Dynamics of semi-flexible polymer solutions in the highly entangled regime
Manlio Tassieri, R. M. L. Evans, Lucian Barbu-Tudoran, G. Nasir Khan, John Trinick, Tom A. Waigh
aa r X i v : . [ c ond - m a t . s o f t ] O c t Dynamics of semi-flexible polymer solutions in the highly entangled regime
Manlio Tassieri † , R. M. L. Evans, Lucian Barbu-Tudoran, Nasir Khan, John Trinick, and Tom A. Waigh ∗ School of Physics and Astronomy, University of Leeds, LS2 9JT, U.K. Institute of Molecular and Cellular Biology, Faculty of Biological Sciences, University of Leeds, LS2 9JT, U.K. Biological Physics, School of Physics and Astronomy, University of Manchester, M60 1QD, U.K. (Dated: August 3, 2007)We present experimental evidence that the effective medium approximation (EMA), developed byD.C. Morse [Phys. Rev. E , 031502, (2001)], provides the correct scaling law of the macroscopicplateau modulus G ∝ ρ / L − / p (where ρ is the contour length per unit volume and L p is thepersistence length) of semi-flexible polymer solutions, in the highly entangled concentration regime.Competing theories, including a self-consistent binary collision approximation (BCA), have insteadpredicted G ∝ ρ / L − / p . We have tested both the EMA and BCA scaling predictions usingactin filament (F-actin) solutions which permit experimental control of L p independently of otherparameters. A combination of passive video particle tracking microrheology and dynamic lightscattering yields independent measurements of the elastic modulus G and L p respectively. Thus wecan distinguish between the two proposed laws, in contrast to previous experimental studies, whichfocus on the (less discriminating) concentration functionality of G . PACS numbers: 83.10.Kn, 83.10.Mj, 36.20.Ey, 87.16.Ka
Despite their importance to soft-matter physics, biol-ogy and industrial processing, the viscoelastic propertiesof semi-flexible polymer solutions are still not well un-derstood and a basic analytical model has not yet beenagreed upon. All current models describing the viscoelas-tic properties of semi-flexible polymer solutions are elab-orations on the early models of Doi and Edwards [2, 3].They developed two full theories of the entangled statefor two extreme cases: completely flexible [2] and rigid-rod [3] polymers. Solutions of semi-flexible polymer,that lie between those extremes, have many regimes ofviscoelastic behavior (requiring many theoretical mod-els [1, 4, 5, 6, 7, 8, 9, 10, 11, 12]), depending on thepolymers’ degree of rigidity (described in terms of persis-tence length L p ), on their contour length L , and on theconcentration (from dilute to highly entangled regimes).We shall focus on highly entangled isotropic solutions ofsemi-flexible polymers, with L/L p ∼
1. In particular, westudy the range of concentration ( ≈ L m is much lessthan L p , and the tube diameter and entanglement lengthare also expected to be much less than L p . This rangeof concentration was defined by Morse [1] as the tightlyentangled regime , and is particularly relevant to manybiological and industrial polymeric fluids. In order to de-scribe the viscoelastic behavior of the polymer network inthis range of concentrations, Morse developed two ana-lytical approximations describing the confinement forcesacting on a randomly chosen test chain embedded in a“thicket” of uncrossable chains: the binary collision ap-proximation (BCA) and effective medium approximation(EMA). In fact, the scaling relation resulting from the † Electronic address: [email protected] ∗ Electronic address: [email protected]
BCA had previously been obtained by several others au-thors [8, 9, 10, 11], but Morse has also estimated the pref-actors. So, prior to the introduction of the EMA, therewas broad agreement regarding the scaling law. The ap-proximations are summarized as follows.(i)
