Yield stress from compaction studies

Compaction studies, because they mimic the tabletting process, offer an ideal method for assessing the mechanical properties of powders. In powder compaction a specific method used to evaluate the average stress of a material during compression relies on the observations of Heckel (1961a,b) who found that for materials that plastically deform, the relative density of a material, D, could be related to the compaction pressure, P, by the equation:

where K and A are constants.

Unfortunately, considerable deviations of the experimental data occur at both low and high pressures owing to particle rearrangement and strain hardening, respectively, but at least over the middle pressure range a straight line relationship exists between In (1/1 — D) and P (Fig. 12).

Equation 31 has been reappraised by Hersey and Rees (1970) who suggested that the reciprocal of K can be regarded as numerically equal to the

5.0 -


Plastic deformation

i ^ 3.0 -


a! £

(Jy =1 y K

Strain hardening

50 100 150 200 250 300 Compaction pressure MPa

Fig. 12 Schematic diagram of the Heckel plot (Heckel, 1961a,b). For definitions of terms, see text.

mean yield stress of the powder. However, as pointed out by Roberts etal. (1989a) this is only a specific case and the reciprocal of K can be regarded as a mean deformation stress, be it a plastic deformation stress (equal to the yield stress) for materials that deform plastically, a fracture deformation stress for materials that undergo fracture or a combination of the two. This approach implies that provided the experiment is carried out on materials close to or below the brittle-ductile transition (Fig. 1) then the reciprocal of K will be numerically equal to the yield stress of the powder.

Historically, two approaches to the analysis of Heckel data have been used and these are generally referred to as 'at pressure' and 'zero pressure' measurements. Pressure/relative density measurements determined during compression are clearly 'at pressure' measurements while those from relative density measurements on the compact after ejection are 'zero pressure' measurements. In the original publication Heckel (1961a) found that for the metals iron, copper, nickel, and tungsten there was no difference between the two measurements but for graphite the 'zero pressure' measurement was higher and could be attributed to the elastic recovery of the compact causing a lower relative density. Support for this hypothesis can be obtained from studies on the pharmaceutical materials dicalcium phosphate dihydrate (8% increase, Paronen, 1987), lactose (30% increase, Fell and Newton, 1971), microcrystalline cellulose (56% increase, Paronen, 1987) and starch (177% increase, Paronen, 1987) where the magnitude of the difference is indirectly related to the modulus of the material, i.e. the larger the increase the lower the modulus.

In the light of the discussion above and the findings that other factors such as punch and die dimensions (York, 1979; Danjo etal., 1989), state Of lubrication (DeBoer etal., 1978; Ragnarsson and Sjogren, 1984), and speed of compaction (Rees and Rue, 1978; Roberts and Rowe, 1985) can have an effect on the measurement, it is not surprising that a great deal of controversy and confusion surrounds the use of data from Heckel plots. However, Roberts and Rowe (1987a) have clearly shown that, provided measurements are carried out 'at pressure' with lubricated punches and dies on material that is below its brittle-ductile transition and at a very slow speed, the reciprocal of K is identical to the yield stress of the material and comparable with that calculated from indentation hardness measurements using equation 5 (Table 7).

Although early measurements were generally performed on either instrumented punches and dies in physical testing machines (Fell and Newton, 1971) or instrumented single-punch tablet machines (DeBoer etal., 1978) with relatively unsophisticated data capture and analysis, recent measurements generally have been carried out using tablet compression simulators.

Table 7 Yield stresses from the Heckel plot compared with those determined by the indentation technique

Yield stress (MPa)

Yield stress (MPa)

Table 7 Yield stresses from the Heckel plot compared with those determined by the indentation technique


Heckel plot




















PTFE, polytetrafluoroethylene.

PTFE, polytetrafluoroethylene.

One such simulator, as used by Roberts and Rowe (1985), is shown in Plate 5. It consists of three units: a hydraulic power pack (A), an electronic control unit (B) and a loading frame (C). The hydraulic system consists of a continuously circulating closed loop of hydraulic fluid supplying three servo valves, two in parallel on the upper actuator and one on the lower. These receive drive signals from the electronic control unit and the profile generator. The two actuators can independently move the two rams and can hold both single punch and rotary punch tooling with conventional dies with a total stroke of 50 mm and 25 mm for the upper and lower rams, respectively. The actuators can achieve compression forces up to 50 kN during a compression cycle in any time interval between 40 ms and 1000 s controlled by a clock generator. The punches are capable of a maximum punch velocity of 300 mm 1 using a sawtooth displacement-time profile with a displacement accuracy of ±10 /¿m and a force accuracy of ±0.025 kN.

Yield stress data for a number of excipients and drugs are shown in Table 8. The trends seen mimic those for indentation hardness with the inorganic carbonates and phosphates showing very high values of yield compared with the polymers, which give low yield values. The variation seen in the drugs may well be due to slight differences in the particle sizes tested.

