Scaling laws of entangled polysaccharides

Journal Pre-proof Scaling Laws of Entangled Polysaccharides Carlos G. Lopez, Lars Voleske, Walter Richtering

PII:

S0144-8617(20)30060-6

DOI:

https://doi.org/10.1016/j.carbpol.2020.115886

Reference:

CARP 115886

To appear in:

Carbohydrate Polymers

Received Date:

16 November 2019

Revised Date:

15 January 2020

Accepted Date:

15 January 2020

Please cite this article as: Carlos G. Lopez, Lars Voleske, Walter Richtering, Scaling Laws of Entangled Polysaccharides, (2020), doi: https://doi.org/10.1016/j.carbpol.2020.115886

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Scaling Laws of Entangled Polysaccharides Carlos G. Lopez, Lars Voleske, Walter Richtering

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Institute of Physical Chemistry, RWTH Aachen University, Landoltweg 2, D-52056 Aachen, Germany

Abstract

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We study the dilute solution properties and entangled dynamics of hydroxypropyl cellulose (HPC), a semiflexible polymer, in aqueous solution. Intrinsic viscosity data are consistent with a polymer in θ solvent with a Kuhn length ' 22 nm. The overlap concentration, estimated as the reciprocal of the intrinsic viscosity scales with the degree of polymerisation as c∗ ∝ N −0.9 . We evaluate different methods for estimating the entanglement cross-over, following to de Gennes scaling and hydrodynamic scaling models, and show that these lead to similar results. Above the entanglement concentration, the specific viscosity, longest relation time and plateau modulus scale as ηsp ' N 3.9 c4.2 , τ ' N 3.9 c2.4 and GP ' N 0 c1.9 . A comparison other polymers suggests that many of the rheological properties displayed by HPC are common to many other polysaccharide systems of varying backbone composition, stiffness and solvent quality, as long as the effect of hyper-entanglements can be neglected. On the other hand, the observed scaling laws differ appreciably from those of synthetic flexible polymer systems in good or θ-solvent.

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Keywords: elsarticle.cls, LATEX, Elsevier, template 2010 MSC: 00-01, 99-00

1. Introduction

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Cellulose based polymers are used in a wide number of industrial applications, for example as flow modifiers in food products, cosmetics or oil-drilling fluids. Above a critical concentration and molar mass, they become entangled, resulting in a slow-down of their dynamics compared to the non-entangled case. Understanding entanglement is key to predicting and improving the performance of polysaccharides in many applications.Sharma et al. (2015); Lopez (2019c); Colby (2010); Haward et al. (2012); Horinaka et al. (2011, 2013, 2018); R´ oz˙ a´ nska et al.; Kwon et al. (2019); Chen et al. (2011, 2018); Lu et al. (2015) Experimental and theoretical work on polymer entanglement has largely focused on flexible systems, where the Kassavalis-Lin-NoolandiLin (1987); Kavas-

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Email address: [email protected] (Carlos G. Lopez)

Preprint submitted to Journal of LATEX Templates

January 15, 2020

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salis & Noolandi (1989, 1987) conjecture coupled with scaling arguments is able to explain many important feature of polymer solutions.Fetters et al. (2007); Colby (2010); Everaers (2012) Horinaka and co-workers have shown that these scaling ideas do not hold for polysaccharides. Horinaka et al. (2011, 2013, 2018, 2012, 2015) The use of ionic liquids as highly viscous solvents has yielded much experimental insight into the rheological properties of cellulose and its derivatives in recent years, but a sound understanding of their entanglement properties is still lacking. Cellulose derivatives typically have Kuhn lengths in the range of 10 − 20 nm, placing them between common synthetic (lK ' 1 − 3 nm) and rigid-rod polymers (lK ∼ 1 µm) in terms of flexibility. Water at room temperature is usually a marginal solvent for most non-ionic cellulose derivatives, which therefore display near unperturbed chain statistics in solution. The purpose of this study is to provide an extensive dataset on the entanglement properties of hydroxypropyl cellulose (HPC) as a function of molar mass and polymer concentration. We first study the conformation of HPC chains in dilute solution and in other to measure its Kuhn length. We then evaluate the entanglement concentration from the viscosity data and compare our results with various models and earlier literature findings. Hydroxypropyl cellulose is produced by reacting activated cellulose with ethylene oxide. The monomer structure of HPC is shown in Figure 1. The degree of substitution (DS) is the average number of hydroxyl groups in the glucose monomer that are hydroxypropylated, out of a maximum of 3. Because hydroxypropylation may occur in a hydroxyl of the side group, the moles of substitution (MS), which refers to the total number of hydroxyproply groups per monomer, can exceed 3.

Figure 1: Monomer structure of HPC. The degree of substitution (DS) is the number of R = (CH2 CH(OH)CH3 )n groups per monomer, out of a maximum of 3. The moles of substitution (MS) is the number of CH2 CH(OH)CH3 groups per monomer.

