Optimal Design Methodology for Composite Materials with Particulate Fillers

Powder Technology, 43 (1985) 27 - 36

Optimal

Design Methodology

27

for Composite Materials with Particulate

Fillers

M. CROSS School of Mathematics, Statistics and Computing, Thames Polytechnic, London SE18 (U.K.) W. H. DOUGLAS School of Dentistry, University of Minnesota, Minneapolis, MN 55455 (U.S.A.) and R. P. FIELDS Dental Products Dept., 3M Company, St. Paul, MN 55101 (U.S.A.) (Received December 20, 1983;in revised form August 16, 1984)

SUMMARY

INTRODUCTION

--macrocomposites or engineered products (e.g. reinforced concrete beams, skis, etc.) If we consider composites to consist of a filler material and fluid binder material, then further classifications of the filler materials yields a wide variety of microcomposites. Specialising further, there are a number of applications for which the filler material is a particulate. The particulates may also be of various shapes and sizes (or more specifically, size distributions). One application of microcomposites employing particulates as the filler is in the manufacture of dental restorative materials. Such materials have been available commercially for some years now and there is an enduring interest in improving their performance characteristics [ 2 - 7]. In an a t t e m p t to evaluate optimal design criteria for dental materials, a detailed theoretical study of the structure of composites has been performed. The results of this study have yielded a methodology for the optimal design of composite materials in which the filler comprises particulates. The objective of this paper is to summarise this methodology and show how it may be applied to the design of composite resins for use as dental restorative materials.

Although composite materials can be classified in m a n y ways, Hill [1] uses a convenient, all-embracing system which has three main categories: --natural composite (e.g. wood, bone, etc.); - - m i c r o c o m p o s i t e materials (e.g. steels, toughened thermoplastics, dental restoratives, etc.);

Classification o f particulate-based composites If a particulate filler material is compacted in a container, then it is expected to be very much stronger in compression and shear than in tension. The natural bonding forces t h a t contribute to the strength of a dry particulate agglomerate have been summarised by R u m p f [8] as

The structure o f composite materials which consist o f a particulate filler and fluid binder are analysed. Composites are shown to have two main matrix structures. In the first, the fluid volume is less than the void volume o f the filler and the composite strength accrues from both the natural bonding forces o f the particulate mass and the bonding action o f the fluid. The second characteristic matrix structure is when the fluid volume exceeds the filler void volume. Here the fluid essentially pushes the particles apart and reduces particle-particle interaction so that the composite strength is based largely upon the bonding action of the fluid alone. The two parameters which govern the structure o f a composite are the proportions o f fluid and filler and the natural voidage o f the latter. A methodology is evolved to develop a composite with a desired matrix structure which includes a mathematical model to predict voidage of particulate size distributions in random mixtures.

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28

particle interlocking, contact potential and/or excess charge, -- Van de Waals forces, -- magnetic attraction. The purpose of a fluid binder is two-fold: - - t o increase the strength of the particulate mass, - - t o provide a degree of fluidity which permits the composite to be compacted within a prescribed irregularly shaped domain. Figure 1 shows the three general states of the well-mixed composite material. Figure l(a) shows the particulate before the addition of fluid. If a small a m o u n t of fluid is added, then it will fit into the interstitial voids, yielding a three-phase (solid-air-fluid) struc--

--electrostatic

(a)

(b)

ture, as shown in Fig. l(b). In this case the original matrix structure of the filler remains intact. As more fluid is added, the point will be reached whereupon its volume is greater than that of the filler's natural interstitial void volume. In this case, the composite structure will be as illustrated in Fig. 1(c). In the case where the fluid addition fits into the interstitial void volume of the particulate material (as in Fig. l(b)), then the composite strength accrues from both - - n a t u r a l bonding forces of the particulate mass, -- bonding forces introduced by the presence of the fluid. In addition, since the integrity of the particle-particle interaction is maintained in the above matrix structure, then both compressive and shear forces are transmitted largely through the particulate structure. By contrast, when the fluid volume exceeds the volume of the particulate voids (as in Fig. l(c)), the particles are effectively pushed apart and the particle-particle interaction is reduced. As a consequence, the natural bonding forces are negated and both the compressive and shear forces are transmitted largely through the fluid phase of the composite structure. If voids are introduced during mixing, then the burden of composite material strength falls even more heavily upon the fluid binder. Suppose that a composite consists of a filler material and fluid binder mixed together in the proportion mf : 1 - mf by mass. Considering the properties of the filler and fluid binder separately first, then the natural voidage of the filler material is e and its specific gravity is pf. Thus, the void volume of the densely packed filler material contributing to the composite is given by Ve -

cmf pf(1

-

-

e)

(1)

Further, if Pb is the specific gravity of the fluid binder, its volume in the composite is given by

(c) Fig. 1. Composite structures. (a), General particle bonding forces; (b), composite structure when fluid binder volume fits into the interstices of filler volume; (c), composite structure when fluid binder volume exceeds interstitial void volume o f filler.