Compaction simulators allow the measurement of yield stress over a wide range of punch velocities (Fig. 13). To compare materials, Roberts and Rowe (1985) proposed the term 'strain rate sensitivity' (SRS) to describe the percentage decrease in yield stress from a punch velocity of 300 mm s_1 to one of 0.033 mm s~'. This was later modified (Roberts and Rowe, 1987a) to a percentage increase in yield stress over the same punch velocities:


Experimental details

Inorganic Calcium phosphate Calcium carbonate Calcium carbonate Magnesium carbonate Dicalcium phosphate dihydrate

Simulator Simulator Hydraulic press Simulator Simulator

Sugars a-Lactose monohydrate a-Lactose monohydrate a-Lactose monohydrate Lactose /3 anhydrous Lactose (spray dried) Lactose (spray dried) Mannitol

Hydraulic press Hydraulic press Simulator Simulator Simulator Simulator Simulator


Sodium chloride Sodium chloride Avicel PH101 Avicel PH101 Avicel PH102 Avicel PH105 Maize starch Stearic acid

Single punch

Simulator Single punch Simulator Simulator Simulator Simulator Hydraulic press

Yield stress

(MPa) References

957 Roberts and Rowe (1987a)

851 Roberts and Rowe (1987a)

610 Ejiofer etai. (1986)

471 Roberts and Rowe (1987a)

431 Roberts and Rowe (1987a)

179 Vromans and Lerk (1988)

183 York (1978)

178 Roberts and Rowe (1986)

149 Roberts and Rowe (1987a)

178 Bateman et at. (1989)

147 Roberts and Rowe (1985)

90 Roberts and Rowe (1987a)

89 Ragnarsson and Sjogren (1985)

89 Roberts etal. (1989a)

50 Humbert-Droz etal. (1982)

46 Roberts and Rowe (1986)

49 Roberts and Rowe (1986)

48 Roberts and Rowe (1986)

40 Roberts and Rowe (1987a)

Polymers PVC/vinyl acetate Polyethylene PTFE


Paracetamol DC

Paracetamol DC

Paracetamol DC






Theophylline (anhydrous)






Simulator Simulator Simulator

Single punch Single punch

Simulator Single punch Single punch Single punch Simulator Hydraulic press Single punch Single punch Single punch Single punch Single punch Simulator

PTFE, polytetrafluoroethylene; PVC, polyvinyl chloride.

70 Roberts and Rowe (1987a) 16 Roberts and Rowe (1987 a) 12 Roberts and Rowe (1987a)

108 Hussain etal. (1991)

81 Humbert-Droz etal. (1982)

109 Roberts and Rowe (1987a) 79 Humbert-Droz etal. (1983) 99 Podczeck and Wenzel (1989)

127 Duberg and Nystrom (1986)

102 Roberts and Rowe (1985)

109 Ramberger and Burger (1985)

75 Podczeck and Wenzel (1989)

25 Humbert-Droz et al. (1983)

73 Duberg and Nystrom (1986)

24 Humbert-Droz etal. (1983)

24 Humbert-Droz etal. (1983)

25 Bateman etal. (1987)

Punch velocity (mm s"1)

Fig. 13 The effect of punch velocity on yield stress for calcium phosphate; A, calcium carbonate; magnesium carbonate; A, a lactose monohydrate; □, lactose 0 anhydrous; V, man-nitol; Avicel PH101; O, maize starch.

Punch velocity (mm s"1)

Fig. 13 The effect of punch velocity on yield stress for calcium phosphate; A, calcium carbonate; magnesium carbonate; A, a lactose monohydrate; □, lactose 0 anhydrous; V, man-nitol; Avicel PH101; O, maize starch.


where ay}0Q and a^o.033 are the yield stresses measured at punch velocities of 300 and 0.033 mm s~\ respectively.

Data on a number of materials are shown in Table 9. Some materials such as the inorganic carbonates/phosphates show little rate dependence while others such as starch and mannitol show a large strain rate dependence

Table 9 Strain rate sensitivities (SRS) of some excipients and drugs

Calcium phosphate 0

Calcium carbonate 0

Heavy magnesium carbonate 0

Paracetamol DC 1.8

Paracetamol 11.9

a-Lactose monohydrate (fine grade) 19.4

Lactose (spray dried) 23.8

Lactose ß anhydrous 25.5

Avicel PH101 63.7

Sodium chloride 66.3

PVC/vinyl acetate 67.5

Mannitol 86.5

Maize starch 97.2

PVC, polyvinyl chloride.

consistent with differences in the time-dependent properties of the material during compaction.

The effects of moisture and particle size of materials will be discussed later in this chapter.

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