3.1. The Ca lon Environments

The Ca ion lies on the mirror plane at x=1/4. Its nvironment is shown in figure 2 and summarized in able 3. The coordination of nine oxygens to Ca consists f three CO3 edges, O(1, 2), O(1, 2) and O(2, 2) nd three apexes, O(1", 2", 2"). The apparent thermal notion of Ca is almost isotropic (table 1, fig. 2).

Corrections as given by Busing and Levy [13] to btain the mean separation between atoms from the bserved atomic positional and thermal parameters were calculated using a program written by Finger [12]. hese corrections are included in tables 3 and 4.

3.2. The CO Group

The details of the CO3 group and its environment re given in table 4 and shown in figure 3. The positions eported by Bragg [3] give C-O distances of 1.26 Å and 30 A, and O-C-O angles of 117°, 117° and 127°, hich, under the circumstances, are all close to those eported here. The C-O distances and O-C-O ngles reported here for the CO; group do not differ rom each other significantly at the 95 percent confience level; the same is true for the O-C-O angles. In iew of the possibility of unknown systematic error, necssarily excluded from the calculations, our results

3

therefore do not preclude trigonality of the CO3 group. Because the oxygens in the CO3 group have very similar environments, little deviation from trigonality is expected. The CO group is nonplanar however. The carbon atom is 0.026(4) Å out of the plane of the oxygen atoms in the CO group in aragonite. This displacement is approximately seven times the standard deviation and is clearly significant. The displacement is towards the nearest Ca layer and is presumably a result of polarization of each oxygen atom by the three bonded Ca ions. If the displacement were caused by Ca . . . C interactions, the C atom would not move towards the Ca layer.

The difference in the O-C-O angles, 119.6° and 120.13°, if real, is consistent with the O(2, 2') edge of the CO group being coordinated slightly more strongly to Ca as judged from the Ca . . . O distances. However, this stronger coordination of O(2) to Ca would also suggest that C-O(2) should be longer than C-O(1). This is not the case. The average value of the C-O distances is 1.286 A. This agrees well with the C-O distances of 1.283(2) A reported [14] for calcite in which 32 symmetry is forced on the CO3 group by spacegroup R3c. If the apparent thermal motions of the atoms in the CO group are attributed to thermal motion rather than to slight positional disorder, there

4.007

4 191

152 146 41 191

5 258

263

2 128-131

6 49

45

b 23.

7 224

-1 213

4

83

A5

7 132 122

6 300

299

We thank W. E. Brown for helpful discussions and P. B. Kingsbury for technical help. The ORTEP program of C. K. Johnson, Oak Ridge National Laboratory, was used to draw the figures. This investigation was supported in part by research grant DE-00572 to the American Dental Association from the National Institute of Dental Research, and is part of the dental research program conducted by the National Bureau of Standards in cooperation with the American Dental Association; the United States Army Medical Research and Development Command; the Dental Sciences Division of the School of Aerospace Medicine, USAF; the National Institute of Dental Research; and the Veterans Administration.

4. References

[1] Palache, C., Berman, H., and Frondel, C., Dana's system of mineralogy, vol. II, 7th ed., J. Wiley and Sons, N.Y. 1963, 182-193.

[2] Busing, W. R., An aid to the analysis of interionic and intermolecular forces in crystals. Abstract M1. American Crystallographic Association winter meeting, Tucson, Arizona (1968).

[3] Bragg, W. L., The structure of aragonite. Proc. Roy. Soc. London, 105, 16-39 (1924). [4] Reference [1], page 184.

[5] Burnham, C. W., Computation of absorption corrections and the significance of end effect. Am. Min., 51, 159–167 (1966). [6] Busing, W. R., Ellison, R. D., Levy, H. A., King, S. P., and Roseberry, R. T., The Oak Ridge Laboratory Computer controlled X-ray Diffractometer. Oak Ridge National Laboratory Report ORNL 4143, January 1968. [7] Alexander, L. E., and Smith, G. S., Single crystal intensity measurements with the three circle counter diffractometer. Acta Cryst., 15, 983-1004 (1962).

[8] Ibers, J. A., Estimates of the standard deviations of the ob served structure factors and of the electron density from intensity data. Acta Cryst., 9, 652–654 (1956).

[9] Tippett, L. H. C., On the extreme individuals and the range of samples taken from a normal population. Biometrika, 17, 364-387 (1925).

[10] International Tables for X-ray Crystallography, 3, p. 202. The Kynoch Press, Birmingham, England, 1962.

[11] Stewart, J. M., Editor. X-ray 67 program system for X-ray crystallography. Technical report 67-58, University of Mary. land, College Park, Maryland 20742, December (1967). [12] Finger, L. W., Determination of cation distribution by least squares refinement of single crystal x-ray data. Carnegie Institute of Washington Year Book, 67, 216–217 (1969). [13] Busing, W. R., and Levy, H. A., The effect of thermal motion on the estimation of bond lengths from diffraction measurements. Acta Cryst., 17, 142–146 (1964).

[14] Chessin, H., Hamilton, W. C., and Post, B., Position and thermal parameters of oxygen atoms in calcite. Acta Cryst..

18, 689-693 (1965).

(Paper 75A1-647)

1. Introduction

In this paper we present the results of a series of periments on a single sample of polyisobutylene lution subjected to various histories of simple shear. he work was done as a part of a larger program to udy the appropriateness of a fairly general constitue equation, the Bernstein, Kearsley, Zapas (BKZ) astic fluid [1], in describing a wide range of nonear viscoelastic behavior for a range of materials. is larger purpose influenced both the choice of test aterial and the choice of experiments. In the first ace, the studies were confined to a single sample so to provide a test of the correlations between experients predicted by the assumed constitutive equation. e sample was chosen to show essentially complete laxation within the time scale of the experiment, ice earlier studies [2] had been made on materials th longer relaxation times. The deformations are rious histories of simple shear, because they are perimentally convenient deformations and because ey are convenient to analyze in terms of the assumed nstitutive equation.

Some measurements of normal stresses were made this sample, but we encountered serious discrepcies when we attempted to correlate them with the ear data. This trouble has been traced to the recently scovered "pressure hole error" [3] and the difficulty solved, but we have postponed publication of the rmal stress data to avoid further delay in the esentation of the shear stress measurements. In the course of these studies some measurement rors, customarily ignored, were found to be surisingly large and some correction methods were rked out and are presented here. A specialized form the BKZ elastic fluid was used in this paper to

Figures in brackets indicate the literature references at the end of this paper.

correlate our experiments with very good success. Although this special form is neither unique nor exact, it has been chosen to be consistent with a wide range of phenomena beyond that described in this paper, and should be of considerable practical use.

2. General Considerations

The class of motions which we shall consider will be limited to simple shearing histories.

In the incompressible isothermal form of the BKZ theory [1, 4] the value of the shearing stress at time t is given by the following relation σ (t) =

*

=

*

T

(2.1)

*

where y(7) is the amount of shear at time 7 and W✩ is a function of strain and time.2 If W is known, the shearing stress σ(t) for any simple shearing flow can be calculated through eq (2.1). Of course, the form of W is not specified by the theory and in general the function W must be determined through experiments. A good idea of the function W can be obtained from examining a single step stress relaxation history in simple shear. In such a deformation the shearing history is given by:

*

Y(T) = 0

Y(T) = y = constant

*

for 7 <0

(2.2)

for T > 0.

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