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Sufficiently substituted grades of HPC (MS & 3.6) display both aqueous and organic solubility. The solubility of HPC in water decreases with increasing temperature, up to phase separation for T & 40 − 50 ◦ C, depending on MS and the regularity of substitution. As is the case with many other cellulose derivatives, HPC self assembles into large supra-molecular structures in water, a feature that has been exploited, for example, for microgel formation.Lu et al. (2000); Xia et al. (2003); Gao et al. (2001). The amphiphilic nature of the cellulose backbone and the hydroxypropyl side-chains make HPC surface active. HPC is used in various industrial formulations as a rheology modifier, binder and film former. For example, in pharmaceutical formulations, it is used as a tablet binder, film coater or as a suspending agent, see Table 1 for some of the properties of commercial HPC samples. The annual production of HPC was estimated at 106 kg in 2000, representing a relatively small value compared to other cellulose derivatives such as carboxymethyl cellulose or hydroxyethyl cellulose. a

molar mass

Value 40-2000 3-4.5 161+ 58MS 2-30000 5-8 40-50

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The aqueous solubility of HPC increases with increasing substitution. Samples with a similar content of hydroxypropyl groups per monomer (i.e. MS) may display markedly different phase behaviour depending on the pattern of substitution.Schagerl¨ of et al. (2006b) Generally, heterogeneously substituted samples precipitate at lower temperatures and display a two-stage profile in their turbidity vs. temperature curves, as opposed to more homogeneously substituted samples where a smooth increase in turbidity with increasing temperature is observed.Schagerl¨ of et al. (2006b,a) The conformational, thermodynamic and transport properties of HPC in dilute solution are consistent with a polymer with moderate stiffness, and a θ temperature around 40◦ C. Wirick et alWirick & Waldman (1970) reported values for the intrinsic viscosity (c[η]), radius of gyration of HPC in ethanol for samples of varying molar mass and polydispersity. Phillies and co-workersPhillies & Quinlan (1995) studied the solution viscosity of HPC in water as a function of polymer concentration, temperature and molar mass, finding that viscosity data were well described by a stretched exponential and a power-law at intermediate and high concentrations respectively. Clasen and KulickeClasen & Kulicke (2001) showed that various rheological parameters for HPC of varying molar masses in aqueous solution could be described as a function of the overlap parameter c[η], following a master-curve similar to those of other cellulose deriva-

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Property Molar Mass (kg/mol) MS M0 a (g/mol) Viscosity at 2wt% aq. sol (mPas) pH (2wt% aq) LCST (◦ C)

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Table 1: Molecular and physicochemical properties of commercial HPC grades. of a monomer

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tives. In aqueous solution at room temperature, HPC forms a liquid crystalline cholesteric phase when the polymer concentration exceeds ' 40 wt%.Werbowyj & Gray (1984) 2. Background Theory

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R ' ξT (N/NT )

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N ≤ NT N ≥ NT

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R ' lK NK ,

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Scaling of semiflexible polymers in solution The conformation of polymers is unperturbed by excluded volume interac2 tions up to the thermal blob size, which can be estimated as: ξT ' lK /B, where lK is the Kuhn length and B is the excluded volume strength, which close to the theta temperature θ, can be approximated by: B ' τ dc where τ =T − θ/T is the reduced temperature and dc is the cross-sectional diameter of the chain. The number of Kuhn monomers in a blob is nk,T = (lK /B)2 The end-to-end distance of a polymer chain is then: (1a)

(1b)

2 0 The volume of one monomer can be approximated pas v0 ' πdc b , and dc can be 0 estimated from the polymer density ρ as dc ' 2 M0 /(ρπb ), where M0 is the molar mass of a monomer. The correlation length or a mesh size of a polymer solution can be calculated as the length-scale at which the self-concentration of a single polymer chain matches the average concentration of the solution. For θ solutions, this yieldsRubinstein et al. (2003); Graessley (2003)

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1 N = N −0.5 R3 (lK b)3/2

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where NK is the number of Kuhn segments in the chain, and NT is the number of monomers in a thermal blob. The overlap concentration in units of number of monomers per unit volume is, for N < NT :

ξ = (b0 lK )−1 c−1

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or in terms of the polymer volume fraction ξ = (π/4)d2c lk−1 φ−1 . The above scaling is expected to hold up to a concentration c∗∗ when the correlation length decreases to the Kuhn length: c∗∗ '

1 2 b0 l K

(2)

For c > c∗∗ chains are rod-like on all length-scales smaller than ξ, and scaling gives: ξ = (b0 c)−1/2 which is identical to the result expected by Dobrynin et al’s model for salt-free polyelectrolyte solutionsDobrynin et al. (1995). 4

Non-entangled dynamics In dilute solution, the specific viscosity of polymer solutions can be approximated by the Huggins equation: ηsp = c[η] + kH (c[η])2 where [η] is the intrinsic viscosity and kH is the Huggins constant. [η] is a measure of hydrodynamic volume of polymer coils and the overlap concentration is therefore often approximated as c∗ = [η]−1 . Dilute polymers coils can be approximated as non-draining and their dynamics are Zimm-like, so that their longest relaxation times and intrinsic viscosities scale as τ ' ηs R3 /kB T and [η] ∼ R3 /M respectively. Above the overlap concentration, scaling assumes Zimm-like dynamics for length-scales smaller than correlation length and Rouse-like dynamics for larger distances. The viscosity of non-entangled solutions is scales as: ηsp,N E ' ηsp (c∗ )ζ(c, T )(c[η])αC

