Chapter 10
Rheology
Rheology is the study of the deformation and flow of matter. A body may deform as a result of a force applied to it and if it is elastic it will return to its original state on removal of the constraining force. As we shall be concerned only with the effects of forces acting uniformly over surfaces, the appropriate measure of these forces is the ‘stress’, which is a force per unit area. The stress may be applied at right angles or tangentially to the surface and the resulting deformation is known as the ‘strain’. The measure of strain is a non dimensional quantity, a fractional increase or decrease, of length or volume. An ideal elastic body is one in which the strain is directly proportional to the stress and we are able to define a modulus of elasticity which is a ratio of the two parameters. Many materials, however, do not behave in this ideal manner and the deformation may increase with time as long as a stress is applied, or the deformation may not recover completely when the stress is removed. Thus, in characterizing the rheological properties of a body, we are usually concerned with the relations between stress, strain and time.
Elastic Solids
Only an elementary treatment will be given in order to introduce the reader to basic concepts. References for further reading are given on p. 160.
Modulus of Rigidity
Consider a small cube of isotropic material which is subjected to equal forces F applied tangentially over the four faces (Fig. 10.1A), i.e. the cube is subjected to shearing stress (an isotropic material is one in which the physical properties do not depend on direction). The cube is in equilibrium under the action of equal and oppositely opposed couples and we may refer its resulting deformation to the face abcd as a reference plane. The deformed figure has the shape A′B′C′D′, abcd. The originally square face ABba has deformed into the rhombus A′B′ba and the amount of deformation can be measured in terms of the angle of shear θ. If L is the length of the side of the cube, the shearing stress p = F/L2 and we can define the modulus of rigidity, η, by the equation:
Fig 10.1 Representation of (A) Modulus of rigidity, (B) Bulk modulus (C) Young’s modulus.
Bulk Modulus
Consider the cubical body in Fig. 10.1B subjected to a uniform hydrostatic pressure, p. If V is the original volume of the body and Δv is the decrease in volume on applying the pressure, then the ‘bulk modulus’ k is defined by the equation:
Young’s Modulus
If the cube is subjected to a tensile stress (F2) or a contractile stress (F1) on two opposite faces (Fig. 10.1C) then Young’s modulus, E, is given by:
where Δl = change in length of body in direction of applied stress and L = original length of side of cube.
Young’s modulus and the bulk modulus are interrelated by the following equation:
where σ is Poisson’s ratio.
This ratio can be defined as the fractional extension at right angles to the applied stress divided by the fractional contraction along the direction of the applied stress. This concept may be visualized if we consider the cube (Fig. 10.1C) subjected to a contractile stress F1, on the two opposite faces; the remaining faces are unconstrained and we would expect extension of the specimen at right angles to the direction of the force. The quantity σ is a characteristic of the material and from the equation above cannot be greater than 0.5.
Measurement of Elasticity
In pharmacy an assessment of the elasticity of gels is important in order to control the quality of gelling agents such as gelatin, agar and pectin. Gelatin gels behave as ideal elastic bodies provided the deformation does not result in the elastic limit being exceeded when irreversible deformation occurs and the gel breaks. Although it is possible to measure the modulus of elasticity of a gel by using cubes of the material and deforming them under a shear stress, it is generally more convenient to use the following methods which have been widely accepted in this country, particularly for gelatin gels.
The Food Industrial Research Association (FIRA) jelly tester consists essentially of a small metal vane mounted on a shaft carrying a scale, the whole assembly being rotatable by application of a torque applied by water running at a pre determined rate into a counterpoised bucket connected to the shaft. The vane is inserted into the jelly sample to a definite depth and a torque is applied until the vane has rotated through a definite angle. The weight of water needed to do this is a measure of the jelly strength. A gel box of standard size is used in which the gel is matured under standard conditions; any slight wall effect will give a higher reading for jelly strength, but for comparative work this is not important.
The Bloom Gelometer described in British Standard 757:1959 is the standard instrument for the determination of jelly strength of gelatin. The jelly strength is expressed as a Bloom number which is equivalent to the weight in grams necessary to produce by means of a plunger 12.7 mm in diameter, a 4 mm depression in a jelly of 6.66 per cent w/w, matured at 0°.
The Bloom number has no exact physical significance but is dependent on the average molecular weight of the gelatin. It has been reported by Ellis (1962) that the solubility of glycerin suppositories is influenced by the jelly strength of gelatin and that the solution time of the present BP glycerin suppository is too prolonged, especially if good quality gelatin is used. The specification for BP gelatin is a jelly strength of not less than 150, no upper limit being specified.
An absolute method for the rigidity modulus of gelatin gels has been described by Saunders and Ward (1954). A cylinder of gelatin set in a glass tube is subjected to shearing stresses by applying a known air pressure at one end of the tube. The rigidity of the gel is calculated from the volume displacement which is measured by causing mercury to move along a calibrated capillary tube. The method was used for measuring rigidities of gelatin gels over the range 5 × 103 to 5 × 105 dyne/cm2 and was suitable for gelatin concentrations up to 10 per cent. Nixon, Georgakopoulas and Carless (1966) using this technique showed that the presence of glycerin up to 40 per cent increased the modulus of rigidity of gelatin gels. No simple relationship was found between Bloom number and rigidity.
Fluids
When a shearing stress is applied to a fluid it deforms irreversibly, i.e. it flows. The term strain was used in the previous section to indicate deformation of a solid and in the case of fluids this term is replaced by rate of change of strain or more usually the term ‘rate of shear’, since the fluid is undergoing flow. Ideal viscous fluids known as Newtonian fluids show a direct relationship be-tween shearing stress and rate of shear, the ratio being known as a coefficient of viscosity. Thus the rate of flow of a Newtonian fluid such as water through a tube is directly proportional to the pressure drop across the tube. Solutions of high molecular weight polymers, emulsions and suspensions show anomalous flow properties and cannot be characterized by a single viscosity value. The term ‘apparent viscosity’ is sometimes used to indicate that the viscosity determined by a particular viscometer is not an absolute value and in such cases it is important that the exact conditions of the measurement are specified.
