Concentration dependence of the effective viscosity of polymer solutions in small pores with repulsive or attractive walls

Concentration Dependence of the Effective Viscosity of Polymer Solutions in Small Pores with Repulsive or Attractive Walls G. CHAUVETEAU,* M. TIRRELL,~- AND A. OMARI~t *Institut Franeais du P~trole, 92506 Rueil Malmaison, France, tDepartment of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, and $Laboratoire de M(canique Physique, I.U.T. de Brest, Rue de Kergoat, 29283 Brest, France Received August 31, 1983; accepted December 23, 1983 Polymer solutions are demonstrated to have apparent viscosities in small pores which depend on pore diameter (or the mean diameter of pore throats in irregular porous media) and which, therefore, can be considerably different from the viscosity of the same solution in an unbounded medium. The apparent viscosities in the pores can be greater or less than in bulk depending upon whether the pore wall is attractive or repulsive for the polymer. Specifically, if there is no adsorption (repulsive wall) we find that the solution viscosity is always less inside the pore than in bulk. On the other hand if the wall is attractive the apparent solution viscosity inside the pore may be greater or less, depending on the concentration of the flowing polymer solution. Data representing these effects are presented for aqueous solutions of hydrolyzed polyacrylamide and xanthan polysaccharide. The data are organized as suggested by a model recently proposed by Chauveteau for polymer solution flow in small pores. INTRODUCTION

2). "Small" here refers to situations where the characteristic size of the macromolecule is not negligible with respect to the size of the pore. Savins (3) published an excellent review of the earliest studies of rheology of polymer solutions in porous media in which he noted that discrepancies between porous media and conventional rheology appear when adsorption of the polymer on the pore wall is a problem. Gogarty (4) observed a reduced viscosity of a polymer solution in a porous medium which h e attributed to a peculiar non-Newtonian effect arising from the complex geometry. Later studies have shown more clearly that the apparent viscosity in the pore may be higher or lower, even under Newtonian flow conditions, depending upon whether the pore wall is attractive or repulsive for the polymer, respectively. Getting accurate and representative data on porous media rheology is a somewhat delicate problem. Precautions must be taken to avoid artifacts such as those caused by the presence of even small amounts of aggregates or particles consisting of polymer which is not mo-

In the vicinity of a solid-liquid interface, the properties of a polymer solution are strongly modified relative to the properties exhibited by the same solution in what we shall call bulk or unbounded conditions. Equilibrium properties such as the local macromolecular conformation and mass concentration, and consequently, dynamic properties such as mass and momentum transport are locally altered. These local alterations have global manifestations in several practical situations, such as in polymer solution flooding of porous media for the purposes of enhancing oil recovery, chromatography of polymers, and various membrane permeation situations. The present study is motivated primarily by the enhanced oil recovery application, where the specific observation of interest, the apparent viscosity of the polymer solution, can be very much different when measured in small pores or small bore porous media than when the viscosity of the same solution is measured in a conventional viscometer (1, 41

Journalof Colloidand InterfaceScience,Vol. 100, No. 1, July 1984

0021-9797/84 $3.00 Copyright© 1984by AcademicPress,Inc. All rightsof reproductionin any form reserved.

42

CHAUVETEAU, TIRRELL, AND OMARI

lecularly dissolved (5). These so-called microgels can partially plug a porous medium and give anomalously high pressure drop data. They can be removed by careful, slow filtration. Another effect, which is interesting in itself but must be avoided in order to get accurate low shear rate porous medium rheological data, is the apparent "shear thickening," which is actually an effect due to the partial elongational character of the flow caused by the converging, diverging, and stagnation zones of any porous medium (6, 7). Avoiding this is simply a matter of making the measurement at sufficiently low flow rate. Low flow rate data in small pores on several polymer solutions without microgels have been obtained previously, at low concentrations (up to 400 tzg/cm 3 by weight polymer). The two polymers studied were a rodlike polysaccharide xanthan (1) and a flexible coil synthetic macromolecule, hydrolyzed polyacrylamide (7). These data were obtained under conditions where the polymer does not adsorb on the pore wall so that the effective pore viscosity was lower than the actual viscosity. Recently a molecular theory of the bead-spring type has been published which was shown to have good predictive capabilities for the viscosity of either flexible or rod-like polymer solutions near a plane repulsive wall at very low concentrations (2). Aubert (8) has very recently extended the model to the case of an attractive wall. These bead-spring models, which are very useful in dilute solution, encounter serious conceptual and practical difficulties in fitting data at higher polymer concentration. Chauveteau has recently introduced a semiempirical model for repulsive walls which is not restricted to dilute solution (1). The purpose of the present work is severalfold. New data are presented on xanthan and polyacrylamide solutions at higher concentrations with both attractive and repulsive walls. The results are organized as suggested by and compared with the Chauveteau model which itself is extended to the case of adsorbing walls. The emphasis is on understanding the Journal of Colloid and Interface Science, Vol. 100, No. 1, July 1984

basic physics of the viscosity modification in small pores. To the extent that this is achieved, viscosity measurements of this type reveal important information on polymer concentration profiles in solutions near solid interfaces. We begin by discussing briefly some established and some new theory for equilibrium concentration profiles and effective viscosity in polymer solutions near walls or in pores, followed by experimental technique, results, and discussion. THEORETICAL CONSIDERATIONS

In dilute solution at equilibrium the theory of macromolecules near inpenetrable but otherwise passive (i.e., neither adsorbing nor longrange repulsive) is fairly well developed. Random flight chains near this type of barrier have been discussed thoroughly by Casassa and coworkers (9-11). The result for the total (averaged) density of segments as a function of distance, z, from the barrier for a large Gaussian coil of n segments can be expressed as CG(x) = Cb

ergx/V-uu) × eft(x/V1 - u)du

[ 1]

where Cb is the (uniform) segment density far from the wall; x is a dimensionless distance scaled by the root meansquare unperturbed end-to-end distance (b0~fn where the segment length is b0): x = (3/2nb2o)l/2z; and, u is a scaled version of the segment index m (0 < m < 1): u = m / n . This equation simply expresses the fact that in order to have the mth segment of an n segment chain at x there must be two random walks, of lengths m and n-m, emanating from x, neither of which cross the barrier. Each of these leads to a single error function probability; the joint probability of the two walks is the product. To get the total density at z we integrate over all choices of the mth segment. The concentration profile corresponding to Eq. [ 1] is shown in Fig. 1. We see that there is a layer near the wall depleted in polymer concentration whose thickness scales with the radius of gyration of the mac-

