# source:sasmodels/sasmodels/models/be_polyelectrolyte.py@ef07e95

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
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1r"""
2Definition
3----------
4This model calculates the structure factor of a polyelectrolyte solution with
5the RPA expression derived by Borue and Erukhimovich\ [#Borue]_.  Note however
6that the fitting procedure here does not follow the notation in that reference
7as 's' and 't' are **not** decoupled. Instead the scattering intensity $I(q)$
8is calculated as
9
10.. math::
11
12    I(q) = K\frac{q^2+k^2}{4\pi L_b\alpha ^2}
13    \frac{1}{1+r_{0}^2(q^2+k^2)(q^2-12hC_a/b^2)} + background
14
15    k^2 = 4\pi L_b(2C_s + \alpha C_a)
16
17    r_{0}^2 = \frac{1}{\alpha \sqrt{C_a} \left( b/\sqrt{48\pi L_b}\right)}
18
19where
20
21$K$ is the contrast factor for the polymer which is defined differently than in
22other models and is given in barns where $1 barn = 10^{-24} cm^2$.  $K$ is
23defined as:
24
25.. math::
26
27    K = a^2
28
29    a = b_p - (v_p/v_s) b_s
30
31where $b_p$ and $b_s$ are sum of the scattering lengths of the atoms
32constituting the monomer of the polymer and the sum of the scattering lengths
33of the atoms constituting the solvent molecules respectively, and $v_p$ and
34$v_s$ are the partial molar volume of the polymer and the solvent respectively
35
36$L_b$ is the Bjerrum length(|Ang|) - **Note:** This parameter needs to be
37kept constant for a given solvent and temperature!
38
39$h$ is the virial parameter (|Ang^3|/mol) - **Note:** See [#Borue]_ for the
40correct interpretation of this parameter.  It incorporates second and third
41virial coefficients and can be Negative.
42
43$b$ is the monomer length(|Ang|), $C_s$ is the concentration of monovalent
44salt(mol/L), $\alpha$ is the ionization degree (ionization degree : ratio of
45charged monomers  to total number of monomers), $C_a$ is the polymer molar
46concentration(mol/L), and $background$ is the incoherent background.
47
48For 2D data the scattering intensity is calculated in the same way as 1D,
49where the $\vec q$ vector is defined as
50
51.. math::
52
53    q = \sqrt{q_x^2 + q_y^2}
54
55References
56----------
57
58.. [#Borue] V Y Borue, I Y Erukhimovich, *Macromolecules*, 21 (1988) 3240
59.. [#] J F Joanny, L Leibler, *Journal de Physique*, 51 (1990) 545
60.. [#] A Moussaid, F Schosseler, J P Munch, S Candau, *J. Journal de Physique
61   II France*, 3 (1993) 573
62.. [#] E Raphael, J F Joanny, *Europhysics Letters*, 11 (1990) 179
63
64Authorship and Verification
65----------------------------
66
67* **Author:** NIST IGOR/DANSE **Date:** pre 2010
69* **Last Reviewed by:** Paul Butler and Richard Heenan **Date:** October 07, 2016
70"""
71
72import numpy as np
73from numpy import inf, pi, sqrt
74
75name = "be_polyelectrolyte"
76title = "Polyelectrolyte with the RPA expression derived by Borue and Erukhimovich"
77description = """
78            Evaluate
79            F(x) = K 1/(4 pi Lb (alpha)^(2)) (q^(2)+k2)/(1+(r02)^(2))
80                 (q^(2)+k2) (q^(2)-(12 h C/b^(2)))
81
82            has 3 internal parameters :
83                   The inverse Debye Length: K2 = 4 pi Lb (2 Cs+alpha C)
84                   r02 =1/alpha/Ca^(0.5) (B/(48 pi Lb)^(0.5))
85                   Ca = 6.022136e-4 C
86            """
87category = "shape-independent"
88
90#   ["name", "units", default, [lower, upper], "type", "description"],
91parameters = [
92    ["contrast_factor",       "barns",   10.0,  [-inf, inf], "", "Contrast factor of the polymer"],
93    ["bjerrum_length",        "Ang",      7.1,  [0, inf],    "", "Bjerrum length"],
94    ["virial_param",          "Ang^3/mol", 12.0,  [-inf, inf], "", "Virial parameter"],
95    ["monomer_length",        "Ang",     10.0,  [0, inf],    "", "Monomer length"],
96    ["salt_concentration",    "mol/L",    0.0,  [-inf, inf], "", "Concentration of monovalent salt"],
97    ["ionization_degree",     "",         0.05, [0, inf],    "", "Degree of ionization"],
98    ["polymer_concentration", "mol/L",    0.7,  [0, inf],    "", "Polymer molar concentration"],
99    ]
101
102
103def Iq(q,
104       contrast_factor=10.0,
105       bjerrum_length=7.1,
106       virial_param=12.0,
107       monomer_length=10.0,