Vb -

1 --mr

(2)

PD

Thus, if Ve > Vb, then the resulting composite structure will be of the form shown in Fig. l(b), whereas if V~ < Vb, the structure will be as shown in Fig. l(c).

29 If we assume t h a t no air is i n t r o d u c e d into the c o m p o s i t e during mixing, t h e n its bulk density will be given b y

TABLE 1 Some basic measured data on three composite materials with particulate fillers

if)

i p~(l~ PB =

Item

Ye ~> VD

mf + 1 - - mf -1 p,

(3)

< vb

Where t h e s t r u c t u r e is t h a t o f Fig. l ( b ) , t h e n it is clear t h a t in this case t h e fluid acts o n l y as a b o n d i n g agent, whilst in the o t h e r structures, t h e fluid forces the particulates apart, i.e. e x p a n d s them. T h e r e f o r e , we m a y classify c o m p o s i t e s as f l u i d - e x p a n d e d ( F E ) or p a r t i c u l a t e - b o n d e d (PB) materials. In the development of many composites (notably for use as d e n t a l restoratives), it is desirable to maintain the integrity of the p a r t i c l e particle i n t e r a c t i o n s t r u c t u r e o f t h e particular filler material, i.e. to develop PB materials which are n o t e x p a n d e d b y the fluid binder, whilst maximising the bulk density o f t h e p r o d u c t . H e n c e , w h e n given a filler: b i n d e r p r o p o r t i o n , mr, it is i m p o r t a n t t o ensure t h a t the natural voidage o f the d e n s e l y p a c k e d filler, b e f o r e mixing, exceeds a certain m i n i m u m value so t h a t Ve > Vb. In o t h e r words, we require e > emi n where p~(1 - - mr) emi n =

pbrnf + p~(1 -- mF)

(4)

By c o m b i n i n g eqns. (3) and (4) we can derive a necessary c o n d i t i o n to ensure a PBs t r u c t u r e d c o m p o s i t e , i.e. PB < Pmax =

PbPf m f p b + (1 -- m~)p~

(5)

A l t h o u g h eqn. (5) is a necessary c o n d i t i o n for a p a r t i c u l a t e - b o n d e d c o m p o s i t e , it is n o t sufficient. If eqn. (5) is o b e y e d , t h e n t h e f a c t o r which d e t e r m i n e s the s t r u c t u r e o f a c o m p o s i t e is w h e t h e r air is i n t r o d u c e d during mixing. Most o f o u r e x p e r i m e n t a l w o r k has involved materials used in the m a n u f a c t u r e o f dental c o m p o s i t e s . Table 1 summarises the essential details o f t h r e e o f the c o m p o s i t e s used in o u r studies. Table 2 shows a comparison o f t h e m e a s u r e d bulk d e n s i t y with the m a x i m u m p e r m i t t e d bulk d e n s i t y o f eqn. (5). These calculations indicate t h a t c o m p o s i t e C is v e r y unlikely t o have a PBstructure. H o w e v e r , according to t h e c u r r e n t

Filler material specific gravity Binder (resin) specific gravity Particle size distribution (wt.% in prescribed fraction) < 2 pm 2 - 7 pm 7 - 15 pm +15 pm Measured bulk density, g/ml Amount of filler, wt.%

Composite material A

B

C

2.65 1.19

2.65 1.19

2.7 1.27

14 26 24 36 1.86 78

36 42 18.5 3.5 2.04 86

15 45 25 0.15 2.18 78.8

TABLE 2 Comparison between measured and predicted maximum bulk density of three composites Item

