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Static properties of a simulated supercooled polymer melt: Structure factors, monomer distributions relative to the center of mass, and triple correlation functions

Martin Aichele, Song-Ho Chong, Jörg Baschnagel, and Matthias Fuchs
Phys. Rev. E 69, 061801 – Published 4 June 2004
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  • INTRODUCTION
  • MODEL AND SIMULATION TECHNIQUE
  • THEORETICAL BACKGROUND
  • RESULTS ON TWO-POINT CORRELATION…
  • RESULTS ON THREE-PARTICLE CORRELATION…
  • SUMMARY AND CONCLUSIONS
  • ACKNOWLEDGEMENTS
  • APPENDICES
    • References

    Abstract

    We analyze structural and conformational properties in a simulated bead-spring model of a nonentangled, supercooled polymer melt. We explore the statics of the model via various structure factors, involving not only the monomers, but also the center of mass (CM). We find that the conformation of the chains and the CM-CM structure factor, which is well described by a recently proposed approximation [Krakoviack et al., Europhys. Lett. 58, 53 (2002)], remain essentially unchanged on cooling toward the critical glass transition temperature Tc of mode-coupling theory. Spatial correlations between monomers on different chains, however, depend on temperature, albeit smoothly. This implies that the glassy behavior of our model cannot result from static intrachain or CM-CM correlations. It must be related to interchain correlations at the monomer level. Additionally, we study the dependence of interchain correlation functions on the position of the monomer along the chain backbone. We find that this site dependence can be well accounted for by a theory based on the polymer reference interaction site model. We also analyze triple correlations by means of the three-monomer structure factors for the melt and for the chains. These structure factors are compared with the convolution approximation that factorizes them into a product of two-monomer structure factors. For the chains this factorization works very well, indicating that chain connectivity does not introduce special triple correlations in our model. For the melt deviations are more pronounced, particularly at wave vectors close to the maximum of the static structure factor.

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    • Received 18 September 2003

    DOI:https://doi.org/10.1103/PhysRevE.69.061801

    ©2004 American Physical Society

    Authors & Affiliations

    Martin Aichele

    • Institut für Physik, Johannes Gutenberg-Universität, 55099 Mainz, Germany and Institut Charles Sadron, 6 rue Boussingault, 67083 Strasbourg, France

    Song-Ho Chong

    • Laboratoire de Physique Mathématique et Théorique, Université Montpellier II, 34095 Montpellier, France

