Rheological characterization of polymer gels and networks

Polymer Gels and Networks 2 (1994) 229-237 ~) 1994 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0966-7822/94/$07-00 ELSEVIER

Rheological Characterization of Polymer Gels and Networks

Simon B. Ross-Murphy Division of Life Sciences, King's College London, Campden Hill Road, Kensington, London W8 7AH, UK

ABSTRACT Although rheological measurements are still not familiar to all workers on gels and networks, a recent definition of the gel state is based upon rheological criteria. Small deformation time, strain and frequency sweeps enable the investigation of gelling properties on a curing system. These experiments have been improved by recent technical advances such as 'multi-wave' experiments, in which the simple sine wave history is replaced by superimposed harmonics, or more complex oscillatory deformations. Large deformation rheometry involves testing a gel sample either in tension/compression or in shear, up to and sometimes past the yield or failure point. Strain, temperature and concentration effects can be investigated using Smith's failure envelope.

INTRODUCTION Polymer gels and networks can be divided into three main classes. The first of these is the covalently cross-linked materials formed by such conventional routes as vulcanization of linear polymer chains, polycondensation of multifunctional oligomeric species or, as with the ubiquitous polyacrylamide gels, by polyaddition of linear (and a small but necessary proportion of branched) unsaturated precursors. Above the gel point the weight average molecular mass of such a sample is infinite, and it will have a finite equilibrium modulus. The second class is the so-called entanglement networks formed simply by the topological interaction of polymer chains either in the melt or in solution, when the product of concentration (C) and molecular weight (relative molecular mass Mr) becomes greater than some critical molecular mass for entanglements Me. There has been much progress in this area since it was recognized that their behaviour can be described by tube (reptation) models) The final category, the so-called physical gels are, arguably, the most 229

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complex of the three. In these materials the polymer chains are physically cross-linked into networks, the cross-links themselves being of small but finite energy, and/or of finite lifetime. 2 The latter include examples formed from both synthetic and biological polymers. Such cross-links usually act not at a point on the chain as do covalent cross-links, but involve more extended 'junction zones'. There are many examples of these including gels formed from both tactic and atactic polystyrenes in appropriate solvents, PVC, PVA, ionomer systems, gelatin, agarose and globular proteins. In many of these, the non-covalent cross-links involve extended quasi-crystalline junction zones of partly helical form. 3

POLYMER GELS AND NETWORKS The present paper will cover only the two categories of covalent and physical gels. Clearly the first point of issue is the often posed question about what constitutes a gel. We will follow the recent definition of Almdal and co-workers 4 that is based upon rheological criteria. They follow previous workers in accepting that a gel is a soft, solid or solid-like material, which consists of two or more components, one of which is a liquid, present in substantial quantity. However, in their definition, a gel must also show a fiat mechanical spectrum in an oscillatory shear experiment. In other words, it should show a shear storage modulus (G') which exhibits a pronounced plateau extending to times of the order of seconds, and a loss modulus (G") which is considerably smaller than the storage modulus in this region. Interestingly they do not require that the gel possesses an infinite relaxation time. They also exclude systems cross-linked in the bulk (sometimes called xerogels). Almdal and co-workers regard only substantially swollen polymer networks as gels.t In their definition, and contrary to some views, entanglement networks in solution can be included. Although these behave as gels only at frequencies higher (timescales shorter) than the lifetime of the topological entanglements, this lifetime can be quite long for samples of very high Mr. Indeed, according to reptation theory, the terminal relaxation time for entangled linear polymers is approximately proportional to Mr3.1 Since the Almdal definition involves rheological measurements, and these are still not very familiar to all workers on gels and networks, this paper will serve as a brief introduction to modern experimental and theoretical methods, with some perhaps more speculative thoughts on future prospects.

SMALL AND L A R G E D E F O R M A T I O N MEASUREMENTS The stress on a material is defined as the force F acting per initial unit area Ao. This tends to produce a deformation, the strain, defined as the ratio of the change in dimension relative to the original dimensions. For a tensile deformation this stress cr produces a strain e equal to dl/i or vice versa. Here t This point is still somewhat contentious, since some workers use the two terms 'polymer gel' and 'polymer network' synonymously.

