On the stability of grafted polymer molecules in elongational flows

ELSEVIER

J. Non-Newtonian

Fluid Mech., 54 (1994) l-10

On the stability of grafted polymer molecules in elongational flows * U.S. Agarwal

‘, R.A. Mashelkar

*

National Chemical Laboratory, Pune 411008, India

Received 22 March 1994

Abstract

In recent literature relating to drag reduction by polymeric additives, it has been purported that grafting side chains is a way of enhancing their shear stability. A careful analysis of the experimental reports shows that grafting in fact reduces the shear stability of long-chain polymers. A simple mechanistic analysis of the fracture of an ideal grafted chain molecule has been performed to support this finding. Some guidelines for the design of stable and effective drag reducing macromolecules are proposed. Keywords:

Shear stability; Grafted polymers; Drag reduction

1. Introdffction

Long-chain polymers are employed as flow modifying additives, e.g. for drag reduction during fluid transport, fire fighting, irrigation, heating and cooling circuits [l-3], slurry transport [4], controlling friction in high friction systems [5], enhanced oil recovery [6], etc. Often the additive performance is determined by the polymer

* Dedicated to Professor Ken Walters FRS on the occasion of his 60th birthday. NCL communication No. 5739. * Corresponding author. ’ Current Address: Chemical Engineering Department, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India.

0377-0257/94/$07.00 Q 1994 - Elsevier Science B.V. All rights reserved S~~rO377-0257~94)01303-~

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molecular weight. Thus, the utility of a polymer may be limited if the chain scission takes place when the medium undergoes intense flow. Significant research has been done to understand the mechanism of breakage of molecules, especially that of linear long-chain molecules in different hydrodynamic fields. It was hoped that this improved understanding would lead to ‘molecular engineering’ of shear stable polymers. Unfortunately, this expectation has not been fulfilled to the desired extent, and somewhat erroneous approaches to designing such macromolecules have pervaded the literature. In this work, we deal with one such approach adopted by the researchers, namely that of creating grafts on a long-chain molecule with the hope that this will increase its shear stability. Through a simple mechanistic analysis of the fracture of an ideal grafted chain molecule modeled as a fully extended linear n-bead backbone with p bead-spring molecular chains of g springs, we show that grafting actually reduces the stability rather than increasing it, as has been erroneously presumed so far. Different chemical approaches have been used in the literature to build shear stable molecules [7-161. One of them involved building associating polymers. It has been experimentally demonstrated that the shear stability of polymers increases if they form interpolymer [ 1,3,14,15] or intrapolymer [ 121 associations via secondary interactions; these in turn could be electrostatic, hydrogen bonding or hydrophobic interactions. The increased shear stability was attributed to the reversible breakage of secondary bonds in preference to the cleavage of the polymer backbone [ 131. It was believed that these secondary bonds are formed in regions of lower shear, making themselves available again for breakage in the intense flow zones. However, by performing a mechanistic analysis of a model, made up of a perfectly ‘zipped’ assembly of fully extended bead-rod chains, of two macromolecules in elongational flow, we showed [ 171 that the enhancement in shear stability of associating polymers is perhaps due to the formation of a structure leading to additional hydrodynamic shielding, rather than due to the breakage and reforming of physical bonds. Let us consider another strategy adopted by the researchers, namely that of grafting. In the recent literature, particularly in relation to drag reduction [ 1,9,11,16,18,19], it has been purported that branching and grafting side chains to the polymer backbone are beneficial routes for enhancing the shear stability of dilute polymer solutions. This has been attributed to the scission of the side chains in preference to the cleavage of the backbone, so that the reduction in polymer molecular weight corresponds to individual branches only, rather than being one-half when the chain scission takes place in the middle of the origin molecule [8,9,11,16,20]. We now present a simple mechanistic analysis to evaluate the effect of grafting on shear stability. This is supported, qualitatively, by a careful analysis and appropriate comparisons of reported experimental results. In section 2, we first briefly review the current theories for shear induced scission of linear polymers, and then extend the analysis to the case of grafted polymers in steady elongational flow. In section 3, we analyze the earlier reported experimental results for degradation of grafted and branched polymers during turbulent drag reduction. The conclusions are summarized in section 4.