(3)

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where scaling expects αC = ν/(3ν − 1) and ν is the solvent quality exponent. The viscosity at the overlap concentration is ηsp (c∗ ) ' 1 + kH , and ζ(c, T ) described the variation of local friction experienced by polymer segments as a function of polymer concentration and temperature, normalised to ζ(0, T ) = 1. A linear dependence of ηsp with N is expected independent of solvent quality. For c > c∗∗ , application of the standard scaling treatment is non-trivial as it is not possible for a rigid segment of a chain to simultaneously exhibit Zimm relaxation up to the correlation length and then Rouse-like relaxation for longer distances with ξ < lK .

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Entanglement concentration and entangled dynamics of flexible polymers At some critical concentration polymers become entangled, and their lateral motion in solution is confined to a tube of diameter a, which leads to a large slowdown in their dynamics. The entanglement concentration of flexible polymers is found to scale as: ce ' Ke N −0.77 where Ke is a system-dependent constant. Above ce , the viscosity of polymers can generally be expressed as: h c iβc h N iβN (4) ηsp /ζ(c, T ) ' ηsp (ce ) ce Ne

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where βN ' 3.4 is expected from reptation theoryRubinstein et al. (2003) and βc is predicted to be 3.9 and 4.7 for flexible polymers in good and θ solvents respectively, in moderate agreement with experimental resultsColby (2010). We interpolate between Eqs. 3 and 4 using the following crossover functions: h  c δc i ηsp = ηN E (c, N ) 1 + (5a) ce h  N δN i ηsp = ηN E (c, N ) 1 + (5b) Ne 5

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Hydrodynamic Scaling

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where ηN E is the ’non-entangled viscosity’, given by Eq. 3. For the following discussion we will assume ηN E varies as a power-law of concentration and molar mass: ηsp,N E ∝ cαC N αβ . The scaling laws for semiflexible polymers are less well understood theoretically, and moderately flexible polymers appear to follow similar relations to Eqs. 5a-b. Highly rigid systems do not display a single power-laws at fixed concentration and molar masses as expected from Eqs. 5, see refs. Ohshima et al. (1995); Enomoto et al. (1985); Takada et al. (1991) for a discussion.

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An alternative to the models discussed above, known as the hydrodynamic scaling modelPhillies (1995); Phillies & Quinlan (1995); Phillies (1992); Phillies & Peczak (1988), was developed by Phillies and co-workers. For the purposed of this study, the main prediction of hydrodynamic scaling (HS) is that the non-entangled viscosity varies as a stretched exponential of the concentration and molar mass: λ µ ηHS (c, N ) = ηs eAc N for c < c+ (6)

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where ηs is the solvent viscosity, and A, λ and µ are system dependent parameters. λ and µ depend on molar mass and concentration respectively, and A depends on both c and N . Reference Phillies (2016) compares Eq. 6 to experimental data for a large number of polymer-solvent systems. Phillies’ model expects a smooth crossover to a power-law behaviour (i.e. Eq. 4) at a concentration of c+ , so that: η(c) = ηHS (c+ , N )(c/c+ )βc fixed N , c > c+ +

+ βN

η(N ) = ηHS (c, N )(N/N )

(7a) (7b)

The aim of this work is to test whether the various models outlined above apply to polysaccharides in near-θ solvents. To this end, we study the conformational and rheological properties of HPC as a function of concentration and molar mass and compare the results with those of other polysaccharide systems.

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fixed c, N > N

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3. Experimental Methods

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Materials and sample preparation: Five HPC samples were donated by NISSO CHEMICAL EUROPE GmbH (Dusseldorf, Germany), the manufacturer’s specifications are given in Table 2. Additional HPC samples were purchased from Sigma-Aldrich. Deionised water was obtained from a miliQ source. Ethanol (Absolute, 99%) was purchased from VWR. Solutions were prepared gravimetrically by mixing the appropriate amount of polymer and solvent, followed by use of a vortex mixer or a roller mixer to promote dissolution. Solutions with high viscosity were allowed to stand for a few days to remove air bubbles.

