Axisymmetric orifice flow for measuring the elongational viscosity of semi-rigid polymer solutions

J. Non-Newtonian Fluid Mech. 110 (2003) 27–43

Axisymmetric orifice flow for measuring the elongational viscosity of semi-rigid polymer solutions A. Mongruel a,∗ , M. Cloitre b a

b

Laboratoire de Rhéologie et Mise en Œuvre des Polymères, Université Pierre et Marie Curie, 60 rue Auber, 94408 Vitry-sur-Seine, France Laboratoire Matière Molle et Chimie, Unité Mixte CNRS ATOFINA, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France Received 7 August 2001; received in revised form 8 November 2002

Abstract We study experimentally semi-dilute solutions of xanthan, a semi-rigid polymer, flowing through an axisymmetric orifice. The pressure drop through the orifice and the size of the secondary vortex upstream of the orifice are measured simultaneously as a function of the flow rate. The results indicate that xanthan solutions behave like suspensions of rigid rods in a Newtonian solvent. Quantitatively, this analogy is supported by a theoretical analysis which combines a macroscopic flow model and the use of a microscopic variable reflecting the contribution of the rods to the bulk stress. With this simple model, it is possible to obtain the elongation viscosity of semi-rigid polymer solutions and estimate unknown macromolecular parameters. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Contraction flow; Rigid rod suspension; Semi-rigid polymer; Elongational viscosity; Vortex growth

1. Introduction Extensional flows are very efficient to orient or to extend fibres and macromolecules, giving rise to large elongation viscosity. In practice, measuring reliably the elongation viscosity remains an experimental challenge. Because of their simple practical implementation, axisymmetric entry flows into contractions are widely used to generate elongation flows. However, they are non-viscosimetric flows, involving complex mixing of shear and extension, and hence they give only an indirect access to the elongation viscosity. In contraction flows, the existence of a large elongation viscosity gives rise to a striking structure of the flow into large vortices near the contraction and to a pressure drop enhancement [1]. The Cogswell method [2] and its subsequent forms [3,4], all relying on the minimisation of the mechanical energy with respect ∗

Corresponding author. Tel.: +33-1-49-60-51-04; fax: +33-1-49-60-70-66. E-mail address: [email protected] (A. Mongruel). 0377-0257/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 0 2 5 7 ( 0 2 ) 0 0 1 7 1 - 4

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to the flow volume, allow to estimate the elongation viscosity from pressure drop measurements. These methods were validated with some success in the case of polymer solutions [4] and polymer melts [4,5]. In a previous work on fibre suspensions flowing through an orifice [6], we developed an alternative analysis of entry flows based on the Stokes equation. This analysis leads to simple scaling laws for the pressure drop and for the size of the vortices as a function of the Trouton ratio. In the case of fibre suspensions, the elongation viscosity can be expressed as a function of microstructural parameters such as fibre orientation distribution and concentration using micromechanical models [7,8]. Combining the two approaches, we obtained a good understanding of the vortex size upstream of the orifice as a function of concentration, length and aspect ratio of the fibres. In the present work, we wish to extend this analysis to solutions of semi-rigid rod polymers. The similarity between rigid fibre suspensions and xanthan solutions, a semi-rigid polymer, has been pointed out long ago for the entry flow into a 4:1 contraction [9]. Microscopic rod-like systems have been experimentally studied in other extensional flow geometries such as the opposing jet device [10,11]. A recent study [12] of the entry flow of semi-dilute xanthan gum solutions into an axisymmetric contraction presented a careful analysis of the vortex size as a function of xanthan concentration, solvent viscosity, flow rate and geometry. So far, pressure drop effects have received little attention and a global understanding of the behaviour of semi-rigid polymer solutions in contraction flows is still lacking. In this paper, we show that both pressure drop and vortex size measurements in orifice flow can be used to give a reliable estimate of the elongation viscosity of semi-rigid polymer solutions. We present the operating conditions of the method, its limitations and drawbacks. The paper is organised as follows. In Section 2, we describe the experimental flow cell, the solutions of xanthan gum used for the study, and their rheological properties in shear. In Section 3, experimental results for the orifice flow of xanthan solutions are presented. Comparison with orifice flow of fibres suspensions shows some analogies between the two systems, but also differences that are pointed out. Section 4 presents the simplified flow model we developed in the case of fibre suspensions to analyse quantitatively orifice flow features. We show how this model can be used in the case of xanthan solutions to finally yield a measure of their elongation viscosity. The shortcomings of the method are discussed in Section 5.