VISCOSITY OF POLYMER SOLUTIONS IN SMALL PORES

43

Note that it is now ~ which scales the thickness of the depletion layer near the wall so that, taking Eqs. [3] and [4] together, we expect that the thickness of the depleted layer near the wall will decrease with increasing concentration. Similar arguments can be made for Rigid Flexible theta conditions leading to different exponents. FIG. 1. Schematic diagram of the cross-channel con- We are more concerned with the good solvent centration profile for a rigid (left) and a flexible (right) situation. macromolecule. The overlap concentration C* is generally defined (16) as the concentration above which "significant" overlap of the pervaded volumes romolecule. O f course, Gaussian statistics are of the polymer coils in solution occurs. Simple only even approximately accurate in theta geometrical arguments and some recent exsolvents, but qualitatively Eq. [ 1] and many periments suggest that C* is reciprocally reof the statements about it above remain valid lated to the intrinsic viscosity, i.e., C*[~] - 1. even in good solvents. It is known, however, from several kinds of The corresponding situation of rigid rod measurements (18) that in order to be suffipolymers near a hard-core repulsive wall has ciently overlapped to observe an effect in dybeen worked out by Auvray (12). The mononamic experiments such as diffusion or rheolmer (or segment) density profile for a rod ogy one must be working at somewhat higher polymer of length L is concentrations, say, C[~] - 3 or 4. It is this CR(Z) = Cb(z/L)(1 - In z/L). [2] latter level that we will take as our approximate operational definition of C*. Here the thickness of the depleted zone scales For rod polymers the concept of overlap is with L (not the radius of gyration as in the somewhat different (19, 20). As one increases case of a coil). The profile itself is nearly linear the concentration in a solution of rods in diexcept when z / L ~ 1, so that the depletion ameter a and length L, one reaches a point is less accentuated in the near wall region than where the number density of rods exceeds for flexible polymers governed by Eq. [ 1]. The l / L 3 and the rods can no longer rotate freely. two profiles are compared in Fig. 1. This is a dynamic restriction. The correlation As the concentration Cb is increased, certain length however in a solution of rods remains changes in the profiles expressed by Eqs. [1 ] constant at L, independent of concentration, and [2] may be expected. In particular, in the so that the thickness of the layer affected by; case of uncharged, flexible polymers in good the presence of the wall also is of order L at solvents (13, 14), above a certain overlap conall concentrations (less than the nematic phase centration, frequently referred to as C*, de transition) (1, 21). Equation [2] gives the conGennes (15) has suggested one may expect a centration profile for all concentrations of inpower law behavior of the density profile: terest to us (12). C(z) = Cb(Z/O ~/3 [3] For adsorbing polymers the situation is somewhat more intricate. Since the rod polyfor z < ~, where ~ is the correlation length, a mers used in the present work do not adsorb sort of mean distance between monomers in on the surfaces studied, we discuss only flexible the solution. The correlation length is itself polymer adsorption. Density profiles in this obviously a decreasing function of Cb which case have been the subject of much recent scaling theories (16) and some experimental interest (22-25). Here, of course, the solution evidence (17) suggest should go like near the wall will be enhanced in concentra= R G ( C b / C * ) 3/4. [4] tion relative to the solution far from the wall. Journal of Colloid and Interface Science, Vol. 100, No. 1, July 1984

CHAUVETEAU, TIRRELL, AND OMARI

44

For our purposes, the case of good solvent is the most interesting one and we mention here only a few of the most pertinent general facts. The concentration in the immediate vicinity of the wall will be determined by the adsorption energy of the polymer segment to the wall. This adsorption energy is in turn related to the relative affinities between the wall, the polymer segments and the solvent molecules. We are primarily interested in the so-called "weak-binding" limit where polymer segments are not strongly bound themselves but the polymer molecule has a sufficient number of segments bound at any one time to keep the entire polymer nearly irreversibly adsorbed. Here one finds (26) that the segment density falls off exponentially from the wall at large distances like C(z) .~

exp(z/O.

[51

Thus, when a weak-binding wall of this type is in contact with a dilute bulk solution there is a characteristic thickness of the adsorbed layer of the order of the radius of gyration of the macromolecule (~ ~ 2RG). As the concentration in the bulk far from the wall is increased we expect that overlap will cause the characteristic thickness of the adsorbed layer to decrease like ~, i.e., as given in Eq. [41. FLOW MODELS One class of models under study is the beadspring type models whose behavior has been worked out in the presence of hard-core repulsive (2) and adsorbing (8) walls. These are one-parameter models, giving the apparent viscosity in the pore as a function of the ratio of polymer radius of gyration to pore radius alone. Even though these are models for infinitely dilute solutions, i.e., C[~/] < 0.1, they have recently been shown (2, 27) to be useful in predicting pore viscosities up to around C[~] 3 to 4, /fthe experimental bulk value of C[~] is used as an input to the model. However, the model should in fact predict C[~] as well as the pore viscosity. If that is done, the dilute solution theory greatly underpredicts Journal of Colloid and Interface Science, Vol. 100, No. 1, July 1984

