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What is the order of the two-dimensional polymer escape transition?

Hsiao-Ping Hsu, Kurt Binder, Leonid I. Klushin, and Alexander M. Skvortsov
Phys. Rev. E 76, 021108 – Published 8 August 2007
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  • INTRODUCTION
  • ESCAPE FOR A 2D GAUSSIAN CHAIN
  • BLOB PICTURE OF A 2D ESCAPE
  • COMPARISON OF THE GAUSSIAN AND BLOB…
  • MONTE CARLO SIMULATION
  • LANDAU THEORY: DISTRIBUTION OF THE ORDER…
  • SUMMARY AND DISCUSSION
  • ACKNOWLEDGEMENTS
  • APPENDICES
    • References

    Abstract

    An end-grafted flexible polymer chain in three-dimensional space between two pistons undergoes an abrupt transition from a confined coil to a flowerlike conformation when the number of monomers in the chain, N, reaches a critical value. In two-dimensional (2D) geometry, excluded-volume interactions between monomers of a chain confined inside a strip of finite length 2L transform the coil conformation into a linear string of blobs. However, the blob picture raises questions about the nature of this escape transition. To check theoretical predictions based on the blob picture we study 2D single-polymer chains with excluded-volume interactions and with one end grafted in the middle of a strip of length 2L and width H by simulating self-avoiding walks on a square lattice with the pruned-enriched Rosenbluth method. We estimate the free energy, the end-to-end distance, the number of imprisoned monomers, the order parameter, and its distribution. It is shown that in the thermodynamic limit of large N and L but finite L∕N, there is a small but finite jump in several average characteristics, including the order parameter. We also present a theoretical description based on the Landau free energy approach, which is in good agreement with the simulation results. Both simulation results and the analytical theory indicate that the 2D escape transition is a weak first-order phase transition.

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    • Received 14 April 2007

    DOI:https://doi.org/10.1103/PhysRevE.76.021108

    ©2007 American Physical Society

    Authors & Affiliations

    Hsiao-Ping Hsu and Kurt Binder

    • Institut für Physik, Johannes Gutenberg-Universität Mainz, D-55099 Mainz, Staudinger Weg 7, Germany

    Leonid I. Klushin

    • American University of Beirut, Department of Physics, Beirut, Lebanon

    Alexander M. Skvortsov

    • Chemical-Pharmaceutical Academy, Prof. Popova 14, 197022 St. Petersburg, Russia

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    Vol. 76, Iss. 2 — August 2007