Viscosity Of Fluids
The viscosity of a fluid is the internal resistance or friction involved in the relative motion of one layer of molecules with respect to the next. In order to maintain a constant rate of flow, a shearing stress has to be continually applied. When there are strong attractions between the molecules of a liquid (van der Waals forces, dipole interactions, etc.), the viscosity will be high; when the attractions are weak, the viscosity will be low. In gases, the molecules are in general so far apart that no appreciable intermolecular force exists and the viscosity can be explained by the kinetic theory of gases. Consequently, the viscosity of gases and liquids changes quite differently with alteration of temperature. At high temperatures, the mutual attractions between molecules of a liquid decrease and the viscosity falls, whereas in the case of a gas, the increased kinetic movement of the gas molecules causes an increase in viscosity.
Flow Characteristics of Newtonian Fluids
Consider a fluid flowing smoothly over a horizontal base AB (Fig. 10.2). Every particle of the liquid moves parallel to the surface in the same direction, i.e. it exhibits laminar or streamline flow.
Fig 10.2 Streamline flow.
The velocity v at a distance x above the base varies from zero where x = 0 to V, where x = h. The velocity gradient dv/dx is a measure of the laminar flow. Consider a thin stratum of liquid Xl Y1, X2 Y2. The more quickly moving liquid above this layer exerts a tangential force tending to accelerate it while the more slowly moving liquid below it will tend to retard it. If the accelerating force from the liquid above X2Y2 exactly balances the retarding force from the liquid below X1Y2 then the layer X1Y1X2Y2 will move in steady motion. Newton deduced that the tangential force F over the area A was given by:
η is the coefficient of dynamic viscosity and has the dimensions of mass × length–1 × time–1. The absolute unit is the poise (P) (g/cm/sec. In SI Units 1P = 0.1 Nsm–2). A liquid has a viscosity of 1 P (100 cP) if a steady tangential force of 1 dyne (10μN) produces a relative velocity of 1 cm/sec between two parallel plates of area 1 cm2, separated by 1 cm and immersed in the liquid. The kinematic viscosity (v) of a fluid is defined by the equation:
where ρ is the density of the fluid. The unit of kinematic viscosity is stokes (S) (For SI units, 1 S = 10–4 m2/sec).
Liquids of equal kinematic viscosities will flow at identical rates through a given tube, when the pressure causing the flow is due solely to the hydrostatic head of the liquid. For any given hydrostatic head, the pressure is proportional to the density. Calculation of the kinematic viscosity is thus readily achieved by comparison with the flow rates of a liquid of known kinematic viscosity, when using a viscometer which is operating under a hydrostatic head. Dynamic viscosities are usually directly measured in other types of viscometers where the flow rate is not dependent on the density of the liquid, e.g. rotational viscometers.
Effect of Temperature
The dependence of the viscosity of a liquid on temperature is expressed approximately by an equation analogous to the Arrhenius equation of chemical kinetics:
where A is a constant depending on the molecular weight and molar volume of the liquid and E is ‘activation energy’ required to move one molecule past another. As a rough guide the viscosity of many liquids decreases by about 2 per cent for each degree rise in temperature.
Streamline and Turbulent Flow
Fluids flowing relatively slowly on a flat base as in Fig. 10.2, or through pipes, travel in an orderly manner in what is called streamline flow. The individual particles flow in straight lines parallel to the axis of the tube. Osbourne Reynolds first demonstrated the nature of streamline motion by carefully injecting a fine stream of dye solution into water flowing slowly through a tube. The unbroken filament of dye was carried along the axis of the tube. As the velocity of the fluid was in creased, the character of the motion changed and the injected dye was seen to swirl along in the tube becoming thoroughly mixed due to disorderly, turbulent flow of the liquid. By carrying out these experiments over a range of flow rates through smooth tubes, Reynolds was able to show that there was a critical transition velocity Vc between streamline and turbulent flow given by the following equation:
For flow in long smooth tubes, Re was approximately 2000, i.e. Re < 2000 results in streamline flow. It should be remembered that the geometry of the entry port leading to the tube can affect the smoothness of flow and this is likely to be most marked when short capillary tubes are used to study flow rates. It is essential that streamline flow is operating in order to apply Newton’s law for the calculation of viscosity. Reynold’s number is a dimensionless quantity, i.e. a pure number.
Determination of Viscosity
Capillary Tube Viscometer
The relationship between the rate of flow of a liquid through a capillary tube, and its viscosity was first given by Poiseuille.The Poiseuille equation for streamline flow is:
where V = volume of liquid (cm3) flowing in time t (sec),
This equation can be used directly for measuring the viscosity of a liquid, but for accurate measurements it is necessary to apply a correction for the kinetic energy gained by the liquid as it accelerates into the capillary tube from the reservoir. The driving pressure in the Poiseuille equation is that necessary to overcome viscous forces only and any additional force which is used to impart kinetic energy to the liquid must be allowed for. However, the correction term is negligible if the viscometer is designed with a large l/r ratio and the rates of flow are kept extremely low. Inaccuracies will also arise due to errors in measuring l and especially r and in practice the Poiseuille equation is generally used for comparative methods, when the rate of flow of the unknown is compared with the rate of flow of a liquid of known viscosity. If these two liquids have approximately the same flow rates then kinetic energy corrections will be self -cancelling.
The most frequently used capillary tube viscometer is the Ostwald type (Fig. 10.3). This consists of a U tube bearing two bulbs X and Y and, in one arm, a capillary CD of suitable bore. The tube is placed vertically in a thermostatically controlled bath (±0.05° or better). Sufficient of the liquid whose viscosity (η1) is to be determined is poured into bulb Y to reach mark E. The liquid is then sucked or blown up to a point 1 cm above A; the time (t1) for the liquid to fall from A to B is measured with a stop watch. The density of the liquid (ρ1) is determined. The viscometer is emptied, rinsed out with suitable solvents, dried and the whole operation repeated with a liquid of known viscosity η2 and density ρ2. The time of flow is t2.