C[~] and therefore also underpredicts the apparent pore viscosities. Continued progress along the molecular theory approach to model these higher concentration results awaits the development of the basic molecular theory of rheology of more concentration polymer solutions, although we note that the dumbbell with anisotropic bead friction tensors, studied recently by de Aguilar and Bird (28), may provide some useful information. A second type of model has been proposed recently by Chauveteau (1). It is a continuum type of model, containing two parameters which define the effect of the wall but which are not explicitly connected in the model to the properties of the macromolecules. As we shall see however, the parameter values can be given some a posteriori interpretation in terms of molecular properties. Moreover, there are two rather important practical advantages of the Chauveteau model. It is quite simple to use and has no fundamental restriction in its formulation regarding concentration range of applicability. The model is discussed fully in Ref. (I). The flow of a polymer solution in a cylindrical pore of radius r is simulated by a coaxial flow of two fluids, one in an annular wall layer of thickness and one in the central core. The central region has relative viscosity ~rb, where the subscript b stands for bulk. This is because in most situations of interest to us the pore is large enough so that there is some central region where the viscosity has the same value as in bulk. The model is not restricted to that situation, however. The fluid in the wall layer of thickness ~ has relative viscosity ~1~. One can quite easily calculate the pressure drop flow rate relationship for Poiseuille flow of two coaxial fluids in a capillary (1) and from that determine the effective apparent viscosity in the pore, nw, as a function of the parameter p =

7/rb/~rw;

nrp = ~w[l - (l/p)(1 - 6/r)4] -1

[6]

keeping in mind that this model is limited to the case where the flow is Newtonian (~rb constant) and 6/r ,~ 1. A more convenient form of Eq. [6] for some purposes is

VISCOSITY OF POLYMER SOLUTIONS IN SMALL PORES ~rp

=

p [ l -- ( l - ~//,)4] + ( l - ~/g)4

[7]

~/rp which shows deafly that ~rb/~rv is a linear function of p.

45

where 3, = a l l - (1 - 6/r)4](Cb[n])2 is often negligible. If in addition 6/r ,~ 1, Eq. [10] can be simplified still further -~rb -7/rp

1 +4(1

- 8) ~ Cb[ 7] r

A. Repulsive Walls In this form the model applies only to the case of repulsive wails, so that the wall layer of thickness 6 will be a layer depleted in polymer concentration and p always be greater than 1. For the most interesting case of hardcore repulsive, but otherwise noninteracting, walls we can begin to make some plausible hypotheses regarding relationship of the parameters ~ and p to molecular properties. The concentration regime of interest will affect these hypotheses regarding parameter selection. A. 1. Dilute solutions. In the dilute regime, for C < C*, drawing on the type of information in Eqs. [1] and [2] and in Fig. 1, we expect the concentration-depleted wall layer to have a thickness ~ comparable to RG for a flexible polymer and to L for a rodlike polymer. Regarding the viscosity parameter p, if we are in quite dilute solutions C[~I] < l, then we expect the viscosity to be given by the Huggins relation, which would lead to nb

1 + Cb[7/] + k'(Cb[~/])2

P = ~w

1 + ~cb[n] + k'(~Cb[n] ~

[8]

where ~ = Cw/Cb, that is the ratio of average concentration in the wall layer to the concentration far from the wall. For C[~] ~ 1, Eq. [8] shows that p is given to a good approximation by a = 1 + (1 - / 3 ) Cb[n] + o/(Cb[?]]) 2

[9]

where a = k' - / ~ +/~2(1/2 - k'). We expect a to be rather small, and again for Cb[~] < 1, the nonlinear term in Eq. [9] may be negligible leading to a simplified form of Eq. [7]: /]rb

~/ro

-

1 +(1

-8)

X [1 -- (1 - ~/r)4]Cb[n] + "y [10]

As we shall demonstrate presently, in our experience this simple two-parameter equation works quite well in dilute solution. The linearity in 8/r and Cb[~] for small values of those parameters is consistent with molecular theory calculations as well (2). For higher concentrations, still less than C*, the Huggins relation must be replaced by the Martin equation (29) ~r = 1 +Cb[n]e kcbt"~

[12]

in determining the value of the parameter p. In practice, for Cb < C*, the characteristic dimension of the macromolecule will determine the depleted layer thickness, 6, and this in turn will imply a certain approximate value of the parameter, 8, to which it must conform if it is to have interpretable physical significance. Specifically, # is the concentration (normalized by Cb) of macromolecules in a volume within a distance ~ away from the wall. One can calculate this quantity from a specific molecular model as, for example, from Eqs. [ 1] or [2] by integrating out to a distance ~. Since C(z) approaches Cb very smoothly, we must choose the depleted layer thickness somewhat arbitrarily. For example, for a rodlike polymer if we choose the point where C = 0.95Cb, then ~ - 0.7L (see Fig. 1) and we find by integration of Eq. [2] that fl - 0.64. Similarly for randomly coiling models if we choose ~ to be one-half the mean end-to-end distance, then ~ = ~f6/2Rc "~ 1.24 Re, then several molecular models (2, 11, 15), for example the bead-spring model (2), give/~ - 0.3. The data we have in the dilute regime, as we shall show, are in quite reasonable agreement with these physically motivated choices of parameters. Journal of Colloid and Interface Science, Vol. 100, No. 1, July 1984

CHAUVETEAU, TIRRELL, AND OMARI

46

A.2. Semidilute solutions. In the case of semidilute solutions (C > C* but still very much less than bulk polymer density) an important qualitative distinction arises between rigid and flexible polymers. For rigid polymers, Eq. [2] gives the concentration profile near a wall even into the semidilute regime. Thus, we expect that for the present flow model the parameters ~ and/3 for rigid macromolecules would remain independent of concentration into the semidilute regime. Equation [7], with typical values of 5 and/3 such as those cited above, can be used over the entire concentration range of interest to us, using in addition Eqs. [8] or [12] depending on the value of the concentration. For flexible polymers in good solVents one expects ~ to decrease with increasing concentration (as mentioned previously in connection with Eq. [4]). We therefore expect a relation like (13) 6 = 1.24RG(C/C*) 3/4

