Journal of Non-Newtonian Fluid Mechanics

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1 J. Non-Newtonian Fluid Mech. 6 (2009) Contents lists available at ScienceDirect Journal of Non-Newtonian Fluid Mechanics journal homepage: Mesoscopic dynamics of polymer chains in high strain rate extensional flows D. Kivotides, V.V. Mitkin, T.G. Theofanous Center for Risk Studies and Safety, Department of Chemical Engineering, University of California, Santa Barbara, CA 937, USA article info abstract Article history: Received July 2008 Received in revised form 22 April 2009 Accepted 23 April 2009 Keywords: Extensional rheology Dilute polymeric flows High strain rates Brownian dynamics Bead-spring model Viscoelastic fluid breakup We take a step towards accessing the physics of viscoelastic liquid breakup in high speed, high strain rate flows by performing Brownian dynamics computations of dilute uniaxial, equibiaxial, and ellipsoidal polymeric extensional flows. Our computational implementation of the bead-spring model, when tailored to the DNA molecule, consistently with recent works of Larson and co-workers, is shown: (a) to predict a coil-stretch transition at Deborah number De 0.5, and (b) to reproduce the experimental longest relaxation time. Furthermore, after adapting the model parameters to represent the polyethylene oxide (PEO) chain (for M = 0 6 Da), we find it possible to reproduce our own experimental data of the longest relaxation time, the transient extensional viscosity of dilute solutions at small Deborah numbers, and a coil-stretch transition at Deborah number De 0.5. Extended to large Deborah numbers, the model predicts that polymer stretching is controlled by: (a) the randomness of the initial conditions that, in combination with rapid kinematically imposed compression, lead to the formation of initially frozen chain-folds, and (b) the speed with which thermo-kinematic processes relax these folds. The slowest fold relaxation occurs during uniaxial extension. As expected, the introduction of stretching along a second direction enhances the efficiency of fold relaxation mechanisms. Even for Deborah numbers (based on the chain longest relaxation time) of the order of one thousand, there is a large variation in the time a polymer needs in order to extend fully, and the effects of Brownian motion cannot be ignored. The computed Trouton ratios and polymer contributions to the total stress as functions of Hencky strain provide information about the relative importance of elastic effects during polymeric liquid stretching. At high strain rates, the steady state elastic stresses increase linearly with the Deborah number, resembling at that stage an anisotropic Newtonian fluid (constant extensional viscosity) Elsevier B.V. All rights reserved.. Introduction The purpose of this paper is to extend recent works [ 7] on computations of molecular-chain dynamics (in solution), at the Brownian level, to large strain rates. While these previous developments are focused on low and medium Deborah numbers, as encountered, for example, in molecular-scale analyses and/or manipulation of single entities, such as DNA molecules [8,9], our interest derives from consideration of breakup in high speed flows as found in atmospheric dissemination of ( thickened ) liquid agents [0]. Our results, for dilute solutions at this stage, show that such an extension is feasible and informative, motivating future investigations of non-dilute cases with significant chain entanglement effects. From the numerics viewpoint, our approach is motivated by the most recent and extensive developments of Brownian dynamics of bead-spring polymer models as detailed in references [ 3, 8]. Corresponding author. address: demos@engineering.ucsb.edu (D. Kivotides). In particular, we employ the Ermak and McCammon method [], along with the semi-implicit predictor corrector integration scheme in references [,3 5]. The later is adapted for excluded volume interactions, which are incorporated in our model. The square root of the diffusion matrix is computed in the Chebyshev approximation, a computationally effective approach, following a line of development detailed in references [2,3,2,6]. Notable are some recent success of such bead-spring models in capturing key polymer physics in diverse applications. One example is found in the synergy of experiments [9 22] and computations [,4 9] investigating the dynamics of single DNA molecules in extensional and shear flows. In very good agreement with experiments (dilute solutions of biomolecules at small or moderate Deborah numbers), these studies have clarified the intrinsic relations between chain conformation, polymer extension rate, and buildup of elastic stresses. Another notable example, in the context of non-biological molecules like polystyrene, is the combination of experiments [23 28] and computations [,29,30] that yielded significant new insights on the role of solvent polymer interaction. In particular, Hsieh et al. [] have shown how (20 bead or more chain models) taking account of hydrodynamic interactions can /$ see front matter 2009 Elsevier B.V. All rights reserved. doi:0.06/j.jnnfm

2 70 D. Kivotides et al. / J. Non-Newtonian Fluid Mech. 6 (2009) Fig.. Visualization of 2.5 mm viscoelastic liquid drops subjected to blasts of various strengths in shock tube. The liquid is a 3.8% solution of Polystyrene-butyl-methacrylate (PSBMA, M = Da) in tributyl-phosphate (TBP). Flow dynamic pressure: Pa (a and b), Pa (c). Instances after shock arrival time: 238 s (a), 333 s (b), 238 s (c). The strain rate of filaments, estimated from sequential frames around these times is approximately 0 4 s. The Reynolds numbers Re = U D d / g (where U is the free stream flow velocity, D d is the initial droplet diameter, and g is the kinematic viscosity of the supersonic gas) are Re = in (a) and (b), and Re = in (c). Since both (a) and (b) depict the same experiment, the scale bar of the former applies also to the latter. reproduce experimental strain stress data with dilute polystyrene solutions without such interactions the solvent is allowed to drain freely, and this results in diminished rise of the Trouton ratio with strain. Similarly good predictions were shown by Li and Larson [30] with dilute polystyrene solutions in both theta and good solvents. They concluded that, since fluid stress rises much faster with strain in a good solvent (as compared with theta solvents), extensional rheometry could serve as an indicator of solvent quality. Our work is a continuation/extension of the aforesaid lines of application of the bead-spring methodology, but in the alternative context of viscoelastic filament elongation and rupture at extremely high strain rates [3 33]. One setting involves an initially spherical drop exposed to the blast behind a strong shock wave (aerobreakup), an example of which is illustrated in Fig.. The remarkable point is that interfacial instabilities and shear lead to the drawing out of filaments, which for sufficiently strong (shocks) flows appear to snap off into pieces. Another setting, contrived as a simplified, model system for such snap offs, involves the creation of an expanding ring-membrane flow structure, an example of which is illustrated in Fig. 2. This flow pattern results from collision of drops (actually micro-liter liquid slugs) upon a conically shaped obstacle in the absence of any other influences (inside an evacuated chamber). The remarkable point here is that the strained liquid ruptures simultaneously at multiple locations that become more numerous as the speed of collision increases beyond a certain critical value. There are some characteristic features of this application area that dictate our modeling approach: (a) The type of polymer elasticity relevant to viscoelastic breakup differs from the elasticity of stiff double-stranded DNA biomolecules. Indeed [34,35], the latter are worm-like chains in contrast to the freely jointed chains that concern us here. Worm-like (Kratky-Porod) chains are special cases of freely rotating chain models [34] and possess uniform flexibility over the whole length of the molecule. In other words, they present bending contour modes even at scales smaller than the Kuhn s length. In contrast, freely jointed chains are rigid at smaller than Kuhn-length scales, and their elasticity is caused by transgauche bond rotations. (b) In addition to uniaxial extensional flows, we need to address biaxial and elipsoidal extensional flows as these become relevant at the forward stagnation point of a liquid drop exposed suddenly to high speed gas flow [32,33], or during coalescence of two drops [36]. Such flows do not appear to have been considered previously. (c) As illustrated in Figs. and 2, our interest is in high Deborah number flows. Based on the longest relaxation time, our working Deborah numbers are of the order of a thousand, or alternatively based on the shortest relaxation time they are of order 0. High Deborah numbers require that we employ Fig. 2. Visualization of ruptures in rapidly strained 8% water solution of PEO (M = Da). The strain is created by injecting 0.5 ml at a velocity of 00 m/s upwards, against the down-facing conical surface at the tip of the probe. At the time around rupture, the radial velocity of the rim is 45 m/s which amounts to an elongational (azimuthally) strain rate of approximately s. The largest diameter of the vertical rod (upper part) is 6.4 mm.