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Light Scattering: Static light scattering measurements were done on a FICA instrument by SLS-Systemtechnik (Denzlingen, Germany) with a laser of wavelength λ = 633 nm. Cylindrical quartz cuvettes with an internal diameter of 20 mm were washed thoroughly with freshly distilled acetone before use. Data were calibrated onto an absolute scale using toluene as a standard. The refractive index increment of HPC in water is taken as dn/dc = 0.14 mL/g.Wittgren & Porsch (2002) Dynamic light scattering measurements were made on an ALV-5000 with a laser of wavelength λ = 633 nm. Disposable glass cuvettes were rinsed with freshly distilled acetone. All light scattering samples were filtered (0.45 µm, regenerated cellulose) prior to measurement and the first few drops of filtered solution were discarded. Rheology: Rheological measurements were performed on a stress controlled Kinexus-Pro rheometer. Cone and plate geometries of diameter 40 and 60 mm, both with an angle of 1◦ were employed. The temperature was controlled with a Peltier plate. A solvent trap was employed to minimise evaporation. Steady shear viscosity measurements were taken after the torque reading was stable to within ' 5% for a period of 10 s. Frequency sweeps were carried out at a fixed shear strain value of 0.01. For the highest concentration of the highest molar mass sample (NI 4140, c ' 7wt%), a strain amplitude sweep at a fixed frequency of 1 Hz, revealed that the linear viscoelastic region extends up to a strain of γ ' 0.4. All data presented in the paper corresponds to measurements for which no sample expulsion from the cone-plane system was observed. Capillary Viscosimetry: An Ubelhode viscometer type 0c was employed for all viscosimetry measurements. Flow times were recorded employing an automatic detection system. Haggenbach-Couette corrections were applied following the manufacturer’s specifications. Density measurements: Density measurements were made on an Anton Paar DMATM 1001 Density Meter over a concentration range of 0.1 wt% < c < 5 wt%. The temperature was fixed at 25.00 ± 0.01 ◦ C. DI water and compressed air was used to clean the instrument after each measurement. UV-Vis: Absorbance measurements were carried out on a Cary 100 Bio UV-Vis spectrophotometer (Agilent) in the 25 < T /◦ C < 65 temperature range at wavelengths of 500, 600 and 700 nm. The heating rate was 0.4 K/min for all measurements. Samples were stirred throughout the measurement. TGA: Thermal gravimetric analysis measurements were performed on a Perkin-Elmer TGA STA6000. HPC powders (30-31 mg) were loaded onto ceramic crucibles. Nitrogen was flowed into the chamber at 40 mL/min to prevent oxidation and the temperature was varied between 20 to 550 ◦ C. Optical Microscopy: An AE 2000 inverted microscope (Motic) with 20x and 40x lenses was employed in bright field and phase contrast mode. Solutions were sandwiched between a glass and a cover slide and the focus was adjusted manually.

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Supplier 40 100 180 700 2500 80

Eq. 8 46 92 143 556 1462 132

417

100

154

47.5

−

−

370

−

45.0

−

1000

−

NI 120 NI 230 NI 390 NI 1380 NI 4140 SA 330

Nisso Nisso Nisso Nisso Nisso SigmaAldrich SigmaAldrich SigmaAldrich SigmaAldrich

SA 390 SA 900 SA 2500

SSL SL LM M VH 80k

3.3 3.8 3.5 3.9 3.6

100k

a

a

370k − 1M

3.4- − 4.4

Mw (kg/mol)

43.8

−

185

Ethanol 0.044 0.54 0.084 0.49 0.160 0.38 0.482 0.403 1.2 − 0.123 0.50

−

− −

−

−

− −

−

−

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0.68 0.66 0.63 0.53 − 0.59

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kH

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Table 2 lists some properties of the samples studied. In addition to these samples, four samples were purchased from Sigma-Aldrich. All samples from Nisso and the two low molar mass samples from Sigma-Aldrich formed clear solutions in water and ethanol. Visible gel-like aggregate were present for solutions of samples the other Sigma-Aldrich samples in both solvents, presumably due to the presence a poorly substituted cellulose fraction. Optical micrographs of solutions of sample SA 2500 confirm the presence of undissolved residues, see the supporting information. The water content for the Nisso samples was evaluated by measuring the mass loss of HPC powders upon heating to T = 120 ◦ C under nitrogen, and found to be ' 4% for all samples. In the following calculations we assume 4% water content is present for all powders and adjust our concentrations accordingly. The mass loss due to thermal degradation of the Nisso samples in the temperature range of 30-550◦ C is plotted in the supporting information. The densities of aqueous solutions of samples N1-N3, N5 and SA1 are plotted as a function of concentration in Figure S1. The solution density for the different samples is found fall into a single curve for all samples studied. Data were fitted to: ρ = ρs + (1 − vρs )c, where ρ is the solution density, ρs is the solvent density, v is the partial specific molar volume and c is the concentration in mass/volume. The partial molar volume is then V = M0 v, where M0 is the molar mass of a monomer in g/mol.

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4.1. General characterisation

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[η] (L/g)

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Table 2: Molecular characteristics of various HPC samples studied. The MS values are the supplier’s specifications. The letters and numbers in the sample name indicate the supplier and degree of polymerisation respectively. a Samples for which we do not have a value of MS, are assumed to have MS = 4.

