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Scale-Dependent Viscosity in Polymer Fluids Alexander Y. Grosberg, Jean-Francois Joanny, Watee Srinin, and Yitzhak Rabin J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b03339 • Publication Date (Web): 26 Apr 2016 Downloaded from http://pubs.acs.org on April 30, 2016

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The Journal of Physical Chemistry

Scale-Dependent Viscosity in Polymer Fluids Alexander Y. Grosberg,∗,†,§ Jean-Fran¸cois Joanny,‡,§ Watee Srinin,† and Yitzhak Rabin¶ New York University, ESPCI-ParisTech, and Bar Ilan University E-mail: [email protected]

∗

To whom correspondence should be addressed Department of Physics and Center for Soft Matter Research, New York University, New York, NY 10003, USA ‡ ESPCI-ParisTech, 10 rue Vauquelin 75005 Paris, France ¶ Department of Physics and Institute for Nanotechnology and Advanced Materials, Bar Ilan University, Ramat Gan, Israel § Physico-Chimie Curie UMR 168, Institut Curie, PSL Research University, 26 rue d’Ulm, 75248 Paris Cedex 05, France †

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Abstract In this communication we use simple physical arguments to construct a ‘phase diagram’ of various frequency and wave vector dependent regimes of effective viscosity for polymer fluids, including non-entangled and entangled melts, semi-dilute solutions without and with hydrodynamic interactions, as well as the more exotic case of a melt of un-concatenated ring polymers.

Introduction Traditional rheology considers the motion of a polymer fluid caused by external forces applied on a macroscopic scale (e.g., on the confining walls, by dragging a large object through the fluid, or by applying a body force such as gravity). While these driving forces can have arbitrary time dependence, in all of the above examples they are nearly spatially uniform, in the sense that they correspond to wavelengths which are much larger than all relevant microscopic (molecular) scales in the fluid. Mathematically, this means that traditional rheology probes the frequency ω-dependent response of the fluid at zero wave vector, q = 0. By contrast, in live cells one frequently encounters the opposite situation, namely, forces are applied on molecular scale, for instance, by molecular motors and other enzymes. Although these non-thermal, active forces are applied locally, they can, and frequently do, cause large scale motions in the surrounding fluid. 1,2 It is, therefore, important to understand the response of the fluid to force applied at finite frequency and wave vector, i.e. to calculate q- and ω-dependent viscosity η(q, ω). The problem is expected to be of particular importance for a polymeric fluid, because polymers possess a wide range of characteristic molecular lengthand time-scales; for instance, a force applied on the scale of a single monomer may result in the motion of the entire polymer chain. Even though we are not aware of any systematic experimental study of q- and ω-dependent viscosity, there are some indirect manifestations of this effect, such as the difference between one- and two-particle microrheology and bulk rheology data on F-actin, 3 and there is also a large body of work on the related subject of 2 ACS Paragon Plus Environment

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drag forces experienced by small particles moving through polymer fluids, particularly by the group in Warsaw, as reviewed recently in; 4 we will return to this question at the end. Theoretically, this problem was first considered by F.Brochard-Wyart and P.-G. de Gennes 5 who wrote down the general Green-Kubo expression for the q-dependent viscosity and estimated it for the Rouse model (see also an earlier work 6 ). A more elaborate calculation of η(q, ω) for an unentangled melt of Rouse chains was carried out by Semenov et al. 7 In the work, 5 the authors also discussed the connection of this problem to size- and shape-dependent diffusion of particles through polymer melt or solution. The analysis of spherical particle diffusion through polymer melts or solutions was further elaborated upon by Cai et al. 8 Note that the connection between q-dependent rheology and particle diffusion is straightforward for a spherical particle, since, if a particle of radius r moves during time t, it probes medium rheology at the wave vector q ∼ 2π/r and frequency ω ∼ 2π/t (in general, this relation involves the complex viscoelastic modulus rather than the viscosity only and thus particle diffusion is influenced by both viscous and elastic effects 9–11 ). Of course, as emphasized in the work Ref., 5 if the particle is not spherical and is characterized by more than one length scale, then this simple argument breaks down, and the entire concept of q-dependent viscosity becomes insufficient; we will not deal with these more complex phenomena. We will also restrict ourselves solely to the linear response or linear rheology regime, assuming that driving forces are not too large. In this article, we will use simple physical arguments to calculate the effective viscosity η(q, ω) in various regimes, for a variety of polymer systems. In all cases we will assume that the force acting on the system is periodically modulated in both time and space; to be specific, let us assume that force acts along x-axis and is a standing wave which is spatially modulated along y-axis: f = f0 [sin ωt] [sin qy] .