U.S. Agarwal, R.A. Mashelkar / J. Non-Newtonian Fluid Mech. 54 (1994) l-10

3

2. Theory

2.1. Fracture of stretched

linear chain molecule

Based on the hydrodynamic forces on a polymer molecule straightened out in the direction of the flow, Frenkel [21] concluded that the molecule is most likely to break at its center. Bueche [22] arrived at the same conclusion for entangled polymer melts. Levinthal and Davidson [23] calculated and experimentally verified the critical flow rates for the fracture of long thin rodlike molecules during laminar flow through a tube. Zimm and co-workers [24,25] also predicted and verified the mid-chain scission of macromolecules. Over the last decade, Ode11 and co-workers [26,27] have significantly enhanced our understanding of the fracture behavior of linear long-chain molecules in steady elongational flows. They consider an isolated polymer molecule, modeled as a bead-rod chain, to be fully extended Fig. l(a) at a flow strength (C) greater than the critical value (C, z z -‘, where r is the longest relaxation time of the chain molecules) [28,29]. They conclude that the tension is at a maximum in the central bonds of a chain, and is given for a chain of n beads (Fig. l(a))

by F, = S, [b
(1)

where i = 6xya is the Stoke’s friction factor of the bead, n being the solvent viscosity and a being the radius of the beads that are separated by the link rods of length b. S, is the shielding factor ( < 1) incorporated to account for the hydrodynamic interaction between the beads. If this force exceeded the bond strength, then chain scission at the central rods would occur. It is notable that the chain fracture is thus predicted to occur at i, NNn -2, i.e. the minimum elongation rate required for the fracture of a given polymer varies as the inverse square of the molecular weight.

--1Relative

DRAG FORCE

Flow

ON BACKSONE

BEADS

(b)

--

DRAG

Fig. 1. The extended bead-rod chain; (b) grafted chain.

chain

model

FORCE

0

ON GRAFTED

for polymer

--

-

chains

‘BEADS’

in steady

elongational

flow: (a) linear

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Although this and other predictions are in conformity with many experiments [27], alternative mechanisms of chain scission during flow have been also presented in the literature, and the mechanism of chain scission seems to be dependent on the flow conditions [l&30,31]. Since the primary purpose of this communication is to highlight the qualitative effect of grafting on polymer shear stability, and thus provide direction for future macromolecular design for enhanced shear stability, we here employ the simplest analysis of Ode11 and Keller [26] briefly reviewed above. 2.2. Fracture

of the ideal grafted

chain molecule

We model the grafted chain molecule in steady extensional flow as a fully extended linear n-bead backbone, on which are grafted (equally spaced) p beadspring molecular chains of g springs (length) c each (Fig. l(b)). Assuming that the grafted chains are small enough, under the given flow conditions each of them can be approximated to a random coil, with friction factor &, = A6nq,cg0.’ [32,33], and equally spaced bg = bn/p apart on the backbone (Fig. l(b)). Here A is a factor of the order of unity and accounts for the intrabead shielding within the grafted coil. The drag on these ‘grafted beads’ would impart additional tension on the backbone central rods given by AF, = S, &,b,ip 2/8 where S2 < 1 accounts for the shielding effect between total tension in the central rods of the backbone is

(2) the grafted

beads. Thus the

where S, < 1 and Sz < A&/S,, if the shielding effect of the main chain the grafted beads on each other is accounted for.

beads and

3. Discussion The experimental results cited herein to support the above analysis are based on turbulent flow (drag reduction). In this case, although the polymer molecules experience significant extensional flow, the chain extension is perhaps far from complete, and the actual flow situation is very complex [ 3,181. However, we expect the underlying physical idea presented in section 2 to be valid, although realizing that only a qualitative comparison is permissible. Comparing Eqs. ( 1) and (3), it is clear that for a given backbone polymer and flow strength, grafting of the side chains would result in additional tension in the central rods of the backbone, thereby enhancing the shear degradation tendency, or reducing the shear stability of a polymer. During drag reduction application of polymeric additives, the efficiency of the added polymer is determined by examining the percentage reduction (abbreviated to DR) in pressure drop for flow-through geometries such as a flow-through orifice,

