Monte Carlo Simulation of Magnetic Properties and Conformation Properties of Magnetic Polymer Chains

Luo Mengbo Huang Jianhua (Department of Physics, Zhejiang University, Hangzhou 310028) (Department of Applied Chemistry, Zhejiang University of Engineering, Hangzhou 310033) Magnetic properties and conformational properties of magnetic polymer chains at different temperatures. Magnetic polymer chains have spontaneous magnetic moments at low temperatures, and the critical temperature of an infinite chain r (= 1.77005/kB. Near the critical temperature, the polymer chain undergoes a collapse phase transition from a stretched random coil to a contracting sphere. .

Analysis of the size, shape, number of neighbors, and energy of the chain showed that the conformational properties of the polymer chain changed significantly from the temperature T (= 1.77), indicating that the phase transition of the polymer Ising chain is an Ising interaction and chain link. The result of synergy of exercise.

2002-12-09 Received, 2G001-16 Revised; National Natural Science Foundation of China (Grant No. 20240014) and Zhejiang Provincial Natural Science Foundation (Grant No. 202030) Funded Project Since the Discovery of Structural Polymer Ferromagnetic Materials in the 1980s The exploration of its magnetic mechanism has become an important issue for physicists, materialists, and chemists. 2~4. The appearance of organic magnets breaks the traditional notion of organics and ferromagnetics, and it is more about the traditional ferromagnetic theory. challenge. Experimentally, many types of ferromagnetic organic polymers have been synthesized and their magnetic properties have been carefully studied. 3~81. Theoretical calculations also show that the possibility of this ferromagnetic organic magnet exists 9~11, but so far. The magnetic nature is still not clear. 1121. Ferromagnetic polymers have the characteristics of low density, low loss, easy processing, easy film formation, etc., and have great application prospects in the fields of communication, energy, information, etc.16121. Research on traditional inorganic magnets When the magnetic properties of magnetic materials are studied, statistical mechanics models such as Ising, XY and Heisenberg models are the main research objects. Using these models, a lot of research has been done on critical phenomena and magnetic responses. 113. However, from the point of view of statistical mechanics, the critical phenomena and conformational changes of magnetic polymers have rarely been studied. 11214. More recently, Gael et al. A statistical model of ferromagnetic polymer chains was used to study the critical phenomena and conformational changes of magnetic chains in good solvents. 114. For the polymer chain of self-avoidance walking (SAW) on simple cubic grids, they considered the links between The Ising interaction of the Hamiltonian is: Here is the spin of the i-th chain on the chain, the coupling coefficient between the nearest neighbor spins, and h represents the external magnetic field. Shows only the sum of nearest neighbor spins. On the simple cubic lattice point, it means that only the two spins that are separated by a lattice constant are considered the Ising interaction. For simplicity, the spin in equation (1) is represented by a quantum number of 1, the coupling coefficient has a unit of energy, h is proportional to the external magnetic field but has a unit of energy, and the Planck constants, Bohr magnetons, and other constants have been It was absorbed with h. For the ferromagnetic interactions, > 0. Using the mean field theory and MonteCa's simulation, they found that the system has a spontaneous magnetic moment at low temperature, and the conformational collapse occurs, and its critical temperature increases with the external magnetic field. Gael et al. used the MonteCao method of motion combined with the importance sampling method of Metropolis to simulate magnetic moments, specific heat, and chain gyration radius of magnetic polymers at a series of temperatures. 1141. The movement of polymer chains at extremely low temperatures is extremely difficult. Starting from the random conformation, the chain conformation may freeze in a certain conformation, and thus cannot obtain the conformational nature of its low temperature. Therefore, Gael et al. used a Monte Carlo simulation of a complex algorithm 1141 >Tp to perform simultaneously, and then exchanged the adjacent temperature (7/and T/+) chain conformations with some probability. We used the Annealing Monte Carlo method to simulate the magnetic and conformational properties of magnetic polymers at different temperatures from high temperature to low temperature. Annealing Monte Carlo method is an effective way to obtain the properties of the system at low temperature, and the result 11516 can be obtained in agreement with the experiment, and the number of simulated temperatures is not limited by the equipment. This paper simulates the conformational and magnetic properties of magnetic polymer chains in good solvents at zero external field (h=0). The results show that magnetic polymer chains have spontaneous magnetic moments at low temperatures, and the criticality of infinite-long chains. The temperature is 1.) and kB is the Boltzmann constant. Near the critical temperature, the polymer chain undergoes a collapsed transition from a stretched random coil to a contracting sphere. The relationship between the magnetic properties and the conformational properties near the critical temperature is also systematically studied. It is pointed out that the phase transition of the Ising chain of the polymer is Isin; the result of the interaction between the interaction and the chain movement.

