Average Helmholtz Free Energy Research Articles

Elastic behavior of comb-like polymer chains

Three-dimensional Monte Carlo simulations of comb-like polymer chains with various backbone lengths N b, arm lengths N a and arm densities m are carried out to study the elastic behavior of comb-like polymer chains. The radius of gyration, the shape factors and bond length in different cases during elastic process are calculated, and it is found that the comb-like polymer molecules with longer backbone or shorter arm are more close to linear chains. But the arm density m affects the chain conformation non-monotonously. Some thermodynamic properties are also studied. Average Helmholtz free energy and elastic force f all increase with elongation ratio λ for all chains.

MECHANICS BEHAVIORS OF ADSORBED THREE-DIMENSIONAL COMPACT CHAINS

Elastic behaviors of a single three-dimensional compact chain adsorbed on the attractive surface are investigated using the pruned-enriched-Rosenbluth method (PERM).The size and shape of the chain,such as mean-square end-to-end distance R~2,mean-square radii of gyration S~2_ xy and S~2_z,and shape factor δ are studied with different adsorption interaction energy and different positions of the chain on which the force is exerted.During the elasticity process,the shape factor δ first decreases with the shape of chains in a sphere,then increases and the chain extends to be a rod shape along the direction of the force.It decreases again,and at the end it remains a constant value.If the chain is far away from the adsorbing surface,all of the conformational properties remain constant,and there would appear the "jump point".On the other hand,some thermodynamic properties such as the average Helmholtz free energy per bond A,average energy per bond U/N,average attraction energy per bond U_a/N,average contact energy per bond U_C/N,elastic force f and energy contribution to elastic force f_U are also calculated during the elasticity process.Different thermodynamic behaviors are obtained when the force acts on different monomers of the chain.Obviously,there exist several long plateaus for elastic force f with strong adsorption interaction,which agrees well with experimental results.We also discuss elastic behaviors of compact chains with weak adsorption interaction,which cannot be obtained from experimental methods.

EFFECTS OF SECONDARY STRUCTURE ON ELASTIC BEHAVIOR OF PROTEIN-LIKE CHAINS

The elastic behavior of protein-like chains was investigated by using the Pruned-Enriched-Rosenbluth Method (PERM). Three typical protein-like chains such as all-α, all-β, and α + β (α/ β) proteins were studied in our modified orientation dependent monomer-monomer interaction (ODI) model. We calculated the ratio of /N and shape factor of protein-like chains in the process of elongation. In the meantime, we discussed the average energy per bond /N, average contact energy per bond c/N, average helical energy per bond h/N and average sheet energy per bond b/N. Three maps of contact formation, α-helix formation, β-sheet formation were depicted. All the results educe a view that the helix structure is the most stable structure, while the other two structures are easy to be destroyed. Besides, the average Helmholtz free energy per bond /N is was presented. The force f obtained from the free energy was also discussed. It was shown that the chain extended itself spontaneously first. The force was studied in the process of elongation. Lastly, the energy contribution to elastic force fu was calculated too. It was noted that fu for all-β chains increased first, and then decreased with x0 increasing.

Elastic Behavior of Polymer Chains

The elastic behavior of the polymer chain was investigated in a three-dimensional off-lattice model. We sample more than 109 conformations of each kind of polymer chain by using a Monte Carlo algorithm, then analyze them with the non-Gaussian theory of rubberlike elasticity, and end with a statistical study. Through observing the effect of the chain flexibility and the stretching ratio on the mean-square end-to-end distance, the average energy, the average Helmholtz free energy, the elastic force, the contribution of energy to the elastic force, and the entropy contribution to elastic force of the polymer chain, we find that a rigid polymer chain is much easier to stretch than a flexible polymer chain. Also, a rigid polymer chain will become difficult to stretch only at a quite high stretching ratio because of the effect of the entropy contribution. These results of our simulation calculation may explain some of the macroscopic phenomena of polymer and biomacromolecular elasticity.

