J. BIOMED. MATEH. RES.
VOL. 9, PP. 35-53 (1975)
The Surface Area of Aggregates Applied to Dental Materials WILLIAM V. LOEBENSTEIN, Dental Research Section, National Bureau of Standards, Washington, D.C.
Summary There is a continuing need for the complete characterization of the physica and chemical properties of dental materials. Among these properties is surface area. The problem is further complicated by the fact that most dental materials are, themselves, mixtures of two or more identifiable components. If the vapor adsorptive properties of these components are different in the mixture from that, which would be expected of them collectively, then interaction is present. Interaction must not be confused with the lack of additivity which results from the limitations of the BET theory applied to mixtures. Equations are derived herein t o estimate the extent of this latter source of variability and to correct for it, giving a “true” surface area for the aggregate. Conversely, the adsorptive properties of either component can be calculated from the properties of the mixture and the remaining component together with the percentage composition. An immediate application can be made in determining the water-vapor adsorptive properties of human dentinal collagen without necessitating its removal from dentin. Any attempt to extract it chemically may produce denaturation or chain rupture thus precluding the possibility of direct determination. In the case of nitrogen adsorption, however, interaction definitely is indicated.
INTRODUCTION The Brunauer, Emmett, and Teller equation (BET)’ has consistently given quite satisfactory results over the years in routine surface area determinations. I t s assumptions and approximation^^-^ for the most part, are well known and as long as its limitations are observed, i t will remain the preferred method over other alternative t h e ~ r i e s . ~ . ’ The fact that the B E T equation may not be appropriate when applied to solid mixtures, or aggregates, all too often escapes even the most careful experimenter. A large error in the calculated or apparent surface area could be inferred by a deviation from linearity 35 Contribution of the National Bureau of Standards. Not subject to copyright.
36
LOEBENSTEIN
of the BET plot; but that deviation might be so slight as to be unnoticed entirely or if discernable, might wrongly be attributed to experimental error. For two or more components each of which possesses a sufficiently different average adsorption energy, the simple B E T equation is inapplicable and cannot be easily rectified. Walker and Zettlemoyers identified the phenomenon in a treatment similar t o Langmuir’s expanded equationg which took account of surface heterogeneities in a unimolecular model. They, as well as others,10-12 developed solutions to the problem which, while correct, were extremely complex and based on techniques such as the simultaneous solution of four nonlinear equations for even the simplest two-component system or a tedious graphical iterative recomputation of adsorption energy distributions. Even as recently as 197212 a practical working solution had not been found. One of the purposes of this paper is t o evolve a workable relationship between the slope and intercept of a n apparent B E T fit for the adsorption data of a n aggregate and to express this in terms of the valid B E T parameters which apply to the individual components comprising the mixture. Although it is admittedly a n approximation, it is sufficiently correct for most applications and decidedly better than no correction a t all. As a corollary, the proposed relationship will identify those applications where the apparent fit requires no correction. A second phenomenon encountered when dealing with adsorbent mixtures is interaction. I n its present context, interaction is defined as the interference of one component with the surface area of another. It would manifest itself if one of the components were contained, to some degree a t least, within the pores of another so as to deny access of the adsorbate and thus give rise to a diminished surface area. No theoretical development is available for predicting the extent of interaction. A second objective of this investigation is t o identify interaction when i t is present. I n the absence of interaction the surface area of the aggregate would be a mass-weighted average of the specific areas of each of its components. Its presence would be verified experimentally if any significant difference existed between the apparent surface area of the aggregate obtained by blind application of the B E T equation and the apparent BET area of a nonintermingled mixture of each of the same components in identical percentages.
AGGREGATES APPLIED TO DENTAL MATERIALS
37
I n testing the relationships derived herein, the experimental applications are confined t o the field of dental materials. They include naturally occurring aggregates such as the tooth itself, as well as man-made aggregates such as polymer-particulate composites similar to those used in dental restorative work. The examples include both water vapor and nitrogen as the adsorbates.
THEORETICAL For a solid adsorbent consisting of a single component, the number of moles, q, of vapor adsorbed per gram a t relative pressure, x, can be expressed by eq. (1) in accordance with the BET free surface equation q =
qmcx (1 - .?.) (1 -
-
1'
+ cs)
where qm is a constant equal to the number of moles of adsorbate needed t o produce a monolayer and therefore is proportional" to the surface area. The other parameter, c, is characteristic of the system and is related exponentially to the difference between the average energy of adsorption of the initial laycr adsorbed and the heat of liquefaction of the adsorbate. With an adsorbent for which the B E T theory is applicable, eq. (1) is valid over a range in I from about 0.1 to 0.3. Let us consider two adsorbents each of which satisfies eq. (1). Two pairs of parameters would tben be obtained, qml and cl, for the first adsorbent and qmz and c2, for the second. If w1 g of the first adsorbent is placed beside (but not mixed with) w2 g of the second in a dual adsorption tube where they are exposed to a n adsorbable vapor a t some relative pressure, 5 , the amount of vapor that would be adsorbed per g of uncombined adsorbent mixture would be (q,7cl q2w2)/(w1 wy)which is the same as
+
+
flql
+
f2q2
=
q
(2)
where fi andf2 are the respective weight fractions of each component. When q, q1, and q 2 in eq. (2) arc replaced by their corresponding
* The proportionality constants used in this work are 98.02 X lo3m*/mol for x 104 m2/mol for H20 a t room temperature.
Nz at -196°C and 7..529
LOEBENSTEIN
38
values according to eq. ( l ) , the following expression results after cancellation of non-zero common factors : QmC
fl9rnlCl
--f
1 - T n . C T
1-Tn.CclT
+l-z+czx
f ~ q m z ~ 2
(3)
I n the general case, eq. (3) can not be an equality, since all elements are constants except x. If i t were an equality a t a particular value of 2, i t could not be equal a t any other value. This limitation of the B E T equation was appreciated by Marino et al.13 In the special case where the c values are all equal, the x’s disappear and an equality does exist, namely: Qm
= fiqm,
+
.f2qm,
(4)
Each qm value is proportional to the surface area, A , with the same proportionality constant as long as the same adsorbate is used a t that temperature. Therefore, eq. (4)is equivalent to eq. (5).
A
=
fiAi
+
f2Az
(5)
Equation (5) is, of course, true and can be derived by simple inference without recourse to any assumed adsorption theory. Deviations from eq. (5) might be appreciable and yet be entirely undetected or they might erroneously be attributed to interaction among the components of a solid mixture rather than to the misapplication of the B E T throry. This delemma can be resolved by separating these two mutually independent sources of variability. The more familiar linear form of eq. (1) is =
a,
+ b,z
(6)
where :
and i = 1 for component 1 and 2 for component 2 while the absence of a subscript signifies the mixture.
AGGREGATES APPLIED TO DENTAL MATERIALS
39
By combining eq. (2) with eq. (7) for all values of 3: within the B E T range and making use of eq. (6) t o eliminate y l and yz, y can be expressed as a function of z:
Y’
a1az
+ (aibz + a z b i ) ~+ bib2z2 + fzad + +
(flu2
(fib2
( 10)
f*b1)5
This function, eq. (lo), is the curve for the aggregate which is so often mistaken for linear that the detection of a curvature is often attributed to experimental error or faulty measurement. By forcing eq. (10) t o a straight line of the form of eq. (6), i t is seen from eq. (8) (without subscripts) that
By the same reasoning the slope of eq. (10) can be obtained by differentiation and equated approximately t o its linear counterpart, eq. (9) (without subscripts). The original parameters may then be substituted back t o eliminate the a’s and b’s. The resultant slope for the mixture, expressed in terms of its components, is
+ z(c1 - l)]‘ + [I- + - 1)l’ + - [I + z(c1 - 1) 1 + z(c2 - 1) fiJlmlCl(C1
c - 1
.
- 1)
[l
2(Cz
fiqmiC1
qmc
f29rn2~2(~2 - 1) f2qm2CZ
1
(12)
The degree to which eq. (10) approximates linearity in any particular combination of components and weight fractions is readily tested by substituting two values for 2 in eq. (12) a t widely different points on the curve yet within the applicable range of the B E T equation. It is surprising how great a difference between c1 and c2 can be tolerated in some instances without reflecting a n appreciable change in slope. In most applications i t would be reasonable when forcing a linear fit to choose the value of x = 0.2. for use in eq. (12) because this is the midpoint of the valid range for all of the data. A small error is introduced by taking the z = 0 value of eq. (10) as equivalent to the extrapolated intercept of a “least-squares” line forced through the experimental points. A somewhat closer agreement would be obtained by the extrapolation of a line passing through a point whose y coordinate at, say, II’ = 0.15 would be found from
40
LOEBENSTEIN
eq. (10) and whose slope was equal to the value determined for eq. (12). Except in extreme cases, however, the extra effort results in little improvement over the simpler use of eq. (11).
