Article pubs.acs.org/Macromolecules
Cite This: Macromolecules XXXX, XXX, XXX−XXX
Nanorheology of Polymer Solutions: A Scaling Theory Yitzhak Rabin*,† and Alexander Y. Grosberg*,‡ †
Department of Physics and Institute for Nanotechnology and Advanced Materials, Bar Ilan University, Ramat Gan, Israel Department of Physics and Center for Soft Matter Research, New York University, 726 Broadway, New York, New York 10003, United States
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‡
ABSTRACT: The current theory of viscoelasticity of polymer fluids deals with their response to time-dependent macroscopic perturbations on arbitrary time scales. We use scaling methods to calculate the elastic and dissipative moduli G′ and G″ on length scales that range between monomer size and macroscopic dimensions for several important models of polymer solutions. We find that while both moduli decrease rapidly with decreasing length scale, G′ is more strongly suppressed than G″ at large wave numbers. Possible experimental tests of our predictions are discussed.
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active “motors” or “swimmers” embedded in the system. Although the spatial distribution of such stresses and strains as well as their time dependence is not trivial, the fact of (approximate) spatial uniformity allows for a simplified formulation of linear response theory in Fourier domain whereby stresses and strains are presented as superpositions of plane waves such that corresponding moduli are only functions of the wavenumber q and frequency ω. We will remain within this approach. The definition of relaxation modulus G(q, ω), consisting of real and imaginary parts, G(q, ω) = G′(q, ω) + iG″(q, ω), follows basic prescriptions of the linear response theory14 as follows. First, we assume that the overall strain amplitude is small enough to remain within the linear response regime. To be specific, we imagine that a material is deformed by shearing along the x direction such that its strain rate is modulated in space along the y direction with some wavenumber q: γ̇ = γ̇0(t)cos(qy). Then the stress will also be modulated with the same wave vector q along the same direction y, and if we know the entire history of strain, specified by γ̇0(t), then we can find stress from
INTRODUCTION Rheology of complex fluids deals with the viscoelastic response of these fluids on time scales that can be either longer or shorter than characteristic molecular time scales. Since rheological experiments use macroscopic flow devices, the usual theory of viscoelasticity1 is concerned with the elastic and viscous properties of the fluid on large scales (vanishing wave numbers) only. However, in order to explore the micrometer-scale dynamics of the cytoplasm and the nucleoplasm of living cells, in recent years there has been a great deal of interest in the microrheology of complex fluids in which one traces the Brownian motion of micrometer-size nanoparticles and uses the generalized Stokes− Einstein relation to obtain the frequency-dependent viscosity of the fluid.2−7 As shown by Brochard and de Gennes8 and more recently by Cai, Panyukov, and Rubinstein,9,10 such measurements provide information about the local viscosity of the fluid on the scale of the tracer particle. More direct attempts to compute the local shear viscosity of polymer solutions and melts have been recently proposed based on linear response theory11 or using simple scaling arguments12 and yielded identical results for melts of unentangled Gaussian chains considered in both papers (the results were also consistent with those of refs 8 and 9). However, while our theoretical analysis satisfactorily addressed the local viscous properties of polymer solutions it left open the question of the local elastic properties of these solutions. Both of these moduli are needed if one wants to understand the nanoscale viscoelasticity of polymer solutions and, in particular, the hydrodynamics of chromatin in the nucleoplasm;13 since chromatin is an important application of our theory and to make our paper self-contained we include in Appendix A a brief summary of the polymer properties of chromatin. In the present study we use a combination of linear polymer rheology and scaling methods to obtain approximate expressions for the storage (elastic) and loss (dissipative) moduli G′ and G″ that govern the linear response of polymer solutions to periodic shearing forces across the entire range of length and time scales (i.e., for all frequencies and wavelengths). In real systems, space- and time-dependent stresses and strains are usually caused by either small passive particles or © XXXX American Chemical Society
t
σ0(t )cos(qy) =
∫−∞ G(q , t − t′)γ0̇ (t′)cos(qy)dt′
(1)
where the integration range is fundamentally restricted by the causality principle. The defining relation (eq 1) is sometimes referred to as the Boltzmann superposition principle (see ref 15 (p 286)). To have a better grasp of G one usually considers a small step strain γ0, such that γ̇0(t) = γ0δ(t), which means every particle in the system is shifted at time t = 0 along x by the distance γ0 cos(qy); in that case, stress at any subsequent time t > 0 is simply proportional to G(q, t), which is the convenient definition of G. As pointed out in the paper,11 there is an important subtlety that one has to keep in mind while using the above definition of the relaxation modulus. From an experimental or a computational point of view, to realize a spatially modulated step Received: June 20, 2019 Revised: August 15, 2019
A
DOI: 10.1021/acs.macromol.9b01272 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules strain experiment, in which the strain rate is γ0 cos(qy)δ(t), one has to locally exert appropriate forces on every volume element (or on every particle) in the system, and it is not enough to exert forces at t = 0. In order to force particles to make a step at t = 0 by γ0 cos(qy) and then stand still one has to continue exerting appropriate forces all of the time after at t > 0. It turns out (see ref 11) that this leads to important consequences in implementing the fluctuation−dissipation relation and relating G(q, t) to the stress−stress correlation function. Here, we will not follow this path and will directly use the modulus definition above. In what follows we will make scaling estimates for the relaxation modulus for a variety of polymer models. We will simultaneously consider both real and imaginary parts of the relaxation modulus, G′(q, ω) and G″(q, ω). Our results for the imaginary part are all in agreement with our earlier paper,12 except that the paper was formulated in terms of viscosity related to the imaginary part of the modulus according to η(q, ω) = G″(q, ω)/ω.
