Academia.eduAcademia.edu

Figure 6 – uploaded by Andrew Masters

See full PDFdownloadDownload figure
ithe Ornstein-Zernike relation was used to Calculate the pair distribution runc- tion, g(1, 2), from our direct correlation function data. Outside the overlap region, the agreement with simulation results is excellent, even for hard spheres close to freezing. Within the overlap region,however, the pair distribution function is not zero as it should be. For spheres, adding more terms in the virial expansion pro- gressively reduces this discrepancy and at 6" order, the core condition is fairly well obeyed. For spheroids, one must consider both the truncation of the virial series and also the truncation of the angular expansion of c(1, 2) in spherical harmonics. We believe that the violation of the core condition is associated with both these truncations and, in a future publication, we will present a systematic study of the effects of including higher order angular terms.

Figure 6 ithe Ornstein-Zernike relation was used to Calculate the pair distribution runc- tion, g(1, 2), from our direct correlation function data. Outside the overlap region, the agreement with simulation results is excellent, even for hard spheres close to freezing. Within the overlap region,however, the pair distribution function is not zero as it should be. For spheres, adding more terms in the virial expansion pro- gressively reduces this discrepancy and at 6" order, the core condition is fairly well obeyed. For spheroids, one must consider both the truncation of the virial series and also the truncation of the angular expansion of c(1, 2) in spherical harmonics. We believe that the violation of the core condition is associated with both these truncations and, in a future publication, we will present a systematic study of the effects of including higher order angular terms.

Related Figures (14)