Fig 10.3 Ostwald viscometer.
From Poiseuille’s law:
(10.1)
where P is due to the force of gravity:
As equal volumes of two liquids flow through the same capillary under a driving pressure proportional to density of the fluid:
(10.2)
and
(10.3)
For the measurement of kinematic viscosity the direct comparison with a liquid of known kinematic viscosity simplifies the procedure since:
(10.4)
The kinematic viscosity of a liquid is thus directly proportional to its flow rate.
The rate of flow through a capillary tube depends on r4, and since Ostwald viscometers can be obtained with r varying from approximately 0.2–2 mm, a range of 104 in viscosity can be covered. Suitable dimensions for a set of five viscometers of the pattern shown in Fig. 10.3 are given in British Standard 188:1957. The No. 0 viscometer can be calibrated with water but the larger sizes require calibration liquids of higher viscosity to ensure streamline flow and also adequate flow times for accurate timing. The Ostwald viscometer is specified for determining the kinematic viscosity of Liquid Paraffin BP.
The measurement of a very viscous liquid in a standard Ostwald viscometer is not convenient due to the difficulty of filling the viscometer with an accurate volume. Drainage errors during the filling procedure may result in overfilling. The suspended level viscometer overcomes this difficulty (Fig. 10.4).
Fig 10.4 Suspended level viscometer.
The additional side arm C ensures that the exit of the capillary is at atmospheric pressure, so that the total amount of liquid in the viscometer does not require to be fixed. The time of flow of the liquid from A to B is measured and the calculation is as before. This instrument is specified in the BP for determining the viscosity of solutions of methyl cellulose.
Falling Sphere Viscometer
A body falling through a viscous medium attains a constant terminal velocity when the accelerating force (gravity less buoyancy) is equal to the retarding force (viscosity). The retarding force (F) according to Stokes is:
(10.5)
for a smooth sphere of radius r cm, moving with a velocity of V cm per sec in a liquid of viscosity η poises. The upthrust on the sphere according to Archimedes’ principle is equal to the weight of liquid, density ρ1 displaced by the sphere, i.e. 4/3πr3ρsg, when ρs is the density of the sphere. The net downward force exerted on the liquid is therefore:
(10.6)
At the terminal velocity, net downward force = net upward force:
(10.7)
(10.8)
(10.9)
where D = diameter of sphere.
This equation, known as Stokes’ law is only valid for streamline flow of a sphere in an infinite fluid. Streamline flow occurs when Reynolds number (Re) is less than 0.2 where Re = DVρ/η.
In the falling sphere viscometer a rust-free ball bearing is timed in falling a measured distance through a cylindrical tube of liquid (Fig. 10.5). For accurate absolute measurements, a correction for the effect of the proximity of the sides of the tube is necessary. The value of η calculated from the equation has to be multiplied by the factor which is approximately (1 + 2.1 r/R) where R is the radius of the tube and where R > 10r. The equation in the BP includes this factor calculated for a sphere of 1.55 mm and tube radius of 25 mm. If the instrument is used for relative determinations, corrections for end and wall effects are eliminated.
Fig 10.5 Falling sphere viscometer.
The chief disadvantage of this viscometer is the large volume of sample required, which needs to be clear or translucent.
Rolling Sphere Viscometer
Whereas the falling sphere viscometer is limited to liquids of high viscosity, the rolling sphere model is capable of covering a range of viscosities from 0.5 to 2,00,000 P. The instrument consists essentially of a short glass tube of large diameter and a closely fitting ball of either glass or steel (Fig. 10.6). The tube is inclined at a definite angle and the ball is timed between two marks. Unlike the previous viscometer, a large ball falls more slowly than a smaller ball as the liquid has to pass between a smaller gap between the ball and the tube. The Stokes’ equation does not apply in these measurements; the viscosity in centipoises is given by the following equation:
where ρs = density of ball, ρ1 = density of liquid, t = time of fall and K = constant for instrument.
Fig 10.6 Rolling sphere viscometer.
The apparatus is calibrated with liquids of known viscosity. The method is capable of high precision provided that adequate temperature control is maintained.
Rotating Cylinder Viscometer
In this instrument, originally proposed by M. Couette in 1890, the sample fluid fills the annular space between two concentric cylinders, either of which may be rotated by a motor; the other cylinder is suspended elastically so that the torsional couple can be measured. In Fig. 10.7, the inner cylinder B is shown suspended from a torsion wire C and the outer one A is rotated. The liquid rotates in concentric laminar flow and exerts a viscous drag on the inner cylinder, causing a deflection proportional to the viscosity. The angular deflection may be measured by a mirror D attached to the wire, reflecting a light beam on to a horizontal graduated scale F. The rate of shear D at any given radius r is given by:
where R1 and R2 are the radii of the inner and outer cylinders, respectively.
Fig 10.7 Concentric cylinder viscometer.
If the annulus is narrow, an average rate of shear can be used. Many commercial viscometers are calibrated on the basis of the rate of shear at the inner cylinder being operative. The shearing stress exerted on the fluid may be calculated from the dimensions of the instrument and the torsion constant of the wire, or by calibrating the instrument using a liquid of known viscosity. If the rate of shear and the viscosity of the liquid is known, the stress constant for the instrument is readily calculated, since:
In the design of these viscometers it is desirable that the viscous drag exerted on the two ends of the viscometer is reduced to a minimum. These ‘end effects’ can be minimized by fitting a stationary guard ring over the upper surface of the inner cylinder or bob, the bottom of which is recessed in order to trap air in the cavity and prevent drag. A deeply recessed upper surface can be used instead of a guard ring and excess fluid will overflow into this, so that the height of the fluid coincides with the height of the inner cylinder. The Rotovisko viscometer uses this principle.
