Relationships Between Properties

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All the properties described above are interrelated with specific ratios between them associated with second-order characteristics such as brittleness, ductility, plasticity and toughness. In this section a number of relationships are reviewed and their usefulness examined.

Of specific interest is the semiempirical equation derived by Marsh (1964) while studying Vickers indentation:

From this relationship (Fig. 19) and a knowledge of the materials studied it is possible to show: first that if E/ay between 10-25 then H/ay is between 1.5-2.0, corresponding to highly elastic materials typical of many polymers; second if E/ay is between 25-30 then H/ay is between 2.0-2.2, corresponding to brittle materials typical of glasses (if H/ay > 2.2 then there is a tendency towards a reduction in brittle behaviour); and third if E/ay > 150 then H/ay > 3.0, corresponding to a rigid-plastic material typical of metals.

This equation is of specific interest as it implies that the ratio H/ay is not always equal to 3 as suggested earlier (equation 5) but is related to both the plasticity and elasticity of the material. This would account for the small differences seen in the calculated yield stress from indentation hardness measurements (using a value of 3) and those from Heckel plots shown in Table 7. This ratio has been extensively examined by Roberts and Rowe (1987a) and for the majority of pharmaceutical materials shown to lie between 2.3 and 3.4.

Fig. 19 Schematic diagram showing the relationship between the hardness (H), yield stress (ay), and Young's modulus of elasticity (E) for a range of materials (adapted from Tabor, 1979).

Fig. 19 Schematic diagram showing the relationship between the hardness (H), yield stress (ay), and Young's modulus of elasticity (E) for a range of materials (adapted from Tabor, 1979).

Equation 48 can be useful in a predictive capacity because it is possible, from a knowledge of two of the properties, to calculate the third with a reasonable degree of accuracy (Roberts and Rowe, 1987a).

A modification of equation 48 allows for the use of both spherical and pyramidal indenters (Johnson, 1970):

r

fee

1 + In

V.

[}ayJ

J

where X is either cot0, where 6 is the semiapical angle for a pyramid, or d/D, where d and D are the diameters of the indent and sphere, respectively.

Using a similar approach to Marsh (1964) and Johnson (1970), Marshall et al. (1982) showed that the ratio H/E is also an important factor in indentation as elastic materials always exhibit a high value whereas plastic materials always exhibit a low value. This ratio has been extrapolated to pharmaceutical materials by Hiestand and Smith (1984) and renamed the 'strain index'; it is described as indicating the relative strain during elastic recovery that follows plastic deformation or the relative potential for strain energy to develop at the tip of a defect. Both these and other workers (Duncan-Hewitt and Weatherly, 1989b) have suggested that the strain index is useful to predict tablettability in terms of capping and lamination but comparison of literature data (Table 15) would tend to dispute this claim as some materials with similar strain indices exhibit different compaction behaviour.

Table 15 The strain index, H/E, for some excipients and drugs

Material

Strain index

Comments

References

Paracetamol Erythromycin Adipic acid Avicel PH101 Starch

Lactose S.D.

Sucrose

Sucrose

Phenacetin

Ibuprofen

Hexamine

NaCl

0.05 Lamination/capping

0.04 Lamination

0.03 Lamination

0.025 Excellent

0.023 Good

0.021 Poor

0.016 Splitting

0.013 Lamination

0.011 Good 0.006 Capping

0.005 Good

Duncan-Hewitt and Hiestand and Smith Duncan-Hewitt and Hiestand and Smith Hiestand and Smith Hiestand and Smith Duncan-Hewitt and Hiestand and Smith Hiestand and Smith Hiestand and Smith Hiestand and Smith Duncan-Hewitt and

Weatherly (1989b) (1984)

Weatherly (1989b)

Weatherly (1989b)

Weatherly (1989b)

Table 16 Brittleness indices for some excipients and drugs

Material

h (MPa)

Kjco (MPam1/2)

Kico (Mm-1/2:

Avicel PH101

168"

0.7569

0.22

Lactose /3 anhydrous

2510

0.7597

0.33

Ibuprofen

35 b

0.1044

0.34

Sodium chloride

213c

0.4769

0.45

Aspirin

sld

0.1561

0.56

Adipic acid

i23c

0.1398

0.88

Paracetamol DC

265"

0.2463

1.08

a-Lactose monohydrate

5l5b

0.3540

1.45

Sucrose

645c

0.2239

2.88

Paracetamol

421c

0.1153

3.65

" Jetzer et al. (1983a) b Leuenberger (1982) c Duncan-Hewitt and Weatherly (1989b) d Ridgway etat. (1969a)