Hypothetical Illustration It is instructive now to consider a hypothetical mixture where no interaction will be permitted between the two components and their surface areas will be identical. Appreciable differences will be assumed for their c values, although not unusually large when compared with many naturally occurring materials. It will be assumed that mol/g and c1 = 10 while c2 = 0.5. The qml = qmz = 10.0 X “true” value of qm for the resultant of a mixture of any proportion mol/g. must, of course, be 10.0 X However, the apparent value obtained from a best fit of the B E T equation will depend upon the composition of the mixture. Consider the amounts of vapor adsorbed per gram for each pure component at, say, 0.14, 0.18, 0.22, and 0.26 relative pressure values. They would be 0.7202, 0.8380, 0.9658, and 1.0518 mmol/g for component 1 and 0.0875, 0.1206, 0.1.585, and 0.2019 mmol/g for component 2, respectively, as may be verified from eq. (1). A nonmixed combination of these two components consisting of 16.5% of component 1 a t the same relative pressures would adsorb 0.1919, 0.2390, 0.2917, and 0.3421 mmol/g, respectively, according to eq. ( 2 ) . When these data are plotted using the linear form of the B E T theory (eqs. (6) and (7)), the resultant curves are shown in Figure 1. Evaluation of qm from eqs. (8) and (9) applied to the combination (as would normally have been donc) would result in a n apparent qm value of less than one half of its weighted average value (eq. (4)). (A least-squares value of 4.74 X mol/g was calculated for qra.) That apparent value, however, can be closely estimated by application of eqs. (11) and (12) a t z = 0.2 t o the parameters of the two pure components. If, in this example the components had been intimately dispersed as in a composite and if the value of qm obtained for the mixture had been appreciably different from 4.74 X lop4 mol/g, interaction would have been definitely indicated.
MATERIALS AND METHODS The preparation of most of the samples used in this investigation has been described p r ~ v i o u s l y . ~The ~ freeze-dried teeth samples were
AGG HEGATES APPLIEI) TO D E N T A L MATERIALS
00
02
01
41
03
X
Fig. 1. BET plots for component 1 (bottom), component 2 (top), and a mixture of the two in a hypothetical experiment in which the error in the computed surface area of the mixture exceeds 50yc.
not even crushed. Immediately upon extraction of each tooth from the patient, i t was first rinsed in water, then quickly frozen by submergence i n liquid nitrogcan. The ttvth were then freeze-dried a t a temperatuw not exceeding 0°C until completely free of water. They were stored in a vacuum desiccator ovw a drying agent* until thrh nitrogen adsorption was carried out. The surface areas available t o nitrogen were obtained volumctrically a t - 196OC by means of an apparatus and technique described in 1951.15 The surface areas obtained gravimetrically from tht. adsorption of water vapor utilized a simple apparatus whose details are discussed in a recent publication.16 The water vapor adsorption experiments were carried out a t room tcmpcrature (23°C). A sample of activated silica gel (28 X 200 mesh) (Davison Chemical Co.) with a BET surface area of 711 m2/g (measured by water * Anhydrouh magnesium perchlorate.
42
LOEBENSTEIN
vapor) was treated with a copolymerizable silane coupling agent prior to being incorporated as a filler in a polymer composite. The silane coupling agent17-lgdesignated as A-174 (Union Carbide Corp.?) consisted of 3-methacryloxypropyltrimethoxysliane in a n amount equal to 50% of the weight of the silica gel. The silane was dissolved in a volume of acetone sufficient t o form a thick slurry in a roundbottom flask. The excess acetone was removed by pumping while rotating the flask until caking was complete. This was followed by an additional evacuation for 12 hr all a t room temperature. The was the addition polymer, designated as BIS-GMA by Bowen,20,21 product of bisphenol A and glycidyl methacrylate with polymerization initiated by the room temperature reaction of benzoyl peroxide with a tertiary-substituted aromatic aminc. Polymerization, both for the polymer alone as well as for composites of known percentages of the silane-coupled silica gel, was carried out in Teflon molds resulting in rod-shaped specimens with dimensions 0.00159 x 0.00159 X 0.0508 m (& X $$ X 2 in.). After curing, these specimens were rinsed with ethanol to remove any trace of unreacted monomer, dried for $ hr in an air over a t 50°C and stored a t room temperature without desiccation.
RESULTS Water Vapor Adsorption a. The system of anorganic whole teeth is composed of anorganic enamel and anorganic dentin. No data are available for the surface area of anorganic enamel calculated from water vapor adsorption. A value of 8.7 m2/g was recently reported22for whole enamel, however, and because of the enhanced adsorptivity of the organic matter23 along with its known percentage c o m p o ~ i t i o na, ~ n ~estimate of about 8 m2/g would appear to be reasonable for the anorganic component. The surface area of anorganic dentin was determined t o be 83.6 m2/g. If the anorganic enamel is present t o the extent of about 7%, the surface area of the anorganic whole teeth would be 78 m2/’gaccording
t Certain commercial materials and equipment are identified in this paper in order to specify adequately the experimental procedure. In no case does such identification imply recommendation or endorsement by the National Bureau of Standards nor does it imply that the material or equipment identified is necessarily the best available for this purpose.
AGGREGATES A P P L I E D TO D E N T A L MATERIALS
43
to eq. (5.). This agrees reasonably well with the measured value of 75 m2/g16 previously reported. It would serve no purpose in this case to calculate an apparent value because the agreement between the weighted average value and the measured value is certainly acceptable for measurements of this nature. The diff erencc between c values caused no problem and the system obviously exhibited aggregate behavior. b. Dentin, itself, is an intimate mixture consisting of approximately 79% of anorganic dentin (dry-weight basis) and 21% dentinal collagen. Thr difficulty in separating out the collagen and, a t the same time, to be certain that no denaturation, rupture of crosslinkages, and/or othrr irreversible change may have occurred precluded its use. Fortunately, however, only small differrnces have been found among the surface areas of various forms of collagrn measured by water vapor.16-23~25-28 For this reason, it was decidrd to apply eqs. (11) and (12) in an attempt to test for aggregate bchavior of a “reconstructed dentin” in which component 1 was a hide collagen whose parameters have been calculated from thr data of Corresponding to a surface area of about 408 m2/g the values of qml and c1 were 5.423 X lov3mol/g and 19.2, respectively, while fi = 0.21.29 The anorganic dentin as component 2 had a surface area of 83.6 m2/g when calculated from qms = 1.110 X mol/g (see Table I). The value of c2 was determined a t 5.3 while f 2 , of course, had to be 0.79, since the sum of the weight fractions is unity. Equation (12) yielded a slope a t ,r = 0.2 of 4.667 X lo2 g/mol which, if obtained experimentally, would h a w been equated to ( c - l)/(qmc). Equation (11) disclosed an effective intercept of 0.377 x lo2 g/mol for l/(qmc). From these values it is immediately verified that the apparent parameters are qm = 19.83 X lop4mol/g and c = 13.4, such that the apparent surface area of thc assumed aggregate is 149 m2/g. This is remarkably close t o the weighted average surface area of this “reconstructed dentin” based on eq. ( 5 ) which is 151,s m2/g, still with the assumption of no interaction. Finally, this assumption is virtually validated by comparing these results with an experimental determination of the apparent surface area from application of eq. (6) to the water vapor adsorption data of human dentin. A BET area of 129 m2/g was obtained corresponding to a qm value of 17.17 x mol/g along with a c value of 13.3. The difference between 152 and 129 is not considered significant here
LOEBKNSTEIN
44
TABLE I Values of the BET Parameters for Components and Mixtures Used with Each Adsorbate
Adsorbate
Adsorbent
qm value (mol/g)
5.423 x 10-3 Water vapor at 23°C hide collagen 1 . i i o x 10-3 anorganic dentin 1.717 X dentin 9.488 X silica gel 2.410 X activated bentonite poly (methyl methacrylate) 4.032 X 15.755 x 10-4 silane-coated silica gel a 4..566 X BJS-GMA (rods) 6.51yo silica gel coniposite 3.309 x 10-4 13.3% silica gel composite 4.406 X
Nitrogen a t
- 196OC
anorganic dentin dentin a freeze-dried teet,h enamel8 hide collagen
1.262 x 1.132 x 8.17 x 4.04 x 6. X
10-3 10-4 10-5 10-5
c value
19.2 .i . 3 13.3 2.31 103. 2.08 5.86 0.845 4.02 4.40 81.7 56.3 220. 6.8 34.2
* Interaction is identified in the applications described.