4. Results and Data Analysis

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1wt% Tc,o v [η] (mL/g) (L/g) Water 59.4 0.791 0.041 56.2 0.79 0.079 50.6 0.803 0.147 49.1 − 0.436 48.7 0.811 1.1 48.1 0.806 0.115

Sample

4.2. Dilute solution conformation

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Static light scattering measurements of dilute HPC solutions in ethanol displayed excess scattering in the low q region, likely arising from supramolecular aggregates. Filtration through 0.45 or 0.2 µm regenerated cellulose membranes reduced the low-angle upturn but it did not remove it completely. Estimating Mw from the scattering signal at higher angles is problematic because data lie mostly in the 1 < qRg < 3 region, where the Zimm or Berry approximations are not valid. Dynamic light scattering results are consistent with SLS measurements in that they show a large decrease in the apparent diffusion coefficient is observed at low angles. Given the difficulties in estimating Mw from light

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Figure 2: Intrinsic Viscosity of HPC as a function of degree of polymerisation in aqueous solution. Line is a fit to Yamakawa and co-workers’ worm-like-chain model.

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scattering measurements, we opt to use a viscosimetric estimate instead. Dilute solution viscosity data were fitted to the Huggins equation and the best fit values of [η] and kH are compiled in Table 2. The following relations were constructed from Wirick et al’s dataWirick & Waldman (1970) for HPC in ethanol: [η]M0 = 0.0828N 1.11 , N ≤ 750

[η]M0 = 0.789N

0.7678

, N > 750

(8a) (8b)

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where [η] is in units g/L. Values of the molar mass calculated from Eq. 8 are listed in Table 2, where the conversion between N and Mw was carried out using M0 = 161+59 MS. Combining our measurements of [η] in aqueous solution with data from refs. Yang & Jamieson (1988); Phillies & Quinlan (1995); Goodwin et al. (2011), the following relationship was found for water at 25 ◦ C:

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[η]M0 = 0.246N 0.912 , for 220 < N ≤ 3700

(9)

Figure 2 plots the intrinsic viscosity of HPC in aqueous solution as a function of degree of polymerisation along with a fit to Yamakawa and co-workers’ model for the intrinsic viscosity of worm-like chains with no excluded volume. This

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4.3. Phase Behaviour

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Cloud points for the different HPC samples in aqueous solutions were determined as the point at which the transmission decreases to 50%. Experiments at wavelengths of λ = 500, 600 and 700 nm yielded similar results, as shown in Fig. S3a. The Sigma-Aldrich samples displayed a two-step absorbance curve, and lower values of Tc than the Nisso samples, see Fig. 3a-b. Both of these features are likely the result of a lower MS and/or a more heterogenous substitution pattern. Schagerl¨ of et al. (2006b)

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model requires three parameter: the effective monomer length (b0 ), the crosssectional diameter of the chain dc and the Kuhn length (lK ). b0 is set to 0.5 nm and dc ' 1.2 nm is estimated from the density measurements, as outlined in the theory section. The Kuhn length is left as a free parameter. We find lK ' 20-22 nm gives a best fit to the combined datasets of this work and refs Phillies & Quinlan (1995); Goodwin et al. (2011); Yang & Jamieson (1988). From the values of Rg , Mw and polydispersity reported by Wirick et alWirick 2 & Waldman (1970) for HPC in ethanol, we compute Rg,z /Nz ' 2 nm2 in the high Nz limit (1000 < Nz < 5500), where Nz is the z-averaged degree of polymerisation. This ratio corresponds to a Kuhn length of 24 nm, which is close to the value estimated for HPC in aqueous solution. GPC-light scattering data by Wittgren and Porsch give a slightly higher value of Rg2 /N ' 2.3, corresponding to a Kuhn length of 28 nm. Using this last value significantly to calculate [η] using Yamakawa’s worm-like chain model significantly over-predicts the viscosity results plotted in Fig. 2. Assuming the theta temperature for HPC in water to be θ ' 318 K gives τ ' 0.07 and B ' 2 ˚ A. The thermal blob size can then be calculated to be ξT ' 2000 nm, each containing ' 104 monomers, which correspond to a molar mass of Mw ' 4 × 106 g/mol. The conformations of all samples considered in this study are therefore expected to be unperturbed by excluded volume.

Figure 3: a: Transmission as a function of temperature for Nisso samples with a concentration of 1 wt%. b: Comparison of Sigma-Aldrich and Nisso samples of similar molar masses.

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Figure 4: a: Viscosity as a function of shear rate for c = 2 wt% solutions of NISSO samples. b: Viscosity as a function of shear rate for NI 4140 solutions. c: Concentration dependence of the zero-shear-rate viscosity for various samples. Lines are fits to the expanded Huggins equation.

4.4. Steady shear rheology

Figure 4 shows the viscosity of HPC solutions with varying molar mass and concentration. As expected, both the viscosity and the onset of pseudoplasticity increase with increasing molar mass and concentration. Viscosity data were fitted to a constant value (η0 ) at low shear rates (i.e. in the Newtonian plateau) and to a power law at high shear rates: η(γ) ˙ = K γ˙ n , where n is the flow index. The inverse shear rate at which these two lines intercept (γ˙c = [η0 /K]1/n ) is taken as the longest relaxation time τ . For samples where the Newtonian plateau is not covered by the measured shear rate range, we obtain η0 from the y-intercept of a η −1 vs. γ n plot. This extrapolation is only required for two of

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Figure 5: Determination of entanglement cross-overs. a: specific viscosity as a function of concentration (symbols) and fits to Eq. 5a (full lines) for data above c∗ , indicated by the dotted line. b: specific viscosity as a function of degree of polymerisation for a fixed concentration, data are from this work and refs. Phillies & Quinlan (1995); Yang & Jamieson (1988); Goodwin et al. (2011). c: Viscosity as a function of concentration and fits to hydrodynamic scaling model, symbols have the same meaning as in part a.