Calculation of direct correlation function for hard particles using a virial expansion
Calculation of direct correlation function for hard particles using a virial expansion
Figure 1. c(1,2) for hard spheres, at 71 = 0.209. Dotted lines are results from virial theory truncated at levels Bz to Bz. Simulation results from [4]. 3. Results
Figure 1. c(1,2) for hard spheres, at 71 = 0.209. Dotted lines are results from virial theory truncated at levels Bz to Bz. Simulation results from [4]. 3. Results
outside the overlap region, the predictions are really rather good.
outside the overlap region, the predictions are really rather good.
en nnn enn on een eee ee ee en OA) oe eee een ee nen ee ee NN OE OIE DLE! SEINE a way off, although it does capture the essential behaviour of the coefficients. Finally, the expansion coefficients of f(1)c(1,2)f(2) were calculated for the sam« system as above. Taking the value of a = 13.92, corresponding to a nematic orde parameter Sy = (P2(w-n)) = 0.8 (where P» is the second Legendre polynomial) from a virial equation of state for a packing fraction of 7 = 0.694, the expansior coefficients of f(1)c(1, 2) f(2) were calculated for this value, and also, for compari son, a = 0, which corresponds to an isotropic system. The values of cooo000 (712) are shown in figure 14. The behavior of cogooo0(T12) for the nematic system is visibl; different from an isotropic system at the same density. At low rj, the magnitud of coooooo(T12) for the nematic system is lower than that of the isotropic system This is expected, due to the high orientational ordering of the system causing mort parallel particles. This in turn lowers the chance of overlapping particles of a short: to mid-range separation. However, at larger separations, the magnitude in the ne. matic system becomes larger than that for the isotropic system, as the orderec nature of the system makes overlaps of particles of this separation more likely. The inversion of these data to obtain h(1,2) for the nematic phase is work in progress
en nnn enn on een eee ee ee en OA) oe eee een ee nen ee ee NN OE OIE DLE! SEINE a way off, although it does capture the essential behaviour of the coefficients. Finally, the expansion coefficients of f(1)c(1,2)f(2) were calculated for the sam« system as above. Taking the value of a = 13.92, corresponding to a nematic orde parameter Sy = (P2(w-n)) = 0.8 (where P» is the second Legendre polynomial) from a virial equation of state for a packing fraction of 7 = 0.694, the expansior coefficients of f(1)c(1, 2) f(2) were calculated for this value, and also, for compari son, a = 0, which corresponds to an isotropic system. The values of cooo000 (712) are shown in figure 14. The behavior of cogooo0(T12) for the nematic system is visibl; different from an isotropic system at the same density. At low rj, the magnitud of coooooo(T12) for the nematic system is lower than that of the isotropic system This is expected, due to the high orientational ordering of the system causing mort parallel particles. This in turn lowers the chance of overlapping particles of a short: to mid-range separation. However, at larger separations, the magnitude in the ne. matic system becomes larger than that for the isotropic system, as the orderec nature of the system makes overlaps of particles of this separation more likely. The inversion of these data to obtain h(1,2) for the nematic phase is work in progress
Figure 6. cooo(r12) at 7 = 0.148 for hard, prolate spheroids of aspect ration 3: 1. Dotted lines are results from virial theory truncated at levels Bz to Bz
Figure 6. cooo(r12) at 7 = 0.148 for hard, prolate spheroids of aspect ration 3: 1. Dotted lines are results from virial theory truncated at levels Bz to Bz
Figure 7. c220(r12) at 7 = 0.148 for hard, prolate spheroids of aspect ration 3: 1. Dotted lines are results from virial theory truncated at levels Bz to Bz
Figure 7. c220(r12) at 7 = 0.148 for hard, prolate spheroids of aspect ration 3: 1. Dotted lines are results from virial theory truncated at levels Bz to Bz
Figure 8. cooo(r12) at 7 = 0.444 for hard, prolate spheroids of aspect ration 3: 1. Dotted lines are results from virial theory truncated at levels Bz to Bz Figure 9. c220(r12) at 7 = 0.444 for hard, prolate spheroids of aspect ration 3: 1. Dotted lines are results from virial theory truncated at levels Bz to Bz
Figure 8. cooo(r12) at 7 = 0.444 for hard, prolate spheroids of aspect ration 3: 1. Dotted lines are results from virial theory truncated at levels Bz to Bz Figure 9. c220(r12) at 7 = 0.444 for hard, prolate spheroids of aspect ration 3: 1. Dotted lines are results from virial theory truncated at levels Bz to Bz
Figure 10. hooo(riz2) at 7 = 0.148 for hard, prolate spheroids of aspect ration 3 : 1. Dotted lines are results from virial theory truncated at levels Bz to By
Figure 10. hooo(riz2) at 7 = 0.148 for hard, prolate spheroids of aspect ration 3 : 1. Dotted lines are results from virial theory truncated at levels Bz to By
Figure 11. h220(riz2) at 7 = 0.148 for hard, prolate spheroids of aspect ration 3 : 1. Dotted lines are results from virial theory truncated at levels Bo to B7
Figure 11. h220(riz2) at 7 = 0.148 for hard, prolate spheroids of aspect ration 3 : 1. Dotted lines are results from virial theory truncated at levels Bo to B7
Figure 12. hooo(riz) at 7 = 0.444 for hard, prolate spheroids of aspect ration 3 : 1. Dotted lines are results from virial theory truncated at levels Bz to Bz Figure 13. h220(riz2) at 7 = 0.444 for hard, prolate spheroids of aspect ration 3 : 1. Dotted lines are results from virial theory truncated at levels Bz to Bz
Figure 12. hooo(riz) at 7 = 0.444 for hard, prolate spheroids of aspect ration 3 : 1. Dotted lines are results from virial theory truncated at levels Bz to Bz Figure 13. h220(riz2) at 7 = 0.444 for hard, prolate spheroids of aspect ration 3 : 1. Dotted lines are results from virial theory truncated at levels Bz to Bz
Figure 14. coooo00(712) at 7 = 0.694 for hard, prolate spheroids of aspect ration 3: 1, for both isotropic and nematic ordering. Dotted lines are results from virial theory truncated at levels Bz to B7 for isotropic ordering, solid lines for nematic ordering.
Figure 14. coooo00(712) at 7 = 0.694 for hard, prolate spheroids of aspect ration 3: 1, for both isotropic and nematic ordering. Dotted lines are results from virial theory truncated at levels Bz to B7 for isotropic ordering, solid lines for nematic ordering.
Related topics:
Molecular PhysicsChemistryStatistical PhysicsPair Distribution FunctionTHEORETICAL AND COMPUTATIONAL CHEMISTRYVirial ExpansionVirial TheoremDirect Correlation Function
580 California St., Suite 400
San Francisco, CA, 94104
© 2025 Academia. All rights reserved