[13]

to be reasonable, it being constructed so that it crosses over smoothly to the dilute solution expression 6 = 1.24RG at C = C*. The parameter/3 should be roughly constant at ~ 0 . 3 although it may increase slightly much like partition coefficients of polymers between pores and bulk increase with increasing concentration (30). The physical reason why the depleted layer decreases with increasing concentration for flexible polymers and remains constant for rigid polymers is that as flexible polymers begin to overlap, the correlation length, that is the distance over which correlations in local concentration exist, is diminished. On the other hand, for rigid rod polymers, overlapped or not, these correlations persist for the length of the rod. This correlation length is also the length scale over which wall effects are propagated into the bulk solutions (15). B. Attractive Walls Attractive walls produce adsorption of the macromolecules. If the thickness of the layer Journal of Colloid and Interface Science,

Vol.100,No.1,July1984

adsorbed on the wall is large enough then this layer can exert a detectable influence on the apparent rheological characteristics of fluid flowing in a small pore (31, 32). The thickness of the layer is determined by several factors including molecular weight, degree of saturation of surface sites by the polymer, and strength of binding. We are most concerned here with the so-called weak binding situation described above. Polymers adsorbed in more compact or dense layers with a smaller thickness exert less rheological influence. In practice, measurement of the thickness of a layer of adsorbed polymer on the inside of a tube has frequently been done inferentially by measuring the apparent hydrodynamic radius of the tube after adsorption on the walls has occurred (31, 32). Similar studies have also been made on polymer adsorption in porous media (7, 33). The apparent viscosity increase Rk (termed the permeability reduction in porous media fluid mechanics) due to a hydrodynamically impermeable adsorbed layer of thickness ~ inside a cylinder (or equivalent cylinder for porous media) of radius r is Rk = (1 -- ~/r)-4. [14] In order to extend the Chauveteau model to cases where adsorbed layers are important, two modifications must be made to Eq. [7]. The bulk viscosity ~rb must be modified by the factor Rk and the radius r must be modified by subtracting ¢, leading to

"~(1-- r-~ 14 [15] or using Eq. [14]:

{[ ( ')41(

--=pl~n)

1--

r-¢

')'}

+

1 ........ r-~

x (1 -

~/r) 4.

[161

In the dilute regime, we expect ~, like ~ to be constant and therefore, referring to Eq. [ 11] (for ~ <~ r), we see that at low concentrations

VISCOSITY OF POLYMER

SOLUTIONS

the effect of the depleted layer grows linearly with concentration while the effect of the adsorbed layer remains constant. In the semidilute regime, C > C*, we expect ~, like 5, to decrease with increasing molecular overlap at high concentrations and we must introduce these variations with concentration, given by Eq. [13] for good solvents, into Eq. [16]. It is important to realize that Eq. [ 16] contains the idea that there can be a depleted layer in addition to, or on top of, an adsorbed layer. We present in a later section experimental evidence that this is a apparently the case. The effect of the adsorbed layer is constant for C < C* and then diminishes above C*, while the effect of the depleted layer continues to increase with concentration. Consequently, in situations where one has effects on both adsorption and depletion, the ratio ~rb/~p takes some value given by Eq. [14] at infinite dilution of the flowing solution then increases with increasing concentration until ~b/nn, = 1, where the effects of the adsorbed and depleted layers will annul one another. We can determine this concentration from a simple quantitative argument. If we take the simplified Eq. [11] and modify it for the effect of adsorption in the case where ~/r g l we obtain 7/r_~b = (1 + 4(1 - / 3 ) -~ Cb[~/])(1 - 4 -~), [17] r ~/rp r which gives ~rb/~rp = 1 when E

Cb[n] = (l --/~)~'

[18]

SO that, if ~ is approximately equal to 25, and we see that for the values of fl for which we agreed earlier we expect nrb/nn, = 1 when Cb[~] 3. At higher concentrations we expect the 77rb/~rp will increase still further, giving an apparent viscosity smaller than in bulk in spite of adsorption. EXPERIMENTAL

CONDITIONS

AND METHODS

A. Polymers Two types of polymers were studied. Both are varieties of polymer which are actually

IN SMALL PORES

47

being used as "pusher" fluids in enhanced oil recovery by polymer flooding. The first is a flexible coil polymer of the hydrolyzed polyacrylamide type (HPAM), specifically, Dow P700, obtained in the form of a powder. In this polymer, approximately 30% of the acrylamide residues are hydrolyzed. The weight average molecular weight is 7 × 106 and the distribution of molecular weights, determined by sedimentation (34), is rather broad. The second type of polymer is a rodlike polymer, the polysaccharide xanthan (XCPS), obtained in the form of fermentation broth from Rhone-Poulenc, for which many of the molecular properties have already been published (l, 5, 35). Xanthan should actually be termed a semirigid polymer. Its viscometric properties are quite consistent with those of rigid rods (or highly elongated prolate ellipsoid) (1, 36) of 0.8 and 1.1 ~tm in the present case. However, when these are combined with molecular weight data on the same samples to calculate the mass per unit length one arrives at a value which at least twice as high as actual mass per unit length from the known chemical structure of xanthan. This seems to indicate clearly some flexibility. Holzwarth's data (36) on the intrinsic viscosity-molecular weight exponent also suggest some flexibility. However, xanthan is clearly very rigid compared to HPAM and the rodlike modeling works well enough for the present purposes. Two samples ofxanthan were studied, one with 3~tw = 2.8 X 10 6, the other with A~tw = 1.8 X 10 6. These have rather narrow, but not monodisperse, distributions with polydispersity about 1.4 as determined independently by hydrodynamic chromatography (37) and electron microscopy (38).