3 D. Kivotides et al. / J. Non-Newtonian Fluid Mech. 6 (2009) numerical methods that do not allow overshooting the maximum chain length constraint. (d) While the basic modeling and numerical approaches seem to be well at hand, particular implementations (to specific polymer solvent pairs) are neither straightforward nor are they ubiquitous. As mentioned above, the DNA and polystyrene chains, the latter in both good and theta solvents, seem to have attracted major attention; the former as a specific interest, the latter as a model system of freely jointed chains. We work within this later class, but the parameter definition needs to be extended as the particular polymers of interest are PSBMA, PIBMA, PBMA, and PEO, all at molecular weights of 4 million [3]. While our end goal is developing an understanding of the non-dilute regime (typically 3 5 w/o), in this paper we focus on dilute solutions of PEO in water (a good solvent). The key quantities/physics of interest are: time required for full chain extension, polymer conformation during the stretching process, transient extensional viscosities, and large strain rate scaling of the steady value zero and variance dt. The symmetric 3 3 tensor is the component of the overall diffusion tensor D that corresponds to beads i and j. Moreover, D = B B T, and is the 3 3 component of B corresponding to beads i and j. The solvent temperature is denoted by T, and k B is Boltzmann s constant. Note that the elastic forceonbeadj is the difference of the elastic forces of the springs that are connected to bead j, i.e., where is the elastic force that spring j (extending between beads j and j + ) exerts on bead j. In order to proceed, we need to specify the diffusion tensor, the constitutive elasticity law for the entropic springs and a formula for the excluded volume interactions. We employ a standard Rotne-Prager-Yamakawa (RPY) [43] diffusion tensor. According to this choice, the spatial derivative in the second term on the right-hand side of Eq. () becomes identically zero. Since we are not going to treat confined systems here, we use the unbounded domain version of the RPY theory [2,45,46], (2) (3) state elastic stresses. Quality assurance and continuity with previous work are addressed by first applying our methods and tools to established results on the DNA molecule [4]. Further, we exemplify the empirical plausibility of our model by showing that it: (a) predicts a coil-stretch transition at Deborah number De 0.5, (b) correctly predicts the experimental longest relaxation time, (c) predicts a plausible average chain size under no flow conditions, and (d) it is in good agreement with our transient extensional viscosity measurements. 2. Mathematical model The mathematical model depicts polymeric fluids as two component systems. The solvent obeys the incompressible Navier- Stokes equations and is kinematically prescribed. The polymers are described by a standard version of the bead-spring model. The latter includes essential thermal fluctuation physics [39 4], but not the physics of chain scission and chain entanglement processes. Hydrodynamic interactions are taken into account via an approximate low Reynolds number hydrodynamics theory that provides a diffusion tensor. By incorporating the latter into the Langevin equation for the beads [39,42], and by performing a first order in time numerical analysis of the resulting equation as shown in reference [], one obtains the Ermak and McCammon scheme []. In particular, every polymeric molecule is discretized into N b beads that are connected by N s = N b non-linear springs. If denotes the position of bead i, then obeys the following Langevin differential equation [] Here, = u T is the transpose kinematic solvent velocity gradient tensor at the bead i location, is the total conservative force acting on bead j, i.e. the sum of elastic and excluded volume forces, (), and is a Gaussian stochastic vector of mean where a denotes the bead radius, s is the solvent dynamic viscosity, I is the unit 3 3 tensor, and.ina computation with N b beads, the diffusion tensor is represented by a3n b 3N b matrix. We have implemented two non-linear elasticity laws: (a) Warner s finitely extensible non-linear elastic law (FENE) [29,54] for use in freely jointed chain cases, and (b) the worm-like chain law (WLC) [48] for use in stiff biomolecular chain (DNA) cases. We define the spring vector for spring j between beads j and j +, and the maximum length a spring can have Q 0 = N K,s b K, where N K,s is the number of Kuhn lengths per spring, and b K is the Kuhn s length of the corresponding ideal freely jointed chain. Moreover, is the length of spring j. The FENE and WLC laws are given by Eqs. (5) and (6) respectively: where H = 3k B T/(N K,s b 2 ) is the elastic spring constant for an ideal K chain after employing the Gaussian approximation in the computation of the free energy [34]. In order to capture good solvent effects, we employ a classical field theoretic model of excluded volume interactions that disfavour bead overlap. Notably, these interactions are not equivalent to spring spring repulsions, and so, since springs are allowed to pass through each other, chain entanglement effects are not treated here. An appropriate law for the potential of excluded volume interaction at bead j located at generated by a bead i located at (4) (5) (6)

4 72 D. Kivotides et al. / J. Non-Newtonian Fluid Mech. 6 (2009) is This relatively soft potential is more practical than the familiar Lennard Jones one, since, as shown in references [49,50], it allows larger numerical time steps. In terms of this potential, the excluded volume force acting on bead j because of bead i is (7) where is a known vector as discussed in references [5,], and C is a constant depending on the numerical time step ıt k B Tıt C =. (4) 3 s ab K Q 0 The numerical method is iterative. At each iteration inside a particular time step, the new values are compared with those from the previous iteration, and their difference is formed. The iteration stops when the error (8) where and ˇ are determined as in Jendrejack et al. [5]. Accordingly, the potential range is of the order of the spring mean-square radius of gyration in an ideal chain Rg 2 ( R2 g =N K,sb 2 K /6 where N K,s is the number of Kuhn lengths per spring), i.e. ˇ = 3/(4 Rg 2 ). The potential strength is ( ) 3/2 3, (9) = 2 v3 k B TN 2 K,s 4 R 2 g where v is a parameter (positive for good solvent) with dimensions of length. In our computations v is chosen to be of the order of the bead radius a. Finally, the solvent velocity field is prescribed in the kinematic approximation u = = r, (0) ) ( f ( + f ), () where is the rate of strain in units of s, and f takes the value f = 0 for planar extensional flow, f = 0.5 for uniaxial extensional flow, f = for equibiaxial extensional flow, and f = 0.5 for ellipsoidal extensional flow. 3. Mathematical methods Since we are interested in high Reynolds number flows and large extensional strain rates, we solve Eq. () with numerical methods that do not allow the polymers to stretch more than their maximum chain length L c. The latter is L c = N K,c b K, where N K,c is the number of Kuhn lengths per chain and, by definition, N K,c = N K,s N s,c with N s,c been the number of springs per chain. We have applied a standard semi-implicit, predictor corrector, first order accurate in time scheme [,3 5]. Here, we have extended the hydrodynamicinteractions version of the scheme in the realm of good solvents by incorporating the effects of excluded volume forces. The details of this extension are similar to the extensive discussion in reference [], and need no further explanation here. In implementation, Q j is updated at each time step via the solution of a cubic algebraic equation that has a unique real solution between zero and Q 0. Following the analysis of references [5,], we have derived the corresponding cubic equations for the FENE (2) and WLC (3) elasticity laws (2) (3) (5) becomes smaller than a prescribed threshold [5]. In all computations performed here, the maximum e I value was e I = 0 3. Smaller e I values resulted in significant computational overhead without any discernible changes in the results. The fluctuation-dissipation and Einstein relations require the decomposition of diffusion matrix D in order to obtain a representation of matrix B. We compute B by either the standard, O(N 3 ) complex, Cholesky decomposition method of [52] or the approximative, O(N 2.25 ) complex, Chebyshev polynomials method of Fixman [2,3,2,6,53]. In the latter approach, B is the square root of D, i.e. B = B T. We control the accuracy of Fixman s method employing a criterion introduced in reference [2] that checks the satisfaction of the fluctuation-dissipation relation. In particular, denoting the Chebyshev approximation of B as B C, the relative error e C of the approximation is e C = (B C dw ) (B C dw) dw dw D dw D dw. (6) The method requires at each time step the smallest and largest eigenvalues of the diffusion matrix. We compute the eigenvalues with the Arnoldi algorithm which is a fast, O(N 2 ) complex, iterative, linear algebraic method, and, following reference [2], we compute at each time step B C employing previously evaluated eigenvalues, checking at the same time whether e C exceeds a prescribed threshold. In case the check fails, we either increase the order of the Chebyshev approximation expansion or recompute the eigenvalues. We have found that the first option is far more economical, and it is worth experimenting with the appropriate order of the Chebyshev approximation, optimizing accordingly the number of calls to the Arnoldi algorithm. 4. Preamble to results Formally, our computational bead-spring model M is a discretemathematical function that receives as input the initial values of polymer dynamical quantities 0, and the values of a finite set of parameters that include the computation stoppage time t, and, by applying the equations of motion and the boundary conditions, produces as output the polymer dynamical quantities t, i.e. M( 0,) t. In our approach, set consists of two subsets: (a) experimentally determined parameters, i.e. the solution temperature T, the solvent density s, the solvent viscosity s, and the number of Kuhn lengths per chain N K,c, (b) effective parameters which are chosen in a heuristic way. The need for effective parameters emanates from the heuristic nature of bead-spring models (they are not derived in a rigorous fashion from non-equilibrium statistical mechanics of the microscopic Hamiltonian dynamics).