The binary collision approximation gives arather detailed description of the interaction of a testchain with individual nearby medium chains, but neglectsany effects arising from the collective elastic relaxationof the network. It yields the following expression for theelastic modulus: G ≈ . k B T ρ / L − / p . (1)(ii) The effective medium approximation startsfrom a very different point of view, by treating the net-work surrounding the test chain as an elastic continuumwith a shear modulus equal to the self-consistently deter-mined plateau modulus of the solution, and the test chainas a thread embedded in this medium. The expressionthus obtained is G ≈ . k B T ρ / L − / p . (2)Comparison of the above scaling predictions raises thequestion of which theoretical approach (if either) bet-ter describes the viscoelastic behaviour of semi-flexiblepolymer solutions in the tightly entangled concentrationregime. Existing experimental measurements of the con-centration dependence of the plateau modulus [1, 13], arenot sufficient to answer that question, because the twoputative values of the exponents are numerically quiteclose ( G ∝ ρ . vs. G ∝ ρ . ), so that they both fit theexperimental data with reasonable accuracy [1].To test the scaling predictions, we experimentally anal-ysed solutions of actin filaments (F-actin), a semi-flexiblepolymer derived from muscle tissue. As we shall show, F-actin has the useful property that its persistence lengthcan be controlled by varying only the ionic propertiesof its solvent, without altering other system parameterssuch as solvent viscosity, polymer concentration or molec-ular weight, thus allowing us to discriminate between theEMA and BCA models. Hence, this is a rare example ofbiology helping to answer questions of interest to physics,rather than vice versa .At low ionic strength in vitro , actin exists in themonomeric (globular) G-actin form. G-actin is roughlyspherical with a diameter of about 5 nm. When the ionicstrength of a G-actin solution is increased to a physiologi-cal value (0.1 M), G-actin self-associates, to form F-actin,which is characterized by a persistence length of 2–20 µ m(depending on the buffer used) [14, 15] and a diameter ofapproximately 8 nm [10, 14]. In order to produce actinfilaments with different mechanical properties ( L p ), twosets of F-actin solutions (named System 1 and 2) wereprepared using two different buffer recipes, which will bereferred to hereafter as F-buffer 1 and
F-buffer 2 . In boththe cases, the initial solutions of G-actin were preparedwith the same buffer recipe (
G-buffer ). Polymerizationwas initialized by increasing the ionic strength of the
G-buffer to that of the
F-buffers . The buffer recipes are asfollows:
G-buffer : 0.2 mM ATP, 0.2 mM CaCl2, 2 mMTris-HCl, 0.5 mM DTT, pH 8.0.
F-buffer 1 : 50 mM KCl,2.0 mM free MgCl2, 5 mM Tris-HCl, 1 mM ATP bufferpH 7.5.
F-buffer 2 : 25 mM KCl, 1.0 mM free MgCl2, 1.0mM EGTA, 10 mM MOPS buffer pH 7.0.In order to obtain independent measurements of thecomplex modulus G ∗ and of L p , the two sets of F-actinsolutions were each investigated by two different tech-niques: passive video particle tracking microrheology(PVPTM) and dynamic light scattering (DLS).The first technique, PVPTM, exploits the relationshipbetween the viscoelastic properties of a fluid under in-vestigation, and the mean-square displacement (MSD) ofprobe particles (of radius a ), suspended in the fluid andexecuting Brownian motion (Fig. 1). This relationship isgiven by the Generalized Stokes-Einstein relation,˜ G ( s ) = k B T /πas h ∆˜ r ( s ) i , (3)where h ∆˜ r ( s ) i and ˜ G ( s ) are the Laplace-transforms of,respectively, the mean square displacement and the timederivative of the shear modulus. Moving from Laplacespace