because of sampling errors in dentin, variation in collagen composition and the fact that small differences may exist in the percentage composition from the average values assumed here. c. An interesting exercise related to the dentin-water vapor system but not involving a “reconstructed dentin” is the use of the dentin parameters and the parameters for anorganic dentin together with cqs. (1 1) and (12) to predict the adsorptive characteristics of dentinal collagen. The assumption of aggregate behavior has already been demonstrated to be reasonably valid from the previous example. These values may be substituted into eq. (12) to obtain: (22.836 - 4.648) X 10-3(c1- 1) 5.776 x 10-3 [1 0.2(Cl - 1 ) ] 2 0 . m x ~ 103 = (22.836 - 4.648) X 2.499 10-3 1 0.2(Cl - 1)
+ +
+
+
I’
(13)
AGGREGATES APPLIEI) TO DENTAL MATERIALS
45
where the subscript 1 applies now t o dentinal collagen. The substitution of (22.836 - 4.648) X lop3corresponding to (qmc - f2qm2c2) in place of flqmlcl is in accordance with eq. (11). Setting
u
=
1
+ 0.2(Cl - 1)
(14)
allows eq. (13) t o be solved as a quadratic in u. Its value may be substituted back in eq. (14) t o obtain c1 which, in turn, may be used in eq. (11) t o solve for qml. These results are qm, = 41.31 X lop4 mol/g corresponding t o a surface area of 311 m2/g and c1 = 21. These values are entirely reasonable for dentinal collagen (assumed t o be more crosslinked than most other types of collagen) and were used in constructing the adsorption isotherm in Figure 2 for comparison with a similar curve for hide collagen. A linear combination of the surface areas for dentinal collagen and anorganic dentin ac-
h
-
CE
\
0
E
E
Y
U
I” 01 0.00
I
0.06
0.12
0.18
0.24
0.30
0.36
X
Fig. 2. Water vapor adsorption isothernu a t 23°C for dentinal collagen (bottom) compared with hide collagen both determined from their respective parameters. The curve for dentinal collagen was constructed from parameters evaluated indirectsly (see text), while that for hide collagen was recalculated from the data of B ~ l l . ~ 3
LOEBENSTEIN
46
cording to cq. ( 5 ) results in only a very slight correction for the anomaly in the B E T equation as applied to aggregates in this in-
stance, 0.21(311)
+ 0.79(83.6) = 131.3 m2/g
(15)
for the weighted average surface area of dentin (available to water vapor) as compared to 129 m2/g from fitting eq. (6). An error this small may justifiably be neglected. d. Water vapor adsorption measurements were made on the sample of silane-coated silica gel and on the rod-shaped specimens of polymerized BIS-GMA. I n addition, and a t the same time, adsorption isotherms were obtained from rod-shaped specimens of two composites, each of which consisted of different prrcentage mixtures of these same two materials. The lower humidity points (i.e. within the accepted B E T range) for each of the four isotherms were fitted reasonably well by eq. (6), the linear form of the B E T theory, to ottain estimates of their apparent surface areas. The high points on each isotherm for the composites (i.e. from 50y0of 100% humidity) were used to obtain an independent estimate of surface area based on the Harkins and Jura (H-J) equation.30 This theory predicts a linear relationship when In (1/z) is plotted as a function of l/q2. The square root of the slope of the resultant best straight line when multiplied b y a constant* yields the H-J surface area. The two composites contained 6.51% and 13.3%, respcctively, of the silane-coated silica gel intimately dispersed in thc BIS-GAIA. For each of these two samples there were now available four different estimates of surface area, one from the H-J equation and the remaining three based on the B E T theory. The first of these is the apparent value obtained from forcing a fit to eq. (6) as previously mentioned. The second is a n estimate of this same apparrnt value resulting from substituting the BET-derived parameters of the two components (filler and resin) into eqs. (11) and (12) at z = 0.2 and corresponding t o f i = 0.0651 for one composite and fi = 0.133 for the other. The final estimate was simply a weighted average of the individual component B E T areas in accordance with eq. (5). These results are summarized in Table 11. It is seen that a considerable difference * The H-J constant for water vapor at 23°C in units consistent with the present work is 5.62 X lo4 m2/mol.
AGGREGATES APPLIED TO DENTAL MATERIALS
47
TABLE I1 Surface Areas Derived from Water Vapor Adsorption by Two Composites with Different Percentage Compositions of Silane-coated Silica Gel (qml = 15.755 X lo-' mol/g, cl = 5.855) and BISGMA rods (qm2= 4.566 X lop4mol/g, c2 = 0.845) ______
6.51% Coated 13.37, Coated Silica Gel in Silica Gel in BIS-GMA BIS-GMA (mZ/g) W/g)
Apparent BET area from fitting experimental data to a conventional BET plot in accordance with eq. (6)
24.9
33.2
Apparent BET area obtained from substituting the BET parameters, qmand c, of the individual components in eqs. (11) and (12)
29.5
33.9
Correct BET area from a weighted average of the BET areas of the components by use of eq. (5)
39.8
45.6
Harkins and Jura area calculated from experimental data
. 37.5
47.8
exists between the apparent, BET areas and the correct values for the composites. The apparent values predicted by eqs. (11) and (12) are consistent with those found by fitting eq. (6) to the experimentally obtained points for the composites. It can be seen that the weighted average BET surface areas are more in agreement with the H-J values. An interesting aspect of this experiment is that the composite filler component, namely, the silane-coated silica gel revealed a B E T area as low as 118.6 m2/g in view of the fact that the corresponding value of the untreated silica gel was in excess of 700 m2/g. Such a dramatic decrease could not possibly be attributable to the anomalous behavior of the BET equation when applied to aggregates, and therefore, represents a clear example of interaction. Even so, this in no way interfered with the aggregate behavior of the coated silica gel and the resin. The agreement between the two apparent B E T areas in each case (i.e. between eq. (6) and eqs. (11) and (12)) confirms the observation that no interaction was evident between these two components of the composites.
48
LOEBENSTEIN
A highly significant result of these experiments is that all of the available surface of the silane-coated silica gel must have been completely accessible to the water molecule after the resin was cured. This could only take place in a resin which is completely pyrmeable to watcr molecules.
Nitrogen Adsorption a. An interesting application involving interaction concerns the surface area available to nitrogen of the componrnts of human dentin. As in the case of water vapor adsorption the absence of data for dentinal collagen made it necessary to use collagen of a different type. In this regard the (1950) results of Zettlcmoyer and coworkers31 relating t o the nitrogen adsorption of collagen derived from cow hide shows a surface of 0.6 mz/g (which would correspond to a qml of 0.006 X lop3mol/g) and a value of 34.2 for (1. The second component is anorganic dentin. The surface area available to nitrogen has varied somewhat depending upon the source of the sample and the temperature a t which i t was outgassed prior to m e a ~ u r e m e n t . ' ~ ~ ~ ~ I n all samples, however, the B E T nitrogen area was in excess of 100 m2/g, whilc some values ranged as high as 150 m2/g. Since this variability is not critical in the present illustration, an average sample was selected with a surface area of 124 m2/g corresponding to qm2 = 1.262 X lop3mol/g and cz = 81.7. Taking x = 0.2 and for the fraction of collagen, fi = 0.21 (as in the example for water vapor adsorption) i t was possible to solve eq. (12) and eq. (11) to obtain the apparent values of qm and c for human dentin. They were qm = 0.998 X mol/g and c = 81.6. This would correspond to an apparent surface area of 98 m2/g. The weighted average area according to eq. (5) would also be 98 m2/g which indicates no discrcpancy resulting from the anomalous effect of the B E T equation applied to this mixture. However, these values are considerably at variance with the BET areas according to eq. (6) when determined from the adsorption isotherm for human dentin, itself. These latter values have consistently been an order of magnitude lower, ranging from about 7 m2/gI4 to about 11 m2/g.33 A discrepancy this large (i.e. by a factor of ten) can only bc attributed to interaction between the components. To deny interaction would require the surface area of dentinal collagen to have a large negative value by virtue of eq. ( 5 ) .
A4GGIlEGATES APPLIED TO DENTAL 3IATERIALS
49
b. Whole teeth may be considered a mixture principally of dentin and enamel disregarding the fact that each of these two components is, in itself, a complex mixture. It would be interesting to determine whether the anomalous behavior of the B E T equation applied to mixtures is appreciable here. A nitrogen surface area (eq. (6)) of 4 m2/g was obtained for enamel resulting from qml = 0.404 X mol/g with c1 = 6.8 while denti@ has also via eq. (6) a B E T area of 11.1 m2/g for the same particle size corresponding to qmz = 1.132 X mol/g with c2 = 56.3. Assuming that the enamel comprises about 5% of the mixture, by weight, all the information is a t hand for applying eqs. (11) and (12). The result is a n apparent qm of 1.095 X mol/g and c value of 55.4. Equation (4), in agreement mol/g with this, predicts a weighted average qm of 1.096 X which corresponds to a surface area of 10.7 m2/g. A sample of freeze-dried teeth possessed a B E T area of 8.0 m2/g. The discrepancy between these latter two values is probably not real because the sample of dentin was a t the uppcr end of the range of surface areas and the freeze-dried teeth were used whole (unground) in the nitrogen adsorption experiment.