(1)

This is shown in a cartoon in figure 1. We begin with the simplest case of an unentangled melt of polymers which can be de3 ACS Paragon Plus Environment


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scribed by Rouse model. In order to develop physical insight about the problem, we will present several different derivations, all of which lead to the same answer which also agrees, in the appropriate limit, with the rigorous (but somewhat non-transparent) calculations of Semenov et al. 7 We then continue to other polymer systems, including entangled melt, semidilute polymer solutions where hydrodynamic interactions are important, and finally to the melt of un-concatenated ring polymers which may be relevant as a model of chromatin in living cells. 12

Unentangled melt Consider first a melt or a concentrated solution of chains short enough to be considered unentangled. Let c be the monomer number concentration.

q-dependent viscosity in steady shear flow For the steady shear, stress is estimated by saying that force ∼ f0 acts on each monomer in the layer of thickness 1/q, yielding σ ∼ f0 c/q. Strain rate is found by assuming that every monomer is driven with the speed ∼ f0 /ζ, and velocity gradient develops over the length scale 1/q, yielding γ˙ ∼ f0 q/ζ, with ζ friction coefficient of one monomer; ζ ∼ ηsolvent a. Thus, using σ = η γ, ˙ one arrives at η = cζ/q 2 .

(2)

Scaling argument for zero frequency case Let us start with the known expression for the bulk (q → 0) viscosity of a fluid of Rouse chains, of N monomers each: η ∼ ζa2 cN = T cN τ0 , with a monomer scale and τ0 = ζa2 /T ; at larger q, it must be then η ∼ T cN τ0 φ (qRG ), and a scaling function φ is found by demanding that at very small distances, q ∼ 1/a, viscosity should become independent of N . This leads


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to η ∼ T cτ0 (qa)−2 ,

(3)

which is the same answer as Eq. (2) above. Here and below, we use energy units for temperature T , assuming Boltzmann constant to be unity, kB = 1.

Rouse modes and Maxwell’s formulation of viscosity As it is well known, we can get the expression for the viscosity in the q = 0 limit using the Maxwell’s formulation of viscosity:

η = Eτ = T cN τ0 ,

(4)

where E = T c/N is the osmotic modulus (thermal energy T times concentration of chains c/N ) and τ = τ0 N 2 is the Rouse relaxation time of the chain. Let us now consider the p−th Rouse mode which is associated with a chain segment of length N/p that has a spatial dimension rp = a(N/p)1/2 . This mode is excited by the application of a sinusoidal force with vavevector q = 1/rp = a−1 (N/p)−1/2 . In this case,

ηp = σp τp = T

c × τ0 (N/p)2 = T cτ0 /(aq)2 , N/p

(5)

which coincides with previous results (2,3). Note that the result is valid as long as the relaxation time of this mode τp = τ0 (aq)4 is shorter than 1/ω.

Blob-based estimate of q- and ω-dependent viscosity Consider Gaussian blobs of the size 1/q, each of them consisting of g monomers such that √ a g ∼ 1/q; thus g = (qa)−2 . Relaxation time of this blob is τq = τ0 g 2 = τ0 (qa)−4 . In the



y f0 f0 f(t,y)=f0 sin(ωt)sin(qy)

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f0 f0

~1/q f0

f0 r

Figure 1: A cartoon showing chain deformation of a Rouse chain in the field of modulated force. regime where this blob fully relaxes during one period of the external drive, τq 1/ω: τ0 ω (qa)4 .

(6)

In this low frequency regime, one can assume that deformation of the blob stretched by the force exerted on its monomers in opposite directions as depicted in figure 1, proceeds nearly as in equilibrium. Introducing the entropic spring constant T /ga2 of a blob, the length of the stretched blob, rblob , is determined by fblob = (T /ga2 )rblob , and since the force is exerted on every monomer, fblob ∼ gf0 ; this yields rblob = r = f0 g 2 a2 /T . The free energy of stretching the blob is rblob fblob ∼ f02 g 3 a2 /T . This amount of energy dissipates when the blob tumbles around, such that its end which was pulled to the right is now pulled to the left, and vice versa. This tumbling happens over the blob relaxation time τq . Therefore, dissipated power per blob is

f02 g 3 a2 . T τq

Since the number of blobs per unit volume is c/g, dissipated power per

unit volume is given by the q-independent expression f02 g 3 a2 c f02 a2 c ˙ W = = . T τq g T τ0

(7)

˙ = σ γ˙ = η γ˙ 2 = σ 2 /η, with This now has to be compared to the macroscopic expression W strain rate γ˙ or stress σ estimated as follows. The relevant velocity corresponds to distance


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r traveled over relaxation time τq , i.e., the velocity is given by r/τq , and since the velocity changes between r/τq and −r/τq over the distance scale about 1/q, we obtain γ˙ = rq/τq . Plugging in the expressions for r and τq , we find 2 ˙ = η f0 a q W T τ0

2 .