to Fourier space is accomplished by substitut-ing the Laplace frequency s with iω , so that the realand imaginary parts of ˜ G ( iω ) correspond to the storageand loss moduli, G ′ ( ω ) and G ′′ ( ω ) respectively, with ω the Fourier frequency (Fig. 2). As probe particles, weused carboxylate-coated polystyrene beads of diameter0.489 µ m. It is well known that microrheology can failto emulate macroscopic results if the probe particle’s sizeand surface chemistry are incorrectly chosen. Microrhe-ological measurements of viscoelastic moduli are oftenfound to match the scaling laws found by macroscopicrheology, whilst disagreeing by a constant factor in ab-solute magnitude. In the present case, we only requirereliable measurements of scaling exponents, not of abso- -2 -1 M S D ( µ m ) τ (s) FIG. 1: Mean-square displacement (MSD) versus lag-time,for 0.489 µ m–diameter beads in F-actin solutions at differentconcentrations [mg/mL] (System 2). The solid line representsthe MSD of the beads suspended in water at 25 o C. -1 G ' , G '' ( P a ) ω (rad/s) G' G'' ~ ω ~ ω FIG. 2: Storage and loss moduli versus frequency, for F-actinsolution at a concentration 1 mg/mL (System 2). The linesare guides for the eye. The same features are also exhibitedby System 1 lute values. Nevertheless, Fig. 3 demonstrates that theabsolute values of our measurements are at least as ac-curate as macroscopic rheometric data in the literature,so that we can be confident of their validity.In all our PVPTM measurements, we found very goodagreement with Morse’s scaling predictions, for both G ′ ( ω, c ) and G ′′ ( ω, c ) as functions of frequency and con-centration. Figure 3 shows the plateau modulus G (de-termined as the value of the storage modulus G ′ ( ω ) at thefrequency for which the ratio G ′′ ( ω ) /G ′ ( ω ) is minimum) versus F-actin concentration, for Systems 1 and 2 in thetightly entangled regime ( ≈ G is not able to discriminatebetween the two predictions (EMA and BCA). However,using the microrheology data, the two theories yield very -2 -1 -4 -3 -2 -1 G ( P a ) c (mg/mL) ~ c ~ c Slope = 1.43 ± 0.05 Slope = 1.35 ± 0.04 G ( P a ) c (mg/mL) FIG. 3: Plateau modulus G versus F-actin concentration,for Systems 1 (circle) and 2 (square). The open trianglesare bulk rheology results taken from ref. [13]. The shortdash and dash–dot lines are linear fits to the data on thelog-log plots (power laws); the dot and dash lines show theBCA ( G ∝ c / ) and EMA ( G ∝ c / ) scaling predictionsrespectively. The inset (System 2) shows the actual existenceof a different power-law regime at concentrations lower thanthe tightly entangled regime (c.f. Ref. [16]). different predictions for the ratio of persistence lengthsin the two systems: BCA = ⇒ L p /L p = (cid:0) G /G (cid:1) = 18 ± EM A = ⇒ L p /L p = (cid:0) G /G (cid:1) = 6 . ± . g (1) ( q, t ) of semi-flexible polymers insemi-dilute solution, where q = [(4 πn ) /λ ] sin( θ/
2) isthe magnitude of the scattering wave-vector, defined asthe difference between the incident and scattered wave-vectors, n is the refractive index of the solvent, λ is thewavelength of the laser in vacuo , and θ is the scatter-ing angle. They showed that, at sufficiently long times( t >> ( qL p ) − / γ − q ), the dynamic structure factor re-duces to a simple stretched exponential, g (1) ( q, t ) = g (1) ( q,
0) exp h − Γ(1 / γ q t ) / / π i , (6)where Γ(1 /
4) = 3 . γ q given by γ q = k B T q / [5 / − ln( qa h )] / πη s L / p , (7)where η s is the solvent viscosity. They also calculated theform of the dynamic structure factor in the limit t → -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4-8-7-6-5-4-3-2-10123 10 -7 -6 -5 -4 -3 -2 -1 ~ t ~ t l n [ - l n ( g ( ) ( q ,t ) / g ( ) ( q , )) ] ln(t) time (s) FIG. 4: Double logarithm of the normalized dynamic struc-ture factor g (1) ( q, t ) /g (1) ( q, versus logarithm of time, at ascattering angle of 90 o , for a solution at F-actin concentra-tion c = 0 . which can be used to measure the microscopic hydrody-namic lateral diameter a h of the polymer chain: g (1) ( q, t ) = g (1) ( q,