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dl is the change in dimension of a cubic sample with sides of length /. The application of a shear stress T to such a sample is more complex, but it can still be regarded as a displacement dl acting parallel to the cube. The resultant shear strain 3, is still given by dl/l. The Young's or tensile modulus E is E = o'/e

(1)

while the shear modulus G is given by G = r/~

(2)

Since strain, being the ratio of two dimensions, is dimensionless, the SI units for both stress and modulus are Nm -2 or Pa. One problem, particularly for the shear modulus, is that measurements are rarely, if ever, performed on a simple cube of material. More useful 'geometries' include parallel plate, cone/plate and Couette (cup/bob), and the relationships between stress, strain and dimensions for these are given in standard texts) When the two terms, small and large deformation, are applied to gels and networks, they have no absolute meaning. For some samples, such as elastomers, the linear viscoelastic strain region, where stress and strain are proportional, i.e. where eqns (1) and (2) apply strictly, is quite large. Indeed it may extend to deformations of greater than four. For other systems the corresponding strain region may extend to values of ca 0.1 (10%) or less. For some colloidal gel networks it may be less than 10 -4 (0"01%). Essentially different instrumentation has been developed for small and large deformation measurements. In the former case specialized dynamic mechanical analysis equipment and oscillatory shear rheometers can be used. For large deformation experiments, devices such as tensile testers are employed. The former are invaluable for making measurements on 'curing' (gelling) systems, while the latter are used more to investigate failure of fully cured systems. Large deformation measurements are also much more relevant to ultimate (failure) properties of the material, whilst small deformation measurements are more easily related to molecular (macromolecular) properties of the component systems.

SMALL D E F O R M A T I O N R H E O M E T R Y Before ca 1982 there were very few measurements of this nature on gels and gelling systems, but the arrival of new and better instrumentation has altered the situation substantially. Nowadays gel cure experiments can be carried out in real time. In these, a fluid (pre-gelled) system is loaded on to, for example, a parallel plate assembly, one of the plates is oscillated at a fixed frequency and fixed maximum strain, and the response of signal (different in phase and amplitude) is monitored. 5 As the gel begins to 'set up' G' and G", the real and imaginary parts of the complex shear modulus (or in tensile deformation the corresponding Young's moduli E' and E") and tan 6 (the ratio G"/G' or E'/E") can be charted against time. For systems allowed to gel under a well-defined thermal regime, there is a characteristic 'cure' curve of log (G', G") against time. For the fluid state, G"

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will be greater than G'. After the start of the cure experiment there is an initial lag time. Subsequently both G" and G' begin to increase, but with G' increasing faster than G". Consequently at a given time there is a cross-over where G' becomes greater than G", sometimes associated with the gel(ation) time. Subsequently G' continues to increase, before levelling off, while G" usually passes through a parabolic maximum, and then decreases to zero. The latter is an effect associated with the relaxation of 'dangling chain ends'. 6 After a further length of time it is appropriate to subject the sample to strain and frequency sweeps to investigate gel properties on the cured system. Most workers use cone/plate, parallel plate or Couette geometries. In principle any of these, or specialized dynamic mechanical equipment employing three point or beam bending test fixtures, may be used. The cone and plate system has the advantage that, provided the cone angle is small enough, say less than 0-1 rad, the nominal strain across the sample may be considered constant. However it does have the disadvantage that the cone truncation can be extremely small (say 50 tzm). Consequently this geometry cannot be used if the temperature is altered, either by heating or cooling depending on the particular system, to induce gel formation. Consequently, parallel plate geometries, including a slightly oversized lipped" bottom plate to hold any protectant oil layer, or Couette systems, have been widely applied. Detection of the gel point itself has often relied on very simple measures. For example it may be judged to be when the sample gelling signal becomes just greater than the background noise, when G' becomes greater than a pre-assigned threshold value, or the time when G' becomes greater than G" (or tan ~ becomes less than 1), sometimes called the cross-over point. However Winter and co-workers 7-9 have suggested that a better criterion is when, in a frequency sweep experiment, both G' and G" show power law behaviour with the same exponent, n. In other words, at the gel point, G ' = K1to n and G" = K2to n. As Winter has shown, the proportionality constants K1 and K2 are not generally equal so this time does not generally correspond with that for the G' vs G" cross-over. Although, as the present author has pointed out, his method is not always useful because of lack of pre-gel signal) ° it has shown the requirement for developing so-called 'on-the-fly' frequency sweep software.

