1 | r""" |
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2 | Definition |
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3 | ---------- |
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4 | |
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5 | This model calculates the SAS signal of a phase separating system |
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6 | undergoing spinodal decomposition. The scattering intensity $I(q)$ is calculated |
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7 | as |
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8 | |
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9 | .. math:: |
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10 | I(q) = I_{max}\frac{(1+\gamma/2)x^2}{\gamma/2+x^{2+\gamma}}+B |
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11 | |
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12 | where $x=q/q_0$, $q_0$ is the peak position, $I_{max}$ is the intensity |
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13 | at $q_0$ (parameterised as the $scale$ parameter), and $B$ is a flat |
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14 | background. The spinodal wavelength, $\Lambda$, is given by $2\pi/q_0$. |
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15 | |
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16 | The definition of $I_{max}$ in the literature varies. Hashimoto *et al* (1991) |
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17 | define it as |
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18 | |
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19 | .. math:: |
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20 | I_{max} = \Lambda^3\Delta\rho^2 |
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21 | |
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22 | whereas Meier & Strobl (1987) give |
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23 | |
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24 | .. math:: |
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25 | I_{max} = V_z\Delta\rho^2 |
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26 | |
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27 | where $V_z$ is the volume per monomer unit. |
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28 | |
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29 | The exponent $\gamma$ is equal to $d+1$ for off-critical concentration |
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30 | mixtures (smooth interfaces) and $2d$ for critical concentration mixtures |
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31 | (entangled interfaces), where $d$ is the dimensionality (ie, 1, 2, 3) of the |
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32 | system. Thus 2 <= $\gamma$ <= 6. A transition from $\gamma=d+1$ to $\gamma=2d$ |
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33 | is expected near the percolation threshold. |
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34 | |
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35 | As this function tends to zero as $q$ tends to zero, in practice it may be |
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36 | necessary to combine it with another function describing the low-angle |
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37 | scattering, or to simply omit the low-angle scattering from the fit. |
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38 | |
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39 | References |
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40 | ---------- |
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41 | |
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42 | .. [#] H. Furukawa. Dynamics-scaling theory for phase-separating unmixing mixtures: Growth rates of droplets and scaling properties of autocorrelation functions. Physica A 123, 497 (1984). |
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43 | .. [#] H. Meier & G. Strobl. Small-Angle X-ray Scattering Study of Spinodal Decomposition in Polystyrene/Poly(styrene-co-bromostyrene) Blends. Macromolecules 20, 649-654 (1987). |
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44 | .. [#] T. Hashimoto, M. Takenaka & H. Jinnai. Scattering Studies of Self-Assembling Processes of Polymer Blends in Spinodal Decomposition. J. Appl. Cryst. 24, 457-466 (1991). |
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45 | |
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46 | Source |
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47 | ------ |
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48 | |
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49 | `spinodal.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/spinodal.py>`_ |
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50 | |
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51 | Authorship and Verification |
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52 | ---------------------------- |
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53 | |
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54 | * **Author:** Dirk Honecker **Date:** Oct 7, 2016 |
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55 | * **Last Modified by:** Steve King **Date:** Oct 25, 2018 |
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56 | * **Last Reviewed by:** Steve King **Date:** Oct 25, 2018 |
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57 | * **Source added by :** Steve King **Date:** March 25, 2019 |
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58 | """ |
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59 | |
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60 | import numpy as np |
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61 | from numpy import inf, errstate |
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62 | |
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63 | name = "spinodal" |
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64 | title = "Spinodal decomposition model" |
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65 | description = """\ |
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66 | I(q) = Imax ((1+gamma/2)x^2)/(gamma/2+x^(2+gamma)) + background |
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67 | |
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68 | List of default parameters: |
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69 | |
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70 | Imax = correlation peak intensity at q_0 |
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71 | background = incoherent background |
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72 | gamma = exponent (see model documentation) |
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73 | q_0 = correlation peak position [1/A] |
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74 | x = q/q_0""" |
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75 | |
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76 | category = "shape-independent" |
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77 | |
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78 | # pylint: disable=bad-whitespace, line-too-long |
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79 | # ["name", "units", default, [lower, upper], "type", "description"], |
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80 | parameters = [["gamma", "", 3.0, [-inf, inf], "", "Exponent"], |
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81 | ["q_0", "1/Ang", 0.1, [-inf, inf], "", "Correlation peak position"] |
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82 | ] |
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83 | # pylint: enable=bad-whitespace, line-too-long |
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84 | |
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85 | def Iq(q, |
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86 | gamma=3.0, |
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87 | q_0=0.1): |
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88 | """ |
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89 | :param q: Input q-value |
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90 | :param gamma: Exponent |
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91 | :param q_0: Correlation peak position |
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92 | :return: Calculated intensity |
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93 | """ |
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94 | with errstate(divide='ignore'): |
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95 | x = q/q_0 |
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96 | inten = ((1 + gamma / 2) * x ** 2) / (gamma / 2 + x ** (2 + gamma)) |
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97 | return inten |
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98 | Iq.vectorized = True # Iq accepts an array of q values |
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99 | |
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100 | def random(): |
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101 | """Return a random parameter set for the model.""" |
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102 | pars = dict( |
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103 | scale=10**np.random.uniform(1, 3), |
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104 | gamma=np.random.uniform(0, 6), |
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105 | q_0=10**np.random.uniform(-3, -1), |
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106 | ) |
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107 | return pars |
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108 | |
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109 | demo = dict(scale=1, background=0, |
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110 | gamma=1, q_0=0.1) |
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