2. Experimental 2.1. Experimental device The flow experiments are conducted in a cylindrical cell with a small axisymmetric orifice (Fig. 1), which has already been described elsewhere [6,13]. This orifice flow is sometimes also termed immersed jet or contraction–expansion flow. It has been far less studied than the axisymmetric 4:1 contraction flow [6,13–15]. With this flow geometry, pressure transducers can be conveniently connected to pressure holes situated upstream and downstream the orifice. Our device consists of a vertical cylindrical cell of inner diameter D0 = 24 mm, in which a thin plate is clamped horizontally, with in this centre a circular die of diameter 2c = 1 mm, thus providing a large contraction ratio D0 /2c = 24. The fluid flows by gravity through the die and experiences a complex flow comprising regions of strong shearing near the walls and strong extension near the centreline [15]. On the centreline, the flow is a non-homogeneous uniaxial extension upstream of the orifice, and non-homogeneous biaxial extension downstream of the orifice [13]. As the flow cell is not thermally regulated, the temperature of the fluid could slightly vary (±2 ◦ C)

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Fig. 1. Schematic diagram of flow through an axisymmetrical orifice, and cylindrical coordinates (ρ, θ) for flow in the converging–diverging region.

around 20 ◦ C from one run to another. The rheological properties of the fluid were then characterised at the same temperature. 2.1.1. Pressure drop and flow rate measurements A flexible tube at the bottom cell outflow can be fixed at different vertical positions, thus imposing the total pressure head between the free surface of the cell and the outflow. The resulting flow rate is measured by weighing with an electronic balance the mass of fluid exiting at the output of the tube at regular time intervals. Two pressure holes are drilled symmetrically with respect to the orifice die, and connected with flexible tubes to a differential pressure transducer (of range 0–100 mbar), to measure the pressure drop through the orifice die, with an absolute accuracy of 1 mbar. The transducers are placed symmetrically with respect to the orifice, at a distance where the flow is parallel and completely developed. 2.1.2. Flow visualisation The whole flow cell is drilled in perspex, thus allowing visualisations of the flow. A HeNe laser beam of low intensity (3 mW) is expanded into a laser plane by crossing a cylindrical lens, this plane of light is carefully focused and intersects exactly the vertical axis of the cell. The optically clear fluids are seeded with iriodine particles of mean diameter 8 ␮m, which reflect the laser light. Observations and measurements in the plane of symmetry of the flow are performed by video-microscopy and image processing. Tracer trajectories are filmed with a CCD camera equipped with a TV zoom lens (18–108 mm, f = 2.5). The camera is connected to a monitor and a video-recorder, and to a PC equipped with an image acquisition card and an image analyser software. On the monitor screen, individual tracers appear as well-defined bright spots. Streamlines of the flow are obtained by superposing individual images recorded at successive time intervals. A 24 × 36 camera is also used to visualise the streamlines by taking photographs with long exposure times. 2.2. Solutions of xanthan in viscous solvents 2.2.1. Preparation Xanthan is a high molecular weight extracellular polysaccharide. The macromolecule is thought to have a considerable stiffness due to its double-helix structure. However, its persistence length is lower

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than its length and it is a semi-rigid rod [10]. We used in our experiments Rhodigel Clear manufactured by Rhˆone-Poulenc (France). The molecular weight Mw of this sample is given by the manufacturer to be of about 4 × 106 . The diameter of the molecule is well defined because of its helicoidal structure; values of dp = 2 nm have been reported [10,12,16]. The length depends on the conformation of the molecule in solution under flow. Lengths measured at equilibrium are of the order of 1 ␮m [12,16,17]. Stock solutions of 0.5 wt.% are prepared by adding a weighed amount of xanthan powder without purification to a known mass of deionised water, with a small amount (0.02 wt.%) of sodium azide to prevent bacterial degradation. Solutions are mixed at room temperature with a magnetic stirrer during 48 h, and then stored at low temperature. The solutions under study are made of a mixture of the xanthan stock solution with a high viscosity Newtonian solvent. The solvent used is a mixture of glucose syrup of density 1.48 and deionised water. A known mass of stock solution is added into a container with prescribed masses of glucose syrup and water, and the container is gently rolled until complete dissolution of xanthan and glucose in water. No salt is added. Two compositions of the glucose in water mixture were used (75 and 85 wt.%, respectively) in order to study the role of solvent viscosity. The addition of xanthan to these solutions did not modify significantly the density of the solutions ρ s (1.32 and 1.38, respectively at 20 ◦ C). The viscosity of the solvents ηs (0.2 and 1.6 Pa s, respectively at 20 ◦ C) are sufficiently high to meet the conditions of low Reynolds number in the flow cell. In practice, the maximum Reynolds number (calculated with apparent shear rate and viscosity at the die) does not exceed a value of 7 at the larger flow rates. The mass concentrations cp (g l−1 ) of xanthan are such that the final solutions are in the semi-dilute regime (see Section 4.2). The range of the semi-dilute regime is known to depend on the conformation of the xanthan molecule in solution, hence on xanthan sample, counterion and solvent ionic strength [17]. Critical concentrations can be determined from intrinsic viscosity measurements. With this method, the semi-dilute regime of xanthan Rhodigel in water with 0.01 M NaCl was found to lie between cp ∼ = 0.1 and 0.78 g l−1 [17]. 2.2.2. Rheological properties in shear The shear viscosity of the xanthan solutions is measured over a wide range of shear rates using a Rheometrics DSR stress-controlled rheometer, equipped with a cone and plate geometry. The cone and plate device is surrounded by an anti-evaporation cell which efficiently prevents drying of the glucose syrup. The temperature in the rheological measurements is the same as in the flow cell (around 20 ◦ C). Fig. 2 shows the curves of shear viscosity η versus shear rate γ˙ for xanthan solutions at different xanthan concentrations, prepared with the two types of solvents. It can be seen that the solutions exhibit a non-negligible shear-thinning behaviour. In contrast to other studies [12], we did not use high viscous solvents that suppress the shear-thinning effect introduced by the polymer, because we had to keep our fluids sufficiently mobile to flow only by gravity trough the orifice cell. In the range of shear rates accessible with the DSR, a zero-shear rate plateau for the viscosity, η0 , is reached only for the lower xanthan concentrations. On the contrary, a second plateau of viscosity η∞ is always reached in the limit of high shear rates accessible with the DSR. The values of η∞ are slightly larger than the viscosity of corresponding pure solvent, ηs , and increase with xanthan concentration. They have been measured for all solutions used and their values for the solutions of Fig. 2 are displayed in Table 1.

