polymer  Polymer models¶
Polymer brushes in a solvent 

Polymer mushrooms in a solvent (volume profile) 

Polymer endtethered to an interface in a solvent 

Generic volume profile function 

Return the thickness of a layer given the microslab z points. 
Layer models for polymer systems.
Analytic Selfconsistent Field (SCF) Brush profile[1][2]
Analytical Selfconsistent Field (SCF) Mushroom Profile[3]
Numerical Selfconsistent Field (SCF) EndTethered Polymer Profile[4][5][6]
 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)[source]¶
Bases:
Layer
Polymer endtethered to an interface in a solvent
Uses a numerical selfconsistent 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. PSCOOH 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:
\[p_l = \frac16 \frac{1+1/Z}{11/Z}\]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()[source]¶
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.
 thickness = None¶
 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)[source]¶
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:
\[\begin{split}V(z) &= \left\{ \begin{array}{ll} V_o & \mbox{if } z <= z_o \\ V_o (1  ((zz_o)/L)^2)^p & \mbox{if } z_o < z < z_o + L \\ 0 & \mbox{if } z >= z_o + L \end{array} \right. \\ V_\sigma(z) &= V(z) \star \frac{e^{\frac{1}{2}(z/\sigma)^2}}{\sqrt{2\pi\sigma^2}} \\ \rho(z) &= \rho_p V_\sigma(z) + \rho_s (1V_\sigma(z))\end{split}\]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()[source]¶
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.
 thickness = None¶
 class refl1d.polymer.PolymerMushroom(thickness=0, interface=0, name='Mushroom', polymer=None, solvent=None, sigma=0, vf=0, delta=0)[source]¶
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()[source]¶
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.
 thickness = None¶
 class refl1d.polymer.VolumeProfile(thickness=0, interface=0, name='VolumeProfile', material=None, solvent=None, profile=None, **kw)[source]¶
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()[source]¶
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.
 thickness = None¶