Bulk density, measured, g/ml Maximum bulk density, eqn. (5), g/ml

Composite material A

B

C

1.86

2.04

2.18

2.087 2.26

2.18

analysis, b o t h c o m p o s i t e s A and B have the p o t e n t i a l t o possess a PB-structure. In o r d e r to evaluate the s t r u c t u r e o f composites A and B, it is n e c e s s a r y to measure the voidage o f the filler materials involved. M e a s u r e m e n t o f the filler material voidage is relatively straightforward (cf. APPENDIX) and t h e results for materials A and B are s h o w n in Table 3. These m e a s u r e m e n t s are c o m p a r e d with the p r e d i c t i o n s o f the minim u m filler voidage necessary t o ensure a PB-structure (cf. eqn. (4)). T h e c o m p a r i s o n s indicate clearly t h a t c o m p o s i t e A is a fluide x p a n d e d material and t h a t o n l y c o m p o s i t e B has a PB-structure (cf. Table 3). F o r a n u m b e r o f c o m p o s i t e applications -and p a r t i c u l a r l y in t h e m a n u f a c t u r e o f d e n t a l restorative m a t e r i a l s - it is desirable to ensure t h a t the c o m p o s i t e has a m a x i m u m bulk d e n s i t y with a p a r t i c u l a t e - b o n d e d s t r u c t u r e . A c o n s t r a i n t in c o m p o s i t e design is t h a t it m u s t remain w o r k a b l e f o r use in d e n t a l filling operations. A l t h o u g h a given filler particle size d i s t r i b u t i o n has s o m e

30 TABLE 3 C o m p a r i s o n o f the m e a s u r e d and c a l c u l a t e d minim u m filler voidage necessary to e n s u r e a liquidbonded composite structure Item

Filler voidage, m e a s u r e d M i n i m u m voidage, eqn. (4)

C o m p o s i t e material A

B

0.324

0.345

0.3456

The conceptual model considers N particulates in a random packing which are distributed by number fraction as f(D), where D is a measure of particle diameter. In the usual way the average particle diameter is given by

= flOW(D) clD

0.262

'natural' fluidity, the proportion of fluid added is the a m o u n t required to ensure a workable composite. Thus, in order to design the optimum composite, it is necessary to evolve a filler material whose size distribution yields a low voidage and a high 'natural' fluidity. Evaluation of composite fluidity can only come through experience on specific materials. However, given a particle size distribution of the filler material, it should prove possible to predict a m a x i m u m bulk density which may yield a particulate-bonded composite, as a function of the filler loading. In order to do this though, it is necessary to be able to quickly evaluate the natural voidage of the compacted filler material.

Suppose that in a packing, Vc(D) is the total volume of space allocated to each particle of size D. Then the total volume of space,

Yw = f

AGE OF P A R T I C U L A T E SIZE D I S T R I B U T I O N S IN R A N D O M P A C K I N G S

Furnas [9] published the first major attempt to predict the minimum voidage of binary mixtures of particulate materials. Various other attempts have been made in the intervening half-century, generally without substantial progress being made. Since the late 1960s a series of papers have been published on simulating the random packing of spheres [ 1 0 - 14]; however, the main thrust of this work did not appear to involve prediction of the packing voidage. Dodds [15] has developed a procedure to predict the voidage of random packings of spheres and his results compare qualitatively with available experimental results. Unfortunately, it is not easy to see how Dodd's approach may be extended to cope with irregularly shaped particles. In what follows, a new general framework for predicting the voidage of random packings of irregularly shaped particulates is described.

V¢(D)Nf(D) dD

(7)

0

Furthermore, if CD is the effective 'volume' equivalent diameter of the particle, then the total volume of solids in the packing is given by

Vs =

-~ ¢3D3Nf(D) dD

(8)

0

If ~ is the voidage of the packing, then it follows that (1 -

M A T H E M A T I C A L M O D E L TO P R E D I C T VOID-

(6)

o

~)Vr = Vs

i.e.

ys

= 1 -- - -

Vr

(9)

For any given mass of particles it is straightforward to evaluate the solid volume. Thus, the main limitation in using eqn. (9) to predict the packing voidage is a knowledge of Vc(D) - - t h e total volume of space allocated to each particle. In order to predict this volume, re-consider the particles in a packing and impose a shell D*/2 over the surface of each particle with diameter D. Associated with the particle is a notional volume with diameter ¢(D + D*). Note that if the particle diameter D > D*, then a volume of solid (~r/6)[¢3(D--D*) 3] is never intersected by any other notional volume, i.e. it belongs to particle D. If D ~
31

%

In this case the volume of the part of a sphere with diameter D intersecting the notional volume is [16]

~1 I' 2 , " - - 7

- x_.J.,, ( - x ' : " ' . _ . - ',LO;

'.L-A ....