5 D. Kivotides et al. / J. Non-Newtonian Fluid Mech. 6 (2009) Hence, the strengths of forces acting on beads at mesoscopic scales do not have first principles connection with more fundamental microscopic quantities. The effective parameters provide a working specification of the various forces as follows: (a) The elastic force is characterized by the maximum spring length Q 0, and the elastic spring constant H H = 3k BT N K,s b 2. (7) K Since N K,s = N K,c /N s,c and N K,c is an experimentally determined parameter, H is fixed by specifying N s,c and b K. These parameters fix also Q 0, and so elastic effects are fully determined. (b) The hydrodynamic interactions are parametrized by the hydrodynamic interaction parameter h h = H 6 s k B T, (8) which is approximately equal to the ratio between the bead radius and the root-mean-square of the distance between two beads that are connected by an equilibrated spring in a theta solvent, and so, for consistency, there must be 0 <h < 0.5. Here, = 6 s a is the friction coefficient. Notably, since at this stage H is already known (via N s,c and b K ), choosing h fixes also the bead radius a. (c) Finally, the excluded volume interactions are parametrized by fixing the v/a ratio. Some quantities of central importance in polymer physics feature in the ensuing discussions. The first is the longest relaxation time scale of a polymer chain. This scale provides an estimate of the time needed by a fully stretched chain in an extensional flow to relax to thermal equilibrium once the strain has been removed. Since all new results presented here refer to good solvents and include hydrodynamic interaction effects, the appropriate time is the Zimm longest relaxation time [34] ( ) Z = sb K vev k B T b 3 N 3 K,c, (9) K where = takes into account excluded volume effects that cause chain-swelling in a good solvent, v ev is the experimental excluded volume, and v ev /b 3 quantifies the quality of the solvent K becoming unity in the athermal solvent case. Notably, Z is smaller than R, the longest relaxation time for the corresponding Rouse, freely draining solvent chain [34] R = sb 3 K k B T N+2. (20) K,c Certainly, smaller parts of a bead-spring chain relax faster than the full chain, and of special importance is the shortest relaxation scale 0 that corresponds to the effective chain monomer, i.e. an individual spring, and is computed from Eq. (9) by replacing N K,c with N K,s. At time-scales smaller than 0, the (computational) chain exhibits purely elastic responses. In extensional flows, these scales define two corresponding Deborah numbers De = and De 0 = 0. When polymers stretch in an extensional flow, elastic stresses build up in the fluid. According to statistical polymeric liquid theory, the total stress tensor of a macromolecular solution has contributions from both polymers and solvent s The polymer contribution is given by the formula [30,57] (2) (22) where = N A c/m is the number of polymers per unit solution volume, c is the polymer mass concentration, i.e. the total mass of the dissolved chains divided by the solution volume, and the angular brackets denote an ensemble average over a sufficient number of molecules. Since the isotropic part of the total polymer stress does not depend on the actual chain con- p values here. figurations, we shall report only anisotropic stress Notably, since all computations presented in this work are performed in unbounded fluid domains, is unspecified. Hence, in order to compare theory with experiment, we multiply the stresses obtained from single-chain calculations with the experimental value in order to obtain a corresponding theoretical stress level. This practice is appropriate for the study of dilute solutions, where essentially each polymer chain evolves in isolation from the other chains. The importance of elastic stresses in the flow is parametrized by the apparent Trouton ratios Tr a i = i E s,i=, 2, (23) where i are the transient extensional viscosities E i E = ii 33,i=, 2. (24) A few remarks regarding general features of our computations are pertinent here. First, due to the stochasticity and non-linearity of Eq. (), individual polymer paths show remarkable variability even when all other controllable flow parameters are kept constant. Hence, only ensemble averages of physical quantities are empirically useful. Such ensemble averages are obtained by repeating computations with the same controllable parameters but different, although statistically equivalent, initial conditions, and different Brownian motion random number sequences. In order to generate appropriate initial conditions, we assume that, before any polymer stretch experiment, the molecule is in thermal equilibrium with the macroscopically stationary solvent. To produce an ensemble of polymers in thermal equilibrium, we evolve a single molecule in a zero velocity solvent flow, and we record its bead-spring configuration every ten Rouse chain times R. Since R is longer than the actual Zimm longest relaxation time of the macromolecule, the recorded polymer configurations are uncorrelated, thermal equilibrium configurations. In the same milieu, we have also verified that employing different random number sequences for the computation of the Wiener process increment in Eq. () results in distinctly variable polymer paths even when the latter emanate from identical initial conditions. Hence, we vary both initial conditions and random number sequences in the various realizations of polymer stretching. In order to avoid very small numbers in the diffusion matrix, we scale our problem so that the diagonal elements of become equal to unity. The scaling units are l = b K, t = 6 s ab 2 K /(k BT), and m = (6 s ab K ) 2 /(k B T), for length, time and mass respectively. The numerical time step ıt needs to ensure the resolution of both chain relaxation and kinematic processes. The kinematic time scale is readily given by I = /, and the smallest chain relaxation scale is 0. On the other hand, since our numerical method is only first order accurate in time, we avoid taking large time steps even if there are no numerical stability problems. Thus, we choose an empirical factor f r = 0 4 for independently restricting t. In the end, t = f r min( 0, I ). We perform three sets of computations: (a) We check the validity of our computational model by comparing its predictions with classical and also recent results in the literature.

6 74 D. Kivotides et al. / J. Non-Newtonian Fluid Mech. 6 (2009) (b) Since our main interest lies in PEO physics, we adapt the beadspring model to the PEO chain, and we test its predictions against our own experiments with dilute solutions. We exemplify the empirical usefulness of the model by showing that it fits (by design) the experimental longest relaxation time and equilibrium mean end to end chain distance, and predicts a standard coil-stretch transition at De 0.5, as well as the experimental extensional viscosity for uniaxial extensional flow for a small De number. (c) Finally, since the flows of interest (Figs. and 2 involve high strain rates, we apply the bead-spring model to this regime, and we obtain significant insight into the rate of polymer stretching and its underlying mechanisms, as well as the level of elastic stresses. 5. Set up and validation of the computational model The availability of experimental results for single DNA chains in an aqueous solvent thickened with added sugar [9 2], and the excellent agreement between these experiments and accompanying computations in reference [4], demand a test of our code against DNA chain results. The bead-spring model parameters for this computation were taken from Larson et al. [4], and are available in Tables and 2. We verified that, consistently with Larson et al., our computational model (a) predicts a coil-stretch transition at Deborah number De 0.5, (b) reproduces the experimental longest relaxation time = 4. s, (c) reproduces average steady state polymer extension values at various Deborah numbers. We are after a PEO (M = 0 6 Da) version of the bead-spring model that is computationally practical and agrees with experiments. Before we discuss how this challenge is met, we note that, in contrast to the DNA case, there are no single-chain PEO experiments in the literature, and in order to establish the required synergy between theory and experiment, we have performed our own measurements of the rheology of dilute aqueous PEO solutions using the filament-thinning technique [37]. Since this type of experiments are now considered standard, we describe them in Appendix A. The various parameters of our PEO version of the bead-spring model are shown in Tables and 2. We explain next how these particular values were chosen. Certainly, we would like to choose the N s,c value as large as computationally feasible, so that we can push the relaxation time scales spectrum of the chain towards high frequencies that are similar to the relevant fast time scales in our laboratory experiments. In this vein, we found that an appropriate choice is N s,c = 40. On the other hand, as discussed previously, we also need to define b K in order to completely determine elastic effects. We specify b K by matching the experimental longest relaxation time of the chain. In this respect, we found that, even with a chain of several hundred beads, the experimental could not be matched by employing the experimentally determined b K value (b K = cm according to reference [56]). We have Table Effective parameters for DNA and PEO computations. The bead radius a is a derived rather than independently specified quantity. Polymer chain N s,c b K ( m) h v/a a 0 4 ( m) DNA PEO Table 2 Experimentally determined parameters for DNA and PEO computations. Polymer chain T (K) s (g cm 3 ) s (g cm s ) N K,c DNA PEO Fig. 3. Experimental longest relaxation time e versus polymer mass concentration c for aqueous solutions of PEO polymers with molecular mass M = 0 6 Da. The overlap polymer mass density corresponding to an athermal solvent is ca = g/cm 3, and the actual estimated overlap polymer mass density is c = g/cm 3. tackled this problem by employing a larger effective b K value in order to increase the computational chain s longest relaxation time and meet measurements. The other two parameters h (0 <h < 0.5) and v, that characterize the strength of hydrodynamic and excluded volume forces can not be chosen freely [29], and their choices must fit the experimental end to end distance of the PEO chain in dilute, aqueous solutions in thermal equilibrium. All other parameters have their standard, experimental values [56]. The elastic springs obey the FENE constitutive law and the length of the computational chain is L = 2 m. Next, we demonstrate the plausibility of this PEO model by showing that it fits (as designed) the longest relaxation time and end to end chain distance, and furthermore, that it predicts a plausible coil-stretch transition, and reasonably approximates experimental extensional viscosities in uniaxial extensional flow. First, we show that, in the infinite dilution limit, our model fits (as designed) the experimental. For this purpose, we have measured (as described in Appendix A) the longest relaxation times of dilute and semi-dilute aqueous PEO solutions at various polymer mass concentrations, shown in Fig. 3. These two regimes are separated by the overlap polymer mass concentration c = M/(N A R 3 ), where R is the average end to end polymer extension. First, we check whether the measurements of Fig. 3 refer to any dilute solutions at all. This we do by computing the smallest c value possible, i.e. the one corresponding to an athermal solvent. In particular, in the scaling formula for the dilute limit chain size [34,35] in a good solvent ( ) 2 vev R = b K b 3 N K,c, (25) K we substitute = 0.588, as well as the experimental N K,c and b K values [56], and set (for athermal solvent) v ev /b 3 K = tofind R a = cm. This value gives ca = g/cm 3. Thus, even in the most unfavourable case, the smallest experimental concentration of Fig. 3 corresponds to a dilute solution. In parallel to the experiments, we have calculated the average relaxation trajectory of previously fully extended PEO macromolecules. We present in Fig. 4 the evolution of x 2 versus time during the final period of a sample relaxation realization, as well asafitforx/l < 0.5 to the average relaxation trajectory that suggests n s. This agrees with an extrapolation of the measurements shown in Fig. 3 to the infinite dilution regime. We can use this (experimental) time in order to find a plau-

