Dynamic properties of hard disk polymer chain inconstrained 2-D space: investigation into a new structure of

polymer sieve

Ziwen Huang1

1Department of Physics, Southeast University, Nanjing 211189, China

Abstract

A new structure of polymer filtrate is of great interest recently, with columns or nanoposts placed on a flatplate. In this paper, we have investigated into the dynamic properties of polymers confined in those columns, suchas its relaxation time and diffusion coefficient, from which we found that such structure can effectively constrainthe motion of long-chained polymers. From the simulation results, we also verify that the model complies with the’tube theory’ proposed by P.G.De Gennes, which was originally proposed to deal with the dynamics of entangledpolymers.

1 Introduction

Since the concept of polymers was raised in the early20th century, polymers and macromolecular have beenuniversal applied around our life. Through the century,numerous polymer physicists did a great deal of workto make clear the principles of polymers.

Recent years, the techniques for controlling or fil-trating polymers with different length become moreand more important. When forced to move on a 2-dimensional space, the physical properties have beenshown by Flory’s theory on so-called ’real chains’,where the unjointed monomers in the chain have inter-action with each other. But recently scientists foundthat the some restrictions on the motion of polymerscould effectively help to impede the long-chain poly-mers’ motion, in which way the polymers could bescreened according to its length.

Scientists proposed a new structure for polymers[1],where columns were placed evenly on a flat plate.We noticed that there are several similarities betweenthis model and ’tube model’ proposed by P.G.DeGennes, used to deal with the phenomena of entangledpolymers[2]. In his model, for instance, the scale ofrelaxation time of polymers is around 3.3[3], with fluc-tuations. Fig 1 shows the schematic graph used in De

Gennes’s tube model.

Figure 1: Tube model: Snake move in a jungle

In this paper, we carefully studied pertinent proper-ties of the polymers in such structure, especially the re-laxation time and coefficient of diffusion. We verifiedthat the motion amid the columns satisfies the descrip-tion of tube model. In addition, the transition of scalefrom free space to constrained space has also been ob-served.

In the simulation, we also found that the scale ofpolymers’ relaxation time amid different intervals ofcolumns varies. The results can be explained by thevariation of the fluctuation of primitive path in tubemodel. We also measured the fluctuation of the prim-itive path of chains with different length.

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2 Model and MethodsThe chain is coarse-grained, which means each

monomer in the chain should be viewed as a clusterof basic component of the polymers. The size of themonomer is dependent on the environment conditions,such as the temperature and the bonding of the basiccomponent of the polymer.

The polymer chain is placed on a flat plate, whereseveral columns were placed evenly on the plate. Toprevent the crossability between chains and columns,the hard-disk potential is adopted. The potential of thesystem can be written as

Uh(|ri − rj |) =

{∞ if |ri − rj | < D

0 if |ri − rj | > D(1)

Un(|ri − rj |) =

{∞ if |ri − rj | > Dm

0 if |ri − rj | < Dm

, i = j ± 1

(2)In the equation, D is the minimum distance the two

centres of the beads can get close to, and Dm is themaximum distance which two adjacent beads can getclose to. i, j mean the ith and jth bead in the chain. riis the ith bead’s position vector. ci is the ith column’sposition vector. In this way, the chain can be connectedwith certain degrees of freedom.

The interaction between column and bead can bewritten as

Un(|ri − ck|) =

{∞ if |ri − ck| < Rb +Rc

0 if |ri − ck| > Rb +Rc

(3)

Rb and Rc are respectively the radius of bead and col-umn, and in our simulation, Rb = D/2. k means thekth column placed on the plate.

The parameters of the size of beads and columnsare carefully chosen to guarantee the uncrossability ofchains and columns.

The schematic of our model is shown in Fig 2.

Figure 2: Tube model: Schematic of the structure’smodel

We adopt Monte Carlo algorithm [4] to simulate themotion of our HD(hard disk) chains amid the columns.In the algorithm, we choose one bead at a time and giveit a random displacement, to simulate the random forcein the solution’s environment. Then calculate the poten-tial of the new configuration. In this case, the potentialcan only be 0 or ∞ . So if the total energy is ∞ ,re-ject the displacement, otherwise accept it. The time ofthe motion of HD chains can be measured as the totalMonte Carlo steps divided by the number of beads ofthe chain.

3 Results and AnalysisBefore showing our simulation results, some impor-

tant concepts and conclusions must be introduced atfirst. In polymer physics, the characteristic time of themotion of a polymer is measured by its relaxation timeτ .