(a)

(b)

"

Fig. 2. Illustration o f the two options showing how the notional v o l u m e s intersect the solid v o l u m e of a particle. (a), D > D* ; (b), D ~< D*.

Furthermore, if C(D) is the co-ordination number of a sphere with diameter D, then the void volume of the notional shell is given by

em(D)Vm(D) Vm(D) = 6 ¢ 3 [ ( D +D*) 3 - (D ~ D*) 3]

(10)

6

where D ~D* =

t

D--D* 0

D>D* O
(11)

If the shell volume Vm(D) is shared amongst n other particles, then the volume of space allocated to a particle with diameter D is Vc(D) =

p

_ rr (D+D)3 - 6D3_C(D)__rr Da

Vm(D ) Ca(D ~ D*) 3 + - n

(12)

Also the solid volume in the total space allocation to particle D is given by

1

12

8 D+/3

(15)

Ouchiyama and Tanaka [ 17 ] have established from simple geometric arguments that a reasonable representation of the co-ordination number is given by 32

C(D) = - ~ [7 -- 8e(D)]

(16)

0

where e(D) is the voidage of a packing with particles of diameter D. The voidage of the packing ~ may now be evaluated from the following procedure: (1) Given sized fractions Di and corresponding voidages e(Di) together with weight fractions W(Di), (2) construct f(D) and use eqn. (6) to evaluate D. (3) Use eqns. (10), (11), (15) and (16) to evaluate era(D). (4) Use eqns. (8), (13) and (14) to evaluate

The problem of predicting the voidage of a packing thus reduces to evaluating two parameters for each particle - - n , the number of notional volumes which share the shell space Vm(D) -- era(D), the voidage of the shell Vm(D) Ouchiyama and Tanaka [16] have derived a solution for the above parameters by making the following assumptions: - - Each particle is a sphere. --Each sphere is surrounded by spheres of average diameter -- The shell width D*/2 is/9/2. - - T h e number of notional volumes which share the shell space Vm(D) is independent of D, i.e. n = ~.

(5) Use eqns. (12) and (7) to evaluate VT. (6) Use eqn. (9) to evaluate g. The framework outlined above has a number of potential advantages over other models. However, perhaps the most powerful is the fact that although assumptions about sphericity have been made in constructing the model its use is n o t constrained to spherical particles. Their shape has merely to be largely convex with a shape factor either independent of or specifiable by size. The reason for this advantage lies in the fact t h a t the model uses as prescribed information the measured voidage of the particulate size intervals. This means that, provided

7r Vm(D ) Vs(D) = ~ ¢3(D ~ D*)a[1 -- Cm(D)] - - r t (13) where era(D) is the voidage of the shell volume Vm(D). Obviously, conservation of mass yields

S Vs(D)N~(D) dD = Vs

(14)

32 t h a t t h e sized intervals are n o t t o o close (e.g. say g r e a t e r t h a n a f a c t o r of 2 or so), the m o d e l s h o u l d be fairly a c c u r a t e . As such, t h e m o d e l p r o v i d e s a practical bridge bet w e e n t h e e m p i r i c a l and e n t i r e l y physical models. T h e a b o v e m o d e l has b e e n t e s t e d against a wide v a r i e t y o f data. Initially, v a l i d a t i o n was p e r f o r m e d using b i n a r y m i x t u r e s o f quartz p a r t i c u l a t e s . T h e basic size d a t a f o r these tests is s h o w n in Table 4, whilst t h e TABLE 4 Basic data on sized quartz for binary mixtures Size range (pm)

Average particle diameter (pm)

