polymer - Polymer models

PolymerBrush	Polymer brushes in a solvent
PolymerMushroom	Polymer mushrooms in a solvent (volume profile)
EndTetheredPolymer	Polymer end-tethered to an interface in a solvent
VolumeProfile	Generic volume profile function
layer_thickness	Return the thickness of a layer given the microslab z points.

Layer models for polymer systems.

Analytic Self-consistent Field (SCF) Brush profile[1][2]

Analytical Self-consistent Field (SCF) Mushroom Profile[3]

Numerical Self-consistent Field (SCF) End-Tethered Polymer Profile[4][5][6]

[1]

Zhulina, EB; Borisov, OV; Pryamitsyn, VA; Birshtein, TM (1991) “Coil-Globule Type Transitions in Polymers. 1. Collapse of Layers of Grafted Polymer Chains”, Macromolecules 24, 140-149.

[2]

Karim, A; Douglas, JF; Horkay, F; Fetters, LJ; Satija, SK (1996) “Comparative swelling of gels and polymer brush layers”, Physica B 221, 331-336. doi:10.1016/0921-4526(95)00946-9

[3] (1,2)

Adamuţi-Trache, M., McMullen, W. E. & Douglas, J. F. Segmental concentration profiles of end-tethered polymers with excluded-volume and surface interactions. J. Chem. Phys. 105, 4798 (1996).

[4] (1,2)

Cosgrove, T., Heath, T., Van Lent, B., Leermakers, F. A. M., & Scheutjens, J. M. H. M. (1987). Configuration of terminally attached chains at the solid/solvent interface: self-consistent field theory and a Monte Carlo model. Macromolecules, 20(7), 1692–1696. doi:10.1021/ma00173a041

[5] (1,2)

De Vos, W. M., & Leermakers, F. A. M. (2009). Modeling the structure of a polydisperse polymer brush. Polymer, 50(1), 305–316. doi:10.1016/j.polymer.2008.10.025

[6] (1,2)

Sheridan, R. J., Orski, S. V., Jones, R. L., Satija, S., & Beers, K. L. (2017). Surface interaction parameter measurement of solvated polymers via model end-tethered chains. [Submitted]

[7]

Vincent, B., Edwards, J., Emmett, S., & Croot, R. (1988). Phase separation in dispersions of weakly-interacting particles in solutions of non-adsorbing polymer. Colloids and Surfaces, 31, 267–298. doi:10.1016/0166-6622(88)80200-2

class refl1d.polymer.EndTetheredPolymer(thickness=0, interface=0, name='EndTetheredPolymer', polymer=None, solvent=None, chi=0, chi_s=0, h_dry=None, l_lat=1, mn=None, m_lat=1, pdi=1, phi_b=0)

Bases: Layer

Polymer end-tethered to an interface in a solvent

Uses a numerical self-consistent field profile.[4][5][6]

Parameters

chi

solvent interaction parameter

chi_s

surface interaction parameter

h_dry

thickness of the neat polymer layer

l_lat

real length per lattice site

mn

Number average molecular weight

m_lat

real mass per lattice segment

pdi

Dispersity (Polydispersity index)

phi_b

volume fraction of free chains in solution. useful for associating grafted films e.g. PS-COOH in Toluene with an SiO2 surface.

thickness

Slab thickness should be greater than the contour length of the polymer

interface

should be zero

material

the polymer material

solvent

the solvent material

Previous layer should not have roughness! Use a spline to simulate it.

According to [7], \(l_\text{lat}\) and \(m_\text{lat}\) should be calculated by the formulas:

\[\begin{split}l_\text{lat} &= \frac{a^2 m/l}{p_l} \\ m_\text{lat} &= \frac{(a m/l)^2}{p_l}\end{split}\]

where \(l\) is the real polymer’s bond length, \(m\) is the real segment mass, and \(a\) is the ratio between molecular weight and radius of gyration at theta conditions. The lattice persistence, \(p_l\), is:

\[p_l = \frac16 \frac{1+1/Z}{1-1/Z}\]

with coordination number \(Z = 6\) for a cubic lattice, \(p_l = .233\).

constraints()

Constraints

find(z)

Find the layer at depth z.

Returns layer, start, end

interface = None

property ismagnetic

layer_parameters()

property magnetism

name = None

parameters()

Returns a dictionary of parameters specific to the layer. These will be added to the dictionary containing interface, thickness and magnetism parameters.

penalty()

Return a penalty value associated with the layer. This should be zero if the parameters are valid, and increasing as the parameters become more invalid. For example, if total volume fraction exceeds unity, then the penalty would be the amount by which it exceeds unity, or if z values must be sorted, then penalty would be the amount by which they are unsorted.

Note that penalties are handled separately from any probability of seeing a combination of layer parameters; the final solution to the problem should not include any penalized points.

profile(z)

render(probe, slabs)

Use the probe to render the layer into a microslab representation.

thickness = None

to_dict()

Return a dictionary representation of the Slab object

class refl1d.polymer.PolymerBrush(thickness=0, interface=0, name='brush', polymer=None, solvent=None, base_vf=None, base=None, length=None, power=None, sigma=None)

Bases: Layer

Polymer brushes in a solvent

Parameters:

thickness

the thickness of the solvent layer

interface

the roughness of the solvent surface

polymer

the polymer material

solvent

the solvent material or vacuum

base_vf

volume fraction (%) of the polymer brush at the interface

base

the thickness of the brush interface (A)

length

the length of the brush above the interface (A)

power

the rate of brush thinning

sigma

rms brush roughness (A)

The materials can either use the scattering length density directly, such as PDMS = SLD(0.063, 0.00006) or they can use chemical composition and material density such as PDMS=Material(“C2H6OSi”, density=0.965).

