Rheology is important in endodontics as it affects the flow of materials within the canal, whether material will be extruded, and whether material will enter lateral canals and dentinal tubules. It is necessary to understand some rheological theory before relating it to the practice of endodontics. The purpose of this article is to outline the historical development of rheology, to define some rheological terms, describe various types of flow and the factors affecting flow. Measurement methods will be described and also the possible source of measurement errors. The basic equations commonly used in endodontic and other dental materials will be given with reference to endodontic and dental material literature.
Rheological studies of endodontic materials: 1. Theory and Basic Equations
Rheology is defined as the science of the deformation and flow of matter(1) .
It is of importance in endodontics as it relates to the flow of materials within the root canal system and along the root canal wall. It will have an effect on whether materials extend to the apical foramen without excessive extrusion, whether they flow into lateral canals and enter into the dentinal tubules.
Rheological terms
In the late 17th Century, the deformation of matter was defined by Hooke’s Law for solids and Newton’s Law for liquids.
Hooke’s Law states that “The power of any spring is in the same proportion with the tension thereof” i.e. the strain produced within a solid is proportional to the stress which is applied to it. The constant of proportionality for this law is the elastic modulus, “G”(2).
Newton’s Law for liquids states that “The resistance which arises from the lack of slipperiness of the parts of a liquid, other things being equal, is proportional to the velocity with which the parts of the liquid are separated from one another”(1). The property referred to here as slipperiness is now known as viscosity. In a Newtonian liquid therefore, the strain within the liquid is directly proportional to the applied stress. The constant of proportionality in this case is the viscosity, “η”.
We now know that there are also liquids, which do not follow Newton’s law and these are known as non-Newtonian fluids. We also know that some materials show a combination of behaviours. Materials can also be compressible, where the density can change, and incompressible, where the density does not change(3). These factors have an effect on the rheology. Under clinical conditions, endodontic materials are incompressible.
Types of flow
Shear flow occurs when the constituent particles or elements flow over or slide along each other.
Extensional (elongational) flow occurs when adjacent elements are pulled away from each other. When this occurs, some elements are also pulled towards each other(2).
If shear and elongation flow are both present in a non-Newtonian fluid, the effects of elongation flow will usually dominate(4).
Both of these flows can be :
Steady when the flow variables are no longer changing with time, as in steady laminar flow, which is flow without turbulence(1).
Dynamic or unsteady when the flow is irregular or intermittent as in start-up flow , end flow, creep and when the motion occurs in steps(3) .
Oscillating when the shear or extensional deformation is performed in, for example, a sinusoidal wave fashion(1).
Viscosity and Elasticity
Viscosity of a fluid is defined as the resistance to external deformation, which is a constant for Newtonian fluid at a given temperature or, for non-Newtonian fluids, depends on the flow rate.
Elasticity is the reversible behaviour of a solid after removal of external stress or strain. An ideal elastic substance will return to its “equilibrium state” when the stresses are removed(5) .
However, there is also viscoelastic behaviour for solids showing both viscosity and elasticity and elastico-viscosity for liquids showing both kinds of behaviour.
Linear and Non-Linear
Both Hooke’s Law for solids and Newton’s Law for liquids are linear laws in that there is direct proportionality between stress and strain (Hookes’ Law) or shear rate (Newton’s Law) within the materials (Fig 1).
Fig.1 An Example of Newtonian Flow (click on image to enlarge)
Viscoelasticity or elastoviscosity for these materials would also be linear.
Many materials however do not exhibit this kind of behaviour and a graph drawn of their stress versus strain would often appear curved. This behaviour would then be described as non-linear. One example of non-linear, non-Newtonian behaviour is shear thinning, where the viscosity reduces with increasing strain rate in steady flow i.e. it becomes easier to flow. Another less common occurrence is shear thickening, where there is increased viscosity with increasing strain rate, i.e. it becomes more difficult to flow (Fig 2).