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Fig. 2. Shear viscosity of xanthan gum solutions in the mixture of glucose syrup in water, for different xanthan concentrations (1 ppm = 10−6 (w/w)). Empty symbols: 75 wt.% glucose in water ((䊊): 150 ppm, (䊐): 200 ppm, (): 250 ppm, ( ): 350 ppm). Full symbols: 85 wt.% glucose in water ((䊉): 100 ppm, (䊏): 350 ppm, (䉱): 500 ppm).

The evolution of viscosity with shear rate can be conveniently characterised by a law of the type Bird–Carreau: η(γ˙ ) = η∞ + (η0 − η∞ )(1 + (λγ˙ )2 )m−1/2 .

(1)

In Eq. (1), we have taken for η∞ the values of viscosity at the plateau reached at high shear rates. We have determined the parameters η0 , λ and m for the solutions used in the experiments, the results are summarised in Table 1, and the resulting curves are drawn on Fig. 2 as continuous lines. The value of m is a measure of the shear-thinning intensity. For a given solvent, it can be seen that m does not vary significantly as the concentration of xanthan becomes larger. For a given xanthan concentration, it increases slightly as the viscosity of the solvent becomes larger. The parameter λ is usually associated to Table 1 Parameters of the Bird–Carreau law (Eq. (1)) for the xanthan solutions of Fig. 2 Solvent

c (ppm)a

η∞ (Pa s)

75 wt.% Glucose syrup in water

150 200 250 350

0.23 0.23 0.23 0.27

85 wt.% Glucose syrup in water

100 350 500

1.59 1.70 1.99

a

1 ppm = 10−6 (w/w).

η0 (Pa s) 0.43 0.54 0.93 5.09 2.9 12.0 22.6

λ (s) 0.82 1.40 7.6 38.7 16.9 114.5 222.8

m 0.47 0.46 0.56 0.45 0.55 0.56 0.56

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the longest relaxation time of the solution; it increases with xanthan concentration and solvent viscosity. The zero-shear viscosity η0 increases with xanthan concentration and solvent viscosity. 3. Axisymmetric orifice flow of xanthan solutions: experiments In this section, we study the behaviour of xanthan solutions in viscous solvents flowing through the axisymmetric orifice. 3.1. Pressure drop versus flow rate relationship Typical variations of the pressure drop through the orifice with the flow rate are shown on Fig. 3. For the range of flow rates tested here, and for each polymer concentration, P(Q) is well represented by a straight line which intersects the vertical axis at P = 0 and a slope P/Q which increases with concentration. This behaviour is identical to that of fibre suspensions in Newtonian solvents [6]. We normalise the slope P/Q measured for xanthan solutions by the slope Pn /Q measured for a Newtonian fluid of viscosity ηs : 3.27ηs Pn = . Q c3

(2)

In Eq. (2), the numerical factor of 3.27 ± 0.25 was determined experimentally and is characteristic of the finite contraction ratio of the flow cell (the Stokes flow through an orifice of infinite contraction ratio predicts instead a factor of 3). A dimensionless pressure drop P/Pn is thus obtained with a precision of ±4%. We have plotted P/Pn on Fig. 4 as a function of xanthan mass concentration, for the two

Fig. 3. Evolution of the pressure drop with flow rate, for xanthan solutions in the 75 wt.% glucose in water solvent, for different xanthan concentrations ((䊊): 100 ppm, (䊐): 200 ppm, (): 500 ppm).

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Fig. 4. Evolution of dimensionless pressure drop with xanthan concentration ((): 75 wt.% glucose in water solvent; (䉲): 85 wt.% glucose in water solvent). Continuous line: prediction of Eqs. (18) and (17) with  = 2.8 ␮m.