7 D. Kivotides et al. / J. Non-Newtonian Fluid Mech. 6 (2009) Fig. 4. Sample final relaxation period of a single, previously fully extended PEO chain with L = 2 m, N s = 40,b K = cm, h = 0.25 and v = cm. The included fit to the average relaxation trajectory for x/l < 0.5 suggests n s. Fig. 5. Average normalized steady state PEO chain extension x /L versus the Deborah number De in uniaxial extensional flow. A coil-stretch transition is predicted at De 0.5. sible value for the v ev /b 3 ratio. Indeed, employing the previously K given formula for the Zimm longest relaxation time, we find that Z = s when v ev/b 3 = The latter value results in a K new, better estimate for the overlap polymer mass concentration c = g/cm 3. In addition, the spring relaxation time becomes Z 0 = s. In all computations performed here, De and De 0 have been defined using these Z and Z time scales that 0 include both hydrodynamic and good solvent effects. We show further that our model fits (as designed) an experimentally plausible chain size. Thus, we compute the average end to end chain distance R by averaging over 340 equilibrium polymer configurations. We find R = cm. We compare this against experiments as follows: (a) we employ the Mark-Houwink- Sakurada equation [] = M 0.65 of reference [58] and find intrinsic viscosity [] = cm 3 /g (for M = 0 6 Da PEO solutions), (b) we employ the scaling relation [34,35][] N A Re 3 /M to find R e = cm, thus R/R e =.74. Taking into account the approximative nature of the relation connecting [] and R, this is a reasonable agreement. Moreover, employing the previously deduced v ev /b 3 = value in the good solvent formula K for R, we find R = cm which is close enough. Finally, we note that the computed R value is consistently smaller than the previously computed chain size in the athermal solvent case R a = cm. To demonstrate the plausibility of the model via its predictions, we have investigated the PEO chain coil-stretch transition. The mean steady state polymer extension x along the stretching axis (normalized with the maximum chain length) is shown in Fig. 5. Here, x denotes the difference between the maximum and minimum chain coordinates rather than the end to end polymer distance. As indicated in Fig. 5, our model predicts a coil-stretch transition in uniaxial extensional flow, and this transition occurs at a plausible De 0.5value [55]. Finally, we show that the model makes reasonable predictions of the experimental extensional viscosity. We have tried two comparisons between theory and experiment: (a) one for the dilute solution concentration c = g/cm 3 which is consistent with the assumptions of our computations, and (b) another for the semi-dilute concentration c = g/cm 3. Although the latter concentration is not, in principle, within the domain of applicability of our model, we would like to gauge the usefulness of our method as a heuristic prediction tool in semi-dilute concentration regimes. There are three important notes: (a) The measured longest relaxation time varies with concentration and differs from the com- putational longest relaxation time that corresponds (by design) to the infinite dilution limit. The variation of the longest relaxation time with concentration in our experiments is in agreement with similar measurements of Bazilevskii et al. [37] and Clasen et al. [38]. Thus care has been taken to compare results in terms of De numbers rather than physical strain rate values. Definitely, this approach is not fully equivalent to a computation with the physical strain rate and the experimentally observed longest relaxation time, but it is a reasonable way of taking into account the observed variation via single-chain calculations. (b) The nominal experimental Deborah number De = 2/3 in the exponential diameter decrease regime is within the coil-stretch transition De number interval of sharp Trouton ratio changes as shown in Figs. 5 and. This makes the task of every comparison between theory and experiment difficult since the outcome is highly sensitive to the approximate nature of both bead-spring model and measurement technique [38,59].However, this choice of De is imposed on us by the particular nature of our experimental method, and we shall see that the model makes reasonably accurate predictions. (c) The computations take into account both the linear and exponential diameter decrease regimes as these are discussed in Appendix A. In both cases, there is no satisfactory agreement between theory and experiment for the nominal experimental value De = 2/3 in the exponential regime. In particular, the computations did not match the experimentally observed Trouton ratio growth rate. Thus, we have gradually increased the De number in the computational exponential diameter decrease regime, in order to find which theoretical De value there fits the measurements. In case (a), Fig. 6 shows that there is a good fit when the theoretical Deborah number is De = Notably, this value is also within the coil-stretch transition regime in Fig. 5. The sensitivity of the theoretical results to the De number (exponential regime) choice is shown in Fig. 7 (left). As discussed in Appendix A, we had to increase (for this case only) the estimated experimental strain rate value in the linear diameter decrease regime by a factor of 20% in order to compensate for asymmetric draining effects. We also note that our results differ from analogous comparisons found in reference [] for polystyrene solutions in that our De number is smaller and within the coil-stretch transition regime, and we have taken into account in our computations the initial linear diameter decrease regime which is absent in the step strain rate experiments employed in the comparison in reference []. The semi-dilute solution case results are shown in Fig. 8. Remarkably, we find a good fit of the De = 2/3 (exponential regime) experimental data for the same theoretical De = 0.76 value that we

8 76 D. Kivotides et al. / J. Non-Newtonian Fluid Mech. 6 (2009) Fig. 6. Experimental (points) and theoretical (solid lines) apparent Trouton ratios Tr a versus Hencky strain for polymer mass concentration c = g/cm 3. The nominal experimental Deborah number in the exponential diameter decrease regime is De = 2/3, and the theoretical De = As explained in Appendix A, the empty-circle symbols correspond to the spurious experimental Trouton ratio values during the initial stages of the experiment. The theoretical results represent averages over 330 realizations. employed in the dilute case. Moreover, since the measurements during the initial experiment stage are now reliable, we have excellent agreement in the linear diameter decrease regime too. Notably, in opposition to the dilute case, there is no evidence of Trouton ratio saturation at high Hencky strains. As depicted in Fig. 7(right), the semi-dilute solution computations show a greater sensitivity of the results on the Deborah number value of the exponential diameter decrease regime. This is inferred by the larger difference between the De = 0.76 and De = 0.72 predictions in the latter case. Fig. 8. Experimental (points) and theoretical (solid lines) apparent Trouton ratios Tr a versus Hencky strain for polymer mass concentration c = g/cm 3. The nominal experimental Deborah number in the exponential diameter decrease regime is De = 2/3, and the theoretical De = The theoretical De numbers match the inferred experimental counterparts in the linear diameter decrease regime. The theoretical results represent averages over 330 realizations. would be of importance in future comparisons between experiment and theory, and (c) these data facilitate the comparison with the corresponding new results for equibiaxial and ellipsoidal flows. Finally, although only mean quantities are measured in ordinary polymeric fluid dynamics experiments, we find it advantageous to analyse single, particularly important molecular evolutions in order to isolate and illustrate the controling factors of polymer stretch rate, and interact with possible future ultra-high strain rate single PEO chain experiments similar to those performed in the context of DNA macromolecules [9 2]. 6. High strain rate extensional flow predictions Next, we investigate the physics of dilute, aqueous PEO solutions under extreme extensional flow conditions. We mainly need to know the rate at which chains extend in such flows, the factors controling their extension rate, and the elastic stress levels sustained by the fluid. This information is an important first step in our effort to understand the high-speed droplet breakup experiments mentioned in the introduction. We compute uniaxial, equibiaxial and ellipsoidal flows. Although the first case has already been extensively discussed in the literature, we do present some key results here because (a) the particular combination of ultra high strain rates with aqueous PEO solutions has not been studied before, (b) the actual anisotropic polymer stress and extensional viscosity values Uniaxial extensional flow We have performed 00 computations of uniaxial extensional flow at De = 540 (De 0 = 0.8) by simultaneously varying Brownian motion random number sequences and initial conditions. Fig. 9 shows the average normalized PEO-chain extension (middle-curve) x /L versus time. The fastest and slowest chain-extension realizations in the computational ensemble are also shown there. The slowest extension rate graph presents a remarkable plateau around x /L 0.5. In this case, full polymer extension takes place at t s which is five times larger than the observed fastest time for full extension. The formation and subsequent relaxation of chain-folds responsible for this pathological behaviour is depicted in Fig. 0. The results of Fig. 9 also show that, even at high strain Fig. 7. Theoretical results for De = 0.76 (upper curve), De = 0.72 (middle curve), and De = 2/3 (lower curve) in the exponential diameter decrease regime for polymer mass concentration c = g/cm 3 (left), and c = g/cm 3 (right). All results represent averages over 330 realizations. In all cases, the initial, linear diameter decrease regime is identical, and only the constant strain rate in the exponential regime is appropriately adjusted.

