First steps toward the construction of a hyperphase

Cent. Eur. J. Phys. • 12(6) • 2014 • 421-426 DOI: 10.2478/s11534-014-0461-z

Central European Journal of Physics

First steps toward the construction of a hyperphase diagram that covers different classes of short polymer chains Research Article

Sid Ahmed Sabeur∗ , Université des Sciences et de la Technologie d’Oran Mohamed Boudiaf, Faculté de Physique, BP 1505 El M’naouer, 31000 Oran, Algeria

Received 21 November 2013; accepted 30 March 2014

Abstract:

We present the results of a multicanonical Monte Carlo study of flexible and wormlike polymer chains, where we investigate how the polymer structures observed during the simulations, mainly coil, liquid, and crystalline structures, can help to construct a hyperphase diagram that covers different polymer classes according to their thermodynamic behavior.

PACS (2008): 87.15.-v,02.70.Uu,05.70.Fh,64.60.Cn Keywords:

Homopolymers • Monte Carlo simulations • phase transition © Versita sp. z o.o.

1.

Introduction

The phase transitions of homopolymer chains have been a topic of considerable interest in recent years [1–3]. A significant amount of work has been carried out to study the behaviour of these kind of many particles systems, mainly using Monte Carlo simulations in generalised ensembles [4]. Different potential models have been used to describe the interactions between monomers, typically the LennardJones (LJ) potential is employed for long range interactions, and the FENE potential for the interactions between bonded monomers [5]. ∗

E-mail: [email protected]

In order to obtain accurate results and reduce computing times, much effort has been made by many groups to develop variants of the Monte Carlo algorithms based on the simulation of a random walk in energy space and the estimation of the density of states, such as multicanonical Monte Carlo method [6, 7] and Wang-Landau algorithm [8]. These new methods enhance the sampling of configurations in systems that exhibit disconnected low energy regions of configuration space. For the potentials cited above, different phase transitions have been observed. As the temperature is decreased, the homopolymer chain undergoes a transition from a coil state to a globular state and a second transition from liquid-globule state to a solid compact globule state at very low temperature [9–11]. Other types of potential models have also been used to study other kinds of transitions like the coil-helix transition and helix-helix transition. It is known that any kind 421

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First steps toward the construction of a hyperphase diagram that covers different classes of short polymer chains

of anisotropic potential or a large stiffness caused by the repulsive part of the potential, can favour the local formation of secondary structures like helices. Interesting results for these kind of transitions include those of Kemp and Chen [12] and more recently those of Bannermann et al. [13], Magee et al. [14] and Vogel et al. [15, 16]. Although the transitions observed in polymer chain collapse processes are known to be of second order, the ground state is non-unique and highly depends on the potential models used in the simulations. It is interesting to combine different potential models in order to construct a hyper phase diagram that classify all the classes of polymers occurring at different values of the temperatures. The paper is organized as follows. In Sec. 2, we describe the potential models used in the present study. A brief background of the method used is provided in Sec. 3. The results are given in Sec. 4 and we conclude by the summary in Sec. 5.

2.

Where ULJ (rij ) = 4ε[(σ /rij )12 − (σ /rij )6 ] is the standard Lennard-Jones potential, rij is the distance between two monomers located at ri and rj (i, j = 1, . . . , N) respectively. Additionally, adjacent monomers are tied together by the finitely extensible non linear elastic (FENE) potential

UFENE (rii+1 ) = −0.5K R 2 ln(1 − [(rii+1 − r0 )/R]2 )

(2)

Following the parametrisation of Ref. [9], the LJ parameters are set to ε = 1, σ = 2−1/6 r0 and the cut-off distance rc = 2.5σ . The two potentials ULJ and UFENE coincide at the minimum distance r0 = 0.7 and the later diverges for r → r0 ± R with R = 0.3. K is a spring constant set to 40. The total energy of the polymer conformation = (r1 , . . . , rN ) is thus given by

Model

tot Uflexible (X ) =

N N−1 X 1 X mod ULJ (rij ) + UFENE (rii+1 ) 2 i,j=1 i=1

(3)

i6=j

First, we have considered the case of a flexible polymer chain consisting of N identical monomers that interact pairwise via a truncated and shifted Lennard-Jones (LJ) potential ULJmod (rij ) = ULJ (min(rij , rc )) − ULJ (rc )

(1)