B. Preparation of Solutions Careful solution preparation is an indispensable feature of accurate measurement of polymer solution properties in small pores. Small bits of material which are not molecularly dispersed but present in the form of aggregates or microgels, perhaps due to crosslinking of some kind, occur frequently and Journal of Colloid and Interface Science, Vol. 100, No. 1, July 1984

48

CHAUVETEAU, TIRRELL, AND OMARI

are difficult to remove by filtration since the particles are readily deformable and thus can squeeze through filters. A workable preparation procedure for removing these microgels has been published (5). The polymers were dissolved by slow agitation followed by filtration of a powder dispersion in the case of HPAM and by rapid dilution of the fermentation broth, followed by filtration for the XCPS samples. The physicochemical conditions of the solution (salinity, pH, temperature, etc.) were chosen so that the molecular conformation of HPAM was that of flexible coil polymer in a good solvent (20 g/liter NaCI, pH 8.0, T -- 30°C) and that of XCPS was a nearly rigid rod (5 g/liter NaCI, pH 7.0, T = 30°C for the 1.8 × 106 sample; 100 g/liter NaCl for the 2.8 × 10 6 sample). Other important molecule characteristics for the polymers studied are the intrinsic viscosity and mean molecular dimensions (presented below as radius of gyration for HPAM and equivalent rod length for XCPS) HPAM: [7] = 3,650 cm3/g, Re = 0.19ttm

(Ref(39))

XCPS: [7] = 4,300 cma/g, L = 0.82 ~tm

Ref(l))

XCPS: [7] = 6,900 cm3/g, L -- 1.13 ~tm

(Ref(5)).

The dimension cited above for the HPAM is the radius of gyration, R~,, calculated via the Flory-Fox equation using intrinsic viscosity data and molecular weight from light scattering data. Other measurements of characteristic dimensions give 0.24 #m via angular dependence of light scattering [41] and 0.15 ~tm for the hydrodynamic radius, RH, via cluasielastic light scattering (41). All of these values are quite consistent, especially in view of the findings of Rudin and co-workers (42) that for a range of polymers RH/RG~ ~- 0.77. With ~ = 1.24Re, this leads to a value of = 0.25 #m which we will use without adJournal of Colloid and Interface Science, Vol. 100, No. 1, July 1984

justment to model pore viscosity results for HPAM. For XCPS on the other hand the viscometricaUy determined effective rod lengths were converted to ~ by 6 = 0.7L as discussed above. All solutions were protected from bacterial and oxidative degradation by the addition of 400 ppm sodium azide. The polymer concentration range studied (170 to 2100 ppm) extends into both the dilute and semi-dilute regimes and is also the range of concentrations employed in enhanced oil recovery. C. Porous Media Several different types of porous media were used: Nuclepore polycarbonate membranes, which have quite nearly circular cylindrical pores available in a range of diameters; columns packed with spherical glass beads of controlled size; columns packed with silicon carbide in granular form, sieved to achieve a modicum of size uniformity and columns packed with sand grains producing sand packs of different porosities and different mean pore diameters. Characterization of these porous media as to mean pore size and shape was done using electron microscopy and fluid mechanical permeability measurements and has been published previously (40). Table I gives some characteristics of the porous media studied. D. Experimental Flow Apparatus The basic flow apparatus has been described elsewhere (1). It is very simple in design but capable of very high precision. Polymer solution is injected into the porous medium with a syringe pump guaranteeing constant flow rate to very high accuracy. The pressure drop associated with flow rate is measured with a two-fluid U-tube manometer containing water-oil, water-mercury, or water-air according to the magnitude of the pressure difference to be measured. In the case of water-oil a surfactant was added to avoid any inaccuracies in the measurement due to capillary pressure. For the experiments with Nuclepore mere-

VISCOSITY OF POLYMER SOLUTIONS IN SMALL PORES TABLE I Some Characteristics of the Porous Media Studied Pore diameter (Electron microscopy) (#m)

Type of porous medium

Nuclepore membranes

10.9 6.6 4.8 2.5 0.69

Pore density (Electron microscopy) (cm-2)

1.06 0.95 4.3 2.1 3.0

--i-0.04 _+ 0.II + 0.07 _ 0.05 + 0.05

× × × × ×

105 105 105 106 107

above. The results are organized in this section in two series of figures where we examine first the case o f nonadsorbing wails and subsequently the effects of adsorbing walls. Within each series o f figures we examine successively the effects of polymer conformation (rodlike or flexible coil), relative dimensions of macromolecule and pore, and concentration o f the flowing polymer. On each figure data on several different porous media are given as indicated.

Glass bead packs Bead diameter Ozm)

400-500 200-250 80-100 40-50 20-30 10-20 8-15

Permeability (#m2)

A. Nonadsorbing Walls Porosity (~)

137 36 8.4 2.4 0.66 0.21 0.11

0.40 0.40 0.40 0.41 0,41 0,41 0,41

Sand packs Grain diameter (#m) 80-120 50-100

5.0 3.4

0.38 0.43

SiC packs Grain diameter (urn) 18 (XCPS) 50 (HPAM)

49

0.08 0.83

0.46 0.38

branes, the apparatus contained membrane holders which would permit stacking of up to 80 membranes in series (1). All apparatus was constructed in such a way that there was no contact between the polymer solutions and metal. For each porous medium-polymer solution measurement, readings were obtained at several flow rates within the Newtonian flow regime and the mean value to r/m is the one reported. Typical standard deviations are less than three percent. RESULTS AND DISCUSSION

A series o f experiments were done in order to assess the validity of the model proposed

A.1. Rigid rod polymers in dilute solution. Figure 2 shows the results for dilute solution o f XCPS in several porous media. The mean pore sizes of the media have been determined as described previously (1, 40). The coordinates used in the figure are those suggested by Eq. [ 11 ]. The solid line is the theoretical prediction of Eq. [ 11 ] with the values of ~ and/~ equal to those suggested above on physical grounds, specifically 6 = 0.7L (with L given in the previous section) and # = 0.64. Clearly, the data are quite linear as predicted up to 1 - (1 - ~/r)4 ~_ 0.75 and a least-squares fit gives/9 = 0.63. At higher values of the abscissa, the curve bends upward as we move into a strongly confined regime. In this regime there is no central region of the pore where C = Cb and thus one of the basic ideas of the "two-

i

i

I

~1

XCPS

0.6

I

~ 04

M w x 10 - 6

1,8

2,8

Nuclepores

•

o

Beads

•

Sand

•

i

,i~~

,/

V

~ ITe~t v 02 ~

od •

-I 0.25

I 0.5

I 0.75

Theory I I

1- (,-~14 FIG. 2. Comparison of bulk and pore relative viscosities for xanthan as a function of the relative molecular and pore size plotted according to Eq. [I 1] of the text. Nonadsorbing pore walls. Journal of Colloid and Interface Science. Vol. 100, No. 1, July 1984

50

CHAUVETEAU, TIRRELL, AND OMARI

fluid" model described here is no longer valid. The curvature is upward toward lower viscosity in the pores, indicating that there is still more to be understood about flow in the strong confinement regime.