(8)

Alternatively, we can estimate the stress by noting that the force ∼ f0 acts on each monomer in the layer of thickness 1/q, yielding σ ∼ f0 c/q; this leads to the same expression (8). Comparing the two expressions (7, 8) for dissipation, and recalling that τ0 = ζa2 /T , we obtain: η(q, ω) =

cζ T cτ0 = 2 . 2 2 aq q

(9)

This is the same result as we obtained before (2, 3), but the present blob argument suggests that the applicability range of this result is given by the condition (6). In the opposite limit τ0 ω (qa)4 , we recover the well known result for the frequencydependent viscosity of the Rouse model, η ∼ T c(τ0 /ω)1/2 . 13 Nevertheless, for completeness, we will present a simple scaling argument below. In the large frequency limit, we define the number of monomers per blob g such that blob relaxation time τ (g) is equal to 1/ω: g ∼ (ωτ0 )−1/2 . Applying the Maxwell’s formulation of viscosity to such blobs yields η ∼ T (c/g) × τ (g) = T c(τ0 /ω)1/2 , as stated above. All the above considerations are valid as long as the number of monomers per blob is smaller than N . When this condition breaks down, because both the frequency and the wave number become small, we recover the usual expression for the static bulk viscosity of the Rouse melt, η ∼ T cN τ0 . The resulting schematic diagram of regimes is shown in figure 2. Of course all cross-overs are smooth.



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1

ωτ0 “classical regime”

η/Tcτ0~1/(τ0ω)1/2

N-2

“new regime”

η/Tcτ0 ~N

η/Tcτ0~1/q2a2 (qa)4 1

N-2

Figure 2: Diagram of regimes, for unentangled system, without hydrodynamic interactions. q and ω are wave number and frequency of the driving force modulation, respectively. a and τ0 are microscopic length and time scales, for one monomer. Dashed lines indicate crossover regions between distinct scaling regimes. For every regime, we indicate the scaling law for viscosity, which characterizes dissipative response of the system to the modulated drive. Viscosity is normalized by T cτ0 , which is the natural viscosity scale achieved when the system is driven at the monomer scale frequency or the monomer scale wave number, with T temperature in energy units, c monomer concentration. Logarithmic scale.

Semi-dilute solution with hydrodynamic interactions We first consider the zero-frequency viscosity. Using the Maxwell’s formulation of viscosity, we can write η ∼ (cT /g)τ (g). where τ (g) is the relaxation time of a blob, where the number √ of monomers per blob, g, is determined by the relation a g ∼ 1/q, which yields g ∼ (qa)−2 . If 1/q is smaller than the “mesh” size ξ (defined as the length scale above which hydrodynamic interactions are screened), then hydrodynamic (HD) interactions are important inside the blob, and τ (g) ∼ τ0 g 3/2 ∼ τ0 /(qa)3 (consistent with Zimm scaling in which the chain relaxation time on length scale ` is proportional to `3 ). This leads to η ∼ cT τ0 /(qa) for qa > a/ξ. If 1/q is larger than the mesh size ξ, then HD interactions are screened and τ (g) is given as the squared number of correlation blobs in the chain section of g monomers, leading to 2 g τ (g) = τξ (ξ/a) = τ0 (a/ξ)(qa)−4 , where τξ ∼ τ0 (ξ/a)3 . This yields η ∼ cT τ0 (a/ξ)(qa)−2 2 for qa < a/ξ. This consideration is valid for finite frequencies as well provided that blob relaxation time


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ωτ0 1

3/4

η/Tcτ0~(a/ξ)1/2(τ0ω)-1/2

1

(ξ/a)N-2 η/Tcτ0 ~ (a/ξ)N

η/Tcτ0~(qa)-1

(a/ξ)