Voidage

0 -2 2 - 4.5 4.5 - 7 22 - 37

1 2.25 5.75 29.5

0.825 0.756 0.5296 0.4037

C o n v e n t i o n a l l y , in a b i n a r y m i x t u r e with d i f f e r i n g characteristic particle sizes, the m i n i m u m voidage occurs in a 7 0 : 3 0 m i x o f t h e large and small m a t e r i a l [ 1 8 ] . Whilst this a p p e a r s to be t h e case f o r the m i x t u r e involving the t w o larger sized particles, it was n o t true f o r the m i x t u r e s involving t h e smaller particle sizes, as illustrated in Fig. 3. This is p r o b a b l y d u e t o t h e f a c t t h a t the v o i d a g e o f the finer m a t e r i a l s is c o n s i d e r a b l y higher t h a n t h o s e at t h e coarse e n d o f t h e size range. It has b e e n suggested t h a t t h e high voidages o f the fine m a t e r i a l s m a y arise f r o m e l e c t r o s t a t i c forces. W h e t h e r this or s o m e o t h e r particle i n t e r a c t i o n is t h e reason f o r t h e large voidages, the fine materials b e h a v e c o n s i s t e n t l y in a b i n a r y m i x . Since the model predictions and experimental results agree so well f o r b i n a r y m i x t u r e s involving t h e fine m a t e r i a l s w i t h high voidage,

4

comparison between measured and predicted voidages f o r s o m e b i n a r y m i x t u r e s is summarised in T a b l e 5. T h e relative e r r o r in each o f t h e s e c o m p a r i s o n s is generally a r o u n d 5%, c o m p a r e d w i t h a range in t h e m e a s u r e d values o f a b o u t 50% o f t h e m i n i m u m value. A l t h o u g h m e a s u r e m e n t s were t a k e n with 7 0 : 3 0 m i x t u r e s , it was felt this r e p r e s e n t e d a reasonable test of the model's predictive capability. As such, it is c o n c l u d e d t h a t t h e m o d e l d e s c r i b e d a b o v e p r o v i d e s an a c c e p t able level o f a c c u r a c y in p r e d i c t i n g b i n a r y mixtures. Even t h o u g h t h e m o d e l p r e d i c t i o n s c o m pare well w i t h e x p e r i m e n t a l m e a s u r e m e n t s , the results t h e m s e l v e s are s o m e w h a t unusual.

0.7 ÷

0.6

~ 0.5 -,4 0.4, 0.3

o

0'.2 '0:0

o:6

V o l p r o p o r t i o n small size ÷

Fig. 3. Experimental measurements of the three binary mixtures of quartz in 70:30 proportions. o, 29.5pro;x, 5.75pm; A, 2.25pm;©, 1.0pro.

TABLE 5 Comparison between measured and predicted voidages for binary mixtures of quartz Test No.

Materials used (average particle diameter)

Proportion

Voidage (measured)

Voidage (calculated)

Percentage error

1

29.5 5.75

0.7 0.3

0.3705

0.38

2.6

2

29.5 2.25

0.7 0.3

0.49

0.52

6.1

3

29.5 1

0.7 0.3

0.56 0.56

0.59 0.59

5.4 5.4

33 it may be concluded that the results illustrated are a genuine feature of such particulates systems. Experiments on ternary mixtures o f various materials have been r epor t e d by a number of workers. Standish and Borger [19] have recorded results for a ternary system of spheres where the ratio o f the m a x i m u m to minimum diameter was about 2. A comparison between Standish and Borger's measurements and the model predictions is shown in Fig. 4. Although the predictions are qualitatively acceptable, i.e. t hey identify the regions of low and high voidage, their quantitative accuracy is subject to a relative error of ab o u t 13%. Jeschar e t al. [18] also reported experiments on ternary spheres where the ratio o f maximum to m i ni m um diameter was 4. A comparison bet w e e n model predictions and measurements, as illustrated in Fig. 5, is much more favourable; generally speaking, the model has a relative error o f rather less than 5%. The final set of available measurements on spheres are those r e p o r t e d by Ridgeway and Tarbuck [20]. The comparison between prediction and measurement here is illustrated in Fig. 6. Although the ratio o f the m a x i m u m t o minimum sphere diameter is also a b o u t 2, both the qualitative and quantitative pictures are relatively well reflected by the model. It is interesting t o not e the i m por t a nc e of the ratio o f largest to smallest sphere size 0

in determining the voidage value and variation in a ternary mixture. Although the quantitative accuracy degrades as dmax/ drain decreases, the broad qualitative predictions o f the model remain reasonable. Standish and Borger [18] also made measurements on a ternary system of coke, using sizes in the ranges (--2 + 3.35, - - 3 . 3 5 + 4.0 and --4. 0 + 6.'35 mm). A comparison between prediction and measurements is shown in Fig. 7. Generally speaking, the 0

100

i ,,--/

\

100

0

0

20

40 %

60 Vol

80

i00

23.m---->

Fig. 5. Comparison between the voidage measurements of Jeschar et al. [18] and model predictions

for ternary mixtures of spheres. , predicted.