DISCUSSION The use of eq. (12) evaluated a t s = 0.2 together with eq. (11) for reconciling and interpreting the results of adsorption data as applied to dental materials as mixed adsorbents in the B E T range has been demonstrated in the present investigation. It is admittedly a tradeoff, sacrificing some accuracy in the interest of obtaining practical equations that can be used by the experimenter. The alternative procedures as proposed by o t h e r ~ ~ J O -involve ' ~ * ~ ~ far more complicated and tedious methods such as the solutions of sets of simultaneous nonlinear equations in four unknowns, iterative graphing techniques, use of very low pressure data, etc. A hypothetical solid mixture was described early in this paper which gave rise to a discrepancy in excess of 50yo from the correct surface area by misapplication of the B E T theory (eq. (6)). In the present work, two different percentage compositions of the same components of a synthetic composite when measured (Table 11) disclosed actual errors between apparent and correct surface areas which amounted t o 30% and 22%, respectively. In a logical extension, it
LOEBENSTEIN
50
would be instructive to explore the changes in percentage (war as a function of the relative composition of a two-component aggrcgatt.. This is shown in Figure 3 corrcsponging to hypothetical mixtures of activated bentonite35 (a clay) in a composite with poly(methy1 methacrylate) subject to water vapor adsorption at 23°C. For the pure polymer and for thc clay the values of qm and c arc given in Table 1. The error curve was calculated for various assumed percentages by comparing values computed from csqs. (11) and (12) with weighted average values, eq. (4) a t corresponding values of fi (for bentonite) and fi (for resin). The resultant error function is quite complex as is evident from the figure. The phenomenon of interaction was encountered in the water vapor adsorption of silane-coated silica gel and in the nitrogen adsorption of dentin. I n the first instancc there was admittedly no way to predict the resultant qm and c values of the coated silica gel. Yet, once measured, i t could be used with eqs. (11) and (12) and considered as a single component of the polymer composite although, in reality, this was a three-component system. I n the second instance, no difficulty was encountcrcd in reconciling thc nitrogen adsorption of
0
4
8
12
I6
20
24
%BENTONITE IN POLYMER MIXTURE
Fig. 3. Error in computed BET area as a function of composition for a binary mixture of activated bentonite and poly(methy1 methacrylate). The adsorbate was water vapor at 23°C.
whole t w t h as comprised of two components, dentin and enamd. Yet, each of thew so-called components was, itself, a mixture of two components with a t least one and probably both of them exhibiting interaction toward nitrogen adsorption. Thv interaction with collagen in dentin was demonstrated hcrr and has bwn recognized in other s t ~ d i e s ~as * , ivell. ~ ~ Enamchl is knownz4 to contain a small amount of organic matter of a kcratin type which probably bt.haves in a similar manner. This, then, would constitute a four-component system. In multicomponent aggregates whrrr interaction is not encountered (such as in the water vapor adsorption on whole it should be possible first t o combine the componcmts in pairs using the method developed here; then thew resultant va1uc.s again combined as pairs and so on until the adsorption parameters were obtained for the entire aggregate. The order in which the components are paired should be immaterial in the absence of interaction. Acrylic resins cured a t ambient tcmperaturcls such as the poly(methyl methacrylates) and t h r BIS-GMA used in this work arc’ similar to the “Pit and fissure sealants” employed in dental rmtorativc. applications. They may seal out bacteria very c4’cctivdy; but they are completely permeable to the water moleeulr. This work has demonstrated that even the silanc, coupling agent uscld on t h r silica gel was reasonably effective in restricting acccss of the water moleculc t o the pores of the silica gel, while the coated silica gel imbeddcd in the polymerized resin imposed no restriction, whatever, on thc easy access of water to its surface.
CONCLUSIONS 1 . Appreciable differences often exist betwwn tht. apparent B E T surface areas of multicomponcnt dental matcrials when determined in the traditional way and the weighted average value derived from the individual components. 2. A practical equation was derived which relates t h r BET parametus of the individual componcnts with thsl apparent parameters of the aggregate. This relationship, whilc inexact, is sufficiently accurate for most applications and is relatively easy to cvaluate. 3. Us(. was made of these findings t o idcntify instanew of t r w interaction which decreased the expected surfacc. area of t h r mixture.
52
LOEBENSTKIN
4. When compared with another applicable isotherm equation (Harkins and Jura), the same aggregate adsorption isotherms yirlded surface areas in agreement with the component weighted average B E T values. 5 . Adsorption of watcr vapor was performed separatrly on each of t h r two components (particlr and rcsin) of the materials which would comprise a composite similar to the type used in dental restorations. Adsorption data were also obtained for the polymerized composite itself. Thc abscnce of interaction confirmed a complete permeability of water vapor through the resin.
References J . .4mer. Chem. SOC.,60, 309 (1938). S. Brunauer, The Adsorption of Gases and Vapors, I-Physical .4dsorption, Princeton University Press, Princeton, N.J., 1943, p. 1.52. H . M . Cassel, J . Phys. Chem., 48, 195 (1944). G. Halsey, J . Chem. Phys., 16, 931 (1948). T. 1,. Hill, in Advances in Catalysis, Vol. IV, Academic Press, h e . , New York, 1952, p. 227. A. W. Adamson, Physical Chemistry ofh'urfaces, Interscience Publishers, Inc., New York, 1960, p. 501. K. B. Anderson and W. E. Hall, J . Amer. Chem. Soc., 70, 1727 (1948). W. C. Walker and A. C. Zettlemoyer, J . Phys. Colloid Chem., 52, 47 (1948). I. Langmuir, J . Amer. Chem. SOC.,40, 1361 (1918); see also Ref. 2, p. 74. A. Takizawa, Kolloid Z., 222, 141 (1968). W. A. Steele, J . Phys. Chem., 61, 1551 (19.57). L. 111. Ilormant and A. W. Adamson, J . Collozd Interface Sci., 38, 28.5 (1972). A. A. hlarino, It. 0. Becker, and C. H. Bachman, Phys. dfed. Riol., 12, 367 (1967). W. V. Loebenstein, J . Dent. Res., 51, 1529 (1972). W. V . Loebenstein and V. It. Deitx, J . Res. Nut. Bur. Stand., 46, 51 (1951). W. V. Loebenstein, J. Dent. Res., 52, 271 (1973). H. A. Clark, E. P. Plueddemann, Mod. Plast., 40, 133, 196 (June, 1963). S. Sterman and J. G. Marsden, Afod. Plast., 40, 135, 177 (July, 1963). S. Sterman and J. B. Toogood, ildhes. Age, 8, 34 (July, 196.5). K. L. Bowen, J . Dent. Res., 44, 690 (1965). U.S. Pat. 3,179, 623 (April 20, 1965) R. L. Bowen. G. €1. Dibdin, J . Dent. Res., 51, 1256 (1972). H. B. Bull, J . ilmer. Chem. Soc., 66, 1499 (1944). F. Brudevold and It. Soremark, in Structural and Chemical Organization of Teeth, Vol. 11, A. E. W. Miles, Ed., Academic Press, New York and London, 1967, p. 248. J. 13. Kanagy, J . Res. Nut. Bur. Stand., 38, 119 (1947).
1. S. Brunauer, P. H. Emmett, and E . Teller,
2. 3. 4. 5.
6. 7. 8. 9. 10. 11. 12. 13. 14. 1.5. 16. 17. 18. 19. 20. 21. 22. 23. 24.
23.
AGGHEGATES APPLIEI) TO DENTAL MATERIALS
53
26. J. R. Kanagy, J . Amer. Leather Chem. Assoc., 45, 12 (19.50). 27. 13. W. Green and K. P. Ang, J . Amer. Chem. SOC.,75, 2733 (19Ti3). 28. H. R. Elden, in International Review of Connective Tissue Research, Vol. 4, D. A. Hall, Ed., Academic Press, New York, 1968, p. 298. 29. J. E. Eastoe, in Structural and Chemical Organization of Teeth, Vol. 11, A. E. W. Miles, Ed., Academic Press, New York and London, 1967, p. 300. 30. A. W. Adamson, Physical Chemistry of Surfaces, Interscience Publishers, Inc., New York, 1960, p. 491. 31. A. C. Zettlemoyer, A. Chand, and E. Gamble, J . Amer. Chem. SOC.,72, 2752 (1950). 32. W. V. Loebenstein, in Adhesive Restorative Dental Materials-I I , Public Health Service, Pub. No. 1494, 196,5, pp. 213-223. 33. G. M. Brauer, Adhesive Restorative Dental Materials-I I , Public Health Service, Pub. No. 1494, 1965, pp. 203-212. 34. L. E. Drain and J. A. Morrison, Trans. Faraday Soc., 48, 316 (1952). 3.5. S. Nagasawa, l’ohoku J . Agr. Res., 4, 7.5 (1953). 36. E. H . Lieberman and S. G. Gilbert, J . Polym. Sci. S y m p . No. 41, 3 3 (1973).