η/Tcτ0~(τ0ω)-1/3

3

η/Tcτ0~(a/ξ)(qa)-2

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(a/ξ)4

N-2

(qa)4 1

Figure 3: Diagram of regimes, for unentangled system, with hydrodynamic interactions. HD interactions are important in the regimes shaded in blue. The notations are the same as in figure 2. Logarithmic scale. is short compared to 1/ω. We now proceed to examine the high frequency limit. As in the Rouse case above, we have to re-define the blobs. Instead of the previously employed 1/q-sized blobs, whose relaxation time is now long compared to 1/ω, we choose blob lengths g such that blob relaxation time is equal to 1/ω, i.e., τ (g) ∼ 1/ω. Unlike the Rouse case, however, here we have to consider two different regimes for τ (g). Although the results can be found in the book, 13 we give a simple argument below. If g < (ξ/a)2 then 1/ω = τ0 g 3/2 and g ∼ (ωτ0 )−2/3 . This gives viscosity η ∼ cT τ0 (ωτ0 )−1/3 (a new scaling with HD interactions). 3 g 2 = If g > (ξ/a)2 then 1/ω = τ (g) ∼ τ0 aξ (ξ/a)2 1/2 1 yielding η ∼ cT τ0 aξ at ωτ0 < (a/ξ)3 . (ωτ0 )1/2

ξ 1/2 1 a (ωτ0 )1/2

and g ∼ (ωτ0 )−1/2 (ξ/a)1/2

Similar to the case of the Rouse melt, the considerations in this section are valid only as long as the number of monomers per blob, g, is smaller than N . When both the frequency and the wave number are small, we find the expression for the static bulk viscosity of an unentangled semi-dilute polymer solution, η ∼ (T cN τ0 )(a/ξ). The results are summarized in the figure 3. 9 ACS Paragon Plus Environment


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Entangled polymer melt In entangled melt, we use standard notations: we assume that there is an important scale Ne , which is the number of monomers along the chain between neighboring entanglements; 1/2

in other words, aNe

is the tube diameter.

Zero frequency case −1/2

Suppose first that the applied stress is constant in time, ω = 0. When qa > Ne

, i.e. 1/q

is less than the tube diameter, the above considerations apply. Each 1/q-blob can relax via the usual Rouse mode, because the relaxation time of these blobs is τq = τ0 (qa)−4 which is less than the Rouse time for the entanglement segments, τe = τ0 Ne2 . The stress due to q-dependent perturbation relaxes long before the chain realizes that it is confined in the tube. Using Maxwell’s formulation of viscosity we have then η/T cτ0 ∼ (qa)−2 , as before. −1/2

In the opposite limit, when qa < Ne

the blobs can not relax via Rouse mechanism

because of topological restrictions. Indeed, the chain relaxes the stress mostly via reptation. As long as the amplitude of applied stress is small enough to remain within the linear response regime, the applied q-dependent stress affects neither the curvilinear diffusion nor the number of entanglement segments. The viscosity is then, as usually, η = Ge τrep , or η T cτ0

∼

N3 , Ne2

where Ge is the plateau modulus and τrep is the reptation time. Thus, viscosity

is q-independent. We should emphasize that this prediction of q-independent viscosity, as all other predictions in this work, are made to scaling accuracy only; that means, we do not exclude the possibility of some weak (non-power law) q-dependence in this regime. What is striking is that there is no smooth continuous crossover in the value of the viscosity as q approaches the onset of entangled regime. In logarithmic scale, this looks like a discontinuity in the η(q) dependence. A better physical feeling for this sharp change is provided by the fact that the complete relaxation of the blob with g = (qa)−2 > Ne monomers occurs roughly at the same time scale as the whole coil, assuming that the only


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mechanism for stress relaxation is reptation. The corrections to this result are due to the contour length fluctuations, and we do not consider them here. Sharp change of friction, of a similar nature, was also predicted for the related problem of a spherical particle diffusing through a polymeric liquid, when particle size approaches tube diameter – see the works by Brochard-Wyart and de Gennes 5 (see, in particular, figure 3 of that work) and by Cai et al. 8 In their later work, 14 Cai et al explore in detail the relaxation mechanism and the friction in this region of sharp transition.

Nonzero frequency case We now return to the time-dependent applied stress, at non-zero ω. Let us start from high frequency, when ωτ0 > (qa)4 , such that the 1/q-blobs do not have enough time to relax during one period 2π/ω, but smaller blobs can still relax during this time, provided they are smaller than Ne . Repeating the same argument as before in the case of an unentangled melt we have

η T cτ0

∼ (τ0 ω)−1/2 . This consideration remains valid, independently of q, as

long as the blob, whose Rouse relaxation requires time 1/ω, remains smaller than Ne , i.e., at ω > τe−1 , or ωτ0 > Ne−2 . In the opposite limit of very small frequencies, smaller than the inverse reptation time, we recover the well known result for the static melt viscosity: η/T cτ0 ' N 3 /Ne2 . −1 With increasing frequency, when ω exceeds the inverse reptation time, ω > τrep (in other

words, ωτ0 > Ne /N 3 ), we enter the range of rubbery plateau. Precise behavior in this range depends sensitively on the details of the model, e.g., classical reptation versus contour length fluctuation (see detailed discussion in the section 9.4.5 and problems 9.8 and 9.36 in the book 13 and in greater detail in the work 15 ). We will restrict ourselves here to the simplest Doi-Edwards reptation theory, in which case the loss modulus goes as ω −1/2 . 13,15 This can also be seen from a simple argument. Think about curvilinear diffusion of the chain along the tube. During the time 2π/ω, the number of entanglement segments visited by the