In Appendix B we show how to compute sin- and cos-Fourier transforms of these functions and obtain their asymptotic behavior α l o ωτ1 ≫ 1 o(ωτ0) , G′(ω) ≈ G0o m o 2 2−α α o o n ω τ1 τ0 , ωτ1 ≪ 1 α l o ωτ1 ≫ 1 o(ωτ0) , G″(ω) ≈ G0o m o 1−α α o o n ωτ1 τ0 , ωτ1 ≪ 1
OVERALL LOGIC If we manage to exert such forces on the system that its velocity history is (2)
with a small distance γ0, then the corresponding force is, up to factors, equal to the relaxation modulus which is G(q, t). We can visualize condition 2 if we imagine a set of permeable planes perpendicular to the y axis and positioned at y = nπ/q, see Figure 1a. At time t = 0, we move planes with odd n distance γ0 in
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UNENTANGLED MELT OF LINEAR CHAINS: ROUSE MODEL Conceptually the simplest example of a polymeric system is the one obeying the Rouse model laws, that is, an unentangled melt of linear chains (in which hydrodynamic interactions are fully screened, see, e.g., ref 15, Chapter 8). As a reminder, short time relaxation of Rouse polymer obeys the t−1/2 power law, which means α = 1/2. We now consider blobs of the size ∼2π/q; as Gaussian chains they have g ≈ (2π/bq)2 (Kuhn) monomers each, and their Rouse relaxation time is τ1 ≈ τ0g2 ≈ τ0(bq)−4. Plugging these results for α and τ1 in eqs 3, 4a, and 4b we find the results that are summarized in the diagram in Figure 2 as the two main regimes. Here c is the concentration, as the number of
Figure 1. Schematics of a polymer system before (a) and after (b) a spatially modulated step strain.
the positive x direction and with even n the same distance in the negative x direction, as shown in Figure 1b. This step strain generates some stress in the polymer system, which exerts the force on our planes. If we insist on keeping these planes immobile we must then apply proper forces to them. We measure how much force we need to exert on those planes in order to keep them at rest. That measured force is, up to trivial factors, G(q, t). Of course, it cannot be overemphasized that all these planes or only represent an eye-catching image of Fourier space; in reality, what we are talking about is the drive by various molecular motors which can be mathematically decomposed into Fourier modes. To address the problem with such a stack of planes we will consider blobs of size 2π/q. These blobs will typically exhibit on short time scales some power law relaxation dynamics, ∼G0(t/τ0)−α, with appropriate power α and appropriate time τ0. This power law relaxation is then cut off at some other longer time τ1. As it is usually done (see, e.g., ref 15, section 8.4.1), we will simply assume that this cutoff can be described by an exponential, i.e. ij t yz G(t ) = G0jjj zzz e−t / τ1 j τ0 z k {
(4b)
These scaling asymptotics are insensitive to the fact that the high-frequency cutoff in eq 3 was taken in the form of a simple exponential; a stretched exponential would lead to identical results in the considered crudest approximation. Note that the relaxation modulus, as any response function, must satisfy certain general conditions14 (Chapter XII), which apply also to the q-dependent modulus considered here. In particular, G′(q, ω) and G″(q, ω), at any fixed q, must be even and odd functions of ω, respectively. To place our subsequent mathematical manipulations on a more intuitive ground, let us mention that the most important characteristic time in the problem τ0 ≃ 6πηb3/T is associated with the diffusion of one Kuhn segment of length b in the medium of (dynamic) viscosity η, while T is the absolute temperature (in energy units; Boltzmann constant is set to unity throughout this paper). For example, assuming viscosity to be that of water (η ≈ 10−2 Poise = 10−9pN·s/nm2) and Kuhn length to be that of chromatin fiber, b ≈ 100 nm, the relaxation time τ0 is in a millisecond range.