To extend this study to the case of wormlike polymer chains, we have also used a directional potential model that encourages the formation of helical structures. This model is taken directly from Ref. [12].

  for σ ≤ rij 0 ˆ j , rij ) = −(ˆ Uhelix (ˆ ui , u ui ˆrij )m − (ˆ uj ˆrij )m for d ≤ rij ≤ σ  ∞ for 0 ≤ rij ≤ d

ˆ i = (~ri − ~ri−1 ) × (~ri+1 − ~ri )/sin(θ) denotes the bond Here u orientation unit vector where ~ri is the position of the ith monomer and θ is the fixed bond angle. For this work, we have used an angle of π/3. ˆrij = (~ri − ~rj )/|~ri − ~rj | represents the unit vector defining the relative position √ between monomers i and j. d = (3/2)a and σ = 45/8a are respectively the excluded-volume diameter and the attractive force range, where a = 0.7 is the bond length between monomers. The values of these parameters for the square well potential are chosen to coincide with the Lennard-Jones potential [17, 18]. The exponent m controls the strength of the anisotropy. Thus, we have set m = 6 to produce perfect helical structures.

(4)

Finally, the total energy of the polymer chain in this case is given by [19]

tot Uhelix (X ) = ε

N−1 X N−1 X

ˆ j , rij ) Uhelix (ˆ ui , u

(5)

i=1 j=i+2

Here, ε is a positive constant parameter that represents the strength of the potential.

3.

Simulation method

We have used the multicanonical Monte Carlo method where the Boltzmann distribution Pcan (E) = g(E)e−βE is deformed artificially in a way to perform a random 422


Sid Ahmed Sabeur,

hOiT =

P −E/kB T E OE H(E)/W (E)e P −E/kB T H(E)/W (E)e E

tot tot U tot (X ) = UFENE + εLJ ULJtot + εhelix Uhelix

1 2

N P i,j=1 i6=j

tot UFENE

ULJmod (rij ).

N−1 P

=

i=1

UFENE (rii+1 )

and

300 iterations have been calculated for the weights estimation with 3 × 105 MC sweeps in each iteration and 3 × 108 MC sweeps have been used in the production run. 10 independent simulations were averaged before calculating thermodynamic properties. An example of a flat histogram obtained at the end of the multicanonical Monte Carlo simulation for s = 2.8, is shown in Fig. 1. 2.0 ×105

1.5 ×105

1.0 ×105

5.0 ×104

0.0 ×100 -40

Figure 1.

N=10,s=2.8 -35

-30

-25 E

-20

-15

-10

Flat multicanonical histogram obtained at the end of the simulation for N=10 and s=2.8. Within the region [−40.0, −10.0] almost perfect flatness is achieved. The energy bins are of size 0.03.

(6)

In order to apply a two-dimensional multicanonical Monte Carlo simulation, we have combined the two potential models described in Sec. 2

Where

and quickly producing conformations which are sufficiently uncorrelated [22].

H(E)

walk in energy space and produce a flat histogram, here g(E) = eS(E) is the density of states that connects the entropy to the energy, where β = 1/kB T is the inverse of the temperature [20]. Generic units are used in which the Boltzmann constant kB ≡ 1. The canonical partition function of the polymer chain at temperature T is given R∞ by Z = Emin dEg(E)e−βE . A detailed analysis of the phase behavior of the polymer chain can be done by precisely estimating the density of states. Thus, the canonical probability is multiplied by a weight factor W (E) which is unknown, “a priori”, and has been determined iteratively. The multicanonical energy distribution will be Pmuca (E) ∝ g(E)e−βE W (E). The iterative procedure starts by setting the weight factor W 0 (E) for all energies, to unity. In the first run, a simulation is performed at infinite temperature under canonical distribution. In each simulation n, (n = 0, 1, 2, . . .), an estimate of weight W n (E) is obtained which yield an estin (E) mate of histogram H n (E). The weights W n+1 (E) = W H n (E) have to be determined iteratively until the multicanonical histogram is almost “flat”. Conformations are sampled with energy values lying in the interval [−40.0, 30.0] and discretised in bins of size 0.03. After having estimated the appropriate weights W (E), a long production run is performed to determine different statistical quantities O which can be obtained by the following equation

(7) ULJtot

=

We have also defined a parameter

s = εhelix /εLJ as the potential strength ratio and during the simulation, εLJ is set to 1 and εhelix is variable between 0 to 3. Some thermodynamic quantities were calculated as functions of temperature T and potential strength ratio s such as average energy P P hE i (s, T )i = E g(E)E i e−βE / E g(E)e−βE and specific heat Cv (s, T ) = (1/N)∂hEi/∂T = (hE 2 i − hEi2 )/NkB T 2 . The energy of the polymer chain is sampled using the pivot algorithm [21] which has been found very efficient

4.