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chain end can be estimated as (N/Ne )(ωτrep )−1/2 . The loss modulus is thus G00 (ω) =

Tc N 1 Ge · · = . N Ne (ωτrep )1/2 (ωτrep )1/2

(10)

Using η ∼ G00 (ω)/ω, we have 1 η 1 ∼ 1/2 · . 3/2 T cτ0 Ne N 3/2 (τ0 ω)

(11)

This approximation is applicable as long as the tube length visited by the chain end during one period remains less than total tube length, i.e., as long as ω remains smaller than the inverse Rouse time of chain fluctuations along the tube. Further increase of frequency, in the range between inverse Rouse time of the whole chain, ∼ N −2 , and inverse Rouse time of the entanglement segment, Ne−2 , is even more sensitive to the model, and we do not attempt to provide a simple scaling argument for it. The detailed discussion of frequency-dependent loss modulus can be found in the work by Likhtman and McLeish 15 from which we reproduced here the left image in figure 4. We indicate all main frequency regimes by color in the Likhtman and McLeish figure (left half of figure 4), and show the corresponding regimes in our diagram by the same colors (right half of the same figure).

Entangled semi-dilute solution of linear chains in good solvent In this section, we briefly consider entangled semi-dilute solution. For simplicity, we only consider very good (athermal) solvent. First of all, diagram in Fig. 4 remains valid on length and time scales larger than the mesh size ξ and mesh relaxation time τξ , with appropriate replacements a → ξ, τ0 → τξ , etc, where τξ ∼ τ0 (ξ/a)3 . On the scale smaller than the mesh size, hydrodynamic interactions are important, which is why τξ indicated above is the 12 ACS Paragon Plus Environment

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1

ωτ0 η/Tcτ0~(ωτ0)-1/2

Ne-2 N-2 Ne/N3

η/Tcτ0~ Ne-1/2N-3/2(ωτ0)-3/2

η/Tcτ0~(qa)-2

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η/Tcτ0~N3/Ne2

N-2

(qa)4 Ne-2

1

Figure 4: Left side is the dependence of loss modulus G00 of an entangled polymer melt on frequency at q = 0, with Z = N/Ne ; the figure is reproduced, with permission, from the work by Likhtman and McLeish. 15 Different lines represent different versions of the reptation theory, accounting for a variety of subtle effects, such as tube length fluctuations, constraint release, etc. Main regimes are indicated by color. Right side is the diagram of regimes for an entangled polymer melt in terms of q and ω. Main regimes at small q are shown in the same colors as in the left figure. Thick gray wedge indicates the place where viscosity changes very rapidly, such that in log scale it looks almost like a discontinuity. Compare with similar observations in the works of Brochard-Wyart and de Gennes 5 (see, in particular, figure 3 of that work) and Cai et al. 8 Note that η ∼ G00 /ω. Logarithmic scale. .



Zimm time corresponding to mesh size ξ. This way we transform the diagram Fig. 4 into the large part of the new diagram which corresponds to qξ < 1 and ωτξ < 1. At larger q and/or larger ω, the dynamics is controlled by hydrodynamic interactions, which is what we examined before in Fig. 3. Thus, combining figures 4 and 3, we arrive at the diagram of scaling regimes for an entangled semi-dilute solution fig. 5. ωτ0 η/η0~(τ0ω)-1/3

τ0/τe

η/η0~ (a/dT)2(τrep/τ0)-1/2(ωτ0)-3/2

τ0/τrep

1

η/η0~(qa)-1

η/η0~(a/ξ)1/2(τ0ω)-1/2

3/4

η/η0~(a/ξ)(qa)-2

1 τ0/τξ

plateau regime

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η/η0~(a/dT)2(τrep/τ0)

(qa)4 (a/dT)4 (a/ξ)4

1

Figure 5: Diagram of scaling regimes for a semi-dilute solution of linear chains. In the diagram, ξ is mesh size, τξ = τ0 (ξ/a)3 is the relaxation time on mesh scale, τe = τξ (Ne /g)2 , Ne /g is the number of correlation volumes per entanglement strand, τrep = τe (N/Ne )3 is 1/2 reptation time, dT ' ξNe is tube diameter, η0 = T cτ0 is the natural monomeric viscosity scale. All other notations are the same as in previous diagrams. Logarithmic scale. .