108       salt_concentration=0.0,
109       ionization_degree=0.05,
110       polymer_concentration=0.7):
111    """
112    :param q:                     Input q-value
113    :param contrast_factor:       Contrast factor of the polymer
114    :param bjerrum_length:        Bjerrum length
115    :param virial_param:          Virial parameter
116    :param monomer_length:        Monomer length
117    :param salt_concentration:    Concentration of monovalent salt
118    :param ionization_degree:     Degree of ionization
119    :param polymer_concentration: Polymer molar concentration
120    :return:                      1-D intensity
121    """
122
123    concentration = polymer_concentration * 6.022136e-4
124
125    k_square = 4.0 * pi * bjerrum_length * (2*salt_concentration +
126                                            ionization_degree * concentration)
127
128    r0_square = 1.0/ionization_degree/sqrt(concentration) * \
129                (monomer_length/sqrt((48.0*pi*bjerrum_length)))
130
131    term1 = contrast_factor/(4.0 * pi * bjerrum_length *
132                             ionization_degree**2) * (q**2 + k_square)
133
134    term2 = 1.0 + r0_square**2 * (q**2 + k_square) * \
135        (q**2 - (12.0 * virial_param * concentration/(monomer_length**2)))
136
137    return term1/term2
138
139Iq.vectorized = True  # Iq accepts an array of q values
140
141def random():
142    # TODO: review random be_polyelectrolyte model generation
143    pars = dict(
144        scale=10000, #background=0,
145        #polymer_concentration=0.7,
146        polymer_concentration=np.random.beta(5, 3), # around 70%
147        #salt_concentration=0.0,
148        # keep salt concentration extremely low
149        # and use explicit molar to match polymer concentration
150        salt_concentration=np.random.beta(1, 100)*6.022136e-4,
151        #contrast_factor=10.0,
152        contrast_fact=np.random.uniform(1, 100),
153        #bjerrum_length=7.1,
154        bjerrum_length=np.random.uniform(1, 10),
155        #virial_param=12.0,
156        virial_param=np.random.uniform(-1000, 30),
157        #monomer_length=10.0,
158        monomer_length=10.0**(4*np.random.beta(1.5, 3)),
159        #ionization_degree=0.05,
160        ionization_degree=np.random.beta(1.5, 4),
161    )
162    return pars
163
164demo = dict(scale=1, background=0.1,
165            contrast_factor=10.0,
166            bjerrum_length=7.1,
167            virial_param=12.0,
168            monomer_length=10.0,
169            salt_concentration=0.0,
170            ionization_degree=0.05,
171            polymer_concentration=0.7)
172
173tests = [
174
175    # Accuracy tests based on content in test/utest_other_models.py
176    [{'contrast_factor':       10.0,
177      'bjerrum_length':         7.1,
178      'virial_param':          12.0,
179      'monomer_length':        10.0,
180      'salt_concentration':     0.0,
181      'ionization_degree':      0.05,
182      'polymer_concentration':  0.7,
183      'background':             0.001,
184     }, 0.001, 0.0948379],
185
186    # Additional tests with larger range of parameters
187    [{'contrast_factor':       10.0,
188      'bjerrum_length':       100.0,
189      'virial_param':           3.0,
190      'monomer_length':         1.0,
191      'salt_concentration':    10.0,
192      'ionization_degree':      2.0,
193      'polymer_concentration': 10.0,
194      'background':             0.0,
195     }, 0.1, -3.75693800588],
196
197    [{'contrast_factor':       10.0,
198      'bjerrum_length':       100.0,
199      'virial_param':           3.0,
200      'monomer_length':         1.0,
201      'salt_concentration':    10.0,
202      'ionization_degree':      2.0,
203      'polymer_concentration': 10.0,
204      'background':           100.0
205     }, 5.0, 100.029142149],
206
207    [{'contrast_factor':     100.0,
208      'bjerrum_length':       10.0,
209      'virial_param':        180.0,
210      'monomer_length':        1.0,
211      'salt_concentration':    0.1,
212      'ionization_degree':     0.5,
213      'polymer_concentration': 0.1,
214      'background':             0.0,
215     }, 200., 1.80664667511e-06],
216    ]
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