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the three most concentrated NI 4140 samples. The concentration dependence of the zero shear-rate viscosity is plotted in Figure 4c, along with fits to the expanded Huggins equation:

where B and m are fit parameters that can be related to the entanglement concentration through: ce ' [B/(1 + kH )]1/(α−m) [η]−1 . Equation 10 is seen to provide a good fit to all datasets across the entire concentration range studied. The best-fit values of B and m are compiled in Table 3.

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ηsp = c[η] + kH (c[η])2 + B(c[η])m

4.5. Determination of entanglement cross-over

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Figure 5a shows the determination of the entanglement concentration (ce ) for HPC with a fixed molar mass using Eq. 5a. Scaling expects αC = 2 for polymers in theta solvent, but the best fit exponent, is found to be αC ' 1.6 instead, see Table 3. Data at fixed concentration and varying molar mass are plotted in Fig 5b, along with fits to Eq. 5b. Combining the best-fit value of αC = 1.6 with the observed dependence of [η] ∝ N 0.9 gives a non-entangled viscosity of ηsp,N E ' (1 + kH )(c[η])αC ∝ N 1.4 , which is inconsistent with the Rouse value of αN = 1 expected by scaling. We have fitted the data both with the theoretical exponent αN = 1 and exponent of βN ' 2.9, as well as with αN = 1.5 and βN ' 2.4, finding that both describe the data well. We discuss the concentration and molar mass dependence of NC and ce respectively in section 4.7. Fits of the hydrodynamic scaling model to the zero-shear-rate viscosity data are presented in Figure 5c. Equations 6-7a track experimental results for the concentration dependence of η0 well for all samples studied. The parameters A and µ, compiled in Table 3, increase with increasing degree of polymerisation of the samples. We are not able to obtain reliable fits of Eqs. 6-7b to the molar mass dependence of the viscosity at a fixed concentration.

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Eq. 6b A λ 0.12 0.71 0.17 0.71 0.25 0.73 0.49 0.78 0.78 0.95 0.15 0.78 0.12 0.84 0.45 0.84 0.92 0.87 0.99 0.88

Eq. 5 ce 0.46 0.18 0.098 0.027 0.0093 0.14 0.11 0.023 0.0076 0.0098

δc 2.5 2.2 2.5 2.5 2.9 2.5 2.5 2.5 2.9 3.0

Eq. 7a c+ (M) βc 0.68 4.3 0.29 3.9 0.13 4.0 0.034 3.9 0.015 4.5 0.22 4.2 0.12 4.2 0.047 4.6 0.018 4.4 0.016 4.2

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NI 120 NI 230 NI 390 NI 1380 NI 4140 SA 330 SA 390 P 990 P 2760 P 3070

Eq. 10a B m 0.025 4.3 0.019 3.9 0.024 4.0 0.056 3.9 0.013 4.5 0.0081 4.2 0.0088 4.2 0.043 4.6 0.043 4.4 0.042 4.2

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Table 3: Rheological parameters for the different samples studied. a Other parameters ([η] and kH ) are reported in Table 2. ’P’ samples are from ref. Phillies & Quinlan (1995). b µ was set 0.

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4.6. Entanglement Density Figure 6a plots the frequency dependence of the storage and loss modulus of sample NI 4140 in solutions of varying concentration. The longest relaxation time (τ ) of solutions can be estimated from the shear viscosity as discussed earlier or from the crossover point in G0 and G00 . These methods show similar trends τ = 1/γC ∼ c2.4 and τ = 1/fC ∼ c2.7 , see the best fit lines in Fig. 6b. While all solutions considered in Fig 6a are above the entanglement crossover, no plateau in G0 is observed due to the limited frequency range studied and the polydispersity of the samples. The crossover point between G0 and G00 , which we refer to as GC can be used as an estimate for the plateau modulus GP . For polydisperse samples, GP exceed GC by over an order of magnitude. Alternatively, the ratio Gη/τ = η0 /τ , can also be used as a measure of the plateau modulus. These two methods, applied to samples NI 4140 and NI 1380 are compared in Fig. 6c and agree well, displaying a power-law of GC ' 5500c1.85 Pa, where c is in moles of repeating units per unit volume. Data by Horinaka et alHorinaka et al. (2015) for a HPC sample of unknown molar mass in ionic liquid BmiAc display GC values in agreement with the results for our samples. The wider frequency range probed in ref. Horinaka et al. (2015) allow the plateau modulus to be estimated as the value of G0 at the frequency when G0 /G00 reaches a local minimum. We refer to the value obtained by this method as GP,m . Horinaka et al’s values for GP,m are larger than GC by an order of magnitude, and display a slightly stronger power-law exponent with concentration: GP,m ' 75000c2 Pa. The degree of polymerisation in an entanglement strand can be estimated from the value of the entanglement plateau as:

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Ne =

kB T c M0 GP

(11)

Combining Eq. 11 with the experimental results in Fig. 6, we obtain Ne ∝ 13

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Figure 6: a: Storage (G0 , hollow symbols) and loss (G00 , full symbols) modulus of NI 4140 vs. frequency. b: relaxation times calculated from the onset of shear thinning (1/γC ) and from the crossover in G0 and G00 (2π/fc ). Data are divided by N 4 to remove influence of molar mass. c: Plateau modulus, approximated from the crossover in G0 and G00 (full black symbols and black crosses), G = η0 /τ (hollow symbols) and G0 at the minimum of G0 /G00 (red crossesHorinaka et al. (2015)). Circles are for NI 4140 and triangles for NI 1380. Lines are best-fit power-laws.

c−0.9 - c−1.1 , which is slightly weaker than the dependence found for flexible polymers in solution (G ∝ c2.2 − c2.3 , Ne ∝ c−1.2 - c−1.3 ). The differences in the exponents observed for our results and for flexible systems may be the result of

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4.7. Concentration regimes The various methods for determining the entanglement crossover are compared in Fig. 7a. Estimates according to Eq. 5a give a power-law of ce ' 66N −1.1 , which is a stronger N −dependence than what has observed for other flexible polymersColby (2010) and cellulose ethers Lopez (2019a). The entanglement crossover determined from Eq. 5b depends only modestly on the choice of αN for the two values selected (1 and 1.5), and shows a weaker N dependence of the entanglement crossover (c ∝ Ne−0.7 − Ne−0.8 ) than the results obtained from Eq. 5a. By contrast, for more flexible systems (e.g. PS/Toluene or NaCMC/0.1M NaCl), Eqs. 5a,b yield consistent (c, N ) datapoints Lopez (2019a,b). The values of c+ obtained from the hydrodynamic scaling model are similar to those obtained from Eq. 5a. Specifically, we find c+ ' (1.6 ± 0.7)ce independent of molar mass, where the error is a 95% confidence interval.

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the approximate methods employed for estimating GP ; for example, different methods to estimate the plateau modulus of NaHy lead to differences of 0.2-0.7 in the GP vs. c exponent Krause et al. (2001); Milas et al. (2002); Oelschlaeger et al. (2013); Nishinari et al. (2002); Calciu-Rusu et al. (2007); Yu et al. (2014).

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Figure 7: a: determination of entanglement crossover according to different methods. Full triangles: Eq. 5b with αC = 1.5, βC = 2.4, hollow triangles: Eq. 5b with αC = 1, βC = 2.9, squares: Eq. 5a, crosses c+ from Eqs. 7a and 7b. Lines are best fit power-laws. b: Concentration regimes for HPC in aqueous solutions. Full circles are overlap concentration, calculated as c∗ = 1/[η], squares are entanglement crossover calculated from Eq. 5a.

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The various concentration regimes for HPC identified are plotted in Fig. 7b. As all samples considered display unperturbed statistics, no semidilute regime is expected. The overlap concentration therefore marks the dilute-concentrated crossover, and follows a power-law of c∗ ' 4.6N −0.90 . For c < c∗ the viscosity is well described by the Huggins equation with an approximately N -independent Huggins constant of kH ' 0.6. The concentrated non-entangled regime spans less than an oder of magnitude in both c and N and we cannot therefore clearly resolve the N and c dependence of the specific viscosity in this region. Our best estimate is ηsp,N E ' (1 + kH )(c[η])1.5 ∝ c1.5 N 1.4 , which disagrees with the scaling prediction of ηsp ∝ N c2 . A precise determination of the entanglement crossover is challenging because the power-law exponent changes observed at a fixed molar mass and varying concentration and those at a fixed concentration and varying molar mass yield conflicting (c, N ) datasets. Above the entanglement crossover a dependence of ηsp ∝ N 3.9 c4 is observed, in line with results for other more flexible polysaccharides as discussed below. Equation 2 gives c∗∗ ' 0.07 M, indicated as a dashed line in in Fig 7b. We observe no clear rheological difference across c∗∗ . 5. Some universal features of polysaccharide rheology