Melt of un-concatenated rings Dynamics of the melt of un-concatenated and unknotted rings was examined by molecular dynamics simulations by Halverso et al 16 and using analytical methods based on the annealed tree model by Smrek and Grosberg. 17 We originally used the results of the latter work in our scaling analysis of q-dependent viscosity. When present paper was already in preparation for submission, the new article appeared by Ge et al, 18 in which authors presented what appears to be a more sophisticated model of ring dynamics, taking into account the phenomenon of tube dilation. Making at present no commitment to one model versus the other, we keep our 14 ACS Paragon Plus Environment

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original analysis (based on the annealed tree model) at the beginning of this section, and add at the end the new consideration based on Ge et al model. 18

Annealed tree model 17 Consider blobs of the size 1/q. For blobs having g monomers each, we have to have 1/q ∼ r(g), where r(g) is the size of a g monomers-long subchain. Assuming g > Ne , the annealed 1/6

tree model gives r(g) ∼ ag 1/3 Ne , leading to

g∼

1 1/2 (qa)3 Ne

=

N √e 3 . (qa Ne )

(12)

1/2

The modulus goes as E ∼ cT /g ∼ cT (qa)3 Ne . Relaxation time of such blob is τ ∼ τ0 g ρ+2 /Neρ , or τ ∼ τ0 Ne2 (g/Ne )ρ+2 , where

ρ = 1/(3νF ) ≈ 0.57 ,

(13)

where νF ≈ 0.588 is the usual Flory exponent for self-avoiding polymers. Using Maxwell’s formulation of viscosity η = Eτ , we arrive at

η∼

cT τ Ne √ 0 3(ρ+1) . (qa Ne )

(14)

−1/6

This is valid as long as g is in the interval N > g > Ne , which means N −1/3 Ne −1/2

Ne

< qa
Ne

p 3(ρ+2) . qa Ne

(15)

√ 3(ρ+2) crosses over smoothly to the , the boundary line ωτ0 ∼ Ne−2 qa Ne

previously established line ωτ0 ∼ (qa)4 for unentangled Rouse chains, as seen in figure 6. In the large frequency regime, on the other side of the cross-over line (15), the blob length g should be defined such that the blob relaxation time equals 1/ω, which means g/Ne ∼ 1/ (τ0 ωNe2 )

1/(ρ+2)

. Then, using again the Maxwell’s formulation of viscosity, the

viscosity is given by η ∼ (cT /g) × 1/ω, or

η∼

cT τ0 Ne (τ0 ωNe2 )(ρ+1)/(ρ+2)

.

(16)

This expression should be valid as long as g is between N and Ne , and at both ends it smoothly crosses over to our previous results. All of the above regimes are shown in the


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diagram 6, where ρ ≈ 0.57 for the annealed tree model.

Ge et al model 18 Again, we consider blobs of the size 1/q. The number of monomers in such blob is given by equation (14), there is no difference between models on this point. The characteristic 1/2

modulus is also unchanged, E ∼ cT /g ∼ cT (qa)3 Ne . The main difference is that, because of tube dilation, the work 18 assumes that chain length between entanglements is a function of time, namely Ne (t) ∼ Ne (t/τe )3/7 . As a result, relaxation time for the loops with g monomers goes as τ (g) ∼ τe

g Ne

7/3 .

(17)

Using Maxwell’s formulation of viscosity η ∼ Eτ for 1/q-loop we arrive then at η 1 ∼ . cT τ0 (qa)4 Ne

(18)

This is valid as long as g is in the interval N > g > Ne . At the large q end of this interval (when 1/q blob becomes comparable to the tube diameter), this formula indeed crosses over smoothly to Rouse result, while at the small q end (when blob becomes comparable to the entire chain) it gives N 4/3 η ∼ 1/3 . cT τ0 Ne

(19)

These results should be valid as long as the 1/q blobs have enough time to relax, that is

ωτ0 (qa)7 Ne5/2 .

(20)

In the opposite limit, we should again re-define blobs such that the relaxation time of the blob τ (g) equals ω1 . Using again the Maxwell’s formulation of viscosity we obtain in this case η 1 ∼ . 1/7 4/7 cT τ0 (ωτ0 ) Ne 17 ACS Paragon Plus Environment

(21)


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Graphic summary of the results for this model is presented by the same Figure 6, except this time one has to use ρ = 1/3.