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vx(t ) = γ0·δ(t )cos(qy)
(4a)
Figure 2. Diagram of scaling regimes for both storage modulus G′(q, ω) and loss modulus G″(q, ω) for unentangled melt of Rouse chains. Notations are as follows: b is Kuhn length, c is concentration as the number of Kuhn segments per unit volume, T is temperature in energy units, and τ0 ≃ 6πηb3/T is a Kuhn segment relaxation time. Underlined symbols are dimensionless quantities: G̲ = G/Tc, q̲ = qb, ω = ωτ0.
−α
(3) B
DOI: 10.1021/acs.macromol.9b01272 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules Kuhn segments per unit volume, and G0 = cT, where T is temperature in energy units. There is also a region of very small q and very small ω, when would-be-blobs are larger than the entire chain; in that case, τ1 should be taken as the relaxation time of the entire chain, τ1 ≈ τ0N2. As usual, at high frequencies we obtain G′ ≈ G″. More interestingly, at smaller frequencies, storage modulus G′ decays very sharply with increasing wavenumber q, as q−6, much faster than the loss modulus G″. That means the system probed at low frequencies even very locally, at the scale much smaller than the coil size, at qb > N−2, still exhibits mostly a viscous, very weakly elastic response. One can get a physical feeling of this result by thinking about blobs of the size ∼2π/q which have sufficient time to tumble, thus intermixing their x, y, and z components over time ∼2π/ω.
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Figure 3. Diagram of scaling regimes for both storage modulus G′ and loss modulus G″ for unentangled solution of Rouse chains with hydrodynamic (HD) interactions. Compared with Figure 2, the new regimes are indicated in blue color. They correspond to the values of wavenumber q and frequency ω such that they probe the scales inside one mesh of the polymer solution, where HD interactions are not screened. ξ is the mesh size, while b is Kuhn segment; in this figure b < ξ; see the opposite limit b > ξ in the text. As before, G̲ = G/Tc, q̲ = qb, ω = ωτ0.
ROLE OF HYDRODYNAMIC INTERACTIONS Previously, we considered the melt of unentangled chains. This is fundamentally the simplest type of polymeric material, but it is not particularly interesting in terms of potential applications: if the substance is a melt then there is by definition no other material except polymer chains, and then there cannot be any room for molecular motors or other agents driving the system at short length scales. We therefore consider now the less pure situation of still unentangled but somewhat less concentrated solution. In that case there is another length scale in the system; we denote it ξ; it can be visualized as the mesh size of the temporary polymer network visible on an instant photograph of all chains. Here in this section we first assume that ξ > b, which means chains are flexible within the scale of one mesh size. Within each mesh diffusion of segments is strongly coupled by hydrodynamic (HD) interactions, while HD interactions are effectively screened on the distances beyond mesh size ξ (see, e.g., ref 16, Chapter 6, or ref 15, section 8.2, or ref 17, section 32). Let us mention that screening of HD interactions beyond ξ is a subtle phenomenon; unlike more familiar electrostatic Debye screening, hydrodynamic screening is not exponential but a power law.18 In the spirit of this work, being interested in only the crudest estimates, we will ignore this complication. Then instead of Figure 2 we obtain a new diagram of possible regimes, Figure 3. Given the screening of HD interactions, the only real novelty in Figure 3 is in qb > b/ξ and ωτ0 > (b/ξ)3, which means that our spatially and temporarily nonuniform drive probes the dynamics inside one mesh; these regimes are marked blue in the diagram in Figure 3. To obtain these new results we should remember that stress relaxation in the (HD screened) Zimm model follows t−2/3 law (α = 2/3, see, e.g., ref 15, section 8.4.2), while the relaxation time of the blob of g ≈ (bq)−2 monomers is τ1 ≈ τ0g3/2. Plugging these results into eqs 3, 4a, and 4b we find the results which are presented in the new (blue) regimes in Figure 3. (As a parenthetical note, we assume here that the chain inside one mesh is Gaussian, not a swollen self-avoiding coil. This is justified because in practice the polymers in question are semiflexible, e.g., a chromatin fiber, for them self-avoidance becomes relevant at so large distances which are unrealistic for the mesh size. However, if need be, there is no difficulty generalizing our results, with α = 1/3ν and τ1 = τ0g3ν, where ν ≈ 0.6 is the standard swelling exponent.) The situation on the length scales larger than ξ is basically the same as before: the “inner” part of the regimes diagram Figure 3, at qb < b/ξ and ωτ0 < (b/ξ)3, remains the same as in Figure 2 in terms of q and ω dependencies. There are however ξ-dependent prefactors which arise essentially from counting. Indeed,
if qb < b/ξ, the blob of the size 1/q still includes g ≈ (qb)−2 Kuhn segments, but its relaxation time is now estimated as follows. This blob contains g/gξ meshes, where gξ = (ξ/b)2. Each mesh has Zimm relaxation time τξ ≈ τ0g3/2 ξ . Then the whole blob has a Rouse relaxation time τ1 ≈ τξ(g/gξ)2. Assembling everything together we obtain the results indicated in Figure 3.
