# 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]

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]

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]

Polymer end-tethered to an interface in a solvent

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

Parameters

chi: solvent interaction parameter surface interaction parameter thickness of the neat polymer layer real length per lattice site Number average molecular weight real mass per lattice segment Dispersity (Polydispersity index) Slab thickness should be greater than the contour length of the polymer should be zero the polymer material 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]

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.