius coefficient defined by pre-exponential factor (A) and Ea/R.
k1 = Arrhenius(1e10, 2e3)

# Arrhenius coefficient defined by value at reference temperature (T0) and Ea/R.
k2 = Arrhenius(1e5, 3e3, T0=350.)

# Arrhenius coefficient including temperature limits, units, symbol, and name.
k3 = Arrhenius(10**7.63, 32.5e3/8.314,
               Tmin=261., Tmax=366.,
               symbol='k_p', unit='L/mol/s', name='styrene')

k4 = Arrhenius(10**7.22, 17.3e3/8.314,
               Tmin=208., Tmax=343.,
               symbol='k_p', unit='L/mol/s', name='butyl acrylate')

# Array of Arrhenius coefficients.
k5 = Arrhenius([1e3, 2e3], [1.8e3, 4e3], T0=298.,
               Tmin=273., Tmax=373.,
               symbol='k_d', unit='1/min', name='(I1, I2)')

# Arrhenius coefficient defined by pre-exponential factor (A) and Ea/R. k1 = Arrhenius(1e10, 2e3) # Arrhenius coefficient defined by value at reference temperature (T0) and Ea/R. k2 = Arrhenius(1e5, 3e3, T0=350.) # Arrhenius coefficient including temperature limits, units, symbol, and name. k3 = Arrhenius(10**7.63, 32.5e3/8.314, Tmin=261., Tmax=366., symbol='k_p', unit='L/mol/s', name='styrene') k4 = Arrhenius(10**7.22, 17.3e3/8.314, Tmin=208., Tmax=343., symbol='k_p', unit='L/mol/s', name='butyl acrylate') # Array of Arrhenius coefficients. k5 = Arrhenius([1e3, 2e3], [1.8e3, 4e3], T0=298., Tmin=273., Tmax=373., symbol='k_d', unit='1/min', name='(I1, I2)')

For Eyring coefficients, we proceed similarly, only with different arguments.

In [3]:

# Eyring coefficient defined by entropy and enthalpy of activation.
k6 = Eyring(1e2, 5e4)

# Eyring coefficient including transmission factor, temperature limits, and name.
k7 = Eyring(DSa=20., DHa=1e4, kappa=0.8,
            Tmin=273., Tmax=373.,
            symbol='k_6', name='A->B')

# Eyring coefficient defined by entropy and enthalpy of activation. k6 = Eyring(1e2, 5e4) # Eyring coefficient including transmission factor, temperature limits, and name. k7 = Eyring(DSa=20., DHa=1e4, kappa=0.8, Tmin=273., Tmax=373., symbol='k_6', name='A->B')

The most important properties of a coefficient can be displayed by evaluating the object directly.

In [4]:

k4

k4

Out[4]:

name:            butyl acrylate
symbol:          k_p
unit:            L/mol/s
Trange [K]:      (208.0, 343.0)
k0 [L/mol/s]:    16595869.074375598
EaR [K]:         2080.827519846043
T0 [K]:          inf

or, equivalently, by using the print() function.

In [5]:

print(k7)

print(k7)

name:            A->B
symbol:          k_6
unit:            1/s
Trange [K]:      (273.0, 373.0)
DSa [J/(mol·K)]: 20.0
DHa [J/mol]:     10000.0
kappa [—]:       0.8

To evaluate a coefficient at a given temperature, we simply call the object with the temperature value (scalar or array-like) as the first positional argument. The temperature unit can be passed as second argument (default is K).

In [6]:

# Evaluate k3 at 25°C
k3(25., 'C')

# Evaluate k3 at 25°C k3(25., 'C')

Out[6]:

86.28385101961442

In [7]:

# Evaluate k3 at 298.15 K
k3(298.15)

# Evaluate k3 at 298.15 K k3(298.15)

Out[7]:

86.28385101961442

In [8]:

# Evaluate k5 at 50°C
k5(50., 'C')

# Evaluate k5 at 50°C k5(50., 'C')

Out[8]:

array([1600.15388708, 5684.89928869])

In [9]:

# Evaluate k6 at multiple temperatures (in °C)
k6([25., 50., 75.], 'C')

# Evaluate k6 at multiple temperatures (in °C) k6([25., 50., 75.], 'C')

Out[9]:

array([1.80718687e+09, 9.32495924e+09, 3.82259806e+10])

Evaluations outside the specified temperature range are allowed, but raise a warning.

In [10]:

k3(100., 'C')

k3(100., 'C')

Warning: `T` input is outside validity range (261.0, 366.0).

Out[10]:

1203.343508225495

Arrhenius coefficients have the special mathematical property that a product or quotient of two Arrhenius coefficients is also an Arrhenius coefficient. The Arrhenius class overloads the *, / and ** operators to consider this intrisic feature.

In [11]:

kp1 = Arrhenius(1e3, 2e3, T0=350., symbol='k_{p1}', name='kp1')
Cm1 = Arrhenius(1e-3, 1e3, T0=300., symbol='C_{m1}', name='Cm1')
f = Arrhenius(1., 5e2, T0=320., symbol='f', name='fudge factor')
kfm1 = Cm1*kp1/f
kfm1

kp1 = Arrhenius(1e3, 2e3, T0=350., symbol='k_{p1}', name='kp1') Cm1 = Arrhenius(1e-3, 1e3, T0=300., symbol='C_{m1}', name='Cm1') f = Arrhenius(1., 5e2, T0=320., symbol='f', name='fudge factor') kfm1 = Cm1*kp1/f kfm1

Out[11]:

name:            Cm1·kp1/fudge factor
symbol:          C_{m1}·k_{p1}/f
unit:            -·-/-
Trange [K]:      (0.0, inf)
k0 [-·-/-]:      1781.336221002905
EaR [K]:         2500.0
T0 [K]:          inf

kfm1 is now a coefficient and can be used as such.

