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., Beers, K. L., et. al (2014). Direct observation of “surface theta” conditions. [in prep]
[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.PolymerBrush(thickness=0, interface=0, name=u'brush', polymer=None, solvent=None, base_vf=None, base=None, length=None, power=None, sigma=None)[source]

Bases: refl1d.model.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
ismagnetic
layer_parameters()
magnetism
parameters()[source]
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)[source]
render(probe, slabs)[source]
thickness = None
class refl1d.polymer.PolymerMushroom(thickness=0, interface=0, name=u'Mushroom', polymer=None, solvent=None, sigma=0, vf=0, delta=0)[source]

Bases: refl1d.model.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
ismagnetic
layer_parameters()
magnetism
parameters()[source]
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)[source]
render(probe, slabs)[source]
thickness = None
class refl1d.polymer.EndTetheredPolymer(thickness=0, interface=0, name=u'EndTetheredPolymer', polymer=None, solvent=None, chi=0, chi_s=0, h_dry=None, l_lat=1, mn=None, m_lat=1, pdi=1)[source]

Bases: refl1d.model.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)
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{eqnarray}
l_text{lat} &=& frac{a^2 m/l}{p_l} \ m_text{lat} &=& frac{(a m/l)^2}{p_l}

end{eqnarray}

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
ismagnetic
layer_parameters()
magnetism
parameters()[source]
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)[source]
render(probe, slabs)[source]
thickness = None
class refl1d.polymer.VolumeProfile(thickness=0, interface=0, name=u'VolumeProfile', material=None, solvent=None, profile=None, **kw)[source]

Bases: refl1d.model.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
ismagnetic
layer_parameters()
magnetism
parameters()[source]
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)[source]
thickness = None
refl1d.polymer.layer_thickness(z)[source]

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.