, Measured;

O. I00

100

% Vol 12ram 2 0 1 / ~ ~

,, \

i /

/i

\'-0

20

40 %

Vol

60

80

1(30

12,Tnml

Fig. 4. Comparison between the voidage measurements of Standish and Borger [19] and model predictions for ternary mixtures of spheres. -- ----, Measured ; , predicted.

20

40

60

80

% Vol 20ram

Fig. 6. Comparison between the voidage measurements of Ridgeway and Tarbuck [20] and model predictions for ternary mixtures of spheres. - - - Measured ;- - , predicted.

0 I00

34

~, Vol 3.71~1

52

0

~ Vol 2.7rtan

/ ,

0

20

40

50 Vol 5.2kh~

80

100

>

Fig. 7. Comparison between the voidage measurements of Standish and Borger [19] and model predictions for ternary mixtures of coke. - - - - - - , Measured; , predicted. m o d e l p r e d i c t i o n s are within 5% relative error o f the m e a s u r e m e n t s ; the qualitative p e r f o r m a n c e o f the m o d e l is good. This result is r a t h e r gratifying because c o k e is distinctly non-spherical ( t h o u g h largely convex) and it is necessary t o ascribe a representative size to each o f the size intervals. The m o d e l has also been tested against o t h e r e x p e r i m e n t a l data [21, 22] and again the predictions c o m p a r e favourably. In fact, the above results provide e n o u g h c o n f i d e n c e to p e r m i t t h e assertion t h a t our m o d e l o f voidage for a r a n d o m packing of sized particulates is a d e q u a t e f o r use in analysing the s t r u c t u r e o f c o m p o s i t e materials.

APPLICATION TO THE DESIGN OF DENTAL COMPOSITES It has been well established b y clinicians t h a t some o f t h e i m p o r t a n t characteristics for dental restorative materials include -- strength in c o m p r e s s i o n and shear -resistance t o wear at the surface -sufficient fluidity b e f o r e setting t o allow insertion o f t h e material into t h e cavity. When applied t o c o m p o s i t e materials composed largely o f a quartz-like filler and liquid resin b o n d i n g agent, this m e a n s t h a t the s t r u c t u r e s h o u l d be p a r t i c u l a t e - b o n d e d and as dense as possible. F u r t h e r m o r e , a case can be m a d e for minimising the average

interparticle distance in o r d e r to reduce surface wear. Essentially, the design p r o b l e m comprises i d e n t i f y i n g a filler size d i s t r i b u t i o n which possesses a voidage and natural fluidity characteristics t h a t p e r m i t t h e use o f a small e n o u g h p r o p o r t i o n o f liquid resin to ensure a particulate-bonded s t r u c t u r e . A l t h o u g h it is n o t c o n c e p t u a l l y difficult to measure the voidage of a particulate m i x t u r e , it is certainly tedious and t i m e - c o n s u m i n g , especially since t h e particles have b e e n ' w e t t e d ' in t h e c o m p o s i t e and this i m p o r t a n t e f f e c t has to be a c c o u n t e d for in m a k i n g the measurem e n t s . As such, the m o d e l described above to p r e d i c t the voidage o f r a n d o m packing c o u l d be particularly useful in identifying filler size distributions which are possible c a n d i d a t e s for resin-bonded structures. F o r e x a m p l e , the sized intervals o f quartz filler material were t r e a t e d in the way o u t l i n e d in the A P P E N D I X to yield ' w e t ' voidages; these are t a b u l a t e d in Table 6. When used to p r e d i c t the voidage o f materials A and B detailed in Table 1, the c o m p a r i s o n s with m e a s u r e m e n t s , cf. Table 7, are certainly g o o d e n o u g h to distinguish b e t w e e n the r e s i n - b o n d e d and t h e r e s i n - e x p a n d e d structure. In fact, the above discussion leads to a s t r a i g h t f o r w a r d m e t h o d o l o g y for the design and assessment of c o m p o s i t e materials for use in dental r e s t o r a t i o n : - - S e l e c t filler material and liquid resin b o n d i n g agent. - - S i z e the filler material and evaluate the ' w e t ' voidage o f each f r a c t i o n . - - S e l e c t a filler size d i s t r i b u t i o n and manuf a c t u r e a c o m p o s i t e t h a t is sufficiently fluid f o r clinical use. -- Use the voidage p r e d i c t i o n m o d e l t o g e t h e r with t h e s t r u c t u r e classification e q u a t i o n s o f TABLE 6 Basic data on wet sized quartz for full size distribution mixtures Size range (pm)