Examples of commercial rotating cylinder viscometers are the Ferranti Portable, Brookfield and the Rotovisko which will be briefly discussed. For a more detailed description of these and other types, the laboratory manual by van Wazer et al. (1963) should be consulted.
This is shown diagrammatically in Fig. 10.8. The rotating outer cylinder is inverted, compared with the conventional Couette type, which permits the determination on bulk fluids into which the cylinder is immersed. The inner cylinder, which is connected to a beryllium copper spring, has its angular displacement indicated by a conventional pointer and scale. The outer cylinder is driven by a synchronous motor with a three or five-speed gear box giving a range of 1 rev/min up to 300 rev/min. Rates of shear can be varied by speed of rotation and also by choice of inner cylinder. The mean shear rates vary from 930 to 0.15 sec–1. Low shear rates are restricted to liquids of high viscosity and high shear rates to liquids of low viscosity in order to obtain readings on the scale. Size of sample required is usually about 100 mL.
Fig 10.8 Ferranti portable viscometer.(A) Rotating shaft; (B) Detachable outer cylinder; (C) Detachable inner cylinder; (D) Guard ring (stationary); (E) Torque spring;(F) Viscosity indicating pointer.
Brookfield Synchrolectric Viscometer
This viscometer has been in general use in the USA since about 1940. It does not give results in terms of absolute shear rates, but has found wide spread use as a comparative instrument. The cylinder that is immersed in the fluid is carried in a rotating spindle driven via a calibrated beryllium copper spring mounted on the motor shaft (Fig. 10.9). The displacement of the spindle due to the viscous drag of the sample is indicated by a pointer attached to the rotating spindle which moves over the scale attached to the motor spindle. Readings are thus complicated by the pointer and scale both revolving, but it is possible to clamp the pointer to the scale and then to slip the motor in order to take a reading. For this reason, it is not a convenient technique for following viscosity changes while shearing the sample. A wide range of viscosity is covered and very low shear rates are possible. Since the bob operates in an ‘infinite sea’ of fluid the calibration data are usually expressed in terms of rev/min and Brookfield reading. Although viscosities of Newtonian liquids can be determined, the ‘apparent’ viscosity of non-Newtonian liquids can only be regarded as comparative measurements.
Fig 10.9 Brookfield viscometer. (A) Driving shaft; (B) Rotating scale; (C) Viscosity indicating pointer; (D) Torque spring; (E) Interchangeable bob; (F) Sample.
Rotovisko (Haake)
This is a very versatile viscometer and will measure viscosities in the range of 5 × 10–3 to 4 × 107 P. The range of rate of shear is 10–2 to 104/sec. A ten-speed gear box drives the inner cylinder via a flexible cable. As in the Brookfield viscometer, the cylinder is not connected directly to the drive but a torsion spring is interposed between the rotor and the drive shaft. The viscous drag on the rotor causes the spring to twist until its torque is balanced by the viscous drag. The resulting angular displacement of the rotor, relative to the drive shaft, is a measure of the shear stress and is measured by means of a slider arm connected to a potentiometer so that the stress value is read from a galvanometer. A variety of outer and inner cylinders is available in order to cover a wide range of viscosity. A continuous record of stress at a constant rate of shear can be obtained by connecting a recording voltmeter to the output terminals.
Cone and Plate Viscometer (Ferranti–Shirley)
Essentially the viscometer consists of a flat plate and a rotating cone with a very obtuse angle (Fig. 10.10). The apex of the cone just touches the plate surface and the fluid sample fills the narrow gap formed by the cone and plate.
Fig 10.10 Cone and plate viscometer (Ferranti–Shirley).Schematic diagram of the recording system.
The usual cone angles are about 0.30 so that only small samples (0.5–1 mL) are required.
The plate is maintained at a constant temperature by circulating water through it and the small volume of sample rapidly reaches an equilibrium temperature.
The rate of shear across the gap for small angles is constant and is given by Ω/ψ; where
The comparison with coaxial viscometers is shown on facing page.
The cone is driven by a variable speed motor through a gear train and a torsion spring is inter posed between the driving shaft and the shaft connected to the cone to measure the shearing stress by means of a potentiometer, in a similar way to the Rotovisko instrument. The rate of shear is capable of being varied over a very wide range, e.g. 0–18000 sec–1 for a 0.3° cone. The stress is indicated on a galvanometer connected to the potentiometer circuit.
The instrument can be modified to allow the simultaneous recording of shear rate and shearing stress during the constant acceleration of the cone, so that flow curves can be obtained automatically on an X–Y recorder. This is achieved by feeding the torque signal from the potentiometer/torsion spring to the X axis (calibrated in dynes/cm2) and a voltage proportional to motor speed is fed into the Y axis (calibrated per sec). The control unit permits different rates of acceleration and deceleration of the cone so that rheological properties of fluids that show time-dependent effects, e.g. thixotropy, can be studied. Automatic recording of flow curves is particularly useful when rapid changes of viscosity occur on shearing, as these may be missed when operating the instrument manually. However, a word of caution is necessary as inertial effects of the cone may produce erroneous results, particu- larly if rapid acceleration—deceleration is applied to liquids of fairly low viscosity. Under these conditions inaccurate hysteresis loops may be obtained with Newtonian liquids.
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The advantages of the cone and plate viscometer can be summarized as follows:
One of the disadvantages of the instrument is that it is unsuitable for coarse suspensions or emulsions owing to the narrow clearance of cone and plate. Usually suspensions of particles of size not greater than 30 µm can be sheared in the standard cone (0.3°).
Extrusion Viscometer
The FIRA/NIRD Extruder, manufactured by H.A. Gaydon & Company, Croydon, originally devised by Prentice of the National Institute for Research in Dairying, was used to investigate the consistency of butter. It was found that instrumental readings could be correlated with such subjective qualities as ‘spreadability’ and ‘firmness’. The instrument is equally suitable for assessing the consistency of ointment bases.