ADVANCES IN SIGNAL PROCESSING SOFTWARE Some commercial instruments (e.g. Rheometrics) already have such 'multiwave' software available. By using this, cure experiments may be carried out, not just at a single frequency, but at several discrete frequencies, by generating harmonics of the nominated fundamental frequency within the time given by the reciprocal of this longest frequency. In this way a series of higher frequency measurements can be performed during the time associated with one wavelength of the lowest frequency. For example a 'two points per decade' sweep from 0.1 to 10 Hz would take at least 10 + 3.16. • • + 1 + 0.316. • • + 0.1 = ~14.54 s, compared to just 10 s in multi-wave mode. Clearly the saving in time would be even greater if more intermediate frequencies were selected. In this way it becomes practical to collect a series of frequency measurements in real time, in order to test the Winter gel point criterion. The principle of the technique is

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described by Holly et al. 11 Figure l(a) and (b) portrays the multi-wave approach. The first three waveforms in the above series are illustrated in Fig. l(a), and the summed signal is shown in Fig. l(b). Clearly there are other potential signal processing techniques that can be employed, and several of these have already been discussed in the rheological literature. One of these is to generate all potential frequencies ('white noise') simultaneously and then to use Fourier analysis to pick out discrete frequencies. Another approach (obviously not suitable for a kinetic gelling system) is to employ the same technique as has been used in Fourier transform IR and NMR instruments. For this the time dependent stress relaxation response, usually an approximately exponential decay, is measured following an 'instantaneous' strain impulse. This time domain signal is then Fourier transformed into the frequency domain. Accuracy is obtained by summing a series of repeated impulse experiments. This approach differs from the usual transformation 5 of a stress relaxation experiment to the frequency domain and

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vice versa by the addition of the sum accumulation stage. Such a method has limitations for gels as the terminal relaxation time will be very (or infinitely) long. A recent approach by Winther and co-workers 12 uses an exponential relaxation time approximation to produce an algorithm up to twice as fast (in real time) as the full Fourier transform method, which is also quite stable with respect to external noise. Grehlinger and colleagues ~3 at Rheometrics have employed the so-called 'arbitrary wave' software option on their instruments, in which a time variable, but non-sinusoidal signal, is employed as the drive response, to demonstrate the relative advantage of other signal processing approaches. In the future we expect such advanced software-generated processing to become even more widely exploited. O T H E R SMALL D E F O R M A T I O N METHODS In case it should be thought that only the most sophisticated equipment is suitable for making gel measurements, it is interesting to see that many home built devices are still being employed. We briefly describe three of these here. The first is the Ward and Sanders U-tube method, 14 in which the gel is set in a simple U-tube manometer, one arm of which is attached to an air line, the other is free. Both may be observed with a travelling microscope. The air pressure exerts a compression stress in the sample and the deformation of the sample can be measured from the differential heights of the manometer arms. E is calculated using the analogue of Poiseuilles equation for capillary flow. By roughening the inner tube wall the method can be used for syneresing gels. 15 In the torsion pendulum, a torsion wire is attached to the inner bob of a Couette system containing the gel. The wire is fixed to a rigid frame, and if a small torque is applied, the whole system will undergo free oscillations, with resonant frequency tot. G' is proportional to (tor- to0)2, where tOo is the resonant frequency in air. More sophisticated systems have been made which undergo driven oscillations, and which are useful for following gelation processes. 16 Such an instrument is probably the best home constructed design for making absolute measurements. Finally the microsphere rheometer is the oscillatory analogue of a sphere falling through a fluid under gravity. A small magnetic ball is placed into the sample. By using external AC and DC coils, the sphere can be positioned and made to oscillate with the frequency to of the AC supply. The maximum deformation can be observed with a travelling microscope, or alternatively tracked using a projection system and photo-diode arrays. Several different designs have been published. 17'1s LARGE DEFORMATION RHEOMETRY Large deformation testing involves examining a gel sample either in tension/compression or shear, up to and sometimes past the yield or failure point. Such measurements are usually of equal (or greater) value to the material scientist since, in applications, gel samples are often subjected to stresses and strains well outside the linear viscoelastic region. The practical