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r

0.21 0

I 0.2

I 0.1 Ce(iv)

I 0.3

I o-4

ions (163moles)

Fig. 2. Variation in shear stability (represented by DR,,/D&) of the acrylamide grafted GG (-) and XG (- - -) polymers with the number of grafted side chains (represented by the amount of Ce( iv) ion initiator employed with one gram of GG (or XG) and 0.14 moles of acrylamide during the grafting reaction). Zero Ce(iv) ions corrsponds to the GG (or XG) homopolymer. The polymer concentration during the drag reduction experiments was 100 ppm. Data from Refs. [7] and [S].

capillary flow, etc. The shear stability of the polymer is determined by examining the redaction in drag reduction efficiency over successive flows through the same geometry. Thus, the ratio DR,/DR, (where DR, and DR, are the drag reduction levels for the first pass and the ith pass) is an indication of the shear stability of a polymer. Singh and co-workers [2,7,8,16] have presented many experimental results of the shear stability of polysaccharide guar gum (GG) and xanthan gum (XG), and their graft.copolymers obtained by introducing acrylamide side chains at many sites, using a ceric ion/nitric acid redox initiator. Although, based on the often cited concept of ‘preferential scission of side chains’, they had anticipated enhancement in the shear stability of the drag reducing polymers on grafting, they found that while GG was very shear stable in their drag reduction experiments (capillary flow) over many runs, GG grafted with acrylamide showed considerable reduction in the drag reduction efficiency due to shear degradation [ 81 (Fig. 2). These results support our mechanism presented above. From Eq. (3), for a given backbone and approximating (g, z s,), the additional contribution to the critical tension varies as (c/u)(p/n)g”.5. The shear degradation is thus expected to increase with increasing number of graft points (p) and graft chain lengths (g). This, in fact, is in agreement with the experiment of Deshmukh and

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Singh [8], who varied p and g in five different runs (by varying the amount of initiator Ce(IV) ions determining p during the grafting of acrylamide on GG), but maintained a constant total acrylamide weight, i.e. the product ng =pg had a fixed value. From the above analysis, it follows that the additional tension in the backbone center would vary as (c/a)(n, /n)g -’ 5, i.e. a reduction of shear stability with smaller g (or larger p) is predicted. As seen in Fig. 2, this was found to be the case at two different concentrations of the polymer analyzed by Deshmukh and Singh [2,8] ! However, when the main chain used was XG, the trend in shear stability with g was less apparent [7] (Fig. 2). This is perhaps because in this case the effect of grafting on shear stability was much smaller (Fig. 2) thereby limiting the resolution. This lesser effect of grafting on the shear stability of XG (as compared to GG) was possibly because of the already bulky pentasaccharide repeating units (large a in Eq. (3) for XG); hence, an increase in drag due to acrylamide grafts is not sufficient to significantly enhance the degradation. Kim et al. [9] prepared a star-shaped macromolecule by grafting polyacrylamide (PAM) chains at seven sites of a small nucleus molecule. Our above analysis shows that the drag tension generated at each branch would be transmitted to the nucleus ‘backbone’. Its center would thus experience the maximum tension and would be most prone to degradation. This is contrary to the authors’ belief [9] that scission in the individual branches is more likely, and would thus result in only a small decrease (of the order of the branch length) in the molecular weight of the parent macromolecule. We conclude that the observed higher shear stability of the branched PAM as compared to linear PAM was actually due to branches being shorter than the backbone of the linear PAM. Similarly, although Herold et al. [34] attributed higher shear stability of poly(decy1 methacrylate) than poly( methyl methacrylate) (PMMA) to the longer side chain in the former providing shielding effect to the main chain, it was most likely due to the shorter backbone (degree of polymerization (DP) = 3200) of the former compared to the PMMA (DP = 6900) used by them. Further, it is not surprising that Malhotra [35] failed to see a dependence on side chain length, since he compared a number of poly(alky1 methacrylate) chains of widely different backbone lengths. It follows from our analysis that for the same backbone length, the shear stability would be highest for linear molecules with the minimum number of side chains. The reported [3] higher stability of GG, XG and carboxymethyl cellulose (CMC) with bulky side groups may in fact be due to their smaller backbone lengths which were not compared. Similarly, the commonly observed [9,36-381 faster degradation of linear polyethylene oxide (PEO) compared to PAM (larger a) may be due to the fact that unequal molecular lengths of these polymers were compared. For example, in the experiments of Kenis [36], the starting molecular weights were 7 x lo6 for PEO and PAM, and hence the longer PEO molecules are in fact expected to degrade faster. On application as drag reducing additives, not only shear degradation but also the drag reduction efficiency of the polymers may be influenced by grafting. Linear polymers having no or few side chains are often found to be the most active drag reducers (e.g. PEO, PAM and polyisobutylene, with extended molecular length (1) to diameter (d) ratio Z/dz 104-lo’), while GG, XG, CMC, DNA (Z/d z lo*-103)

