Changeset 2c1bbcdd in sasmodels
- Timestamp:
- Mar 20, 2016 11:08:31 AM (9 years ago)
- Branches:
- master, core_shell_microgels, costrafo411, magnetic_model, release_v0.94, release_v0.95, ticket-1257-vesicle-product, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
- Children:
- 3a45c2c
- Parents:
- a154ad16
- File:
-
- 1 edited
-
sasmodels/models/raspberry.py (modified) (7 diffs)
Legend:
- Unmodified
- Added
- Removed
-
sasmodels/models/raspberry.py
r5745f0b r2c1bbcdd 3 3 ---------- 4 4 5 The figure below shows a schematic of a large droplet surrounded by several smaller particles6 forming a structure similar to that of Pickering emulsions.5 The figure below shows a schematic of a large droplet surrounded by several 6 smaller particles forming a structure similar to that of Pickering emulsions. 7 7 8 8 .. figure:: img/raspberry_geometry.jpg … … 10 10 Schematic of the raspberry model 11 11 12 In order to calculate the form factor of the entire complex, the self-correlation of the large droplet, 13 the self-correlation of the particles, the correlation terms between different particles 14 and the cross terms between large droplet and small particles all need to be calculated. 12 In order to calculate the form factor of the entire complex, the self- 13 correlation of the large droplet, the self-correlation of the particles, the 14 correlation terms between different particles and the cross terms between large 15 droplet and small particles all need to be calculated. 15 16 16 Consider two infinitely thin shells of radii R2 and R2 separated by distance r. The general 17 structure of the equation is then the form factor of the two shells multiplied by the phase 18 factor that accounts for the separation of their centers. 17 Consider two infinitely thin shells of radii R1 and R2 separated by distance r. 18 The general structure of the equation is then the form factor of the two shells 19 multiplied by the phase factor that accounts for the separation of their 20 centers. 19 21 20 22 .. math:: … … 22 24 S(q) = \frac{sin(qR_1)}{qR_1}\frac{sin(qR_2)}{qR_2}\frac{sin(qr)}{qr} 23 25 24 In this case, the large droplet and small particles are solid spheres rather than thin shells. Thus 25 the two terms must be integrated over $R_L$ and $R_S$ respectively using the weighting function of 26 a sphere. We then obtain the functions for the form of the two spheres: 26 In this case, the large droplet and small particles are solid spheres rather 27 than thin shells. Thus the two terms must be integrated over $R_L$ and $R_S$ 28 respectively using the weighting function of a sphere. We then obtain the 29 functions for the form of the two spheres: 27 30 28 31 .. math:: 29 32 30 \Psi_L = \int_0^{R_L}(4\pi R^2_L)\frac{sin(qR_L)}{qR_L}dR_L = \frac{3[sin(qR_L)-qR_Lcos(qR_L)]}{(qR_L)^2} 33 \Psi_L = \int_0^{R_L}(4\pi R^2_L)\frac{sin(qR_L)}{qR_L}dR_L = 34 \frac{3[sin(qR_L)-qR_Lcos(qR_L)]}{(qR_L)^2} 31 35 32 36 .. math:: 33 37 34 \Psi_S = \int_0^{R_S}(4\pi R^2_S)\frac{sin(qR_S)}{qR_S}dR_S = \frac{3[sin(qR_S)-qR_Lcos(qR_S)]}{(qR_S)^2} 38 \Psi_S = \int_0^{R_S}(4\pi R^2_S)\frac{sin(qR_S)}{qR_S}dR_S = 39 \frac{3[sin(qR_S)-qR_Lcos(qR_S)]}{(qR_S)^2} 35 40 36 41 The cross term between the large droplet and small particles is given by: … … 42 47 43 48 .. math:: 44 S_{SS} = \Psi_S^2\bigl[\frac{sin(q(R_L+\delta R_S))}{q(R_L+\delta\ R_S)}\bigr]^2 49 S_{SS} = \Psi_S^2\biggl[\frac{sin(q(R_L+\delta