Representative particle diameter (tim)

Measured voidage (wet)

0 -2 2-7 7 - 15 15+

1.5 4 12 17.5

0.514 0.400 0.416 0.412

35 TABLE 7 Comparison between measured and predicted voidages of the filler materials used in composites A and B Item

Material A

Material B

Measured voidage Predicted voidage Relative error, %

0.324 0.304 --6

0.345 0.36 +4.3

the section o n Classification o f particulatebased c o m p o s i t e s to identify w h e t h e r o r n o t the c o m p o s i t e is p a r t i c u l a t e - b o n d e d . - - D e v e l o p a series o f c o m p o s i t e materials t o provide t h e basis f r o m which to design the o p t i m u m c o m p o s i t e s t r u c t u r e satisfying - - desired p a r t i c u l a t e b o n d e d s t r u c t u r e , - - m a x i m u m b u l k density, - - s u f f i c i e n t fluidity o f present c o m p o s i t e for clinical use, - - f i l l e r size d i s t r i b u t i o n and resin v o l u m e which minimises surface wear.

CONCLUSION The objective o f the w o r k r e p o r t e d a b o v e has been t o i d e n t i f y a r o u t e to design a dental c o m p o s i t e material with t h e o p t i m u m properties to maximise its clinical p e r f o r m a n c e . It is s h o w n t h a t c o m p o s i t e materials with particulate fillers m a y have t w o distinct m a t r i x structures. In the first, t h e v o l u m e of binding fluid is less t h a n the void v o l u m e o f the c o m p a c t e d filler prior to mixing. Here b o t h t h e n a t u r a l b o n d i n g forces o f the particulate mass and the action o f the fluid contribute to the composite's mechanical properties. I n t h e s e c o n d case, t h e binding fluid v o l u m e exceeds t h a t o f t h e void v o l u m e o f the n a t u r a l c o m p a c t e d filler. In this case, the fluid effectively pushes t h e particles apart and t h e c o m p o s i t e strength d e p e n d s a l m o s t solely o n t h e b o n d i n g a c t i o n o f the fluid since t h e p a r t i c l e - p a r t i c l e i n t e r a c t i o n is r e d u c e d . Expressions have been derived to classify the m a t r i x s t r u c t u r e o f a given c o m p o s i t e . In o r d e r t o clearly i d e n t i f y the c o m p o s i t e m a t r i x s t r u c t u r e , it is necessary t o evaluate the natural voidage o f c o m p a c t e d filler material. A m a t h e m a t i c a l m o d e l t o p r e d i c t the voidage o f a c o m p a c t e d mass o f particles

is p r e s e n t e d which is generally within 5% relative error on a variety of materials. Utilisation o f this m o d e l , with t h e f o r m u l a devised to classify the c o m p o s i t e m a t r i x structure, provides a r o u t e to i d e n t i f y t h e o p t i m u m design f o r dental materials.

ACKNOWLEDGEMENT T h e a u t h o r s wish to t h a n k the referee for his useful c o m m e n t s , w h i c h helped to improve the paper considerably.