The basic principle of the extruder is extremely simple. A sample is removed from the bulk material in a cylindrical borer about 4 cm long and 1.25 cm diameter. The sample is then forced from this cylinder through a circular orifice by a plunger moving at a uniform speed. The thrust required to extrude the material is recorded on a moving chart throughout the test and this chart serves as a permanent record of the firmness of the material. Should any inhomogeneities be present in the sample caused, for example, by the presence of agglomerates of solid particles or air bubbles, the recorded thrust will fluctuate and the nature of the inhomogeneity can often be deter mined by an examination of the extruded thread of material.
Creep Testing
The rheological evaluation of viscoelastic semi solids (i.e. which possess viscous and elastic proper ties) can be done by a creep test where a stress is suddenly applied and maintained constant over a period of time. The resultant deformation (strain) of the material with time is known as the ‘creep curve’. One type of apparatus used for this measurement consists of two concentric cylinders, the inner cylinder being capable of rotation on application of a torsional couple. The viscoelastic material to be tested fills the annular gap between the two cylinders. To carry out the measurement, a small fixed tangential stress is applied to the material through the inner cylinder and the slow rotation of this cylinder is followed by means of an optical mirror or suitable sensing device (Davies, 1969). A typical creep curve is shown in Fig. 10.11 and can be analysed using the theory of linear viscoelasticity. Three ‘separate regions can be identified in this figure: A–B represents an instantaneous elastic component; B–C represents viscoelastic flow; C–D is associated with viscous flow. On removal of the applied stress, the instantaneous elasticity is observed (region D–E) but the material does not completely recover due to some energy being dissipated in viscous flow. The viscoelastic properties of ointment bases have been studied by Davies (1969) and by Barry and Grace (1971), the latter workers being interested in the rheology of soft paraffin studied by continuous shear and creep testing. The rheology of pharmaceutical and cosmetic semisolids has been reviewed by Barry (1974).
Fig 10.11 Creep curve of viscoelastic material (constant stress applied).
Flow Properties of Non-Newtonian Liquid
The viscosity of a Newtonian liquid is completely defined by a single figure, the coefficient of viscosity. This coefficient is the ratio of shearing stress rate of shear and provided that streamline flow is maintained, it is of course independent of the actual rate of shear. Many colloidal dispersions, suspen- sions and emulsions cannot be characterized by a viscosity coefficient and it is necessary to determine the relationship between stress and rate of shear over a wide range of shear rate. The resulting plot of stress versus rate of shear is known as a flow curve and examples are shown in Fig. 10.12. Viscometers that are suitable for these determinations are the coaxial and cone-plate viscometers in which the rate of shear can be varied. Fig. 10.12A shows the flow curves for two Newtonian liquids of different viscosities.
Fig 10.12 Flow curves of Newtonian and non-Newtonian fluids.
Pseudoplastic Flow
Solutions of many polymeric substances, e.g. cellulose ethers, tragacanth, alginates, etc. do not show a direct relationship between stress and shear rate (Fig. 10.12B). At low shear rates the ratio of stress –shear rate (i.e. apparent viscosity) is higher than at the greater shear rates. The flow curve is seen to straighten out at high rates of shear so that the solution reaches a limiting viscosity: This effect is ascribed to molecular interactions resulting in a three-dimensional network structure of solute molecules. For flow to occur this structure must be broken down, and as the rate of shear is increased the molecules will tend to become orientated in the streamlines of the liquid and offer less resistance to flow. Many attempts have been made by rheologists to represent the pseudoplastic curve by equations that have a theoretical basis, but with only limited success. The empirical power law
has been found to represent the flow curve of some pseudoplastic materials. D is the rate of shear and S the shearing stress. The exponent is indicative of non-Newtonian flow. If N has a value of unity the equation reduces to the simple Newtonian equation of flow. The greater the values of N above unity the greater the pseudoplastic behaviour of the material is a constant characteristic of the solution and does not have the same physical significance as a coefficient of viscosity. The equation may be written in logarithmic form:
The plot of log S against log D will give a straight line of slope N and intercept of log η′ on the log S axis. The advantages of obtaining an equation to fit a flow curve is that the curve may be more conveniently expressed in terms of the parameters N and η′ and this simplifies the compilation of data. For instance, these parameters could be obtained for different concentrations of a polymer in solution, and if a correlation between log η′ and concentration could be established, the flow properties of a solution of any other concentration could then be calculated. Kabre et al. (1964) applied this approach to the characterization of the viscosity of a number of natural and synthetic gums.
Plastic Flow
Whereas a Newtonian or pseudoplastic liquid will yield under the application of the smallest shearing force, a plastic material fails to flow until a certain shearing stress has been applied. This is typical of concentrated suspensions of solids, ointments and gels and is thought to result from the aggregation of particles forming a continuous but not orientated, structure throughout the mass. Flow cannot take place until the applied stress exceeds the force of flocculation which causes the particles to adhere together. When flow is initiated the particle layers move relative to the adjacent layers, and the particle contacts are broken but reform again on removal of the stress. Thus the internal structure of the substance is not permanently altered. This behaviour is expressed in Fig. 10.12C. Flow does not commence until the stress reaches the yield value f (dynes/cm2) above which point the shear rate becomes directly proportional to the shear stress and the reciprocal of the slope of graph gives the plastic viscosity U in poises.
This is expressed by the equation first given by Bingham:
A plastic curve obtained from a coaxial cylinder viscometer is nonlinear over the lower portion of the curve and the yield value f is usually obtained by extrapolation.