types of solvent used. The dimensionless pressure drop increases with increasing xanthan concentration. We see that the data obtained with the same xanthan concentration but different solvents coincide within a good precision. 3.2. Vortex size A typical flow pattern of xanthan solutions flowing through the orifice is shown in Fig. 5a. For xanthan solutions, like for solutions of flexible polymers, a development of a vortex is observed upstream the orifice. The upstream vortex size is denoted Lv , where Lv is taken as the vertical distance between the plate and the first streamline departing from the cell wall (see Fig. 1). The flow of xanthan solutions may not be perfectly symmetrical with respect to the vertical axis of the cell (on an axial cut of the flow, two vortex of slightly different heights were sometimes observed) particularly at high flow rates and large xanthan concentrations. In those instances, the asymmetry being not greater than 5%, we take the average of the left and right vortex size as the vortex size. Fig. 5b shows for comparison the streamlines obtained in the same flow cell for fibre suspensions in the semi-dilute regime. The main difference between xanthan solutions and fibres suspensions is that, strikingly, a downstream vortex is observed in the case of fibre suspensions. Another difference is the straight boundaries of the vortex observed for fibre suspensions [6] (while upstream vortices are slightly convex in xanthan solutions). Moreover, for fibre suspensions the flow is always symmetrical with respect to the vertical axis of the cell, and the vortex size is independent on flow rate. The evolution of the relative size Lv /2R with flow rate is shown in Fig. 6 for different xanthan mass concentrations and the two solvents used in the experiments. For the less viscous solvent, at a given xanthan concentration, the vortex size does not depend on the flow rate, in the range of flow rates investigated. For

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Fig. 5. Example of streamlines obtained by long exposure time (120 s) photography, for the orifice flow: (a) of a solution of xanthane of 75 ppm in the 75 wt.% glucose in water solvent, flow rate is 0.10 cm3 s−1 ; (b) of a suspension of fibres in the semi-dilute regime ( = 2 mm, d = 13 ␮m, Φ = 0.0015) in a mixture of glucose in water, flow rate is 0.047 cm3 s−1 .

the more viscous solvent, however, a slight increase of Lv /2R with the flow rate is observed. An increase of vortex size with increasing flow rate is also reported [12] when the solvent viscosity is large. However, this flow rate dependence remains very moderate compared to that observed in the case of flexible polymer solutions [14,15]. Consequently, we will consider that, in the range of flow conditions tested here, the vortex size of xanthan solutions depends mainly on concentration. We shall take the average of the vortex sizes measured over the range of flow rates investigated here to characterise the vortex size at a given xanthan concentration. The vortex sizes measured in entry flow [12] at a given xanthan concentration are systematically smaller than those measured in this work (smaller contraction ratio and smaller xanthan molecules are used in [12], these differences are consistent with smaller vortices).

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Fig. 6. Evolution of the dimensionless vortex size, Lv /2R, with flow rate, for xanthan solutions of different concentrations (empty symbols: 75 wt.% glucose in water, xanthan concentrations from bottom to up: 25, 50, 75, 150, 200 and 350 ppm; full symbols: 85 wt.% glucose in water, xanthan concentrations from bottom to up: 100, 350 and 500 ppm). Continuous lines are guides for the eyes.

Fig. 7. Evolution of the average dimensionless vortex size, Lv /2R, with xanthan concentration. ((): 75 wt.% glucose in water; (䉲): 85 wt.% glucose in water). Continuous line: prediction of Eqs. (13) and (17) with  = 2.8 ␮m.

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Fig. 7 shows the evolution of vortex size with xanthan concentration. On this plot, error bars constructed by taking the maximum and minimum values measured at a given flow rate do not exceed 5%. Interestingly, we see that the mean vortex size measured with two different solvents for the same xanthan concentration coincide within a good accuracy. The vortex size first increases with xanthan concentration, and then it remains practically constant. 4. Analysis of experimental results The results of the above section show that xanthan solutions upstream the orifice flow exhibit qualitatively the same macroscopic behaviour as fibre suspensions in Newtonian solvents. Downstream of the orifice, however, no vortex is observed in the case of xanthan solutions, in contrast to fibre suspensions. In this section, we present a flow model applicable to rigid rod suspensions, and we show how this model can be modified in order to analyse the experimental results for xanthan. 4.1. Micromechanical model for the orifice flow of a rigid rod suspension 4.1.1. Contribution of particles to the bulk stress A general expression for the bulk stress tensor in an unbound suspension of non-Brownian fibres of given orientation is [7,8] σ = −pI + 2ηs E + ηf (a a a a : E), (3) ¯¯¯¯ where p is the pressure, ηs the shear viscosity of the Newtonian suspending fluid, E the rate of strain tensor and a the rod orientation vector. Coefficient ηf has the dimension of a viscosity, and is a measure ¯ of the contribution of the rods to the bulk stress. It is a function of microstructural parameters such as fibre geometry, concentration and orientation. In the following, fibres have a length , a diameter d, an aspect ratio r = /d and the particle volume fraction is Φ. In the case of semi-dilute suspensions (1/r < Φ 1/r 2 ) of fibres having a preferential orientation along the direction of elongation, Batchelor [7] derived the following expression for ηf : Φr 2 4 . ηf = ηs 3 ln(π/Φ)

(4)

This simple expression was recovered to the leading order by subsequent theoretical works [8]. In the following, we will denote as β the group of variables involving the microstructural parameters: β=

Φr 2 . ln(π/Φ)

(5)

Eq. (3) can be used to calculate the stress in fibre suspensions submitted to various types of extensional flows. It is convenient to define the extensional viscosity as ηE =