9 D. Kivotides et al. / J. Non-Newtonian Fluid Mech. 6 (2009) at smaller De values. Finally, Fig. (right) shows the evolution of Tr a with Hencky strain for De = 540 (De 0 = 0.8) Equibiaxial extensional flow Fig. 9. Average (middle curve) normalized polymer extension /L versus time for aqueous PEO solution in uniaxial extensional flow with Deborah number De = 540 (De 0 = 0.8). The fastest and slowest chain-extensions amongst a computational ensemble of 00 realizations are also shown. rates, Brownian motion effects are very important. This conclusion was further supported by additional computations performed at De 080 (De 0 =.6). In order to obtain stress tensor values of realistic order of magnitude, we choose to be equal to the overlap polymer number density = molecules/cm 3. Fig. (left) shows the steady state apparent Trouton ratio Tr a versus De. Evidently, for high strain rates with De 3, Tr a approaches a constant value and the liquid resembles an anisotropic Newtonian fluid with the corresponding terminal extensional viscosity. By theoretical analysis of simplified models of breakup processes, Renardy [60] and Fontelos & Li [6] have shown that, in this case, the rupture of a viscoelastic thread could occur via a self-similar necking process in which the filament radius decays linearly with time, in contrast to a power law decay for viscoelastic fluid rupture First, we note that an equibiaxial, extensional, solvent flow field (as specified by Eq. ()) has no preferred direction on the x y plane. That is, at the same distance r along every polar direction, there is an outgoing velocity (relative to the origin of the coordinate system) of constant magnitude equal to r. This suggests that all polar directions are equiprobable during polymer extension. We quantify polymer alignment with the x-axis as follows: (a) we project every spring vector on the x y plane, (b) we compute the angle between this projection and either the positive or negative x-axis so that 0 /2, (c) we average over the springs that comprise the chain and obtain the mean value s over a single chain, (d) we average s over all performed realizations of chain extensions to obtain the macroscopically useful average over many chains m. Therefore, according to this definition, it must be m = /4 rad in equibiaxial extension. This should be the case not only during the steady state, but throughout the evolution of the average polymer chain. In order to directly compare equibiaxial and uniaxial extensional flow results, we have performed 00 equibiaxial extensional flow computations at De = 540 (De 0 = 0.8) with different (equilibrium) initial conditions and random number sequences. Fig. 2 (right) shows that, as dictated by the symmetry of the problem, the average (over both springs and realizations) angle of alignment between the polymer and the x-axis m is to a very good approximation constant and equal to m = 45 at all times. The evolution of the average normalized polymer extension l /L is shown on Fig. 2(left). The fastest and slowest chain-extensions amongst a computational ensemble of 00 realizations are also shown there. A comparison with the corresponding results of Fig. 9 indicates that the polymer stretches faster in the equibiaxial case. Fig. 0. (Colour online.) PEO-chain stretching configurations at De = 540 (De 0 = 0.8) for the slowest extension case of Fig. 9. Time increases from the upper-left graph towards the lower-right graph as follows: (a) t a = 0s, (b) t b = s, (c) t c = s, (d) t d = s. The first two graphs depict the whole molecule, while the last two are detailed views. Initially, (a) and (b), the chain is stretched and compressed fast enough to present frozen reversals as shown in (c). At the plateau of Fig. 9, the chain consists of two approximately equal parallel segments. The latter unfold by sliding relative to each other as shown in (d). Fig.. Left: Steady state apparent Trouton ratio Tr a versus Deborah number De for aqueous PEO solutions with overlap polymer mass density in uniaxial extensional flow. Right: Apparent Trouton ratio Tr a versus Hencky strain for PEO solution subject to uniaxial extensional strain at De = 540 (De0 = 0.8).

10 78 D. Kivotides et al. / J. Non-Newtonian Fluid Mech. 6 (2009) Fig. 2. Left: Average (middle curve) normalized polymer extension l /L versus time. The fastest and slowest chain-extensions amongst a computational ensemble of 00 realizations are also shown. Right: Average (over both springs and realizations) angle m (in degrees) between spring-vectors and the x-axis versus time. Results refer to aqueous PEO solutions subject to equibiaxial extensional strain at De = 540 (De 0 = 0.8). Fig. 3. (Colour online). Clarification of a typical extension-rate plateau observed in a particular realization of a PEO-chain in equibiaxial stretching flowatde = 540 (De 0 = 0.8). Top-left: Polymer extensions along the two stretching axes x/l (middle curve) and y/l (lower curve), and maximum bead-distance within the polymer l/l (upper curve) versus time during a slow polymer extension process. Top-right: Corresponding average (over springs) angle s (in degrees) between spring-vectors and the x-axis versus time. The graphs in Fig. 3(a) (f) show the corresponding chain configurations. Time increases from the middle-left graph towards the lower-right graph as follows: (a) t a = 0 s, (b) t b = s, (c) t c = s, (d) t d = s, (e) t e = s. (f) t f = s. The vertical line signifies the x-axis. The other line denotes the y-axis. In order to depict the whole polymer, the spatial scale changes from one graph to another. The polymer collapses onto the x y plane and subsequently folds into two segments forming an approximately s 60 o angle with the x-axis. Due to equibiaxial stretching, the fold relaxes eventually into a straight line configuration that forms a s 25 o angle with the x-axis.

11 D. Kivotides et al. / J. Non-Newtonian Fluid Mech. 6 (2009) Fig. 4. Left: Apparent Trouton ratio Tr a versus Hencky strain. Right: Anisotropic polymer contribution to the stress 2 p versus Hencky strain. The curves are averages over a 22 sample of 00 realizations. Results refer to aqueous PEO solutions subject to equibiaxial extensional strain at De = 540 (De 0 = 0.8). Since, on the average, the fully stretched polymer forms angle m = 45 0 with the extension plane axes, Tr a and p have similar values. This is plausible, since, in the latter case, the flow equally favours an infinite number of directions for the full extension of the polymer, while in the uniaxial case there is only one possible such direction. In order to understand better the controling factors of polymer extension rate, we analyse typical plateaus (similar to that depicted in Fig. 2) that appear in pathologically slow chain extensions. Let us first define, in addition to polymer extension x along the x-axis and polymer extension y along the y-axis, the maximum distance l between any two beads in the chain. The latter is the appropriate diagnostic of polymer stretching in equibiaxial stretching solvent velocity fields. Fig. 3 (top, left) shows x/l (intermediate curve), y/l (lower curve) and l/l (upper plateau) versus time in a sample polymer extension in equibiaxial flow that presents a distinct plateau around the value l = 0.5L. At later times, y/l remains fairly constant and approximately equal to y/l = 0.4 whilst x/l shows significant dynamics. Moreover, Fig. 3(top, right) shows that during the period of the y/l plateau, s changes from a value s 60 that indicates a tendency for polymer alignment with the y-axis to s 25 that indicates a tendency for alignment with the x-axis. This phenomenology is explained in Fig. 3(a) (f). In particular, Fig. 3(a) and (b) shows how the initially three-dimensional chain configuration collapses onto the x y plane due to compression along Fig. 5. Projection on the x y plane of a family of streamlines that start along a circle positioned on a plane normal to the z axis at a certain distance from the origin. Left: Ellipsoidal extensional flow for which f = 0.5 in the rate of strain matrix of Eq. (). Right: Extensional flow with f = 0.2. Fig. 6. Left: Average (middle curve) normalized polymer extension x /L along the x-axis versus time. The fastest and slowest chain-extensions amongst a computational ensemble of 00 realizations are also shown. Right: Average (over springs and realizations) angle m (in degrees) between spring-vectors and the x-axis versus time. Results refer to aqueous PEO solutions subject to ellipsoidal extensional strain at De = 540 (De 0 = 0.8).