Discussion Using simple physical arguments, we investigated various regimes of q- and ω-dependent viscosity for solutions and melts of both linear and ring polymers. For a melt of unentangled Rouse chains our results coincide with those obtained previously by Brochard-Wyart and de Gennes 5 and by Semenov et al. 7 To the best of our knowledge, all other results are new. As we mentioned in the introduction, our results can be re-formulated in terms of spherical particle diffusion, assuming that the particle is not permeable to polymers, but does not interact with them otherwise (i.e., non-sticky particle). To make it quantitative, we can make use of the generalized Stokes-Einstein formula which is known in micro-rheology literature: 9–11

r˜2 (s) =

T . ˜ πrp sG(s)

(22)

Here, h˜ r2 (t)i is the mean square displacement of a particle over time t, h˜ r2 (s)i is the Laplace ˜ transform of this quantity, with s the Laplace variable, and G(s) is the generalized modulus ˜ in Laplace representation, rp is the particle radius. To apply this, one has to determine G(s), including both storage and loss moduli. This is easy when the response is predominantly dissipative, which means storage modulus is much smaller than loss modulus. If that is the case, and taking also into account that viscosity of the polymer is much larger than that ˜ of the solvent, then we can directly estimate G(s) based on our results for η(q, ω) using ˜ G(s) → ωη(q, ω)|q∼2π/d,

ω=s .

Importantly, this simple procedure fails when storage modulus is not negligible, which is the case in the plateau regime for a melt or semi-dilute solution of linear polymers; those are frequency regimes between inverse reptation time and inverse entanglement time, 1/τrep < ω < 1/τe , at q smaller than the inverse tube diameter q < 1/dT (see diagrams in figures 4 18 ACS Paragon Plus Environment

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and 5). Another subtle case is when a particle is smaller than the mesh size in a polymer solution, rp < ξ. In this case, the particle is coupled to the polymer through hydrodynamic interaction only. In this situation, moving particle is very different from the basic model of our paper, which is embodied in formula (1), and which assumes that force is exerted on monomers only and not (directly) on the solvent. This is why for these very small particles their motion, in terms of a polymer response, is not similar to a force modulated with wave vector ˜ q = 2π/rp > 2π/ξ. In other words, in this case we should remember that G(s) in formula (22) is actually a sum of contributions from polymer (which dominates in all cases except small particles) and q- and ω- independent contribution from the (Newtonian) solvent (which dominates for a small particle). In the light of this discussion, all of our results for solutions and melts of linear polymers can be mapped onto the results of the work by Cai et al. 8 The problem of a particle motion through a polymeric medium was studied in a series of works by the group in Warsaw, as reviewed in the recent paper; 4 see also references therein. These authors examined only the zero-frequency mobility for the relatively concentrated polymeric systems, the relevant comparison would be with our results in section on entangled semi-dilute polymer solutions, specifically, with the horizontal axis (ω = 0) in the figure 5. Although direct quantitative comparison between theory and experiment is impossible (see below), there is a qualitative agreement. Based on their numerous measurements, authors in the work 4 formulated an interpolation formula describing effective viscosity experienced by a particle of radius rp as a function of rp ; to translate it to our language, we should assume q ∼ 2π/rp . Although this formula has no theoretical underpinning, it serves as a useful summary of the data. At small rp , or large q, this formula indicates a power law (rpa , with a adjustable parameter) correction to the constant viscosity, consistent with our q −2 result. At very large rp , it predicts saturation at some viscosity independent of rp , consistent with our q-independent prediction at small q. And there is a rather sharp switch between these two regimes, described by a stretched exponential function in the interpolation



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formula – consistent with our statement of sharp change indicated by a gray wedge area in the diagram figure 5. More specific quantitative comparison is impossible at this point, for several reasons. First, different values of rp are obtained in experiment by observing particles of different nature – proteins, dyes, colloidal spheres. These particles, especially smaller ones, interact with polymers in various non-trivial ways, so that the dissipation in their motion is not only due to hydrodynamic response of the polymer fluid, but also due to their stickiness. Second, some of these particles are charged, which brings up another set of complications related to electrophoretic mobility (see e.g. refs. 19,20 ). Third, there is a relatively limited number of data points in terms of changing rp (or q) – one would need (much) more to reliably establish the q-dependence. Moreover, most of the particles tested fall in the cross-over region. According to our predictions the q-dependence of the viscosity of polymer fluids becomes important only on truly microscopic length scales and therefore a direct test of these predictions would require the application of forces on polymer fluids on sub-micrometer length scales. While this rules out most currently available optical and mechanical methods, a possible solution is offered by molecular (protein) motors that can produce hydrodynamic forces on DNA and other bio-macromolecules on truly nanometric scales (see e.g. refs. 2,21–23 ). Other interesting extension of this work would be to analyze hydrodynamic waves on the surface of a polymer fluid, potentially improving upon the consideration, 24,25 and also the dynamics of squeezing the solvent from a polymer gel. 26 Also interesting would be to analyze q-dependent response of polyelectrolytes.