Fig.2 Viscosity v Shear Rate
The terms pseudoplastic (shear thinning) and dilatant (shear thickening) were used previously but are now considered to be misleading, as they are not all-inclusive(6). This is because, for example, there is not always an increase in volume (dilatancy) when shear thickening is present. Most materials used in endodontics are fairly complex and would be expected to show non-linear, non-Newtonian behaviour.(Fig 3)
Fig 3 Shear thinning behaviour of an endodontic sealer
Thixotropy is a time-dependant decrease in apparent viscosity under constant shear stress or shear rate followed by a gradual recovery when the shear stress or shear rate is removed(1). It results from the structural breakdown of the material under stress6 . This may be an important property of some dental materials but many studies to demonstrate thixotropy show a varying stress and strain as well as time. The change cannot be attributed to time alone and a false positive for thixotropy is obtained(7). A more accurate representation of thixotropy could be shown by a graph of viscosity against time at a constant stress, which is then discontinued at time, t (Fig 4).
Fig.4 Thixotropy
The hysteresis loop demonstrates structural breakdown of a material, where stress and shear rate are varying(6)(Fig 5). The down curve shows that the material requires time to re-structure after the stress is discontinued, or is not rebuilding at all(6).
Fig.5 Hysteresis Loop
For many years, some fluids appeared to have a yield stress. It appeared that for some very shear thinning materials, there was no flow taking place at stresses below the yield stress(2). In fact now with modern controlled stress rheometers, very low rotation rates can be achieved and it has been demonstrated that some flow is actually taking place. However for some other materials there may be a low shear rate below which the materials may not appear to flow(6).
A Bingham body has the behaviour of a solid up to the yield stress after which it starts to flow and the rate of shear is directly proportional to the shear stress minus the yield stress (1,3). This property is important in some foods e.g spreading with a knife(3) and is also important when we want a dental material to remain static until instrumented .
Factors affecting flow.
All materials will show variation in flow with varying temperature and varying pressure(2). Increasing the temperature will enable constituent particles to move more rapidly relative to each other and so make the material flow more easily. A similar effect can be achieved by varying the particle size distribution of the material, by having the particles more spherical and by the addition of a superplasticiser to the continuous phase. The more concentrated the material or the greater its density the greater the viscosity and therefore there is reduction in flow.
Increasing pressure also produces reduced flow. Humidity, which affects atmospheric pressure, will also affect flow unless steps are taken to minimise this effect. Flow of materials is very much affected by the geometry of the container, measuring device or machinery surrounding the material.
It follows therefore that the flow of endodontic materials will be affected by the geometry of the delivery system, the geometry of the root canal and the physical conditions within the canal. Significant effects on the rheology of endodontic materials will be found also from their chemical composition, the molecular weight of macromolecules, their molecular weight distribution and the molecular architecture(3). There is a critical molecular weight for polymer entanglement and the effect on the rheology of polymeric materials alters above and below this value(3).
Measurement
Many viscometers are available for the measurement of viscosity(8), which are used as a scale for comparison with materials measured by the same apparatus at the same flow conditions. Some simple measurements can be made using flow cups and a viscosity-related measure given as the volume of material flowing through the flow cup for a specific time. However, flow between two plates, according to the requirements of EN ISO 6876:2002(9), is the method most recorded in endodontic studies(10,11,12,13,14,15,16).
Pressure can be exerted to a material within a capillary tube and the amount of material extruded in a specific time will give a value related to viscosity(17,18). The volumetric flow and velocity of a material under pressure will also give a value related to viscosity(19). These values will vary with varying angles of the flow cup or internal dimensions of the capillary so the specifics of geometry must always be quoted. The values will also vary with the temperature. Great care must be taken in comparing the results of studies using apparatus with different geometries or different temperatures.
Other geometries which could be used for measurement of material flow properties are rotating vane, ram and piston and concentric cylinders.
Controlled stress rheometers are now available which measure not only viscosity but also normal stresses, loss and storage moduli and tan delta . These rheometers can begin at very low stress and run for very long periods of time (e.g. until a material has set), and achieve very high levels of shear stress and shear rate depending on the viscosity of the material. The results are often given as the log value as the stress and shear rate may be of vastly differing magnitude. These rheometers are also temperature controlled.
In these controlled stress rheometers, the geometry may be cone and plate (Fig 6 ), parallel plate and re-entrant (reversed) cone and plate. Choice of instrument often depends on the particle size of the material, requiring a minimum gap size, and whether it is of high or low viscosity.
Fig.6 Cone and Plate Geometry
Note that only with cone and plate geometry is the applied shear stress the same throughout the entire testing sample, and results are therefore most accurate(20).