Results: Specific heat N=10

Fig. 2 shows the specific heat Cv (s, T ) as a function of temperature for different values of the potential strength ratio s. With increasing potential strength ratio, the pronounced peak at low temperature corresponding to the coil-globule transition, shifts slightly to lower temperatures. The maxima of the peaks decrease with increasing s. The helix-helix transition is visible as a small shoulder below the freezing peak. The shoulders indicating these transitions become larger with increasing potentials strength ratio. The conformational phase diagram is shown in Fig. 3 and is separated into four major regions: random coils and stiff rods (A and A∗ ), liquid globules (B), compact globules (C G) and compact helices (C H). For clarity, visual representation of homopolymer conformations corresponding to these regions, are depicted in Table 1. At high temperature and near the flexible edge (low s values), the conformations are random-coil, this corresponds to phase A. By increasing the potential strength ratio s in the A∗ 423



Table 1.

Visual representation of conformations being thermodynamically relevant in the respective pseudophase regions shown in Fig. 3

Phase

Type

A

random coil

A*

stiff rod

B

liquid globule

CH

compact helix

CG

compact globule

2

s=0.6 s=0.8 s=1.2 s=1.8 s=2.2 s=2.4

1.8 1.6 1.4 1.2 Cv(T)/N

Representative conformation

1 0.8 0.6 0.4 0.2 0 -0.2

0

Figure 2.

0.5

1

1.5

2

2.5 E

3

3.5

4

4.5

5

Specific heat curves for different values of potential strength ratios s in the case of a polymer chain with N = 10. Error bars were obtained from independent simulations and represent less than 3% almost everywhere.

phase, the polymer chain has a stiff rod shape due to the restricted entropic freedom. In the C G regime at low s and low T , there is a maximum number of pairwise monomer-monomer contacts. Therefore the polymer structure is solid and several symmetrical structures [10] can occur within the globule when s is close to the flexible limit (s = 0). At low s and intermediate T in phase B, the polymer chain behaves like a liquid where conformations are unstructured but have a compact shape. At high s and low T , the polymer chain crystallizes abruptly from “vapor” phase A to solid ordered helical structures in phase C H. Recently, Vogel et al. [15, 16] have investigated the conformational phase diagram of flexible polymers with explicit thickness. They have identified four principal pseudophases in thickness-temperature parameter space la-

Figure 3.

Specific heat C v(s, T )/N for N = 10 as a function of potential strength ration s and temperature T . Dark colors represent stable regions. Bright colors are maxima representing phase transition, mainly coil-helix and helix-helix transitions. The transitions separate the following conformational phases: A, random coil; A∗ stiff rod, B, liquid globule; C G, compact globule; C H, compact helix

beled respectively α for helices, β for sheets, γ for rings, and δ for stiff rods. Although they have simulated a different model, we can explore some similarities and differences in the phase diagrams. The structural change of sprawled random coils to a rodlike conformation in δ region due to the increase of thickness is quite similar to the passage from A to A∗ phases when the potential strength ratio s is increased. In contrast to the conformations obtained for low potential strength values in our case, only secondary-structure segments like helices and sheets were dominant for low thickness values and the globular arrangements were not relevant. This difference confirms that the tube polymer concept restricts the conformational space significantly and it is suitable only for

424


Sid Ahmed Sabeur,

the classification of the secondary structures of specific polymers or proteins. Therefore, using a variant of the wormlike potential model seems to be a better choice for classifying a large variety of polymers as described in Ref. [23].

some structures like helices are not very long in nature, but the challenge remains, namely optimizing the method to simulate larger chain lengths. Furthermore, it is already challenging to get accurate data for the 10-mer in the entire parameter space. Also, due to the finite nature of the polymer chain investigated in this study, the transitions observed are not pure phase transitions and a microcanonical inflection-point analysis [24, 25] is necessary in a future work.

-10 〈E〉

Acknowledgments

-20 -30 -40 0

0.5

1 s

Figure 4.