U.S. Agarwal, R.A. Mashelkar 1 J. Non-Newtonian Fluid Mech. 54 (1994) I-IO

7

are less effective [ l-3,36,39,40]. Further, polymers lacking linear structure such as branched polysaccharides, gum arabic, and dextrans, are found to be ineffective [38,39]. Similarly, Gramain and Barreill [41] found that star- and comb-shaped polystyrene were not drag reducing, while linear polystyrene of the same total molecular weight was a highly effective drag reducing agent. In many instances it has been found that the drag reducing efficiency of polymers depends not on the total molecular weight, but on the maximum contour length [ 1,2,42,43]. Hence, it appears that the grafting of side chains, at the cost of reduced backbone length, is not the optimum route for enhancing drag reduction efficiency. Gryte et al. [44] found this to be the case for PAA grafted PEO. However, Deshmukh and Singh [7,8] found that by grafting flexible PAM on the main chain, drag reduction efficiencies of XG and GG polysaccharides could be enhanced at low concentrations of the polymer. This may be because the comparison is not made at constant total molecular weight, and the increased molecular weight or volume [ 1,2] could have been responsible for the enhancement in drag reduction efficiency at low concentration, although at the cost of reduced shear stability. Finally, our analysis has implications even on the shear stability of the zipped structures formed by interpolymer associations between linear chains. In the system described by Malik and Mashelkar [45], a proton-donating polymer (copolymer of dodecylacrylate and methacrylic acid) and a proton-accepting polymer (a terpolymer comprising dodecylacrylate, styrene and vinyl pyridine) were used as a mixture. It was shown that the extent of drag reduction went up significantly (by 200-600%) in comparison with the individual component polymers. The shear stability also went up significantly. Such systems would give rise to an interpolymer hydrogen bonded network between the proton acceptor and proton donor chains shown schematically in Fig. 4(a). It has been proposed (see [ 171 for references) that, in flow regions of low deformation rates, such reversible secondary bonds are formed at random active sites along the chains (Fig. 4(b)). In regions of high deformation rates, the weaker secondary bonds offer themselves as ‘sacrifice’ in preference to main chain covalent bonds, thereby protecting the latter. However, we feel that structures as depicted in Fig. 4(b) are actually analogous to irregularly branched or grafted polymers. The analysis presented in this work has shown that such branching or grafting will give rise to additional frictional drag tension at the backbone centre. This will lead to an enhancement of backbone degradation when compared to the individual molecules. Even if the secondary bonds can break in preference to the backbone, the individual molecules so formed cannot have a higher shear stability than the individual component polymers. Thus the claims of the mechanisms presented in earlier works do not seem to be substantiated. Agarwal and Mashelkar [ 171 have examined such ‘zipped’ assemblies in steady elongational flow and explained the enhanced stability as being due to the distribution (near the vulnerable chain centre) of drag tension into two parallel ‘zipped’ chains, and reduction of the drag force as being due to the enhanced hydrodynamic shielding. A limitation of the present work needs to be mentioned. The results cited in this work are based on experiments on drag reduction in turbulent flows. Here, although the polymer molecules experience significant extensional flow, the chain