The relaxation time can be measured by correlationfunction

C(t) ≡ < R(t) · R(0) >

< R(0) · R(0) >(4)

where R ≡ rN − r1. N is the total number of beads inthe chain. When time is long enough, it has been provedthat C(t) v exp(-t/τ ) [3] and lnC(t) v -t/τ . So the relax-ation time can be easily derived if C(t) is obtained fromthe simulation.

To measure the diffusion properties, We can use Ein-stein’s diffusion law to calculate the diffusion coeffi-cient.

In tube theory, it’s universally acknowledged thatthe scale should be ranging from 3.2 to 3.4. But inour model, we found that the scale has more complex

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curves. In our simulation, we set the distance of thenearest columns to be 6, 13 and 20, respectively, to seewhat will happen to the relaxation time.

Fig 3 and Fig 4 illustrate the relaxation time and dif-fusion coefficient amid columns with different intervals.

Figure 3: Relaxation time with different column inter-vals

Figure 4: Diffusion coefficient with different columnintervals

From Fig 3, it’s easy to find that the three uppercurves are almost parallel to each other and the slopeof them is very close to 3. And compared with the mo-tion on free plate with no columns, which is plotted red,the the scale of the former three curves are bigger andtheir relaxation time is considerably longer. In specific,the nearer the column is, the longer the relaxation timewill be.

Similar results can also be observed in the simulationof diffusion coefficient.

But to further understand the data in detail, the relax-ation time is plotted below after divided by the cubic ofthe number of monomers, to investigate the differenceof scale more carefully.

Figure 5: Relaxation time in detail. The first figure iscolumns with 20 interval and the second is 13 interval

From these two graphs we can find an interesting re-sult that the scale of the relaxation time will go un-der 3, about 2.6, and then rise to about 3.2 in the firstgraph and 3.3 in the second. The result can be simplyexplained by the fact that when the chain is too shortcompared with the interval of the columns, the columnshave slight influence on the motion of those polymerchains. With shorter length of polymer chains, we cansee that the curve is going nearer to the curve of freeplate.

And when the chain is longer, tube theory, as men-tioned above, will has effect on the chains.

In tube theory, the relaxation time satisfy the relation

τ v N3(1− µ< ∆l >

< l >) (5)

In the relation above, l is the primitive path, and ∆l

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Figure 6: relaxation time with different column inter-vals

is the fluctuation of l. In tube theory, this length meansthe length of the ’primitive’ tube, before relaxation be-gins. When the chain is not too long compared with thecolumn interval, ∆l/l is comparable to 1, so the slopeis slightly higher than 3.

We also measured the case where the column intervalis 6, in this way the length of the chain is several timesthe distance between the columns. The fluctuation willbe much smaller, compared with the formal two cases,so the scale is very near to 3. And from fig 6, we canobserve that the scale of shorter chains amid longer in-terval of columns is very near to the free space’s simu-lation.l and ∆l is also carefully measured in our simula-

tion(the results are not shown in this paper for incom-plete work), which proves that the variation of fluctua-tions of primitive chains.

4 Conclusion

The structure is very useful in impeding the motionof relatively long chains, not only for the difference inabsolute value of dynamic properties, but also for thescale variation. The characteristic time on constrainedspace is around 10 times the time on free plate, and thescale of relaxation time in column environment is over3 and in free space is only 2.6, The same is with itsdiffusion coefficient, dropping from -1 on free plate toaround -1.6.

Additionally, it’s verified in this paper that the tubetheory is effective in explaining the scale transition,with the primitive path and its fluctuation. It’s veri-

fied that the transition is dependent on the fluctuationof primitive path described in equation[5].

The results could be further applied in the manufac-ture of polymer sieve of this structure. If the transitionof the scale can be carefully measured and controlled inapplications, according to specific conditions, the tech-nique of controlling and filtrating of different length ofpolymers will be much more flexible.

References[1] Han J., Craighead, H.G.Science 288,1026 (2000)

[2] P. G. de Gennes, ”Scaling Concepts in PolymerPhysics” Cornell Press (1979)

[3] M. Rubinstein and R.H.Colby ”Polymerphysics”Oxford(2003)

[4] Kurt Binder, ”Monte Carlo and Molecular Dy-namic simulations in Polymer Physics”OxfordUniversity Press 288,1026 (1995)

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