1 | r""" |
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2 | *This model was implemented by an interested user!* |
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3 | |
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4 | Unlike a concentrated polymer solution, the fine-scale polymer distribution |
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5 | in a gel involves at least two characteristic length scales, |
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6 | a shorter correlation length ( $a1$ ) to describe the rapid fluctuations |
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7 | in the position of the polymer chains that ensure thermodynamic equilibrium, |
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8 | and a longer distance (denoted here as $a2$ ) needed to account for the static |
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9 | accumulations of polymer pinned down by junction points or clusters of such |
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10 | points. The latter is derived from a simple Guinier function. |
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11 | |
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12 | |
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13 | Definition |
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14 | ---------- |
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15 | |
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16 | The scattered intensity $I(q)$ is calculated as |
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17 | |
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18 | .. math:: |
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19 | |
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20 | I(Q) = I(0)_L \frac{1}{\left( 1+\left[ ((D+1/3)Q^2a_{1}^2 |
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21 | \right]\right)^{D/2}} + I(0)_G exp\left( -Q^2a_{2}^2\right) + B |
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22 | |
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23 | where |
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24 | |
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25 | .. math:: |
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26 | |
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27 | a_{2}^2 \approx \frac{R_{g}^2}{3} |
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28 | |
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29 | Note that the first term reduces to the Ornstein-Zernicke equation |
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30 | when $D = 2$; ie, when the Flory exponent is 0.5 (theta conditions). |
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31 | In gels with significant hydrogen bonding $D$ has been reported to be |
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32 | ~2.6 to 2.8. |
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33 | |
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34 | |
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35 | .. figure:: img/gel_fit_1d.gif |
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36 | |
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37 | 1D plot using the default values (with 300 data points). |
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38 | |
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39 | Reference |
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40 | --------- |
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41 | |
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42 | Mitsuhiro Shibayama, Toyoichi Tanaka, Charles C Han, |
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43 | *J. Chem. Phys.* 1992, 97 (9), 6829-6841 |
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44 | |
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45 | Simon Mallam, Ferenc Horkay, Anne-Marie Hecht, Adrian R Rennie, Erik Geissler, |
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46 | *Macromolecules* 1991, 24, 543-548 |
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47 | |
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48 | """ |
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49 | |
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50 | from numpy import inf |
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51 | |
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52 | name = "gel_fit" |
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53 | title = "Fitting using fine-scale polymer distribution in a gel." |
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54 | description = """\ |
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55 | Structure factor for interacting particles: |
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56 | |
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57 | Shibayama-Geissler Two-Length Scale Fit for Gels (GelFit) |
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58 | |
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59 | Shibayama; Tanaka; Han J Chem Phys (1992), 97(9), 6829-6841 |
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60 | Mallam; Horkay; Hecht; Rennie; Geissler, Macromol (1991), 24, 543 |
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61 | """ |
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62 | category = "shape-independent" |
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63 | |
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64 | # pylint: disable=bad-whitespace, line-too-long |
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65 | # ["name", "units", default, [lower, upper], "type","description"], |
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66 | parameters = [["guinier_scale", "cm^{-1}", 1.7, [-inf, inf], "", "Guinier length scale"], |
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67 | ["lorentzian_scale", "cm^{-1}", 3.5, [-inf, inf], "", "Lorentzian length scale"], |
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68 | ["gyration_radius", "Ang", 104.0, [2, inf], "", "Radius of gyration"], |
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69 | ["fractal_exp", "", 2.0, [0, inf], "", "Fractal exponent"], |
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70 | ["cor_length", "Ang", 16.0, [0, inf], "", "Correlation length"] |
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71 | ] |
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72 | # pylint: enable=bad-whitespace, line-too-long |
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73 | |
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74 | source = ["gel_fit.c"] |
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75 | |
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76 | demo = dict(background=0.01, |
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77 | guinier_scale=1.7, |
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78 | lorentzian_scale=3.5, |
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79 | gyration_radius=104, |
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80 | fractal_exp=2.0, |
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81 | cor_length=16.0) |
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82 | |
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83 | oldname = 'GelFitModel' |
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84 | oldpars = dict(guinier_scale='gScale', |
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85 | lorentzian_scale='lScale', |
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86 | gyration_radius='radius', |
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87 | fractal_exp='FractalExp', |
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88 | cor_length='zeta') |
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89 | |
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90 | tests = [[{'guinier_scale': 1.0, |
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91 | 'lorentzian_scale': 1.0, |
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92 | 'gyration_radius': 10.0, |
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93 | 'fractal_exp': 10.0, |
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94 | 'cor_length': 20.0 |
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95 | }, 0.1, 0.716532], |
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96 | |
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97 | [{'guinier_scale': 4.0, |
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98 | 'lorentzian_scale': 10.0, |
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99 | 'gyration_radius': 500.0, |
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100 | 'fractal_exp': 1.0, |
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101 | 'cor_length': 20.0, |
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102 | 'background': 20.0, |
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103 | }, 5.0, 20.1224653026], |
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104 | ] |
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