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Estimating the N dependence of the semidilute, non-entangled viscosity of polysaccharides is challenging because ce /c∗ . 5 is generally observed. One method, applied in this study to HPC, is to combine the [η] vs. N powerlaw with Eq. 3, where αC is obtained from steady shear rheology data in the region c∗ < c < ce . This method yields αN ' 1.3−1.5 for several polysaccharide systemsKrause et al. (2001); Lopez et al. (2016); Del Giudice et al. (2017), which is inconsistent with scaling. One exception to the preceding considerations is carboxymethyl cellulose in salt-free water, where αN can be measured directly by plotting ηsp vs. N for two orders of magnitude in N in the non-entangled regime Lopez et al. (2016). This method gives αN ' 1.4, similar to the values estimated for HPC and other polysaccharides using the Eq. 3, and in line with that observed for some flexible polyelectrolytes Lopez & Richtering (2019). Another common feature of polysaccharide rheology is the low values of ce [η] ' 3 − 5 observed for various systems, in contrast with flexible synthetic polymers, where ce [η] ' 10 is usually observed. Polysaccharides therefore require less chain overlap (fewer binary interchain contacts) to entangle in solution. Polysaccharide systems with significantly different properties display similar [η]ce values. For example HPC in water and NaCMC in 0.1 M NaCl both display [η]ce ' 4.5. HPC has a longer Kuhn length (' 22 nm) and water is a near-theta solvent; 0.1 M NaCl on the other is a good solvent for NaCMC, and its Kuhn length is significantly smaller (11 nm). The observed power-laws for the specific viscosity with concentration for HPC in entangled solution are very close to those reported for other polysaccharide/solvent systems of varying Kuhn length, solvent quality or backbone composition. For example, HEC Del Giudice et al. (2017); Castelain et al. (1987), NaHyKrause et al. (2001); Fouissac et al. (1993), NaCMCLopez et al.

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(2016), AlginateStorz et al. (2010) or galactomans Sittikijyothin et al. (2005) all display a power-law in the range of ηsp ∝ c3.8−4.4 for c > c3 . Higher values are typically observed for systems with attractive interactions between chains, which lead to the formation of hyper-entanglements at some concentration cHE & ce . Lopez et al. (2018); Doyle et al. (2009); Albuquerque et al. (2014). Data for nonpolysaccharide semiflexible polymers display a similar concentration-dependence of the specific viscosity, for example poly(n-hexyl isocyanate) in DCM Ohshima et al. (1995) gives ηsp ∝ c3.6 . The entangled viscosities of rigid polymers such as Xanthan (lK ' 120 nm) or Schizophyllan (lK ' 220 nm) do not display powerlaw behaviour with concentration (or molar mass); if a power-law is fitted over a limited concentration range, exponents of ηsp ∝ c5−7 are observed Enomoto et al. (1985); Takada et al. (1991). The relation between NC and Ne for polysaccharide systems remains an open question. For flexible polymers, NC ∼ 2Ne and NC ∼ 0.5Ne are observed in the melt and in solution respectivelyLopez (2019b). For NaCMC in solution, a similar relation of NC ∼ 0.6Ne observed. The situation for HPC is less clear, if we use estimates for NC based on ηsp vs. N plots (triangles in Fig. 7), the same relation as for flexible polymers is recovered. If instead we use Eq. 5a, NC ' Ne results. Direct measurements of the entanglement plateau for c ' 0.02 − 0.05M would help resolve this problem, as the two methods for estimating NC converge in this region. 6. Conclusions

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We have studied the dilute solution and entanglement properties of HPC in aqueous solution, a model polyaccharide-solvent system. The Kuhn length is found to be lK ' 22 nm in aqueous and ethanol solution. The concentration and molar mass dependence of the viscosity can be well described by either an expanded Huggins equation, a smooth cross-over between two power laws or a stretched exponential and a power-law, as predicted by different models. The entanglement concentration obtained from different methods varies by up to a factor of ' 2 − 3, highlighting the difficulties in obtaining reliable exponents for ce vs. N . The molar mass and concentration dependence of the specific viscosity in the non-entangled regime shows appreciable differences with the Rouse prediction for flexible polymers. More specifically, the data suggest a dependence of ηsp ∼ N 1.5 for c∗ < c < ce , which is stronger than linear dependence expected by the Rouse model. The origin of this discrepancy is likely related to the relatively high stiffness of HPC. The large Kuhn length of HPC presumably also accounts for the exponent of ηsp ' N 3.9 observed in entangled solution, which differs from the reptation exponent of ηsp ∼ N 3.4 observed for flexible polymer. Samples with different substituent distribution, as identified from their turbidity/temperature curves display similar rheological behaviour in water at room temperature. A comparison with other polysaccharides reveals many similarities in the entanglement properties of various systems, and shows that these vary only weakly for systems of differing solvent quality and/or stiff-

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ness, but the behaviour of polysaccharides differs noticeably from that of flexible synthetic polymers. 7. Acknowledgements

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We thank Nisso Chemical Europe GmbH for kindly providing HPC samples, Wenjing Xu (RWTH, Aachen) for help with the UV-Vis measurements, professor George Phillies (Worcester University) for providing us with the tabulated data of his work and Will Sharratt and Joao Cabral (Imperial College London) for access to and help with the density measurements.

Albuquerque, P. B., Barros Jr, W., Santos, G. R., Correia, M. T., Mour˜ao, P. A., Teixeira, J. A., & Carneiro-da Cunha, M. G. (2014). Characterization and rheological study of the galactomannan extracted from seeds of cassia grandis. Carbohydrate polymers, 104 , 127–134.

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