8

U.S. Agarwal,

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1 J. Non-Newtonian

Fluid Mech. 54 (1994) I&IO

l.O-

---_ \ o

0.8-

f5 \

\

\

\

\

\

\

s 6

o-7-

0.6

t O.$

I

0.1

I

I

0.2

0.3

I

0.4

Ce(iv) 10”s (163moles) Fig. 3. Variation in Fig. 2.

extension complex. bead-row adopted turbulent

in shear stability

at 400 ppm concentration

of polymer.

The conditions

are the same as

may not be complete and the actual flow situation may be rather more For simplicity, we have carried out the analysis on fully extended molecules in steady extensional flows in keeping with the approach in the past. We expect that the results would be qualitatively valid for flow.

4. Conclusions There has been no previous attempt to theoretically analyze the effect of grafted/branched structures on the shear stability of polymers. Our work makes a preliminary effort in this direction. We show that the previous explanations based on preferential scission of the side chain do not seem to have any validity. Considering the wide range of processes in which shear degradation of polymers is of concern, and the continuing interest in finding polymers that are more stable, the present work is helpful in outlining appropriate molecular parameters for the evaluation of shear stability and, hopefully, even in designing shear stable polymers. Backbone and graft lengths, and bulkiness of the repeat unit, are important considerations when evaluating the effect of grafting on shear stability, and when ignored may lead to erroneous conclusions. The optimum route to effcient and stable drag reducing agents seems to be through structures with (i) chains with

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(a)

(b) Fig. 4. (a) The interpolymer hydrogen bonding between proton acceptor and proton The equilvalent macromolecule with a randomly branched or grafted structure.

donor

chains.

(b)

maximum length and minimum transverse dimension or (ii) the vulnerable central portion of the chain protected by local strengthening. Work is in progress to design such molecules by innovative chemical means.

Acknowledgment It is a great pleasure to contribute to this volume dedicated to Ken Walters. One of us (R.A.M.) has been fortunate to have known Ken for over two decades now; Ken’s inspiring work on secondary flows in elastic liquids in the early seventies got him started in engineering rheology. Ken’s work has continued to inspire generations of rheologists and non-Newtonian fluid mechanicists. We wish him decades of productivity, and personal and professional happiness.

References [l] S.E. Morgan and C.L. McCormick, Prog. Polym. Sci., 15 (1990) 103. [2] R.P. Singh, in Encyclopedia of Fluid Mechanics, Vol. 9, Polymer Houston, 1990, p. 425.