1 Modeling and Simulation Monte Carlo simulations are performed on simple cubic grids. The polymer chain segments can only fall on the simple cubic lattice points. The two chain links combined by chemical bonds can only fall on the adjacent two lattice points. That is, the key length is equal to the lattice constant. A polymer chain with a chain length of n consists of n+1 equal-quality links, each with spin i=1, where +1 represents the positive direction of the spin to the z-axis, and 1 represents spin. Point to the negative direction of the z axis. In addition to considering self-avoidance between links, Ising interactions between spins are also considered. The Hamiltonian of the system is represented by formula (1). Although this Ising interaction is not always attractive, we found from the results that the net effect at low temperatures is the mutual attraction between the chains and causes the collapse of the chain. We know that the long-range attraction between chain links (such as the attractive part of the inter-molecular van der Waals potential) can also cause the chain collapse phase transition 171, so for the sake of discussion, we do not consider the remote attraction between chain links. The study considers the effect of Ising interaction on the conformational properties of polymer chains. Only one polymer chain was considered in the simulation to simulate a magnetic polymer chain in a rare solvent.

We used the dynamic Monte Carlo method to generate samples. The method is to generate new conformations through the movement of macromolecular segments. And during the chain conformation movement, the temperature of the system is continuously reduced to simulate the annealing process. First, a polymer chain with a chain length of n is randomly generated at a high temperature, and the spin of each chain is randomly given. Subsequently, a new chain conformation is generated with reference to the motion method proposed by Gurler et al. 1181, and the new conformation is obtained by 90 motions of the terminal bond, 180 rotations of the double bond, and 90 crank motions of the triple bond. Since the spin interactions are considered and different chain conformations have different energies, we have also applied the Metropolis importance sampling method to effectively generate new conformations. That is, the acceptance probability of new conformations is: where AE is the energy difference between the old and new conformations. . Because there is an association between the old and new constellations, the two conformations are not independent in a short time. In order to eliminate this association, we began to record data after the macromolecule chain experienced Brownian motion at time t. The time interval for each recorded data was also t. The time interval t was taken as the relaxation time of the SAW chain. t=0.25n213. The unit of time here is the Monte Carlo step (MCS). All links are tried to move once in one MCS time, and the spin direction is also tried to be reversed once. Each temperature is moved for 500 r, 500 independent constellations are obtained, and then the temperature of the system is lowered. At this time, the initial conformation of the simulation is the final conformation of the previous temperature.

The simulation results also averaged over 100 initial high temperature regime images, that is, we produced 50,000 independent constellations for either temperature.

The temperature T in the simulation is in units. In addition, the length unit is taken as the base length of a simple cubic grid point, and we also take a value of 1. For a simple cubic lattice point, we simulate the chain length from 40, 2 Results and discussion 21 Spontaneous magnetization As we all know, one-dimensional Ising model does not exist spontaneously. The magnetic moment, but the two-dimensional and three-dimensional Ising models all have spontaneous magnetic moments at low temperature, that is, there is a phase transition from magnetic disorder to magnetic order. 113. This paper first studied the spontaneous magnetization of magnetic polymer chains. The magnetization of the system is taken as the order parameter, but since h = 0, the statistical mean value of M<M> = 0, so we use the mean square value of M <M2> instead. The relationship of <M2> with temperature is given. We found that <M2> is close to 0 at high temperatures, but is approximately 1 at low temperatures, indicating that there is a spontaneous magnetization phenomenon in the Ising model of ferromagnetic polymer chains, or that there is a phase transition from magnetic disorder to magnetic order. We define the steepest change in the <M2>-T curve as the maximum temperature value defined as the critical temperature T. Then, the critical temperature T (. With the increase of the chain length, see. We estimate that when the chain length is infinite, the critical temperature is about 1.770.05, which is consistent with the simulation results of Garel et al. (1.80 0.04)114 This also shows that our annealing Monte Carlo simulation method is effective.