R_S))}{q(R_L+\delta\ R_S)} 50 \biggr]^2 45 51 46 52 The number of small particles per large droplet, $N_p$, is given by: … … 50 56 N_p = \frac{\phi_S\phi_{surface}V_L}{\phi_L V_S} 51 57 52 where $\phi_S$ is the volume fraction of small particles in the sample, $\phi_{surface}$ is the 53 fraction of the small particles that are adsorbed to the large droplets, $\phi_L$ is the volume fraction 54 of large droplets in the sample, and $V_S$ and $V_L$ are the volumes of individual small particles and 58 where $\phi_S$ is the volume fraction of small particles in the sample, 59 $\phi_{surface}$ is the fraction of the small particles that are adsorbed to 60 the large droplets, $\phi_L$ is the volume fraction of large droplets in the 61 sample, and $V_S$ and $V_L$ are the volumes of individual small particles and 55 62 large droplets respectively. 56 63 57 The form factor of the entire complex can now be calculated including the excess scattering length 58 densities of the components $\Delta\rho_L$ and $\Delta\rho_S$, where $\Delta\rho_x = |\rho_x-\rho_{solvent}|$ : 64 The form factor of the entire complex can now be calculated including the excess 65 scattering length densities of the components $\Delta\rho_L$ and $\Delta\rho_S$, 66 where $\Delta\rho_x = |\rho_x-\rho_{solvent}|$ : 59 67 60 68 .. math:: 61 69 62 P_{LS} = \frac{1}{M^2}\bigl[(\Delta\rho_L)^2V_L^2\Psi_L^2+N_p(\Delta\rho_S)^2V_S^2\Psi_S^2 63 + N_p(1-N_p)(\Delta\rho_S)^2V_S^2S_{SS} + 2N_p\Delta\rho_L\Delta\rho_SV_LV_SS_{LS} 70 P_{LS} = \frac{1}{M^2}\bigl[(\Delta\rho_L)^2V_L^2\Psi_L^2 71 +N_p(\Delta\rho_S)^2V_S^2\Psi_S^2 72 + N_p(1-N_p)(\Delta\rho_S)^2V_S^2S_{SS} 73 + 2N_p\Delta\rho_L\Delta\rho_SV_LV_SS_{LS}\bigr] 64 74 65 75 where M is the total scattering length of the whole complex : … … 68 78 M = \Delta\rho_LV_L + N_p\Delta\rho_SV_S 69 79 70 In a real system, there will ususally be an excess of small particles such that some fraction remain unbound. 71 Therefore the overall scattering intensity is given by: 80 In a real system, there will ususally be an excess of small particles such that 81 some fraction remain unbound. Therefore the overall scattering intensity is 82 given by: 72 83 73 84 .. math:: 74 I(Q) = I_{LS}(Q) + I_S(Q) = (\phi_L(\Delta\rho_L)^2V_L + \phi_S\phi_{surface}N_p(\Delta\rho_S)^2V_S)P_{LS} 85 I(Q) = I_{LS}(Q) + I_S(Q) = (\phi_L(\Delta\rho_L)^2V_L + 86 \phi_S\phi_{surface}N_p(\Delta\rho_S)^2V_S)P_{LS} 75 87 + \phi_S(1-\phi_{surface})(\Delta\rho_S)^2V_S\Psi_S^2 76 88 77 A useful parameter to extract is the fraction of the surface area of the large droplets that is covered by small 78 particles. This can be calculated from the model parameters as: 89 A useful parameter to extract is the fraction of the surface area of the large 90 droplets that is covered by small particles. This can be calculated from the 91 model parameters as: 79 92 80 93 .. math:: … … 85 98 ---------- 86 99 87 K Larson-Smith, A Jackson, and D C Pozzo, *Small angle scattering model for Pickering emulsions and raspberry* 88 *particles*, *Journal of Colloid and Interface Science*, 343(1) (2010) 36-41 100 K Larson-Smith, A Jackson, and D C Pozzo, *Small angle scattering model for 101 Pickering emulsions and raspberry particles*, *Journal of Colloid and Interface 102 Science*, 343(1) (2010) 36-41 89 103 90 104 **Author:** Andrew Jackson **on:** 2008
Note: See TracChangeset
for help on using the changeset viewer.