Other modern rheometers available are capillary rheometers and concentric cylinder rheometers. In the capillary rheometer, the stress within the material varies from zero at the centre to a maximum at the capillary wall. So a mathematical model has to be constructed for calculating viscosity. The stress varies also within concentric cylinders but if the gap is small enough, it can be assumed that the stress is uniform.
Experimental Sources of Error
Wall slip
When measuring particulate sample, very smooth internal walls of a capillary or the very smooth surface of a cone and plate apparatus can result in a thin, particle free layer of material at the surface or apparatus/material interface. This leads to increase in overall flow rate and subsequently to reduce the measurement reading of apparent viscosity. This error can be avoided by using a textured surface with certain roughness. In practical terms, this particle free layer may affect the flow and adhesion of endodontic materials at very smooth walls or gutta percha points.
Secondary flows
These may occur in a very dense material, when the density has a greater effect than viscosity on the flow. They result in readings which are higher than would be expected. In pipe or capillary flow, proper laminar flow does not occur until after a sufficient length has been traversed. At the entrance and exit, therefore, there may be exhibited transit flow behaviour, which would give false readings. Inertial effects depend on the density and velocity of a material and, if these are high, will cause turbulent flow, giving false readings for viscosity
Zero errors
Rheometers need to be properly set to zero by testing with a standard viscosity material, usually a Newtonian liquid. It is also advisable to calibrate them regularly with a standard Newtonian liquid.
Basic Equations used in endodontic and material studies
Viscosity (eta) is equal to shear stress (tau) divided by shear rate (gamma dot). Rheological studies, therefore, are involved with determining shear stress and shear rate to obtain a value of viscosity. For Newtonian fluids, viscosity is constant and independent of flow rate. For non-Newtonian materials, the viscosity is not constant(7).
In capillary flow, the shear stress at the capillary wall (τw) is given by
where ΔP is the change in pressure along the capillary wall, r is the capillary internal radius and L is the length of the capillary.
The shear rate at the capillary wall is given by
where Q is the volumetric flow rate and r is the capillary radius.
From the shear stress and shear rate at the capillary wall, the apparent viscosity (ηa) is calculated.
For non-Newtonian fluids, shear stress (τ-tau) is not a linear function of shear rate (ẙ-gamma dot) and for many of them, can be described by the power law
where k and n are constants.
From this power law shown above, the shear stress at the capillary wall (τw) is given by
Therefore
Plotting log τw versus ẙw gives a line with slope of “n”.
When n = 1, the material is Newtonian. If n>1, the material is shear thickening and if n<1>
For Newtonian materials
For non-Newtonian materials, where n does not equal 1, Rabinowitch’s correction can be used (3,22). The true shear rate at the capillary wall is then given by
From equation above, apparent viscosity at capillary wall (ηa) is then given by
The equations above are also suitable for a capillary extrusion rheometer(17).
Similarly, the viscosity of material flow in a ram and cylinder extrusion viscometer with an exit tube can be given by
where α is the capillary radius and P is the pressure which is given by the force recorded divided by the ram area(23) .
In a rotational viscometer with spindle geometry, the shear stress may be calculated from the measured torque
where S is the shear stress, m is the torque, R is the spindle radius and h is the depth of immersion of the spindle. The slope of the log angular velocity of the spindle(omega) vs log shear stress (S) then gives the reciprocal of the power law exponent (n). As before if n=1 , the material is Newtonian , if n>1 it is shear thickening and if n<1>Flow through an orifice , for example a flow cup, or from a wide tube to a narrower tube, will give values for extensional or elongational viscosity (25) so that 
where lambda is the elongational viscosity, eta is the shear viscosity, P0 is the pressure drop through the orifice and ẙ is the shear rate which will depend on the flow rate and the radius of the orifice.
Phase Volume or Volume Fraction(phi), defined as the ratio of the particulate volume and the total volume of sample(1), affects the rheological properties of suspension fluids and relates to the zero-shear viscosity by
where k1 is a constant related to material density and k2 represents a crowding factor(26) .