0) exp (cid:0) − γ t (cid:1) (8)with initial decay rate γ = k B T q [5 / − ln( qa h )] / π η s . (9)The validity of Eqs. (6–8) requires [17] the conditions a h << q − << L p ∼ L and q − << L m , all of whichare satisfied by all of our samples, since a h ≈ q − ≈ −
33 nm (where only the high- q data will beused in the final result), while L p and L are of order ofmicrons and L m ≈ . − . µ m [11]. It is clear fromFig. 4 that Kroy and Frey’s calculations [17] are appli-cable to our experimental system, despite the scatter ofdata at early times. Indeed, g (1) ( q, t ) decays initially as asimple exponential (identified by the line with unit slopein Fig. 4) and exhibits stretched exponential behavior atlonger times (line with slope 3 / a h was obtained fromthe data by evaluating the initial decay rate, γ , via alinear fit to the logarithm of the normalized dynamicstructure factor, in a time window between 10 − s and4 × − s. From Eq. (9) the non-dimensionalized ini-tial decay rate, Γ = 6 π η s γ / ( k B T q ), was then fittedby [5 / − ln( qa h )] (Fig. 5) to obtain a h . For all the F-actin solutions investigated (as with other large polymers[18]), we found very good agreement between the indirectmeasurement of the lateral hydrodynamic diameter (av-eraged value a h = 9 ± a h = 9 . ± . C oun t s Diameter (nm) d = 9.2 ± 0.1 nm Γ = γ π η s / ( k B T q ) q ( µ m -1 ) a h = 9.7 ± 3.6 nm FIG. 5: Normalized initial decay rate Γ versus scatter-ing wave-vector q , and the best fit of Eq. (9) using a h asa free parameter, for a solution at F-actin concentration c = 0 . L p ( µ m ) q ( µ m -1 ) L p = 1 2 .2 ± 0 .8 µ m FIG. 6: Persistence length L p versus wave-vector q for a solu-tion at F-actin concentration c = 0 . a h = 9 . With the hydrodynamic diameter now determined, itcan be used in Eqs. (6) and (7) to estimate the persis-tence length L p . We adopt an average value a h = 9 . q , with L p as the onlyfitting parameter. The results for System 1 are shown inFig. 6. One might expect the result to be independentof q but, as noted in Ref. [17], the equations apply onlyto scattering vectors large enough to resolve single fila-ments. For scattering vectors smaller than the inversemesh size L m , the structure factor is averaged over sev-eral filaments and Eqs. (6) and (7) no longer hold. It istherefore the large- q asymptote that marks the true per-sistence length in Fig. 6. We thus consider only averagedvalues of L p for q ≥ µ m − , as reported for the two sys-tems in Table I, where three independent measurementof the ratio of persistence lengths are all in agreement.In conclusion, the true ratio of persistence lengths inthe two systems is consistent with the value predictedby Morse’s EMA approximation (equation 2) [1], but ismore than four standard deviations away from that pre-dicted by the more established BCA scaling law (equa-tion 1). We have thus resolved the controversy, and es-tablished that the effective medium approximation moreaccurately models the tightly-entangled regime of semi-flexible polymer solutions. The strength of this resultrests on the clear agreement between two completely in-dependent experimental techniques.ACKNOWLEDGMENTS: We thank Peter Olmstedand Tanniemola Liverpool for helpful conversations. Thework was funded by the EPSRC, and RMLE is fundedby the Royal Society. Actin concentration L p L p L p /L p (mg/mL) ( µ m) ( µ m)0.1 10 ± . ± .
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15 5 . ± . ± . ± .
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