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    Images

    • Figure 1
      Figure 1
      Schematic drawings of a flexible polymer chain of length N grafted in the middle of the strip of length 2L and width H, in a blob picture: (a) As the chain is imprisoned inside the strip, it forms a sequence of nb blobs. (b) As the chain length is larger than the maximum chain length N* of a chain in an imprisoned state, the chain partially escapes from the strip and forms an escaped state. An escaped state consists of a “stem” containing N* monomers and a “crown” containing N−N* monomers.Reuse & Permissions
    • Figure 2
      Figure 2
      Theoretical predictions for various averaged chain characteristics plotted against L∕N at constant strip width H: (a) the free energy per monomer F∕N, (b) the lateral force fL, (c) the compression force per monomer fH∕N, (d) the fraction of imprisoned monomers Nimp∕N, (e) the order parameter S, and (f) the end-to-end distance per monomer R∕N. Gaussian model results are shown by dotted lines, blob model results by solid lines. The chain is in an imprisoned state for LN>(LN)**[(LN)*] and in an escaped state for LN<(LN)**[(LN)*] for the Gaussian chain model [blob picture].Reuse & Permissions
    • Figure 3
      Figure 3
      Schematic drawing of a polymer chain growing as a self-avoiding walk inside a finite strip and grafted at (x=0,y=0). Monomers are allowed to sit on the lattice sites, except for the lattice sites representing the walls {−L⩽x⩽L,y=0} and {−L⩽x⩽L,y=H}. The first monomer is attached with a bond to the grafting site marked by a cross. Lengths are measured in units of the lattice spacing.Reuse & Permissions
    • Figure 4
      Figure 4
      (a) Free energy relative to a free chain, F(N,L,H)=−ln[Z(N,L,H)∕Z0(N)], plotted against N3∕4H−1 for L=6400 and H=17, 33, 65, and 129. The dashed curve is Fimp(N,L,H)=1.944NH−4∕3 and gives the best fit of the data.Reuse & Permissions
    • Figure 5
      Figure 5
      The log-log plot of the excess free energy of the escaped chain, Fesc(N,L,H), plotted against L∕H for various values of L and H. The dashed line is Fesc(N,L,H)=2.03L∕H and gives the best fit of the data.Reuse & Permissions
    • Figure 6
      Figure 6
      Free energy relative to a free chain divided by N, F(N,L,H)∕N, plotted against L∕N for various values of L and H. The dashed line extrapolated to zero is 2.03L∕(NH).Reuse & Permissions
    • Figure 7
      Figure 7
      Average end-to-end distance divided by N, ⟨x⟩∕N, plotted against L∕N for various values of L and H. The dashed line is ⟨x⟩∕N=L∕N.Reuse & Permissions
    • Figure 8
      Figure 8
      Average fraction of the imprisoned monomers, ⟨Nimp⟩∕N, plotted against L∕N for L=800 and various values of H. The jump becomes more prominent as H decreases. The ends of the extrapolated straight dashed lines to L∕N=0 all approach zero.Reuse & Permissions
    • Figure 9
      Figure 9
      Average order parameter S plotted against L∕N for various values of L and H.Reuse & Permissions
    • Figure 10
      Figure 10
      Average fraction of the imprisoned monomers, ⟨Nimp⟩∕N (solid line), and two partial contributions ⟨Nimp⟩1∕N due to confined configurations and ⟨Nimp⟩2∕N due to escaped configurations (dotted lines) as functions of L∕N near the transition point (L∕N)tr=0.381 for L=3200 and H=17.Reuse & Permissions
    • Figure 11
      Figure 11
      Variances of the number of imprisoned monomers divided by N, (a) σ12(Nimp)∕N, for the imprisoned state, and (b) σ22(Nimp)∕N, for the escaped state, plotted against L∕N. The height of peaks increases with L for a fixed value of H.Reuse & Permissions
    • Figure 12
      Figure 12
      Transition points (N∕L)tr versus H. The dashed line is (N∕L)tr=1.025(35)H1∕3 and gives the best fit of the data.Reuse & Permissions
    • Figure 13
      Figure 13
      Based on the Landau theory, the theoretical predictions of the average values of (a) the fraction of imprisoned monomers, Nimp∕N, and (b) the order parameter S are plotted against L∕N. The chain is in an imprisoned state for L∕N>(3a∕25∕3)(A∕B)1∕6(a∕H)1∕3 and in an escaped state for L∕N<(3a∕25∕3)(A∕B)1∕6(a∕H)1∕3.Reuse & Permissions
    • Figure 14
      Figure 14
      The Landau free energy divided by N, Φ(N,L,H,s)∕N, plotted against s for various values of L and H=28. The predicted Landau free energy functions ΦP(s)=Φimp, Eq. (28), in the imprisoned regime, and ΦP(s)=Φesc, Eq. (32), in the escaped regime, are also plotted (dashed lines).Reuse & Permissions
    • Figure 15
      Figure 15
      The Landau free energy divided by N, Φ(N,L,H,s)∕N, plotted against s for various values of L and H=28. The two minima of Φ(N,L,H,s)∕N are determined by fitting g(s)=gimp(s), Eq. (41), in the imprisoned regime, and g(s)=gesc(s), Eq. (42), in the escaped regime, going through those lower points around the two minima, respectively.Reuse & Permissions
    • Figure 16
      Figure 16
      The reduced jump of the order parameter ΔS∕Sesc plotted against L−1.Reuse & Permissions
    • Figure 17
      Figure 17
      The square root of the variance σ1(m) for the imprisoned states, σ2(m) for the escaped chains, 1−⟨m⟩, σ(m) of the chain in either an imprisoned or escaped state, and the difference Δσ=σ1(m)−σ2(m) against L∕N.Reuse & Permissions
    • Figure 18
      Figure 18
      Variance due to the imprisoned configuration multiplied by N, Nσ12(m), plotted against L∕N for L=3200 and H=17. The solid curve is the best fit of Eq. (A8), Nm12∕4cosh2[(t−ttr)a], with the height of the peak A1(L,H)=N(m1)2∕4≈2099.74, the full width at half maximum (FWHM) Γimp(L,H)≈1.7627∕a=0.0035, and the position of the peak ttr,1=(L∕N)tr,1=0.3810. The FWHM are given by the distance between points on the curve shown at which the corresponding height reaches half height of the peaks (half maximum).Reuse & Permissions
    • Figure 19
      Figure 19
      FWHM Γα(L,H) for the imprisoned state (α=1) and for the escaped state (α=2) against L−1. The dashed curves are a1,H(H∕L)+b1,H(H∕L)2 and give the best fit of the data. Values of a1,H and b1,H are listed in Table .Reuse & Permissions
    • Figure 20
      Figure 20
      Inverse of the height of the peaks for the imprisoned state, (a) A1−1(L,H), and the escaped state, (b) A2−1(L,H), plotted against L−1. The dashed curves are (a) a2,H(4H−1∕3∕L) and (b) c2,H(H∕L)+d2,H(H∕L)2, and give the best fit of the data. Values of a2,H, c2,H, and d2,H are listed in Table .Reuse & Permissions
    • Figure 21
      Figure 21
      The relative reduction in the number of imprisoned monomers, Δm, plotted against H∕L.Reuse & Permissions
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