σ33 − (1/2)(σ11 + σ22 ) , E33

(6)

where the indice 3 denotes the principal axis of deformation. Two forms of rate of strain tensor of interest here are the uniaxial and the biaxial extensions, giving rise to stable orientation of the fibre either parallel

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(uniaxial extension) or perpendicular (biaxial extension) to the streamlines. From Eq. (3), taking the rod orientation vector a either parallel or perpendicular to the principal axis of extension, we obtain the ¯ the extensional viscosity: following expressions for
 ηf ηE = 3ηs 1 + in uniaxial extension, (7) 3ηs
 1 ηf ηE = 3ηs 1 + in biaxial extension. (8) 4 3ηs Note that the perpendicular orientation of fibres brings a factor 1/4 in Eq. (8) and not 1/2 as was erroneously mentioned in [6]. 4.1.2. Simple model for the orifice flow of a rigid rod suspension In this paragraph, we briefly present the macroscopic flow model we developed for the orifice flow (see [6] for more details). The flow through an orifice involves a complex mixing of elongation and shear. In fibre suspensions, the flow upstream (respectively downstream) of the orifice divides into a converging (respectively diverging) conical region, and a lateral toroidal vortex (Fig. 5b). The velocity in this vortex is negligible compared to that in the conical region. This allows to carry out simple approximations. We restrict our attention to the conical regions and we suppose that: (i) the velocity field is purely radial; (ii) the velocity u(ρ,θ ) vanishes on the boundaries of this region (for θ0 = arctan(R/Lv )). Writing the mechanical equilibrium ∇ · σ = 0 in spherical coordinates (ρ, θ , φ), eliminating the pressure term by cross-multiplying by the curl operator: ∇x(∇ · σ ) = 0, and using the definition of extensional viscosity (Eq. (6)), leads to 
  ∂ ηs 1 ∂u ∂ 2 u ηE ∂ 2 u + 2 + = 0. (9) ∂θ ρ 2 tan θ ∂θ 3 ∂θ 2 ∂θ Eq. (9) holds locally for any values of shear viscosity ηs and elongation viscosity ηE . In the limit where the angle of the central conical region is small, Eq. (9) reduces to
 ηs ∂ 2 u ∼ ηE ∂ 2 u . (10) = ρ 2 ∂θ 2 3 ∂θ 2 A small angle of the central conical region is compatible with experimental observations in the semi-dilute regime where significant vortex sizes are observed. Eq. (10) reflects the mechanical equilibrium between shear and elongation stresses in the flowing region. When both ηs and ηE are supposed to be constants (independent of the deformation rate), simple scaling laws for the dimensionless vortex size and the pressure drop can be derived from Eq. (10):
 Lv ηE 1/2 ∝ , (11) 2R 3ηs

 ηs Q ηE 1/2 P ∝ . (12) c3 3ηs Note that in this model, the same reasoning holds for upstream and downstream of the orifice leading to the same scaling, provided that ηE is taken as the elongation viscosity either in uniaxial or biaxial extension.

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4.1.3. Comparison with entry flow models This idea of relating directly macroscopic measured quantities such as pressure drop and vortex size to the rheological properties of the fluid is not new. In the case of the entry flow of polymers into an axisymmetric contraction, a method of minimisation of mechanical energy has been extensively developed that gives good estimations for the elongational viscosity [2–4]. We examine, for instance, the results of Binding’s analysis developed for the entry flow into an axisymmetric contraction of a generalised Newtonian fluid having power law behaviour both in shear and in elongation. In the case where the rheological behaviour is independent of deformation, and for large contraction ratio, this analysis reduces to expressions for the dimensionless upstream vortex size and for the dimensionless entry pressure drop that are, in order of magnitude, similar to ours [3]. In particular, it gives the same dependences with Trouton ratio as Eqs. (11) and (12). Moreover, this analysis explicitly shows that the exponent 1/2 of the Trouton ratio comes from a constant elongational viscosity, and that the independence of Lv /2R on flow rate comes from a thinning behaviour that is the same in elongation as in shear. 4.1.4. From the microstructure to the global variables For the sake of analytical tractability, we assume that upstream and downstream of the orifice, fibres are subjected respectively to uniaxial extension and biaxial extension. We then find that the dimensionless vortex size should vary as
 Lv 4 1/2 = 0.28 1 + β upstream of the orifice, (13) 2R 9
 1 1/2 Lv = 0.28 1 + β downstream of the orifice. (14) 2R 9 The pre-factor 0.28 (±0.05) is the dimensionless vortex size measured for a Newtonian fluid [6]. Eqs. (13) and (14) predict that the upstream vortex should be twice as large as the downstream vortex. Such a tendency can be indeed observed on Fig. 5b. We also obtain predictions for the total pressure drop through the orifice summing up the contributions due to upstream and downstream flows:

