Bond orientation autocorrelation function

Important information on local rearrangements is provided by tlte bond orientation autocorrelation functions (BCFs). The 1 and 2" d ee BCFs are... 

The relationship between the structure of a polymer chain and it dynamics has long been a focus for work in polymer science. It is on the local level that the dynamics of a polymer chain are most directly linked to the monomer structure. The techniques of time-resolved optical spectroscopy provide a uniquely detailed picture of local segmental motions. This is accomplished through the direct observation of the time dependence of the orientation autocorrelation function of a bond in the polymer chain. Optical techniques include fluorescence anisotropy decay experiments (J ) and transient absorption measurements(7 ). A common feature of these methods is the use of polymer chains with chromophore labels attached. The transition dipole of the attached chromophore defines the vector whose reorientation is observed in the experiment. A common labeling scheme is to bond the chromophore into the polymer chain such that the transition dipole is rigidly affixed either para 1 lei (1-7) or perpendicular(8,9) to the chain backbone. 

The transient absorption method utilized in the experiments reported here is the transient holographic grating technique(7,10). In the transient grating experiment, a pair of polarized excitation pulses is used to create the anisotropic distribution of excited state transition dipoles. The motions of the polymer backbone are monitored by a probe pulse which enters the sample at some chosen time interval after the excitation pulses and probes the orientational distribution of the transition dipoles at that time. By changing the time delay between the excitation and probe pulses, the orientation autocorrelation function of a transition dipole rigidly associated with a backbone bond can be determined. In the present context, the major advantage of the transient grating measurement in relation to typical fluorescence measurements is the fast time resolution (- 50 psec in these experiments). In transient absorption techniques the time resolution is limited by laser pulse widths and not by the speed of electronic detectors. Fast time resolution is necessary for the experiments reported here because of the sub-nanosecond time scales for local motions in very flexible polymers such as polyisoprene. 

In this paper, we report measurements of the orientation autocorrelation function of a backbone bond in dilute solutions of anthracene-labeled polyisoprene. The anthracene chromophore was covalently bonded into the chain such that the transition dipole for the lowest electronic excited state lies along the chain backbone. This assures that only backbone motions are detected. 

In this paper, we have shown the utility of time-resolved optical techniques for the investigation of local segmental motions in polymer chains on a sub-nanosecond time scale. Detailed information about chain motions is contained in the time dependence of the orientation autocorrelation function of a backbone bond. 

The expression for the autocorrelation function derived by Hall and Helfand can be identified, in a generalized diffusion and loss equation, with the orientation cross-correlation function of two neighbouring bonds inside the polymer chain. To account for motional coupling of non-neighbouring bonds, resulting for example from the presence of side-chains, Viovy et al (VMB) [5] have introduced cross-correlation functions of a pair of bonds separated by j bonds into the orientation autocorrelation function. These functions are written... 

Another expression for the orientation autocorrelation function of chains undergoing three-bond jumps on a tetrahedral lattice has been developed by Jones and Stockmayer [7]. Analysis [10] of the derived orientation autocorrelation function has shown that this function can be considered as a particular case of expression (6.4). 

Here 6(t) is the angle between the C-H bond vector at time zero and time t. This particular correlation function is referred to as a second order orientation autocorrelation function. 

ABSTRACT - The fluorescence anisotropy decay (FAD) technique is first described, then the different expressions ich have been proposed for the orientation autocorrelation function (OACF) of polymer chains are presented. Typical FAD curves of dilute and concentrated solutions of polystyrene labelled with an anthracene group in the middle of the chain are compared to the various OACF expressions and discussed. In the case of bulk polybutadiene, FAD results obtained either on anthracene labelled chains or on 9,10 dialkylanthracene probes free in the polymer matrix, show that the same type of OACF as for polymer solutions can account for the experimental data. Besides, the temperature dependence of the correlation time of the labelled polybutadiene appears to agree with the WLF equation derived from macroscopic viscoelastic measurements, proving that the segmental motions of about 20 bonds which lead to the FAD of labelled polybutadiene participate in the glass transition processes of this polymer. 

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

To proceed, we assume that the orientations of the dipole moments m along the GGS bonds (defined by dj) are random and imcorrelated [30-32]. This key feature simplifies the situation considerably. Then, for the autocorrelation function of Af(f), after averaging over all possible distributions of bonds orientations, with the average denoted below by ( lorient (here we follow [31] closely), we have ... 

For the dipolar relaxation mechanism, the spin lattice relaxation time is sensitive to the reorientation dynamics of the CH bond vectors. Different orientations of the CH bond result in slightly different magnetic fields at the carbon nucleus and the modulation of this field allows the spin flips to occur. When we define ecn as the unit vector along a CH bond, the second Legendre polynomial of its autocorrelation function is given by... 

In this expression, m is a unit vector characterizing the orientation of the C-X bond with respect to Z-axis (perpendicular to the interface) and the average is taken over all possible time origins. The rotational autocorrelation functions provide information about the rotational motions and the characteristic time scales. The results for the three systems are compared in Fig. 11.8. They exhibit a more or less exponential form with a faster decay in the order MeCl > MeCN > MeOH. The associated relaxation times may be estimated by fitting these curves by a biexponential function, as described recently for OH groups of water at the air/water interface [69] ... 

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