Dilatancy
If starch is made into a paste with cold water, it can be stirred slowly but tends to solidify if the stirring rate is increased. This effect, common to many concentrated deflocculated suspensions can be explained on the basis of packing characteristics of particles. Deflocculated particles pack into a mass of minimum volume and only a small quantity of liquid is needed to fill the voids between the particles and there is just sufficient to enable the particles to move past each other slowly. When the suspension is sheared rapidly the particles are forced out of their close packing and the void space is increased, i.e. system dilates. There is insufficient liquid to fill the voids completely, the paste appears to ‘dry out’ and offers considerable resistance to further shearing (Fig. 10.12D). On resting, it liquefies as the particles move back into close packing. Dilatancy is also encountered when a dilute deflocculated suspension settles out on storage. The sediment is dilatant and resists any energetic attempts at stirring or shaking. This effect is known as ‘caking’ or ‘claying’ of suspensions and is best avoided by formulating the suspension so that the particles are slightly flocculated.
Thixotropy
In the description of the different types of non-Newtonian behaviour, it was implied that the viscosity of a fluid might vary with shear rate; it was independent of the length of time that the shear rate was applied, and at the same shear rate would always produce the same viscosity. Most non-Newtonian materials, e.g. particles or macromolecules are colloidal in nature; they may not immediately adapt to the new shearing conditions. Therefore, when such a material is subjected to a particular shear rate, shear stress and consequently viscosity will decrease with time. Furthermore, once the shear stress has been removed, even if the structure that has been broken down is reversible, it may not instantly return to its original structure (rheological ground state). Because of this reason, a characteristic ‘hysteresis loop’ is formed, which shows that a breakdown in the structure has occurred after applying the shearing stress, and the area within the loop may be used as an index of the degree of breakdown. This phenomenon, i.e. breaking and rebuilding of the structure after applying and consequently withdrawing the shear stress, is termed thixotropy.
Thixotropy is defined as a reversible isothermal transition from gel to sol. The word comes from Greek words thixis (from thinganein, which means ‘to touch’) and tropy (from tropos, which means ‘of turning’, ‘changeable’, ‘to turn’, or in simple word ‘to change by touch’). Thixotropy may only be applied to shear thinning system, i.e. plastic and pseudoplastic system. In the case of plastic and pseudoplastic materials, the down curve will be displaced to the right of the up curve, while in case of Newtonian system, the resultant curve is identical with and superimposed on the up curve and Fig. 10.13 illustrates the flow diagrams of such systems.
Fig 10.13 Thixotropy in plastic and pseudoplastic flow system.
It is believed that thixotropic systems are usually composed of asymmetric particles or macromolecules, which are capable of interacting by numerous secondary bonds to produce a loose three-dimensional structure, and confer some degree of rigidity on the system, which resembles a gel. The energy imparted during shearing disrupts these bonds, which causes the interparticle links to be broken; the network therefore disintegrates and the viscosity falls, and a gel–sol transformation occurs. When the shear stress is eventually removed, the structure will tend to reform, although the process is not immediate and will increase with time as the molecules return to the original state under the influence of Brownian motion. For such cases, the rheogram depends on the rate at which shear is increased or decreased and the length of time a sample is suggested to any one rate of shear.
It was seen that a thixotropic material is similar to a pseudoplastic material in one manner; it exhibits lower viscosity as shear increases. However, the thixotropic profile differs substantially from the pseudoplastic profile in the manner in which viscosity recovers from the release of shear. Where viscosity in a pseudoplastic material recovers immediately on release of shear, in a thixotropic material there is a measurable delay in the recovery of viscosity. This delay allows for a period of flow and levelling or film consolidation to occur. Such a delay also allows the graphing of a hysteresis loop in which the viscosity is plotted as a function of both as shear is applied and released. This hysteresis loop allows to display number of properties that are of a system’s rheology, which includes the degree of thixotropy, the absolute viscosity values at a given shear rate, and the speed of viscosity recovery.
Applications of Thixotropy
Thixotropy is a desirable property in liquid pharmaceutical system; ideally it should have a high consistency in the container yet pour or spread easily. If emulsion or suspension, it implies low viscosity, which causes either rapid settling of solid particles in suspension or rapid creaming of emulsion. As we know, according to Stock’s equation (Eq. 10.9) the rate of sedimentation is proportional to viscosity. Frequent settling of solid produces sediment difficult to redisperse, whereas creaming is the first step towards coalescence. Addition of thixotropic agents, i.e. bentonite magma, clay, colloidal silicon dioxide, microcrystallin cellulose, etc. into suspension or emulsion provides a high viscosity. High viscosity retards sedimentation or creaming as well as flow below the yield stress. When other applicant is desired to pour some of the suspension or emulsion from container, it is shaken well at sufficient shear stress, which breaks the thixotropic structure and lowers the apparent viscosity. Back on shelf the viscosity again slowly increases again and the structure is rebuilt because of the Brownian motion. A similar pattern is required with lotion, cream, ointment and parenteral suspensions for intramuscular depot therapy.
Concentrated parenteral suspension containing about 40–70% w/v of procaine penicillin G in water has a high inherent thixotropy. Breakdown of the structure occurred when the suspension was caused to pass through the hypodermic needle. Consequently, rheological structure was rebuilt and a depot of drug formed at the site of injection in the muscles from which drug was slowly released and made available for absorption.
Measurement of Thixotropy
Thixotropy can be quantitatively represented by the area of the hysteresis loop by a coefficient of thixotropic breakdown or by the decay of shearing stress or apparent viscosity as a function of time at constant rate of shear. Structural breakdown or thixotropy can be determined by two approaches. In the first approach, thixotropy is determined by the structural breakdown with time at a constant rate of shear. The shear rate of a thixotropic material is increased in a constant manner from point a to point b and is then decreased at the same rate back to point e. This forms the hysteresis loop abe. If sample was taken to point b and the shear rate held constant for a certain period of time, i.e. for t1 second, the shearing stress and the consistency would decrease to an extent depending on the time of shear, the rate of shear and the degree of structure in the sample. This would lead to hysteresis loop abce. If the sample has been held at the same rate of shear for t2 second, this will give abcde loop (Fig. 10.14). On the basis of such rheogram, a thixotropic coefficient B, the rate of breakdown with time at constant shear rate, is calculated as:
Fig 10.14 Structural breakdown with time of a plastic system possessing thixotropy when subjected to constant rates of shear with different time values.