A.2. Rigid rod polymers: Effect of concentration. The influence of concentration with the XCPS polymers (sample 1) on the apparent viscosity in porous media has been studied in a column packed with silicon carbide grains of approximately 18 # m in diameter (40). The data are shown in Fig. 3. We see that the agreement between experiment and calculation, using Eqs. [7] and [12], persists well above the overlap concentration C* - 3.5/[n]. In fact, the same values of 6 and fl as used in Fig. 2 in dilute solution fit the data here at m u c h higher concentration. This is in accord with the idea that the correlation length ~, and therefore 6, remain constant well into the molecular overlap concentration regime in solutions of rod polymers. A.3. Flexible polymers in dilute solutions. The plot analogous to Fig. 2 is shown for flexible polymers in dilute solution (C[n] = 1.24) (HPAM) in flow through Nuclepore m e m branes in Fig. 4. ( H P A M adsorbs on all the other porous media in this study.) We see again the linearity of these data as indicated by Eq. [ 11 ]. The values of 6 and/5 used for the theoretical curve shown are 6 = 0.25 /~m and = 0.3 exactly those suggested above on physical grounds. I

i

i

i

i

i

L

I

I

~

0.4

Data Theory 0.2

I 02

O0

I 0.4

= 1.24RG(C/C*) -3/4

i

I

I

I

I

I

I

I

I

1

2

3

4

5

6

7

8

FIG. 3. Comparisonof bulk and pore relativeviscosities for xanthan as a function of concentration in SiC packs. Nonadsorbing pore walls. Vol.

100,

No.

1, July

/

22

1.5

~c/

/

-J. Dat . . . .a ....

c*r~l

Theory ~ constant

I

l

c[n]

c[-, 7]

Journal of Colloid and Interface Science,

r /

HPAM

I "

[19]

rationalized above, is shown and demonstrates that above C* it is necessary to take into account the diminution o f the thickness of the

Nuclepores

0

L 0.8

A.4. Flexible polymers: Effect on concentration. Figure 5 shows data on flow of H P A M through Nuclepore m e m b r a n e s with pores of diameter 6.6/~m. Thus, 6/r in the dilute regime is 0.25/6.6 = 0.038. We see that the data at low concentrations are linear as expected and as seen previously. The solid line is the theory with ~ and fl constant as one expects in dilute solution. It can be seen clearly in Fig. 5 that the data do not conform to the linear relation above C* = 3.5/[~]. A second theoretical curve (dashed line) which uses

I

Theory ~ constan

I 0.6

Fxo. 4. Comparisonof bulk and pore relativeviscosities for hydrolyzedImlyacrylamideas a function of the relative molecular and pore size plotted according to Eq. [ 11] of the text. Nonadsorbing pore walls.

2.5

- -

I

HPAM

I

4

I

0.6

1984

FIG. 5. Comparisonof bulk and pore relativeviscosities for hydrolyzed polyacrylamideas a function of concentration in Nuclepore membranes. Nonadsorbing pore walls.

VISCOSITY OF POLYMER SOLUTIONS IN SMALL PORES depleted layer in order to have an accurate prediction. We do not suggest that these results can be taken as a definite test of the scaling exponent, - 3 / 4 , in Eq. [ 19], that awaits a more thorough investigation, but it certainly does give good results for this limited amount of data. B. Adsorbing Walls

r = ot(8k/~) 1/2.

The coefficient a is exactly one for a capillary bundle and depends upon the form of the porous medium. It has been found experimentally (1) to be very near one for packs on uniform beads and grains; for consolidated media such as sandstone a ~ 3 has been found (40). For the two porous media studied here we obtained Sand pack

B.1. Flexible polymers in the dilute regime. In the experiments with adsorbed layers on the pore walls, the adsorbed layer was created by exposing the pore wall surface before the experiment started to a polymer solution with concentration and exposure time sufficient to saturate the adsorbing surface. Only HPAM was used in the study of adsorbed layers. Saturation is defined operationally as the point where no further change in the amount of adsorption could be detected. Gentle flow of H P A M solution of the concentration used through the porous media produces saturation in one to three days. The thickness of the adsorbed layer (and therefore the conclusions stated above regarding saturation) is determined by measuring the reduction in the mean hydrodynamic radius of the pore via water flow (43). Two kinds of porous media were used in these experiments with adsorbed layers: silicon carbide packed columns and packs of Fontainbleau sand. It is essential to prescribe the means for defining and determining the mean pore radius in porous media such as these. What is readily measured in a porous medium is the permeability, k, which is the proportionality constant between the pressure gradient and the flow velocity. On dimensional arguments alone we expect the characteristic or average pore throat radius to be proportional to V-k. If we make the usual assumption that the real porous medium can be represented by an equivalent bundle of uniform, circular capillary pores. For such a bundle of equivalent permeability, the pore radii and the porosity, ff~, are related by [20]

51

k = 4.5

].t2;

tI) = 0.38;

r = 9.7 u m SiC

k = 0.83/~m2;

• = 0.39;

r = 4.1 ~tm, before exposing them to any polymer. Two polymer solutions of different concentrations were studied in the dilute range: C*[~/] = 1.24 and 2.48. Each was exposed to the porous media for sufficient time to saturate the adsorbed layer. The permeability to water was then rechecked. The reduction in permeability was used to determine the thickness of the adsorbed layer via Eq. [ 14]. The results were Sand pack

SiC

Rk = 1.18; R k = 1.45;

~ = 0.41 /zm ~ = 0.40 t~m.