Fig 7 A Custom-made Capillary Rheometer
Conclusion
Rheology is a difficult but fascinating subject. Dentistry can gain much from theoretical rheologists. Knowledge of dental rheology can guide formulation design of dental materials, improve their handling properties and their long term effectiveness. A subsequent article in this series will discuss further the experimental methods, the results and clinical applications of rheological studies of endodontic materials.
References
1. Barnes HA, Hutton JF, and Walters K (2001). 'Introduction to Rheology.' (Elsevier: Amsterdam.)
2. Barnes HA (2000). 'A Handbook of Elementary Rheology.' (University of Wales Institute of Non-Newtonian Fluid Mechanics: Aberystwyth, Wales.)
3. Morrison FA (2001) “Understanding Rheology” (OUP:Oxford)
4. Petrie CJS (1979) Elongational flows : aspects of the behaviour of model elasticoviscous fluids (Pitman:London)
5. Frederickson AG(1964) “Principles and Applications of Rheology” Prentice –Hall: New York
6. Tattersall GH and Banfill PFG(1983) “ The Rheology of Fresh Concrete” (Pitman Books Ltd:London)
7. Whorlow RW (1992) “Rheological techniques” Prentice-Hall:Gale
8. Barnes HA (2002). 'Viscosity.' (University of Wales, Institute of Non-Newtonian Fluid Mechanics, Aberystwyth: Aberystwyth,Wales.)
9. EN ISO 6876:2002 International Standard (2002) Dental root canal sealing materials.
10. McComb, D. and Smith, D. C. (1976). Comparison of physical properties of polycarboxylate-based and conventional root canal sealers. Journal of Endodontics 2, 228-235.
11. Ørstavik D (1982). Seating of gutta-percha points:effect of sealers with varying film thickness. Journal of Endodontics 8, 213-218.
12. Ørstavik, D. (1983). Physical properties of root canal sealers: measurement of flow, working time, and compressive strength. International Endodontic Journal 16, 99-107.
13. Siqueira, F. J., Jr., Fraga, R. C., and Garcia, P. F. (1995). Evaluation of sealing ability, pH and flow rate of three calcium hydroxide-based sealers. Endodontics and Dental Traumatology 11, 225-228.
14. Siqueira JF Jr, Favieri A, Gahyva SM, Moraes SR, Lima KC and Lopes HP (2000) Antimicrobial activity and flow rate of newer and established root canal sealers. Journal of Endodontics 26(5):274-7
15. Gambarini G, Testarelli L, PongioneG, Gerosa R, Galiani M. Radiographic and rheological properties of a new endodontic sealer. Aust Endod J 2006; 32:31-34
16. Asgary S, Shahabi S, Jafarzadeh T, Amini S, Kheirieh S. The properties of a new endodontic material J Endod 2008; 34:990-993
17. Vermilyea, S. G., Huget, E. F., and De Simon, L. B. (1979). Extrusion rheometry of fluid materials. Journal of Dental Research 58, 1691-1695.
18. Uhrich, J. M., Moser, J. B., and Heuer, M. A. (1978). The rheology of selected root canal sealer cements. Journal of Endodontics 4, 373-379.
19. Lacey S, Pitt Ford TR, Watson TF and Sherriff M ( 2005) A study of the rheological properties of endodontic sealers. International Endodontic Journal 38:499-504
20. McKennel, R. (1960). 'The measurement and control of viscosity and related flow properties.' (Ferranti Ltd: Manchester, UK.)
21. Vermilyea, S. G., De Simon, L. B., and Huget, E. F. (1978). The rheologic properties of endodontic sealers. Oral Surgery,Oral Medicine,Oral Pathology 46, 711-716
22. Combe, E. C. and Moser, J. B. (1976). An apparatus for measuring the rheological properties of dental materials. Journal of Dental Research 55, 223-228.
23. Braden, M. (1967). Rheology of dental composition (impression compound). Journal of Dental Research 46, 620-622.
24. Vermilyea, S., Powers, J. M., and Craig, R. G. (1977). Rotational viscometry of a zinc phosphate and a zinc polyacrylate cement. Journal of Dental Research 56, 762-767.
25. McCabe JF and Bowman AJ (1981) The Rheological Properties of Dental Impression Materials . British Dental Journal 151 179-183
26. Cook, W. D. (1983). Dental polyelectrolyte cements: II: effect of powder/liquid ratio on their rheology. Biomaterials 4, 21-24.
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