Zero-order scalar model

This model embodies the assmmtion that the actual intensity within the resist film I(x, z) can be approximated by  

This is referred to as the separability assumption in that the aerial image and the standing wave can be calculated independently and then multiplied together to yield the total intensity within the resist.  

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The scalar models for calculating aerial image intensity neglects the effects of polarization of light. This is a legitimate approximation for numerical apertures less than 0.7. Scalar models are the simplest of the aerial models and come in three main types the zero-order, first-order, and high-NA scalar models. 

Here scalar order parameter, has the interpretation of a normalized difference between the oil and water concentrations go is the strength of surfactant and /o is the parameter describing the stability of the microemulsion and is proportional to the chemical potential of the surfactant. The constant go is solely responsible for the creation of internal surfaces in the model. The microemulsion or the lamellar phase forms only when go is negative. The function/(<))) is the bulk free energy and describes the coexistence of the pure water phase (4> = —1), pure oil phase (4> = 1), and microemulsion (< ) = 0), provided that/o = 0 (in the mean-held approximation). One can easily calculate the correlation function (4>(r)(0)) — (4>(r) (4>(0)) in various bulk homogeneous phases. In the microemulsion this function oscillates, indicating local correlations between water-rich and oil-rich domains. In the pure water or oil phases it should decay monotonically to zero. This does occur, provided that g2 > 4 /TT/o — go- Because of the < ), —<(> (oil-water) symmetry of the model, the interface between the oil-rich and water-rich domains is given by... 

In order to understand the physical basis for turbulent-diffusivity-based models for the scalar flux, we first consider a homogeneous turbulent flow with zero mean velocity gradient18 and a uniform mean scalar gradient (Taylor 1921). In this flow, velocity fluctuations of characteristic size... 

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