Normalized correlation function decay constants

The normalized correlation function can be related to the distribution of decay constants as shown by Equation 2. 

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... 

Fig. 24.2. Single-molecule recording of T4 lysozyme conformational motions and enzymatic reaction turnovers of hydrolysis of an E. coli B cell wall in real time, (a) This panel shows a pair of trajectories from a fluorescence donor tetramethyl-rhodamine blue) and acceptor Texas Red (red) pair in a single-T4 lysozyme in the presence of E. coli cells of 2.5mg/mL at pH 7.2 buffer. Anticorrelated fluctuation features are evident. (b) The correlation functions (C (t)) of donor ( A/a (0) Aid (f)), blue), acceptor ((A/a (0) A/a (t)), red), and donor-acceptor cross-correlation function ((A/d (0) A/d (t)), black), deduced from the single-molecule trajectories in (a). They are fitted with the same decay rate constant of 180 40s. A long decay component of 10 2s is also evident in each autocorrelation function. The first data point (not shown) of each correlation function contains the contribution from the measurement noise and fluctuations faster than the time resolution. The correlation functions are normalized, and the (A/a (0) A/a (t)) is presented with a shift on the y axis to enhance the view, (c) A pair of fluorescence trajectories from a donor (blue) and acceptor (red) pair in a T4 lysozyme protein without substrates present. The acceptor was photo-bleached at about 8.5 s. (d) The correlation functions (C(t)) of donor ((A/d (0) A/d (t)), blue), acceptor ((A/a (0) A/a (t)), red) derived from the trajectories in (c). The autocorrelation function only shows a spike at t = 0 and drops to zero at t > 0, which indicates that only uncorrelated measurement noise and fluctuation faster than the time resolution recorded (Adapted with permission from [12]. Copyright 2003 American Chemical Society)...

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