Acknowledgements The work of AYG and YR on this project was supported in part by a grant from the USIsrael Binational Science Foundation. YR would like to acknowledge the hospitality of the Center for Soft Matter Research at New York University where part of this work was done.


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AYG acknowledges the hospitality of both the Curie Institute and ESPCI where part of this work was performed. AYG also would like to thank P.Pincus for an illuminating discussion.

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(9) Mason, T. G.; Weitz, D. A. Optical Measurements of Frequency-Dependent Linear Viscoelastic Moduli of Complex Fluids. Phys. Rev. Lett. 1995, 74, 1250 – 1253. (10) Mason, T. G.; Gang, H.; Weitz, D. A. Diffusing-wave-spectroscopy Measurements of Viscoelasticity of Complex Fluids. J. Opt. Soc. Am. A 1997, 14, 139–149. (11) Mason, T. G. Estimating the Viscoelastic Moduli of Complex Fluids Using the Generalized StokesEinstein Equation. Rheologica Acta 2000, 39, 371–378. (12) Halverson, J. D.; Smrek, J.; Kremer, K.; Grosberg, A. Y. From a Melt of Rings to Chromosome Territories: the Role of Topological Constraints in Genome Folding. Rep. Prog. Phys. 2014, 77, 022601. (13) Rubinstein, M.; Colby, R. Polymer Physics; Oxford University Press, 2005. (14) Cai, L.-H.; Panyukov, S.; Rubinstein, M. Hopping Diffusion of Nanoparticles in Polymer Matrices. Macromolecules 2015, 48, 847–862. (15) Likhtman, A. E.; McLeish, T. C. B. Quantitative Theory for Linear Dynamics of Linear Entangled Polymers. Macromolecules 2002, 35, 6332–6343. (16) Jonathan D. Halverson, W. B. L., and; Grest, G. S.; Grosberg, A. Y.; Kremer, K. Molecular Dynamics Simulation Study of Nonconcatenated Ring Polymers in a Melt. II. Dynamics. J. Chem. Phys. 2011, 134, 204905. (17) Smrek, J.; Grosberg, A. Y. Understanding the Dynamics of Rings in the Melt in Terms of the Annealed Tree Model. J. Phys.: Cond. Matt. 2015, 27, 064117. (18) Ge, T.; Panyukov, S.; Rubinstein, M. Self-Similar Conformations and Dynamics in Entangled Melts and Solutions of Nonconcatenated Ring Polymers. Macromolecules 2016, 49, 708 – 722.


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(19) Long, D.; Viovy, J.-L.; Ajdari, A. Simultaneous Action of Electric Fields and Nonelectric Forces on a Polyelectrolyte: Motion and Deformation. Phys. Rev. Lett. 1996, 76, 3858– 3861. (20) Grosberg, A. Y.; Rabin, Y. DNA Capture into a Nanopore: Interplay of Diffusion and Electrohydrodynamics. J. Chem. Phys. 2010, 133, 165102. (21) MacKintosh, F. C.; Levine, A. J. Nonequilibrium Mechanics and Dynamics of MotorActivated Gels. Phys. Rev. Lett. 2008, 100, 018104. (22) Woodhouse, F. G.; Goldstein, R. E. Spontaneous Circulation of Confined Active Suspensions. Phys. Rev. Lett. 2012, 109, 168105. (23) Zidovska, A.; A.Weitz, D.; J.Mitchison, T. Micron-scale coherence in chromatin interphase dynamics. Proc. Nat. Ac. Sci. USA 2013, 110, 15555–15560. (24) Harden, J. L.; Pincus, P. A.; Pleiner, H. Surface Hydrodynamic Modes on Concentrated Polymer Solutions and Gels. Symposium V – Macromolecular Liquids. 1989; pp 383 – 391. (25) Harden, J.; Pleiner, H.; Pincus, P. Hydrodynamic Surface Modes on Concentrated Polymer Solutions and Gels. J. Chem. Phys. 1991, 94, 5208 – 5221. (26) Fredrickson, G. H.; Pincus, P. A. Drainage of Compressed Polymer Layers: Dynamics of a “Squeezed Sponge”. Langmuir 1991, 7, 786 – 795.



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f0

f0

f0 f0 f0

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Table of Contents Graphic


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Scale-Dependent Viscosity in Polymer Fluids - American Chemical

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