Received Ifarch 27, 1974 Revised July 1, 1974
VOL. 9, PP. 35-53 (1975)
The Surface Area of Aggregates Applied to Dental Materials WILLIAM V. LOEBENSTEIN, Dental Research Section, National Bureau of Standards, Washington, D.C.
Summary There is a continuing need for the complete characterization of the physica and chemical properties of dental materials. Among these properties is surface area. The problem is further complicated by the fact that most dental materials are, themselves, mixtures of two or more identifiable components. If the vapor adsorptive properties of these components are different in the mixture from that, which would be expected of them collectively, then interaction is present. Interaction must not be confused with the lack of additivity which results from the limitations of the BET theory applied to mixtures. Equations are derived herein t o estimate the extent of this latter source of variability and to correct for it, giving a “true” surface area for the aggregate. Conversely, the adsorptive properties of either component can be calculated from the properties of the mixture and the remaining component together with the percentage composition. An immediate application can be made in determining the water-vapor adsorptive properties of human dentinal collagen without necessitating its removal from dentin. Any attempt to extract it chemically may produce denaturation or chain rupture thus precluding the possibility of direct determination. In the case of nitrogen adsorption, however, interaction definitely is indicated.
INTRODUCTION The Brunauer, Emmett, and Teller equation (BET)’ has consistently given quite satisfactory results over the years in routine surface area determinations. I t s assumptions and approximation^^-^ for the most part, are well known and as long as its limitations are observed, i t will remain the preferred method over other alternative t h e ~ r i e s . ~ . ’ The fact that the B E T equation may not be appropriate when applied to solid mixtures, or aggregates, all too often escapes even the most careful experimenter. A large error in the calculated or apparent surface area could be inferred by a deviation from linearity 35 Contribution of the National Bureau of Standards. Not subject to copyright.
36
LOEBENSTEIN
of the BET plot; but that deviation might be so slight as to be unnoticed entirely or if discernable, might wrongly be attributed to experimental error. For two or more components each of which possesses a sufficiently different average adsorption energy, the simple B E T equation is inapplicable and cannot be easily rectified. Walker and Zettlemoyers identified the phenomenon in a treatment similar t o Langmuir’s expanded equationg which took account of surface heterogeneities in a unimolecular model. They, as well as others,10-12 developed solutions to the problem which, while correct, were extremely complex and based on techniques such as the simultaneous solution of four nonlinear equations for even the simplest two-component system or a tedious graphical iterative recomputation of adsorption energy distributions. Even as recently as 197212 a practical working solution had not been found. One of the purposes of this paper is t o evolve a workable relationship between the slope and intercept of a n apparent B E T fit for the adsorption data of a n aggregate and to express this in terms of the valid B E T parameters which apply to the individual components comprising the mixture. Although it is admittedly a n approximation, it is sufficiently correct for most applications and decidedly better than no correction a t all. As a corollary, the proposed relationship will identify those applications where the apparent fit requires no correction. A second phenomenon encountered when dealing with adsorbent mixtures is interaction. I n its present context, interaction is defined as the interference of one component with the surface area of another. It would manifest itself if one of the components were contained, to some degree a t least, within the pores of another so as to deny access of the adsorbate and thus give rise to a diminished surface area. No theoretical development is available for predicting the extent of interaction. A second objective of this investigation is t o identify interaction when i t is present. I n the absence of interaction the surface area of the aggregate would be a mass-weighted average of the specific areas of each of its components. Its presence would be verified experimentally if any significant difference existed between the apparent surface area of the aggregate obtained by blind application of the B E T equation and the apparent BET area of a nonintermingled mixture of each of the same components in identical percentages.
AGGREGATES APPLIED TO DENTAL MATERIALS
37
I n testing the relationships derived herein, the experimental applications are confined t o the field of dental materials. They include naturally occurring aggregates such as the tooth itself, as well as man-made aggregates such as polymer-particulate composites similar to those used in dental restorative work. The examples include both water vapor and nitrogen as the adsorbates.
THEORETICAL For a solid adsorbent consisting of a single component, the number of moles, q, of vapor adsorbed per gram a t relative pressure, x, can be expressed by eq. (1) in accordance with the BET free surface equation q =
qmcx (1 - .?.) (1 -
-
1'
+ cs)
where qm is a constant equal to the number of moles of adsorbate needed t o produce a monolayer and therefore is proportional" to the surface area. The other parameter, c, is characteristic of the system and is related exponentially to the difference between the average energy of adsorption of the initial laycr adsorbed and the heat of liquefaction of the adsorbate. With an adsorbent for which the B E T theory is applicable, eq. (1) is valid over a range in I from about 0.1 to 0.3. Let us consider two adsorbents each of which satisfies eq. (1). Two pairs of parameters would tben be obtained, qml and cl, for the first adsorbent and qmz and c2, for the second. If w1 g of the first adsorbent is placed beside (but not mixed with) w2 g of the second in a dual adsorption tube where they are exposed to a n adsorbable vapor a t some relative pressure, 5 , the amount of vapor that would be adsorbed per g of uncombined adsorbent mixture would be (q,7cl q2w2)/(w1 wy)which is the same as
+
+
flql
+
f2q2
=
q
(2)
where fi andf2 are the respective weight fractions of each component. When q, q1, and q 2 in eq. (2) arc replaced by their corresponding
* The proportionality constants used in this work are 98.02 X lo3m*/mol for x 104 m2/mol for H20 a t room temperature.
Nz at -196°C and 7..529
LOEBENSTEIN
38
values according to eq. ( l ) , the following expression results after cancellation of non-zero common factors : QmC
fl9rnlCl
--f
1 - T n . C T
1-Tn.CclT
+l-z+czx
f ~ q m z ~ 2
(3)
I n the general case, eq. (3) can not be an equality, since all elements are constants except x. If i t were an equality a t a particular value of 2, i t could not be equal a t any other value. This limitation of the B E T equation was appreciated by Marino et al.13 In the special case where the c values are all equal, the x’s disappear and an equality does exist, namely: Qm
= fiqm,
+
.f2qm,
(4)
Each qm value is proportional to the surface area, A , with the same proportionality constant as long as the same adsorbate is used a t that temperature. Therefore, eq. (4)is equivalent to eq. (5).
A
=
fiAi
+
f2Az
(5)
Equation (5) is, of course, true and can be derived by simple inference without recourse to any assumed adsorption theory. Deviations from eq. (5) might be appreciable and yet be entirely undetected or they might erroneously be attributed to interaction among the components of a solid mixture rather than to the misapplication of the B E T throry. This delemma can be resolved by separating these two mutually independent sources of variability. The more familiar linear form of eq. (1) is =
a,
+ b,z
(6)
where :
and i = 1 for component 1 and 2 for component 2 while the absence of a subscript signifies the mixture.
AGGREGATES APPLIED TO DENTAL MATERIALS
39
By combining eq. (2) with eq. (7) for all values of 3: within the B E T range and making use of eq. (6) t o eliminate y l and yz, y can be expressed as a function of z:
Y’
a1az
+ (aibz + a z b i ) ~+ bib2z2 + fzad + +
(flu2
(fib2
( 10)
f*b1)5
This function, eq. (lo), is the curve for the aggregate which is so often mistaken for linear that the detection of a curvature is often attributed to experimental error or faulty measurement. By forcing eq. (10) t o a straight line of the form of eq. (6), i t is seen from eq. (8) (without subscripts) that
By the same reasoning the slope of eq. (10) can be obtained by differentiation and equated approximately t o its linear counterpart, eq. (9) (without subscripts). The original parameters may then be substituted back t o eliminate the a’s and b’s. The resultant slope for the mixture, expressed in terms of its components, is
+ z(c1 - l)]‘ + [I- + - 1)l’ + - [I + z(c1 - 1) 1 + z(c2 - 1) fiJlmlCl(C1
c - 1
.
- 1)
[l
2(Cz
fiqmiC1
qmc
f29rn2~2(~2 - 1) f2qm2CZ
1
(12)
The degree to which eq. (10) approximates linearity in any particular combination of components and weight fractions is readily tested by substituting two values for 2 in eq. (12) a t widely different points on the curve yet within the applicable range of the B E T equation. It is surprising how great a difference between c1 and c2 can be tolerated in some instances without reflecting a n appreciable change in slope. In most applications i t would be reasonable when forcing a linear fit to choose the value of x = 0.2. for use in eq. (12) because this is the midpoint of the valid range for all of the data. A small error is introduced by taking the z = 0 value of eq. (10) as equivalent to the extrapolated intercept of a “least-squares” line forced through the experimental points. A somewhat closer agreement would be obtained by the extrapolation of a line passing through a point whose y coordinate at, say, II’ = 0.15 would be found from
40
LOEBENSTEIN
eq. (10) and whose slope was equal to the value determined for eq. (12). Except in extreme cases, however, the extra effort results in little improvement over the simpler use of eq. (11).