In which U1 and U2 is plastic viscosity of two down curves obtained after shearing at a constant rate for t1 and t2 sec, respectively. The choice of rate is arbitrary. A more meaningful though time-consuming method for characterizing thixotropic behaviour is to measure the fall in stress with time at several rates of shear.
The second approach is to determine the structural breakdown due to increasing shear shown in Fig. 10.15, in which two hysteresis loops are obtained having different maximum rates of shear, V1 and V2. In this case, a thixotropic coefficient M, the loss in shearing stress per unit increase in shear rate, is obtained from
where U1 and U2 are the plastic viscosities for two separate down curves having maximum shearing rates of V1 and V2, respectively. With this technique, the rates of shears V1and V2 are chosen arbitrarily. The value of M will depend on the rates of shear chosen since these shear rates will affect the down curves and hence the value of U that are calculated.
Fig 10.15 Structural breakdown of a plastic system possessing thixotropy when subjected to increasing shear rates.
Bulges and Spurs: In some cases different type of thixotropic rheogram is obtained for examples bulges and spurs type rheogram. A concentrated aqueous bentonite gel (10–15% by weight) produces a characteristic bulge in the up curve of a hysteresis loop. This is due to the presence of crystalline plates in a pattern of ‘house-of-cards’ structure, which causes the swelling of the bentonite magmas (Fig. 10.16A). In another case—procaine penicillin gel formulation for IM injections, the hysteresis loop is in the form of spur-like protrusion. The structure showed a high-yield value, Y, which traces out a bowed up curve due to sharp breakdown of three-dimensional structure at slow shear rate, as shown in Fig. 10.16B. It was found that penicillin gels having definite Y values were very thixotropic, forming intramuscular depots on injection that afforded prolonged blood level of the drug.
Fig 10.16 Rheogram of a thixotropic material showing (A) Bulge and (B) Spur in the hysteresis loop.
Negative Thixotropy: Negative thixotropy, also known as antithixotropy. It is a rheological phenomenon that represents an increase rather than a decrease in consistency on the down curve. This increase in thickness or resistance to flow with increased tie of shear was observed by Chong et al. (1960). This effect can be observed in a wide range of polymers and solvents. In case of negative thixotropy, e.g. when magnesium magma was treated alternatively with increasing and decreasing rates of shear, the magma continuously thickened and then finally reached an equilibrium state in which further cycles of increasing and decreasing shear rates no longer increased the consistency of the magma (Fig 10.17). The equilibrium system was found to be gel like and providing great suspendability, yet it was readily pourable. In this case the equilibrium state is the sol.
Fig 10.17 Rheogram of magnesia magma showing antithixotropy behaviour.
The origin of negative thixotropy is the tendency of a polymer to aggregate in a solvent, giving rise to temporary link aggregates or crystallites or, in other words, it results from an increased collision frequency of dispersed particles or polymer molecules in suspension, which increased interparticle bonding with time. This tendency to aggregate is usually considered to vary along the polymer chain and it is also a change of the small to large number of big particle or floccules, while at rest large floccules break and gain their original structure.
Phenomenon of negative thixotropy is quite different from the dilatant flow in a manner as dilatant systems are deflocculated and contain more than 50% by volume of solid dispersed phase, whereas antithixotropy systems consist of 1–10% solid contents and are flocculated.
Rheopaxy
Once the links among suspended particles or the entanglements among dissolved polymer chains have been broken by shear, their restoration by Brownian motion is slow if the suspension or solutions are viscous. In such cases, slow-flow gentle agitation or moderate and rhythmic vibration may accelerate the rebuilding of the structure, i.e. the restoration of the links between particles or macromolecules by Brownian motion. Low shear rates hasten the regaining of the apparent viscosity or onset of gelation in thixotropic sols. Increase in apparent viscosity or the advent of a yield value by gentle agitation is called rheopaxy and it is well known that in a rheopectic system gel is the equilibrium form.
Selection of Viscometer
Newtonian Liquids
For the accurate measurement of liquids of low viscosity, the capillary tube or the Hoeppler viscometer are suitable. These are simple instruments to use and to calibrate. They provide only one shear rate (i.e. single-point instrument) and this is adequate as the viscosity of a Newtonian liquid is independent of shear rate.
Non-Newtonian Liquids
The majority of emulsions and suspensions are non-Newtonian and, therefore, the limitations of single point measurement should be appreciated. This aspect has been emphasized by Martin et al. (1964). Fig. 10.18 shows the flow curves for two emulsions, A and B, where a measurement at shear rate D1 would indicate that B had a higher viscosity than A, while the opposite result would be obtained at shear rate D3. At shear rate D2, the emulsions would apparently have the same viscosity. This shows that even for comparative work single point instruments can be misleading, particularly if no information is available on the type of flow behaviour. Where a simple test for ‘viscosity’ is required for quality control, it may be sufficient to measure the ‘apparent viscosity’ at a rate of shear which approximates to the rate of shear to which the product is subjected during usage. Henderson et al. (1961) have calculated the approximate rates of shear encountered in such operations as milling, pouring from bottles and extrusion from orifices (Table 10.1).
Fig 10.18 Effect of shear rate on viscosity determination of emulsions A and B.
Table 10.1 Approximate rates of shear for various processes (after Henderson et al., 1961) |
|
| Process | Rate of shear (sec–1) |
| Rubbing on ointment tile | 150 |
| Roller mill | 1000–12000 |
| Hypodermic needle | Up to 10000 |
| Nasal spray (plastic squeeze bottle) | 2000 |
| Pouring from bottle | Less than 100 |
The assessment of thixotropic products requires rotational viscometry in order to plot up and down curves. Lack of reproducibility may be due to lack of standardized pretreatment of samples before measurements.