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difficulties are, however, still quite significant if really meaningful data are to be obtained over a wide range of strains. For example, although it is easy to set up an experiment to compress a cylinder of gel, the response of the sample is often dominated by friction between the sample surface and the compressing plates. If the Poisson's ratio of the gel is close to 0.5 (totally incompressible) uniaxial compression say in the z-plane is equal to biaxial extension in the x-and y-planes. However this extension can only take place if the surfaces of the gel are very well lubricated. Usually there is a component of friction and the sample begins to 'barrel' or 'bulge'. Tensile measurements are just as difficult to perform because merely gripping weaker gel samples with pneumatic or screw clamps will induce preferential failure at the clamping point. One approach to overcome this problem is to cut notches in the centre of the sample to encourage failure at this specific point. Large deformation measurements in shear are also difficult to perform because of de-adherence of the sample from the test fixture. In a conventional parallel plate or cone/plate assembly, de-adherence of the sample from the fixture, 'slippage', is very ditiicult to avoid. Of course this can be reduced by roughening the surfaces of the fixtures. However, because of this very roughness, the calculation of real stresses and strains is often difficult. Work by Bagley and his co-workers t9 is particularly interesting, because it compares the large deformation behaviour of gels under both lubricated and unlubricated conditions in uniaxial compression. The lubricated surfaces employed paraffin oil and Teflon covered platens, while the unlubricated experiment used cyanoacrylate adhesive to fix the gel to aluminium foil. Under these circumstances the stress-strain profiles for both bonded and lubricated compression agreed, but stress and strain at fracture are much lower in bonded compression (by factors of around three for the samples measured). One method that we and others have found quite valuable is to extend either rings or, even better, 'race track' shaped gel samples over dowel pins so that tensile measurements can be made without the need to use clamps. E° Our own work 21 was carried out on gelatin gels, which are comparatively easy to measure, as the extension ratio at failure can be as much as three, but Kohyama and co-workers 2e have used the method successfully for konjac mannan gels. Another approach is to measure cylindrical gel samples in tension, but rather than clamping them, superglue the ends on to sandpaper supports, themselves attached to clamped metal plates. Stading and coworkers 23 have used this technique for both notched and un-notched /3lactoglobulin gels. THE F A I L U R E ENVELOPE A P P R O A C H Since failure is a statistical process, instead of several experiments being carried out on a single sample as in the small deformation technique, many replicate experiments need to be employed on nominally identical samples. The parameters of major interest are the initial strain modulus as measured in small deformation, the overall shape of the stress-strain profile, the (tensile) yield or failure stress tra, failure strain es and their ratio sometimes called the failure modulus EB. It is also important to see how these are affected by changes in temperature or strain rate.

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One strategy is simply to measure a mean and standard deviation of the various parameters. It is then usual to apply the methods of fracture mechanics to calculate, for example, surface energy and crack size effects. However most gels are not glassy materials but rubbery, and so they tend to fail by rupture rather than fracture. There are fewer methods for investigating failure in such systems, but we have found one of the most useful approaches is to investigate strain and concentration effects by employing reduced variables as in the Smith failure envelope technique. 24 His method was originally applied to the rupture of bulk elastomers, but can be just as useful for gels. In its simple form it merely requires data to be collected for say 10 replicates at each of three or more (logarithmically) different strain rates, and at several different temperatures. A plot of OrB VS eB is constructed for all replicates and all strain rates, as shown in Fig. 2(a). Increasing the gel concentration is then analogous to a

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Rheological characterization of polymer gels and networks

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lowering of the temperature in the original Smith approach. This implies that the nominal strain applied to the higher concentration gel samples is greater by some, so far unestablished, strain magnification factor. This can be found empirically by multiplying the higher concentration samples by a constant factor (equivalent to sliding the data along the x-axis) so that they can be superposed into an envelope of failure points. Here the superposed data are shown in Fig. 2(b). Although this approach seems arbitrary, it has a reasonably sound theoretical basis, allied to the fact that now the failure behaviour of all concentration samples at all strain rates can be calculated from the new plot. More details of the approach and its justification are given elsewhere. 2°'25

ACKNOWLEDGEMENTS SBRM thanks Professor K. Kajiwara, Professor Y. Osada and the Society of Polymer Science Japan Gels Group for their generous hospitality and the opportunity to attend their joint meeting with the Polymer Networks Group.

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