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SEMIFLEXIBLE CHAINS PROBED WITHIN KUHN LENGTH We now consider the situation when the wavenumber is so large that we probe dynamics inside one Kuhn length. That means we need to extend the diagram of Figure 2 beyond qb = 1 to somewhat larger q, as shown in Figure 4. The elasticity and
Figure 4. Diagram of scaling regimes for both storage G′(q, ω) and loss G″(q, ω) modulus for a concentrated system of semiflexible chains. New regimes are in pink. As before, G̲ = G/Tc, q̲ = qb, ω = ωτ0. Here, b is Kuhn segment, c is the number of Kuhn segments per unit volume, d < b is the shortest length along the chain, something like chain length per one base pair 0.3 nm for dsDNA, or fiber thickness. Similar diagram, with the same behavior of moduli in all regimes, is also valid in the case when chains within one mesh are nearly straight (ξ < b); in that case, the qb > 1 (pink) region consists of qb < b/ξ (qξ < 1) when we probe the system within one persistence length but still over many meshes and qb > b/ξ, when we probe the interior of one mesh. We do not separate these regimes by different colors, because moduli scale in the same way in both. C
DOI: 10.1021/acs.macromol.9b01272 Macromolecules XXXX, XXX, XXX−XXX
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entanglement length, a phenomenological parameter interpreted as the number of monomers per one topological constraint. In the most interesting case, when qb < N−1/2 , the situation e can be understood as follows. First, the blob of the size 1/q has now the number of monomers proportional to q−3; more accurately, it is estimated from (g/Ne)1/3 × bN1/2 e ≈ 1/q, yielding g ≈ (qb)−3N−1/2 (indeed, bN1/2 is the size of one section e e between entanglements, because polymer remains Gaussian up to Ne monomers, and g/Ne is the number of such entanglement sections arranged with fractal dimension df = 3). Second, the relaxation dynamics of such crumpled chains was extensively studied, and there are several models in the literature.24,25 Here, in the main text of this paper, we present the simplest results which correspond to the model of Ge et al.25 In Appendix C, we include a more general discussion of various models and show that all of the main results are qualitatively insensitive to the choice of the model. In the simplest case,25 the relaxation time of the blob of g > Ne monomers and the stress relaxation power law are estimated as
dynamics of semiflexible chains was studied in great detail in the context of actin, microtubules, and other intracellular filaments (see review article19 and numerous references therein). Here, as everywhere in this paper, we restrict ourselves to the simplest estimates based on eqs 3, 4a, and 4b. For that we use the result cited, e.g., in ref 15 (p 334), which is relaxation modulus G(t) ∝ t−3/4 or α = 3/4 for semiflexible polymers. This means that at high frequencies and small q we get G′/Tc ≈ G″/Tc ≈ (ωτ0)3/4, where c as before is the number of Kuhn segments per unit volume. This is basically the result of the work20 where q = 0 was assumed. At large q, when we probe inside one Kuhn segment, we have to introduce a “blob” of the size 1/q. In this case, the blob has length λ(q) ≈ 1/q. The relaxation time of such piece is (see formula 8.108 in ref 15) τ0(qb)−4, and we should use it as τ1 in the above integrals. This gives G′ ≈ Tc[τ1/τ0]5/4(ωτ0)2 ≈ Tc(qb)−5(ωτ0)2 and G″ ≈ Tc[τ1/τ0]1/4(ωτ0) ≈ Tc(ωτ0)(qb)−1. Crossover between these two regimes is the same ωτ0 ≈ (qb)4 as in the Gaussian case. These regimes are summarized in Figure 4. The novel regimes, characteristic of semiflexible polymers, are marked pink. It is possible (and perhaps even likely for chromatin (see Appendix A) that the mesh size in the system is smaller than the Kuhn length. In that case, the qξ > 1 region, where we probe parts of the system with unscreened HD interactions (blue in Figure 3), is located at qb > 1. In fact, on length scales smaller than the persistence length b, hydrodynamic interaction introduces only logarithmic corrections to the friction coefficient and therefore can be neglected at our level of approximation, both below and even more so above the mesh size ξ. Thus, when ξ < b, hydrodynamics can be neglected everywhere (unless semiflexible chains run parallel over long distances, forming a nematic phase, which we do not consider here), and accordingly, both moduli G′ and G″ scale in the same way inside and outside the scale of mesh size, so they follow Figure 4. The latter conclusion should be compared to the results of simulations presented in ref 21 where hydrodynamic screening was ignored, leading to the long-ranged hydrodynamic effects.