Flow

Engineering,

Gulf,

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[3] W.M. Kulicke, M. Kotter and H. Grager, Adv. Polym. Sci., 89 (1989) 1. [4] R.H. Sellin, Proc. Inst. Civ. Eng. Part 1, 86 (1989) 381. [5] J.F.Hutton, in K. Walters (Ed.), Rheometry: Industrial Applications, Wiley, New York. 1980, p. 119. [6] A.B. Metzner, in D.O. Shah and R.S. Schechter (Eds.), Improved Oil Recovery by Surfactant and Polymer Floodmg, Academic Press, New York, 1977, p. 439. [7] S.R. Deshmukh and R.P. Singh. J. Appl. Polym. Sci., 32 (1986) 6163. [8] S.R. Deshmukh and R.P. Singh, J. Appl. Polym. Sci., 33 (1987) 1963. [9] O.K. Kim, R.C. Little, R.L. Patterson and R.Y. Ting, Nature, 250 ( 1974) 408. [lo] G.W. Ver Strate, E.N. Kresege and W.W. Graessley, Ger. Patent 3,013,318, October 30, 1980. [ 1 l] J. Roovers, m Encyclopedia of Polymer Science and Engineering, Vol. 2, Wiley, New York, 1986, p. 478. [ 121 R.Y. Ting and R.C. Little, Nature Phys. Sci., 241 (1973) 42. [ 131 D.N. Schulz, K. Kitano, I. Duvdevani, R.M. Kowalik and H.J.A. Eckert, in D.N. Schulz and J.E. Glass (Eds.), Polymers as Rheology Modifiers, ACS Symp. Ser. 462, ACS. Washington, DC, 1991, p. 176. [14] S. Malik, S.N. Shintre and R. A. Mashelkar, U.S. Patent No. 5080121, 1991. [ 151 S. Malik, S.N. Shintre and R. A. Mashelkar, Macromolecules, 26 (1993) 55. [16] S. Ungeheuer. H.W. Bewersdorff and R.P. Smgh, J. Appl. Polym. Sci., 37 (1989) 2933. [ 171 U.S. Agarwal and R.A. Mashelkar, J. Chem. Phys.. in press. [18] T.Q. Nguyen and H.H. Kausch, Adv. Polym. Sci., 100 (1992) 73. [19] S.U.S. Choi, Y.I. Cho and K.E. Kasza, J. Non-Newtonian Fluid Mech., 41 (1992) 289. [20] S.H. Agarwal, R.F. Jenkins and R.S. Porter, J. Appl. Polyp. Sci.. 27 ( 1982) 113. [21] J. Frenkel, Acta Physicochim. URSS, 14 (1944) 51. [22] F. Bueche, J. Appl. Polym. Sci., 4 (1960) 101. [23] C. Levinthal and P.F. Davidson, J. Mol. Biol., 3 (1961) 674. [24] R.E. Harrington and B.H. Zimm, J. Phys. Chem., 69 (1965) 161. [25] R.E. Adam and B.H. Zimm, Nucl. Acid Res., 4 (1977) 1513. [26] J.A. Ode11 and A. Keller, J. Polym. Sci., Polym. Phys. Ed., 24 ( 1986) 1889. 23 (1990) 3092. v71 J.A. Odell, A.J. Muller, K.A. Narh and A. Keller, Macromolecules, WI A. Peterlin, Pure Appl. Chem., 12 (1966) 563. 1291 P.G. De Gennes, J. Chem. Phys., 60 (1974) 5030. Fluid Mech., 30 (1988) 119. 1301 Y. Rabin, J. Non-Newtonian 1311 M. Boudlin, W.M. Kulicke and H. Kehler. Colloid. Polym. Sci., 266 (1988) 793. 1321 J.G. Kirkwood and J. Riseman, J. Chem. Phys., 16 (1948) 565. [331 P. Debye and A.M. Beuche, J. Chem. Phys., 16 (1948) 573. [341 F.K. Herold, G.V. Schulz and B.A. Wolf, Polym. Commun., 27 (1986) 59. [351 S.L. Malhotra. J. Macromol. Sci., Chem., 23 (1986) 729. [361 P.R. Kenis, J. Appl. Polym. Sci.. 15 (1971) 607. I371 D.H. Fisher and F. Rodriguez, J. Appl. Polym. Sci., 15 (1971) 2975. 1381 J.W. Hoyt, Trends Biotechnol., 3 (1985) 17. of Polymer Science and Engineering, Vol. 5, Wiley, New York, 1986, [391 J.W. Hoyt, in Encyclopedia p. 129. ACR-112, Office of Naval t401 J.W. Hoyt and A.G. Fabula, 5th Symp. Naval Hydrodynamics, Research, 1964, p. 947. [411 P. Gramain and J. Barreill, Rheol. Acta, 17 (1978) 303. ~421 E.W. Merrill, K.A. Smith, H. Shin and H.S. Mickley. Trans. Sot. Rheol., IO (1966) 335. [431 CA. Parker and A.H. Hedley, J. Appl. Polym. Sci., 18 (1974) 3403. 1441 C.G. Gryte, C. Koroneos, A. Agarwal and A. Hochberg, Polym. Eng. Sci., 20 (1989) 478. I451 S. Malik and R.A. Mashelkar, Chem. Eng. Sci., in press.