Hypothetical Illustration It is instructive now to consider a hypothetical mixture where no interaction will be permitted between the two components and their surface areas will be identical. Appreciable differences will be assumed for their c values, although not unusually large when compared with many naturally occurring materials. It will be assumed that mol/g and c1 = 10 while c2 = 0.5. The qml = qmz = 10.0 X “true” value of qm for the resultant of a mixture of any proportion mol/g. must, of course, be 10.0 X However, the apparent value obtained from a best fit of the B E T equation will depend upon the composition of the mixture. Consider the amounts of vapor adsorbed per gram for each pure component at, say, 0.14, 0.18, 0.22, and 0.26 relative pressure values. They would be 0.7202, 0.8380, 0.9658, and 1.0518 mmol/g for component 1 and 0.0875, 0.1206, 0.1.585, and 0.2019 mmol/g for component 2, respectively, as may be verified from eq. (1). A nonmixed combination of these two components consisting of 16.5% of component 1 a t the same relative pressures would adsorb 0.1919, 0.2390, 0.2917, and 0.3421 mmol/g, respectively, according to eq. ( 2 ) . When these data are plotted using the linear form of the B E T theory (eqs. (6) and (7)), the resultant curves are shown in Figure 1. Evaluation of qm from eqs. (8) and (9) applied to the combination (as would normally have been donc) would result in a n apparent qm value of less than one half of its weighted average value (eq. (4)). (A least-squares value of 4.74 X mol/g was calculated for qra.) That apparent value, however, can be closely estimated by application of eqs. (11) and (12) a t z = 0.2 t o the parameters of the two pure components. If, in this example the components had been intimately dispersed as in a composite and if the value of qm obtained for the mixture had been appreciably different from 4.74 X lop4 mol/g, interaction would have been definitely indicated.
MATERIALS AND METHODS The preparation of most of the samples used in this investigation has been described p r ~ v i o u s l y . ~The ~ freeze-dried teeth samples were
AGG HEGATES APPLIEI) TO D E N T A L MATERIALS
00
02
01
41
03
X
Fig. 1. BET plots for component 1 (bottom), component 2 (top), and a mixture of the two in a hypothetical experiment in which the error in the computed surface area of the mixture exceeds 50yc.
not even crushed. Immediately upon extraction of each tooth from the patient, i t was first rinsed in water, then quickly frozen by submergence i n liquid nitrogcan. The ttvth were then freeze-dried a t a temperatuw not exceeding 0°C until completely free of water. They were stored in a vacuum desiccator ovw a drying agent* until thrh nitrogen adsorption was carried out. The surface areas available t o nitrogen were obtained volumctrically a t - 196OC by means of an apparatus and technique described in 1951.15 The surface areas obtained gravimetrically from tht. adsorption of water vapor utilized a simple apparatus whose details are discussed in a recent publication.16 The water vapor adsorption experiments were carried out a t room tcmpcrature (23°C). A sample of activated silica gel (28 X 200 mesh) (Davison Chemical Co.) with a BET surface area of 711 m2/g (measured by water * Anhydrouh magnesium perchlorate.
42
LOEBENSTEIN
vapor) was treated with a copolymerizable silane coupling agent prior to being incorporated as a filler in a polymer composite. The silane coupling agent17-lgdesignated as A-174 (Union Carbide Corp.?) consisted of 3-methacryloxypropyltrimethoxysliane in a n amount equal to 50% of the weight of the silica gel. The silane was dissolved in a volume of acetone sufficient t o form a thick slurry in a roundbottom flask. The excess acetone was removed by pumping while rotating the flask until caking was complete. This was followed by an additional evacuation for 12 hr all a t room temperature. The was the addition polymer, designated as BIS-GMA by Bowen,20,21 product of bisphenol A and glycidyl methacrylate with polymerization initiated by the room temperature reaction of benzoyl peroxide with a tertiary-substituted aromatic aminc. Polymerization, both for the polymer alone as well as for composites of known percentages of the silane-coupled silica gel, was carried out in Teflon molds resulting in rod-shaped specimens with dimensions 0.00159 x 0.00159 X 0.0508 m (& X $$ X 2 in.). After curing, these specimens were rinsed with ethanol to remove any trace of unreacted monomer, dried for $ hr in an air over a t 50°C and stored a t room temperature without desiccation.
RESULTS Water Vapor Adsorption a. The system of anorganic whole teeth is composed of anorganic enamel and anorganic dentin. No data are available for the surface area of anorganic enamel calculated from water vapor adsorption. A value of 8.7 m2/g was recently reported22for whole enamel, however, and because of the enhanced adsorptivity of the organic matter23 along with its known percentage c o m p o ~ i t i o na, ~ n ~estimate of about 8 m2/g would appear to be reasonable for the anorganic component. The surface area of anorganic dentin was determined t o be 83.6 m2/g. If the anorganic enamel is present t o the extent of about 7%, the surface area of the anorganic whole teeth would be 78 m2/’gaccording
t Certain commercial materials and equipment are identified in this paper in order to specify adequately the experimental procedure. In no case does such identification imply recommendation or endorsement by the National Bureau of Standards nor does it imply that the material or equipment identified is necessarily the best available for this purpose.
AGGREGATES A P P L I E D TO D E N T A L MATERIALS
43
to eq. (5.). This agrees reasonably well with the measured value of 75 m2/g16 previously reported. It would serve no purpose in this case to calculate an apparent value because the agreement between the weighted average value and the measured value is certainly acceptable for measurements of this nature. The diff erencc between c values caused no problem and the system obviously exhibited aggregate behavior. b. Dentin, itself, is an intimate mixture consisting of approximately 79% of anorganic dentin (dry-weight basis) and 21% dentinal collagen. Thr difficulty in separating out the collagen and, a t the same time, to be certain that no denaturation, rupture of crosslinkages, and/or othrr irreversible change may have occurred precluded its use. Fortunately, however, only small differrnces have been found among the surface areas of various forms of collagrn measured by water vapor.16-23~25-28 For this reason, it was decidrd to apply eqs. (11) and (12) in an attempt to test for aggregate bchavior of a “reconstructed dentin” in which component 1 was a hide collagen whose parameters have been calculated from thr data of Corresponding to a surface area of about 408 m2/g the values of qml and c1 were 5.423 X lov3mol/g and 19.2, respectively, while fi = 0.21.29 The anorganic dentin as component 2 had a surface area of 83.6 m2/g when calculated from qms = 1.110 X mol/g (see Table I). The value of c2 was determined a t 5.3 while f 2 , of course, had to be 0.79, since the sum of the weight fractions is unity. Equation (12) yielded a slope a t ,r = 0.2 of 4.667 X lo2 g/mol which, if obtained experimentally, would h a w been equated to ( c - l)/(qmc). Equation (11) disclosed an effective intercept of 0.377 x lo2 g/mol for l/(qmc). From these values it is immediately verified that the apparent parameters are qm = 19.83 X lop4mol/g and c = 13.4, such that the apparent surface area of thc assumed aggregate is 149 m2/g. This is remarkably close t o the weighted average surface area of this “reconstructed dentin” based on eq. ( 5 ) which is 151,s m2/g, still with the assumption of no interaction. Finally, this assumption is virtually validated by comparing these results with an experimental determination of the apparent surface area from application of eq. (6) to the water vapor adsorption data of human dentin. A BET area of 129 m2/g was obtained corresponding to a qm value of 17.17 x mol/g along with a c value of 13.3. The difference between 152 and 129 is not considered significant here
LOEBKNSTEIN
44
TABLE I Values of the BET Parameters for Components and Mixtures Used with Each Adsorbate
Adsorbate
Adsorbent
qm value (mol/g)
5.423 x 10-3 Water vapor at 23°C hide collagen 1 . i i o x 10-3 anorganic dentin 1.717 X dentin 9.488 X silica gel 2.410 X activated bentonite poly (methyl methacrylate) 4.032 X 15.755 x 10-4 silane-coated silica gel a 4..566 X BJS-GMA (rods) 6.51yo silica gel coniposite 3.309 x 10-4 13.3% silica gel composite 4.406 X
Nitrogen a t
- 196OC
anorganic dentin dentin a freeze-dried teet,h enamel8 hide collagen
1.262 x 1.132 x 8.17 x 4.04 x 6. X
10-3 10-4 10-5 10-5
c value
19.2 .i . 3 13.3 2.31 103. 2.08 5.86 0.845 4.02 4.40 81.7 56.3 220. 6.8 34.2
* Interaction is identified in the applications described.