τg = τe(g /Ne)7/3
(5)
G(t ) = Ge(t /τe)−3/7
(6)
τ0N2e
where τe = is the Rouse relaxation time of one entanglement section and Ge = Tc/Ne. Collecting everything together yields the answers summarized in the diagram in Figure 6 (see a more detailed description of the algebra involved in Appendix C). Independently of the details of the model, our result suggests that the storage part of shear modulus (G′) decays very rapidly with increasing wavenumber q, as q−11 or even a bit sharper (see Appendix C). It suggests that these systems behave with respect to a shear as purely viscous at all scales but the very largest ones. This conclusion corroborates the intuitive feeling that a crumpled rope is unable to support any significant elasticity.
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DISCUSSION AND OUTLOOK Using a combination of standard linear viscoelasticity and scaling methods we calculated the q- and ω-dependent storage and loss moduli G′(q, ω) and G″(q, ω) of polymer solutions for various polymer models. In particular, we studied unentangled (Rouse) melts and solutions (with hydrodynamic interactions) of linear Gaussian chains, solutions of semiflexible polymers, and territorial melts of unconcatenated rings. In all cases we find the following. (1) In the low-frequency limit G′ and G″ scale as ω2 and ω, respectively, in agreement with the expectation that fluids display finite static viscosity (G″/ω) and vanishing static elasticity. This observation is valid for all values of q and is model independent. (2) In all models the two moduli coincide in the highfrequency, finite wavenumber limit. (3) For intermediate and high wavenumbers G′ decays much faster than G″ with q. The conclusion that at small scales the behavior of polymer solutions is largely viscous with elasticity playing a minor role concurs with expectations based on our understanding of polymer gels, namely, that they behave as fluids on length scales smaller than the mesh size.26 (4) Both G′ and G″ decay faster with q in the intermediatethan in the high-q regime. Our other detailed predictions are system dependent and are shown in the corresponding diagrams.
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MELT OF SELF-SIMILAR CRUMPLED CHAINS (OR UNCONCATENATED RINGS) We skip the case of entangled melt or concentrated solution of linear chains. We examined the q-dependent viscosity in this case12 and could do storage modulus too, but it is rather cumbersome and does not seem relevant for any applications, so we do not include it. The reason why this case is more cumbersome is that the dynamics is not self-similar, as it involves reptation over the entire chain length as a prerequisite for stress relaxation. Instead, we now consider another important case when typical chain conformations are fractals but with fractal dimension df = 3 instead of the Gaussian 2. This is relevant for a number of socalled crumpled or fractal globule models, their most obvious feature being territoriality, and the most important possible application being interphase chromatin.22,23 Unlike previous cases in Figures 3 and 4 where new regimes were added at large q and large ω, here interesting new regimes occur at small q and ω, a fact directly related to the above-mentioned territoriality. Indeed, these new regimes, marked yellow in Figure 6, correspond to the situation when the (q, ω) drive probes the chain on the scale where it has fractal dimension df = 3, that is, inside its territory. Only at qb > N−1/2 or 1/q < bN1/2 e e , when topological constraints are unimportant, do we obtain the same behavior as for the Rouse system in Figure 2. Here Ne is the so-called D
DOI: 10.1021/acs.macromol.9b01272 Macromolecules XXXX, XXX, XXX−XXX
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estimates are only averages; in fact, different regions in the nucleus have different densities (euchromatin versus heterochromatin); nevertheless, averages give the right overall idea of the order of magnitude. A more delicate issue is the estimate of entanglement length Ne for chromatin fiber. Present-day best estimates range from 4 to 1300 kbp (from 1 to 300 Kuhn segments).23,31,32 Regarding the rheologic properties of the nucleoplasm, we refer the reader to a recent article33 and references therein.