12 80 D. Kivotides et al. / J. Non-Newtonian Fluid Mech. 6 (2009) the z-axis. In Fig. 3(c), the polymer chain is (modulo thermal fluctuations) a two dimensional curve, that, due to the fast collapse dynamics, retains some of the randomness present in the initial conditions. The latter is evident in the polymer contour fluctuations of Fig. 3(c). As time proceeds, the combined thermo-kinematic effects smooth the polymer that folds into two equal, almost parallel segments that form an approximately s 60 angle with the x-axis. The two equal segments chain configuration shown in Fig. 3(d) explain the plateau of Fig. 3(top, left) at l/l 0.5. In the uniaxial case, the similar fold of Fig. 0 relaxed by having its two segments slide relative to each other whilst retaining, due to the two compressive flow directions, their orientation in space. In the equibiaxial case, the mechanism of relaxation, as shown in Fig. 3(e), is different. Small deviations from perfect alignment are now amplified, and the chain settles to a permanent orientation only after the fold has relaxed fully. The final state depicted in Fig. 3(f) forms a s 25 angle with the x-axis. It is conceivable that Brownian motion induced chain fluctuations could cause a slow rotation of the chain s orientation with time. However, from t. 0 5 s, when the polymer is first stretched fully, to t s, when the computation ends, s presents only random fluctuations around a mean value without any discernible drift. Finally, Fig. 3 clarifies also the constancy of y/l over a period of strong x/l dynamics. During the early stages of the y/l plateau, the polymer is folded in two segments and s 60 so that y 0.5L sin 60. During steady state, the chain is fully stretched and s 25 so that y L sin 25. Hence, since sin sin 60, the y-axis chain extensions in the two cases are the same whilst the corresponding polymer configurations differ significantly. Finally, we have computed Tr a (Fig. 4, left), and p (Fig. 4, right) evolutions. As expected, the Trouton ratio reaches its steady state more quickly than the corresponding uniaxial extension case in Fig. (right). Fig. 7. Evolution of (a) average polymer extension along the x-axis in uniaxial extensional flow (lower, dashed curve), (b) average polymer extension along the x-axis in ellipsoidal extensional flow (middle, solid curve), and (c) average polymer extension (along any direction) in equibiaxial extensional flow (upper, dotted curve). Results refer to aqueous PEO solutions, and Deborah number De = 540 (De 0 = 0.8). The averaging was performed over 00 flow realizations Ellipsoidal extensional flow Many extensional flows involve asymmetric stretching along two orthogonal directions. In the experiments mentioned in the introduction for example, such cases could arise due to deviations of the initial droplet geometry from the geometry of the perfect sphere. Since every degree of asymmetry defines a separate flow, there are an infinite number of asymmetric-stretching flows. For illustration, we choose to study here the canonical ellipsoidal case [28], where the fluid is stretched along one direction with twice the Fig. 8. (Colour online.) PEO-chain ellipsoidal stretching configurations at De = 540 (De 0 = 0.8) for the slowest, sample extension process presented in Fig. 6. Time increases from the upper-left graph towards the lower-right graph as follows: (a) t a = 0 s, (b) t b = s, (c) t c = s, (d) t d = s, (e) t e = s. (f) t f = s. The vertical line signifies the x-axis. The other line denotes the y axis. In order to depict the whole polymer, the spatial scale changes from one graph to another. The polymer collapses onto the x y plane and subsequently evolves into a two-fold shape, with two peaks along the axis of strongest extension and a connecting valley in between them. Later, the valley is tranformed into a peak, and the polymer becomes a circular arc before stretching fully whilst reorientating itself along the direction of the strongest extension rate.

13 D. Kivotides et al. / J. Non-Newtonian Fluid Mech. 6 (2009) Fig. 9. Left: Apparent Trouton ratios Tr a (upper curve) and Tra (lower curve) versus Hencky strain. In agreement with Newtonian fluid mechanics, the initial 2 Tra value is Tr a = 5, and the initial Tra value is 2 Tra = 4. The steady state 2 Tra value is identical to the steady state Tra value in the uniaxial case. The maximum Tra value is approximately 2 two orders of magnitude smaller than the maximum Tr a value. Right: Anisotropic polymer contribution to stress p (upper curve) and p (lower curve). Results refer to 22 aqueous PEO solutions subject to ellipsoidal extensional strain at De = 540 (De 0 = 0.8). The averaging was performed over 00 flow realizations. strength with which it is stretched along the other. Fig. 5 shows projections on the x y plane of a family of streamlines that start along a circle positioned on a plane normal to the z-axis at a certain distance from the origin. The left graph presents streamlines in an ellipsoidal extensional flow for which f = 0.5 in the rate of strain matrix of Eq. (), while the right presents streamlines in an extensional flow with f = 0.2. The ellipsoidal flow is an interpolation between the equibiaxial extensional flow where the streamlines are along the radial direction, and the planar extensional flow (f = 0) where they remain on the x z plane. Analysis of a sample of 00 flow realizations at De = 540 (De 0 = 0.8) has shown that, as expected, the polymers always align eventually with the x-axis. Fig. 6 (right) shows the rate with which the average angle m of alignment between the polymer and the x-axis tends to zero. Initially, for smaller times than 0, there is a linear decrease of m. Throughout this regime, the elastic energy stored in the springs is not yet dynamically significant. The linear decay regime is followed by a regime of even steeper m decay. This suggests that, in this intermediate regime, the growth of elastic energy favours the alignment between polymers and major stretching axis. Finally, for small m values, m < 0, there is a slower m decay. According to Fig. 7, the polymers extend in the ellipsoidal case slightly faster than in the uniaxial case. This is because, in the uniaxial case, the rapid collapse of the polymers along the x-axis results in frozen chain reversals that require time in order to unravel, whilst, in the ellipsoidal case, the additional extensional strain along the y-axis allows a faster fold unraveling. A notable feature of Fig. 6(left) is that individual polymer stretching processes can differ from the mean process much more distinctly in the ellipsoidal flow case than in the other two flow cases. The polymer configurations during the pathological slowest case of Fig. 6 are depicted in Fig. 8. As shown there, the relatively long period during which the polymer s extension along the x-axis is small and approximately constant corresponds to the evolution of the chain from a two fold configuration into a circular arc. Our results indicate that such two fold configurations are unique to the ellipsoidal flow case. Fig. 9 shows Tr a and p evolutions. As expected, Tr a and p steady state values are identical to the ones for uniaxial extensional flow. The smallest stretching rate along the y-axis causes the maximum values of Tr a 2 and p to be two orders of magnitude smaller 22 than the corresponding maximum values of Tr a and p. 7. Conclusion In order to probe the physics of viscoelastic liquid break up in high speed, high strain rate flows, we have elaborated on a link between mesoscopic polymer dynamics and macroscopic viscoelastic fluid behaviour. By modeling the solvent as continuum fluid and the polymers as bead-spring chains, we have computed dilute polymeric flows with homogeneous strain rate fields. The analysis was based on a computational model of the PEO polymer that (a) predicts a coil-stretch transition at De 0.5, (b) matches the experimental longest relaxation time, (c) predicts a plausible average chain end to end distance under no flow conditions, and (d) agrees with experimental transient extensional viscosity data at small De numbers. At high De numbers, three prototype extensional flows of dilute aqueous PEO solutions were considered: uniaxial, equibiaxial and ellipsoidal strain rate fields. The predictions of the model suggest that polymer stretching is controlled by (a) the randomness of the initial conditions that in combination with rapid kinematic compression lead to the formation of initially frozen chain-folds, and (b) the speed with which thermo-kinematic processes relax these folds. Due to kinematic compression along two space directions, the slowest fold relaxation occurs during uniaxial extension. The introduction of stretching along a second direction enhances the efficiency of fold relaxation mechanisms. However, one needs to consider ratios of stretching rates along the two axes well beyond the one half ratio of ellipsoidal flow and towards the unity ratio of equibiaxial flow in order to observe a distinct effect. Even for De numbers as high as De 080 (De 0 =.6), there is a large variation of the time a polymer needs in order to extend fully, and the effects of Brownian motion can not be ignored. This is due to the aforesaid synergy between thermal and kinematic effects in the context of fast chain-fold relaxation. The computed Trouton ratios and polymer contributions to the total stress as functions of Hencky strain provide information about the relative importance of elastic effects during stretching processes in polymer solutions. At high strain rates (De > 4), the steady state elastic stresses increase linearly with De. This suggests that the extensional viscosity approaches a terminal constant value resembling a material property, and that the fluid behaves like an anisotropic Newtonian fluid. In the ellipsoidal flow case, the maximum extensional viscosity corresponding to the direction of weaker stretching is two orders of magnitude smaller than the steady state extensional viscocity corresponding to the direction of major stretching. In future extension of this work, we plan perform a systematic study of concentration and chain-entanglement effects [62] by computing polymer dynamics in confined systems, with appropriate treatment of boundary conditions [63,6] and bond crossings prevention [64,65]. Equally desirable would be the introduction of multi-fluid effects by modeling material interfaces and surfacetension processes. Moreover, for sufficiently dense polymer tangles, the back-reaction of the polymers on the large scale solvent flow