REFERENCES 1 D. Hill, An Introduction to Composite Materials, Cambridge University Press, Cambridge, England, 1981. 2 J. W. Osborne, E. N. Gale and G. W. Ferguson, J. Prosthet. Dent., 30 (1973) 795. 3 P. L. Fan and J. M. Powers, J. Dent. Res., 59 (1980) 2066. 4 K.-J. Soderholm, J. Dent. Res., 60 (1981)1867. 5 K. F. Leinfelder, T. B. Sluder, C. L. Stockwell, W. D. Strickland and J. T. Wall, J. Prosthet. Dent., 33 (1975) 407. 6 K. D. Jorgensen, Scand. J. Dent. Res., 88 (1980) 557. 7 D. Dogan, W. H. Douglas and M. Cross, Structured comparison o f two composite resins, presented at IADR Meeting, New Orleans (February 1981). 8 H. Rumpf, in K. V. S. Sastry (ed.), Agglomeration 77, AIME, New York, 1977, p. 97. 9 C. C. Furnas, Ind. and Eng. Chem., 23 (1931) 1052. 10 D. J. Adams and J. Matheson, J. Chem. Phys., 56 (1972) 1989. 11 W. S. Jodrey and E. M. Tory, Simulation, 32 (1979) 1. 12 S. Debbas and H. Rumpf, Chem. Eng. Sci., 21 (1966) 583. 13 .4. D. Scott and D. M. Kilgour, British J. A. Phys., (D) (1969)863. 14 J. L. Finney, Proc. Roy. Soc. A, 319 (1970) 479. 15 J. A. Dodds, J. Coll. Int. Sci., 77 (1980) 317. 16 N. Ouchiyama and T. Tanaka, Ind. Eng. Chem. Fundam., 20 (1981) 66. 17 N. Ouchiyama and T. Tanaka, Ind. Eng. Chem. Fundam., 19 (1980) 338. 18 R. Jeschar, in N. Standish (ed.), Blast Furnace Aerodynamics, Aust. IMM Press, Wollongong, N.S.W., 1975, p. 136. 19 N. Standish and D. E. Borger, Powder Technol., 22 (1979) 121. 20 K. Ridgeway and K. J. Tarbuck, Chem. Proc. Eng., 49 (1968) 103. 21 N. Standish and P. J. Leyshan, Powder Technol., 30 (1981) 119. 22 N. Standish and D. N. Collins, Powder Technol., 36 (1983) 55.

36

APPENDIX MEASUREMENT VOIDAGE

OF

FILLER

MATERIAL

T h e general p r o c e d u r e to e v a l u a t e t h e d r y voidage o f a densely p a c k e d filler m a t e r i a l involved t h e f o l l o w i n g steps: (i) Measure a p r e s c r i b e d mass M o f m a t e r i a l into a c y l i n d r i c a l c o n t a i n e r . (ii) Place t h e c y l i n d e r o n a v i b r a t o r y table to achieve a d e n s e l y p a c k e d d r y bed. (iii) Measure t h e c o m p a c t e d d r y v o l u m e V. I f t h e specific gravity o f t h e filler m a t e r i a l is Ps, t h e n t h e voidage o f t h e p a c k e d b e d is given b y M e

-- 1

-

- -

(At)

PsV The procedure to calculate the wet voidage is slightly different. For example, to calculate the voidages for the sized fractions of the material in Table 6, the following procedures were used:

(i) T h e finely g r o u n d q u a r t z m a t e r i a l was s e p a r a t e d i n t o the f o u r size ranges indicated, using a l a b o r a t o r y - s i z e A l p i n e Particle Classifier. (ii) T h e mass o f 25 ml o f e a c h size f r a c t i o n was d e t e r m i n e d b y placing t h e material in a 50 m l g r a d u a t e d c y l i n d e r a n d vibrating on a v i b r a t o r y t a b l e to achieve a d e n s e l y p a c k e d d r y b e d . ( I n c r e m e n t s o f filler w e r e a d d e d a n d v i b r a t e d until its c o m p a c t e d v o l u m e was e x a c t l y 25 ml.) The a m o u n t o f filler was t h e n weighed. (iii) E x a c t l y 20 ml o f w a t e r was a d d e d to e a c h size f r a c t i o n , t h e m i x t u r e was t h o r o u g h l y stirred, and c o v e r e d t o p r e v e n t e v a p o r a t i o n o f water. (iv) T h e c o n t e n t s of t h e c y l i n d e r w e r e all o w e d t o settle for 21 d a y s , t h e n t h e d a t a d e t a i l e d in T a b l e A1 w e r e collected. H e r e eqn. (A1) was used to calculate t h e voidage, e x c e p t t h a t t h e v o l u m e V was the m e a s u r e d c o m p a c t e d v o l u m e o f t h e particulate mass in t h e cylinder.

TABLE A1 Data determined from the experimental measurements Size fraction (um) 0 -2 2 -7 7 - 15 15+

Mass of 25 ml of Total volume of Clear liquid Volume of water 'dry' packed filler water + filler volume filler mix

Filler void Calculated filler volume voidage

(g)

(ml)

(ml)

(ml)

(ml)

(%)

22.2 32.0 37.2 39.9

29.0 32.0 34.0 35.0

10.5 12.0 10.0 9.5

18.5 20.0 24.0 25.5

9.5 8.0 10.0 10.5

51.4 40.0 41.6 41.2