When we discuss the relaxation moduli at short length scales, it is intuitively appealing to think that those are very much fluctuation dependent. To this end we should emphasize that the scaling theory developed in our work is not a mean field theory; it does not neglect fluctuations, rather it is all about fluctuations. We can compare this situation with any of the classical scaling theories, such as, e.g., Pincus blobs in a stretched chain, where Pincus blobs are exactly the scale below which fluctuations remain unaffected and above which fluctuations are suppressed by the external force.16 Similarly in our system, blobs (either those of the size 2π/q or those with relaxation time 2π/ω, whichever is relevant) represent the crossover between modes of the system whose fluctuations are unaffected and other modes whose fluctuations are ironed out by the applied force. Another due comment is about the fact that in living systems many of the small length scale (large q) dynamical properties discussed here are actually driven by active processes, for instance, by molecular motors. One may wonder to which extent the activity may affect the relaxation processes. We emphasize in this regard that such influence, although possible in general, definitely goes beyond linear response. If the drive is weak enough it does not affect the response. Finally, we would like to comment on the relevance of our results to the present and future experimental studies. Our results on melts of crumpled polymer rings may be relevant to understanding the dynamics of the nucleoplasm as observed by displacement correlation spectroscopy.13,21,27 A direct experimental test of our predictions is highly nontrivial since it involves perturbing and observing the viscoelastic fluid on the nanoscale. One such method may be to use fluorescent nanoparticles of variable size and perform time-resolved fluorescence polarization anisotropy experiments28,29 (see also a related discussion by Cai et al.9).
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APPENDIX B
Evaluating Integrals for Equations 4a and 4b
Let H(t ) =
−α
() t τ0
e−t/ τ1, eq 3. We want to find asymptotic
expressions for H′(ω) = ω
∫0
H″(ω) = ω
∫0
∞
sin ωtH(t )dt
(7a)
∞
cos ωtH(t )dt
(7b)
Convergence is guaranteed at 0 < α < 1. Case ωτ1 ≪ 1. Since integration is effectively cut off at t ≈ τ1 by the exponential, at these low frequencies we can replace sin ωt ≃ ωt and cos ωt ≃ 1 yielding H′(ω) ≃ ω 2τ12 − ατ0α
H″(ω) ≃ ωτ11 − ατ0α
∫0
∫0
∞
∞
x 1 − α e −x dx
x −α e −x dx
(8a) (8b)
Case ωτ1 ≫ 1. In the high-frequency regime, rapidly oscillating sin ωt and cos ωt help the integral converge without exponential cut-off and we can drop the exponential altogether, yielding
APPENDIX A
H′(ω) ≃ (ωτ0)α
∫0
H″(ω) ≃ (ωτ0)α
∫0
Chromatin Fiber As a Polymer: A Brief Summary
Chromatin is a functional form of DNA as it is present in the nucleus of a living cell. Chromatin fiber represents a complex of double-helical DNA with proteins (called histones) in which DNA wraps around protein particles, called histone octamers, to form the so-called nucleosomes. Chromatin fiber thus resembles a chain of beads on a string, with overall thickness close to d ≈ 10 nm (which is why it is sometimes referred to as a 10 nm fiber). We refer the reader to the standard molecular biology textbooks for further details, e.g., Alberts et al.30 As an example, diploid human genome includes about 6.6 × 109 base pairs which amounts to about 2 m worth of dsDNA. The linear density of chromatin fiber is usually assumed to be about 40 −120bp/nm, which means the total length L of the fiber in one human cell nucleus is between 5 and 15 cm. This length is distributed between 23 pairs of chromosomes, each containing a few millimeters worth of chromatin fiber. Chromatin fiber persistence length is not smaller than that of dsDNA, i.e., Kuhn segment is at least about 100 nm. All chromatin is packed in the nucleus of about 10μm diameter, or V ≈ 103 μm3 volume. That means the average density, measured as the fiber length per unit volume in the nucleus, is about L/V ≈ 10−4 nm−2. This corresponds to average volume fraction of fiber in the nucleus about ϕ ≈ Ld2/V ≈ 0.02 and mesh size about ξ = (V/L)1/2 ≈ 100 nm, the same order as Kuhn segment. The latter result suggests that chromatin fiber is pretty rigid on the scale of an average mesh. Of course, these
∞
∞
x−α sin x dx
(9a)
x−α cos x dx
(9b)
To summarize and dropping numerical coefficients (represented by the convergent integrals above), we get eqs 4a and 4b in the main text.
