摘要
珠鍊在通道中作布朗運動的觀測過程,我們追蹤所定位的珠子確切位置並可確切分析內部擷取的資訊:珠鏈前後端點距離、利用前後端點距離所求得鬆弛時間和擴散係數等,珠鏈處於通道內的特性與文獻 [1][2][3][4]所提在Odijk 區間下的行為進行比較;對於前後端點平均距離
Abstract The positions of each individual bead of the chain are tracked during the random motion. The statistical and dynamical properties of vibrated bead chains confined in thin channels. In the Odijk regime, these properties are the same as those of the measured and/or expected for long chain polymers. A phenomenogical form of the free energy, which is different from the Flory-type free energy, of the bead chain can explain the measured distribution of the end-to-end distance. The diffusion constant and the relaxation time In a confined environment, the conformation of a semi-flexible long chain polymer is elongated, and the dynamic properties of the polymer molecule, such as the diffusion constant and the relaxation time, are changed due to the interaction with the confining walls. Thus the behaviors of a confined polymer are much complicated compared to those of the freely relaxed polymers. However, the situation can be much simplified for the strongly confined regime (the Odijk regime), where at least one of the dimensions of the confining structure is smaller than the persistence length of the long chain polymer. There will be a transition region between the Odijk regime and the regime where the de Gennes regime, where the blob theory can be applied. In this study, we shall focus on the statistical and dynamical properties of polymer chain in the Odijk regime. The Flory free energy can account for many statistical and dynamical properties of long chain polymers in unconfined space. There are two terms in the Flory free energy, one is the bending elastic energy, which is proportional to the square of end-to-end distance, the other is the repulsion energy due to the excluded-volume interaction between the monomers of the chain, which is inversely proportional to L. However, in the Odijk regime, the Flory-type free energy can not explain the average end-to-end distance as a function of the width of the confining channel, and the distribution function of the end-to-end distance (L). One obvious argument is that in the strongly confined situation (Odijk regime), the semi-flexible long chain polymer is highly extended, and the interaction between other monomers (except the nearest neighbors) is not possible. The only interaction is the random collision with the confining wall. Thus, the second term in the Flory-type free energy should be zero, and the applicability of the Flory-type free energy in the Odijk regime is questionable. A new form free energy was recently proposed. Wang and Gao found that the extension of the strongly confined polymer is the same as the extension of an unconfined polymer by an effective force. Based on this finding, Su el. al. derived that the free energy can be written as F(x)/kBT = A/(L-x) – B x, where A and B depend on the contour length and the persistence length of the polymer, and on the size of the confining channel. The first term of this free energy is actually equal to the free energy used by Reisner et.al.. And the second term is an effective force to stretch the the confined polymer. The free energy in (1) was used to explain the extension and the distribution of the extension as a function of the channel width, and also the relaxation time, of the confined DNA molecule. In this work, we find the statistical and dynamical properties of the confined bead chains can also be accounted for with the above form of free energy. Granular bead chain has long been a model system for the long chain molecules in many theoretical studies of the statistical and dynamical properties of the molecules. Recent experimental results(16,17) showed that the vertically vibrated granular bead chains have the properties of the long chain molecules, however, only in the projection to the two-dimensional (2D) space. In this study, the confined DNA molecule in random motion is simulated with a strongly vibrated long bead chain. The chain consists of 2.1 mm diameter hollow metal beads connected to each other with a loose link that sets the maximum distance between two adjacent beads to 2.8 mm. The mass of each individual bead is mb = 35 mg and the bead chain is semi-flexible with a persistence length of 28 mm (i.e., the length of 10 beads). The confining channels have rectangular cross sections of dimension D mm x 8 mm. The height of the channels is fixed to 8 mm so that the crossing of the chain will not occur. And the condition for the excluded volume effect will hold in all the experiments. The horizontal motion of the beads is influenced by the random collision with the walls of the channel and their nearest neighbors. Our experiments are carried out in channel with width D that varies from 3 mm to 120 mm. The length of the channels is fixed at 60 cm. The channels are mounted on a 60 60 cm2 vibration platform that oscillates sinusoidally in the vertical direction at 19.7 Hz. The vertical vibration is driven by a heavy-duty electromagnetic shaker. In the measurements of the mobility of the randomly moving bead chain, the platform is tilted slightly, so that the gravitational force can be used as the drive for the damped motion. The speeds during the downward slides are found to be constant terminal speed of the confined chains. A fast digital camera (frame rate up to 200 fps) is used to record the images of the moving bead chains. Then the positions of the individual beads are tracked for further analysis. For each set of measurement, more than 2 x 104 images are analyzed to get good statistics. We also measure the velocities of beads inside and outside the channel. From the statistics of the bead velocities, we obtain the effective granular temperature T which is defined as with v being the velocity of the beads.
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