A comprehensive treatise on commercial viscometers is given by van Wazer et al. (1963). There is no single viscometer that is likely to be satisfactory for pharmaceutical products that may vary in consistency from water to stiff gels. In selecting a viscometer the following points should be studied:
1. If low shear rates are required, is the instrument sensitive enough to measure the corresponding stress?
2. Are the units of rate of shear and shearing stress measured in absolute units or comparative units?
3. Size of sample available, e.g. a cone and plate may require only a 1 mL sample, while a coaxial viscometer may require several hundred millilitres.
4. Can the sample be easily maintained at a constant temperature?
5. Can the instrument be easily cleaned? This is essential when a large number of samples are to be examined.
Rheology of Suspensions
The addition of a disperse phase (solid or liquid) to a liquid increases the viscosity due to the disturbance of the stream lines of the liquid around the particles. Einstein (1906–1911) developed a theoretical expression relating the viscosity of a suspension η, to the volume fraction ϕ of the particles present:
where η0 is the viscosity of the vehicle. This equation is strictly applicable only to very dilute suspensions of rigid spherical particles. In order to extend the equation to higher concentrations, equations of the type:
have been developed where k1 and k2 are constants for the system.
These equations are of little application to pharmaceutical suspensions as the theoretical treatments have been largely confined to Newtonian flow. The effects of particle shape, particle size distribution and in particular the degree of flocculation of the particles, playa predominant role in determining the flow properties of pharmaceutical suspensions, particularly if the concentration of disperse phase is high. The more uniform the particles, the greater the viscosity of the suspension for a given concentration of solid. Flocculation gives rise to plastic flow properties often combined with thixotropy. If deflocculating agents are added, e.g. surface-active agents, peptizing electrolytes, the yield value is reduced or disappears altogether and the flow becomes pseudoplastic, or even Newtonian, depending on the extent of deflocculation. Fig. 10.19 shows the variation of flow properties of suspensions such as kaolin and barium sulphate in water as the concentration is increased. Curve D shows how the viscosity is reduced in the presence of sodium citrate which deflocculates the suspension.
Fig 10.19 Flow curves of kaolin suspensions in water. (A) 20 per cent w/v; (B) 30 per cent w/v; (C) 40 per cent w/v; (D) 40 per cent w/v + 10 per cent sodium citrate.
A similar result is observed with dispersions of zinc oxide in liquid paraffin, where the addition of a little oleic acid produces a similar deflocculation effect.
Rheology of Emulsions
When droplets of one liquid are emulsified in an immiscible liquid, the viscosity of the emulsion is greater than that of the vehicle, as one would expect from the Einstein equation. This equation is obeyed by very dilute emulsions of small globule size, indicating that the globules are behaving like solid particles. The factors that have been discussed under the section Rheology of Suspensions apply to emulsions except that the particle shape factor is unlikely to play an important part, since the globules are mainly spherical.
Fig. 10.20 shows the variation of apparent viscosity with volume concentration of disperse phase in a water-in-oil emulsion. The emulsion may invert to an unstable oil-in- water emulsion of low viscosity when the water is increased above a critical amount. The point of inversion depends on the concentration and nature of the emulsifying agent.
Fig 10.20 Variation of apparent viscosity with volume concentration (per cent) of water in a water-in-oil emulsion. Maximum viscosity at X-inversion beyond this point.
The theoretical maximum volume of uniform spheres that can be packed into a given volume is 74 per cent v/v but emulsion droplets are not uniform so that small drops can pack between the larger ones. They can also distort so that emulsions containing as much as 90 per cent v/v internal phase are possible. The viscosity of the internal phase has little or no effect on the viscosity of the concentrated emulsions. The important factors influencing the viscosity of emulsions are:
The review by Sherman (1964) should be con sulted for detailed information.
Oil-in-water emulsions stabilized by cetyl and stearyl alcohols and water-soluble surface agents have been studied by Axon (1956) who showed that increasing quantities of cetyl alcohol produced an increase in both the yield value and the plastic viscosity of some emulsions. Talman et al. (1967) studied the rheology of similar emulsions and suggested that the large increase in viscosity as the cetostearyl alcohol was increased, was due to the formation of a gel in the aqueous phase due to the interaction of the alcohol with sodium lauryl sulphate. It seems likely that such gel formation would play a dominant role in the viscosity of emulsions containing emulsifying waxes. Baqy and Shotton (1967) have studied the thixo tropic and viscoelastic properties of the sodium lauryl sulphate–ethyl alcohol–water gel system in the absence of an emulsified phase.
Rheology of Suspending Agents
The majority of these produce pseudoplastic solutions and have definite advantages over Newtonian liquids such as glycerin or syrups. The high viscosity of pseudoplastics at low rates of shear enables them to stabilize insoluble particles against rapid sedimentation and the shear thinning charac teristics enable easy pouring from the bottle. These agents include methyl cellulose, hydroxyethyl cellulose, sodium carboxy cellulose, alginates and clays such as bentonite. Meyer and Cohen (1959) re ported that permanent suspensions could be prepared using carboxyvinyl–polymer (Carbopol 934 BF, Goodrich Chemical Co.), which exhibits plastic flow properties. Provided a certain minimum yield value was present, suspensions were observed to be permanent. However, such suspensions should be examined critically since the ‘permanency’ may be affected by deterioration of suspending efficiency due to chemical deterioration, pH changes, etc. Hiestand (1964) refers to suspending agents that possess yield values, as ‘structured vehicles’ on account of the structure of the suspending or gelling material. Particles may also remain in sus pension due not only to the structure in the vehicle but also to the flocculation of the particles which produces a scaffold structure that extends throughout the suspension. Many long chain polymers induce flocculation of particles—due to adsorption of segments of the polymer on to them, the remaining segments being free to project into the vehicle and adsorbing on to adjacent particles. Thus polymer bridge flocculation may be as least as important as the structure of the vehicle itself in preventing suspensions from sedimenting.
Martin et al. (1964) have reviewed the properties of suspending agents and also the effect of processing on the rheology of gums and suspensions.
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