Self-avoiding polymer trapped inside a cylindrical pore: Flory free energy and unexpected dynamics

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 79, Issue 6, 2009, Pages

  Jung, Youngkyun ^a   Jun, Suckjoon ^b   Ha, Bae Yeun ^c  

^a Korea Institute of Science and Technology Information (KISTI)   (South Korea)

^b HARVARD UNIVERSITY   (United States)

^c UNIVERSITY OF WATERLOO   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

CHAIN RELAXATION; CYLINDRICAL PORES; DYNAMIC BEHAVIORS; FINITE SIZE EFFECT; HYDRODYNAMIC INTERACTION; MOLECULAR DYNAMICS SIMULATIONS; NUMERICAL RESULTS; RELAXATION DYNAMICS; SELF-AVOIDING CHAINS; SELF-AVOIDING POLYMERS; SPRING CONSTANTS;

HYDRODYNAMICS; RELAXATION PROCESSES;

FLUID DYNAMICS;

EID: 67649544689     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.79.061912     Document Type: Article

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33. We also note that this compactness of the 1-d case agrees with our mean-field picture [cf. Eq. 4]. To see this, we calculate the relative end-to-end distance fluctuation of the chain using Eq. 4, δ R / R 0 ∼ k eff -1/2 R 0 -1 ∼ (D/a) 11/6 N-1/2. (This is consistent with the scaling analysis.) Thus, for (D/a) 11/3 N, the overall chain fluctuations become negligible. On the other hand, the noncompactness of higher spatial dimensions makes the effect of fluctuations more serious in that any mean-field approach is less reliable. For instance, Eq. 1 for d=3 erroneously implies that δR/ RF ∼ N-1/10 →0 for large N. The correct relation is δR∼ RF -the chain size in this compact noncompact case is set by the end fluctuation. We believe Eq. 5 suffers from a similar inconsistency.
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38. This should be, however, understood with caution: near the full stretch, the FECs are sensitive to intermolecular interactions that hold adjacent monomers together and hence are model dependent. Nevertheless, note that the FECs eventually become D independent.
39. In our analysis with a constant ρ, we combined the scaling result for R 0 (R 0 1.1× D-2/3 N) with ḡ (k); to this end, the numerical prefactor was chosen to ensure the best fit to our simulation data for R 0 (when N=256 was used). In our "hybrid" analysis with numerically determined ρ (r) (diamonds in cyan in Fig. 4), however, both the numerical prefactor and the D exponent were determined from the data. As described later in the main text, the resulting D exponent (0.67) is in excellent agreement with the scaling result. Thus the discrepancy between diamonds in cyan (as well as in blue) and the red line mainly comes from the difference in the functional behavior of ρ (r).
40. Note that ξchain given in Eq. 7 is the friction for diffusion of a confined chain in a narrow capillary, more precisely the friction for longitudinal diffusion of its center of mass. [The wall effect is through the no-slip boundary condition mentioned above Eq. 7.] Imagine that the center of mass, denoted as RCM, is subject to a constant external force in the longitudinal direction, fext. Obviously, the force balance leads to fext = ξchain d RCM /dt, if the inertia term is ignored. One can argue that this is also the friction for the slowest-relaxation mode of the confined chain (τR). The only difference is that the force experienced by the slowest mode is a harmonic force (see Ref.). In the case of a Rouse chain (an ideal noninteracting chain), the friction coefficient of chain diffusion can readily be shown to coincide with that for the slowest-relaxation mode (see for instance Ref.).
41. What remains unclear is why the scaling regime for keff is reached only slowly. While a complete picture is still elusive, we believe that this can be attributed to the fact that keff is related to the variation in free energy with R around its minimum. The blob-scaling picture hinges on the (hidden) assumption that each blob deforms independently of each other; thus the free energy cost per each blob can be obtained accordingly. It is conceivable that this assumption works better for larger Nb (No. of blobs) and D; at the opposite extreme limit of D a, blob deformations are expected to be strongly correlated. This may explain the discrepancy and the sensitivity of keff to finite-size effects. On the other hand, such quantities as R 0 and g (R ) are essentially determined by free-energy minima, not variations; they are expected to enter the scaling regime more easily, as demonstrated in this work (see also Ref. for relevant discussions).