These parameters combine in the following profile formula:

\[\begin{split}V(z) &= \left\{ \begin{array}{ll} V_o & \mbox{if } z <= z_o \\ V_o (1 - ((z-z_o)/L)^2)^p & \mbox{if } z_o < z < z_o + L \\ 0 & \mbox{if } z >= z_o + L \end{array} \right. \\ V_\sigma(z) &= V(z) \star \frac{e^{-\frac{1}{2}(z/\sigma)^2}}{\sqrt{2\pi\sigma^2}} \\ \rho(z) &= \rho_p V_\sigma(z) + \rho_s (1-V_\sigma(z))\end{split}\]

where \(V_\sigma(z)\) is volume fraction convoluted with brush roughness \(\sigma\) and \(\rho(z)\) is the complex scattering length density of the profile.

constraints()

Constraints

find(z)

Find the layer at depth z.

Returns layer, start, end

interface = None

property ismagnetic

layer_parameters()

property magnetism

name = None

parameters()

Returns a dictionary of parameters specific to the layer. These will be added to the dictionary containing interface, thickness and magnetism parameters.

penalty()

Return a penalty value associated with the layer. This should be zero if the parameters are valid, and increasing as the parameters become more invalid. For example, if total volume fraction exceeds unity, then the penalty would be the amount by which it exceeds unity, or if z values must be sorted, then penalty would be the amount by which they are unsorted.

Note that penalties are handled separately from any probability of seeing a combination of layer parameters; the final solution to the problem should not include any penalized points.

profile(z)

render(probe, slabs)

Use the probe to render the layer into a microslab representation.

thickness = None

to_dict()

Return a dictionary representation of the Slab object

class refl1d.polymer.PolymerMushroom(thickness=0, interface=0, name='Mushroom', polymer=None, solvent=None, sigma=0, vf=0, delta=0)

Bases: Layer

Polymer mushrooms in a solvent (volume profile)

Parameters:

delta | real scalar

interaction parameter

vf | real scalar

not quite volume fraction (dimensionless grafting density)

sigma | real scalar

convolution roughness (A)

Using analytical SCF methods for gaussian chains, which are scaled by the radius of gyration of the equivalent free polymer as an approximation to results of renormalization group methods.[3]

Solutions are only strictly valid for vf << 1.

constraints()

Constraints

find(z)

Find the layer at depth z.

Returns layer, start, end

interface = None

property ismagnetic

layer_parameters()

property magnetism

name = None

parameters()

Returns a dictionary of parameters specific to the layer. These will be added to the dictionary containing interface, thickness and magnetism parameters.

penalty()

Return a penalty value associated with the layer. This should be zero if the parameters are valid, and increasing as the parameters become more invalid. For example, if total volume fraction exceeds unity, then the penalty would be the amount by which it exceeds unity, or if z values must be sorted, then penalty would be the amount by which they are unsorted.

Note that penalties are handled separately from any probability of seeing a combination of layer parameters; the final solution to the problem should not include any penalized points.

profile(z)

render(probe, slabs)

Use the probe to render the layer into a microslab representation.

thickness = None

to_dict()

Return a dictionary representation of the Slab object

class refl1d.polymer.VolumeProfile(thickness=0, interface=0, name='VolumeProfile', material=None, solvent=None, profile=None, **kw)

Bases: Layer

Generic volume profile function

Parameters:

thickness

the thickness of the solvent layer

interface

the roughness of the solvent surface

material

the polymer material

solvent

the solvent material

profile

the profile function, suitably parameterized

The materials can either use the scattering length density directly, such as PDMS = SLD(0.063, 0.00006) or they can use chemical composition and material density such as PDMS=Material(“C2H6OSi”, density=0.965).

These parameters combine in the following profile formula:

sld = material.sld * profile + solvent.sld * (1 - profile)

The profile function takes a depth z and returns a density rho.

For volume profiles, the returned rho should be the volume fraction of the material. For SLD profiles, rho should be complex scattering length density of the material.

Fitting parameters are the available named arguments to the function. The first argument must be z, which is the array of depths at which the profile is to be evaluated. It is guaranteed to be increasing, with step size 2*z[0].

Initial values for the function parameters can be given using name=value. These values can be scalars or fitting parameters. The function will be called with the current parameter values as arguments. The layer thickness can be computed as :func: layer_thickness.

constraints()

Constraints

find(z)

Find the layer at depth z.

Returns layer, start, end

interface = None

property ismagnetic

layer_parameters()

property magnetism

name = None

parameters()

Returns a dictionary of parameters specific to the layer. These will be added to the dictionary containing interface, thickness and magnetism parameters.

penalty()

Return a penalty value associated with the layer. This should be zero if the parameters are valid, and increasing as the parameters become more invalid. For example, if total volume fraction exceeds unity, then the penalty would be the amount by which it exceeds unity, or if z values must be sorted, then penalty would be the amount by which they are unsorted.

Note that penalties are handled separately from any probability of seeing a combination of layer parameters; the final solution to the problem should not include any penalized points.

render(probe, slabs)

Use the probe to render the layer into a microslab representation.

thickness = None

to_dict()

Return a dictionary representation of the Slab object

refl1d.polymer.layer_thickness(z)

Return the thickness of a layer given the microslab z points.

The z points are at the centers of the bins. we can use the recurrence that boundary b[k] = z[k-1] + (z[k-1] - b[k-1]) to compute the total length of the layer.