 P 1 4 1/2 1 1 1/2 = + . (15) 1+ β 1+ β Pn 2 9 2 9 A test of this prediction for fibres suspensions will be presented in Section 4.2.1. 4.2. Predictions for xanthan solutions In this section, we apply the model developed in the case of fibre suspensions to the xanthan solutions. This requires several assumptions. In the upstream converging region, xanthan molecules are supposed completely extended by the flow and aligned parallel to the streamlines. Downstream of the orifice, the case of xanthan is more complicated than the case of fibres, because the conformation of xanthan molecules in biaxial extension is not known. This conformation modification past the orifice is at the origin of the striking macroscopic difference between xanthan and fibres behaviour, namely the absence of downstream vortex for xanthan. Therefore, in the present analysis, a “Newtonian-like” behaviour is postulated for xanthan solutions downstream the orifice. Another assumption for xanthan solutions is that

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the shear viscosity ηs is supposed to be independent of shear rate. These approximations will be discussed in Section 5. The key point is now to relate the volume fraction Φ to the molecular parameters of xanthan molecules, i.e. the rod length , the rod diameter d and the polymer concentration cp : Φ=

πd 2 NA cp , 4Mw

(16)

where NA is the Avogadro number. Substituting Eq. (16) into Eq. (5) gives an analytical expression for the microscopic variable β of xanthan solutions: β=

π3 NA cp . 4Mw ln(4Mw /NA d 2 cp )

(17)

4.2.1. Analysis of pressure drop data Following the hypothesis of xanthan molecules behaving like rigid rods upstream of the orifice, Eq. (12) together with Eq. (7) describe the associated upstream contribution to the total pressure drop. The contribution of the downstream flow to the total pressure drop is described by Eq. (12) with ηE = 3ηs . Summing up the two contributions, this leads to a new expression of the pressure drop through the orifice, valid for semi-rigid rods:
 P 1 4 1/2 1 = + . 1+ β (18) Pn 2 9 2 Eq. (18) together with Eq. (17) provide now an analytical expression for the pressure drop associated with the flow of xanthan solutions through the orifice. The rod diameter and the molecular weight of xanthan being known (d = 2 nm, Mw = 4.106 g mol−1 ), the rod length  is the only unknown parameter. In the following, it will be left free as a fitting parameter. In Fig. 4, we observe that our theoretical prediction describes nicely the variations of P/Pn , with the polymer concentration. The rod length leading to the best agreement between prediction and data is  = 2.8 ␮m. This length has to be changed by ±7% (±0.2 ␮m) to give a significant discrepancy between the data and the prediction. The order of magnitude of  found in our experiments seems reasonable. Comparisons with xanthan lengths reported in the literature have to be made with caution, as xanthan lots, solvent conditions and techniques of measurements differ from one work to another [16]. For instance, from extensional viscosity measurements with an opposing nozzles device [10], a length of 3.5 ␮m on a xanthan gum of larger molecular weight (Mw = 10 × 106 ) was determined. Measurements by light scattering [12,16] provided an hydrodynamic length at equilibrium of 1.25 ␮m on a xanthan gum of smaller molecular weight (Mw = 2.4 × 106 ). Given the good agreement between our prediction and experiments, we can use these results to estimate the elongation viscosity of our xanthan solutions. With the molecular parameters found, the microscopic variable β is calculated at a given xanthan concentration (Eq. (17)), and hence the dimensionless elongation viscosity ηE /3ηs (Eq. (7)). The values obtained for the dimensionless elongation viscosity in uniaxial extension at different xanthan concentrations are shown in Table 2. We show also in Table 2 the results for the elongation viscosity in uniaxial extension of xanthan solutions from [10]: they are comparable to ours, even if the macromolecular parameters are slightly different. On the contrary, the elongational viscosity measured in [12,16] with an opposed jet extensional rheometer was significantly smaller than ours, but the length of the xanthan molecule is also smaller. Having found an order of

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Table 2 Non-dimensional elongation viscosity ηE /3ηs in uniaxial extension for xanthan solutions in the semi-dilute regime cp (ppm)a

25 50 75 100 150 200 250 350 500

ηE /3ηs b

ηE /3ηs c

75 wt.% Glucose in water

85 wt.% Glucose in water

90 wt.% Glycerine in water

80 wt.% Glycerine in water

4.4 8.2 12.3 16.5 25.3 34.4 43.8 63.3 94.0

– – – 17.3 – – – 66.5 98.7

– 4.6 7.6 10.7 – – – – –

– 4.5 8.3 15.7 – – – – –

1 ppm = 10−6 (w/w). This work (Mw = 4 × 106 ): estimations from Eqs. (4) and (7), with molecular parameters deduced from the analysis of pressure drop measurements ( = 2.8 ␮m). c Comparison with results of [10] ( = 3.5 ␮m, Mw = 10 × 106 ): direct measurements by the opposing nozzles in two solvents (mixtures of glycerine in water). a