" Jetzer et al. (1983a) b Leuenberger (1982) c Duncan-Hewitt and Weatherly (1989b) d Ridgway etat. (1969a)

By considering that the brittleness of a material is a measure of the relative susceptibility of that material to the two competing mechanical responses of deformation and fracture, Lawn and Marshall (1979) proposed a brittleness index defined as the ratio of indentation hardness to the critical stress intensity factor with units /¿m~l/2. Table 16 shows data for a variety of materials; h is obtained from indentation hardness and k,co from single edge notched beam tests (Roberts etal., 1993). Duncan-Hewitt and Weatherly (1989b) also used this index suggesting that materials with a high value tend to undergo fragmentation during compaction. The results in Table 16 for a wider range of materials tend to support this approach with materials (e.g. Avicel PH101, ibuprofen) known to be essentially ductile in behaviour exhibiting very low values and materials (e.g. sucrose, paracetamol) known to be brittle in behaviour exhibiting very high values.

It is interesting to compare the values of the brittleness index given in Table 16 with values in the literature for medium-strength steel (0.1), polymethyl methacrylate (0.14), alumina (3.0) and glass (8.8). Based on this classification all pharmaceutical materials can be considered to be brittle or semi-brittle in nature.

A relationship already proposed earlier (equation 6) between the critical stress intensity factor and Young's modulus of elasticity provides a comparative measure of the fracture toughness (R). Data on a wide range of

Table 17 Fracture toughness values for some excipients and drugs

Young's Critical stress Fracture modulus intensity factor toughness

Young's Critical stress Fracture modulus intensity factor toughness

Table 17 Fracture toughness values for some excipients and drugs

Material

(GPa)

(MPam1/2)

(J m-2

Celluloses

Avicel PHI05

9.4"

1.33a

214.5

Avicel PH105

10.1®

1.33a

175.1

Avicel PH101

7.8C

0.76'

643.3

Avicel PH101

10.3d

1.21d

142.1

Avicel PH101

9.2"

0.87a

106.5

Avicel PH101

9.0®

0.87a

84.1

Avicel PH101

7.8C

2.24'

74.1

Avicel PHI02

8.7®

0.76a

95.2

Avicel PHI02

8.2"

0.76a

70.4

Emcocel

7.1"

0.80a

90.1

Emcocel

9.0®

0.80a

94.0

Emcocel (90M)

8.9a

0.83a

77.4

Emcocel (90M)

9.4®

0.83a

68.1

Unimac (MG100)

8.0"

1.05a

137.8

Unimac (MG100)

8.8®

1.05a

72.7

Unimac (MG200)

7.3"

1.05a

79.1

Unimac (MG200)

8.0®

1.05a

56.1

Sugars

a-Lactose monohydrate

24.le

0.35'

5.1

Sorbitol instant

45.(/

0.4/

4.9

Sucrose

32.3*

0.08*

0.2

Drugs

Ibuprofen

5.0e

0.101

2.0

Aspirin

1.5'

0.16'

3.4

Paracetamol DC

w.r

0.25'

5.3

Paracetamol

U.T

0.12'

1.2

Paracetamol

8.4*

0.05*

0.3

Others

Sodium chloride

43.0*

0.50*

5.8

Adipic acid

4.1*

0.02*

0.1

c Roberts et al. (1989b) > Roberts and Rowe (1989)

d Mashadi and Newton (1987b) e Roberts etal. (1991) 1 Mashadi and Newton (1987a) * Duncan-Hewitt and Weatherly (1989b)

c Roberts et al. (1989b) > Roberts and Rowe (1989)

d Mashadi and Newton (1987b) e Roberts etal. (1991) 1 Mashadi and Newton (1987a) * Duncan-Hewitt and Weatherly (1989b)

materials using beam specimens to generate values for modulus of elasticity and critical stress intensity factors are shown in Table 17. It can be seen that fracture toughness can vary over two orders of magnitude, with materials exhibiting high values being regarded as tough.

Lawn and Wilshaw (1975) and Atkins and Mai (1985) have used fracture toughness as a means of classifying materials, e.g.

• For highly brittle materials where crack propagation occurs via the reversible fracture of cohesive bonds the fracture energy is equal to the surface free energy R = 0.5-5 J m"2.

• For semi-brittle materials where there is some plastic flow at crack tips R = 5-50 J m-2.

• For non-brittle materials where blunting of crack tips occur R can be in excess of 5 x 104Jm-2.

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