Elastic behavior of single polymer chains adsorbed on rough surfaces

In this paper, elastic behaviors of single polymer chains adsorbed on the rough surfaces with a substrate and some periodically tactic pillars are investigated by the pruned-enriched-Rosenbluth method (PERM). In our simulation, a single polymer chain is firstly adsorbed on the substrate and then pulled along the z-axis direction, which is vertical to the substrate. We investigate the chain size and shape of polymer chains, such as mean-square radii of gyration per bond 〈 S 2〉 xy / N, 〈 S 2〉 z / N and shape factor 〈 δ ∗〉 in order to show how the size and shape of adsorbed polymer chains change during the desorption process. Due to the occurrences of separation of the chains from the substrate, farther adsorption on the upper surfaces of pillars and complete separation from the whole rough surfaces in the elastic process, the changes of 〈 S 2〉 xy / N, 〈 S 2〉 z / N and 〈 δ ∗〉 during the process are complicated. On the other hand, some thermodynamic properties such as average energy per bond, average Helmholtz free energy per bond, elastic force f are investigated, and our aim is to study the elastic behaviors of polymer chains adsorbed on the rough surface during the elasticity process. Elastic force f has some plateaus during the desorption process for strong adsorption interaction. If there is no adsorption interaction, the chains can get away from the rough surfaces spontaneously. These investigations can provide some insights into the elastic behaviors of polymer chains adsorbed on the rough surface.

Translocation of a proteinlike chain through a finite channel

We use the pruned-enriched-Rosenbluth method and the modified orientation-dependent monomer-monomer interaction model to study the translocation of a proteinlike chain through a finite channel. The mean-square radius of gyration per bond <S2>/N and shape factor <delta*> of proteinlike chains with different secondary structures transporting through a finite channel with different channel radii R=1, 2, 3, 4, and 20 are investigated in the translocation. The average Helmholtz free energy per bond A/N and the mechanical force f are also presented. A/N remains unchanged when X(0)<0 and X(0)>1, and decreases monotonously when 0.5<X(0)<0.1. Here X(0)=X/N identical with 2X/L,X is the position of the first monomer, N is chain length, and L is channel length. No free energy barrier is found in our calculation. f is negative and has a plateaulike behavior. The plateau becomes narrow and the value of f increases as R increases. The total energy per bond <U>/N is also calculated in the process of translocation. An energy barrier is shown. The proteinlike chains must cross this energy barrier when they escape from the channel. The position of the maximum of <U>/N depends on the secondary structures and the channel radius. We also discuss the average contact energy per bond <U>c/N, the average alpha-helical energy per bond <U>h/N, and the average beta-sheet energy per bond <U>b/N.

Elastic behavior of compact polymer chain transporting through an infinite adsorption channel

The elastic behavior of a single compact chain transporting through an infinite adsorption channel is investigated using the pruned-enriched-Rosenbluth method (PERM). In our model, single compact chain is fixed with one of its first monomer at a position in an infinite adsorption channel, is then pulled slowly along the direction of z-axis. We first calculate the chain size and shape of compact chains transporting through an infinite channel, such as mean-square end-to-end distance per bond 〈 R 2〉/ N, mean-square radii of gyration per bond 〈 S 2〉/ N, 〈 S 2〉 xy / N and 〈 S 2〉 z/ N, and shape factor 〈 δ〉 for the changes in the size and shape of compact chains during the translocation process. If there are strong adsorption interactions between the monomers and the channel, some special behaviors for the size and shape of compact chains are obtained during the process. On the other hand, some thermodynamics properties are also investigated, such as average energy bond, average Helmholtz free energy per bond, elastic force per bond f and energy contribution to elastic force per bond f E. During the translocation process, elastic force f is less than zero, and has some plateaus in some special regions for strong adsorption interaction, which may explain how the adsorption interaction drives chains through narrow channels or pores in many biological systems because f<0 means that the translocation process need no external force on the chains. These investigations can provide some insights into the mechanics of proteins infiltrating through membrane. In the meantime, by recording and comparing these force-extension curves, we may also investigate the complex interactions between biopolymers (such as protein, RNA, and DNA) and the membranes, and determine indirectly the complicated structure of the channel.

Direct prediction of HP sequences of compact polymer chains from elastic force

In this paper, a new method is proposed to predict HP sequences of compact polymer chains from elastic force based on the PERM (pruned-enriched-Rosenbluth method) simulation. Two different HP sequences are selected here, and we let them pass through a nanopore after acting of elastic force, which only allow single but not double strands to pass. Some thermodynamics properties of compact polymer chains are investigated during the translocation process, such as average Helmholtz free energy 〈 A〉, average energy per bond 〈 U〉, and average contact energy per bond 〈 U c〉. We find that the curves of them change non-monotonously with different steps, which inversely can be used to distinguish H and P accurately. The most important parameter of them is elastic force f because it can be measured directly using single-molecule force spectroscopy (SMFS) in experiment. Through recording and comparing force-extension curves, we can determine the HP sequences accurately. This method is also applied to determine DNA sequence directly, and to study the contact interactions in proteins and the protein folding.