because of sampling errors in dentin, variation in collagen composition and the fact that small differences may exist in the percentage composition from the average values assumed here. c. An interesting exercise related to the dentin-water vapor system but not involving a “reconstructed dentin” is the use of the dentin parameters and the parameters for anorganic dentin together with cqs. (1 1) and (12) to predict the adsorptive characteristics of dentinal collagen. The assumption of aggregate behavior has already been demonstrated to be reasonably valid from the previous example. These values may be substituted into eq. (12) to obtain: (22.836 - 4.648) X 10-3(c1- 1) 5.776 x 10-3 [1 0.2(Cl - 1 ) ] 2 0 . m x ~ 103 = (22.836 - 4.648) X 2.499 10-3 1 0.2(Cl - 1)
+ +
+
+
I’
(13)
AGGREGATES APPLIEI) TO DENTAL MATERIALS
45
where the subscript 1 applies now t o dentinal collagen. The substitution of (22.836 - 4.648) X lop3corresponding to (qmc - f2qm2c2) in place of flqmlcl is in accordance with eq. (11). Setting
u
=
1
+ 0.2(Cl - 1)
(14)
allows eq. (13) t o be solved as a quadratic in u. Its value may be substituted back in eq. (14) t o obtain c1 which, in turn, may be used in eq. (11) t o solve for qml. These results are qm, = 41.31 X lop4 mol/g corresponding t o a surface area of 311 m2/g and c1 = 21. These values are entirely reasonable for dentinal collagen (assumed t o be more crosslinked than most other types of collagen) and were used in constructing the adsorption isotherm in Figure 2 for comparison with a similar curve for hide collagen. A linear combination of the surface areas for dentinal collagen and anorganic dentin ac-
h
-
CE
\
0
E
E
Y
U
I” 01 0.00
I
0.06
0.12
0.18
0.24
0.30
0.36
X
Fig. 2. Water vapor adsorption isothernu a t 23°C for dentinal collagen (bottom) compared with hide collagen both determined from their respective parameters. The curve for dentinal collagen was constructed from parameters evaluated indirectsly (see text), while that for hide collagen was recalculated from the data of B ~ l l . ~ 3
LOEBENSTEIN
46
cording to cq. ( 5 ) results in only a very slight correction for the anomaly in the B E T equation as applied to aggregates in this in-
stance, 0.21(311)
+ 0.79(83.6) = 131.3 m2/g
(15)
for the weighted average surface area of dentin (available to water vapor) as compared to 129 m2/g from fitting eq. (6). An error this small may justifiably be neglected. d. Water vapor adsorption measurements were made on the sample of silane-coated silica gel and on the rod-shaped specimens of polymerized BIS-GMA. I n addition, and a t the same time, adsorption isotherms were obtained from rod-shaped specimens of two composites, each of which consisted of different prrcentage mixtures of these same two materials. The lower humidity points (i.e. within the accepted B E T range) for each of the four isotherms were fitted reasonably well by eq. (6), the linear form of the B E T theory, to ottain estimates of their apparent surface areas. The high points on each isotherm for the composites (i.e. from 50y0of 100% humidity) were used to obtain an independent estimate of surface area based on the Harkins and Jura (H-J) equation.30 This theory predicts a linear relationship when In (1/z) is plotted as a function of l/q2. The square root of the slope of the resultant best straight line when multiplied b y a constant* yields the H-J surface area. The two composites contained 6.51% and 13.3%, respcctively, of the silane-coated silica gel intimately dispersed in thc BIS-GAIA. For each of these two samples there were now available four different estimates of surface area, one from the H-J equation and the remaining three based on the B E T theory. The first of these is the apparent value obtained from forcing a fit to eq. (6) as previously mentioned. The second is a n estimate of this same apparrnt value resulting from substituting the BET-derived parameters of the two components (filler and resin) into eqs. (11) and (12) at z = 0.2 and corresponding t o f i = 0.0651 for one composite and fi = 0.133 for the other. The final estimate was simply a weighted average of the individual component B E T areas in accordance with eq. (5). These results are summarized in Table 11. It is seen that a considerable difference * The H-J constant for water vapor at 23°C in units consistent with the present work is 5.62 X lo4 m2/mol.
AGGREGATES APPLIED TO DENTAL MATERIALS
47
TABLE I1 Surface Areas Derived from Water Vapor Adsorption by Two Composites with Different Percentage Compositions of Silane-coated Silica Gel (qml = 15.755 X lo-' mol/g, cl = 5.855) and BISGMA rods (qm2= 4.566 X lop4mol/g, c2 = 0.845) ______
6.51% Coated 13.37, Coated Silica Gel in Silica Gel in BIS-GMA BIS-GMA (mZ/g) W/g)
Apparent BET area from fitting experimental data to a conventional BET plot in accordance with eq. (6)
24.9
33.2
Apparent BET area obtained from substituting the BET parameters, qmand c, of the individual components in eqs. (11) and (12)
29.5
33.9
Correct BET area from a weighted average of the BET areas of the components by use of eq. (5)
39.8
45.6
Harkins and Jura area calculated from experimental data
. 37.5
47.8
exists between the apparent, BET areas and the correct values for the composites. The apparent values predicted by eqs. (11) and (12) are consistent with those found by fitting eq. (6) to the experimentally obtained points for the composites. It can be seen that the weighted average BET surface areas are more in agreement with the H-J values. An interesting aspect of this experiment is that the composite filler component, namely, the silane-coated silica gel revealed a B E T area as low as 118.6 m2/g in view of the fact that the corresponding value of the untreated silica gel was in excess of 700 m2/g. Such a dramatic decrease could not possibly be attributable to the anomalous behavior of the BET equation when applied to aggregates, and therefore, represents a clear example of interaction. Even so, this in no way interfered with the aggregate behavior of the coated silica gel and the resin. The agreement between the two apparent B E T areas in each case (i.e. between eq. (6) and eqs. (11) and (12)) confirms the observation that no interaction was evident between these two components of the composites.
48
LOEBENSTEIN
A highly significant result of these experiments is that all of the available surface of the silane-coated silica gel must have been completely accessible to the water molecule after the resin was cured. This could only take place in a resin which is completely pyrmeable to watcr molecules.
Nitrogen Adsorption a. An interesting application involving interaction concerns the surface area available to nitrogen of the componrnts of human dentin. As in the case of water vapor adsorption the absence of data for dentinal collagen made it necessary to use collagen of a different type. In this regard the (1950) results of Zettlcmoyer and coworkers31 relating t o the nitrogen adsorption of collagen derived from cow hide shows a surface of 0.6 mz/g (which would correspond to a qml of 0.006 X lop3mol/g) and a value of 34.2 for (1. The second component is anorganic dentin. The surface area available to nitrogen has varied somewhat depending upon the source of the sample and the temperature a t which i t was outgassed prior to m e a ~ u r e m e n t . ' ~ ~ ~ ~ I n all samples, however, the B E T nitrogen area was in excess of 100 m2/g, whilc some values ranged as high as 150 m2/g. Since this variability is not critical in the present illustration, an average sample was selected with a surface area of 124 m2/g corresponding to qm2 = 1.262 X lop3mol/g and cz = 81.7. Taking x = 0.2 and for the fraction of collagen, fi = 0.21 (as in the example for water vapor adsorption) i t was possible to solve eq. (12) and eq. (11) to obtain the apparent values of qm and c for human dentin. They were qm = 0.998 X mol/g and c = 81.6. This would correspond to an apparent surface area of 98 m2/g. The weighted average area according to eq. (5) would also be 98 m2/g which indicates no discrcpancy resulting from the anomalous effect of the B E T equation applied to this mixture. However, these values are considerably at variance with the BET areas according to eq. (6) when determined from the adsorption isotherm for human dentin, itself. These latter values have consistently been an order of magnitude lower, ranging from about 7 m2/gI4 to about 11 m2/g.33 A discrepancy this large (i.e. by a factor of ten) can only bc attributed to interaction between the components. To deny interaction would require the surface area of dentinal collagen to have a large negative value by virtue of eq. ( 5 ) .
A4GGIlEGATES APPLIED TO DENTAL 3IATERIALS
49
b. Whole teeth may be considered a mixture principally of dentin and enamel disregarding the fact that each of these two components is, in itself, a complex mixture. It would be interesting to determine whether the anomalous behavior of the B E T equation applied to mixtures is appreciable here. A nitrogen surface area (eq. (6)) of 4 m2/g was obtained for enamel resulting from qml = 0.404 X mol/g with c1 = 6.8 while denti@ has also via eq. (6) a B E T area of 11.1 m2/g for the same particle size corresponding to qmz = 1.132 X mol/g with c2 = 56.3. Assuming that the enamel comprises about 5% of the mixture, by weight, all the information is a t hand for applying eqs. (11) and (12). The result is a n apparent qm of 1.095 X mol/g and c value of 55.4. Equation (4), in agreement mol/g with this, predicts a weighted average qm of 1.096 X which corresponds to a surface area of 10.7 m2/g. A sample of freeze-dried teeth possessed a B E T area of 8.0 m2/g. The discrepancy between these latter two values is probably not real because the sample of dentin was a t the uppcr end of the range of surface areas and the freeze-dried teeth were used whole (unground) in the nitrogen adsorption experiment.