The equal values obtained indicate that a similar adsorbed layer is produced in each case which is what one expects in the weak binding limit. We also note that the values obtained for E - 2R~ are also quite reasonable physically. In Fig. 6 we plot the data for the two concentrations according to the coordinates suggested by Eq. [15], with r modified by the factor ~. Also on the same plot are data for HPAM in Nuclepore membranes where we saw no adsorption. In all three cases the value 6 = 0.25 ~tm was used for plotting purposes. We again see the linear relationship expected from the two-fluid model and also as expected, ~/rb/On, is proportional to C[~]. The result which is the most remarkable however is that the data both with and without adsorbed layers lie on exactly the same lines. This means that there is apparently an effect of a depleted layer of polymer concentration Journal of Colloid and Interface Science, Vol. 100, No. 1, July 1984

52

CHAUVETEAU, TIRRELL, AND OMARI I

1.6

,,

1.4

I

I

I

HPAM

c < c ~

that even above C*, there are apparently simultaneous effects of adsorption, creating a bound layer, which in turn, somehow acts as a hard-core repulsive wall for polymer molecules in solution flowing over it. CONCLUSIONS

1.2

/ 1.(;

~' ~ I 0.1

• SiC • Sand I 0.2

I 0.3

res Non-adsorptiol Adsorption Adsorption I 0.4

FIG. 6. Data on relative bulk and pore viscosities of HPAM at low concentrationboth in presence(A, V) and absence (O) of adsorption.

on top of the adsorbed layer. In other words, the thickness e of the adsorbed layer determined hydrodynamically appears to affect a polymer solution flowing over it exactly as would a hard-core repulsive wall. Whether this is indeed the case awaits further study and is discussed further below.

B.2. Flexible polymers: Effects of concentration. All experiments at higher concentration, into the semidilute regime of the flowing polymer solution, were conducted in the sandstone porous medium. The range of concentrations studied was from 340 #g/cm 3 (C[n] = 1.24) to 2080 #g/cm 3 (C[~/]) = 8.00)). Figure 7 shows the data on ~rb/~/n, as a function of concentration. We see again the anticipated linearity up to C*[~/], above which there is a departure. Three theoretical lines are drawn. One is the line representing Eq. [ 15] with constant parameters & and e. With the short dashed line, we show the predictions of Eq. [ 15] but with ~ varying with concentration as (C/C*)-3/4; we see that apparently underprediets the data at C > C*. The long dashed line shows the theory with both 5 and e varying as (C/C*) -3/4, as is expected from scaling theory, and the prediction is much better. We certainly do not claim that this is proof of the scaling exponent, only for the fact that the range concentration profile disturbance by the wall diminishes with increasing concentration Cb and above C*. Clearly too, Fig. 7 shows Journal of Colloid and Interface Science, Vol. 100, No. 1, July 1984

The contribution we feel has been made by the work presented in this communication has been primarily to show, through the use of a two-fluid model, which is both conceptually simple and easy to use, how a wide variety o f different experiments on flow of polymer solutions in porous media can be synthesized into a coherent and physically reasonable picture. Data were presented here on flexible and rigid macromolecules, over a wide concentration range, in four chemically and structurally different porous media, as a function of molecular and pore size, with both repulsive and attractive walls. While none individually has an overabundance of data, taken together Figs. 2-7 present a cogent, logical and consistent picture. The picture that emerges presents several profound, fundamental physical facts. First and most importantly, in porous media polymer rheology (at low deformation rates), the main influences on the flow properties inside the pore are characteristic size scales: pore size, of course, and the range o f the concentration inhomogeneity caused by the wall (the molecular size in dilute solution). There is a great insensitivity to the details of

1.75

~ x

HPAM Sand Pack

/

~

.. e

1.~

1.25 C'['r/J

1o~

zI

~

b-----

6I

Theory ~and aC-5/4 i 10

c[~] RG.7. Relativebulkandporeviscositiesof HPAMas a function of concentration with adsorption.

VISCOSITY OF POLYMER SOLUTIONS IN SMALL PORES any particular geometry or chemical nature once these size scales are determined. The point is that these size scales tell most of the story and therefore, turning things around, narrow channel flow is a good means to study such size scales (44). This is not to diminish the value of undertaking more rigorous and exact theoretical studies. We have shown clearly the anticipated rheological effects of wall zones of depletion and adsorption, and that the characteristic sizes of these zones is that of the macromolecule itself in dilute solution. Above some overlap concentration, the thicknesses of these layers diminish with increasing concentration for flexible polymers while remaining constant for rodlike polymers. The least expected results of all those presented here are those of Figs. 6 and 7, where we set what is apparently the effect of a concentration depleted layer on top of an adsorbed layer. This picture envisions a n o n m o n o t o n i c concentration profile near an adsorbing wall exposed to a polymer solution of bulk concentration Cb. In other words, for this view to be correct, we must have a concentration above Cb at the wall dropping down over a distance e to a value well below Cb, and then increasing again over a distance 6 back up to Cb. This sort o f n o n m o n o t o n i c density profile has been documented for low molecular weight fluids near solid surfaces, and has been shown to be caused by the repulsive forces between the molecules (45, 46). This occurs even for molecules whose interactions are essentially those of hard spheres (45). In a good solvent such as we have used, interactions among the polymer molecules may be effectively repulsive enough to behave in some ways similar to hard spheres. This might be a plausible picture in dilute solutions where ~ and - 0(/~); however, it is dearly inconsistent to assert this picture at high concentrations when we are also saying that e and 6 are diminishing with increasing molecular overlap. We expect that this behavior would not be seen in theta solvent where the binary interactions are annulled. In fact, we feel that it is