Elastic behavior of adsorbed single compact chains

Elastic behaviors of short single two-dimensional compact chains adsorbed on the attractive surface are investigated in this paper by using the enumeration calculation method. In our model a single compact chain is fixed with one of its end at a position above the impenetrable surface, and then it is pulled away from the attractive surface slowly through elastic force acting. We investigate the chain size and shape of adsorbed compact chains, such as mean-square end-to-end distance per bond 〈 R 2〉/ N mean-square radii of gyration per bond 〈 S 2〉 x / N and 〈 S 2〉 y / N, shape factors 〈 δ〉, and fraction of adsorbed monomers f a in order to illuminate how the size and shape of adsorbed compact chains change during the process of tensile elongation. Especially for strong attraction interaction there are some special behaviors in the chain size and shape during this process. If there exits adsorption interaction, single compact chain is first almost pulled down to the adsorption surface and then moves in the direction of force until to leave the adsorption surface. These changes become more obvious with strong adsorption interaction. Our calculation can show this elastic process of adsorbed compact chains visually and simply. On the other hand, some thermodynamics properties are also studied here. We use average energy per bond, average Helmholtz free energy per bond, elastic force f and energy contribution to elastic force f u to study the elastic behavior of adsorbed single compact chains in the process of tensile elongation. Elastic force f has a long plateau during the tensile elongation for strong adsorption interaction, which agrees well with experimental and theoretical ones. These investigations can provide some insights into the elastic behaviors of adsorbed protein chains.

Elastic behavior of ring polymer chains

AbstractIn this article, the conformational properties and elastic behaviors of ring polymers in the process of tensile elongation are investigated with the Monte Carlo method and the bond fluctuation model. The ratio of the mean‐square diameter &lt;d2&gt; to the mean‐square radius of gyration &lt;S2&gt; increases with the elongation ratio, λ, and the instantaneous shape of ring polymers is more symmetric than that of linear chains in the process of tensile elongation. Here &lt;d2&gt; for ring polymers rather than the mean‐square end‐to‐end distance &lt;R2&gt; for linear polymers is defined as the average of squared distances between two segments separated by N/2 bonds, where N represents the total number of bonds. Local quantities, that is, the mean‐square bond length &lt;b2&gt; and the mean bond angle &lt;θ&gt; increase with λ, especially for short ring chains. The &lt;d2&gt; and &lt;S2&gt; have the same relationship with the chain length, N, that is, &lt;d2&gt; ∼ N1.130±0.020 and &lt;S2&gt; ∼ N1.160±0.013 for a different λ. Some thermodynamics properties are also addressed here. The average energy per bond &lt;U&gt; decreases with λ and the average Helmholtz free energy and elastic force f increase with λ, especially for short ring chains. Comparisons with linear chains are also made. These investigations may provide insight into the elastic behaviors of ring polymers. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 223–232, 2005

Elastic behavior of adsorbed polymer chains

Elastic behaviors of single polymer chains adsorbed on the attractive surface are first investigated using Monte Carlo simulation method based on the bond fluctuation model. We investigate the chain size and shape of adsorbed chains, such as mean-square radius of gyration S2, mean-square bond length b2, shape factors sf(i) and delta*, and the orientation of chain segments P2<cos theta>, to illuminate how the shape of polymer chains changes during the process of tensile elongation. There are some special behaviors of the chain size and shape at the beginning of elongation, especially for strong attraction interaction. For example, mean fraction of adsorbed segments decreases abruptly in the region of small elongation ratio and then decreases slowly with increasing elongation ratio. In fact, the chain size and shape also changes abruptly for small elongation ratio with strong attraction interaction. Some thermodynamics properties are also investigated here. Average Helmholtz free energy increases fast for elongation ratio lambda<1.15, especially with strong attraction, and increases slowly for lambda>1.15. Similar behaviors are obtained for average energy per bond. Elastic force (f ) and energy contribution to force (f(U)) are also studied, and we find that elastic force decreases abruptly for lambda<1.15, and there is a minimum of elastic force for strong attraction interaction, then increases very slowly with increasing elongation ratio. However, there are different behaviors for weak attraction interaction. For energy contribution to force (f(U)), there is a maximum value for strong attraction interaction in the region of lambda<1.15. Some comparisons with the atomic force microscopy experiments are also made. These investigations may provide some insights into the elastic behaviors of adsorbed polymer chains.