DISCUSSION The use of eq. (12) evaluated a t s = 0.2 together with eq. (11) for reconciling and interpreting the results of adsorption data as applied to dental materials as mixed adsorbents in the B E T range has been demonstrated in the present investigation. It is admittedly a tradeoff, sacrificing some accuracy in the interest of obtaining practical equations that can be used by the experimenter. The alternative procedures as proposed by o t h e r ~ ~ J O -involve ' ~ * ~ ~ far more complicated and tedious methods such as the solutions of sets of simultaneous nonlinear equations in four unknowns, iterative graphing techniques, use of very low pressure data, etc. A hypothetical solid mixture was described early in this paper which gave rise to a discrepancy in excess of 50yo from the correct surface area by misapplication of the B E T theory (eq. (6)). In the present work, two different percentage compositions of the same components of a synthetic composite when measured (Table 11) disclosed actual errors between apparent and correct surface areas which amounted t o 30% and 22%, respectively. In a logical extension, it
LOEBENSTEIN
50
would be instructive to explore the changes in percentage (war as a function of the relative composition of a two-component aggrcgatt.. This is shown in Figure 3 corrcsponging to hypothetical mixtures of activated bentonite35 (a clay) in a composite with poly(methy1 methacrylate) subject to water vapor adsorption at 23°C. For the pure polymer and for thc clay the values of qm and c arc given in Table 1. The error curve was calculated for various assumed percentages by comparing values computed from csqs. (11) and (12) with weighted average values, eq. (4) a t corresponding values of fi (for bentonite) and fi (for resin). The resultant error function is quite complex as is evident from the figure. The phenomenon of interaction was encountered in the water vapor adsorption of silane-coated silica gel and in the nitrogen adsorption of dentin. I n the first instancc there was admittedly no way to predict the resultant qm and c values of the coated silica gel. Yet, once measured, i t could be used with eqs. (11) and (12) and considered as a single component of the polymer composite although, in reality, this was a three-component system. I n the second instance, no difficulty was encountcrcd in reconciling thc nitrogen adsorption of
0
4
8
12
I6
20
24
%BENTONITE IN POLYMER MIXTURE
Fig. 3. Error in computed BET area as a function of composition for a binary mixture of activated bentonite and poly(methy1 methacrylate). The adsorbate was water vapor at 23°C.
whole t w t h as comprised of two components, dentin and enamd. Yet, each of thew so-called components was, itself, a mixture of two components with a t least one and probably both of them exhibiting interaction toward nitrogen adsorption. Thv interaction with collagen in dentin was demonstrated hcrr and has bwn recognized in other s t ~ d i e s ~as * , ivell. ~ ~ Enamchl is knownz4 to contain a small amount of organic matter of a kcratin type which probably bt.haves in a similar manner. This, then, would constitute a four-component system. In multicomponent aggregates whrrr interaction is not encountered (such as in the water vapor adsorption on whole it should be possible first t o combine the componcmts in pairs using the method developed here; then thew resultant va1uc.s again combined as pairs and so on until the adsorption parameters were obtained for the entire aggregate. The order in which the components are paired should be immaterial in the absence of interaction. Acrylic resins cured a t ambient tcmperaturcls such as the poly(methyl methacrylates) and t h r BIS-GMA used in this work arc’ similar to the “Pit and fissure sealants” employed in dental rmtorativc. applications. They may seal out bacteria very c4’cctivdy; but they are completely permeable to the water moleeulr. This work has demonstrated that even the silanc, coupling agent uscld on t h r silica gel was reasonably effective in restricting acccss of the water moleculc t o the pores of the silica gel, while the coated silica gel imbeddcd in the polymerized resin imposed no restriction, whatever, on thc easy access of water to its surface.
CONCLUSIONS 1 . Appreciable differences often exist betwwn tht. apparent B E T surface areas of multicomponcnt dental matcrials when determined in the traditional way and the weighted average value derived from the individual components. 2. A practical equation was derived which relates t h r BET parametus of the individual componcnts with thsl apparent parameters of the aggregate. This relationship, whilc inexact, is sufficiently accurate for most applications and is relatively easy to cvaluate. 3. Us(. was made of these findings t o idcntify instanew of t r w interaction which decreased the expected surfacc. area of t h r mixture.
52
LOEBENSTKIN
4. When compared with another applicable isotherm equation (Harkins and Jura), the same aggregate adsorption isotherms yirlded surface areas in agreement with the component weighted average B E T values. 5 . Adsorption of watcr vapor was performed separatrly on each of t h r two components (particlr and rcsin) of the materials which would comprise a composite similar to the type used in dental restorations. Adsorption data were also obtained for the polymerized composite itself. Thc abscnce of interaction confirmed a complete permeability of water vapor through the resin.
References J . .4mer. Chem. SOC.,60, 309 (1938). S. Brunauer, The Adsorption of Gases and Vapors, I-Physical .4dsorption, Princeton University Press, Princeton, N.J., 1943, p. 1.52. H . M . Cassel, J . Phys. Chem., 48, 195 (1944). G. Halsey, J . Chem. Phys., 16, 931 (1948). T. 1,. Hill, in Advances in Catalysis, Vol. IV, Academic Press, h e . , New York, 1952, p. 227. A. W. Adamson, Physical Chemistry ofh'urfaces, Interscience Publishers, Inc., New York, 1960, p. 501. K. B. Anderson and W. E. Hall, J . Amer. Chem. Soc., 70, 1727 (1948). W. C. Walker and A. C. Zettlemoyer, J . Phys. Colloid Chem., 52, 47 (1948). I. Langmuir, J . Amer. Chem. SOC.,40, 1361 (1918); see also Ref. 2, p. 74. A. Takizawa, Kolloid Z., 222, 141 (1968). W. A. Steele, J . Phys. Chem., 61, 1551 (19.57). L. 111. Ilormant and A. W. Adamson, J . Collozd Interface Sci., 38, 28.5 (1972). A. A. hlarino, It. 0. Becker, and C. H. Bachman, Phys. dfed. Riol., 12, 367 (1967). W. V. Loebenstein, J . Dent. Res., 51, 1529 (1972). W. V . Loebenstein and V. It. Deitx, J . Res. Nut. Bur. Stand., 46, 51 (1951). W. V. Loebenstein, J. Dent. Res., 52, 271 (1973). H. A. Clark, E. P. Plueddemann, Mod. Plast., 40, 133, 196 (June, 1963). S. Sterman and J. G. Marsden, Afod. Plast., 40, 135, 177 (July, 1963). S. Sterman and J. B. Toogood, ildhes. Age, 8, 34 (July, 196.5). K. L. Bowen, J . Dent. Res., 44, 690 (1965). U.S. Pat. 3,179, 623 (April 20, 1965) R. L. Bowen. G. €1. Dibdin, J . Dent. Res., 51, 1256 (1972). H. B. Bull, J . ilmer. Chem. Soc., 66, 1499 (1944). F. Brudevold and It. Soremark, in Structural and Chemical Organization of Teeth, Vol. 11, A. E. W. Miles, Ed., Academic Press, New York and London, 1967, p. 248. J. 13. Kanagy, J . Res. Nut. Bur. Stand., 38, 119 (1947).
1. S. Brunauer, P. H. Emmett, and E . Teller,
2. 3. 4. 5.
6. 7. 8. 9. 10. 11. 12. 13. 14. 1.5. 16. 17. 18. 19. 20. 21. 22. 23. 24.
23.
AGGHEGATES APPLIEI) TO DENTAL MATERIALS
53
26. J. R. Kanagy, J . Amer. Leather Chem. Assoc., 45, 12 (19.50). 27. 13. W. Green and K. P. Ang, J . Amer. Chem. SOC.,75, 2733 (19Ti3). 28. H. R. Elden, in International Review of Connective Tissue Research, Vol. 4, D. A. Hall, Ed., Academic Press, New York, 1968, p. 298. 29. J. E. Eastoe, in Structural and Chemical Organization of Teeth, Vol. 11, A. E. W. Miles, Ed., Academic Press, New York and London, 1967, p. 300. 30. A. W. Adamson, Physical Chemistry of Surfaces, Interscience Publishers, Inc., New York, 1960, p. 491. 31. A. C. Zettlemoyer, A. Chand, and E. Gamble, J . Amer. Chem. SOC.,72, 2752 (1950). 32. W. V. Loebenstein, in Adhesive Restorative Dental Materials-I I , Public Health Service, Pub. No. 1494, 196,5, pp. 213-223. 33. G. M. Brauer, Adhesive Restorative Dental Materials-I I , Public Health Service, Pub. No. 1494, 1965, pp. 203-212. 34. L. E. Drain and J. A. Morrison, Trans. Faraday Soc., 48, 316 (1952). 3.5. S. Nagasawa, l’ohoku J . Agr. Res., 4, 7.5 (1953). 36. E. H . Lieberman and S. G. Gilbert, J . Polym. Sci. S y m p . No. 41, 3 3 (1973).
Received Ifarch 27, 1974 Revised July 1, 1974
