Orientation autocorrelation function

Here n is the number of attached protons at a given carbon atom and (fin and c are the proton and 13 C resonance frequencies. The constant K assumes a value of 2.29 x 109 s-2 and 2.42 x 109 s-2 for sp3 and sp2 nuclei, respectively. The orientational autocorrelation function is obtained from the simulation trajectory using the relationship... 

It is interesting to note that in dielectric relaxation experiments, the orientation autocorrelation function that is involved in the theory is... 

The dynamic RIS model is used to calculate the conformational and first and second orientational autocorrelation functions for PE. Various sequence lengths and directions in the chain are considered. 

Models for the Orientation Autocorrelation Function of an Isolated Chain 102... 

The calculation of orientational autocorrelation functions from the free rotator Eq. (14) which describes the rotational Brownian motion of a sphere is relatively easy because Sack [19] has shown how the one-sided Fourier transform of the orientational autocorrelation functions (here the longitudinal and transverse autocorrelation functions) may be expressed as continued fractions. The corresponding calculation from Eq. (15) for the three-dimensional rotation in a potential is very difficult because of the nonlinear relation between and p [33] arising from the kinematic equation, Eq. (7). 

The complex susceptibility x( ) yielded by Eq. (9), combined with Eq. (22) when the small oscillation approximation is abandoned, may be calculated using the shift theorem for Fourier transforms combined with the matrix continued fraction solution for the fixed center of oscillation cosine potential model treated in detail in Ref. 25. Thus we shall merely outline that solution as far as it is needed here and refer the reader to Ref. 25 for the various matrix manipulations, and so on. On considering the orientational autocorrelation function of the surroundings ps(t) and expanding the double exponential, we have... 

The local dynamics of polymers in solution have been extensively studied during the last decade. On the other hand, for polymer melts many questions are still unanswered, such as, for example, the nature of the orientation autocorrelation function (OACF) which is involved, and the relationship of the segmental motions occuring at high frequency in the melt with the elementary processes responsible for the glass-rubber transition. 

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. 

Time-resolved optical experiments rely on a short pulse of polarized light from a laser, synchrotron, or flash lamp to photoselect chromophores which have their transition dipoles oriented in the same direction as the polarization of the exciting light. This non-random orientational distribution of excited state transition dipoles will randomize in time due to motions of the polymer chains to which the chromophores are attached. The precise manner in which the oriented distribution randomizes depends upon the detailed character of the molecular motions taking place and is described by the orientation autocorrelation function. This randomization of the orientational distribution can be observed either through time-resolved polarized fluorescence (as in fluorescence anisotropy decay experiments) or through time-resolved polarized absorption. 

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. 

Our experimental measurements of the orientation autocorrelation function on sub-nanosecond time scales are consistent with the theoretical models for backbone motions proposed by Hall and Helfand(ll) and by Bendler and Yaris(12). The correlation functions observed in three different solvents at various temperatures have the same shape within experimental error. This implies that the fundamental character of the local segmental dynamics is the same in the different environments investigated. Analysis of the temperature dependence of the correlation function yields an activation energy of 7 kJ/mole for local segmental motions. 

The experimental anisotropy contains information about molecular motion, but is independent of the excited state lifetime. Equation 4 indicates that the orientation autocorrelation function can be obtained directly and unambiguously (within the multiplicative constant r(0)) from the results of a transient grating experiment. 

Theoretical Models for Local Segmental Motions. The second order orientation autocorrelation function measured in this experiment is defined by ... 

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 orientation autocorrelation function P2[cos 0(t)] is given by r(t) and reflects the motion undergone by the fluorescent chromophore (2,14). A number of models for Brownian motion have been proposed (14) but in the simple case of a rigid sphere, r(t) is described by a single exponential decay where Tf., the rotational correlation time is related to the hydrodynamic volume of the sphere and the viscosity of the medium through the Stokes-Einstein relation (14,16). More complex motions of fluorophores necessitate the development of models which fit the functional form of r(t) experimentally obtained (14). 

The quantity (3 cos co(t) — l)/2 is the orientation autocorrelation function it represents the probability that a molecule having a certain orientation at time zero is oriented at co with respect to its initial orientation. The quantity (3x — 1)/2 is the Legendre polynomial of order 2, Piix), and Eq. (5.32) is sometimes written in the following form... 

The fluorophore was modeled by two beads that are attached as a short pendant side-chain (tag). Both the absorption and emission dipole moments of the fluorophore are defined by the direction of the tag (parallel), as indicated by the vector in Fig. 19, and the fluorescence anisotropy was calculated from its orientation autocorrelation function. For simplicity, we assumed that the reorientaional motion of the fluorophore is the only source of fluorescence depolarization. We neglected energy transfer and other processes that might occur in real systems. The fluorescence anisotropy decays were interpreted using the mean relaxation time, defined as ... 

Rigid, isotropic rotor. If we suppose that the rotor makes random jumps to new, random orientations then its orientational autocorrelation function will decay in the same first-order way as radioactive fission of nuclei. Therefore, in this case... 

In polymers, due to the constraint resulting from the connectivity of the chain, the local motions are usually too complicated to be described by a single isotropic correlation time x, as discussed in chapter 4. Indeed, fluorescence anisotropy decay experiments, which directly yield the orientation autocorrelation function, have shown that the experimental data obtained on anthracene-labelled polybutadiene and polyisoprene in solution or in the melt cannot be represented by simple motional models. To account for the connectivity of the polymer backbone, specific autocorrelation functions, based on models in which conformational changes propagate along the chain according to a damped diffusional process, have been derived for local chain... 

Among the various expressions that are based on a conformational jump model and have been proposed for the orientation autocorrelation function of a polymer chain, G t), the formula derived by Hall and Helfand (HH) [4] leads to a very good agreement with fluorescence anisotropy decay data. It is written as... 

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). 

From a practical point of view, all the above expressions for the orientation autocorrelation function lead to very similar numerical results for NMR spin-lattice relaxation time () calculations. 

The Dejean-Laupretre-Monnerie (DLM) orientation autocorrelation function is based on the above description. It takes into account independent damped conformational jumps, described by the Hall-Helfand autocorrelation function, and librations of the internuclear vectors represented, as proposed by Howarth [17] (see chapter 4) by a random anisotropic fast reorientation of the CH vector inside a cone of half-angle and axis the rest position of the internuclear vector. The resulting orientation autocorrelation function can be written as... 

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. 

Helfand has characterized local segmental dynamics in polyethylene using orientation autocorrelation functions and a set of molecule fixed vectors [28]. 

In time-resolved fluorescence anisotropy studies the orientational motion of a fluorescent label rigidly affixed to a point on the chain, or that of a probe dissolved in the system, are studied. These experiments yield directly the time decay of the second orientational autocorrelation function, MaCx), associated with the unit vector m(x) along the transition moment of the chromophore at time X. M2(x) is given by... 

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