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APPENDIX C
Details of Calculations and Other Models for Territorial Polymers
The purpose of this Appendix is to present in some greater details the derivation of our results for crumpled territorial polymers presented in Figure 5 and also formulate similar results for other models of territorial polymers suggested in the literature. We do not attempt here to retell or explain the details of the relevant works24,25 and only cite the results. We will follow the general framework suggested by Ge et al.25 in which territorial polymer is characterized by two fractal dimensions, that of the chain itself, which is df = 3 on the scale above entanglement length Ne, and that of the primitive path, dp. We remind that the concept of primitive path goes back to the foundations of topologically constrained polymers and reptation model and refer the reader to the standard textbook about it.1 The “annealed tree” model24 suggests dp ≈ 1/ν ≈ 5/3 ≈ 1.7, while the “loopy globule” model of ref 25 uses dp = 1. The most subtle aspect of the dynamics has to do with “tube dilation”,25 E
DOI: 10.1021/acs.macromol.9b01272 Macromolecules XXXX, XXX, XXX−XXX
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Now the formula is exactly in the form of eq 3, and we can plug it in the results in eqs 4a and 4b. First, consider very small q and very small ω. Then τ in the exponential cutoff must be relaxation time of the entire ring, i.e.
τ ji N zy = Ne2jjj zzz j Ne z τ0 (14) k { Note that we write here τ/τ0 instead of τ/τe in formula 10, thus picking up another two powers of Ne. Now we plug everything in formula 4a, lower line, and obtain 2 + (1 − θ )ρ + θ /3
ij N yz G′(ω) = Ne2α − 1 + 2(2 − α)jjj zzz 2 j Ne z Tc(ωτ0) k {
[2 + (1 − θ )ρ + θ /3](2 − α)
i yz = N jj zzz j Ne z k { 3j jN
Figure 5. Diagram of scaling regimes for both G′ and G″ for territorial polymers exemplified as a melt of unconcatenated rings. Regimes marked yellow are new compared with Figure 2; in these regimes we probe the system at the length and time scales of a territory. As before, G̲ = G/Tc, q̲ = qb, ω = ωτ0. Ne is the entanglement length. Crossover line between two intraterritorial (yellow marked) regimes is ωτ0 = (qb)7N3/2 e . Results in this diagram are formulated for the “loopy globule” model of Ge et al.25 Essentially similar diagrams for other models are presented in Appendix C, Figures 6, 7, and 8.
(3 − 2θ − κ )ρ + 2θ /3
(15)
which is the answer indicated in Figure 6. Similarly, we use the upper line of formula 4a to establish the result for G′(ω) at larger
which is one way to describe the fact that on short time scales motion of each segment of the chain is restricted much more severely than on the long time scales. The degree of tube dilation can be quantified by two parameters, θ and κ (0 ≤ θ, κ ≤ 1): the most complete tube dilation25 corresponds to θ = κ = 1, complete lack of tube dilation25 means θ = κ = 0, and annealed tree model24 corresponds to θ = 0 and κ = 1. In terms of these parameters, the relaxation time of the blob of g > Ne monomers is estimated as τg = τe(g /Ne)2 + (1 − θ)ρ + θ /3
(10)
where ρ = dp/df and τe = τ0N2e is the Rouse relaxation time of one entanglement section. Together with the above estimate of g we get the relaxation time that should now be inserted in eqs 4a and 4b as τ1. The stress relaxation modulus was also calculated in the same articles24,25 with the result G(t ) = Ge(t /τe)−1 − (1 − κ)ρ /2 + (1 − θ)ρ + θ /3 e−t / τ
Figure 6. Diagram of scaling regimes for both G′ and G″ for territorial polymers exemplified as a melt of unconcatenated rings. The regimes marked yellow are new compared with Figure 2; in these regimes we probe the system at the length and time scales of a territory. As before, G̲ = G/Tc, q̲ = qb, ω = ωτ0. Ne is the entanglement length, while ρ = dp/df the ratio of fractal dimensions of the primitive path dp and of the chain df = 3. Although ρ is significantly different for the “annealed tree” model24 (ρ ≈ 0.57) and for the “loopy globule” model25 (ρ = 1/3), the appearance of this phase diagram is very nearly the same. Parameters θ and κ quantify the degree of tube dilation (see in the text),