1.5

2

2.5

3

4.55 3.54 3 2.5 T 1.52 1 00.5

Mean energy for polymer chain with N = 10 as a function of temperature T and potential strength ratio s.

Fig. 4 illustrates the mean energy as a function of temperature and for different values of potential strength ratios s. The minimum values of the mean energy are in good agreement with the transitions identified from the specific heat. At low s and low T in the C G regime, the mean energy fluctuates around −20 corresponding to a compact globule ground state. For higher s values, the minimum values of the mean energy are much lower near −40 corresponding to a solid helical structure.

5.

Summary

We have combined two potential models to investigate the phase behavior of different classes of polymers from flexible to semi-flexible depending on the potential strength ratio s and temperature T for a N = 10 length polymer chain. We have used for this purpose a 2D multicanonical Monte Carlo simulation where we have obtained some results for the thermodynamic properties of short polymer chains. The hybrid potential model used in this study demonstrates its ability to capture the phase behavior of different polymer classes and also bridges the gap between recent studies on flexible and semi-flexible polymers, since the structural and thermodynamic properties deviate from those seen using a standard wormlike model [12]. We believe that this approach can give more insights for understanding polymer phase behavior. In the present work, we have focused on the analysis of the thermodynamic properties of short polymer chains, since

This work has been supported by the Algerian Ministry of High Education and Scientific Research through the CNEPRU project No. D01920100054. The author also wish to thank Professor Michael Bachmann for helpful comments and suggestions.

References [1] D. T. Seaton, T. W¨ ust, D. P. Landau, Comput. Phys. Commun. 180, 587 (2009) [2] T. Vogel, M. Bachmann, W. Janke, Phys. Rev. E 76, 061803 (2007) [3] M. P. Taylor, W. Paul, K. Binder, J. Chem. Phys. 131, 114907 (2009) [4] D. P. Landau, K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, Cambridge and New York, 2005) [5] K. Binder, Monte Carlo and Molecular Dynamics Simulations in Polymer Science (Oxford University Press, Oxford and New York 1995) [6] B. A. Berg, T. Neuhaus, Phys. Rev. Lett. 68, 9 (1992) [7] W. Janke, Physica A 254, 164 (1998) [8] F. Wang, D. P. Landau, Phys. Rev. Lett. 86, 2050 (2001) [9] S. Schnabel, T. Vogel, M. Bachmann, W. Janke, Chem. Phys. Lett. 476, 201 (2009) [10] S. Schnabel, M. Bachmann, W. Janke, J. Chem. Phys. 131, 124904 (2009) [11] D. T. Seaton, T. W¨ ust, D. P. Landau, Phys. Rev. E 81, 011802 (2010) [12] J. P. Kemp, Z. Y. Chen, Phys. Rev. Lett. 81, 3880 (1998) [13] M. N. Bannerman, J. E. Magee, L. Lue, Phys. Rev. E 80, 021801 (2009) [14] J. E. Magee, V. R. Vasquez, L. Lue, Phys. Rev. Lett. 96, 207802 (2006) [15] T. Vogel, T. Neuhaus, M. Bachmann, W. Janke, EPLEurophys. Lett. 85, 10003 (2009) [16] T. Vogel, T. Neuhaus, M. Bachmann, W. Janke, Phys. 425



Rev. E 80, 011802 (2009) [17] Y. Zou, C. K. Hall, M. Karplus, Phys. Rev. Lett. 77, 2822 (1996) [18] H. Li, C. Tang, N. S. Wingreen, Phys. Rev. Lett. 79, 765 (1997) [19] W. Browne, P. L. Geissler, J. Chem. Phys. 133, 145102 (2010) [20] M. M¨oddel, M. Bachmann, W. Janke, J. Phys. Chem. B 113, 3314 (2009) [21] N. Madras, A. D. Sokal, J. Stat. Phys. 50, 109 (1988)

[22] J. P. Kemp, J. Z. Y. Chen, Biomacromolecules 2, 389 (2001) [23] D. T. Seaton, S. Schnabel, D. P. Landau, M. Bachmann, Phys. Rev. Lett. 110, 028103 (2013) [24] C. Junghans, M. Bachmann, W. Janke, Phys. Rev. Lett. 97, 218103 (2006) [25] C. Junghans, M. Bachmann, W. Janke, J. Chem. Phys. 128, 085103 (2008)

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