Elastic behavior of non-Gaussian polymethylene chains

In this paper, elastic behaviors of non-Gaussian polymethylene (PM) chains with chain length N=100 are investigated by rotational isomeric state model. Here the tetrahedral lattice of PM chain and the non-local interaction of Sutherland potential are adopted. In the metropolis movement of PM chain, a four-bond movement model is used. The average energy and average Helmholtz free energy with various elongation ratios λ are calculated by Monte Carlo simulation method. The average energy increases with elongation ratio λ and the average Helmholtz free energy decreases with elongation ratio λ. The elastic force f and the energy contribution to elastic force f u can be obtained from f=∂〈 A〉/∂ r and f=∂〈 U〉/∂ r. We find that the elastic force f increases with elongation ratio λ and the energy contribution f u decreases with elongation ratio λ, and f u is less than zero. The ratio f u/ f is close to −0.21 for λ⩽1.25, and −0.04 to −0.35 for λ>1.25 at T=364 K. In our calculation, the rubber elasticity may be discussed in terms of the chemical structure of polymer chains.

A molecular study on the reinforcement of polymethylene elastomers

Monte Carlo simulations are carried out on filled networks of polymethylene (PM), which are modeled as composites of PM chains and three-dimensional cubic lattices of filler particles. Calculations are carried out for PM chains with various chain lengths n and various cubic unit dimensions a. The elastic behavior is investigated by using a realistic rotational-isomeric-state (RIS) model and enumeration calculation method. The average conformations, such as a priori probability P η and the segmental orientation function 〈 P 2(cos ζ)〉 of PM chains are also calculated. In the process of tensile deformation, the a priori probability P t increases with elongation ratio λ, however, it decreases with increasing cubic unit dimensions a. The segmental orientation distribution function 〈 P 2(cos ζ)〉 of deformed PM chains decreases with increasing cubic unit dimensions a, especially in the region of large deformation. Average Helmholtz free energy per bond becomes small when increasing cubic unit dimensions a, and average energy per bond becomes large when increasing cubic unit dimensions a. We find that the elastic force increases with elongation ratio for small λ, and abruptly for large λ. In the meantime, the energy contribution to elastic force is negative and significant. It is also shown that the elastic force and the energy contribution to elastic force is almost the same with various cubic unit dimensions a. The ratio f u / f ranges from −0.4 to −0.6 at T=425 K. The reinforcement effects on the Helmholtz free energy 〈 A〉 and energy 〈 U〉 are important; however, the effect on the elastic force is insignificant. Our calculation may provide some insight into the macroscopic phenomena of rubber elasticity.

Elastic behavior of bimodal poly(dimethylsiloxane) networks

AbstractThe rubberlike elastic behavior of bimodal poly(dimethylsiloxane) (PDMS) networks was investigated by the Monte Carlo simulation method and enumeration calculation method on the basis of the rotational‐isomeric‐state (RIS) model. These bimodal PDMS networks consist of short chains (chain length from 10 to 20) as well as long chains (chain length equal to 150). For long PDMS chains, through generating many PDMS conformations in the equilibrium state using the Monte Carlo simulation method we can obtain the average Helmholtz free energy and the average energy. For short PDMS chains with chain lengths from 10 to 20, as the total number of conformations is only from 6.56 × 103 to 3.87 × 108, we adopt the enumeration calculation method. The deformation is partitioned nonaffinely between the long and short chains, and this partitioning can be determined by requiring the free energy of the deformed network to be minimized. Chain dimensions and thermodynamic statistical properties of bimodal PDMS networks at various elongation ratios are discussed. We find that elastic force f increases with elongation ratio λ; the energy contribution fu to elastic force is significant, and the ratio of ranges from 0.15 to 0.36 at T = 343 K. In the meantime, elastic force f increases with the average energy 〈U〉. The energy change in the process of tensile elongation is taken over, which has been ignored in previous theories. Our calculations may provide some insights into the phenomena of rubberlike elasticity of bimodal networks. © 2001 John Wiley &amp; Sons, Inc. J Polym Sci Part B: Polym Phys 40: 105–114, 2002

A Monte Carlo Study of the Elastic Behavior of Polymethylene Chains

The rubber elastic behaviors of long polymethylene (PM) chains are investigated using Monte Carlo simulations and considering the rotational-isomeric state model. Through the Monte Carlo method we can generate many PM chains in the equilibrium states, and obtain the average Helmholtz free energy and average energy . Chain dimensions and thermodynamic statistical properties of long PM chains under various elongatiion ratios λ are also calculated. We find that the elastic force f increases with elongation ratio λ, and that energy contribution f u to the elastic force is negative and significant. The ratio f u /f ranges from -0.37 to -0.32 at T=300 K and from -0.53 to -0.40 at T=413 K, and decreases with increasing temperature, which agrees with the experimental data. Our calculations may provide some insinghts into the macroscopic phenomena of rubber elasticity. Conformations of (a) an isolated chain and (b) a chain with a force acting on it.