53

likely that 8 and c would really decrease with increasing concentration even below C - C* due to these interactions. It seems to us that experiments comparing rheology in the presence of adsorbed layers in both good and theta solvents may be useful in deciding these issues. In addition to the fundamental interest of the work presented here for understanding the physics of polymers in confined and dimensionally restricted situations, it has significance for applications in polymer flooding for enhanced oil recovery as well. Clearly polymer solution properties in small pores can be radically different than those measured in routine laboratory instruments. The data presented here, interpreted in light of the two-fluid model, allow one to predict porous media rheology from conventional viscometry with the additional inputs of pore size, polymer size, and pore wall-polymer interaction. ACKNOWLEDGMENTS The authors thank Philipe DeLaplaceand Ren6Tabary for makinga large fraction of the measurements reported here. Financial support from l'Association pour la Recherche sur les Techniqued'Exploitationdu P&role(ARTEP), composedof the Soci&6Nationale Elf-Aquitaine, the Compagnie Francaise des P&roles and the Institut Francais du P&role, the Alfred P. Sloan Foundation, and from the U. S. National Science Foundation (Polymers Program, NSF-DMR-8115733) is much appreciated. REFERENCES 1. Chauveteau, G., J. Rheol. 26, 111 (1982). 2. Aubert, J. I-I., and Tirrell, M., J. Chem. Phys. 77, 553 (1982). 3. Savins, J. G., Ind. Eng. Chem. 61, l0 (1969). 4. Gogarty,W. B., Soc. Pet. Eng. Paper 1566-A(1967). 5. Chauveteau,G., and Kohler, N., Soc. Pet. Eng. Paper 9295 (1980). 6. James, D. F., and Saringer, J. H., J. Non-Newtonian Fluid Mech. 11, 317 (1982). 7. Chauveteau, G., Soc. Pet. Eng. Paper 10060 (1981). 8. Aubert,J. H., J. Colloid Interface Sci. 96, 135 (1983). 9. Casassa, E. F., J. Polym. Sci. Part B 5, 773 (1967). 10. Casassa, E. F., and Tagami, Y., Macromolecules 2, 14 (1969). 11. Casassa, E. F., Macromolecules, in press. 12. Auvray, L., J. Phys. (Orsay, Ft.) 42, 79 (1981). 13. de Gennes, P. G., "Scaling Concepts in Polymer Physics." CornellUniv. Press, Ithaca, N. Y., 1979. Journal of Colloid and Interface Science, Vol. 100, No. 1, July 1984

54

CHAUVETEAU, TIRRELL, AND OMARI

14. Hesselink, F. T., J. Colloid Interface Sci. 60, 448 (1977). 15. de Gennes, P. G., Macromolecules 14, 1637 (1981). 16. Edwards, S. F., Proc. Phys. Soc. London 85, 613 (1965). 17. Daoud, M., Cotton, J. P., Farnoux, B., Jannink, G., Sarma, G., Benoit, H., Duplessix, R., Picot, C., and de Gennes, P. G., Macromolecules 8, 804 (1975). 18. Graessley, W. W., Polymer 21, 258 (1980). 19. Doi, M., and Edwards, S. F., J. Chem. Soc. Faraday Trans. II 74, 560 (1978). 20. Doi, M., and Edwards, S. F., J. Chem. Soc. Faraday Trans. H 74, 918 (1978). 21. Berry, D. H., and Russel, W. B., J. Fluid Mech., in press. 22. Whittington, S. G., Advan. Chem. Phys. 32, 1 (1982). 23. Dolan, A. K., and Edwards, S. F., Proc. Roy. Soc. London Ser A. 343, 427 (1975). 24. Jones, I. S., and Richmond, P., J. Chem. Soc., Faraday Trans. H 73, 1062 (1977). 25. Scheutjens, J. M. H., and Fleer, G. J., J. Phys. Chem. 83, 1619 (1979). 26. de Gennes, P. G., Rep. Progr. Phys. 32, 187 (1969). 27. Aubert, J. H., Tirrell, M., and Chauveteau, G., Amer. Chem. Soc. Polym. Div. Prepr. 23(2), 15 (1982). 28. de Aguilar, J., and Bird, R. B., J. Non-Newtonian FluidMech. 13, 149 (1983). 29. Utracki, L., and Simha, R., J. Polym. Sci. Part A 1, 1089 (1963). 30. Aubert, J. H,, and Tirrell, M., J. Liq. Chromatogr. 6, 219 (1983).

Journalof Colloidand InterfaceScience.Vol. 100,No. 1, July 1984

31. Rowland, F. W., and Eirich, F. R., J. Polym. Sci. Part A-1 4, 2401 (1966). 32. Varoqui, R., and Dejardin, P., J. Chem. Phys. 66, 4395 (1977). 33. Gramain, P., and Myard, P., Macromolecules 14, 180 (1981). 34. Seright, R. S., and Maerker, J. M., Amer. Chem. Soc. Polym. Div. Prepr. 22(2), 30 (1981). 35. Kohler, N., and Chauveteau, G., J. Pet. Technol. 33, 349 (1981). 36. Holzwarth, G. M., ACS Syrup. Set. 150, 15 (1981). 37. Lecourtier, J., and Chauveteau, G., Macromolecules 17, in press. 38. Wellington, S., Amer. Chem. Soc. Polym. Div. Prepr. 22(2), 63 (1981). 40. Chauveteau, G., and Zaitoun, A., "Proceedings European Symposium on Enhanced Oil Recovery, Bournemouth (1981)." 41. Muller, G., Universit6 de Rouen, private communication. 42. Kok, C. M., and Rudin, A., Makromol. Chem. Rapid Commun. 2, 655 (1981). 43. Fuller, G. G., J. Polym. Sci., Polym. Phys. Ed. 21, 151 (1983). 44. de Gennes, P. G., Macromolecules 15, 492 (1982). 45. Christenson, H. K., Horn, R. G., and Israelachvili, J. N., J. Colloid Interface Sci. 88, 79 (1982). 46. Snook, I. K., and van Megen, W., J. Chem. Phys. 72, 2907 (1980). 47. Graessley, W. W., Advan. Polym. Sci. 16, 1 (1974).