Unlike Ising models such as one-dimensional and two-dimensional models, the number of adjacent links in a polymer chain changes due to the movement of the links. If the Ising model of the polymer chain does not consider the Ising interactions between the remote links, then there are only two neighbors for each link, which, like the one-dimensional Ising model, does not have spontaneous magnetic moments. Here the remote link refers to two links that are not nearest neighbors in the chain order and does not refer to the distance in space. There is a spontaneous magnetic moment in the Ising model of the polymer chain, which indicates that the number of neighbors of the chain link is greater than 2. The variation of the mean neighboring number <Nmi> of the chain link with temperature is given. It can be seen that, above the critical temperature, the average number of neighbors is about 25, which is not much different from the number of neighbors of the one-dimensional Ising model. However, near the critical temperature, the number of neighbors suddenly increases, and the addition of the neighbors makes the phase change become may. For a chain with a chain length of n = 300, the number of neighbors in a narrow temperature range is reached: the number of neighbors of the dimensional Ising model.

2.2 Conformation changes with temperature The physical quantity that characterizes the conformation of the chain is mainly the mean square end-to-end distance. First, the mean square radius of gyration, CS2>, is calculated as a function of temperature. The radius of gyration can be obtained by the radius of gyration, where s =) Kxi, y, zi) is the matrix of the position vector of the i-th link away from the center of mass in the chain, and col is the row matrix, and is the transposed matrix. The chain of length n is n + 1 links, numbered 01 until n. The radius of gyration S2 is (a) The mean square radius of rotation of the polymer Ising chain with different chain lengths is given <S2>/n. With temperature changes, we found that there is a significant decrease in the mean square radius of gyration near the critical temperature. Using the scale relationship <S2>an2vs, we calculate the scale index at different temperatures vs. When the temperature is much higher than the critical temperature (such as T=4.0), we have to use vsM6 to indicate that the polymer chain is A stretched random configuration; and when the temperature is much lower than the critical temperature (such as T = 0.6), we get vsQ Ming polymer chain is a compact sphere. The value of the scale index vs indicates that the chain has undergone a collapse phase transition from a high temperature random coil to a low temperature compact sphere. From (a), as the chain length increases, the phase transition becomes more obvious, and the transition temperature gradually approaches Tc=1.77. The common magnetization intensity is zero, and the effect on the conformation is also small. In this regard, we can see from the relationship between the average energy of each link (E) and temperature. Even at high temperatures, <E> is obviously less than 0. But at this time, <M2> is equal to about 0, indicating different Small area spins are still random. As the temperature decreases, the regions where the spin alignments are consistent expand and the energy of the system is lower. The movement of the link changes the number of neighbors and the number of neighbors (the chain is denser) further reduces the energy of the system. The synergy between the two causes the magnetization of the system to be large and the size of the chain to decrease. And at a certain temperature, this reduction in energy is sufficient to overcome the thermal motion energy (/bT), resulting in phase transitions. It can be seen that the phase transition of the polymer Ising chain is the result of the interaction of the Ising interaction and the link motion.

In this paper, the magnetic properties and conformational properties of the magnetic polymer chains with the nearest neighbor Ising interacted on a simple cubic lattice site are simulated from the high temperature to the low temperature using the Annealed Monte Carlo method. The mean square magnetization <M2> experienced a change from 0 at high temperature to 1 at low temperature, indicating that the spontaneous magnetization of magnetic polymer chains exists at low temperatures, and the critical temperature T (. = 1.770.05) of an infinite chain is estimated. Near the critical temperature, the size and shape of the polymer chains changed significantly. From the simulation of the size index and the average non-spherical factor, the polymer chain experienced a collapse from a stretched random coil to a tight sphere. The analysis of the size, shape, number of neighbors, and energy shows that the conformational properties of the polymer chains with different chain lengths change from the temperature T (= 1.77), which indicates that the phase of the polymer ising chain The change is the result of the synergistic effect of the Ising interaction and the chain conformational movement.