Diffusion Coefficients¶

Overview¶

The polykin.properties.diffusion module provides methods to correlate and estimate mutual and self-diffusion coefficients in liquid mixtures.

Class/Function	Solution type	Calculated property
VrentasDudaBinary	Binary polymer solution	Mutual and self-diffusion coefficient
DL_Wilke_Chang	Binary liquid solution	Infinite-dilution diffusion coefficient
DL_Hayduk_Minhas	Binary liquid solution	Infinite-dilution diffusion coefficient
DV_Wilke_Lee	Binary gas solution	Mutual diffusion coefficient

Information about the corresponding arguments and their units are documented in the respective docstrings.

Polymer solution¶

Vrentas-Duda model¶

In [1]:

# %pip install polykin
from polykin.properties.diffusion import VrentasDudaBinary

# %pip install polykin from polykin.properties.diffusion import VrentasDudaBinary

The Vrentas-Duda model is perhaps the most widely used method to describe the effect of temperature and solvent content on the diffusivity of polymer solutions. For this demonstration, let's consider the binary system toluene / poly(vinyl acetate) along with the free-volume parameters reported in Table 4 of Zielinski & Duda, 1992. To instantiate a model object, we call the respective class constructor with the parameters specific to this method.

In [2]:

d = VrentasDudaBinary(D0=4.82e-4, E=0., v1star=0.917, v2star=0.728, z=0.82,
                      K11=1.45e-3, K12=4.33e-4, K21=-86.32, K22=-258.2, X=0.5,
                      unit='cm²/s',
                      name='Tol(1)/PVAc(2)')

d = VrentasDudaBinary(D0=4.82e-4, E=0., v1star=0.917, v2star=0.728, z=0.82, K11=1.45e-3, K12=4.33e-4, K21=-86.32, K22=-258.2, X=0.5, unit='cm²/s', name='Tol(1)/PVAc(2)')

In [3]:

d

d

Out[3]:

name: Tol(1)/PVAc(2)
unit: cm²/s
D0:   0.000482
E:    0.0
v1*:  0.917
v2*:  0.728
ξ:    0.82
K11:  0.00145
K12:  0.000433
K21:  -86.32
K22:  -258.2
Tg1:  0.0
Tg2:  0.0
𝛾:    1.0
𝜒:    0.5

To evaluate the solvent self-diffusion coefficient or the mutual diffusion coefficient, we call the instance directly with the solvent mass fraction and the temperature as mandatory arguments (scalar or array-like). The temperature unit can be passed as optional argument (default is K). By default, the method delivers the mutual diffusion coefficient.

In [4]:

# D at w1=0.2 and T=25°C.
d(0.2, 25., Tunit='C')

# D at w1=0.2 and T=25°C. d(0.2, 25., Tunit='C')

Out[4]:

3.789815380454319e-08

In [5]:

# D at w1=[0.2, 0.3, 0.4] and T=298.15K.
d([0.2, 0.3, 0.4], 298.15)

# D at w1=[0.2, 0.3, 0.4] and T=298.15K. d([0.2, 0.3, 0.4], 298.15)

Out[5]:

array([3.78981538e-08, 2.14599443e-07, 4.51290147e-07])

To get the solvent self-diffusion coefficient instead of the mutual diffusion coefficient, we just need to flip the keyword argument selfd.

In [6]:

# D1 at w1=0.2 and T=50°C.
d(0.2, 25., Tunit='C', selfd=True)

# D1 at w1=0.2 and T=50°C. d(0.2, 25., Tunit='C', selfd=True)

Out[6]:

7.401983164949841e-08

The class comes with a convenient built-in method called plot() to enable a rapid visualization of the effects of composition and temperature. Let's reproduce Figure 6 of Zielinski & Duda, 1992.

In [7]:

d.plot(T=[40., 80., 110.], Tunit='C', ylim=(1e-11, 1e-5))

d.plot(T=[40., 80., 110.], Tunit='C', ylim=(1e-11, 1e-5))

We can also generate the analogous plot for the solvent self-diffusion coefficient.

In [8]:

d.plot(T=[40., 80., 110.], Tunit='C', selfd=True,
       w1range=(0., 1.), ylim=(1e-11, 1e-4))

d.plot(T=[40., 80., 110.], Tunit='C', selfd=True, w1range=(0., 1.), ylim=(1e-11, 1e-4))

Notice how the solvent self-diffusion coefficient is a monotonically increasing function of the solvent content, unlike the mutual diffusion coefficient.