(11)
where Ge = Tc/Ne. At t ≪ τg, the power law result applies to the blobs of our interest while at a later time it is exponentially cut off. Thus, the exponent α (G(t) ∝ t−α) is directly read out of formula 11 as well as the prefactor G0. To make further progress from this point we rewrite eq 11 in greater detail
α=
ω, the intermediate regime of frequencies in Figure 6. Also in a similar way, eq 4b produces our results for G″(ω). Once we have results for G′(ω) and G″(ω) at very small q, which corresponds to distances larger than the territory size, and remembering that at qb > N−1/2 , which corresponds to distances e smaller than entanglement, we have our old familiar Rouse result (Figure 2) and can reconstruct the intermediate behavior between these two limits simply by continuity: write, say, G′ as (qb)xNye and adjust x and y to produce smooth crossovers. These results can also be independently derived in the same way as
Tc / Ne
Tc (τ0/τe)−αÖÆ (t /τ0)−α e−t / τ Ne ´ÖÖÖÖÖÖÖÖÖÖ≠2ÖÖÖÖÖÖÖÖÖ Ne α ´ÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖ≠ÖÖÖÖÖÖÖÖÖÖÖÖÖÖ ÖÆ 2α− 1 G0 = TcNe
(12)
1 − (1 − κ )ρ 2 + (1 − θ )ρ + θ /3
(13)
where α=
Crossover line between two new (intraterritorial,
yellow marked) regimes is ωτ0 = (qb)3[2+(1−θ)ρ+θ/3]N1e + 3(1−θ)ρ/2+θ/2.
G (t ) = ß Ge (t /τe)−α e−t / τ =
1 − (1 − κ )ρ . 2 + (1 − θ)ρ + θ / 3
F
DOI: 10.1021/acs.macromol.9b01272 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules ORCID
Alexander Y. Grosberg: 0000-0002-4230-8690 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS Y.R.’s work was supported by grants from the Israel Science Foundation and from the Israeli Centers for Research Excellence program of the Planning and Budgeting Committee. The work of A.Y.G. was supported partially by the MRSEC Program of the National Science Foundation under Award Number DMR1420073. A.Y.G. thanks the Aspen Center for Physics, where part of this work was done with the support of the National Science Foundation under Grant No. PHY-1607611.
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Figure 7. Same diagram as Figure 6 (for territorial polymers exemplified as a melt of unconcatenated rings) specified for the “annealed tree” model of ref 24 with θ = 0, κ = 1, and ρ = 5/9 (in reality, in this model ρ ≈ 1/3ν ≈ 0.57 ≈ 5/9; we show in the diagram the 5/9 approximation). Crossover line between two new (intraterritorial, yellow marked) regimes is ωτ0 = (qb)23/3N11/6 e . As before, G̲ = G/Tc, q̲ = qb, ω̲ = ωτ0.
Figure 8. Same diagram as Figure 6 (for territorial polymers exemplified as a melt of unconcatenated rings) specified for the “loopy globule” model of ref 25 with no tube dilation, corresponding to θ = κ = 0 and ρ = 1/3. Crossover line between two intraterritorial (yellow marked) regimes is ωτ0 = (qb)7N3/2 e . As before, G̲ = G/Tc, q̲ = qb, ω = ωτ0.
above, with the only difference that instead of relaxation time of the entire ring, eq 14, we should use relaxation time of the 1/q-sized blob eq 10. This results in the most general diagram presented in Figure 6. Figure 5 in the main text as well as Figures 7 and 8 represent concrete realizations of the general diagram Figure 6 for various specific models suggested in the literature, namely, the “loopy globule” model25 with complete tube dilation (θ = κ = 1, Figure 5), “loopy globule”25 with no tube dilation (θ = κ = 0, Figure 8), and “annealed tree” model24 (θ = 0 and κ = 1, Figure 7). The results of our previous paper12 for q-dependent viscosity were all formulated for the case θ = 0, κ = 1.
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DOI: 10.1021/acs.macromol.9b01272 Macromolecules XXXX, XXX, XXX−XXX
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DOI: 10.1021/acs.macromol.9b01272 Macromolecules XXXX, XXX, XXX−XXX