14 82 D. Kivotides et al. / J. Non-Newtonian Fluid Mech. 6 (2009) needs also to be taken into account. We plan to develop genuinely fully coupled numerical schemes in order to capture such physics. Finally, in the related topic of worm-like micellar solutions, Rothstein [66] performed extensional rheology experiments with emphasis on filament rupture. He concluded that micellar tubes can be usefully modeled by FENE type elasticity. Moreover, since micellar tubes are held together by hydrophobic bonds rather than covalent polymer bonds, they are capable of rupture and recombination [67 69]. Hence, incorporating reconnections [70,7] in the model employed here would allow a theoretical study of extensional rheometry of worm-like micellar fluids. Acknowledgments We are grateful to Dr. Alexei Rozhkov (Russian Academy of Science) for useful discussions, and to the anonymous reviewers for their constructive questions and comments. This work was supported by the Joint Science and Technology Office, Defense Threat Reduction Agency (JSTO/DTRA), and the National Ground Intelligence Center (NGIC). Appendix A. Filament-thinning rheometry: methods employed The particular quantities of interest for our calculations are the longest relaxation time of the Polyethylene Oxide (PEO) molecule in water, and the Trouton ratio as a function of Hencky strain in elongation. The choice of this system was dictated by our need to find a readily available and environmentally benign system that could be made to simulate the rheology, especially the rupture behavior, of certain Polysterene Butyl Methacrylate (PSBMA) solutions in Tributyl Phosphate (TBP) [3,32]. This need was met empirically, and this in turn created a need for a deep understanding of the microstructural mechanics in this model system a direction of work which this paper is meant to initiate. The characterizations we seek are not available in the literature for aqueous PEO solutions, and although the measurement approach we employ, based on strains created by the elongational flow of a filament (the filament-thinning rheometer, FTR, idea), is quite well documented [72,73], it was felt that complete clarity demanded some brief elaboration. As a point of reference we cite the work of Larson and coworkers in reference []; it was conducted in the same vein as here. It makes use of the experimental study of reference [23] who advanced the FTR idea so as to access Deborah numbers larger than the 2/3 value demonstrated by the original work on FTR [72]. More specifically, the data shown in reference [23] are for the Polystyrene system and cover the range.3 <De<.3. A key difference is that in reference [23] the FTR operates at a constant strain rate (within operating limits of the apparatus), while in the capillaritycontrolled mode used here there is a transition from a linear to an exponential regime in the rate of thinning. This transition results in a sudden jump of the Trouton ratio (due to a sudden drop in strain rate), then followed by a rather well defined (exponential) rate of growth for the rest of the filament life this period is characterized by a constant strain rate with De = 2/3, which thusly yields also the longest relaxation time (in a single-mode Oldroyd B model interpretation). Another key difference from reference [23] is that our system, especially in the desired extreme of infinite dilution, is much less viscous than Polystyrene, which creates certain operational difficulties as described below. As we will see these differences avail of certain opportunities for comparison with computations which are complementary to those in the constant rate FTR utilized in references [,23]. Quite simply our experiment involves the sudden separation of two, 5 mm in diameter discs, from an initial separation of.0 mm to a final distance of 4 mm. This separation is made with a controlled speed over approximately 7 ms, and subject to capillary forces the filament thins out by draining towards the two ends as illustrated by sample sequences in Figs. A. and A.2. These images were obtained from high resolution (85 pixels per mm) video records at 20 khz, and the reduced data consist of the variation of filament diameter with time, as illustrated in Figs. A.3(a) and A.4(a). The sudden transitions evident in these figures demarcate the transition from a linear to an exponential regime, and the latter has been shown to be characterized (based on the Oldroyd B model) by a constant Deborah number value of 2/3. Thus an estimate of the filament-thinning rate in this regime can be related to a strain rate, and thereby one can obtain the value of the longest relaxation time. The apparent (elongational) viscosity can be found from the stress in the axial direction. This is readily obtained from the surface tension of the solution (it was measured to coincide to that of pure water in our case), and the elongational strain rate. The relevant formulae are: e = d ln (d (t)), (A.) 3 dt app = ḋ, (A.2) 2w (d (t) = 0 wt), t < t, 2 (A.3) (3 e ), t t, d 0 wt, ( ) t < t, d(t) = (d 0 wt )exp t t 3 e, t t (A.4), { 2lnd0 2lnd(t), t < t, (t) = 2lnd 0 2lnd(t ) + 2(t t ) 3 e, t t (A.5), where e is the longest relaxation time, d is the filament diameter, t is the time, app is the apparent viscosity, w is the rate of diameter decrease in the linear regime, is a surface tension of tested liquid, is the Hencky strain, d 0 is the diameter at t = 0, and t is the Fig. A.. Selected images of filament thinning in FTR; PEO 0.025%: (a) at the moment of maximum separation, taken as time zero; (b d) at times t = 5, 8, and ms. Note that time zero in this sequence corresponds to time 3.3 ms before the beginning of time in Fig. A.3, and the measured longest relaxation time is 0.88 ms.

15 D. Kivotides et al. / J. Non-Newtonian Fluid Mech. 6 (2009) Fig. A.2. Selected images of filament thinning in FTR; PEO 0.%: (a) at the moment of maximum separation, taken as time zero; (b d) at times t = 4, 7, and ms. Note that time zero in this sequence corresponds to time.3 ms before the beginning of time in Fig. A.4, and the measured longest relaxation time is.7 ms. Fig. A.3. Filament-thinning transient (a) for a M PEO solution of 0.025%; (b and c) corresponding Deborah number and Trouton ratio transients. The arrows in (a) mark the times of the last three images in Fig. A.. The same picture shows also the linear and exponential data-fits. Note: the stepwise variation of diameter near the end of the exponential regime is due to the resolution limit employed in visualization, but this has a minimal effect on the determination of the rate of thinning which needed in data reduction. transition time between the linear and exponential regimes. We note that this transition happens because initially, and for small Hencky strain values, the polymer chains have not yet stretched significantly, so one observes the dynamics of a Newtonian viscous fluid which is responsible for the linear diameter decrease regime of Eq. (A.4). At times close to t, the extensional viscosity starts deviating significantly from its Newtonian fluid value, and the viscoelastic physics become apparent by causing the exponential diameter decrease regime of Eq. (A.4). Thus the t time scale is set by the rate of chain extension. In implementation, the derivative is calculated from analytical fits to the two regions of the transient, Eq. (A.4), as illustrated in Figs. A.3 and A.4(a). Tested liquids were PEO, M molar mass (supplied by Aldrich), solutions in deionized water with concentrations from 0.025% to 4%. A solution of concentration 0.5% was used to prepare solutions with smaller concentrations. Water and PEO were mixed by gentle stirring over 24 h and were used within two days to avoid any degradation. The results of the Deborah number and Trouton ratio transients for the two sample cases detailed here are shown in Figs. A.3(b and c) and A.4(b and c). The relaxation times over the full range of concentrations covered are shown in Fig. 3 of the main text. It is noted that the data with the lowest concentration solution exhibit somewhat erratic behavior; namely, a small kink in the trend at about ms before the transition, and a near-constant apparent viscosity over the duration of the linear regime. This is due to the very low viscosity of this solution it is insufficient to damp oscillations of the liquid mass retained on the lower plate (these are transmitted and affect filament stretching during most of the linear period), and moreover it is responsible for asymmetric draining during and immediately after the separation time, which is also evident in Fig. A.(b). While this gives rise to errors (in the linear regime only) that are hard to quantify, as we show in the main text (Section 5) a 20% increase in the estimated rate of strain from the experiment provides sufficient preconditioning for predicting both the magnitude of the jump and the following exponential growth in Trouton ratio quite adequately. We can also see that these errors are rather insignificant at a concentration of 0.%, even though the development is not as perfect as found at still higher viscosities, as for example with the % solution shown in Fig. A.5. Even though the lowest viscosity case is right up to the limits of FTR, it is important here in that it is well into the infinite dilution limit. Moreover the computations show that variations in Trouton ratio with changes in Fig. A.4. Filament-thinning transient (a) for a M PEO solution of 0.%; (b and c) corresponding Deborah number and Trouton ratio transients. The arrows in (a) mark the times of the last three images in Fig. A.2. The same picture shows also the linear and exponential data-fits. Note: the stepwise variation of diameter near the end of the exponential regime is due to the resolution limit employed in visualization, but this has a minimal effect on the determination of the rate of thinning which needed in data reduction. The measured longest relaxation time is.7 ms.