Binary liquid solutions¶

Wilke-Chang and Haduk-Minhas correlations¶

In [9]:

from polykin.properties.diffusion import DL_Wilke_Chang, DL_Hayduk_Minhas

from polykin.properties.diffusion import DL_Wilke_Chang, DL_Hayduk_Minhas

The Wilke-Chang and Haduk-Minhas correlations are among the most used to estimate the infinite-dilution diffusion coefficient of a solute in a solvent. Let's apply both methods to estimate the diffusion coefficient of vinyl chloride in liquid water.

In [10]:

DL_Wilke_Chang(
    T=298.,         # temperature
    MA=62.5e-3,     # molar mass of vinyl chloride
    MB=18.0e-3,     # molar mass of water
    rhoA=910.,      # density of vinyl chloride at the normal boiling point
    viscB=0.89e-3,  # viscosity of water at solution temperature
    phi=2.6         # association factor for water (see docstring)
    )

DL_Wilke_Chang( T=298., # temperature MA=62.5e-3, # molar mass of vinyl chloride MB=18.0e-3, # molar mass of water rhoA=910., # density of vinyl chloride at the normal boiling point viscB=0.89e-3, # viscosity of water at solution temperature phi=2.6 # association factor for water (see docstring) )

Out[10]:

1.339916480081557e-09

In [11]:

DL_Hayduk_Minhas(
    T=298.,           # temperature
    method='aqueous', # equation for aqueous solutions
    MA=62.5e-3,       # molar mass of vinyl chloride
    rhoA=910.,        # density of vinyl chloride at the normal boiling point
    viscB=0.89e-3     # viscosity of water at solution temperature
    )

DL_Hayduk_Minhas( T=298., # temperature method='aqueous', # equation for aqueous solutions MA=62.5e-3, # molar mass of vinyl chloride rhoA=910., # density of vinyl chloride at the normal boiling point viscB=0.89e-3 # viscosity of water at solution temperature )

Out[11]:

1.2579698249278908e-09

Binary gas mixtures¶

Wilke-Lee correlation¶

In [12]:

from polykin.properties.diffusion import DV_Wilke_Lee

from polykin.properties.diffusion import DV_Wilke_Lee

The Wilke-Lee correlation is general and reliable method to estimate the diffusivity of a binary gas solution. Let's apply it to estimate the diffusion coefficient of vinyl chloride in air.

In [13]:

DV_Wilke_Lee(
    T=298.,       # temperature
    P=1e5,        # pressure
    MA=62.5e-3,   # molar mass of vinyl chloride
    MB=29.0e-3,   # molar mass of air
    rhoA=910.,    # density of vinyl chloride at the normal boiling point
    rhoB=None,    # air (see docstring)
    TA=260.,      # normal boiling point of vinyl chloride
    TB=None,      # air (see docstring)
    )

DV_Wilke_Lee( T=298., # temperature P=1e5, # pressure MA=62.5e-3, # molar mass of vinyl chloride MB=29.0e-3, # molar mass of air rhoA=910., # density of vinyl chloride at the normal boiling point rhoB=None, # air (see docstring) TA=260., # normal boiling point of vinyl chloride TB=None, # air (see docstring) )

Out[13]:

1.1652957293550837e-05

And, for comparison, the diffusion coefficient of vinyl chloride through water vapor.

In [14]:

DV_Wilke_Lee(
    T=298.,       # temperature
    P=1e5,        # pressure
    MA=62.5e-3,   # molar mass of vinyl chloride
    MB=18.0e-3,   # molar mass of water
    rhoA=910.,    # density of vinyl chloride at the normal boiling point
    rhoB=959.,    # density of water at the normal boiling point
    TA=260.,      # normal boiling point of vinyl chloride
    TB=373.,      # normal boiling point of water
    )

DV_Wilke_Lee( T=298., # temperature P=1e5, # pressure MA=62.5e-3, # molar mass of vinyl chloride MB=18.0e-3, # molar mass of water rhoA=910., # density of vinyl chloride at the normal boiling point rhoB=959., # density of water at the normal boiling point TA=260., # normal boiling point of vinyl chloride TB=373., # normal boiling point of water )

Out[14]:

1.1035218654621124e-05