In [12]:

kfm1([25., 50.], 'C')

kfm1([25., 50.], 'C')

Out[12]:

array([0.40660136, 0.77784632])

Plots¶

Arrhenius and Eyring come with a convenient method called plot(), enabling a rapid visualization of the corresponding kinetic coefficients.

In [13]:

# Default ('linear') plot over the validity range 
k3.plot(Tunit='K')

# Default ('linear') plot over the validity range k3.plot(Tunit='K')

In [14]:

# Arrhenius plot over the validity range 
k3.plot(kind='Arrhenius')

# Arrhenius plot over the validity range k3.plot(kind='Arrhenius')

In [15]:

# If no temperature range is known, the plot range defaults to 0-100°C
kfm1.plot()

# If no temperature range is known, the plot range defaults to 0-100°C kfm1.plot()

Lastly, the function plotequations() can be used to overlay multiple coefficients on the same plot. The optional keywords arguments are those of the plot() method.

In [16]:

from polykin import plotequations

from polykin import plotequations

In [17]:

_ = plotequations([k3, k4], kind='Arrhenius')

_ = plotequations([k3, k4], kind='Arrhenius')

Chain-length dependent coefficients¶

PropagationHalfLength¶

In [18]:

from polykin.kinetics import PropagationHalfLength

from polykin.kinetics import PropagationHalfLength

This class implements the chain-length-dependent $k_p$ equation proposed by Smith et al. (2005). The class only describes the chain-length effect; the temperature effect must be described separtely (e.g., by Arrhenius or Eyring).

In [19]:

# CLD-kp
kpi = PropagationHalfLength(kp=k3, C=10, ihalf=0.5, name='kp(T,i) of styrene')

# CLD-kp kpi = PropagationHalfLength(kp=k3, C=10, ihalf=0.5, name='kp(T,i) of styrene')

In [20]:

# kp at 50°C for a trimeric radical
kpi(50, 3, Tunit='C')

# kp at 50°C for a trimeric radical kpi(50, 3, Tunit='C')

Out[20]:

371.75986615653215

In [21]:

# kp at 50°C for a series of oligomer lengths
kpi(50, [1, 2, 3, 10], Tunit='C')

# kp at 50°C for a series of oligomer lengths kpi(50, [1, 2, 3, 10], Tunit='C')

Out[21]:

array([2379.2631434 ,  773.26052161,  371.75986616,  237.93448289])

TerminationCompositeModel¶

In [22]:

from polykin.kinetics import TerminationCompositeModel

from polykin.kinetics import TerminationCompositeModel

This class implements the chain-length-dependent composite termination model proposed by Smith & Russel (2003). The class only describes the chain-length effect; the temperature effect must be described separtely (e.g., by Arrhenius or Eyring).

In [23]:

# kt(T) of monomeric radicals
kt11 = Arrhenius(1e9, 2e3, T0=298., symbol='k_t(T,1,1)', unit='L/mol/s',
                 name='kt11 of Y')

# CLD-kt
ktij = TerminationCompositeModel(kt11, icrit=30, name='ktij of Y')

# kt(T) of monomeric radicals kt11 = Arrhenius(1e9, 2e3, T0=298., symbol='k_t(T,1,1)', unit='L/mol/s', name='kt11 of Y') # CLD-kt ktij = TerminationCompositeModel(kt11, icrit=30, name='ktij of Y')

In [24]:

# kt at 25°C between radicals with chain lengths 150 and 200
ktij(25., 150, 200, Tunit='C')

# kt at 25°C between radicals with chain lengths 150 and 200 ktij(25., 150, 200, Tunit='C')

Out[24]:

129008375.03821689

The internal functions are vectorized, so the arguments can also be arrays. For example, we can create a contour plot of $k_t(i,j)$ with just a few lines of code.

In [25]:

import numpy as np
import matplotlib.pyplot as plt

# Generate mesh of chain-lengths
length_range = np.arange(1, 60)
mesh = np.meshgrid(length_range, length_range)

# Evaluate kt for mesh
kt_result = ktij(T=25., i=mesh[0], j=mesh[1], Tunit='C')

# Plot
plt.contourf(length_range, length_range, kt_result)
plt.colorbar()
plt.xlabel(r"$i$")
plt.ylabel(r"$j$")
_ = plt.title(r"$k_t(i,j)$")

import numpy as np import matplotlib.pyplot as plt # Generate mesh of chain-lengths length_range = np.arange(1, 60) mesh = np.meshgrid(length_range, length_range) # Evaluate kt for mesh kt_result = ktij(T=25., i=mesh[0], j=mesh[1], Tunit='C') # Plot plt.contourf(length_range, length_range, kt_result) plt.colorbar() plt.xlabel(r"$i$") plt.ylabel(r"$j$") _ = plt.title(r"$k_t(i,j)$")