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The [[Ornstein-Zernike relation | Ornstein-Zernike]] equations for mixtures of monomers with coincident oligomers | The [[Ornstein-Zernike relation | Ornstein-Zernike]] equations for mixtures (MOZ) of monomers with coincident oligomers | ||
(coincident dimers, trimers,...,''n''-mers). | (coincident dimers, trimers,...,''n''-mers). | ||
:<math>h_{an}(r) - c_{an}(r) = \int h_{aa} (r')~\rho_a ~c_{an}(|r - r'|) | :<math>h_{an}({\mathbf r}) - c_{an}({\mathbf r}) = \int h_{aa} ({\mathbf r'})~\rho_a ~c_{an}(|{\mathbf r} - {\mathbf r'}|) {\rm d}{\mathbf r'}~~~~~n=a,2,3,...</math> | ||
Since all oligomers are at infinite dilution, the OZ's for all <math>n>1</math> are decoupled. The first member is for the bulk monomer fluid ''a'' (with size <math>\sigma</math> and | Since all oligomers are at infinite dilution, the OZ's for all <math>n>1</math> are decoupled. The first member is for the bulk monomer fluid ''a'' (with size <math>\sigma</math> and | ||
energy <math>\epsilon</math>) | energy <math>\epsilon</math>) | ||
:<math>h_{aa}(r) - c_{aa}(r) = \int h_{aa} (r')~\rho_a ~c_{aa}(|r - r'|) | :<math>h_{aa}({\mathbf r}) - c_{aa}({\mathbf r}) = \int h_{aa} ({\mathbf r'})~\rho_a ~c_{aa}(|{\mathbf r} - {\mathbf r'}|) {\rm d}{\mathbf r'}</math> | ||
For a coincident dimer (<math>n=2</math>) of size <math>\sigma</math> and | For a coincident dimer (<math>n=2</math>) of size <math>\sigma</math> and | ||
energy <math>2\epsilon</math> at infinite dilution in the bulk ''a''-monomers: | energy <math>2\epsilon</math> at infinite dilution in the bulk ''a''-monomers: | ||
:<math>h_{a2}(r) - c_{a2}(r) = \int h_{aa} (r')~\rho_a ~c_{a2}(|r - r'|) | :<math>h_{a2}({\mathbf r}) - c_{a2}({\mathbf r}) = \int h_{aa} ({\mathbf r'})~\rho_a ~c_{a2}(|{\mathbf r} - {\mathbf r'}|) {\rm d}{\mathbf r'}</math> | ||
For a coincident trimer (<math>n=3</math>) of size <math>\sigma</math> and | For a coincident trimer (<math>n=3</math>) of size <math>\sigma</math> and | ||
energy <math>3\epsilon</math> at infinite dilution in the bulk ''a''-monomers: | energy <math>3\epsilon</math> at infinite dilution in the bulk ''a''-monomers: | ||
:<math>h_{a3}(r) - c_{a3}(r) = \int h_{aa} (r')~\rho_a ~c_{a3}(|r - r'|) | :<math>h_{a3}({\mathbf r}) - c_{a3}({\mathbf r}) = \int h_{aa} ({\mathbf r'})~\rho_a ~c_{a3}(|{\mathbf r} - {\mathbf r'}|) {\rm d}{\mathbf r'}</math> | ||
==References== | |||
[[Category: Integral equations]] | [[Category: Integral equations]] | ||
Latest revision as of 16:17, 10 July 2007
The Ornstein-Zernike equations for mixtures (MOZ) of monomers with coincident oligomers (coincident dimers, trimers,...,n-mers).
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_{an}({\mathbf r}) - c_{an}({\mathbf r}) = \int h_{aa} ({\mathbf r'})~\rho_a ~c_{an}(|{\mathbf r} - {\mathbf r'}|) {\rm d}{\mathbf r'}~~~~~n=a,2,3,...}
Since all oligomers are at infinite dilution, the OZ's for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n>1} are decoupled. The first member is for the bulk monomer fluid a (with size Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} and energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} )
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_{aa}({\mathbf r}) - c_{aa}({\mathbf r}) = \int h_{aa} ({\mathbf r'})~\rho_a ~c_{aa}(|{\mathbf r} - {\mathbf r'}|) {\rm d}{\mathbf r'}}
For a coincident dimer (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=2} ) of size Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} and energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\epsilon} at infinite dilution in the bulk a-monomers:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_{a2}({\mathbf r}) - c_{a2}({\mathbf r}) = \int h_{aa} ({\mathbf r'})~\rho_a ~c_{a2}(|{\mathbf r} - {\mathbf r'}|) {\rm d}{\mathbf r'}}
For a coincident trimer (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=3} ) of size Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} and energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\epsilon} at infinite dilution in the bulk a-monomers:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_{a3}({\mathbf r}) - c_{a3}({\mathbf r}) = \int h_{aa} ({\mathbf r'})~\rho_a ~c_{a3}(|{\mathbf r} - {\mathbf r'}|) {\rm d}{\mathbf r'}}