b

magnitude for the length of the xanthan molecules, we can now recalculate the range of the different concentration regimes in our experiments. For both solvents used here, the range of semi-dilute regime corresponds to xanthan concentrations comprised between cp = 2.3 × 10−4 and 0.32 g l−1 . This checks out that our experiments are situated in the semi-dilute regime, except for the most concentrated solution (500 ppm). Finally, in Fig. 8, we present a summary of experimental results for the pressure drop obtained for the two systems under study: the values of P/Pn for fibre suspensions [6] and for xanthan solutions (this work) are plotted together, versus the microscopic variable β. The continuous line is the prediction of Eq. (18), and fits well the xanthan data, as a consequence of the analysis just carried out earlier. The dashed line is the prediction of Eq. (15) valid for fibres suspensions. It has to be pointed out that the data obtained for fibres suspensions prepared with the longest fibres available ( = 2 mm ) give the best agreement with Eq. (15). It is likely that for these fibres having the larger aspect ratio, the slender body approximation under which Eq. (5) was established best holds. 4.2.2. Analysis of vortex size data Using the molecular parameters obtained from the analysis of pressure drop, and substituting via Eq. (17) into Eq. (13), we obtain the prediction for dimensionless vortex size upstream of the orifice plotted as a continuous line on Fig. 7. In contrast to the evolution of the pressure drop, this prediction does not describe correctly the measured vortex size over the whole range of xanthan concentrations. Very good agreement is found between prediction and measurements for concentrations cp ≤ 0.2 g l−1 , whereas the prediction overestimates the measured vortex size for larger concentrations. We have checked that values of  different from 2.8 ␮m did not improve the quality of the prediction. This discrepancy between the model predictions and the measurements is the indication for shortcomings of our model at large xanthan concentrations. These are discussed in Section 5.

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Fig. 8. Master curves showing the evolution of the dimensionless pressure drop with the variable β for fibre suspensions in Newtonian solvents ((䊊):  = 0.5 mm, d = 13 ␮m; ( ):  = 1 mm, d = 13 ␮m; (䊐):  = 2 mm, d = 13 ␮m; (䉫):  = 1 mm, d = 19 ␮m) and for xanthan gum solutions ((): 75 wt.% glucose in water solvent, (䉲): 85 wt.% in water solvent). Continuous line: prediction of Eq. (18). Dashed line: prediction of Eq. (15).

5. Discussion of approximations and concluding remarks We found in the above sections that the data obtained in orifice flow with xanthan solutions in thick solvents are well described by a simple flow model adapted from a model developed in the case of fibre suspensions in Newtonian solvents. However, some assumptions and particular departures from this analogy have to be addressed with more details. This discussion holds only for the flow upstream of the orifice where xanthan molecules are in extension. 5.1. Approximation of complete alignment: effect of rotational diffusion In a suspension of Brownian rods, the degree of rod alignment in the flow is a result of the competition between Brownian rotation, which tends to globally rearrange the rods towards isotropy, and alignment by the flow. This competition can be quantified by the comparison of two characteristic times: the inverse of the rod rotational diffusion coefficient Dr , and the inverse of the applied shear rate γ˙ (or elongational rate ε˙ ). Complete alignment is achieved when γ˙ , ε˙  Dr . The values of rotational diffusion coefficient Dr have to be compared with the deformation rates generated by the orifice flow. The Brownian rotation coefficient Dr0 of a single prolate ellipsoid (of axes a and b with a  b) in the limit of vanishing concentration, is given by Dr0 = [2 ln(2a/b) − 1]3kB T /2πηs a 3 with kB the Boltzmann constant and T the absolute temperature [18]. Taking for a and b, the length  and diameter

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d of the xanthan molecules, we calculate a value of Dr0 of the order 10−2 and 10−3 s−1 , respectively for the two solvents used in the experiments. In a semi-dilute solution, the rotation of each rod is restricted by the presence of its neighbours. The theory [18] predicts that the rotational diffusion coefficient Dr of a single rod in a semi-dilute solution decreases with increasing rod concentration, when the rods orientation is isotropic. The theory further predicts [11,18] that the diffusion is in turn affected by the flow, an increasing alignment of the rods tends to release the constraint to rotation, and therefore to increase Brownian rotational diffusion. Nevertheless, it is likely that in our experiments, Dr is smaller than Dr0 . In orifice flow, maximum deformation rates occur at the orifice and can be estimated from an apparent strain rate (4Q)/(πc3 ). Taking the range of flow rates Q experienced in the flow cell (between 10−2 and 1 cm3 s−1 ), we obtain a range of apparent strain rates comprised between 102 and 104 s−1 , so that at the orifice, Dr is always much smaller than the calculated apparent elongation rate, even at low xanthan concentrations. Far upstream from the orifice, however, in the region where the converging flow approaches the Poiseuille flow, the characteristic strain rate is divided by a factor 104 compared to that at the orifice, so that, for small values of flow rate it may fall into the range of magnitude of Dr . In view of this, we conclude that in our experiments, the elongational rates are sufficiently high to always provide complete alignment of the rods in the upstream vicinity of the orifice. In the regions the most remote from the orifice, however, alignment may not be completed. 5.2. Discussion of the model The previous paragraph shows that the degree of alignment of the xanthan molecules upstream of the orifice is a function of the distance from the orifice. An additional effect comes from the fact that xanthan molecules are semi-rigid molecules and that their degree of extension can also vary with the distance from the orifice. The model developed here relies on two approximations. First, the shear viscosity ηs of the solutions is assumed to be a constant, independent on shear rate. Second, the elongation viscosity ηE is obtained in the case of complete alignment. These two approximations are true near the orifice but not far from it. The consequences are negligible when dealing with the pressure drop, that is dominated by its value in the vicinity of the orifice. On the contrary, the shape and size of the vortex are determined in a non-local way, and the variations of rheological properties with deformation rate play a role. More precisely, far upstream of the orifice, shear rate decreases and thus ηs may significantly increase for the more concentrated solutions (Fig. 2), so that the equilibrium reflected by Eq. (10) for a given value of ηE can occur at smaller ∂ 2 u/∂ 2 θ , i.e. smaller vortex sizes. This explains why a further vortex size development is not needed to counterbalance the increase of elongational viscosity with xanthan concentration. With the same reasoning, it is also possible to explain why a slight increase of vortex size with the flow rate can be detected for a given xanthan concentration. 5.3. Concluding remarks We have studied the flow through an orifice of solutions of xanthan gum in thick solvents. We show, both on an experimental basis and with the use of simple flow modelling, that the analogy between semi-rigid polymer solutions and macroscopic fibres suspensions, that was pointed out in the case of entry flow into an axisymmetric contraction, is not complete in the case of the orifice flow. The main difference arises in