16 84 D. Kivotides et al. / J. Non-Newtonian Fluid Mech. 6 (2009) Fig. A.5. Selected images of filament thinning in FTR; PEO %: (a) at the moment of maximum separation, taken as time zero; (b d) at times t = 20, 50, and 80 ms. concentration right up to near the dilute limit (0.25%) are coherently captured by our model. References [] C. Hsieh, L. Li, R.G. Larson, Modeling hydrodynamic interaction in Brownian dynamics: simulations of extensional flows of dilute solutions of DNA and polystyrene, J. Non-Newtonian Fluid Mech. (2003) 47. [2] R.M. Jedrejack, M.D. Graham, J.J. de Pablo, Hydrodynamic interactions in long chain polymers: application of the Chebyshev polynomial approximation in stochastic simulations, J. Chem. Phys. 3 (2000) [3] M. Kröger, A. Alba, M. Laso, H.C. Öttinger, Variance reduced Brownian simulation of a bead-spring chain under steady shear flow considering hydrodynamic interaction effects, J. Chem. Phys. 3 (2000) [4] R.G. Larson, H. Hu, D.E. Smith, S. Chu, Brownian dynamics simulations of a DNA molecule in an extensional flow field, J. Rheol. 43 (999) 267. [5] J.S. Hur, E.S.G. Shaqfeh, R.G. Larson, Brownian dynamics simulations of single DNA molecules in shear flow, J. Rheol. 44 (2000) 73. [6] R.M. Jendrejack, E.T. Dimalanta, D.C. Schwartz, M.D. Graham, J.J. de Pablo, DNA dynamics in a microchannel, Phys. Rev. Lett. 9 (2003) [7] C. Stoltz, J.J. de Pablo, M.D. Graham, Concentration dependence of shear and extensional rheology of polymer solutions: Brownian dynamics simulations, J. Rheol. 50 (2006) 37. [8] R.G. Larson, Hydrodynamics of a DNA molecule in a flow field, Phys. Rev. E 55 (997) 794. [9] Y.-L. Chen, M.D. Graham, J.J. de Pablo, G.C. Randall, M. Gupta, P.S. Doyle, Conformation and dynamics of single DNA molecules in parallel-plate slit microchannels, Phys. Rev. E 70 (2004) (R). [0] T.G. Theofanous, R.R. Nourgaliev, G.J. Li, T.N. Dinh, Compressible multihydrodynamics (CMH): breakup, mixing, and dispersal, of liquids/solids in high speed flows, in: A. Prosperetti, S. Balachandar (Eds.), Computational Approaches to Disperse Multiphase Flow, Springer, Berlin, [] D.L. Ermak, J.A. McCammon, Brownian dynamics with hydrodynamic interactions, J. Chem. Phys. 69 (978) 352. [2] M. Fixman, Construction of Langevin forces in the simulation of hydrodynamic interaction, Macromolecules 9 (986) 204. [3] H.C. Öttinger, Stochastic Processes in Polymeric Fluids, Springer, Berlin, 996. [4] M. Herrchen, H.C. Öttinger, A detailed comparison of various FENE dumbbell models, J. Non-Newtonian Fluid Mech. 68 (997) 7. [5] M. Somasi, B. Khomami, N.J. Woo, J.S. Hur, E.S.G. Shaqfeh, Brownian dynamics simulations of bead-rod and bead-spring chains: numerical algorithms and coarse-graining issues, J. Non-Newtonian Fluid Mech. 08 (2002) 227. [6] A.J. Banchio, J.F. Brady, Accelerated Stokesian dynamics: Brownian motion, J. Chem. Phys. 8 (2003) [7] P.S. Grassia, E.J. Hinch, L.C. Nitsche, Computer simulations of Brownian motion of complex systems, J. Fluid Mech. 282 (995) 373. [8] P.S. Grassia, E.J. Hinch, Computer simulations of polymer chain relaxation via Brownian motion, J. Fluid Mech. 308 (996) 255. [9] T.T. Perkins, D.E. Smith, S. Chu, Single polymer dynamics in an elongational flow, Science 276 (997) 206. [20] D.E. Smith, S. Chu, Response of flexible polymers to a sudden elongational flow, Science 28 (998) 335. [2] C.M. Schroeder, H.P. Babcock, E.S.G. Shaqfeh, S. Chu, Observation of polymer conformation hysteresis inelongational flow, Science 30 (2003) 55. [22] B. Ladoux, P.S. Doyle, Stretching tethered DNA chains in shear flow, Europhys. Lett. 52 (2000) 5. [23] N.V. Orr, T. Sridhar, Probing the dynamics of polymer solutions in extensional flow using step strain experiments, J. Non-Newtonian Fluid Mech. 82 (999) 203. [24] T. Sridhar, D.A. Nguyen, G.G. Fuller, Birefrigence and stress growth in uniaxial extension of polymer solutions, J. Non-Newtonian Fluid Mech. 90 (2000) 299. [25] R.K. Gupta, D.A. Nguyen, T. Sridhar, Extensional viscosity of dilute polystyrene solutions: effect of concentration and molecular weight, Phys. Fluids 2 (2000) 296. [26] G.H. McKinley, T. Sridhar, Filament-stretching rheometry of complex fluids, Annu. Rev. Fluid Mech. 34 (2002) 375. [27] C.J.S. Petrie, One hundred years of extensional flow, J. Non-Newtonian Fluid Mech.37(2006). [28] C.J.S. Petrie, Extensional viscosity: a critical discussion, J. Non-Newtonian Fluid Mech. 37 (2006) 5. [29] L. Li, R. Larson, T. Sridhar, Brownian dynamics simulations of dilute polystyrene solutions, J. Rheol. 44 (2000) 29. [30] L. Li, R. Larson, Excluded volume effects on the birefringence and stress of dilute polymer solutions in extensional flow, Rheol. Acta 39 (2000) 49. [3] V.V. Mitkin, A.N. Rohzkov, T. G. Theofanous, Pulse jets, rims and elastic-liquid sheets: rheology at high strain rates and rupture criteria, Proceedings of XVth International Congress on Rheology, 2008 (ICR Paper#CF-2). [32] C-L Ng, T.G. Theofanous, Modes of Aero-breakup with viscoelastic liquids, Proceedings of XVth International Congress on Rheology, 2008 (ICR Paper#CF-2). [33] T.G. Theofanous, G.J. Li, The physics of aerobreakup, Phys. Fluids 20 (2008) [34] M. Rubinstein, R.H. Colby, Polymer Physics, Oxford University Press, Oxford, [35] T. Kawakatsu, Statistical Physics of Polymers, Springer, Berlin, [36] A.N. Rozhkov, V.V. Mitkin, T.G. Theofanous, The dynamics of visco-elastic bridges in drop coalescence, in: XXII ICTAM, [37] A.V. Bazilevskii, V.M. Entov, A.N. Rozhkov, Breakup of an Oldroyd liquid bridge as a method for testing the rheological properties of polymer solutions, Polym. Sci., Ser. A (200) 76. [38] C. Clasen, J.P. Plog, W.-M. Kulicke, M. Owens, C. Macosko, L.E. Scriven, M. Verani, G.H. McKinley, How dilute are dilute solutions in extensional flows? J. Rheol. 50 (2006) 849. [39] I. Snook, The Langevin and Generalised Langevin Approach to the Dynamics of Atomic, Polymeric and Colloidal Systems, Elsevier Science, Amsterdam, [40] R.M. Mazo, Brownian Motion: Fluctuations, Dynamics, and Applications, Oxford University Press, Oxford, [4] W.T. Coffey, Yu.P. Kalmykov, J.T. Waldron, The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering, World Scientific, Singapore, [42] M. Gitterman, The Noisy Oscillator, World Scientific, Singapore, [43] J. Rotne, S. Prager, Variational treatment of hydrodynamic interactions in polymers, J. Chem. Phys. (969) 483. [45] R.M. Jedrejack, M.D. Graham, J.J. de Pablo, Stochastic simulations of DNA in flow: dynamics and the effects of hydrodynamic interactions, J. Chem. Phys. 6 (2002) [46] C. Hsieh, S. Jain, R.G. Larson, Brownian dynamics simulations with stiff finitely extensible nonlinear elastic-fraenkel springs as approximations to rods in bead-rod models, J. Chem. Phys. 24 (2006) [48] R.M. Jendrejack, D.C. Schwartz, M.D. Graham, J.J. de Pablo, Effect of confinement on DNA dynamics in microfluidic devices, J. Chem. Phys. 9 (2003) 65. [49] A. Rey, J.J. Freire, J.G. de la Torre, Brownian dynamics simulation of flexible polymer chains with excluded volume and hydrodynamic interactions. A comparison with Monte Carlo and theoretical results, Polymer 33 (992) [50] J.G.H. Cifre, J.G. de la Torre, Steady-state behavior of dilute polymers in elongational flow. Dependence of the critical elongational rate on chain length, hydrodynamic interaction, and excluded volume, J. Rheol. 43 (999) 339. [5] R.M. Jendrejack, D.C. Schwartz, J.J. de Pablo, M.D. Graham, Shear-induced migration in flowing polymer solutions: simulation of long-chain DNA in microchannels, J. Chem. Phys. 20 (2004) 253. [52] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes in FORTRAN 77: The Art of Scientific Computing, Cambridge University Press, Cambridge, 992. [53] M. Kröger, Models for Polymeric and Anisotropic Liquids, Springer, Berlin, [54] R.G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press, Oxford, 999. [55] R.G. Larson, J.J. Magda, Coil-stretch transitions in mixed shear and extensional flows of dilute polymer solutions, Macromolecules 22 (989) [56] L.E. Rodd, T.P. Scott, D.V. Boger, J.J. Cooper-White, G.H. McKinley, The inertioelastic planar entry flow of low-viscosity elastic fluids in micro-fabricated geometries, J. Non-Newtonian Fluid Mech. 29 (2005). [57] R.B. Bird, C.F. Curtiss, R.C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, Wiley-Interscience, New York, 989.