    Jörg Baschnagel*

    • Institut Charles Sadron, 6 rue Boussingault, 67083 Strasbourg, France

    Matthias Fuchs

    • Fachbereich Physik, Universität Konstanz, 78457 Konstanz, Germany

    • *Corresponding author. Email address: baschnag@ics.u-strasbg.fr

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    Issue

    Vol. 69, Iss. 6 — June 2004

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    Images

    • Figure 1
      Figure 1
      Schematic illustration of the correlation functions defined in Sec. 3B. ria is the position of the ath monomer of chain i and Ri the position of the chain’s center of mass. wcd and wa,C [Eqs. (11, 18)] denote intrachain structure factors, hcd and ha,C denote [Eqs. (12, 19)] interchain structure factors. SC [Eq. (16)] is the structure factor of the centers of mass.Reuse & Permissions
    • Figure 2
      Figure 2
      Collective static structure factor S(q) of the melt vs the modulus of the wave vector q [Eq. (13)]. The temperatures shown are: T=1 (dashed line), 0.7, 0.65, 0.6, 0.55, 0.52, 0.5, 0.48, 0.47, and 0.46 (solid line). Inset: First maximum of S(q), S(qmax), vs T (qmax≈7.15). The dashed horizontal line indicates the Hansen-Verlet criterion for the glass transition of hard spheres within the ideal MCT [S(qmax)≈3.54] [38].Reuse & Permissions
    • Figure 3
      Figure 3
      Polymer static structure factor w(q) [Eq. (14)] at the lowest and the highest temperature, T=0.46 (solid line) and T=1.0 (dashed line), respectively. The simulation data are compared to the Debye formula [Eq. (28), dotted line] and to the large-q approximation, Eq. (27) (rb=0.9609, dash-dotted line). The inset magnifies of the small-q behavior of w(q).Reuse & Permissions
    • Figure 4
      Figure 4
      Distinct contribution to the static structure factor ρmh(q) [Eq. (13)] at T=0.46, 0.65, and 1.0. Inset: Comparison of ρmh(q) with S(q)−1 and −wD(q) at T=0.46 (see text for further details).Reuse & Permissions
    • Figure 5
      Figure 5
      Examination of the equivalent-site approximation, Eq. (31), at T=0.47. The solid lines in both panels denote c(q) determined from the simulation results for S(q) and w(q) via Eq. (33). The dashed lines represent c11(q) (upper panel) and c15(q) (lower panel), the dotted lines c22(q), c33(q), c44(q), c55(q) (upper panel) and c25(q), c35(q), c45(q), c55(q) (lower panel). The site-site direct correlation functions cab(q) are calculated from the simulation results for Sab(q) and wab(q) via Eq. (30). The insets magnify the region close to qmax.Reuse & Permissions
    • Figure 6
      Figure 6
      Static structure factor wab(q) of the monomer pair (a,b) at T=0.47. a and b are monomers of the same chain. Note that wab(q) depends on ∣a−b∣ only. The simulation data for ∣a−b∣=1 (a=1,b=2 and a=5,b=6) are compared with Eq. (26) (circles). For separations ∣a−b∣=1,…,5 the Gaussian approximation, Eq. (34), is also shown (dotted lines).Reuse & Permissions
    • Figure 7
      Figure 7
      Interchain static structure factor ρhab(q) at T=0.47 for different pairs (a,b) [Eq. (13)]. ρhab(q) depends on (a,b) at qmax≃7.15 and for q≲5.5. The correlation of two chain ends (a=b=1) behaves differently in comparison to all other curves. The average over all monomers ρh(q) is also included.Reuse & Permissions
    • Figure 8
      Figure 8
      Comparison of the site-site intermolecular pair-correlation functions hab(q) determined from the simulation data at T=0.47 (circles) and from the PRISM theory (gray solid lines) [46]. Some curves are shifted vertically for clarity. Note that h21(q)=h210(q) in the PRISM theory (see text for details).Reuse & Permissions
    • Figure 9
      Figure 9
      Static structure factor of the CM SC(q) at all simulated temperatures, i.e., T=0.46,…,1 (solid lines). The dashed line shows Eq. (37) calculated from the simulation data for wm,C(q), w(q), and h(q) at T=1. The dotted horizontal line indicates the limit of SC(q) for q→0 [Eq. (38)]. kBTρmκT was read off from the small-q behavior of S(q) (see Fig. 2). The dotted vertical line shows 1∕Rg≃0.69 (Rg≃1.45).Reuse & Permissions
    • Figure 10
      Figure 10
      wm,C(q)2∕w(q)2 vs q at T=0.46 (solid line) and T=1 (dashed line). As w(q) does not depend on temperature (see Fig. 3), the figure indicates that the monomer-CM correlation wm,C is also independent of T. The dotted line shows the Gaussian approximation, Eq. (39). The dotted vertical line indicates q=1∕Rg≃0.69 (Rg≃1.45).Reuse & Permissions
    • Figure 11
      Figure 11
      Site-resolved structure factors resulting from monomer-CM correlations [Eqs. (17)–(19)]: wa,C(q) (intrachain part), ρha,C(q) (interchain part), and Sa,C(q) (all chains). These structure factors are (almost) independent of T. The data shown were obtained at T=0.47. Circles indicate the Gaussian approximation [Eq. (40)] for wa,C(q) at a=1 (chain end) and a=5 (middle monomer). The thickness of the lines and the symbols increases from a=1 to a=5.Reuse & Permissions
    • Figure 12
      Figure 12
      Comparison of the monomer-CM intermolecular pair-correlation function ha,C(q) determined from the simulation data at T=0.47 for a=1 and a=5 (circles) and from the PRISM theory [Eq. (41), solid lines]. The data for h1,C(q) are shifted vertically for clarity.Reuse & Permissions
    • Figure 13
      Figure 13
      Comparison of the three-monomer structure factor (thin lines) for the melt S3(q,q,q) and for the polymers w3(q,q,q), with the respective convolution approximations (thick lines) S(q)3 and w(q)3 at T=0.47. The simulation results for the triple correlations are not smoothed. The lower statistical accuracy of S3(q,q,q) compared to S(q)3 is clearly visible, especially at large q.Reuse & Permissions
    • Figure 14
      Figure 14
      w3[q,q,p=q2(1−cosφ)] (top) and S3[q,q,p=q2(1−cosφ)] (bottom) vs cosφ for some selected q values at T=0.47. The simulation data for w3 and S3 are represented by thin lines, the convolution approximation [Eq. (24) with c3=0] by thick lines. Note that the data for S3(q) at q=7.1 are rescaled by a factor of 0.1.Reuse & Permissions
    • Figure 15
      Figure 15
      Direct correlation functions vs q at T=0.47. The correlation functions cm,m(q), cm,CM(q), and cCM,CM(q) are calculated from Eq. (A8). The results of this calculation are numerically reliable for q≳1 [57]. Additionally, the figure shows the monomer-monomer direct correlation c(q) given by Eq. (33) and the CM-CM direct correlation function cC(q) defined from SC(q) by ρcC(q)=1−1∕SC(q). The monomer-monomer direct correlation functions cm,m(q) and c(q) are indistinguishable from one another (for q≳1 [57]). For q→0, ρmc(q) and ρcC(q) tend to the limits [see Eqs. (32, 38)]: Nρmc(q→0)=1−N∕kBTρmκT=ρcC(q→0) (≈−840 for T=0.47).Reuse & Permissions
    • Figure 16
      Figure 16
      Average monomer-CM static structure factor Sm,C(q) [Eq. (21)] vs q. The solid line represents the simulation results at T=0.47. The dashed line is the PRISM prediction Sm,CPRISM(q)=wm,C(q)S(q)∕w(q), where the right-hand side was calculated from the simulation. Although the simulated Sm,C(q) is fairly noisy, the comparison still suggests that there are systematic deviations between Sm,C(q) and Sm,CPRISM(q) for q≲6.5. These deviations come from the assumption of Eq. (A9): the direct correlation functions involving the CM become truly 0 only for q≳6.5. The deviations might be responsible for the differences found between the simulated SC(q) and Eq. (A10) at q≈3 [see Fig. 9; we could not test this hypothesis due to insufficient statistics of Sm,C(q)]. However, they do not appear to hamper the good agreement between the simulation results and the PRISM predictions in Figs. 9, 12.Reuse & Permissions
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