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the flow downstream of the orifice, as a result of the change of the xanthan molecule conformation when subjected subsequently to an uniaxial extension and to a biaxial extension. Besides of this striking difference, the analogy remains valid upstream of the orifice, where xanthan solutions behave like suspensions of macroscopic fibres in Newtonian solvents. This is a consequence of the strong extensional components of upstream flow, that completely extends and orients the xanthan molecules in the upstream vicinity of the orifice, so that they behave like non-Brownian rigid rods. The analogy is made quantitative by using the microscopic variable of Batchelor’s theory for suspensions of completely oriented rods, combined with macroscopic flow models. Measurements of global flow features only, such as pressure drop and vortex size, lead to the determination of the elongation viscosity of the solutions, and to a correct estimation of the length of the xanthan molecules. References [1] G.G. Lipscomb, M.M. Denn, D.U. Hur, D.V. Boger, The flow of fiber suspensions in complex geometries, J. Non-Newtonian Fluid Mech. 26 (1988) 297–325. [2] F.N. Cogswell, Converging flow of polymer melts in extrusion dies, Polym. Eng. Sci. 12 (1972) 64–73. [3] D.M. Binding, An approximate analysis for contraction and converging flows, J. Non-Newtonian Fluid Mech. 27 (1988) 173–189. [4] M.E. Mackay, G. Astarita, Analysis of entry flow to determine elongation flow properties revisited, J. Non-Newtonian Fluid Mech. 70 (1997) 219–235. [5] S.A. McGlashan, M.E. Mackay, Comparison of entry flow techniques for measuring elongation flow properties, J. Non-Newtonian Fluid Mech. 85 (1999) 213–227. [6] A. Mongruel, M. Cloitre, Extensional flow of semi-dilute suspensions of rod-like particles through an orifice, Phys. Fluids 7 (11) (1995) 1–7. [7] G.K. Batchelor, The stress generated in a non-dilute suspension of elongated particles by pure straining motion, J. Fluid Mech. 46 (1971) 813–829. [8] E.S.G. Shaqfeh, G.H. Fredrickson, The hydrodynamic stress in a suspension of rods, Phys. Fluids A 2 (1990) 7–24. [9] R.J. Binnington, D.V. Boger, Entry flow of semi-rigid rod solutions, J. Non-Newtonian Fluid Mech. 26 (1987) 115–123. [10] G.G. Fuller, C.A. Cathey, B. Hubbard, B.E. Zebrowski, Extensional viscosity measurements for low-viscosity fluids, J. Rheol. 31 (1987) 235–249. [11] C.A. Cathey, G.G. Fuller, Uniaxial and biaxial extensional viscosity measurements of dilute and semi-dilute solutions of rigid-rod polymers, J. Non-Newtonian Fluid Mech. 30 (1988) 303–316. [12] M.A. Zirnsack, D.V. Boger, Axisymmetric entry flow of semi-dilute xanthan gum solutions: prediction and experiment, J. Non-Newtonian Fluid Mech. 79 (1998) 105–136. [13] M. Cloitre, A. Mongruel, Dynamics of non-Brownian rodlike particles in a non-uniform elongational flow, Phys. Fluids 11 (1999) 773–777. [14] U. Cartalos, J.M. Piau, Creeping flow regimes of low concentration polymer solutions in thick solvents through an orifice die, J. Non-Newtonian Fluid Mech. 45 (1992) 231–285. [15] J.P. Rothstein, G.H. McKinley, The axisymmetric contraction–expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop, J. Non-Newtonian Fluid Mech. 98 (2001) 33–63. [16] M.A. Zirnsack, D.V. Boger, V. Tirtaatmadja, Steady shear and dynamic rheological properties of xanthan gum solutions in viscous solvents, J. Rheol. 43 (1999) 627–650. [17] M. Milas, M. Rinaudo, M. Knipper, J.L. Schuppiser, Flow and viscoelastic properties of xanthan gum solutions, Macromolecules 23 (1990) 2506–2511. [18] M. Do¨ı, S.F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford, 1986.